Finite Group Databasebeta

← #29748041

Subgroups — #29748041

Order 15616.

910 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450178C211yesmaximal, characteristic, complemented
2#21448835C211yesmaximal, characteristic, complemented
3#21447131C211yesmaximal, characteristic, complemented
4#21447001C211yesmaximal, characteristic, complemented
5#21446954C211yesmaximal, characteristic, complemented
6#21446817C211yesmaximal, characteristic
7#21446769C211yesmaximal, characteristic, complemented
8#21446500C211yesmaximal, characteristic, complemented
9#21446437C211yesmaximal, characteristic, complemented
10#21446422C211yesmaximal, characteristic, complemented
11#21446388C211yesmaximal, characteristic, complemented
12#21446168C211yesmaximal, characteristic, complemented
13#21446136C211yesmaximal, characteristic, complemented
14#21445633C211yesmaximal, characteristic, complemented
15C61×#1823C211yesmaximal, characteristic, complemented
16C22×D488C2211yescharacteristic
17C2×#14519776C2211yescharacteristic
18C2×#14519763—21no—
19C2×#14519751C2211yescharacteristic
20C2×#14519751C2211yescharacteristic
21C2×#14519739C2211yescharacteristic
22C2×C4×C22⋊D61—21no—
23C2×#14519710C2211yescharacteristic
24C2×#14519706C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30C2×#14519697C2211yescharacteristic
31C2×C22⋊#92418C2211yescharacteristic
32C2×C8×Dic61C2211yescharacteristic
33#16839677—21no—
34#16839675—21no—
35#16839663C2211yescharacteristic
36#16839663C2211yescharacteristic
37#16839663C2211yescharacteristic
38#16839658C2211yescharacteristic
39#16839652—21no—
40#16839651—21no—
41#16839641—21no—
42#16839633—21no—
43C4⋊C4×Dic61C2211yescharacteristic
44C4⋊C4×Dic61C2211yescharacteristic
45C4×C22⋊Dic61C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C22×C4⋊C4×C61C2211yescharacteristic
53C2×C22⋊C8×C61C2211yescharacteristic
54C61×#380C2211yescharacteristic
55C61×#340C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C22×Dic122—21no—
60C22×D244C2311yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C22⋊D61—21no—
63C22×C22⋊D61—21no—
64C22×C4×D61—21no—
65C2×D488—22no—
66C2×D488—41no—
67C2×D488—41no—
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70#14519776—22no—
71#14519776—41no—
72#14519763—44no—
73#14519751—22no—
74#14519751—22no—
75#14519751—22no—
76#14519751—41no—
77#14519739—22no—
78#14519739—41no—
79C4×C22⋊D61—44no—
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×C22⋊Dic61D411yescharacteristic
85C2×C22⋊Dic61D411yescharacteristic
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198—21no—
94C2×#11758198—21no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61C2311yescharacteristic
98C2×C4×Dic61C2311yescharacteristic
99#14519710—22no—
100#14519706—41no—
101#14519699—22no—
102#14519699—22no—
103#14519697—22no—
104#14519697—22no—
105#14519697—41no—
106#14519697—41no—
107C22⋊#92418—41no—
108C8×Dic61—41no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684D411yescharacteristic
113#14519684D411yescharacteristic
114C23×C4×C61—21no—
115C23×C4×C61C2311yescharacteristic
116C22×C8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118C2×C4⋊C4×C61—22no—
119C2×C4⋊C4×C61—21no—
120C2×C4⋊C4×C61—41no—
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C22⋊C4×C61—21no—
126C22⋊C8×C61—41no—
127#95×C61—21no—
128C24×C61C2×D411yescharacteristic
129C23×D61—21no—
130C23×D61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—42no—
142C22×C4×C61—42no—
143C22×C4×C61—81no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2411yescharacteristic
146C2×C22⋊D61—24no—
147C2×C22⋊D61—42no—
148C2×C22⋊D61—42no—
149C2×C22⋊D61—42no—
150C2×C22⋊D61—42no—
151C2×C22⋊D61—81no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—21no—
169C2×D244#4112yes—
170C2×C4×D61—42no—
171C2×C4×D61—42no—
172C2×C4×D61—81no—
173C2×Dic122—42no—
174C2×Dic122—42no—
175C2×Dic122—81no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—42no—
190C4⋊C4×C61#4112yes—
191C22⋊C4×C61—42no—
192C22⋊Dic61—41no—
193C22⋊Dic61D814yescomplemented
194#11758198—24no—
195#11758198—42no—
196#11758198—42no—
197#11758198—42no—
198#11758198—42no—
199#11758198—42no—
200#11758198—42no—
201#11758198—81no—
202#11758198—81no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61#4112yes—
206C4⋊Dic61#4112yes—
207#11758196—42no—
208#11758196—42no—
209C4×Dic61—21no—
210C4×Dic61—21no—
211C4×Dic61—21no—
212C4×Dic61—21no—
213C4×Dic61—42no—
214C4×Dic61—42no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221D488—44no—
222C23×C61—21no—
223C23×C61—21no—
224C23×C61—21no—
225C23×C61—21no—
226C23×C61—21no—
227C23×C61—21no—
228C23×C61C2×D812yes—
229C23×C61C22×D411yescharacteristic
230C22×D61—22no—
231C22×D61—41no—
232C22×D61—41no—
233C22×D61—41no—
234C22×D61—41no—
235C22×D61—41no—
236C22×D61—41no—
237C22×D61—41no—
238C22×D61—41no—
239C22×D61—41no—
240C22×D61—41no—
241C22×D61—41no—
242C22×D61—41no—
243C22×D61—41no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—22no—
247C2×C4×C61—21no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—84no—
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C61C2×#4111yescharacteristic
268C2×C4×C61H(5,𝔽2)11yescharacteristic
269C2×C4×C61H(5,𝔽2)11yescharacteristic
270C22⋊D61—48no—
271C22⋊D61—84no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—22no—
277C2×Dic61—21no—
278C2×Dic61—21no—
279C2×Dic61—22no—
280C2×Dic61—21no—
281C2×Dic61—21no—
282C2×Dic61—21no—
283C2×Dic61—21no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61C2×D812yes—
289C2×Dic61C2×D812yes—
290D244—22no—
291D244—22no—
292D244—41no—
293D244—41no—
294D244—41no—
295D244—41no—
296D244—41no—
297D244—41no—
298C4×D61—84no—
299Dic122—84no—
300C8×C61—41no—
301C8×C61—41no—
302#92418—41no—
303#92418—41no—
304#26720—611nomaximal, Sylow
305C22×C61—22no—
306C22×C61—22no—
307C22×C61—22no—
308C22×C61—41no—
309C22×C61—41no—
310C22×C61—41no—
311C22×C61—41no—
312C22×C61#52511yescharacteristic
313C22×C61#54611yescharacteristic
314C22×C61#54611yescharacteristic
315C22×C61C22×D811yescharacteristic
316C22×C61#57511yescharacteristic
317C22×C61#57711yescharacteristic
318C22×C61#57711yescharacteristic
319C2×D61—42no—
320C2×D61—42no—
321C2×D61—42no—
322C2×D61—81no—
323C2×D61—81no—
324C2×D61—81no—
325C2×D61—81no—
326C2×D61—81no—
327C2×D61—81no—
328C2×D61—81no—
329C2×D61—81no—
330C2×D61—81no—
331C2×D61—81no—
332C2×D61—81no—
333C4×C61—41no—
334C4×C61—42no—
335C4×C61—81no—
336C4×C61—81no—
337C4×C61—81no—
338C4×C61#52911yescharacteristic
339C4×C61#53811yescharacteristic
340C4×C61#55511yescharacteristic
341C4×C61#55911yescharacteristic
342Dic61—22no—
343Dic61—22no—
344Dic61—42no—
345Dic61—42no—
346Dic61—41no—
347Dic61—41no—
348#2491—611no—
349#2361—611no—
350#2054—611no—
351#2033—611no—
352#2023—611no—
353#1999—611no—
354#1988—611no—
355#1934—611no—
356#1925—611no—
357#1923—611no—
358#1918—611no—
359#1879—611no—
360#1874—611no—
361#1823D6111yescharacteristic, complemented
362C4×#102—611no—
363C2×C61—42no—
364C2×C61#245911yescharacteristic
365C2×C61#310011yescharacteristic
366C2×C61#312511yescharacteristic
367C2×C61#329411yescharacteristic
368C2×C61#329411yescharacteristic
369C2×C61#332111yescharacteristic
370C2×C61#332611yescharacteristic
371D61—82no—
372D61—161no—
373C22×D8—611no—
374C2×C4⋊Q8—611no—
375C2×#124—1221no—
376C2×#121—611no—
377C2×#121—611no—
378C2×C4×D4—611no—
379C2×C4×D4—1221no—
380C22×C4⋊C4C2×D6111yescharacteristic
381C2×#107—611no—
382C2×C4⋊C8—611no—
383C2×#102—611no—
384C2×#102—611no—
385C2×#102—611no—
386C2×#102—611no—
387C2×#102—611no—
388C2×#102—611no—
389C2×C22⋊C8—611no—
390C2×C22⋊C8C2×D6111yescharacteristic
391C2×C4×C8—611no—
392#394—1221no—
393#393—1221no—
394#390—611no—
395#390—611no—
396#390—611no—
397#389—611no—
398#388—1221no—
399#388—1221no—
400#386—1221no—
401#385—1221no—
402#380C2×D6111yescharacteristic
403C4×C4⋊C4—611no—
404C4×C4⋊C4—611no—
405C4×C22⋊C4—611no—
406#340—611no—
407#340—611no—
408#340—611no—
409#340—611no—
410#340C2×D6111yescharacteristic
411#340C2×D6111yescharacteristic
412#340C2×D6111yescharacteristic
413#340C2×D6111yescharacteristic
414#336—611no—
415#336—611no—
416C61#2672011yescharacteristic, complemented, Sylow
417C22×Q8—1221no—
418C22×D4—611no—
419C22×D4—611no—
420C22×D4—1221no—
421C22×D4—1221no—
422C23×C4—21no—
423C23×C4—1221no—
424C23×C4C22×D6111yescharacteristic
425C2×D8—1222no—
426C2×D8—2441no—
427C2×D8—2441no—
428C22×C8—611no—
429C22×C8—611no—
430C22×C8C22×D6111yescharacteristic
431C22×C8C22×D6111yescharacteristic
432C4⋊Q8—1222no—
433C4⋊Q8—2441no—
434#124—2444no—
435#121—1222no—
436#121—1222no—
437#121—1222no—
438#121—2441no—
439C4×D4—1222no—
440C4×D4—2441no—
441C4×D4—2444no—
442C2×C4⋊C4—22no—
443C2×C4⋊C4—21no—
444C2×C4⋊C4—41no—
445C2×C4⋊C4—611no—
446C2×C4⋊C4—611no—
447C2×C4⋊C4—1221no—
448C2×C4⋊C4—1221no—
449C2×C4⋊C4C22×D6111yescharacteristic
450C2×C4⋊C4C22×D6111yescharacteristic
451C2×C4⋊C4C22×D6111yescharacteristic
452C2×C4⋊C4C22×D6111yescharacteristic
453C2×C22⋊C4—21no—
454C2×C22⋊C4—611no—
455C2×C22⋊C4—611no—
456C2×C22⋊C4—1221no—
457C2×C22⋊C4—1221no—
458C2×C22⋊C4—1221no—
459C2×C22⋊C4—1221no—
460C2×C22⋊C4—1221no—
461C2×C22⋊C4—1221no—
462C2×C22⋊C4—1221no—
463C2×C22⋊C4—1221no—
464C2×C22⋊C4—1221no—
465C2×C42—611no—
466C2×C42—611no—
467C2×C42—1221no—
468C2×C42—1221no—
469#107—1222no—
470C4⋊C8—2441no—
471#102—1222no—
472#102—1222no—
473#102—1222no—
474#102—1222no—
475#102—2441no—
476#102—2441no—
477C22⋊C8—41no—
478C22⋊C8—2441no—
479C4×C8—2441no—
480#95—21no—
481#95—611no—
482#95—611no—
483#95—1221no—
484#95—1221no—
485#95—1221no—
486C24—1221no—
487C24—1221no—
488C24D4×D6111yescharacteristic
489C2×Q8—4881no—
490C2×Q8—2442no—
491C2×Q8—2442no—
492C2×D4—612no—
493C2×D4—1224no—
494C2×D4—1221no—
495C2×D4—1221no—
496C2×D4—1221no—
497C2×D4—1221no—
498C2×D4—1221no—
499C2×D4—1221no—
500C2×D4—1221no—
501C2×D4—1221no—
502C2×D4—1221no—
503C2×D4—1221no—
504C2×D4—1221no—
505C2×D4—4881no—
506C2×D4—2442no—
507C2×D4—2442no—
508C2×D4—2442no—
509C2×D4—2442no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—21no—
519C22×C4—21no—
520C22×C4—42no—
521C22×C4—42no—
522C22×C4—81no—
523C22×C4—611no—
524C22×C4—611no—
525C22×C4—611no—
526C22×C4—611no—
527C22×C4—1221no—
528C22×C4—1221no—
529C22×C4—4881no—
530C22×C4—2442no—
531C22×C4—2442no—
532C22×C4D4×D6111yescharacteristic
533C22×C4C23×D6111yescharacteristic
534D8—2444no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—21no—
541C2×C8—1221no—
542C2×C8—1221no—
543C2×C8—1221no—
544C2×C8—1221no—
545C2×C8—1221no—
546C2×C8—1221no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—21no—
554C4⋊C4—42no—
555C4⋊C4—612no—
556C4⋊C4—612no—
557C4⋊C4—1221no—
558C4⋊C4—1221no—
559C4⋊C4—2442no—
560C4⋊C4—2442no—
561C4⋊C4#1175822212yes—
562C22⋊C4—42no—
563C22⋊C4—614no—
564C22⋊C4—1224no—
565C22⋊C4—4881no—
566C22⋊C4—4881no—
567C22⋊C4—2442no—
568C22⋊C4—2442no—
569C22⋊C4—2442no—
570C22⋊C4—2442no—
571C22⋊C4—2442no—
572C22⋊C4—2442no—
573C22⋊C4—2441no—
574C42—1221no—
575C42—1221no—
576C42—1221no—
577C42—1221no—
578C42—2442no—
579C42—2442no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—1222no—
587C23—2441no—
588C23—2441no—
589C23—2441no—
590C23—2441no—
591C23—2441no—
592C23—2441no—
593C23—2441no—
594C23—2441no—
595C23—2441no—
596C23—2441no—
597C23—2441no—
598C23—2441no—
599C23—2441no—
600C23D8×D6112yes—
601C23C2×D4×D6111yescharacteristic
602Q8—4884no—
603D4—1222no—
604D4—1222no—
605D4—4884no—
606D4—2448no—
607D4—2441no—
608D4—2441no—
609D4—2441no—
610D4—2441no—
611D4—2441no—
612D4—2441no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—84no—
633C2×C4—612no—
634C2×C4—612no—
635C2×C4—1221no—
636C2×C4—1221no—
637C2×C4—1221no—
638C2×C4—1221no—
639C2×C4—1222no—
640C2×C4—1221no—
641C2×C4—1221no—
642C2×C4—1221no—
643C2×C4—1221no—
644C2×C4—1221no—
645C2×C4—1221no—
646C2×C4—1222no—
647C2×C4—4884no—
648C2×C4—2441no—
649C2×C4—2441no—
650C2×C4—2441no—
651C2×C4—2441no—
652C2×C4#41×D6111yescharacteristic
653C2×C4#41×D6111yescharacteristic
654C2×C4#41×D6111yescharacteristic
655C2×C4C2×#1175822211yescharacteristic
656C2×C4#1451982211yescharacteristic
657C2×C4#1451982311yescharacteristic
658C8—41no—
659C8—41no—
660C8—2441no—
661C8—2441no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—4881no—
670C22—4881no—
671C22—4881no—
672C22—4881no—
673C22—4881no—
674C22—4881no—
675C22—4881no—
676C22—4881no—
677C22—4881no—
678C22—4881no—
679C22—4881no—
680C22—2442no—
681C22—2442no—
682C22—2442no—
683C22#1684027011yescharacteristic
684C22#1684040211yescharacteristic
685C22#1684040411yescharacteristic
686C22C2×D8×D6111yescharacteristic
687C22#1684056111yescharacteristic
688C22#1684056811yescharacteristic
689C22#1684057111yescharacteristic
690C4—41no—
691C4—42no—
692C4—81no—
693C4—81no—
694C4—81no—
695C4—1222no—
696C4—1222no—
697C4—2441no—
698C4—2441no—
699C4—2442no—
700C4—2442no—
701C4#1684029011yescharacteristic
702C4#1684034811yescharacteristic
703C4#1684047511yescharacteristic
704C4#1684049611yescharacteristic
705C2—42no—
706C2—9761no—
707C2—4882no—
708C2#2144975311yescharacteristic
709C2#2145581611yescharacteristic
710C2#2145609111yescharacteristic
711C2#2145762811yescharacteristic
712C2#2145763011yescharacteristic
713C2#2145800611yescharacteristic
714C2#2145807511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1823C2
socleC23×C61C22×D4
prime radical#21446817C2
hypercenter#1823D61
hypocenterC61#26720
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #21450178C2semidirect
2. #21448835C2semidirect
3. #21447131C2semidirect
4. #21447001C2semidirect
5. #21446954C2semidirect
7. #21446769C2semidirect
8. #21446500C2semidirect
9. #21446437C2semidirect
10. #21446422C2semidirect
11. #21446388C2semidirect
12. #21446168C2semidirect
13. #21446136C2semidirect
14. #21445633C2semidirect
15. C61×#1823C2semidirect
193. C22⋊Dic61D8semidirect
361. #1823D61semidirect
416. C61#26720semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214501782
#214488352
#214471312
#214470012
#214469542
#214468172
#214467692
#214465002
#214464372
#214464222
#214463882
#214461682
#214461362
#214456332
C61×#18232
#2672061