Finite Group Databasebeta

← #29697309

Subgroups — #29697309

Order 15616.

1038 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839762C211yesmaximal, characteristic
2#21448687C211yesmaximal, characteristic, complemented
3#21448648C211yesmaximal, characteristic
4#21446497C211yesmaximal, characteristic, complemented
5#21446457C211yesmaximal, characteristic, complemented
6#21446456C211yesmaximal, characteristic, complemented
7#21446079C211yesmaximal, characteristic, complemented
8#21446067C211yesmaximal, characteristic, complemented
9#21445908C211yesmaximal, characteristic, complemented
10#21445851C211yesmaximal, characteristic, complemented
11#21445758C211yesmaximal, characteristic
12#21445645C211yesmaximal, characteristic, complemented
13#21445633C211yesmaximal, characteristic, complemented
14#21445604C211yesmaximal, characteristic, complemented
15C61×#1935C211yesmaximal, characteristic, complemented
16C22×#11758226C2211yescharacteristic
17C22×#11758189C2211yescharacteristic
18C2×C4×D244C2211yescharacteristic
19C2×#14519739C2211yescharacteristic
20C2×#14519735—21no—
21C2×C4⋊C4×D61C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24#16839762C418yescomplemented
25C2×#14519701C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519699C2211yescharacteristic
28C2×#14519699C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30C2×#14519697C2211yescharacteristic
31C2×#14519689C2211yescharacteristic
32C2×C8×Dic61C2211yescharacteristic
33#16839669C2211yescharacteristic
34#16839661C2211yescharacteristic
35#16839649—21no—
36#16839643—21no—
37#16839640—21no—
38#16839640—21no—
39#16839624—21no—
40#16839624—21no—
41#16839619C2211yescharacteristic
42#16839619C2211yescharacteristic
43C4⋊C4×Dic61C2211yescharacteristic
44C4×C4⋊Dic61C2211yescharacteristic
45C4×#11758198C2211yescharacteristic
46#95×D61—21no—
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×#103×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×#97×C61C2211yescharacteristic
56C61×#391C2211yescharacteristic
57C61×#389C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C2×#11758226—22no—
61C2×#11758226—22no—
62C2×#11758226—22no—
63C2×#11758226—41no—
64C2×#11758226—41no—
65C2×#11758226C2×C414yes—
66C22×D244—21no—
67C22×D244C2311yescharacteristic
68C22×C4×D61—21no—
69C22×C4×D61—21no—
70C22×C4×D61—21no—
71C22×C4×D61—21no—
72C22×C4×D61C2311yescharacteristic
73C2×#11758189—41no—
74C2×#11758189—41no—
75C2×#11758189C2×C414yescomplemented
76C22×#92418C2311yescharacteristic
77C22×#92418C2311yescharacteristic
78C4×D244—22no—
79C4×D244—41no—
80#14519739—22no—
81#14519739—41no—
82#14519735—44no—
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×#11758196—21no—
88C4⋊C4×D61—22no—
89C4⋊C4×D61C2×C414yescomplemented
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198—21no—
94C2×#11758198D411yescharacteristic
95C2×#11758198D411yescharacteristic
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61C2311yescharacteristic
99#14519710—41no—
100#14519708—41no—
101#14519701—41no—
102#14519701C2×C414yescomplemented
103#14519699—22no—
104#14519699C2×C414yes—
105#14519697—22no—
106#14519697C2×C414yes—
107#14519689—22no—
108C8×Dic61—41no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684D411yescharacteristic
117#14519684D411yescharacteristic
118C22×Q8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C42×C61C2311yescharacteristic
126#103×C61—41no—
127#103×C61C2×C414yescomplemented
128#97×C61—41no—
129#95×C61—21no—
130#95×C61—21no—
131#95×C61—21no—
132C23×D61—21no—
133C23×D61C2×D411yescharacteristic
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61C22×C412yes—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2411yescharacteristic
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153#11758226—28no—
154#11758226—44no—
155#11758226—44no—
156#11758226—44no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—24no—
161C2×D244—42no—
162C2×D244—42no—
163C2×D244C22×C412yes—
164C2×D244#4112yes—
165C2×D244#4112yes—
166C2×C4×D61—21no—
167C2×C4×D61—21no—
168C2×C4×D61—21no—
169C2×C4×D61—21no—
170C2×C4×D61—24no—
171C2×C4×D61—24no—
172C2×C4×D61—42no—
173C2×C4×D61—42no—
174C2×C4×D61—42no—
175C2×C4×D61—42no—
176C2×C4×D61—42no—
177C2×C4×D61—42no—
178C2×C4×D61—42no—
179C2×C4×D61—42no—
180C2×C4×D61—81no—
181C2×C4×D61—81no—
182C2×C4×D61C22×C412yes—
183C2×C4×D61#4112yes—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C2×C8×C61—21no—
188C2×C8×C61—21no—
189C2×C8×C61C22×C412yes—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—42no—
196C4⋊C4×C61C22×C412yes—
197C42×C61—21no—
198C42×C61—21no—
199#11758198—22no—
200#11758198—41no—
201#11758198—42no—
202#11758198—42no—
203#11758198—42no—
204#11758198—81no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—42no—
211C4⋊Dic61C22×C412yes—
212#11758196—24no—
213C4×Dic61—21no—
214C4×Dic61—21no—
215C4×Dic61—42no—
216C4×Dic61—42no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418C22×C412yes—
223#11758189—44no—
224C23×C61C22×D411yescharacteristic
225C22×D61—22no—
226C22×D61—22no—
227C22×D61—22no—
228C22×D61—21no—
229C22×D61—21no—
230C22×D61—21no—
231C22×D61—21no—
232C22×D61—21no—
233C22×D61—21no—
234C22×D61—41no—
235C22×D61—41no—
236C22×D61—41no—
237C22×D61—41no—
238C22×D61C4×D412yes—
239Q8×C61—22no—
240Q8×C61—22no—
241Q8×C61—41no—
242Q8×C61—41no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
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×C61C23×C411yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61H(5,𝔽2)11yescharacteristic
266C2×C4×C61H(5,𝔽2)11yescharacteristic
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—22no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61C4×D412yes—
284C2×Dic61AGL(1,ℤ/8)12yes—
285D244—22no—
286D244—22no—
287D244—22no—
288D244—48no—
289D244#11714yescomplemented
290C4×D61—22no—
291C4×D61—22no—
292C4×D61—22no—
293C4×D61—22no—
294C4×D61—48no—
295C4×D61—48no—
296C4×D61—84no—
297C4×D61—84no—
298C8×C61—22no—
299C8×C61—41no—
300#92418—22no—
301#92418—41no—
302#22582—611nomaximal, Sylow
303C22×C61C2×C4×D411yescharacteristic
304C22×C61#54611yescharacteristic
305C22×C61#54611yescharacteristic
306C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
307C22×C61#57511yescharacteristic
308C22×C61#57711yescharacteristic
309C22×C61#57811yescharacteristic
310C2×D61—22no—
311C2×D61—24no—
312C2×D61—22no—
313C2×D61—22no—
314C2×D61—41no—
315C2×D61—42no—
316C2×D61—42no—
317C2×D61—42no—
318C2×D61—42no—
319C2×D61—41no—
320C2×D61—41no—
321C2×D61—42no—
322C2×D61—42no—
323C2×D61—41no—
324C4×C61—41no—
325C4×C61—42no—
326C4×C61—42no—
327C4×C61—41no—
328C4×C61—81no—
329C4×C61—81no—
330C4×C61C2×#11711yescharacteristic
331C4×C61#51811yescharacteristic
332C4×C61#55111yescharacteristic
333C4×C61#55111yescharacteristic
334Dic61—22no—
335Dic61—22no—
336Dic61—42no—
337Dic61—41no—
338Dic61—81no—
339Dic61—81no—
340C2×#417—611no—
341#2344—611no—
342#2339—611no—
343#1935D6111yescharacteristic, complemented
344#1933—611no—
345#1928—611no—
346#1927—611no—
347#1864—611no—
348#1862—611no—
349#1824—611no—
350#1809—611no—
351#1788—611no—
352#1762—611no—
353C4×#102—611no—
354#1753—611no—
355C2×C61#234211yescharacteristic
356C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
357C2×C61#294711yescharacteristic
358C2×C61#330611yescharacteristic
359C2×C61#331511yescharacteristic
360C2×C61#332311yescharacteristic
361C2×C61#333011yescharacteristic
362D61—42no—
363D61—44no—
364C22×#41—611no—
365C22×M16—611no—
366C2×C4×D4—611no—
367C2×C4×D4—611no—
368C2×#117—1221no—
369C22×C4⋊C4—611no—
370C2×#107—611no—
371C2×#106—611no—
372#417—618no—
373C2×#103—611no—
374C2×#103—611no—
375C2×#103C2×D6111yescharacteristic
376C2×#103C2×D6111yescharacteristic
377C2×#102—611no—
378C2×#102—611no—
379C2×#102—611no—
380C2×#102—611no—
381C2×C22⋊C8—611no—
382C2×#97C2×D6111yescharacteristic
383C2×C4×C8—611no—
384#391—611no—
385#391C2×D6111yescharacteristic
386#389—611no—
387#389C2×D6111yescharacteristic
388#387—1221no—
389#387—1221no—
390#386—1221no—
391#386—1221no—
392#381—1221no—
393#381—1221no—
394#380—611no—
395#380—611no—
396C4×C4⋊C4—611no—
397C4×C4⋊C4—611no—
398C4×C22⋊C4—611no—
399C2×#95—1221no—
400#340—611no—
401#340—611no—
402#340—611no—
403#340—611no—
404#340C2×D6111yescharacteristic
405#340C2×D6111yescharacteristic
406#336—611no—
407#336—611no—
408C61#2258211yescharacteristic, complemented, Sylow
409C2×#41—614no—
410C2×#41—1222no—
411C2×#41—1222no—
412C2×#41—1222no—
413C2×#41—2441no—
414C2×#41—2441no—
415C22×Q8C22×D6111yescharacteristic
416C22×D4—611no—
417C22×D4—1221no—
418C23×C4—611no—
419C23×C4—1221no—
420C23×C4—1221no—
421C23×C4—1221no—
422C23×C4—1221no—
423C2×M16—614no—
424C2×M16—2441no—
425C2×M16—2441no—
426C22×C8—611no—
427C22×C8—611no—
428C22×C8C22×D6111yescharacteristic
429C22×C8C22×D6111yescharacteristic
430C4×D4—1222no—
431C4×D4—1222no—
432C4×D4—2441no—
433C4×D4—2441no—
434#117—2444no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—611no—
437C2×C4⋊C4—614no—
438C2×C4⋊C4—611no—
439C2×C4⋊C4—611no—
440C2×C4⋊C4—1222no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4C22×D6111yescharacteristic
444C2×C4⋊C4C22×D6111yescharacteristic
445C2×C4⋊C4C22×D6111yescharacteristic
446C2×C22⋊C4—611no—
447C2×C22⋊C4—611no—
448C2×C22⋊C4—1221no—
449C2×C22⋊C4—1221no—
450C2×C22⋊C4—1221no—
451C2×C22⋊C4—1221no—
452C2×C42—611no—
453C2×C42—1221no—
454C2×C42—1221no—
455C2×C42C22×D6111yescharacteristic
456#107—2441no—
457#106—2441no—
458#103—41no—
459#103—614no—
460#103—2441no—
461#103C4×D6114yescomplemented
462#102—614no—
463#102—614no—
464#102—1222no—
465#102—1222no—
466C22⋊C8—1222no—
467#97—41no—
468C4×C8—2441no—
469#95—21no—
470#95—21no—
471#95—21no—
472#95—611no—
473#95—611no—
474#95—1221no—
475#95—1221no—
476#95—1221no—
477#95—1221no—
478#95—1221no—
479#95—1221no—
480#95—1221no—
481C24—611no—
482C24—1221no—
483#41—1228no—
484#41—2444no—
485#41—2444no—
486#41—2444no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—21no—
492C2×Q8C2×C4×D6112yes—
493C2×D4—612no—
494C2×D4—612no—
495C2×D4—612no—
496C2×D4—1224no—
497C2×D4—1221no—
498C2×D4—1221no—
499C2×D4—1221no—
500C2×D4—2442no—
501C2×D4—2442no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—611no—
507C22×C4—611no—
508C22×C4—612no—
509C22×C4—612no—
510C22×C4—611no—
511C22×C4—1224no—
512C22×C4—1224no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—1221no—
516C22×C4—1221no—
517C22×C4—1221no—
518C22×C4—1221no—
519C22×C4—1221no—
520C22×C4—4881no—
521C22×C4—4881no—
522C22×C4—2442no—
523C22×C4—2442no—
524C22×C4—2442no—
525C22×C4—2442no—
526C22×C4—2442no—
527C22×C4—2442no—
528C22×C4—2442no—
529C22×C4—2442no—
530C22×C4D4×D6111yescharacteristic
531C22×C4D4×D6111yescharacteristic
532C22×C4C23×D6111yescharacteristic
533M16—2444no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—612no—
540C2×C8—1221no—
541C2×C8—1221no—
542C2×C8—1221no—
543C2×C8—1221no—
544C2×C8—1221no—
545C2×C8C2×C4×D6112yes—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—42no—
552C4⋊C4—612no—
553C4⋊C4—1224no—
554C4⋊C4—1221no—
555C4⋊C4—1221no—
556C4⋊C4—1221no—
557C4⋊C4—1221no—
558C4⋊C4—1221no—
559C4⋊C4—2442no—
560C4⋊C4C2×C4×D6112yes—
561C22⋊C4—1222no—
562C22⋊C4—4881no—
563C22⋊C4—2441no—
564C22⋊C4—2442no—
565C22⋊C4—2442no—
566C22⋊C4—2442no—
567C42—21no—
568C42—21no—
569C42—1221no—
570C42—1221no—
571C42—2442no—
572C42—2442no—
573C23—612no—
574C23—1222no—
575C23—1221no—
576C23—1221no—
577C23—1221no—
578C23—1221no—
579C23—1222no—
580C23—1221no—
581C23—1222no—
582C23—1221no—
583C23—2441no—
584C23—2441no—
585C23—2441no—
586C23—2441no—
587C23C2×D4×D6111yescharacteristic
588Q8—22no—
589Q8—22no—
590Q8—41no—
591Q8—41no—
592D4—614no—
593D4—1222no—
594D4—1222no—
595D4—1222no—
596D4—2448no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—612no—
616C2×C4—612no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—1221no—
620C2×C4—1221no—
621C2×C4—1222no—
622C2×C4—1222no—
623C2×C4—1222no—
624C2×C4—1222no—
625C2×C4—1222no—
626C2×C4—1221no—
627C2×C4—1221no—
628C2×C4—1221no—
629C2×C4—4884no—
630C2×C4—4884no—
631C2×C4—2448no—
632C2×C4—2448no—
633C2×C4—2441no—
634C2×C4—2441no—
635C2×C4—2441no—
636C2×C4—2441no—
637C2×C4—2441no—
638C2×C4—2441no—
639C2×C4—2441no—
640C2×C4—2441no—
641C2×C4C22×C4×D6111yescharacteristic
642C2×C4#41×D6111yescharacteristic
643C2×C4#41×D6111yescharacteristic
644C2×C4#41×D6111yescharacteristic
645C2×C4#1451982211yescharacteristic
646C2×C4#1451982311yescharacteristic
647C8—22no—
648C8—41no—
649C8—1222no—
650C8—2441no—
651C22—1222no—
652C22—1222no—
653C22—1222no—
654C22—1224no—
655C22—2442no—
656C22—2442no—
657C22—2442no—
658C22—2441no—
659C22—2442no—
660C22—2441no—
661C22—2441no—
662C22—2442no—
663C22—2441no—
664C22—2442no—
665C22C4×D4×D6111yescharacteristic
666C22#1684040111yescharacteristic
667C22#1684040511yescharacteristic
668C22AGL(1,ℤ/8)×D6111yescharacteristic
669C22#1684056111yescharacteristic
670C22#1684057111yescharacteristic
671C22#1684057811yescharacteristic
672C4—42no—
673C4—41no—
674C4—41no—
675C4—42no—
676C4—81no—
677C4—81no—
678C4—1222no—
679C4—1222no—
680C4—4881no—
681C4—4881no—
682C4—2442no—
683C4—2441no—
684C4#117×D6111yescharacteristic
685C4#1684023411yescharacteristic
686C4#1684043711yescharacteristic
687C4#1684044011yescharacteristic
688C2—2442no—
689C2—2444no—
690C2#2144866711yescharacteristic
691C2#2145462411yescharacteristic
692C2#2145466111yescharacteristic
693C2#2145777911yescharacteristic
694C2#2145791411yescharacteristic
695C2#2145804111yescharacteristic
696C2#2145811911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1935C2
socleC23×C61C22×D4
prime radicalC22×#11758226C22
cosocleC22×C4×C61C24
hypercenter#1935D61
hypocenterC61#22582
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1935

Complements

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

NH ≅ G/Ntype
2. #21448687C2semidirect
4. #21446497C2semidirect
5. #21446457C2semidirect
6. #21446456C2semidirect
7. #21446079C2semidirect
8. #21446067C2semidirect
9. #21445908C2semidirect
10. #21445851C2semidirect
12. #21445645C2semidirect
13. #21445633C2semidirect
14. #21445604C2semidirect
15. C61×#1935C2semidirect
24. #16839762C4semidirect
75. C2×#11758189C2×C4semidirect
89. C4⋊C4×D61C2×C4semidirect
102. #14519701C2×C4semidirect
127. #103×C61C2×C4semidirect
289. D244#117semidirect
343. #1935D61semidirect
408. C61#22582semidirect
461. #103C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168397622
#214486872
#214486482
#214464972
#214464572
#214464562
#214460792
#214460672
#214459082
#214458512
#214457582
#214456452
#214456332
#214456042
C61×#19352
#2258261