Finite Group Databasebeta

← #29779650

Subgroups — #29779650

Order 15616.

850 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21451644C211yesmaximal, characteristic, complemented
2#21448412C211yesmaximal, characteristic
3#21447241C211yesmaximal, characteristic, complemented
4#21447236C211yesmaximal, characteristic, complemented
5#21446959C211yesmaximal, characteristic, complemented
6#21446954C211yesmaximal, characteristic, complemented
7#21446517C211yesmaximal, characteristic, complemented
8#21446507C211yesmaximal, characteristic, complemented
9#21446500C211yesmaximal, characteristic, complemented
10#21446496C211yesmaximal, characteristic, complemented
11#21445905C211yesmaximal, characteristic, complemented
12#21445760C211yesmaximal, characteristic
13#21445757C211yesmaximal, characteristic
14#21445587C211yesmaximal, characteristic, complemented
15C61×#1824C211yesmaximal, characteristic, complemented
16C22×#11758222C2211yescharacteristic
17C2×#14519776C2211yescharacteristic
18C2×#14519736C2211yescharacteristic
19C2×#14519702C2211yescharacteristic
20C2×#14519702C2211yescharacteristic
21C2×#14519700C2211yescharacteristic
22C2×#14519700C2211yescharacteristic
23C2×#14519699C2211yescharacteristic
24C2×#14519699C2211yescharacteristic
25C2×#14519697C2211yescharacteristic
26C2×#14519697C2211yescharacteristic
27C2×C22⋊#92418C2211yescharacteristic
28C2×#14519687C2211yescharacteristic
29C2×C8×Dic61C2211yescharacteristic
30#16839704—21no—
31#16839698—21no—
32#16839683—21no—
33#16839674—21no—
34#16839667C2211yescharacteristic
35#16839667C2211yescharacteristic
36#16839663C2211yescharacteristic
37#16839663C2211yescharacteristic
38#16839661C2211yescharacteristic
39#16839658C2211yescharacteristic
40#16839641—21no—
41#16839641—21no—
42#16839627C2211yescharacteristic
43#16839622C2211yescharacteristic
44#16839619—21no—
45#16839619—21no—
46#16839524C2211yescharacteristic
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
59C2×#11758222—22no—
60C2×#11758222—22no—
61C2×#11758222—22no—
62C2×#11758222—22no—
63C2×#11758222—41no—
64C2×#11758222—41no—
65C22×Dic122—21no—
66C22×Dic122C2311yescharacteristic
67C22×D244C2311yescharacteristic
68C22×C22⋊D61—21no—
69C22×C4×D61—21no—
70C22×#92418C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72#14519776—41no—
73#14519776—41no—
74#14519736—41no—
75#14519736—41no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61D411yescharacteristic
78C2×C4⋊Dic61D411yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61D411yescharacteristic
85C2×C22⋊Dic61D411yescharacteristic
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×C4×Dic61C2311yescharacteristic
91C2×C4×Dic61C2311yescharacteristic
92#14519702—22no—
93#14519702—41no—
94#14519700—22no—
95#14519700—41no—
96#14519699—22no—
97#14519699—41no—
98#14519697—22no—
99#14519697—41no—
100C22⋊#92418—22no—
101#14519687—41no—
102C8×Dic61—41no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112C23×C4×C61—21no—
113C23×C4×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C22×C8×C61C2311yescharacteristic
116C2×C4⋊C4×C61—22no—
117C2×C4⋊C4×C61—22no—
118C2×C4⋊C4×C61—21no—
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C22⋊C4×C61—21no—
124C22⋊C8×C61—22no—
125#95×C61—21no—
126C24×C61C2×D411yescharacteristic
127C23×D61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—42no—
137C22×C4×C61—81no—
138C22×C4×C61—81no—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61#4112yes—
141C22×C4×C61#4112yes—
142C22×C4×C61C2411yescharacteristic
143C2×C22⋊D61—24no—
144C2×C22⋊D61—42no—
145C2×C22⋊D61—42no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153#11758222—44no—
154#11758222—44no—
155#11758222—44no—
156#11758222—44no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244#4112yes—
163C2×C4×D61—24no—
164C2×C4×D61—42no—
165C2×C4×D61—42no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—42no—
172C2×Dic122—81no—
173C2×Dic122—81no—
174C2×Dic122#4112yes—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C4⋊C4×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—
190C22⋊C4×C61—81no—
191C22⋊Dic61—41no—
192C22⋊Dic61—41no—
193C22⋊Dic61—81no—
194#11758198—81no—
195#11758198—81no—
196#11758198—81no—
197#11758198—81no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—41no—
203C4⋊Dic61—41no—
204C4⋊Dic61—81no—
205#11758196—42no—
206#11758196—81no—
207C4×Dic61—21no—
208C4×Dic61—21no—
209C4×Dic61—21no—
210C4×Dic61—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C23×C61—21no—
218C23×C61—21no—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—21no—
222C23×C61—21no—
223C23×C61#12412yes—
224C23×C61C22×D411yescharacteristic
225C22×D61—22no—
226C22×D61—41no—
227C22×D61—41no—
228C22×D61—41no—
229C22×D61—41no—
230C22×D61—41no—
231C22×D61—41no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—84no—
251C2×C4×C61#12412yes—
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61H(5,𝔽2)11yescharacteristic
257C2×C4×C61H(5,𝔽2)11yescharacteristic
258C22⋊D61—48no—
259C2×Dic61—22no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280D244—22no—
281D244—22no—
282D244—41no—
283D244—41no—
284C4×D61—48no—
285Dic122—22no—
286Dic122—22no—
287Dic122—41no—
288Dic122—41no—
289Dic122—84no—
290C8×C61—41no—
291C8×C61—41no—
292#92418—41no—
293#92418—41no—
294#29120—611nomaximal, Sylow
295C22×C61—22no—
296C22×C61—22no—
297C22×C61—22no—
298C22×C61—41no—
299C22×C61—41no—
300C22×C61—41no—
301C22×C61—41no—
302C22×C61#52511yescharacteristic
303C22×C61C2×#12411yescharacteristic
304C22×C61#53411yescharacteristic
305C22×C61#57611yescharacteristic
306C22×C61#57711yescharacteristic
307C22×C61#57711yescharacteristic
308C22×C61#57811yescharacteristic
309C2×D61—42no—
310C2×D61—42no—
311C2×D61—42no—
312C2×D61—81no—
313C2×D61—81no—
314C2×D61—81no—
315C2×D61—81no—
316C4×C61—22no—
317C4×C61—81no—
318C4×C61—81no—
319C4×C61—81no—
320C4×C61—81no—
321C4×C61#55111yescharacteristic
322C4×C61#55111yescharacteristic
323C4×C61#55111yescharacteristic
324C4×C61#55111yescharacteristic
325Dic61—42no—
326Dic61—41no—
327Dic61—41no—
328Dic61—41no—
329Dic61—41no—
330Dic61—81no—
331Dic61—81no—
332#2628—611no—
333#2311—611no—
334#2074—611no—
335#2073—611no—
336#2024—611no—
337#2023—611no—
338#1936—611no—
339#1935—611no—
340#1934—611no—
341#1933—611no—
342#1824D6111yescharacteristic, complemented
343#1823—611no—
344#1788—611no—
345#1788—611no—
346#1750—611no—
347C2×C61—42no—
348C2×C61#256111yescharacteristic
349C2×C61#312411yescharacteristic
350C2×C61#312511yescharacteristic
351C2×C61#313611yescharacteristic
352C2×C61#313711yescharacteristic
353C2×C61#319811yescharacteristic
354C2×C61#320111yescharacteristic
355D61—82no—
356C22×#41—611no—
357C2×C4⋊Q8—611no—
358C2×#117—611no—
359C22×C4⋊C4C2×D6111yescharacteristic
360C2×#103—611no—
361C2×#103—611no—
362C2×#103—611no—
363C2×#103—611no—
364C2×#102—611no—
365C2×#102—611no—
366C2×#102—611no—
367C2×#102—611no—
368C2×C22⋊C8—611no—
369C2×C22⋊C8C2×D6111yescharacteristic
370C2×#97—611no—
371C2×C4×C8—611no—
372#400—1221no—
373#399—1221no—
374#395—1221no—
375#393—1221no—
376#391—611no—
377#391—611no—
378#390—611no—
379#390—611no—
380#389—611no—
381#389—611no—
382#386—1221no—
383#386—1221no—
384#382—611no—
385#380—611no—
386#380—1221no—
387#380—1221no—
388#380C2×D6111yescharacteristic
389#340—611no—
390#340—611no—
391#340—611no—
392#340—611no—
393#340C2×D6111yescharacteristic
394#340C2×D6111yescharacteristic
395#340C2×D6111yescharacteristic
396#340C2×D6111yescharacteristic
397#336—611no—
398#336—611no—
399C61#2912011yescharacteristic, complemented, Sylow
400C2×#41—1222no—
401C2×#41—1222no—
402C2×#41—1222no—
403C2×#41—1222no—
404C2×#41—2441no—
405C2×#41—2441no—
406C22×Q8—611no—
407C22×Q8—1221no—
408C22×D4—611no—
409C22×D4—1221no—
410C23×C4—21no—
411C23×C4—1221no—
412C23×C4C22×D6111yescharacteristic
413C22×C8—611no—
414C22×C8—611no—
415C22×C8C22×D6111yescharacteristic
416C22×C8C22×D6111yescharacteristic
417C4⋊Q8—2441no—
418C4⋊Q8—2441no—
419#117—2441no—
420#117—2441no—
421C2×C4⋊C4—22no—
422C2×C4⋊C4—22no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—611no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4—1221no—
431C2×C4⋊C4C22×D6111yescharacteristic
432C2×C4⋊C4C22×D6111yescharacteristic
433C2×C4⋊C4C22×D6111yescharacteristic
434C2×C4⋊C4C22×D6111yescharacteristic
435C2×C22⋊C4—21no—
436C2×C22⋊C4—611no—
437C2×C22⋊C4—611no—
438C2×C22⋊C4—1221no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1221no—
443C2×C42—611no—
444C2×C42—611no—
445#103—1222no—
446#103—1222no—
447#103—2441no—
448#103—2441no—
449#102—1222no—
450#102—1222no—
451#102—2441no—
452#102—2441no—
453C22⋊C8—22no—
454C22⋊C8—1222no—
455#97—2441no—
456C4×C8—2441no—
457#95—21no—
458#95—1221no—
459#95—1221no—
460#95—1221no—
461#95—1221no—
462#95—1221no—
463#95—1221no—
464#95—1221no—
465#95—1221no—
466#95—1221no—
467C24—1221no—
468C24D4×D6111yescharacteristic
469#41—2444no—
470#41—2444no—
471#41—2444no—
472#41—2444no—
473C2×Q8—612no—
474C2×Q8—1221no—
475C2×Q8—1221no—
476C2×Q8—1221no—
477C2×Q8—1221no—
478C2×Q8—1221no—
479C2×Q8—4881no—
480C2×Q8—4881no—
481C2×Q8—2442no—
482C2×D4—612no—
483C2×D4—1224no—
484C2×D4—1221no—
485C2×D4—1221no—
486C2×D4—1221no—
487C2×D4—1221no—
488C2×D4—1221no—
489C2×D4—2442no—
490C2×D4—2442no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—42no—
500C22×C4—81no—
501C22×C4—81no—
502C22×C4—611no—
503C22×C4—611no—
504C22×C4—611no—
505C22×C4—611no—
506C22×C4—1224no—
507C22×C4—1221no—
508C22×C4—1221no—
509C22×C4—1221no—
510C22×C4—2442no—
511C22×C4—2442no—
512C22×C4#1175822212yes—
513C22×C4D4×D6111yescharacteristic
514C22×C4#1175822412yes—
515C22×C4C23×D6111yescharacteristic
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—1221no—
523C2×C8—1221no—
524C2×C8—1221no—
525C2×C8—1221no—
526C2×C8—1221no—
527C2×C8—1221no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—42no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4—1221no—
541C4⋊C4—4881no—
542C4⋊C4—4881no—
543C4⋊C4—2441no—
544C4⋊C4—2441no—
545C4⋊C4—2442no—
546C22⋊C4—81no—
547C22⋊C4—4881no—
548C22⋊C4—4881no—
549C22⋊C4—4881no—
550C22⋊C4—4881no—
551C22⋊C4—4881no—
552C22⋊C4—2441no—
553C22⋊C4—2441no—
554C42—1221no—
555C42—1221no—
556C42—1221no—
557C42—1221no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—1222no—
565C23—2441no—
566C23—2441no—
567C23—2441no—
568C23—2441no—
569C23—2441no—
570C23—2441no—
571C23#1451976312yes—
572C23C2×D4×D6111yescharacteristic
573Q8—1222no—
574Q8—1222no—
575Q8—4884no—
576Q8—2441no—
577Q8—2441no—
578D4—1222no—
579D4—1222no—
580D4—2448no—
581D4—2441no—
582D4—2441no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—84no—
602C2×C4—1222no—
603C2×C4—1221no—
604C2×C4—1221no—
605C2×C4—1221no—
606C2×C4—1221no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—1221no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—2448no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4#1451976312yes—
625C2×C4#41×D6111yescharacteristic
626C2×C4#41×D6111yescharacteristic
627C2×C4C2×#1175822411yescharacteristic
628C2×C4C2×#1175822211yescharacteristic
629C2×C4#1451982311yescharacteristic
630C2×C4#1451982311yescharacteristic
631C8—41no—
632C8—41no—
633C8—2441no—
634C8—2441no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—4881no—
643C22—4881no—
644C22—4881no—
645C22—4881no—
646C22—2442no—
647C22—2442no—
648C22—2442no—
649C22#1684027011yescharacteristic
650C22C2×#1451976311yescharacteristic
651C22#1684032311yescharacteristic
652C22#1684056611yescharacteristic
653C22#1684056811yescharacteristic
654C22#1684057211yescharacteristic
655C22#1684057511yescharacteristic
656C4—22no—
657C4—81no—
658C4—81no—
659C4—81no—
660C4—81no—
661C4—4881no—
662C4—4881no—
663C4—2441no—
664C4—2441no—
665C4—2441no—
666C4—2441no—
667C4—2442no—
668C4#1684043511yescharacteristic
669C4#1684043511yescharacteristic
670C4#1684043611yescharacteristic
671C4#1684043611yescharacteristic
672C2—42no—
673C2—4882no—
674C2#2145097611yescharacteristic
675C2#2145608411yescharacteristic
676C2#2145609911yescharacteristic
677C2#2145616911yescharacteristic
678C2#2145618211yescharacteristic
679C2#2145667611yescharacteristic
680C2#2145670611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1824C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
hypercenter#1824D61
hypocenterC61#29120
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
1. #21451644C2semidirect
3. #21447241C2semidirect
4. #21447236C2semidirect
5. #21446959C2semidirect
6. #21446954C2semidirect
7. #21446517C2semidirect
8. #21446507C2semidirect
9. #21446500C2semidirect
10. #21446496C2semidirect
11. #21445905C2semidirect
14. #21445587C2semidirect
15. C61×#1824C2semidirect
342. #1824D61semidirect
399. C61#29120semidirect

Maximal subgroups

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

subgroupindex
#214516442
#214484122
#214472412
#214472362
#214469592
#214469542
#214465172
#214465072
#214465002
#214464962
#214459052
#214457602
#214457572
#214455872
C61×#18242
#2912061