Finite Group Databasebeta

← #29748070

Subgroups — #29748070

Order 15616.

848 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21448833C211yesmaximal, characteristic
2#21447134C211yesmaximal, characteristic, complemented
3#21447008C211yesmaximal, characteristic
4#21446952C211yesmaximal, characteristic, complemented
5#21446820C211yesmaximal, characteristic
6#21446768C211yesmaximal, characteristic, complemented
7#21446498C211yesmaximal, characteristic, complemented
8#21446450C211yesmaximal, characteristic, complemented
9#21446427C211yesmaximal, characteristic, complemented
10#21446397C211yesmaximal, characteristic, complemented
11#21446156C211yesmaximal, characteristic, complemented
12#21446127C211yesmaximal, characteristic, complemented
13#21445903C211yesmaximal, characteristic, complemented
14C4×#14519697C211yesmaximal, characteristic, complemented
15C61×#2492C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C2×#14519751C2211yescharacteristic
18C2×C4×D244C2211yescharacteristic
19C2×#14519737—21no—
20C22×C4⋊Dic61C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×C4⋊#92418C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519697C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×C22⋊#92418C2211yescharacteristic
30C2×C22⋊#92418C2211yescharacteristic
31C2×C4×#92418C2211yescharacteristic
32#16839664C2211yescharacteristic
33#16839660C2211yescharacteristic
34#16839654—21no—
35#16839639—21no—
36#16839636—21no—
37C4×C4⋊Dic61C2211yescharacteristic
38#16839524C2211yescharacteristic
39#16839524C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839517C2211yescharacteristic
47#16839517C2211yescharacteristic
48C2×C4⋊Q8×C61C2211yescharacteristic
49C2×C22⋊Q8×C61C2211yescharacteristic
50C61×#400—21no—
51C61×#393—21no—
52C61×#391—21no—
53C61×#391C2211yescharacteristic
54C61×#391C2211yescharacteristic
55C61×#388—21no—
56C61×#380C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C4×C22⋊C4×C61C2211yescharacteristic
59C22×D244C2311yescharacteristic
60C22×C22⋊D61—21no—
61C23×Dic61—21no—
62C22×C4×D61—21no—
63C2×#11758201—22no—
64C2×#11758201—41no—
65C2×#11758201—41no—
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70#14519751—22no—
71#14519751—22no—
72C4×D244—22no—
73C4×D244—41no—
74#14519737—44no—
75C2×C4⋊Dic61—22no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—41no—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758196—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×C4×Dic61—21no—
87#14519708—22no—
88C4⋊#92418—41no—
89#14519701—22no—
90#14519701—22no—
91#14519701—41no—
92#14519697—22no—
93#14519697—22no—
94#14519697—41no—
95C22⋊#92418—41no—
96C22⋊#92418—41no—
97C4×#92418—41no—
98#14519684—21no—
99#14519684—21no—
100C22×Q8×C61—21no—
101C22×Q8×C61C2311yescharacteristic
102C23×C4×C61C2311yescharacteristic
103C4⋊Q8×C61—22no—
104C4⋊Q8×C61—41no—
105C22⋊Q8×C61—22no—
106C22⋊Q8×C61—41no—
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61—21no—
114C2×C22⋊C4×C61D411yescharacteristic
115C2×C22⋊C4×C61D411yescharacteristic
116C2×C42×C61—21no—
117C2×C42×C61C2311yescharacteristic
118C2×C42×C61C2311yescharacteristic
119#95×C61—21no—
120#95×C61—21no—
121#95×C61—21no—
122#95×C61—21no—
123#95×C61—21no—
124#95×C61—21no—
125#95×C61D411yescharacteristic
126#95×C61D411yescharacteristic
127C24×C61C2×D411yescharacteristic
128C23×D61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—42no—
136C2×Q8×C61—42no—
137C2×Q8×C61—81no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2411yescharacteristic
154C2×C22⋊D61—24no—
155C2×C22⋊D61—42no—
156C2×C22⋊D61—42no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—42no—
161C22×Dic61—42no—
162C22×Dic61—81no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244#4112yes—
169C2×C4×D61—42no—
170C2×C4×D61—42no—
171C2×C4×D61—81no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—42no—
178C4⋊C4×C61—42no—
179C4⋊C4×C61#4112yes—
180C22⋊C4×C61—41no—
181C22⋊C4×C61—42no—
182C22⋊C4×C61—81no—
183C22⋊C4×C61SD1614yescomplemented
184C42×C61—21no—
185C42×C61—21no—
186C42×C61—21no—
187C42×C61—21no—
188C42×C61—81no—
189#11758201—44no—
190#11758198—24no—
191#11758198—42no—
192#11758198—42no—
193#11758198—81no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—42no—
199C4⋊Dic61#4112yes—
200C4⋊Dic61#4112yes—
201#11758196—42no—
202C4×Dic61—42no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C23×C61—21no—
216C23×C61—21no—
217C23×C61—21no—
218C23×C61—21no—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61C2×SD1612yes—
222C23×C61C22×D411yescharacteristic
223C22×D61—22no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—41no—
228C22×D61—41no—
229C22×D61—41no—
230Q8×C61—41no—
231Q8×C61—41no—
232Q8×C61—41no—
233Q8×C61—41no—
234Q8×C61—84no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—22no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
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—41no—
264C2×C4×C61C2×SD1612yes—
265C2×C4×C61C2×SD1612yes—
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C61C2×#4111yescharacteristic
268C2×C4×C61C2×#4111yescharacteristic
269C2×C4×C61C2×#4111yescharacteristic
270C2×C4×C61H(5,𝔽2)11yescharacteristic
271C2×C4×C61H(5,𝔽2)11yescharacteristic
272C22⋊D61—48no—
273C2×Dic61—22no—
274C2×Dic61—22no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—84no—
283D244—22no—
284D244—22no—
285D244—41no—
286D244—41no—
287C4×D61—84no—
288#92418—41no—
289#92418—41no—
290#92418—41no—
291#92418—41no—
292#26722—611nomaximal, Sylow
293C22×C61—22no—
294C22×C61—22no—
295C22×C61—22no—
296C22×C61—41no—
297C22×C61—41no—
298C22×C61—41no—
299C22×C61—41no—
300C22×C61#52511yescharacteristic
301C22×C61#54611yescharacteristic
302C22×C61#54611yescharacteristic
303C22×C61C22×SD1611yescharacteristic
304C22×C61#57511yescharacteristic
305C22×C61#57611yescharacteristic
306C22×C61#57811yescharacteristic
307C2×D61—42no—
308C2×D61—42no—
309C2×D61—42no—
310C2×D61—81no—
311C2×D61—81no—
312C2×D61—81no—
313C2×D61—81no—
314C4×C61—22no—
315C4×C61—22no—
316C4×C61—41no—
317C4×C61—41no—
318C4×C61—41no—
319C4×C61—42no—
320C4×C61—81no—
321C4×C61—81no—
322C4×C61—81no—
323C4×C61#52911yescharacteristic
324C4×C61#53811yescharacteristic
325C4×C61#55511yescharacteristic
326C4×C61#55911yescharacteristic
327Dic61—42no—
328Dic61—42no—
329Dic61—81no—
330#2492D6111yescharacteristic, complemented
331#2361—611no—
332#2055—611no—
333#2034—611no—
334#2023—611no—
335#2000—611no—
336#1987—611no—
337#1933—611no—
338#1927—611no—
339#1923—611no—
340#1919—611no—
341#1877—611no—
342#1872—611no—
343#1823—611no—
344C4×#102—611no—
345C2×C61—42no—
346C2×C61#245911yescharacteristic
347C2×C61#310111yescharacteristic
348C2×C61#312411yescharacteristic
349C2×C61#329511yescharacteristic
350C2×C61#329811yescharacteristic
351C2×C61#331311yescharacteristic
352C2×C61#332211yescharacteristic
353D61—82no—
354C22×SD16—611no—
355C2×C4⋊Q8C2×D6111yescharacteristic
356C2×C22⋊Q8C2×D6111yescharacteristic
357C2×#121—611no—
358C2×C4×D4—611no—
359C2×C4×D4—1221no—
360C22×C4⋊C4—611no—
361C2×#106—611no—
362C2×C4⋊C8—611no—
363C2×#103—611no—
364C2×#103—611no—
365C2×#103—611no—
366C2×#102—611no—
367C2×#102—611no—
368C2×#102—611no—
369C2×C22⋊C8—611no—
370C2×C22⋊C8—611no—
371C2×C4×C8—611no—
372#400—21no—
373#393—21no—
374#391—21no—
375#391C2×D6111yescharacteristic
376#391C2×D6111yescharacteristic
377#390—611no—
378#389—611no—
379#388—21no—
380#388—1221no—
381#386—1221no—
382#385—1221no—
383#380C2×D6111yescharacteristic
384C4×C4⋊C4—611no—
385C4×C4⋊C4C2×D6111yescharacteristic
386C4×C22⋊C4C2×D6111yescharacteristic
387#340—611no—
388#340—611no—
389#340—611no—
390#340—611no—
391#340—611no—
392#340—611no—
393#340—611no—
394#340—611no—
395#336—611no—
396#336—611no—
397C61#2672211yescharacteristic, complemented, Sylow
398C22×Q8—21no—
399C22×Q8C22×D6111yescharacteristic
400C22×D4—611no—
401C22×D4—1221no—
402C23×C4—1221no—
403C23×C4—1221no—
404C23×C4C22×D6111yescharacteristic
405C2×SD16—1222no—
406C2×SD16—2441no—
407C2×SD16—2441no—
408C22×C8—611no—
409C22×C8—611no—
410C22×C8—611no—
411C22×C8—611no—
412C4⋊Q8—22no—
413C4⋊Q8—41no—
414C22⋊Q8—22no—
415C22⋊Q8—41no—
416#121—1222no—
417#121—1222no—
418C4×D4—1222no—
419C4×D4—2441no—
420C4×D4—2444no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—611no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—611no—
426C2×C4⋊C4—1222no—
427C2×C4⋊C4—1221no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—2441no—
430C2×C4⋊C4C22×D6111yescharacteristic
431C2×C4⋊C4C22×D6111yescharacteristic
432C2×C4⋊C4C22×D6111yescharacteristic
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—1221no—
436C2×C22⋊C4—1221no—
437C2×C22⋊C4—1221no—
438C2×C22⋊C4—1221no—
439C2×C22⋊C4C22⋊D6111yescharacteristic
440C2×C22⋊C4C22⋊D6111yescharacteristic
441C2×C42—21no—
442C2×C42—1221no—
443C2×C42C22×D6111yescharacteristic
444C2×C42C22×D6111yescharacteristic
445#106—1222no—
446C4⋊C8—2441no—
447#103—1222no—
448#103—1222no—
449#103—2441no—
450#102—1222no—
451#102—1222no—
452#102—2441no—
453C22⋊C8—2441no—
454C22⋊C8—2441no—
455C4×C8—2441no—
456#95—21no—
457#95—21no—
458#95—21no—
459#95—21no—
460#95—21no—
461#95—21no—
462#95—1221no—
463#95—1221no—
464#95C22⋊D6111yescharacteristic
465#95C22⋊D6111yescharacteristic
466C24—1221no—
467C24C2×C22⋊D6111yescharacteristic
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—42no—
475C2×Q8—42no—
476C2×Q8—81no—
477C2×D4—612no—
478C2×D4—1224no—
479C2×D4—1221no—
480C2×D4—1221no—
481C2×D4—1221no—
482C2×D4—1221no—
483C2×D4—1221no—
484C2×D4—2442no—
485C2×D4—2442no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—1221no—
499C22×C4—4881no—
500C22×C4—4881no—
501C22×C4—2442no—
502C22×C4—2442no—
503C22×C4—2442no—
504C22×C4—2442no—
505C22×C4C2×C22⋊D6111yescharacteristic
506C22×C4C2×C22⋊D6111yescharacteristic
507C22×C4C2×C22⋊D6111yescharacteristic
508C22×C4C2×C22⋊D6111yescharacteristic
509C22×C4C2×C22⋊D6111yescharacteristic
510C22×C4C23×D6111yescharacteristic
511SD16—2444no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—1221no—
523C2×C8—1221no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—612no—
532C4⋊C4—612no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—2442no—
538C4⋊C4—2442no—
539C4⋊C4#1175822412yes—
540C22⋊C4—41no—
541C22⋊C4—42no—
542C22⋊C4—81no—
543C22⋊C4—1224no—
544C22⋊C4—4881no—
545C22⋊C4—2442no—
546C22⋊C4—2442no—
547C22⋊C4#1175820114yescomplemented
548C42—21no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—81no—
553C42—2442no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—1222no—
561C23—2441no—
562C23—2441no—
563C23—2441no—
564C23—2441no—
565C23—2441no—
566C23—2441no—
567C23C2×#1175820112yes—
568C23C22×C22⋊D6111yescharacteristic
569Q8—41no—
570Q8—41no—
571Q8—41no—
572Q8—41no—
573Q8—84no—
574D4—1222no—
575D4—1222no—
576D4—2448no—
577D4—2441no—
578D4—2441no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
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—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—1222no—
609C2×C4—1222no—
610C2×C4—4884no—
611C2×C4—4884no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4C2×#1175820112yes—
620C2×C4C2×#1175820112yes—
621C2×C4#41×D6111yescharacteristic
622C2×C4#41×D6111yescharacteristic
623C2×C4#41×D6111yescharacteristic
624C2×C4C2×#1175822411yescharacteristic
625C2×C4#1451982211yescharacteristic
626C2×C4#1451982311yescharacteristic
627C8—2441no—
628C8—2441no—
629C8—2441no—
630C8—2441no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—4881no—
639C22—4881no—
640C22—4881no—
641C22—4881no—
642C22—2442no—
643C22—2442no—
644C22—2442no—
645C22#1684026911yescharacteristic
646C22#1684040011yescharacteristic
647C22#1684040611yescharacteristic
648C22C22×#1175820111yescharacteristic
649C22#1684056011yescharacteristic
650C22#1684056511yescharacteristic
651C22#1684057711yescharacteristic
652C4—22no—
653C4—22no—
654C4—41no—
655C4—41no—
656C4—42no—
657C4—41no—
658C4—81no—
659C4—81no—
660C4—81no—
661C4—4881no—
662C4—2442no—
663C4—2442no—
664C4#1684029111yescharacteristic
665C4#1684034611yescharacteristic
666C4#1684047411yescharacteristic
667C4#1684049411yescharacteristic
668C2—42no—
669C2—4882no—
670C2#2144975211yescharacteristic
671C2#2145582311yescharacteristic
672C2#2145607511yescharacteristic
673C2#2145763711yescharacteristic
674C2#2145768411yescharacteristic
675C2#2145787911yescharacteristic
676C2#2145802011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2492C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
hypercenter#2492D61
hypocenterC61#26722
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2492

Complements

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

NH ≅ G/Ntype
2. #21447134C2semidirect
4. #21446952C2semidirect
6. #21446768C2semidirect
7. #21446498C2semidirect
8. #21446450C2semidirect
9. #21446427C2semidirect
10. #21446397C2semidirect
11. #21446156C2semidirect
12. #21446127C2semidirect
13. #21445903C2semidirect
14. C4×#14519697C2semidirect
15. C61×#2492C2semidirect
183. C22⋊C4×C61SD16semidirect
330. #2492D61semidirect
397. C61#26722semidirect
547. C22⋊C4#11758201semidirect

Maximal subgroups

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

subgroupindex
#214488332
#214471342
#214470082
#214469522
#214468202
#214467682
#214464982
#214464502
#214464272
#214463972
#214461562
#214461272
#214459032
C4×#145196972
C61×#24922
#2672261