Finite Group Databasebeta

← #29748234

Subgroups — #29748234

Order 15616.

804 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450184C211yesmaximal, characteristic
2#21448850C211yesmaximal, characteristic, complemented
3#21447132C211yesmaximal, characteristic, complemented
4#21447019C211yesmaximal, characteristic, complemented
5#21446958C211yesmaximal, characteristic, complemented
6#21446765C211yesmaximal, characteristic, complemented
7#21446504C211yesmaximal, characteristic, complemented
8#21446442C211yesmaximal, characteristic, complemented
9#21446422C211yesmaximal, characteristic, complemented
10#21446393C211yesmaximal, characteristic, complemented
11#21446176C211yesmaximal, characteristic, complemented
12#21446142C211yesmaximal, characteristic
13#21445915C211yesmaximal, characteristic, complemented
14#103×Dic61C211yesmaximal, characteristic, complemented
15C61×#2000C211yesmaximal, characteristic
16C2×#14519776C2211yescharacteristic
17C2×#14519764—21no—
18C2×Q8×Dic61C2211yescharacteristic
19C2×#14519736C2211yescharacteristic
20C2×#14519710C2211yescharacteristic
21C2×#14519706C2211yescharacteristic
22C2×#14519701C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×C22⋊#92418C2211yescharacteristic
27C2×C8×Dic61C2211yescharacteristic
28#16839676—21no—
29#16839674—21no—
30#16839669C2211yescharacteristic
31#16839665C2211yescharacteristic
32#16839665C2211yescharacteristic
33#16839661C2211yescharacteristic
34#16839656—21no—
35#16839655—21no—
36#16839655—21no—
37#16839648—21no—
38#16839622C2211yescharacteristic
39C4⋊C4×Dic61—21no—
40C4⋊C4×Dic61C2211yescharacteristic
41C4⋊C4×Dic61C2211yescharacteristic
42C4×C22⋊Dic61C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C22×SD16×C61C2211yescharacteristic
53C2×C22⋊Q8×C61C2211yescharacteristic
54C2×#121×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×C22⋊C8×C61C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C22×Dic122—21no—
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62#14519776—22no—
63#14519776—41no—
64#14519764—44no—
65Q8×Dic61—22no—
66Q8×Dic61—41no—
67#14519736—22no—
68#14519736—41no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61D411yescharacteristic
83C2×C22⋊Dic61D411yescharacteristic
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61C2311yescharacteristic
89C2×C4×Dic61C2311yescharacteristic
90#14519710—22no—
91#14519706—41no—
92#14519701—22no—
93#14519701—41no—
94#14519698—22no—
95#14519698—41no—
96C22⋊#92418—41no—
97C8×Dic61—41no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684D411yescharacteristic
105#14519684D411yescharacteristic
106C22×Q8×C61C2311yescharacteristic
107C22×D4×C61—21no—
108C22×D4×C61C2311yescharacteristic
109C23×C4×C61C2311yescharacteristic
110C2×SD16×C61—22no—
111C2×SD16×C61—41no—
112C2×SD16×C61—41no—
113C22×C8×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C22⋊Q8×C61—22no—
116C22⋊Q8×C61—22no—
117#121×C61—22no—
118#121×C61—41no—
119C2×C4⋊C4×C61—21no—
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C22⋊C4×C61—21no—
123C2×C22⋊C4×C61—21no—
124#103×C61—22no—
125#102×C61—22no—
126C22⋊C8×C61—41no—
127C24×C61—21no—
128C24×C61C2×D411yescharacteristic
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61#4112yes—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—42no—
142C2×D4×C61—42no—
143C2×D4×C61—81no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2411yescharacteristic
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163C2×Dic122—42no—
164C2×Dic122—42no—
165C2×Dic122—81no—
166SD16×C61—44no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C4⋊C4×C61—24no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61#4112yes—
177C4⋊C4×C61#4112yes—
178C22⋊C4×C61—24no—
179C22⋊C4×C61—42no—
180C22⋊Dic61—22no—
181C22⋊Dic61—22no—
182C22⋊Dic61—42no—
183C22⋊Dic61—42no—
184C22⋊Dic61—42no—
185C22⋊Dic61—42no—
186C22⋊Dic61—42no—
187C22⋊Dic61—81no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—42no—
196C4⋊Dic61—42no—
197C4⋊Dic61#4112yes—
198#11758196—42no—
199C4×Dic61—21no—
200C4×Dic61—21no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—42no—
204C4×Dic61—42no—
205C4×Dic61—42no—
206C4×Dic61—42no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C23×C61—21no—
214C23×C61—21no—
215C23×C61—21no—
216C23×C61—21no—
217C23×C61—21no—
218C23×C61—21no—
219C23×C61—21no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61C22×D411yescharacteristic
228Q8×C61—22no—
229Q8×C61—22no—
230Q8×C61—41no—
231Q8×C61—41no—
232D4×C61—41no—
233D4×C61—41no—
234D4×C61—41no—
235D4×C61—41no—
236D4×C61—84no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—22no—
242C2×C4×C61—21no—
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—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61H(5,𝔽2)11yescharacteristic
258C2×C4×C61H(5,𝔽2)11yescharacteristic
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—22no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61#13512yes—
282C2×Dic61AGL(1,ℤ/8)12yes—
283Dic122—84no—
284C8×C61—41no—
285C8×C61—41no—
286#92418—41no—
287#92418—41no—
288#26733—611nomaximal, Sylow
289C22×C61—41no—
290C22×C61—41no—
291C22×C61—41no—
292C22×C61—41no—
293C22×C61—41no—
294C22×C61—41no—
295C22×C61—41no—
296C22×C61—81no—
297C22×C61—81no—
298C22×C61—81no—
299C22×C61—81no—
300C22×C61—81no—
301C22×C61—81no—
302C22×C61—81no—
303C22×C61#52511yescharacteristic
304C22×C61#54611yescharacteristic
305C22×C61#54611yescharacteristic
306C22×C61C2×#13511yescharacteristic
307C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
308C22×C61#57711yescharacteristic
309C22×C61#57811yescharacteristic
310C4×C61—42no—
311C4×C61—41no—
312C4×C61—42no—
313C4×C61—42no—
314C4×C61#52911yescharacteristic
315C4×C61#53811yescharacteristic
316C4×C61#55511yescharacteristic
317C4×C61#55911yescharacteristic
318Dic61—22no—
319Dic61—22no—
320Dic61—41no—
321Dic61—42no—
322Dic61—41no—
323Dic61—81no—
324Dic61—81no—
325Dic61—81no—
326#2491—611no—
327#2362—611no—
328#2055—611no—
329#2036—611no—
330#2024—611no—
331#2000D6111yescharacteristic
332#1987—611no—
333#1934—611no—
334#1925—611no—
335#1923—611no—
336#1918—611no—
337#1880—611no—
338#1875—611no—
339#1825—611no—
340C4×#103—611no—
341C2×C61—81no—
342C2×C61—161no—
343C2×C61#245911yescharacteristic
344C2×C61#310611yescharacteristic
345C2×C61#312111yescharacteristic
346C2×C61#330011yescharacteristic
347C2×C61#330311yescharacteristic
348C2×C61#330811yescharacteristic
349C2×C61#332411yescharacteristic
350C22×SD16C2×D6111yescharacteristic
351C2×C4⋊Q8—611no—
352C2×#124—1221no—
353C2×C22⋊Q8C2×D6111yescharacteristic
354C2×#121C2×D6111yescharacteristic
355C2×C4×Q8—611no—
356C2×#117—611no—
357C2×#107—611no—
358C2×C4⋊C8—611no—
359C2×#103—611no—
360C2×#103—611no—
361C2×#103C2×D6111yescharacteristic
362C2×#102—611no—
363C2×#102—611no—
364C2×#102C2×D6111yescharacteristic
365C2×C22⋊C8—611no—
366C2×C22⋊C8C2×D6111yescharacteristic
367C2×C4×C8—611no—
368#394—1221no—
369#393—1221no—
370#391—611no—
371#390—611no—
372#390—611no—
373#389—611no—
374#388—1221no—
375#388—1221no—
376#388—1221no—
377#387—1221no—
378#380—611no—
379C4×C4⋊C4—611no—
380C4×C4⋊C4—611no—
381C4×C4⋊C4—1221no—
382C4×C22⋊C4—611no—
383#340—611no—
384#340—611no—
385#340—611no—
386#340—611no—
387#340—611no—
388#340—611no—
389#340—611no—
390#340C2×D6111yescharacteristic
391#336—611no—
392#336—611no—
393C61#2673311yescharacteristic, complemented, Sylow
394C22×Q8—1221no—
395C22×Q8C22×D6111yescharacteristic
396C22×D4—21no—
397C22×D4C22×D6111yescharacteristic
398C23×C4C22×D6111yescharacteristic
399C2×SD16—22no—
400C2×SD16—41no—
401C2×SD16—41no—
402C22×C8—611no—
403C22×C8—611no—
404C22×C8C22×D6111yescharacteristic
405C22×C8C22×D6111yescharacteristic
406C4⋊Q8—1222no—
407C4⋊Q8—2441no—
408#124—2444no—
409C22⋊Q8—22no—
410C22⋊Q8—22no—
411#121—22no—
412#121—41no—
413C4×Q8—1222no—
414C4×Q8—2441no—
415#117—1222no—
416#117—2441no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—611no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4C22×D6111yescharacteristic
426C2×C4⋊C4C22×D6111yescharacteristic
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—611no—
430C2×C22⋊C4—611no—
431C2×C22⋊C4—1221no—
432C2×C22⋊C4—1221no—
433C2×C22⋊C4—1221no—
434C2×C22⋊C4—1221no—
435C2×C22⋊C4—1221no—
436C2×C22⋊C4—1221no—
437C2×C42—611no—
438C2×C42—611no—
439C2×C42—1221no—
440C2×C42—1221no—
441C2×C42—1221no—
442C2×C42—1221no—
443#107—1222no—
444C4⋊C8—2441no—
445#103—22no—
446#103—1222no—
447#103—2441no—
448#102—22no—
449#102—1222no—
450#102—2441no—
451C22⋊C8—41no—
452C22⋊C8—2441no—
453C4×C8—2441no—
454#95—611no—
455#95—611no—
456#95—1221no—
457#95—1221no—
458#95—1221no—
459#95—1221no—
460#95—1221no—
461#95—1221no—
462C24—21no—
463C24D4×D6111yescharacteristic
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—4881no—
470C2×Q8—2442no—
471C2×Q8—2442no—
472C2×Q8#1175822412yes—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—42no—
480C2×D4—42no—
481C2×D4—81no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—611no—
492C22×C4—611no—
493C22×C4—611no—
494C22×C4—611no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—1221no—
499C22×C4D4×D6111yescharacteristic
500C22×C4C23×D6111yescharacteristic
501SD16—44no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—1221no—
514C4⋊C4—24no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—612no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4—2442no—
526C4⋊C4—2442no—
527C4⋊C4—2442no—
528C4⋊C4#1175822412yes—
529C4⋊C4#1175822412yes—
530C22⋊C4—24no—
531C22⋊C4—42no—
532C22⋊C4—1222no—
533C22⋊C4—1222no—
534C22⋊C4—4881no—
535C22⋊C4—2442no—
536C22⋊C4—2442no—
537C22⋊C4—2442no—
538C22⋊C4—2442no—
539C22⋊C4—2442no—
540C42—1221no—
541C42—1221no—
542C42—1221no—
543C42—1221no—
544C42—2442no—
545C42—2442no—
546C42—2442no—
547C42—2442no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23C2×D4×D6111yescharacteristic
563Q8—22no—
564Q8—22no—
565Q8—41no—
566Q8—41no—
567Q8—4884no—
568D4—41no—
569D4—41no—
570D4—41no—
571D4—41no—
572D4—84no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—612no—
590C2×C4—612no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—1222no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4#41×D6111yescharacteristic
614C2×C4C2×#1175822411yescharacteristic
615C2×C4C2×#1175822411yescharacteristic
616C2×C4C2×#1175822411yescharacteristic
617C2×C4#1451982211yescharacteristic
618C2×C4#1451982211yescharacteristic
619C8—41no—
620C8—41no—
621C8—2441no—
622C8—2441no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22#1684026811yescharacteristic
638C22#1684039911yescharacteristic
639C22#1684040111yescharacteristic
640C22#135×D6111yescharacteristic
641C22AGL(1,ℤ/8)×D6111yescharacteristic
642C22#1684056811yescharacteristic
643C22#1684057811yescharacteristic
644C4—42no—
645C4—41no—
646C4—42no—
647C4—42no—
648C4—1222no—
649C4—1222no—
650C4—4881no—
651C4—4881no—
652C4—4881no—
653C4—2441no—
654C4—2442no—
655C4—2441no—
656C4#1684029311yescharacteristic
657C4#1684034711yescharacteristic
658C4#1684047611yescharacteristic
659C4#1684049811yescharacteristic
660C2—81no—
661C2—161no—
662C2#2144975111yescharacteristic
663C2#2145587211yescharacteristic
664C2#2145604111yescharacteristic
665C2#2145769911yescharacteristic
666C2#2145774211yescharacteristic
667C2#2145780511yescharacteristic
668C2#2145804311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2000C2
socleC23×C61C22×D4
prime radicalC2×#121×C61C22
hypercenter#2000D61
hypocenterC61#26733
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2000

Complements

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

NH ≅ G/Ntype
2. #21448850C2semidirect
3. #21447132C2semidirect
4. #21447019C2semidirect
5. #21446958C2semidirect
6. #21446765C2semidirect
7. #21446504C2semidirect
8. #21446442C2semidirect
9. #21446422C2semidirect
10. #21446393C2semidirect
11. #21446176C2semidirect
13. #21445915C2semidirect
14. #103×Dic61C2semidirect
393. C61#26733semidirect

Maximal subgroups

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

subgroupindex
#214501842
#214488502
#214471322
#214470192
#214469582
#214467652
#214465042
#214464422
#214464222
#214463932
#214461762
#214461422
#214459152
#103×Dic612
C61×#20002
#2673361