Finite Group Databasebeta

← #29748073

Subgroups — #29748073

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#21450195C211yesmaximal, characteristic, complemented
2#21448835C211yesmaximal, characteristic
3#21447133C211yesmaximal, characteristic, complemented
4#21447009C211yesmaximal, characteristic
5#21446954C211yesmaximal, characteristic, complemented
6#21446820C211yesmaximal, characteristic
7#21446765C211yesmaximal, characteristic, complemented
8#21446492C211yesmaximal, characteristic, complemented
9#21446455C211yesmaximal, characteristic, complemented
10#21446428C211yesmaximal, characteristic, complemented
11#21446396C211yesmaximal, characteristic, complemented
12#21446150C211yesmaximal, characteristic, complemented
13#21445905C211yesmaximal, characteristic, complemented
14#21445633C211yesmaximal, characteristic, complemented
15C61×#1872C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C2×#14519776C2211yescharacteristic
18C2×#14519751C2211yescharacteristic
19C2×#14519739C2211yescharacteristic
20C2×C4×C22⋊D61—21no—
21C2×#14519709C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519699C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519697C2211yescharacteristic
27C2×C22⋊#92418C2211yescharacteristic
28C2×C8×Dic61C2211yescharacteristic
29#16839700—21no—
30#16839674—21no—
31#16839669C2211yescharacteristic
32#16839669C2211yescharacteristic
33#16839667—21no—
34#16839663C2211yescharacteristic
35#16839658C2211yescharacteristic
36#16839656—21no—
37#16839652—21no—
38#16839641—21no—
39#16839633—21no—
40#16839622C2211yescharacteristic
41C4⋊C4×Dic61C2211yescharacteristic
42C4⋊C4×Dic61C2211yescharacteristic
43C4×C22⋊Dic61C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×C22⋊Q8×C61C2211yescharacteristic
53C22×C4⋊C4×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C2×C22⋊C8×C61C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C22×Dic122—21no—
60C22×D244C2311yescharacteristic
61C22×C22⋊D61—21no—
62C22×C4×D61—21no—
63C2×#11758201—22no—
64C2×#11758201—41no—
65C2×#11758201—41no—
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519776—22no—
69#14519776—41no—
70#14519751—22no—
71#14519751—22no—
72#14519739—22no—
73#14519739—41no—
74C4×C22⋊D61—44no—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61D411yescharacteristic
82C2×C22⋊Dic61D411yescharacteristic
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61C2311yescharacteristic
90C2×C4×Dic61C2311yescharacteristic
91#14519709—22no—
92#14519706—41no—
93#14519701—22no—
94#14519699—22no—
95#14519699—41no—
96#14519697—22no—
97C22⋊#92418—41no—
98C8×Dic61—41no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684D411yescharacteristic
108#14519684D411yescharacteristic
109C22×Q8×C61C2311yescharacteristic
110C23×C4×C61—21no—
111C23×C4×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C22⋊Q8×C61—22no—
115C22⋊Q8×C61—41no—
116C2×C4⋊C4×C61—22no—
117C2×C4⋊C4×C61—21no—
118C2×C4⋊C4×C61—21no—
119C2×C4⋊C4×C61—41no—
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C22⋊C4×C61—21no—
124#103×C61—22no—
125#103×C61—41no—
126C22⋊C8×C61—41no—
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—
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—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—42no—
146C22×C4×C61—42no—
147C22×C4×C61—81no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2411yescharacteristic
150C2×C22⋊D61—24no—
151C2×C22⋊D61—42no—
152C2×C22⋊D61—42no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159C22×Dic61C2×D411yescharacteristic
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244#4112yes—
166C2×C4×D61—42no—
167C2×C4×D61—42no—
168C2×C4×D61—81no—
169C2×Dic122—42no—
170C2×Dic122—42no—
171C2×Dic122—81no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—42no—
183C4⋊C4×C61—42no—
184C4⋊C4×C61#4112yes—
185C4⋊C4×C61#4112yes—
186C22⋊C4×C61—42no—
187C22⋊Dic61—41no—
188C22⋊Dic61—81no—
189C22⋊Dic61SD1614yescomplemented
190#11758201—44no—
191#11758198—24no—
192#11758198—42no—
193#11758198—42no—
194#11758198—81no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61#4112yes—
201#11758196—42no—
202#11758196—42no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C4×Dic61—21no—
207C4×Dic61—42no—
208C4×Dic61—81no—
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—
234C2×C4×C61—21no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
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—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—84no—
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C61H(5,𝔽2)11yescharacteristic
259C2×C4×C61H(5,𝔽2)11yescharacteristic
260C22⋊D61—48no—
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—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—
280C2×Dic61C2×SD1612yes—
281C2×Dic61C2×SD1612yes—
282D244—22no—
283D244—22no—
284D244—41no—
285D244—41no—
286C4×D61—84no—
287Dic122—84no—
288C8×C61—41no—
289C8×C61—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—41no—
315C4×C61—42no—
316C4×C61—42no—
317C4×C61—81no—
318C4×C61—81no—
319C4×C61#52911yescharacteristic
320C4×C61#53811yescharacteristic
321C4×C61#55511yescharacteristic
322C4×C61#55911yescharacteristic
323Dic61—22no—
324Dic61—22no—
325Dic61—42no—
326Dic61—41no—
327Dic61—41no—
328Dic61—81no—
329Dic61—81no—
330#2492—611no—
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#1872D6111yescharacteristic, complemented
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⋊Q8—611no—
356C2×C22⋊Q8C2×D6111yescharacteristic
357C2×#121—611no—
358C2×C4×D4—611no—
359C2×C4×D4—1221no—
360C22×C4⋊C4C2×D6111yescharacteristic
361C2×#106—611no—
362C2×C4⋊C8—611no—
363C2×#103—611no—
364C2×#103C2×D6111yescharacteristic
365C2×#103C2×D6111yescharacteristic
366C2×#102—611no—
367C2×#102—611no—
368C2×#102—611no—
369C2×C22⋊C8—611no—
370C2×C22⋊C8C2×D6111yescharacteristic
371C2×C4×C8—611no—
372#400—1221no—
373#393—1221no—
374#391—611no—
375#391—611no—
376#391—1221no—
377#390—611no—
378#389—611no—
379#388—1221no—
380#388—1221no—
381#386—1221no—
382#385—1221no—
383#380—611no—
384C4×C4⋊C4—611no—
385C4×C4⋊C4—611no—
386C4×C22⋊C4—611no—
387#340—611no—
388#340—611no—
389#340—611no—
390#340—611no—
391#340—611no—
392#340—611no—
393#340C2×D6111yescharacteristic
394#340C2×D6111yescharacteristic
395#336—611no—
396#336—611no—
397C61#2672211yescharacteristic, complemented, Sylow
398C22×Q8—1221no—
399C22×Q8C22×D6111yescharacteristic
400C22×D4—611no—
401C22×D4—1221no—
402C23×C4—21no—
403C23×C4—1221no—
404C23×C4C22×D6111yescharacteristic
405C2×SD16—1222no—
406C2×SD16—2441no—
407C2×SD16—2441no—
408C22×C8—611no—
409C22×C8—611no—
410C22×C8C22×D6111yescharacteristic
411C22×C8C22×D6111yescharacteristic
412C4⋊Q8—1222no—
413C4⋊Q8—2441no—
414C22⋊Q8—22no—
415C22⋊Q8—41no—
416#121—1222no—
417#121—1222no—
418C4×D4—1222no—
419C4×D4—2441no—
420C4×D4—2444no—
421C2×C4⋊C4—22no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—41no—
425C2×C4⋊C4—611no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4C22×D6111yescharacteristic
431C2×C4⋊C4C22×D6111yescharacteristic
432C2×C4⋊C4C22×D6111yescharacteristic
433C2×C22⋊C4—21no—
434C2×C22⋊C4—611no—
435C2×C22⋊C4—611no—
436C2×C22⋊C4—1221no—
437C2×C22⋊C4—1221no—
438C2×C22⋊C4—1221no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C42—611no—
442C2×C42—611no—
443C2×C42—1221no—
444C2×C42—1221no—
445#106—1222no—
446C4⋊C8—2441no—
447#103—22no—
448#103—41no—
449#103—1222no—
450#102—1222no—
451#102—1222no—
452#102—2441no—
453C22⋊C8—41no—
454C22⋊C8—2441no—
455C4×C8—2441no—
456#95—611no—
457#95—611no—
458#95—1221no—
459#95—1221no—
460#95—1221no—
461#95—1221no—
462#95—1221no—
463#95—1221no—
464#95—1221no—
465#95—1221no—
466C24—1221no—
467C24D4×D6111yescharacteristic
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—4881no—
475C2×Q8—2442no—
476C2×Q8—2442no—
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—42no—
497C22×C4—42no—
498C22×C4—81no—
499C22×C4—611no—
500C22×C4—611no—
501C22×C4—611no—
502C22×C4—611no—
503C22×C4—1221no—
504C22×C4—1221no—
505C22×C4—1221no—
506C22×C4—4881no—
507C22×C4—2442no—
508C22×C4—2442no—
509C22×C4D4×D6111yescharacteristic
510C22×C4C23×D6111yescharacteristic
511SD16—2444no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
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—42no—
529C4⋊C4—42no—
530C4⋊C4—612no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—2442no—
537C4⋊C4—2442no—
538C4⋊C4#1175822212yes—
539C4⋊C4#1175822412yes—
540C22⋊C4—42no—
541C22⋊C4—614no—
542C22⋊C4—1224no—
543C22⋊C4—4881no—
544C22⋊C4—4881no—
545C22⋊C4—2442no—
546C22⋊C4—2442no—
547C22⋊C4—2441no—
548C42—1221no—
549C42—1221no—
550C42—1221no—
551C42—1221no—
552C42—4881no—
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—
567C23SD16×D6112yes—
568C23C2×D4×D6111yescharacteristic
569Q8—41no—
570Q8—41no—
571Q8—41no—
572Q8—41no—
573Q8—4884no—
574D4—1222no—
575D4—1222no—
576D4—2448no—
577D4—2441no—
578D4—2441no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
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—84no—
599C2×C4—612no—
600C2×C4—612no—
601C2×C4—1221no—
602C2×C4—1221no—
603C2×C4—1221no—
604C2×C4—1221no—
605C2×C4—1222no—
606C2×C4—1221no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—1221no—
611C2×C4—1221no—
612C2×C4—4884no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4#41×D6111yescharacteristic
622C2×C4#41×D6111yescharacteristic
623C2×C4C2×#1175822411yescharacteristic
624C2×C4C2×#1175822211yescharacteristic
625C2×C4#1451982211yescharacteristic
626C2×C4#1451982211yescharacteristic
627C8—41no—
628C8—41no—
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#1684027011yescharacteristic
646C22#1684039911yescharacteristic
647C22#1684040211yescharacteristic
648C22C2×SD16×D6111yescharacteristic
649C22#1684056111yescharacteristic
650C22#1684056611yescharacteristic
651C22#1684057511yescharacteristic
652C4—41no—
653C4—42no—
654C4—42no—
655C4—81no—
656C4—81no—
657C4—1222no—
658C4—1222no—
659C4—4881no—
660C4—4881no—
661C4—2441no—
662C4—2442no—
663C4—2441no—
664C4#1684029411yescharacteristic
665C4#1684034411yescharacteristic
666C4#1684047611yescharacteristic
667C4#1684049511yescharacteristic
668C2—42no—
669C2—4882no—
670C2#2144975511yescharacteristic
671C2#2145582411yescharacteristic
672C2#2145608411yescharacteristic
673C2#2145763611yescharacteristic
674C2#2145768011yescharacteristic
675C2#2145787811yescharacteristic
676C2#2145802111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1872C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
hypercenter#1872D61
hypocenterC61#26722
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
1. #21450195C2semidirect
3. #21447133C2semidirect
5. #21446954C2semidirect
7. #21446765C2semidirect
8. #21446492C2semidirect
9. #21446455C2semidirect
10. #21446428C2semidirect
11. #21446396C2semidirect
12. #21446150C2semidirect
13. #21445905C2semidirect
14. #21445633C2semidirect
15. C61×#1872C2semidirect
189. C22⋊Dic61SD16semidirect
342. #1872D61semidirect
397. C61#26722semidirect

Maximal subgroups

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

subgroupindex
#214501952
#214488352
#214471332
#214470092
#214469542
#214468202
#214467652
#214464922
#214464552
#214464282
#214463962
#214461502
#214459052
#214456332
C61×#18722
#2672261