Finite Group Databasebeta

← #29748026

Subgroups — #29748026

Order 15616.

680 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21449615C211yesmaximal, characteristic
2#21448915C211yesmaximal, characteristic, complemented
3#21447311C211yesmaximal, characteristic, complemented
4#21447056C211yesmaximal, characteristic
5#21446970C211yesmaximal, characteristic, complemented
6#21446966C211yesmaximal, characteristic
7#21446748C211yesmaximal, characteristic
8#21446514C211yesmaximal, characteristic
9#21446439C211yesmaximal, characteristic
10#21446438C211yesmaximal, characteristic
11#21446413C211yesmaximal, characteristic, complemented
12#21446384C211yesmaximal, characteristic, complemented
13#21446071C211yesmaximal, characteristic, complemented
14#21445651C211yesmaximal, characteristic, complemented
15C61×#1928C211yesmaximal, characteristic, complemented
16C2×#14519769C2211yescharacteristic
17C2×#14519766C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19C2×#14519710C2211yescharacteristic
20C2×#14519706C2211yescharacteristic
21C2×#14519701C2211yescharacteristic
22C2×#14519701C2211yescharacteristic
23C2×#14519699C2211yescharacteristic
24C2×#14519699C2211yescharacteristic
25C2×#14519697C2211yescharacteristic
26C2×#14519697C2211yescharacteristic
27C2×#14519687C2211yescharacteristic
28#16839699—21no—
29#16839689—21no—
30#16839664C2211yescharacteristic
31#16839663C2211yescharacteristic
32#16839661C2211yescharacteristic
33#16839659—21no—
34#16839658C2211yescharacteristic
35#16839658C2211yescharacteristic
36#16839654—21no—
37#16839654—21no—
38#16839649—21no—
39#16839646—21no—
40C4⋊C4×Dic61C2211yescharacteristic
41C4⋊C4×Dic61C2211yescharacteristic
42C4×#11758196—21no—
43C4×C4⋊Dic61C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839517C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×#107×C61C2211yescharacteristic
53C2×#103×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C61×#391C2211yescharacteristic
56C4×C4⋊C4×C61C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×D244C2311yescharacteristic
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62#14519769—22no—
63#14519769—22no—
64#14519766—22no—
65#14519766—41no—
66Q8×Dic61—22no—
67Q8×Dic61—41no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61D411yescharacteristic
71C2×C4⋊Dic61D411yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61C2311yescharacteristic
89C2×C4×Dic61C2311yescharacteristic
90#14519710—41no—
91#14519706—41no—
92#14519701—22no—
93#14519701—22no—
94#14519699—22no—
95#14519699—22no—
96#14519697—22no—
97#14519697—41no—
98#14519687—41no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684D411yescharacteristic
105#14519684D411yescharacteristic
106C22×Q8×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C42×C61—21no—
113C2×C42×C61C2311yescharacteristic
114#107×C61—22no—
115#103×C61—22no—
116#103×C61—22no—
117#95×C61—21no—
118#95×C61—21no—
119#95×C61—21no—
120C23×D61—21no—
121C2×Q8×C61—21no—
122C2×Q8×C61—21no—
123C2×Q8×C61—21no—
124C2×Q8×C61—21no—
125C2×Q8×C61—21no—
126C2×Q8×C61#4112yes—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2411yescharacteristic
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61C2×D411yescharacteristic
138C22×Dic61C2×D411yescharacteristic
139C22×Dic61C2×D411yescharacteristic
140C22×Dic61C2×D411yescharacteristic
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244—21no—
147C2×C8×C61—21no—
148C2×C8×C61—21no—
149C2×C8×C61—21no—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61#4112yes—
159C42×C61—21no—
160C42×C61—21no—
161C42×C61—81no—
162#11758198—42no—
163#11758198—42no—
164#11758198—42no—
165#11758198—42no—
166#11758198—81no—
167#11758198—81no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—41no—
173C4⋊Dic61—41no—
174C4⋊Dic61—42no—
175C4⋊Dic61—81no—
176C4⋊Dic61#4112yes—
177C4⋊Dic61#4112yes—
178#11758196—24no—
179#11758196—24no—
180#11758196—42no—
181#11758196—81no—
182C4×Dic61—21no—
183C4×Dic61—21no—
184C4×Dic61—21no—
185C4×Dic61—21no—
186C4×Dic61—42no—
187C4×Dic61—42no—
188C4×Dic61—81no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—21no—
194C2×#92418—21no—
195C23×C61C22×D411yescharacteristic
196C22×D61—41no—
197C22×D61—41no—
198C22×D61—41no—
199C22×D61—41no—
200C22×D61—41no—
201C22×D61—41no—
202C22×D61—41no—
203Q8×C61—22no—
204Q8×C61—22no—
205Q8×C61—41no—
206Q8×C61—41no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C61C2×#4111yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C61H(5,𝔽2)11yescharacteristic
230C2×C4×C612−1+411yescharacteristic
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252D244—41no—
253D244—41no—
254D244—41no—
255D244—41no—
256C8×C61—41no—
257C8×C61—41no—
258#92418—41no—
259#92418—41no—
260#26719—611nomaximal, Sylow
261C22×C61#52511yescharacteristic
262C22×C61#54611yescharacteristic
263C22×C61#54811yescharacteristic
264C22×C61#57511yescharacteristic
265C22×C61#57511yescharacteristic
266C22×C61#57611yescharacteristic
267C22×C61#57811yescharacteristic
268C2×D61—81no—
269C2×D61—81no—
270C2×D61—81no—
271C2×D61—81no—
272C2×D61—81no—
273C2×D61—81no—
274C2×D61—81no—
275C4×C61—41no—
276C4×C61—42no—
277C4×C61—41no—
278C4×C61—42no—
279C4×C61—81no—
280C4×C61—81no—
281C4×C61#52911yescharacteristic
282C4×C61#54211yescharacteristic
283C4×C61#55211yescharacteristic
284C4×C61#55911yescharacteristic
285Dic61—42no—
286Dic61—42no—
287Dic61—41no—
288Dic61—41no—
289Dic61—41no—
290Dic61—41no—
291Dic61—81no—
292#2448—611no—
293#2370—611no—
294#2086—611no—
295#2042—611no—
296#2026—611no—
297#2025—611no—
298#1983—611no—
299#1936—611no—
300#1928D6111yescharacteristic, complemented
301#1925—611no—
302#1925—611no—
303#1921—611no—
304#1917—611no—
305#1863—611no—
306#1763—611no—
307C2×C61#245811yescharacteristic
308C2×C61#311111yescharacteristic
309C2×C61#312411yescharacteristic
310C2×C61#330911yescharacteristic
311C2×C61#332711yescharacteristic
312C2×C61#337011yescharacteristic
313C2×C61#337711yescharacteristic
314D61—161no—
315C2×#125—611no—
316C2×#124—611no—
317C2×C4×Q8—611no—
318C2×#107—611no—
319C2×#107C2×D6111yescharacteristic
320C2×C4⋊C8—611no—
321C2×#103—611no—
322C2×#103—611no—
323C2×#103C2×D6111yescharacteristic
324C2×#103C2×D6111yescharacteristic
325C2×#102—611no—
326C2×#102—611no—
327C2×#102—611no—
328C2×#102—611no—
329C2×#97—611no—
330#399—1221no—
331#397—1221no—
332#391C2×D6111yescharacteristic
333#390—611no—
334#390—611no—
335#389—611no—
336#389—611no—
337#389—611no—
338#389—1221no—
339#388—1221no—
340#388—1221no—
341#387—1221no—
342#387—1221no—
343C4×C4⋊C4—611no—
344C4×C4⋊C4—611no—
345C4×C4⋊C4—611no—
346C4×C4⋊C4—1221no—
347C4×C4⋊C4C2×D6111yescharacteristic
348#340—611no—
349#340—611no—
350#340—611no—
351#340—611no—
352#340—611no—
353#340C2×D6111yescharacteristic
354#336—611no—
355#336—611no—
356#336—611no—
357#336C2×D6111yescharacteristic
358C61#2671911yescharacteristic, complemented, Sylow
359C22×Q8C22×D6111yescharacteristic
360C22×D4—611no—
361C22×C8—611no—
362C22×C8—611no—
363C22×C8C22×D6111yescharacteristic
364C22×C8C22×D6111yescharacteristic
365#125—1222no—
366#125—1222no—
367#124—1222no—
368#124—2441no—
369C4×Q8—1222no—
370C4×Q8—2441no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—611no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4C22×D6111yescharacteristic
383C2×C4⋊C4C22×D6111yescharacteristic
384C2×C4⋊C4C22×D6111yescharacteristic
385C2×C22⋊C4—1221no—
386C2×C22⋊C4—1221no—
387C2×C22⋊C4—1221no—
388C2×C22⋊C4—1221no—
389C2×C22⋊C4—1221no—
390C2×C22⋊C4—1221no—
391C2×C42—21no—
392C2×C42—611no—
393C2×C42—611no—
394C2×C42—1221no—
395C2×C42—1221no—
396C2×C42—1221no—
397C2×C42C22×D6111yescharacteristic
398#107—22no—
399#107—2441no—
400C4⋊C8—2441no—
401#103—22no—
402#103—22no—
403#103—1222no—
404#103—1222no—
405#102—1222no—
406#102—1222no—
407#102—1222no—
408#102—2441no—
409#97—2441no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—611no—
414#95—611no—
415#95—1221no—
416#95—1221no—
417#95—1221no—
418#95—1221no—
419#95—1221no—
420C24—1221no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8#1175822212yes—
427C2×D4—1221no—
428C2×D4—1221no—
429C2×D4—1221no—
430C2×D4—1221no—
431C2×D4—1221no—
432C2×D4—1221no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—611no—
438C22×C4—611no—
439C22×C4—611no—
440C22×C4—611no—
441C22×C4—1221no—
442C22×C4—1221no—
443C22×C4—1221no—
444C22×C4D4×D6111yescharacteristic
445C22×C4D4×D6111yescharacteristic
446C22×C4C23×D6111yescharacteristic
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—1221no—
454C2×C8—1221no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—1221no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—612no—
465C4⋊C4—612no—
466C4⋊C4—1224no—
467C4⋊C4—1224no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—4881no—
473C4⋊C4—4881no—
474C4⋊C4—2441no—
475C4⋊C4—2441no—
476C4⋊C4—2442no—
477C4⋊C4—2442no—
478C4⋊C4#1175822412yes—
479C22⋊C4—4881no—
480C22⋊C4—4881no—
481C22⋊C4—2442no—
482C22⋊C4—2442no—
483C22⋊C4—2442no—
484C22⋊C4—2442no—
485C42—21no—
486C42—21no—
487C42—81no—
488C42—1221no—
489C42—1221no—
490C42—1221no—
491C42—1221no—
492C42—4881no—
493C42—2442no—
494C42—2442no—
495C23—2441no—
496C23—2441no—
497C23—2441no—
498C23—2441no—
499C23—2441no—
500C23—2441no—
501C23—2441no—
502C23C2×D4×D6111yescharacteristic
503Q8—22no—
504Q8—22no—
505Q8—41no—
506Q8—41no—
507D4—2441no—
508D4—2441no—
509D4—2441no—
510D4—2441no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—1222no—
530C2×C4—1222no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4#41×D6111yescharacteristic
551C2×C4#41×D6111yescharacteristic
552C2×C4C2×#1175822411yescharacteristic
553C2×C4C2×#1175822211yescharacteristic
554C2×C4#1451982311yescharacteristic
555C2×C4#1451982411yescharacteristic
556C8—41no—
557C8—41no—
558C8—2441no—
559C8—2441no—
560C22—4881no—
561C22—4881no—
562C22—4881no—
563C22—4881no—
564C22—4881no—
565C22—4881no—
566C22—4881no—
567C22#1684027011yescharacteristic
568C22#1684040811yescharacteristic
569C22#1684041811yescharacteristic
570C22#1684055911yescharacteristic
571C22#1684056111yescharacteristic
572C22#1684056611yescharacteristic
573C22#1684057511yescharacteristic
574C4—42no—
575C4—41no—
576C4—42no—
577C4—41no—
578C4—81no—
579C4—81no—
580C4—4881no—
581C4—2442no—
582C4—2441no—
583C4—2441no—
584C4—2441no—
585C4—2441no—
586C4—2442no—
587C4#1684029211yescharacteristic
588C4#1684037811yescharacteristic
589C4#1684044511yescharacteristic
590C4#1684049411yescharacteristic
591C2—9761no—
592C2#2144973811yescharacteristic
593C2#2145593111yescharacteristic
594C2#2145608411yescharacteristic
595C2#2145783011yescharacteristic
596C2#2145809211yescharacteristic
597C2#2145857611yescharacteristic
598C2#2145865411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1928C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1928D61
hypocenterC61#26719
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1928

Complements

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

NH ≅ G/Ntype
2. #21448915C2semidirect
3. #21447311C2semidirect
5. #21446970C2semidirect
11. #21446413C2semidirect
12. #21446384C2semidirect
13. #21446071C2semidirect
14. #21445651C2semidirect
15. C61×#1928C2semidirect
300. #1928D61semidirect
358. C61#26719semidirect

Maximal subgroups

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

subgroupindex
#214496152
#214489152
#214473112
#214470562
#214469702
#214469662
#214467482
#214465142
#214464392
#214464382
#214464132
#214463842
#214460712
#214456512
C61×#19282
#2671961