Finite Group Databasebeta

← #29753084

Subgroups — #29753084

Order 15616.

700 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#21450241C211yesmaximal, characteristic
2#21447227C211yesmaximal, characteristic, complemented
3#21447200C211yesmaximal, characteristic, complemented
4#21447019C211yesmaximal, characteristic
5#21447019C211yesmaximal, characteristic
6#21446577C211yesmaximal, characteristic, complemented
7#21446554C211yesmaximal, characteristic, complemented
8#21446434C211yesmaximal, characteristic
9#21446408C211yesmaximal, characteristic, complemented
10#21446390C211yesmaximal, characteristic, complemented
11#21446073C211yesmaximal, characteristic, complemented
12#21446018C211yesmaximal, characteristic
13#21445914C211yesmaximal, characteristic
14#21445649C211yesmaximal, characteristic, complemented
15C61×#2362C211yesmaximal, characteristic
16C2×#14519770—21no—
17C2×#14519710C2211yescharacteristic
18C2×#14519710C2211yescharacteristic
19C2×#14519710C2211yescharacteristic
20C2×#14519708C2211yescharacteristic
21C2×C4⋊#92418C2211yescharacteristic
22C2×#14519701C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×C22⋊#92418C2211yescharacteristic
27C2×C22⋊#92418C2211yescharacteristic
28C2×#14519688C2211yescharacteristic
29#16839702—21no—
30#16839687—21no—
31#16839660C2211yescharacteristic
32#16839660C2211yescharacteristic
33#16839647—21no—
34#16839632C2211yescharacteristic
35#16839622C2211yescharacteristic
36#16839622C2211yescharacteristic
37C4×C4⋊Dic61C2211yescharacteristic
38#16839524C2211yescharacteristic
39#16839524C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839517C2211yescharacteristic
47#16839517C2211yescharacteristic
48C2×C22⋊Q8×C61C2211yescharacteristic
49C2×C4×Q8×C61C2211yescharacteristic
50C2×#117×C61C2211yescharacteristic
51C61×#391C2211yescharacteristic
52C61×#389C2211yescharacteristic
53C61×#388—21no—
54C61×#388—21no—
55C61×#387—21no—
56C4×C4⋊C4×C61—21no—
57C4×C4⋊C4×C61C2211yescharacteristic
58C4×C22⋊C4×C61C2211yescharacteristic
59C22×#92418C2311yescharacteristic
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63#14519770—44no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61C2311yescharacteristic
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×#11758196—21no—
71C2×#11758196—21no—
72C2×C22⋊Dic61—21no—
73C2×C22⋊Dic61—21no—
74C2×C4×Dic61—21no—
75#14519710—22no—
76#14519710—22no—
77#14519710—22no—
78#14519708—41no—
79C4⋊#92418—22no—
80#14519701—22no—
81#14519701—22no—
82#14519701—22no—
83#14519701—41no—
84C22⋊#92418—22no—
85C22⋊#92418—41no—
86#14519688—41no—
87#14519684—21no—
88#14519684—21no—
89#14519684—21no—
90#14519684—21no—
91#14519684—21no—
92C22×Q8×C61C2311yescharacteristic
93C23×C4×C61C2311yescharacteristic
94C22⋊Q8×C61—22no—
95C22⋊Q8×C61—22no—
96C4×Q8×C61—22no—
97C4×Q8×C61—41no—
98#117×C61—22no—
99#117×C61—41no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C22⋊C4×C61—21no—
107C2×C22⋊C4×C61—21no—
108C2×C22⋊C4×C61D411yescharacteristic
109C2×C22⋊C4×C61D411yescharacteristic
110C2×C42×C61—21no—
111C2×C42×C61—21no—
112C2×C42×C61—21no—
113C2×C42×C61C2311yescharacteristic
114C2×C42×C61C2311yescharacteristic
115#95×C61—21no—
116#95×C61—21no—
117#95×C61—21no—
118#95×C61—21no—
119#95×C61D411yescharacteristic
120#95×C61D411yescharacteristic
121C24×C61C2×D411yescharacteristic
122C2×Q8×C61—21no—
123C2×Q8×C61—21no—
124C2×Q8×C61—21no—
125C2×Q8×C61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61#4112yes—
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—21no—
137C22×C4×C61—21no—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2411yescharacteristic
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C4⋊C4×C61—24no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—42no—
154C4⋊C4×C61—42no—
155C4⋊C4×C61#4112yes—
156C4⋊C4×C61#4112yes—
157C22⋊C4×C61—24no—
158C22⋊C4×C61—22no—
159C22⋊C4×C61—41no—
160C22⋊C4×C61—42no—
161C42×C61—21no—
162C42×C61—21no—
163C42×C61—21no—
164C42×C61—21no—
165C42×C61—42no—
166C42×C61—42no—
167C42×C61—42no—
168C22⋊Dic61—42no—
169C22⋊Dic61—42no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—42no—
179C4⋊Dic61—42no—
180#11758196—42no—
181#11758196—42no—
182C4×Dic61—42no—
183C2×#92418—21no—
184C2×#92418—21no—
185C2×#92418—21no—
186C2×#92418—21no—
187C2×#92418—21no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—21no—
194C2×#92418#4112yes—
195C23×C61—21no—
196C23×C61—21no—
197C23×C61—21no—
198C23×C61—21no—
199C23×C61—21no—
200C23×C61—21no—
201C23×C61—21no—
202C23×C61C22×D411yescharacteristic
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—22no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61AGL(1,ℤ/8)12yes—
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61H(5,𝔽2)11yescharacteristic
243C2×C4×C612−1+411yescharacteristic
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256#92418—22no—
257#92418—41no—
258#92418—41no—
259#92418—41no—
260#27058—611nomaximal, Sylow
261C22×C61—41no—
262C22×C61—41no—
263C22×C61—41no—
264C22×C61—41no—
265C22×C61—41no—
266C22×C61—41no—
267C22×C61—41no—
268C22×C61#52511yescharacteristic
269C22×C61#54611yescharacteristic
270C22×C61#54711yescharacteristic
271C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
272C22×C61#57511yescharacteristic
273C22×C61#57811yescharacteristic
274C22×C61#57811yescharacteristic
275C4×C61—22no—
276C4×C61—41no—
277C4×C61—42no—
278C4×C61—41no—
279C4×C61—42no—
280C4×C61—41no—
281C4×C61—42no—
282C4×C61—41no—
283C4×C61—81no—
284C4×C61#52911yescharacteristic
285C4×C61#53911yescharacteristic
286C4×C61#55311yescharacteristic
287C4×C61#55611yescharacteristic
288Dic61—81no—
289Dic61—81no—
290Dic61—81no—
291Dic61—81no—
292#2496—611no—
293#2362D6111yescharacteristic
294#2072—611no—
295#2066—611no—
296#2036—611no—
297#2036—611no—
298#1947—611no—
299#1943—611no—
300#1924—611no—
301#1921—611no—
302#1918—611no—
303#1863—611no—
304#1854—611no—
305#1825—611no—
306#1763—611no—
307C2×C61—81no—
308C2×C61#246611yescharacteristic
309C2×C61#311511yescharacteristic
310C2×C61#312711yescharacteristic
311C2×C61#331311yescharacteristic
312C2×C61#333011yescharacteristic
313C2×C61#334311yescharacteristic
314C2×C61#334911yescharacteristic
315C2×#126—1221no—
316C2×C22⋊Q8C2×D6111yescharacteristic
317C2×C4×Q8C2×D6111yescharacteristic
318C2×#117C2×D6111yescharacteristic
319C2×#107—611no—
320C2×#107—611no—
321C2×#107—611no—
322C2×#106—611no—
323C2×C4⋊C8—611no—
324C2×#103—611no—
325C2×#103—611no—
326C2×#103—611no—
327C2×#103—611no—
328C2×C22⋊C8—611no—
329C2×C22⋊C8—611no—
330C2×#97—611no—
331#400—1221no—
332#397—1221no—
333#391C2×D6111yescharacteristic
334#389—611no—
335#389—611no—
336#389C2×D6111yescharacteristic
337#388—21no—
338#388—21no—
339#387—21no—
340#387—1221no—
341#384—611no—
342#380—611no—
343#380—611no—
344C4×C4⋊C4—21no—
345C4×C4⋊C4—611no—
346C4×C4⋊C4C2×D6111yescharacteristic
347C4×C22⋊C4C2×D6111yescharacteristic
348#340—611no—
349#340—611no—
350#340—611no—
351#340—611no—
352#340—611no—
353#340—611no—
354#340—611no—
355#340—611no—
356#336—611no—
357#336—611no—
358C61#2705811yescharacteristic, complemented, Sylow
359C22×Q8C22×D6111yescharacteristic
360C23×C4C22×D6111yescharacteristic
361C22×C8—611no—
362C22×C8—611no—
363C22×C8—611no—
364C22×C8—611no—
365#126—2444no—
366C22⋊Q8—22no—
367C22⋊Q8—22no—
368C4×Q8—22no—
369C4×Q8—41no—
370#117—22no—
371#117—41no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—611no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4C22×D6111yescharacteristic
384C2×C4⋊C4C22×D6111yescharacteristic
385C2×C4⋊C4C22×D6111yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—1221no—
389C2×C22⋊C4—1221no—
390C2×C22⋊C4C22⋊D6111yescharacteristic
391C2×C22⋊C4C22⋊D6111yescharacteristic
392C2×C42—21no—
393C2×C42—21no—
394C2×C42—21no—
395C2×C42—1221no—
396C2×C42C22×D6111yescharacteristic
397C2×C42C22×D6111yescharacteristic
398#107—1222no—
399#107—1222no—
400#107—1222no—
401#106—2441no—
402C4⋊C8—1222no—
403#103—1222no—
404#103—1222no—
405#103—1222no—
406#103—2441no—
407C22⋊C8—1222no—
408C22⋊C8—2441no—
409#97—2441no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—1221no—
415#95—1221no—
416#95—1221no—
417#95—1221no—
418#95—1221no—
419#95C22⋊D6111yescharacteristic
420#95C22⋊D6111yescharacteristic
421C24C2×C22⋊D6111yescharacteristic
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8#1175822212yes—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—1221no—
439C22×C4—1221no—
440C22×C4—1221no—
441C22×C4—1221no—
442C22×C4C2×C22⋊D6111yescharacteristic
443C22×C4C2×C22⋊D6111yescharacteristic
444C22×C4C2×C22⋊D6111yescharacteristic
445C22×C4C2×C22⋊D6111yescharacteristic
446C22×C4C2×C22⋊D6111yescharacteristic
447C22×C4C23×D6111yescharacteristic
448C2×C8—612no—
449C2×C8—1221no—
450C2×C8—1221no—
451C2×C8—1221no—
452C2×C8—1221no—
453C2×C8—1221no—
454C2×C8—1221no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1221no—
460C4⋊C4—24no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—2442no—
476C4⋊C4—2442no—
477C4⋊C4—2442no—
478C4⋊C4—2442no—
479C4⋊C4#1175822212yes—
480C4⋊C4#1175822412yes—
481C22⋊C4—24no—
482C22⋊C4—22no—
483C22⋊C4—41no—
484C22⋊C4—42no—
485C22⋊C4—2442no—
486C22⋊C4—2442no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—42no—
492C42—42no—
493C42—42no—
494C42—2442no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23C22×C22⋊D6111yescharacteristic
503Q8—22no—
504Q8—22no—
505Q8—41no—
506Q8—41no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4#1451980212yes—
550C2×C4#41×D6111yescharacteristic
551C2×C4C2×#1175822411yescharacteristic
552C2×C4C2×#1175822211yescharacteristic
553C2×C4C2×#1175822211yescharacteristic
554C2×C4#1451982211yescharacteristic
555C2×C4#1451982411yescharacteristic
556C8—1222no—
557C8—2441no—
558C8—2441no—
559C8—2441no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22#1684027311yescharacteristic
568C22#1684040611yescharacteristic
569C22#1684041111yescharacteristic
570C22C2×#1451980211yescharacteristic
571C22#1684056011yescharacteristic
572C22#1684057711yescharacteristic
573C22#1684057711yescharacteristic
574C4—22no—
575C4—42no—
576C4—41no—
577C4—42no—
578C4—41no—
579C4—41no—
580C4—42no—
581C4—41no—
582C4—81no—
583C4—4881no—
584C4—4881no—
585C4—4881no—
586C4—4881no—
587C4#1684029211yescharacteristic
588C4#1684035411yescharacteristic
589C4#1684046211yescharacteristic
590C4#1684048211yescharacteristic
591C2—81no—
592C2#2144984911yescharacteristic
593C2#2145597011yescharacteristic
594C2#2145611911yescharacteristic
595C2#2145788511yescharacteristic
596C2#2145812411yescharacteristic
597C2#2145825911yescharacteristic
598C2#2145834311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2362C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#2362D61
hypocenterC61#27058
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2362

Complements

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

NH ≅ G/Ntype
2. #21447227C2semidirect
3. #21447200C2semidirect
6. #21446577C2semidirect
7. #21446554C2semidirect
9. #21446408C2semidirect
10. #21446390C2semidirect
11. #21446073C2semidirect
14. #21445649C2semidirect
358. C61#27058semidirect

Maximal subgroups

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

subgroupindex
#214502412
#214472272
#214472002
#214470192
#214465772
#214465542
#214464342
#214464082
#214463902
#214460732
#214460182
#214459142
#214456492
C61×#23622
#2705861