Finite Group Databasebeta

← #29733372

Subgroups — #29733372

Order 15616.

1126 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840132C211yesmaximal, characteristic, complemented
2C2×#16839972C211yesmaximal, characteristic, complemented
3#21450772C211yesmaximal, characteristic
4#21446827C212yesmaximal, complemented
5#21446827C212yesmaximal, complemented
6#21446777C211yesmaximal, characteristic, complemented
7#21446618C211yesmaximal, characteristic
8#21446131C212yesmaximal, complemented
9#21446126C212yesmaximal, complemented
10#21445649C211yesmaximal, characteristic, complemented
11C2×C61×#552C211yesmaximal, characteristic
12C22×#11758202C2212yes—
13C2×#14519777C2211yescharacteristic
14C2×#14519756C2212yes—
15C2×#14519756C2212yes—
16#16840132—24no—
17#16839972—24no—
18C2×C4⋊#92418C2211yescharacteristic
19C2×#14519701C2212yes—
20C2×#14519700C2212yes—
21C2×#14519700C2212yes—
22C2×#14519700C2212yes—
23C2×C22⋊#92418C2212yes—
24C2×C22⋊#92418C2212yes—
25C2×#14519688C2211yescharacteristic
26#16839698—21no—
27#16839698—21no—
28#16839665—21no—
29#16839631—21no—
30C4×C4⋊Dic61C2211yescharacteristic
31#16839524C2212yes—
32#16839524C2212yes—
33#16839524C2212yes—
34C61×#552—24no—
35C61×#552—24no—
36C2×#126×C61—21no—
37C2×#125×C61—21no—
38C2×#125×C61C2211yescharacteristic
39C2×#123×C61—21no—
40C2×C22⋊Q8×C61C2211yescharacteristic
41C2×C22⋊Q8×C61C2211yescharacteristic
42C2×C4×Q8×C61C2211yescharacteristic
43C2×C4×D4×C61C2211yescharacteristic
44C2×#117×C61—21no—
45C2×#117×C61C2211yescharacteristic
46C22×C4⋊C4×C61C2211yescharacteristic
47C22×Dic122C2312yes—
48C2×#11758202—24no—
49C2×#11758202—42no—
50C2×#11758202—42no—
51C22×#92418C2312yes—
52C22×#92418C2312yes—
53#14519777—22no—
54#14519777—22no—
55#14519756—24no—
56#14519756—24no—
57#14519756—42no—
58#14519756—42no—
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61C2312yes—
62C2×#11758196—22no—
63C2×#11758196—22no—
64C2×C22⋊Dic61—22no—
65C2×C22⋊Dic61—22no—
66C2×C4×Dic61—21no—
67C4⋊#92418—22no—
68#14519701—24no—
69#14519700—24no—
70#14519700—24no—
71#14519700—24no—
72C22⋊#92418—42no—
73C22⋊#92418—42no—
74#14519688—22no—
75#14519684—21no—
76C22×Q8×C61C2311yescharacteristic
77C22×D4×C61D411yescharacteristic
78C23×C4×C61—21no—
79C23×C4×C61C2311yescharacteristic
80C23×C4×C61C2311yescharacteristic
81#126×C61—28no—
82#125×C61—28no—
83#125×C61—22no—
84#125×C61D414yes—
85#123×C61—28no—
86C22⋊Q8×C61—22no—
87C22⋊Q8×C61—22no—
88C22⋊Q8×C61—22no—
89C22⋊Q8×C61—22no—
90C4×Q8×C61—22no—
91C4×Q8×C61D414yes—
92C4×D4×C61—22no—
93C4×D4×C61—22no—
94#117×C61—28no—
95#117×C61—22no—
96#117×C61—22no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—22no—
104C2×C4⋊C4×C61—22no—
105C2×C4⋊C4×C61D411yescharacteristic
106C2×C4⋊C4×C61C2311yescharacteristic
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C4⋊C4×C61C2311yescharacteristic
109C2×C22⋊C4×C61—21no—
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61D411yescharacteristic
113C2×C22⋊C4×C61D411yescharacteristic
114C2×C42×C61—21no—
115C2×C42×C61—21no—
116C2×C42×C61C2311yescharacteristic
117C24×C61C2×D411yescharacteristic
118C24×C61C2×D411yescharacteristic
119C2×Q8×C61—21no—
120C2×Q8×C61—22no—
121C2×Q8×C61—22no—
122C2×Q8×C61C2×D412yes—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—41no—
126C22×C4×C61—24no—
127C22×C4×C61—21no—
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—22no—
136C22×C4×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61—42no—
140C22×C4×C61—42no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2411yescharacteristic
146C22×Dic61—22no—
147C22×Dic61—22no—
148C2×Dic122—22no—
149C2×Dic122—22no—
150C2×Dic122—22no—
151C2×Dic122—22no—
152C2×Dic122—22no—
153C2×Dic122—22no—
154C4⋊C4×C61—24no—
155C4⋊C4×C61—24no—
156C4⋊C4×C61—24no—
157C4⋊C4×C61—24no—
158C4⋊C4×C61—24no—
159C4⋊C4×C61—24no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61C2×D412yes—
165C4⋊C4×C61C2×D412yes—
166C4⋊C4×C61C2×D412yes—
167C22⋊C4×C61—24no—
168C22⋊C4×C61—24no—
169C22⋊C4×C61—24no—
170C22⋊C4×C61—22no—
171C22⋊C4×C61—22no—
172C42×C61—24no—
173C42×C61—24no—
174C42×C61C2×D412yes—
175C42×C61C2×D412yes—
176C22⋊Dic61—44no—
177C22⋊Dic61—44no—
178#11758202—48no—
179C4⋊Dic61—22no—
180C4⋊Dic61—42no—
181C4⋊Dic61—42no—
182C4⋊Dic61#4114yes—
183#11758196—44no—
184#11758196—44no—
185C4×Dic61—42no—
186C2×#92418—22no—
187C2×#92418—22no—
188C2×#92418—22no—
189C2×#92418—22no—
190C2×#92418—22no—
191C2×#92418—22no—
192C23×C61—21no—
193C23×C61—22no—
194C23×C61—21no—
195C23×C61—22no—
196C23×C61—21no—
197C23×C61—22no—
198C23×C61—21no—
199C23×C61—21no—
200C23×C61—22no—
201C23×C61#13712yes—
202C23×C61C22×D411yescharacteristic
203Q8×C61—22no—
204Q8×C61—22no—
205Q8×C61—42no—
206D4×C61—44no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—22no—
210C2×C4×C61—21no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—21no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—48no—
219C2×C4×C61—42no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—42no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61#12012yes—
232C2×C4×C61#12012yes—
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C2×#4112yes—
236C2×C4×C61H(5,𝔽2)12yes—
237C2×Dic61—24no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243Dic122—42no—
244Dic122—42no—
245Dic122—42no—
246Dic122—42no—
247#92418—42no—
248#92418—42no—
249#25584—611nomaximal, Sylow
250C22×C61—22no—
251C22×C61—24no—
252C22×C61—41no—
253C22×C61—41no—
254C22×C61—42no—
255C22×C61—41no—
256C22×C61—41no—
257C22×C61—41no—
258C22×C61—42no—
259C22×C61—42no—
260C22×C61C2×#12011yescharacteristic
261C22×C61#54612yes—
262C22×C61C2×#13711yescharacteristic
263C22×C61#57511yescharacteristic
264C22×C61#57812yes—
265C4×C61—22no—
266C4×C61—22no—
267C4×C61—41no—
268C4×C61—41no—
269C4×C61—42no—
270C4×C61—42no—
271C4×C61—42no—
272C4×C61—42no—
273C4×C61#53211yescharacteristic
274C4×C61#53511yescharacteristic
275C4×C61#55012yes—
276Dic61—44no—
277Dic61—82no—
278C2×#552D6111yescharacteristic
279C2×#491—611no—
280C2×#462—611no—
281#2542—611no—
282#2001—612no—
283#2001—612no—
284#1989—611no—
285#1955—611no—
286#1873—612no—
287#1872—612no—
288#1763—611no—
289C2×C61—42no—
290C2×C61—81no—
291C2×C61#243111yescharacteristic
292C2×C61#300711yescharacteristic
293C2×C61#301111yescharacteristic
294C2×C61#331212yes—
295C2×C61#331512yes—
296C22×Q16—612no—
297#552—24no—
298#552—24no—
299C2×C4⋊Q8—611no—
300C2×#126—21no—
301C2×#125—21no—
302C2×#125C2×D6111yescharacteristic
303C2×#123—21no—
304C2×C22⋊Q8—612no—
305C2×C22⋊Q8—612no—
306C2×C22⋊Q8C2×D6111yescharacteristic
307C2×C22⋊Q8C2×D6111yescharacteristic
308C2×C4×Q8C2×D6111yescharacteristic
309C2×C4×D4C2×D6111yescharacteristic
310C2×#117—21no—
311C2×#117C2×D6111yescharacteristic
312C22×C4⋊C4C2×D6111yescharacteristic
313#491—1224no—
314#462—1224no—
315C2×C4⋊C8—611no—
316C2×#103—612no—
317C2×#103—612no—
318C2×#103—612no—
319C2×#103—612no—
320C2×C22⋊C8—612no—
321C2×C22⋊C8—612no—
322C2×#97—611no—
323#399—1221no—
324#399—1221no—
325#390—1221no—
326#384—1221no—
327C4×C4⋊C4—611no—
328#340—612no—
329#340—612no—
330#340—612no—
331C61#2558411yescharacteristic, complemented, Sylow
332C22×Q8—612no—
333C22×Q8C22×D6111yescharacteristic
334C22×D4C22⋊D6111yescharacteristic
335C23×C4—21no—
336C23×C4C22×D6111yescharacteristic
337C23×C4C22×D6111yescharacteristic
338C2×Q16—1224no—
339C2×Q16—2442no—
340C2×Q16—2442no—
341C22×C8—612no—
342C22×C8—612no—
343C4⋊Q8—1222no—
344C4⋊Q8—1222no—
345#126—28no—
346#125—28no—
347#125—22no—
348#125C22⋊D6114yes—
349#123—28no—
350C22⋊Q8—22no—
351C22⋊Q8—22no—
352C22⋊Q8—22no—
353C22⋊Q8—22no—
354C22⋊Q8—1224no—
355C22⋊Q8—1224no—
356C22⋊Q8—2442no—
357C22⋊Q8—2442no—
358C4×Q8—22no—
359C4×Q8C22⋊D6114yes—
360C4×D4—22no—
361C4×D4—22no—
362#117—28no—
363#117—22no—
364#117—22no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—612no—
374C2×C4⋊C4—1222no—
375C2×C4⋊C4—1222no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4C22⋊D6111yescharacteristic
379C2×C4⋊C4C22×D6111yescharacteristic
380C2×C4⋊C4C22×D6111yescharacteristic
381C2×C4⋊C4C22×D6111yescharacteristic
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—1222no—
386C2×C22⋊C4—1222no—
387C2×C22⋊C4C22⋊D6111yescharacteristic
388C2×C22⋊C4C22⋊D6111yescharacteristic
389C2×C42—21no—
390C2×C42—21no—
391C2×C42—1221no—
392C2×C42C22×D6111yescharacteristic
393C4⋊C8—1222no—
394#103—1224no—
395#103—1224no—
396#103—1224no—
397#103—1224no—
398C22⋊C8—2442no—
399C22⋊C8—2442no—
400#97—1222no—
401#95—1221no—
402C24C2×C22⋊D6111yescharacteristic
403C24C2×C22⋊D6111yescharacteristic
404C2×Q8—21no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—1222no—
408C2×Q8—1222no—
409C2×Q8—1222no—
410C2×Q8—1222no—
411C2×Q8—1222no—
412C2×Q8—1222no—
413C2×Q8C2×C22⋊D6112yes—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—41no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4C2×C22⋊D6111yescharacteristic
435C22×C4C2×C22⋊D6111yescharacteristic
436C22×C4C2×C22⋊D6111yescharacteristic
437C22×C4C2×C22⋊D6111yescharacteristic
438C22×C4C23×D6111yescharacteristic
439Q16—2448no—
440C2×C8—1222no—
441C2×C8—1222no—
442C2×C8—1222no—
443C2×C8—1222no—
444C2×C8—1222no—
445C2×C8—1222no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—614no—
457C4⋊C4—1222no—
458C4⋊C4—2444no—
459C4⋊C4—2444no—
460C4⋊C4—2442no—
461C4⋊C4—2442no—
462C4⋊C4C2×C22⋊D6112yes—
463C4⋊C4C2×C22⋊D6112yes—
464C4⋊C4C2×C22⋊D6112yes—
465C22⋊C4—24no—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—2444no—
471C22⋊C4—2444no—
472C42—24no—
473C42—24no—
474C42—2442no—
475C42C2×C22⋊D6112yes—
476C42C2×C22⋊D6112yes—
477C23—21no—
478C23—21no—
479C23—22no—
480C23—22no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—22no—
485C23—22no—
486C23#1451980712yes—
487C23C22×C22⋊D6111yescharacteristic
488Q8—22no—
489Q8—22no—
490Q8—42no—
491Q8—2442no—
492Q8—2442no—
493Q8—2442no—
494Q8—2442no—
495D4—44no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—48no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—1224no—
521C2×C4—2442no—
522C2×C4—2442no—
523C2×C4—2442no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4#1451974612yes—
527C2×C4#1451974612yes—
528C2×C4C22×C22⋊D6111yescharacteristic
529C2×C4C22×C22⋊D6111yescharacteristic
530C2×C4#41×D6112yes—
531C2×C4#1451982312yes—
532C8—2442no—
533C8—2442no—
534C22—24no—
535C22—22no—
536C22—41no—
537C22—41no—
538C22—42no—
539C22—41no—
540C22—42no—
541C22—41no—
542C22—41no—
543C22—42no—
544C22C2×#1451974611yescharacteristic
545C22#1684040012yes—
546C22C2×#1451980711yescharacteristic
547C22#1684056011yescharacteristic
548C22#1684057712yes—
549C4—22no—
550C4—22no—
551C4—42no—
552C4—41no—
553C4—41no—
554C4—42no—
555C4—42no—
556C4—42no—
557C4—4882no—
558C4—2444no—
559C4#1684030811yescharacteristic
560C4#1684032611yescharacteristic
561C4#1684043112yes—
562C2—42no—
563C2—81no—
564C2#2144943411yescharacteristic
565C2#2145507911yescharacteristic
566C2#2145511411yescharacteristic
567C2#2145786412yes—
568C2#2145790912yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#552C2
socleC23×C61C22×D4
prime radicalC22×D4×C61D4
hypercenterC2×#552D61
hypocenterC61#25584
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#552

Complements

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

NH ≅ G/Ntype
1. C2×#16840132C2semidirect
2. C2×#16839972C2semidirect
4. #21446827C2semidirect
5. #21446827C2semidirect
6. #21446777C2semidirect
8. #21446131C2semidirect
9. #21446126C2semidirect
10. #21445649C2semidirect
331. C61#25584semidirect

Maximal subgroups

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

subgroupindex
C2×#168401322
C2×#168399722
#214507722
#214468272
#214467772
#214466182
#214461312
#214461262
#214456492
C2×C61×#5522
#2558461