Finite Group Databasebeta

← #29685342

Subgroups — #29685342

Order 15616.

1122 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459208C211yesmaximal, characteristic
2#21454456C211yesmaximal, characteristic, complemented
3C4×#14519734C211yesmaximal, characteristic, complemented
4#21446799C211yesmaximal, characteristic, complemented
5#21446796C211yesmaximal, characteristic, complemented
6#21446652C211yesmaximal, characteristic, complemented
7#21446264C212yesmaximal, complemented
8#21446259C212yesmaximal, complemented
9#21445631C212yesmaximal, complemented
10#104×Dic61C212yesmaximal, complemented
11C61×#1962C211yesmaximal, characteristic, complemented
12#16840381C414yescomplemented
13#16840373C412yescomplemented
14#16840366C412yescomplemented
15#16840357—22no—
16#16840349—22no—
17#16840319C2211yescharacteristic
18#16840292—24no—
19#16840269C2212yes—
20#41×Dic61C2212yes—
21C2×#14519734C2211yescharacteristic
22C2×#14519734C2211yescharacteristic
23C2×C42×D61C2211yescharacteristic
24#16839822—24no—
25#16839786C2211yescharacteristic
26C2×#14519704C2212yes—
27C2×#14519704C2212yes—
28#16839725C2212yes—
29M16×Dic61C2212yes—
30C4×#11758196—22no—
31C4×#11758198—22no—
32#16839608C2211yescharacteristic
33#16839608C2211yescharacteristic
34C42×Dic61C2211yescharacteristic
35#16839523C2212yes—
36#16839523C2212yes—
37#16839522C2212yes—
38#16839522C2212yes—
39C61×#532C2211yescharacteristic
40C61×#423C2211yescharacteristic
41C2×#104×C61C2212yes—
42C2×#104×C61C2212yes—
43C43×C61C2211yescharacteristic
44C22×C4×D61—21no—
45C22×C4×D61C2311yescharacteristic
46C2×#11758194C2312yes—
47#14519777C2×C412yescomplemented
48#14519772—24no—
49#14519770—24no—
50#14519769C2×C412yescomplemented
51#14519767—24no—
52#14519763—24no—
53#14519761—24no—
54#14519758—22no—
55#14519758—24no—
56#14519756—22no—
57#14519754—24no—
58#14519753C2×C414yes—
59#14519749—24no—
60#14519748C2×C414yes—
61#14519745—22no—
62Q8×Dic61C2×C414yescomplemented
63C4×Dic122—22no—
64D4×Dic61—24no—
65D4×Dic61C2×C414yescomplemented
66C4×C22⋊D61—24no—
67#14519736—22no—
68#14519736C2312yes—
69#14519734—22no—
70#14519734—22no—
71#14519734C2×C412yes—
72#14519734C2×C412yes—
73#14519734D414yes—
74C2×#11758196—21no—
75C2×#11758196D412yes—
76C2×#11758198—21no—
77C2×#11758198D412yes—
78C2×C4×Dic61—21no—
79C2×C4×Dic61—22no—
80C2×C4×Dic61—22no—
81C2×C4×Dic61C2311yescharacteristic
82C2×C4×Dic61C2311yescharacteristic
83C2×C4×Dic61C2311yescharacteristic
84C42×D61—22no—
85C42×D61—22no—
86C4⋊#92418—22no—
87#14519704—24no—
88#14519704—24no—
89#14519689—24no—
90#14519687—22no—
91C8×Dic61—22no—
92#14519684—22no—
93#14519684—22no—
94C2×#41×C61C2312yes—
95C2×M16×C61C2312yes—
96C4⋊Q8×C61C2×C411yescharacteristic, complemented
97#127×C61C2×C411yescharacteristic, complemented
98#124×C61C2×C412yescomplemented
99#121×C61—22no—
100C4×D4×C61—22no—
101C2×C42×C61—22no—
102C2×C42×C61C2311yescharacteristic
103C2×C42×C61C2311yescharacteristic
104C2×C42×C61C2311yescharacteristic
105C4⋊C8×C61—22no—
106#104×C61—24no—
107#104×C61—24no—
108C23×D61C2×D411yescharacteristic
109#41×C61—22no—
110#41×C61—22no—
111#41×C61—22no—
112#41×C61—22no—
113C2×Q8×C61C22×C412yes—
114C2×D4×C61—22no—
115C2×D4×C61—22no—
116C2×D4×C61C22×C412yes—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—22no—
120C22×C4×C61C2×D412yes—
121C22×C4×C61C2411yescharacteristic
122C2×C22⋊D61—24no—
123C22×Dic61—21no—
124C22×Dic61—21no—
125C22×Dic61—24no—
126C22×Dic61C2×D411yescharacteristic
127C22×Dic61C2×D412yes—
128C2×C4×D61—21no—
129C2×C4×D61—21no—
130C2×C4×D61—21no—
131C2×C4×D61—21no—
132C2×C4×D61—24no—
133C2×C4×D61—44no—
134C2×C4×D61C22×C412yes—
135C2×C4×D61C2×D412yes—
136C2×Dic122—22no—
137M16×C61—22no—
138M16×C61—22no—
139C2×C8×C61—22no—
140C4⋊C4×C61—22no—
141C22⋊C4×C61—24no—
142C42×C61—21no—
143C42×C61—21no—
144C42×C61—44no—
145C42×C61C22×C411yescharacteristic
146C42×C61C22×C411yescharacteristic
147C42×C61C2×D412yes—
148C42×C61#4112yes—
149C42×C61#4112yes—
150C22⋊Dic61—22no—
151C22⋊Dic61—24no—
152C22⋊Dic61—24no—
153#11758198—24no—
154#11758198—24no—
155#11758198C22⋊C414yes—
156#11758198C22⋊C414yes—
157C4⋊Dic61—22no—
158C4⋊Dic61—24no—
159C4⋊Dic61C22×C414yes—
160#11758196—24no—
161#11758196—24no—
162#11758196—24no—
163#11758196C22⋊C414yes—
164#11758196C22⋊C414yes—
165C4×Dic61—24no—
166C4×Dic61—22no—
167C4×Dic61—21no—
168C4×Dic61—21no—
169C4×Dic61—24no—
170C4×Dic61—44no—
171C4×Dic61C22×C412yes—
172C4×Dic61C2×D412yes—
173C4×Dic61#4112yes—
174C4×Dic61#4112yes—
175#11758194—22no—
176#11758194—22no—
177C2×#92418—22no—
178C23×C61—22no—
179C23×C61C22×D411yescharacteristic
180C22×D61—21no—
181C22×D61—21no—
182C22×D61—42no—
183C22×D61C2×C22⋊C412yes—
184Q8×C61—22no—
185Q8×C61—22no—
186D4×C61—22no—
187D4×C61—22no—
188D4×C61—42no—
189D4×C61—42no—
190D4×C61—82no—
191C2×C4×C61—24no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—42no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61C2×C22⋊C412yes—
200C2×C4×C61C2×C22⋊C412yes—
201C2×C4×C61C23×C411yescharacteristic
202C2×C4×C61C22×D411yescharacteristic
203C2×C4×C61C2×#4111yescharacteristic
204C2×C4×C61C2×#4111yescharacteristic
205C2×C4×C61C2×#4111yescharacteristic
206C2×C4×C61C2×#4111yescharacteristic
207C22⋊D61—84no—
208C2×Dic61—21no—
209C2×Dic61—21no—
210C2×Dic61—24no—
211C2×Dic61—22no—
212C2×Dic61—22no—
213C2×Dic61—42no—
214C2×Dic61—42no—
215C2×Dic61—42no—
216C2×Dic61—84no—
217C2×Dic61C2×C22⋊C414yes—
218C2×Dic61C2×C22⋊C412yes—
219C4×D61—22no—
220C4×D61—22no—
221C4×D61—22no—
222C4×D61—22no—
223C4×D61—48no—
224Dic122—82no—
225C8×C61—42no—
226#92418—42no—
227#21419—611nomaximal, Sylow
228C22×C61—22no—
229C22×C61—44no—
230C22×C61—42no—
231C22×C61C22×C22⋊C411yescharacteristic
232C22×C61#52511yescharacteristic
233C22×C61#52511yescharacteristic
234C2×D61—24no—
235C2×D61—22no—
236C2×D61—42no—
237C2×D61—42no—
238C4×C61—24no—
239C4×C61—22no—
240C4×C61—22no—
241C4×C61—42no—
242C4×C61—42no—
243C4×C61C4×#4111yescharacteristic
244C4×C61C4×#4111yescharacteristic
245C4×C61#53211yescharacteristic
246C4×C61#53211yescharacteristic
247Dic61—24no—
248Dic61—22no—
249Dic61—42no—
250Dic61—42no—
251Dic61—44no—
252#3455—611no—
253#2921—611no—
254C4×#117—611no—
255#1995—611no—
256#1994—611no—
257#1963—611no—
258#1962D6111yescharacteristic, complemented
259#1897—612no—
260#1896—612no—
261#1759—612no—
262C4×#104—612no—
263C2×C61—21no—
264C2×C61—21no—
265C2×C61—82no—
266C2×C61#229711yescharacteristic
267C2×C61#290112yes—
268D61—42no—
269#542—614no—
270#541—612no—
271#540—612no—
272#539—1222no—
273#538—1222no—
274#533—611no—
275#532C2×D6111yescharacteristic
276#529—1224no—
277#525—612no—
278C4×#41—612no—
279C2×#117—611no—
280C2×#117—611no—
281C22×C42—611no—
282#433—1224no—
283#423—611no—
284#423C2×D6111yescharacteristic
285C2×#104—612no—
286C2×#104—612no—
287C2×#104C2×D6112yes—
288C2×#104C2×D6112yes—
289#407—612no—
290C4×M16—612no—
291C4×C4⋊C4—1222no—
292C4×C22⋊C4—1222no—
293#376—611no—
294#376—611no—
295C43—611no—
296C43C2×D6111yescharacteristic
297#339—612no—
298#339—612no—
299#339—612no—
300#339—612no—
301C61#2141911yescharacteristic, complemented, Sylow
302C2×#41C22×D6112yes—
303C23×C4—611no—
304C23×C4—1221no—
305C2×M16—612no—
306C2×M16C22×D6112yes—
307C4⋊Q8—612no—
308C4⋊Q8C4×D6111yescharacteristic, complemented
309#127C4×D6111yescharacteristic, complemented
310#126—1224no—
311#126—1224no—
312#125—612no—
313#125—1224no—
314#124—1224no—
315#124C4×D6112yescomplemented
316#123—1224no—
317#123—1224no—
318#123—1222no—
319C22⋊Q8—614no—
320C22⋊Q8—1224no—
321C22⋊Q8—1222no—
322#121—22no—
323#121—614no—
324#121—1224no—
325#120—1222no—
326C4×Q8—614no—
327C4×Q8—1222no—
328C4×D4—22no—
329C4×D4—614no—
330C4×D4—1224no—
331C4×D4—1224no—
332#117—612no—
333#117—612no—
334#117—614no—
335#117—612no—
336#117—1222no—
337#117—1222no—
338#117—1222no—
339C2×C4⋊C4—612no—
340C2×C4⋊C4—1221no—
341C2×C22⋊C4—612no—
342C2×C22⋊C4—1221no—
343C2×C42—22no—
344C2×C42—611no—
345C2×C42—611no—
346C2×C42—611no—
347C2×C42—1222no—
348C2×C42—1222no—
349C2×C42—1221no—
350C2×C42—1222no—
351C2×C42—1222no—
352C2×C42C22×D6111yescharacteristic
353C2×C42C22×D6111yescharacteristic
354C2×C42C22×D6111yescharacteristic
355C4⋊C8—22no—
356C4⋊C8—1222no—
357#104—24no—
358#104—24no—
359#104—1224no—
360#104—1224no—
361C22⋊C8—1224no—
362#97—1222no—
363C4×C8—1222no—
364#95—1222no—
365#95—1222no—
366C24—611no—
367#41—22no—
368#41—22no—
369#41—22no—
370#41—22no—
371C2×Q8—1222no—
372C2×Q8C2×C4×D6112yes—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—1224no—
376C2×D4C2×C4×D6112yes—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—611no—
381C22×C4—612no—
382C22×C4—612no—
383C22×C4—612no—
384C22×C4—1224no—
385C22×C4—1224no—
386C22×C4—1221no—
387C22×C4—1221no—
388C22×C4—1221no—
389C22×C4—1221no—
390C22×C4—1221no—
391C22×C4—1221no—
392C22×C4—2444no—
393C22×C4D4×D6112yes—
394C22×C4C23×D6111yescharacteristic
395M16—22no—
396M16—22no—
397M16—1222no—
398M16—1222no—
399C2×C8—22no—
400C2×C8—1222no—
401C4⋊C4—22no—
402C4⋊C4—614no—
403C4⋊C4—614no—
404C4⋊C4—614no—
405C4⋊C4—1224no—
406C4⋊C4—1224no—
407C4⋊C4—1224no—
408C4⋊C4—1224no—
409C4⋊C4—1222no—
410C22⋊C4—24no—
411C22⋊C4—614no—
412C22⋊C4—614no—
413C22⋊C4—1224no—
414C22⋊C4—1224no—
415C22⋊C4—1224no—
416C22⋊C4—1224no—
417C22⋊C4—1222no—
418C42—21no—
419C42—21no—
420C42—44no—
421C42—612no—
422C42—612no—
423C42—612no—
424C42—612no—
425C42—1224no—
426C42—1224no—
427C42—1222no—
428C42—1221no—
429C42—1221no—
430C42—2444no—
431C42C2×C4×D6111yescharacteristic
432C42C2×C4×D6111yescharacteristic
433C42D4×D6112yes—
434C42#1175822412yes—
435C42#1175822412yes—
436C23—22no—
437C23—612no—
438C23—1221no—
439C23—1221no—
440C23—2442no—
441C23C2×D4×D6111yescharacteristic
442Q8—22no—
443Q8—22no—
444Q8—4882no—
445D4—22no—
446D4—22no—
447D4—42no—
448D4—42no—
449D4—82no—
450D4—4884no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—614no—
460C2×C4—612no—
461C2×C4—1221no—
462C2×C4—1221no—
463C2×C4—1222no—
464C2×C4—1222no—
465C2×C4—1222no—
466C2×C4—1222no—
467C2×C4—1224no—
468C2×C4—1222no—
469C2×C4—1222no—
470C2×C4—4884no—
471C2×C4—2448no—
472C2×C4—2442no—
473C2×C4—2442no—
474C2×C4—2442no—
475C2×C4C22⋊C4×D6112yes—
476C2×C4C22⋊C4×D6112yes—
477C2×C4C22×C4×D6111yescharacteristic
478C2×C4C2×D4×D6111yescharacteristic
479C2×C4#41×D6111yescharacteristic
480C2×C4#41×D6111yescharacteristic
481C2×C4C2×#1175822411yescharacteristic
482C2×C4C2×#1175822411yescharacteristic
483C8—42no—
484C8—2442no—
485C22—22no—
486C22—44no—
487C22—42no—
488C22—1224no—
489C22—1222no—
490C22—2442no—
491C22—2442no—
492C22C2×C22⋊C4×D6111yescharacteristic
493C22#1684027211yescharacteristic
494C22#1684027211yescharacteristic
495C4—24no—
496C4—22no—
497C4—22no—
498C4—42no—
499C4—42no—
500C4—1224no—
501C4—1222no—
502C4—2442no—
503C4—2442no—
504C4—2444no—
505C4C4×#1175822411yescharacteristic
506C4C4×#1175822411yescharacteristic
507C4#1684031011yescharacteristic
508C4#1684031111yescharacteristic
509C2—21no—
510C2—21no—
511C2—82no—
512C2—2442no—
513C2#2144830611yescharacteristic
514C2#2145436312yes—
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1962C2
socleC22×C61C22×C22⋊C4
prime radical#21459208C2
cosocleC22×C4×C61C24
hypercenter#1962D61
hypocenterC61#21419
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1962

Complements

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

NH ≅ G/Ntype
2. #21454456C2semidirect
3. C4×#14519734C2semidirect
4. #21446799C2semidirect
5. #21446796C2semidirect
6. #21446652C2semidirect
7. #21446264C2semidirect
8. #21446259C2semidirect
9. #21445631C2semidirect
10. #104×Dic61C2semidirect
11. C61×#1962C2semidirect
12. #16840381C4semidirect
13. #16840373C4semidirect
14. #16840366C4semidirect
47. #14519777C2×C4semidirect
50. #14519769C2×C4semidirect
62. Q8×Dic61C2×C4semidirect
65. D4×Dic61C2×C4semidirect
96. C4⋊Q8×C61C2×C4semidirect
97. #127×C61C2×C4semidirect
98. #124×C61C2×C4semidirect
258. #1962D61semidirect
301. C61#21419semidirect
308. C4⋊Q8C4×D61semidirect
309. #127C4×D61semidirect
315. #124C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214592082
#214544562
C4×#145197342
#214467992
#214467962
#214466522
#214462642
#214462592
#214456312
#104×Dic612
C61×#19622
#2141961