Finite Group Databasebeta

← #29734240

Subgroups — #29734240

Order 15616.

984 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456219C211yesmaximal, characteristic, complemented
2#21455641C211yesmaximal, characteristic, complemented
3#21454179C211yesmaximal, characteristic, complemented
4#21447045C212yesmaximal, complemented
5#21447040C212yesmaximal, complemented
6#21446813C211yesmaximal, characteristic, complemented
7#21446648C211yesmaximal, characteristic, complemented
8#21446052C212yesmaximal, complemented
9#21446045C212yesmaximal, complemented
10C4×#14519704C211yesmaximal, characteristic, complemented
11C61×#3512C211yesmaximal, characteristic
12C2×#14519769C2211yescharacteristic
13#16840130—21no—
14#16840126—21no—
15#16840116—22no—
16#16840069—22no—
17#16840062—22no—
18#16839805C2212yes—
19#16839786C2211yescharacteristic
20C2×#14519704C2211yescharacteristic
21C2×#14519704C2212yes—
22C2×#14519704C2211yescharacteristic
23C2×#14519704C2212yes—
24#16839763C2212yes—
25#16839763C2212yes—
26#16839763C2212yes—
27C4×#11758194C2211yescharacteristic
28#16839657—24no—
29#16839631C2212yes—
30#16839627C2212yes—
31C42×Dic61C2211yescharacteristic
32#16839531C2212yes—
33#16839531C2212yes—
34#16839521C2211yescharacteristic
35#16839521C2211yescharacteristic
36C61×#553—22no—
37C61×#553—24no—
38C61×#552—22no—
39C61×#541—21no—
40C61×#540—21no—
41C61×#537C2212yes—
42C61×#532C2211yescharacteristic
43C61×#529—22no—
44C2×#125×C61C2211yescharacteristic
45C61×#520C2212yes—
46C4×#41×C61C2211yescharacteristic
47C2×#11758194C2311yescharacteristic
48C2×#11758194C2311yescharacteristic
49C2×#11758194C2312yes—
50#14519769—22no—
51#14519769—21no—
52#14519769—21no—
53C2×C4⋊Dic61C2312yes—
54C2×#11758196—22no—
55C2×#11758196—24no—
56C2×C4×Dic61—22no—
57C2×C4×Dic61C2311yescharacteristic
58C2×C4×Dic61C2311yescharacteristic
59#14519710—22no—
60#14519708—22no—
61C4⋊#92418—22no—
62#14519704—22no—
63#14519704—24no—
64#14519704—22no—
65#14519704—24no—
66#14519701—22no—
67#14519701—22no—
68#14519701—22no—
69#14519698—22no—
70#14519698—22no—
71#14519698—22no—
72#14519688—21no—
73C4×#92418—21no—
74#14519684—24no—
75C2×#41×C61C2312yes—
76C2×#41×C61C2311yescharacteristic
77C4⋊Q8×C61D411yescharacteristic
78#127×C61D411yescharacteristic
79#126×C61—24no—
80#126×C61—24no—
81#125×C61—24no—
82#125×C61—21no—
83#125×C61—22no—
84#125×C61—21no—
85#124×C61D412yes—
86#123×C61—24no—
87#123×C61—24no—
88C22⋊Q8×C61—22no—
89C22⋊Q8×C61—22no—
90#121×C61—24no—
91#121×C61—22no—
92#121×C61—22no—
93#121×C61—22no—
94#121×C61—24no—
95C4×Q8×C61—22no—
96C4×Q8×C61D412yes—
97C4×D4×C61—24no—
98C4×D4×C61—22no—
99C4×D4×C61—22no—
100C4×D4×C61—24no—
101C4×D4×C61—22no—
102C4×D4×C61D412yes—
103#117×C61—24no—
104#117×C61—22no—
105#117×C61—21no—
106#117×C61C2311yescharacteristic
107C2×C4⋊C4×C61—22no—
108C2×C4⋊C4×C61D414yes—
109C2×C4⋊C4×C61C2312yes—
110C2×C42×C61—21no—
111C2×C42×C61C2311yescharacteristic
112#41×C61—22no—
113#41×C61—22no—
114#41×C61—21no—
115#41×C61—21no—
116#41×C61—22no—
117#41×C61—22no—
118#41×C61—21no—
119#41×C61—21no—
120C2×Q8×C61C2×D412yes—
121C2×Q8×C61C2×D411yescharacteristic
122C2×D4×C61—21no—
123C2×D4×C61—22no—
124C2×D4×C61—24no—
125C2×D4×C61—22no—
126C2×D4×C61C2×D412yes—
127C2×D4×C61C2×D411yescharacteristic
128C22×C4×C61—24no—
129C22×C4×C61—21no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61C2×D414yes—
133C22×C4×C61C2×D412yes—
134C22×C4×C61C2411yescharacteristic
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—22no—
138C4⋊C4×C61—24no—
139C4⋊C4×C61—24no—
140C4⋊C4×C61—24no—
141C4⋊C4×C61—24no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61C2×D412yes—
147C22⋊C4×C61—24no—
148C22⋊C4×C61—24no—
149C22⋊C4×C61—24no—
150C22⋊C4×C61—21no—
151C22⋊C4×C61—22no—
152C22⋊C4×C61—22no—
153C22⋊C4×C61—22no—
154C42×C61—24no—
155C42×C61—22no—
156C42×C61—21no—
157C42×C61C2×D411yescharacteristic
158C42×C61C2×D411yescharacteristic
159C42×C61C2×D412yes—
160C4⋊Dic61—22no—
161C4⋊Dic61—22no—
162#11758196—48no—
163#11758196—42no—
164#11758196—42no—
165C4×Dic61—21no—
166C4×Dic61—21no—
167C4×Dic61—44no—
168C4×Dic61#4112yes—
169C4×Dic61#4112yes—
170#11758194—22no—
171#11758194—22no—
172#11758194—21no—
173#11758194—21no—
174#11758194—21no—
175#11758194—21no—
176C2×#92418—21no—
177C2×#92418—21no—
178C2×#92418—22no—
179C23×C61—22no—
180C23×C61—21no—
181C23×C61C22×D411yescharacteristic
182Q8×C61—21no—
183Q8×C61—21no—
184Q8×C61—22no—
185Q8×C61—22no—
186D4×C61—21no—
187D4×C61—21no—
188D4×C61—22no—
189D4×C61—22no—
190D4×C61—42no—
191D4×C61—42no—
192D4×C61—41no—
193D4×C61—41no—
194D4×C61—82no—
195D4×C61—84no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—21no—
199C2×C4×C61—24no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—44no—
206C2×C4×C61—82no—
207C2×C4×C61—84no—
208C2×C4×C61#12012yes—
209C2×C4×C61#12012yes—
210C2×C4×C61C22×D411yescharacteristic
211C2×C4×C61C22×D411yescharacteristic
212C2×C4×C61C2×#4111yescharacteristic
213C2×C4×C61C2×#4111yescharacteristic
214C2×C4×C612−1+411yescharacteristic
215C2×C4×C612−1+411yescharacteristic
216C2×Dic61—24no—
217C2×Dic61—22no—
218C2×Dic61—22no—
219C2×Dic61—42no—
220C2×Dic61—44no—
221C2×Dic61—42no—
222#92418—42no—
223#92418—41no—
224#92418—41no—
225#25668—611nomaximal, Sylow
226C22×C61—22no—
227C22×C61—42no—
228C22×C61—41no—
229C22×C61—42no—
230C22×C61—42no—
231C22×C61—42no—
232C22×C61C2×#12011yescharacteristic
233C22×C61#54811yescharacteristic
234C22×C61#54811yescharacteristic
235C4×C61—22no—
236C4×C61—22no—
237C4×C61—42no—
238C4×C61—41no—
239C4×C61—44no—
240C4×C61—42no—
241C4×C61#53211yescharacteristic
242C4×C61#53611yescharacteristic
243C4×C61#54711yescharacteristic
244C4×C61#54711yescharacteristic
245Dic61—24no—
246Dic61—42no—
247Dic61—82no—
248#3512D6111yescharacteristic
249#3142—611no—
250#3080—611no—
251#2859—611no—
252#2040—612no—
253#2039—612no—
254#1998—611no—
255#1962—611no—
256#1860—612no—
257#1859—612no—
258C4×#104—611no—
259C2×C61—21no—
260C2×C61—21no—
261C2×C61—82no—
262C2×C61—81no—
263C2×C61#243211yescharacteristic
264C2×C61#301812yes—
265#553—22no—
266#553—24no—
267#552—22no—
268#541—21no—
269#540—21no—
270#537C2×D6112yes—
271#532C2×D6111yescharacteristic
272#529—22no—
273C2×#125—611no—
274C2×#125C2×D6111yescharacteristic
275#520C2×D6112yes—
276C4×#41C2×D6111yescharacteristic
277#491—1221no—
278#490—1221no—
279#488—1222no—
280#479—1222no—
281#478—1222no—
282#428—612no—
283#423—611no—
284C2×#104—612no—
285C2×#104—612no—
286C2×#104—611no—
287C2×#104—611no—
288#417—612no—
289#417—612no—
290#417—612no—
291C4×M16—611no—
292#389—1224no—
293#384—612no—
294#382—612no—
295C43—611no—
296#343—612no—
297#343—612no—
298#339—611no—
299#339—611no—
300C61#2566811yescharacteristic, complemented, Sylow
301C2×#41C22×D6111yescharacteristic
302C2×#41C22×D6112yes—
303C2×M16—611no—
304C2×M16—611no—
305C2×M16—612no—
306C4⋊Q8C22⋊D6111yescharacteristic
307#127C22⋊D6111yescharacteristic
308#126—24no—
309#126—24no—
310#125—24no—
311#125—21no—
312#125—21no—
313#125—22no—
314#125—1222no—
315#125—1221no—
316#125—1221no—
317#124C22⋊D6112yes—
318#123—24no—
319#123—24no—
320C22⋊Q8—22no—
321C22⋊Q8—22no—
322#121—24no—
323#121—22no—
324#121—22no—
325#121—22no—
326#121—24no—
327C4×Q8—22no—
328C4×Q8C22⋊D6112yes—
329C4×D4—24no—
330C4×D4—22no—
331C4×D4—22no—
332C4×D4—24no—
333C4×D4—22no—
334C4×D4C22⋊D6112yes—
335#117—24no—
336#117—22no—
337#117—21no—
338#117C22×D6111yescharacteristic
339C2×C4⋊C4—22no—
340C2×C4⋊C4—612no—
341C2×C4⋊C4—1222no—
342C2×C4⋊C4—1224no—
343C2×C4⋊C4C22⋊D6114yes—
344C2×C4⋊C4C22×D6112yes—
345C2×C42—21no—
346C2×C42—611no—
347C2×C42—611no—
348C2×C42—1222no—
349C2×C42C22×D6111yescharacteristic
350#107—1222no—
351#106—1222no—
352C4⋊C8—1222no—
353#104—1224no—
354#104—1224no—
355#104—1222no—
356#104—1222no—
357#103—1222no—
358#103—1222no—
359#103—1222no—
360#102—1222no—
361#102—1222no—
362#102—1222no—
363#97—1221no—
364C4×C8—1221no—
365#95—1224no—
366#41—22no—
367#41—22no—
368#41—21no—
369#41—21no—
370#41—22no—
371#41—22no—
372#41—21no—
373#41—21no—
374C2×Q8C2×C22⋊D6112yes—
375C2×Q8C2×C22⋊D6111yescharacteristic
376C2×D4—21no—
377C2×D4—22no—
378C2×D4—24no—
379C2×D4—22no—
380C2×D4C2×C22⋊D6112yes—
381C2×D4C2×C22⋊D6111yescharacteristic
382C22×C4—24no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—1221no—
387C22×C4—1221no—
388C22×C4—1222no—
389C22×C4C2×C22⋊D6114yes—
390C22×C4C2×C22⋊D6112yes—
391C22×C4C23×D6111yescharacteristic
392M16—1222no—
393M16—1222no—
394M16—1221no—
395M16—1221no—
396M16—1221no—
397M16—1221no—
398C2×C8—1221no—
399C2×C8—1221no—
400C2×C8—1222no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—1222no—
410C4⋊C4—1222no—
411C4⋊C4—2448no—
412C4⋊C4—2442no—
413C4⋊C4—2442no—
414C4⋊C4C2×C22⋊D6112yes—
415C22⋊C4—24no—
416C22⋊C4—24no—
417C22⋊C4—24no—
418C22⋊C4—21no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C42—24no—
423C42—21no—
424C42—22no—
425C42—612no—
426C42—612no—
427C42—1221no—
428C42—1221no—
429C42—2444no—
430C42C2×C22⋊D6111yescharacteristic
431C42C2×C22⋊D6111yescharacteristic
432C42C2×C22⋊D6112yes—
433C23—22no—
434C23—21no—
435C23C22×C22⋊D6111yescharacteristic
436Q8—21no—
437Q8—21no—
438Q8—22no—
439Q8—22no—
440D4—21no—
441D4—21no—
442D4—22no—
443D4—22no—
444D4—42no—
445D4—42no—
446D4—41no—
447D4—41no—
448D4—82no—
449D4—84no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—24no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—44no—
460C2×C4—82no—
461C2×C4—84no—
462C2×C4—1224no—
463C2×C4—1222no—
464C2×C4—1222no—
465C2×C4—2444no—
466C2×C4—2442no—
467C2×C4—2442no—
468C2×C4#1451974612yes—
469C2×C4#1451974612yes—
470C2×C4C22×C22⋊D6111yescharacteristic
471C2×C4C22×C22⋊D6111yescharacteristic
472C2×C4#41×D6111yescharacteristic
473C2×C4#41×D6111yescharacteristic
474C2×C4#1451982411yescharacteristic
475C2×C4#1451982411yescharacteristic
476C8—2442no—
477C8—2441no—
478C8—2441no—
479C22—22no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—41no—
485C22C2×#1451974611yescharacteristic
486C22#1684041911yescharacteristic
487C22#1684041911yescharacteristic
488C4—22no—
489C4—22no—
490C4—44no—
491C4—42no—
492C4—42no—
493C4—41no—
494C4—1224no—
495C4—4882no—
496C4—2442no—
497C4#1684030811yescharacteristic
498C4#1684033311yescharacteristic
499C4#1684041111yescharacteristic
500C4#1684041111yescharacteristic
501C2—21no—
502C2—21no—
503C2—82no—
504C2—81no—
505C2#2144944311yescharacteristic
506C2#2145516812yes—
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3512C2
socleC22×C61C2×#120
hypercenter#3512D61
hypocenterC61#25668
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3512

Complements

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

NH ≅ G/Ntype
1. #21456219C2semidirect
2. #21455641C2semidirect
3. #21454179C2semidirect
4. #21447045C2semidirect
5. #21447040C2semidirect
6. #21446813C2semidirect
7. #21446648C2semidirect
8. #21446052C2semidirect
9. #21446045C2semidirect
10. C4×#14519704C2semidirect
300. C61#25668semidirect

Maximal subgroups

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

subgroupindex
#214562192
#214556412
#214541792
#214470452
#214470402
#214468132
#214466482
#214460522
#214460452
C4×#145197042
C61×#35122
#2566861