Finite Group Databasebeta

← #29667974

Subgroups — #29667974

Order 15616.

710 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456768C211yesmaximal, characteristic
2#21456763C211yesmaximal, characteristic
3#21456404C211yesmaximal, characteristic
4#21456362C211yesmaximal, characteristic
5#21456360C211yesmaximal, characteristic
6#500×D61C211yesmaximal, characteristic
7#21445499C211yesmaximal, characteristic, complemented
8#21445496C211yesmaximal, characteristic, complemented
9#21445495C211yesmaximal, characteristic, complemented
10#21445436C211yesmaximal, characteristic, complemented
11#21445435C211yesmaximal, characteristic, complemented
12#21445415C211yesmaximal, characteristic, complemented
13#21445414C211yesmaximal, characteristic, complemented
14#21444098C211yesmaximal, characteristic
15C61×#1723C211yesmaximal, characteristic, complemented
16#16840317C2211yescharacteristic
17#16840317C2211yescharacteristic
18C4⋊Q8×D61C2211yescharacteristic
19C4⋊Q8×D61C2211yescharacteristic
20#16840170—22no—
21#16840169—22no—
22#16840167C2211yescharacteristic
23#16840167C2211yescharacteristic
24#16840166—22no—
25#16840165C2211yescharacteristic
26#16840165C2211yescharacteristic
27#16840165C2211yescharacteristic
28#16840165C2211yescharacteristic
29#16840165C2211yescharacteristic
30#16840160C2211yescharacteristic
31#16840158C2211yescharacteristic
32#16840158C2211yescharacteristic
33#16840142C2211yescharacteristic
34#16840141C2211yescharacteristic
35#16840141C2211yescharacteristic
36#16840141C2211yescharacteristic
37#16840114C2211yescharacteristic
38#16840113C2211yescharacteristic
39#16840113C2211yescharacteristic
40#16840112C2211yescharacteristic
41#16840002—22no—
42#16839971—22no—
43#16839970—22no—
44#16839919—22no—
45#16839916—22no—
46#16839854—22no—
47#16839813C2211yescharacteristic
48#16839803—22no—
49#16839801—22no—
50#107×D61—22no—
51C4×#11758189C2211yescharacteristic
52C4×C8×D61C2211yescharacteristic
53#16839484C2211yescharacteristic
54C8×#92418C2211yescharacteristic
55C8×#92418C2211yescharacteristic
56C61×#500C2211yescharacteristic
57C61×#500C2211yescharacteristic
58C61×#498C2211yescharacteristic
59C61×#494C2211yescharacteristic
60C61×#487C2211yescharacteristic
61C61×#487C2211yescharacteristic
62C61×#322C2211yescharacteristic
63C2×Q8×D61—21no—
64C2×Q8×D61—21no—
65C2×Dic244—22no—
66C2×#11758202—22no—
67C2×#11758202—22no—
68C2×#11758202—22no—
69C2×#11758189D412yes—
70C2×C8×D61D412yes—
71#14519777C2311yescharacteristic
72#14519777C2311yescharacteristic
73#14519777C2311yescharacteristic
74#14519777C2311yescharacteristic
75#14519776—21no—
76#14519776—21no—
77#14519776—21no—
78#14519776—21no—
79#14519775—21no—
80#14519775—21no—
81#14519768—21no—
82#14519768—21no—
83#14519757—21no—
84#14519757—21no—
85#14519753—21no—
86#14519753—21no—
87Q8×Dic61—21no—
88Q8×Dic61—21no—
89#14519742—21no—
90#14519742—21no—
91#14519735—21no—
92#14519735—21no—
93C4⋊C4×D61—21no—
94C4⋊C4×D61—21no—
95C4⋊C4×D61—21no—
96C4⋊C4×D61—21no—
97C42×D61C2311yescharacteristic
98#14519711—22no—
99#14519711—22no—
100#14519710—22no—
101#14519710—22no—
102#14519710—22no—
103#14519710—22no—
104#14519710—22no—
105#14519709—22no—
106#14519708—22no—
107#14519708—22no—
108#14519706D412yes—
109#14519702—22no—
110#14519702—22no—
111#14519701—22no—
112#14519700—22no—
113#14519689D412yes—
114#14519687D412yes—
115C4×#92418C2311yescharacteristic
116C4×#92418C2311yescharacteristic
117C4×#92418C2311yescharacteristic
118C8×Dic61D412yes—
119C2×Q16×C61—22no—
120C4⋊Q8×C61C2311yescharacteristic
121C4⋊Q8×C61C2311yescharacteristic
122C4⋊Q8×C61C2311yescharacteristic
123C4⋊Q8×C61C2311yescharacteristic
124#107×C61—22no—
125#107×C61—22no—
126#106×C61—22no—
127#103×C61—22no—
128#103×C61—22no—
129C4×C8×C61C2311yescharacteristic
130C4×C8×C61C2311yescharacteristic
131C4×C8×C61C2311yescharacteristic
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136Q8×D61—82no—
137Q8×D61—82no—
138C2×C4×D61—41no—
139C2×C4×D61—41no—
140C2×C4×D61C2×D411yescharacteristic
141C2×C4×D61C2×D411yescharacteristic
142C2×C4×D61C2×D411yescharacteristic
143C2×Dic122—21no—
144C2×Dic122—21no—
145C2×Dic122—21no—
146C2×Dic122—21no—
147C2×Dic122—41no—
148C2×Dic122—41no—
149Q16×C61—82no—
150C2×C8×C61C2×D412yes—
151C2×C8×C61C2×D412yes—
152C2×C8×C61C2×D412yes—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C42×C61C2411yescharacteristic
162#11758202—44no—
163#11758202—82no—
164#11758202—82no—
165#11758198—41no—
166#11758198—41no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—41no—
176C4⋊Dic61—41no—
177#11758196—41no—
178#11758196—41no—
179#11758196—41no—
180#11758196—41no—
181#11758196—41no—
182#11758196—41no—
183C4×Dic61—41no—
184C4×Dic61—41no—
185C4×Dic61C2×D411yescharacteristic
186C4×Dic61C2×D411yescharacteristic
187C4×Dic61C2×D411yescharacteristic
188C2×#92418C2×D412yes—
189C2×#92418C2×D412yes—
190C2×#92418C2×D412yes—
191Dic244—82no—
192#11758189Q1618yescomplemented
193C8×D61—24no—
194C22×D61#12711yescharacteristic
195Q8×C61—42no—
196Q8×C61—42no—
197Q8×C61—42no—
198Q8×C61—42no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61C22×D411yescharacteristic
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C22×D411yescharacteristic
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61#12711yescharacteristic
211C2×Dic61#12711yescharacteristic
212C2×Dic61#12711yescharacteristic
213C4×D61—22no—
214C4×D61—22no—
215C4×D61—82no—
216C4×D61—82no—
217C4×D61C2×Q1614yes—
218Dic122—42no—
219Dic122—42no—
220Dic122—42no—
221Dic122—42no—
222Dic122—82no—
223Dic122—82no—
224C8×C61—22no—
225C8×C61—22no—
226C8×C61C2×Q1614yes—
227#92418—22no—
228#92418C2×Q1614yes—
229#92418#13714yes—
230#19633—611nomaximal, Sylow
231C22×C61C2×#12711yescharacteristic
232C2×D61—21no—
233C2×D61—21no—
234C2×D61#49412yes—
235C4×C61—81no—
236C4×C61—81no—
237C4×C61—81no—
238C4×C61—81no—
239C4×C61C22×Q1612yes—
240C4×C61C2×#13712yes—
241C4×C61#57612yes—
242Dic61—21no—
243Dic61—21no—
244Dic61—81no—
245Dic61—81no—
246Dic61—81no—
247Dic61—81no—
248Dic61#49412yes—
249#3209—611no—
250#3209—611no—
251#3170—611no—
252#3165—611no—
253#3165—611no—
254C2×#500—611no—
255#1723—611no—
256#1723—611no—
257#1723—611no—
258#1723D6111yescharacteristic, complemented
259#1713—611no—
260#1713—611no—
261#1707—611no—
262#1707—611no—
263C8×M16—611no—
264C2×C61C2×#49411yescharacteristic
265C2×C61#315211yescharacteristic
266C2×C61#315511yescharacteristic
267D61—22no—
268#533—611no—
269#533—611no—
270C2×C4⋊Q8—611no—
271C2×C4⋊Q8—611no—
272#501—1222no—
273#501—1222no—
274#500—611no—
275#500—611no—
276#500—611no—
277#500—611no—
278#500—611no—
279#500—611no—
280#500—611no—
281#500—1222no—
282#500C2×D6111yescharacteristic
283#500C2×D6111yescharacteristic
284#498—611no—
285#498—611no—
286#498—611no—
287#498C2×D6111yescharacteristic
288#494—611no—
289#494—611no—
290#494—611no—
291#494—611no—
292#494C2×D6111yescharacteristic
293#487—611no—
294#487—611no—
295#487—611no—
296#487—611no—
297#487C2×D6111yescharacteristic
298#487C2×D6111yescharacteristic
299#467—1222no—
300#462—1222no—
301#462—1222no—
302#451—1222no—
303#451—1222no—
304C4×Q16—1222no—
305#431—611no—
306#428—1222no—
307#428—1222no—
308C2×#107—1222no—
309C4×M16—611no—
310C2×C4×C8—611no—
311#322—611no—
312#322C2×D6111yescharacteristic
313C82—611no—
314C82—611no—
315C61#1963311yescharacteristic, complemented, Sylow
316C22×Q8—1221no—
317C22×Q8—1221no—
318C2×Q16—22no—
319C2×Q16—1222no—
320C2×Q16—1222no—
321C2×Q16—1222no—
322C2×Q16—1222no—
323C2×M16—612no—
324C22×C8—612no—
325C4⋊Q8—611no—
326C4⋊Q8—611no—
327C4⋊Q8—611no—
328C4⋊Q8—611no—
329C4⋊Q8—1221no—
330C4⋊Q8—1221no—
331C4⋊Q8—1221no—
332C4⋊Q8—1221no—
333C4⋊Q8—1221no—
334C4⋊Q8—1221no—
335C4⋊Q8C22×D6111yescharacteristic
336C4⋊Q8C22×D6111yescharacteristic
337C4⋊Q8C22×D6111yescharacteristic
338C4⋊Q8C22×D6111yescharacteristic
339#125—1221no—
340#125—1221no—
341C22⋊Q8—1221no—
342C22⋊Q8—1221no—
343C22⋊Q8—1221no—
344C22⋊Q8—1221no—
345C4×Q8—1221no—
346C4×Q8—1221no—
347C4×Q8—1221no—
348C4×Q8—1221no—
349#117—1221no—
350#117—1221no—
351C2×C4⋊C4—1221no—
352C2×C4⋊C4—1221no—
353C2×C4⋊C4—1221no—
354C2×C4⋊C4—1221no—
355C2×C42—611no—
356#107—22no—
357#107—22no—
358#107—1222no—
359#107—1222no—
360#107—1222no—
361#107—1222no—
362#107—1222no—
363#107—1222no—
364#107—1222no—
365#106—22no—
366#106—1222no—
367#106—1222no—
368#106—1222no—
369C4⋊C8—612no—
370#103—22no—
371#103—22no—
372#103—1222no—
373#103—1222no—
374#103—1222no—
375#103—1222no—
376C22⋊C8—612no—
377#97—612no—
378C4×C8—612no—
379C4×C8—611no—
380C4×C8—611no—
381C4×C8—611no—
382C4×C8C22×D6111yescharacteristic
383C4×C8C22×D6111yescharacteristic
384C4×C8C22×D6111yescharacteristic
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—1221no—
390C2×Q8—1221no—
391C2×Q8—1221no—
392C2×Q8—1221no—
393C2×Q8—4882no—
394C2×Q8—4882no—
395C2×Q8—2441no—
396C2×Q8—2441no—
397C22×C4—611no—
398C22×C4—611no—
399C22×C4—611no—
400C22×C4—2441no—
401C22×C4—2441no—
402Q16—82no—
403Q16—4882no—
404Q16—4882no—
405Q16—4882no—
406Q16—2444no—
407M16—618no—
408C2×C8—612no—
409C2×C8—612no—
410C2×C8—612no—
411C2×C8—1224no—
412C2×C8D4×D6112yes—
413C2×C8D4×D6112yes—
414C2×C8D4×D6112yes—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—1221no—
424C4⋊C4—1221no—
425C4⋊C4—1221no—
426C4⋊C4—1221no—
427C4⋊C4—1221no—
428C4⋊C4—1221no—
429C4⋊C4—1221no—
430C4⋊C4—1221no—
431C4⋊C4—2441no—
432C4⋊C4—2441no—
433C4⋊C4—2441no—
434C4⋊C4—2441no—
435C4⋊C4—2441no—
436C4⋊C4—2441no—
437C4⋊C4—2441no—
438C4⋊C4—2441no—
439C22⋊C4—2441no—
440C22⋊C4—2441no—
441C42—611no—
442C42—611no—
443C42—611no—
444C42—2441no—
445C42—2441no—
446C42C23×D6111yescharacteristic
447C23—611no—
448Q8—42no—
449Q8—42no—
450Q8—42no—
451Q8—42no—
452Q8—4882no—
453Q8—4882no—
454Q8—2442no—
455Q8—2442no—
456Q8—2442no—
457Q8—2442no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—611no—
463C2×C4—614no—
464C2×C4—611no—
465C2×C4—611no—
466C2×C4—1222no—
467C2×C4—1222no—
468C2×C4—4882no—
469C2×C4—4882no—
470C2×C4—2441no—
471C2×C4—2441no—
472C2×C4—2441no—
473C2×C4—2441no—
474C2×C4C2×D4×D6111yescharacteristic
475C2×C4C2×D4×D6111yescharacteristic
476C2×C4C2×D4×D6111yescharacteristic
477C8—22no—
478C8—22no—
479C8—614no—
480C8—614no—
481C8—1222no—
482C8Q16×D6114yes—
483C22—612no—
484C22—1221no—
485C22—1221no—
486C22#127×D6111yescharacteristic
487C4—81no—
488C4—81no—
489C4—81no—
490C4—81no—
491C4—612no—
492C4—1221no—
493C4—1221no—
494C4—4881no—
495C4—4881no—
496C4—4881no—
497C4—4881no—
498C4C2×Q16×D6112yes—
499C4#137×D6112yes—
500C4#1684056612yes—
501C2—1222no—
502C2#494×D6111yescharacteristic
503C2#2145627511yescharacteristic
504C2#2145630311yescharacteristic
subgroupG/H
centerC22#127×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1723C2
socleC22×C61C2×#127
prime radicalC22×D61#127
hypercenter#1723D61
hypocenterC61#19633
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1723

Complements

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

NH ≅ G/Ntype
7. #21445499C2semidirect
8. #21445496C2semidirect
9. #21445495C2semidirect
10. #21445436C2semidirect
11. #21445435C2semidirect
12. #21445415C2semidirect
13. #21445414C2semidirect
15. C61×#1723C2semidirect
192. #11758189Q16semidirect
258. #1723D61semidirect
315. C61#19633semidirect

Maximal subgroups

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

subgroupindex
#214567682
#214567632
#214564042
#214563622
#214563602
#500×D612
#214454992
#214454962
#214454952
#214454362
#214454352
#214454152
#214454142
#214440982
C61×#17232
#1963361