Finite Group Databasebeta

← #29667966

Subgroups — #29667966

Order 15616.

750 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#21456971C211yesmaximal, characteristic
2#21456966C211yesmaximal, characteristic
3#21456404C211yesmaximal, characteristic
4#21456362C211yesmaximal, characteristic
5#21456360C211yesmaximal, characteristic
6#494×D61C211yesmaximal, characteristic
7#21445498C211yesmaximal, characteristic, complemented
8#21445496C211yesmaximal, characteristic, complemented
9#21445494C211yesmaximal, characteristic, complemented
10#21445436C211yesmaximal, characteristic, complemented
11#21445436C211yesmaximal, characteristic, complemented
12#21445415C211yesmaximal, characteristic, complemented
13#21445414C211yesmaximal, characteristic, complemented
14#21444098C211yesmaximal, characteristic
15C61×#1723C211yesmaximal, characteristic, complemented
16C2×Q16×D61—22no—
17#16840317C2211yescharacteristic
18#16840317C2211yescharacteristic
19C4⋊Q8×D61C2211yescharacteristic
20C4⋊Q8×D61C2211yescharacteristic
21#16840170—22no—
22#16840169—22no—
23#16840167C2211yescharacteristic
24#16840167C2211yescharacteristic
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#16840140—22no—
38#16840114C2211yescharacteristic
39#16840113C2211yescharacteristic
40#16840113C2211yescharacteristic
41#16840112C2211yescharacteristic
42#16840099—22no—
43#16840096—22no—
44#16840056—22no—
45#16840055—22no—
46#16840002—22no—
47#16839854—22no—
48#16839813C2211yescharacteristic
49#16839803—22no—
50#16839801—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—
69Q16×D61—48no—
70C2×#11758189D412yes—
71C2×C8×D61D412yes—
72#14519777C2311yescharacteristic
73#14519777C2311yescharacteristic
74#14519777C2311yescharacteristic
75#14519777C2311yescharacteristic
76#14519776—21no—
77#14519776—21no—
78#14519776—21no—
79#14519776—21no—
80#14519775—21no—
81#14519775—21no—
82#14519768—21no—
83#14519768—21no—
84#14519757—21no—
85#14519757—21no—
86#14519753—21no—
87#14519753—21no—
88Q8×Dic61—21no—
89Q8×Dic61—21no—
90#14519742—21no—
91#14519742—21no—
92#14519735—21no—
93#14519735—21no—
94C4⋊C4×D61—21no—
95C4⋊C4×D61—21no—
96C4⋊C4×D61—21no—
97C4⋊C4×D61—21no—
98C42×D61C2311yescharacteristic
99#14519711—22no—
100#14519711—22no—
101#14519710—22no—
102#14519710—22no—
103#14519710—22no—
104#14519710—22no—
105#14519710—22no—
106#14519709—22no—
107#14519708—22no—
108#14519708—22no—
109#14519706D412yes—
110#14519702—22no—
111#14519702—22no—
112#14519701—22no—
113#14519700—22no—
114#14519689D412yes—
115#14519687D412yes—
116C4×#92418C2311yescharacteristic
117C4×#92418C2311yescharacteristic
118C4×#92418C2311yescharacteristic
119C8×Dic61D412yes—
120C2×Q16×C61—22no—
121C4⋊Q8×C61C2311yescharacteristic
122C4⋊Q8×C61C2311yescharacteristic
123C4⋊Q8×C61C2311yescharacteristic
124C4⋊Q8×C61C2311yescharacteristic
125#107×C61—22no—
126#107×C61—22no—
127#106×C61—22no—
128#103×C61—22no—
129#103×C61—22no—
130C4×C8×C61C2311yescharacteristic
131C4×C8×C61C2311yescharacteristic
132C4×C8×C61C2311yescharacteristic
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137Q8×D61—44no—
138Q8×D61—44no—
139C2×C4×D61—41no—
140C2×C4×D61—41no—
141C2×C4×D61C2×D411yescharacteristic
142C2×C4×D61C2×D411yescharacteristic
143C2×C4×D61C2×D411yescharacteristic
144C2×Dic122—21no—
145C2×Dic122—21no—
146C2×Dic122—21no—
147C2×Dic122—21no—
148C2×Dic122—41no—
149C2×Dic122—41no—
150Q16×C61—44no—
151C2×C8×C61C2×D412yes—
152C2×C8×C61C2×D412yes—
153C2×C8×C61C2×D412yes—
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—
161C4⋊C4×C61—21no—
162C42×C61C2411yescharacteristic
163#11758202—44no—
164#11758202—44no—
165#11758202—44no—
166#11758198—41no—
167#11758198—41no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—41no—
177C4⋊Dic61—41no—
178#11758196—41no—
179#11758196—41no—
180#11758196—41no—
181#11758196—41no—
182#11758196—41no—
183#11758196—41no—
184C4×Dic61—41no—
185C4×Dic61—41no—
186C4×Dic61C2×D411yescharacteristic
187C4×Dic61C2×D411yescharacteristic
188C4×Dic61C2×D411yescharacteristic
189C2×#92418C2×D412yes—
190C2×#92418C2×D412yes—
191C2×#92418C2×D412yes—
192Dic244—44no—
193#11758189Q1618yescomplemented
194C8×D61—24no—
195C22×D61#12711yescharacteristic
196Q8×C61—42no—
197Q8×C61—42no—
198Q8×C61—42no—
199Q8×C61—42no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C22×D411yescharacteristic
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—41no—
211C2×Dic61#12711yescharacteristic
212C2×Dic61#12711yescharacteristic
213C2×Dic61#12711yescharacteristic
214C4×D61—22no—
215C4×D61—22no—
216C4×D61—82no—
217C4×D61—82no—
218C4×D61C2×Q1614yes—
219Dic122—42no—
220Dic122—42no—
221Dic122—42no—
222Dic122—42no—
223Dic122—82no—
224Dic122—82no—
225C8×C61—22no—
226C8×C61—22no—
227C8×C61C2×Q1614yescomplemented
228#92418—22no—
229#92418C2×Q1614yescomplemented
230#92418AGL(1,ℤ/8)14yes—
231#19632—611nomaximal, Sylow
232C22×C61C2×#12711yescharacteristic
233C2×D61—21no—
234C2×D61—21no—
235C2×D61#49412yes—
236C4×C61—81no—
237C4×C61—81no—
238C4×C61—81no—
239C4×C61—81no—
240C4×C61C22×Q1612yes—
241C4×C61C2×AGL(1,ℤ/8)12yes—
242C4×C61#57812yes—
243Dic61—21no—
244Dic61—21no—
245Dic61—81no—
246Dic61—81no—
247Dic61—81no—
248Dic61—81no—
249Dic61#49412yes—
250#3230—611no—
251#3230—611no—
252#3170—611no—
253#3165—611no—
254#3165—611no—
255C2×#494—611no—
256#1723—611no—
257#1723—611no—
258#1723—611no—
259#1723D6111yescharacteristic, complemented
260#1713—611no—
261#1713—611no—
262#1707—611no—
263#1707—611no—
264C8×M16—611no—
265C2×C61C2×#49411yescharacteristic
266C2×C61#315111yescharacteristic
267C2×C61#315611yescharacteristic
268D61—22no—
269C22×Q16—1222no—
270#533—611no—
271#533—611no—
272C2×C4⋊Q8—611no—
273C2×C4⋊Q8—611no—
274#501—1222no—
275#501—1222no—
276#500—611no—
277#500—611no—
278#500—611no—
279#500—611no—
280#500—611no—
281#500—611no—
282#500—611no—
283#500C2×D6111yescharacteristic
284#500C2×D6111yescharacteristic
285#498—611no—
286#498—611no—
287#498—611no—
288#498C2×D6111yescharacteristic
289#494—611no—
290#494—611no—
291#494—611no—
292#494—611no—
293#494—1222no—
294#494C2×D6111yescharacteristic
295#487—611no—
296#487—611no—
297#487—611no—
298#487—611no—
299#487C2×D6111yescharacteristic
300#487C2×D6111yescharacteristic
301#484—1222no—
302#484—1222no—
303#477—1222no—
304#477—1222no—
305#467—1222no—
306C4×Q16—1222no—
307#431—611no—
308#428—1222no—
309#428—1222no—
310C4×M16—611no—
311C2×C4×C8—611no—
312#322—611no—
313#322C2×D6111yescharacteristic
314C82—611no—
315C82—611no—
316C61#1963211yescharacteristic, complemented, Sylow
317C22×Q8—1221no—
318C22×Q8—1221no—
319C2×Q16—22no—
320C2×Q16—1222no—
321C2×Q16—1222no—
322C2×Q16—1222no—
323C2×Q16—1222no—
324C2×Q16—2448no—
325C2×M16—612no—
326C22×C8—612no—
327C4⋊Q8—611no—
328C4⋊Q8—611no—
329C4⋊Q8—611no—
330C4⋊Q8—611no—
331C4⋊Q8—1221no—
332C4⋊Q8—1221no—
333C4⋊Q8—1221no—
334C4⋊Q8—1221no—
335C4⋊Q8—1221no—
336C4⋊Q8—1221no—
337C4⋊Q8C22×D6111yescharacteristic
338C4⋊Q8C22×D6111yescharacteristic
339C4⋊Q8C22×D6111yescharacteristic
340C4⋊Q8C22×D6111yescharacteristic
341#125—1221no—
342#125—1221no—
343C22⋊Q8—1221no—
344C22⋊Q8—1221no—
345C22⋊Q8—1221no—
346C22⋊Q8—1221no—
347C4×Q8—1221no—
348C4×Q8—1221no—
349C4×Q8—1221no—
350C4×Q8—1221no—
351#117—1221no—
352#117—1221no—
353C2×C4⋊C4—1221no—
354C2×C4⋊C4—1221no—
355C2×C4⋊C4—1221no—
356C2×C4⋊C4—1221no—
357C2×C42—611no—
358#107—22no—
359#107—22no—
360#107—1222no—
361#107—1222no—
362#107—1222no—
363#107—1222no—
364#107—1222no—
365#107—1222no—
366#107—1222no—
367#106—22no—
368#106—1222no—
369#106—1222no—
370#106—1222no—
371C4⋊C8—612no—
372#103—22no—
373#103—22no—
374#103—1222no—
375#103—1222no—
376#103—1222no—
377#103—1222no—
378C22⋊C8—612no—
379#97—612no—
380C4×C8—612no—
381C4×C8—611no—
382C4×C8—611no—
383C4×C8—611no—
384C4×C8C22×D6111yescharacteristic
385C4×C8C22×D6111yescharacteristic
386C4×C8C22×D6111yescharacteristic
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—1221no—
392C2×Q8—1221no—
393C2×Q8—1221no—
394C2×Q8—1221no—
395C2×Q8—2441no—
396C2×Q8—2441no—
397C2×Q8—2444no—
398C2×Q8—2444no—
399C22×C4—611no—
400C22×C4—611no—
401C22×C4—611no—
402C22×C4—2441no—
403C22×C4—2441no—
404Q16—44no—
405Q16—2444no—
406Q16—2444no—
407Q16—2444no—
408Q16—2444no—
409M16—618no—
410C2×C8—612no—
411C2×C8—612no—
412C2×C8—612no—
413C2×C8—1224no—
414C2×C8D4×D6112yes—
415C2×C8D4×D6112yes—
416C2×C8D4×D6112yes—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—1221no—
426C4⋊C4—1221no—
427C4⋊C4—1221no—
428C4⋊C4—1221no—
429C4⋊C4—1221no—
430C4⋊C4—1221no—
431C4⋊C4—1221no—
432C4⋊C4—1221no—
433C4⋊C4—2441no—
434C4⋊C4—2441no—
435C4⋊C4—2441no—
436C4⋊C4—2441no—
437C4⋊C4—2441no—
438C4⋊C4—2441no—
439C4⋊C4—2441no—
440C4⋊C4—2441no—
441C22⋊C4—2441no—
442C22⋊C4—2441no—
443C42—611no—
444C42—611no—
445C42—611no—
446C42—2441no—
447C42—2441no—
448C42C23×D6111yescharacteristic
449C23—611no—
450Q8—42no—
451Q8—42no—
452Q8—42no—
453Q8—42no—
454Q8—4882no—
455Q8—4882no—
456Q8—2442no—
457Q8—2442no—
458Q8—2442no—
459Q8—2442no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—611no—
465C2×C4—614no—
466C2×C4—611no—
467C2×C4—611no—
468C2×C4—1222no—
469C2×C4—1222no—
470C2×C4—4882no—
471C2×C4—4882no—
472C2×C4—2441no—
473C2×C4—2441no—
474C2×C4—2441no—
475C2×C4—2441no—
476C2×C4C2×D4×D6111yescharacteristic
477C2×C4C2×D4×D6111yescharacteristic
478C2×C4C2×D4×D6111yescharacteristic
479C8—22no—
480C8—22no—
481C8—614no—
482C8—614no—
483C8—1222no—
484C8Q16×D6114yescomplemented
485C22—612no—
486C22—1221no—
487C22—1221no—
488C22#127×D6111yescharacteristic
489C4—81no—
490C4—81no—
491C4—81no—
492C4—81no—
493C4—612no—
494C4—1221no—
495C4—1221no—
496C4—4881no—
497C4—4881no—
498C4—4881no—
499C4—4881no—
500C4C2×Q16×D6112yes—
501C4AGL(1,ℤ/8)×D6112yes—
502C4#1684057812yes—
503C2—1222no—
504C2#494×D6111yescharacteristic
505C2#2145626811yescharacteristic
506C2#2145631111yescharacteristic
subgroupG/H
centerC22#127×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1723C2
socleC22×C61C2×#127
prime radicalC22×D61#127
hypercenter#1723D61
hypocenterC61#19632
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. #21445498C2semidirect
8. #21445496C2semidirect
9. #21445494C2semidirect
10. #21445436C2semidirect
11. #21445436C2semidirect
12. #21445415C2semidirect
13. #21445414C2semidirect
15. C61×#1723C2semidirect
193. #11758189Q16semidirect
227. C8×C61C2×Q16semidirect
229. #92418C2×Q16semidirect
259. #1723D61semidirect
316. C61#19632semidirect
484. C8Q16×D61semidirect

Maximal subgroups

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

subgroupindex
#214569712
#214569662
#214564042
#214563622
#214563602
#494×D612
#214454982
#214454962
#214454942
#214454362
#214454152
#214454142
#214440982
C61×#17232
#1963261