Finite Group Databasebeta

← #29667982

Subgroups — #29667982

Order 15616.

698 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456966C211yesmaximal, characteristic
2#21456763C211yesmaximal, characteristic
3#21456406C211yesmaximal, characteristic
4#21456362C211yesmaximal, characteristic
5#21456360C211yesmaximal, characteristic
6#487×D61C211yesmaximal, characteristic
7#21445498C211yesmaximal, characteristic, complemented
8#21445496C211yesmaximal, characteristic, complemented
9#21445495C211yesmaximal, characteristic, complemented
10#21445428C211yesmaximal, characteristic, complemented
11#21445427C211yesmaximal, characteristic, complemented
12#21445414C211yesmaximal, characteristic, complemented
13#21445413C211yesmaximal, 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#16840165C2211yescharacteristic
25#16840165C2211yescharacteristic
26#16840165C2211yescharacteristic
27#16840160C2211yescharacteristic
28#16840158C2211yescharacteristic
29#16840158C2211yescharacteristic
30#16840142C2211yescharacteristic
31#16840141C2211yescharacteristic
32#16840114C2211yescharacteristic
33#16840114C2211yescharacteristic
34#16840114C2211yescharacteristic
35#16840113C2211yescharacteristic
36#16840113C2211yescharacteristic
37#16840112C2211yescharacteristic
38#16840112C2211yescharacteristic
39#16840112C2211yescharacteristic
40#16840111—22no—
41#16840096—22no—
42#16840055—22no—
43#16840004—22no—
44#16839970—22no—
45#16839916—22no—
46#16839856—22no—
47#16839813C2211yescharacteristic
48#16839803—22no—
49#16839801—22no—
50#103×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×#11758189D412yes—
68C2×C8×D61D412yes—
69#14519777C2311yescharacteristic
70#14519777C2311yescharacteristic
71#14519777C2311yescharacteristic
72#14519777C2311yescharacteristic
73#14519776—21no—
74#14519776—21no—
75#14519776—21no—
76#14519776—21no—
77#14519775—21no—
78#14519775—21no—
79#14519768—21no—
80#14519768—21no—
81#14519757—21no—
82#14519757—21no—
83#14519753—21no—
84#14519753—21no—
85Q8×Dic61—21no—
86Q8×Dic61—21no—
87#14519742—21no—
88#14519742—21no—
89#14519735—21no—
90#14519735—21no—
91C4⋊C4×D61—21no—
92C4⋊C4×D61—21no—
93C4⋊C4×D61—21no—
94C4⋊C4×D61—21no—
95C42×D61C2311yescharacteristic
96#14519711—22no—
97#14519711—22no—
98#14519710—22no—
99#14519710—22no—
100#14519710—22no—
101#14519709—22no—
102#14519708—22no—
103#14519708—22no—
104#14519706D412yes—
105#14519702—22no—
106#14519702—22no—
107#14519701—22no—
108#14519701—22no—
109#14519701—22no—
110#14519700—22no—
111#14519700—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—44no—
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—82no—
163#11758198—41no—
164#11758198—41no—
165C4⋊Dic61—21no—
166C4⋊Dic61—21no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—41no—
174C4⋊Dic61—41no—
175#11758196—41no—
176#11758196—41no—
177#11758196—41no—
178#11758196—41no—
179#11758196—41no—
180#11758196—41no—
181C4×Dic61—41no—
182C4×Dic61—41no—
183C4×Dic61C2×D411yescharacteristic
184C4×Dic61C2×D411yescharacteristic
185C4×Dic61C2×D411yescharacteristic
186C2×#92418C2×D412yes—
187C2×#92418C2×D412yes—
188C2×#92418C2×D412yes—
189Dic244—44no—
190#11758189Q1618yescomplemented
191C8×D61—42no—
192C22×D61#12711yescharacteristic
193Q8×C61—42no—
194Q8×C61—42no—
195Q8×C61—42no—
196Q8×C61—42no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61C22×D411yescharacteristic
202C2×C4×C61C22×D411yescharacteristic
203C2×C4×C61C22×D411yescharacteristic
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61#12711yescharacteristic
209C2×Dic61#12711yescharacteristic
210C2×Dic61#12711yescharacteristic
211C4×D61—22no—
212C4×D61—22no—
213C4×D61—82no—
214C4×D61—82no—
215C4×D61C2×Q1614yes—
216Dic122—42no—
217Dic122—42no—
218Dic122—42no—
219Dic122—42no—
220Dic122—82no—
221Dic122—82no—
222C8×C61—22no—
223C8×C61—22no—
224C8×C61C2×Q1614yes—
225#92418—22no—
226#92418—22no—
227#92418C2×Q1614yes—
228#19634—611nomaximal, Sylow
229C22×C61C2×#12711yescharacteristic
230C2×D61—21no—
231C2×D61—21no—
232C2×D61#49412yes—
233C4×C61—81no—
234C4×C61—81no—
235C4×C61—81no—
236C4×C61—81no—
237C4×C61C22×Q1612yes—
238C4×C61#57512yes—
239C4×C61#57712yes—
240Dic61—21no—
241Dic61—21no—
242Dic61—81no—
243Dic61—81no—
244Dic61—81no—
245Dic61—81no—
246Dic61#49412yes—
247#3230—611no—
248#3209—611no—
249#3170—611no—
250#3165—611no—
251#3165—611no—
252C2×#487—611no—
253#1723—611no—
254#1723—611no—
255#1723—611no—
256#1723D6111yescharacteristic, complemented
257#1710—611no—
258#1710—611no—
259#1707—611no—
260#1707—611no—
261C8×M16—611no—
262C2×C61C2×#49411yescharacteristic
263C2×C61#315311yescharacteristic
264C2×C61#315411yescharacteristic
265D61—22no—
266#533—611no—
267#533—611no—
268C2×C4⋊Q8—611no—
269C2×C4⋊Q8—611no—
270#501—1222no—
271#501—1222no—
272#500—611no—
273#500—611no—
274#500—611no—
275#500—611no—
276#500—611no—
277#500C2×D6111yescharacteristic
278#500C2×D6111yescharacteristic
279#498—611no—
280#498—611no—
281#498—611no—
282#498C2×D6111yescharacteristic
283#494—611no—
284#494—611no—
285#494C2×D6111yescharacteristic
286#487—611no—
287#487—611no—
288#487—611no—
289#487—611no—
290#487—611no—
291#487—611no—
292#487—611no—
293#487—611no—
294#487—1222no—
295#487C2×D6111yescharacteristic
296#487C2×D6111yescharacteristic
297#484—1222no—
298#477—1222no—
299#467—1222no—
300#462—1222no—
301#451—1222no—
302C4×Q16—1222no—
303#431—611no—
304#428—1222no—
305#428—1222no—
306C2×#103—1222no—
307C4×M16—611no—
308C2×C4×C8—611no—
309#322—611no—
310#322C2×D6111yescharacteristic
311C82—611no—
312C82—611no—
313C61#1963411yescharacteristic, complemented, Sylow
314C22×Q8—1221no—
315C22×Q8—1221no—
316C2×Q16—22no—
317C2×Q16—1222no—
318C2×Q16—1222no—
319C2×M16—612no—
320C22×C8—612no—
321C4⋊Q8—611no—
322C4⋊Q8—611no—
323C4⋊Q8—611no—
324C4⋊Q8—611no—
325C4⋊Q8—1221no—
326C4⋊Q8—1221no—
327C4⋊Q8—1221no—
328C4⋊Q8—1221no—
329C4⋊Q8—1221no—
330C4⋊Q8—1221no—
331C4⋊Q8C22×D6111yescharacteristic
332C4⋊Q8C22×D6111yescharacteristic
333C4⋊Q8C22×D6111yescharacteristic
334C4⋊Q8C22×D6111yescharacteristic
335#125—1221no—
336#125—1221no—
337C22⋊Q8—1221no—
338C22⋊Q8—1221no—
339C22⋊Q8—1221no—
340C22⋊Q8—1221no—
341C4×Q8—1221no—
342C4×Q8—1221no—
343C4×Q8—1221no—
344C4×Q8—1221no—
345#117—1221no—
346#117—1221no—
347C2×C4⋊C4—1221no—
348C2×C4⋊C4—1221no—
349C2×C4⋊C4—1221no—
350C2×C4⋊C4—1221no—
351C2×C42—611no—
352#107—22no—
353#107—22no—
354#107—1222no—
355#107—1222no—
356#107—1222no—
357#107—1222no—
358#107—1222no—
359#106—22no—
360#106—1222no—
361#106—1222no—
362#106—1222no—
363C4⋊C8—612no—
364#103—22no—
365#103—22no—
366#103—1222no—
367#103—1222no—
368#103—1222no—
369#103—1222no—
370#103—1222no—
371#103—1222no—
372#103—1222no—
373#103—1222no—
374C22⋊C8—612no—
375#97—612no—
376C4×C8—612no—
377C4×C8—611no—
378C4×C8—611no—
379C4×C8—611no—
380C4×C8C22×D6111yescharacteristic
381C4×C8C22×D6111yescharacteristic
382C4×C8C22×D6111yescharacteristic
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—1221no—
388C2×Q8—1221no—
389C2×Q8—1221no—
390C2×Q8—1221no—
391C2×Q8—4882no—
392C2×Q8—2441no—
393C2×Q8—2441no—
394C2×Q8—2444no—
395C22×C4—611no—
396C22×C4—611no—
397C22×C4—611no—
398C22×C4—2441no—
399C22×C4—2441no—
400Q16—82no—
401Q16—4882no—
402Q16—2444no—
403M16—618no—
404C2×C8—612no—
405C2×C8—612no—
406C2×C8—612no—
407C2×C8—2442no—
408C2×C8D4×D6112yes—
409C2×C8D4×D6112yes—
410C2×C8D4×D6112yes—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—1221no—
420C4⋊C4—1221no—
421C4⋊C4—1221no—
422C4⋊C4—1221no—
423C4⋊C4—1221no—
424C4⋊C4—1221no—
425C4⋊C4—1221no—
426C4⋊C4—1221no—
427C4⋊C4—2441no—
428C4⋊C4—2441no—
429C4⋊C4—2441no—
430C4⋊C4—2441no—
431C4⋊C4—2441no—
432C4⋊C4—2441no—
433C4⋊C4—2441no—
434C4⋊C4—2441no—
435C22⋊C4—2441no—
436C22⋊C4—2441no—
437C42—611no—
438C42—611no—
439C42—611no—
440C42—2441no—
441C42—2441no—
442C42C23×D6111yescharacteristic
443C23—611no—
444Q8—42no—
445Q8—42no—
446Q8—42no—
447Q8—42no—
448Q8—4882no—
449Q8—4882no—
450Q8—2442no—
451Q8—2442no—
452Q8—2442no—
453Q8—2442no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—611no—
459C2×C4—614no—
460C2×C4—611no—
461C2×C4—611no—
462C2×C4—1222no—
463C2×C4—1222no—
464C2×C4—4882no—
465C2×C4—4882no—
466C2×C4—2441no—
467C2×C4—2441no—
468C2×C4—2441no—
469C2×C4—2441no—
470C2×C4C2×D4×D6111yescharacteristic
471C2×C4C2×D4×D6111yescharacteristic
472C2×C4C2×D4×D6111yescharacteristic
473C8—22no—
474C8—22no—
475C8—614no—
476C8—1222no—
477C8—1222no—
478C8Q16×D6114yes—
479C22—612no—
480C22—1221no—
481C22—1221no—
482C22#127×D6111yescharacteristic
483C4—81no—
484C4—81no—
485C4—81no—
486C4—81no—
487C4—612no—
488C4—1221no—
489C4—1221no—
490C4—4881no—
491C4—4881no—
492C4—4881no—
493C4—4881no—
494C4C2×Q16×D6112yes—
495C4#1684056112yes—
496C4#1684057212yes—
497C2—1222no—
498C2#494×D6111yescharacteristic
499C2#2145628211yescharacteristic
500C2#2145629211yescharacteristic
subgroupG/H
centerC22#127×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1723C2
socleC22×C61C2×#127
prime radicalC22×D61#127
hypercenter#1723D61
hypocenterC61#19634
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. #21445495C2semidirect
10. #21445428C2semidirect
11. #21445427C2semidirect
12. #21445414C2semidirect
13. #21445413C2semidirect
15. C61×#1723C2semidirect
190. #11758189Q16semidirect
256. #1723D61semidirect
313. C61#19634semidirect

Maximal subgroups

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

subgroupindex
#214569662
#214567632
#214564062
#214563622
#214563602
#487×D612
#214454982
#214454962
#214454952
#214454282
#214454272
#214454142
#214454132
#214440982
C61×#17232
#1963461