Finite Group Databasebeta

← #29667835

Subgroups — #29667835

Order 15616.

712 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456405C211yesmaximal, characteristic, complemented
2#21456401C211yesmaximal, characteristic, complemented
3#21456374C211yesmaximal, characteristic
4#21456329C211yesmaximal, characteristic, complemented
5#21456141C211yesmaximal, characteristic
6#21445493C211yesmaximal, characteristic, complemented
7#21445488C211yesmaximal, characteristic, complemented
8#21445480C211yesmaximal, characteristic, complemented
9#21445469C211yesmaximal, characteristic, complemented
10#21445466C211yesmaximal, characteristic, complemented
11#21445462C211yesmaximal, characteristic, complemented
12#21445453C211yesmaximal, characteristic, complemented
13#21445438C211yesmaximal, characteristic, complemented
14C2×#16839483C211yesmaximal, characteristic, complemented
15C61×#3167C211yesmaximal, characteristic
16#16840319C2211yescharacteristic
17#16840319C2211yescharacteristic
18#16840167C2211yescharacteristic
19#16840165C2211yescharacteristic
20#16840165C2211yescharacteristic
21#16840165C2211yescharacteristic
22#16840161—22no—
23#16840158C2211yescharacteristic
24#16840158C2211yescharacteristic
25#16840158C2211yescharacteristic
26#16840142C2211yescharacteristic
27#16840141C2211yescharacteristic
28#16840134C2211yescharacteristic
29#16840116—22no—
30#16840114C2211yescharacteristic
31#16840114C2211yescharacteristic
32#16840113C2211yescharacteristic
33#16840113C2211yescharacteristic
34#16840112C2211yescharacteristic
35#16840112C2211yescharacteristic
36#16840112C2211yescharacteristic
37#16840108C2211yescharacteristic
38#16840108C2211yescharacteristic
39#16840108C2211yescharacteristic
40#16840003—22no—
41#16840001—22no—
42#16839989—22no—
43C4×Dic244—22no—
44C4×#11758202—22no—
45C4×#11758200—22no—
46#16839798—22no—
47#16839758—22no—
48C2×C4×#92418C2211yescharacteristic
49C2×C4×#92418C2211yescharacteristic
50#16839483C2211yescharacteristic
51#16839483C2211yescharacteristic
52#16839483C2211yescharacteristic
53#16839483C2211yescharacteristic
54C61×#533C2211yescharacteristic
55C61×#532C2211yescharacteristic
56C61×#498C2211yescharacteristic
57C61×#492C2211yescharacteristic
58C61×#487C2211yescharacteristic
59C61×#486C2211yescharacteristic
60C61×#465—22no—
61C4×SD16×C61—22no—
62C2×C4×C8×C61C2211yescharacteristic
63C2×Dic244—22no—
64C2×#11758202—22no—
65C2×#11758200—22no—
66C22×#92418D412yes—
67C22×#92418D412yes—
68#14519777C2311yescharacteristic
69#14519777C2311yescharacteristic
70#14519777C2311yescharacteristic
71#14519777C2311yescharacteristic
72#14519769—21no—
73#14519769—21no—
74#14519756—21no—
75#14519756—21no—
76#14519756—21no—
77#14519756—21no—
78C4×Dic122—21no—
79C4×Dic122—21no—
80C4×Dic122—21no—
81C4×Dic122—21no—
82#14519736—21no—
83#14519736—21no—
84#14519711—22no—
85#14519710—22no—
86#14519710—22no—
87#14519710—22no—
88#14519708—22no—
89#14519708—22no—
90#14519708—22no—
91#14519702—22no—
92#14519702—22no—
93#14519701—22no—
94#14519701—22no—
95#14519701—22no—
96#14519700—22no—
97#14519700—22no—
98#14519698—22no—
99#14519698—22no—
100#14519698—22no—
101C4×#92418D412yes—
102C4×#92418D412yes—
103C4×#92418C2311yescharacteristic
104C4×#92418C2311yescharacteristic
105C4×#92418C2311yescharacteristic
106C4×#92418C2311yescharacteristic
107C2×#41×C61—21no—
108C2×SD16×C61—22no—
109C22×C8×C61D412yes—
110C4⋊Q8×C61C2311yescharacteristic
111C4⋊Q8×C61C2311yescharacteristic
112C4⋊Q8×C61C2311yescharacteristic
113#127×C61C2311yescharacteristic
114#125×C61—21no—
115#124×C61—21no—
116C22⋊Q8×C61—21no—
117C22⋊Q8×C61—21no—
118#121×C61—21no—
119#121×C61—21no—
120C4×Q8×C61—21no—
121C4×Q8×C61—21no—
122C4×D4×C61—21no—
123C4×D4×C61—21no—
124#117×C61—21no—
125C2×C42×C61C2311yescharacteristic
126#106×C61—22no—
127#103×C61—22no—
128#102×C61—22no—
129C4×C8×C61D412yes—
130C4×C8×C61C2311yescharacteristic
131C4×C8×C61C2311yescharacteristic
132#41×C61—44no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—41no—
140C22×C4×C61—41no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C2×Dic122—21no—
145C2×Dic122—21no—
146C2×Dic122—21no—
147C2×Dic122—21no—
148SD16×C61—44no—
149C2×C8×C61—24no—
150C2×C8×C61C2×D412yes—
151C2×C8×C61C2×D412yes—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—41no—
159C4⋊C4×C61—41no—
160C22⋊C4×C61—41no—
161C22⋊C4×C61—41no—
162C22⋊C4×C61—41no—
163C42×C61—41no—
164C42×C61C2×D411yescharacteristic
165C42×C61C2×D411yescharacteristic
166C42×C61C2×D411yescharacteristic
167C42×C61C2411yescharacteristic
168C22⋊Dic61—41no—
169C22⋊Dic61—41no—
170#11758202—44no—
171#11758200—44no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180#11758196—41no—
181#11758196—41no—
182#11758196—41no—
183#11758196—41no—
184C4×Dic61—41no—
185C4×Dic61—41no—
186C2×#92418—24no—
187C2×#92418—42no—
188C2×#92418C2×D412yes—
189C2×#92418C2×D412yes—
190C2×#92418C2×D412yes—
191C2×#92418C2×D412yes—
192Dic244—44no—
193C23×C61—41no—
194C23×C61#12711yescharacteristic
195Q8×C61—42no—
196Q8×C61—42no—
197Q8×C61—42no—
198D4×C61—42no—
199D4×C61—42no—
200D4×C61—42no—
201D4×C61—82no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—82no—
208C2×C4×C61#12711yescharacteristic
209C2×C4×C61#12711yescharacteristic
210C2×C4×C61#12711yescharacteristic
211C2×C4×C61#13714yes—
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C22×D411yescharacteristic
215C2×Dic61—41no—
216C2×Dic61—41no—
217C2×Dic61—41no—
218C2×Dic61—41no—
219Dic122—42no—
220Dic122—42no—
221Dic122—42no—
222Dic122—42no—
223C8×C61—22no—
224C8×C61#13514yes—
225#92418—22no—
226#92418—22no—
227#92418—22no—
228#92418—22no—
229#19620—611nomaximal, Sylow
230C22×C61—21no—
231C22×C61—21no—
232C22×C61—81no—
233C22×C61—81no—
234C22×C61—81no—
235C22×C61#49712yes—
236C22×C61C2×#12711yescharacteristic
237C4×C61—21no—
238C4×C61—21no—
239C4×C61—81no—
240C4×C61—81no—
241C4×C61—81no—
242C4×C61#49712yes—
243C4×C61C2×#13512yes—
244C4×C61C2×#13712yes—
245C4×C61#57512yes—
246Dic61—81no—
247Dic61—81no—
248Dic61—81no—
249Dic61—81no—
250#3170—611no—
251#3170—611no—
252#3167—611no—
253#3167D6111yescharacteristic
254#3160—611no—
255#3131—611no—
256#1723—611no—
257#1722—611no—
258#1720—611no—
259#1719—611no—
260#1718—611no—
261#1718—611no—
262#1716—611no—
263#1714—611no—
264C2×#322—611no—
265C2×C61—22no—
266C2×C61—161no—
267C2×C61#314711yescharacteristic
268C2×C61C2×#49711yescharacteristic
269C2×C61#315011yescharacteristic
270#533—611no—
271#533—611no—
272#533C2×D6111yescharacteristic
273#532C2×D6111yescharacteristic
274#500—611no—
275#500—611no—
276#500—611no—
277#500—611no—
278#499—1222no—
279#498—611no—
280#498—611no—
281#498—611no—
282#498C2×D6111yescharacteristic
283#494—611no—
284#494—611no—
285#492—611no—
286#492C2×D6111yescharacteristic
287#488—1222no—
288#487—611no—
289#487—611no—
290#487—611no—
291#487—611no—
292#487—611no—
293#487—611no—
294#487—611no—
295#487C2×D6111yescharacteristic
296#486—611no—
297#486—611no—
298#486—611no—
299#486C2×D6111yescharacteristic
300#467—1222no—
301#467—1222no—
302#465—22no—
303#465—1222no—
304C4×Q16—1222no—
305C4×Q16—1222no—
306C4×SD16—22no—
307C4×SD16—1222no—
308#427—1222no—
309#416—1222no—
310C2×C4×C8—611no—
311C2×C4×C8—611no—
312C2×C4×C8C2×D6111yescharacteristic
313#322—611no—
314#322—611no—
315#322—611no—
316#322—611no—
317C61#1962011yescharacteristic, complemented, Sylow
318C2×#41—21no—
319C2×Q16—1222no—
320C2×Q16—1222no—
321C2×SD16—22no—
322C2×SD16—1222no—
323C22×C8—612no—
324C22×C8—612no—
325C22×C8C22⋊D6112yes—
326C4⋊Q8—611no—
327C4⋊Q8—611no—
328C4⋊Q8—611no—
329C4⋊Q8—611no—
330C4⋊Q8C22×D6111yescharacteristic
331C4⋊Q8C22×D6111yescharacteristic
332C4⋊Q8C22×D6111yescharacteristic
333#127C22×D6111yescharacteristic
334#125—21no—
335#125—1221no—
336#125—1221no—
337#124—21no—
338C22⋊Q8—21no—
339C22⋊Q8—21no—
340C22⋊Q8—1221no—
341C22⋊Q8—1221no—
342C22⋊Q8—1221no—
343C22⋊Q8—1221no—
344#121—21no—
345#121—21no—
346C4×Q8—21no—
347C4×Q8—21no—
348C4×Q8—1221no—
349C4×Q8—1221no—
350C4×Q8—1221no—
351C4×Q8—1221no—
352C4×D4—21no—
353C4×D4—21no—
354#117—21no—
355#117—1221no—
356#117—1221no—
357C2×C42C22×D6111yescharacteristic
358#107—1222no—
359#107—1222no—
360#107—1222no—
361#107—1222no—
362#106—22no—
363#106—1222no—
364#106—1222no—
365#106—1222no—
366#103—22no—
367#103—1222no—
368#103—1222no—
369#103—1222no—
370#103—1222no—
371#103—1222no—
372#103—1222no—
373#103—1222no—
374#102—22no—
375#102—1222no—
376#102—1222no—
377#102—1222no—
378C4×C8—612no—
379C4×C8—612no—
380C4×C8—611no—
381C4×C8—611no—
382C4×C8—611no—
383C4×C8—611no—
384C4×C8C22⋊D6112yes—
385C4×C8C22×D6111yescharacteristic
386C4×C8C22×D6111yescharacteristic
387#41—44no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—1221no—
392C2×Q8—1221no—
393C2×Q8—1221no—
394C2×Q8—1221no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—41no—
399C22×C4—41no—
400C22×C4D4×D6111yescharacteristic
401C22×C4D4×D6111yescharacteristic
402C22×C4C2×C22⋊D6111yescharacteristic
403Q16—2444no—
404Q16—2444no—
405SD16—44no—
406SD16—2444no—
407C2×C8—24no—
408C2×C8—612no—
409C2×C8—612no—
410C2×C8—612no—
411C2×C8—612no—
412C2×C8—1224no—
413C2×C8—2442no—
414C2×C8C2×C22⋊D6112yes—
415C2×C8C2×C22⋊D6112yes—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—1221no—
425C4⋊C4—1221no—
426C4⋊C4—1221no—
427C4⋊C4—1221no—
428C4⋊C4—1221no—
429C4⋊C4—1221no—
430C4⋊C4—1221no—
431C4⋊C4—1221no—
432C4⋊C4—2441no—
433C4⋊C4—2441no—
434C4⋊C4—2441no—
435C4⋊C4—2441no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—2441no—
440C22⋊C4—2441no—
441C42—41no—
442C42—2441no—
443C42—2441no—
444C42D4×D6111yescharacteristic
445C42D4×D6111yescharacteristic
446C42C2×C22⋊D6111yescharacteristic
447C42C23×D6111yescharacteristic
448C23—41no—
449C23#1451977311yescharacteristic
450Q8—42no—
451Q8—42no—
452Q8—42no—
453Q8—2442no—
454Q8—2442no—
455Q8—2442no—
456Q8—2442no—
457D4—42no—
458D4—42no—
459D4—42no—
460D4—82no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—82no—
467C2×C4—2441no—
468C2×C4—2441no—
469C2×C4—2441no—
470C2×C4—2441no—
471C2×C4#1451977311yescharacteristic
472C2×C4#1451977311yescharacteristic
473C2×C4#1451977311yescharacteristic
474C2×C4#1451980614yes—
475C2×C4C2×D4×D6111yescharacteristic
476C2×C4C2×D4×D6111yescharacteristic
477C2×C4C22×C22⋊D6111yescharacteristic
478C8—22no—
479C8—1222no—
480C8—1222no—
481C8—1222no—
482C8—1222no—
483C8#1451979714yes—
484C22—21no—
485C22—21no—
486C22—81no—
487C22—81no—
488C22—81no—
489C22#1684015312yes—
490C22C2×#1451977311yescharacteristic
491C4—21no—
492C4—21no—
493C4—81no—
494C4—81no—
495C4—81no—
496C4—4881no—
497C4—4881no—
498C4—4881no—
499C4—4881no—
500C4#1684015312yes—
501C4C2×#1451979712yes—
502C4C2×#1451980612yes—
503C4#1684055912yes—
504C2—22no—
505C2—161no—
506C2#2145624511yescharacteristic
507C2C2×#1684015311yescharacteristic
508C2#2145626211yescharacteristic
subgroupG/H
centerC2×C4#14519773
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3167C2
socleC22×C61C2×#127
prime radicalC61×#532C22
hypercenter#3167D61
hypocenterC61#19620
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
1. #21456405C2semidirect
2. #21456401C2semidirect
4. #21456329C2semidirect
6. #21445493C2semidirect
7. #21445488C2semidirect
8. #21445480C2semidirect
9. #21445469C2semidirect
10. #21445466C2semidirect
11. #21445462C2semidirect
12. #21445453C2semidirect
13. #21445438C2semidirect
14. C2×#16839483C2semidirect
317. C61#19620semidirect

Maximal subgroups

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

subgroupindex
#214564052
#214564012
#214563742
#214563292
#214561412
#214454932
#214454882
#214454802
#214454692
#214454662
#214454622
#214454532
#214454382
C2×#168394832
C61×#31672
#1962061