Finite Group Databasebeta

← #29836420

Subgroups — #29836420

Order 15616.

476 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456471C211yesmaximal, characteristic, complemented
2#21456459C211yesmaximal, characteristic
3#21447981C211yesmaximal, characteristic, complemented
4#21447979C211yesmaximal, characteristic, complemented
5#21447970C211yesmaximal, characteristic, complemented
6#21447959C211yesmaximal, characteristic, complemented
7#21447934C211yesmaximal, characteristic
8#21447925C211yesmaximal, characteristic, complemented
9#21447915C211yesmaximal, characteristic
10#21447884C211yesmaximal, characteristic, complemented
11#21447883C211yesmaximal, characteristic, complemented
12#21447867C211yesmaximal, characteristic, complemented
13#21447863C211yesmaximal, characteristic, complemented
14#21447577C211yesmaximal, characteristic, complemented
15C61×#2216C211yesmaximal, characteristic, complemented
16#16840385—21no—
17#16840372—21no—
18C2×#14519723C2211yescharacteristic
19C2×#14519722C2211yescharacteristic
20C2×#14519720C2211yescharacteristic
21C2×#14519717C2211yescharacteristic
22#16840166C2211yescharacteristic
23#16840163C2211yescharacteristic
24#16840147C2211yescharacteristic
25#16840140C2211yescharacteristic
26#16840129—21no—
27#16840123—21no—
28#16840018—21no—
29#16840010—21no—
30#16840003C2211yescharacteristic
31#16839998C2211yescharacteristic
32#16839866—21no—
33#16839865—21no—
34Q16×Dic61C2211yescharacteristic
35Q16×Dic61C2211yescharacteristic
36#16839856C2211yescharacteristic
37#16839844C2211yescharacteristic
38#16839717C2211yescharacteristic
39#16839590C2211yescharacteristic
40#16839589C2211yescharacteristic
41#16839587C2211yescharacteristic
42#16839580C2211yescharacteristic
43#16839580C2211yescharacteristic
44#16839580C2211yescharacteristic
45#16839580C2211yescharacteristic
46#16839567C2211yescharacteristic
47#16839566C2211yescharacteristic
48#16839566C2211yescharacteristic
49#16839565C2211yescharacteristic
50#16839564C2211yescharacteristic
51#16839562C2211yescharacteristic
52C2×Q32×C61C2211yescharacteristic
53C22×C16×C61C2211yescharacteristic
54C61×#467C2211yescharacteristic
55C61×#467C2211yescharacteristic
56C61×#366C2211yescharacteristic
57C61×#358C2211yescharacteristic
58C61×#358C2211yescharacteristic
59C2×Dic244C2311yescharacteristic
60C2×#11758202—21no—
61C2×#11758201—21no—
62C2×D488C2311yescharacteristic
63C2×#11758194D411yescharacteristic
64#14519776—21no—
65#14519775—21no—
66#14519772—41no—
67#14519770—41no—
68#14519770—41no—
69#14519768—21no—
70#14519768—41no—
71#14519767—41no—
72#14519766—21no—
73#14519763—41no—
74#14519759—41no—
75#14519757—41no—
76#14519756—21no—
77#14519751—21no—
78Q8×Dic61—21no—
79Q8×Dic61—21no—
80#14519742—21no—
81#14519739—21no—
82#14519736D411yescharacteristic
83#14519723—41no—
84#14519722—41no—
85#14519720—41no—
86#14519717—41no—
87C2×#11758185C2311yescharacteristic
88C2×#11758185C2311yescharacteristic
89#14519711C2311yescharacteristic
90#14519711C2311yescharacteristic
91#14519710—21no—
92#14519708—21no—
93#14519702—21no—
94#14519701—21no—
95#14519701—21no—
96#14519700—21no—
97#14519699—21no—
98#14519697—21no—
99#14519687D411yescharacteristic
100#14519687D411yescharacteristic
101C8×Dic61C2311yescharacteristic
102C8×Dic61C2311yescharacteristic
103C2×Q16×C61C2311yescharacteristic
104C2×Q16×C61C2311yescharacteristic
105C22×C8×C61C2311yescharacteristic
106C22⋊Q8×C61—21no—
107C22⋊Q8×C61—21no—
108Q32×C61—41no—
109C2×C16×C61—41no—
110C2×C16×C61C2311yescharacteristic
111C2×C16×C61C2311yescharacteristic
112#107×C61C2311yescharacteristic
113#107×C61C2311yescharacteristic
114#103×C61—21no—
115#103×C61—21no—
116C2×Q8×C61—21no—
117C2×Q8×C61—21no—
118C22×C4×C61C2×D411yescharacteristic
119C2×C22⋊D61—41no—
120C2×D244—21no—
121C2×C4×D61—41no—
122C2×Dic122—21no—
123C2×Dic122—41no—
124Q16×C61—21no—
125Q16×C61—21no—
126Q16×C61—21no—
127Q16×C61—21no—
128C2×C8×C61—21no—
129C2×C8×C61—21no—
130C2×C8×C61C2×D411yescharacteristic
131C2×C8×C61C2411yescharacteristic
132C4⋊C4×C61—21no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—41no—
135C4⋊C4×C61—41no—
136C22⋊C4×C61—41no—
137C22⋊C4×C61—41no—
138C22⋊Dic61—41no—
139C22⋊Dic61D811yescharacteristic
140C22⋊Dic61D811yescharacteristic
141#11758202—42no—
142#11758201—42no—
143#11758198—41no—
144#11758198—41no—
145#11758198—41no—
146#11758198—41no—
147C4⋊Dic61—21no—
148C4⋊Dic61—21no—
149C4⋊Dic61—41no—
150C4⋊Dic61—41no—
151C4⋊Dic61—41no—
152C4⋊Dic61D811yescharacteristic
153C4⋊Dic61D811yescharacteristic
154#11758196—41no—
155#11758196—41no—
156#11758196—41no—
157#11758196—41no—
158#11758196—41no—
159#11758196—41no—
160C4×Dic61—41no—
161C4×Dic61—41no—
162C4×Dic61—41no—
163C4×Dic61C2×D411yescharacteristic
164C4×Dic61C2×D411yescharacteristic
165#11758194—41no—
166C2×#92418C2×D411yescharacteristic
167C2×#92418C2×D411yescharacteristic
168Dic244—21no—
169Dic244—21no—
170D488—21no—
171D488—21no—
172C16×C61—21no—
173C16×C61—21no—
174#11758185—21no—
175#11758185—21no—
176C23×C61C2×D811yescharacteristic
177C22×D61—41no—
178Q8×C61—41no—
179Q8×C61—41no—
180Q8×C61—41no—
181Q8×C61—41no—
182C2×C4×C61—21no—
183C2×C4×C61—21no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61C2×D811yescharacteristic
189C2×C4×C61C22×D411yescharacteristic
190C22⋊D61—161no—
191C2×Dic61—41no—
192C2×Dic61—41no—
193C2×Dic61—41no—
194C2×Dic61C2×D811yescharacteristic
195C2×Dic61C2×D811yescharacteristic
196C2×Dic61C2×D811yescharacteristic
197C2×Dic61C2×D811yescharacteristic
198D244—41no—
199D244—41no—
200C4×D61—161no—
201Dic122—41no—
202Dic122—41no—
203Dic122—161no—
204C8×C61—21no—
205C8×C61H(5,𝔽2)11yescharacteristic
206C8×C612−1+411yescharacteristic
207#92418—21no—
208#92418—21no—
209#33824—611nomaximal, Sylow
210C22×C61—21no—
211C22×C61—21no—
212C22×C61—21no—
213C22×C61C22×D811yescharacteristic
214C2×D61—81no—
215C2×D61—81no—
216C2×D61—81no—
217C4×C61—21no—
218C4×C61—81no—
219C4×C61—81no—
220C4×C61—81no—
221C4×C61—81no—
222C4×C61#53711yescharacteristic
223C4×C61#57511yescharacteristic
224Dic61—21no—
225Dic61—21no—
226Dic61—21no—
227Dic61—21no—
228Dic61—81no—
229Dic61—81no—
230Dic61—81no—
231#3177—611no—
232#3175—611no—
233#2229—611no—
234#2228—611no—
235#2227—611no—
236#2225—611no—
237#2220—611no—
238#2219—611no—
239#2217—611no—
240#2216D6111yescharacteristic, complemented
241#2211—611no—
242#2211—611no—
243#2209—611no—
244#2208—611no—
245#2151—611no—
246C2×C61—41no—
247C2×C61#324511yescharacteristic
248C2×C61#341611yescharacteristic
249C2×C61#341711yescharacteristic
250D61—161no—
251#542—1221no—
252#541—1221no—
253C2×Q32—611no—
254C2×Q32C2×D6111yescharacteristic
255C2×SD32—611no—
256C2×SD32—611no—
257C2×M32—611no—
258C22×C16C2×D6111yescharacteristic
259#500—611no—
260#499—611no—
261#495—611no—
262#494—611no—
263#491—1221no—
264#489—1221no—
265#470—1221no—
266#468—1221no—
267#467—611no—
268#467C2×D6111yescharacteristic
269#467C2×D6111yescharacteristic
270#466—611no—
271#441—1221no—
272#440—1221no—
273C4×Q16—611no—
274C4×Q16—611no—
275C4×Q16—611no—
276C4×D8—611no—
277#405—611no—
278#367—611no—
279#367—611no—
280#366—611no—
281#366C2×D6111yescharacteristic
282C4⋊C16—611no—
283C4⋊C16—611no—
284C4⋊C16—611no—
285C4⋊C16—611no—
286#358—611no—
287#358—611no—
288#358—611no—
289#358—611no—
290#358C2×D6111yescharacteristic
291#358C2×D6111yescharacteristic
292#357—611no—
293#357—611no—
294C61#3382411yescharacteristic, complemented, Sylow
295C2×Q16—611no—
296C2×Q16—1221no—
297C2×Q16C22×D6111yescharacteristic
298C2×Q16C22×D6111yescharacteristic
299C2×SD16—1221no—
300C2×D8—611no—
301C2×M16—611no—
302C22×C8C22×D6111yescharacteristic
303C4⋊Q8—1221no—
304C4⋊Q8—1221no—
305#126—2441no—
306#126—2441no—
307#126—2441no—
308#125—1221no—
309#125—2441no—
310#125—2441no—
311#124—1221no—
312#124—2441no—
313#123—2441no—
314C22⋊Q8—21no—
315C22⋊Q8—21no—
316C22⋊Q8—1221no—
317C22⋊Q8—2441no—
318#121—1221no—
319C4×Q8—1221no—
320C4×Q8—1221no—
321C4×Q8—1221no—
322C4×D4—1221no—
323#117—611no—
324Q32—41no—
325Q32—2441no—
326SD32—2441no—
327SD32—2441no—
328M32—2441no—
329C2×C16—41no—
330C2×C16—611no—
331C2×C16—611no—
332C2×C16C22×D6111yescharacteristic
333C2×C16C22×D6111yescharacteristic
334#107—611no—
335#107—611no—
336#107—1221no—
337#107C22×D6111yescharacteristic
338#107C22×D6111yescharacteristic
339#106—1221no—
340#103—21no—
341#103—21no—
342#103—1221no—
343#103—1221no—
344#103—1221no—
345#103—1221no—
346#102—1221no—
347#102—1221no—
348#97—611no—
349#97—611no—
350C4×C8—611no—
351C4×C8—611no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—1221no—
355C2×Q8—2441no—
356C2×D4—1221no—
357C2×D4—2441no—
358C22×C4—2441no—
359C22×C4D4×D6111yescharacteristic
360Q16—21no—
361Q16—21no—
362Q16—21no—
363Q16—21no—
364Q16—1221no—
365Q16—1221no—
366Q16—2442no—
367SD16—2442no—
368D8—1221no—
369D8—1221no—
370M16—2441no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—611no—
374C2×C8—611no—
375C2×C8D4×D6111yescharacteristic
376C2×C8C23×D6111yescharacteristic
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—611no—
382C4⋊C4—611no—
383C4⋊C4—1221no—
384C4⋊C4—1221no—
385C4⋊C4—2441no—
386C4⋊C4—2441no—
387C4⋊C4—2441no—
388C4⋊C4—2441no—
389C4⋊C4—2441no—
390C4⋊C4—2441no—
391C4⋊C4—2441no—
392C4⋊C4—2441no—
393C4⋊C4—2441no—
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4—611no—
397C22⋊C4—611no—
398C22⋊C4—2441no—
399C22⋊C4—2441no—
400C22⋊C4—2441no—
401C22⋊C4—2441no—
402C22⋊C4—2441no—
403C42—611no—
404C42—611no—
405C42—2441no—
406C42—2441no—
407C42—2441no—
408C16—21no—
409C16—21no—
410C16—1221no—
411C16—1221no—
412C23—2441no—
413C23D8×D6111yescharacteristic
414Q8—41no—
415Q8—41no—
416Q8—41no—
417Q8—41no—
418Q8—9761no—
419Q8—2441no—
420Q8—2441no—
421D4—9761no—
422D4—2441no—
423D4—2441no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—611no—
431C2×C4—611no—
432C2×C4—611no—
433C2×C4—611no—
434C2×C4—9761no—
435C2×C4—2441no—
436C2×C4—2441no—
437C2×C4—2441no—
438C2×C4D8×D6111yescharacteristic
439C2×C4C2×D4×D6111yescharacteristic
440C8—21no—
441C8—1221no—
442C8—1221no—
443C8#1451982211yescharacteristic
444C8#1451982411yescharacteristic
445C22—21no—
446C22—21no—
447C22—21no—
448C22—4881no—
449C22—4881no—
450C22—4881no—
451C22C2×D8×D6111yescharacteristic
452C4—21no—
453C4—81no—
454C4—81no—
455C4—81no—
456C4—81no—
457C4—1221no—
458C4—1221no—
459C4—1221no—
460C4—1221no—
461C4—4881no—
462C4—4881no—
463C4—4881no—
464C4#1684033911yescharacteristic
465C4#1684055911yescharacteristic
466C2—41no—
467C2—9761no—
468C2#2145712511yescharacteristic
469C2#2145896211yescharacteristic
470C2#2145896811yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2216C2
socleC22×C61C22×D8
prime radical#16839998C22
hypercenter#2216D61
hypocenterC61#33824
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2216

Complements

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

NH ≅ G/Ntype
1. #21456471C2semidirect
3. #21447981C2semidirect
4. #21447979C2semidirect
5. #21447970C2semidirect
6. #21447959C2semidirect
8. #21447925C2semidirect
10. #21447884C2semidirect
11. #21447883C2semidirect
12. #21447867C2semidirect
13. #21447863C2semidirect
14. #21447577C2semidirect
15. C61×#2216C2semidirect
240. #2216D61semidirect
294. C61#33824semidirect

Maximal subgroups

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

subgroupindex
#214564712
#214564592
#214479812
#214479792
#214479702
#214479592
#214479342
#214479252
#214479152
#214478842
#214478832
#214478672
#214478632
#214475772
C61×#22162
#3382461