Finite Group Databasebeta

← #46848387

Subgroups — #46848387

Order 34688.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774587C211yesmaximal, characteristic, complemented
2D4×D1084C211yesmaximal, characteristic
3#31774262C211yesmaximal, characteristic, complemented
4#31774222C211yesmaximal, characteristic, complemented
5#31774211C211yesmaximal, characteristic, complemented
6#31774198C211yesmaximal, characteristic, complemented
7#31774143C211yesmaximal, characteristic, complemented
8#31774126C211yesmaximal, characteristic, complemented
9#31774098C211yesmaximal, characteristic, complemented
10#31774050C211yesmaximal, characteristic, complemented
11#31774050C211yesmaximal, characteristic, complemented
12#31773999C211yesmaximal, characteristic, complemented
13#31773903C211yesmaximal, characteristic, complemented
14C2×#22576082C211yesmaximal, characteristic, complemented
15#435×C271C211yesmaximal, characteristic, complemented
16C22×D1084C2211yescharacteristic
17C22×D1084C2211yescharacteristic
18C2×D4×D271—21no—
19C2×#16864643C2211yescharacteristic
20C2×#16864643C2211yescharacteristic, complemented
21C2×#16864643C2211yescharacteristic, complemented
22#22576157C2211yescharacteristic
23#22576153—21no—
24#22576152C2211yescharacteristic, complemented
25#22576150—21no—
26#22576141—21no—
27#22576139C2211yescharacteristic, complemented
28#22576139C2211yescharacteristic, complemented
29#22576135—21no—
30#22576134C2211yescharacteristic
31#22576134C2211yescharacteristic
32#22576130—21no—
33#22576130—21no—
34C4×Dic542C2211yescharacteristic, complemented
35C4×D1084C2211yescharacteristic
36#22576119C2211yescharacteristic, complemented
37#22576116—21no—
38C2×C4⋊Dic271C2211yescharacteristic, complemented
39#22576094C2211yescharacteristic, complemented
40#22576094C2211yescharacteristic, complemented
41#22576092C2211yescharacteristic, complemented
42#22576085C2211yescharacteristic, complemented
43#22576082C2211yescharacteristic
44#22576082C2211yescharacteristic
45#22576082C2211yescharacteristic
46#22576082C2211yescharacteristic, complemented
47#22576082C2211yescharacteristic, complemented
48#22576082C2211yescharacteristic, complemented
49#22576082C2211yescharacteristic, complemented
50#22576082C2211yescharacteristic, complemented
51#22576082C2211yescharacteristic, complemented
52C2×M16×C271C2211yescharacteristic, complemented
53C22×C8×C271C2211yescharacteristic, complemented
54C4×D4×C271C2211yescharacteristic
55C4⋊C8×C271C2211yescharacteristic, complemented
56C22⋊C8×C271C2211yescharacteristic, complemented
57C22⋊C8×C271C2211yescharacteristic, complemented
58#97×C271C2211yescharacteristic, complemented
59C23×D271—21no—
60C23×D271—21no—
61C2×Dic542C2311yescharacteristic
62C2×D1084—22no—
63C2×D1084—22no—
64C2×D1084—21no—
65C2×D1084—21no—
66C2×D1084—21no—
67C2×D1084—21no—
68C2×D1084—41no—
69C2×D1084—41no—
70C2×D1084C2311yescharacteristic
71C2×D1084C2311yescharacteristic
72C2×D1084C2311yescharacteristic
73C2×C22⋊D271—21no—
74C2×C22⋊D271—21no—
75D4×D271—44no—
76C22×Dic271—21no—
77C2×C4×D271—21no—
78#16864643—22no—
79#16864643—41no—
80#16864643—41no—
81C4⋊Dic271—21no—
82C4⋊Dic271C2311yescharacteristic
83C4⋊Dic271C2311yescharacteristic
84C4⋊Dic271C2311yescharacteristic
85C4⋊Dic271C2311yescharacteristic
86Dic271⋊C4—21no—
87Dic271⋊C4—21no—
88Dic271⋊C4—21no—
89Dic271⋊C4—21no—
90Dic271⋊C4—21no—
91C22⋊Dic271—21no—
92C22⋊Dic271—21no—
93C22⋊Dic271—21no—
94(C2×D271)⋊C4—21no—
95(C2×D271)⋊C4—21no—
96(C2×D271)⋊C4—21no—
97C4×Dic271—21no—
98C4×Dic271—21no—
99C2×D4×C271D411yescharacteristic
100C22×C4×C271C2311yescharacteristic
101C22×C4×C271C2311yescharacteristic
102M16×C271—41no—
103C2×C8×C271—41no—
104C2×C8×C271C2311yescharacteristic
105C2×C8×C271C2311yescharacteristic
106C2×C8×C271C2311yescharacteristic
107C2×C8×C271C2311yescharacteristic
108C4⋊C4×C271D411yescharacteristic
109C22⋊C4×C271D411yescharacteristic
110C22⋊C4×C271D411yescharacteristic
111C42×C271C2311yescharacteristic
112C22×D271—21no—
113C22×D271—21no—
114C22×D271—21no—
115C22×D271—42no—
116C22×D271—42no—
117C22×D271—42no—
118C22×D271—42no—
119C22×D271—81no—
120C22×D271—81no—
121Dic542—21no—
122Dic542—21no—
123D1084—21no—
124D1084—21no—
125D1084—21no—
126D1084—21no—
127D1084—21no—
128D1084—42no—
129D1084—42no—
130D1084—42no—
131D1084—81no—
132D1084#4112yes—
133C22⋊D271—42no—
134C22⋊D271—42no—
135C2×Dic271—21no—
136C2×Dic271—21no—
137C2×Dic271—21no—
138C2×Dic271—21no—
139C2×Dic271—21no—
140C2×Dic271—42no—
141C4×D271—42no—
142C23×C271C2×D411yescharacteristic
143C23×C271C2×D411yescharacteristic
144D4×C271—22no—
145C2×C4×C271—21no—
146C2×C4×C271—21no—
147C2×C4×C271—21no—
148C2×C4×C271—21no—
149C2×C4×C271C2×D411yescharacteristic
150C2×C4×C271C2×D411yescharacteristic
151C2×C4×C271C2×D411yescharacteristic
152C2×C4×C271C2×D411yescharacteristic
153C2×C4×C271C2411yescharacteristic
154C8×C271—21no—
155C8×C271—21no—
156C8×C271—21no—
157C8×C271—21no—
158C22×C271—21no—
159C22×C271—21no—
160C22×C271—21no—
161C22×C271—21no—
162C22×C271—21no—
163C22×C271AGL(1,ℤ/8)12yes—
164C22×C271C22×D411yescharacteristic
165C2×D271—22no—
166C2×D271—41no—
167C2×D271—41no—
168C2×D271—41no—
169C2×D271—41no—
170C2×D271—41no—
171C2×D271—41no—
172C2×D271—41no—
173C2×D271—41no—
174C2×D271—84no—
175C2×D271—84no—
176C4×C271—21no—
177C4×C271—21no—
178C4×C271—21no—
179C4×C271—21no—
180C4×C271C2×#4111yescharacteristic
181C4×C271H(5,𝔽2)11yescharacteristic
182Dic271—41no—
183Dic271—41no—
184Dic271—41no—
185Dic271—41no—
186Dic271—41no—
187C2×C271—22no—
188C2×C271—41no—
189C2×C271#54611yescharacteristic
190C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C271#57711yescharacteristic
192D271—42no—
193D271—81no—
194D271—81no—
195C271#331011yescharacteristic, complemented, Sylow
196#3310—2711nomaximal, Sylow
197#552—2711no—
198D42—2711no—
199#490—2711no—
200#482—2711no—
201#480—2711no—
202#478—2711no—
203#468—2711no—
204#465—2711no—
205#460—2711no—
206#450—2711no—
207#450—2711no—
208#440—2711no—
209#435D27111yescharacteristic, complemented
210#418—2711no—
211C2×#102—2711no—
212C22×D4—2711no—
213C22×D4—2711no—
214C22×D4—5421no—
215C2×SD16—2711no—
216C2×SD16—2711no—
217C2×SD16—2711no—
218C2×M16C2×D27111yescharacteristic, complemented
219C22×C8C2×D27111yescharacteristic, complemented
220#127—2711no—
221#126—5421no—
222#125—2711no—
223#125—5421no—
224#123—5421no—
225C22⋊Q8—2711no—
226C22⋊Q8—2711no—
227#121—2711no—
228#121—2711no—
229#121—5421no—
230#120—5421no—
231#120—5421no—
232C4×Q8—2711no—
233C4×D4—2711no—
234C4×D4C2×D27111yescharacteristic
235#117—2711no—
236#117—5421no—
237C2×C4⋊C4—2711no—
238#107—2711no—
239#107—2711no—
240#106—2711no—
241C4⋊C8C2×D27111yescharacteristic, complemented
242#103—2711no—
243#102—2711no—
244#102—2711no—
245#102—2711no—
246#102—2711no—
247#102—2711no—
248#102—2711no—
249#102—2711no—
250#102—2711no—
251#102—2711no—
252C22⋊C8C2×D27111yescharacteristic, complemented
253C22⋊C8C2×D27111yescharacteristic, complemented
254#97C2×D27111yescharacteristic, complemented
255C24—5421no—
256C24—5421no—
257C2×Q8—2711no—
258C2×D4—2711no—
259C2×D4—2711no—
260C2×D4—2711no—
261C2×D4—5422no—
262C2×D4—5422no—
263C2×D4—5421no—
264C2×D4—5421no—
265C2×D4—5421no—
266C2×D4—5421no—
267C2×D4—5421no—
268C2×D4—5421no—
269C2×D4—10841no—
270C2×D4—10841no—
271C2×D4—10844no—
272C2×D4D108411yescharacteristic
273C22×C4—5421no—
274C22×C4—5421no—
275C22×C4C22×D27111yescharacteristic
276C22×C4C22×D27111yescharacteristic
277SD16—5422no—
278SD16—10841no—
279SD16—10841no—
280M16—41no—
281C2×C8—41no—
282C2×C8C22×D27111yescharacteristic
283C2×C8C22×D27111yescharacteristic
284C2×C8C22×D27111yescharacteristic
285C2×C8C22×D27111yescharacteristic
286C4⋊C4—2711no—
287C4⋊C4—2711no—
288C4⋊C4—2711no—
289C4⋊C4—2711no—
290C4⋊C4—5421no—
291C4⋊C4—5421no—
292C4⋊C4—5421no—
293C4⋊C4—5421no—
294C4⋊C4—5421no—
295C4⋊C4—5421no—
296C4⋊C4D108411yescharacteristic
297C22⋊C4—5421no—
298C22⋊C4—5421no—
299C22⋊C4—5421no—
300C22⋊C4—5421no—
301C22⋊C4—5421no—
302C22⋊C4—5421no—
303C22⋊C4D108411yescharacteristic
304C22⋊C4D108411yescharacteristic
305C42—5421no—
306C42—5421no—
307C42C22×D27111yescharacteristic
308C23—5421no—
309C23—5421no—
310C23—5421no—
311C23—10842no—
312C23—10842no—
313C23—10842no—
314C23—10842no—
315C23—21681no—
316C23—21681no—
317C23C2×D108411yescharacteristic
318C23C2×D108411yescharacteristic
319Q8—5421no—
320Q8—5421no—
321D4—22no—
322D4—2712no—
323D4—5421no—
324D4—5421no—
325D4—5421no—
326D4—5421no—
327D4—5421no—
328D4—10842no—
329D4—10842no—
330D4—10842no—
331D4—10842no—
332D4—10842no—
333D4—21681no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—5421no—
339C2×C4—5421no—
340C2×C4—5421no—
341C2×C4—5421no—
342C2×C4—5421no—
343C2×C4—10842no—
344C2×C4—10842no—
345C2×C4C2×D108411yescharacteristic
346C2×C4C2×D108411yescharacteristic
347C2×C4C2×D108411yescharacteristic
348C2×C4C2×D108411yescharacteristic
349C2×C4C23×D27111yescharacteristic
350C8—21no—
351C8—21no—
352C8—21no—
353C8—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—5422no—
360C22—10841no—
361C22—10841no—
362C22—10841no—
363C22—10841no—
364C22—10841no—
365C22—10841no—
366C22—10841no—
367C22—10841no—
368C22—21684no—
369C22—21684no—
370C22#2257618812yes—
371C22C22×D108411yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—10841no—
377C4—10841no—
378C4—10841no—
379C4—10841no—
380C4—10841no—
381C4#41×D27111yescharacteristic
382C4#2257620511yescharacteristic
383C2—22no—
384C2—41no—
385C2—10842no—
386C2—21681no—
387C2—21681no—
388C2#3177454211yescharacteristic
389C2C2×#2257618811yescharacteristic
390C2#3177470911yescharacteristic
subgroupG/H
centerC22C22×D1084
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#435×C271C2
socleC22×C271C22×D4
prime radicalD4×D1084C2
hypercenter#435D271
hypocenterC271#3310
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #31774587C2semidirect
3. #31774262C2semidirect
4. #31774222C2semidirect
5. #31774211C2semidirect
6. #31774198C2semidirect
7. #31774143C2semidirect
8. #31774126C2semidirect
9. #31774098C2semidirect
10. #31774050C2semidirect
11. #31774050C2semidirect
12. #31773999C2semidirect
13. #31773903C2semidirect
14. C2×#22576082C2semidirect
15. #435×C271C2semidirect
20. C2×#16864643C22semidirect
21. C2×#16864643C22semidirect
24. #22576152C22semidirect
27. #22576139C22semidirect
28. #22576139C22semidirect
34. C4×Dic542C22semidirect
36. #22576119C22semidirect
38. C2×C4⋊Dic271C22semidirect
39. #22576094C22semidirect
40. #22576094C22semidirect
41. #22576092C22semidirect
42. #22576085C22semidirect
46. #22576082C22semidirect
47. #22576082C22semidirect
48. #22576082C22semidirect
49. #22576082C22semidirect
50. #22576082C22semidirect
51. #22576082C22semidirect
52. C2×M16×C271C22semidirect
53. C22×C8×C271C22semidirect
55. C4⋊C8×C271C22semidirect
56. C22⋊C8×C271C22semidirect
57. C22⋊C8×C271C22semidirect
58. #97×C271C22semidirect
195. C271#3310semidirect
209. #435D271semidirect
218. C2×M16C2×D271semidirect
219. C22×C8C2×D271semidirect
241. C4⋊C8C2×D271semidirect
252. C22⋊C8C2×D271semidirect
253. C22⋊C8C2×D271semidirect
254. #97C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317745872
D4×D10842
#317742622
#317742222
#317742112
#317741982
#317741432
#317741262
#317740982
#317740502
#317739992
#317739032
C2×#225760822
#435×C2712
#3310271