Finite Group Databasebeta

← #46848384

Subgroups — #46848384

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#31774585C211yesmaximal, characteristic, complemented
2#31774532C211yesmaximal, characteristic
3#31774219C211yesmaximal, characteristic, complemented
4#31774209C211yesmaximal, characteristic, complemented
5#31774194C211yesmaximal, characteristic, complemented
6#31774142C211yesmaximal, characteristic, complemented
7#31774125C211yesmaximal, characteristic, complemented
8#31774094C211yesmaximal, characteristic, complemented
9#31774049C211yesmaximal, characteristic, complemented
10#31774049C211yesmaximal, characteristic, complemented
11#31773999C211yesmaximal, characteristic, complemented
12#31773970C211yesmaximal, characteristic, complemented
13#31773900C211yesmaximal, characteristic, complemented
14#102×D271C211yesmaximal, characteristic, complemented
15#490×C271C211yesmaximal, characteristic, complemented
16C2×D4×D271—21no—
17C2×D4×D271C2211yescharacteristic
18C2×D4×D271C2211yescharacteristic
19C2×#16864643C2211yescharacteristic
20C2×#16864642C2211yescharacteristic, complemented
21C2×#16864642C2211yescharacteristic, complemented
22C2×#16864637C2211yescharacteristic, complemented
23C2×C8×D271C2211yescharacteristic, complemented
24#22576155—21no—
25#22576151—21no—
26#22576145—21no—
27#22576140C2211yescharacteristic, complemented
28#22576140C2211yescharacteristic, complemented
29#22576131—21no—
30#22576131C2211yescharacteristic
31#22576131C2211yescharacteristic
32#22576128—21no—
33#22576128—21no—
34C4×Dic542C2211yescharacteristic, complemented
35C4×D1084C2211yescharacteristic
36C4×D1084C2211yescharacteristic
37#22576118—21no—
38#22576118C2211yescharacteristic, complemented
39C4⋊C4×D271C2211yescharacteristic, complemented
40#22576094C2211yescharacteristic, complemented
41#22576093C2211yescharacteristic, complemented
42#22576091C2211yescharacteristic, complemented
43C4⋊(C271⋊C8)C2211yescharacteristic, complemented
44#22576085C2211yescharacteristic, complemented
45#22576082C2211yescharacteristic
46#22576081C2211yescharacteristic, complemented
47#22576081C2211yescharacteristic, complemented
48#22576080C2211yescharacteristic
49#22576080C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic, complemented
51(C2×D271)⋊C8C2211yescharacteristic, complemented
52#127×C271C2211yescharacteristic
53#125×C271C2211yescharacteristic, complemented
54#102×C271C2211yescharacteristic, complemented
55#102×C271C2211yescharacteristic, complemented
56#102×C271C2211yescharacteristic, complemented
57#102×C271C2211yescharacteristic, complemented
58#97×C271C2211yescharacteristic, complemented
59C23×D271—21no—
60C23×D271—21no—
61C2×Dic542C2311yescharacteristic
62C2×D1084D411yescharacteristic
63C2×D1084C2311yescharacteristic
64C2×C22⋊D271—21no—
65C2×C22⋊D271—21no—
66C2×C22⋊D271—21no—
67C2×C22⋊D271—21no—
68D4×D271—22no—
69D4×D271—22no—
70D4×D271—41no—
71D4×D271—41no—
72D4×D271—44no—
73C2×C4×D271—21no—
74C2×C4×D271—21no—
75C2×C4×D271C2311yescharacteristic
76C2×C4×D271C2311yescharacteristic
77#16864643—22no—
78#16864642—41no—
79#16864642—41no—
80#16864637—41no—
81C2×C271⋊C8C2311yescharacteristic
82C2×C271⋊C8C2311yescharacteristic
83C8×D271—41no—
84C4⋊Dic271—21no—
85C4⋊Dic271—21no—
86C4⋊Dic271D411yescharacteristic
87C4⋊Dic271C2311yescharacteristic
88C4⋊Dic271C2311yescharacteristic
89Dic271⋊C4—21no—
90Dic271⋊C4—21no—
91C22⋊Dic271—21no—
92C22⋊Dic271—21no—
93(C2×D271)⋊C4—21no—
94(C2×D271)⋊C4—21no—
95(C2×D271)⋊C4—21no—
96(C2×D271)⋊C4—21no—
97(C2×D271)⋊C4D411yescharacteristic
98(C2×D271)⋊C4D411yescharacteristic
99C4×Dic271—21no—
100C4×Dic271—21no—
101C2×D4×C271—21no—
102C2×D4×C271—21no—
103C2×D4×C271C2311yescharacteristic
104C2×D4×C271C2311yescharacteristic
105C2×C8×C271C2311yescharacteristic
106C2×C8×C271C2311yescharacteristic
107C4⋊C4×C271—21no—
108C4⋊C4×C271—21no—
109C4⋊C4×C271C2311yescharacteristic
110C4⋊C4×C271C2311yescharacteristic
111C42×C271C2311yescharacteristic
112C22×D271—21no—
113C22×D271—42no—
114C22×D271—42no—
115C22×D271—42no—
116C22×D271—42no—
117C22×D271—81no—
118C22×D271—81no—
119C22×D271C2×D411yescharacteristic
120C22×D271C2×D411yescharacteristic
121Dic542—21no—
122Dic542—21no—
123D1084—22no—
124D1084—21no—
125D1084#4112yes—
126C22⋊D271—42no—
127C22⋊D271—42no—
128C22⋊D271—42no—
129C22⋊D271—42no—
130C2×Dic271—21no—
131C2×Dic271—21no—
132C2×Dic271—21no—
133C2×Dic271C2×D411yescharacteristic
134C2×Dic271C2×D411yescharacteristic
135C4×D271—21no—
136C4×D271—21no—
137C4×D271—21no—
138C4×D271—21no—
139C4×D271—42no—
140C4×D271—42no—
141C271⋊C8—21no—
142C271⋊C8—21no—
143C23×C271—21no—
144C23×C271—21no—
145D4×C271—21no—
146D4×C271—21no—
147D4×C271—21no—
148D4×C271—21no—
149D4×C271—42no—
150D4×C271—81no—
151C2×C4×C271—21no—
152C2×C4×C271—21no—
153C2×C4×C271C2×D411yescharacteristic
154C2×C4×C271C2×D411yescharacteristic
155C2×C4×C271C2411yescharacteristic
156C8×C271—21no—
157C8×C271—21no—
158C22×C271—41no—
159C22×C271—41no—
160C22×C271—41no—
161C22×C271—41no—
162C22×C271—41no—
163C22×C271—41no—
164C22×C271C22×D411yescharacteristic
165C2×D271—21no—
166C2×D271—21no—
167C2×D271—21no—
168C2×D271—22no—
169C2×D271—21no—
170C2×D271—21no—
171C2×D271—41no—
172C2×D271—41no—
173C2×D271—84no—
174C2×D271—84no—
175C2×D271AGL(1,ℤ/8)12yes—
176C4×C271—21no—
177C4×C271—21no—
178C4×C271—41no—
179C4×C271—41no—
180C4×C271C2×#4111yescharacteristic
181C4×C271H(5,𝔽2)11yescharacteristic
182Dic271—21no—
183Dic271—21no—
184Dic271—41no—
185Dic271—41no—
186Dic271—41no—
187C2×C271—81no—
188C2×C271—81no—
189C2×C271#54611yescharacteristic
190C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C271#57711yescharacteristic
192D271—22no—
193D271—41no—
194D271—42no—
195C271#331011yescharacteristic, complemented, Sylow
196#3310—2711nomaximal, Sylow
197#552—2711no—
198D42—2711no—
199#490D27111yescharacteristic, complemented
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#435—2711no—
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×M16—2711no—
219C22×C8—2711no—
220#127C2×D27111yescharacteristic
221#126—5421no—
222#125—5421no—
223#125C2×D27111yescharacteristic, complemented
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×D4—2711no—
235#117—2711no—
236#117—5421no—
237C2×C4⋊C4—2711no—
238#107—2711no—
239#107—2711no—
240#106—2711no—
241C4⋊C8—2711no—
242#103—2711no—
243#102—2711no—
244#102—2711no—
245#102—2711no—
246#102—2711no—
247#102—2711no—
248#102C2×D27111yescharacteristic, complemented
249#102C2×D27111yescharacteristic, complemented
250#102C2×D27111yescharacteristic, complemented
251#102C2×D27111yescharacteristic, complemented
252C22⋊C8—2711no—
253C22⋊C8—2711no—
254#97C2×D27111yescharacteristic, complemented
255C24—5421no—
256C24—5421no—
257C2×Q8—2711no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—2711no—
261C2×D4—2711no—
262C2×D4—5421no—
263C2×D4—5421no—
264C2×D4—5422no—
265C2×D4—5422no—
266C2×D4—5421no—
267C2×D4—5421no—
268C2×D4—10841no—
269C2×D4—10841no—
270C2×D4—10844no—
271C2×D4C22×D27111yescharacteristic
272C2×D4C22×D27111yescharacteristic
273C22×C4—2711no—
274C22×C4—2711no—
275C22×C4—5421no—
276C22×C4—5421no—
277SD16—5422no—
278SD16—10841no—
279SD16—10841no—
280M16—10841no—
281C2×C8—2711no—
282C2×C8—2711no—
283C2×C8—10841no—
284C2×C8C22×D27111yescharacteristic
285C2×C8C22×D27111yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—2711no—
289C4⋊C4—2711no—
290C4⋊C4—2711no—
291C4⋊C4—5421no—
292C4⋊C4—5421no—
293C4⋊C4—5421no—
294C4⋊C4—5421no—
295C4⋊C4C22×D27111yescharacteristic
296C4⋊C4C22×D27111yescharacteristic
297C22⋊C4—2711no—
298C22⋊C4—2711no—
299C22⋊C4—5421no—
300C22⋊C4—5421no—
301C22⋊C4—5421no—
302C22⋊C4—5421no—
303C22⋊C4—5421no—
304C22⋊C4—5421no—
305C42—5421no—
306C42—5421no—
307C42C22×D27111yescharacteristic
308C23—21no—
309C23—21no—
310C23—2711no—
311C23—2711no—
312C23—5421no—
313C23—10842no—
314C23—10842no—
315C23—10842no—
316C23—10842no—
317C23—21681no—
318C23—21681no—
319Q8—5421no—
320Q8—5421no—
321D4—21no—
322D4—21no—
323D4—21no—
324D4—21no—
325D4—42no—
326D4—81no—
327D4—2712no—
328D4—5422no—
329D4—5421no—
330D4—10842no—
331D4—10842no—
332D4—10842no—
333D4—10842no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—2711no—
337C2×C4—2711no—
338C2×C4—5421no—
339C2×C4—5421no—
340C2×C4—5421no—
341C2×C4—5421no—
342C2×C4—5421no—
343C2×C4—5421no—
344C2×C4—5421no—
345C2×C4—10842no—
346C2×C4—10842no—
347C2×C4D4×D27111yescharacteristic
348C2×C4D4×D27111yescharacteristic
349C2×C4C23×D27111yescharacteristic
350C8—21no—
351C8—21no—
352C8—5421no—
353C8—5421no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—2712no—
361C22—5421no—
362C22—5421no—
363C22—5421no—
364C22—5422no—
365C22—5421no—
366C22—5421no—
367C22—10841no—
368C22—10841no—
369C22—21684no—
370C22—21684no—
371C22C2×D4×D27111yescharacteristic
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—5421no—
377C4—5421no—
378C4—10841no—
379C4—10841no—
380C4—10841no—
381C4#41×D27111yescharacteristic
382C4#2257620611yescharacteristic
383C2—81no—
384C2—81no—
385C2—5422no—
386C2—10841no—
387C2—10842no—
388C2#3177454011yescharacteristic
389C2AGL(1,ℤ/8)×D27111yescharacteristic
390C2#3177470711yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#490×C271C2
socleC22×C271C22×D4
prime radical#31774532C2
hypercenter#490D271
hypocenterC271#3310
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #31774585C2semidirect
3. #31774219C2semidirect
4. #31774209C2semidirect
5. #31774194C2semidirect
6. #31774142C2semidirect
7. #31774125C2semidirect
8. #31774094C2semidirect
9. #31774049C2semidirect
10. #31774049C2semidirect
11. #31773999C2semidirect
12. #31773970C2semidirect
13. #31773900C2semidirect
14. #102×D271C2semidirect
15. #490×C271C2semidirect
20. C2×#16864642C22semidirect
21. C2×#16864642C22semidirect
22. C2×#16864637C22semidirect
23. C2×C8×D271C22semidirect
27. #22576140C22semidirect
28. #22576140C22semidirect
34. C4×Dic542C22semidirect
38. #22576118C22semidirect
39. C4⋊C4×D271C22semidirect
40. #22576094C22semidirect
41. #22576093C22semidirect
42. #22576091C22semidirect
43. C4⋊(C271⋊C8)C22semidirect
44. #22576085C22semidirect
46. #22576081C22semidirect
47. #22576081C22semidirect
50. (C2×D271)⋊C8C22semidirect
51. (C2×D271)⋊C8C22semidirect
53. #125×C271C22semidirect
54. #102×C271C22semidirect
55. #102×C271C22semidirect
56. #102×C271C22semidirect
57. #102×C271C22semidirect
58. #97×C271C22semidirect
195. C271#3310semidirect
199. #490D271semidirect
223. #125C2×D271semidirect
248. #102C2×D271semidirect
249. #102C2×D271semidirect
250. #102C2×D271semidirect
251. #102C2×D271semidirect
254. #97C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317745852
#317745322
#317742192
#317742092
#317741942
#317741422
#317741252
#317740942
#317740492
#317739992
#317739702
#317739002
#102×D2712
#490×C2712
#3310271