Finite Group Databasebeta

← #46848170

Subgroups — #46848170

Order 34688.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774547C211yesmaximal, characteristic, complemented
2D4×C22⋊D271C211yesmaximal, characteristic, complemented
3#31774242C211yesmaximal, characteristic, complemented
4#31774211C211yesmaximal, characteristic, complemented
5#31774209C211yesmaximal, characteristic, complemented
6#31774175C211yesmaximal, characteristic, complemented
7#31774131C211yesmaximal, characteristic, complemented
8#31774130C211yesmaximal, characteristic, complemented
9#31774090C211yesmaximal, characteristic, complemented
10#31774029C211yesmaximal, characteristic, complemented
11D8×Dic271C211yesmaximal, characteristic, complemented
12(C22⋊D271)⋊C8C211yesmaximal, characteristic, complemented
13C2×#22576081C211yesmaximal, characteristic, complemented
14#102×D271C211yesmaximal, characteristic, complemented
15#447×C271C211yesmaximal, characteristic, complemented
16C2×#16864653—21no—
17C22×C22⋊D271—21no—
18C2×D4×D271C2211yescharacteristic
19C2×#16864639C2211yescharacteristic, complemented
20C22×C271⋊C8C2211yescharacteristic, complemented
21C2×C8×D271C2211yescharacteristic, complemented
22#22576159C2211yescharacteristic, complemented
23#22576156C2211yescharacteristic
24#22576142—21no—
25#22576141—21no—
26#22576137—21no—
27#22576134C2211yescharacteristic
28#22576132—21no—
29#22576131C2211yescharacteristic
30#22576131C2211yescharacteristic, complemented
31#22576129—21no—
32#22576128—21no—
33D4×Dic271C2211yescharacteristic
34D4×Dic271C2211yescharacteristic, complemented
35C4×C22⋊D271C2211yescharacteristic
36C2×C4⋊Dic271C2211yescharacteristic, complemented
37C4⋊C4×D271C2211yescharacteristic, complemented
38#22576094C2211yescharacteristic, complemented
39#22576094C2211yescharacteristic, complemented
40#22576093C2211yescharacteristic, complemented
41Dic271⋊C8C2211yescharacteristic, complemented
42#22576082C2211yescharacteristic, complemented
43#22576082C2211yescharacteristic, complemented
44#22576081C2211yescharacteristic
45#22576081C2211yescharacteristic
46#22576081C2211yescharacteristic, complemented
47#22576081C2211yescharacteristic, complemented
48#22576081C2211yescharacteristic, complemented
49#22576080C2211yescharacteristic, complemented
50(C2×D271)⋊C8C2211yescharacteristic, complemented
51C8×Dic271C2211yescharacteristic, complemented
52C22×D4×C271C2211yescharacteristic
53C2×D8×C271C2211yescharacteristic
54C2×D8×C271C2211yescharacteristic, complemented
55#121×C271C2211yescharacteristic, complemented
56#102×C271C2211yescharacteristic
57#102×C271C2211yescharacteristic, complemented
58C22⋊C8×C271C2211yescharacteristic, complemented
59C23×D271—21no—
60#16864653—44no—
61C2×Dic542—21no—
62C2×D1084C2311yescharacteristic
63C2×C22⋊D271—21no—
64C2×C22⋊D271—21no—
65C2×C22⋊D271—21no—
66C2×C22⋊D271—21no—
67C2×C22⋊D271—21no—
68C2×C22⋊D271—21no—
69C2×C22⋊D271—44no—
70C2×C22⋊D271D411yescharacteristic
71D4×D271—22no—
72D4×D271—41no—
73C22×Dic271—21no—
74C22×Dic271—21no—
75C22×Dic271—21no—
76C2×C4×D271—21no—
77C2×C4×D271C2311yescharacteristic
78#16864639—41no—
79C2×C271⋊C8—41no—
80C2×C271⋊C8C2311yescharacteristic
81C2×C271⋊C8C2311yescharacteristic
82C8×D271—41no—
83C4⋊Dic271—21no—
84C4⋊Dic271C2311yescharacteristic
85C4⋊Dic271C2311yescharacteristic
86C4⋊Dic271C2311yescharacteristic
87Dic271⋊C4—21no—
88Dic271⋊C4—21no—
89Dic271⋊C4D411yescharacteristic
90C22⋊Dic271—21no—
91C22⋊Dic271—21no—
92C22⋊Dic271—21no—
93C22⋊Dic271—21no—
94C22⋊Dic271D411yescharacteristic
95(C2×D271)⋊C4—21no—
96(C2×D271)⋊C4D411yescharacteristic
97C4×Dic271C2311yescharacteristic
98C24×C271—21no—
99C2×D4×C271—22no—
100C2×D4×C271—21no—
101C2×D4×C271—21no—
102C2×D4×C271—41no—
103C2×D4×C271C2311yescharacteristic
104C2×D4×C271C2311yescharacteristic
105C2×D4×C271C2311yescharacteristic
106C22×C4×C271C2311yescharacteristic
107D8×C271—22no—
108D8×C271—41no—
109C2×C8×C271C2311yescharacteristic
110C2×C8×C271C2311yescharacteristic
111C4⋊C4×C271C2311yescharacteristic
112C22⋊C4×C271—21no—
113C22×D271—21no—
114C22×D271—42no—
115C22×D271—42no—
116C22×D271—81no—
117C22×D271C2×D411yescharacteristic
118Dic542—42no—
119D1084—21no—
120D1084—21no—
121C22⋊D271—42no—
122C22⋊D271—42no—
123C22⋊D271—42no—
124C22⋊D271—42no—
125C22⋊D271—42no—
126C22⋊D271—81no—
127C22⋊D271D814yescomplemented
128C2×Dic271—21no—
129C2×Dic271—21no—
130C2×Dic271—21no—
131C2×Dic271—42no—
132C2×Dic271—42no—
133C2×Dic271—42no—
134C2×Dic271C2×D411yescharacteristic
135C2×Dic271C2×D411yescharacteristic
136C2×Dic271C2×D411yescharacteristic
137C4×D271—21no—
138C4×D271—21no—
139C4×D271—42no—
140C271⋊C8—21no—
141C271⋊C8—21no—
142C23×C271—21no—
143C23×C271—21no—
144C23×C271—21no—
145C23×C271—42no—
146C23×C271—42no—
147C23×C271—81no—
148C23×C271C2×D411yescharacteristic
149D4×C271—21no—
150D4×C271—21no—
151D4×C271—21no—
152D4×C271—21no—
153D4×C271—21no—
154D4×C271—42no—
155D4×C271—42no—
156D4×C271#4112yescomplemented
157C2×C4×C271—21no—
158C2×C4×C271—21no—
159C2×C4×C271—21no—
160C2×C4×C271C2×D411yescharacteristic
161C2×C4×C271C2411yescharacteristic
162C8×C271—21no—
163C8×C271—21no—
164C22×C271—21no—
165C22×C271—21no—
166C22×C271—22no—
167C22×C271—41no—
168C22×C271—41no—
169C22×C271—41no—
170C22×C271—41no—
171C22×C271—41no—
172C22×C271—41no—
173C22×C271—41no—
174C22×C271—41no—
175C22×C271—84no—
176C22×C271C2×D812yes—
177C22×C271C22×D411yescharacteristic
178C2×D271—21no—
179C2×D271—21no—
180C2×D271—41no—
181C2×D271—41no—
182C2×D271—41no—
183C2×D271—84no—
184C2×D271C2×D812yes—
185C4×C271—21no—
186C4×C271—41no—
187C4×C271C2×#4111yescharacteristic
188C4×C271H(5,𝔽2)11yescharacteristic
189Dic271—21no—
190Dic271—21no—
191Dic271—41no—
192Dic271—41no—
193Dic271—41no—
194Dic271C2×D812yes—
195C2×C271—22no—
196C2×C271—42no—
197C2×C271—81no—
198C2×C271—81no—
199C2×C271#54611yescharacteristic
200C2×C271C22×D811yescharacteristic
201C2×C271#57711yescharacteristic
202D271—22no—
203D271—81no—
204C271#329411yescharacteristic, complemented, Sylow
205#3294—2711nomaximal, Sylow
206#547—2711no—
207D42—2711no—
208#486—2711no—
209#480—2711no—
210#480—2711no—
211#474—2711no—
212#466—2711no—
213#466—2711no—
214#459—2711no—
215#447—2711no—
216#447D27111yescharacteristic, complemented
217C4×D8—2711no—
218C8×D4—2711no—
219C2×#102—2711no—
220C2×#102—2711no—
221C2×#41—5421no—
222C22×D4—2711no—
223C22×D4—5421no—
224C22×D4C2×D27111yescharacteristic
225C2×D8—2711no—
226C2×D8C2×D27111yescharacteristic
227C2×D8C2×D27111yescharacteristic, complemented
228C22×C8—2711no—
229C22×C8—2711no—
230C4⋊Q8—2711no—
231#127—2711no—
232#123—5421no—
233#123—5421no—
234C22⋊Q8—5421no—
235#121—2711no—
236#121—2711no—
237#121—2711no—
238#121—5421no—
239#121C2×D27111yescharacteristic, complemented
240#120—5421no—
241#120—5421no—
242C4×D4—2711no—
243C4×D4—2711no—
244C4×D4—2711no—
245C2×C4⋊C4—2711no—
246C2×C4⋊C4—2711no—
247#107—2711no—
248#107—2711no—
249#107—2711no—
250C4⋊C8—2711no—
251#102—2711no—
252#102—2711no—
253#102—2711no—
254#102—2711no—
255#102—2711no—
256#102—2711no—
257#102—2711no—
258#102—2711no—
259#102C2×D27111yescharacteristic
260#102C2×D27111yescharacteristic, complemented
261C22⋊C8—2711no—
262C22⋊C8C2×D27111yescharacteristic, complemented
263C4×C8—2711no—
264C24—21no—
265C24—5421no—
266#41—10844no—
267C2×Q8—5421no—
268C2×D4—22no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—41no—
272C2×D4—2711no—
273C2×D4—2711no—
274C2×D4—5421no—
275C2×D4—5421no—
276C2×D4—5421no—
277C2×D4—5422no—
278C2×D4—5421no—
279C2×D4—5421no—
280C2×D4—5421no—
281C2×D4—10841no—
282C2×D4—10844no—
283C2×D4C22×D27111yescharacteristic
284C2×D4C22×D27111yescharacteristic
285C2×D4C22×D27111yescharacteristic
286C22×C4—2711no—
287C22×C4—5421no—
288C22×C4—5421no—
289C22×C4—5421no—
290C22×C4—5421no—
291C22×C4C22×D27111yescharacteristic
292D8—22no—
293D8—41no—
294D8—10841no—
295C2×C8—2711no—
296C2×C8—2711no—
297C2×C8—10841no—
298C2×C8—10841no—
299C2×C8C22×D27111yescharacteristic
300C2×C8C22×D27111yescharacteristic
301C4⋊C4—2711no—
302C4⋊C4—2711no—
303C4⋊C4—2711no—
304C4⋊C4—2711no—
305C4⋊C4—5421no—
306C4⋊C4—5421no—
307C4⋊C4—5421no—
308C4⋊C4C22×D27111yescharacteristic
309C22⋊C4—21no—
310C22⋊C4—2711no—
311C22⋊C4—2711no—
312C22⋊C4—5421no—
313C22⋊C4—5421no—
314C22⋊C4—5421no—
315C22⋊C4—5421no—
316C22⋊C4—5421no—
317C42—2711no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—42no—
322C23—42no—
323C23—81no—
324C23—2711no—
325C23—5421no—
326C23—10842no—
327C23—10842no—
328C23—21681no—
329C23D4×D27111yescharacteristic
330Q8—10842no—
331D4—21no—
332D4—21no—
333D4—21no—
334D4—21no—
335D4—21no—
336D4—42no—
337D4—42no—
338D4—2714no—
339D4—5421no—
340D4—5421no—
341D4—10842no—
342D4—10842no—
343D4—10842no—
344D4—10842no—
345D4—10842no—
346D4—21681no—
347D4#1686465312yescomplemented
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—2711no—
352C2×C4—2711no—
353C2×C4—2711no—
354C2×C4—5421no—
355C2×C4—5421no—
356C2×C4—5421no—
357C2×C4—5421no—
358C2×C4—5421no—
359C2×C4—10842no—
360C2×C4—10842no—
361C2×C4—10842no—
362C2×C4—10842no—
363C2×C4D4×D27111yescharacteristic
364C2×C4C23×D27111yescharacteristic
365C8—21no—
366C8—21no—
367C8—5421no—
368C8—5421no—
369C22—21no—
370C22—21no—
371C22—22no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—84no—
381C22—2712no—
382C22—5421no—
383C22—5421no—
384C22—10841no—
385C22—10841no—
386C22—10841no—
387C22—21684no—
388C22D8×D27112yes—
389C22C2×D4×D27111yescharacteristic
390C4—21no—
391C4—41no—
392C4—2712no—
393C4—5421no—
394C4—5421no—
395C4—10841no—
396C4—10841no—
397C4—10841no—
398C4C2×#1686465311yescharacteristic
399C4#2257620511yescharacteristic
400C2—22no—
401C2—42no—
402C2—81no—
403C2—81no—
404C2—5422no—
405C2—21681no—
406C2#3177453411yescharacteristic
407C2C2×D8×D27111yescharacteristic
408C2#3177470711yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#447×C271C2
socleC22×C271C22×D4
hypercenter#447D271
hypocenterC271#3294
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #31774547C2semidirect
2. D4×C22⋊D271C2semidirect
3. #31774242C2semidirect
4. #31774211C2semidirect
5. #31774209C2semidirect
6. #31774175C2semidirect
7. #31774131C2semidirect
8. #31774130C2semidirect
9. #31774090C2semidirect
10. #31774029C2semidirect
11. D8×Dic271C2semidirect
12. (C22⋊D271)⋊C8C2semidirect
13. C2×#22576081C2semidirect
14. #102×D271C2semidirect
15. #447×C271C2semidirect
19. C2×#16864639C22semidirect
20. C22×C271⋊C8C22semidirect
21. C2×C8×D271C22semidirect
22. #22576159C22semidirect
30. #22576131C22semidirect
34. D4×Dic271C22semidirect
36. C2×C4⋊Dic271C22semidirect
37. C4⋊C4×D271C22semidirect
38. #22576094C22semidirect
39. #22576094C22semidirect
40. #22576093C22semidirect
41. Dic271⋊C8C22semidirect
42. #22576082C22semidirect
43. #22576082C22semidirect
46. #22576081C22semidirect
47. #22576081C22semidirect
48. #22576081C22semidirect
49. #22576080C22semidirect
50. (C2×D271)⋊C8C22semidirect
51. C8×Dic271C22semidirect
54. C2×D8×C271C22semidirect
55. #121×C271C22semidirect
57. #102×C271C22semidirect
58. C22⋊C8×C271C22semidirect
127. C22⋊D271D8semidirect
156. D4×C271#41semidirect
204. C271#3294semidirect
216. #447D271semidirect
227. C2×D8C2×D271semidirect
239. #121C2×D271semidirect
260. #102C2×D271semidirect
262. C22⋊C8C2×D271semidirect
347. D4#16864653semidirect

Maximal subgroups

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

subgroupindex
#317745472
D4×C22⋊D2712
#317742422
#317742112
#317742092
#317741752
#317741312
#317741302
#317740902
#317740292
D8×Dic2712
(C22⋊D271)⋊C82
C2×#225760812
#102×D2712
#447×C2712
#3294271