Finite Group Databasebeta

← #46846715

Subgroups — #46846715

Order 34688.

404 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774524C211yesmaximal, characteristic, complemented
2#31774465C211yesmaximal, characteristic
3#31774262C211yesmaximal, characteristic, complemented
4#31774260C211yesmaximal, characteristic, complemented
5#31774258C211yesmaximal, characteristic, complemented
6#31774253C211yesmaximal, characteristic, complemented
7#31774251C211yesmaximal, characteristic, complemented
8#31774245C211yesmaximal, characteristic, complemented
9#31774242C211yesmaximal, characteristic, complemented
10#31773900C211yesmaximal, characteristic, complemented
11#31773897C211yesmaximal, characteristic, complemented
12#31773892C211yesmaximal, characteristic, complemented
13#102×D271C211yesmaximal, characteristic, complemented
14#31773854C211yesmaximal, characteristic, complemented
15#489×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×D4×D271C2211yescharacteristic
18C2×#16864637C2211yescharacteristic
19C2×C8×D271C2211yescharacteristic
20#22576159C2211yescharacteristic, complemented
21#22576158—21no—
22#22576157C2211yescharacteristic
23#22576156C2211yescharacteristic
24#22576152C2211yescharacteristic, complemented
25#22576151C2211yescharacteristic, complemented
26#22576150—21no—
27#22576149C2211yescharacteristic
28#22576138—21no—
29#22576136—21no—
30#22576133—21no—
31#22576133—21no—
32#22576130—21no—
33#22576130—21no—
34#22576118C2211yescharacteristic
35#22576117C2211yescharacteristic
36C4⋊C4×D271C2211yescharacteristic
37#22576084C2211yescharacteristic, complemented
38#22576084C2211yescharacteristic, complemented
39#22576082C2211yescharacteristic
40#22576082C2211yescharacteristic
41#22576082C2211yescharacteristic, complemented
42#22576082C2211yescharacteristic, complemented
43#22576081C2211yescharacteristic
44#22576081C2211yescharacteristic
45#22576080C2211yescharacteristic
46#22576080C2211yescharacteristic
47#22576080C2211yescharacteristic, complemented
48#22576080C2211yescharacteristic, complemented
49#22576070C2211yescharacteristic, complemented
50C4×C271⋊C8C2211yescharacteristic, complemented
51C8×Dic271C2211yescharacteristic, complemented
52C4⋊Q8×C271C2211yescharacteristic, complemented
53#124×C271C2211yescharacteristic
54#103×C271C2211yescharacteristic, complemented
55#103×C271C2211yescharacteristic, complemented
56#102×C271C2211yescharacteristic
57#102×C271C2211yescharacteristic
58#97×C271C2211yescharacteristic, complemented
59C23×D271—21no—
60#16864655—22no—
61#16864655—22no—
62C2×Dic542—21no—
63C2×D1084—21no—
64C2×D1084—21no—
65C2×D1084—21no—
66C2×D1084C2311yescharacteristic
67C2×D1084C2311yescharacteristic
68C2×C22⋊D271—21no—
69C2×C22⋊D271—21no—
70C2×C22⋊D271—21no—
71D4×D271—22no—
72D4×D271—22no—
73C2×C4×D271—21no—
74C2×C4×D271—21no—
75C2×C4×D271C2311yescharacteristic
76#16864637—22no—
77C2×C271⋊C8C2311yescharacteristic
78C2×C271⋊C8C2311yescharacteristic
79C8×D271—22no—
80C4⋊Dic271—21no—
81C4⋊Dic271C2311yescharacteristic
82C4⋊Dic271C2311yescharacteristic
83Dic271⋊C4—21no—
84Dic271⋊C4—21no—
85Dic271⋊C4—21no—
86Dic271⋊C4—21no—
87Dic271⋊C4—21no—
88Dic271⋊C4D411yescharacteristic
89Dic271⋊C4D411yescharacteristic
90(C2×D271)⋊C4—21no—
91(C2×D271)⋊C4—21no—
92(C2×D271)⋊C4—21no—
93(C2×D271)⋊C4D411yescharacteristic
94(C2×D271)⋊C4D411yescharacteristic
95C4×Dic271—21no—
96C4×Dic271C2311yescharacteristic
97C2×Q8×C271—21no—
98C2×Q8×C271C2311yescharacteristic
99C2×D4×C271C2311yescharacteristic
100C2×C8×C271C2311yescharacteristic
101C2×C8×C271C2311yescharacteristic
102C4⋊C4×C271—21no—
103C4⋊C4×C271C2311yescharacteristic
104C4⋊C4×C271C2311yescharacteristic
105C22⋊C4×C271—21no—
106C22⋊C4×C271—21no—
107C42×C271C2311yescharacteristic
108C22×D271—21no—
109C22×D271—21no—
110C22×D271—42no—
111C22×D271—81no—
112C22×D271—81no—
113C22×D271C2×D411yescharacteristic
114Dic542—81no—
115D1084—21no—
116D1084—21no—
117D1084—21no—
118D1084—21no—
119D1084—42no—
120D1084—81no—
121D1084—81no—
122C22⋊D271—42no—
123C22⋊D271—81no—
124C22⋊D271—81no—
125C2×Dic271—21no—
126C2×Dic271—21no—
127C2×Dic271C2×D411yescharacteristic
128C2×Dic271C2×D411yescharacteristic
129C2×Dic271C2×D411yescharacteristic
130C4×D271—42no—
131C4×D271—81no—
132C4×D271#4112yes—
133C4×D271#4112yes—
134C271⋊C8—21no—
135C271⋊C8—21no—
136C23×C271—21no—
137Q8×C271—21no—
138Q8×C271—21no—
139Q8×C271—81no—
140D4×C271—21no—
141D4×C271—21no—
142C2×C4×C271—21no—
143C2×C4×C271—21no—
144C2×C4×C271—21no—
145C2×C4×C271C2×D411yescharacteristic
146C2×C4×C271C2×D411yescharacteristic
147C2×C4×C271C2411yescharacteristic
148C8×C271—21no—
149C8×C271—21no—
150C22×C271—41no—
151C22×C271—41no—
152C22×C271—41no—
153C22×C271C22×D411yescharacteristic
154C2×D271—21no—
155C2×D271—21no—
156C2×D271—41no—
157C2×D271—41no—
158C2×D271—41no—
159C2×D271—41no—
160C2×D271—41no—
161C2×D271—41no—
162C2×D271—84no—
163C2×D271#12412yes—
164C4×C271—21no—
165C4×C271—21no—
166C4×C271—41no—
167C4×C271—41no—
168C4×C271—41no—
169C4×C271C2×#4111yescharacteristic
170C4×C271C2×#4111yescharacteristic
171Dic271—21no—
172Dic271—21no—
173Dic271—41no—
174Dic271—41no—
175Dic271#12412yes—
176C2×C271—81no—
177C2×C271C2×#12411yescharacteristic
178C2×C271#57611yescharacteristic
179C2×C271#57711yescharacteristic
180D271—22no—
181D271—81no—
182D271—81no—
183C271#313611yescharacteristic, complemented, Sylow
184#3136—2711nomaximal, Sylow
185#544—2711no—
186#535—2711no—
187#490—2711no—
188#490—2711no—
189#489—2711no—
190#489D27111yescharacteristic, complemented
191#488—2711no—
192#488—2711no—
193#486—2711no—
194#486—2711no—
195#418—2711no—
196#417—2711no—
197#416—2711no—
198C2×#102—2711no—
199#405—2711no—
200C2×#41—2711no—
201C22×D4—2711no—
202C2×M16—2711no—
203C22×C8—2711no—
204C4⋊Q8—2711no—
205C4⋊Q8—5421no—
206C4⋊Q8C2×D27111yescharacteristic, complemented
207#127—2711no—
208#127—2711no—
209#125—2711no—
210#125—2711no—
211#125—5421no—
212#124—2711no—
213#124C2×D27111yescharacteristic
214C22⋊Q8—5421no—
215C22⋊Q8—5421no—
216#121—5421no—
217#121—5421no—
218#120—5421no—
219#120—5421no—
220#117—2711no—
221#117—2711no—
222C2×C4⋊C4—2711no—
223#103—2711no—
224#103—2711no—
225#103C2×D27111yescharacteristic, complemented
226#103C2×D27111yescharacteristic, complemented
227#102—2711no—
228#102—2711no—
229#102—2711no—
230#102—2711no—
231#102—2711no—
232#102—2711no—
233#102—2711no—
234#102—2711no—
235#102—2711no—
236#102—2711no—
237#102C2×D27111yescharacteristic
238#102C2×D27111yescharacteristic
239#97—2711no—
240#97C2×D27111yescharacteristic, complemented
241C4×C8—2711no—
242C4×C8—2711no—
243C24—5421no—
244#41—5422no—
245#41—5422no—
246C2×Q8—21no—
247C2×Q8—5421no—
248C2×Q8C22×D27111yescharacteristic
249C2×D4—2711no—
250C2×D4—2711no—
251C2×D4—5421no—
252C2×D4—5422no—
253C2×D4—5422no—
254C2×D4—5421no—
255C2×D4—5421no—
256C2×D4—5421no—
257C2×D4—5421no—
258C2×D4—5421no—
259C2×D4C22×D27111yescharacteristic
260C22×C4—2711no—
261C22×C4—5421no—
262C22×C4—5421no—
263M16—5422no—
264C2×C8—2711no—
265C2×C8—2711no—
266C2×C8—5422no—
267C2×C8C22×D27111yescharacteristic
268C2×C8C22×D27111yescharacteristic
269C4⋊C4—21no—
270C4⋊C4—2711no—
271C4⋊C4—2711no—
272C4⋊C4—2711no—
273C4⋊C4—2711no—
274C4⋊C4—5421no—
275C4⋊C4—5421no—
276C4⋊C4—5421no—
277C4⋊C4—5421no—
278C4⋊C4—5421no—
279C4⋊C4—5421no—
280C4⋊C4C22×D27111yescharacteristic
281C4⋊C4C22×D27111yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—2711no—
285C22⋊C4—2711no—
286C22⋊C4—5421no—
287C22⋊C4—5421no—
288C22⋊C4—5421no—
289C42—2711no—
290C42—5421no—
291C42C22×D27111yescharacteristic
292C23—21no—
293C23—2711no—
294C23—5421no—
295C23—5421no—
296C23—10842no—
297C23—21681no—
298C23—21681no—
299Q8—21no—
300Q8—21no—
301Q8—81no—
302Q8—21681no—
303D4—21no—
304D4—21no—
305D4—5421no—
306D4—5421no—
307D4—5421no—
308D4—5421no—
309D4—10842no—
310D4—10842no—
311D4—21681no—
312D4—21681no—
313D4—21681no—
314D4—21681no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—2711no—
319C2×C4—2712no—
320C2×C4—2712no—
321C2×C4—2711no—
322C2×C4—2711no—
323C2×C4—5421no—
324C2×C4—5421no—
325C2×C4—10842no—
326C2×C4—21681no—
327C2×C4D4×D27111yescharacteristic
328C2×C4D4×D27111yescharacteristic
329C2×C4C23×D27111yescharacteristic
330C8—21no—
331C8—21no—
332C8—5421no—
333C8—5421no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—2712no—
338C22—5421no—
339C22—5421no—
340C22—10841no—
341C22—10841no—
342C22—10841no—
343C22—10841no—
344C22—10841no—
345C22—10841no—
346C22—21684no—
347C22C2×D4×D27111yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—2712no—
354C4—5421no—
355C4—5421no—
356C4—10841no—
357C4—10841no—
358C4#41×D27111yescharacteristic
359C4#41×D27111yescharacteristic
360C2—81no—
361C2—5422no—
362C2—21681no—
363C2—21681no—
364C2#124×D27111yescharacteristic
365C2#3177470111yescharacteristic
366C2#3177470611yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#489×C271C2
socleC22×C271C22×D4
prime radical#31774465C2
hypercenter#489D271
hypocenterC271#3136
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #31774524C2semidirect
3. #31774262C2semidirect
4. #31774260C2semidirect
5. #31774258C2semidirect
6. #31774253C2semidirect
7. #31774251C2semidirect
8. #31774245C2semidirect
9. #31774242C2semidirect
10. #31773900C2semidirect
11. #31773897C2semidirect
12. #31773892C2semidirect
13. #102×D271C2semidirect
14. #31773854C2semidirect
15. #489×C271C2semidirect
20. #22576159C22semidirect
24. #22576152C22semidirect
25. #22576151C22semidirect
37. #22576084C22semidirect
38. #22576084C22semidirect
41. #22576082C22semidirect
42. #22576082C22semidirect
47. #22576080C22semidirect
48. #22576080C22semidirect
49. #22576070C22semidirect
50. C4×C271⋊C8C22semidirect
51. C8×Dic271C22semidirect
52. C4⋊Q8×C271C22semidirect
54. #103×C271C22semidirect
55. #103×C271C22semidirect
58. #97×C271C22semidirect
183. C271#3136semidirect
190. #489D271semidirect
206. C4⋊Q8C2×D271semidirect
225. #103C2×D271semidirect
226. #103C2×D271semidirect
240. #97C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317745242
#317744652
#317742622
#317742602
#317742582
#317742532
#317742512
#317742452
#317742422
#317739002
#317738972
#317738922
#102×D2712
#317738542
#489×C2712
#3136271