Finite Group Databasebeta

← #46848221

Subgroups — #46848221

Order 34688.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774597C211yesmaximal, characteristic, complemented
2#31774540C211yesmaximal, characteristic, complemented
3#31774251C211yesmaximal, characteristic, complemented
4#31774231C211yesmaximal, characteristic, complemented
5#31774220C211yesmaximal, characteristic, complemented
6#31774198C211yesmaximal, characteristic, complemented
7#31774130C211yesmaximal, characteristic, complemented
8#31774123C211yesmaximal, characteristic, complemented
9#31774041C211yesmaximal, characteristic, complemented
10#31774029C211yesmaximal, characteristic, complemented
11C4×#16864639C211yesmaximal, characteristic, complemented
12#31773962C211yesmaximal, characteristic, complemented
13#31773892C211yesmaximal, characteristic, complemented
14#102×D271C211yesmaximal, characteristic, complemented
15#463×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×D4×D271C2211yescharacteristic
18C2×#16864639C2211yescharacteristic
19C2×C8×D271C2211yescharacteristic
20C2×C8×D271C2211yescharacteristic, complemented
21#22576153—21no—
22#22576152C2211yescharacteristic, complemented
23#22576143—21no—
24#22576142—21no—
25#22576136C2211yescharacteristic
26#22576133—21no—
27#22576133—21no—
28#22576131C2211yescharacteristic
29#22576131C2211yescharacteristic
30#22576131C2211yescharacteristic, complemented
31#22576130—21no—
32C4×D1084C2211yescharacteristic
33C4×D1084C2211yescharacteristic
34C4×C22⋊D271—21no—
35#22576118C2211yescharacteristic, complemented
36C4⋊C4×D271C2211yescharacteristic
37C22⋊C4×D271—21no—
38#22576094C2211yescharacteristic, complemented
39#22576093C2211yescharacteristic, complemented
40#22576092C2211yescharacteristic, complemented
41#22576084C2211yescharacteristic, complemented
42#22576084C2211yescharacteristic, complemented
43#22576082C2211yescharacteristic
44#22576082C2211yescharacteristic
45#22576081C2211yescharacteristic
46#22576081C2211yescharacteristic
47#22576081C2211yescharacteristic, complemented
48#22576080C2211yescharacteristic
49(C2×D271)⋊C8C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic, complemented
51C4×C271⋊C8C2211yescharacteristic, complemented
52C2×SD16×C271C2211yescharacteristic, complemented
53C2×D8×C271C2211yescharacteristic
54#124×C271C2211yescharacteristic
55C4×D4×C271C2211yescharacteristic, complemented
56C4⋊C8×C271C2211yescharacteristic, complemented
57#103×C271C2211yescharacteristic, complemented
58#102×C271C2211yescharacteristic
59C23×D271—21no—
60#16864655—22no—
61#16864655—41no—
62C2×D1084—21no—
63C2×D1084D411yescharacteristic
64C2×D1084C2311yescharacteristic
65C2×C22⋊D271—21no—
66C2×C22⋊D271—21no—
67C2×C22⋊D271—21no—
68C2×C22⋊D271—21no—
69D4×D271—22no—
70D4×D271—41no—
71C2×C4×D271—21no—
72C2×C4×D271—21no—
73C2×C4×D271—21no—
74C2×C4×D271C2311yescharacteristic
75C2×C4×D271C2311yescharacteristic
76#16864639—22no—
77C2×C271⋊C8C2311yescharacteristic
78C2×C271⋊C8C2311yescharacteristic
79C8×D271—41no—
80C8×D271—41no—
81C4⋊Dic271D411yescharacteristic
82C4⋊Dic271C2311yescharacteristic
83C4⋊Dic271C2311yescharacteristic
84C4⋊Dic271C2311yescharacteristic
85Dic271⋊C4—21no—
86Dic271⋊C4—21no—
87Dic271⋊C4—21no—
88Dic271⋊C4—21no—
89C22⋊Dic271—21no—
90C22⋊Dic271—21no—
91C22⋊Dic271—21no—
92(C2×D271)⋊C4—21no—
93(C2×D271)⋊C4—21no—
94(C2×D271)⋊C4—21no—
95(C2×D271)⋊C4D411yescharacteristic
96(C2×D271)⋊C4D411yescharacteristic
97C4×Dic271—21no—
98C2×Q8×C271C2311yescharacteristic
99C2×D4×C271C2311yescharacteristic
100C2×D4×C271C2311yescharacteristic
101C22×C4×C271—21no—
102SD16×C271—41no—
103D8×C271—41no—
104C2×C8×C271C2311yescharacteristic
105C2×C8×C271C2311yescharacteristic
106C4⋊C4×C271C2311yescharacteristic
107C22⋊C4×C271—21no—
108C22⋊C4×C271—21no—
109C22⋊C4×C271—21no—
110C42×C271C2311yescharacteristic
111C22×D271—21no—
112C22×D271—42no—
113C22×D271—42no—
114C22×D271—81no—
115C22×D271C2×D411yescharacteristic
116C22×D271C2×D411yescharacteristic
117D1084—22no—
118D1084—21no—
119D1084—42no—
120D1084#4112yes—
121C22⋊D271—42no—
122C22⋊D271—42no—
123C22⋊D271—81no—
124C22⋊D271—81no—
125C2×Dic271—21no—
126C2×Dic271—21no—
127C2×Dic271—21no—
128C2×Dic271C2×D411yescharacteristic
129C2×Dic271C2×D411yescharacteristic
130C4×D271—21no—
131C4×D271—21no—
132C4×D271—21no—
133C4×D271—21no—
134C4×D271—42no—
135C4×D271—81no—
136C4×D271—81no—
137C271⋊C8—21no—
138C271⋊C8—21no—
139C23×C271—21no—
140C23×C271—21no—
141Q8×C271—21no—
142Q8×C271—21no—
143D4×C271—21no—
144D4×C271—21no—
145D4×C271—21no—
146D4×C271—21no—
147C2×C4×C271—21no—
148C2×C4×C271—21no—
149C2×C4×C271—81no—
150C2×C4×C271C2×D411yescharacteristic
151C2×C4×C271C2×D411yescharacteristic
152C2×C4×C271C2411yescharacteristic
153C8×C271—21no—
154C8×C271—21no—
155C22×C271—41no—
156C22×C271—41no—
157C22×C271—41no—
158C22×C271—41no—
159C22×C271—41no—
160C22×C271—41no—
161C22×C271C22×D411yescharacteristic
162C2×D271—21no—
163C2×D271—21no—
164C2×D271—21no—
165C2×D271—22no—
166C2×D271—21no—
167C2×D271—21no—
168C2×D271—41no—
169C2×D271—41no—
170C2×D271—84no—
171C2×D271#13512yes—
172C4×C271—21no—
173C4×C271—21no—
174C4×C271—41no—
175C4×C271—41no—
176C4×C271C2×#4111yescharacteristic
177C4×C271H(5,𝔽2)11yescharacteristic
178Dic271—21no—
179Dic271—21no—
180Dic271—41no—
181Dic271—41no—
182Dic271—41no—
183C2×C271—81no—
184C2×C271—81no—
185C2×C271#54611yescharacteristic
186C2×C271C2×#13511yescharacteristic
187C2×C271#57611yescharacteristic
188D271—22no—
189D271—41no—
190D271—42no—
191C271#329711yescharacteristic, complemented, Sylow
192#3297—2711nomaximal, Sylow
193#553—2711no—
194#546—2711no—
195#488—2711no—
196#484—2711no—
197#482—2711no—
198#478—2711no—
199#466—2711no—
200#465—2711no—
201#463D27111yescharacteristic, complemented
202#449—2711no—
203#447—2711no—
204C4×D8—2711no—
205C8×D4—2711no—
206#416—2711no—
207C2×#102—2711no—
208C2×#41—2711no—
209C22×D4—2711no—
210C2×SD16C2×D27111yescharacteristic, complemented
211C2×D8—2711no—
212C2×D8C2×D27111yescharacteristic
213C22×C8—2711no—
214C22×C8—2711no—
215#126—5421no—
216#125—2711no—
217#124C2×D27111yescharacteristic
218#123—5421no—
219#123—5421no—
220C22⋊Q8—2711no—
221#121—2711no—
222#121—2711no—
223#121—2711no—
224#121—5421no—
225#121—5421no—
226#120—5421no—
227C4×D4—2711no—
228C4×D4—2711no—
229C4×D4—5421no—
230C4×D4C2×D27111yescharacteristic, complemented
231#117—2711no—
232C2×C4⋊C4—2711no—
233C2×C22⋊C4—5421no—
234#107—2711no—
235#107—2711no—
236#106—2711no—
237C4⋊C8C2×D27111yescharacteristic, complemented
238#103—2711no—
239#103—2711no—
240#103C2×D27111yescharacteristic, complemented
241#102—2711no—
242#102—2711no—
243#102—2711no—
244#102—2711no—
245#102—2711no—
246#102—2711no—
247#102C2×D27111yescharacteristic
248C22⋊C8—2711no—
249C22⋊C8—2711no—
250C4×C8—2711no—
251C24—5421no—
252#41—5422no—
253#41—10841no—
254C2×Q8C22×D27111yescharacteristic
255C2×D4—2711no—
256C2×D4—2711no—
257C2×D4—5421no—
258C2×D4—5421no—
259C2×D4—5422no—
260C2×D4—5421no—
261C2×D4—5421no—
262C2×D4—5421no—
263C2×D4—10841no—
264C2×D4C22×D27111yescharacteristic
265C2×D4C22×D27111yescharacteristic
266C22×C4—21no—
267C22×C4—2711no—
268C22×C4—2711no—
269C22×C4—5421no—
270C22×C4—5421no—
271C22×C4—5421no—
272SD16—41no—
273D8—41no—
274D8—5422no—
275C2×C8—2711no—
276C2×C8—2711no—
277C2×C8—10841no—
278C2×C8—10841no—
279C2×C8C22×D27111yescharacteristic
280C2×C8C22×D27111yescharacteristic
281C4⋊C4—2711no—
282C4⋊C4—2711no—
283C4⋊C4—2711no—
284C4⋊C4—2711no—
285C4⋊C4—5421no—
286C4⋊C4—5421no—
287C4⋊C4—5421no—
288C4⋊C4—5421no—
289C4⋊C4C22×D27111yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—2711no—
294C22⋊C4—2711no—
295C22⋊C4—5421no—
296C22⋊C4—5421no—
297C22⋊C4—5421no—
298C22⋊C4—5421no—
299C22⋊C4—5421no—
300C22⋊C4—5421no—
301C42—5421no—
302C42C22×D27111yescharacteristic
303C23—21no—
304C23—21no—
305C23—2711no—
306C23—2711no—
307C23—5421no—
308C23—10842no—
309C23—10842no—
310C23—21681no—
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—2712no—
318D4—5422no—
319D4—5421no—
320D4—10842no—
321D4—10842no—
322D4—10842no—
323D4—21681no—
324D4—21681no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—81no—
328C2×C4—2711no—
329C2×C4—2711no—
330C2×C4—5421no—
331C2×C4—5421no—
332C2×C4—5421no—
333C2×C4—5421no—
334C2×C4—5421no—
335C2×C4—5421no—
336C2×C4—5421no—
337C2×C4—10842no—
338C2×C4—21681no—
339C2×C4—21681no—
340C2×C4D4×D27111yescharacteristic
341C2×C4D4×D27111yescharacteristic
342C2×C4C23×D27111yescharacteristic
343C8—21no—
344C8—21no—
345C8—5421no—
346C8—5421no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—2712no—
354C22—5421no—
355C22—5421no—
356C22—5421no—
357C22—5422no—
358C22—5421no—
359C22—5421no—
360C22—10841no—
361C22—10841no—
362C22—21684no—
363C22C2×D4×D27111yescharacteristic
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—5421no—
369C4—5421no—
370C4—10841no—
371C4—10841no—
372C4—10841no—
373C4#41×D27111yescharacteristic
374C4#2257620511yescharacteristic
375C2—81no—
376C2—81no—
377C2—5422no—
378C2—10841no—
379C2—10842no—
380C2#3177454411yescharacteristic
381C2#135×D27111yescharacteristic
382C2#3177470111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#463×C271C2
socleC22×C271C22×D4
hypercenter#463D271
hypocenterC271#3297
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #31774597C2semidirect
2. #31774540C2semidirect
3. #31774251C2semidirect
4. #31774231C2semidirect
5. #31774220C2semidirect
6. #31774198C2semidirect
7. #31774130C2semidirect
8. #31774123C2semidirect
9. #31774041C2semidirect
10. #31774029C2semidirect
11. C4×#16864639C2semidirect
12. #31773962C2semidirect
13. #31773892C2semidirect
14. #102×D271C2semidirect
15. #463×C271C2semidirect
20. C2×C8×D271C22semidirect
22. #22576152C22semidirect
30. #22576131C22semidirect
35. #22576118C22semidirect
38. #22576094C22semidirect
39. #22576093C22semidirect
40. #22576092C22semidirect
41. #22576084C22semidirect
42. #22576084C22semidirect
47. #22576081C22semidirect
50. (C2×D271)⋊C8C22semidirect
51. C4×C271⋊C8C22semidirect
52. C2×SD16×C271C22semidirect
55. C4×D4×C271C22semidirect
56. C4⋊C8×C271C22semidirect
57. #103×C271C22semidirect
191. C271#3297semidirect
201. #463D271semidirect
210. C2×SD16C2×D271semidirect
230. C4×D4C2×D271semidirect
237. C4⋊C8C2×D271semidirect
240. #103C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317745972
#317745402
#317742512
#317742312
#317742202
#317741982
#317741302
#317741232
#317740412
#317740292
C4×#168646392
#317739622
#317738922
#102×D2712
#463×C2712
#3297271