Finite Group Databasebeta

← #46848207

Subgroups — #46848207

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#31774594C211yesmaximal, characteristic, complemented
2#31774535C211yesmaximal, characteristic, complemented
3#31774253C211yesmaximal, characteristic, complemented
4#31774231C211yesmaximal, characteristic, complemented
5#31774222C211yesmaximal, characteristic, complemented
6#31774199C211yesmaximal, characteristic, complemented
7#31774131C211yesmaximal, characteristic, complemented
8#31774123C211yesmaximal, characteristic, complemented
9#31774114C211yesmaximal, characteristic, complemented
10#31774029C211yesmaximal, characteristic, complemented
11D8×Dic271C211yesmaximal, characteristic, complemented
12(C22⋊D271)⋊C8C211yesmaximal, characteristic, complemented
13#31773893C211yesmaximal, characteristic, complemented
14#102×D271C211yesmaximal, characteristic, complemented
15#449×C271C211yesmaximal, characteristic, complemented
16C2×D4×D271C2211yescharacteristic
17C2×#16864639C2211yescharacteristic
18C22×C271⋊C8C2211yescharacteristic, complemented
19C2×C8×D271C2211yescharacteristic
20#22576153—21no—
21#22576151C2211yescharacteristic, complemented
22#22576149C2211yescharacteristic
23#22576143—21no—
24#22576141—21no—
25#22576136C2211yescharacteristic
26#22576134C2211yescharacteristic
27#22576132—21no—
28#22576131—21no—
29#22576131C2211yescharacteristic
30#22576128—21no—
31D4×Dic271C2211yescharacteristic
32D4×Dic271C2211yescharacteristic, complemented
33C4×C22⋊D271C2211yescharacteristic
34(C22⋊D271)⋊C4—21no—
35#22576119C2211yescharacteristic, complemented
36C4⋊C4×D271C2211yescharacteristic
37C2×(C2×D271)⋊C4—21no—
38#22576094C2211yescharacteristic, complemented
39#22576093C2211yescharacteristic, complemented
40#22576092C2211yescharacteristic, complemented
41Dic271⋊C8C2211yescharacteristic, complemented
42#22576084C2211yescharacteristic, complemented
43#22576084C2211yescharacteristic, complemented
44#22576082C2211yescharacteristic
45#22576082C2211yescharacteristic
46#22576081C2211yescharacteristic
47#22576081C2211yescharacteristic
48#22576081C2211yescharacteristic, complemented
49#22576080C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51C8×Dic271C2211yescharacteristic, complemented
52C2×#41×C271C2211yescharacteristic
53C2×SD16×C271C2211yescharacteristic, complemented
54C2×D8×C271C2211yescharacteristic
55#121×C271C2211yescharacteristic, complemented
56#103×C271C2211yescharacteristic, complemented
57#102×C271C2211yescharacteristic
58C22⋊C8×C271C2211yescharacteristic, complemented
59C23×D271—21no—
60C2×D1084C2311yescharacteristic
61C2×C22⋊D271—21no—
62C2×C22⋊D271—21no—
63C2×C22⋊D271—21no—
64C2×C22⋊D271D411yescharacteristic
65D4×D271—22no—
66D4×D271—41no—
67C22×Dic271—21no—
68C22×Dic271—21no—
69C2×C4×D271—21no—
70C2×C4×D271C2311yescharacteristic
71#16864639—41no—
72C2×C271⋊C8—41no—
73C2×C271⋊C8C2311yescharacteristic
74C2×C271⋊C8C2311yescharacteristic
75C8×D271—41no—
76C4⋊Dic271—21no—
77C4⋊Dic271C2311yescharacteristic
78C4⋊Dic271C2311yescharacteristic
79C4⋊Dic271C2311yescharacteristic
80Dic271⋊C4—21no—
81Dic271⋊C4—21no—
82Dic271⋊C4—21no—
83Dic271⋊C4D411yescharacteristic
84C22⋊Dic271—21no—
85C22⋊Dic271—21no—
86C22⋊Dic271—21no—
87C22⋊Dic271—21no—
88C22⋊Dic271D411yescharacteristic
89(C2×D271)⋊C4—21no—
90(C2×D271)⋊C4—21no—
91(C2×D271)⋊C4—21no—
92(C2×D271)⋊C4—21no—
93(C2×D271)⋊C4D411yescharacteristic
94C4×Dic271—21no—
95C4×Dic271C2311yescharacteristic
96#41×C271—22no—
97#41×C271—41no—
98C2×Q8×C271C2311yescharacteristic
99C2×D4×C271—21no—
100C2×D4×C271—21no—
101C2×D4×C271C2311yescharacteristic
102C2×D4×C271C2311yescharacteristic
103C22×C4×C271—21no—
104C22×C4×C271C2311yescharacteristic
105SD16×C271—41no—
106D8×C271—22no—
107C2×C8×C271C2311yescharacteristic
108C2×C8×C271C2311yescharacteristic
109C4⋊C4×C271C2311yescharacteristic
110C22⋊C4×C271—21no—
111C22×D271—21no—
112C22×D271—42no—
113C22×D271—42no—
114C22×D271—81no—
115C22×D271C2×D411yescharacteristic
116D1084—21no—
117D1084—21no—
118C22⋊D271—22no—
119C22⋊D271—42no—
120C22⋊D271—42no—
121C22⋊D271—81no—
122C2×Dic271—21no—
123C2×Dic271—21no—
124C2×Dic271—21no—
125C2×Dic271—81no—
126C2×Dic271—81no—
127C2×Dic271C2×D411yescharacteristic
128C2×Dic271C2×D411yescharacteristic
129C2×Dic271C2×D411yescharacteristic
130C4×D271—21no—
131C4×D271—21no—
132C4×D271—81no—
133C271⋊C8—21no—
134C271⋊C8—21no—
135C23×C271—21no—
136C23×C271—21no—
137C23×C271C2×D411yescharacteristic
138Q8×C271—21no—
139Q8×C271—21no—
140D4×C271—21no—
141D4×C271—21no—
142D4×C271—21no—
143D4×C271—42no—
144D4×C271—81no—
145D4×C271#4112yes—
146C2×C4×C271—21no—
147C2×C4×C271—21no—
148C2×C4×C271—21no—
149C2×C4×C271—21no—
150C2×C4×C271—42no—
151C2×C4×C271C2×D411yescharacteristic
152C2×C4×C271C2411yescharacteristic
153C8×C271—21no—
154C8×C271—21no—
155C22×C271—21no—
156C22×C271—22no—
157C22×C271—21no—
158C22×C271—21no—
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—41no—
168C2×D271—41no—
169C2×D271—41no—
170C2×D271—84no—
171C2×D271#13512yes—
172C4×C271—21no—
173C4×C271—41no—
174C4×C271—41no—
175C4×C271C2×#4111yescharacteristic
176C4×C271H(5,𝔽2)11yescharacteristic
177Dic271—21no—
178Dic271—21no—
179Dic271—21no—
180Dic271—41no—
181Dic271—41no—
182Dic271—41no—
183C2×C271—42no—
184C2×C271—41no—
185C2×C271—81no—
186C2×C271#54611yescharacteristic
187C2×C271C2×#13511yescharacteristic
188C2×C271#57611yescharacteristic
189D271—22no—
190D271—81no—
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#463—2711no—
202#449D27111yescharacteristic, complemented
203#447—2711no—
204C4×D8—2711no—
205C8×D4—2711no—
206#416—2711no—
207C2×#102—2711no—
208C2×#41C2×D27111yescharacteristic
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#124—2711no—
218#123—5421no—
219#123—5421no—
220C22⋊Q8—2711no—
221#121—2711no—
222#121—2711no—
223#121—5421no—
224#121—5421no—
225#121C2×D27111yescharacteristic, complemented
226#120—5421no—
227C4×D4—2711no—
228C4×D4—2711no—
229C4×D4—2711no—
230C4×D4—5421no—
231#117—2711no—
232C2×C4⋊C4—2711no—
233C2×C22⋊C4—5421no—
234#107—2711no—
235#107—2711no—
236#106—2711no—
237C4⋊C8—2711no—
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⋊C8C2×D27111yescharacteristic, complemented
250C4×C8—2711no—
251C24—5421no—
252#41—22no—
253#41—41no—
254C2×Q8C22×D27111yescharacteristic
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—2711no—
258C2×D4—2711no—
259C2×D4—5421no—
260C2×D4—5422no—
261C2×D4—5421no—
262C2×D4—5421no—
263C2×D4—10841no—
264C2×D4C22×D27111yescharacteristic
265C2×D4C22×D27111yescharacteristic
266C22×C4—21no—
267C22×C4—2711no—
268C22×C4—5421no—
269C22×C4—5421no—
270C22×C4—5421no—
271C22×C4C22×D27111yescharacteristic
272SD16—41no—
273D8—22no—
274D8—10841no—
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—2711no—
292C22⋊C4—2711no—
293C22⋊C4—5421no—
294C22⋊C4—5421no—
295C22⋊C4—5421no—
296C22⋊C4—5421no—
297C22⋊C4—5421no—
298C22⋊C4—5421no—
299C22⋊C4—5421no—
300C22⋊C4—5421no—
301C42—2711no—
302C42—5421no—
303C23—21no—
304C23—21no—
305C23—2711no—
306C23—5421no—
307C23—10842no—
308C23—10842no—
309C23—21681no—
310C23D4×D27111yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—42no—
317D4—81no—
318D4—5422no—
319D4—5421no—
320D4—5421no—
321D4—10842no—
322D4—10842no—
323D4—21681no—
324D4#1686465312yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—42no—
330C2×C4—2711no—
331C2×C4—2711no—
332C2×C4—2711no—
333C2×C4—5421no—
334C2×C4—5421no—
335C2×C4—5421no—
336C2×C4—5421no—
337C2×C4—5421no—
338C2×C4—21681no—
339C2×C4—21681no—
340C2×C4—21681no—
341C2×C4D4×D27111yescharacteristic
342C2×C4C23×D27111yescharacteristic
343C8—21no—
344C8—21no—
345C8—5421no—
346C8—5421no—
347C22—21no—
348C22—22no—
349C22—21no—
350C22—21no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—2712no—
357C22—5421no—
358C22—5421no—
359C22—10841no—
360C22—10841no—
361C22—10841no—
362C22—21684no—
363C22C2×D4×D27111yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—5421no—
368C4—5421no—
369C4—5421no—
370C4—10841no—
371C4—10841no—
372C4—10841no—
373C4C2×#1686465311yescharacteristic
374C4#2257620511yescharacteristic
375C2—41no—
376C2—42no—
377C2—81no—
378C2—5422no—
379C2—21681no—
380C2#3177453411yescharacteristic
381C2#135×D27111yescharacteristic
382C2#3177470111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#449×C271C2
socleC22×C271C22×D4
hypercenter#449D271
hypocenterC271#3297
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #31774594C2semidirect
2. #31774535C2semidirect
3. #31774253C2semidirect
4. #31774231C2semidirect
5. #31774222C2semidirect
6. #31774199C2semidirect
7. #31774131C2semidirect
8. #31774123C2semidirect
9. #31774114C2semidirect
10. #31774029C2semidirect
11. D8×Dic271C2semidirect
12. (C22⋊D271)⋊C8C2semidirect
13. #31773893C2semidirect
14. #102×D271C2semidirect
15. #449×C271C2semidirect
18. C22×C271⋊C8C22semidirect
21. #22576151C22semidirect
32. D4×Dic271C22semidirect
35. #22576119C22semidirect
38. #22576094C22semidirect
39. #22576093C22semidirect
40. #22576092C22semidirect
41. Dic271⋊C8C22semidirect
42. #22576084C22semidirect
43. #22576084C22semidirect
48. #22576081C22semidirect
51. C8×Dic271C22semidirect
53. C2×SD16×C271C22semidirect
55. #121×C271C22semidirect
56. #103×C271C22semidirect
58. C22⋊C8×C271C22semidirect
191. C271#3297semidirect
202. #449D271semidirect
210. C2×SD16C2×D271semidirect
225. #121C2×D271semidirect
240. #103C2×D271semidirect
249. C22⋊C8C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317745942
#317745352
#317742532
#317742312
#317742222
#317741992
#317741312
#317741232
#317741142
#317740292
D8×Dic2712
(C22⋊D271)⋊C82
#317738932
#102×D2712
#449×C2712
#3297271