Finite Group Databasebeta

← #3296×C269 #46808403

Subgroups — #3296×C269 #46808403

Order 34432.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#552×C269C211yesmaximal, characteristic, complemented
2#546×C269C211yesmaximal, characteristic
3#488×C269C211yesmaximal, characteristic, complemented
4#483×C269C211yesmaximal, characteristic, complemented
5#482×C269C211yesmaximal, characteristic, complemented
6#478×C269C211yesmaximal, characteristic, complemented
7#467×C269C211yesmaximal, characteristic, complemented
8#465×C269C211yesmaximal, characteristic, complemented
9#464×C269C211yesmaximal, characteristic, complemented
10#452×C269C211yesmaximal, characteristic, complemented
11#450×C269C211yesmaximal, characteristic, complemented
12C4×SD16×C269C211yesmaximal, characteristic, complemented
13C8×D4×C269C211yesmaximal, characteristic, complemented
14#416×C269C211yesmaximal, characteristic, complemented
15C2×#102×C269C211yesmaximal, characteristic, complemented
16C2×#41×C269C2211yescharacteristic
17C22×D4×C269C2211yescharacteristic
18C2×Q16×C269C2211yescharacteristic, complemented
19C2×SD16×C269C2211yescharacteristic
20C2×SD16×C269C2211yescharacteristic
21C22×C8×C269C2211yescharacteristic
22C22×C8×C269C2211yescharacteristic, complemented
23#126×C269—21no—
24#125×C269—21no—
25#125×C269C2211yescharacteristic, complemented
26#124×C269C2211yescharacteristic
27#123×C269—21no—
28#123×C269—21no—
29C22⋊Q8×C269C2211yescharacteristic
30C22⋊Q8×C269C2211yescharacteristic
31C22⋊Q8×C269C2211yescharacteristic, complemented
32#121×C269—21no—
33#121×C269C2211yescharacteristic
34#120×C269—21no—
35C4×Q8×C269C2211yescharacteristic, complemented
36C4×D4×C269C2211yescharacteristic
37C4×D4×C269C2211yescharacteristic
38#117×C269—21no—
39#117×C269C2211yescharacteristic, complemented
40C2×C4⋊C4×C269C2211yescharacteristic
41C2×C22⋊C4×C269—21no—
42#107×C269C2211yescharacteristic, complemented
43#106×C269C2211yescharacteristic, complemented
44#106×C269C2211yescharacteristic, complemented
45C4⋊C8×C269C2211yescharacteristic, complemented
46#103×C269C2211yescharacteristic, complemented
47#103×C269C2211yescharacteristic, complemented
48#103×C269C2211yescharacteristic, complemented
49#103×C269C2211yescharacteristic, complemented
50#102×C269C2211yescharacteristic
51#102×C269C2211yescharacteristic
52#102×C269C2211yescharacteristic
53#102×C269C2211yescharacteristic
54#102×C269C2211yescharacteristic
55#102×C269C2211yescharacteristic
56C22⋊C8×C269C2211yescharacteristic
57C22⋊C8×C269C2211yescharacteristic, complemented
58C4×C8×C269C2211yescharacteristic, complemented
59C24×C269—21no—
60#41×C269—22no—
61#41×C269—41no—
62C2×Q8×C269C2311yescharacteristic
63C2×Q8×C269C2311yescharacteristic
64C2×D4×C269—22no—
65C2×D4×C269—21no—
66C2×D4×C269—21no—
67C2×D4×C269—21no—
68C2×D4×C269—41no—
69C2×D4×C269D411yescharacteristic
70C2×D4×C269C2311yescharacteristic
71C2×D4×C269C2311yescharacteristic
72C22×C4×C269—21no—
73C22×C4×C269—21no—
74C22×C4×C269—21no—
75C22×C4×C269C2311yescharacteristic
76C22×C4×C269C2311yescharacteristic
77Q16×C269—41no—
78SD16×C269—22no—
79SD16×C269—41no—
80C2×C8×C269—41no—
81C2×C8×C269—41no—
82C2×C8×C269C2311yescharacteristic
83C2×C8×C269C2311yescharacteristic
84C2×C8×C269C2311yescharacteristic
85C2×C8×C269C2311yescharacteristic
86C4⋊C4×C269—21no—
87C4⋊C4×C269—21no—
88C4⋊C4×C269—21no—
89C4⋊C4×C269—21no—
90C4⋊C4×C269—21no—
91C4⋊C4×C269—21no—
92C4⋊C4×C269—21no—
93C4⋊C4×C269D411yescharacteristic
94C4⋊C4×C269C2311yescharacteristic
95C4⋊C4×C269C2311yescharacteristic
96C4⋊C4×C269C2311yescharacteristic
97C4⋊C4×C269C2311yescharacteristic
98C22⋊C4×C269—21no—
99C22⋊C4×C269—21no—
100C22⋊C4×C269—21no—
101C22⋊C4×C269—21no—
102C22⋊C4×C269—21no—
103C22⋊C4×C269—21no—
104C22⋊C4×C269—21no—
105C22⋊C4×C269—21no—
106C22⋊C4×C269D411yescharacteristic
107C22⋊C4×C269D411yescharacteristic
108C42×C269—21no—
109C42×C269—21no—
110C42×C269C2311yescharacteristic
111C23×C269—21no—
112C23×C269—21no—
113C23×C269—42no—
114C23×C269—42no—
115C23×C269—81no—
116C23×C269C2×D411yescharacteristic
117C23×C269C2×D411yescharacteristic
118Q8×C269—21no—
119Q8×C269—21no—
120Q8×C269—21no—
121Q8×C269—21no—
122D4×C269—22no—
123D4×C269—21no—
124D4×C269—21no—
125D4×C269—21no—
126D4×C269—42no—
127D4×C269—42no—
128D4×C269—81no—
129D4×C269#4112yes—
130C2×C4×C269—21no—
131C2×C4×C269—21no—
132C2×C4×C269—21no—
133C2×C4×C269—21no—
134C2×C4×C269—21no—
135C2×C4×C269—21no—
136C2×C4×C269—21no—
137C2×C4×C269—21no—
138C2×C4×C269—21no—
139C2×C4×C269—21no—
140C2×C4×C269—42no—
141C2×C4×C269—42no—
142C2×C4×C269—81no—
143C2×C4×C269C2×D411yescharacteristic
144C2×C4×C269C2×D411yescharacteristic
145C2×C4×C269C2×D411yescharacteristic
146C2×C4×C269C2×D411yescharacteristic
147C2×C4×C269C2411yescharacteristic
148C8×C269—21no—
149C8×C269—21no—
150C8×C269—21no—
151C8×C269—21no—
152C22×C269—21no—
153C22×C269—21no—
154C22×C269—21no—
155C22×C269—22no—
156C22×C269—21no—
157C22×C269—21no—
158C22×C269—41no—
159C22×C269—41no—
160C22×C269—41no—
161C22×C269—41no—
162C22×C269—41no—
163C22×C269—84no—
164C22×C269#13512yes—
165C22×C269C22×D411yescharacteristic
166C4×C269—21no—
167C4×C269—21no—
168C4×C269—21no—
169C4×C269—21no—
170C4×C269—41no—
171C4×C269—41no—
172C4×C269—41no—
173C4×C269—41no—
174C4×C269—41no—
175C4×C269—41no—
176C4×C269C2×#4111yescharacteristic
177C4×C269H(5,𝔽2)11yescharacteristic
178C2×C269—22no—
179C2×C269—41no—
180C2×C269—42no—
181C2×C269—81no—
182C2×C269#54611yescharacteristic
183C2×C269C2×#13511yescharacteristic
184C2×C269#57711yescharacteristic
185C269#329611yescharacteristic, direct product factor, Sylow
186#3296C26911yesmaximal, characteristic, direct product factor, Sylow
187#552C2×C26911yescharacteristic, complemented
188#546C2×C26911yescharacteristic
189#488C2×C26911yescharacteristic, complemented
190#483C2×C26911yescharacteristic, complemented
191#482C2×C26911yescharacteristic, complemented
192#478C2×C26911yescharacteristic, complemented
193#467C2×C26911yescharacteristic, complemented
194#465C2×C26911yescharacteristic, complemented
195#464C2×C26911yescharacteristic, complemented
196#452C2×C26911yescharacteristic, complemented
197#450C2×C26911yescharacteristic, complemented
198C4×SD16C2×C26911yescharacteristic, complemented
199C8×D4C2×C26911yescharacteristic, complemented
200#416C2×C26911yescharacteristic, complemented
201C2×#102C2×C26911yescharacteristic, complemented
202C2×#41C22×C26911yescharacteristic
203C22×D4C22×C26911yescharacteristic
204C2×Q16C22×C26911yescharacteristic, complemented
205C2×SD16C22×C26911yescharacteristic
206C2×SD16C22×C26911yescharacteristic
207C22×C8C22×C26911yescharacteristic
208C22×C8C22×C26911yescharacteristic, complemented
209#126—21no—
210#125—21no—
211#125C22×C26911yescharacteristic, complemented
212#124C22×C26911yescharacteristic
213#123—21no—
214#123—21no—
215C22⋊Q8C22×C26911yescharacteristic
216C22⋊Q8C22×C26911yescharacteristic
217C22⋊Q8C22×C26911yescharacteristic, complemented
218#121—21no—
219#121C22×C26911yescharacteristic
220#120—21no—
221C4×Q8C22×C26911yescharacteristic, complemented
222C4×D4C22×C26911yescharacteristic
223C4×D4C22×C26911yescharacteristic
224#117—21no—
225#117C22×C26911yescharacteristic, complemented
226C2×C4⋊C4C22×C26911yescharacteristic
227C2×C22⋊C4—21no—
228#107C22×C26911yescharacteristic, complemented
229#106C22×C26911yescharacteristic, complemented
230#106C22×C26911yescharacteristic, complemented
231C4⋊C8C22×C26911yescharacteristic, complemented
232#103C22×C26911yescharacteristic, complemented
233#103C22×C26911yescharacteristic, complemented
234#103C22×C26911yescharacteristic, complemented
235#103C22×C26911yescharacteristic, complemented
236#102C22×C26911yescharacteristic
237#102C22×C26911yescharacteristic
238#102C22×C26911yescharacteristic
239#102C22×C26911yescharacteristic
240#102C22×C26911yescharacteristic
241#102C22×C26911yescharacteristic
242C22⋊C8C22×C26911yescharacteristic
243C22⋊C8C22×C26911yescharacteristic, complemented
244C4×C8C22×C26911yescharacteristic, complemented
245C24—21no—
246#41—22no—
247#41—41no—
248C2×Q8C23×C26911yescharacteristic
249C2×Q8C23×C26911yescharacteristic
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—41no—
255C2×D4D4×C26911yescharacteristic
256C2×D4C23×C26911yescharacteristic
257C2×D4C23×C26911yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4C23×C26911yescharacteristic
262C22×C4C23×C26911yescharacteristic
263Q16—41no—
264SD16—22no—
265SD16—41no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8C23×C26911yescharacteristic
269C2×C8C23×C26911yescharacteristic
270C2×C8C23×C26911yescharacteristic
271C2×C8C23×C26911yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4D4×C26911yescharacteristic
280C4⋊C4C23×C26911yescharacteristic
281C4⋊C4C23×C26911yescharacteristic
282C4⋊C4C23×C26911yescharacteristic
283C4⋊C4C23×C26911yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4D4×C26911yescharacteristic
293C22⋊C4D4×C26911yescharacteristic
294C42—21no—
295C42—21no—
296C42C23×C26911yescharacteristic
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23C2×D4×C26911yescharacteristic
303C23C2×D4×C26911yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—21no—
308D4—22no—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—42no—
313D4—42no—
314D4—81no—
315D4#41×C26912yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—81no—
329C2×C4C2×D4×C26911yescharacteristic
330C2×C4C2×D4×C26911yescharacteristic
331C2×C4C2×D4×C26911yescharacteristic
332C2×C4C2×D4×C26911yescharacteristic
333C2×C4C24×C26911yescharacteristic
334C8—21no—
335C8—21no—
336C8—21no—
337C8—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—22no—
342C22—21no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—84no—
350C22#135×C26912yes—
351C22C22×D4×C26911yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4C2×#41×C26911yescharacteristic
363C4H(5,𝔽2)×C26911yescharacteristic
364C2—22no—
365C2—41no—
366C2—42no—
367C2—81no—
368C2#546×C26911yescharacteristic
369C2C2×#135×C26911yescharacteristic
370C2#577×C26911yescharacteristic
subgroupG/H
center, socleC22×C269C22×D4
commutator, Frattini, cosocleC2×C4C24×C269
prime radical#546×C269C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C269 ◁ C4×D4×C269 ◁ G

Complements

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

NH ≅ G/Ntype
1. #552×C269C2semidirect
3. #488×C269C2semidirect
4. #483×C269C2semidirect
5. #482×C269C2semidirect
6. #478×C269C2semidirect
7. #467×C269C2semidirect
8. #465×C269C2semidirect
9. #464×C269C2semidirect
10. #452×C269C2semidirect
11. #450×C269C2semidirect
12. C4×SD16×C269C2semidirect
13. C8×D4×C269C2semidirect
14. #416×C269C2semidirect
15. C2×#102×C269C2semidirect
18. C2×Q16×C269C22semidirect
22. C22×C8×C269C22semidirect
25. #125×C269C22semidirect
31. C22⋊Q8×C269C22semidirect
35. C4×Q8×C269C22semidirect
39. #117×C269C22semidirect
42. #107×C269C22semidirect
43. #106×C269C22semidirect
44. #106×C269C22semidirect
45. C4⋊C8×C269C22semidirect
46. #103×C269C22semidirect
47. #103×C269C22semidirect
48. #103×C269C22semidirect
49. #103×C269C22semidirect
57. C22⋊C8×C269C22semidirect
58. C4×C8×C269C22semidirect
185. C269#3296direct (N × H)
186. #3296C269direct (N × H)
187. #552C2×C269semidirect
189. #488C2×C269semidirect
190. #483C2×C269semidirect
191. #482C2×C269semidirect
192. #478C2×C269semidirect
193. #467C2×C269semidirect
194. #465C2×C269semidirect
195. #464C2×C269semidirect
196. #452C2×C269semidirect
197. #450C2×C269semidirect
198. C4×SD16C2×C269semidirect
199. C8×D4C2×C269semidirect
200. #416C2×C269semidirect
201. C2×#102C2×C269semidirect
204. C2×Q16C22×C269semidirect
208. C22×C8C22×C269semidirect
211. #125C22×C269semidirect
217. C22⋊Q8C22×C269semidirect
221. C4×Q8C22×C269semidirect
225. #117C22×C269semidirect
228. #107C22×C269semidirect
229. #106C22×C269semidirect
230. #106C22×C269semidirect
231. C4⋊C8C22×C269semidirect
232. #103C22×C269semidirect
233. #103C22×C269semidirect
234. #103C22×C269semidirect
235. #103C22×C269semidirect
243. C22⋊C8C22×C269semidirect
244. C4×C8C22×C269semidirect

Maximal subgroups

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

subgroupindex
#552×C2692
#546×C2692
#488×C2692
#483×C2692
#482×C2692
#478×C2692
#467×C2692
#465×C2692
#464×C2692
#452×C2692
#450×C2692
C4×SD16×C2692
C8×D4×C2692
#416×C2692
C2×#102×C2692
#3296269