Finite Group Databasebeta

← #46813664

Subgroups — #46813664

Order 34432.

476 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31770504C211yesmaximal, characteristic, complemented
2#31770504C211yesmaximal, characteristic, complemented
3#31770498C211yesmaximal, characteristic
4#31770498C211yesmaximal, characteristic
5#31770498C211yesmaximal, characteristic
6#31770491C211yesmaximal, characteristic, complemented
7#31770491C211yesmaximal, characteristic, complemented
8#31770491C211yesmaximal, characteristic, complemented
9#31770480C211yesmaximal, characteristic
10#31770480C211yesmaximal, characteristic
11#31770471C211yesmaximal, characteristic, complemented
12C4⋊C4×Dic269C211yesmaximal, characteristic, complemented
13C4×C4⋊Dic269C211yesmaximal, characteristic, complemented
14C22⋊C4×Dic269C211yesmaximal, characteristic
15C2×#126×C269C211yesmaximal, characteristic
16C23×Dic269C2211yescharacteristic
17C2×C4⋊Dic269C2211yescharacteristic
18C2×C4⋊Dic269C2211yescharacteristic
19C2×C4⋊Dic269C2211yescharacteristic
20C2×C4⋊Dic269C2211yescharacteristic
21C2×C4⋊Dic269C2211yescharacteristic
22C2×C4⋊Dic269C2211yescharacteristic
23C2×C4⋊Dic269C2211yescharacteristic
24C2×C22⋊Dic269C2211yescharacteristic
25C2×C22⋊Dic269C2211yescharacteristic
26C2×C22⋊Dic269C2211yescharacteristic
27C2×C4×Dic269C2211yescharacteristic
28C2×C4×Dic269C2211yescharacteristic
29C2×C4×Dic269C2211yescharacteristic
30C2×C4×Dic269C2211yescharacteristic
31C2×C4×Dic269C2211yescharacteristic
32#22575360C2211yescharacteristic
33#22575360C2211yescharacteristic
34#22575360C2211yescharacteristic
35#22575360C2211yescharacteristic
36#22575360C2211yescharacteristic
37#22575360C2211yescharacteristic
38#22575360C2211yescharacteristic
39#22575360C2211yescharacteristic
40#22575360C2211yescharacteristic
41#22575360C2211yescharacteristic
42#22575360C2211yescharacteristic
43#22575360C2211yescharacteristic
44#126×C269C418yescomplemented
45C2×C4⋊C4×C269C2211yescharacteristic
46C2×C4⋊C4×C269C2211yescharacteristic
47C2×C4⋊C4×C269C2211yescharacteristic
48C2×C22⋊C4×C269C2211yescharacteristic
49C2×C22⋊C4×C269C2211yescharacteristic
50C2×C22⋊C4×C269C2211yescharacteristic
51C2×C42×C269C2211yescharacteristic
52C22×Dic269—22no—
53C22×Dic269—41no—
54C22×Dic269—41no—
55C22×Dic269C2311yescharacteristic
56C22×Dic269C2311yescharacteristic
57C22×Dic269C2311yescharacteristic
58C22×Dic269C2311yescharacteristic
59C22×Dic269C2311yescharacteristic
60C22×Dic269C2311yescharacteristic
61C22×Dic269C2311yescharacteristic
62C22×Dic269C2311yescharacteristic
63C4⋊Dic269—22no—
64C4⋊Dic269—22no—
65C4⋊Dic269—22no—
66C4⋊Dic269—41no—
67C4⋊Dic269—41no—
68C4⋊Dic269—41no—
69C4⋊Dic269—41no—
70C22⋊Dic269—22no—
71C22⋊Dic269—41no—
72C22⋊Dic269—41no—
73C4×Dic269—22no—
74C4×Dic269—41no—
75C4×Dic269—41no—
76C4×Dic269—41no—
77C4×Dic269—41no—
78C24×C269C2311yescharacteristic
79C22×C4×C269C2311yescharacteristic
80C22×C4×C269C2311yescharacteristic
81C22×C4×C269C2311yescharacteristic
82C22×C4×C269C2311yescharacteristic
83C22×C4×C269C2311yescharacteristic
84C22×C4×C269C2311yescharacteristic
85C4⋊C4×C269C2×C414yescomplemented
86C4⋊C4×C269C2×C414yescomplemented
87C4⋊C4×C269C2×C414yescomplemented
88C22⋊C4×C269C2×C414yes—
89C22⋊C4×C269C2×C414yes—
90C22⋊C4×C269C2×C414yes—
91C42×C269C2×C414yescomplemented
92C2×Dic269—21no—
93C2×Dic269—21no—
94C2×Dic269—21no—
95C2×Dic269—21no—
96C2×Dic269—21no—
97C2×Dic269—21no—
98C2×Dic269—21no—
99C2×Dic269—21no—
100C2×Dic269—21no—
101C2×Dic269—21no—
102C2×Dic269—21no—
103C2×Dic269—21no—
104C2×Dic269—21no—
105C2×Dic269—21no—
106C2×Dic269—21no—
107C2×Dic269—21no—
108C2×Dic269—21no—
109C2×Dic269—21no—
110C2×Dic269—21no—
111C2×Dic269—21no—
112C2×Dic269—21no—
113C2×Dic269—21no—
114C2×Dic269—44no—
115C2×Dic269#4112yes—
116C2×Dic269#4112yes—
117C23×C269—21no—
118C23×C269—21no—
119C23×C269—21no—
120C23×C269—21no—
121C23×C269—21no—
122C23×C269—21no—
123C23×C269C22×C412yes—
124C23×C269C2411yescharacteristic
125C2×C4×C269—21no—
126C2×C4×C269—21no—
127C2×C4×C269—21no—
128C2×C4×C269—21no—
129C2×C4×C269—21no—
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×C269C22×C412yes—
138C2×C4×C269C22×C412yes—
139C2×C4×C269C22×C412yes—
140C2×C4×C269C22×C412yes—
141C2×C4×C269C22×C412yes—
142C2×C4×C269C22×C412yes—
143C22×C269—22no—
144C22×C269—22no—
145C22×C269—22no—
146C22×C269—41no—
147C22×C269—41no—
148C22×C269—41no—
149C22×C269—41no—
150C22×C269C23×C411yescharacteristic
151C22×C269C2×#4111yescharacteristic
152C22×C269C2×#4111yescharacteristic
153C22×C269H(5,𝔽2)11yescharacteristic
154C22×C269H(5,𝔽2)11yescharacteristic
155C22×C2692−1+411yescharacteristic
156C22×C2692−1+411yescharacteristic
157C4×C269—22no—
158C4×C269—22no—
159C4×C269—22no—
160C4×C269—22no—
161C4×C269—22no—
162C4×C269—22no—
163Dic269—22no—
164Dic269—22no—
165Dic269—41no—
166Dic269—41no—
167Dic269—41no—
168Dic269—41no—
169Dic269—41no—
170Dic269—41no—
171C2×C269—42no—
172C2×C269C4×#4111yescharacteristic
173C2×C269#52011yescharacteristic
174C2×C269#52011yescharacteristic
175C2×C269#53811yescharacteristic
176C2×C269#53911yescharacteristic
177C2×C269#54111yescharacteristic
178C2×C269#54211yescharacteristic
179C269#233411yescharacteristic, complemented, Sylow
180#2334—2691nomaximal, Sylow
181C2×#126D26911yescharacteristic
182#389—2691no—
183#389—2691no—
184#388—2691no—
185#388—2691no—
186#388—2691no—
187#387—2691no—
188#387—2691no—
189#387—2691no—
190#385—2691no—
191#385—2691no—
192#382—2691no—
193C4×C4⋊C4—2691no—
194C4×C4⋊C4—2691no—
195C4×C22⋊C4—2691no—
196C23×C4—2691no—
197#126Dic26918yescomplemented
198C2×C4⋊C4—2691no—
199C2×C4⋊C4—2691no—
200C2×C4⋊C4—2691no—
201C2×C4⋊C4—2691no—
202C2×C4⋊C4—2691no—
203C2×C4⋊C4—2691no—
204C2×C4⋊C4—2691no—
205C2×C4⋊C4C2×D26911yescharacteristic
206C2×C4⋊C4C2×D26911yescharacteristic
207C2×C4⋊C4C2×D26911yescharacteristic
208C2×C22⋊C4—2691no—
209C2×C22⋊C4—2691no—
210C2×C22⋊C4—2691no—
211C2×C22⋊C4C2×D26911yescharacteristic
212C2×C22⋊C4C2×D26911yescharacteristic
213C2×C22⋊C4C2×D26911yescharacteristic
214C2×C42—2691no—
215C2×C42—2691no—
216C2×C42—2691no—
217C2×C42—2691no—
218C2×C42—2691no—
219C2×C42C2×D26911yescharacteristic
220#95—2691no—
221#95—2691no—
222#95—2691no—
223#95—2691no—
224#95—2691no—
225#95—2691no—
226#95—2691no—
227#95—2691no—
228#95—2691no—
229#95—2691no—
230#95—2691no—
231#95—2691no—
232C24C22×D26911yescharacteristic
233C22×C4—2691no—
234C22×C4—2691no—
235C22×C4—2691no—
236C22×C4—2691no—
237C22×C4—2691no—
238C22×C4—2691no—
239C22×C4—2691no—
240C22×C4—2691no—
241C22×C4—5382no—
242C22×C4—10761no—
243C22×C4—10761no—
244C22×C4C22×D26911yescharacteristic
245C22×C4C22×D26911yescharacteristic
246C22×C4C22×D26911yescharacteristic
247C22×C4C22×D26911yescharacteristic
248C22×C4C22×D26911yescharacteristic
249C22×C4C22×D26911yescharacteristic
250C4⋊C4—5382no—
251C4⋊C4—5382no—
252C4⋊C4—5382no—
253C4⋊C4—10761no—
254C4⋊C4—10761no—
255C4⋊C4—10761no—
256C4⋊C4—10761no—
257C4⋊C4C2×Dic26914yescomplemented
258C4⋊C4C2×Dic26914yescomplemented
259C4⋊C4C2×Dic26914yescomplemented
260C22⋊C4—5382no—
261C22⋊C4—10761no—
262C22⋊C4—10761no—
263C22⋊C4C2×Dic26914yes—
264C22⋊C4C2×Dic26914yes—
265C22⋊C4C2×Dic26914yes—
266C42—5382no—
267C42—10761no—
268C42—10761no—
269C42—10761no—
270C42—10761no—
271C42C2×Dic26914yescomplemented
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23C22×Dic26912yes—
279C23C23×D26911yescharacteristic
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—2692no—
293C2×C4—2692no—
294C2×C4—5381no—
295C2×C4—5381no—
296C2×C4—5381no—
297C2×C4—5381no—
298C2×C4—5381no—
299C2×C4—5381no—
300C2×C4—5381no—
301C2×C4—5381no—
302C2×C4—5381no—
303C2×C4—5381no—
304C2×C4—5381no—
305C2×C4—5381no—
306C2×C4—5381no—
307C2×C4—5381no—
308C2×C4—5381no—
309C2×C4—5381no—
310C2×C4—5381no—
311C2×C4—5381no—
312C2×C4—5381no—
313C2×C4—5381no—
314C2×C4—5381no—
315C2×C4—5381no—
316C2×C4—10764no—
317C2×C4C22×Dic26912yes—
318C2×C4C22×Dic26912yes—
319C2×C4C22×Dic26912yes—
320C2×C4C22×Dic26912yes—
321C2×C4C22×Dic26912yes—
322C2×C4C22×Dic26912yes—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22C23×Dic26911yescharacteristic
331C22#41×D26911yescharacteristic
332C22#41×D26911yescharacteristic
333C22#2257549911yescharacteristic
334C22#2257549911yescharacteristic
335C22#2257550011yescharacteristic
336C22#2257550011yescharacteristic
337C4—22no—
338C4—22no—
339C4—22no—
340C4—22no—
341C4—22no—
342C4—22no—
343C4—5382no—
344C4—5382no—
345C4—10761no—
346C4—10761no—
347C4—10761no—
348C4—10761no—
349C4—10761no—
350C4—10761no—
351C2—42no—
352C2#41×Dic26911yescharacteristic
353C2#3177108811yescharacteristic
354C2#3177108811yescharacteristic
355C2#3177119411yescharacteristic
356C2#3177119611yescharacteristic
357C2#3177121211yescharacteristic
358C2#3177122111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D269
commutatorC22×C269C23×C4
FittingC2×#126×C269C2
socle, cosocleC23×C269C24
prime radicalC24×C269C23
hypercenterC2×#126D269
hypocenterC269#2334
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. #31770504C2semidirect
2. #31770504C2semidirect
6. #31770491C2semidirect
7. #31770491C2semidirect
8. #31770491C2semidirect
11. #31770471C2semidirect
12. C4⋊C4×Dic269C2semidirect
13. C4×C4⋊Dic269C2semidirect
44. #126×C269C4semidirect
85. C4⋊C4×C269C2×C4semidirect
86. C4⋊C4×C269C2×C4semidirect
87. C4⋊C4×C269C2×C4semidirect
91. C42×C269C2×C4semidirect
179. C269#2334semidirect
197. #126Dic269semidirect
257. C4⋊C4C2×Dic269semidirect
258. C4⋊C4C2×Dic269semidirect
259. C4⋊C4C2×Dic269semidirect
271. C42C2×Dic269semidirect

Maximal subgroups

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

subgroupindex
#317705042
#317704982
#317704912
#317704802
#317704712
C4⋊C4×Dic2692
C4×C4⋊Dic2692
C22⋊C4×Dic2692
C2×#126×C2692
#2334269