Finite Group Databasebeta

← #3258×C269 #46808365

Subgroups — #3258×C269 #46808365

Order 34432.

334 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#542×C269C211yesmaximal, characteristic
2#541×C269C211yesmaximal, characteristic, complemented
3#479×C269C211yesmaximal, characteristic, complemented
4#478×C269C211yesmaximal, characteristic, complemented
5#477×C269C211yesmaximal, characteristic, complemented
6#475×C269C211yesmaximal, characteristic, complemented
7#470×C269C211yesmaximal, characteristic, complemented
8#468×C269C211yesmaximal, characteristic
9#467×C269C211yesmaximal, characteristic, complemented
10#465×C269C211yesmaximal, characteristic
11#464×C269C211yesmaximal, characteristic, complemented
12#464×C269C211yesmaximal, characteristic, complemented
13#462×C269C211yesmaximal, characteristic, complemented
14#461×C269C211yesmaximal, characteristic, complemented
15#424×C269C211yesmaximal, characteristic, complemented
16C2×Q16×C269C2211yescharacteristic
17C2×Q16×C269C2211yescharacteristic
18C2×SD16×C269C2211yescharacteristic
19C2×SD16×C269C2211yescharacteristic
20C2×M16×C269C2211yescharacteristic
21C22×C8×C269C2211yescharacteristic
22C4⋊Q8×C269C2211yescharacteristic
23C4⋊Q8×C269C2211yescharacteristic
24#126×C269—21no—
25#126×C269—21no—
26#126×C269—21no—
27#125×C269—21no—
28#125×C269—21no—
29#125×C269C2211yescharacteristic
30#124×C269—21no—
31#124×C269C2211yescharacteristic
32#123×C269—21no—
33C22⋊Q8×C269—21no—
34C22⋊Q8×C269C2211yescharacteristic
35C22⋊Q8×C269C2211yescharacteristic
36C22⋊Q8×C269C2211yescharacteristic
37#121×C269C2211yescharacteristic
38C4×Q8×C269C2211yescharacteristic
39C4×Q8×C269C2211yescharacteristic
40C4×Q8×C269C2211yescharacteristic
41C4×D4×C269C2211yescharacteristic
42#117×C269C2211yescharacteristic
43#107×C269C2211yescharacteristic
44#107×C269C2211yescharacteristic
45#106×C269C2211yescharacteristic
46#106×C269C2211yescharacteristic
47C4⋊C8×C269C2211yescharacteristic
48C4⋊C8×C269C2211yescharacteristic
49C4⋊C8×C269C2211yescharacteristic
50C4⋊C8×C269C2211yescharacteristic
51#103×C269C2211yescharacteristic
52#103×C269C2211yescharacteristic
53#103×C269C2211yescharacteristic
54#103×C269C2211yescharacteristic
55#103×C269C2211yescharacteristic
56#103×C269C2211yescharacteristic
57#102×C269C2211yescharacteristic
58#102×C269C2211yescharacteristic
59C2×Q8×C269—21no—
60C2×Q8×C269C2311yescharacteristic
61C2×Q8×C269C2311yescharacteristic
62C2×Q8×C269C2311yescharacteristic
63C2×D4×C269—21no—
64C2×D4×C269C2311yescharacteristic
65C22×C4×C269—21no—
66C22×C4×C269C2311yescharacteristic
67Q16×C269—41no—
68Q16×C269—41no—
69SD16×C269—41no—
70SD16×C269—41no—
71M16×C269—41no—
72C2×C8×C269—41no—
73C2×C8×C269C2311yescharacteristic
74C2×C8×C269C2311yescharacteristic
75C2×C8×C269C2311yescharacteristic
76C2×C8×C269C2311yescharacteristic
77C4⋊C4×C269—21no—
78C4⋊C4×C269—21no—
79C4⋊C4×C269—21no—
80C4⋊C4×C269—21no—
81C4⋊C4×C269—21no—
82C4⋊C4×C269—21no—
83C4⋊C4×C269—21no—
84C4⋊C4×C269—21no—
85C4⋊C4×C269—21no—
86C4⋊C4×C269—21no—
87C4⋊C4×C269—21no—
88C4⋊C4×C269D411yescharacteristic
89C4⋊C4×C269D411yescharacteristic
90C4⋊C4×C269C2311yescharacteristic
91C4⋊C4×C269C2311yescharacteristic
92C4⋊C4×C269C2311yescharacteristic
93C4⋊C4×C269C2311yescharacteristic
94C22⋊C4×C269—21no—
95C22⋊C4×C269—21no—
96C22⋊C4×C269—21no—
97C22⋊C4×C269—21no—
98C22⋊C4×C269—21no—
99C22⋊C4×C269—21no—
100C22⋊C4×C269—21no—
101C22⋊C4×C269D411yescharacteristic
102C22⋊C4×C269D411yescharacteristic
103C42×C269—21no—
104C42×C269—21no—
105C42×C269—21no—
106C42×C269C2311yescharacteristic
107C42×C269C2311yescharacteristic
108C23×C269—21no—
109C23×C269C2×D411yescharacteristic
110Q8×C269—21no—
111Q8×C269—21no—
112Q8×C269—21no—
113Q8×C269—21no—
114Q8×C269—21no—
115Q8×C269—21no—
116Q8×C269—81no—
117D4×C269—21no—
118D4×C269—21no—
119D4×C269—81no—
120C2×C4×C269—21no—
121C2×C4×C269—21no—
122C2×C4×C269—21no—
123C2×C4×C269—21no—
124C2×C4×C269—21no—
125C2×C4×C269—21no—
126C2×C4×C269—21no—
127C2×C4×C269—21no—
128C2×C4×C269—21no—
129C2×C4×C269—81no—
130C2×C4×C269C2×D411yescharacteristic
131C2×C4×C269C2×D411yescharacteristic
132C2×C4×C269C2×D411yescharacteristic
133C2×C4×C269C2×D411yescharacteristic
134C2×C4×C269C2×D411yescharacteristic
135C2×C4×C269C2411yescharacteristic
136C8×C269—21no—
137C8×C269—21no—
138C8×C269—21no—
139C8×C269—21no—
140C22×C269—21no—
141C22×C269—21no—
142C22×C269—21no—
143C22×C269—41no—
144C22×C269—41no—
145C22×C269—41no—
146C22×C269C22×D411yescharacteristic
147C4×C269—21no—
148C4×C269—21no—
149C4×C269—21no—
150C4×C269—21no—
151C4×C269—21no—
152C4×C269—41no—
153C4×C269—41no—
154C4×C269—41no—
155C4×C269—41no—
156C4×C269—41no—
157C4×C269—41no—
158C4×C269—41no—
159C4×C269H(5,𝔽2)11yescharacteristic
160C4×C2692−1+411yescharacteristic
161C2×C269—41no—
162C2×C269—81no—
163C2×C269#53711yescharacteristic
164C2×C269#57711yescharacteristic
165C2×C269#57811yescharacteristic
166C269#325811yescharacteristic, direct product factor, Sylow
167#3258C26911yesmaximal, characteristic, direct product factor, Sylow
168#542C2×C26911yescharacteristic
169#541C2×C26911yescharacteristic, complemented
170#479C2×C26911yescharacteristic, complemented
171#478C2×C26911yescharacteristic, complemented
172#477C2×C26911yescharacteristic, complemented
173#475C2×C26911yescharacteristic, complemented
174#470C2×C26911yescharacteristic, complemented
175#468C2×C26911yescharacteristic
176#467C2×C26911yescharacteristic, complemented
177#465C2×C26911yescharacteristic
178#464C2×C26911yescharacteristic, complemented
179#464C2×C26911yescharacteristic, complemented
180#462C2×C26911yescharacteristic, complemented
181#461C2×C26911yescharacteristic, complemented
182#424C2×C26911yescharacteristic, complemented
183C2×Q16C22×C26911yescharacteristic
184C2×Q16C22×C26911yescharacteristic
185C2×SD16C22×C26911yescharacteristic
186C2×SD16C22×C26911yescharacteristic
187C2×M16C22×C26911yescharacteristic
188C22×C8C22×C26911yescharacteristic
189C4⋊Q8C22×C26911yescharacteristic
190C4⋊Q8C22×C26911yescharacteristic
191#126—21no—
192#126—21no—
193#126—21no—
194#125—21no—
195#125—21no—
196#125C22×C26911yescharacteristic
197#124—21no—
198#124C22×C26911yescharacteristic
199#123—21no—
200C22⋊Q8—21no—
201C22⋊Q8C22×C26911yescharacteristic
202C22⋊Q8C22×C26911yescharacteristic
203C22⋊Q8C22×C26911yescharacteristic
204#121C22×C26911yescharacteristic
205C4×Q8C22×C26911yescharacteristic
206C4×Q8C22×C26911yescharacteristic
207C4×Q8C22×C26911yescharacteristic
208C4×D4C22×C26911yescharacteristic
209#117C22×C26911yescharacteristic
210#107C22×C26911yescharacteristic
211#107C22×C26911yescharacteristic
212#106C22×C26911yescharacteristic
213#106C22×C26911yescharacteristic
214C4⋊C8C22×C26911yescharacteristic
215C4⋊C8C22×C26911yescharacteristic
216C4⋊C8C22×C26911yescharacteristic
217C4⋊C8C22×C26911yescharacteristic
218#103C22×C26911yescharacteristic
219#103C22×C26911yescharacteristic
220#103C22×C26911yescharacteristic
221#103C22×C26911yescharacteristic
222#103C22×C26911yescharacteristic
223#103C22×C26911yescharacteristic
224#102C22×C26911yescharacteristic
225#102C22×C26911yescharacteristic
226C2×Q8—21no—
227C2×Q8C23×C26911yescharacteristic
228C2×Q8C23×C26911yescharacteristic
229C2×Q8C23×C26911yescharacteristic
230C2×D4—21no—
231C2×D4C23×C26911yescharacteristic
232C22×C4—21no—
233C22×C4C23×C26911yescharacteristic
234Q16—41no—
235Q16—41no—
236SD16—41no—
237SD16—41no—
238M16—41no—
239C2×C8—41no—
240C2×C8C23×C26911yescharacteristic
241C2×C8C23×C26911yescharacteristic
242C2×C8C23×C26911yescharacteristic
243C2×C8C23×C26911yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4D4×C26911yescharacteristic
256C4⋊C4D4×C26911yescharacteristic
257C4⋊C4C23×C26911yescharacteristic
258C4⋊C4C23×C26911yescharacteristic
259C4⋊C4C23×C26911yescharacteristic
260C4⋊C4C23×C26911yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—21no—
264C22⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4D4×C26911yescharacteristic
269C22⋊C4D4×C26911yescharacteristic
270C42—21no—
271C42—21no—
272C42—21no—
273C42C23×C26911yescharacteristic
274C42C23×C26911yescharacteristic
275C23—21no—
276C23C2×D4×C26911yescharacteristic
277Q8—21no—
278Q8—21no—
279Q8—21no—
280Q8—21no—
281Q8—21no—
282Q8—21no—
283Q8—81no—
284D4—21no—
285D4—21no—
286D4—81no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—81no—
297C2×C4C2×D4×C26911yescharacteristic
298C2×C4C2×D4×C26911yescharacteristic
299C2×C4C2×D4×C26911yescharacteristic
300C2×C4C2×D4×C26911yescharacteristic
301C2×C4C2×D4×C26911yescharacteristic
302C2×C4C24×C26911yescharacteristic
303C8—21no—
304C8—21no—
305C8—21no—
306C8—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22C22×D4×C26911yescharacteristic
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4H(5,𝔽2)×C26911yescharacteristic
327C42−1+4×C26911yescharacteristic
328C2—41no—
329C2—81no—
330C2#537×C26911yescharacteristic
331C2#577×C26911yescharacteristic
332C2#578×C26911yescharacteristic
subgroupG/H
center, socleC22×C269C22×D4
commutator, Frattini, cosocleC2×C4C24×C269
prime radical#121×C269C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C269 ◁ #117×C269 ◁ G

Complements

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

NH ≅ G/Ntype
2. #541×C269C2semidirect
3. #479×C269C2semidirect
4. #478×C269C2semidirect
5. #477×C269C2semidirect
6. #475×C269C2semidirect
7. #470×C269C2semidirect
9. #467×C269C2semidirect
11. #464×C269C2semidirect
12. #464×C269C2semidirect
13. #462×C269C2semidirect
14. #461×C269C2semidirect
15. #424×C269C2semidirect
166. C269#3258direct (N × H)
167. #3258C269direct (N × H)
169. #541C2×C269semidirect
170. #479C2×C269semidirect
171. #478C2×C269semidirect
172. #477C2×C269semidirect
173. #475C2×C269semidirect
174. #470C2×C269semidirect
176. #467C2×C269semidirect
178. #464C2×C269semidirect
179. #464C2×C269semidirect
180. #462C2×C269semidirect
181. #461C2×C269semidirect
182. #424C2×C269semidirect

Maximal subgroups

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

subgroupindex
#542×C2692
#541×C2692
#479×C2692
#478×C2692
#477×C2692
#475×C2692
#470×C2692
#468×C2692
#467×C2692
#465×C2692
#464×C2692
#462×C2692
#461×C2692
#424×C2692
#3258269