Finite Group Databasebeta

← #46821133

Subgroups — #46821133

Order 34432.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771215C211yesmaximal, characteristic, complemented
2#31771079C211yesmaximal, characteristic
3#31770946C211yesmaximal, characteristic, complemented
4#31770908C211yesmaximal, characteristic, complemented
5#31770904C211yesmaximal, characteristic, complemented
6#31770809C211yesmaximal, characteristic, complemented
7#31770809C211yesmaximal, characteristic, complemented
8#31770758C211yesmaximal, characteristic
9#31770755C211yesmaximal, characteristic
10#31770708C211yesmaximal, characteristic, complemented
11#31770704C211yesmaximal, characteristic, complemented
12#31770692C211yesmaximal, characteristic, complemented
13#31770690C211yesmaximal, characteristic, complemented
14#31770663C211yesmaximal, characteristic, complemented
15#485×C269C211yesmaximal, characteristic, complemented
16C22×D1076C2211yescharacteristic
17C2×#16864391C2211yescharacteristic
18C2×#16864391C2211yescharacteristic
19C2×#16864389C2211yescharacteristic
20C2×#16864389C2211yescharacteristic
21#22575451C2211yescharacteristic, complemented
22#22575446—21no—
23#22575446—21no—
24#22575444—21no—
25#22575444C2211yescharacteristic, complemented
26#22575443—21no—
27#22575432C2211yescharacteristic
28Q8×Dic269C2211yescharacteristic, complemented
29#22575418C2211yescharacteristic, complemented
30C4×D1076—21no—
31C4×D1076—21no—
32#22575415C2211yescharacteristic
33#22575415C2211yescharacteristic
34#22575415C2211yescharacteristic
35#22575415C2211yescharacteristic
36#22575412C2211yescharacteristic
37C22⋊C4×D269—21no—
38C22⋊C4×D269—21no—
39#22575386C2211yescharacteristic, complemented
40#22575384C2211yescharacteristic, complemented
41#22575382C2211yescharacteristic, complemented
42#22575382C2211yescharacteristic, complemented
43#22575376C2211yescharacteristic, complemented
44#22575376C2211yescharacteristic, complemented
45#22575375C2211yescharacteristic
46#22575375C2211yescharacteristic
47#22575373C2211yescharacteristic
48#22575373C2211yescharacteristic
49C22⋊#14524130C2211yescharacteristic
50#22575363C2211yescharacteristic, complemented
51C8×Dic269C2211yescharacteristic, complemented
52C22⋊Q8×C269C2211yescharacteristic
53#117×C269C2211yescharacteristic
54#107×C269C2211yescharacteristic, complemented
55#106×C269C2211yescharacteristic, complemented
56#103×C269C2211yescharacteristic, complemented
57#103×C269C2211yescharacteristic, complemented
58C22⋊C8×C269C2211yescharacteristic
59C23×D269—21no—
60C2×Dic538C2311yescharacteristic
61C2×D1076—22no—
62C2×D1076—22no—
63C2×D1076—21no—
64C2×D1076C2311yescharacteristic
65C2×D1076C2311yescharacteristic
66C2×C4×D269—21no—
67C2×C4×D269—21no—
68C2×C4×D269—21no—
69C2×C4×D269—21no—
70#16864391—22no—
71#16864391—22no—
72#16864389—22no—
73#16864389—22no—
74C2×#14524130C2311yescharacteristic
75C2×#14524130C2311yescharacteristic
76C4⋊Dic269—21no—
77C4⋊Dic269—21no—
78C4⋊Dic269D411yescharacteristic
79C4⋊Dic269D411yescharacteristic
80C4⋊Dic269C2311yescharacteristic
81#16864381—21no—
82#16864381—21no—
83#16864381—21no—
84#16864381—21no—
85#16864381—21no—
86C22⋊Dic269—21no—
87C22⋊Dic269D411yescharacteristic
88C22⋊Dic269D411yescharacteristic
89#16864379—21no—
90#16864379—21no—
91#16864379—21no—
92#16864379—21no—
93C4×Dic269—21no—
94C4×Dic269—21no—
95C4×Dic269C2311yescharacteristic
96C4×Dic269C2311yescharacteristic
97C2×Q8×C269C2311yescharacteristic
98C22×C4×C269C2311yescharacteristic
99C2×C8×C269C2311yescharacteristic
100C2×C8×C269C2311yescharacteristic
101C4⋊C4×C269—21no—
102C4⋊C4×C269C2311yescharacteristic
103C4⋊C4×C269C2311yescharacteristic
104C4⋊C4×C269C2311yescharacteristic
105C22⋊C4×C269—21no—
106C22⋊C4×C269—21no—
107C42×C269—21no—
108C22×D269—21no—
109C22×D269—21no—
110C22×D269—24no—
111C22×D269—81no—
112C22×D269—81no—
113Dic538—21no—
114Dic538—21no—
115D1076—24no—
116D1076—21no—
117D1076—21no—
118D1076#4112yes—
119D1076#4112yes—
120C2×Dic269—21no—
121C2×Dic269—21no—
122C2×Dic269C2×D411yescharacteristic
123C2×Dic269C2×D411yescharacteristic
124C2×Dic269C2×D411yescharacteristic
125C2×Dic269C2×D411yescharacteristic
126C4×D269—81no—
127C4×D269—81no—
128C4×D269—81no—
129C4×D269—81no—
130#14524130—21no—
131#14524130—21no—
132C23×C269C2×D411yescharacteristic
133Q8×C269—21no—
134Q8×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—21no—
141C2×C4×C269C2×D411yescharacteristic
142C2×C4×C269C2411yescharacteristic
143C8×C269—21no—
144C8×C269—21no—
145C22×C269—21no—
146C22×C269—21no—
147C22×C269—21no—
148C22×C269C22×D411yescharacteristic
149C2×D269—22no—
150C2×D269—22no—
151C2×D269—41no—
152C2×D269—41no—
153C2×D269—41no—
154C2×D269—41no—
155C2×D269—84no—
156C4×C269—21no—
157C4×C269—41no—
158C4×C269—41no—
159C4×C269—41no—
160C4×C269—41no—
161C4×C269C2×#4111yescharacteristic
162C4×C269C2×#4111yescharacteristic
163Dic269—21no—
164Dic269—21no—
165Dic269—21no—
166Dic269—21no—
167Dic269—41no—
168Dic269—41no—
169C2×C269—41no—
170C2×C269#52511yescharacteristic
171C2×C269#57711yescharacteristic
172C2×C269#57711yescharacteristic
173D269—42no—
174D269—42no—
175C269#312311yescharacteristic, complemented, Sylow
176#3123—2691nomaximal, Sylow
177#541—2691no—
178#518—2691no—
179#485—2691no—
180#485D26911yescharacteristic, complemented
181#478—2691no—
182#478—2691no—
183#461—2691no—
184#461—2691no—
185#450—2691no—
186#450—2691no—
187#440—2691no—
188#440—2691no—
189C4×SD16—2691no—
190C4×SD16—2691no—
191#433—2691no—
192C22×D4—2691no—
193C2×SD16—2691no—
194C2×SD16—2691no—
195C2×SD16—2691no—
196C2×SD16—2691no—
197C4⋊Q8—2691no—
198#126—5381no—
199#126—5381no—
200#125—2691no—
201#125—5381no—
202#125—5381no—
203C22⋊Q8—2691no—
204C22⋊Q8C2×D26911yescharacteristic
205C4×Q8—2691no—
206C4×Q8—2691no—
207C4×D4—2691no—
208C4×D4—2691no—
209C4×D4—2691no—
210C4×D4—2691no—
211C4×D4—5381no—
212C4×D4—5381no—
213#117—2691no—
214#117C2×D26911yescharacteristic
215C2×C22⋊C4—5381no—
216C2×C22⋊C4—5381no—
217#107—2691no—
218#107C2×D26911yescharacteristic, complemented
219#106—2691no—
220#106C2×D26911yescharacteristic, complemented
221C4⋊C8—2691no—
222C4⋊C8—2691no—
223#103—2691no—
224#103—2691no—
225#103C2×D26911yescharacteristic, complemented
226#103C2×D26911yescharacteristic, complemented
227#102—2691no—
228#102—2691no—
229#102—2691no—
230#102—2691no—
231C22⋊C8—2691no—
232C22⋊C8C2×D26911yescharacteristic
233#97—2691no—
234C4×C8—2691no—
235C24—5381no—
236C2×Q8—2691no—
237C2×Q8C22×D26911yescharacteristic
238C2×D4—2691no—
239C2×D4—2691no—
240C2×D4—5382no—
241C2×D4—5382no—
242C2×D4—5381no—
243C22×C4—5381no—
244C22×C4—5381no—
245C22×C4—5381no—
246C22×C4—5381no—
247C22×C4C22×D26911yescharacteristic
248SD16—5382no—
249SD16—5382no—
250SD16—5382no—
251SD16—5382no—
252C2×C8—2691no—
253C2×C8—2691no—
254C2×C8C22×D26911yescharacteristic
255C2×C8C22×D26911yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—2691no—
258C4⋊C4—2691no—
259C4⋊C4—2691no—
260C4⋊C4—5381no—
261C4⋊C4—5381no—
262C4⋊C4—5381no—
263C4⋊C4—5381no—
264C4⋊C4—5381no—
265C4⋊C4—5381no—
266C4⋊C4—5381no—
267C4⋊C4C22×D26911yescharacteristic
268C4⋊C4C22×D26911yescharacteristic
269C4⋊C4C22×D26911yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—2691no—
273C22⋊C4—2691no—
274C22⋊C4—5381no—
275C22⋊C4—5381no—
276C22⋊C4—5381no—
277C22⋊C4—5381no—
278C22⋊C4—5381no—
279C42—21no—
280C42—2691no—
281C42—2691no—
282C42—5381no—
283C42—5381no—
284C23—5384no—
285C23—5381no—
286C23—5381no—
287C23—21521no—
288C23—21521no—
289C23D4×D26911yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—5381no—
293Q8—5381no—
294D4—2692no—
295D4—2692no—
296D4—5384no—
297D4—5381no—
298D4—5381no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—2691no—
306C2×C4—2691no—
307C2×C4—2691no—
308C2×C4—2691no—
309C2×C4—5381no—
310C2×C4—5381no—
311C2×C4—21521no—
312C2×C4—21521no—
313C2×C4—21521no—
314C2×C4—21521no—
315C2×C4D4×D26911yescharacteristic
316C2×C4C23×D26911yescharacteristic
317C8—21no—
318C8—21no—
319C8—5381no—
320C8—5381no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—5382no—
325C22—5382no—
326C22—10761no—
327C22—10761no—
328C22—10761no—
329C22—10761no—
330C22—21524no—
331C22C2×D4×D26911yescharacteristic
332C4—21no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—5381no—
338C4—5381no—
339C4—5381no—
340C4—5381no—
341C4—10761no—
342C4—10761no—
343C4#41×D26911yescharacteristic
344C4#41×D26911yescharacteristic
345C2—41no—
346C2—10762no—
347C2—10762no—
348C2#3177111311yescharacteristic
349C2#3177141211yescharacteristic
350C2#3177141411yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#485×C269C2
socleC22×C269C22×D4
prime radicalC22×D1076C22
hypercenter#485D269
hypocenterC269#3123
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
1. #31771215C2semidirect
3. #31770946C2semidirect
4. #31770908C2semidirect
5. #31770904C2semidirect
6. #31770809C2semidirect
7. #31770809C2semidirect
10. #31770708C2semidirect
11. #31770704C2semidirect
12. #31770692C2semidirect
13. #31770690C2semidirect
14. #31770663C2semidirect
15. #485×C269C2semidirect
21. #22575451C22semidirect
25. #22575444C22semidirect
28. Q8×Dic269C22semidirect
29. #22575418C22semidirect
39. #22575386C22semidirect
40. #22575384C22semidirect
41. #22575382C22semidirect
42. #22575382C22semidirect
43. #22575376C22semidirect
44. #22575376C22semidirect
50. #22575363C22semidirect
51. C8×Dic269C22semidirect
54. #107×C269C22semidirect
55. #106×C269C22semidirect
56. #103×C269C22semidirect
57. #103×C269C22semidirect
175. C269#3123semidirect
180. #485D269semidirect
218. #107C2×D269semidirect
220. #106C2×D269semidirect
225. #103C2×D269semidirect
226. #103C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317712152
#317710792
#317709462
#317709082
#317709042
#317708092
#317707582
#317707552
#317707082
#317707042
#317706922
#317706902
#317706632
#485×C2692
#3123269