Finite Group Databasebeta

← #46822736

Subgroups — #46822736

Order 34432.

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#31771299C211yesmaximal, characteristic, complemented
2#31771243C211yesmaximal, characteristic, complemented
3#31770961C211yesmaximal, characteristic, complemented
4#31770939C211yesmaximal, characteristic, complemented
5#31770929C211yesmaximal, characteristic, complemented
6#31770903C211yesmaximal, characteristic, complemented
7#31770841C211yesmaximal, characteristic, complemented
8#31770837C211yesmaximal, characteristic, complemented
9#31770823C211yesmaximal, characteristic, complemented
10#31770750C211yesmaximal, characteristic, complemented
11#31770740C211yesmaximal, characteristic, complemented
12#31770685C211yesmaximal, characteristic, complemented
13C8×C22⋊D269C211yesmaximal, characteristic, complemented
14#102×D269C211yesmaximal, characteristic, complemented
15#416×C269C211yesmaximal, characteristic, complemented
16C2×D4×D269C2211yescharacteristic
17C2×#16864389C2211yescharacteristic, complemented
18C2×D2152C2211yescharacteristic
19C2×#16864387C2211yescharacteristic
20C2×C8×D269C2211yescharacteristic
21#22575447—21no—
22#22575444C2211yescharacteristic, complemented
23#22575442C2211yescharacteristic
24#22575436—21no—
25#22575435—21no—
26#22575429C2211yescharacteristic
27#22575428C2211yescharacteristic
28#22575427C2211yescharacteristic
29#22575427C2211yescharacteristic, complemented
30#22575426—21no—
31#22575425—21no—
32#22575421—21no—
33C4×D1076—21no—
34D4×Dic269C2211yescharacteristic
35#22575415C2211yescharacteristic, complemented
36C4×C22⋊D269C2211yescharacteristic
37C4⋊C4×D269C2211yescharacteristic
38C2×#16864379—21no—
39#22575387C2211yescharacteristic, complemented
40#22575386C2211yescharacteristic, complemented
41#22575384C2211yescharacteristic, complemented
42#22575382C2211yescharacteristic, complemented
43#22575377C2211yescharacteristic, complemented
44#22575375C2211yescharacteristic
45#22575375C2211yescharacteristic, complemented
46#22575374C2211yescharacteristic
47#22575373C2211yescharacteristic
48#22575373C2211yescharacteristic
49C22⋊#14524130C2211yescharacteristic, complemented
50#22575365C2211yescharacteristic
51C8×Dic269C2211yescharacteristic, complemented
52C2×#41×C269C2211yescharacteristic
53C22×C8×C269C2211yescharacteristic, complemented
54#117×C269C2211yescharacteristic, complemented
55#103×C269C2211yescharacteristic, complemented
56#103×C269C2211yescharacteristic, complemented
57#102×C269C2211yescharacteristic
58#102×C269C2211yescharacteristic
59C23×D269—21no—
60C2×D1076—21no—
61C2×D1076C2311yescharacteristic
62C2×D1076C2311yescharacteristic
63C2×C22⋊D269—21no—
64C2×C22⋊D269—21no—
65C2×C22⋊D269—21no—
66C2×C22⋊D269D411yescharacteristic
67D4×D269—22no—
68D4×D269—41no—
69C22×Dic269—21no—
70C2×C4×D269—21no—
71C2×C4×D269—21no—
72C2×C4×D269C2311yescharacteristic
73#16864389—41no—
74D2152—41no—
75#16864387—22no—
76C2×#14524130C2311yescharacteristic
77C2×#14524130C2311yescharacteristic
78C8×D269—41no—
79C4⋊Dic269—21no—
80C4⋊Dic269C2311yescharacteristic
81C4⋊Dic269C2311yescharacteristic
82#16864381—21no—
83#16864381—21no—
84#16864381—21no—
85#16864381D411yescharacteristic
86C22⋊Dic269—21no—
87C22⋊Dic269D411yescharacteristic
88#16864379—21no—
89#16864379—21no—
90#16864379—21no—
91#16864379—21no—
92#16864379—21no—
93#16864379—21no—
94#16864379—21no—
95#16864379D411yescharacteristic
96C4×Dic269C2311yescharacteristic
97#41×C269—22no—
98#41×C269—41no—
99C2×Q8×C269C2311yescharacteristic
100C2×D4×C269—21no—
101C2×D4×C269C2311yescharacteristic
102C22×C4×C269—21no—
103C22×C4×C269C2311yescharacteristic
104C2×C8×C269—41no—
105C2×C8×C269C2311yescharacteristic
106C2×C8×C269C2311yescharacteristic
107C4⋊C4×C269C2311yescharacteristic
108C4⋊C4×C269C2311yescharacteristic
109C22⋊C4×C269—21no—
110C42×C269—21no—
111C22×D269—21no—
112C22×D269—21no—
113C22×D269—42no—
114C22×D269—42no—
115C22×D269—81no—
116C22×D269C2×D411yescharacteristic
117D1076—21no—
118D1076—21no—
119D1076—21no—
120D1076—21no—
121D1076—42no—
122C22⋊D269—22no—
123C22⋊D269—42no—
124C22⋊D269—81no—
125C22⋊D269—81no—
126C2×Dic269—21no—
127C2×Dic269—21no—
128C2×Dic269—81no—
129C2×Dic269C2×D411yescharacteristic
130C2×Dic269C2×D411yescharacteristic
131C2×Dic269C2×D411yescharacteristic
132C4×D269—21no—
133C4×D269—21no—
134C4×D269—81no—
135C4×D269—81no—
136#14524130—21no—
137#14524130—21no—
138C23×C269—21no—
139C23×C269C2×D411yescharacteristic
140Q8×C269—21no—
141Q8×C269—21no—
142D4×C269—21no—
143D4×C269—42no—
144D4×C269#4112yes—
145C2×C4×C269—21no—
146C2×C4×C269—21no—
147C2×C4×C269—21no—
148C2×C4×C269—21no—
149C2×C4×C269—21no—
150C2×C4×C269—42no—
151C2×C4×C269C2×D411yescharacteristic
152C2×C4×C269C2411yescharacteristic
153C8×C269—21no—
154C8×C269—21no—
155C22×C269—21no—
156C22×C269—22no—
157C22×C269—21no—
158C22×C269—21no—
159C22×C269—41no—
160C22×C269—41no—
161C22×C269C22×D411yescharacteristic
162C2×D269—21no—
163C2×D269—21no—
164C2×D269—41no—
165C2×D269—41no—
166C2×D269—41no—
167C2×D269—41no—
168C2×D269—41no—
169C2×D269—41no—
170C2×D269—84no—
171C2×D269#13512yes—
172C4×C269—21no—
173C4×C269—41no—
174C4×C269—41no—
175C4×C269—41no—
176C4×C269C2×#4111yescharacteristic
177C4×C269H(5,𝔽2)11yescharacteristic
178Dic269—21no—
179Dic269—21no—
180Dic269—21no—
181Dic269—41no—
182Dic269—41no—
183C2×C269—42no—
184C2×C269—41no—
185C2×C269#54611yescharacteristic
186C2×C269C2×#13511yescharacteristic
187C2×C269#57611yescharacteristic
188D269—22no—
189D269—81no—
190D269—81no—
191C269#329711yescharacteristic, complemented, Sylow
192#3297—2691nomaximal, Sylow
193#553—2691no—
194#546—2691no—
195#488—2691no—
196#484—2691no—
197#482—2691no—
198#478—2691no—
199#466—2691no—
200#465—2691no—
201#463—2691no—
202#449—2691no—
203#447—2691no—
204C4×D8—2691no—
205C8×D4—2691no—
206#416D26911yescharacteristic, complemented
207C2×#102—2691no—
208C2×#41C2×D26911yescharacteristic
209C22×D4—2691no—
210C2×SD16—2691no—
211C2×D8—2691no—
212C2×D8—2691no—
213C22×C8—2691no—
214C22×C8C2×D26911yescharacteristic, complemented
215#126—5381no—
216#125—2691no—
217#124—2691no—
218#123—5381no—
219#123—5381no—
220C22⋊Q8—2691no—
221#121—2691no—
222#121—2691no—
223#121—2691no—
224#121—5381no—
225#121—5381no—
226#120—5381no—
227C4×D4—2691no—
228C4×D4—2691no—
229C4×D4—2691no—
230C4×D4—5381no—
231#117C2×D26911yescharacteristic, complemented
232C2×C4⋊C4—2691no—
233C2×C22⋊C4—5381no—
234#107—2691no—
235#107—2691no—
236#106—2691no—
237C4⋊C8—2691no—
238#103—2691no—
239#103C2×D26911yescharacteristic, complemented
240#103C2×D26911yescharacteristic, complemented
241#102—2691no—
242#102—2691no—
243#102—2691no—
244#102—2691no—
245#102—2691no—
246#102C2×D26911yescharacteristic
247#102C2×D26911yescharacteristic
248C22⋊C8—2691no—
249C22⋊C8—2691no—
250C4×C8—2691no—
251C24—5381no—
252#41—22no—
253#41—41no—
254C2×Q8C22×D26911yescharacteristic
255C2×D4—21no—
256C2×D4—2691no—
257C2×D4—2691no—
258C2×D4—2691no—
259C2×D4—5381no—
260C2×D4—5381no—
261C2×D4—5382no—
262C2×D4—5381no—
263C2×D4—5381no—
264C2×D4—10761no—
265C2×D4C22×D26911yescharacteristic
266C22×C4—21no—
267C22×C4—2691no—
268C22×C4—5381no—
269C22×C4—5381no—
270C22×C4—5381no—
271C22×C4C22×D26911yescharacteristic
272SD16—10761no—
273D8—5382no—
274D8—10761no—
275C2×C8—41no—
276C2×C8—2691no—
277C2×C8—2691no—
278C2×C8—10761no—
279C2×C8C22×D26911yescharacteristic
280C2×C8C22×D26911yescharacteristic
281C4⋊C4—2691no—
282C4⋊C4—2691no—
283C4⋊C4—2691no—
284C4⋊C4—5381no—
285C4⋊C4—5381no—
286C4⋊C4—5381no—
287C4⋊C4—5381no—
288C4⋊C4C22×D26911yescharacteristic
289C4⋊C4C22×D26911yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—2691no—
292C22⋊C4—2691no—
293C22⋊C4—5381no—
294C22⋊C4—5381no—
295C22⋊C4—5381no—
296C22⋊C4—5381no—
297C22⋊C4—5381no—
298C22⋊C4—5381no—
299C22⋊C4—5381no—
300C22⋊C4—5381no—
301C42—21no—
302C42—2691no—
303C23—21no—
304C23—2691no—
305C23—5381no—
306C23—5381no—
307C23—10762no—
308C23—10762no—
309C23—21521no—
310C23D4×D26911yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—42no—
315D4—5382no—
316D4—5381no—
317D4—5381no—
318D4—5381no—
319D4—5381no—
320D4—10762no—
321D4—10762no—
322D4—21521no—
323D4—21521no—
324D4#1686440212yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—2691no—
332C2×C4—2691no—
333C2×C4—2691no—
334C2×C4—5381no—
335C2×C4—5381no—
336C2×C4—5381no—
337C2×C4—5381no—
338C2×C4—21521no—
339C2×C4—21521no—
340C2×C4—21521no—
341C2×C4D4×D26911yescharacteristic
342C2×C4C23×D26911yescharacteristic
343C8—21no—
344C8—21no—
345C8—5381no—
346C8—5381no—
347C22—21no—
348C22—21no—
349C22—22no—
350C22—21no—
351C22—41no—
352C22—41no—
353C22—2692no—
354C22—5381no—
355C22—5381no—
356C22—10761no—
357C22—10761no—
358C22—10761no—
359C22—10761no—
360C22—10761no—
361C22—10761no—
362C22—21524no—
363C22C2×D4×D26911yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—5381no—
369C4—5381no—
370C4—5381no—
371C4—10761no—
372C4—10761no—
373C4C2×#1686440211yescharacteristic
374C4#2257549811yescharacteristic
375C2—41no—
376C2—42no—
377C2—5382no—
378C2—21521no—
379C2—21521no—
380C2#3177124511yescharacteristic
381C2#135×D26911yescharacteristic
382C2#3177140711yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#416×C269C2
socleC22×C269C22×D4
hypercenter#416D269
hypocenterC269#3297
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #31771299C2semidirect
2. #31771243C2semidirect
3. #31770961C2semidirect
4. #31770939C2semidirect
5. #31770929C2semidirect
6. #31770903C2semidirect
7. #31770841C2semidirect
8. #31770837C2semidirect
9. #31770823C2semidirect
10. #31770750C2semidirect
11. #31770740C2semidirect
12. #31770685C2semidirect
13. C8×C22⋊D269C2semidirect
14. #102×D269C2semidirect
15. #416×C269C2semidirect
17. C2×#16864389C22semidirect
22. #22575444C22semidirect
29. #22575427C22semidirect
35. #22575415C22semidirect
39. #22575387C22semidirect
40. #22575386C22semidirect
41. #22575384C22semidirect
42. #22575382C22semidirect
43. #22575377C22semidirect
45. #22575375C22semidirect
49. C22⋊#14524130C22semidirect
51. C8×Dic269C22semidirect
53. C22×C8×C269C22semidirect
54. #117×C269C22semidirect
55. #103×C269C22semidirect
56. #103×C269C22semidirect
191. C269#3297semidirect
206. #416D269semidirect
214. C22×C8C2×D269semidirect
231. #117C2×D269semidirect
239. #103C2×D269semidirect
240. #103C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317712992
#317712432
#317709612
#317709392
#317709292
#317709032
#317708412
#317708372
#317708232
#317707502
#317707402
#317706852
C8×C22⋊D2692
#102×D2692
#416×C2692
#3297269