Finite Group Databasebeta

← #46822704

Subgroups — #46822704

Order 34432.

462 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771313C211yesmaximal, characteristic, complemented
2#31771239C211yesmaximal, characteristic
3#31770950C211yesmaximal, characteristic, complemented
4#31770925C211yesmaximal, characteristic, complemented
5#31770921C211yesmaximal, characteristic, complemented
6#31770894C211yesmaximal, characteristic, complemented
7#31770837C211yesmaximal, characteristic, complemented
8#31770803C211yesmaximal, characteristic, complemented
9#31770755C211yesmaximal, characteristic, complemented
10#31770754C211yesmaximal, characteristic, complemented
11#31770690C211yesmaximal, characteristic, complemented
12C8×C22⋊D269C211yesmaximal, characteristic, complemented
13C2×#22575375C211yesmaximal, characteristic, complemented
14#102×D269C211yesmaximal, characteristic, complemented
15#465×C269C211yesmaximal, characteristic, complemented
16C22×D1076C2211yescharacteristic
17C2×D4×D269—21no—
18C2×D4×D269C2211yescharacteristic
19C2×#16864389C2211yescharacteristic
20C2×#16864389C2211yescharacteristic, complemented
21C2×C8×D269C2211yescharacteristic, complemented
22#22575452C2211yescharacteristic, complemented
23#22575449C2211yescharacteristic
24#22575443—21no—
25#22575433—21no—
26#22575430—21no—
27#22575429C2211yescharacteristic, complemented
28#22575428C2211yescharacteristic
29#22575426—21no—
30#22575423—21no—
31#22575421—21no—
32Q8×Dic269C2211yescharacteristic, complemented
33#22575415C2211yescharacteristic
34C4×C22⋊D269C2211yescharacteristic
35#22575413—21no—
36C2×C4⋊Dic269C2211yescharacteristic, complemented
37C4⋊C4×D269C2211yescharacteristic, complemented
38#22575384C2211yescharacteristic, complemented
39#22575384C2211yescharacteristic, complemented
40#22575382C2211yescharacteristic, complemented
41#22575375C2211yescharacteristic
42#22575375C2211yescharacteristic
43#22575375C2211yescharacteristic, complemented
44#22575375C2211yescharacteristic, complemented
45#22575374C2211yescharacteristic, complemented
46#22575374C2211yescharacteristic, complemented
47#22575373C2211yescharacteristic
48#22575373C2211yescharacteristic, complemented
49C22⋊#14524130C2211yescharacteristic, complemented
50#22575365C2211yescharacteristic, complemented
51C8×Dic269C2211yescharacteristic, complemented
52C2×SD16×C269C2211yescharacteristic, complemented
53C22×C8×C269C2211yescharacteristic, complemented
54C22⋊Q8×C269C2211yescharacteristic, complemented
55#121×C269C2211yescharacteristic
56#106×C269C2211yescharacteristic, complemented
57#103×C269C2211yescharacteristic, complemented
58#102×C269C2211yescharacteristic, complemented
59C23×D269—21no—
60C23×D269—21no—
61C2×Dic538—21no—
62C2×D1076—22no—
63C2×D1076—21no—
64C2×D1076—21no—
65C2×D1076—41no—
66C2×D1076C2311yescharacteristic
67C2×D1076C2311yescharacteristic
68C2×C22⋊D269—21no—
69C2×C22⋊D269—21no—
70C2×C22⋊D269—21no—
71C2×C22⋊D269D411yescharacteristic
72D4×D269—22no—
73D4×D269—41no—
74D4×D269—44no—
75C22×Dic269—21no—
76C2×C4×D269—21no—
77C2×C4×D269—21no—
78C2×C4×D269C2311yescharacteristic
79#16864389—22no—
80#16864389—41no—
81C2×#14524130C2311yescharacteristic
82C2×#14524130C2311yescharacteristic
83C8×D269—41no—
84C4⋊Dic269—21no—
85C4⋊Dic269—21no—
86C4⋊Dic269C2311yescharacteristic
87C4⋊Dic269C2311yescharacteristic
88#16864381—21no—
89#16864381—21no—
90#16864381—21no—
91#16864381D411yescharacteristic
92C22⋊Dic269D411yescharacteristic
93#16864379—21no—
94#16864379—21no—
95#16864379—21no—
96#16864379D411yescharacteristic
97C4×Dic269—21no—
98C4×Dic269C2311yescharacteristic
99C2×Q8×C269C2311yescharacteristic
100C2×D4×C269—21no—
101C2×D4×C269C2311yescharacteristic
102C22×C4×C269C2311yescharacteristic
103SD16×C269—41no—
104C2×C8×C269—41no—
105C2×C8×C269C2311yescharacteristic
106C2×C8×C269C2311yescharacteristic
107C4⋊C4×C269—21no—
108C4⋊C4×C269C2311yescharacteristic
109C4⋊C4×C269C2311yescharacteristic
110C22⋊C4×C269—21no—
111C22⋊C4×C269—21no—
112C22×D269—21no—
113C22×D269—21no—
114C22×D269—42no—
115C22×D269—42no—
116C22×D269—42no—
117C22×D269—42no—
118C22×D269—81no—
119C22×D269—81no—
120C22×D269C2×D411yescharacteristic
121Dic538—81no—
122D1076—21no—
123D1076—21no—
124D1076—21no—
125D1076—42no—
126D1076—42no—
127D1076#4112yes—
128C22⋊D269—42no—
129C22⋊D269—42no—
130C22⋊D269—81no—
131C22⋊D269SD1614yescomplemented
132C2×Dic269—21no—
133C2×Dic269—21no—
134C2×Dic269—81no—
135C2×Dic269C2×D411yescharacteristic
136C2×Dic269C2×D411yescharacteristic
137C2×Dic269C2×D411yescharacteristic
138C4×D269—21no—
139C4×D269—21no—
140C4×D269—42no—
141C4×D269—81no—
142#14524130—21no—
143#14524130—21no—
144C23×C269—21no—
145C23×C269C2×D411yescharacteristic
146Q8×C269—21no—
147Q8×C269—21no—
148D4×C269—21no—
149D4×C269—21no—
150D4×C269—42no—
151C2×C4×C269—21no—
152C2×C4×C269—21no—
153C2×C4×C269—21no—
154C2×C4×C269—21no—
155C2×C4×C269—21no—
156C2×C4×C269C2×D411yescharacteristic
157C2×C4×C269C2411yescharacteristic
158C8×C269—21no—
159C8×C269—21no—
160C22×C269—21no—
161C22×C269—21no—
162C22×C269—41no—
163C22×C269—41no—
164C22×C269—41no—
165C22×C269C2×SD1612yes—
166C22×C269C22×D411yescharacteristic
167C2×D269—21no—
168C2×D269—21no—
169C2×D269—22no—
170C2×D269—41no—
171C2×D269—41no—
172C2×D269—41no—
173C2×D269—41no—
174C2×D269—41no—
175C2×D269—84no—
176C2×D269—84no—
177C2×D269C2×SD1612yes—
178C4×C269—21no—
179C4×C269—41no—
180C4×C269—41no—
181C4×C269—41no—
182C4×C269C2×#4111yescharacteristic
183C4×C269H(5,𝔽2)11yescharacteristic
184Dic269—21no—
185Dic269—21no—
186Dic269—41no—
187Dic269—41no—
188Dic269C2×SD1612yes—
189C2×C269—22no—
190C2×C269—81no—
191C2×C269#54611yescharacteristic
192C2×C269C22×SD1611yescharacteristic
193C2×C269#57611yescharacteristic
194D269—22no—
195D269—42no—
196D269—81no—
197C269#329511yescharacteristic, complemented, Sylow
198#3295—2691nomaximal, Sylow
199#554—2691no—
200D42—2691no—
201#486—2691no—
202#481—2691no—
203#481—2691no—
204#476—2691no—
205#465—2691no—
206#465D26911yescharacteristic, complemented
207#460—2691no—
208#450—2691no—
209#450—2691no—
210C4×SD16—2691no—
211C8×D4—2691no—
212C2×#102—2691no—
213C2×#102—2691no—
214C22×D4—2691no—
215C22×D4—2691no—
216C22×D4—5381no—
217C2×SD16—2691no—
218C2×SD16—2691no—
219C2×SD16C2×D26911yescharacteristic, complemented
220C22×C8—2691no—
221C22×C8C2×D26911yescharacteristic, complemented
222C4⋊Q8—2691no—
223#127—2691no—
224#125—5381no—
225C22⋊Q8—2691no—
226C22⋊Q8—5381no—
227C22⋊Q8—5381no—
228C22⋊Q8C2×D26911yescharacteristic, complemented
229#121—2691no—
230#121—5381no—
231#121C2×D26911yescharacteristic
232#120—5381no—
233#120—5381no—
234C4×Q8—2691no—
235C4×D4—2691no—
236C4×D4—2691no—
237C4×D4—5381no—
238C2×C4⋊C4—2691no—
239C2×C4⋊C4—2691no—
240#106—2691no—
241#106—2691no—
242#106C2×D26911yescharacteristic, complemented
243C4⋊C8—2691no—
244#103C2×D26911yescharacteristic, complemented
245#102—2691no—
246#102—2691no—
247#102—2691no—
248#102—2691no—
249#102—2691no—
250#102—2691no—
251#102—2691no—
252#102—2691no—
253#102C2×D26911yescharacteristic, complemented
254C22⋊C8—2691no—
255C22⋊C8—2691no—
256C4×C8—2691no—
257C24—5381no—
258C24—5381no—
259C2×Q8—5381no—
260C2×Q8C22×D26911yescharacteristic
261C2×D4—21no—
262C2×D4—2691no—
263C2×D4—2691no—
264C2×D4—2691no—
265C2×D4—5382no—
266C2×D4—5381no—
267C2×D4—5381no—
268C2×D4—5381no—
269C2×D4—5381no—
270C2×D4—5382no—
271C2×D4—5381no—
272C2×D4—10761no—
273C2×D4—10761no—
274C2×D4—10764no—
275C2×D4C22×D26911yescharacteristic
276C22×C4—2691no—
277C22×C4—5381no—
278C22×C4—5381no—
279C22×C4—5381no—
280C22×C4C22×D26911yescharacteristic
281SD16—41no—
282SD16—5382no—
283SD16—10761no—
284C2×C8—41no—
285C2×C8—2691no—
286C2×C8—2691no—
287C2×C8—10761no—
288C2×C8C22×D26911yescharacteristic
289C2×C8C22×D26911yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—2691no—
292C4⋊C4—2691no—
293C4⋊C4—2691no—
294C4⋊C4—5381no—
295C4⋊C4—5381no—
296C4⋊C4—5381no—
297C4⋊C4—5381no—
298C4⋊C4—5381no—
299C4⋊C4C22×D26911yescharacteristic
300C4⋊C4C22×D26911yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—2691no—
304C22⋊C4—2691no—
305C22⋊C4—5381no—
306C22⋊C4—5381no—
307C22⋊C4—5381no—
308C42—2691no—
309C42—5381no—
310C23—21no—
311C23—2691no—
312C23—5381no—
313C23—5381no—
314C23—10762no—
315C23—10762no—
316C23—10762no—
317C23—10762no—
318C23—21521no—
319C23—21521no—
320C23D4×D26911yescharacteristic
321Q8—21no—
322Q8—21no—
323Q8—21521no—
324D4—21no—
325D4—21no—
326D4—42no—
327D4—2694no—
328D4—2692no—
329D4—5381no—
330D4—5381no—
331D4—5381no—
332D4—10762no—
333D4—10762no—
334D4—10762no—
335D4—10762no—
336D4—21521no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—2691no—
343C2×C4—2691no—
344C2×C4—2691no—
345C2×C4—5381no—
346C2×C4—5381no—
347C2×C4—5381no—
348C2×C4—5381no—
349C2×C4—10762no—
350C2×C4—21521no—
351C2×C4—21521no—
352C2×C4D4×D26911yescharacteristic
353C2×C4C23×D26911yescharacteristic
354C8—21no—
355C8—21no—
356C8—5381no—
357C8—5381no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—2692no—
364C22—5381no—
365C22—5381no—
366C22—5382no—
367C22—10761no—
368C22—10761no—
369C22—10761no—
370C22—10761no—
371C22—10761no—
372C22—21524no—
373C22—21524no—
374C22SD16×D26912yes—
375C22C2×D4×D26911yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—2692no—
381C4—5381no—
382C4—5381no—
383C4—10761no—
384C4—10761no—
385C4#41×D26911yescharacteristic
386C4#2257549811yescharacteristic
387C2—22no—
388C2—81no—
389C2—5382no—
390C2—10762no—
391C2—21521no—
392C2#3177124411yescharacteristic
393C2C2×SD16×D26911yescharacteristic
394C2#3177140711yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#465×C269C2
socleC22×C269C22×D4
prime radical#31771239C2
hypercenter#465D269
hypocenterC269#3295
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #31771313C2semidirect
3. #31770950C2semidirect
4. #31770925C2semidirect
5. #31770921C2semidirect
6. #31770894C2semidirect
7. #31770837C2semidirect
8. #31770803C2semidirect
9. #31770755C2semidirect
10. #31770754C2semidirect
11. #31770690C2semidirect
12. C8×C22⋊D269C2semidirect
13. C2×#22575375C2semidirect
14. #102×D269C2semidirect
15. #465×C269C2semidirect
20. C2×#16864389C22semidirect
21. C2×C8×D269C22semidirect
22. #22575452C22semidirect
27. #22575429C22semidirect
32. Q8×Dic269C22semidirect
36. C2×C4⋊Dic269C22semidirect
37. C4⋊C4×D269C22semidirect
38. #22575384C22semidirect
39. #22575384C22semidirect
40. #22575382C22semidirect
43. #22575375C22semidirect
44. #22575375C22semidirect
45. #22575374C22semidirect
46. #22575374C22semidirect
48. #22575373C22semidirect
49. C22⋊#14524130C22semidirect
50. #22575365C22semidirect
51. C8×Dic269C22semidirect
52. C2×SD16×C269C22semidirect
53. C22×C8×C269C22semidirect
54. C22⋊Q8×C269C22semidirect
56. #106×C269C22semidirect
57. #103×C269C22semidirect
58. #102×C269C22semidirect
131. C22⋊D269SD16semidirect
197. C269#3295semidirect
206. #465D269semidirect
219. C2×SD16C2×D269semidirect
221. C22×C8C2×D269semidirect
228. C22⋊Q8C2×D269semidirect
242. #106C2×D269semidirect
244. #103C2×D269semidirect
253. #102C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317713132
#317712392
#317709502
#317709252
#317709212
#317708942
#317708372
#317708032
#317707552
#317707542
#317706902
C8×C22⋊D2692
C2×#225753752
#102×D2692
#465×C2692
#3295269