Finite Group Databasebeta

← #46822901

Subgroups — #46822901

Order 34432.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771289C211yesmaximal, characteristic, complemented
2#31771239C211yesmaximal, characteristic
3#31770968C211yesmaximal, characteristic, complemented
4#31770926C211yesmaximal, characteristic, complemented
5#31770920C211yesmaximal, characteristic, complemented
6#31770908C211yesmaximal, characteristic, complemented
7#31770837C211yesmaximal, characteristic, complemented
8#31770803C211yesmaximal, characteristic, complemented
9#31770755C211yesmaximal, characteristic, complemented
10#31770754C211yesmaximal, characteristic, complemented
11#31770708C211yesmaximal, characteristic, complemented
12#31770674C211yesmaximal, characteristic, complemented
13#31770611C211yesmaximal, characteristic, complemented
14#102×D269C211yesmaximal, characteristic, complemented
15#468×C269C211yesmaximal, characteristic, complemented
16C22×D1076C2211yescharacteristic
17C2×D4×D269—21no—
18C2×D4×D269C2211yescharacteristic
19C2×#16864389C2211yescharacteristic
20C2×#16864389C2211yescharacteristic, complemented
21C2×C8×D269C2211yescharacteristic, complemented
22#22575449C2211yescharacteristic
23#22575446—21no—
24#22575444C2211yescharacteristic, complemented
25#22575443—21no—
26#22575435—21no—
27#22575429C2211yescharacteristic, complemented
28#22575428C2211yescharacteristic
29#22575426—21no—
30#22575423—21no—
31#22575421—21no—
32Q8×Dic269C2211yescharacteristic, complemented
33#22575415C2211yescharacteristic
34C4×C22⋊D269C2211yescharacteristic
35#22575412C2211yescharacteristic, complemented
36#22575411—21no—
37C4⋊C4×D269C2211yescharacteristic, complemented
38#22575386C2211yescharacteristic, 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
51#22575363C2211yescharacteristic, complemented
52C2×SD16×C269C2211yescharacteristic, complemented
53C2×M16×C269C2211yescharacteristic, complemented
54C22⋊Q8×C269C2211yescharacteristic, complemented
55#121×C269C2211yescharacteristic
56#107×C269C2211yescharacteristic, complemented
57#103×C269C2211yescharacteristic, complemented
58#102×C269C2211yescharacteristic, complemented
59C23×D269—21no—
60C23×D269—21no—
61C2×D1076—22no—
62C2×D1076—21no—
63C2×D1076—21no—
64C2×D1076—41no—
65C2×D1076C2311yescharacteristic
66C2×D1076C2311yescharacteristic
67C2×C22⋊D269—21no—
68C2×C22⋊D269—21no—
69C2×C22⋊D269—21no—
70C2×C22⋊D269D411yescharacteristic
71D4×D269—22no—
72D4×D269—41no—
73D4×D269—44no—
74C2×C4×D269—21no—
75C2×C4×D269—21no—
76C2×C4×D269C2311yescharacteristic
77#16864389—22no—
78#16864389—41no—
79C2×#14524130C2311yescharacteristic
80C2×#14524130C2311yescharacteristic
81C8×D269—41no—
82C4⋊Dic269—21no—
83C4⋊Dic269—21no—
84C4⋊Dic269C2311yescharacteristic
85C4⋊Dic269C2311yescharacteristic
86#16864381—21no—
87#16864381—21no—
88#16864381—21no—
89#16864381D411yescharacteristic
90C22⋊Dic269—21no—
91C22⋊Dic269D411yescharacteristic
92#16864379—21no—
93#16864379—21no—
94#16864379—21no—
95#16864379D411yescharacteristic
96C4×Dic269—21no—
97C4×Dic269—21no—
98C4×Dic269C2311yescharacteristic
99C2×Q8×C269C2311yescharacteristic
100C2×D4×C269—21no—
101C2×D4×C269C2311yescharacteristic
102C22×C4×C269C2311yescharacteristic
103SD16×C269—41no—
104M16×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
121D1076—21no—
122D1076—21no—
123D1076—21no—
124D1076—42no—
125D1076—42no—
126D1076#4112yes—
127C22⋊D269—22no—
128C22⋊D269—42no—
129C22⋊D269—42no—
130C22⋊D269—81no—
131C2×Dic269—21no—
132C2×Dic269—21no—
133C2×Dic269C2×D411yescharacteristic
134C2×Dic269C2×D411yescharacteristic
135C2×Dic269C2×D411yescharacteristic
136C4×D269—21no—
137C4×D269—21no—
138C4×D269—42no—
139C4×D269—42no—
140#14524130—21no—
141#14524130—21no—
142C23×C269—21no—
143C23×C269C2×D411yescharacteristic
144Q8×C269—21no—
145Q8×C269—21no—
146D4×C269—21no—
147D4×C269—21no—
148D4×C269—42no—
149C2×C4×C269—21no—
150C2×C4×C269—21no—
151C2×C4×C269—21no—
152C2×C4×C269—21no—
153C2×C4×C269—21no—
154C2×C4×C269C2×D411yescharacteristic
155C2×C4×C269C2411yescharacteristic
156C8×C269—21no—
157C8×C269—21no—
158C22×C269—21no—
159C22×C269—21no—
160C22×C269—21no—
161C22×C269—41no—
162C22×C269—41no—
163C22×C269—41no—
164C22×C269C22×D411yescharacteristic
165C2×D269—21no—
166C2×D269—21no—
167C2×D269—22no—
168C2×D269—41no—
169C2×D269—41no—
170C2×D269—41no—
171C2×D269—41no—
172C2×D269—41no—
173C2×D269—84no—
174C2×D269—84no—
175C2×D269AGL(1,ℤ/8)12yes—
176C4×C269—21no—
177C4×C269—41no—
178C4×C269—41no—
179C4×C269—41no—
180C4×C269C2×#4111yescharacteristic
181C4×C269H(5,𝔽2)11yescharacteristic
182Dic269—21no—
183Dic269—21no—
184Dic269—21no—
185Dic269—41no—
186Dic269—41no—
187C2×C269—41no—
188C2×C269—81no—
189C2×C269#54611yescharacteristic
190C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C269#57711yescharacteristic
192D269—22no—
193D269—42no—
194D269—81no—
195C269#331011yescharacteristic, complemented, Sylow
196#3310—2691nomaximal, Sylow
197#552—2691no—
198D42—2691no—
199#490—2691no—
200#482—2691no—
201#480—2691no—
202#478—2691no—
203#468D26911yescharacteristic, complemented
204#465—2691no—
205#460—2691no—
206#450—2691no—
207#450—2691no—
208#440—2691no—
209#435—2691no—
210#418—2691no—
211C2×#102—2691no—
212C22×D4—2691no—
213C22×D4—2691no—
214C22×D4—5381no—
215C2×SD16—2691no—
216C2×SD16—2691no—
217C2×SD16C2×D26911yescharacteristic, complemented
218C2×M16C2×D26911yescharacteristic, complemented
219C22×C8—2691no—
220#127—2691no—
221#126—5381no—
222#125—2691no—
223#125—5381no—
224#123—5381no—
225C22⋊Q8—2691no—
226C22⋊Q8C2×D26911yescharacteristic, complemented
227#121—2691no—
228#121—5381no—
229#121C2×D26911yescharacteristic
230#120—5381no—
231#120—5381no—
232C4×Q8—2691no—
233C4×D4—2691no—
234C4×D4—2691no—
235#117—2691no—
236#117—5381no—
237C2×C4⋊C4—2691no—
238#107—2691no—
239#107C2×D26911yescharacteristic, complemented
240#106—2691no—
241C4⋊C8—2691no—
242#103C2×D26911yescharacteristic, complemented
243#102—2691no—
244#102—2691no—
245#102—2691no—
246#102—2691no—
247#102—2691no—
248#102—2691no—
249#102—2691no—
250#102—2691no—
251#102C2×D26911yescharacteristic, complemented
252C22⋊C8—2691no—
253C22⋊C8—2691no—
254#97—2691no—
255C24—5381no—
256C24—5381no—
257C2×Q8C22×D26911yescharacteristic
258C2×D4—21no—
259C2×D4—2691no—
260C2×D4—2691no—
261C2×D4—2691no—
262C2×D4—5382no—
263C2×D4—5381no—
264C2×D4—5381no—
265C2×D4—5381no—
266C2×D4—5381no—
267C2×D4—5382no—
268C2×D4—5381no—
269C2×D4—10761no—
270C2×D4—10761no—
271C2×D4—10764no—
272C2×D4C22×D26911yescharacteristic
273C22×C4—2691no—
274C22×C4—5381no—
275C22×C4—5381no—
276C22×C4C22×D26911yescharacteristic
277SD16—41no—
278SD16—5382no—
279SD16—10761no—
280M16—41no—
281C2×C8—2691no—
282C2×C8—2691no—
283C2×C8—10761no—
284C2×C8C22×D26911yescharacteristic
285C2×C8C22×D26911yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—2691no—
288C4⋊C4—2691no—
289C4⋊C4—2691no—
290C4⋊C4—5381no—
291C4⋊C4—5381no—
292C4⋊C4—5381no—
293C4⋊C4—5381no—
294C4⋊C4—5381no—
295C4⋊C4C22×D26911yescharacteristic
296C4⋊C4C22×D26911yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—2691no—
300C22⋊C4—2691no—
301C22⋊C4—5381no—
302C22⋊C4—5381no—
303C22⋊C4—5381no—
304C22⋊C4—5381no—
305C42—2691no—
306C42—5381no—
307C42—5381no—
308C23—21no—
309C23—2691no—
310C23—5381no—
311C23—5381no—
312C23—10762no—
313C23—10762no—
314C23—10762no—
315C23—10762no—
316C23—21521no—
317C23—21521no—
318C23D4×D26911yescharacteristic
319Q8—21no—
320Q8—21no—
321D4—21no—
322D4—21no—
323D4—42no—
324D4—2692no—
325D4—5382no—
326D4—5381no—
327D4—5381no—
328D4—5381no—
329D4—10762no—
330D4—10762no—
331D4—10762no—
332D4—10762no—
333D4—21521no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—2691no—
340C2×C4—2691no—
341C2×C4—2691no—
342C2×C4—5381no—
343C2×C4—5381no—
344C2×C4—5381no—
345C2×C4—5381no—
346C2×C4—10762no—
347C2×C4—10762no—
348C2×C4D4×D26911yescharacteristic
349C2×C4C23×D26911yescharacteristic
350C8—21no—
351C8—21no—
352C8—5381no—
353C8—5381no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—2692no—
361C22—5381no—
362C22—5381no—
363C22—5382no—
364C22—10761no—
365C22—10761no—
366C22—10761no—
367C22—10761no—
368C22—10761no—
369C22—21524no—
370C22—21524no—
371C22C2×D4×D26911yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—5381no—
377C4—5381no—
378C4—5381no—
379C4—10761no—
380C4—10761no—
381C4#41×D26911yescharacteristic
382C4#2257549811yescharacteristic
383C2—41no—
384C2—81no—
385C2—5382no—
386C2—10762no—
387C2—21521no—
388C2#3177124411yescharacteristic
389C2AGL(1,ℤ/8)×D26911yescharacteristic
390C2#3177141411yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#468×C269C2
socleC22×C269C22×D4
prime radical#31771239C2
hypercenter#468D269
hypocenterC269#3310
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #31771289C2semidirect
3. #31770968C2semidirect
4. #31770926C2semidirect
5. #31770920C2semidirect
6. #31770908C2semidirect
7. #31770837C2semidirect
8. #31770803C2semidirect
9. #31770755C2semidirect
10. #31770754C2semidirect
11. #31770708C2semidirect
12. #31770674C2semidirect
13. #31770611C2semidirect
14. #102×D269C2semidirect
15. #468×C269C2semidirect
20. C2×#16864389C22semidirect
21. C2×C8×D269C22semidirect
24. #22575444C22semidirect
27. #22575429C22semidirect
32. Q8×Dic269C22semidirect
35. #22575412C22semidirect
37. C4⋊C4×D269C22semidirect
38. #22575386C22semidirect
39. #22575384C22semidirect
40. #22575382C22semidirect
43. #22575375C22semidirect
44. #22575375C22semidirect
45. #22575374C22semidirect
46. #22575374C22semidirect
48. #22575373C22semidirect
49. C22⋊#14524130C22semidirect
50. #22575365C22semidirect
51. #22575363C22semidirect
52. C2×SD16×C269C22semidirect
53. C2×M16×C269C22semidirect
54. C22⋊Q8×C269C22semidirect
56. #107×C269C22semidirect
57. #103×C269C22semidirect
58. #102×C269C22semidirect
195. C269#3310semidirect
203. #468D269semidirect
217. C2×SD16C2×D269semidirect
218. C2×M16C2×D269semidirect
226. C22⋊Q8C2×D269semidirect
239. #107C2×D269semidirect
242. #103C2×D269semidirect
251. #102C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317712892
#317712392
#317709682
#317709262
#317709202
#317709082
#317708372
#317708032
#317707552
#317707542
#317707082
#317706742
#317706112
#102×D2692
#468×C2692
#3310269