Finite Group Databasebeta

← #46824553

Subgroups — #46824553

Order 34432.

1658 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22575499C211yesmaximal, characteristic, complemented
2C2×#41×D269C211yesmaximal, characteristic, complemented
3#31771274C211yesmaximal, characteristic, complemented
4#31771267C211yesmaximal, characteristic, complemented
5#31771262C211yesmaximal, characteristic, complemented
6#31771252C213yesmaximal, complemented
7#31771251C213yesmaximal, complemented
8#31771240C213yesmaximal, complemented
9#31771239C213yesmaximal, complemented
10#31771184C213yesmaximal, complemented
11#31771168C213yesmaximal, complemented
12#31771114C213yesmaximal, complemented
13#121×D269C213yesmaximal, complemented
14#31771087C211yesmaximal, characteristic, complemented
15#550×C269C211yesmaximal, characteristic, complemented
16#22575499—28no—
17C2×#16864403C2211yescharacteristic
18C2×#16864403C2211yescharacteristic, complemented
19C2×#16864402C2213yescomplemented
20C2×#16864402C2213yescomplemented
21C2×#16864401C2213yescomplemented
22#41×D269—28no—
23C2×Q8×D269C2211yescharacteristic, complemented
24C22×D1076C2213yescomplemented
25C2×D4×D269C2213yescomplemented
26C2×D4×D269C2213yescomplemented
27C2×D4×D269C2213yescomplemented
28C2×D4×D269C2216yescomplemented
29C22×C4×D269C2213yescomplemented
30#22575449C2213yescomplemented
31#22575441C2213yescomplemented
32#22575440C2213yescomplemented
33#22575438C2213yescomplemented
34#22575435C2213yescomplemented
35#22575433C2213yescomplemented
36#22575432C2213yescomplemented
37#22575431C2211yescharacteristic, complemented
38#22575430C2213yescomplemented
39#22575428C2213yescomplemented
40#22575427C2213yescomplemented
41#22575426C2216yescomplemented
42#22575426C2213yescomplemented
43#22575425C2213yescomplemented
44#22575425C2216yescomplemented
45#22575424C2216yescomplemented
46#22575424C2213yescomplemented
47#22575424C2213yescomplemented
48#22575423C2213yescomplemented
49#22575421C2216yescomplemented
50#22575421C2216yescomplemented
51#22575421C2213yescomplemented
52Q8×Dic269C2211yescharacteristic, complemented
53C4×D1076C2213yescomplemented
54C4×D1076C2213yescomplemented
55#22575415C2213yescomplemented
56C4×C22⋊D269C2213yescomplemented
57C4×C22⋊D269C2213yescomplemented
58#22575410C2213yescomplemented
59C4⋊C4×D269C2213yescomplemented
60C22⋊C4×D269C2216yescomplemented
61C2×#41×C269C2211yescharacteristic, complemented
62C2×#41×C269C2211yescharacteristic, complemented
63#127×C269C2213yescomplemented
64#121×C269C2213yescomplemented
65#121×C269C2213yescomplemented
66C4×Q8×C269C2211yescharacteristic, complemented
67C4×D4×C269C2213yescomplemented
68C23×D269C2313yes—
69C23×D269C2316yes—
70#16864403—24no—
71#16864403D418yescomplemented
72#16864402—212no—
73#16864402—212no—
74#16864401—212no—
75C2×Dic538C2311yescharacteristic
76C2×Dic538C2313yes—
77Q8×D269—24no—
78C2×D1076—212no—
79C2×D1076C2313yes—
80C2×D1076C2313yes—
81C2×D1076C2313yes—
82C2×D1076C2313yes—
83C2×C22⋊D269C2313yes—
84C2×C22⋊D269C2316yes—
85C2×C22⋊D269C2313yes—
86C2×C22⋊D269C2313yes—
87C2×C22⋊D269C2313yes—
88C2×C22⋊D269C2316yes—
89D4×D269—212no—
90D4×D269—212no—
91D4×D269—212no—
92D4×D269—412no—
93C22×Dic269C2313yes—
94C2×C4×D269—212no—
95C2×C4×D269C2313yes—
96C2×C4×D269C2313yes—
97C2×C4×D269C2313yes—
98C2×C4×D269C2311yescharacteristic
99C2×C4×D269C2316yes—
100C2×C4×D269C2313yes—
101C4⋊Dic269C2313yes—
102C4⋊Dic269C2313yes—
103#16864381C2313yes—
104#16864381C2311yescharacteristic
105#16864381C2311yescharacteristic
106#16864381C2313yes—
107C22⋊Dic269C2316yes—
108C22⋊Dic269C2313yes—
109C22⋊Dic269C2313yes—
110#16864379C2313yes—
111#16864379C2316yes—
112#16864379C2313yes—
113#16864379C2313yes—
114#16864379C2311yescharacteristic
115#16864379C2311yescharacteristic
116#16864379C2313yes—
117C4×Dic269C2313yes—
118#41×C269—24no—
119#41×C269—24no—
120C2×Q8×C269C2311yescharacteristic
121C2×D4×C269C2313yes—
122C2×D4×C269C2313yes—
123C2×D4×C269C2313yes—
124C2×D4×C269C2316yes—
125C22×C4×C269C2313yes—
126C22×C4×C269C2313yes—
127C4⋊C4×C269C2313yes—
128C4⋊C4×C269C2311yescharacteristic
129C22⋊C4×C269C2313yes—
130C22⋊C4×C269C2313yes—
131C42×C269C2313yes—
132C22×D269—26no—
133C22×D269—26no—
134C22×D269—212no—
135C22×D269—43no—
136C22×D269—46no—
137C22×D269—46no—
138C22×D269C2413yes—
139C22×D269C2413yes—
140C22×D269C2411yescharacteristic
141Dic538—26no—
142Dic538—22no—
143D1076—26no—
144D1076—26no—
145D1076—26no—
146D1076C2×D4112yescomplemented
147C22⋊D269—26no—
148C22⋊D269—212no—
149C22⋊D269—26no—
150C22⋊D269—26no—
151C22⋊D269—46no—
152C22⋊D269—43no—
153C2×Dic269—26no—
154C2×Dic269C2411yescharacteristic
155C2×Dic269C2413yes—
156C2×Dic269C2413yes—
157C2×Dic269C2411yescharacteristic
158C2×Dic269C2411yescharacteristic
159C4×D269—26no—
160C4×D269—26no—
161C4×D269—26no—
162C4×D269—22no—
163C4×D269—46no—
164C4×D269C2×D4112yescomplemented
165C23×C269C2413yes—
166C23×C269C2413yes—
167Q8×C269C2×D414yescomplemented
168D4×C269—26no—
169D4×C269—26no—
170D4×C269—43no—
171D4×C269—46no—
172C2×C4×C269—26no—
173C2×C4×C269—26no—
174C2×C4×C269C2411yescharacteristic
175C2×C4×C269C2413yes—
176C2×C4×C269C2411yescharacteristic
177C2×C4×C269C2411yescharacteristic
178C2×C4×C269C2413yes—
179C22×C269—23no—
180C22×C269—23no—
181C22×C269—23no—
182C22×C269—23no—
183C22×C269—23no—
184C22×C269—23no—
185C22×C269C2511yescharacteristic
186C2×D269—21no—
187C2×D269—23no—
188C2×D269—21no—
189C2×D269—23no—
190C2×D269—23no—
191C2×D269—23no—
192C2×D269—23no—
193C2×D269—412no—
194C2×D269—812no—
195C2×D269C22×D416yes—
196C2×D269H(5,𝔽2)12yescomplemented
197C4×C269—21no—
198C4×C269—21no—
199C4×C269—23no—
200C4×C269—21no—
201C4×C269C22×D416yes—
202Dic269—21no—
203Dic269—23no—
204Dic269—23no—
205Dic269—21no—
206Dic269C22×D412yes—
207C2×C269—43no—
208C2×C269—43no—
209C2×C269C23×D411yescharacteristic
210C2×C269C2×H(5,𝔽2)11yescharacteristic
211C2×C269#58511yescharacteristic
212D269—22no—
213D269—26no—
214D269—43no—
215C269#348811yescharacteristic, complemented, Sylow
216#3488—2691nomaximal, Sylow
217C2×H(5,𝔽2)—2691no—
218C22×#41—2691no—
219#550—2691no—
220#550D26911yescharacteristic, complemented
221#548—2691no—
222#548—2691no—
223#546—2693no—
224#546—2693no—
225D42—2693no—
226D42—2693no—
227#537—2693no—
228#535—2693no—
229#525—2693no—
230C2×#121—2693no—
231#520—2691no—
232H(5,𝔽2)—5388no—
233C2×#41—2691no—
234C2×#41—2691no—
235C2×#41—2693no—
236C2×#41—2693no—
237C2×#41—2693no—
238C2×#41—5388no—
239C2×#41C2×D26911yescharacteristic, complemented
240C2×#41C2×D26911yescharacteristic, complemented
241C22×Q8—2691no—
242C22×D4—2693no—
243C22×D4—2693no—
244C22×D4—2693no—
245C22×D4—2696no—
246C22×D4—2693no—
247C23×C4—2693no—
248#127—2693no—
249#127C2×D26913yescomplemented
250#124—2693no—
251#124—2693no—
252#123—2693no—
253#123—2693no—
254C22⋊Q8—2693no—
255C22⋊Q8—2691no—
256C22⋊Q8—2693no—
257C22⋊Q8—2693no—
258#121—2696no—
259#121—2693no—
260#121—2693no—
261#121—2696no—
262#121—2693no—
263#121—2693no—
264#121—2693no—
265#121—2693no—
266#121—2696no—
267#121C2×D26913yescomplemented
268#121C2×D26913yescomplemented
269#120—2693no—
270#120—2696no—
271#120—2696no—
272#120—2693no—
273C4×Q8—2691no—
274C4×Q8C2×D26911yescharacteristic, complemented
275C4×D4—2693no—
276C4×D4—2693no—
277C4×D4—2693no—
278C4×D4—2693no—
279C4×D4—2693no—
280C4×D4C2×D26913yescomplemented
281#117—2693no—
282C2×C4⋊C4—2693no—
283C2×C22⋊C4—2696no—
284C24—2693no—
285C24—2696no—
286#41—24no—
287#41—24no—
288#41—2698no—
289#41—53812no—
290#41—53812no—
291#41—53812no—
292#41—5384no—
293C2×Q8—2691no—
294C2×Q8—2693no—
295C2×Q8—5384no—
296C2×Q8C22×D26911yescharacteristic
297C2×D4—2693no—
298C2×D4—2696no—
299C2×D4—2693no—
300C2×D4—2693no—
301C2×D4—2693no—
302C2×D4—2696no—
303C2×D4—2693no—
304C2×D4—2693no—
305C2×D4—2693no—
306C2×D4—2693no—
307C2×D4—53812no—
308C2×D4—53812no—
309C2×D4—53812no—
310C2×D4—53812no—
311C2×D4—107612no—
312C2×D4C22×D26913yes—
313C2×D4C22×D26913yes—
314C2×D4C22×D26913yes—
315C2×D4C22×D26916yes—
316C22×C4—2693no—
317C22×C4—2693no—
318C22×C4—2693no—
319C22×C4—2693no—
320C22×C4—2693no—
321C22×C4—2691no—
322C22×C4—2696no—
323C22×C4—53812no—
324C22×C4C22×D26913yes—
325C22×C4C22×D26913yes—
326C4⋊C4—2693no—
327C4⋊C4—2693no—
328C4⋊C4—2691no—
329C4⋊C4—2691no—
330C4⋊C4—2693no—
331C4⋊C4—2693no—
332C4⋊C4C22×D26913yes—
333C4⋊C4C22×D26911yescharacteristic
334C22⋊C4—2696no—
335C22⋊C4—2693no—
336C22⋊C4—2693no—
337C22⋊C4—2693no—
338C22⋊C4—2696no—
339C22⋊C4—2693no—
340C22⋊C4—2693no—
341C22⋊C4—2691no—
342C22⋊C4—2691no—
343C22⋊C4—2693no—
344C22⋊C4C22×D26913yes—
345C22⋊C4C22×D26913yes—
346C42—2693no—
347C42C22×D26913yes—
348C23—2693no—
349C23—2691no—
350C23—2693no—
351C23—5386no—
352C23—5386no—
353C23—53812no—
354C23—10763no—
355C23—10766no—
356C23—10766no—
357C23C23×D26913yes—
358C23C23×D26913yes—
359Q8—5382no—
360Q8—5386no—
361Q8D4×D26914yescomplemented
362D4—26no—
363D4—26no—
364D4—46no—
365D4—43no—
366D4—26912no—
367D4—5386no—
368D4—5386no—
369D4—5386no—
370D4—5386no—
371D4—53812no—
372D4—5386no—
373D4—5386no—
374D4—10766no—
375D4—10763no—
376C2×C4—26no—
377C2×C4—26no—
378C2×C4—2691no—
379C2×C4—26912no—
380C2×C4—2691no—
381C2×C4—2693no—
382C2×C4—2693no—
383C2×C4—2691no—
384C2×C4—5382no—
385C2×C4—5386no—
386C2×C4—5386no—
387C2×C4—5386no—
388C2×C4—5386no—
389C2×C4—10766no—
390C2×C4C23×D26911yescharacteristic
391C2×C4C23×D26913yes—
392C2×C4C23×D26911yescharacteristic
393C2×C4C23×D26911yescharacteristic
394C2×C4C23×D26913yes—
395C22—23no—
396C22—23no—
397C22—23no—
398C22—23no—
399C22—23no—
400C22—23no—
401C22—2696no—
402C22—2692no—
403C22—5381no—
404C22—5383no—
405C22—5381no—
406C22—5383no—
407C22—5383no—
408C22—5383no—
409C22—5383no—
410C22—107612no—
411C22—215212no—
412C22C24×D26911yescharacteristic
413C4—21no—
414C4—21no—
415C4—23no—
416C4—21no—
417C4—2692no—
418C4—5381no—
419C4—5383no—
420C4—5383no—
421C4—5381no—
422C4C2×D4×D26916yes—
423C2—43no—
424C2—43no—
425C2—5382no—
426C2—5386no—
427C2—10763no—
428C2C22×D4×D26911yescharacteristic
429C2H(5,𝔽2)×D26911yescharacteristic
430C2#3177144111yescharacteristic
subgroupG/H
center, FrattiniC22C24×D269
commutator, socle, cosocleC22×C269C25
Fitting#550×C269C2
hypercenter#550D269
hypocenterC269#3488
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C22 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#22575499C2semidirect
2. C2×#41×D269C2semidirect
3. #31771274C2semidirect
4. #31771267C2semidirect
5. #31771262C2semidirect
6. #31771252C2semidirect
7. #31771251C2semidirect
8. #31771240C2semidirect
9. #31771239C2semidirect
10. #31771184C2semidirect
11. #31771168C2semidirect
12. #31771114C2semidirect
13. #121×D269C2semidirect
14. #31771087C2semidirect
15. #550×C269C2semidirect
18. C2×#16864403C22semidirect
19. C2×#16864402C22semidirect
20. C2×#16864402C22semidirect
21. C2×#16864401C22semidirect
23. C2×Q8×D269C22semidirect
24. C22×D1076C22semidirect
25. C2×D4×D269C22semidirect
26. C2×D4×D269C22semidirect
27. C2×D4×D269C22semidirect
28. C2×D4×D269C22semidirect
29. C22×C4×D269C22semidirect
30. #22575449C22semidirect
31. #22575441C22semidirect
32. #22575440C22semidirect
33. #22575438C22semidirect
34. #22575435C22semidirect
35. #22575433C22semidirect
36. #22575432C22semidirect
37. #22575431C22semidirect
38. #22575430C22semidirect
39. #22575428C22semidirect
40. #22575427C22semidirect
41. #22575426C22semidirect
42. #22575426C22semidirect
43. #22575425C22semidirect
44. #22575425C22semidirect
45. #22575424C22semidirect
46. #22575424C22semidirect
47. #22575424C22semidirect
48. #22575423C22semidirect
49. #22575421C22semidirect
50. #22575421C22semidirect
51. #22575421C22semidirect
52. Q8×Dic269C22semidirect
53. C4×D1076C22semidirect
54. C4×D1076C22semidirect
55. #22575415C22semidirect
56. C4×C22⋊D269C22semidirect
57. C4×C22⋊D269C22semidirect
58. #22575410C22semidirect
59. C4⋊C4×D269C22semidirect
60. C22⋊C4×D269C22semidirect
61. C2×#41×C269C22semidirect
62. C2×#41×C269C22semidirect
63. #127×C269C22semidirect
64. #121×C269C22semidirect
65. #121×C269C22semidirect
66. C4×Q8×C269C22semidirect
67. C4×D4×C269C22semidirect
71. #16864403D4semidirect
146. D1076C2×D4semidirect
164. C4×D269C2×D4semidirect
167. Q8×C269C2×D4semidirect
196. C2×D269H(5,𝔽2)semidirect
215. C269#3488semidirect
220. #550D269semidirect
239. C2×#41C2×D269semidirect
240. C2×#41C2×D269semidirect
249. #127C2×D269semidirect
267. #121C2×D269semidirect
268. #121C2×D269semidirect
274. C4×Q8C2×D269semidirect
280. C4×D4C2×D269semidirect
361. Q8D4×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754992
C2×#41×D2692
#317712742
#317712672
#317712622
#317712522
#317712512
#317712402
#317712392
#317711842
#317711682
#317711142
#121×D2692
#317710872
#550×C2692
#3488269