Finite Group Databasebeta

← #46824374

Subgroups — #46824374

Order 34432.

1648 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
1C22×#16864403C211yesmaximal, characteristic, complemented
2C2×#41×D269C211yesmaximal, characteristic, complemented
3#31771274C211yesmaximal, characteristic, complemented
4Q8×C22⋊D269C211yesmaximal, characteristic, complemented
5#31771267C211yesmaximal, characteristic, complemented
6#31771257C213yesmaximal, complemented
7#31771251C213yesmaximal, complemented
8#31771248C213yesmaximal, complemented
9#31771239C213yesmaximal, complemented
10#31771154C213yesmaximal, complemented
11#31771118C213yesmaximal, complemented
12#31771115C213yesmaximal, complemented
13C4×#16864403C211yesmaximal, characteristic, complemented
14C4×D4×D269C213yesmaximal, complemented
15#548×C269C211yesmaximal, characteristic, complemented
16C2×#16864403—28no—
17C2×#16864403C2211yescharacteristic
18C2×#16864403C2213yescomplemented
19C2×#16864403C2213yescomplemented
20C2×#16864403C2211yescharacteristic, complemented
21C2×#16864402C2213yescomplemented
22C2×#16864401C2213yescomplemented
23#41×D269—28no—
24C2×Q8×D269C2211yescharacteristic, complemented
25C22×D1076C2213yescomplemented
26C2×D4×D269C2213yes—
27C2×D4×D269C2213yescomplemented
28C22×C4×D269C2213yescomplemented
29C22×C4×D269C2213yescomplemented
30#22575451C2213yescomplemented
31#22575449C2213yescomplemented
32#22575441C2213yescomplemented
33#22575438C2216yescomplemented
34#22575435C2216yescomplemented
35#22575433C2213yescomplemented
36#22575432C2213yescomplemented
37#22575429C2213yescomplemented
38#22575428C2213yescomplemented
39#22575427C2213yescomplemented
40#22575426C2216yescomplemented
41#22575426C2216yescomplemented
42#22575424C2213yescomplemented
43#22575423C2216yescomplemented
44#22575421C2216yescomplemented
45Q8×Dic269C2211yescharacteristic, complemented
46C4×D1076C2213yescomplemented
47C4×D1076C2213yescomplemented
48D4×Dic269C2213yescomplemented
49#22575415C2213yescomplemented
50#22575415C2213yescomplemented
51C4×C22⋊D269C2213yes—
52C4×C22⋊D269C2213yescomplemented
53#22575413C2211yescharacteristic
54#22575413C2213yescomplemented
55#22575413C2216yescomplemented
56#22575411C2213yescomplemented
57C4⋊C4×D269C2213yescomplemented
58C22⋊C4×D269C2213yescomplemented
59C22⋊C4×D269C2213yescomplemented
60C42×D269C2213yescomplemented
61C2×#41×C269C2211yescharacteristic, complemented
62C22×Q8×C269C2211yescharacteristic, complemented
63#124×C269C2213yescomplemented
64C22⋊Q8×C269C2213yescomplemented
65#121×C269C2213yescomplemented
66C4×Q8×C269C2211yescharacteristic, complemented
67C4×D4×C269C2213yescomplemented
68C23×D269C2313yes—
69C23×D269C2313yes—
70#16864403—212no—
71#16864403—212no—
72#16864403—24no—
73#16864403D418yescomplemented
74#16864402—212no—
75#16864401—212no—
76C2×Dic538C2313yes—
77Q8×D269—24no—
78C2×D1076—212no—
79C2×D1076C2313yes—
80C2×D1076C2316yes—
81C2×D1076C2313yes—
82C2×D1076C2313yes—
83C2×C22⋊D269C2313yes—
84C2×C22⋊D269C2316yes—
85C2×C22⋊D269C2316yes—
86C2×C22⋊D269C2311yescharacteristic
87D4×D269—212no—
88D4×D269—212no—
89C22×Dic269C2311yescharacteristic
90C22×Dic269C2313yes—
91C2×C4×D269—212no—
92C2×C4×D269—212no—
93C2×C4×D269C2313yes—
94C2×C4×D269C2313yes—
95C2×C4×D269C2313yes—
96C2×C4×D269C2311yescharacteristic
97C2×C4×D269C2313yes—
98C2×C4×D269C2313yes—
99C2×C4×D269C2313yes—
100C2×C4×D269C2313yes—
101C2×C4×D269C2313yes—
102C4⋊Dic269C2313yes—
103C4⋊Dic269C2313yes—
104#16864381C2311yescharacteristic
105#16864381C2313yes—
106#16864381C2316yes—
107C22⋊Dic269C2313yes—
108C22⋊Dic269C2313yes—
109#16864379C2311yescharacteristic
110#16864379C2313yes—
111#16864379C2313yes—
112#16864379C2313yes—
113#16864379C2316yes—
114#16864379C2316yes—
115C4×Dic269C2311yescharacteristic
116C4×Dic269C2313yes—
117C4×Dic269C2313yes—
118#41×C269—24no—
119C2×Q8×C269—24no—
120C2×Q8×C269C2313yes—
121C2×Q8×C269C2311yescharacteristic
122C2×D4×C269C2313yes—
123C2×D4×C269C2313yes—
124C22×C4×C269C2313yes—
125C22×C4×C269C2313yes—
126C4⋊C4×C269C2313yes—
127C4⋊C4×C269C2313yes—
128C22⋊C4×C269C2311yescharacteristic
129C22⋊C4×C269C2313yes—
130C22⋊C4×C269C2316yes—
131C42×C269C2313yes—
132C22×D269—26no—
133C22×D269—26no—
134C22×D269—26no—
135C22×D269—26no—
136C22×D269—43no—
137C22×D269—43no—
138C22×D269C2413yes—
139C22×D269C2413yes—
140C22×D269C2411yescharacteristic
141Dic538—26no—
142D1076—212no—
143D1076—26no—
144D1076—26no—
145D1076C2×D4112yescomplemented
146C22⋊D269—212no—
147C22⋊D269—26no—
148C22⋊D269—46no—
149C22⋊D269#4114yescomplemented
150C2×Dic269—22no—
151C2×Dic269—26no—
152C2×Dic269C2413yes—
153C2×Dic269C2413yes—
154C2×Dic269C2411yescharacteristic
155C2×Dic269C2411yescharacteristic
156C2×Dic269C2411yescharacteristic
157C4×D269—26no—
158C4×D269—26no—
159C4×D269—26no—
160C4×D269—26no—
161C4×D269—22no—
162C4×D269—26no—
163C4×D269—26no—
164C4×D269—43no—
165C4×D269C2×D4112yescomplemented
166C23×C269C2413yes—
167C23×C269C2411yescharacteristic
168Q8×C269—26no—
169Q8×C269C2×D414yescomplemented
170D4×C269—26no—
171D4×C269—26no—
172C2×C4×C269—26no—
173C2×C4×C269—26no—
174C2×C4×C269C2411yescharacteristic
175C2×C4×C269C2413yes—
176C2×C4×C269C2413yes—
177C2×C4×C269C2411yescharacteristic
178C2×C4×C269C2413yes—
179C22×C269—21no—
180C22×C269—21no—
181C22×C269—23no—
182C22×C269—23no—
183C22×C269—23no—
184C22×C269C2×#4112yescomplemented
185C22×C269C2511yescharacteristic
186C2×D269—21no—
187C2×D269—23no—
188C2×D269—23no—
189C2×D269—21no—
190C2×D269—23no—
191C2×D269—23no—
192C2×D269—23no—
193C2×D269—412no—
194C2×D269—412no—
195C2×D269C22×D416yes—
196C2×D269C2×#4112yescomplemented
197C4×C269—21no—
198C4×C269—23no—
199C4×C269—23no—
200C4×C269—21no—
201C4×C269C22×D416yes—
202Dic269—21no—
203Dic269—23no—
204Dic269—23no—
205Dic269C22×D412yes—
206Dic269C2×#4112yescomplemented
207C2×C269—22no—
208C2×C269—43no—
209C2×C269C23×D411yescharacteristic
210C2×C269C22×#4111yescharacteristic
211C2×C269#58511yescharacteristic
212D269—22no—
213D269—26no—
214D269—43no—
215C269D4×#4111yescharacteristic, complemented, Sylow
216D4×#41—2691nomaximal, Sylow
217C22×#41—2691no—
218C22×#41—2691no—
219#550—2691no—
220D4×Q8—2691no—
221#548—2691no—
222#548D26911yescharacteristic, complemented
223#547—2693no—
224#546—2693no—
225#546—2693no—
226D42—2693no—
227#532—2693no—
228#525—2693no—
229#525—2693no—
230C4×#41—2691no—
231C2×C4×D4—2693no—
232C2×#41—2691no—
233C2×#41—2693no—
234C2×#41—2693no—
235C2×#41—2691no—
236C2×#41—2693no—
237C2×#41—2693no—
238C2×#41—5388no—
239C2×#41—5388no—
240C2×#41C2×D26911yescharacteristic, complemented
241C22×Q8—2691no—
242C22×Q8C2×D26911yescharacteristic, complemented
243C22×D4—2693no—
244C22×D4—2693no—
245C22×D4—2693no—
246C23×C4—2693no—
247C23×C4—2693no—
248C4⋊Q8—2693no—
249#127—2693no—
250#124—2693no—
251#124C2×D26913yescomplemented
252#123—2696no—
253#123—2696no—
254C22⋊Q8—2693no—
255C22⋊Q8—2693no—
256C22⋊Q8—2693no—
257C22⋊Q8C2×D26913yescomplemented
258#121—2693no—
259#121—2693no—
260#121—2693no—
261#121—2696no—
262#121—2696no—
263#121C2×D26913yescomplemented
264#120—2696no—
265#120—2696no—
266C4×Q8—2691no—
267C4×Q8C2×D26911yescharacteristic, complemented
268C4×D4—2693no—
269C4×D4—2693no—
270C4×D4—2693no—
271C4×D4—2693no—
272C4×D4—2693no—
273C4×D4—2691no—
274C4×D4—2693no—
275C4×D4—2696no—
276C4×D4—2693no—
277C4×D4—2693no—
278C4×D4C2×D26913yescomplemented
279#117—2693no—
280C2×C4⋊C4—2693no—
281C2×C22⋊C4—2693no—
282C2×C22⋊C4—2693no—
283C2×C42—2693no—
284C24—2693no—
285C24—2693no—
286#41—24no—
287#41—2698no—
288#41—53812no—
289#41—53812no—
290#41—53812no—
291#41—53812no—
292#41—5384no—
293C2×Q8—24no—
294C2×Q8—2693no—
295C2×Q8—5384no—
296C2×Q8C22×D26913yes—
297C2×Q8C22×D26911yescharacteristic
298C2×D4—2691no—
299C2×D4—2693no—
300C2×D4—2696no—
301C2×D4—2696no—
302C2×D4—2693no—
303C2×D4—2693no—
304C2×D4—2696no—
305C2×D4—2693no—
306C2×D4—53812no—
307C2×D4—53812no—
308C2×D4—53812no—
309C2×D4C22×D26913yes—
310C2×D4C22×D26913yes—
311C22×C4—2693no—
312C22×C4—2691no—
313C22×C4—2693no—
314C22×C4—2693no—
315C22×C4—2693no—
316C22×C4—2693no—
317C22×C4—2691no—
318C22×C4—2693no—
319C22×C4—2693no—
320C22×C4—2693no—
321C22×C4—2693no—
322C22×C4—53812no—
323C22×C4—53812no—
324C22×C4C22×D26913yes—
325C22×C4C22×D26913yes—
326C4⋊C4—2691no—
327C4⋊C4—2693no—
328C4⋊C4—2693no—
329C4⋊C4—2693no—
330C4⋊C4—2696no—
331C4⋊C4C22×D26913yes—
332C4⋊C4C22×D26913yes—
333C22⋊C4—2693no—
334C22⋊C4—2693no—
335C22⋊C4—2691no—
336C22⋊C4—2693no—
337C22⋊C4—2693no—
338C22⋊C4—2693no—
339C22⋊C4—2696no—
340C22⋊C4—2696no—
341C22⋊C4C22×D26911yescharacteristic
342C22⋊C4C22×D26913yes—
343C22⋊C4C22×D26916yes—
344C42—2691no—
345C42—2693no—
346C42—2693no—
347C42C22×D26913yes—
348C23—2691no—
349C23—2693no—
350C23—2693no—
351C23—5386no—
352C23—5386no—
353C23—5386no—
354C23—5386no—
355C23—10763no—
356C23—10763no—
357C23C23×D26913yes—
358C23C23×D26911yescharacteristic
359Q8—26no—
360Q8—5386no—
361Q8D4×D26914yescomplemented
362D4—26no—
363D4—26no—
364D4—2694no—
365D4—26912no—
366D4—53812no—
367D4—5386no—
368D4—5386no—
369D4—53812no—
370D4—5386no—
371D4—10766no—
372C2×C4—26no—
373C2×C4—26no—
374C2×C4—2691no—
375C2×C4—2691no—
376C2×C4—26912no—
377C2×C4—2693no—
378C2×C4—2691no—
379C2×C4—2693no—
380C2×C4—5382no—
381C2×C4—5386no—
382C2×C4—5386no—
383C2×C4—5386no—
384C2×C4—5386no—
385C2×C4—5382no—
386C2×C4—5386no—
387C2×C4—5386no—
388C2×C4—5386no—
389C2×C4—10763no—
390C2×C4C23×D26911yescharacteristic
391C2×C4C23×D26913yes—
392C2×C4C23×D26911yescharacteristic
393C2×C4C23×D26913yes—
394C2×C4C23×D26913yes—
395C22—21no—
396C22—21no—
397C22—23no—
398C22—23no—
399C22—23no—
400C22—2696no—
401C22—2692no—
402C22—5381no—
403C22—5383no—
404C22—5383no—
405C22—5381no—
406C22—5383no—
407C22—5383no—
408C22—5383no—
409C22—107612no—
410C22—107612no—
411C22#41×D26912yescomplemented
412C22C24×D26911yescharacteristic
413C4—21no—
414C4—23no—
415C4—23no—
416C4—21no—
417C4—2692no—
418C4—2692no—
419C4—5381no—
420C4—5383no—
421C4—5383no—
422C4C2×D4×D26916yes—
423C2—22no—
424C2—43no—
425C2—5382no—
426C2—5386no—
427C2—10763no—
428C2C22×D4×D26911yescharacteristic
429C2C2×#41×D26911yescharacteristic
430C2#3177144211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D269
commutator, socle, cosocleC22×C269C25
Fitting#548×C269C2
hypercenter#548D269
hypocenterC269D4×#41
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C22×#16864403C2semidirect
2. C2×#41×D269C2semidirect
3. #31771274C2semidirect
4. Q8×C22⋊D269C2semidirect
5. #31771267C2semidirect
6. #31771257C2semidirect
7. #31771251C2semidirect
8. #31771248C2semidirect
9. #31771239C2semidirect
10. #31771154C2semidirect
11. #31771118C2semidirect
12. #31771115C2semidirect
13. C4×#16864403C2semidirect
14. C4×D4×D269C2semidirect
15. #548×C269C2semidirect
18. C2×#16864403C22semidirect
19. C2×#16864403C22semidirect
20. C2×#16864403C22semidirect
21. C2×#16864402C22semidirect
22. C2×#16864401C22semidirect
24. C2×Q8×D269C22semidirect
25. C22×D1076C22semidirect
27. C2×D4×D269C22semidirect
28. C22×C4×D269C22semidirect
29. C22×C4×D269C22semidirect
30. #22575451C22semidirect
31. #22575449C22semidirect
32. #22575441C22semidirect
33. #22575438C22semidirect
34. #22575435C22semidirect
35. #22575433C22semidirect
36. #22575432C22semidirect
37. #22575429C22semidirect
38. #22575428C22semidirect
39. #22575427C22semidirect
40. #22575426C22semidirect
41. #22575426C22semidirect
42. #22575424C22semidirect
43. #22575423C22semidirect
44. #22575421C22semidirect
45. Q8×Dic269C22semidirect
46. C4×D1076C22semidirect
47. C4×D1076C22semidirect
48. D4×Dic269C22semidirect
49. #22575415C22semidirect
50. #22575415C22semidirect
52. C4×C22⋊D269C22semidirect
54. #22575413C22semidirect
55. #22575413C22semidirect
56. #22575411C22semidirect
57. C4⋊C4×D269C22semidirect
58. C22⋊C4×D269C22semidirect
59. C22⋊C4×D269C22semidirect
60. C42×D269C22semidirect
61. C2×#41×C269C22semidirect
62. C22×Q8×C269C22semidirect
63. #124×C269C22semidirect
64. C22⋊Q8×C269C22semidirect
65. #121×C269C22semidirect
66. C4×Q8×C269C22semidirect
67. C4×D4×C269C22semidirect
73. #16864403D4semidirect
145. D1076C2×D4semidirect
149. C22⋊D269#41semidirect
165. C4×D269C2×D4semidirect
169. Q8×C269C2×D4semidirect
184. C22×C269C2×#41semidirect
196. C2×D269C2×#41semidirect
206. Dic269C2×#41semidirect
215. C269D4×#41semidirect
222. #548D269semidirect
240. C2×#41C2×D269semidirect
242. C22×Q8C2×D269semidirect
251. #124C2×D269semidirect
257. C22⋊Q8C2×D269semidirect
263. #121C2×D269semidirect
267. C4×Q8C2×D269semidirect
278. C4×D4C2×D269semidirect
361. Q8D4×D269semidirect
411. C22#41×D269semidirect

Maximal subgroups

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

subgroupindex
C22×#168644032
C2×#41×D2692
#317712742
Q8×C22⋊D2692
#317712672
#317712572
#317712512
#317712482
#317712392
#317711542
#317711182
#317711152
C4×#168644032
C4×D4×D2692
#548×C2692
D4×#41269