Finite Group Databasebeta

← #46849853

Subgroups — #46849853

Order 34688.

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×#16864655C211yesmaximal, characteristic, complemented
2C2×#41×D271C211yesmaximal, characteristic, complemented
3#31774566C211yesmaximal, characteristic, complemented
4Q8×C22⋊D271C211yesmaximal, characteristic, complemented
5#31774559C211yesmaximal, characteristic, complemented
6#31774549C213yesmaximal, complemented
7#31774543C213yesmaximal, complemented
8#31774540C213yesmaximal, complemented
9#31774531C213yesmaximal, complemented
10#31774446C213yesmaximal, complemented
11#31774410C213yesmaximal, complemented
12#31774407C213yesmaximal, complemented
13C4×#16864655C211yesmaximal, characteristic, complemented
14C4×D4×D271C213yesmaximal, complemented
15#548×C271C211yesmaximal, characteristic, complemented
16C2×#16864655—28no—
17C2×#16864655C2211yescharacteristic
18C2×#16864655C2213yescomplemented
19C2×#16864655C2211yescharacteristic, complemented
20C2×#16864655C2213yescomplemented
21C2×#16864654C2213yescomplemented
22C2×#16864653C2213yescomplemented
23#41×D271—28no—
24C2×Q8×D271C2211yescharacteristic, complemented
25C22×D1084C2213yescomplemented
26C2×D4×D271C2213yes—
27C2×D4×D271C2213yescomplemented
28C22×C4×D271C2213yescomplemented
29C22×C4×D271C2213yescomplemented
30#22576158C2213yescomplemented
31#22576156C2213yescomplemented
32#22576148C2213yescomplemented
33#22576145C2216yescomplemented
34#22576142C2216yescomplemented
35#22576140C2213yescomplemented
36#22576139C2213yescomplemented
37#22576136C2213yescomplemented
38#22576135C2213yescomplemented
39#22576134C2213yescomplemented
40#22576133C2216yescomplemented
41#22576133C2216yescomplemented
42#22576131C2213yescomplemented
43#22576130C2216yescomplemented
44#22576128C2216yescomplemented
45Q8×Dic271C2211yescharacteristic, complemented
46C4×D1084C2213yescomplemented
47C4×D1084C2213yescomplemented
48D4×Dic271C2213yescomplemented
49#22576122C2213yescomplemented
50#22576122C2213yescomplemented
51C4×C22⋊D271C2213yes—
52C4×C22⋊D271C2213yescomplemented
53(C22⋊D271)⋊C4C2211yescharacteristic
54(C22⋊D271)⋊C4C2213yescomplemented
55(C22⋊D271)⋊C4C2216yescomplemented
56#22576118C2213yescomplemented
57C4⋊C4×D271C2213yescomplemented
58C22⋊C4×D271C2213yescomplemented
59C22⋊C4×D271C2213yescomplemented
60C42×D271C2213yescomplemented
61C2×#41×C271C2211yescharacteristic, complemented
62C22×Q8×C271C2211yescharacteristic, complemented
63#124×C271C2213yescomplemented
64C22⋊Q8×C271C2213yescomplemented
65#121×C271C2213yescomplemented
66C4×Q8×C271C2211yescharacteristic, complemented
67C4×D4×C271C2213yescomplemented
68C23×D271C2313yes—
69C23×D271C2313yes—
70#16864655—212no—
71#16864655—212no—
72#16864655—24no—
73#16864655D418yescomplemented
74#16864654—212no—
75#16864653—212no—
76C2×Dic542C2313yes—
77Q8×D271—24no—
78C2×D1084—212no—
79C2×D1084C2313yes—
80C2×D1084C2316yes—
81C2×D1084C2313yes—
82C2×D1084C2313yes—
83C2×C22⋊D271C2316yes—
84C2×C22⋊D271C2313yes—
85C2×C22⋊D271C2316yes—
86C2×C22⋊D271C2311yescharacteristic
87D4×D271—212no—
88D4×D271—212no—
89C22×Dic271C2311yescharacteristic
90C22×Dic271C2313yes—
91C2×C4×D271—212no—
92C2×C4×D271—212no—
93C2×C4×D271C2313yes—
94C2×C4×D271C2313yes—
95C2×C4×D271C2313yes—
96C2×C4×D271C2313yes—
97C2×C4×D271C2313yes—
98C2×C4×D271C2311yescharacteristic
99C2×C4×D271C2313yes—
100C2×C4×D271C2313yes—
101C2×C4×D271C2313yes—
102C4⋊Dic271C2313yes—
103C4⋊Dic271C2313yes—
104Dic271⋊C4C2313yes—
105Dic271⋊C4C2316yes—
106Dic271⋊C4C2311yescharacteristic
107C22⋊Dic271C2313yes—
108C22⋊Dic271C2313yes—
109(C2×D271)⋊C4C2316yes—
110(C2×D271)⋊C4C2316yes—
111(C2×D271)⋊C4C2313yes—
112(C2×D271)⋊C4C2313yes—
113(C2×D271)⋊C4C2313yes—
114(C2×D271)⋊C4C2311yescharacteristic
115C4×Dic271C2313yes—
116C4×Dic271C2311yescharacteristic
117C4×Dic271C2313yes—
118#41×C271—24no—
119C2×Q8×C271—24no—
120C2×Q8×C271C2313yes—
121C2×Q8×C271C2311yescharacteristic
122C2×D4×C271C2313yes—
123C2×D4×C271C2313yes—
124C22×C4×C271C2313yes—
125C22×C4×C271C2313yes—
126C4⋊C4×C271C2313yes—
127C4⋊C4×C271C2313yes—
128C22⋊C4×C271C2316yes—
129C22⋊C4×C271C2313yes—
130C22⋊C4×C271C2311yescharacteristic
131C42×C271C2313yes—
132C22×D271—26no—
133C22×D271—26no—
134C22×D271—26no—
135C22×D271—26no—
136C22×D271—43no—
137C22×D271—43no—
138C22×D271C2413yes—
139C22×D271C2413yes—
140C22×D271C2411yescharacteristic
141Dic542—26no—
142D1084—212no—
143D1084—26no—
144D1084—26no—
145D1084C2×D4112yescomplemented
146C22⋊D271—212no—
147C22⋊D271—26no—
148C22⋊D271—46no—
149C22⋊D271#4114yescomplemented
150C2×Dic271—22no—
151C2×Dic271—26no—
152C2×Dic271C2413yes—
153C2×Dic271C2413yes—
154C2×Dic271C2411yescharacteristic
155C2×Dic271C2411yescharacteristic
156C2×Dic271C2411yescharacteristic
157C4×D271—26no—
158C4×D271—26no—
159C4×D271—26no—
160C4×D271—26no—
161C4×D271—22no—
162C4×D271—26no—
163C4×D271—26no—
164C4×D271—43no—
165C4×D271C2×D4112yescomplemented
166C23×C271C2413yes—
167C23×C271C2411yescharacteristic
168Q8×C271—26no—
169Q8×C271C2×D414yescomplemented
170D4×C271—26no—
171D4×C271—26no—
172C2×C4×C271—26no—
173C2×C4×C271—26no—
174C2×C4×C271C2411yescharacteristic
175C2×C4×C271C2413yes—
176C2×C4×C271C2413yes—
177C2×C4×C271C2411yescharacteristic
178C2×C4×C271C2413yes—
179C22×C271—21no—
180C22×C271—21no—
181C22×C271—23no—
182C22×C271—23no—
183C22×C271—23no—
184C22×C271C2×#4112yescomplemented
185C22×C271C2511yescharacteristic
186C2×D271—21no—
187C2×D271—23no—
188C2×D271—23no—
189C2×D271—21no—
190C2×D271—23no—
191C2×D271—23no—
192C2×D271—23no—
193C2×D271—412no—
194C2×D271—412no—
195C2×D271C22×D416yes—
196C2×D271C2×#4112yescomplemented
197C4×C271—21no—
198C4×C271—23no—
199C4×C271—23no—
200C4×C271—21no—
201C4×C271C22×D416yes—
202Dic271—21no—
203Dic271—23no—
204Dic271—23no—
205Dic271C22×D412yes—
206Dic271C2×#4112yescomplemented
207C2×C271—22no—
208C2×C271—43no—
209C2×C271C23×D411yescharacteristic
210C2×C271C22×#4111yescharacteristic
211C2×C271#58511yescharacteristic
212D271—22no—
213D271—26no—
214D271—43no—
215C271D4×#4111yescharacteristic, complemented, Sylow
216D4×#41—2711nomaximal, Sylow
217C22×#41—2711no—
218C22×#41—2711no—
219#550—2711no—
220D4×Q8—2711no—
221#548—2711no—
222#548D27111yescharacteristic, complemented
223#547—2713no—
224#546—2713no—
225#546—2713no—
226D42—2713no—
227#532—2713no—
228#525—2713no—
229#525—2713no—
230C4×#41—2711no—
231C2×C4×D4—2713no—
232C2×#41—2711no—
233C2×#41—2713no—
234C2×#41—2713no—
235C2×#41—2711no—
236C2×#41—2713no—
237C2×#41—2713no—
238C2×#41—5428no—
239C2×#41—5428no—
240C2×#41C2×D27111yescharacteristic, complemented
241C22×Q8—2711no—
242C22×Q8C2×D27111yescharacteristic, complemented
243C22×D4—2713no—
244C22×D4—2713no—
245C22×D4—2713no—
246C23×C4—2713no—
247C23×C4—2713no—
248C4⋊Q8—2713no—
249#127—2713no—
250#124—2713no—
251#124C2×D27113yescomplemented
252#123—2716no—
253#123—2716no—
254C22⋊Q8—2713no—
255C22⋊Q8—2713no—
256C22⋊Q8—2713no—
257C22⋊Q8C2×D27113yescomplemented
258#121—2713no—
259#121—2713no—
260#121—2713no—
261#121—2716no—
262#121—2716no—
263#121C2×D27113yescomplemented
264#120—2716no—
265#120—2716no—
266C4×Q8—2711no—
267C4×Q8C2×D27111yescharacteristic, complemented
268C4×D4—2713no—
269C4×D4—2713no—
270C4×D4—2713no—
271C4×D4—2713no—
272C4×D4—2713no—
273C4×D4—2711no—
274C4×D4—2713no—
275C4×D4—2716no—
276C4×D4—2713no—
277C4×D4—2713no—
278C4×D4C2×D27113yescomplemented
279#117—2713no—
280C2×C4⋊C4—2713no—
281C2×C22⋊C4—2713no—
282C2×C22⋊C4—2713no—
283C2×C42—2713no—
284C24—2713no—
285C24—2713no—
286#41—24no—
287#41—2718no—
288#41—54212no—
289#41—54212no—
290#41—54212no—
291#41—54212no—
292#41—5424no—
293C2×Q8—24no—
294C2×Q8—2713no—
295C2×Q8—5424no—
296C2×Q8C22×D27113yes—
297C2×Q8C22×D27111yescharacteristic
298C2×D4—2711no—
299C2×D4—2713no—
300C2×D4—2716no—
301C2×D4—2716no—
302C2×D4—2713no—
303C2×D4—2713no—
304C2×D4—2716no—
305C2×D4—2713no—
306C2×D4—54212no—
307C2×D4—54212no—
308C2×D4—54212no—
309C2×D4C22×D27113yes—
310C2×D4C22×D27113yes—
311C22×C4—2713no—
312C22×C4—2711no—
313C22×C4—2713no—
314C22×C4—2713no—
315C22×C4—2713no—
316C22×C4—2713no—
317C22×C4—2711no—
318C22×C4—2713no—
319C22×C4—2713no—
320C22×C4—2713no—
321C22×C4—2713no—
322C22×C4—54212no—
323C22×C4—54212no—
324C22×C4C22×D27113yes—
325C22×C4C22×D27113yes—
326C4⋊C4—2711no—
327C4⋊C4—2713no—
328C4⋊C4—2713no—
329C4⋊C4—2713no—
330C4⋊C4—2716no—
331C4⋊C4C22×D27113yes—
332C4⋊C4C22×D27113yes—
333C22⋊C4—2713no—
334C22⋊C4—2713no—
335C22⋊C4—2711no—
336C22⋊C4—2713no—
337C22⋊C4—2713no—
338C22⋊C4—2713no—
339C22⋊C4—2716no—
340C22⋊C4—2716no—
341C22⋊C4C22×D27111yescharacteristic
342C22⋊C4C22×D27113yes—
343C22⋊C4C22×D27116yes—
344C42—2711no—
345C42—2713no—
346C42—2713no—
347C42C22×D27113yes—
348C23—2711no—
349C23—2713no—
350C23—2713no—
351C23—5426no—
352C23—5426no—
353C23—5426no—
354C23—5426no—
355C23—10843no—
356C23—10843no—
357C23C23×D27113yes—
358C23C23×D27111yescharacteristic
359Q8—26no—
360Q8—5426no—
361Q8D4×D27114yescomplemented
362D4—26no—
363D4—26no—
364D4—2714no—
365D4—27112no—
366D4—54212no—
367D4—5426no—
368D4—5426no—
369D4—54212no—
370D4—5426no—
371D4—10846no—
372C2×C4—26no—
373C2×C4—26no—
374C2×C4—2711no—
375C2×C4—2711no—
376C2×C4—27112no—
377C2×C4—2713no—
378C2×C4—2711no—
379C2×C4—2713no—
380C2×C4—5422no—
381C2×C4—5426no—
382C2×C4—5426no—
383C2×C4—5426no—
384C2×C4—5426no—
385C2×C4—5422no—
386C2×C4—5426no—
387C2×C4—5426no—
388C2×C4—5426no—
389C2×C4—10843no—
390C2×C4C23×D27111yescharacteristic
391C2×C4C23×D27113yes—
392C2×C4C23×D27111yescharacteristic
393C2×C4C23×D27113yes—
394C2×C4C23×D27113yes—
395C22—21no—
396C22—21no—
397C22—23no—
398C22—23no—
399C22—23no—
400C22—2712no—
401C22—2716no—
402C22—5423no—
403C22—5423no—
404C22—5421no—
405C22—5421no—
406C22—5423no—
407C22—5423no—
408C22—5423no—
409C22—108412no—
410C22—108412no—
411C22#41×D27112yescomplemented
412C22C24×D27111yescharacteristic
413C4—23no—
414C4—21no—
415C4—21no—
416C4—23no—
417C4—2712no—
418C4—2712no—
419C4—5423no—
420C4—5421no—
421C4—5423no—
422C4C2×D4×D27116yes—
423C2—22no—
424C2—43no—
425C2—5422no—
426C2—5426no—
427C2—10843no—
428C2C22×D4×D27111yescharacteristic
429C2C2×#41×D27111yescharacteristic
430C2#3177473411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D271
commutator, socle, cosocleC22×C271C25
Fitting#548×C271C2
hypercenter#548D271
hypocenterC271D4×#41
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C22×#16864655C2semidirect
2. C2×#41×D271C2semidirect
3. #31774566C2semidirect
4. Q8×C22⋊D271C2semidirect
5. #31774559C2semidirect
6. #31774549C2semidirect
7. #31774543C2semidirect
8. #31774540C2semidirect
9. #31774531C2semidirect
10. #31774446C2semidirect
11. #31774410C2semidirect
12. #31774407C2semidirect
13. C4×#16864655C2semidirect
14. C4×D4×D271C2semidirect
15. #548×C271C2semidirect
18. C2×#16864655C22semidirect
19. C2×#16864655C22semidirect
20. C2×#16864655C22semidirect
21. C2×#16864654C22semidirect
22. C2×#16864653C22semidirect
24. C2×Q8×D271C22semidirect
25. C22×D1084C22semidirect
27. C2×D4×D271C22semidirect
28. C22×C4×D271C22semidirect
29. C22×C4×D271C22semidirect
30. #22576158C22semidirect
31. #22576156C22semidirect
32. #22576148C22semidirect
33. #22576145C22semidirect
34. #22576142C22semidirect
35. #22576140C22semidirect
36. #22576139C22semidirect
37. #22576136C22semidirect
38. #22576135C22semidirect
39. #22576134C22semidirect
40. #22576133C22semidirect
41. #22576133C22semidirect
42. #22576131C22semidirect
43. #22576130C22semidirect
44. #22576128C22semidirect
45. Q8×Dic271C22semidirect
46. C4×D1084C22semidirect
47. C4×D1084C22semidirect
48. D4×Dic271C22semidirect
49. #22576122C22semidirect
50. #22576122C22semidirect
52. C4×C22⋊D271C22semidirect
54. (C22⋊D271)⋊C4C22semidirect
55. (C22⋊D271)⋊C4C22semidirect
56. #22576118C22semidirect
57. C4⋊C4×D271C22semidirect
58. C22⋊C4×D271C22semidirect
59. C22⋊C4×D271C22semidirect
60. C42×D271C22semidirect
61. C2×#41×C271C22semidirect
62. C22×Q8×C271C22semidirect
63. #124×C271C22semidirect
64. C22⋊Q8×C271C22semidirect
65. #121×C271C22semidirect
66. C4×Q8×C271C22semidirect
67. C4×D4×C271C22semidirect
73. #16864655D4semidirect
145. D1084C2×D4semidirect
149. C22⋊D271#41semidirect
165. C4×D271C2×D4semidirect
169. Q8×C271C2×D4semidirect
184. C22×C271C2×#41semidirect
196. C2×D271C2×#41semidirect
206. Dic271C2×#41semidirect
215. C271D4×#41semidirect
222. #548D271semidirect
240. C2×#41C2×D271semidirect
242. C22×Q8C2×D271semidirect
251. #124C2×D271semidirect
257. C22⋊Q8C2×D271semidirect
263. #121C2×D271semidirect
267. C4×Q8C2×D271semidirect
278. C4×D4C2×D271semidirect
361. Q8D4×D271semidirect
411. C22#41×D271semidirect

Maximal subgroups

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

subgroupindex
C22×#168646552
C2×#41×D2712
#317745662
Q8×C22⋊D2712
#317745592
#317745492
#317745432
#317745402
#317745312
#317744462
#317744102
#317744072
C4×#168646552
C4×D4×D2712
#548×C2712
D4×#41271