Finite Group Databasebeta

← C2×#31771212 #46824248

Subgroups — C2×#31771212 #46824248

Order 34432.

1004 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771212C2116yesmaximal, direct product factor
2C2×#22575452C211yesmaximal, characteristic, complemented
3C2×#22575446C211yesmaximal, characteristic
4C2×#22575446C211yesmaximal, characteristic
5C2×#22575446C211yesmaximal, characteristic
6C2×#22575445C211yesmaximal, characteristic, complemented
7C2×#22575444C211yesmaximal, characteristic, complemented
8C2×#22575443C211yesmaximal, characteristic, complemented
9C2×#22575443C211yesmaximal, characteristic, complemented
10C2×#22575443C211yesmaximal, characteristic, complemented
11C2×#22575430C211yesmaximal, characteristic
12C2×#22575430C211yesmaximal, characteristic
13C2×C4×Dic538C211yesmaximal, characteristic, complemented
14C2×#22575418C211yesmaximal, characteristic, complemented
15C2×#22575409C211yesmaximal, characteristic
16C2×#126×C269C211yesmaximal, characteristic
17C22×Dic538C2211yescharacteristic
18C22×Dic538C2211yescharacteristic
19C23×Dic269C2211yescharacteristic
20#22575452C2218yescomplemented
21#22575446C2218yes—
22#22575446C2218yes—
23#22575446C2218yes—
24#22575445C2218yescomplemented
25#22575444C2218yescomplemented
26#22575443C2218yescomplemented
27#22575443C2218yescomplemented
28#22575443C2218yescomplemented
29#22575430C2218yes—
30#22575430C2218yes—
31C4×Dic538C2218yescomplemented
32#22575418C2218yescomplemented
33#22575409C2218yes—
34C2×C4⋊Dic269C2211yescharacteristic
35C2×C4⋊Dic269C2211yescharacteristic
36C2×C4⋊Dic269C2211yescharacteristic
37C2×C4⋊Dic269C2211yescharacteristic
38C2×C4⋊Dic269C2211yescharacteristic
39C2×#16864381C2211yescharacteristic
40C2×#16864381C2211yescharacteristic
41C2×#16864381C2211yescharacteristic
42C2×#16864381C2211yescharacteristic
43C2×#16864381C2211yescharacteristic
44C2×#16864381C2211yescharacteristic
45C2×#16864381C2211yescharacteristic
46C2×#16864381C2211yescharacteristic
47C2×#16864381C2211yescharacteristic
48C2×#16864381C2211yescharacteristic
49C2×#16864381C2211yescharacteristic
50C2×#16864381C2211yescharacteristic
51C2×C22⋊Dic269C2211yescharacteristic
52C2×C22⋊Dic269C2211yescharacteristic
53C2×C22⋊Dic269C2211yescharacteristic
54C2×C4×Dic269C2211yescharacteristic
55C2×C4×Dic269C2211yescharacteristic
56C2×C4×Dic269C2211yescharacteristic
57C2×C4×Dic269C2211yescharacteristic
58C2×C4×Dic269C2211yescharacteristic
59#126×C269C2218yes—
60C2×C4⋊C4×C269C2211yescharacteristic
61C2×C4⋊C4×C269C2211yescharacteristic
62C2×C4⋊C4×C269C2211yescharacteristic
63C2×C22⋊C4×C269C2211yescharacteristic
64C2×C22⋊C4×C269C2211yescharacteristic
65C2×C22⋊C4×C269C2211yescharacteristic
66C2×C42×C269C2211yescharacteristic
67C2×Dic538—24no—
68C2×Dic538—24no—
69C2×Dic538C2314yes—
70C2×Dic538C2314yes—
71C22×Dic269—24no—
72C22×Dic269C2311yescharacteristic
73C22×Dic269C2311yescharacteristic
74C22×Dic269C2311yescharacteristic
75C22×Dic269C2311yescharacteristic
76C22×Dic269C2311yescharacteristic
77C22×Dic269C2311yescharacteristic
78C22×Dic269C2311yescharacteristic
79C22×Dic269C2314yes—
80C22×Dic269C2311yescharacteristic
81C4⋊Dic269C2314yes—
82C4⋊Dic269C2314yes—
83C4⋊Dic269C2314yes—
84C4⋊Dic269C2314yes—
85C4⋊Dic269C2314yes—
86#16864381C2314yes—
87#16864381C2314yes—
88#16864381C2314yes—
89#16864381C2314yes—
90#16864381C2314yes—
91#16864381C2314yes—
92#16864381C2314yes—
93#16864381C2314yes—
94#16864381C2314yes—
95#16864381C2314yes—
96#16864381C2314yes—
97#16864381C2314yes—
98C22⋊Dic269C2314yes—
99C22⋊Dic269C2314yes—
100C22⋊Dic269C2314yes—
101C4×Dic269C2314yes—
102C4×Dic269C2314yes—
103C4×Dic269C2314yes—
104C4×Dic269C2314yes—
105C4×Dic269C2314yes—
106C24×C269C2311yescharacteristic
107C22×C4×C269C2311yescharacteristic
108C22×C4×C269C2311yescharacteristic
109C22×C4×C269C2311yescharacteristic
110C22×C4×C269C2311yescharacteristic
111C22×C4×C269C2311yescharacteristic
112C22×C4×C269C2311yescharacteristic
113C4⋊C4×C269C2314yes—
114C4⋊C4×C269C2314yes—
115C4⋊C4×C269C2314yes—
116C22⋊C4×C269C2314yes—
117C22⋊C4×C269C2314yes—
118C22⋊C4×C269C2314yes—
119C42×C269C2314yes—
120Dic538—28no—
121Dic538—28no—
122C2×Dic269—22no—
123C2×Dic269—22no—
124C2×Dic269—22no—
125C2×Dic269—22no—
126C2×Dic269—22no—
127C2×Dic269—22no—
128C2×Dic269—22no—
129C2×Dic269—28no—
130C2×Dic269#4114yes—
131C2×Dic269C2412yes—
132C2×Dic269C2412yes—
133C2×Dic269C2412yes—
134C2×Dic269C2412yes—
135C2×Dic269C2412yes—
136C2×Dic269C2412yes—
137C2×Dic269C2412yes—
138C2×Dic269C2412yes—
139C23×C269—22no—
140C23×C269—22no—
141C23×C269—22no—
142C23×C269C2412yes—
143C23×C269C2411yescharacteristic
144C2×C4×C269—22no—
145C2×C4×C269—22no—
146C2×C4×C269—22no—
147C2×C4×C269—22no—
148C2×C4×C269—22no—
149C2×C4×C269—22no—
150C2×C4×C269C2412yes—
151C2×C4×C269C2412yes—
152C2×C4×C269C2412yes—
153C2×C4×C269C2412yes—
154C2×C4×C269C2412yes—
155C2×C4×C269C2412yes—
156C22×C269—22no—
157C22×C269—22no—
158C22×C269—22no—
159C22×C269—44no—
160C22×C269C2×#4112yes—
161C22×C2692−1+412yes—
162C22×C2692−1+412yes—
163C22×C269C2511yescharacteristic
164C4×C269—22no—
165C4×C269—22no—
166C4×C269—22no—
167C4×C269—22no—
168C4×C269—22no—
169C4×C269—22no—
170Dic269—22no—
171Dic269—22no—
172Dic269—22no—
173Dic269—22no—
174Dic269—22no—
175Dic269—22no—
176Dic269—22no—
177Dic269C2×#4114yes—
178C2×C269—42no—
179C2×C269#54114yescomplemented
180C2×C269C22×#4111yescharacteristic
181C2×C269C2×2−1+411yescharacteristic
182C2×C269C2×2−1+411yescharacteristic
183C269C2×#54111yescharacteristic, complemented, Sylow
184C2×#541—2691nomaximal, Sylow
185#541—26916no—
186C2×C4⋊Q8—2691no—
187C2×#126—2691no—
188C2×#126—2691no—
189C2×#126—2691no—
190C2×#126D26911yescharacteristic
191C2×#125—2691no—
192C2×#125—2691no—
193C2×#125—2691no—
194C2×#125—2691no—
195C2×#125—2691no—
196C2×C22⋊Q8—2691no—
197C2×C22⋊Q8—2691no—
198C2×C4×Q8—2691no—
199C2×C4×Q8—2691no—
200C2×#117—2691no—
201C22×Q8—2691no—
202C22×Q8—2691no—
203C23×C4—2691no—
204C4⋊Q8—2698no—
205#126—2698no—
206#126—2698no—
207#126—2698no—
208#126C2×D26918yes—
209#125—2698no—
210#125—2698no—
211#125—2698no—
212#125—2698no—
213#125—2698no—
214C22⋊Q8—2698no—
215C22⋊Q8—2698no—
216C4×Q8—2698no—
217C4×Q8—2698no—
218#117—2698no—
219C2×C4⋊C4—2691no—
220C2×C4⋊C4—2691no—
221C2×C4⋊C4—2691no—
222C2×C4⋊C4—2691no—
223C2×C4⋊C4—2691no—
224C2×C4⋊C4—2691no—
225C2×C4⋊C4—2691no—
226C2×C4⋊C4—2691no—
227C2×C4⋊C4—2691no—
228C2×C4⋊C4—2691no—
229C2×C4⋊C4—2691no—
230C2×C4⋊C4—2691no—
231C2×C4⋊C4—2691no—
232C2×C4⋊C4—2691no—
233C2×C4⋊C4—2691no—
234C2×C4⋊C4—2691no—
235C2×C4⋊C4—2691no—
236C2×C4⋊C4C2×D26911yescharacteristic
237C2×C4⋊C4C2×D26911yescharacteristic
238C2×C4⋊C4C2×D26911yescharacteristic
239C2×C22⋊C4—2691no—
240C2×C22⋊C4—2691no—
241C2×C22⋊C4—2691no—
242C2×C22⋊C4C2×D26911yescharacteristic
243C2×C22⋊C4C2×D26911yescharacteristic
244C2×C22⋊C4C2×D26911yescharacteristic
245C2×C42—2691no—
246C2×C42—2691no—
247C2×C42—2691no—
248C2×C42—2691no—
249C2×C42—2691no—
250C2×C42C2×D26911yescharacteristic
251C24C22×D26911yescharacteristic
252C2×Q8—2694no—
253C2×Q8—2694no—
254C2×Q8—5384no—
255C2×Q8—5384no—
256C22×C4—2691no—
257C22×C4—2694no—
258C22×C4—2691no—
259C22×C4—2691no—
260C22×C4—2691no—
261C22×C4—2691no—
262C22×C4—2691no—
263C22×C4—2691no—
264C22×C4—2691no—
265C22×C4—5384no—
266C22×C4C22×D26911yescharacteristic
267C22×C4C22×D26911yescharacteristic
268C22×C4C22×D26911yescharacteristic
269C22×C4C22×D26911yescharacteristic
270C22×C4C22×D26911yescharacteristic
271C22×C4C22×D26911yescharacteristic
272C4⋊C4—2694no—
273C4⋊C4—2694no—
274C4⋊C4—2694no—
275C4⋊C4—2694no—
276C4⋊C4—2694no—
277C4⋊C4—2694no—
278C4⋊C4—2694no—
279C4⋊C4—2694no—
280C4⋊C4—2694no—
281C4⋊C4—2694no—
282C4⋊C4—2694no—
283C4⋊C4—2694no—
284C4⋊C4—2694no—
285C4⋊C4—2694no—
286C4⋊C4—2694no—
287C4⋊C4—2694no—
288C4⋊C4—2694no—
289C4⋊C4C22×D26914yes—
290C4⋊C4C22×D26914yes—
291C4⋊C4C22×D26914yes—
292C22⋊C4—2694no—
293C22⋊C4—2694no—
294C22⋊C4—2694no—
295C22⋊C4C22×D26914yes—
296C22⋊C4C22×D26914yes—
297C22⋊C4C22×D26914yes—
298C42—2694no—
299C42—2694no—
300C42—2694no—
301C42—2694no—
302C42—2694no—
303C42C22×D26914yes—
304C23—22no—
305C23—22no—
306C23—22no—
307C23C23×D26912yes—
308C23C23×D26911yescharacteristic
309Q8—5388no—
310Q8—5388no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—2694no—
318C2×C4—2692no—
319C2×C4—2692no—
320C2×C4—2692no—
321C2×C4—2692no—
322C2×C4—2692no—
323C2×C4—2692no—
324C2×C4—2692no—
325C2×C4—2692no—
326C2×C4—5388no—
327C2×C4—5382no—
328C2×C4—5382no—
329C2×C4—5382no—
330C2×C4—5382no—
331C2×C4—5382no—
332C2×C4—5382no—
333C2×C4—5382no—
334C2×C4C23×D26912yes—
335C2×C4C23×D26912yes—
336C2×C4C23×D26912yes—
337C2×C4C23×D26912yes—
338C2×C4C23×D26912yes—
339C2×C4C23×D26912yes—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—44no—
344C22#41×D26912yes—
345C22#2257550012yes—
346C22#2257550012yes—
347C22C24×D26911yescharacteristic
348C4—22no—
349C4—22no—
350C4—22no—
351C4—22no—
352C4—22no—
353C4—22no—
354C4—2694no—
355C4—5382no—
356C4—5382no—
357C4—5382no—
358C4—5382no—
359C4—5382no—
360C4—5382no—
361C4—5382no—
362C2—42no—
363C2#3177121214yesdirect product factor
364C2C2×#41×D26911yescharacteristic
365C2C2×#2257550011yescharacteristic
366C2C2×#2257550011yescharacteristic
subgroupG/H
centerC23C23×D269
commutator, cosocleC22×C269C25
FrattiniC22C24×D269
FittingC2×#126×C269C2
socleC23×C269C24
prime radicalC24×C269C23
hypercenterC2×#126D269
hypocenterC269C2×#541
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. #31771212C2direct (N × H)
2. C2×#22575452C2semidirect
6. C2×#22575445C2semidirect
7. C2×#22575444C2semidirect
8. C2×#22575443C2semidirect
9. C2×#22575443C2semidirect
10. C2×#22575443C2semidirect
13. C2×C4×Dic538C2semidirect
14. C2×#22575418C2semidirect
20. #22575452C22semidirect
24. #22575445C22semidirect
25. #22575444C22semidirect
26. #22575443C22semidirect
27. #22575443C22semidirect
28. #22575443C22semidirect
31. C4×Dic538C22semidirect
32. #22575418C22semidirect
179. C2×C269#541semidirect
183. C269C2×#541semidirect
363. C2#31771212direct (N × H)

Maximal subgroups

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

subgroupindex
#317712122
C2×#225754522
C2×#225754462
C2×#225754452
C2×#225754442
C2×#225754432
C2×#225754302
C2×C4×Dic5382
C2×#225754182
C2×#225754092
C2×#126×C2692
C2×#541269