Finite Group Databasebeta

← C2×#31771288 #46824390

Subgroups — C2×#31771288 #46824390

Order 34432.

1124 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#31771288C2116yesmaximal, direct product factor
2C2×#22575448C211yesmaximal, characteristic, complemented
3C2×#22575447C211yesmaximal, characteristic, complemented
4C2×#22575445C211yesmaximal, characteristic, complemented
5C2×#22575443C211yesmaximal, characteristic, complemented
6C2×#22575443C211yesmaximal, characteristic, complemented
7C2×#22575437C211yesmaximal, characteristic
8C2×#22575436C211yesmaximal, characteristic
9C2×#22575431C211yesmaximal, characteristic, complemented
10C2×#22575430C211yesmaximal, characteristic, complemented
11C2×C4×Dic538C211yesmaximal, characteristic, complemented
12C2×#22575413C211yesmaximal, characteristic
13C2×#22575410C211yesmaximal, characteristic, complemented
14C2×#22575409C211yesmaximal, characteristic, complemented
15C22×#16864381C211yesmaximal, characteristic, complemented
16C2×#117×C269C211yesmaximal, characteristic, complemented
17C22×Dic538C2211yescharacteristic
18C22×C22⋊D269C2211yescharacteristic
19C23×Dic269C2211yescharacteristic
20C23×Dic269C2211yescharacteristic
21C22×C4×D269C2211yescharacteristic
22#22575448C2218yescomplemented
23#22575447C2218yescomplemented
24#22575445C2218yescomplemented
25#22575443C2218yescomplemented
26#22575443C2218yescomplemented
27#22575437C2218yes—
28#22575436C2218yes—
29#22575431C2218yescomplemented
30#22575430C2218yescomplemented
31C4×Dic538C2218yescomplemented
32#22575413C2218yes—
33#22575410C2218yescomplemented
34#22575409C2218yescomplemented
35C2×C4⋊Dic269C2211yescharacteristic
36C2×C4⋊Dic269C2211yescharacteristic
37C2×C4⋊Dic269C2211yescharacteristic
38C2×#16864381C2211yescharacteristic
39C2×#16864381C2211yescharacteristic
40C2×#16864381C2211yescharacteristic
41C2×#16864381C2211yescharacteristic
42C2×#16864381C2211yescharacteristic
43C2×#16864381C2211yescharacteristic
44C2×#16864381C2211yescharacteristic
45C2×#16864381C2211yescharacteristic
46C2×#16864381C2211yescharacteristic
47C2×#16864381C2211yescharacteristic
48C2×#16864381C2211yescharacteristic
49C2×#16864381C2218yescomplemented
50C2×C22⋊Dic269C2211yescharacteristic
51C2×#16864379C2211yescharacteristic
52C2×#16864379C2211yescharacteristic
53C2×#16864379C2211yescharacteristic
54C2×#16864379C2211yescharacteristic
55C2×#16864379C2211yescharacteristic
56C2×C4×Dic269C2211yescharacteristic
57C2×C4×Dic269C2211yescharacteristic
58C2×C4×Dic269C2211yescharacteristic
59C23×C4×C269C2211yescharacteristic
60#117×C269C2218yescomplemented
61C2×C4⋊C4×C269C2211yescharacteristic
62C2×C4⋊C4×C269C2211yescharacteristic
63C2×C22⋊C4×C269C2211yescharacteristic
64C2×C22⋊C4×C269C2211yescharacteristic
65C2×C42×C269C2211yescharacteristic
66C2×C42×C269C2211yescharacteristic
67C23×D269C2311yescharacteristic
68C2×Dic538—24no—
69C2×Dic538C2314yes—
70C2×C22⋊D269—24no—
71C2×C22⋊D269C2314yes—
72C22×Dic269—24no—
73C22×Dic269—24no—
74C22×Dic269C2311yescharacteristic
75C22×Dic269C2311yescharacteristic
76C22×Dic269C2311yescharacteristic
77C22×Dic269C2311yescharacteristic
78C22×Dic269C2311yescharacteristic
79C22×Dic269C2311yescharacteristic
80C22×Dic269C2314yes—
81C22×Dic269C2314yes—
82C22×Dic269C2311yescharacteristic
83C2×C4×D269—24no—
84C2×C4×D269C2314yes—
85C4⋊Dic269C2314yes—
86C4⋊Dic269C2314yes—
87C4⋊Dic269C2314yes—
88#16864381C2314yes—
89#16864381C2314yes—
90#16864381C2314yes—
91#16864381C2314yes—
92#16864381C2314yes—
93#16864381C2314yes—
94#16864381C2314yes—
95#16864381C2314yes—
96#16864381C2314yes—
97#16864381C2314yes—
98#16864381C2314yes—
99C22⋊Dic269C2314yes—
100#16864379C2314yes—
101#16864379C2314yes—
102#16864379C2314yes—
103#16864379C2314yes—
104#16864379C2314yes—
105C4×Dic269C2314yes—
106C4×Dic269C2314yes—
107C4×Dic269C2314yes—
108C24×C269C2311yescharacteristic
109C22×C4×C269—24no—
110C22×C4×C269C2311yescharacteristic
111C22×C4×C269C2311yescharacteristic
112C22×C4×C269C2311yescharacteristic
113C22×C4×C269C2311yescharacteristic
114C22×C4×C269C2311yescharacteristic
115C22×C4×C269C2311yescharacteristic
116C22×C4×C269C2314yes—
117C4⋊C4×C269C2314yes—
118C4⋊C4×C269C2314yes—
119C22⋊C4×C269C2314yes—
120C22⋊C4×C269C2314yes—
121C42×C269C2314yes—
122C42×C269C2314yes—
123C22×D269—22no—
124C22×D269—22no—
125C22×D269—22no—
126C22×D269C2412yes—
127Dic538—28no—
128C22⋊D269—28no—
129C2×Dic269—22no—
130C2×Dic269—28no—
131C2×Dic269—22no—
132C2×Dic269—22no—
133C2×Dic269—22no—
134C2×Dic269—22no—
135C2×Dic269—22no—
136C2×Dic269—28no—
137C2×Dic269#4114yes—
138C2×Dic269C2412yes—
139C2×Dic269C2412yes—
140C2×Dic269C2412yes—
141C2×Dic269C2412yes—
142C2×Dic269C2412yes—
143C2×Dic269C2412yes—
144C2×Dic269C2412yes—
145C4×D269—28no—
146C23×C269—22no—
147C23×C269—22no—
148C23×C269#4114yes—
149C23×C269C2412yes—
150C23×C269C2411yescharacteristic
151C2×C4×C269—28no—
152C2×C4×C269—22no—
153C2×C4×C269—22no—
154C2×C4×C269—22no—
155C2×C4×C269—22no—
156C2×C4×C269—22no—
157C2×C4×C269—22no—
158C2×C4×C269C2412yes—
159C2×C4×C269C2412yes—
160C2×C4×C269C2412yes—
161C2×C4×C269C2412yes—
162C2×C4×C269C2412yes—
163C2×C4×C269C2412yes—
164C22×C269—28no—
165C22×C269—22no—
166C22×C269—22no—
167C22×C269C2×#4112yes—
168C22×C269C2×#4114yes—
169C22×C269C2×#4112yes—
170C22×C2692−1+412yes—
171C22×C269C2511yescharacteristic
172C2×D269—22no—
173C2×D269—22no—
174C2×D269—22no—
175C2×D269—44no—
176C4×C269—22no—
177C4×C269—22no—
178C4×C269—22no—
179C4×C269—22no—
180C4×C269—22no—
181C4×C269—22no—
182Dic269—22no—
183Dic269—22no—
184Dic269—22no—
185Dic269—22no—
186Dic269—22no—
187Dic269—22no—
188Dic269C2×#4114yes—
189C2×C269—24no—
190C2×C269#55214yescomplemented
191C2×C269C22×#4111yescharacteristic
192C2×C269C22×#4111yescharacteristic
193C2×C269C2×2−1+411yescharacteristic
194D269—42no—
195C269C2×#55211yescharacteristic, complemented, Sylow
196C2×#552—2691nomaximal, Sylow
197#552—26916no—
198C2×#126—2691no—
199C2×#126—2691no—
200C2×#125—2691no—
201C2×#125—2691no—
202C2×#125—2691no—
203C2×#123—2691no—
204C2×#123—2691no—
205C2×C22⋊Q8—2691no—
206C2×C22⋊Q8—2691no—
207C2×C4×Q8—2691no—
208C2×C4×D4—2691no—
209C2×#117—2691no—
210C2×#117—2691no—
211C2×#117D26911yescharacteristic, complemented
212C22×C4⋊C4—2691no—
213C22×Q8—2691no—
214C22×D4—2691no—
215C23×C4—2691no—
216C23×C4—2691no—
217C23×C4—2691no—
218C23×C4C2×D26911yescharacteristic
219#126—2698no—
220#126—2698no—
221#125—2698no—
222#125—2698no—
223#125—2698no—
224#123—2698no—
225#123—2698no—
226C22⋊Q8—2698no—
227C22⋊Q8—2698no—
228C4×Q8—2698no—
229C4×D4—2698no—
230#117—2698no—
231#117—2698no—
232#117C2×D26918yescomplemented
233C2×C4⋊C4—2691no—
234C2×C4⋊C4—2691no—
235C2×C4⋊C4—2691no—
236C2×C4⋊C4—2691no—
237C2×C4⋊C4—2698no—
238C2×C4⋊C4—2691no—
239C2×C4⋊C4—2691no—
240C2×C4⋊C4—2691no—
241C2×C4⋊C4—2691no—
242C2×C4⋊C4—2691no—
243C2×C4⋊C4—2691no—
244C2×C4⋊C4—2691no—
245C2×C4⋊C4—2691no—
246C2×C4⋊C4—2691no—
247C2×C4⋊C4—2691no—
248C2×C4⋊C4C2×D26911yescharacteristic
249C2×C4⋊C4C2×D26911yescharacteristic
250C2×C22⋊C4—2691no—
251C2×C22⋊C4—2691no—
252C2×C22⋊C4—2691no—
253C2×C22⋊C4—2691no—
254C2×C22⋊C4—2691no—
255C2×C22⋊C4—2691no—
256C2×C22⋊C4C2×D26911yescharacteristic
257C2×C22⋊C4C2×D26911yescharacteristic
258C2×C42—2691no—
259C2×C42—2691no—
260C2×C42—2691no—
261C2×C42C2×D26911yescharacteristic
262C2×C42C2×D26911yescharacteristic
263C24—2691no—
264C24C22×D26911yescharacteristic
265C2×Q8—2694no—
266C2×Q8—5384no—
267C2×D4—2694no—
268C2×D4—5384no—
269C22×C4—24no—
270C22×C4—2691no—
271C22×C4—2694no—
272C22×C4—2694no—
273C22×C4—2691no—
274C22×C4—2691no—
275C22×C4—2691no—
276C22×C4—2691no—
277C22×C4—2691no—
278C22×C4—2691no—
279C22×C4—2694no—
280C22×C4—5384no—
281C22×C4—5384no—
282C22×C4—5384no—
283C22×C4C22×D26914yes—
284C22×C4C22×D26911yescharacteristic
285C22×C4C22×D26911yescharacteristic
286C22×C4C22×D26911yescharacteristic
287C22×C4C22×D26911yescharacteristic
288C22×C4C22×D26911yescharacteristic
289C22×C4C22×D26911yescharacteristic
290C4⋊C4—2694no—
291C4⋊C4—2694no—
292C4⋊C4—2694no—
293C4⋊C4—2694no—
294C4⋊C4—2694no—
295C4⋊C4—2694no—
296C4⋊C4—2694no—
297C4⋊C4—2694no—
298C4⋊C4—2694no—
299C4⋊C4—2694no—
300C4⋊C4—2694no—
301C4⋊C4—2694no—
302C4⋊C4—2694no—
303C4⋊C4—2694no—
304C4⋊C4C22×D26914yes—
305C4⋊C4C22×D26914yes—
306C22⋊C4—2694no—
307C22⋊C4—2694no—
308C22⋊C4—2694no—
309C22⋊C4—2694no—
310C22⋊C4—2694no—
311C22⋊C4—2694no—
312C22⋊C4C22×D26914yes—
313C22⋊C4C22×D26914yes—
314C42—2694no—
315C42—2694no—
316C42—2694no—
317C42C22×D26914yes—
318C42C22×D26914yes—
319C23—22no—
320C23—22no—
321C23—2692no—
322C23—5382no—
323C23—5382no—
324C23—5382no—
325C23#1686440214yes—
326C23C23×D26912yes—
327C23C23×D26911yescharacteristic
328Q8—5388no—
329D4—5388no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—28no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—2694no—
338C2×C4—2692no—
339C2×C4—2692no—
340C2×C4—2692no—
341C2×C4—2692no—
342C2×C4—2692no—
343C2×C4—2692no—
344C2×C4—2692no—
345C2×C4—5388no—
346C2×C4—5388no—
347C2×C4—5382no—
348C2×C4—5382no—
349C2×C4—5388no—
350C2×C4—5382no—
351C2×C4—5382no—
352C2×C4—5382no—
353C2×C4—5382no—
354C2×C4C23×D26912yes—
355C2×C4C23×D26912yes—
356C2×C4C23×D26912yes—
357C2×C4C23×D26912yes—
358C2×C4C23×D26912yes—
359C2×C4C23×D26912yes—
360C22—22no—
361C22—22no—
362C22—28no—
363C22—5382no—
364C22—5382no—
365C22—5382no—
366C22—10764no—
367C22#41×D26912yes—
368C22C2×#1686440214yes—
369C22C2×#1686440212yes—
370C22#2257550012yes—
371C22C24×D26911yescharacteristic
372C4—22no—
373C4—22no—
374C4—22no—
375C4—22no—
376C4—22no—
377C4—22no—
378C4—2694no—
379C4—5382no—
380C4—5382no—
381C4—5382no—
382C4—5382no—
383C4—5382no—
384C4—5382no—
385C2—24no—
386C2—10762no—
387C2#3177128814yesdirect product factor
388C2C2×#41×D26911yescharacteristic
389C2C22×#1686440211yescharacteristic
390C2C2×#2257550011yescharacteristic
subgroupG/H
centerC23C23×D269
commutator, cosocleC22×C269C25
FrattiniC22C24×D269
FittingC2×#117×C269C2
socleC23×C269C24
prime radicalC22×C22⋊D269C22
hypercenterC2×#117D269
hypocenterC269C2×#552
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. #31771288C2direct (N × H)
2. C2×#22575448C2semidirect
3. C2×#22575447C2semidirect
4. C2×#22575445C2semidirect
5. C2×#22575443C2semidirect
6. C2×#22575443C2semidirect
9. C2×#22575431C2semidirect
10. C2×#22575430C2semidirect
11. C2×C4×Dic538C2semidirect
13. C2×#22575410C2semidirect
14. C2×#22575409C2semidirect
15. C22×#16864381C2semidirect
16. C2×#117×C269C2semidirect
22. #22575448C22semidirect
23. #22575447C22semidirect
24. #22575445C22semidirect
25. #22575443C22semidirect
26. #22575443C22semidirect
29. #22575431C22semidirect
30. #22575430C22semidirect
31. C4×Dic538C22semidirect
33. #22575410C22semidirect
34. #22575409C22semidirect
49. C2×#16864381C22semidirect
60. #117×C269C22semidirect
190. C2×C269#552semidirect
195. C269C2×#552semidirect
211. C2×#117D269semidirect
232. #117C2×D269semidirect
387. C2#31771288direct (N × H)

Maximal subgroups

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

subgroupindex
#317712882
C2×#225754482
C2×#225754472
C2×#225754452
C2×#225754432
C2×#225754372
C2×#225754362
C2×#225754312
C2×#225754302
C2×C4×Dic5382
C2×#225754132
C2×#225754102
C2×#225754092
C22×#168643812
C2×#117×C2692
C2×#552269