Finite Group Databasebeta

← C2×#31771356 #46825016

Subgroups — C2×#31771356 #46825016

Order 34432.

1024 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#31771356C2116yesmaximal, direct product factor
2C2×#22575452C211yesmaximal, characteristic, complemented
3C2×#22575451C211yesmaximal, characteristic, complemented
4C2×#22575448C211yesmaximal, characteristic, complemented
5C2×#22575448C211yesmaximal, characteristic, complemented
6C2×#22575446C211yesmaximal, characteristic, complemented
7C2×#22575446C211yesmaximal, characteristic, complemented
8C2×#22575444C211yesmaximal, characteristic, complemented
9C2×#22575443C211yesmaximal, characteristic, complemented
10C2×#22575439C211yesmaximal, characteristic
11C2×#22575436C211yesmaximal, characteristic
12C2×#22575435C211yesmaximal, characteristic
13C2×#22575432C211yesmaximal, characteristic, complemented
14C2×#22575431C211yesmaximal, characteristic, complemented
15C2×#22575429C211yesmaximal, characteristic, complemented
16C2×C22⋊Q8×C269C211yesmaximal, characteristic, complemented
17C22×Dic538C2211yescharacteristic
18C22×Dic538C2211yescharacteristic
19C22×C22⋊D269C2211yescharacteristic
20C22×C4×D269C2211yescharacteristic
21#22575452C2218yescomplemented
22#22575451C2218yescomplemented
23#22575448C2218yescomplemented
24#22575448C2218yescomplemented
25#22575446C2218yescomplemented
26#22575446C2218yescomplemented
27#22575444C2218yescomplemented
28#22575443C2218yescomplemented
29#22575439C2218yes—
30#22575436C2218yes—
31#22575435C2218yes—
32#22575432C2218yescomplemented
33#22575431C2218yescomplemented
34#22575429C2218yescomplemented
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×C22⋊Dic269C2211yescharacteristic
49C2×C22⋊Dic269C2211yescharacteristic
50C2×C22⋊Dic269C2211yescharacteristic
51C2×#16864379C2211yescharacteristic
52C2×#16864379C2211yescharacteristic
53C2×#16864379C2211yescharacteristic
54C2×#16864379C2211yescharacteristic
55C2×#16864379C2211yescharacteristic
56C2×C4×Dic269C2211yescharacteristic
57C2×C4×Dic269C2211yescharacteristic
58C2×C4×Dic269C2211yescharacteristic
59C22×Q8×C269C2211yescharacteristic
60C23×C4×C269C2211yescharacteristic
61C22⋊Q8×C269C2218yescomplemented
62C2×C4⋊C4×C269C2211yescharacteristic
63C2×C4⋊C4×C269C2211yescharacteristic
64C2×C4⋊C4×C269C2211yescharacteristic
65C2×C22⋊C4×C269C2211yescharacteristic
66C2×C22⋊C4×C269C2211yescharacteristic
67C23×D269C2311yescharacteristic
68C2×Dic538—42no—
69C2×Dic538—42no—
70C2×Dic538C2314yes—
71C2×Dic538C2314yes—
72C2×C22⋊D269—42no—
73C2×C22⋊D269C2314yes—
74C22×Dic269C2311yescharacteristic
75C22×Dic269C2311yescharacteristic
76C22×Dic269C2311yescharacteristic
77C22×Dic269C2311yescharacteristic
78C22×Dic269C2311yescharacteristic
79C22×Dic269C2311yescharacteristic
80C22×Dic269C2311yescharacteristic
81C2×C4×D269—42no—
82C2×C4×D269C2314yes—
83C4⋊Dic269C2314yes—
84C4⋊Dic269C2314yes—
85C4⋊Dic269C2314yes—
86C4⋊Dic269C2314yes—
87#16864381C2314yes—
88#16864381C2314yes—
89#16864381C2314yes—
90#16864381C2314yes—
91#16864381C2314yes—
92#16864381C2314yes—
93#16864381C2314yes—
94#16864381C2314yes—
95#16864381C2314yes—
96C22⋊Dic269C2314yes—
97C22⋊Dic269C2314yes—
98C22⋊Dic269C2314yes—
99#16864379C2314yes—
100#16864379C2314yes—
101#16864379C2314yes—
102#16864379C2314yes—
103#16864379C2314yes—
104C4×Dic269C2314yes—
105C4×Dic269C2314yes—
106C4×Dic269C2314yes—
107C24×C269C2311yescharacteristic
108C2×Q8×C269—42no—
109C2×Q8×C269C2314yes—
110C22×C4×C269—42no—
111C22×C4×C269C2311yescharacteristic
112C22×C4×C269C2311yescharacteristic
113C22×C4×C269C2311yescharacteristic
114C22×C4×C269C2311yescharacteristic
115C22×C4×C269C2311yescharacteristic
116C22×C4×C269C2311yescharacteristic
117C22×C4×C269C2314yes—
118C4⋊C4×C269C2314yes—
119C4⋊C4×C269C2314yes—
120C4⋊C4×C269C2314yes—
121C22⋊C4×C269C2314yes—
122C22⋊C4×C269C2314yes—
123C22×D269—22no—
124C22×D269—22no—
125C22×D269—22no—
126C22×D269C2412yes—
127Dic538—44no—
128Dic538—44no—
129C22⋊D269—44no—
130C2×Dic269—22no—
131C2×Dic269—22no—
132C2×Dic269—22no—
133C2×Dic269—22no—
134C2×Dic269—22no—
135C2×Dic269—22no—
136C2×Dic269—22no—
137C2×Dic269C2412yes—
138C2×Dic269C2412yes—
139C2×Dic269C2412yes—
140C2×Dic269C2412yes—
141C2×Dic269C2412yes—
142C2×Dic269C2412yes—
143C2×Dic269C2412yes—
144C4×D269—44no—
145C23×C269—22no—
146C23×C269—22no—
147C23×C269—22no—
148C23×C269C2412yes—
149C23×C269C2411yescharacteristic
150Q8×C269—44no—
151C2×C4×C269—22no—
152C2×C4×C269—22no—
153C2×C4×C269—22no—
154C2×C4×C269—22no—
155C2×C4×C269—22no—
156C2×C4×C269—22no—
157C2×C4×C269—44no—
158C2×C4×C269C2412yes—
159C2×C4×C269C2412yes—
160C2×C4×C269C2412yes—
161C2×C4×C269C2412yes—
162C2×C4×C269C2412yes—
163C2×C4×C269C2412yes—
164C22×C269—22no—
165C22×C269—22no—
166C22×C269—22no—
167C22×C269—44no—
168C22×C269H(5,𝔽2)12yes—
169C22×C2692−1+412yes—
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—
188Dic269—22no—
189C2×C269—42no—
190C2×C269#56314yescomplemented
191C2×C269C2×H(5,𝔽2)11yescharacteristic
192C2×C269C2×2−1+411yescharacteristic
193C2×C269C2×2−1+411yescharacteristic
194D269—42no—
195C269C2×#56311yescharacteristic, complemented, Sylow
196C2×#563—2691nomaximal, Sylow
197#563—26916no—
198C2×C4⋊Q8—2691no—
199C2×C4⋊Q8—2691no—
200C2×#126—2691no—
201C2×#126—2691no—
202C2×#126—2691no—
203C2×#126—2691no—
204C2×#125—2691no—
205C2×#125—2691no—
206C2×#124—2691no—
207C2×#123—2691no—
208C2×#123—2691no—
209C2×C22⋊Q8—2691no—
210C2×C22⋊Q8—2691no—
211C2×C22⋊Q8—2691no—
212C2×C22⋊Q8D26911yescharacteristic, complemented
213C22×Q8—2691no—
214C22×Q8—2691no—
215C22×Q8C2×D26911yescharacteristic
216C22×D4—2691no—
217C23×C4—2691no—
218C23×C4C2×D26911yescharacteristic
219C4⋊Q8—2698no—
220C4⋊Q8—2698no—
221#126—2698no—
222#126—2698no—
223#126—2698no—
224#126—2698no—
225#125—2698no—
226#125—2698no—
227#124—2698no—
228#123—2698no—
229#123—2698no—
230C22⋊Q8—2698no—
231C22⋊Q8—2698no—
232C22⋊Q8—2698no—
233C22⋊Q8C2×D26918yescomplemented
234C2×C4⋊C4—2691no—
235C2×C4⋊C4—2691no—
236C2×C4⋊C4—2691no—
237C2×C4⋊C4—2691no—
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⋊C4C2×D26911yescharacteristic
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⋊C4—2691no—
257C2×C22⋊C4—2691no—
258C2×C22⋊C4C2×D26911yescharacteristic
259C2×C22⋊C4C2×D26911yescharacteristic
260C2×C42—2691no—
261C2×C42—2691no—
262C2×C42—2691no—
263C24—2691no—
264C24C22×D26911yescharacteristic
265C2×Q8—42no—
266C2×Q8—2694no—
267C2×Q8—2694no—
268C2×Q8—10762no—
269C2×Q8—10762no—
270C2×Q8C22×D26914yes—
271C2×D4—2694no—
272C2×D4—10762no—
273C22×C4—42no—
274C22×C4—2691no—
275C22×C4—2691no—
276C22×C4—2691no—
277C22×C4—2691no—
278C22×C4—2691no—
279C22×C4—2691no—
280C22×C4—2691no—
281C22×C4—2694no—
282C22×C4—10762no—
283C22×C4C22×D26911yescharacteristic
284C22×C4C22×D26911yescharacteristic
285C22×C4C22×D26911yescharacteristic
286C22×C4C22×D26911yescharacteristic
287C22×C4C22×D26911yescharacteristic
288C22×C4C22×D26911yescharacteristic
289C22×C4C22×D26914yes—
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⋊C4C22×D26914yes—
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⋊C4—2694no—
313C22⋊C4—2694no—
314C22⋊C4C22×D26914yes—
315C22⋊C4C22×D26914yes—
316C42—2694no—
317C42—2694no—
318C42—2694no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—2692no—
323C23—5382no—
324C23—5382no—
325C23—5382no—
326C23C23×D26912yes—
327C23C23×D26911yescharacteristic
328Q8—44no—
329Q8—10764no—
330Q8—10764no—
331D4—10764no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—44no—
339C2×C4—2692no—
340C2×C4—2692no—
341C2×C4—2692no—
342C2×C4—2692no—
343C2×C4—2692no—
344C2×C4—2692no—
345C2×C4—2692no—
346C2×C4—5382no—
347C2×C4—5382no—
348C2×C4—5382no—
349C2×C4—5382no—
350C2×C4—5382no—
351C2×C4—5382no—
352C2×C4—5382no—
353C2×C4—10764no—
354C2×C4C23×D26912yes—
355C2×C4C23×D26912yes—
356C2×C4C23×D26912yes—
357C2×C4C23×D26912yes—
358C2×C4C23×D26912yes—
359C2×C4C23×D26912yes—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—44no—
364C22—5382no—
365C22—5382no—
366C22—5382no—
367C22—10764no—
368C22#2257549812yes—
369C22#2257550012yes—
370C22#2257550112yes—
371C22C24×D26911yescharacteristic
372C4—22no—
373C4—22no—
374C4—22no—
375C4—22no—
376C4—22no—
377C4—22no—
378C4—5382no—
379C4—5382no—
380C4—5382no—
381C4—5382no—
382C4—5382no—
383C4—5382no—
384C4—5382no—
385C2—42no—
386C2—10762no—
387C2#3177135614yesdirect product factor
388C2C2×#2257549811yescharacteristic
389C2C2×#2257550011yescharacteristic
390C2C2×#2257550111yescharacteristic
subgroupG/H
centerC23C23×D269
commutator, cosocleC22×C269C25
FrattiniC22C24×D269
FittingC2×C22⋊Q8×C269C2
socleC23×C269C24
prime radicalC22×C22⋊D269C22
hypercenterC2×C22⋊Q8D269
hypocenterC269C2×#563
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #31771356C2direct (N × H)
2. C2×#22575452C2semidirect
3. C2×#22575451C2semidirect
4. C2×#22575448C2semidirect
5. C2×#22575448C2semidirect
6. C2×#22575446C2semidirect
7. C2×#22575446C2semidirect
8. C2×#22575444C2semidirect
9. C2×#22575443C2semidirect
13. C2×#22575432C2semidirect
14. C2×#22575431C2semidirect
15. C2×#22575429C2semidirect
16. C2×C22⋊Q8×C269C2semidirect
21. #22575452C22semidirect
22. #22575451C22semidirect
23. #22575448C22semidirect
24. #22575448C22semidirect
25. #22575446C22semidirect
26. #22575446C22semidirect
27. #22575444C22semidirect
28. #22575443C22semidirect
32. #22575432C22semidirect
33. #22575431C22semidirect
34. #22575429C22semidirect
61. C22⋊Q8×C269C22semidirect
190. C2×C269#563semidirect
195. C269C2×#563semidirect
212. C2×C22⋊Q8D269semidirect
233. C22⋊Q8C2×D269semidirect
387. C2#31771356direct (N × H)

Maximal subgroups

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

subgroupindex
#317713562
C2×#225754522
C2×#225754512
C2×#225754482
C2×#225754462
C2×#225754442
C2×#225754432
C2×#225754392
C2×#225754362
C2×#225754352
C2×#225754322
C2×#225754312
C2×#225754292
C2×C22⋊Q8×C2692
C2×#563269