Finite Group Databasebeta

← C2×#31774580 #46849869

Subgroups — C2×#31774580 #46849869

Order 34688.

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#31774580C2116yesmaximal, direct product factor
2C2×#22576155C211yesmaximal, characteristic, complemented
3C2×#22576154C211yesmaximal, characteristic, complemented
4C2×#22576152C211yesmaximal, characteristic, complemented
5C2×#22576150C211yesmaximal, characteristic, complemented
6C2×#22576150C211yesmaximal, characteristic, complemented
7C2×#22576144C211yesmaximal, characteristic
8C2×#22576143C211yesmaximal, characteristic
9C2×#22576138C211yesmaximal, characteristic, complemented
10C2×#22576137C211yesmaximal, characteristic, complemented
11C2×C4×Dic542C211yesmaximal, characteristic, complemented
12C2×(C22⋊D271)⋊C4C211yesmaximal, characteristic
13C2×#22576117C211yesmaximal, characteristic, complemented
14C2×#22576116C211yesmaximal, characteristic, complemented
15C22×Dic271⋊C4C211yesmaximal, characteristic, complemented
16C2×#117×C271C211yesmaximal, characteristic, complemented
17C22×Dic542C2211yescharacteristic
18C22×C22⋊D271C2211yescharacteristic
19C23×Dic271C2211yescharacteristic
20C23×Dic271C2211yescharacteristic
21C22×C4×D271C2211yescharacteristic
22#22576155C2218yescomplemented
23#22576154C2218yescomplemented
24#22576152C2218yescomplemented
25#22576150C2218yescomplemented
26#22576150C2218yescomplemented
27#22576144C2218yes—
28#22576143C2218yes—
29#22576138C2218yescomplemented
30#22576137C2218yescomplemented
31C4×Dic542C2218yescomplemented
32(C22⋊D271)⋊C4C2218yes—
33#22576117C2218yescomplemented
34#22576116C2218yescomplemented
35C2×C4⋊Dic271C2211yescharacteristic
36C2×C4⋊Dic271C2211yescharacteristic
37C2×C4⋊Dic271C2211yescharacteristic
38C2×Dic271⋊C4C2211yescharacteristic
39C2×Dic271⋊C4C2211yescharacteristic
40C2×Dic271⋊C4C2211yescharacteristic
41C2×Dic271⋊C4C2211yescharacteristic
42C2×Dic271⋊C4C2211yescharacteristic
43C2×Dic271⋊C4C2211yescharacteristic
44C2×Dic271⋊C4C2211yescharacteristic
45C2×Dic271⋊C4C2211yescharacteristic
46C2×Dic271⋊C4C2211yescharacteristic
47C2×Dic271⋊C4C2211yescharacteristic
48C2×Dic271⋊C4C2211yescharacteristic
49C2×Dic271⋊C4C2218yescomplemented
50C2×C22⋊Dic271C2211yescharacteristic
51C2×(C2×D271)⋊C4C2211yescharacteristic
52C2×(C2×D271)⋊C4C2211yescharacteristic
53C2×(C2×D271)⋊C4C2211yescharacteristic
54C2×(C2×D271)⋊C4C2211yescharacteristic
55C2×(C2×D271)⋊C4C2211yescharacteristic
56C2×C4×Dic271C2211yescharacteristic
57C2×C4×Dic271C2211yescharacteristic
58C2×C4×Dic271C2211yescharacteristic
59C23×C4×C271C2211yescharacteristic
60#117×C271C2218yescomplemented
61C2×C4⋊C4×C271C2211yescharacteristic
62C2×C4⋊C4×C271C2211yescharacteristic
63C2×C22⋊C4×C271C2211yescharacteristic
64C2×C22⋊C4×C271C2211yescharacteristic
65C2×C42×C271C2211yescharacteristic
66C2×C42×C271C2211yescharacteristic
67C23×D271C2311yescharacteristic
68C2×Dic542—24no—
69C2×Dic542C2314yes—
70C2×C22⋊D271—24no—
71C2×C22⋊D271C2314yes—
72C22×Dic271—24no—
73C22×Dic271—24no—
74C22×Dic271C2314yes—
75C22×Dic271C2314yes—
76C22×Dic271C2311yescharacteristic
77C22×Dic271C2311yescharacteristic
78C22×Dic271C2311yescharacteristic
79C22×Dic271C2311yescharacteristic
80C22×Dic271C2311yescharacteristic
81C22×Dic271C2311yescharacteristic
82C22×Dic271C2311yescharacteristic
83C2×C4×D271—24no—
84C2×C4×D271C2314yes—
85C4⋊Dic271C2314yes—
86C4⋊Dic271C2314yes—
87C4⋊Dic271C2314yes—
88Dic271⋊C4C2314yes—
89Dic271⋊C4C2314yes—
90Dic271⋊C4C2314yes—
91Dic271⋊C4C2314yes—
92Dic271⋊C4C2314yes—
93Dic271⋊C4C2314yes—
94Dic271⋊C4C2314yes—
95Dic271⋊C4C2314yes—
96Dic271⋊C4C2314yes—
97Dic271⋊C4C2314yes—
98Dic271⋊C4C2314yes—
99C22⋊Dic271C2314yes—
100(C2×D271)⋊C4C2314yes—
101(C2×D271)⋊C4C2314yes—
102(C2×D271)⋊C4C2314yes—
103(C2×D271)⋊C4C2314yes—
104(C2×D271)⋊C4C2314yes—
105C4×Dic271C2314yes—
106C4×Dic271C2314yes—
107C4×Dic271C2314yes—
108C24×C271C2311yescharacteristic
109C22×C4×C271—24no—
110C22×C4×C271C2314yes—
111C22×C4×C271C2311yescharacteristic
112C22×C4×C271C2311yescharacteristic
113C22×C4×C271C2311yescharacteristic
114C22×C4×C271C2311yescharacteristic
115C22×C4×C271C2311yescharacteristic
116C22×C4×C271C2311yescharacteristic
117C4⋊C4×C271C2314yes—
118C4⋊C4×C271C2314yes—
119C22⋊C4×C271C2314yes—
120C22⋊C4×C271C2314yes—
121C42×C271C2314yes—
122C42×C271C2314yes—
123C22×D271—22no—
124C22×D271—22no—
125C22×D271—22no—
126C22×D271C2412yes—
127Dic542—28no—
128C22⋊D271—28no—
129C2×Dic271—22no—
130C2×Dic271—28no—
131C2×Dic271—22no—
132C2×Dic271—22no—
133C2×Dic271—22no—
134C2×Dic271—22no—
135C2×Dic271—22no—
136C2×Dic271—28no—
137C2×Dic271#4114yes—
138C2×Dic271C2412yes—
139C2×Dic271C2412yes—
140C2×Dic271C2412yes—
141C2×Dic271C2412yes—
142C2×Dic271C2412yes—
143C2×Dic271C2412yes—
144C2×Dic271C2412yes—
145C4×D271—28no—
146C23×C271—22no—
147C23×C271—22no—
148C23×C271#4114yes—
149C23×C271C2412yes—
150C23×C271C2411yescharacteristic
151C2×C4×C271—28no—
152C2×C4×C271—22no—
153C2×C4×C271—22no—
154C2×C4×C271—22no—
155C2×C4×C271—22no—
156C2×C4×C271—22no—
157C2×C4×C271—22no—
158C2×C4×C271C2412yes—
159C2×C4×C271C2412yes—
160C2×C4×C271C2412yes—
161C2×C4×C271C2412yes—
162C2×C4×C271C2412yes—
163C2×C4×C271C2412yes—
164C22×C271—28no—
165C22×C271—22no—
166C22×C271—22no—
167C22×C271C2×#4112yes—
168C22×C271C2×#4114yes—
169C22×C271C2×#4112yes—
170C22×C2712−1+412yes—
171C22×C271C2511yescharacteristic
172C2×D271—22no—
173C2×D271—22no—
174C2×D271—22no—
175C2×D271—44no—
176C4×C271—22no—
177C4×C271—22no—
178C4×C271—22no—
179C4×C271—22no—
180C4×C271—22no—
181C4×C271—22no—
182Dic271—22no—
183Dic271—22no—
184Dic271—22no—
185Dic271—22no—
186Dic271—22no—
187Dic271—22no—
188Dic271C2×#4114yes—
189C2×C271—24no—
190C2×C271#55214yescomplemented
191C2×C271C22×#4111yescharacteristic
192C2×C271C22×#4111yescharacteristic
193C2×C271C2×2−1+411yescharacteristic
194D271—42no—
195C271C2×#55211yescharacteristic, complemented, Sylow
196C2×#552—2711nomaximal, Sylow
197#552—27116no—
198C2×#126—2711no—
199C2×#126—2711no—
200C2×#125—2711no—
201C2×#125—2711no—
202C2×#125—2711no—
203C2×#123—2711no—
204C2×#123—2711no—
205C2×C22⋊Q8—2711no—
206C2×C22⋊Q8—2711no—
207C2×C4×Q8—2711no—
208C2×C4×D4—2711no—
209C2×#117—2711no—
210C2×#117—2711no—
211C2×#117D27111yescharacteristic, complemented
212C22×C4⋊C4—2711no—
213C22×Q8—2711no—
214C22×D4—2711no—
215C23×C4—2711no—
216C23×C4—2711no—
217C23×C4—2711no—
218C23×C4C2×D27111yescharacteristic
219#126—2718no—
220#126—2718no—
221#125—2718no—
222#125—2718no—
223#125—2718no—
224#123—2718no—
225#123—2718no—
226C22⋊Q8—2718no—
227C22⋊Q8—2718no—
228C4×Q8—2718no—
229C4×D4—2718no—
230#117—2718no—
231#117—2718no—
232#117C2×D27118yescomplemented
233C2×C4⋊C4—2711no—
234C2×C4⋊C4—2711no—
235C2×C4⋊C4—2711no—
236C2×C4⋊C4—2711no—
237C2×C4⋊C4—2718no—
238C2×C4⋊C4—2711no—
239C2×C4⋊C4—2711no—
240C2×C4⋊C4—2711no—
241C2×C4⋊C4—2711no—
242C2×C4⋊C4—2711no—
243C2×C4⋊C4—2711no—
244C2×C4⋊C4—2711no—
245C2×C4⋊C4—2711no—
246C2×C4⋊C4—2711no—
247C2×C4⋊C4—2711no—
248C2×C4⋊C4C2×D27111yescharacteristic
249C2×C4⋊C4C2×D27111yescharacteristic
250C2×C22⋊C4—2711no—
251C2×C22⋊C4—2711no—
252C2×C22⋊C4—2711no—
253C2×C22⋊C4—2711no—
254C2×C22⋊C4—2711no—
255C2×C22⋊C4—2711no—
256C2×C22⋊C4C2×D27111yescharacteristic
257C2×C22⋊C4C2×D27111yescharacteristic
258C2×C42—2711no—
259C2×C42—2711no—
260C2×C42—2711no—
261C2×C42C2×D27111yescharacteristic
262C2×C42C2×D27111yescharacteristic
263C24—2711no—
264C24C22×D27111yescharacteristic
265C2×Q8—2714no—
266C2×Q8—5424no—
267C2×D4—2714no—
268C2×D4—5424no—
269C22×C4—24no—
270C22×C4—2711no—
271C22×C4—2714no—
272C22×C4—2714no—
273C22×C4—2711no—
274C22×C4—2711no—
275C22×C4—2711no—
276C22×C4—2711no—
277C22×C4—2711no—
278C22×C4—2711no—
279C22×C4—2714no—
280C22×C4—5424no—
281C22×C4—5424no—
282C22×C4—5424no—
283C22×C4C22×D27114yes—
284C22×C4C22×D27111yescharacteristic
285C22×C4C22×D27111yescharacteristic
286C22×C4C22×D27111yescharacteristic
287C22×C4C22×D27111yescharacteristic
288C22×C4C22×D27111yescharacteristic
289C22×C4C22×D27111yescharacteristic
290C4⋊C4—2714no—
291C4⋊C4—2714no—
292C4⋊C4—2714no—
293C4⋊C4—2714no—
294C4⋊C4—2714no—
295C4⋊C4—2714no—
296C4⋊C4—2714no—
297C4⋊C4—2714no—
298C4⋊C4—2714no—
299C4⋊C4—2714no—
300C4⋊C4—2714no—
301C4⋊C4—2714no—
302C4⋊C4—2714no—
303C4⋊C4—2714no—
304C4⋊C4C22×D27114yes—
305C4⋊C4C22×D27114yes—
306C22⋊C4—2714no—
307C22⋊C4—2714no—
308C22⋊C4—2714no—
309C22⋊C4—2714no—
310C22⋊C4—2714no—
311C22⋊C4—2714no—
312C22⋊C4C22×D27114yes—
313C22⋊C4C22×D27114yes—
314C42—2714no—
315C42—2714no—
316C42—2714no—
317C42C22×D27114yes—
318C42C22×D27114yes—
319C23—22no—
320C23—22no—
321C23—2712no—
322C23—5422no—
323C23—5422no—
324C23—5422no—
325C23#1686465414yes—
326C23C23×D27112yes—
327C23C23×D27111yescharacteristic
328Q8—5428no—
329D4—5428no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—28no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—2714no—
338C2×C4—2712no—
339C2×C4—2712no—
340C2×C4—2712no—
341C2×C4—2712no—
342C2×C4—2712no—
343C2×C4—2712no—
344C2×C4—2712no—
345C2×C4—5428no—
346C2×C4—5428no—
347C2×C4—5422no—
348C2×C4—5422no—
349C2×C4—5428no—
350C2×C4—5422no—
351C2×C4—5422no—
352C2×C4—5422no—
353C2×C4—5422no—
354C2×C4C23×D27112yes—
355C2×C4C23×D27112yes—
356C2×C4C23×D27112yes—
357C2×C4C23×D27112yes—
358C2×C4C23×D27112yes—
359C2×C4C23×D27112yes—
360C22—22no—
361C22—22no—
362C22—28no—
363C22—5422no—
364C22—5422no—
365C22—5422no—
366C22—10844no—
367C22#41×D27112yes—
368C22C2×#1686465414yes—
369C22C2×#1686465412yes—
370C22#2257620712yes—
371C22C24×D27111yescharacteristic
372C4—22no—
373C4—22no—
374C4—22no—
375C4—22no—
376C4—22no—
377C4—22no—
378C4—2714no—
379C4—5422no—
380C4—5422no—
381C4—5422no—
382C4—5422no—
383C4—5422no—
384C4—5422no—
385C2—24no—
386C2—10842no—
387C2#3177458014yesdirect product factor
388C2C2×#41×D27111yescharacteristic
389C2C22×#1686465411yescharacteristic
390C2C2×#2257620711yescharacteristic
subgroupG/H
centerC23C23×D271
commutator, cosocleC22×C271C25
FrattiniC22C24×D271
FittingC2×#117×C271C2
socleC23×C271C24
prime radicalC22×C22⋊D271C22
hypercenterC2×#117D271
hypocenterC271C2×#552
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. #31774580C2direct (N × H)
2. C2×#22576155C2semidirect
3. C2×#22576154C2semidirect
4. C2×#22576152C2semidirect
5. C2×#22576150C2semidirect
6. C2×#22576150C2semidirect
9. C2×#22576138C2semidirect
10. C2×#22576137C2semidirect
11. C2×C4×Dic542C2semidirect
13. C2×#22576117C2semidirect
14. C2×#22576116C2semidirect
15. C22×Dic271⋊C4C2semidirect
16. C2×#117×C271C2semidirect
22. #22576155C22semidirect
23. #22576154C22semidirect
24. #22576152C22semidirect
25. #22576150C22semidirect
26. #22576150C22semidirect
29. #22576138C22semidirect
30. #22576137C22semidirect
31. C4×Dic542C22semidirect
33. #22576117C22semidirect
34. #22576116C22semidirect
49. C2×Dic271⋊C4C22semidirect
60. #117×C271C22semidirect
190. C2×C271#552semidirect
195. C271C2×#552semidirect
211. C2×#117D271semidirect
232. #117C2×D271semidirect
387. C2#31774580direct (N × H)

Maximal subgroups

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

subgroupindex
#317745802
C2×#225761552
C2×#225761542
C2×#225761522
C2×#225761502
C2×#225761442
C2×#225761432
C2×#225761382
C2×#225761372
C2×C4×Dic5422
C2×(C22⋊D271)⋊C42
C2×#225761172
C2×#225761162
C22×Dic271⋊C42
C2×#117×C2712
C2×#552271