Finite Group Databasebeta

← C2×#3516 #62785

Subgroups — C2×#3516 #62785

Order 256.

3565 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3516C2132yesmaximal, direct product factor
2C2×#557C211yesmaximal, characteristic, complemented
3C2×#556C213yesmaximal, complemented
4C2×#554C213yesmaximal, complemented
5C2×#553C213yesmaximal, complemented
6C2×#552C213yesmaximal, complemented
7C2×D4×Q8C211yesmaximal, characteristic, complemented
8C2×#548C211yesmaximal, characteristic, complemented
9C2×#548C211yesmaximal, characteristic, complemented
10C2×#542C213yesmaximal, complemented
11C2×#541C213yesmaximal, complemented
12C2×#529C213yesmaximal, complemented
13C2×#529C213yesmaximal, complemented
14C2×#520C211yesmaximal, characteristic, complemented
15C2×#519C211yesmaximal, characteristic, complemented
16C22×C4×Q8C211yesmaximal, characteristic, complemented
17C22×#41C2211yescharacteristic
18C23×Q8C2211yescharacteristic
19C23×Q8C2211yescharacteristic
20#557C22116yescomplemented
21#556C22148yescomplemented
22#554C22148yescomplemented
23#553C22148yescomplemented
24#552C22148yescomplemented
25D4×Q8C22116yescomplemented
26#548C22116yescomplemented
27#548C22116yescomplemented
28#542C22148yescomplemented
29#541C22148yescomplemented
30C2×C4⋊Q8C2213yes—
31C2×C4⋊Q8C2213yes—
32#529C22148yescomplemented
33#529C22148yescomplemented
34C2×#126C2216yes—
35C2×#126C2213yes—
36C2×#126C2213yes—
37C2×#126C2216yes—
38C2×#125C2213yes—
39C2×#125C2213yes—
40C2×#125C2216yes—
41C2×#125C2213yes—
42C2×#124C2213yes—
43C2×#124C2213yes—
44C2×#124C2213yes—
45C2×#123C2211yescharacteristic
46C2×#123C2213yes—
47C2×#123C2216yes—
48C2×C22⋊Q8C2213yes—
49C2×C22⋊Q8C2213yes—
50C2×C22⋊Q8C2213yes—
51C2×C22⋊Q8C2213yes—
52C2×C22⋊Q8C2213yes—
53C2×C22⋊Q8C2216yes—
54C2×#121C2213yes—
55C2×#121C2213yes—
56C2×#121C2213yes—
57#520C22116yescomplemented
58#519C22116yescomplemented
59C2×C4×Q8C2211yescharacteristic
60C2×C4×Q8C2213yes—
61C2×C4×Q8C2211yescharacteristic
62C2×C4×Q8C2213yes—
63C2×C4×Q8C2213yes—
64C2×C4×Q8C2211yescharacteristic
65C2×C4×Q8C2213yes—
66C2×C4×Q8C2211yescharacteristic
67C2×C4×Q8C2213yes—
68C2×C4×Q8C22116yescomplemented
69C2×C4×D4C2213yes—
70C2×C4×D4C2213yes—
71C2×C4×D4C2213yes—
72C2×C4×D4C2213yes—
73C2×C4×D4C2213yes—
74C2×C4×D4C2216yes—
75C2×#117C2213yes—
76C2×#117C2213yes—
77C2×#117C2213yes—
78C2×#117C2216yes—
79C22×C4⋊C4C2213yes—
80C22×C4⋊C4C2213yes—
81C22×C42C2213yes—
82C2×#41—28no—
83C2×#41C2318yes—
84C22×Q8—28no—
85C22×Q8—28no—
86C22×Q8C2313yes—
87C22×Q8C2313yes—
88C22×Q8C2318yes—
89C22×Q8C2318yes—
90C22×Q8C2313yes—
91C22×Q8C2311yescharacteristic
92C22×D4C2313yes—
93C22×D4C2311yescharacteristic
94C22×D4C2313yes—
95C22×D4C2313yes—
96C23×C4C2313yes—
97C23×C4C2313yes—
98C23×C4C2313yes—
99C23×C4C2311yescharacteristic
100C23×C4C2313yes—
101C23×C4C2316yes—
102C4⋊Q8C23124yes—
103C4⋊Q8C23124yes—
104#126C23148yes—
105#126C23124yes—
106#126C23124yes—
107#126C23148yes—
108#125C23124yes—
109#125C23124yes—
110#125C23148yes—
111#125C23124yes—
112#124C23124yes—
113#124C23124yes—
114#124C23124yes—
115#123C2318yes—
116#123C23124yes—
117#123C23148yes—
118C22⋊Q8C23124yes—
119C22⋊Q8C23124yes—
120C22⋊Q8C23124yes—
121C22⋊Q8C23124yes—
122C22⋊Q8C23124yes—
123C22⋊Q8C23148yes—
124#121C23124yes—
125#121C23124yes—
126#121C23124yes—
127C4×Q8C2318yes—
128C4×Q8C23124yes—
129C4×Q8C2318yes—
130C4×Q8C23124yes—
131C4×Q8C23124yes—
132C4×Q8C2318yes—
133C4×Q8C23124yes—
134C4×Q8C2318yes—
135C4×Q8C23124yes—
136C4×D4C23124yes—
137C4×D4C23124yes—
138C4×D4C23124yes—
139C4×D4C23124yes—
140C4×D4C23124yes—
141C4×D4C23148yes—
142#117C23124yes—
143#117C23124yes—
144#117C23124yes—
145#117C23148yes—
146C2×C4⋊C4C2313yes—
147C2×C4⋊C4C2313yes—
148C2×C4⋊C4C2313yes—
149C2×C4⋊C4C2313yes—
150C2×C4⋊C4C2316yes—
151C2×C4⋊C4C23124yes—
152C2×C4⋊C4C23124yes—
153C2×C4⋊C4C2311yescharacteristic
154C2×C4⋊C4C2313yes—
155C2×C4⋊C4C2313yes—
156C2×C4⋊C4C2313yes—
157C2×C4⋊C4C2313yes—
158C2×C4⋊C4C2316yes—
159C2×C4⋊C4C2313yes—
160C2×C4⋊C4C2313yes—
161C2×C4⋊C4C2316yes—
162C2×C4⋊C4C2313yes—
163C2×C4⋊C4C2311yescharacteristic
164C2×C4⋊C4C2313yes—
165C2×C22⋊C4C2313yes—
166C2×C22⋊C4C2311yescharacteristic
167C2×C22⋊C4C2313yes—
168C2×C22⋊C4C2313yes—
169C2×C22⋊C4C2313yes—
170C2×C22⋊C4C2316yes—
171C2×C22⋊C4C2311yescharacteristic
172C2×C22⋊C4C2313yes—
173C2×C22⋊C4C2311yescharacteristic
174C2×C22⋊C4C2316yes—
175C2×C22⋊C4C2313yes—
176C2×C22⋊C4C2313yes—
177C2×C42C2313yes—
178C2×C42C2316yes—
179C2×C42C2313yes—
180C2×C42C2313yes—
181C2×C42C2313yes—
182C2×C42C2313yes—
183C2×C42C2313yes—
184C2×C42C23124yes—
185C24C2411yescharacteristic
186C24C2413yes—
187C24C2411yescharacteristic
188#41—232no—
189C2×Q8—232no—
190C2×Q8—232no—
191C2×Q8—212no—
192C2×Q8—212no—
193C2×Q8—46no—
194C2×Q8#4118yes—
195C2×Q8C24112yes—
196C2×Q8C24112yes—
197C2×Q8C24112yes—
198C2×Q8C2414yes—
199C2×D4—212no—
200C2×D4—24no—
201C2×D4—212no—
202C2×D4—46no—
203C2×D4C24112yes—
204C2×D4C2414yes—
205C2×D4C24112yes—
206C2×D4C24112yes—
207C22×C4—24no—
208C22×C4—212no—
209C22×C4—212no—
210C22×C4—212no—
211C22×C4—412no—
212C22×C4—46no—
213C22×C4C2414yes—
214C22×C4C24112yes—
215C22×C4C24112yes—
216C22×C4C24112yes—
217C22×C4C2411yescharacteristic
218C22×C4C2411yescharacteristic
219C22×C4C2413yes—
220C22×C4C2411yescharacteristic
221C22×C4C2413yes—
222C22×C4C2411yescharacteristic
223C22×C4C2413yes—
224C22×C4C2413yes—
225C22×C4C2413yes—
226C22×C4C2413yes—
227C22×C4C2411yescharacteristic
228C22×C4C24112yes—
229C22×C4C24124yes—
230C22×C4C2413yes—
231C4⋊C4C24112yes—
232C4⋊C4C24112yes—
233C4⋊C4C24112yes—
234C4⋊C4C24112yes—
235C4⋊C4C24124yes—
236C4⋊C4C2414yes—
237C4⋊C4C24112yes—
238C4⋊C4C24112yes—
239C4⋊C4C24112yes—
240C4⋊C4C24112yes—
241C4⋊C4C24124yes—
242C4⋊C4C24112yes—
243C4⋊C4C24112yes—
244C4⋊C4C24124yes—
245C4⋊C4C24112yes—
246C4⋊C4C2414yes—
247C4⋊C4C24112yes—
248C22⋊C4C24112yes—
249C22⋊C4C2414yes—
250C22⋊C4C24112yes—
251C22⋊C4C24112yes—
252C22⋊C4C24112yes—
253C22⋊C4C24124yes—
254C22⋊C4C2414yes—
255C22⋊C4C24112yes—
256C22⋊C4C2414yes—
257C22⋊C4C24124yes—
258C22⋊C4C24112yes—
259C22⋊C4C24112yes—
260C42C24112yes—
261C42C24112yes—
262C42C24112yes—
263C42C24112yes—
264C42C24124yes—
265C42C24112yes—
266C42C24112yes—
267C23—22no—
268C23—22no—
269C23—22no—
270C23—26no—
271C23—22no—
272C23—26no—
273C23—26no—
274C23—22no—
275C232−1+414yes—
276C23C2512yes—
277C23C2516yes—
278C23C2512yes—
279C23C2511yescharacteristic
280Q8—224no—
281Q8—224no—
282Q8—412no—
283Q8C2×#41116yes—
284D4—224no—
285D4—28no—
286D4—224no—
287D4—412no—
288C2×C4—22no—
289C2×C4—26no—
290C2×C4—26no—
291C2×C4—28no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—26no—
295C2×C4—22no—
296C2×C4—26no—
297C2×C4—26no—
298C2×C4—26no—
299C2×C4—22no—
300C2×C4—224no—
301C2×C4—224no—
302C2×C4—224no—
303C2×C4—424no—
304C2×C4—412no—
305C2×C4C2×#41112yes—
306C2×C4C2512yes—
307C2×C4C2512yes—
308C2×C4C2516yes—
309C2×C4C2512yes—
310C2×C4C2516yes—
311C2×C4C2512yes—
312C2×C4C2516yes—
313C2×C4C2516yes—
314C2×C4C2516yes—
315C2×C4C2516yes—
316C2×C4C2512yes—
317C2×C4C2516yes—
318C22—22no—
319C22—22no—
320C22—28no—
321C22—22no—
322C22—26no—
323C22—22no—
324C22—26no—
325C22—26no—
326C22—22no—
327C22—44no—
328C22—412no—
329C22C22×#4112yes—
330C22C2×2−1+414yes—
331C22C2×2−1+412yes—
332C22#58512yes—
333C22C2611yescharacteristic
334C4—22no—
335C4—26no—
336C4—26no—
337C4—22no—
338C4—22no—
339C4—26no—
340C4—22no—
341C4—26no—
342C4—26no—
343C4—26no—
344C4—22no—
345C4C22×#41112yes—
346C2—24no—
347C2—42no—
348C2—46no—
349C2#351614yesdirect product factor
350C2C23×#4111yescharacteristic
351C2C22×2−1+411yescharacteristic
352C2C2×#58511yescharacteristic
subgroupG/H
center, socleC23C25
commutator, Frattini, cosocleC22C26
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3516C2direct (N × H)
2. C2×#557C2semidirect
3. C2×#556C2semidirect
4. C2×#554C2semidirect
5. C2×#553C2semidirect
6. C2×#552C2semidirect
7. C2×D4×Q8C2semidirect
8. C2×#548C2semidirect
9. C2×#548C2semidirect
10. C2×#542C2semidirect
11. C2×#541C2semidirect
12. C2×#529C2semidirect
13. C2×#529C2semidirect
14. C2×#520C2semidirect
15. C2×#519C2semidirect
16. C22×C4×Q8C2semidirect
20. #557C22semidirect
21. #556C22semidirect
22. #554C22semidirect
23. #553C22semidirect
24. #552C22semidirect
25. D4×Q8C22semidirect
26. #548C22semidirect
27. #548C22semidirect
28. #542C22semidirect
29. #541C22semidirect
32. #529C22semidirect
33. #529C22semidirect
57. #520C22semidirect
58. #519C22semidirect
68. C2×C4×Q8C22semidirect
349. C2#3516direct (N × H)

Maximal subgroups

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

subgroupindex
#35162
C2×#5572
C2×#5562
C2×#5542
C2×#5532
C2×#5522
C2×D4×Q82
C2×#5482
C2×#5422
C2×#5412
C2×#5292
C2×#5202
C2×#5192
C22×C4×Q82