Finite Group Databasebeta

← C2×#37604 #7633778

Subgroups — C2×#37604 #7633778

Order 512.

6106 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37604C2132yesmaximal, direct product factor
2C2×#2845C211yesmaximal, characteristic, complemented
3C2×#2819C213yesmaximal, complemented
4C2×#2458C216yesmaximal, complemented
5C2×#2464C216yesmaximal, complemented
6C2×#2661C216yesmaximal, complemented
7C2×#2653C216yesmaximal, complemented
8C2×C4×#124C213yesmaximal, complemented
9C22×#124C2213yes—
10C22×#123C2216yes—
11C22×#121C2216yes—
12C22×C4×Q8C2213yes—
13C22×C4×D4C2213yes—
14#2845C22116yescomplemented
15#2819C22148yescomplemented
16#2661C22196yescomplemented
17#2653C22196yescomplemented
18#2464C22196yescomplemented
19#2458C22196yescomplemented
20C2×#401C2212yes—
21C2×#398C2216yes—
22C2×#398C2216yes—
23C2×#397C2216yes—
24C2×#394C2216yes—
25C2×#394C2216yes—
26C2×#394C2216yes—
27C2×#393C2216yes—
28C2×#393C2216yes—
29C2×#392C2212yes—
30C2×#392C2216yes—
31C2×#392C2216yes—
32C4×#124C22148yescomplemented
33C2×#391C2213yes—
34C2×#390C2213yes—
35C2×#390C2216yes—
36C2×#388C2216yes—
37C2×#388C2216yes—
38C2×#388C2216yes—
39C2×#388C2216yes—
40C2×#387C2216yes—
41C2×#386C2216yes—
42C2×#385C2216yes—
43C2×#382C2213yes—
44C2×C4×C22⋊C4C2216yes—
45C2×C4×C22⋊C4C2216yes—
46C2×C43C2211yescharacteristic
47C23×Q8C2313yes—
48C23×D4C2313yes—
49C23×D4C2316yes—
50C23×D4C2316yes—
51C24×C4C2316yes—
52C2×#124—224no—
53C2×#124C23124yes—
54C2×#123—424no—
55C2×#123C23148yes—
56C2×#121—424no—
57C2×#121C23148yes—
58C2×C4×Q8—412no—
59C2×C4×Q8C23124yes—
60C2×C4×D4—412no—
61C2×C4×D4C23124yes—
62C22×C4⋊C4C2316yes—
63C22×C4⋊C4C2313yes—
64C22×C4⋊C4C2313yes—
65C22×C4⋊C4C2316yes—
66C22×C4⋊C4C2316yes—
67C22×C22⋊C4C2316yes—
68C22×C22⋊C4C2316yes—
69C22×C22⋊C4C2316yes—
70C22×C22⋊C4C2316yes—
71C22×C22⋊C4C2316yes—
72C22×C22⋊C4C2316yes—
73C22×C22⋊C4C2316yes—
74C22×C22⋊C4C2316yes—
75C22×C22⋊C4C2316yes—
76C22×C42C2313yes—
77C22×C42C2316yes—
78C22×C42C2313yes—
79C22×C42C2311yescharacteristic
80#401C23116yes—
81#398C23148yes—
82#398C23148yes—
83#397C23148yes—
84#394C23148yes—
85#394C23148yes—
86#394C23148yes—
87#393C23148yes—
88#393C23148yes—
89#392C23148yes—
90#392C23148yes—
91#392C23116yes—
92#391C23124yes—
93#390C23124yes—
94#390C23148yes—
95#388C23148yes—
96#388C23148yes—
97#388C23148yes—
98#388C23148yes—
99#387C23148yes—
100#386C23148yes—
101#385C23148yes—
102#382C23124yes—
103C4×C22⋊C4C23148yes—
104C4×C22⋊C4C23148yes—
105C2×#95C2316yes—
106C2×#95C2316yes—
107C2×#95C2312yes—
108C2×#95C2316yes—
109C2×#95C2316yes—
110C2×#95C2316yes—
111C2×#95C2312yes—
112C2×#95C2316yes—
113C43C2318yes—
114C25C2416yes—
115C22×Q8—212no—
116C22×Q8—46no—
117C22×Q8—46no—
118C22×Q8C24112yes—
119C22×D4—212no—
120C22×D4—46no—
121C22×D4—46no—
122C22×D4—412no—
123C22×D4—412no—
124C22×D4—412no—
125C22×D4—412no—
126C22×D4—412no—
127C22×D4—412no—
128C22×D4C24112yes—
129C22×D4C24124yes—
130C22×D4C24124yes—
131C23×C4—224no—
132C23×C4—412no—
133C23×C4—412no—
134C23×C4C2416yes—
135C23×C4C2411yescharacteristic
136C23×C4C2416yes—
137C23×C4C2416yes—
138C23×C4C2413yes—
139C23×C4C2413yes—
140C23×C4C24124yes—
141#124—296no—
142#123—496no—
143#121—496no—
144C4×Q8—448no—
145C4×D4—448no—
146C2×C4⋊C4—224no—
147C2×C4⋊C4—46no—
148C2×C4⋊C4—46no—
149C2×C4⋊C4—412no—
150C2×C4⋊C4—412no—
151C2×C4⋊C4C24124yes—
152C2×C4⋊C4C24112yes—
153C2×C4⋊C4C24112yes—
154C2×C4⋊C4C24124yes—
155C2×C4⋊C4C24124yes—
156C2×C22⋊C4—224no—
157C2×C22⋊C4—224no—
158C2×C22⋊C4—224no—
159C2×C22⋊C4—224no—
160C2×C22⋊C4—412no—
161C2×C22⋊C4—412no—
162C2×C22⋊C4—412no—
163C2×C22⋊C4—412no—
164C2×C22⋊C4—412no—
165C2×C22⋊C4C24124yes—
166C2×C22⋊C4C24124yes—
167C2×C22⋊C4C24124yes—
168C2×C22⋊C4C24124yes—
169C2×C22⋊C4C24124yes—
170C2×C22⋊C4C24124yes—
171C2×C22⋊C4C24124yes—
172C2×C22⋊C4C24124yes—
173C2×C22⋊C4C24124yes—
174C2×C42—212no—
175C2×C42—412no—
176C2×C42—46no—
177C2×C42#4118yes—
178C2×C42C24112yes—
179C2×C42C24124yes—
180C2×C42C24112yes—
181C2×C42C2414yes—
182#95C24124yes—
183#95C24124yes—
184#95C24124yes—
185#95C24124yes—
186#95C24124yes—
187#95C2418yes—
188#95C24124yes—
189#95C2418yes—
190C24—212no—
191C24—212no—
192C24—212no—
193C24—212no—
194C24—212no—
195C24—212no—
196C24—212no—
197C24C25112yes—
198C24C2511yescharacteristic
199C2×Q8—224no—
200C2×Q8—412no—
201C2×Q8—412no—
202C2×Q8—448no—
203C2×D4—224no—
204C2×D4—412no—
205C2×D4—412no—
206C2×D4—448no—
207C2×D4—424no—
208C2×D4—424no—
209C2×D4—424no—
210C2×D4—424no—
211C2×D4—424no—
212C2×D4—424no—
213C2×D4—848no—
214C2×D4—848no—
215C22×C4—26no—
216C22×C4—26no—
217C22×C4—26no—
218C22×C4—26no—
219C22×C4—212no—
220C22×C4—212no—
221C22×C4—212no—
222C22×C4—212no—
223C22×C4—212no—
224C22×C4—212no—
225C22×C4—26no—
226C22×C4—212no—
227C22×C4—212no—
228C22×C4—212no—
229C22×C4—248no—
230C22×C4—424no—
231C22×C4—424no—
232C22×C4—848no—
233C22×C4C2×#41112yes—
234C22×C4H(5,𝔽2)112yes—
235C22×C4C2512yes—
236C22×C4C25112yes—
237C22×C4C25112yes—
238C22×C4C25112yes—
239C22×C4C2516yes—
240C22×C4C2516yes—
241C4⋊C4—248no—
242C4⋊C4—412no—
243C4⋊C4—412no—
244C4⋊C4—424no—
245C4⋊C4—424no—
246C22⋊C4—248no—
247C22⋊C4—248no—
248C22⋊C4—248no—
249C22⋊C4—248no—
250C22⋊C4—424no—
251C22⋊C4—424no—
252C22⋊C4—424no—
253C22⋊C4—424no—
254C22⋊C4—424no—
255C42—224no—
256C42—412no—
257C42—424no—
258C42C2×#41116yes—
259C23—212no—
260C23—212no—
261C23—212no—
262C23—212no—
263C23—212no—
264C23—212no—
265C23—212no—
266C23—424no—
267C23—424no—
268C23—424no—
269C23—424no—
270C23—424no—
271C23—424no—
272C23—424no—
273C23C22×#4112yes—
274C23C2×H(5,𝔽2)16yes—
275C23#58516yes—
276C23C2611yescharacteristic
277Q8—448no—
278D4—448no—
279D4—848no—
280D4—848no—
281C2×C4—26no—
282C2×C4—26no—
283C2×C4—26no—
284C2×C4—26no—
285C2×C4—224no—
286C2×C4—224no—
287C2×C4—212no—
288C2×C4—212no—
289C2×C4—212no—
290C2×C4—212no—
291C2×C4—212no—
292C2×C4—212no—
293C2×C4—26no—
294C2×C4—212no—
295C2×C4—212no—
296C2×C4—212no—
297C2×C4—424no—
298C2×C4—424no—
299C2×C4—424no—
300C2×C4—44no—
301C2×C4—848no—
302C2×C4C22×#41112yes—
303C2×C4C2×H(5,𝔽2)112yes—
304C22—412no—
305C22—412no—
306C22—412no—
307C22—412no—
308C22—412no—
309C22—412no—
310C22—412no—
311C22—848no—
312C22#3460112yes—
313C22#3558112yes—
314C22#356714yes—
315C22C23×#4111yescharacteristic
316C22C22×H(5,𝔽2)13yes—
317C22C2×#58513yes—
318C4—212no—
319C4—212no—
320C4—412no—
321C4—412no—
322C4—42no—
323C4—412no—
324C2—812no—
325C2#3760418yesdirect product factor
326C2C2×#346013yes—
327C2C2×#355813yes—
328C2C2×#356711yescharacteristic
subgroupG/H
center, socleC24C25
commutator, Frattini, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37604C2direct (N × H)
2. C2×#2845C2semidirect
3. C2×#2819C2semidirect
4. C2×#2458C2semidirect
5. C2×#2464C2semidirect
6. C2×#2661C2semidirect
7. C2×#2653C2semidirect
8. C2×C4×#124C2semidirect
14. #2845C22semidirect
15. #2819C22semidirect
16. #2661C22semidirect
17. #2653C22semidirect
18. #2464C22semidirect
19. #2458C22semidirect
32. C4×#124C22semidirect
325. C2#37604direct (N × H)

Maximal subgroups

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

subgroupindex
#376042
C2×#28452
C2×#28192
C2×#24582
C2×#24642
C2×#26612
C2×#26532
C2×C4×#1242