Finite Group Databasebeta

← C2×#52993 #8093064

Subgroups — C2×#52993 #8093064

Order 512.

6322 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#52993C2132yesmaximal, direct product factor
2C2×#2760C212yesmaximal, complemented
3C2×#2693C216yesmaximal, complemented
4C2×#2841C216yesmaximal, complemented
5C2×#2834C212yesmaximal, complemented
6C2×#2486C213yesmaximal, complemented
7C2×#2417C216yesmaximal, complemented
8C2×#2409C213yesmaximal, complemented
9C2×#2645C212yesmaximal, complemented
10C2×#2397C211yesmaximal, characteristic
11C24×Q8C2211yescharacteristic
12C22×#123C2216yes—
13C22×C22⋊Q8C2216yes—
14C22×C4×D4C2213yes—
15C22×C4×D4C2213yes—
16C23×C4⋊C4C2213yes—
17#2841C22196yescomplemented
18#2834C22132yescomplemented
19#2760C22132yescomplemented
20#2693C22196yescomplemented
21#2645C22132yescomplemented
22#2486C22148yescomplemented
23#2417C22196yescomplemented
24#2409C22148yescomplemented
25C2×#401C2214yes—
26#2397C22116yes—
27C2×#400C22112yes—
28C2×#399C2211yescharacteristic
29C2×#399C2213yes—
30C2×#398C22112yes—
31C2×#398C2216yes—
32C2×#397C2216yes—
33C2×#396C22112yes—
34C2×#395C2216yes—
35C2×#393C2216yes—
36C2×#392C2213yes—
37C2×#392C2211yescharacteristic
38C2×#391C2216yes—
39C2×#389C2216yes—
40C2×#386C2216yes—
41C2×#386C22112yes—
42C2×#385C2216yes—
43C2×#381C2216yes—
44C2×#380C22112yes—
45C2×#380C2213yes—
46C2×#380C2213yes—
47C2×#376C2211yescharacteristic
48C23×Q8—412no—
49C23×Q8C2312yes—
50C23×Q8C2312yes—
51C23×Q8C2318yes—
52C23×D4C2313yes—
53C23×D4C2313yes—
54C24×C4C2311yescharacteristic
55C24×C4C2313yes—
56C24×C4C2316yes—
57C24×C4C2312yes—
58C2×#123—424no—
59C2×#123C23148yes—
60C2×C22⋊Q8—424no—
61C2×C22⋊Q8C23148yes—
62C2×C4×D4—412no—
63C2×C4×D4—412no—
64C2×C4×D4C23124yes—
65C2×C4×D4C23124yes—
66C22×C4⋊C4—412no—
67C22×C4⋊C4C2316yes—
68C22×C4⋊C4C2313yes—
69C22×C4⋊C4C23112yes—
70C22×C4⋊C4C2316yes—
71C22×C4⋊C4C23112yes—
72C22×C4⋊C4C2313yes—
73C22×C4⋊C4C23124yes—
74C22×C22⋊C4C2316yes—
75C22×C22⋊C4C2313yes—
76C22×C22⋊C4C2316yes—
77C22×C22⋊C4C2313yes—
78C22×C22⋊C4C23112yes—
79C22×C22⋊C4C2316yes—
80C22×C42C2313yes—
81#401C23132yes—
82#400C23196yes—
83#399C2318yes—
84#399C23124yes—
85#398C23196yes—
86#398C23148yes—
87#397C23148yes—
88#396C23196yes—
89#395C23148yes—
90#393C23148yes—
91#392C23124yes—
92#392C2318yes—
93#391C23148yes—
94#389C23148yes—
95#386C23148yes—
96#386C23196yes—
97#385C23148yes—
98#381C23148yes—
99#380C23196yes—
100#380C23124yes—
101#380C23124yes—
102#376C2318yes—
103C2×#95C2313yes—
104C2×#95C2311yescharacteristic
105C2×#95C2316yes—
106C2×#95C23112yes—
107C2×#95C23112yes—
108C2×#95C2316yes—
109C2×#95C23112yes—
110C25C2411yescharacteristic
111C25C2413yes—
112C22×Q8—224no—
113C22×Q8—412no—
114C22×Q8—448no—
115C22×Q8—464no—
116C22×Q8C2418yes—
117C22×Q8C2418yes—
118C22×D4—212no—
119C22×D4—212no—
120C22×D4—224no—
121C22×D4—46no—
122C22×D4—46no—
123C22×D4C24112yes—
124C22×D4C24112yes—
125C23×C4—224no—
126C23×C4—212no—
127C23×C4—224no—
128C23×C4—212no—
129C23×C4—412no—
130C23×C4—412no—
131C23×C4—412no—
132C23×C4C24112yes—
133C23×C4C2414yes—
134C23×C4C2413yes—
135C23×C4C24112yes—
136C23×C4C2411yescharacteristic
137C23×C4C2414yes—
138C23×C4C2413yes—
139C23×C4C2413yes—
140C23×C4C24124yes—
141C23×C4C2418yes—
142C23×C4C2411yescharacteristic
143#123—496no—
144C22⋊Q8—496no—
145C4×D4—448no—
146C4×D4—448no—
147C2×C4⋊C4—224no—
148C2×C4⋊C4—448no—
149C2×C4⋊C4—412no—
150C2×C4⋊C4—46no—
151C2×C4⋊C4—46no—
152C2×C4⋊C4—424no—
153C2×C4⋊C4—424no—
154C2×C4⋊C4C24124yes—
155C2×C4⋊C4C24124yes—
156C2×C4⋊C4C24112yes—
157C2×C4⋊C4C24112yes—
158C2×C4⋊C4C24148yes—
159C2×C4⋊C4C24148yes—
160C2×C22⋊C4—224no—
161C2×C22⋊C4—412no—
162C2×C22⋊C4—46no—
163C2×C22⋊C4—46no—
164C2×C22⋊C4—424no—
165C2×C22⋊C4—412no—
166C2×C22⋊C4C24124yes—
167C2×C22⋊C4C24124yes—
168C2×C22⋊C4C24112yes—
169C2×C22⋊C4C24112yes—
170C2×C22⋊C4C24148yes—
171C2×C22⋊C4C24124yes—
172C2×C42—212no—
173C2×C42C24112yes—
174#95C24112yes—
175#95C2414yes—
176#95C24124yes—
177#95C24124yes—
178#95C24148yes—
179#95C24148yes—
180#95C24148yes—
181C24—212no—
182C24—26no—
183C24—26no—
184C24—26no—
185C24—26no—
186C24—212no—
187C24H(5,𝔽2)14yes—
188C242−1+4112yes—
189C24C2512yes—
190C24C2516yes—
191C24C2511yescharacteristic
192C2×Q8—248no—
193C2×Q8—424no—
194C2×Q8—4128no—
195C2×Q8—432no—
196C2×Q8—432no—
197C2×D4—224no—
198C2×D4—224no—
199C2×D4—248no—
200C2×D4—412no—
201C2×D4—412no—
202C2×D4—448no—
203C2×D4—448no—
204C22×C4—26no—
205C22×C4—26no—
206C22×C4—224no—
207C22×C4—224no—
208C22×C4—26no—
209C22×C4—212no—
210C22×C4—26no—
211C22×C4—224no—
212C22×C4—26no—
213C22×C4—224no—
214C22×C4—26no—
215C22×C4—212no—
216C22×C4—248no—
217C22×C4—224no—
218C22×C4—248no—
219C22×C4—224no—
220C22×C4—424no—
221C22×C4—424no—
222C22×C4—424no—
223C22×C4—432no—
224C22×C4—448no—
225C22×C4—416no—
226C22×C4—848no—
227C22×C4H(5,𝔽2)112yes—
228C22×C4C2516yes—
229C22×C4C2516yes—
230C22×C4C25124yes—
231C22×C4C2512yes—
232C22×C4C2518yes—
233C22×C4C2516yes—
234C22×C4C2512yes—
235C4⋊C4—248no—
236C4⋊C4—424no—
237C4⋊C4—412no—
238C4⋊C4—412no—
239C4⋊C4—448no—
240C4⋊C4—448no—
241C22⋊C4—248no—
242C22⋊C4—424no—
243C22⋊C4—412no—
244C22⋊C4—412no—
245C22⋊C4—448no—
246C22⋊C4—424no—
247C42—224no—
248C23—212no—
249C23—26no—
250C23—26no—
251C23—26no—
252C23—26no—
253C23—212no—
254C23—224no—
255C23—224no—
256C23—248no—
257C23—44no—
258C23—412no—
259C23—424no—
260C23—412no—
261C23—412no—
262C23C2×H(5,𝔽2)14yes—
263C23C2×H(5,𝔽2)16yes—
264C23C2×H(5,𝔽2)12yes—
265C23C2×2−1+4112yes—
266C23C2×2−1+416yes—
267C23C2611yescharacteristic
268Q8—432no—
269Q8—432no—
270D4—448no—
271D4—448no—
272C2×C4—26no—
273C2×C4—26no—
274C2×C4—224no—
275C2×C4—224no—
276C2×C4—26no—
277C2×C4—212no—
278C2×C4—26no—
279C2×C4—224no—
280C2×C4—26no—
281C2×C4—224no—
282C2×C4—26no—
283C2×C4—212no—
284C2×C4—432no—
285C2×C4—412no—
286C2×C4—44no—
287C2×C4—416no—
288C2×C4—448no—
289C2×C4—412no—
290C2×C4—412no—
291C2×C4—448no—
292C2×C4—416no—
293C2×C4—848no—
294C2×C4#560116yes—
295C2×C4C2×H(5,𝔽2)112yes—
296C22—212no—
297C22—212no—
298C22—224no—
299C22—448no—
300C22—416no—
301C22—412no—
302C22—46no—
303C22—46no—
304C22—42no—
305C22—46no—
306C22C2×#56014yes—
307C22#3536112yes—
308C22#3538112yes—
309C22C22×H(5,𝔽2)11yescharacteristic
310C22C22×H(5,𝔽2)13yes—
311C22C22×2−1+413yes—
312C4—424no—
313C4—46no—
314C4—46no—
315C4—46no—
316C4—42no—
317C4—48no—
318C4C2×#56018yes—
319C2—412no—
320C2—44no—
321C2#5299318yesdirect product factor
322C2C22×#56011yescharacteristic
323C2C2×#353613yes—
324C2C2×#353813yes—
subgroupG/H
center, socleC24C25
commutator, Frattini, cosocleC23C26
prime radicalC2×#2397C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #52993C2direct (N × H)
2. C2×#2760C2semidirect
3. C2×#2693C2semidirect
4. C2×#2841C2semidirect
5. C2×#2834C2semidirect
6. C2×#2486C2semidirect
7. C2×#2417C2semidirect
8. C2×#2409C2semidirect
9. C2×#2645C2semidirect
17. #2841C22semidirect
18. #2834C22semidirect
19. #2760C22semidirect
20. #2693C22semidirect
21. #2645C22semidirect
22. #2486C22semidirect
23. #2417C22semidirect
24. #2409C22semidirect
321. C2#52993direct (N × H)

Maximal subgroups

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

subgroupindex
#529932
C2×#27602
C2×#26932
C2×#28412
C2×#28342
C2×#24862
C2×#24172
C2×#24092
C2×#26452
C2×#23972