Finite Group Databasebeta

← C2×#60667 #10575277

Subgroups — C2×#60667 #10575277

Order 512.

6702 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3503C212yesmaximal
2#60667C2132yesmaximal, direct product factor
3C2×#3165C216yesmaximal, complemented
4C2×#3164C2112yesmaximal, complemented
5C2×#3160C213yesmaximal, complemented
6C22×#500C213yesmaximal, complemented
7C2×#2913C214yesmaximal
8C2×C4×#131C211yesmaximal, characteristic
9C22×#131C2212yes—
10#3503—216no—
11#3503C22132yes—
12C2×Q82—22no—
13C2×#557—22no—
14C2×#554—26no—
15C2×D4×Q8—26no—
16C2×#544C22112yes—
17C2×#533C2216yes—
18C22×C4⋊Q8C2216yes—
19C2×#520C2214yes—
20C2×C4×#41C2211yescharacteristic
21#3165C22196yescomplemented
22#3164C221192yescomplemented
23C2×#501C22124yes—
24#3160C22148yescomplemented
25C2×#500C2213yes—
26C2×#500C22112yes—
27C2×#500C2216yes—
28C2×#500C2211yescharacteristic
29C2×#500C22148yescomplemented
30C2×#499C22112yes—
31C2×#498C2216yes—
32#2913C22164yes—
33C2×#428C22124yes—
34C2×#427C22112yes—
35C22×#107C22112yes—
36C4×#131C22116yes—
37C2×#405C2216yes—
38C2×C4×M16C2213yes—
39C22×C4×C8C2213yes—
40C22×#41C2311yescharacteristic
41C23×Q8—26no—
42C2×#131Q8132yescomplemented
43C2×#131C23116yes—
44C22×M16C2316yes—
45C23×C8C2316yes—
46Q82—232no—
47#557—232no—
48#554—296no—
49D4×Q8—296no—
50#544—248no—
51#544C23196yes—
52#533—224no—
53#533C23148yes—
54C2×C4⋊Q8—212no—
55C2×C4⋊Q8—26no—
56C2×C4⋊Q8—26no—
57C2×C4⋊Q8—224no—
58C2×C4⋊Q8C2312yes—
59C2×C4⋊Q8C2316yes—
60C2×C4⋊Q8C23112yes—
61C2×C4⋊Q8C23148yes—
62C2×#125—212no—
63C2×#125C23112yes—
64C2×C22⋊Q8—26no—
65C2×C22⋊Q8—26no—
66C2×C22⋊Q8—224no—
67#520—216no—
68#520C23132yes—
69C4×#41D418yes—
70C4×#41C2318yes—
71C2×C4×Q8—24no—
72C2×C4×Q8—26no—
73C2×C4×Q8—26no—
74C2×C4×Q8D411yescharacteristic
75C2×C4×Q8D411yescharacteristic
76C2×C4×D4—212no—
77C2×C4×D4D413yes—
78C2×C4×D4D413yes—
79C2×#117C23112yes—
80C2×#117C2313yes—
81C22×C4⋊C4—26no—
82C22×C4⋊C4C23112yes—
83C22×C42C2313yes—
84#501C231192yes—
85#500C23124yes—
86#500C23196yes—
87#500C23148yes—
88#500C2318yes—
89#499C23196yes—
90#498C23148yes—
91#428C231192yes—
92#427C23196yes—
93C2×#107C2314yes—
94C2×#107C23112yes—
95C2×#107C23124yes—
96C2×#107C23196yes—
97C2×#106C23124yes—
98#405C23148yes—
99C4×M16C23124yes—
100C2×#97C2316yes—
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2313yes—
103C2×C4×C8C2316yes—
104C2×C4×C8C23124yes—
105C2×#41C2×D414yes—
106C2×#41C2×Q8116yes—
107C2×#41C2414yes—
108C22×Q8—22no—
109C22×Q8—26no—
110C22×Q8—212no—
111C22×Q8—248no—
112C22×Q8—824no—
113C22×Q8C2×D411yescharacteristic
114C22×D4C2×D413yes—
115C23×C4—212no—
116C23×C4C2×D413yes—
117C23×C4C2×D413yes—
118C23×C4C2413yes—
119#131C2×Q81128yescomplemented
120C2×M16C2×Q8148yescomplemented
121C2×M16C24124yes—
122C22×C8C2×Q8148yescomplemented
123C22×C8C2412yes—
124C22×C8C2416yes—
125C22×C8C24124yes—
126C4⋊Q8—296no—
127C4⋊Q8—212no—
128C4⋊Q8—224no—
129C4⋊Q8—248no—
130C4⋊Q8—248no—
131C4⋊Q8—24no—
132C4⋊Q8C24124yes—
133C4⋊Q8C24148yes—
134C4⋊Q8C2418yes—
135#125—296no—
136#125—224no—
137#125C24148yes—
138C22⋊Q8—248no—
139C22⋊Q8—248no—
140C22⋊Q8—2192no—
141C4×Q8—232no—
142C4×Q8—248no—
143C4×Q8—248no—
144C4×Q8C2×D418yes—
145C4×Q8C2×D418yes—
146C4×D4—296no—
147C4×D4C2×D4124yes—
148C4×D4C2×D4124yes—
149#117—224no—
150#117C2×D4112yes—
151#117C24148yes—
152#117C24112yes—
153C2×C4⋊C4—26no—
154C2×C4⋊C4—212no—
155C2×C4⋊C4—248no—
156C2×C4⋊C4—212no—
157C2×C4⋊C4—224no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—212no—
160C2×C4⋊C4—212no—
161C2×C4⋊C4C2×D413yes—
162C2×C4⋊C4C2×D413yes—
163C2×C4⋊C4C24112yes—
164C2×C4⋊C4C24148yes—
165C2×C4⋊C4C2414yes—
166C2×C22⋊C4—212no—
167C2×C22⋊C4C2×D416yes—
168C2×C42—212no—
169C2×C42C2×D413yes—
170C2×C42C2×D413yes—
171C2×C42C2×D4112yes—
172C2×C42C2413yes—
173C2×C42C24112yes—
174C2×C42C2411yescharacteristic
175#107C24148yes—
176#107C24196yes—
177#107C24116yes—
178#106C24196yes—
179#97C24124yes—
180C4×C8C2414yes—
181C4×C8C24112yes—
182C4×C8C24124yes—
183C24C22×D413yes—
184#41C4⋊Q8132yes—
185#41C22×Q8132yes—
186C2×Q8—28no—
187C2×Q8—224no—
188C2×Q8—248no—
189C2×Q8—424no—
190C2×Q8—48no—
191C2×Q8—824no—
192C2×Q8—896no—
193C2×Q8C4⋊Q818yes—
194C2×Q8C22×D414yes—
195C2×D4C4⋊Q8124yes—
196C2×D4C22×D4112yes—
197C22×C4—212no—
198C22×C4—212no—
199C22×C4—24no—
200C22×C4—212no—
201C22×C4—248no—
202C22×C4—824no—
203C22×C4C22×D4112yes—
204C22×C4C22×D4112yes—
205C22×C4C22×D413yes—
206C22×C4C22×D413yes—
207C22×C4C22×D416yes—
208C22×C4C22×D413yes—
209C22×C4C22×D411yescharacteristic
210C22×C4C22×D411yescharacteristic
211C22×C4C22×Q8124yes—
212C22×C4C2516yes—
213C22×C4C2511yescharacteristic
214M16C22×Q8196yescomplemented
215C2×C8C22×Q818yes—
216C2×C8C22×Q8124yes—
217C2×C8C22×Q8196yescomplemented
218C2×C8C2514yes—
219C2×C8C25112yes—
220C4⋊C4—224no—
221C4⋊C4—212no—
222C4⋊C4—248no—
223C4⋊C4—248no—
224C4⋊C4—28no—
225C4⋊C4—248no—
226C4⋊C4—248no—
227C4⋊C4—24no—
228C4⋊C4C22×D4112yes—
229C4⋊C4C22×D4112yes—
230C4⋊C4C2518yes—
231C4⋊C4C25124yes—
232C22⋊C4—248no—
233C22⋊C4C22×D4124yes—
234C42—248no—
235C42C22×D412yes—
236C42C22×D416yes—
237C42C22×D4112yes—
238C42C22×D4112yes—
239C42C2512yes—
240C42C2516yes—
241C23—26no—
242C23—26no—
243C23C2×C4⋊Q8112yes—
244C23C23×D416yes—
245C23C23×D411yescharacteristic
246Q8—448no—
247Q8—416no—
248Q8—848no—
249Q8C2×C4⋊Q8116yes—
250D4C2×C4⋊Q8148yes—
251C2×C4—26no—
252C2×C4—26no—
253C2×C4—224no—
254C2×C4—224no—
255C2×C4—28no—
256C2×C4—224no—
257C2×C4—424no—
258C2×C4—48no—
259C2×C4—848no—
260C2×C4C2×C4⋊Q8112yes—
261C2×C4C2×C4⋊Q8124yes—
262C2×C4#57614yes—
263C2×C4#57814yes—
264C2×C4C23×D416yes—
265C2×C4C23×D416yes—
266C2×C4C23×D416yes—
267C2×C4C23×D411yescharacteristic
268C2×C4C23×D412yes—
269C2×C4C23×D412yes—
270C2×C4C23×Q814yes—
271C2×C4C23×Q8124yes—
272C2×C4C2611yescharacteristic
273C8C23×Q818yes—
274C8C23×Q8124yes—
275C22—26no—
276C22—26no—
277C22—224no—
278C22C22×C4⋊Q8112yes—
279C22C22×C4⋊Q812yes—
280C22C2×#57612yes—
281C22C2×#57812yes—
282C22C24×D411yescharacteristic
283C4—26no—
284C4—26no—
285C4—424no—
286C4—48no—
287C4C22×C4⋊Q812yes—
288C4C22×C4⋊Q8112yes—
289C4C2×#57614yes—
290C4C2×#57814yes—
291C4C24×Q812yes—
292C2—212no—
293C2#6066714yesdirect product factor
294C2C23×C4⋊Q811yescharacteristic
295C2C22×#57611yescharacteristic
296C2C22×#57811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC2×C4C26
prime radicalC22×#41C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60667C2direct (N × H)
3. C2×#3165C2semidirect
4. C2×#3164C2semidirect
5. C2×#3160C2semidirect
6. C22×#500C2semidirect
21. #3165C22semidirect
22. #3164C22semidirect
24. #3160C22semidirect
29. C2×#500C22semidirect
42. C2×#131Q8semidirect
119. #131C2×Q8semidirect
120. C2×M16C2×Q8semidirect
122. C22×C8C2×Q8semidirect
214. M16C22×Q8semidirect
217. C2×C8C22×Q8semidirect
293. C2#60667direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#35032
#606672
C2×#31652
C2×#31642
C2×#31602
C22×#5002
C2×#29132
C2×C4×#1312