Finite Group Databasebeta

← C2×#61883 #10578555

Subgroups — C2×#61883 #10578555

Order 512.

7296 conjugacy classes of subgroups.

The subgroup lattice has 323 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3497C211yesmaximal, characteristic
2C22×#557C211yesmaximal, characteristic, complemented
3#61883C2132yesmaximal, direct product factor
4C2×#3392C218yesmaximal, complemented
5C2×#3359C218yesmaximal, complemented
6C2×#3206C216yesmaximal, complemented
7C2×#3103C216yesmaximal, complemented
8C2×#2982C211yesmaximal, characteristic, complemented
9C22×2−1+4—21no—
10#3497—28no—
11#3497C22116yes—
12C2×#557—24no—
13C2×#557—28no—
14C2×#557C2218yes—
15C2×#557C22116yescomplemented
16C2×#550C2214yes—
17C2×D4×Q8C2214yes—
18C2×#548—24no—
19C2×#537—23no—
20C2×#532C2216yes—
21C22×C4⋊Q8C2213yes—
22C22×#125—23no—
23C22×C4×Q8—21no—
24C22×C4×Q8C2213yes—
25C22×C4×Q8C2211yescharacteristic
26#3392C221128yescomplemented
27#3359C221128yescomplemented
28#3206C22196yescomplemented
29C2×#492C22112yes—
30C2×#487C22112yes—
31#3103C22196yescomplemented
32C2×#481C2214yes—
33C2×#475C22112yes—
34C2×#460C22112yes—
35C2×#461C22124yes—
36C2×#448C22112yes—
37#2982C22116yescomplemented
38C2×C8×Q8C2218yes—
39C2×C4×SD16C22124yes—
40C2×#431C2216yes—
41C2×2−1+4—216no—
42C2×2−1+4—416no—
43C22×#41—24no—
44C22×#41—26no—
45C23×Q8—24no—
46C23×Q8D411yescharacteristic
47C23×Q8C2313yes—
48C22×SD16C23124yes—
49#557—232no—
50#557—264no—
51#557C23164yes—
52#550—216no—
53#550C23132yes—
54D4×Q8—216no—
55D4×Q8C23132yes—
56#548—264no—
57#537—248no—
58#532—224no—
59#532C23148yes—
60C2×C4⋊Q8—26no—
61C2×C4⋊Q8—212no—
62C2×C4⋊Q8C23112yes—
63C2×C4⋊Q8C23124yes—
64C2×C4⋊Q8C2316yes—
65C2×#127C2316yes—
66C2×#125—224no—
67C2×#125—248no—
68C2×#124—26no—
69C2×C22⋊Q8—212no—
70C2×#121—26no—
71C2×#121—212no—
72C2×#121C2314yes—
73C2×C4×Q8—28no—
74C2×C4×Q8—216no—
75C2×C4×Q8—26no—
76C2×C4×Q8—212no—
77C2×C4×Q8D414yes—
78C2×C4×Q8D418yes—
79C2×C4×Q8C23112yes—
80C2×C4×Q8C23124yes—
81C2×C4×Q8C2314yes—
82C2×C4×Q8C2318yes—
83C2×C4×D4—26no—
84C2×C4×D4C23112yes—
85C22×C4⋊C4—23no—
86C22×C4⋊C4—23no—
87C22×C4⋊C4—23no—
88C22×C4⋊C4D413yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C42—23no—
91C22×C42C2313yes—
92#492C23196yes—
93#487C23196yes—
94#481C23132yes—
95#475C23196yes—
96#461C231192yes—
97#460C23196yes—
98#448C23196yes—
99C8×Q8C23164yes—
100C4×SD16C231192yes—
101#431C23148yes—
102C2×#106C2318yes—
103C2×C4⋊C8C23112yes—
104C2×#103C23124yes—
105C2×#102C2318yes—
106C2×C22⋊C8C2314yes—
107C2×C4×C8C23112yes—
1082−1+4—4128no—
109C2×#41—232no—
110C2×#41—248no—
111C2×#41—432no—
112C2×#41—448no—
113C22×Q8—232no—
114C22×Q8—26no—
115C22×Q8—23no—
116C22×Q8—26no—
117C22×Q8—412no—
118C22×Q8—432no—
119C22×Q8SD16116yescomplemented
120C22×Q8C2×D414yes—
121C22×Q8C2×D418yes—
122C22×Q8C2416yes—
123C22×Q8C24112yes—
124C22×D4—24no—
125C22×D4—26no—
126C22×D4—26no—
127C22×D4C2414yes—
128C23×C4—23no—
129C23×C4—21no—
130C23×C4—212no—
131C23×C4C2×D413yes—
132C23×C4C2×D413yes—
133C23×C4C2411yescharacteristic
134C2×SD16—448no—
135C2×SD16C24196yes—
136C22×C8C2418yes—
137C4⋊Q8—248no—
138C4⋊Q8—224no—
139C4⋊Q8—212no—
140C4⋊Q8C24148yes—
141C4⋊Q8C24124yes—
142#127—212no—
143#127C24124yes—
144#125—2192no—
145#124—248no—
146C22⋊Q8—296no—
147#121—248no—
148#121—28no—
149#121—296no—
150#121C24116yes—
151C4×Q8—264no—
152C4×Q8—248no—
153C4×Q8—224no—
154C4×Q8C2×D4132yes—
155C4×Q8C2×D4116yes—
156C4×Q8C24148yes—
157C4×Q8C24116yes—
158C4×D4—248no—
159C4×D4—224no—
160C4×D4C24148yes—
161C2×C4⋊C4—224no—
162C2×C4⋊C4—212no—
163C2×C4⋊C4—224no—
164C2×C4⋊C4—212no—
165C2×C4⋊C4—224no—
166C2×C4⋊C4—212no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—26no—
170C2×C4⋊C4C2×D4112yes—
171C2×C4⋊C4C2×D4124yes—
172C2×C4⋊C4C2412yes—
173C2×C4⋊C4C2414yes—
174C2×C4⋊C4C2414yes—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—212no—
177C2×C42—224no—
178C2×C42—212no—
179C2×C42C2×D416yes—
180C2×C42C2×D4112yes—
181C2×C42C2416yes—
182C2×C42C24112yes—
183#106C24132yes—
184C4⋊C8C24148yes—
185#103C24196yes—
186#102C24132yes—
187C22⋊C8C24116yes—
188C4×C8C24148yes—
189C24—24no—
190C24C22×D411yescharacteristic
191#41—4128no—
192#41—4192no—
193C2×Q8—212no—
194C2×Q8—26no—
195C2×Q8—224no—
196C2×Q8—224no—
197C2×Q8—424no—
198C2×Q8—412no—
199C2×Q8—424no—
200C2×Q8—4128no—
201C2×Q8—424no—
202C2×Q8C2×SD16132yes—
203C2×Q8C2×SD16164yescomplemented
204C2×Q8C22×D4116yes—
205C2×Q8C25112yes—
206C2×D4—216no—
207C2×D4—24no—
208C2×D4—224no—
209C2×D4—224no—
210C2×D4—216no—
211C2×D4—416no—
212C2×D4—424no—
213C2×D4—812no—
214C2×D4C2518yes—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—212no—
218C22×C4—22no—
219C22×C4—26no—
220C22×C4—24no—
221C22×C4—248no—
222C22×C4—46no—
223C22×C4—44no—
224C22×C4—412no—
225C22×C4—448no—
226C22×C4C2×SD16124yes—
227C22×C4C22×D412yes—
228C22×C4C22×D411yescharacteristic
229C22×C4C22×D416yes—
230C22×C4C22×D4112yes—
231C22×C4C22×D4112yes—
232C22×C4C22×D416yes—
233C22×C4C2512yes—
234C22×C4C2511yescharacteristic
235SD16—496no—
236C2×C8—216no—
237C2×C8C25116yes—
238C4⋊C4—248no—
239C4⋊C4—248no—
240C4⋊C4—248no—
241C4⋊C4—24no—
242C4⋊C4—22no—
243C4⋊C4—224no—
244C4⋊C4—24no—
245C4⋊C4C22×D4148yes—
246C4⋊C4C2514yes—
247C4⋊C4C2518yes—
248C22⋊C4—216no—
249C22⋊C4—248no—
250C42—248no—
251C42C22×D4124yes—
252C42C22×D4112yes—
253C42C25112yes—
254C23—22no—
255C23—22no—
256C23—28no—
257C23—48no—
258C23—48no—
259C23—48no—
260C23C22×SD1614yes—
261C23C23×D412yes—
262C23C23×D411yescharacteristic
263Q8—224no—
264Q8—424no—
265Q8—412no—
266Q8—448no—
267Q8C22×SD16164yes—
268D4—216no—
269D4—432no—
270D4—48no—
271D4—448no—
272D4—824no—
273C2×C4—212no—
274C2×C4—22no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—212no—
279C2×C4—28no—
280C2×C4—412no—
281C2×C4—48no—
282C2×C4—424no—
283C2×C4—44no—
284C2×C4—412no—
285C2×C4—48no—
286C2×C4—496no—
287C2×C4C22×SD16124yes—
288C2×C4C22×SD16148yes—
289C2×C4C23×D412yes—
290C2×C4C23×D4112yes—
291C2×C4C23×D411yescharacteristic
292C2×C4C23×D4112yes—
293C2×C4#58514yes—
294C2×C4C2611yescharacteristic
295C8—216no—
296C22—22no—
297C22—22no—
298C22—28no—
299C22—48no—
300C22—48no—
301C22—48no—
302C22—816no—
303C22#348712yes—
304C22C23×SD1614yes—
305C22C23×SD1612yes—
306C22#358612yes—
307C22C24×D411yescharacteristic
308C4—22no—
309C4—212no—
310C4—412no—
311C4—48no—
312C4—44no—
313C4#348712yes—
314C4C23×SD16124yes—
315C4C2×#58512yes—
316C2—24no—
317C2—88no—
318C2#6188314yesdirect product factor
319C2C2×#348711yescharacteristic
320C2C24×SD1611yescharacteristic
321C2C2×#358611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC2×C4C26
prime radicalC2×#3497C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#557C2semidirect
3. #61883C2direct (N × H)
4. C2×#3392C2semidirect
5. C2×#3359C2semidirect
6. C2×#3206C2semidirect
7. C2×#3103C2semidirect
8. C2×#2982C2semidirect
15. C2×#557C22semidirect
26. #3392C22semidirect
27. #3359C22semidirect
28. #3206C22semidirect
31. #3103C22semidirect
37. #2982C22semidirect
119. C22×Q8SD16semidirect
203. C2×Q8C2×SD16semidirect
318. C2#61883direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#34972
C22×#5572
#618832
C2×#33922
C2×#33592
C2×#32062
C2×#31032
C2×#29822