Finite Group Databasebeta

← C2×#11512 #480281

Subgroups — C2×#11512 #480281

Order 512.

1686 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 1686 conjugacy classes of subgroups — too many to draw. Only its 250 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#384C211yesmaximal, characteristic
2C2×#1463C212yesmaximal
3#11512C218yesmaximal, direct product factor
4C2×#1325C214yesmaximal, complemented
5C23×C4⋊C4—22no—
6C23×C4⋊C4C411yescharacteristic
7C23×C4⋊C4C2211yescharacteristic
8C23×C42C411yescharacteristic
9C2×#407—22no—
10C2×#384—22no—
11C2×#384—24no—
12C2×#384C414yes—
13C2×#384C412yes—
14C2×#384C2212yes—
15C2×#384C2214yes—
16#1463—24no—
17#1463C2218yes—
18C2×#324—22no—
19C2×#324C2214yes—
20#1325C22116yescomplemented
21C24×C4—21no—
22C24×C4—21no—
23C24×C4C2×C411yescharacteristic
24C24×C4D412yes—
25C22×M16—42no—
26C22×C4⋊C4—28no—
27C22×C4⋊C4—24no—
28C22×C4⋊C4—22no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4—24no—
31C22×C4⋊C4—28no—
32C22×C4⋊C4—41no—
33C22×C4⋊C4—41no—
34C22×C4⋊C4—42no—
35C22×C4⋊C4—42no—
36C22×C4⋊C4—41no—
37C22×C4⋊C4—41no—
38C22×C4⋊C4—416no—
39C22×C4⋊C4C2×C414yes—
40C22×C4⋊C4C2×C412yes—
41C22×C4⋊C4C2×C412yes—
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4D412yes—
44C22×C4⋊C4D414yes—
45C22×C4⋊C4C2311yescharacteristic
46C22×C4⋊C4C2312yes—
47C22×C42—22no—
48C22×C42—21no—
49C22×C42—28no—
50C22×C42C2×C414yes—
51C22×C42C2×C412yes—
52#407—28no—
53#407—44no—
54C2×C22⋊C8—24no—
55C2×C22⋊C8—42no—
56#384—28no—
57#384—24no—
58#384—24no—
59#384—28no—
60#384C2×C414yes—
61#384C2×C418yes—
62#384C2314yes—
63#324—28no—
64#324—24no—
65#324C2318yes—
66C25C22⋊C411yescharacteristic
67C23×C4—28no—
68C23×C4—28no—
69C23×C4—28no—
70C23×C4—22no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—24no—
74C23×C4—22no—
75C23×C4—24no—
76C23×C4—24no—
77C23×C4—22no—
78C23×C4—22no—
79C23×C4—24no—
80C23×C4—24no—
81C23×C4—22no—
82C23×C4—48no—
83C23×C4—48no—
84C23×C4—42no—
85C23×C4—41no—
86C23×C4—41no—
87C23×C4C22⋊C412yes—
88C23×C4C22⋊C411yescharacteristic
89C23×C4C22⋊C414yes—
90C23×C4C22⋊C412yes—
91C23×C4C22×C412yes—
92C23×C4C22×C411yescharacteristic
93C23×C4C2×D412yes—
94C23×C4C2×D414yes—
95C2×M16—48no—
96C2×M16—88no—
97C22×C8—44no—
98C2×C4⋊C4—28no—
99C2×C4⋊C4—216no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—28no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—44no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—44no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—44no—
116C2×C4⋊C4—44no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—432no—
120C2×C4⋊C4—416no—
121C2×C4⋊C4—416no—
122C2×C4⋊C4—432no—
123C2×C4⋊C4—44no—
124C2×C4⋊C4—44no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4C22⋊C412yes—
128C2×C4⋊C4C22⋊C414yes—
129C2×C4⋊C4C22×C414yes—
130C2×C4⋊C4C22×C411yescharacteristic
131C2×C4⋊C4C22×C412yes—
132C2×C4⋊C4C2×D412yes—
133C2×C4⋊C4C2×D414yes—
134C2×C4⋊C4C2411yescharacteristic
135C2×C42—28no—
136C2×C42—216no—
137C2×C42—24no—
138C2×C42—22no—
139C2×C42—24no—
140C2×C42—216no—
141C2×C42M1618yescomplemented
142C2×C42C22×C414yes—
143C22⋊C8—28no—
144C22⋊C8—44no—
145C22⋊C8—48no—
146C24—24no—
147C24—24no—
148C24—22no—
149C24—22no—
150C24#9912yes—
151C24#9911yescharacteristic
152C24#9911yescharacteristic
153C24C2×C22⋊C412yes—
154C24C2×C22⋊C411yescharacteristic
155C22×C4—216no—
156C22×C4—216no—
157C22×C4—28no—
158C22×C4—264no—
159C22×C4—216no—
160C22×C4—28no—
161C22×C4—216no—
162C22×C4—28no—
163C22×C4—28no—
164C22×C4—24no—
165C22×C4—22no—
166C22×C4—28no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—22no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—24no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—22no—
188C22×C4—24no—
189C22×C4—48no—
190C22×C4—48no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—48no—
194C22×C4—416no—
195C22×C4—48no—
196C22×C4—44no—
197C22×C4—42no—
198C22×C4—44no—
199C22×C4—42no—
200C22×C4—84no—
201C22×C4—88no—
202C22×C4—88no—
203C22×C4—84no—
204C22×C4—84no—
205C22×C4#10114yes—
206C22×C4C2×C22⋊C411yescharacteristic
207C22×C4C2×C22⋊C412yes—
208C22×C4C2×C22⋊C412yes—
209C22×C4C2×C22⋊C414yes—
210C22×C4C2×M1614yes—
211C22×C4C23×C411yescharacteristic
212C22×C4C22×D412yes—
213M16—816no—
214C2×C8—48no—
215C2×C8—88no—
216C4⋊C4—42no—
217C4⋊C4—44no—
218C4⋊C4—42no—
219C4⋊C4—41no—
220C4⋊C4—42no—
221C4⋊C4—44no—
222C4⋊C4—41no—
223C4⋊C4—44no—
224C4⋊C4—416no—
225C4⋊C4—432no—
226C4⋊C4—416no—
227C4⋊C4—44no—
228C4⋊C4—48no—
229C42—216no—
230C42—24no—
231C42—28no—
232C42C2×M1618yescomplemented
233C23—216no—
234C23—216no—
235C23—22no—
236C23—22no—
237C23—24no—
238C23—22no—
239C23—24no—
240C23—22no—
241C23—24no—
242C23—22no—
243C23—24no—
244C23—22no—
245C23—24no—
246C23—22no—
247C23—22no—
248C23#35314yes—
249C23#35614yes—
250C23#40712yes—
251C23C2×#9912yes—
252C23C2×#9912yes—
253C23C2×#9911yescharacteristic
254C23C2×#9911yescharacteristic
255C23C2×#10112yes—
256C23C22×C22⋊C411yescharacteristic
257C2×C4—28no—
258C2×C4—264no—
259C2×C4—28no—
260C2×C4—232no—
261C2×C4—28no—
262C2×C4—28no—
263C2×C4—28no—
264C2×C4—28no—
265C2×C4—22no—
266C2×C4—28no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—24no—
276C2×C4—22no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—44no—
280C2×C4—48no—
281C2×C4—48no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—42no—
286C2×C4—48no—
287C2×C4—44no—
288C2×C4—82no—
289C2×C4—84no—
290C2×C4—84no—
291C2×C4—82no—
292C2×C4—88no—
293C2×C4—84no—
294C2×C4—88no—
295C2×C4—84no—
296C2×C4#40712yes—
297C2×C4C2×#10114yes—
298C2×C4C22×M1612yes—
299C8—88no—
300C22—28no—
301C22—28no—
302C22—216no—
303C22—21no—
304C22—22no—
305C22—21no—
306C22—21no—
307C22—22no—
308C22—21no—
309C22—22no—
310C22—22no—
311C22—24no—
312C22—28no—
313C22—28no—
314C22—24no—
315C22#146312yes—
316C22#146312yes—
317C22C2×#35314yes—
318C22C2×#35312yes—
319C22C2×#35614yes—
320C22C2×#35612yes—
321C22C2×#40711yescharacteristic
322C22C22×#9911yescharacteristic
323C22C22×#10111yescharacteristic
324C4—216no—
325C4—22no—
326C4—22no—
327C4—24no—
328C4—42no—
329C4—42no—
330C4—44no—
331C4—84no—
332C2—21no—
333C2—21no—
334C2—24no—
335C2—24no—
336C2—22no—
337C2#1151214yesdirect product factor
338C2C2×#146311yescharacteristic
339C2C22×#35311yescharacteristic
340C2C22×#35611yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11512C2direct (N × H)
4. C2×#1325C2semidirect
20. #1325C22semidirect
141. C2×C42M16semidirect
232. C42C2×M16semidirect
337. C2#11512direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#3842
C2×#14632
#115122
C2×#13252