Finite Group Databasebeta

← #480121

Subgroups — #480121

Order 512.

1008 conjugacy classes of subgroups.

The subgroup lattice has 333 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34209C211yesmaximal, characteristic
2#17352C212yesmaximal
3#11482C214yesmaximal, complemented
4#11478C214yesmaximal, complemented
5#11447C212yesmaximal
6#11434C212yesmaximal
7C22×C4⋊Q8C411yescharacteristic
8C2×#519C411yescharacteristic
9C2×#519C2211yescharacteristic
10#2982—22no—
11#2365—22no—
12#2365—22no—
13#2344—24no—
14#2327C412yes—
15#2327C2212yes—
16#2310C411yescharacteristic
17#2310C2211yescharacteristic
18C4×C4⋊Q8C412yes—
19C4×C4⋊Q8C2212yes—
20#2285C2211yescharacteristic
21C4×#117C411yescharacteristic
22#1461—22no—
23#1461C2218yes—
24#1458—24no—
25#1458C2214yes—
26#1325C2214yes—
27#1324C2214yes—
28#1319C2218yes—
29C23×Q8C2×C411yescharacteristic
30C23×Q8C2×C411yescharacteristic
31C2×C4⋊Q8—22no—
32C2×C4⋊Q8—24no—
33C2×C4⋊Q8C818yescomplemented
34C2×C4⋊Q8C2×C412yes—
35C2×C4⋊Q8C2×C412yes—
36#519—44no—
37#519—44no—
38C2×C4×Q8—21no—
39C2×C4×Q8—21no—
40C2×C4×Q8C2×C412yes—
41C2×C4×Q8C2×C412yes—
42C2×C4×Q8C2×C412yes—
43C2×C4×Q8D412yes—
44C2×C4×Q8D412yes—
45C2×C4×Q8C2312yes—
46C2×#117—22no—
47C2×#117C2×C411yescharacteristic
48C2×#117C2×C411yescharacteristic
49C2×#117C2×C411yescharacteristic
50C2×#117D414yes—
51C2×#117C2311yescharacteristic
52C22×C4⋊C4—22no—
53C22×C42C2×C411yescharacteristic
54C8×Q8—48no—
55#431—28no—
56#431—24no—
57#391—22no—
58#391—22no—
59#391C2×C412yes—
60#391C2×C412yes—
61#391C2×C412yes—
62#391C2312yes—
63#389—24no—
64#389—24no—
65#387—24no—
66#387—24no—
67#387—24no—
68#387—24no—
69#384C2×C411yescharacteristic
70#384C2311yescharacteristic
71#382C2×C411yescharacteristic
72#382C2311yescharacteristic
73#381—22no—
74#381—22no—
75#380—22no—
76C4×C4⋊C4—21no—
77C4×C4⋊C4—21no—
78C4×C4⋊C4—24no—
79C4×C4⋊C4—24no—
80C4×C4⋊C4—22no—
81C4×C22⋊C4—22no—
82#376C2×C412yes—
83C43C2×C412yes—
84#324—28no—
85#324C2318yes—
86#323—24no—
87C22×Q8—21no—
88C22×Q8—22no—
89C22×Q8—42no—
90C22×Q8—42no—
91C22×Q8—42no—
92C22×Q8—42no—
93C22×Q8C22⋊C412yes—
94C22×Q8C2×C814yes—
95C22×Q8C2×C814yes—
96C22×Q8M1614yes—
97C22×Q8C22×C412yes—
98C22×Q8C22×C412yes—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4C22⋊C412yes—
102C23×C4C22×C411yescharacteristic
103C4⋊Q8—216no—
104C4⋊Q8—24no—
105C4⋊Q8—24no—
106C4⋊Q8C2×C818yes—
107C4⋊Q8C2×C818yes—
108C4×Q8—28no—
109C4×Q8—44no—
110C4×Q8—44no—
111C4×Q8—44no—
112C4×Q8—44no—
113C4×Q8—44no—
114C4×Q8—44no—
115C4×Q8—44no—
116#117—24no—
117#117—42no—
118#117—48no—
119#117—48no—
120#117—42no—
121#117—42no—
122C2×C4⋊C4—216no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4C22⋊C411yescharacteristic
131C2×C4⋊C4C22⋊C414yes—
132C2×C4⋊C4C22⋊C411yescharacteristic
133C2×C4⋊C4C22×C411yescharacteristic
134C2×C4⋊C4C22×C411yescharacteristic
135C2×C4⋊C4C22×C411yescharacteristic
136C2×C4⋊C4C2×D414yes—
137C2×C4⋊C4C2411yescharacteristic
138C2×C22⋊C4—21no—
139C2×C22⋊C4—21no—
140C2×C22⋊C4C22⋊C412yes—
141C2×C22⋊C4C2×D412yes—
142C2×C42—21no—
143C2×C42—22no—
144C2×C42—22no—
145C2×C42—22no—
146C2×C42—21no—
147C2×C42—22no—
148C2×C42—22no—
149C2×C42C22⋊C414yes—
150C2×C42C2×C814yes—
151C2×C42C22×C412yes—
152C2×C42C22×C411yescharacteristic
153C2×C42C22×C411yescharacteristic
154C2×C42C22×C412yes—
155C2×C42C2×D414yes—
156C2×C42C2×D412yes—
157C4⋊C8—48no—
158C4⋊C8—44no—
159C22⋊C8—28no—
160C4×C8—48no—
161C4×C8—44no—
162#95—22no—
163#95—22no—
164#95—22no—
165#95—24no—
166#95—24no—
167#95—24no—
168#95—22no—
169#95—24no—
170C24C2×C22⋊C411yescharacteristic
171C2×Q8—22no—
172C2×Q8—24no—
173C2×Q8—24no—
174C2×Q8—24no—
175C2×Q8—42no—
176C2×Q8—42no—
177C2×Q8—48no—
178C2×Q8—48no—
179C2×Q8—48no—
180C2×Q8—48no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—42no—
185C2×Q8C22⋊C818yes—
186C2×Q8C22×C814yes—
187C2×Q8C22×C814yes—
188C2×Q8C2×M1614yes—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—22no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4C22⋊C818yes—
206C22×C4C2×C22⋊C414yes—
207C22×C4C2×C22⋊C412yes—
208C22×C4C2×C22⋊C412yes—
209C22×C4C2×C22⋊C412yes—
210C22×C4C2×C22⋊C411yescharacteristic
211C22×C4C22×C812yes—
212C22×C4C2×M1612yes—
213C22×C4C23×C411yescharacteristic
214C22×C4C22×D412yes—
215C2×C8—48no—
216C4⋊C4—216no—
217C4⋊C4—216no—
218C4⋊C4—22no—
219C4⋊C4—28no—
220C4⋊C4—41no—
221C4⋊C4—44no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—44no—
226C4⋊C4—44no—
227C4⋊C4—44no—
228C4⋊C4—44no—
229C4⋊C4—42no—
230C4⋊C4#9914yes—
231C22⋊C4—24no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C42—21no—
236C42—21no—
237C42—24no—
238C42—28no—
239C42—24no—
240C42—24no—
241C42—44no—
242C42—42no—
243C42—44no—
244C42—44no—
245C42—44no—
246C42—44no—
247C42—42no—
248C42C22×C812yes—
249C42C22×C812yes—
250C23—21no—
251C23—21no—
252C23—41no—
253C23—41no—
254C23C2×C22⋊C812yes—
255C23C22×C22⋊C411yescharacteristic
256Q8—216no—
257Q8—44no—
258Q8—44no—
259Q8—44no—
260Q8—44no—
261Q8—48no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—24no—
267C2×C4—21no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—22no—
272C2×C4—24no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—42no—
277C2×C4—48no—
278C2×C4—48no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—48no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4C2×C22⋊C818yes—
295C2×C4C2×C22⋊C812yes—
296C2×C4C2×#9914yes—
297C2×C4C2×#9912yes—
298C2×C4#41312yes—
299C2×C4C23×C811yescharacteristic
300C2×C4C22×M1611yescharacteristic
301C8—88no—
302C22—21no—
303C22—21no—
304C22—22no—
305C22—42no—
306C22—42no—
307C22—41no—
308C22—41no—
309C22C22×C22⋊C811yescharacteristic
310C22C22×#9911yescharacteristic
311C22C2×#41311yescharacteristic
312C4—24no—
313C4—24no—
314C4—21no—
315C4—21no—
316C4—28no—
317C4—42no—
318C4—42no—
319C4—42no—
320C4—42no—
321C4—44no—
322C4—82no—
323C4—81no—
324C4—81no—
325C4#145914yes—
326C2—21no—
327C2—21no—
328C2—42no—
329C2C2×#145911yescharacteristic
330C2#3336011yescharacteristic
331C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC2×C4C23×C8
Frattini, cosocleC2×C4⋊C4C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#519 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11482C2semidirect
4. #11478C2semidirect
33. C2×C4⋊Q8C8semidirect

Maximal subgroups

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

subgroupindex
#342092
#173522
#114822
#114782
#114472
#114342