Finite Group Databasebeta

← #480307

Subgroups — #480307

Order 512.

1068 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34183C211yesmaximal, characteristic
2#17370C212yesmaximal
3#11514C214yesmaximal, complemented
4#11512C214yesmaximal, complemented
5#11450C212yesmaximal, complemented
6#11447C212yesmaximal, complemented
7C2×#520C411yescharacteristic
8C2×#520C2211yescharacteristic
9C2×C4×#41C411yescharacteristic
10#2975—22no—
11Q8×C4⋊C4—22no—
12D4×C4⋊C4—24no—
13D4×C4⋊C4—22no—
14#2317C412yes—
15#2317C411yescharacteristic
16#2317C2211yescharacteristic
17#2317C2212yes—
18#2285C412yes—
19#2285C2211yescharacteristic
20C2×#384C411yescharacteristic
21C2×#384C2212yes—
22#1465—24no—
23#1463C2218yes—
24#1461—22no—
25#1458C2214yes—
26#1325C2214yes—
27#1325C2218yes—
28#1324C2214yes—
29C22×#41C2×C411yescharacteristic
30#520—44no—
31#520—44no—
32C4×#41—24no—
33C4×#41—24no—
34C2×C4×Q8—21no—
35C2×C4×Q8—21no—
36C2×C4×Q8D412yes—
37C2×C4×D4—22no—
38C2×C4×D4—21no—
39C2×C4×D4—22no—
40C2×C4×D4—21no—
41C2×C4×D4D414yes—
42C2×C4×D4D412yes—
43C2×#117C2×C411yescharacteristic
44C2×#117C2×C412yes—
45C2×#117C2×C412yes—
46C2×#117C2×C412yes—
47C2×#117C2×C411yescharacteristic
48C2×#117C2311yescharacteristic
49C22×C4⋊C4—22no—
50C22×C4⋊C4—24no—
51C22×C4⋊C4C2×C412yes—
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C42C2×C412yes—
56C22×C42C2×C411yescharacteristic
57#436—48no—
58#431—24no—
59#407—28no—
60#389—22no—
61#389—24no—
62#389—24no—
63#389—22no—
64#385—22no—
65#385—24no—
66#385—24no—
67#385—22no—
68#384—22no—
69#384—24no—
70#384C2×C412yes—
71#384C2×C412yes—
72#384C2×C412yes—
73#384C2×C411yescharacteristic
74#384C2312yes—
75#384C2311yescharacteristic
76#382C2×C411yescharacteristic
77#382C2×C412yes—
78#382C2×C412yes—
79#382C2311yescharacteristic
80#380—24no—
81#380—22no—
82C4×C4⋊C4—24no—
83C4×C4⋊C4—22no—
84#324—28no—
85#324C2318yes—
86#323—24no—
87C2×#41—24no—
88C2×#41—24no—
89C2×#41—22no—
90C2×#41—22no—
91C22×Q8C22⋊C411yescharacteristic
92C22×D4C22⋊C412yes—
93C22×D4C22⋊C411yescharacteristic
94C23×C4—22no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—22no—
98C23×C4C22⋊C412yes—
99C23×C4C22×C412yes—
100C23×C4C22×C411yescharacteristic
101C23×C4C2×D414yes—
102C2×M16—48no—
103C4×Q8—28no—
104C4×Q8—44no—
105C4×Q8—44no—
106C4×D4—216no—
107C4×D4—28no—
108C4×D4—28no—
109C4×D4—48no—
110C4×D4—48no—
111C4×D4—44no—
112#117—22no—
113#117—24no—
114#117—24no—
115#117—42no—
116#117—44no—
117#117—44no—
118#117—42no—
119#117M1614yescomplemented
120C2×C4⋊C4—22no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—44no—
132C2×C4⋊C4—44no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—88no—
135C2×C4⋊C4—84no—
136C2×C4⋊C4C22⋊C412yes—
137C2×C4⋊C4C22×C411yescharacteristic
138C2×C4⋊C4C22×C412yes—
139C2×C4⋊C4C22×C411yescharacteristic
140C2×C4⋊C4C22×C412yes—
141C2×C4⋊C4C22×C411yescharacteristic
142C2×C4⋊C4C2×D414yes—
143C2×C4⋊C4C2411yescharacteristic
144C2×C22⋊C4—22no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4C22⋊C414yes—
149C2×C22⋊C4C2×D412yes—
150C2×C42—22no—
151C2×C42—21no—
152C2×C42—22no—
153C2×C42—24no—
154C2×C42—24no—
155C2×C42—22no—
156C2×C42—21no—
157C2×C42C22⋊C414yes—
158C2×C42M1614yescomplemented
159C2×C42C22×C412yes—
160C2×C42C22×C411yescharacteristic
161C2×C42C22×C411yescharacteristic
162C2×C42C2×D412yes—
163C4⋊C8—44no—
164C22⋊C8—28no—
165C22⋊C8—48no—
166C4×C8—44no—
167#95—24no—
168#95—24no—
169#95—22no—
170#95—22no—
171C24C2×C22⋊C411yescharacteristic
172C24C2×C22⋊C412yes—
173#41—216no—
174#41—216no—
175C2×Q8—22no—
176C2×Q8—22no—
177C2×Q8#10114yes—
178C2×D4—24no—
179C2×D4—24no—
180C2×D4—22no—
181C2×D4—24no—
182C2×D4—22no—
183C2×D4#9914yes—
184C22×C4—21no—
185C22×C4—24no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—24no—
190C22×C4—22no—
191C22×C4—28no—
192C22×C4—22no—
193C22×C4—21no—
194C22×C4—24no—
195C22×C4—28no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—42no—
200C22×C4—44no—
201C22×C4—42no—
202C22×C4—44no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—44no—
206C22×C4—84no—
207C22×C4—84no—
208C22×C4—82no—
209C22×C4—82no—
210C22×C4C2×C22⋊C412yes—
211C22×C4C2×C22⋊C412yes—
212C22×C4C2×C22⋊C414yes—
213C22×C4C2×C22⋊C411yescharacteristic
214C22×C4C2×M1612yes—
215C22×C4C2×M1612yes—
216C22×C4C23×C411yescharacteristic
217C22×C4C22×D412yes—
218M16—168no—
219C2×C8—48no—
220C4⋊C4—28no—
221C4⋊C4—24no—
222C4⋊C4—24no—
223C4⋊C4—42no—
224C4⋊C4—41no—
225C4⋊C4—44no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—42no—
229C4⋊C4—41no—
230C4⋊C4—44no—
231C4⋊C4—44no—
232C4⋊C4—48no—
233C4⋊C4—44no—
234C4⋊C4—42no—
235C4⋊C4—84no—
236C4⋊C4—84no—
237C4⋊C4—84no—
238C22⋊C4—28no—
239C22⋊C4—24no—
240C22⋊C4—28no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—44no—
244C42—28no—
245C42—21no—
246C42—21no—
247C42—22no—
248C42—24no—
249C42—24no—
250C42—44no—
251C42—44no—
252C42—42no—
253C42C2×M1612yes—
254C42C2×M1614yes—
255C42C2×M1612yes—
256C23—22no—
257C23—21no—
258C23—21no—
259C23—22no—
260C23—42no—
261C23—42no—
262C23—41no—
263C23—41no—
264C23#40712yes—
265C23C2×#9914yes—
266C23C22×C22⋊C411yescharacteristic
267Q8—28no—
268D4—216no—
269D4—28no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—28no—
273C2×C4—28no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—48no—
287C2×C4—416no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—44no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—88no—
302C2×C4—162no—
303C2×C4—164no—
304C2×C4—162no—
305C2×C4#40712yes—
306C2×C4C2×#9912yes—
307C2×C4C2×#10114yes—
308C2×C4C2×#10112yes—
309C2×C4C22×M1611yescharacteristic
310C2×C4C22×M1611yescharacteristic
311C8—88no—
312C22—21no—
313C22—21no—
314C22—24no—
315C22—24no—
316C22—22no—
317C22—24no—
318C22—42no—
319C22—44no—
320C22—44no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22C2×#40711yescharacteristic
325C22C22×#9911yescharacteristic
326C22C22×#10111yescharacteristic
327C4—24no—
328C4—21no—
329C4—21no—
330C4—24no—
331C4—44no—
332C4—42no—
333C4—42no—
334C4—44no—
335C4—42no—
336C4—81no—
337C4—82no—
338C4—81no—
339C4#146314yes—
340C2—21no—
341C2—21no—
342C2—44no—
343C2—42no—
344C2C2×#146311yescharacteristic
345C2#3335911yescharacteristic
346C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#520 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11514C2semidirect
4. #11512C2semidirect
5. #11450C2semidirect
6. #11447C2semidirect
119. #117M16semidirect
158. C2×C42M16semidirect

Maximal subgroups

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

subgroupindex
#341832
#173702
#115142
#115122
#114502
#114472