Finite Group Databasebeta

← #480779

Subgroups — #480779

Order 512.

797 conjugacy classes of subgroups.

The subgroup lattice has 347 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35668C211yesmaximal, characteristic
2#17371C212yesmaximal
3#11573C214yesmaximal, complemented
4#11571C214yesmaximal, complemented
5#11515C212yesmaximal
6#11513C212yesmaximal
7C2×#544C411yescharacteristic
8C2×#519C2211yescharacteristic
9#2983—22no—
10#2568C2212yes—
11#2567C412yes—
12#2565C412yes—
13#2565C2212yes—
14#2453—22no—
15#2451—22no—
16#2438—22no—
17#2438—22no—
18#2317C2211yescharacteristic
19#2310C411yescharacteristic
20#2296C411yescharacteristic
21C2×#391C411yescharacteristic
22#2285C2211yescharacteristic
23#1467—22no—
24#1467C2214yes—
25#1464—22no—
26#1464C2214yes—
27#1461C2214yes—
28#1458—22no—
29#1325C2214yes—
30#1324C2218yes—
31#1324C2214yes—
32C23×Q8C2×C411yescharacteristic
33#544—44no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8C2×C412yes—
36C2×#125—21no—
37C2×#125C2×C412yes—
38C2×C22⋊Q8—22no—
39#519—44no—
40C2×C4×Q8C2×C412yes—
41C2×C4×Q8D412yes—
42C2×C4×Q8D412yes—
43C2×C4×Q8C2312yes—
44C2×#117C2×C411yescharacteristic
45C2×#117C2×C411yescharacteristic
46C2×#117C2×C411yescharacteristic
47C2×#117D412yes—
48C2×#117D412yes—
49C2×#117C2311yescharacteristic
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C42C2×C411yescharacteristic
52#446—48no—
53#432—24no—
54#432—24no—
55#431—24no—
56#400—22no—
57#400—22no—
58#398—22no—
59#398—22no—
60#398—22no—
61#398—22no—
62#395—22no—
63#395—22no—
64#391—22no—
65#391—22no—
66#391—22no—
67#391—22no—
68#391C2×C412yes—
69#391C2×C412yes—
70#391C2×C412yes—
71#391C2×C412yes—
72#389—21no—
73#389—21no—
74#389—22no—
75#389—22no—
76#389—22no—
77#387—22no—
78#387—22no—
79#387—22no—
80#387—22no—
81#385—22no—
82#385—22no—
83#384C2×C412yes—
84#384C2×C411yescharacteristic
85#384C2311yescharacteristic
86#382C2×C411yescharacteristic
87#382C2312yes—
88#382C2311yescharacteristic
89#381—21no—
90#381—21no—
91#380—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4C2×C412yes—
94C4×C4⋊C4C2×C412yes—
95C2×#95—21no—
96C2×#95—21no—
97#324—24no—
98#324—24no—
99#324C2318yes—
100#323—24no—
101C22×Q8—24no—
102C22×Q8—22no—
103C22×Q8—42no—
104C22×Q8—42no—
105C22×Q8C22⋊C412yes—
106C22×Q8C22×C412yes—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4C22⋊C411yescharacteristic
110C23×C4C22⋊C411yescharacteristic
111C23×C4C22×C411yescharacteristic
112C4⋊Q8—44no—
113C4⋊Q8—44no—
114#125—44no—
115#125—44no—
116C22⋊Q8—48no—
117C4×Q8—28no—
118C4×Q8—44no—
119C4×Q8—44no—
120C4×Q8—44no—
121#117—24no—
122#117—42no—
123#117—42no—
124#117—44no—
125#117—44no—
126#117—42no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4C22⋊C411yescharacteristic
140C2×C4⋊C4C22⋊C412yes—
141C2×C4⋊C4C22⋊C412yes—
142C2×C4⋊C4C22⋊C411yescharacteristic
143C2×C4⋊C4C22×C412yes—
144C2×C4⋊C4C22×C411yescharacteristic
145C2×C4⋊C4C22×C411yescharacteristic
146C2×C4⋊C4C22×C411yescharacteristic
147C2×C4⋊C4C2×D412yes—
148C2×C4⋊C4C2×D412yes—
149C2×C4⋊C4C2411yescharacteristic
150C2×C22⋊C4—21no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4C22⋊C412yes—
153C2×C22⋊C4C2×D412yes—
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—42no—
157C2×C42C22⋊C412yes—
158C2×C42C22⋊C412yes—
159C2×C42C22×C412yes—
160C2×C42C22×C411yescharacteristic
161C2×C42C22×C411yescharacteristic
162C2×C42C22×C412yes—
163C2×C42C2×D412yes—
164C2×C42C2×D412yes—
165C2×C42C2×D412yes—
166C4⋊C8—44no—
167C4⋊C8—44no—
168C4⋊C8—44no—
169C22⋊C8—28no—
170#97—44no—
171#97—44no—
172C4×C8—44no—
173#95—22no—
174#95—22no—
175#95—22no—
176#95—22no—
177#95—22no—
178#95—22no—
179#95—22no—
180#95—22no—
181#95—22no—
182#95—22no—
183#95—22no—
184#95—22no—
185#95—22no—
186#95—22no—
187C24C2×C22⋊C411yescharacteristic
188C2×Q8—24no—
189C2×Q8—24no—
190C2×Q8—24no—
191C2×Q8—24no—
192C2×Q8—42no—
193C2×Q8—42no—
194C2×Q8—416no—
195C2×Q8—44no—
196C2×Q8—82no—
197C2×Q8—82no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—81no—
216C22×C4—81no—
217C22×C4—81no—
218C22×C4#10114yes—
219C22×C4C2×C22⋊C412yes—
220C22×C4C2×C22⋊C411yescharacteristic
221C22×C4C2×C22⋊C412yes—
222C22×C4C2×C22⋊C412yes—
223C22×C4C2×C22⋊C412yes—
224C22×C4C2×C22⋊C412yes—
225C22×C4C23×C411yescharacteristic
226C22×C4C22×D412yes—
227C2×C8—48no—
228C4⋊C4—22no—
229C4⋊C4—24no—
230C4⋊C4—24no—
231C4⋊C4—41no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—41no—
235C4⋊C4—42no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—44no—
239C4⋊C4—44no—
240C4⋊C4—42no—
241C4⋊C4—44no—
242C4⋊C4—44no—
243C4⋊C4—42no—
244C4⋊C4—44no—
245C4⋊C4—82no—
246C4⋊C4—82no—
247C4⋊C4—82no—
248C4⋊C4—82no—
249C4⋊C4#9914yes—
250C22⋊C4—24no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C42—24no—
255C42—24no—
256C42—24no—
257C42—24no—
258C42—22no—
259C42—22no—
260C42—24no—
261C42—42no—
262C42—42no—
263C42—42no—
264C42—42no—
265C23—21no—
266C23—21no—
267C23—41no—
268C23—41no—
269C23C2×#10112yes—
270C23C22×C22⋊C411yescharacteristic
271Q8—48no—
272Q8—48no—
273Q8—164no—
274C2×C4—28no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—42no—
289C2×C4—44no—
290C2×C4—41no—
291C2×C4—44no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—84no—
310C2×C4—162no—
311C2×C4C2×#9914yes—
312C2×C4C2×#9912yes—
313C2×C4C2×#10114yes—
314C2×C4#56812yes—
315C8—88no—
316C22—21no—
317C22—21no—
318C22—22no—
319C22—42no—
320C22—42no—
321C22—41no—
322C22—41no—
323C22#287911yescharacteristic
324C22C22×#9911yescharacteristic
325C22C22×#10111yescharacteristic
326C4—24no—
327C4—22no—
328C4—22no—
329C4—24no—
330C4—42no—
331C4—42no—
332C4—42no—
333C4—42no—
334C4—42no—
335C4—82no—
336C4—81no—
337C4—81no—
338C4—82no—
339C4—81no—
340C2—21no—
341C2—21no—
342C2—42no—
343C2#1739711yescharacteristic
344C2#3336011yescharacteristic
345C2#3337011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×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. #11573C2semidirect
4. #11571C2semidirect

Maximal subgroups

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

subgroupindex
#356682
#173712
#115732
#115712
#115152
#115132