Finite Group Databasebeta

← #6408076

Subgroups — #6408076

Order 512.

1216 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#564C211yesmaximal, characteristic
2#23372C212yesmaximal
3#23371C214yesmaximal
4#23370C214yesmaximal
5#23370C214yesmaximal
6#23369C211yesmaximal, characteristic
7#21675C214yesmaximal
8#21674C214yesmaximal
9#21674C214yesmaximal
10#21673C212yesmaximal
11C8×#125C211yesmaximal, characteristic
12C2×#564C411yescharacteristic
13#2382C414yes—
14#2382C414yes—
15#2382C414yes—
16#2382C412yes—
17#2382C411yescharacteristic
18C4×#125C2211yescharacteristic
19C4×#125C2214yes—
20C4×#125C2214yes—
21C4×#125C2214yes—
22C4×#125C2212yes—
23#1940C2214yes—
24#1940C2214yes—
25#1940C2214yes—
26#1940C2214yes—
27#1939C2214yes—
28#1939C2212yes—
29#1939C2212yes—
30#1939C2214yes—
31#1939C2212yes—
32#1917C2214yes—
33#1917C2214yes—
34#1917C2214yes—
35#1917C2214yes—
36#1917C2214yes—
37#1917C2214yes—
38#1917C2214yes—
39#1917C2214yes—
40#1916C2214yes—
41#1916C2214yes—
42#1916C2214yes—
43#1916C2214yes—
44#1916C2214yes—
45#1916C2214yes—
46#1916C2214yes—
47#1831C2212yes—
48#1831C2214yes—
49#1831C2214yes—
50#1831C2214yes—
51#1831C2211yescharacteristic
52#1771C2214yes—
53#1771C2214yes—
54#1771C2214yes—
55#1771C2214yes—
56#1770C2212yes—
57#1770C2212yes—
58#1770C2214yes—
59C8×C4⋊C4C2212yes—
60C8×C4⋊C4C2214yes—
61C4×#97C2214yes—
62C42×C8C2211yescharacteristic
63#564—28no—
64C2×#125C2×C414yes—
65C2×#125C2×C414yes—
66C2×#125C2×C414yes—
67C2×#125C2×C412yes—
68C2×#125C2×C411yescharacteristic
69C2×C4⋊C8C2312yes—
70C2×C4⋊C8C2314yes—
71C2×C4⋊C8C2314yes—
72C2×C4⋊C8C2314yes—
73C2×C4⋊C8C2314yes—
74C2×C4⋊C8C2314yes—
75C2×C4⋊C8C2312yes—
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2314yes—
78C2×#97C2314yes—
79C2×#97C2314yes—
80C2×#97C2314yes—
81C2×#97C2314yes—
82C2×C4×C8C2312yes—
83C2×C4×C8C2311yescharacteristic
84C2×C4×C8C2314yes—
85C2×C4×C8C2311yescharacteristic
86C2×C4×C8C2312yes—
87C2×C4×C8C2314yes—
88#389C2×C414yes—
89#389C2×C414yes—
90#389C2×C414yes—
91#389C2×C414yes—
92#389C2×C414yes—
93#389C2×C412yes—
94#389C2×C412yes—
95#389C2×C412yes—
96#389C2×C414yes—
97#387C2×C414yes—
98#387C2×C414yes—
99#387C2×C414yes—
100#387C2×C414yes—
101#387C2×C414yes—
102#387C2×C414yes—
103#387C2×C414yes—
104#387C2×C414yes—
105#387C2×C414yes—
106#387C2×C414yes—
107#387C2×C414yes—
108#387C2×C414yes—
109#387C2×C414yes—
110#387C2×C414yes—
111#387C2×C414yes—
112#382C2×C414yes—
113#382C2×C414yes—
114#382C2×C414yes—
115#382C2×C411yescharacteristic
116#382C2×C412yes—
117C4×C4⋊C4C2312yes—
118C4×C4⋊C4C2314yes—
119C4×C4⋊C4C2314yes—
120C4×C4⋊C4C2314yes—
121C4×C4⋊C4C2314yes—
122C4×C4⋊C4C2314yes—
123C4×C4⋊C4C2312yes—
124C4×C4⋊C4C2314yes—
125C4×C4⋊C4C2312yes—
126C43C2311yescharacteristic
127C43C2314yes—
128#336C2314yes—
129#336C2314yes—
130#336C2314yes—
131#336C2314yes—
132#336C2314yes—
133#336C2314yes—
134#336C2314yes—
135#336C2314yes—
136#336C2314yes—
137#336C2314yes—
138#336C2314yes—
139#336C2314yes—
140#336C2314yes—
141#336C2314yes—
142#336C2314yes—
143C22×C8C2411yescharacteristic
144C22×C8C2412yes—
145C22×C8C2414yes—
146C22×C8C2414yes—
147C22×C8C2414yes—
148C22×C8C2411yescharacteristic
149#125—216no—
150#125—216no—
151#125—216no—
152#125—28no—
153#125M1618yes—
154C2×C4⋊C4C22×C414yes—
155C2×C4⋊C4C22×C414yes—
156C2×C4⋊C4C22×C414yes—
157C2×C4⋊C4C22×C414yes—
158C2×C4⋊C4C22×C414yes—
159C2×C4⋊C4C22×C412yes—
160C2×C4⋊C4C22×C412yes—
161C2×C4⋊C4C22×C414yes—
162C2×C4⋊C4C22×C414yes—
163C2×C4⋊C4C22×C414yes—
164C2×C4⋊C4C22×C414yes—
165C2×C4⋊C4C22×C414yes—
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C22×C414yes—
168C2×C4⋊C4C22×C412yes—
169C2×C4⋊C4C22×C412yes—
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C22×C414yes—
172C2×C42C22×C414yes—
173C2×C42C22×C414yes—
174C2×C42C22×C414yes—
175C2×C42C22×C411yescharacteristic
176C2×C42C22×C412yes—
177C2×C42C22×C414yes—
178C2×C42C22×C411yescharacteristic
179C2×C42C2414yes—
180C2×C42C2411yescharacteristic
181C2×C42C2412yes—
182C2×C42C2414yes—
183C2×C42C2414yes—
184C4⋊C8—42no—
185C4⋊C8—44no—
186C4⋊C8—44no—
187C4⋊C8—42no—
188C4⋊C8—42no—
189C4⋊C8—44no—
190C4⋊C8—44no—
191C4⋊C8—44no—
192C4⋊C8—44no—
193#97—28no—
194#97—28no—
195#97—44no—
196#97—44no—
197C4×C8—24no—
198C4×C8—28no—
199C4×C8—22no—
200C4×C8—41no—
201C4×C8—42no—
202C4×C8—44no—
203#95C22×C414yes—
204#95C22×C414yes—
205#95C22×C414yes—
206#95C22×C414yes—
207#95C22×C414yes—
208#95C22×C414yes—
209#95C22×C414yes—
210#95C22×C414yes—
211#95C22×C414yes—
212#95C22×C414yes—
213#95C22×C414yes—
214#95C22×C414yes—
215#95C22×C414yes—
216#95C22×C414yes—
217#95C22×C414yes—
218C22×C4C23×C411yescharacteristic
219C22×C4C23×C412yes—
220C22×C4C23×C414yes—
221C22×C4C23×C414yes—
222C22×C4C23×C414yes—
223C22×C4C23×C414yes—
224C22×C4C23×C414yes—
225C22×C4C23×C414yes—
226C22×C4C23×C411yescharacteristic
227C22×C4C23×C412yes—
228C22×C4C2511yescharacteristic
229C2×C8—21no—
230C2×C8—22no—
231C2×C8—24no—
232C2×C8—24no—
233C2×C8—24no—
234C2×C8—22no—
235C2×C8—22no—
236C2×C8—22no—
237C2×C8—24no—
238C2×C8—24no—
239C2×C8—24no—
240C2×C8—24no—
241C2×C8—24no—
242C2×C8—24no—
243C2×C82−1+414yes—
244C2×C82−1+412yes—
245C4⋊C4—28no—
246C4⋊C4—28no—
247C4⋊C4—28no—
248C4⋊C4—24no—
249C4⋊C4—28no—
250C4⋊C4—28no—
251C4⋊C4—24no—
252C4⋊C4—42no—
253C4⋊C4—44no—
254C4⋊C4—42no—
255C4⋊C4—44no—
256C4⋊C4—42no—
257C4⋊C4—44no—
258C4⋊C4—44no—
259C4⋊C4—44no—
260C4⋊C4—44no—
261C4⋊C4C2×M1618yes—
262C4⋊C4C2×M16116yes—
263C42—28no—
264C42—21no—
265C42—22no—
266C42—24no—
267C42—24no—
268C42—24no—
269C42—21no—
270C42—22no—
271C42—24no—
272C42—24no—
273C42—24no—
274C42—41no—
275C42—42no—
276C42—44no—
277C42—44no—
278C42—44no—
279C42C2×M1614yes—
280C23C24×C411yescharacteristic
281C2×C4—21no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—24no—
286C2×C4—22no—
287C2×C4—24no—
288C2×C4—21no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—22no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—22no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4C22×M1618yes—
308C2×C4C22×M1612yes—
309C2×C4C22×M1614yes—
310C2×C4C2×2−1+411yescharacteristic
311C2×C4C2×2−1+412yes—
312C2×C4#58511yescharacteristic
313C2×C4#58512yes—
314C8—42no—
315C8—41no—
316C8—44no—
317C8—44no—
318C8—44no—
319C8#56414yescomplemented
320C22C4×2−1+411yescharacteristic
321C22C4×2−1+412yes—
322C22C23×M1611yescharacteristic
323C22#357311yescharacteristic
324C22#357312yes—
325C4—22no—
326C4—28no—
327C4—24no—
328C4—44no—
329C4—41no—
330C4—44no—
331C4—44no—
332C4—42no—
333C4—44no—
334C4—44no—
335C4C2×#56411yescharacteristic
336C4#353112yes—
337C4#353111yescharacteristic
338C2C4×#56411yescharacteristic
339C2#6033311yescharacteristic
340C2#6033811yescharacteristic
341C2#6034312yes—
342C2#6034612yes—
subgroupG/H
center, Frattini, cosocleC22×C4C25
commutator, socle, prime radicalC23C24×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
319. C8#564semidirect

Maximal subgroups

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

subgroupindex
C4×#5642
#233722
#233712
#233702
#233692
#216752
#216742
#216732
C8×#1252