Finite Group Databasebeta

← #6408078

Subgroups — #6408078

Order 512.

1104 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36701C211yesmaximal, characteristic
2#23373C214yesmaximal
3#23373C214yesmaximal
4#23373C214yesmaximal
5#23372C211yesmaximal, characteristic
6#23369C212yesmaximal
7#21679C214yesmaximal
8#21678C214yesmaximal
9#21678C214yesmaximal
10#21673C212yesmaximal
11C8×#125C211yesmaximal, characteristic
12#2845C411yescharacteristic
13#2839C414yes—
14#2839C414yes—
15#2839C414yes—
16#2822C411yescharacteristic
17#2822C412yes—
18#2402C2214yes—
19#2402C2214yes—
20#2402C2214yes—
21C4×#125C2212yes—
22C4×#125C2211yescharacteristic
23#1940C2214yes—
24#1940C2214yes—
25#1939C2214yes—
26#1939C2214yes—
27#1939C2212yes—
28#1939C2212yes—
29#1939C2212yes—
30#1939C2214yes—
31#1939C2214yes—
32#1917C2214yes—
33#1917C2214yes—
34#1917C2214yes—
35#1917C2214yes—
36#1916C2214yes—
37#1916C2214yes—
38#1916C2214yes—
39#1916C2214yes—
40#1916C2214yes—
41#1916C2214yes—
42#1916C2214yes—
43#1916C2214yes—
44#1916C2214yes—
45#1916C2214yes—
46#1916C2214yes—
47#1832C2214yes—
48#1832C2214yes—
49#1832C2214yes—
50#1831C2211yescharacteristic
51#1831C2212yes—
52#1771C2214yes—
53#1771C2214yes—
54#1771C2214yes—
55#1771C2214yes—
56#1770C2212yes—
57#1770C2212yes—
58#1770C2214yes—
59C8×C4⋊C4C2212yes—
60C8×C4⋊C4C2214yes—
61#1726C2214yes—
62C42×C8C2211yescharacteristic
63C2×#125C2×C412yes—
64C2×#125C2×C411yescharacteristic
65C2×C4⋊C8C2312yes—
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2312yes—
68C2×C4⋊C8C2314yes—
69C2×C4⋊C8C2314yes—
70C2×C4⋊C8C2314yes—
71C2×C4⋊C8C2314yes—
72C2×C4⋊C8C2314yes—
73C2×C4⋊C8C2314yes—
74C2×#97C2314yes—
75C2×#97C2314yes—
76C2×#97C2314yes—
77C2×C4×C8C2312yes—
78C2×C4×C8C2311yescharacteristic
79C2×C4×C8C2314yes—
80C2×C4×C8C2312yes—
81C2×C4×C8C2311yescharacteristic
82C2×C4×C8C2314yes—
83C2×C4×C8C2314yes—
84#400C2×C414yes—
85#400C2×C414yes—
86#400C2×C414yes—
87#400C2×C414yes—
88#400C2×C414yes—
89#400C2×C414yes—
90#400C2×C414yes—
91#400C2×C414yes—
92#400C2×C414yes—
93#400C2×C414yes—
94#400C2×C414yes—
95#400C2×C414yes—
96#398C2×C414yes—
97#398C2×C414yes—
98#398C2×C414yes—
99#398C2×C414yes—
100#398C2×C414yes—
101#398C2×C414yes—
102#398C2×C414yes—
103#398C2×C414yes—
104#398C2×C414yes—
105#398C2×C414yes—
106#398C2×C414yes—
107#398C2×C414yes—
108#389C2×C412yes—
109#389C2×C412yes—
110#389C2×C412yes—
111#387C2×C414yes—
112#387C2×C414yes—
113#387C2×C414yes—
114#382C2×C412yes—
115#382C2×C411yescharacteristic
116C4×C4⋊C4C2312yes—
117C4×C4⋊C4C2312yes—
118C4×C4⋊C4C2314yes—
119C4×C4⋊C4C2314yes—
120C4×C4⋊C4C2314yes—
121C4×C4⋊C4C2312yes—
122C4×C4⋊C4C2314yes—
123C4×C4⋊C4C2314yes—
124C4×C4⋊C4C2314yes—
125#376C2314yes—
126C43C2311yescharacteristic
127#336C2314yes—
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—
142C22×C8C2414yes—
143C22×C8C2412yes—
144C22×C8C2411yescharacteristic
145C22×C8C2414yes—
146C22×C8C2414yes—
147C22×C8C2411yescharacteristic
148#125—28no—
149#125—24no—
150C2×C4⋊C4C22×C414yes—
151C2×C4⋊C4C22×C414yes—
152C2×C4⋊C4C22×C414yes—
153C2×C4⋊C4C22×C412yes—
154C2×C4⋊C4C22×C412yes—
155C2×C4⋊C4C22×C412yes—
156C2×C4⋊C4C22×C412yes—
157C2×C4⋊C4C22×C412yes—
158C2×C4⋊C4C22×C412yes—
159C2×C4⋊C4C22×C414yes—
160C2×C4⋊C4C22×C414yes—
161C2×C4⋊C4C22×C414yes—
162C2×C4⋊C4C22×C414yes—
163C2×C4⋊C4C22×C414yes—
164C2×C4⋊C4C22×C414yes—
165C2×C4⋊C4C22×C414yes—
166C2×C4⋊C4C22×C414yes—
167C2×C4⋊C4C22×C414yes—
168C2×C42C22×C411yescharacteristic
169C2×C42C22×C412yes—
170C2×C42C22×C411yescharacteristic
171C2×C42C2412yes—
172C2×C42C2411yescharacteristic
173C2×C42C2414yes—
174C2×C42C2414yes—
175C2×C42C2414yes—
176C4⋊C8—42no—
177C4⋊C8—42no—
178C4⋊C8—44no—
179C4⋊C8—44no—
180C4⋊C8—44no—
181C4⋊C8—44no—
182C4⋊C8—42no—
183C4⋊C8—44no—
184C4⋊C8—44no—
185#97—28no—
186#97—28no—
187#97—44no—
188C4×C8—24no—
189C4×C8—28no—
190C4×C8—22no—
191C4×C8—42no—
192C4×C8—41no—
193C4×C8—44no—
194C4×C8—44no—
195#95C22×C414yes—
196#95C22×C414yes—
197#95C22×C414yes—
198#95C22×C414yes—
199#95C22×C414yes—
200#95C22×C414yes—
201#95C22×C414yes—
202#95C22×C414yes—
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—
214C22×C4C23×C411yescharacteristic
215C22×C4C23×C412yes—
216C22×C4C23×C414yes—
217C22×C4C23×C414yes—
218C22×C4C23×C414yes—
219C22×C4C23×C414yes—
220C22×C4C23×C414yes—
221C22×C4C23×C414yes—
222C22×C4C23×C411yescharacteristic
223C22×C4C23×C412yes—
224C22×C4C2511yescharacteristic
225C2×C8—21no—
226C2×C8—22no—
227C2×C8—24no—
228C2×C8—24no—
229C2×C8—24no—
230C2×C8—22no—
231C2×C8—24no—
232C2×C8—22no—
233C2×C8—22no—
234C2×C8—24no—
235C2×C8—24no—
236C2×C8—24no—
237C2×C8—24no—
238C2×C8—24no—
239C2×C82−1+414yes—
240C2×C82−1+412yes—
241C4⋊C4—28no—
242C4⋊C4—28no—
243C4⋊C4—28no—
244C4⋊C4—24no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—44no—
251C4⋊C4—44no—
252C4⋊C4—44no—
253C4⋊C4—44no—
254C4⋊C4—44no—
255C4⋊C4—44no—
256C4⋊C4—44no—
257C4⋊C4—44no—
258C4⋊C4—44no—
259C42—21no—
260C42—22no—
261C42—24no—
262C42—24no—
263C42—24no—
264C42—21no—
265C42—22no—
266C42—24no—
267C42—24no—
268C42—24no—
269C42—41no—
270C42—42no—
271C42#13114yes—
272C23C24×C411yescharacteristic
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—24no—
278C2×C4—21no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4C2×#13112yes—
301C2×C4C2×#13114yes—
302C2×C4C2×2−1+411yescharacteristic
303C2×C4C2×2−1+412yes—
304C2×C4#58511yescharacteristic
305C2×C4#58512yes—
306C8—42no—
307C8—44no—
308C8—41no—
309C8—44no—
310C8—44no—
311C8#56414yes—
312C22C4×2−1+411yescharacteristic
313C22C4×2−1+412yes—
314C22C22×#13111yescharacteristic
315C22#357311yescharacteristic
316C22#357312yes—
317C4—22no—
318C4—24no—
319C4—44no—
320C4—41no—
321C4—44no—
322C4—44no—
323C4—44no—
324C4—42no—
325C4—44no—
326C4—44no—
327C4C2×#56411yescharacteristic
328C4#353112yes—
329C4#353111yescharacteristic
330C2C4×#56411yescharacteristic
331C2#6033411yescharacteristic
332C2#6033811yescharacteristic
333C2#6033912yes—
334C2#6034612yes—
subgroupG/H
center, Frattini, cosocleC22×C4C25
commutator, socle, prime radicalC23C24×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#367012
#233732
#233722
#233692
#216792
#216782
#216732
C8×#1252