Finite Group Databasebeta

← #6374076

Subgroups — #6374076

Order 512.

1552 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34552C211yesmaximal, characteristic
2#21717C214yesmaximal, complemented
3#21677C212yesmaximal, complemented
4#21676C212yesmaximal, complemented
5#21606C218yesmaximal
6#21355C212yesmaximal
7#21345C214yesmaximal, complemented
8#20811C211yesmaximal, characteristic
9#20520C214yesmaximal, complemented
10#20480C212yesmaximal
11#20331C211yesmaximal, characteristic
12C2×#446C2218yes—
13C2×#432C2212yes—
14C2×#405C2214yes—
15C2×C4×M16C2211yescharacteristic
16C22×#97C2211yescharacteristic
17#2402C414yes—
18#2402C2211yescharacteristic
19#2402C2211yescharacteristic
20#2399C414yes—
21#2399C2212yes—
22#2396C414yes—
23#2396C414yes—
24#2396C2214yes—
25C42×Q8C2214yes—
26C4×#117C2212yes—
27C4×#117C2211yescharacteristic
28#1984C2218yes—
29#1940C2214yes—
30#1940C2214yes—
31#1939C2218yes—
32#1917C2218yes—
33#1917C2218yes—
34#1850C2218yes—
35#1843C2212yes—
36#1832C2212yes—
37#1798C2214yes—
38#1771C2218yes—
39#1771C2218yes—
40#1770C2214yes—
41C8×C4⋊C4C2214yes—
42C8×C4⋊C4C2218yes—
43#1767C2214yes—
44#1767C2214yes—
45#1767C2214yes—
46#1753C2218yes—
47#1745C2214yes—
48#1726C2212yes—
49#1726C2212yes—
50C4×#97C2214yes—
51C4×#97C2212yes—
52C4×#97C2212yes—
53C22×M16C2312yes—
54C23×C8C2312yes—
55C2×C4×Q8C2×C414yes—
56C2×C4×Q8C2×C414yes—
57C2×C4×Q8C2314yes—
58C2×#117C2×C414yes—
59C2×#117C2312yes—
60C22×C42C2311yescharacteristic
61#446—232no—
62#432—28no—
63C2×C4⋊C8C2318yes—
64C2×C4⋊C8C2314yes—
65C2×C4⋊C8C2318yes—
66C2×C4⋊C8C2318yes—
67C2×C22⋊C8C2314yes—
68#405—216no—
69C4×M16Q818yescomplemented
70C2×#97—24no—
71C2×#97C2312yes—
72C2×#97C2312yes—
73C2×#97C2314yes—
74C2×#97C2314yes—
75C2×#97C2318yes—
76C2×#97C2311yescharacteristic
77C2×#97C2311yescharacteristic
78C2×#97C2314yes—
79C2×C4×C8C2314yes—
80C2×C4×C8C2314yes—
81C2×C4×C8C2318yes—
82C2×C4×C8C2312yes—
83#400C2×C414yes—
84#400C2×C414yes—
85#400C2×C414yes—
86#399C2×C414yes—
87#398C2×C414yes—
88#398C2×C414yes—
89#398C2×C414yes—
90#398C2×C414yes—
91#398C2×C414yes—
92#398C2×C414yes—
93#396C2×C414yes—
94#396C2×C414yes—
95#395C2×C414yes—
96#395C2×C414yes—
97#395C2×C414yes—
98#393C2×C414yes—
99C4×C4⋊C4C2×C414yes—
100C4×C4⋊C4C2×C414yes—
101C4×C4⋊C4C2×C414yes—
102C4×C4⋊C4C2×C414yes—
103C4×C4⋊C4C2×C414yes—
104C4×C4⋊C4C2×C414yes—
105C4×C4⋊C4C2×C414yes—
106C4×C4⋊C4C2×C414yes—
107C4×C4⋊C4C2314yes—
108C4×C4⋊C4C2314yes—
109C4×C4⋊C4C2314yes—
110C4×C4⋊C4C2312yes—
111C4×C4⋊C4C2312yes—
112C4×C22⋊C4C2×C412yes—
113C4×C22⋊C4C2×C412yes—
114C4×C22⋊C4C2×C414yes—
115C4×C22⋊C4C2314yes—
116#376C2×C412yes—
117#376C2×C412yes—
118#376C2311yescharacteristic
119#376C2311yescharacteristic
120C43C2312yes—
121C43C2314yes—
122#336C2318yes—
123#336C2318yes—
124#336C2318yes—
125#336C2318yes—
126#336C2318yes—
127C22×Q8C22×C414yes—
128C23×C4C22×C412yes—
129C23×C4C2411yescharacteristic
130C2×M16—28no—
131C2×M16C2×Q818yes—
132C22×C8—24no—
133C22×C8—28no—
134C22×C8C2418yes—
135C22×C8C2412yes—
136C22×C8C2414yes—
137C22×C8C2412yes—
138C4×Q8—216no—
139C4×Q8—48no—
140C4×Q8—48no—
141#117—28no—
142#117—48no—
143C2×C4⋊C4C22×C414yes—
144C2×C4⋊C4C22×C414yes—
145C2×C4⋊C4C22×C414yes—
146C2×C4⋊C4C22×C414yes—
147C2×C4⋊C4C22×C414yes—
148C2×C4⋊C4C22×C414yes—
149C2×C4⋊C4C22×C414yes—
150C2×C4⋊C4C22×C414yes—
151C2×C4⋊C4C22×C414yes—
152C2×C4⋊C4C22×C414yes—
153C2×C4⋊C4C22×C414yes—
154C2×C4⋊C4C22×C414yes—
155C2×C22⋊C4C22×C414yes—
156C2×C22⋊C4C22×C414yes—
157C2×C22⋊C4C22×C414yes—
158C2×C42C22×C414yes—
159C2×C42C22×C414yes—
160C2×C42C22×C414yes—
161C2×C42C22×C414yes—
162C2×C42C22×C414yes—
163C2×C42C22×C414yes—
164C2×C42C22×C412yes—
165C2×C42C2×Q814yes—
166C2×C42#4114yes—
167C2×C42C2412yes—
168C2×C42C2414yes—
169C2×C42C2414yes—
170C2×C42C2411yescharacteristic
171C2×C42C2411yescharacteristic
172C2×C42C2412yes—
173C4⋊C8—216no—
174C4⋊C8—216no—
175C4⋊C8—28no—
176C4⋊C8—216no—
177C22⋊C8—28no—
178#97—28no—
179#97—216no—
180#97—24no—
181#97—24no—
182#97—28no—
183#97C2×Q814yes—
184#97C2×Q814yes—
185#97#41116yes—
186C4×C8—28no—
187C4×C8—216no—
188C4×C8—28no—
189C4×C8C2×Q818yes—
190#95C22×C412yes—
191#95C22×C412yes—
192#95C22×C412yes—
193#95C22×C412yes—
194#95C22×C414yes—
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—
204C24C23×C411yescharacteristic
205C2×Q8—28no—
206C2×Q8—28no—
207C2×Q8—44no—
208C22×C4—24no—
209C22×C4—44no—
210C22×C4C4×Q818yes—
211C22×C4C23×C414yes—
212C22×C4C23×C414yes—
213C22×C4C23×C414yes—
214C22×C4C23×C414yes—
215C22×C4C23×C411yescharacteristic
216C22×C4C23×C412yes—
217C22×C4C23×C412yes—
218C22×C4C23×C414yes—
219C22×C4C23×C414yes—
220C22×C4C22×Q812yes—
221C22×C4C2×#4112yes—
222C22×C4C2511yescharacteristic
223M16—216no—
224C2×C8—216no—
225C2×C8—24no—
226C2×C8—24no—
227C2×C8—24no—
228C2×C8—28no—
229C2×C8—28no—
230C2×C8—28no—
231C2×C8C22×Q814yes—
232C2×C8C22×Q814yes—
233C2×C8C2×#4118yes—
234C2×C8C2×#4118yes—
235C4⋊C4—28no—
236C4⋊C4—28no—
237C4⋊C4—28no—
238C4⋊C4—28no—
239C4⋊C4—28no—
240C4⋊C4—28no—
241C4⋊C4—28no—
242C4⋊C4—28no—
243C4⋊C4—44no—
244C4⋊C4—44no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C22⋊C4—28no—
248C22⋊C4—28no—
249C22⋊C4—44no—
250C42—28no—
251C42—24no—
252C42—28no—
253C42—28no—
254C42—44no—
255C42—44no—
256C42—44no—
257C42—44no—
258C42—44no—
259C42C4×Q818yes—
260C42C22×Q812yes—
261C42C22×Q812yes—
262C42C2×#4112yes—
263C42C2×#4112yes—
264C42C2×#4118yes—
265C23—22no—
266C23—24no—
267C23C2×C4×Q812yes—
268C23C24×C411yescharacteristic
269Q8—416no—
270C2×C4—24no—
271C2×C4—22no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—48no—
292C2×C4C2×C4×Q812yes—
293C2×C4C2×C4×Q814yes—
294C2×C4C2×C4×Q814yes—
295C2×C4C4×#4118yes—
296C2×C4#56818yes—
297C2×C4C23×Q811yescharacteristic
298C2×C4C22×#4114yes—
299C2×C4C22×#4111yescharacteristic
300C8—24no—
301C8—28no—
302C8—48no—
303C8#53318yes—
304C22—24no—
305C22—22no—
306C22—44no—
307C22C22×C4×Q811yescharacteristic
308C22C2×C4×#4112yes—
309C22C2×#56814yes—
310C4—24no—
311C4—24no—
312C4—28no—
313C4—28no—
314C4—22no—
315C4—44no—
316C4—44no—
317C4—44no—
318C4—44no—
319C4C2×#53312yes—
320C4#344412yes—
321C2—42no—
322C2C4×#53311yescharacteristic
323C2#6019212yes—
324C2#6021114yes—
subgroupG/H
center, Frattini, cosocleC22×C4C25
commutator, socleC23C24×C4
prime radicalC24C23×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
2. #21717C2semidirect
3. #21677C2semidirect
4. #21676C2semidirect
7. #21345C2semidirect
9. #20520C2semidirect
69. C4×M16Q8semidirect

Maximal subgroups

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

subgroupindex
#345522
#217172
#216772
#216762
#216062
#213552
#213452
#208112
#205202
#204802
#203312