Finite Group Databasebeta

← #124158

Subgroups — #124158

Order 512.

472 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C8×#121C211yesmaximal, characteristic
2C4×C22⋊C16C211yesmaximal, characteristic, complemented
3C2×#1374C211yesmaximal, characteristic, complemented
4#7118C212yesmaximal, complemented
5#7102C212yesmaximal, complemented
6C23×C16—21no—
7C2×C8×D4—21no—
8C2×C8×D4C411yescharacteristic
9C22×C4×C8C2211yescharacteristic
10C4×#121C411yescharacteristic
11C2×C22⋊C16—21no—
12C2×C4×C16—21no—
13#1954C2211yescharacteristic
14#1939C2211yescharacteristic
15#1923—21no—
16#1869—21no—
17#1797C411yescharacteristic
18C8×C4⋊C4C411yescharacteristic
19C8×C22⋊C4—21no—
20#1374—21no—
21#1374C2212yes—
22#1374C2212yes—
23C23×C8—21no—
24C23×C8D411yescharacteristic
25C23×C8D411yescharacteristic
26C2×#121C811yescharacteristic
27C2×C4×D4—21no—
28C2×C4×D4C2×C411yescharacteristic
29C22×C42C2×C411yescharacteristic
30C22×C16—24no—
31C22×C16—22no—
32C22×C16—22no—
33C22×C16—42no—
34C22×C16—42no—
35C8×D4—42no—
36C8×D4—42no—
37C8×D4—42no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8C2×C411yescharacteristic
40C2×C4⋊C8C2×C411yescharacteristic
41C2×C22⋊C8—21no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—21no—
44C2×C22⋊C8—21no—
45C2×C4×C8—21no—
46C2×C4×C8—42no—
47C2×C4×C8C2×C411yescharacteristic
48C2×C4×C8D411yescharacteristic
49C2×C4×C8D411yescharacteristic
50C2×C4×C8C2311yescharacteristic
51#390C811yescharacteristic
52#389C811yescharacteristic
53#388—21no—
54#386—21no—
55#380C811yescharacteristic
56C4×C4⋊C4C2×C411yescharacteristic
57C4×C22⋊C4—21no—
58C22⋊C16—24no—
59C4×C16—42no—
60#336—21no—
61#336—21no—
62#336—21no—
63C22×D4—21no—
64C22×D4C2×C811yescharacteristic
65C23×C4—21no—
66C23×C4C22⋊C411yescharacteristic
67C23×C4C2×C811yescharacteristic
68C23×C4M1611yescharacteristic
69C22×C8—22no—
70C22×C8—22no—
71C22×C8—22no—
72C22×C8—22no—
73C22×C8—21no—
74C22×C8—21no—
75C22×C8—41no—
76C22×C8—42no—
77C22×C8—42no—
78C22×C8—42no—
79C22×C8C22⋊C411yescharacteristic
80C22×C8C22⋊C411yescharacteristic
81C22×C8D812yes—
82C22×C8SD1612yes—
83C22×C8C2×D411yescharacteristic
84C22×C8C2×D411yescharacteristic
85#121—22no—
86#121C1614yescomplemented
87C4×D4—41no—
88C4×D4—41no—
89C4×D4—42no—
90C4×D4—42no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4C2×C811yescharacteristic
93C2×C4⋊C4C2×C811yescharacteristic
94C2×C4⋊C4C2×C811yescharacteristic
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C42—21no—
101C2×C42—41no—
102C2×C42—41no—
103C2×C42C22⋊C411yescharacteristic
104C2×C42C2×C811yescharacteristic
105C2×C42M1611yescharacteristic
106C2×C42C22×C411yescharacteristic
107C2×C16—24no—
108C2×C16—24no—
109C2×C16—42no—
110C2×C16—42no—
111C2×C16—48no—
112C2×C16—42no—
113C2×C16—42no—
114C4⋊C8—41no—
115C4⋊C8—41no—
116C4⋊C8—41no—
117C4⋊C8—41no—
118C22⋊C8—42no—
119C22⋊C8—42no—
120C22⋊C8—41no—
121C22⋊C8—41no—
122C22⋊C8—42no—
123C4×C8—22no—
124C4×C8—22no—
125C4×C8—41no—
126C4×C8—41no—
127C4×C8—42no—
128#95—21no—
129#95—21no—
130#95—21no—
131C24—21no—
132C24C22⋊C811yescharacteristic
133C2×D4—24no—
134C2×D4—21no—
135C2×D4—41no—
136C2×D4—41no—
137C2×D4—42no—
138C2×D4—42no—
139C2×D4C2×C1612yescomplemented
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—22no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4C22⋊C811yescharacteristic
159C22×C4C22⋊C811yescharacteristic
160C22×C4C22⋊C811yescharacteristic
161C22×C4#10212yes—
162C22×C4#10412yes—
163C22×C4C2×C1612yes—
164C22×C4M3212yes—
165C22×C4C2×C22⋊C411yescharacteristic
166C22×C4C22×C811yescharacteristic
167C22×C4C2×M1611yescharacteristic
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—24no—
173C2×C8—24no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—44no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—84no—
185C2×C8#9912yes—
186C2×C8#10212yes—
187C2×C8#10412yes—
188C2×C8C2×D811yescharacteristic
189C2×C8C2×SD1611yescharacteristic
190C2×C8#13512yes—
191C4⋊C4—21no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—42no—
195C4⋊C4C2×C1612yescomplemented
196C22⋊C4—24no—
197C22⋊C4—42no—
198C22⋊C4—42no—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—41no—
206C42—41no—
207C42—41no—
208C42—41no—
209C16—44no—
210C16—44no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—22no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23D4⋊C812yes—
223C23C22⋊C1612yes—
224C23#35012yes—
225C23C2×C22⋊C811yescharacteristic
226D4—42no—
227D4—42no—
228D4—48no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—44no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4D4⋊C812yes—
261C2×C4C22⋊C1612yes—
262C2×C4#35012yes—
263C2×C4C2×#9911yescharacteristic
264C2×C4#41311yescharacteristic
265C2×C4C2×#10211yescharacteristic
266C2×C4#41611yescharacteristic
267C2×C4C2×#10411yescharacteristic
268C2×C4C2×#10411yescharacteristic
269C2×C4C22×C1611yescharacteristic
270C2×C4C2×M3211yescharacteristic
271C8—21no—
272C8—21no—
273C8—22no—
274C8—41no—
275C8—81no—
276C8—81no—
277C8#32714yes—
278C22—22no—
279C22—22no—
280C22—22no—
281C22—22no—
282C22—22no—
283C22—22no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22—81no—
291C22D4⋊C1614yes—
292C22#145911yescharacteristic
293C22C2×D4⋊C811yescharacteristic
294C22#147611yescharacteristic
295C22C2×C22⋊C1611yescharacteristic
296C22#211511yescharacteristic
297C22C2×#35011yescharacteristic
298C22C2×#35011yescharacteristic
299C4—22no—
300C4—22no—
301C4—21no—
302C4—21no—
303C4—22no—
304C4—42no—
305C4—41no—
306C4—81no—
307C4—81no—
308C4C2×#32711yescharacteristic
309C4#150111yescharacteristic
310C4#152411yescharacteristic
311C4#154011yescharacteristic
312C2—24no—
313C2—82no—
314C2#756511yescharacteristic
315C2#1146511yescharacteristic
316C2#1146811yescharacteristic
317C2C2×D4⋊C1611yescharacteristic
318C2#1158611yescharacteristic
319C2#1161311yescharacteristic
320C2#1161811yescharacteristic
subgroupG/H
centerC22×C8C22⋊C4
commutatorC2×C4C22×C16
Frattini, cosocleC2×C4×C8C23
socleC23C2×C22⋊C8
prime radicalC2×#121C8
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C8 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×C22⋊C16C2semidirect
3. C2×#1374C2semidirect
4. #7118C2semidirect
5. #7102C2semidirect
86. #121C16semidirect
139. C2×D4C2×C16semidirect
195. C4⋊C4C2×C16semidirect

Maximal subgroups

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

subgroupindex
C8×#1212
C4×C22⋊C162
C2×#13742
#71182
#71022