Finite Group Databasebeta

← #128296

Subgroups — #128296

Order 512.

662 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C8×C22⋊C4C211yesmaximal, characteristic
2#12599C212yesmaximal, complemented
3C2×#1314C212yesmaximal
4#7183C212yesmaximal, complemented
5C24×C8C411yescharacteristic
6C22×C22⋊C8C2211yescharacteristic
7C22×C4×C8C411yescharacteristic
8C22×C4×C8C2211yescharacteristic
9C2×C4×C22⋊C4C411yescharacteristic
10C2×C22⋊C16C2214yes—
11C2×#346—22no—
12C8×C22⋊C4—22no—
13C8×C22⋊C4—22no—
14C2×#336C411yescharacteristic
15C2×#336C2211yescharacteristic
16#1374—22no—
17#1314C418yescomplemented
18C24×C4C2×C411yescharacteristic
19C23×C8—21no—
20C23×C8—21no—
21C23×C8—22no—
22C23×C8—42no—
23C23×C8—42no—
24C23×C8C814yes—
25C23×C8C2×C411yescharacteristic
26C23×C8C2×C411yescharacteristic
27C23×C8C2×C411yescharacteristic
28C23×C8C2311yescharacteristic
29C22×C22⋊C4C812yes—
30C22×C42C2×C411yescharacteristic
31C22×C42C2×C411yescharacteristic
32C22×C16—24no—
33C2×C22⋊C8—22no—
34C2×C22⋊C8—22no—
35C2×C22⋊C8C2×C414yes—
36C2×C4×C8—21no—
37C2×C4×C8—21no—
38C2×C4×C8—42no—
39C2×C4×C8—42no—
40C2×C4×C8D412yes—
41C2×C4×C8D412yes—
42C4×C22⋊C4—22no—
43C4×C22⋊C4—22no—
44C2×#95C812yes—
45C22⋊C16—24no—
46C22⋊C16C2×C418yes—
47#346—44no—
48#336—22no—
49#336—22no—
50C25C2×C811yescharacteristic
51C23×C4—21no—
52C23×C4—21no—
53C23×C4—21no—
54C23×C4—21no—
55C23×C4—41no—
56C23×C4—41no—
57C23×C4—41no—
58C23×C4—41no—
59C23×C4C4212yes—
60C23×C4C2×C811yescharacteristic
61C23×C4C2×C812yes—
62C23×C4C2×C812yes—
63C23×C4C2×C812yes—
64C23×C4C22×C411yescharacteristic
65C22×C8—22no—
66C22×C8—22no—
67C22×C8—21no—
68C22×C8—21no—
69C22×C8—21no—
70C22×C8—21no—
71C22×C8—24no—
72C22×C8—24no—
73C22×C8—41no—
74C22×C8—41no—
75C22×C8—42no—
76C22×C8—42no—
77C22×C8—48no—
78C22×C8—48no—
79C22×C8—42no—
80C22×C8—42no—
81C22×C8—41no—
82C22×C8—41no—
83C22×C8—41no—
84C22×C8—41no—
85C22×C8C4212yes—
86C22×C8C22⋊C411yescharacteristic
87C22×C8C22⋊C411yescharacteristic
88C22×C8C2×C812yes—
89C22×C8C2×C812yes—
90C22×C8M1612yes—
91C22×C8C22×C412yes—
92C22×C8C2×D412yes—
93C22×C8#4112yes—
94C2×C22⋊C4—24no—
95C2×C22⋊C4—24no—
96C2×C22⋊C4—24no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—41no—
100C2×C42—41no—
101C2×C42—41no—
102C2×C42—41no—
103C2×C42C22⋊C412yes—
104C2×C42M1612yes—
105C2×C16—28no—
106C2×C16—44no—
107C2×C16—44no—
108C22⋊C8—28no—
109C4×C8—24no—
110C4×C8—42no—
111C4×C8—42no—
112C4×C8—42no—
113#95—24no—
114C24—21no—
115C24—21no—
116C24—22no—
117C24—41no—
118C24—41no—
119C24—42no—
120C24C4×C812yes—
121C24M3214yes—
122C24C22×C811yescharacteristic
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—42no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—42no—
157C22×C4C4×C812yes—
158C22×C4C22⋊C814yes—
159C22×C4M3214yes—
160C22×C4C2×C4211yescharacteristic
161C22×C4C2×C22⋊C411yescharacteristic
162C22×C4#11711yescharacteristic
163C22×C4C22×C811yescharacteristic
164C22×C4C2×M1611yescharacteristic
165C22×C4C2×M1611yescharacteristic
166C2×C8—22no—
167C2×C8—21no—
168C2×C8—22no—
169C2×C8—21no—
170C2×C8—24no—
171C2×C8—24no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8—48no—
177C2×C8—42no—
178C2×C8—42no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—48no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8C22⋊C812yes—
187C2×C8C22⋊C812yes—
188C2×C8#9912yes—
189C2×C8#10012yes—
190C2×C8C4×D414yes—
191C22⋊C4—216no—
192C42—22no—
193C42—22no—
194C42—42no—
195C42—42no—
196C42—41no—
197C42—41no—
198C16—48no—
199C23—22no—
200C23—21no—
201C23—21no—
202C23—22no—
203C23—24no—
204C23—24no—
205C23—22no—
206C23—42no—
207C23—42no—
208C23—44no—
209C23—44no—
210C23—42no—
211C23—42no—
212C23—44no—
213C23—44no—
214C23—42no—
215C23—41no—
216C23—41no—
217C23#34614yescomplemented
218C23C2×C4×C811yescharacteristic
219C23C2×C22⋊C811yescharacteristic
220C23#43111yescharacteristic
221C23C2×M3212yes—
222C23C2×M3212yes—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—24no—
234C2×C4—24no—
235C2×C4—24no—
236C2×C4—24no—
237C2×C4—22no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—44no—
254C2×C4C4×C22⋊C411yescharacteristic
255C2×C4C4×M1611yescharacteristic
256C2×C4C2×C22⋊C811yescharacteristic
257C2×C4C2×#9911yescharacteristic
258C2×C4#41011yescharacteristic
259C2×C4C2×#10011yescharacteristic
260C2×C4#41311yescharacteristic
261C2×C4#43111yescharacteristic
262C8—22no—
263C8—24no—
264C8—42no—
265C8—42no—
266C8#32314yes—
267C22—21no—
268C22—21no—
269C22—22no—
270C22—22no—
271C22—21no—
272C22—21no—
273C22—22no—
274C22—22no—
275C22—24no—
276C22—44no—
277C22—44no—
278C22—48no—
279C22—42no—
280C22—44no—
281C22—42no—
282C22—44no—
283C22#145911yescharacteristic
284C22#146011yescharacteristic
285C22C4×C22⋊C811yescharacteristic
286C22C2×#34611yescharacteristic
287C22C4×M3211yescharacteristic
288C22#211212yes—
289C22#211512yes—
290C22#216312yes—
291C4—21no—
292C4—21no—
293C4—44no—
294C4—42no—
295C4—44no—
296C4C2×#32311yescharacteristic
297C4#145811yescharacteristic
298C4C4×#9911yescharacteristic
299C4C4×#10011yescharacteristic
300C2—21no—
301C2—21no—
302C2—22no—
303C2—44no—
304C2C4×#32311yescharacteristic
305C2#1146612yes—
306C2#1148812yes—
307C2#1259911yescharacteristic
308C2#1260211yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC23C2×C4×C8
Frattini, cosocleC23×C8C23
socleC23C2×C22⋊C8
prime radicalC25C2×C8
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12599C2semidirect
4. #7183C2semidirect
17. #1314C4semidirect
217. C23#346semidirect

Maximal subgroups

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

subgroupindex
C2×C8×C22⋊C42
#125992
C2×#13142
#71832