Finite Group Databasebeta

← #128293

Subgroups — #128293

Order 512.

754 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 174 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×C4×C22⋊C8C211yesmaximal, characteristic
2#12599C212yesmaximal, complemented
3C2×#1314C212yesmaximal
4#7183C212yesmaximal, complemented
5C23×C42C411yescharacteristic
6C22×C22⋊C8C411yescharacteristic
7C22×C22⋊C8C2211yescharacteristic
8C22×C4×C8C411yescharacteristic
9C22×C4×C8C2211yescharacteristic
10C2×C22⋊C16C2214yes—
11C2×#346—22no—
12C4×C22⋊C8—22no—
13C4×C22⋊C8—22no—
14C2×#336C411yescharacteristic
15C2×#336C2211yescharacteristic
16#1374—22no—
17#1314C418yescomplemented
18C24×C4C812yes—
19C24×C4C2×C411yescharacteristic
20C23×C8C2×C411yescharacteristic
21C23×C8C2×C411yescharacteristic
22C23×C8C2×C411yescharacteristic
23C23×C8C2311yescharacteristic
24C22×C42—21no—
25C22×C42—21no—
26C22×C42—22no—
27C22×C42—42no—
28C22×C42—42no—
29C22×C42C812yes—
30C22×C42C2×C411yescharacteristic
31C22×C42C2×C411yescharacteristic
32C22×C16—24no—
33C2×C22⋊C8—22no—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8C814yes—
38C2×C22⋊C8C2×C414yes—
39C2×C4×C8—21no—
40C2×C4×C8—21no—
41C2×C4×C8—42no—
42C2×C4×C8—42no—
43C2×C4×C8D412yes—
44C2×C4×C8D412yes—
45C22⋊C16—24no—
46C22⋊C16C2×C418yescomplemented
47#346—44no—
48#336—22no—
49#336—22no—
50C25C2×C811yescharacteristic
51C23×C4—24no—
52C23×C4—22no—
53C23×C4—22no—
54C23×C4—24no—
55C23×C4—21no—
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—21no—
59C23×C4—44no—
60C23×C4—44no—
61C23×C4—41no—
62C23×C4—41no—
63C23×C4—41no—
64C23×C4—41no—
65C23×C4C4212yes—
66C23×C4C2×C812yes—
67C23×C4C2×C811yescharacteristic
68C23×C4C2×C812yes—
69C23×C4C2×C812yes—
70C23×C4C22×C411yescharacteristic
71C22×C8—21no—
72C22×C8—21no—
73C22×C8—21no—
74C22×C8—21no—
75C22×C8—41no—
76C22×C8—41no—
77C22×C8—41no—
78C22×C8—41no—
79C22×C8—41no—
80C22×C8—41no—
81C22×C8—41no—
82C22×C8—41no—
83C22×C8C4212yes—
84C22×C8C22⋊C412yes—
85C22×C8C2×C812yes—
86C22×C8C2×C812yes—
87C22×C8M1612yes—
88C22×C8C22×C412yes—
89C22×C8C2×D412yes—
90C22×C8#4112yes—
91C2×C42—22no—
92C2×C42—22no—
93C2×C42—22no—
94C2×C42—22no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—48no—
98C2×C42—42no—
99C2×C42—42no—
100C2×C42—42no—
101C2×C42—42no—
102C2×C42—44no—
103C2×C42—41no—
104C2×C42—41no—
105C2×C42C22⋊C411yescharacteristic
106C2×C42C22⋊C411yescharacteristic
107C2×C42M1612yes—
108C2×C16—28no—
109C2×C16—44no—
110C2×C16—44no—
111C22⋊C8—28no—
112C22⋊C8—28no—
113C4×C8—24no—
114C4×C8—42no—
115C4×C8—42no—
116C4×C8—42no—
117C24—21no—
118C24—21no—
119C24—22no—
120C24—41no—
121C24—42no—
122C24—41no—
123C24C4×C812yes—
124C24M3214yes—
125C24C22×C811yescharacteristic
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—24no—
129C22×C4—24no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—28no—
135C22×C4—28no—
136C22×C4—22no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—44no—
145C22×C4—42no—
146C22×C4—416no—
147C22×C4—416no—
148C22×C4—44no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—41no—
168C22×C4C4×C812yes—
169C22×C4C22⋊C812yes—
170C22×C4C22⋊C812yes—
171C22×C4M3214yes—
172C22×C4C2×C4211yescharacteristic
173C22×C4C2×C22⋊C411yescharacteristic
174C22×C4#11711yescharacteristic
175C22×C4C22×C811yescharacteristic
176C22×C4C2×M1611yescharacteristic
177C22×C4C2×M1611yescharacteristic
178C2×C8—22no—
179C2×C8—22no—
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—22no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8C22⋊C814yes—
197C2×C8C4×D414yes—
198C42—24no—
199C42—21no—
200C42—21no—
201C42—44no—
202C42—42no—
203C42—44no—
204C42—41no—
205C42—41no—
206C42#9912yes—
207C42#10012yes—
208C16—48no—
209C23—22no—
210C23—21no—
211C23—21no—
212C23—22no—
213C23—24no—
214C23—24no—
215C23—22no—
216C23—42no—
217C23—44no—
218C23—42no—
219C23—44no—
220C23—44no—
221C23—42no—
222C23—42no—
223C23—44no—
224C23—42no—
225C23—41no—
226C23—41no—
227C23#34614yescomplemented
228C23C2×C4×C811yescharacteristic
229C23C2×C22⋊C811yescharacteristic
230C23#43111yescharacteristic
231C23C2×M3212yes—
232C23C2×M3212yes—
233C2×C4—28no—
234C2×C4—28no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—416no—
252C2×C4—416no—
253C2×C4—44no—
254C2×C4—44no—
255C2×C4—44no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—44no—
267C2×C4#32314yes—
268C2×C4#34914yes—
269C2×C4C4×C22⋊C411yescharacteristic
270C2×C4C4×M1611yescharacteristic
271C2×C4C2×C22⋊C811yescharacteristic
272C2×C4C2×#9911yescharacteristic
273C2×C4#41011yescharacteristic
274C2×C4C2×#10011yescharacteristic
275C2×C4#41311yescharacteristic
276C2×C4#43111yescharacteristic
277C8—24no—
278C8—24no—
279C8—42no—
280C8—42no—
281C22—21no—
282C22—21no—
283C22—22no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—21no—
288C22—21no—
289C22—24no—
290C22—44no—
291C22—44no—
292C22—48no—
293C22—42no—
294C22—44no—
295C22—44no—
296C22—42no—
297C22C2×#32311yescharacteristic
298C22#145811yescharacteristic
299C22C4×C22⋊C811yescharacteristic
300C22C2×#34611yescharacteristic
301C22C4×M3211yescharacteristic
302C22#211212yes—
303C22C2×#34912yes—
304C22#216312yes—
305C4—24no—
306C4—24no—
307C4—21no—
308C4—21no—
309C4—44no—
310C4—42no—
311C4#145911yescharacteristic
312C4#146011yescharacteristic
313C4C4×#9911yescharacteristic
314C4C4×#10011yescharacteristic
315C2—21no—
316C2—21no—
317C2—22no—
318C2—44no—
319C2C4×#32311yescharacteristic
320C2#1141812yes—
321C2#1146012yes—
322C2#1259911yescharacteristic
323C2C4×#34911yescharacteristic
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
46. C22⋊C16C2×C4semidirect
227. C23#346semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊C82
#125992
C2×#13142
#71832