Finite Group Databasebeta

← #162888

Subgroups — #162888

Order 512.

1080 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 1080 conjugacy classes of subgroups — too many to draw. Only its 226 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×#2861C211yesmaximal, characteristic
2C4×#432C212yesmaximal, complemented
3#7676C218yesmaximal, complemented
4#7629C212yesmaximal, complemented
5#7626C212yesmaximal
6C22×#125C411yescharacteristic
7C22×#124C411yescharacteristic
8C2×#432—22no—
9#2861—22no—
10#2861—22no—
11#2861C414yes—
12#2861C2214yes—
13C2×#391C411yescharacteristic
14C2×#391C2211yescharacteristic
15C2×#390C411yescharacteristic
16C2×#390C2211yescharacteristic
17C2×#388—22no—
18C2×#388—22no—
19C2×C43C2211yescharacteristic
20#1767—22no—
21C4×C4⋊C8C2214yescomplemented
22#1749C2214yes—
23C4×C22⋊C8—22no—
24#1745C2214yescomplemented
25C4×#97—22no—
26#1272C2218yescomplemented
27#1270C2218yes—
28C23×Q8C2×C411yescharacteristic
29C23×D4C2×C411yescharacteristic
30C2×#125—22no—
31C2×#125—44no—
32C2×#125C2×C414yes—
33C2×#124—22no—
34C2×#124—44no—
35C2×#124C2×C414yes—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C22⋊C4—21no—
41C22×C22⋊C4—21no—
42C22×C22⋊C4—22no—
43C22×C22⋊C4—22no—
44C22×C42C2×C411yescharacteristic
45C22×C42C2×C411yescharacteristic
46C22×C42D412yes—
47C22×C42D412yes—
48C22×C42C2311yescharacteristic
49#432—216no—
50C2×C4⋊C8—24no—
51C2×C22⋊C8—24no—
52C2×#97—24no—
53C2×#97—24no—
54C2×C4×C8—24no—
55#391—21no—
56#391—21no—
57#391—22no—
58#391C2×C412yes—
59#391C2×C414yes—
60#391C2312yes—
61#390—21no—
62#390—21no—
63#390—22no—
64#390C2×C412yes—
65#390C2×C414yes—
66#390C2312yes—
67#388—24no—
68#388—24no—
69#388—24no—
70#388—24no—
71#388—24no—
72#388—24no—
73#388—24no—
74#388—24no—
75C2×#95—22no—
76C2×#95—22no—
77C43C2×C412yes—
78C43D412yes—
79C43D412yes—
80C43C2312yes—
81#336—24no—
82#336C2318yes—
83C25—21no—
84C22×Q8—21no—
85C22×Q8—42no—
86C22×Q8—42no—
87C22×Q8—42no—
88C22×Q8C22×C412yes—
89C22×D4—21no—
90C22×D4—42no—
91C22×D4—42no—
92C22×D4—42no—
93C22×D4C22×C412yes—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4C22⋊C411yescharacteristic
98C23×C4C22⋊C411yescharacteristic
99C23×C4C22⋊C412yes—
100C23×C4C22×C411yescharacteristic
101C23×C4C2×D412yes—
102C22×C8—28no—
103#125—48no—
104#125—48no—
105#124—48no—
106#124—48no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—44no—
114C2×C4⋊C4—44no—
115C2×C4⋊C4C22×C412yes—
116C2×C4⋊C4C22×C412yes—
117C2×C22⋊C4—24no—
118C2×C22⋊C4—24no—
119C2×C22⋊C4—24no—
120C2×C22⋊C4—24no—
121C2×C22⋊C4—24no—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—48no—
124C2×C22⋊C4—44no—
125C2×C22⋊C4—44no—
126C2×C22⋊C4—48no—
127C2×C42—24no—
128C2×C42—28no—
129C2×C42—42no—
130C2×C42—44no—
131C2×C42—42no—
132C2×C42C22⋊C412yes—
133C2×C42C22⋊C412yes—
134C2×C42C22⋊C412yes—
135C2×C42C22⋊C412yes—
136C2×C42C22⋊C412yes—
137C2×C42C22⋊C411yescharacteristic
138C2×C42C22⋊C411yescharacteristic
139C2×C42C22×C412yes—
140C2×C42C22×C412yes—
141C2×C42C22×C411yescharacteristic
142C2×C42C2×D412yes—
143C2×C42C2×D412yes—
144C2×C42C2×D412yes—
145C2×C42C2×D412yes—
146C2×C42C2×D412yes—
147C2×C42C2411yescharacteristic
148C4⋊C8—216no—
149C22⋊C8—216no—
150#97—216no—
151#97—48no—
152C4×C8—48no—
153#95—24no—
154#95—24no—
155#95—24no—
156#95—24no—
157C24—22no—
158C24—44no—
159C24—42no—
160C24—42no—
161C24—42no—
162C24—44no—
163C24C2×C22⋊C411yescharacteristic
164C2×Q8—24no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—88no—
171C2×D4—24no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—88no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—44no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—44no—
199C22×C4#10014yes—
200C22×C4#10418yes—
201C22×C4C2×C22⋊C412yes—
202C22×C4C2×C22⋊C412yes—
203C22×C4C2×C22⋊C411yescharacteristic
204C22×C4C2×C22⋊C412yes—
205C22×C4C2×C22⋊C412yes—
206C22×C4C2×C22⋊C412yes—
207C22×C4C23×C411yescharacteristic
208C22×C4C22×D412yes—
209C2×C8—216no—
210C2×C8—48no—
211C2×C8—48no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—48no—
217C4⋊C4—48no—
218C22⋊C4—48no—
219C22⋊C4—48no—
220C22⋊C4—48no—
221C22⋊C4—48no—
222C22⋊C4—48no—
223C22⋊C4—48no—
224C42—24no—
225C42—24no—
226C42—24no—
227C42—24no—
228C42—24no—
229C42—24no—
230C42—24no—
231C42—24no—
232C42—24no—
233C42—24no—
234C42—41no—
235C42—41no—
236C42—42no—
237C42—42no—
238C42—41no—
239C42—41no—
240C23—21no—
241C23—21no—
242C23—22no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—44no—
247C23—44no—
248C23—84no—
249C23—84no—
250C23—88no—
251C23—88no—
252C23—84no—
253C23C2×#10012yes—
254C23C2×#10414yes—
255C23C22×C22⋊C411yescharacteristic
256Q8—84no—
257Q8—84no—
258D4—84no—
259D4—84no—
260C2×C4—28no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—44no—
280C2×C4—48no—
281C2×C4—48no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—84no—
291C2×C4—84no—
292C2×C4—84no—
293C2×C4#41012yes—
294C2×C4C2×#10014yes—
295C2×C4C2×#10414yes—
296C2×C4C2×#10414yes—
297C2×C4#56812yes—
298C2×C4#57514yes—
299C8—416no—
300C22—22no—
301C22—24no—
302C22—24no—
303C22—22no—
304C22—82no—
305C22—84no—
306C22—84no—
307C22—82no—
308C22—82no—
309C22—168no—
310C22#152514yes—
311C22#287911yescharacteristic
312C22C2×#41011yescharacteristic
313C22C22×#10011yescharacteristic
314C22#289612yes—
315C22C22×#10412yes—
316C4—24no—
317C4—24no—
318C4—24no—
319C4—24no—
320C4—22no—
321C4—22no—
322C4—42no—
323C4—42no—
324C4—82no—
325C4—82no—
326C4—82no—
327C2—24no—
328C2—162no—
329C2#1739311yescharacteristic
330C2#1750212yes—
331C2#1753712yes—
332C2C2×#152511yescharacteristic
333C2#1775411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#432C2semidirect
3. #7676C2semidirect
4. #7629C2semidirect
21. C4×C4⋊C8C22semidirect
24. #1745C22semidirect
26. #1272C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28612
C4×#4322
#76762
#76292
#76262