Finite Group Databasebeta

← #129201

Subgroups — #129201

Order 512.

2070 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 2070 conjugacy classes of subgroups — too many to draw. Only its 190 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
1C23×C22⋊C8C211yesmaximal, characteristic
2#12679C212yesmaximal, complemented
3C2×#1314C214yesmaximal, complemented
4C25×C4C411yescharacteristic
5C24×C8C411yescharacteristic
6C24×C8C2211yescharacteristic
7C22×C22⋊C8—28no—
8C22×C22⋊C8—24no—
9C22×C22⋊C8C412yes—
10C22×C22⋊C8C2212yes—
11C2×C22⋊C16—24no—
12C2×C22⋊C16C2214yescomplemented
13#1314—28no—
14C26C811yescharacteristic
15C24×C4—24no—
16C24×C4—22no—
17C24×C4—24no—
18C24×C4—22no—
19C24×C4—44no—
20C24×C4—44no—
21C24×C4C812yes—
22C24×C4C811yescharacteristic
23C24×C4C2×C411yescharacteristic
24C24×C4C2×C412yes—
25C23×C8—22no—
26C23×C8—24no—
27C23×C8—24no—
28C23×C8—44no—
29C23×C8—44no—
30C23×C8C2×C411yescharacteristic
31C23×C8C2×C412yes—
32C23×C8D414yes—
33C23×C8C2311yescharacteristic
34C22×C16—24no—
35C2×C22⋊C8—232no—
36C2×C22⋊C8—24no—
37C2×C22⋊C8—24no—
38C2×C22⋊C8—24no—
39C2×C22⋊C8—24no—
40C2×C22⋊C8—24no—
41C2×C22⋊C8D418yes—
42C22⋊C16—28no—
43C22⋊C16—48no—
44C25—24no—
45C25—22no—
46C25—24no—
47C25—22no—
48C25—44no—
49C25—44no—
50C25C1614yescomplemented
51C25C2×C811yescharacteristic
52C25C2×C812yes—
53C23×C4—24no—
54C23×C4—24no—
55C23×C4—22no—
56C23×C4—22no—
57C23×C4—28no—
58C23×C4—216no—
59C23×C4—28no—
60C23×C4—24no—
61C23×C4—22no—
62C23×C4—24no—
63C23×C4—24no—
64C23×C4—24no—
65C23×C4—22no—
66C23×C4—22no—
67C23×C4—22no—
68C23×C4—22no—
69C23×C4—416no—
70C23×C4—416no—
71C23×C4—432no—
72C23×C4—48no—
73C23×C4—48no—
74C23×C4—44no—
75C23×C4—44no—
76C23×C4—48no—
77C23×C4—44no—
78C23×C4—44no—
79C23×C4—44no—
80C23×C4—44no—
81C23×C4—44no—
82C23×C4C22⋊C412yes—
83C23×C4C22⋊C414yes—
84C23×C4C2×C811yescharacteristic
85C23×C4C2×C812yes—
86C23×C4M1612yes—
87C23×C4M1614yes—
88C23×C4C22×C411yescharacteristic
89C22×C8—28no—
90C22×C8—28no—
91C22×C8—22no—
92C22×C8—22no—
93C22×C8—22no—
94C22×C8—28no—
95C22×C8—42no—
96C22×C8—44no—
97C22×C8—42no—
98C22×C8—416no—
99C22×C8—416no—
100C22×C8—44no—
101C22×C8—44no—
102C22×C8—44no—
103C22×C8C22⋊C412yes—
104C22×C8C22⋊C414yes—
105C22×C8C2×D412yes—
106C22×C8C2×D414yes—
107C2×C16—48no—
108C2×C16—44no—
109C22⋊C8—216no—
110C22⋊C8—216no—
111C24—24no—
112C24—24no—
113C24—22no—
114C24—22no—
115C24—28no—
116C24—28no—
117C24—216no—
118C24—28no—
119C24—216no—
120C24—22no—
121C24—22no—
122C24—22no—
123C24—22no—
124C24—44no—
125C24—416no—
126C24—44no—
127C24—416no—
128C24—44no—
129C24—416no—
130C24—432no—
131C24—416no—
132C24—44no—
133C24—44no—
134C24—44no—
135C24C22⋊C812yes—
136C24C22⋊C814yes—
137C24C2×C1612yes—
138C24C2×C1614yes—
139C24M3212yes—
140C24M3214yes—
141C24C22×C811yescharacteristic
142C22×C4—28no—
143C22×C4—28no—
144C22×C4—216no—
145C22×C4—24no—
146C22×C4—22no—
147C22×C4—24no—
148C22×C4—24no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—24no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—216no—
157C22×C4—28no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—24no—
161C22×C4—24no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—416no—
165C22×C4—464no—
166C22×C4—416no—
167C22×C4—48no—
168C22×C4—48no—
169C22×C4—416no—
170C22×C4—416no—
171C22×C4—48no—
172C22×C4—416no—
173C22×C4—44no—
174C22×C4—416no—
175C22×C4—44no—
176C22×C4—44no—
177C22×C4—44no—
178C22×C4—48no—
179C22×C4—416no—
180C22×C4—44no—
181C22×C4—44no—
182C22×C4—48no—
183C22×C4—44no—
184C22×C4—44no—
185C22×C4—44no—
186C22×C4C22⋊C812yes—
187C22×C4C22⋊C814yes—
188C22×C4#9914yes—
189C22×C4#10014yes—
190C22×C4C2×C22⋊C412yes—
191C22×C4C2×C22⋊C411yescharacteristic
192C22×C4C2×M1611yescharacteristic
193C22×C4C2×M1612yes—
194C2×C8—24no—
195C2×C8—22no—
196C2×C8—24no—
197C2×C8—24no—
198C2×C8—44no—
199C2×C8—416no—
200C2×C8—44no—
201C2×C8—416no—
202C2×C8—44no—
203C2×C8—44no—
204C2×C8—44no—
205C2×C8#12014yes—
206C16—84no—
207C23—28no—
208C23—28no—
209C23—216no—
210C23—28no—
211C23—28no—
212C23—24no—
213C23—22no—
214C23—24no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—232no—
221C23—24no—
222C23—24no—
223C23—24no—
224C23—24no—
225C23—24no—
226C23—24no—
227C23—44no—
228C23—44no—
229C23—48no—
230C23—48no—
231C23—416no—
232C23—44no—
233C23—464no—
234C23—464no—
235C23—416no—
236C23—44no—
237C23—416no—
238C23—48no—
239C23—416no—
240C23—416no—
241C23—48no—
242C23—416no—
243C23—44no—
244C23—44no—
245C23#32314yes—
246C23C22⋊C1614yes—
247C23C22⋊C1618yescomplemented
248C23#34914yes—
249C23C2×C22⋊C812yes—
250C23C2×C22⋊C811yescharacteristic
251C23C22×C1611yescharacteristic
252C23C2×M3211yescharacteristic
253C23C2×M3212yes—
254C2×C4—28no—
255C2×C4—216no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—22no—
264C2×C4—24no—
265C2×C4—22no—
266C2×C4—24no—
267C2×C4—432no—
268C2×C4—48no—
269C2×C4—48no—
270C2×C4—416no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—48no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4#32314yes—
278C2×C4#37911yescharacteristic
279C2×C4#40712yes—
280C2×C4#40711yescharacteristic
281C2×C4C2×#9912yes—
282C2×C4C2×#10012yes—
283C8—24no—
284C8—42no—
285C8—44no—
286C22—216no—
287C22—216no—
288C22—24no—
289C22—22no—
290C22—24no—
291C22—24no—
292C22—22no—
293C22—24no—
294C22—22no—
295C22—22no—
296C22—24no—
297C22—24no—
298C22—24no—
299C22—28no—
300C22—48no—
301C22—464no—
302C22—48no—
303C22—48no—
304C22—48no—
305C22—44no—
306C22—44no—
307C22—48no—
308C22—48no—
309C22#131418yescomplemented
310C22C2×#32312yes—
311C22#177911yescharacteristic
312C22C2×C22⋊C1612yes—
313C22C2×C22⋊C1611yescharacteristic
314C22#211212yes—
315C22#211211yescharacteristic
316C22C2×#34912yes—
317C4—24no—
318C4—22no—
319C4—44no—
320C4#146212yes—
321C4#178111yescharacteristic
322C4#178311yescharacteristic
323C2—28no—
324C2—22no—
325C2—22no—
326C2—48no—
327C2#829811yescharacteristic
328C2C2×#131412yes—
329C2#1141812yes—
330C2#1267911yescharacteristic
331C2#1268111yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC23C22×C16
Frattini, cosocleC23×C8C23
socleC23C2×C22⋊C8
prime radicalC26C8
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C24×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12679C2semidirect
3. C2×#1314C2semidirect
12. C2×C22⋊C16C22semidirect
50. C25C16semidirect
247. C23C22⋊C16semidirect
309. C22#1314semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊C82
#126792
C2×#13142