Finite Group Databasebeta

← #129694

Subgroups — #129694

Order 512.

609 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 609 conjugacy classes of subgroups — too many to draw. Only its 120 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×#2984C211yesmaximal, characteristic
2#11642C212yesmaximal, complemented
3#11639C212yesmaximal, complemented
4#11460C211yesmaximal, characteristic
5#11459C211yesmaximal, characteristic
6C22×C4×Q8C411yescharacteristic
7#2984—24no—
8#2984—24no—
9C2×#446—21no—
10C2×#446—21no—
11C2×#446C411yescharacteristic
12C2×#446C2211yescharacteristic
13C2×C8×Q8C411yescharacteristic
14C2×C8×Q8C2211yescharacteristic
15C2×#432—21no—
16C2×#432C411yescharacteristic
17C2×#431—21no—
18C2×#431C2211yescharacteristic
19#2163C2212yes—
20#2162C2212yes—
21#1338—24no—
22Q8⋊C16—24no—
23#1322—21no—
24#1322—21no—
25#1314—21no—
26#1314—21no—
27C23×Q8C811yescharacteristic
28C2×C4×Q8—22no—
29C2×C4×Q8—22no—
30C2×C4×Q8—22no—
31C2×C4×Q8—22no—
32C2×C4×Q8—21no—
33C2×C4×Q8—21no—
34C2×C4×Q8C811yescharacteristic
35C2×C4×Q8C811yescharacteristic
36C2×C4×Q8C2×C411yescharacteristic
37C2×C4×Q8C2×C411yescharacteristic
38C22×C4⋊C4—21no—
39C22×C4⋊C4C811yescharacteristic
40C22×C42—21no—
41C22×C42C2×C411yescharacteristic
42#446—28no—
43#446—28no—
44#446—22no—
45#446—22no—
46#446—22no—
47#446D414yes—
48C8×Q8—22no—
49C8×Q8—22no—
50C8×Q8—22no—
51C8×Q8D414yes—
52#432—28no—
53#432—22no—
54#432—22no—
55#431—28no—
56#431—22no—
57#431—22no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8C2×C411yescharacteristic
63C2×C4⋊C8C2×C411yescharacteristic
64C2×C4⋊C8D411yescharacteristic
65C2×C4⋊C8D411yescharacteristic
66C2×C22⋊C8—21no—
67C2×C22⋊C8D411yescharacteristic
68C2×C22⋊C8D411yescharacteristic
69C2×#97—21no—
70C2×#97—21no—
71C2×#97—21no—
72C2×C4×C8—21no—
73C2×C4×C8—21no—
74C2×C4×C8C2×C411yescharacteristic
75C2×C4×C8C2311yescharacteristic
76C4⋊C16—22no—
77C4⋊C16—22no—
78C22⋊C16—22no—
79C22⋊C16—22no—
80#346—22no—
81C4×C16—22no—
82C22×Q8—22no—
83C22×Q8—22no—
84C22×Q8—21no—
85C22×Q8—44no—
86C22×Q8C2×C811yescharacteristic
87C22×Q8C2×C811yescharacteristic
88C23×C4—21no—
89C23×C4—21no—
90C23×C4C22⋊C411yescharacteristic
91C23×C4C2×C811yescharacteristic
92C23×C4M1611yescharacteristic
93C22×C8—21no—
94C22×C8—21no—
95C22×C8C22⋊C411yescharacteristic
96C22×C8C22⋊C411yescharacteristic
97C22×C8C2×D411yescharacteristic
98C22×C8C2×D411yescharacteristic
99C4×Q8—24no—
100C4×Q8—24no—
101C4×Q8—24no—
102C4×Q8—24no—
103C4×Q8—22no—
104C4×Q8—22no—
105C4×Q8—22no—
106C4×Q8—22no—
107C4×Q8—21no—
108C4×Q8—21no—
109C4×Q8—21no—
110C4×Q8—21no—
111C4×Q8C22⋊C412yes—
112C4×Q8C22⋊C412yes—
113C4×Q8M1612yes—
114C4×Q8M1612yes—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4C2×C811yescharacteristic
125C2×C4⋊C4C2×C811yescharacteristic
126C2×C42—24no—
127C2×C42—24no—
128C2×C42—21no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—21no—
134C2×C42—21no—
135C2×C42—21no—
136C2×C42C22⋊C411yescharacteristic
137C2×C42C2×C811yescharacteristic
138C2×C42M1611yescharacteristic
139C2×C42C22×C411yescharacteristic
140C2×C16—42no—
141C2×C16—42no—
142C4⋊C8—24no—
143C4⋊C8—24no—
144C4⋊C8—24no—
145C4⋊C8—24no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—21no—
149C4⋊C8—21no—
150C4⋊C8C22⋊C412yes—
151C4⋊C8C22⋊C412yes—
152C22⋊C8—24no—
153C22⋊C8—22no—
154C22⋊C8—22no—
155#97—24no—
156#97—24no—
157#97—24no—
158C4×C8—24no—
159C4×C8—24no—
160C4×C8—21no—
161C4×C8—21no—
162C4×C8C2×D412yes—
163C4×C8C2×D412yes—
164C24C22⋊C811yescharacteristic
165C2×Q8—24no—
166C2×Q8—24no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—42no—
170C2×Q8—44no—
171C2×Q8—48no—
172C2×Q8—44no—
173C2×Q8—44no—
174C2×Q8C22⋊C812yes—
175C2×Q8C22⋊C812yes—
176C22×C4—21no—
177C22×C4—24no—
178C22×C4—24no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—41no—
190C22×C4—44no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4C22⋊C811yescharacteristic
197C22×C4C22⋊C811yescharacteristic
198C22×C4C22⋊C811yescharacteristic
199C22×C4C2×C22⋊C411yescharacteristic
200C22×C4C22×C811yescharacteristic
201C22×C4C2×M1611yescharacteristic
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8#12012yes—
213C2×C8#12012yes—
214C4⋊C4—24no—
215C4⋊C4—24no—
216C4⋊C4—24no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4C22⋊C812yes—
224C4⋊C4C22⋊C812yes—
225C42—24no—
226C42—24no—
227C42—22no—
228C42—22no—
229C42—21no—
230C42—21no—
231C42—21no—
232C42—22no—
233C42—22no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42C2×C22⋊C411yescharacteristic
238C42C2×C22⋊C411yescharacteristic
239C42C2×M1611yescharacteristic
240C42C2×M1611yescharacteristic
241C16—82no—
242C16—82no—
243C23—21no—
244C23—21no—
245C23—41no—
246C23—41no—
247C23#35012yes—
248C23C2×C22⋊C811yescharacteristic
249Q8—24no—
250Q8—22no—
251Q8—42no—
252Q8—48no—
253Q8—42no—
254Q8#34914yes—
255C2×C4—24no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—48no—
274C2×C4—42no—
275C2×C4—48no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4#37911yescharacteristic
287C2×C4C2×C22⋊C811yescharacteristic
288C2×C4C2×C22⋊C811yescharacteristic
289C2×C4#40711yescharacteristic
290C2×C4#40711yescharacteristic
291C2×C4#40711yescharacteristic
292C8—22no—
293C8—22no—
294C8—41no—
295C8—41no—
296C8—42no—
297C8—42no—
298C22—21no—
299C22—21no—
300C22—22no—
301C22—42no—
302C22—42no—
303C22—41no—
304C22—41no—
305C22#177911yescharacteristic
306C22C2×#35011yescharacteristic
307C22#211711yescharacteristic
308C4—21no—
309C4—24no—
310C4—21no—
311C4—21no—
312C4—21no—
313C4—41no—
314C4—42no—
315C4—41no—
316C4—41no—
317C4—42no—
318C4—42no—
319C4#179611yescharacteristic
320C4#179611yescharacteristic
321C4C2×#34912yes—
322C2—21no—
323C2—21no—
324C2—42no—
325C2#832311yescharacteristic
326C2#1268111yescharacteristic
327C2#1268311yescharacteristic
subgroupG/H
centerC2×C4#379
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC22#1779
prime radicalC24C22⋊C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11642C2semidirect
3. #11639C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29842
#116422
#116392
#114602
#114592