Finite Group Databasebeta

← #190153

Subgroups — #190153

Order 512.

1033 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 1033 conjugacy classes of subgroups — too many to draw. Only its 232 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
1#34057C211yesmaximal, characteristic
2#20726C212yesmaximal
3#8755C214yesmaximal, complemented
4#8752C212yesmaximal
5#8751C212yesmaximal
6#8116C214yesmaximal, complemented
7C22×C4×D4C2211yescharacteristic
8#2920C2214yes—
9#2318—24no—
10#2315C412yes—
11#2313C2212yes—
12#2312C412yes—
13#2308C2211yescharacteristic
14#2308C2211yescharacteristic
15#2290—24no—
16#2283C2212yes—
17#2280C412yes—
18C2×C4×C22⋊C4C412yes—
19#1821C2214yes—
20#1800—24no—
21#1783C2214yes—
22C4×#100—24no—
23#1306C2218yes—
24#1306C2218yes—
25C23×D4C2311yescharacteristic
26C24×C4C2×C411yescharacteristic
27C24×C4C2×C411yescharacteristic
28C2×C4×D4—24no—
29C2×C4×D4—24no—
30C2×C4×D4D414yes—
31C2×C4×D4D414yes—
32C2×#117—22no—
33C22×C4⋊C4C2311yescharacteristic
34C22×C22⋊C4C2×C411yescharacteristic
35C22×C22⋊C4C2×C411yescharacteristic
36C22×C22⋊C4C2×C412yes—
37C22×C22⋊C4C2×C412yes—
38C22×C42—22no—
39C22×C42C2311yescharacteristic
40#432—24no—
41#432—24no—
42#431—24no—
43#431—24no—
44#423—24no—
45C2×#100—28no—
46#407C2318yes—
47C4×M16—24no—
48#388—24no—
49#388—24no—
50#387—24no—
51#387—24no—
52#386—24no—
53#386—24no—
54#385—24no—
55#385—24no—
56#383—22no—
57#382—22no—
58#381—22no—
59#381C2312yes—
60#381C2312yes—
61#380—22no—
62#380C2×C414yes—
63#380C2×C414yes—
64#379C2×C412yes—
65#379C2×C412yes—
66C4×C4⋊C4—22no—
67C4×C22⋊C4—22no—
68C4×C22⋊C4—22no—
69C4×C22⋊C4—22no—
70C4×C22⋊C4C2×C414yes—
71C4×C22⋊C4C2×C414yes—
72#376—22no—
73C2×#95—22no—
74#343—28no—
75C25C22×C411yescharacteristic
76C25C22×C411yescharacteristic
77C22×D4—24no—
78C22×D4—44no—
79C22×D4C2×D414yes—
80C23×C4—22no—
81C23×C4—22no—
82C23×C4—22no—
83C23×C4—22no—
84C23×C4—22no—
85C23×C4—22no—
86C23×C4—82no—
87C23×C4—82no—
88C23×C4C22⋊C412yes—
89C23×C4C22⋊C412yes—
90C23×C4C22⋊C412yes—
91C23×C4C22⋊C412yes—
92C23×C4C22×C411yescharacteristic
93C23×C4C22×C411yescharacteristic
94C23×C4C22×C411yescharacteristic
95C23×C4C22×C411yescharacteristic
96C23×C4C2411yescharacteristic
97C2×M16—28no—
98C4×D4—28no—
99C4×D4—28no—
100C4×D4—28no—
101C4×D4—28no—
102#117—84no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4C2×D414yes—
107C2×C22⋊C4—24no—
108C2×C22⋊C4—24no—
109C2×C22⋊C4—22no—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—44no—
116C2×C22⋊C4—44no—
117C2×C22⋊C4C22⋊C414yes—
118C2×C22⋊C4C22⋊C414yes—
119C2×C22⋊C4C22×C414yes—
120C2×C22⋊C4C22×C414yes—
121C2×C42—24no—
122C2×C42—24no—
123C2×C42—84no—
124C2×C42C2×D412yes—
125C2×C42C2×D412yes—
126C2×C42#4114yes—
127C2×C42#4114yes—
128C4⋊C8—44no—
129C4⋊C8—44no—
130#100—416no—
131C22⋊C8—28no—
132C22⋊C8—28no—
133#97—44no—
134C4×C8—44no—
135#95—24no—
136#95—24no—
137#95—24no—
138#95—24no—
139#95—22no—
140#95—22no—
141#95—22no—
142#95—22no—
143C24—22no—
144C24—24no—
145C24—22no—
146C24—42no—
147C24—81no—
148C24—81no—
149C24—81no—
150C24—81no—
151C24C2×C22⋊C412yes—
152C24C2×C22⋊C412yes—
153C24C23×C411yescharacteristic
154C2×D4—28no—
155C2×D4—28no—
156C2×D4—44no—
157C2×D4—416no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—24no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—44no—
171C22×C4—44no—
172C22×C4—44no—
173C22×C4—44no—
174C22×C4—84no—
175C22×C4—84no—
176C22×C4—81no—
177C22×C4—84no—
178C22×C4—84no—
179C22×C4—82no—
180C22×C4—82no—
181C22×C4—82no—
182C22×C4—82no—
183C22×C4—82no—
184C22×C4—82no—
185C22×C4—81no—
186C22×C4—81no—
187C22×C4C2×C22⋊C411yescharacteristic
188C22×C4C2×C22⋊C412yes—
189C22×C4C2×C22⋊C412yes—
190C22×C4C2×C22⋊C411yescharacteristic
191C22×C4C2×C22⋊C412yes—
192C22×C4#11712yes—
193C22×C4#11714yes—
194C22×C4#11712yes—
195C22×C4C22×D412yes—
196C22×C4C2×#4114yes—
197M16—416no—
198C2×C8—48no—
199C4⋊C4—28no—
200C4⋊C4—84no—
201C4⋊C4—84no—
202C22⋊C4—28no—
203C22⋊C4—28no—
204C22⋊C4—44no—
205C22⋊C4—44no—
206C22⋊C4—44no—
207C22⋊C4—44no—
208C22⋊C4—82no—
209C22⋊C4—82no—
210C22⋊C4—82no—
211C22⋊C4—82no—
212C42—48no—
213C42—84no—
214C42#12314yes—
215C42#12314yes—
216C42#12314yes—
217C42#12314yes—
218C23—28no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—24no—
224C23—41no—
225C23—41no—
226C23—42no—
227C23—42no—
228C23—44no—
229C23—42no—
230C23—42no—
231C23—82no—
232C23—82no—
233C23—82no—
234C23—82no—
235C23—88no—
236C23—88no—
237C23#41014yes—
238C23C22×C22⋊C411yescharacteristic
239C23C2×#11712yes—
240D4—416no—
241C2×C4—24no—
242C2×C4—22no—
243C2×C4—24no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—44no—
254C2×C4—44no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—84no—
264C2×C4—84no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—88no—
268C2×C4—88no—
269C2×C4#38114yes—
270C2×C4#38111yescharacteristic
271C2×C4#38112yes—
272C2×C4#38111yescharacteristic
273C2×C4C2×#12314yes—
274C8—88no—
275C22—21no—
276C22—21no—
277C22—22no—
278C22—24no—
279C22—22no—
280C22—22no—
281C22—44no—
282C22—44no—
283C22—44no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—42no—
289C22—84no—
290C22—88no—
291C22—88no—
292C22C2×#38111yescharacteristic
293C22C2×#41012yes—
294C4—24no—
295C4—21no—
296C4—22no—
297C4—24no—
298C4—42no—
299C4—42no—
300C4—84no—
301C4—84no—
302C4#182912yes—
303C2—22no—
304C2—42no—
305C2—44no—
306C2—82no—
307C2#2068811yescharacteristic
308C2C2×#182911yescharacteristic
309C2#2071311yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8755C2semidirect
6. #8116C2semidirect

Maximal subgroups

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

subgroupindex
#340572
#207262
#87552
#87522
#87512
#81162