Finite Group Databasebeta

← #190188

Subgroups — #190188

Order 512.

1055 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 1055 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#17362C212yesmaximal
3#8758C214yesmaximal, complemented
4#8756C214yesmaximal, complemented
5#8108C214yesmaximal, complemented
6C22×C4×D4C2211yescharacteristic
7C2×#407C2214yes—
8#2318—24no—
9#2315C2212yes—
10#2313C2212yes—
11#2312C412yes—
12#2308C411yescharacteristic
13#2308C411yescharacteristic
14#2290—24no—
15#2283C412yes—
16#2280C2212yes—
17C2×C4×C22⋊C4C412yes—
18#1465—24no—
19#1463C2214yes—
20#1462C2214yes—
21#1459—24no—
22#1305C22116yes—
23C23×D4C2×C411yescharacteristic
24C24×C4C2×C411yescharacteristic
25C24×C4C2311yescharacteristic
26C22×M16—24no—
27C2×C4×D4—24no—
28C2×C4×D4—24no—
29C2×C4×D4D414yes—
30C2×C4×D4D414yes—
31C2×#117—22no—
32C22×C4⋊C4C2311yescharacteristic
33C22×C22⋊C4C2×C411yescharacteristic
34C22×C22⋊C4C2×C412yes—
35C22×C22⋊C4C2×C412yes—
36C22×C22⋊C4C2311yescharacteristic
37C22×C42—22no—
38C22×C42C2×C411yescharacteristic
39#407—24no—
40#407—24no—
41#407—24no—
42#407C2318yes—
43C2×C22⋊C8—24no—
44C2×C22⋊C8—24no—
45#388—24no—
46#388—24no—
47#387—24no—
48#387—24no—
49#386—24no—
50#386—24no—
51#385—24no—
52#385—24no—
53#383—22no—
54#382—22no—
55#381—22no—
56#381C2×C412yes—
57#381C2×C412yes—
58#380—22no—
59#380C2×C414yes—
60#380C2314yes—
61#379C2×C412yes—
62#379C2×C412yes—
63C4×C4⋊C4—22no—
64C4×C22⋊C4—22no—
65C4×C22⋊C4—22no—
66C4×C22⋊C4—22no—
67C4×C22⋊C4C2×C414yes—
68C4×C22⋊C4C2×C414yes—
69#376—22no—
70C2×#95—22no—
71#324—28no—
72#323—28no—
73C25C22×C411yescharacteristic
74C25C22×C411yescharacteristic
75C22×D4—24no—
76C22×D4—44no—
77C22×D4C22⋊C414yes—
78C23×C4—22no—
79C23×C4—22no—
80C23×C4—22no—
81C23×C4—22no—
82C23×C4—82no—
83C23×C4—82no—
84C23×C4C22⋊C412yes—
85C23×C4C22⋊C412yes—
86C23×C4C22×C411yescharacteristic
87C23×C4C22×C411yescharacteristic
88C23×C4C22×C411yescharacteristic
89C23×C4C22×C411yescharacteristic
90C23×C4C2×D412yes—
91C23×C4C2×D412yes—
92C23×C4#4114yes—
93C23×C4#4114yes—
94C23×C4C2411yescharacteristic
95C2×M16—28no—
96C2×M16—44no—
97C2×M16—88no—
98C22×C8—44no—
99C4×D4—28no—
100C4×D4—28no—
101C4×D4—28no—
102C4×D4—28no—
103#117—84no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4C2×D414yes—
108C2×C22⋊C4—24no—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—44no—
117C2×C22⋊C4—44no—
118C2×C22⋊C4C22⋊C414yes—
119C2×C22⋊C4C22×C414yes—
120C2×C22⋊C4C22×C414yes—
121C2×C22⋊C4C2×D414yes—
122C2×C42—22no—
123C2×C42—22no—
124C2×C42—24no—
125C2×C42—24no—
126C2×C42—84no—
127C2×C42C22⋊C412yes—
128C2×C42C22⋊C412yes—
129C22⋊C8—28no—
130C22⋊C8—28no—
131C22⋊C8—44no—
132C22⋊C8—44no—
133#95—24no—
134#95—24no—
135#95—24no—
136#95—22no—
137#95—22no—
138#95—22no—
139#95—22no—
140#95—24no—
141C24—22no—
142C24—24no—
143C24—42no—
144C24—81no—
145C24—81no—
146C24—81no—
147C24—81no—
148C24C2×C22⋊C412yes—
149C24C2×C22⋊C412yes—
150C24#11714yes—
151C24C23×C411yescharacteristic
152C2×D4—28no—
153C2×D4—28no—
154C2×D4—44no—
155C2×D4—416no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—24no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—44no—
165C22×C4—48no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—44no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—84no—
172C22×C4—82no—
173C22×C4—84no—
174C22×C4—81no—
175C22×C4—84no—
176C22×C4—82no—
177C22×C4—82no—
178C22×C4—82no—
179C22×C4—82no—
180C22×C4—82no—
181C22×C4—81no—
182C22×C4—81no—
183C22×C4C2×C22⋊C412yes—
184C22×C4C2×C22⋊C411yescharacteristic
185C22×C4C2×C22⋊C412yes—
186C22×C4C2×C22⋊C412yes—
187C22×C4C2×C22⋊C411yescharacteristic
188C22×C4#11712yes—
189C22×C4#11712yes—
190C22×C4#12314yes—
191C22×C4#12314yes—
192C22×C4#12314yes—
193C22×C4#12314yes—
194C22×C4C22×D412yes—
195C22×C4C2×#4114yes—
196M16—88no—
197M16—88no—
198C2×C8—48no—
199C2×C8—88no—
200C4⋊C4—28no—
201C4⋊C4—84no—
202C4⋊C4—84no—
203C22⋊C4—28no—
204C22⋊C4—28no—
205C22⋊C4—44no—
206C22⋊C4—44no—
207C22⋊C4—44no—
208C22⋊C4—44no—
209C22⋊C4—82no—
210C22⋊C4—82no—
211C22⋊C4—82no—
212C22⋊C4—82no—
213C42—22no—
214C42—24no—
215C42—22no—
216C42—48no—
217C42—84no—
218C23—28no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—24no—
223C23—41no—
224C23—41no—
225C23—42no—
226C23—42no—
227C23—44no—
228C23—44no—
229C23—44no—
230C23—42no—
231C23—42no—
232C23—82no—
233C23—82no—
234C23—88no—
235C23—88no—
236C23#38114yes—
237C23#41314yes—
238C23C22×C22⋊C411yescharacteristic
239C23C2×#11712yes—
240D4—416no—
241C2×C4—22no—
242C2×C4—28no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—42no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—44no—
254C2×C4—44no—
255C2×C4—44no—
256C2×C4—44no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—84no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—88no—
270C2×C4—88no—
271C2×C4#38111yescharacteristic
272C2×C4#38112yes—
273C2×C4#38111yescharacteristic
274C2×C4C2×#12314yes—
275C8—88no—
276C22—21no—
277C22—21no—
278C22—22no—
279C22—24no—
280C22—24no—
281C22—22no—
282C22—24no—
283C22—22no—
284C22—24no—
285C22—44no—
286C22—44no—
287C22—44no—
288C22—42no—
289C22—42no—
290C22—84no—
291C22—88no—
292C22—88no—
293C22C2×#38111yescharacteristic
294C22C2×#41312yes—
295C4—24no—
296C4—21no—
297C4—22no—
298C4—42no—
299C4—42no—
300C4—42no—
301C4—84no—
302C4—84no—
303C4#182812yes—
304C2—22no—
305C2—42no—
306C2—44no—
307C2—44no—
308C2#2069411yescharacteristic
309C2C2×#182811yescharacteristic
310C2#2071311yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8758C2semidirect
4. #8756C2semidirect
5. #8108C2semidirect

Maximal subgroups

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

subgroupindex
#340572
#173622
#87582
#87562
#81082