Finite Group Databasebeta

← #190119

Subgroups — #190119

Order 512.

1806 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 1806 conjugacy classes of subgroups — too many to draw. Only its 250 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#34044C211yesmaximal, characteristic
2#20584C212yesmaximal
3#8748C214yesmaximal, complemented
4#8747C214yesmaximal, complemented
5#8102C214yesmaximal, complemented
6C23×#41C2211yescharacteristic
7C22×#117C412yescomplemented
8C2×#410—24no—
9#2311—24no—
10#2310C412yescomplemented
11#2309C2212yes—
12#2308C412yescomplemented
13#2308C2212yes—
14#2297—24no—
15#2280C412yescomplemented
16#2280C2214yes—
17#2280C2212yes—
18#1782—24no—
19#1781C2214yes—
20#1781C2214yes—
21#1304C22116yes—
22C22×#41—28no—
23C22×#41—48no—
24C22×#41D414yes—
25C22×#41D414yes—
26C23×Q8C2×C411yescharacteristic
27C23×D4C2×C411yescharacteristic
28C23×D4C2311yescharacteristic
29C23×D4C2311yescharacteristic
30C24×C4C2×C411yescharacteristic
31C24×C4C2×C411yescharacteristic
32C24×C4C2311yescharacteristic
33C2×#117—22no—
34C2×#117—22no—
35C2×#117—24no—
36C2×#117—48no—
37C2×#117C2×C414yes—
38C2×#117C2×C414yes—
39C22×C4⋊C4—22no—
40C22×C22⋊C4C2×C414yes—
41C22×C42—22no—
42#410—416no—
43C2×#99—28no—
44C2×#99—28no—
45#391—28no—
46#390—28no—
47#386—28no—
48#386—28no—
49#381—24no—
50#381C2×C414yes—
51#381C2×C414yes—
52#380—24no—
53#380—24no—
54#380—28no—
55#379C2×C414yes—
56#379C2318yes—
57#379C2314yes—
58C4×C22⋊C4—24no—
59C4×C22⋊C4—24no—
60C4×C22⋊C4—28no—
61C25C22×C411yescharacteristic
62C25C22×C411yescharacteristic
63C25C2411yescharacteristic
64C2×#41—216no—
65C2×#41—28no—
66C2×#41—28no—
67C2×#41—216no—
68C2×#41—48no—
69C2×#41—48no—
70C2×#41—48no—
71C2×#41—464no—
72C22×Q8—24no—
73C22×Q8—44no—
74C22×Q8C22⋊C414yes—
75C22×D4—24no—
76C22×D4—24no—
77C22×D4—24no—
78C22×D4—44no—
79C22×D4—44no—
80C22×D4—44no—
81C22×D4C22⋊C414yes—
82C22×D4C2×D414yes—
83C22×D4C2×D414yes—
84C23×C4—28no—
85C23×C4—24no—
86C23×C4—24no—
87C23×C4—24no—
88C23×C4—44no—
89C23×C4—44no—
90C23×C4C22⋊C412yes—
91C23×C4C22⋊C412yes—
92C23×C4C22⋊C412yes—
93C23×C4C22⋊C412yes—
94C23×C4C22×C411yescharacteristic
95C23×C4C22×C411yescharacteristic
96C23×C4C22×C411yescharacteristic
97C23×C4C22×C411yescharacteristic
98C23×C4C2×D412yes—
99C23×C4C2×D412yes—
100C23×C4#4118yes—
101#117—48no—
102#117—48no—
103#117—48no—
104#117—48no—
105#117—416no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—48no—
109C2×C4⋊C4—44no—
110C2×C22⋊C4—24no—
111C2×C22⋊C4—28no—
112C2×C22⋊C4—28no—
113C2×C22⋊C4—28no—
114C2×C22⋊C4—216no—
115C2×C22⋊C4—48no—
116C2×C22⋊C4C22×C418yes—
117C2×C42—24no—
118C2×C42—24no—
119C2×C42—48no—
120C2×C42—44no—
121#99—416no—
122#99—416no—
123#95—28no—
124#95—28no—
125#95—48no—
126C24—22no—
127C24—24no—
128C24—22no—
129C24—42no—
130C24—42no—
131C24—82no—
132C24—82no—
133C24—82no—
134C24C2×C22⋊C412yes—
135C24C2×C22⋊C412yes—
136C24C23×C411yescharacteristic
137C24C22×D412yes—
138C24C2×#4114yes—
139#41—432no—
140#41—432no—
141#41—464no—
142C2×Q8—28no—
143C2×Q8—28no—
144C2×Q8—44no—
145C2×Q8—416no—
146C2×D4—28no—
147C2×D4—28no—
148C2×D4—28no—
149C2×D4—28no—
150C2×D4—28no—
151C2×D4—28no—
152C2×D4—44no—
153C2×D4—44no—
154C2×D4—44no—
155C2×D4—416no—
156C2×D4—416no—
157C2×D4—416no—
158C22×C4—232no—
159C22×C4—22no—
160C22×C4—24no—
161C22×C4—24no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—28no—
168C22×C4—22no—
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—28no—
172C22×C4—28no—
173C22×C4—416no—
174C22×C4—44no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—44no—
178C22×C4—416no—
179C22×C4—48no—
180C22×C4—48no—
181C22×C4—88no—
182C22×C4#9918yes—
183C22×C4C2×C22⋊C412yes—
184C22×C4C2×C22⋊C412yes—
185C22×C4C2×C22⋊C411yescharacteristic
186C22×C4C2×C22⋊C412yes—
187C22×C4C2×C22⋊C411yescharacteristic
188C22×C4#11714yes—
189C22×C4#11714yes—
190C22×C4#12314yes—
191C22×C4#12314yes—
192C22×C4#12318yes—
193C4⋊C4—48no—
194C4⋊C4—48no—
195C4⋊C4—88no—
196C22⋊C4—48no—
197C22⋊C4—48no—
198C22⋊C4—416no—
199C22⋊C4—88no—
200C22⋊C4—88no—
201C22⋊C4—816no—
202C42—48no—
203C42—48no—
204C42—88no—
205C23—28no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—24no—
210C23—22no—
211C23—24no—
212C23—42no—
213C23—44no—
214C23—48no—
215C23—44no—
216C23—44no—
217C23—44no—
218C23—44no—
219C23—84no—
220C23—84no—
221C23—88no—
222C23—88no—
223C23—88no—
224C23C2×#9914yes—
225C23C22×C22⋊C411yescharacteristic
226C23C2×#11712yes—
227C23C2×#12314yes—
228Q8—416no—
229D4—416no—
230D4—416no—
231D4—416no—
232C2×C4—28no—
233C2×C4—216no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—24no—
237C2×C4—22no—
238C2×C4—28no—
239C2×C4—28no—
240C2×C4—28no—
241C2×C4—216no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—44no—
245C2×C4—48no—
246C2×C4—48no—
247C2×C4—44no—
248C2×C4—48no—
249C2×C4—416no—
250C2×C4—48no—
251C2×C4—816no—
252C2×C4—88no—
253C2×C4—816no—
254C2×C4#38112yes—
255C2×C4#38114yes—
256C2×C4#38112yes—
257C2×C4C2×#9914yes—
258C2×C4C2×#9914yes—
259C22—22no—
260C22—22no—
261C22—24no—
262C22—24no—
263C22—28no—
264C22—24no—
265C22—44no—
266C22—44no—
267C22—44no—
268C22—44no—
269C22—42no—
270C22—44no—
271C22—84no—
272C22—84no—
273C22—88no—
274C22—88no—
275C22—88no—
276C22C2×#38111yescharacteristic
277C22C22×#9912yes—
278C4—22no—
279C4—24no—
280C4—24no—
281C4—42no—
282C4—88no—
283C4—168no—
284C4#181914yes—
285C2—22no—
286C2—44no—
287C2—42no—
288C2—44no—
289C2—82no—
290C2C2×#181911yescharacteristic
291C2#2071012yes—
subgroupG/H
center, socleC22C2×#381
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×#41C22
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8748C2semidirect
4. #8747C2semidirect
5. #8102C2semidirect
7. C22×#117C4semidirect
10. #2310C4semidirect
12. #2308C4semidirect
15. #2280C4semidirect

Maximal subgroups

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

subgroupindex
#340442
#205842
#87482
#87472
#81022