Finite Group Databasebeta

← #190099

Subgroups — #190099

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#8746C214yesmaximal, complemented
5#8100C214yesmaximal, 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—24no—
23C22×#41—24no—
24C22×#41—48no—
25C22×#41D414yes—
26C22×#41D414yes—
27C23×Q8C2×C411yescharacteristic
28C23×D4C2×C411yescharacteristic
29C23×D4C2311yescharacteristic
30C23×D4C2311yescharacteristic
31C24×C4C2×C411yescharacteristic
32C24×C4C2×C411yescharacteristic
33C24×C4C2311yescharacteristic
34C2×#117—22no—
35C2×#117—22no—
36C2×#117—24no—
37C2×#117—48no—
38C2×#117C2×C414yes—
39C2×#117C2×C414yes—
40C22×C4⋊C4—22no—
41C22×C22⋊C4C2×C414yes—
42C22×C42—22no—
43#410—416no—
44C2×#99—28no—
45C2×#99—28no—
46#391—28no—
47#390—28no—
48#386—28no—
49#386—28no—
50#381—24no—
51#381C2×C414yes—
52#381C2×C414yes—
53#380—24no—
54#380—24no—
55#380—28no—
56#379C2×C414yes—
57#379C2318yes—
58#379C2314yes—
59C4×C22⋊C4—24no—
60C4×C22⋊C4—24no—
61C4×C22⋊C4—28no—
62C25C22×C411yescharacteristic
63C25C22×C411yescharacteristic
64C25C2411yescharacteristic
65C2×#41—216no—
66C2×#41—28no—
67C2×#41—216no—
68C2×#41—28no—
69C2×#41—48no—
70C2×#41—48no—
71C2×#41—48no—
72C2×#41—432no—
73C2×#41—432no—
74C22×Q8—24no—
75C22×Q8—44no—
76C22×Q8C22⋊C414yes—
77C22×D4—24no—
78C22×D4—24no—
79C22×D4—24no—
80C22×D4—44no—
81C22×D4—44no—
82C22×D4—44no—
83C22×D4C22⋊C414yes—
84C22×D4C2×D414yes—
85C22×D4C2×D414yes—
86C23×C4—28no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—24no—
92C23×C4—44no—
93C23×C4—44no—
94C23×C4C22⋊C412yes—
95C23×C4C22⋊C412yes—
96C23×C4C22⋊C412yes—
97C23×C4C22⋊C412yes—
98C23×C4C22×C411yescharacteristic
99C23×C4C22×C411yescharacteristic
100C23×C4C22×C411yescharacteristic
101C23×C4C22×C411yescharacteristic
102C23×C4C2×D412yes—
103C23×C4C2×D412yes—
104C23×C4#4114yes—
105C23×C4#4114yes—
106#117—48no—
107#117—48no—
108#117—48no—
109#117—48no—
110#117—416no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—48no—
114C2×C4⋊C4—44no—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—28no—
117C2×C22⋊C4—28no—
118C2×C22⋊C4—28no—
119C2×C22⋊C4—216no—
120C2×C22⋊C4—48no—
121C2×C22⋊C4C22×C418yes—
122C2×C42—24no—
123C2×C42—24no—
124C2×C42—48no—
125C2×C42—44no—
126#99—416no—
127#99—416no—
128#95—28no—
129#95—28no—
130#95—48no—
131C24—22no—
132C24—24no—
133C24—22no—
134C24—42no—
135C24—42no—
136C24—82no—
137C24—82no—
138C24—82no—
139C24C2×C22⋊C412yes—
140C24C2×C22⋊C412yes—
141C24C23×C411yescharacteristic
142C24C22×D412yes—
143C24C2×#4114yes—
144#41—432no—
145#41—432no—
146#41—464no—
147C2×Q8—28no—
148C2×Q8—28no—
149C2×Q8—44no—
150C2×Q8—416no—
151C2×D4—28no—
152C2×D4—28no—
153C2×D4—28no—
154C2×D4—28no—
155C2×D4—28no—
156C2×D4—28no—
157C2×D4—44no—
158C2×D4—44no—
159C2×D4—44no—
160C2×D4—416no—
161C2×D4—416no—
162C2×D4—416no—
163C22×C4—216no—
164C22×C4—216no—
165C22×C4—22no—
166C22×C4—24no—
167C22×C4—24no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—28no—
175C22×C4—22no—
176C22×C4—24no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—28no—
180C22×C4—28no—
181C22×C4—48no—
182C22×C4—48no—
183C22×C4—44no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—44no—
187C22×C4—48no—
188C22×C4—48no—
189C22×C4—48no—
190C22×C4—48no—
191C22×C4—88no—
192C22×C4#9918yes—
193C22×C4C2×C22⋊C412yes—
194C22×C4C2×C22⋊C412yes—
195C22×C4C2×C22⋊C411yescharacteristic
196C22×C4C2×C22⋊C412yes—
197C22×C4C2×C22⋊C411yescharacteristic
198C22×C4#11712yes—
199C22×C4#11712yes—
200C22×C4#11714yes—
201C22×C4#12314yes—
202C22×C4#12314yes—
203C22×C4#12318yes—
204C4⋊C4—48no—
205C4⋊C4—48no—
206C4⋊C4—88no—
207C22⋊C4—48no—
208C22⋊C4—48no—
209C22⋊C4—416no—
210C22⋊C4—88no—
211C22⋊C4—88no—
212C22⋊C4—816no—
213C42—48no—
214C42—48no—
215C42—88no—
216C23—28no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—24no—
221C23—22no—
222C23—24no—
223C23—42no—
224C23—44no—
225C23—48no—
226C23—44no—
227C23—44no—
228C23—44no—
229C23—44no—
230C23—84no—
231C23—84no—
232C23—88no—
233C23—88no—
234C23—88no—
235C23C2×#9914yes—
236C23C22×C22⋊C411yescharacteristic
237C23C2×#11712yes—
238C23C2×#12314yes—
239Q8—416no—
240D4—416no—
241D4—416no—
242D4—416no—
243C2×C4—28no—
244C2×C4—216no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—28no—
250C2×C4—28no—
251C2×C4—24no—
252C2×C4—24no—
253C2×C4—28no—
254C2×C4—28no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—44no—
258C2×C4—48no—
259C2×C4—48no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—416no—
264C2×C4—48no—
265C2×C4—816no—
266C2×C4—88no—
267C2×C4—816no—
268C2×C4#38112yes—
269C2×C4#38114yes—
270C2×C4#38111yescharacteristic
271C2×C4#38111yescharacteristic
272C2×C4C2×#9914yes—
273C2×C4C2×#9914yes—
274C22—22no—
275C22—22no—
276C22—24no—
277C22—24no—
278C22—28no—
279C22—24no—
280C22—44no—
281C22—44no—
282C22—44no—
283C22—44no—
284C22—42no—
285C22—44no—
286C22—84no—
287C22—84no—
288C22—88no—
289C22—88no—
290C22—88no—
291C22C2×#38111yescharacteristic
292C22C22×#9912yes—
293C4—22no—
294C4—24no—
295C4—24no—
296C4—42no—
297C4—88no—
298C4—168no—
299C4#181912yes—
300C4#181912yes—
301C2—22no—
302C2—44no—
303C2—42no—
304C2—44no—
305C2—82no—
306C2C2×#181911yescharacteristic
307C2#2071012yes—
subgroupG/H
centerC2×C4#381
commutatorC24C23×C4
Frattini, cosocleC25C24
socleC22C2×#381
prime radicalC23×#41C22
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8748C2semidirect
4. #8746C2semidirect
5. #8100C2semidirect
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
#87462
#81002