Finite Group Databasebeta

← #190109

Subgroups — #190109

Order 512.

1784 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 1784 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#34174C211yesmaximal, characteristic
2#20584C212yesmaximal
3#8749C214yesmaximal, complemented
4#8748C214yesmaximal, complemented
5#8102C212yesmaximal, complemented
6#8100C212yesmaximal, complemented
7C23×#41C2211yescharacteristic
8C2×#519C412yescomplemented
9C2×#410—24no—
10#2341—24no—
11#2341—24no—
12#2309C412yescomplemented
13#2309C2212yes—
14#2309C2212yes—
15#2280C414yescomplemented
16#2280C2214yes—
17#2280C2211yescharacteristic
18#2280C2211yescharacteristic
19#1782—24no—
20#1781C2218yes—
21#1304C2214yes—
22#1304C2218yes—
23#1304C2214yes—
24C22×#41—22no—
25C22×#41—24no—
26C22×#41—22no—
27C22×#41—48no—
28C22×#41D414yes—
29C22×#41D414yes—
30C23×Q8C2×C411yescharacteristic
31C23×D4C2×C411yescharacteristic
32C23×D4C2312yes—
33C24×C4C2×C412yes—
34C24×C4C2311yescharacteristic
35#519—48no—
36C2×C4×Q8—24no—
37C2×C4×Q8—22no—
38C2×C4×Q8—22no—
39C2×#117—24no—
40C2×#117C2×C412yes—
41C2×#117C2×C412yes—
42C2×#117C2×C414yes—
43C2×#117C2×C414yes—
44#410—416no—
45C2×#99—216no—
46#390—24no—
47#390—24no—
48#390—28no—
49#390—28no—
50#381—24no—
51#381C2×C412yes—
52#381C2×C412yes—
53#380—28no—
54#380—22no—
55#380—22no—
56#380—24no—
57#380—24no—
58#379C2×C414yes—
59#379C2×C414yes—
60#379C2318yes—
61#379C2311yescharacteristic
62#379C2312yes—
63#379C2311yescharacteristic
64C4×C22⋊C4—28no—
65C4×C22⋊C4—22no—
66C4×C22⋊C4—22no—
67C4×C22⋊C4—24no—
68C4×C22⋊C4—24no—
69C25C22×C412yes—
70C25C2411yescharacteristic
71C2×#41—216no—
72C2×#41—28no—
73C2×#41—28no—
74C2×#41—28no—
75C2×#41—28no—
76C2×#41—48no—
77C2×#41—416no—
78C2×#41—416no—
79C2×#41—432no—
80C2×#41—416no—
81C22×Q8—28no—
82C22×Q8—22no—
83C22×Q8—22no—
84C22×Q8C22⋊C412yes—
85C22×Q8C22⋊C412yes—
86C22×D4—24no—
87C22×D4—24no—
88C22×D4—22no—
89C22×D4—22no—
90C22×D4—44no—
91C22×D4—48no—
92C22×D4C22⋊C412yes—
93C22×D4C22⋊C412yes—
94C22×D4C2×D418yes—
95C23×C4—22no—
96C23×C4—24no—
97C23×C4—22no—
98C23×C4—48no—
99C23×C4—44no—
100C23×C4C22⋊C418yes—
101C23×C4C22×C412yes—
102C23×C4C22×C411yescharacteristic
103C23×C4C22×C411yescharacteristic
104C23×C4C2×D412yes—
105C23×C4C2×D412yes—
106C23×C4#4112yes—
107C23×C4#4114yes—
108C23×C4#4112yes—
109C4×Q8—416no—
110C4×Q8—48no—
111C4×Q8—48no—
112#117—44no—
113#117—48no—
114#117—44no—
115#117—48no—
116#117—416no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—44no—
122C2×C22⋊C4—28no—
123C2×C22⋊C4—28no—
124C2×C22⋊C4—28no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—28no—
129C2×C22⋊C4—28no—
130C2×C22⋊C4C22×C414yes—
131C2×C22⋊C4C22×C412yes—
132C2×C22⋊C4C22×C412yes—
133C2×C42—22no—
134C2×C42—24no—
135C2×C42—24no—
136C2×C42—22no—
137C2×C42—44no—
138#99—432no—
139#95—24no—
140#95—24no—
141#95—44no—
142#95—44no—
143C24—22no—
144C24—22no—
145C24—44no—
146C24—42no—
147C24—82no—
148C24—84no—
149C24C2×C22⋊C414yes—
150C24C23×C411yescharacteristic
151C24C22×D412yes—
152C24C2×#4112yes—
153C24C2×#4112yes—
154#41—432no—
155#41—432no—
156#41—432no—
157#41—432no—
158C2×Q8—216no—
159C2×Q8—216no—
160C2×Q8—24no—
161C2×Q8—24no—
162C2×Q8—28no—
163C2×Q8—24no—
164C2×Q8#9918yes—
165C2×D4—216no—
166C2×D4—24no—
167C2×D4—24no—
168C2×D4—216no—
169C2×D4—28no—
170C2×D4—44no—
171C2×D4—48no—
172C2×D4—416no—
173C2×D4—416no—
174C2×D4—48no—
175C2×D4—48no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—28no—
179C22×C4—24no—
180C22×C4—28no—
181C22×C4—28no—
182C22×C4—28no—
183C22×C4—216no—
184C22×C4—24no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—44no—
188C22×C4—48no—
189C22×C4—44no—
190C22×C4—42no—
191C22×C4—48no—
192C22×C4—44no—
193C22×C4—48no—
194C22×C4—416no—
195C22×C4—48no—
196C22×C4—48no—
197C22×C4C2×C22⋊C412yes—
198C22×C4C2×C22⋊C411yescharacteristic
199C22×C4C2×C22⋊C414yes—
200C22×C4C2×C22⋊C411yescharacteristic
201C22×C4#11712yes—
202C22×C4#11712yes—
203C22×C4#11711yescharacteristic
204C22×C4#11712yes—
205C22×C4#11711yescharacteristic
206C22×C4#12314yes—
207C22×C4#12314yes—
208C22×C4#12314yes—
209C22×C4#12314yes—
210C4⋊C4—44no—
211C4⋊C4—48no—
212C4⋊C4—48no—
213C4⋊C4—44no—
214C4⋊C4—88no—
215C22⋊C4—44no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—416no—
219C22⋊C4—82no—
220C22⋊C4—82no—
221C22⋊C4—84no—
222C22⋊C4—88no—
223C22⋊C4—88no—
224C22⋊C4—88no—
225C22⋊C4—88no—
226C42—44no—
227C42—48no—
228C42—48no—
229C42—44no—
230C42—88no—
231C23—22no—
232C23—24no—
233C23—24no—
234C23—28no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—44no—
239C23—44no—
240C23—42no—
241C23—42no—
242C23—44no—
243C23—44no—
244C23—48no—
245C23—84no—
246C23—84no—
247C23—88no—
248C23—816no—
249C23C22×C22⋊C411yescharacteristic
250C23C2×#11711yescharacteristic
251C23C2×#11711yescharacteristic
252C23C2×#12314yes—
253Q8—216no—
254Q8—216no—
255D4—432no—
256D4—48no—
257D4—48no—
258C2×C4—28no—
259C2×C4—28no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—28no—
265C2×C4—28no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—416no—
274C2×C4—416no—
275C2×C4—48no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—88no—
282C2×C4—88no—
283C2×C4—88no—
284C2×C4#38112yes—
285C2×C4#38112yes—
286C2×C4#38112yes—
287C2×C4#38112yes—
288C2×C4C2×#9918yes—
289C2×C4C2×#9914yes—
290C22—22no—
291C22—21no—
292C22—21no—
293C22—24no—
294C22—24no—
295C22—24no—
296C22—22no—
297C22—22no—
298C22—44no—
299C22—44no—
300C22—48no—
301C22—84no—
302C22—88no—
303C22—84no—
304C22—84no—
305C22—88no—
306C22—88no—
307C22C2×#38111yescharacteristic
308C22C22×#9912yes—
309C4—22no—
310C4—22no—
311C4—22no—
312C4—28no—
313C4—84no—
314C4—82no—
315C4—82no—
316C4—168no—
317C4#181914yes—
318C2—22no—
319C2—44no—
320C2—41no—
321C2—41no—
322C2—84no—
323C2C2×#181911yescharacteristic
324C2#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. #8749C2semidirect
4. #8748C2semidirect
5. #8102C2semidirect
6. #8100C2semidirect
8. C2×#519C4semidirect
12. #2309C4semidirect
15. #2280C4semidirect

Maximal subgroups

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

subgroupindex
#341742
#205842
#87492
#87482
#81022
#81002