Finite Group Databasebeta

← #190069

Subgroups — #190069

Order 512.

936 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 936 conjugacy classes of subgroups — too many to draw. Only its 226 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#34346C211yesmaximal, characteristic
2#21800C211yesmaximal, characteristic
3#21794C211yesmaximal, characteristic
4#8739C214yesmaximal, complemented
5#8738C214yesmaximal, complemented
6#7945C212yesmaximal, complemented
7#7944C212yesmaximal, complemented
8C2×#532C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2381—21no—
12#2380—21no—
13#2379—22no—
14#2377—24no—
15#2297C2214yes—
16#2290C418yescomplemented
17#2283C2212yes—
18#2283C2212yes—
19#1967—22no—
20#1966—21no—
21#1965—21no—
22#1964—24no—
23#1785C2212yes—
24#1785C2212yes—
25#1782C2214yes—
26#1284C2218yes—
27#1284C2214yes—
28#1284C2214yes—
29C22×#41C2312yes—
30#532—22no—
31#532—22no—
32#532—22no—
33#532—22no—
34C2×C4⋊Q8D411yescharacteristic
35C2×#127D411yescharacteristic
36C2×#124D412yes—
37C2×#121D414yes—
38C2×C4×D4C2×C414yes—
39C2×#117C2×C414yes—
40C22×C42C2×C414yes—
41C22×C42C2311yescharacteristic
42#432—22no—
43#432—22no—
44#431—22no—
45#431—22no—
46#423—21no—
47#423—21no—
48C2×#101—22no—
49C2×#101—22no—
50C2×#100—22no—
51C2×#100—22no—
52C2×#99—24no—
53C2×#99—24no—
54#407C2312yes—
55#407C2311yescharacteristic
56#407C2311yescharacteristic
57C4×M16—21no—
58C4×M16—21no—
59#391—24no—
60#390—24no—
61#388—28no—
62#387—28no—
63#386—28no—
64#386—24no—
65#386—24no—
66#385—28no—
67#383—22no—
68#383—22no—
69#382—22no—
70#382—22no—
71#381—22no—
72#381C2314yes—
73#380—22no—
74#380—22no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4C2×C414yes—
77C4×C22⋊C4—22no—
78C4×C22⋊C4C2×C418yes—
79C4×C22⋊C4C2314yes—
80#376—22no—
81#376C2×C414yes—
82C2×#41—24no—
83C2×#41—24no—
84C2×#41—82no—
85C2×#41—82no—
86C22×Q8C2×D412yes—
87C22×D4C22⋊C414yes—
88C22×D4C2×D414yes—
89C22×D4C2×D412yes—
90C23×C4—24no—
91C23×C4C22×C414yes—
92C23×C4C2×D412yes—
93C23×C4C2411yescharacteristic
94C2×M16—22no—
95C2×M16—21no—
96C2×M16—21no—
97C4⋊Q8—21no—
98C4⋊Q8—21no—
99C4⋊Q8—22no—
100#127—21no—
101#127—21no—
102#127—22no—
103#124—22no—
104#124—22no—
105#124—24no—
106#121—24no—
107#121—24no—
108#121—24no—
109#121—24no—
110C4×D4—24no—
111C4×D4—24no—
112C4×D4—24no—
113C4×D4—24no—
114#117—84no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4C22⋊C414yes—
119C2×C22⋊C4—24no—
120C2×C22⋊C4—24no—
121C2×C22⋊C4—28no—
122C2×C22⋊C4—24no—
123C2×C22⋊C4C22⋊C418yes—
124C2×C42—24no—
125C2×C42—42no—
126C2×C42—84no—
127C2×C42C22×C418yes—
128C2×C42C22×C412yes—
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42#4112yes—
132C2×C42#4112yes—
133C2×C42#4112yes—
134C2×C42#4112yes—
135C4⋊C8—42no—
136C4⋊C8—42no—
137#101—82no—
138#101—82no—
139#100—82no—
140#100—82no—
141#99—84no—
142#99—84no—
143C22⋊C8—22no—
144C22⋊C8—22no—
145C22⋊C8—24no—
146#97—41no—
147#97—41no—
148C4×C8—41no—
149C4×C8—41no—
150#95—24no—
151#95—24no—
152#95—28no—
153#95—24no—
154#95—24no—
155#95—24no—
156#95—24no—
157#95—42no—
158#95—42no—
159C24C2×C22⋊C414yes—
160C24C22×D411yescharacteristic
161#41—88no—
162#41—88no—
163C2×Q8—22no—
164C2×Q8—22no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×D4—28no—
168C2×D4—22no—
169C2×D4—22no—
170C2×D4—24no—
171C2×D4—24no—
172C2×D4—44no—
173C2×D4—44no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—84no—
177C22×C4—28no—
178C22×C4—28no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4—42no—
182C22×C4—44no—
183C22×C4—82no—
184C22×C4—84no—
185C22×C4—88no—
186C22×C4C2×C22⋊C414yes—
187C22×C4C2×C22⋊C412yes—
188C22×C4C2×C22⋊C412yes—
189C22×C4#12312yes—
190C22×C4#12312yes—
191C22×C4#12312yes—
192C22×C4#12312yes—
193C22×C4C23×C411yescharacteristic
194C22×C4C22×D411yescharacteristic
195C22×C4C2×#4112yes—
196C22×C4C2×#4112yes—
197M16—42no—
198M16—42no—
199M16—42no—
200M16—42no—
201C2×C8—42no—
202C2×C8—41no—
203C2×C8—41no—
204C4⋊C4—24no—
205C4⋊C4—24no—
206C4⋊C4—84no—
207C4⋊C4—84no—
208C4⋊C4—84no—
209C22⋊C4—28no—
210C22⋊C4—28no—
211C22⋊C4—48no—
212C22⋊C4—84no—
213C22⋊C4—84no—
214C22⋊C4—88no—
215C42—48no—
216C42—84no—
217C42—84no—
218C42#11712yes—
219C42#11712yes—
220C42#11712yes—
221C42#11712yes—
222C42#12311yescharacteristic
223C42#12311yescharacteristic
224C42#12311yescharacteristic
225C42#12311yescharacteristic
226C42#12312yes—
227C42#12312yes—
228C23—22no—
229C23—24no—
230C23—24no—
231C23—24no—
232C23—42no—
233C23—48no—
234C23C22×C22⋊C411yescharacteristic
235C23C2×#12312yes—
236Q8—82no—
237Q8—82no—
238D4—82no—
239D4—82no—
240D4—84no—
241D4—84no—
242D4—88no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—416no—
257C2×C4—48no—
258C2×C4—48no—
259C2×C4—48no—
260C2×C4—816no—
261C2×C4—84no—
262C2×C4—84no—
263C2×C4—84no—
264C2×C4#38112yes—
265C2×C4#38112yes—
266C2×C4#38112yes—
267C2×C4#38112yes—
268C2×C4C2×#11712yes—
269C2×C4C2×#12311yescharacteristic
270C2×C4C2×#12311yescharacteristic
271C8—82no—
272C8—81no—
273C8—81no—
274C22—22no—
275C22—22no—
276C22—22no—
277C22—22no—
278C22—44no—
279C22—48no—
280C22—82no—
281C22—84no—
282C22—84no—
283C22C2×#38111yescharacteristic
284C22#288312yes—
285C4—21no—
286C4—21no—
287C4—22no—
288C4—22no—
289C4—22no—
290C4—22no—
291C4—22no—
292C4—44no—
293C4—88no—
294C4—164no—
295C2—22no—
296C2—42no—
297C2—84no—
298C2#2069011yescharacteristic
299C2#2070812yes—
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ 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
4. #8739C2semidirect
5. #8738C2semidirect
6. #7945C2semidirect
7. #7944C2semidirect
16. #2290C4semidirect

Maximal subgroups

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

subgroupindex
#343462
#218002
#217942
#87392
#87382
#79452
#79442