Finite Group Databasebeta

← #106774

Subgroups — #106774

Order 512.

914 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 188 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#34179C211yesmaximal, characteristic
2#20694C211yesmaximal, characteristic
3#20685C211yesmaximal, characteristic
4#8108C212yesmaximal
5#7925C212yesmaximal
6C22×#117C2211yescharacteristic
7C2×#407C2211yescharacteristic
8C2×#407C2212yes—
9C22×C22⋊C8C2211yescharacteristic
10#2347—24no—
11#2346—24no—
12#2316—21no—
13#2312C414yes—
14#2283—22no—
15#2280C2212yes—
16#2274—21no—
17#1822—22no—
18#1821—21no—
19#1821—21no—
20#1818C414yescomplemented
21#1817—22no—
22#1742C414yescomplemented
23#1729—22no—
24#1305—24no—
25#1282—24no—
26C24×C4C2311yescharacteristic
27C22×M16—21no—
28C22×M16C2×C414yes—
29C23×C8—21no—
30C2×#117—22no—
31C2×#117—22no—
32C2×#117—22no—
33C2×#117—22no—
34C2×#117—22no—
35C2×#117C2×C412yes—
36C2×#117C2×C412yes—
37C2×#117D412yes—
38C2×#117Q812yes—
39C22×C4⋊C4D411yescharacteristic
40C22×C4⋊C4D411yescharacteristic
41C22×C22⋊C4C2×C412yes—
42C22×C42D412yes—
43#407—22no—
44#407—22no—
45#407—22no—
46#407—22no—
47#407—22no—
48#407—22no—
49C2×C22⋊C8—22no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—22no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—22no—
55C2×C22⋊C8C2×C412yes—
56C2×C22⋊C8C2×C412yes—
57#389—28no—
58#388—216no—
59#383—24no—
60#382—24no—
61#381—22no—
62#380—24no—
63#380—22no—
64#380D414yes—
65#379C2×C414yes—
66C4×C4⋊C4—28no—
67C4×C4⋊C4—24no—
68C4×C22⋊C4—24no—
69C4×C22⋊C4—22no—
70C4×C22⋊C4D414yes—
71#376—24no—
72#343—24no—
73#343—24no—
74#336—28no—
75C25C22×C411yescharacteristic
76C23×C4—44no—
77C23×C4C22⋊C412yes—
78C23×C4C22⋊C412yes—
79C23×C4C22×C411yescharacteristic
80C23×C4C22×C411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C23×C4C2×D412yes—
85C23×C4C2×D412yes—
86C23×C4C2×Q811yescharacteristic
87C23×C4#4112yes—
88C23×C4#4112yes—
89C2×M16—24no—
90C2×M16—24no—
91C2×M16—44no—
92C2×M16—42no—
93C2×M16—48no—
94C22×C8—22no—
95C22×C8—22no—
96C22×C8—24no—
97C22×C8—44no—
98C22×C8—42no—
99#117—24no—
100#117—24no—
101#117—24no—
102#117—24no—
103#117—24no—
104#117—24no—
105#117—24no—
106#117—24no—
107#117—48no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—28no—
114C2×C4⋊C4C22⋊C412yes—
115C2×C4⋊C4C22⋊C412yes—
116C2×C4⋊C4C4⋊C412yes—
117C2×C4⋊C4C4⋊C412yes—
118C2×C22⋊C4—24no—
119C2×C22⋊C4—24no—
120C2×C22⋊C4—24no—
121C2×C22⋊C4—24no—
122C2×C22⋊C4C4214yes—
123C2×C22⋊C4C22⋊C418yes—
124C2×C22⋊C4C4⋊C414yes—
125C2×C42—22no—
126C2×C42—22no—
127C2×C42—22no—
128C2×C42—22no—
129C2×C42—24no—
130C2×C42—28no—
131C2×C42C22⋊C414yes—
132C2×C42C4⋊C414yes—
133C22⋊C8—24no—
134C22⋊C8—24no—
135C22⋊C8—41no—
136C22⋊C8—41no—
137C22⋊C8—42no—
138C22⋊C8—42no—
139C22⋊C8—42no—
140C22⋊C8—41no—
141C22⋊C8—41no—
142C22⋊C8—42no—
143#95—28no—
144#95—28no—
145#95—24no—
146#95—24no—
147C24—42no—
148C24—42no—
149C24—81no—
150C24C2×C4211yescharacteristic
151C24C2×C22⋊C411yescharacteristic
152C24C2×C22⋊C412yes—
153C24C2×C4⋊C411yescharacteristic
154C24#11712yes—
155C22×C4—24no—
156C22×C4—24no—
157C22×C4—24no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—28no—
163C22×C4—44no—
164C22×C4—44no—
165C22×C4—44no—
166C22×C4—44no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—44no—
171C22×C4—82no—
172C22×C4#9514yes—
173C22×C4#9514yes—
174C22×C4C2×C22⋊C411yescharacteristic
175C22×C4C2×C22⋊C411yescharacteristic
176C22×C4C2×C4⋊C411yescharacteristic
177C22×C4C2×C4⋊C411yescharacteristic
178C22×C4C4×D412yes—
179C22×C4C4×D412yes—
180C22×C4C4×D412yes—
181C22×C4C4×D412yes—
182C22×C4#12012yes—
183C22×C4#12011yescharacteristic
184C22×C4#12011yescharacteristic
185C22×C4#12112yes—
186C22×C4#12112yes—
187C22×C4C22⋊Q812yes—
188C22×C4C22⋊Q812yes—
189C22×C4#12312yes—
190C22×C4#12311yescharacteristic
191C22×C4#12311yescharacteristic
192M16—48no—
193M16—84no—
194M16—84no—
195C2×C8—44no—
196C2×C8—44no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—44no—
200C2×C8—84no—
201C2×C8—84no—
202C4⋊C4—24no—
203C4⋊C4—24no—
204C4⋊C4—24no—
205C4⋊C4—24no—
206C4⋊C4—48no—
207C4⋊C4—88no—
208C22⋊C4—28no—
209C22⋊C4—28no—
210C22⋊C4—48no—
211C22⋊C4—48no—
212C22⋊C4—84no—
213C42—24no—
214C42—24no—
215C42—24no—
216C42—24no—
217C42—48no—
218C42—88no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—24no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—41no—
231C23—42no—
232C23—41no—
233C23—42no—
234C23—88no—
235C23C2×#9511yescharacteristic
236C23C4×C22⋊C412yes—
237C23#37911yescharacteristic
238C23#38012yes—
239C23#38111yescharacteristic
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—48no—
264C2×C4—48no—
265C2×C4—48no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4#38511yescharacteristic
269C2×C4#38511yescharacteristic
270C2×C4#38512yes—
271C2×C4#38611yescharacteristic
272C2×C4#38611yescharacteristic
273C2×C4#38612yes—
274C2×C4#41014yes—
275C8—82no—
276C8—82no—
277C8—84no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—22no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—21no—
288C22—21no—
289C22—22no—
290C22—22no—
291C22—44no—
292C22—44no—
293C22—84no—
294C22—82no—
295C22—84no—
296C22C22⋊#9511yescharacteristic
297C22#174012yes—
298C22#179112yes—
299C4—21no—
300C4—21no—
301C4—22no—
302C4—21no—
303C4—21no—
304C4—44no—
305C4—44no—
306C4—88no—
307C4#189914yes—
308C2—22no—
309C2—22no—
310C2—41no—
311C2—42no—
312C2—42no—
313C2#1073611yescharacteristic
314C2#1074711yescharacteristic
315C2#1075611yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC24C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
20. #1818C4semidirect
22. #1742C4semidirect

Maximal subgroups

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

subgroupindex
#341792
#206942
#206852
#81082
#79252