Finite Group Databasebeta

← #119468

Subgroups — #119468

Order 512.

902 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 902 conjugacy classes of subgroups — too many to draw. Only its 138 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
1C2×#1821C211yesmaximal, characteristic
2#14342C211yesmaximal, characteristic
3#13924C211yesmaximal, characteristic
4#8015C212yesmaximal
5C2×#1284C212yesmaximal
6C23×C4⋊C4C2211yescharacteristic
7C2×#407C2212yes—
8C2×#381C2212yes—
9C2×#380C2211yescharacteristic
10C2×C4×C22⋊C4C2211yescharacteristic
11#1821—24no—
12C2×#343—22no—
13#1445—24no—
14#1445—22no—
15C4⋊#95C414yescomplemented
16C22⋊#95—22no—
17C22⋊#95—22no—
18#1294—24no—
19#1284—24no—
20C24×C4D412yes—
21C24×C4C2311yescharacteristic
22C22×M16—22no—
23C22×C4⋊C4—22no—
24C22×C4⋊C4—22no—
25C22×C4⋊C4—22no—
26C22×C4⋊C4—22no—
27C22×C4⋊C4—22no—
28C22×C4⋊C4—22no—
29C22×C4⋊C4C2×C412yes—
30C22×C4⋊C4C2×C412yes—
31C22×C4⋊C4D412yes—
32C22×C22⋊C4—21no—
33C22×C22⋊C4—21no—
34C22×C22⋊C4—22no—
35C22×C42C2×C412yes—
36#407—22no—
37#407—22no—
38#407—22no—
39#407D414yes—
40C2×C22⋊C8—22no—
41C2×C22⋊C8—22no—
42#381—22no—
43#381—22no—
44#381—22no—
45#381—22no—
46#380—22no—
47#380—22no—
48C4×C22⋊C4—22no—
49C4×C22⋊C4—22no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—22no—
53C2×#95—28no—
54C2×#95—28no—
55C2×#95D414yes—
56#343—28no—
57C25C2×D411yescharacteristic
58C23×C4—22no—
59C23×C4—22no—
60C23×C4—24no—
61C23×C4—24no—
62C23×C4—22no—
63C23×C4—22no—
64C23×C4—24no—
65C23×C4—44no—
66C23×C4—84no—
67C23×C4C22⋊C414yes—
68C23×C4C22×C411yescharacteristic
69C23×C4C2×D411yescharacteristic
70C23×C4C2×D412yes—
71C23×C4C2×D412yes—
72C23×C4#4112yes—
73C23×C4#4112yes—
74C23×C4#4112yes—
75C23×C4#4112yes—
76C2×M16—24no—
77C2×M16—42no—
78C2×M16—48no—
79C22×C8—42no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4—24no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—44no—
98C2×C22⋊C4—24no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4—42no—
101C2×C22⋊C4—48no—
102C2×C22⋊C4—44no—
103C2×C22⋊C4—41no—
104C2×C22⋊C4—42no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—42no—
107C2×C42—24no—
108C2×C42—44no—
109C22⋊C8—24no—
110C22⋊C8—24no—
111C22⋊C8—42no—
112C22⋊C8—42no—
113#95—24no—
114#95—24no—
115#95—41no—
116#95—42no—
117#95—42no—
118#95—41no—
119#95—42no—
120#95—88no—
121#95—88no—
122C24—24no—
123C24—22no—
124C24—22no—
125C24—42no—
126C24C2×C22⋊C411yescharacteristic
127C24#12011yescharacteristic
128C24#12112yes—
129C24#12311yescharacteristic
130C24#12312yes—
131C22×C4—24no—
132C22×C4—24no—
133C22×C4—24no—
134C22×C4—24no—
135C22×C4—24no—
136C22×C4—24no—
137C22×C4—44no—
138C22×C4—44no—
139C22×C4—44no—
140C22×C4—44no—
141C22×C4—44no—
142C22×C4—44no—
143C22×C4—48no—
144C22×C4—44no—
145C22×C4—48no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—44no—
149C22×C4—44no—
150C22×C4—44no—
151C22×C4—48no—
152C22×C4—44no—
153C22×C4—48no—
154C22×C4—44no—
155C22×C4—42no—
156C22×C4—44no—
157C22×C4—44no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—84no—
161C22×C4—88no—
162C22×C4—88no—
163C22×C4#11711yescharacteristic
164C22×C4#11711yescharacteristic
165C22×C4C4×D412yes—
166C22×C4C4×D412yes—
167C22×C4#12011yescharacteristic
168C22×C4#12112yes—
169C22×C4#12112yes—
170C22×C4#12112yes—
171C22×C4#12312yes—
172C22×C4#12311yescharacteristic
173C22×C4#12312yes—
174C22×C4#12312yes—
175C22×C4#12412yes—
176C22×C4#12412yes—
177C22×C4#12612yes—
178C22×C4#12612yes—
179M16—48no—
180M16—84no—
181C2×C8—44no—
182C2×C8—84no—
183C4⋊C4—44no—
184C4⋊C4—44no—
185C4⋊C4—44no—
186C4⋊C4—44no—
187C4⋊C4—84no—
188C22⋊C4—48no—
189C22⋊C4—84no—
190C22⋊C4—84no—
191C22⋊C4—82no—
192C22⋊C4—82no—
193C42—84no—
194C23—21no—
195C23—21no—
196C23—22no—
197C23—22no—
198C23—22no—
199C23—24no—
200C23—21no—
201C23—21no—
202C23—24no—
203C23—22no—
204C23—22no—
205C23—24no—
206C23—24no—
207C23—42no—
208C23—42no—
209C23—48no—
210C23—48no—
211C23#38111yescharacteristic
212C23#38611yescharacteristic
213C23#38611yescharacteristic
214C23#39412yes—
215C23#39712yes—
216C23#41014yes—
217C23#45412yes—
218C23#45612yes—
219C2×C4—24no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—24no—
223C2×C4—24no—
224C2×C4—24no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—44no—
236C2×C4—48no—
237C2×C4—48no—
238C2×C4—48no—
239C2×C4—44no—
240C2×C4—82no—
241C2×C4—84no—
242C2×C4—88no—
243C2×C4—88no—
244C2×C4—82no—
245C2×C4—84no—
246C2×C4—84no—
247C2×C4—84no—
248C2×C4—88no—
249C2×C4—88no—
250C2×C4—88no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—84no—
255C2×C4#38812yes—
256C2×C4#38812yes—
257C2×C4#39412yes—
258C2×C4#39712yes—
259C8—84no—
260C22—21no—
261C22—21no—
262C22—22no—
263C22—21no—
264C22—21no—
265C22—22no—
266C22—24no—
267C22—24no—
268C22—24no—
269C22—24no—
270C22—22no—
271C22—22no—
272C22—22no—
273C22—48no—
274C22—48no—
275C22—42no—
276C22—42no—
277C22#144011yescharacteristic
278C22#182012yes—
279C22#188611yescharacteristic
280C22#188811yescharacteristic
281C22#189912yes—
282C22#189912yes—
283C22#191112yes—
284C22#203814yes—
285C4—22no—
286C4—22no—
287C4—22no—
288C4—22no—
289C4—84no—
290C4—84no—
291C4—164no—
292C4—162no—
293C2—22no—
294C2—24no—
295C2—22no—
296C2—42no—
297C2#946112yes—
298C2#946512yes—
299C2#1115511yescharacteristic
300C2#1116211yescharacteristic
301C2#1119711yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. C4⋊#95C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18212
#143422
#139242
#80152
C2×#12842