Finite Group Databasebeta

← #117642

Subgroups — #117642

Order 512.

778 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 778 conjugacy classes of subgroups — too many to draw. Only its 134 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#21949C211yesmaximal, characteristic
2C2×#1826C212yesmaximal
3#20694C211yesmaximal, characteristic
4C2×#1297C211yesmaximal, characteristic
5#7929C212yesmaximal
6C22×#117C2211yescharacteristic
7C2×#407C2211yescharacteristic
8C2×#407C2212yes—
9C2×#407C2211yescharacteristic
10C2×C4×C22⋊C4C2212yes—
11#1855—24no—
12#1826—24no—
13#1826—24no—
14#1822—22no—
15#1821C414yescomplemented
16#1811—22no—
17#1742—22no—
18C2×#339—24no—
19#1297—22no—
20#1282—24no—
21C24×C4C2311yescharacteristic
22C22×M16—21no—
23C22×M16D414yes—
24C22×M16D412yes—
25C2×#117—21no—
26C2×#117—21no—
27C2×#117—22no—
28C2×#117—22no—
29C2×#117—22no—
30C2×#117—22no—
31C2×#117D412yes—
32C2×#117Q812yes—
33C22×C4⋊C4C2×C412yes—
34C22×C22⋊C4—21no—
35C22×C22⋊C4—22no—
36C22×C42—21no—
37C22×C42—22no—
38C2×#108—24no—
39#407—22no—
40#407—22no—
41#407—22no—
42#407—21no—
43#407—22no—
44#407—22no—
45#407—22no—
46#407C2×C412yes—
47#407C2×C412yes—
48#407D412yes—
49C2×C22⋊C8—22no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—22no—
52C2×C22⋊C8—22no—
53C4×C22⋊C4—22no—
54C4×C22⋊C4—22no—
55C4×C22⋊C4—22no—
56C4×C22⋊C4D414yes—
57C2×#95—22no—
58#343—28no—
59#341—24no—
60#339—216no—
61C25#4111yescharacteristic
62C23×C4—22no—
63C23×C4—44no—
64C23×C4—42no—
65C23×C4C22×C411yescharacteristic
66C23×C4C2×D411yescharacteristic
67C23×C4C2×D412yes—
68C23×C4C2×D411yescharacteristic
69C23×C4C2×D411yescharacteristic
70C23×C4C2×D412yes—
71C23×C4C2×Q811yescharacteristic
72C23×C4#4112yes—
73C23×C4#4112yes—
74C23×C4#4111yescharacteristic
75C2×M16—22no—
76C2×M16—22no—
77C2×M16—24no—
78C2×M16—24no—
79C2×M16—41no—
80C2×M16—44no—
81C2×M16—48no—
82C2×M16—44no—
83C2×M16C22⋊C414yes—
84C22×C8—22no—
85C22×C8—24no—
86C22×C8—41no—
87#117—24no—
88#117—24no—
89#117—24no—
90#117—24no—
91#117—41no—
92#117—42no—
93#117—42no—
94#117—41no—
95#117—41no—
96#117—41no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4C4⋊C414yes—
103C2×C22⋊C4—24no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—48no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4—42no—
109C2×C22⋊C4—42no—
110C2×C42—22no—
111C2×C42—22no—
112C2×C42—24no—
113C2×C42—42no—
114C2×C42—42no—
115C2×C42—48no—
116C2×C42—42no—
117#108—84no—
118C22⋊C8—24no—
119C22⋊C8—24no—
120C22⋊C8—24no—
121C22⋊C8—42no—
122C22⋊C8—42no—
123C22⋊C8—42no—
124C22⋊C8—42no—
125C22⋊C8—42no—
126#95—24no—
127#95—42no—
128C24—44no—
129C24—42no—
130C24#11711yescharacteristic
131C24#12111yescharacteristic
132C24C22⋊Q811yescharacteristic
133C24#12312yes—
134C24#12412yes—
135C22×C4—28no—
136C22×C4—24no—
137C22×C4—24no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—44no—
142C22×C4—44no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—44no—
146C22×C4—48no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—44no—
150C22×C4—84no—
151C22×C4—84no—
152C22×C4—84no—
153C22×C4C2×C22⋊C411yescharacteristic
154C22×C4C2×C4⋊C411yescharacteristic
155C22×C4C4×D412yes—
156C22×C4C4×D412yes—
157C22×C4#12012yes—
158C22×C4#12011yescharacteristic
159C22×C4#12112yes—
160C22×C4#12112yes—
161C22×C4#12111yescharacteristic
162C22×C4#12112yes—
163C22×C4C22⋊Q812yes—
164C22×C4C22⋊Q812yes—
165C22×C4C22⋊Q811yescharacteristic
166C22×C4#12311yescharacteristic
167C22×C4#12312yes—
168C22×C4#12411yescharacteristic
169C22×C4#12612yes—
170C22×C4#12711yescharacteristic
171M16—44no—
172M16—44no—
173M16—48no—
174M16—48no—
175M16—82no—
176M16—82no—
177C2×C8—24no—
178C2×C8—44no—
179C2×C8—42no—
180C2×C8—44no—
181C2×C8—82no—
182C2×C8—84no—
183C4⋊C4—24no—
184C4⋊C4—24no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C22⋊C4—28no—
188C22⋊C4—44no—
189C22⋊C4—48no—
190C22⋊C4—42no—
191C22⋊C4—44no—
192C22⋊C4—42no—
193C42—48no—
194C42—42no—
195C42—44no—
196C42—42no—
197C42—84no—
198C23—24no—
199C23—24no—
200C23—24no—
201C23—24no—
202C23—22no—
203C23—41no—
204C23—42no—
205C23—42no—
206C23—41no—
207C23—42no—
208C23—42no—
209C23—42no—
210C23—88no—
211C23#38011yescharacteristic
212C23#38812yes—
213C23#39311yescharacteristic
214C23#39412yes—
215C23#39911yescharacteristic
216C2×C4—22no—
217C2×C4—24no—
218C2×C4—24no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—24no—
222C2×C4—24no—
223C2×C4—24no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—48no—
234C2×C4—44no—
235C2×C4—84no—
236C2×C4—84no—
237C2×C4—84no—
238C2×C4—816no—
239C2×C4—84no—
240C2×C4—84no—
241C2×C4#38512yes—
242C2×C4#38611yescharacteristic
243C2×C4#39011yescharacteristic
244C2×C4#39211yescharacteristic
245C2×C4#39612yes—
246C2×C4#39711yescharacteristic
247C2×C4#45814yes—
248C8—44no—
249C8—82no—
250C8—84no—
251C22—22no—
252C22—22no—
253C22—22no—
254C22—22no—
255C22—21no—
256C22—21no—
257C22—22no—
258C22—22no—
259C22—22no—
260C22—22no—
261C22—48no—
262C22—44no—
263C22—84no—
264C22—82no—
265C22—82no—
266C22#143911yescharacteristic
267C22#193214yes—
268C22#203012yes—
269C4—21no—
270C4—22no—
271C4—22no—
272C4—21no—
273C4—88no—
274C4—84no—
275C4#189012yes—
276C4#202212yes—
277C2—24no—
278C2—22no—
279C2—42no—
280C2—41no—
281C2—41no—
282C2#1107011yescharacteristic
283C2#1114612yes—
subgroupG/H
centerC2×C4#392
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
socleC22#1439
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #1821C4semidirect

Maximal subgroups

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

subgroupindex
#219492
C2×#18262
#206942
C2×#12972
#79292