Finite Group Databasebeta

← #117621

Subgroups — #117621

Order 512.

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

normal subgroups only

This group has 714 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
2#20694C211yesmaximal, characteristic
3#20685C212yesmaximal
4#8057C212yesmaximal
5C2×#1298C211yesmaximal, characteristic
6C22×#117C2211yescharacteristic
7C2×#407C2211yescharacteristic
8C2×#407C2212yes—
9C2×#407C2211yescharacteristic
10C22×C22⋊C8C2212yes—
11#1855—24no—
12#1822—22no—
13#1821C414yes—
14#1818—24no—
15#1817—24no—
16#1811—22no—
17#1742—22no—
18#1729—24no—
19#1298—22no—
20#1298—24no—
21C24×C4C2311yescharacteristic
22C22×M16—21no—
23C22×M16D414yes—
24C22×M16D412yes—
25C23×C8—22no—
26C2×#117—22no—
27C2×#117—21no—
28C2×#117—22no—
29C2×#117—21no—
30C2×#117—21no—
31C2×#117—21no—
32C2×#117—22no—
33C2×#117—22no—
34C22×C4⋊C4C2×C412yes—
35C22×C22⋊C4Q812yes—
36C22×C42D412yes—
37C2×#108—24no—
38#407—22no—
39#407—22no—
40#407—22no—
41#407—22no—
42#407—21no—
43#407—22no—
44#407—22no—
45#407C2×C412yes—
46#407C2×C412yes—
47#407D412yes—
48C2×C22⋊C8—22no—
49C2×C22⋊C8—22no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—22no—
52C2×C22⋊C8—22no—
53C2×C22⋊C8—22no—
54C2×C22⋊C8—22no—
55C2×C22⋊C8—22no—
56C2×C22⋊C8—22no—
57C2×C22⋊C8D414yes—
58#343—28no—
59#341—24no—
60#336—216no—
61C25C2×Q811yescharacteristic
62C23×C4—44no—
63C23×C4C4⋊C414yes—
64C23×C4C22×C411yescharacteristic
65C23×C4C2×D411yescharacteristic
66C23×C4C2×D412yes—
67C23×C4C2×D411yescharacteristic
68C23×C4C2×D412yes—
69C23×C4C2×D411yescharacteristic
70C23×C4#4111yescharacteristic
71C23×C4#4111yescharacteristic
72C23×C4#4112yes—
73C23×C4#4112yes—
74C2×M16—22no—
75C2×M16—22no—
76C2×M16—24no—
77C2×M16—24no—
78C2×M16—48no—
79C2×M16—41no—
80C2×M16—44no—
81C2×M16—44no—
82C2×M16C22⋊C414yes—
83C22×C8—22no—
84C22×C8—24no—
85C22×C8—24no—
86C22×C8—48no—
87C22×C8—42no—
88C22×C8—42no—
89C22×C8—41no—
90#117—41no—
91#117—42no—
92#117—42no—
93#117—42no—
94#117—42no—
95#117—41no—
96#117—42no—
97#117—41no—
98#117—42no—
99#117—41no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C22⋊C4—24no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—24no—
110C2×C42—24no—
111C2×C42—22no—
112C2×C42—22no—
113C2×C42—24no—
114#108—84no—
115C22⋊C8—24no—
116C22⋊C8—24no—
117C22⋊C8—24no—
118C22⋊C8—24no—
119C22⋊C8—24no—
120C22⋊C8—42no—
121C22⋊C8—42no—
122C22⋊C8—42no—
123C22⋊C8—42no—
124C22⋊C8—42no—
125C22⋊C8—42no—
126C22⋊C8—42no—
127C24—42no—
128C24—44no—
129C24C2×C4⋊C411yescharacteristic
130C24C22⋊Q812yes—
131C24C22⋊Q811yescharacteristic
132C24C22⋊Q812yes—
133C24C22⋊Q811yescharacteristic
134C22×C4—28no—
135C22×C4—24no—
136C22×C4—24no—
137C22×C4—24no—
138C22×C4—24no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—44no—
143C22×C4—44no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—44no—
147C22×C4—48no—
148C22×C4C2×C22⋊C411yescharacteristic
149C22×C4#11711yescharacteristic
150C22×C4C4×D412yes—
151C22×C4C4×D412yes—
152C22×C4#12012yes—
153C22×C4#12011yescharacteristic
154C22×C4#12112yes—
155C22×C4#12112yes—
156C22×C4#12111yescharacteristic
157C22×C4#12112yes—
158C22×C4#12111yescharacteristic
159C22×C4#12311yescharacteristic
160C22×C4#12312yes—
161C22×C4#12312yes—
162C22×C4#12412yes—
163C22×C4#12411yescharacteristic
164C22×C4#12612yes—
165C22×C4#12711yescharacteristic
166M16—44no—
167M16—44no—
168M16—48no—
169M16—82no—
170M16—82no—
171M16—84no—
172C2×C8—24no—
173C2×C8—44no—
174C2×C8—48no—
175C2×C8—48no—
176C2×C8—42no—
177C2×C8—44no—
178C2×C8—44no—
179C2×C8—84no—
180C2×C8—84no—
181C2×C8—82no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C22⋊C4—42no—
187C22⋊C4—44no—
188C22⋊C4—42no—
189C42—42no—
190C42—44no—
191C42—42no—
192C23—24no—
193C23—24no—
194C23—24no—
195C23—24no—
196C23—22no—
197C23—41no—
198C23—42no—
199C23—42no—
200C23—41no—
201C23—42no—
202C23—42no—
203C23—42no—
204C23—88no—
205C23#38011yescharacteristic
206C23#38512yes—
207C23#39311yescharacteristic
208C23#39612yes—
209C23#39911yescharacteristic
210C2×C4—24no—
211C2×C4—22no—
212C2×C4—24no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—24no—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—22no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—48no—
228C2×C4—48no—
229C2×C4—44no—
230C2×C4—84no—
231C2×C4#38611yescharacteristic
232C2×C4#38812yes—
233C2×C4#39011yescharacteristic
234C2×C4#39211yescharacteristic
235C2×C4#39412yes—
236C2×C4#39711yescharacteristic
237C2×C4#45814yes—
238C8—44no—
239C8—84no—
240C8—82no—
241C8—84no—
242C22—22no—
243C22—22no—
244C22—22no—
245C22—21no—
246C22—22no—
247C22—21no—
248C22—22no—
249C22—22no—
250C22—22no—
251C22—22no—
252C22—44no—
253C22—48no—
254C22—84no—
255C22—82no—
256C22—82no—
257C22#143911yescharacteristic
258C22#185814yes—
259C22#203012yes—
260C4—21no—
261C4—22no—
262C4—22no—
263C4—21no—
264C4—84no—
265C4#189012yes—
266C4#202212yes—
267C2—24no—
268C2—22no—
269C2—41no—
270C2—42no—
271C2—41no—
272C2#1107011yescharacteristic
273C2#1113712yes—
subgroupG/H
centerC2×C4#392
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
socleC22#1439
prime radicalC25C2×Q8
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#219492
#206942
#206852
#80572
C2×#12982