Finite Group Databasebeta

← #104715

Subgroups — #104715

Order 512.

589 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 589 conjugacy classes of subgroups — too many to draw. Only its 136 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#17073C211yesmaximal, characteristic
2#11373C211yesmaximal, characteristic
3#11357C212yesmaximal
4#7025C212yesmaximal
5#7025C211yesmaximal, characteristic
6C2×#398—22no—
7C2×#384C2211yescharacteristic
8C2×#340—22no—
9C2×#340—22no—
10C2×#340—22no—
11C2×#336C2212yes—
12C2×#336C2211yescharacteristic
13#1444—22no—
14#1441—22no—
15C4⋊#95C2212yes—
16C4⋊#95C2211yescharacteristic
17C23×C8—22no—
18C23×C8—21no—
19C22×C4⋊C4—21no—
20C22×C4⋊C4—22no—
21C22×C4⋊C4—22no—
22C22×C4⋊C4—22no—
23C22×C4⋊C4D412yes—
24C22×C4⋊C4D412yes—
25C22×C4⋊C4D412yes—
26C22×C42C2311yescharacteristic
27#398—44no—
28#398—44no—
29#384—22no—
30#384—21no—
31#384Q812yes—
32C2×#95—22no—
33C2×#95—22no—
34C2×#95—22no—
35C2×#95—22no—
36C2×#95—22no—
37#340—44no—
38#340—44no—
39#340—44no—
40#336—22no—
41#336—22no—
42#336—22no—
43C23×C4—22no—
44C23×C4—22no—
45C23×C4—22no—
46C23×C4—41no—
47C23×C4C2×D412yes—
48C23×C4C2×D411yescharacteristic
49C22×C8—24no—
50C22×C8—22no—
51C22×C8—41no—
52C22×C8—42no—
53C22×C8—42no—
54C22×C8—42no—
55C22×C8—42no—
56C22×C8—42no—
57C22×C8—41no—
58C22×C8—42no—
59C22×C8—42no—
60C22×C8—42no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—24no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—42no—
68C2×C4⋊C4—42no—
69C2×C4⋊C4—42no—
70C2×C4⋊C4—42no—
71C2×C4⋊C4—42no—
72C2×C4⋊C4—42no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—44no—
80C2×C4⋊C4—44no—
81C2×C4⋊C4—44no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—82no—
85C2×C4⋊C4—82no—
86C2×C4⋊C4—82no—
87C2×C4⋊C4D814yes—
88C2×C4⋊C4SD1614yes—
89C2×C42—42no—
90C2×C42C2×Q811yescharacteristic
91C2×C42#4112yes—
92C2×C42#4111yescharacteristic
93#95—44no—
94#95—42no—
95#95—42no—
96#95—42no—
97#95—42no—
98#95—42no—
99#95—42no—
100#95—82no—
101C24#12011yescharacteristic
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—24no—
110C22×C4—24no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4—82no—
132C22×C4—82no—
133C22×C4—81no—
134C22×C4—81no—
135C22×C4C22⋊Q812yes—
136C22×C4C22⋊Q811yescharacteristic
137C22×C4#12412yes—
138C22×C4#12411yescharacteristic
139C22×C4C2×D812yes—
140C22×C4C2×SD1612yes—
141C22×C4#13512yes—
142C22×C4AGL(1,ℤ/8)12yes—
143C22×C4#13712yes—
144C22×C4#13712yes—
145C2×C8—44no—
146C2×C8—44no—
147C2×C8—44no—
148C2×C8—42no—
149C2×C8—44no—
150C2×C8—82no—
151C2×C8—82no—
152C2×C8—82no—
153C2×C8—82no—
154C2×C8—81no—
155C2×C8—82no—
156C2×C8—81no—
157C4⋊C4—44no—
158C4⋊C4—44no—
159C4⋊C4—44no—
160C4⋊C4—44no—
161C4⋊C4—44no—
162C4⋊C4—44no—
163C4⋊C4—82no—
164C4⋊C4—82no—
165C4⋊C4—82no—
166C4⋊C4—82no—
167C4⋊C4—88no—
168C4⋊C4—88no—
169C4⋊C4—82no—
170C4⋊C4—82no—
171C42—41no—
172C42—41no—
173C42—42no—
174C23#39311yescharacteristic
175C23#44712yes—
176C23#44812yes—
177C23#45212yes—
178C23#45311yescharacteristic
179C23#45411yescharacteristic
180C23#45512yes—
181C23#45612yes—
182C23#45812yes—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—44no—
204C2×C4—44no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—44no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—84no—
225C2×C4—82no—
226C2×C4—161no—
227C2×C4—161no—
228C2×C4—162no—
229C2×C4#47412yes—
230C2×C4#47612yes—
231C2×C4#47912yes—
232C2×C4#48612yes—
233C2×C4#48811yescharacteristic
234C2×C4#48912yes—
235C2×C4#49111yescharacteristic
236C8—84no—
237C8—82no—
238C22#160712yes—
239C22#160812yes—
240C22#161012yes—
241C22#161112yes—
242C22#161512yes—
243C22#161812yes—
244C22#161911yescharacteristic
245C22#162311yescharacteristic
246C22#162712yes—
247C22#164112yes—
248C22#164312yes—
249C22#164712yes—
250C22#165312yes—
251C22#166112yes—
252C22#166211yescharacteristic
253C22#166311yescharacteristic
254C22#202312yes—
255C22#202611yescharacteristic
256C22#202711yescharacteristic
257C22#202812yes—
258C22#203011yescharacteristic
259C4—22no—
260C4—22no—
261C4—22no—
262C4—22no—
263C4—21no—
264C4—22no—
265C4—21no—
266C4—84no—
267C4—84no—
268C4—162no—
269C4—162no—
270C2#778712yes—
271C2#779012yes—
272C2#779812yes—
273C2#781412yes—
274C2#1034212yes—
275C2#1034711yescharacteristic
276C2#1034811yescharacteristic
277C2#1035612yes—
278C2#1036111yescharacteristic
subgroupG/H
center, socle, prime radicalC24#120
commutator, Frattini, cosocleC22×C42C23
LCSG ▷ C22×C42 ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#170732
#113732
#113572
#70252