Finite Group Databasebeta

← #102246

Subgroups — #102246

Order 512.

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

normal subgroups only

This group has 438 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#33225C211yesmaximal, characteristic
2#9898C211yesmaximal, characteristic
3#9898C211yesmaximal, characteristic
4#9692C211yesmaximal, characteristic
5#9692C211yesmaximal, characteristic
6#8784C211yesmaximal, characteristic
7#8261C211yesmaximal, characteristic
8C2×#487—22no—
9#2868C2211yescharacteristic
10#2868C2211yescharacteristic
11#1956—21no—
12#1956—21no—
13#1956C414yes—
14#1842C2211yescharacteristic
15#1842C2211yescharacteristic
16C4×C4⋊C8C2211yescharacteristic
17C4×C4⋊C8C2211yescharacteristic
18C42×C8C2211yescharacteristic
19C2×#332—22no—
20C2×#332—22no—
21C2×#322—21no—
22#1296—22no—
23#1296—22no—
24#1277—22no—
25#1273—21no—
26C2×C4⋊Q8—21no—
27C2×C4⋊Q8—21no—
28C2×C4⋊Q8—21no—
29C2×C4⋊Q8—21no—
30C2×C4⋊Q8—21no—
31C2×C4⋊Q8C2×C412yes—
32#487—44no—
33#487—44no—
34C2×C4⋊C8—21no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8D412yes—
41C2×C4⋊C8D412yes—
42C2×#103—24no—
43C2×#103—24no—
44C2×C4×C8—21no—
45C2×C4×C8—21no—
46C2×C4×C8C2×C412yes—
47C2×C4×C8D412yes—
48C2×C4×C8D411yescharacteristic
49C2×C4×C8D411yescharacteristic
50#384—21no—
51#384—21no—
52#384—21no—
53#384—21no—
54#384—21no—
55#384—21no—
56#384—21no—
57#384C2×C412yes—
58C43C2311yescharacteristic
59#336—21no—
60#336—21no—
61#332—44no—
62#332—44no—
63#322—42no—
64C22×Q8—22no—
65C22×Q8—41no—
66C22×C8—22no—
67C22×C8—22no—
68C22×C8—41no—
69C22×C8—41no—
70C22×C8C22⋊C412yes—
71C22×C8C22⋊C412yes—
72C4⋊Q8—44no—
73C4⋊Q8—42no—
74C4⋊Q8—42no—
75C4⋊Q8—41no—
76C4⋊Q8—41no—
77C4⋊Q8—42no—
78C4⋊Q8—42no—
79C4⋊Q8—42no—
80C4⋊Q8—41no—
81C4⋊Q8—42no—
82C4⋊Q8—42no—
83C4⋊Q8—42no—
84C4⋊Q8—41no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C42C22×C411yescharacteristic
98C2×C42C2×D411yescharacteristic
99C2×C42C2×D411yescharacteristic
100C2×C42C2×D411yescharacteristic
101C2×C42C2×D411yescharacteristic
102C2×C42#4111yescharacteristic
103C2×C42#4111yescharacteristic
104C4⋊C8—22no—
105C4⋊C8—22no—
106C4⋊C8—22no—
107C4⋊C8—42no—
108C4⋊C8—42no—
109C4⋊C8—42no—
110C4⋊C8—42no—
111C4⋊C8—42no—
112C4⋊C8—42no—
113C4⋊C8SD1614yes—
114#103—48no—
115#103—48no—
116C4×C8—21no—
117C4×C8—21no—
118C4×C8—21no—
119C4×C8—22no—
120C4×C8—22no—
121C4×C8—21no—
122C4×C8—42no—
123C4×C8—41no—
124C4×C8—41no—
125C4×C8—41no—
126C4×C8—41no—
127C2×Q8—44no—
128C2×Q8—42no—
129C2×Q8—42no—
130C2×Q8—44no—
131C2×Q8—81no—
132C2×Q8—81no—
133C2×Q8—81no—
134C2×Q8—81no—
135C2×Q8—81no—
136C2×Q8—81no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4C2×C22⋊C411yescharacteristic
140C22×C4C4×D411yescharacteristic
141C22×C4C4×D411yescharacteristic
142C22×C4#12111yescharacteristic
143C22×C4#12111yescharacteristic
144C22×C4#12411yescharacteristic
145C22×C4#12711yescharacteristic
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—24no—
153C2×C8—42no—
154C2×C8—42no—
155C2×C8—42no—
156C2×C8—42no—
157C2×C8—42no—
158C2×C8—81no—
159C2×C8—81no—
160C2×C8—81no—
161C2×C8—81no—
162C2×C8—81no—
163C2×C8—81no—
164C4⋊C4—44no—
165C4⋊C4—44no—
166C4⋊C4—44no—
167C4⋊C4—82no—
168C4⋊C4—82no—
169C4⋊C4—81no—
170C4⋊C4—81no—
171C4⋊C4—81no—
172C4⋊C4—81no—
173C4⋊C4—82no—
174C4⋊C4—82no—
175C4⋊C4—82no—
176C4⋊C4—82no—
177C4⋊C4—81no—
178C4⋊C4—81no—
179C42—22no—
180C42—22no—
181C42—22no—
182C42C2×SD1612yes—
183C42#13511yescharacteristic
184C42#13511yescharacteristic
185C42#13512yes—
186C42AGL(1,ℤ/8)11yescharacteristic
187C42AGL(1,ℤ/8)12yes—
188C42AGL(1,ℤ/8)12yes—
189C42AGL(1,ℤ/8)12yes—
190C42#13711yescharacteristic
191C42#13712yes—
192C23#39011yescharacteristic
193Q8—84no—
194Q8—84no—
195Q8—161no—
196Q8—161no—
197Q8—161no—
198Q8—161no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—82no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4#41312yes—
212C2×C4#41611yescharacteristic
213C2×C4#41711yescharacteristic
214C2×C4#41812yes—
215C2×C4C4×SD1612yes—
216C2×C4#44012yes—
217C2×C4#46311yescharacteristic
218C2×C4#46311yescharacteristic
219C2×C4#46411yescharacteristic
220C2×C4#46411yescharacteristic
221C2×C4#46512yes—
222C2×C4#46812yes—
223C2×C4#47312yes—
224C2×C4#47312yes—
225C2×C4#49012yes—
226C2×C4#49211yescharacteristic
227C2×C4#49511yescharacteristic
228C2×C4#49611yescharacteristic
229C2×C4#49611yescharacteristic
230C2×C4#49611yescharacteristic
231C2×C4#49711yescharacteristic
232C8—42no—
233C8—42no—
234C8—44no—
235C8—82no—
236C8—161no—
237C8—161no—
238C22#196111yescharacteristic
239C22#196711yescharacteristic
240C22#196811yescharacteristic
241C22#197311yescharacteristic
242C22#197911yescharacteristic
243C22#197911yescharacteristic
244C22#198311yescharacteristic
245C4—22no—
246C4—22no—
247C4—22no—
248C4—162no—
249C4—162no—
250C4#155312yes—
251C4#155712yes—
252C4#167912yes—
253C4#168012yes—
254C4#169912yes—
255C4#170012yes—
256C4#171411yescharacteristic
257C4#171511yescharacteristic
258C4#171711yescharacteristic
259C4#171911yescharacteristic
260C2#971011yescharacteristic
261C2#973611yescharacteristic
262C2#974811yescharacteristic
263C2#977911yescharacteristic
264C2#979711yescharacteristic
265C2#980911yescharacteristic
266C2#982911yescharacteristic
subgroupG/H
center, socle, prime radicalC23#390
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#332252
#98982
#96922
#87842
#82612