Finite Group Databasebeta

← #102270

Subgroups — #102270

Order 512.

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

normal subgroups only

This group has 422 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#9896C211yesmaximal, characteristic
5#9896C211yesmaximal, 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×#322—21no—
20#1296—22no—
21#1296—22no—
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#322—42no—
62C22×Q8—22no—
63C22×Q8—41no—
64C22×C8—22no—
65C22×C8—22no—
66C22×C8—41no—
67C22×C8—41no—
68C22×C8C22⋊C412yes—
69C22×C8C22⋊C412yes—
70C4⋊Q8—44no—
71C4⋊Q8—41no—
72C4⋊Q8—41no—
73C4⋊Q8—42no—
74C4⋊Q8—42no—
75C4⋊Q8—41no—
76C4⋊Q8—42no—
77C4⋊Q8—42no—
78C4⋊Q8—41no—
79C4⋊Q8—82no—
80C4⋊Q8—82no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C42C22×C411yescharacteristic
94C2×C42C2×D411yescharacteristic
95C2×C42C2×D411yescharacteristic
96C2×C42C2×D411yescharacteristic
97C2×C42C2×D411yescharacteristic
98C2×C42#4111yescharacteristic
99C2×C42#4111yescharacteristic
100C4⋊C8—22no—
101C4⋊C8—22no—
102C4⋊C8—22no—
103C4⋊C8—42no—
104C4⋊C8—42no—
105C4⋊C8—42no—
106C4⋊C8—42no—
107C4⋊C8—42no—
108C4⋊C8—42no—
109C4⋊C8Q1614yes—
110#103—48no—
111#103—48no—
112C4×C8—21no—
113C4×C8—21no—
114C4×C8—21no—
115C4×C8—22no—
116C4×C8—22no—
117C4×C8—21no—
118C4×C8—42no—
119C4×C8—41no—
120C4×C8—41no—
121C4×C8—41no—
122C4×C8—41no—
123C2×Q8—44no—
124C2×Q8—42no—
125C2×Q8—42no—
126C2×Q8—44no—
127C2×Q8—82no—
128C2×Q8—82no—
129C2×Q8—82no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4C2×C22⋊C411yescharacteristic
133C22×C4C4×D411yescharacteristic
134C22×C4C4×D411yescharacteristic
135C22×C4#12111yescharacteristic
136C22×C4#12111yescharacteristic
137C22×C4#12411yescharacteristic
138C22×C4#12711yescharacteristic
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—24no—
146C2×C8—42no—
147C2×C8—42no—
148C2×C8—42no—
149C2×C8—42no—
150C2×C8—42no—
151C2×C8—81no—
152C2×C8—81no—
153C2×C8—81no—
154C2×C8—81no—
155C2×C8—81no—
156C2×C8—81no—
157C4⋊C4—44no—
158C4⋊C4—82no—
159C4⋊C4—82no—
160C4⋊C4—82no—
161C4⋊C4—82no—
162C4⋊C4—81no—
163C4⋊C4—81no—
164C4⋊C4—81no—
165C4⋊C4—81no—
166C4⋊C4—82no—
167C4⋊C4—82no—
168C4⋊C4—82no—
169C4⋊C4—81no—
170C4⋊C4—81no—
171C4⋊C4—81no—
172C4⋊C4—81no—
173C42—22no—
174C42—22no—
175C42—22no—
176C42C2×Q1612yes—
177C42#13511yescharacteristic
178C42#13511yescharacteristic
179C42#13512yes—
180C42AGL(1,ℤ/8)11yescharacteristic
181C42AGL(1,ℤ/8)12yes—
182C42AGL(1,ℤ/8)12yes—
183C42#13711yescharacteristic
184C42#13712yes—
185C42#13712yes—
186C23#39011yescharacteristic
187Q8—84no—
188Q8—84no—
189Q8—164no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4#41312yes—
203C2×C4#41611yescharacteristic
204C2×C4#41711yescharacteristic
205C2×C4#41812yes—
206C2×C4C4×Q1612yes—
207C2×C4#44112yes—
208C2×C4#46311yescharacteristic
209C2×C4#46311yescharacteristic
210C2×C4#46411yescharacteristic
211C2×C4#46411yescharacteristic
212C2×C4#46712yes—
213C2×C4#47012yes—
214C2×C4#47112yes—
215C2×C4#47112yes—
216C2×C4#49012yes—
217C2×C4#49411yescharacteristic
218C2×C4#49511yescharacteristic
219C2×C4#49611yescharacteristic
220C2×C4#49611yescharacteristic
221C2×C4#49711yescharacteristic
222C2×C4#49711yescharacteristic
223C8—42no—
224C8—42no—
225C8—44no—
226C8—82no—
227C8—161no—
228C8—161no—
229C22#196111yescharacteristic
230C22#196711yescharacteristic
231C22#196911yescharacteristic
232C22#197111yescharacteristic
233C22#197711yescharacteristic
234C22#197711yescharacteristic
235C22#198311yescharacteristic
236C4—22no—
237C4—22no—
238C4—22no—
239C4—162no—
240C4—162no—
241C4#155312yes—
242C4#155712yes—
243C4#168312yes—
244C4#168412yes—
245C4#170312yes—
246C4#170412yes—
247C4#172011yescharacteristic
248C4#172011yescharacteristic
249C4#172311yescharacteristic
250C4#172311yescharacteristic
251C2#971011yescharacteristic
252C2#973711yescharacteristic
253C2#974511yescharacteristic
254C2#977711yescharacteristic
255C2#979511yescharacteristic
256C2#980311yescharacteristic
257C2#983511yescharacteristic
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
#98962
#87842
#82612