Finite Group Databasebeta

← #102259

Subgroups — #102259

Order 512.

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

normal subgroups only

This group has 402 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#33289C211yesmaximal, characteristic
2#9898C211yesmaximal, characteristic
3#9896C211yesmaximal, characteristic
4#9694C211yesmaximal, characteristic
5#9691C211yesmaximal, characteristic
6#8784C211yesmaximal, characteristic
7#8261C211yesmaximal, characteristic
8#2868C2211yescharacteristic
9#2868C2211yescharacteristic
10#1990—21no—
11#1990—21no—
12#1990C414yes—
13#1842C2211yescharacteristic
14#1842C2211yescharacteristic
15#1834—22no—
16C4×C4⋊C8C2211yescharacteristic
17C4×C4⋊C8C2211yescharacteristic
18C42×C8C2211yescharacteristic
19C2×#333—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—
32C2×C4⋊C8—21no—
33C2×C4⋊C8—21no—
34C2×C4⋊C8—21no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8D412yes—
39C2×C4⋊C8D412yes—
40C2×C4×C8—21no—
41C2×C4×C8—21no—
42C2×C4×C8C2×C412yes—
43C2×C4×C8D412yes—
44C2×C4×C8D411yescharacteristic
45C2×C4×C8D411yescharacteristic
46#384—21no—
47#384—21no—
48#384—21no—
49#384—21no—
50#384—21no—
51#384—21no—
52#384—21no—
53#384C2×C412yes—
54C43C2311yescharacteristic
55#340—24no—
56#340—24no—
57#336—21no—
58#336—21no—
59#333—44no—
60#332—44no—
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—42no—
72C4⋊Q8—42no—
73C4⋊Q8—41no—
74C4⋊Q8—41no—
75C4⋊Q8—42no—
76C4⋊Q8—42no—
77C4⋊Q8—41no—
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⋊C8D814yes—
110C4×C8—21no—
111C4×C8—21no—
112C4×C8—21no—
113C4×C8—21no—
114C4×C8—22no—
115C4×C8—22no—
116C4×C8—42no—
117C4×C8—41no—
118C4×C8—41no—
119C4×C8—41no—
120C4×C8—41no—
121C2×Q8—44no—
122C2×Q8—82no—
123C2×Q8—82no—
124C2×Q8—81no—
125C2×Q8—81no—
126C2×Q8—82no—
127C2×Q8—82no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4C2×C22⋊C411yescharacteristic
131C22×C4C4×D411yescharacteristic
132C22×C4C4×D411yescharacteristic
133C22×C4#12111yescharacteristic
134C22×C4#12111yescharacteristic
135C22×C4#12411yescharacteristic
136C22×C4#12711yescharacteristic
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—24no—
144C2×C8—42no—
145C2×C8—42no—
146C2×C8—42no—
147C2×C8—42no—
148C2×C8—42no—
149C2×C8—81no—
150C2×C8—81no—
151C2×C8—81no—
152C2×C8—81no—
153C2×C8—81no—
154C2×C8—81no—
155C4⋊C4—44no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—44no—
159C4⋊C4—82no—
160C4⋊C4—82no—
161C4⋊C4—82no—
162C4⋊C4—82no—
163C4⋊C4—81no—
164C4⋊C4—81no—
165C4⋊C4—82no—
166C4⋊C4—82no—
167C4⋊C4—81no—
168C4⋊C4—81no—
169C4⋊C4—81no—
170C4⋊C4—81no—
171C42—22no—
172C42—22no—
173C42—22no—
174C42C2×D812yes—
175C42#13511yescharacteristic
176C42#13511yescharacteristic
177C42#13512yes—
178C42AGL(1,ℤ/8)11yescharacteristic
179C42#13711yescharacteristic
180C42#13712yes—
181C42#13712yes—
182C42#13712yes—
183C42#13712yes—
184C23#39011yescharacteristic
185Q8—164no—
186Q8—162no—
187Q8—162no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4#41312yes—
201C2×C4#41611yescharacteristic
202C2×C4#41711yescharacteristic
203C2×C4#41912yes—
204C2×C4C4×D812yes—
205C2×C4#44112yes—
206C2×C4#46311yescharacteristic
207C2×C4#46311yescharacteristic
208C2×C4#46411yescharacteristic
209C2×C4#46411yescharacteristic
210C2×C4#46612yes—
211C2×C4#47012yes—
212C2×C4#47112yes—
213C2×C4#47112yes—
214C2×C4#49112yes—
215C2×C4#49311yescharacteristic
216C2×C4#49511yescharacteristic
217C2×C4#49711yescharacteristic
218C2×C4#49711yescharacteristic
219C2×C4#49711yescharacteristic
220C2×C4#49711yescharacteristic
221C8—42no—
222C8—42no—
223C8—44no—
224C8—82no—
225C8—161no—
226C8—161no—
227C22#196011yescharacteristic
228C22#196711yescharacteristic
229C22#197011yescharacteristic
230C22#197111yescharacteristic
231C22#197711yescharacteristic
232C22#197711yescharacteristic
233C22#198311yescharacteristic
234C4—22no—
235C4—22no—
236C4—22no—
237C4—162no—
238C4—162no—
239C4#155412yes—
240C4#155912yes—
241C4#168112yes—
242C4#168212yes—
243C4#170312yes—
244C4#170412yes—
245C4#171811yescharacteristic
246C4#171811yescharacteristic
247C4#172211yescharacteristic
248C4#172211yescharacteristic
249C2#970911yescharacteristic
250C2#973811yescharacteristic
251C2#974511yescharacteristic
252C2#977611yescharacteristic
253C2#979611yescharacteristic
254C2#980311yescharacteristic
255C2#983111yescharacteristic
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
#332892
#98982
#98962
#96942
#96912
#87842
#82612