Finite Group Databasebeta

← #122433

Subgroups — #122433

Order 512.

269 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 269 conjugacy classes of subgroups — too many to draw. Only its 74 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#21718C211yesmaximal, characteristic
2#11494C211yesmaximal, characteristic
3#11493C211yesmaximal, characteristic
4#7066C212yesmaximal
5#7061C212yesmaximal
6C2×#446C411yescharacteristic
7C2×C8×Q8C2211yescharacteristic
8#2399C411yescharacteristic
9#2163—21no—
10#2162—21no—
11#1985C411yescharacteristic
12#1985C2211yescharacteristic
13#1939—21no—
14#1939—21no—
15#1843C2211yescharacteristic
16#1771—21no—
17#1771—21no—
18#1726C411yescharacteristic
19#1322—21no—
20#1322—21no—
21#1322C2212yes—
22#1322C2212yes—
23C2×C4×Q8C2×C411yescharacteristic
24C2×C4×Q8C2×C411yescharacteristic
25#446—41no—
26#446—41no—
27C8×Q8—41no—
28C8×Q8—41no—
29C2×C4⋊C8—21no—
30C2×C4⋊C8—21no—
31C2×C4⋊C8—21no—
32C2×C4⋊C8C2×C411yescharacteristic
33C2×C4⋊C8D411yescharacteristic
34C2×C4⋊C8D411yescharacteristic
35C2×C4⋊C8C2311yescharacteristic
36C2×#97—21no—
37C2×#97—21no—
38C2×#97C2×C411yescharacteristic
39C2×C4×C8—21no—
40C2×C4×C8C2×C411yescharacteristic
41C2×C4×C8D411yescharacteristic
42C2×C4×C8D411yescharacteristic
43#398—21no—
44#398C811yescharacteristic
45#398C811yescharacteristic
46#395—21no—
47#395C811yescharacteristic
48#395C811yescharacteristic
49C4×C4⋊C4—21no—
50C4×C4⋊C4—21no—
51#376C2×C411yescharacteristic
52C4⋊C16—41no—
53C4⋊C16—41no—
54C22⋊C16—22no—
55C22⋊C16—22no—
56#346—41no—
57C4×C16—41no—
58#336—21no—
59#336—21no—
60#336—21no—
61#336—21no—
62C22×Q8C2×C811yescharacteristic
63C22×Q8M1611yescharacteristic
64C22×C8—21no—
65C22×C8—21no—
66C22×C8C22⋊C411yescharacteristic
67C22×C8C22⋊C411yescharacteristic
68C22×C8C2×D411yescharacteristic
69C22×C8C2×D411yescharacteristic
70C4×Q8—41no—
71C4×Q8—41no—
72C4×Q8—41no—
73C4×Q8—41no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4C2×C811yescharacteristic
80C2×C4⋊C4C2×C811yescharacteristic
81C2×C4⋊C4C2×C811yescharacteristic
82C2×C4⋊C4M1611yescharacteristic
83C2×C42—21no—
84C2×C42—21no—
85C2×C42C22⋊C411yescharacteristic
86C2×C42C22⋊C411yescharacteristic
87C2×C42C22×C411yescharacteristic
88C2×C16—42no—
89C2×C16—42no—
90C4⋊C8—41no—
91C4⋊C8—41no—
92C4⋊C8—41no—
93C4⋊C8—41no—
94C4⋊C8—42no—
95C4⋊C8—42no—
96C4⋊C8—42no—
97#97—24no—
98#97—41no—
99#97—41no—
100#97—41no—
101C4×C8—21no—
102C4×C8—21no—
103C4×C8—21no—
104C4×C8—21no—
105C4×C8—21no—
106C4×C8—21no—
107C4×C8—42no—
108#95—21no—
109#95—21no—
110#95—21no—
111#95—21no—
112#95—21no—
113#95C2×C811yescharacteristic
114#95C2×C811yescharacteristic
115C2×Q8—22no—
116C2×Q8—41no—
117C2×Q8—41no—
118C2×Q8—41no—
119C2×Q8—81no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4C22⋊C811yescharacteristic
125C22×C4C22⋊C811yescharacteristic
126C22×C4C22⋊C811yescharacteristic
127C22×C4C22⋊C811yescharacteristic
128C22×C4C2×C22⋊C411yescharacteristic
129C22×C4C22×C811yescharacteristic
130C22×C4C2×M1611yescharacteristic
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—41no—
137C2×C8—41no—
138C2×C8—41no—
139C2×C8—41no—
140C2×C8—41no—
141C2×C8—41no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8#9911yescharacteristic
145C2×C8#9911yescharacteristic
146C4⋊C4—41no—
147C4⋊C4—41no—
148C4⋊C4—41no—
149C4⋊C4—41no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—81no—
154C4⋊C4—81no—
155C42—22no—
156C42—22no—
157C42—21no—
158C42—21no—
159C42—21no—
160C42—21no—
161C42—21no—
162C42—21no—
163C42—41no—
164C42—41no—
165C16—82no—
166C16—82no—
167C23C2×C22⋊C811yescharacteristic
168Q8—44no—
169Q8—82no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4#35012yes—
199C2×C4C2×#9911yescharacteristic
200C2×C4#41311yescharacteristic
201C8—21no—
202C8—42no—
203C8—41no—
204C8—41no—
205C8—41no—
206C8—81no—
207C8#35612yes—
208C22—21no—
209C22—21no—
210C22#145911yescharacteristic
211C22C2×#35011yescharacteristic
212C22#211711yescharacteristic
213C4—22no—
214C4—21no—
215C4—42no—
216C4—42no—
217C4—41no—
218C4—41no—
219C4—41no—
220C4—41no—
221C4—41no—
222C4—81no—
223C4—81no—
224C4#212611yescharacteristic
225C4C2×#35611yescharacteristic
226C2—21no—
227C2—21no—
228C2#1147011yescharacteristic
229C2#1147811yescharacteristic
230C2#1148411yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4⋊C8C23
socleC22#1459
prime radicalC23C2×C22⋊C8
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#217182
#114942
#114932
#70662
#70612