Finite Group Databasebeta

← #122306

Subgroups — #122306

Order 512.

319 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 319 conjugacy classes of subgroups — too many to draw. Only its 86 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#20755C211yesmaximal, characteristic
2#11440C211yesmaximal, characteristic
3C2×#1322C211yesmaximal, characteristic
4#7062C214yesmaximal, complemented
5C2×#423C411yescharacteristic
6C22×C4⋊C8C2211yescharacteristic
7C4×#117C411yescharacteristic
8#2112—21no—
9C2×C22⋊C16—21no—
10#1832—21no—
11#1832—21no—
12#1831C412yes—
13#1831C2212yes—
14#1729—21no—
15#1729—21no—
16#1322—21no—
17#1322—21no—
18#1322C2212yes—
19#1322C2212yes—
20C22×M16—21no—
21C23×C8D411yescharacteristic
22C23×C8D411yescharacteristic
23C2×#117—21no—
24C22×C42C2×C411yescharacteristic
25C2×M32—41no—
26C22×C16—41no—
27#423—41no—
28#423—41no—
29C2×C4⋊C8—21no—
30C2×C4⋊C8—41no—
31C2×C4⋊C8—41no—
32C2×C4⋊C8C2×C412yes—
33C2×C4⋊C8C2×C411yescharacteristic
34C2×C4⋊C8D411yescharacteristic
35C2×C4⋊C8D411yescharacteristic
36C2×C4⋊C8C2311yescharacteristic
37C4×C4⋊C4—21no—
38C4×C4⋊C4C811yescharacteristic
39C4×C4⋊C4C811yescharacteristic
40C4×C22⋊C4—21no—
41C4×C22⋊C4C812yes—
42#376—21no—
43C43C2×C412yes—
44C22⋊C16—22no—
45C22⋊C16—22no—
46C22⋊C16—41no—
47C22⋊C16—41no—
48#336—22no—
49#336—22no—
50#336—22no—
51#336—22no—
52C23×C4C22⋊C411yescharacteristic
53C23×C4C2×C811yescharacteristic
54C23×C4M1611yescharacteristic
55C2×M16—44no—
56C2×M16—41no—
57C2×M16—41no—
58C22×C8—21no—
59C22×C8—21no—
60C22×C8—21no—
61C22×C8—21no—
62C22×C8—22no—
63C22×C8—42no—
64C22×C8—42no—
65C22×C8C22⋊C411yescharacteristic
66C22×C8C22⋊C411yescharacteristic
67C22×C8C2×D411yescharacteristic
68C22×C8C2×D411yescharacteristic
69#117—42no—
70#117—42no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C22⋊C4—22no—
74C2×C42—21no—
75C2×C42—21no—
76C2×C42—21no—
77C2×C42—21no—
78C2×C42—22no—
79C2×C42—22no—
80C2×C42C22⋊C411yescharacteristic
81C2×C42C2×C812yes—
82C2×C42C2×C811yescharacteristic
83C2×C42C2×C812yes—
84C2×C42M1611yescharacteristic
85C2×C42C22×C411yescharacteristic
86M32—82no—
87C2×C16—42no—
88C2×C16—42no—
89C2×C16—82no—
90C4⋊C8—41no—
91C4⋊C8—41no—
92C4⋊C8—41no—
93C4⋊C8—41no—
94C4⋊C8—42no—
95C4⋊C8—42no—
96#95—21no—
97#95—21no—
98#95—22no—
99C24C22⋊C811yescharacteristic
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—41no—
110C22×C4—81no—
111C22×C4C22⋊C811yescharacteristic
112C22×C4C22⋊C811yescharacteristic
113C22×C4C22⋊C811yescharacteristic
114C22×C4#9911yescharacteristic
115C22×C4#9911yescharacteristic
116C22×C4#10111yescharacteristic
117C22×C4#10111yescharacteristic
118C22×C4C2×C22⋊C411yescharacteristic
119C22×C4C22×C811yescharacteristic
120C22×C4C2×M1611yescharacteristic
121M16—82no—
122M16—82no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—42no—
128C2×C8—42no—
129C2×C8—41no—
130C2×C8—41no—
131C2×C8—41no—
132C2×C8—41no—
133C2×C8—44no—
134C2×C8—42no—
135C2×C8—81no—
136C2×C8—81no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C22⋊C4—44no—
140C42—21no—
141C42—21no—
142C42—21no—
143C42—21no—
144C42—21no—
145C42—21no—
146C42—21no—
147C42—21no—
148C42—24no—
149C42—42no—
150C42—42no—
151C42—42no—
152C42—82no—
153C16—82no—
154C16—82no—
155C23—22no—
156C23—21no—
157C23—21no—
158C23—41no—
159C23#32411yescharacteristic
160C23#32411yescharacteristic
161C23C2×C22⋊C811yescharacteristic
162C2×C4—22no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—81no—
187C2×C4—81no—
188C2×C4#32411yescharacteristic
189C2×C4#32411yescharacteristic
190C2×C4#35412yes—
191C2×C4#35512yes—
192C2×C4C2×#9911yescharacteristic
193C2×C4C2×#10111yescharacteristic
194C2×C4#50412yes—
195C8—41no—
196C8—41no—
197C8—41no—
198C8—41no—
199C8—82no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—42no—
208C22—42no—
209C22#132412yes—
210C22C2×#32411yescharacteristic
211C22#211311yescharacteristic
212C22#211511yescharacteristic
213C4—22no—
214C4—21no—
215C4—21no—
216C4—41no—
217C4—41no—
218C4—41no—
219C4—82no—
220C4—82no—
221C4#132412yes—
222C4C2×#35411yescharacteristic
223C4C2×#35511yescharacteristic
224C2—21no—
225C2—21no—
226C2—22no—
227C2—41no—
228C2#1144111yescharacteristic
229C2C2×#132411yescharacteristic
230C2#1145611yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4⋊C8C23
socleC22C2×#324
prime radicalC24C22⋊C8
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #7062C2semidirect

Maximal subgroups

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

subgroupindex
#207552
#114402
C2×#13222
#70622