Finite Group Databasebeta

← #122302

Subgroups — #122302

Order 512.

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

normal subgroups only

This group has 368 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#20808C211yesmaximal, characteristic
2#11460C212yesmaximal
3#7063C212yesmaximal, complemented
4#7062C212yesmaximal, complemented
5C2×#431C411yescharacteristic
6C2×#431C2211yescharacteristic
7C2×C43C411yescharacteristic
8#2163—22no—
9#1842C411yescharacteristic
10#1842C2211yescharacteristic
11#1831C411yescharacteristic
12#1831C2211yescharacteristic
13#1797—22no—
14C4×C4⋊C8—22no—
15#1322C2214yes—
16#1314—22no—
17C22×C42—21no—
18C22×C42—21no—
19C22×C42C811yescharacteristic
20C22×C42C811yescharacteristic
21C22×C42C2×C411yescharacteristic
22#431—41no—
23#431—41no—
24#431—41no—
25#431—41no—
26C2×C4⋊C8—22no—
27C2×C4⋊C8—22no—
28C2×C4⋊C8C2×C411yescharacteristic
29C2×C4⋊C8C2×C411yescharacteristic
30C2×C4⋊C8C2×C411yescharacteristic
31C2×C4⋊C8C2311yescharacteristic
32C2×C22⋊C8—21no—
33C2×C22⋊C8D412yes—
34C2×C4×C8—21no—
35C2×C4×C8D412yes—
36C43—21no—
37C43—21no—
38C43C811yescharacteristic
39C43C811yescharacteristic
40C43C2×C411yescharacteristic
41C43C2×C411yescharacteristic
42C4⋊C16—42no—
43C22⋊C16—24no—
44#346—42no—
45#336—22no—
46#336—22no—
47C23×C4—21no—
48C23×C4—21no—
49C23×C4C22⋊C411yescharacteristic
50C23×C4C2×C811yescharacteristic
51C23×C4M1611yescharacteristic
52C22×C8—21no—
53C22×C8—21no—
54C22×C8C22⋊C412yes—
55C22×C8C2×D412yes—
56C2×C42—21no—
57C2×C42—21no—
58C2×C42—21no—
59C2×C42—21no—
60C2×C42—21no—
61C2×C42—21no—
62C2×C42—21no—
63C2×C42—22no—
64C2×C42—22no—
65C2×C42—21no—
66C2×C42—21no—
67C2×C42—21no—
68C2×C42—21no—
69C2×C42—21no—
70C2×C42—21no—
71C2×C42—21no—
72C2×C42—42no—
73C2×C42—42no—
74C2×C42—44no—
75C2×C42—42no—
76C2×C42C22⋊C411yescharacteristic
77C2×C42C2×C811yescharacteristic
78C2×C42C2×C811yescharacteristic
79C2×C42C2×C811yescharacteristic
80C2×C42C2×C811yescharacteristic
81C2×C42C2×C811yescharacteristic
82C2×C42M1611yescharacteristic
83C2×C42C22×C411yescharacteristic
84C2×C16—44no—
85C4⋊C8—41no—
86C4⋊C8—41no—
87C4⋊C8—41no—
88C4⋊C8—41no—
89C4⋊C8—44no—
90C4⋊C8—44no—
91C22⋊C8—42no—
92C22⋊C8—42no—
93C4×C8—22no—
94C4×C8—22no—
95C4×C8—41no—
96C4×C8—41no—
97C24C22⋊C811yescharacteristic
98C22×C4—22no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—24no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4—41no—
117C22×C4—41no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4C22⋊C811yescharacteristic
121C22×C4C22⋊C811yescharacteristic
122C22×C4C22⋊C811yescharacteristic
123C22×C4C2×C22⋊C411yescharacteristic
124C22×C4C22×C811yescharacteristic
125C22×C4C2×M1611yescharacteristic
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—42no—
129C2×C8—42no—
130C2×C8—42no—
131C2×C8—42no—
132C2×C8—41no—
133C2×C8—41no—
134C42—22no—
135C42—22no—
136C42—24no—
137C42—24no—
138C42—21no—
139C42—21no—
140C42—21no—
141C42—21no—
142C42—21no—
143C42—21no—
144C42—21no—
145C42—21no—
146C42—41no—
147C42—41no—
148C42—42no—
149C42—42no—
150C42—42no—
151C42—42no—
152C42—41no—
153C42—41no—
154C42—42no—
155C42—42no—
156C42#9911yescharacteristic
157C42#9911yescharacteristic
158C42#10111yescharacteristic
159C42#10111yescharacteristic
160C42M3212yes—
161C42M3212yes—
162C16—84no—
163C23—21no—
164C23—21no—
165C23—41no—
166C23—41no—
167C23#34912yes—
168C23C2×C22⋊C811yescharacteristic
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—48no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—48no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4#32411yescharacteristic
204C2×C4#32411yescharacteristic
205C2×C4#32411yescharacteristic
206C2×C4#32411yescharacteristic
207C2×C4C2×#9911yescharacteristic
208C2×C4C2×#10111yescharacteristic
209C2×C4C2×M3211yescharacteristic
210C2×C4C2×M3211yescharacteristic
211C8—42no—
212C8—42no—
213C8—81no—
214C8—81no—
215C22—21no—
216C22—21no—
217C22—22no—
218C22—42no—
219C22—42no—
220C22—41no—
221C22—41no—
222C22C2×#32411yescharacteristic
223C22#211211yescharacteristic
224C22C2×#34911yescharacteristic
225C4—24no—
226C4—21no—
227C4—21no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—21no—
232C4—42no—
233C4—42no—
234C4—42no—
235C4—41no—
236C4—41no—
237C4#132712yes—
238C4#212611yescharacteristic
239C4#213211yescharacteristic
240C2—21no—
241C2—21no—
242C2—42no—
243C2#1144011yescharacteristic
244C2#1144711yescharacteristic
245C2C2×#132711yescharacteristic
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 ◁ C2×C42 ◁ C2×#431 ◁ G
LECSG ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7063C2semidirect
4. #7062C2semidirect

Maximal subgroups

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

subgroupindex
#208082
#114602
#70632
#70622