Finite Group Databasebeta

← #135255

Subgroups — #135255

Order 512.

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

normal subgroups only

This group has 470 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
1C2×C4×#107C211yesmaximal, characteristic
2#12599C211yesmaximal, characteristic
3#8595C211yesmaximal, characteristic
4#7178C214yesmaximal, complemented
5C22×#107C411yescharacteristic
6C22×#107C411yescharacteristic
7C22×C4×C8C2211yescharacteristic
8C2×C4×C4⋊C4—21no—
9C2×C22⋊C16—21no—
10C2×#346C2212yes—
11#1797—21no—
12C4×#107—21no—
13C4×#107—21no—
14C4×#107C412yes—
15C4×#107C2212yes—
16C2×#340—21no—
17C2×#334C2212yes—
18#1374—21no—
19#1277—21no—
20C23×C8C2×C411yescharacteristic
21C23×C8D411yescharacteristic
22C22×C4⋊C4—21no—
23C22×C4⋊C4—21no—
24C22×C42—41no—
25C22×C42D411yescharacteristic
26C22×C16—22no—
27C2×#107—21no—
28C2×#107—21no—
29C2×#107—22no—
30C2×#107—24no—
31C2×#107C814yescomplemented
32C2×#107C2×C412yes—
33C2×#107C2×C412yes—
34C2×C4⋊C8—22no—
35C2×C22⋊C8—41no—
36C2×C4×C8—24no—
37C2×C4×C8C2×C411yescharacteristic
38C2×C4×C8D411yescharacteristic
39C2×C4×C8D411yescharacteristic
40C2×C4×C8C2311yescharacteristic
41C4×C4⋊C4—22no—
42C4×C4⋊C4—22no—
43C4×C4⋊C4—41no—
44C4×C4⋊C4—41no—
45C2×#95—41no—
46C22⋊C16—24no—
47#346—24no—
48#340—22no—
49#340—22no—
50#336—41no—
51#334—24no—
52C23×C4—41no—
53C23×C4—41no—
54C23×C4C22⋊C411yescharacteristic
55C23×C4D811yescharacteristic
56C23×C4SD1611yescharacteristic
57C22×C8—22no—
58C22×C8—22no—
59C22×C8—22no—
60C22×C8—22no—
61C22×C8—41no—
62C22×C8—42no—
63C22×C8C22⋊C411yescharacteristic
64C22×C8C22⋊C411yescharacteristic
65C22×C8C2×C812yes—
66C22×C8M1612yes—
67C22×C8C22×C411yescharacteristic
68C22×C8C2×D411yescharacteristic
69C2×C4⋊C4—24no—
70C2×C4⋊C4—24no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—42no—
77C2×C42—22no—
78C2×C42—22no—
79C2×C42—42no—
80C2×C42—42no—
81C2×C42—84no—
82C2×C42C22⋊C411yescharacteristic
83C2×C42D811yescharacteristic
84C2×C42SD1611yescharacteristic
85C2×C42C2×D411yescharacteristic
86C2×C16—24no—
87C2×C16—42no—
88C2×C16—42no—
89#107—24no—
90#107—24no—
91#107—22no—
92#107—22no—
93#107C2×C814yes—
94#107M1614yes—
95C4⋊C8—42no—
96C4⋊C8—42no—
97C22⋊C8—82no—
98C4×C8—22no—
99C4×C8—22no—
100C4×C8—22no—
101C4×C8#4114yes—
102#95—42no—
103#95—42no—
104C24#10211yescharacteristic
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—41no—
115C22×C4—44no—
116C22×C4—44no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4—82no—
120C22×C4—82no—
121C22×C4—82no—
122C22×C4—82no—
123C22×C4C22⋊C812yes—
124C22×C4#10211yescharacteristic
125C22×C4#10211yescharacteristic
126C22×C4#10211yescharacteristic
127C22×C4#10412yes—
128C22×C4C2×C22⋊C411yescharacteristic
129C22×C4C2×D811yescharacteristic
130C22×C4C2×SD1611yescharacteristic
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—24no—
136C2×C8—24no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—44no—
141C2×C8—82no—
142C2×C8—82no—
143C2×C8—82no—
144C2×C8C22⋊C812yes—
145C2×C8#11712yes—
146C2×C8#12312yes—
147C2×C8C22×C811yescharacteristic
148C2×C8C2×M1612yes—
149C2×C8C2×M1611yescharacteristic
150C4⋊C4—24no—
151C4⋊C4—24no—
152C4⋊C4—44no—
153C4⋊C4—44no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C42—21no—
157C42—21no—
158C42—22no—
159C42—22no—
160C42—84no—
161C42—84no—
162C42#12312yes—
163C42#13712yes—
164C16—44no—
165C23—21no—
166C23—21no—
167C23—21no—
168C23—21no—
169C23D4⋊C812yes—
170C23#36012yes—
171C23#36212yes—
172C23C2×#10211yescharacteristic
173C2×C4—28no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—44no—
190C2×C4—44no—
191C2×C4—44no—
192C2×C4—88no—
193C2×C4—88no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4D4⋊C812yes—
199C2×C4#36112yes—
200C2×C4#38111yescharacteristic
201C2×C4C2×C22⋊C811yescharacteristic
202C2×C4#40711yescharacteristic
203C2×C4#41911yescharacteristic
204C2×C4C2×#10411yescharacteristic
205C2×C4#42111yescharacteristic
206C2×C4#48012yes—
207C2×C4#48312yes—
208C8—22no—
209C8—22no—
210C8—41no—
211C8—162no—
212C8#43112yes—
213C8#43212yes—
214C22—22no—
215C22—22no—
216C22—22no—
217C22—22no—
218C22—22no—
219C22—22no—
220C22#133414yes—
221C22C2×D4⋊C811yescharacteristic
222C22#148611yescharacteristic
223C22#182311yescharacteristic
224C22C2×#36011yescharacteristic
225C22#214511yescharacteristic
226C22C2×#36111yescharacteristic
227C22C2×#36211yescharacteristic
228C4—22no—
229C4—22no—
230C4—22no—
231C4—22no—
232C4—41no—
233C4—84no—
234C4—84no—
235C4#147911yescharacteristic
236C4#149111yescharacteristic
237C4#181711yescharacteristic
238C4#182611yescharacteristic
239C2—24no—
240C2#870611yescharacteristic
241C2C2×#133411yescharacteristic
242C2#1159911yescharacteristic
243C2#1166311yescharacteristic
244C2#1166811yescharacteristic
245C2#1287011yescharacteristic
246C2#1287111yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC2×C8C22×C8
Frattini, cosocleC2×C4×C8C23
prime radicalC24#102
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7178C2semidirect
31. C2×#107C8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1072
#125992
#85952
#71782