Finite Group Databasebeta

← #128473

Subgroups — #128473

Order 512.

516 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 106 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
1C8×#127C211yesmaximal, characteristic
2C4×#353C211yesmaximal, characteristic
3#11475C211yesmaximal, characteristic
4#8160C212yesmaximal, complemented
5#7205C212yesmaximal, complemented
6C2×C8×D4—22no—
7C2×C8×D4C411yescharacteristic, complemented
8C2×C8×D4C2211yescharacteristic
9C4×#127C2211yescharacteristic
10C2×#353C412yes—
11#1954—22no—
12#1954C411yescharacteristic, complemented
13#1954C2211yescharacteristic
14#1842C411yescharacteristic, complemented
15C4×#99C2212yes—
16#1752—22no—
17C42×C8C411yescharacteristic, complemented
18#1459C2212yes—
19#1393C412yes—
20#1325C412yescomplemented
21#1318C412yescomplemented
22#1282—22no—
23C23×C8—21no—
24C23×C8D412yes—
25C2×#127C2×C411yescharacteristic
26C2×C4×D4—22no—
27C2×C4×D4C2×C411yescharacteristic
28C2×C4×D4C2311yescharacteristic
29C8×D4—41no—
30C8×D4—41no—
31C8×D4—41no—
32C8×D4—41no—
33C8×D4—48no—
34C2×C4⋊C8—22no—
35C2×C4⋊C8C2×C411yescharacteristic
36C2×C4⋊C8C2×C411yescharacteristic
37C2×#99C2×C414yes—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8—22no—
41C2×C22⋊C8—21no—
42C2×C22⋊C8D412yes—
43C2×#97—42no—
44C2×C4×C8—21no—
45C2×C4×C8—21no—
46C2×C4×C8C2×C411yescharacteristic
47C2×C4×C8C2×C411yescharacteristic
48#390—22no—
49#390C2×C411yescharacteristic
50#390C2×C411yescharacteristic
51#384C2×C411yescharacteristic
52C4×C22⋊C4—22no—
53C43C2×C411yescharacteristic
54#353—24no—
55#342—22no—
56#336—42no—
57#324C2×C414yes—
58#323—22no—
59C22×D4—22no—
60C22×D4C4211yescharacteristic
61C22×D4C22×C411yescharacteristic
62C23×C4—21no—
63C23×C4C2×D412yes—
64C22×C8—21no—
65C22×C8—22no—
66C22×C8—22no—
67C22×C8—21no—
68C22×C8—44no—
69C22×C8—44no—
70C22×C8—42no—
71C22×C8—41no—
72C22×C8—41no—
73C22×C8C22⋊C412yes—
74C22×C8C22⋊C411yescharacteristic
75C22×C8C22⋊C411yescharacteristic
76#127—22no—
77#127M1614yes—
78C4×D4—41no—
79C4×D4—41no—
80C4×D4—44no—
81C4×D4—44no—
82C4×D4—41no—
83C4×D4—41no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4C4211yescharacteristic
86C2×C4⋊C4C22×C411yescharacteristic
87C2×C22⋊C4—22no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—24no—
91C2×C22⋊C4#4112yes—
92C2×C42—21no—
93C2×C42—21no—
94C2×C42—42no—
95C2×C42C4211yescharacteristic
96C2×C42C4211yescharacteristic
97C2×C42C22×C411yescharacteristic
98C4⋊C8—41no—
99C4⋊C8—41no—
100C4⋊C8—44no—
101#99—24no—
102#99—24no—
103C22⋊C8—24no—
104C22⋊C8—42no—
105C22⋊C8—44no—
106C22⋊C8—44no—
107C22⋊C8—42no—
108C22⋊C8—42no—
109#97—84no—
110C4×C8—24no—
111C4×C8—21no—
112C4×C8—21no—
113C4×C8—41no—
114C4×C8—42no—
115#95—42no—
116C24—21no—
117C24C4×D412yes—
118C2×D4—28no—
119C2×D4—21no—
120C2×D4—41no—
121C2×D4—41no—
122C2×D4—41no—
123C2×D4—41no—
124C2×D4—44no—
125C2×D4—44no—
126C2×D4#9712yescomplemented
127C2×D4C2×M1612yes—
128C2×D4#13112yes—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—44no—
141C22×C4C2×C4211yescharacteristic
142C22×C4C2×C22⋊C411yescharacteristic
143C22×C4#11711yescharacteristic
144C22×C4C4×D412yes—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—21no—
148C2×C8—22no—
149C2×C8—41no—
150C2×C8—41no—
151C2×C8—44no—
152C2×C8—41no—
153C2×C8—42no—
154C2×C8—44no—
155C2×C8—42no—
156C2×C8—41no—
157C2×C8—82no—
158C2×C8—82no—
159C2×C8—82no—
160C2×C8—82no—
161C2×C8—82no—
162C2×C8#9911yescharacteristic
163C2×C8#9911yescharacteristic
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—44no—
167C22⋊C4—42no—
168C22⋊C4—44no—
169C22⋊C4—44no—
170C22⋊C4—42no—
171C22⋊C4—44no—
172C22⋊C4—44no—
173C42—22no—
174C42—22no—
175C42—21no—
176C42—21no—
177C42—21no—
178C42—41no—
179C42—42no—
180C42—82no—
181C42—82no—
182C42#9712yescomplemented
183C23—22no—
184C23—22no—
185C23—22no—
186C23—42no—
187C23—42no—
188C23—42no—
189C23—42no—
190C23—41no—
191C23—41no—
192C23C4×C22⋊C411yescharacteristic
193C23#43514yes—
194D4—42no—
195D4—42no—
196D4—42no—
197D4—416no—
198D4—42no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—84no—
222C2×C4—84no—
223C2×C4—84no—
224C2×C4C2×#9711yescharacteristic
225C2×C4#40511yescharacteristic
226C2×C4C2×#9911yescharacteristic
227C2×C4#41011yescharacteristic
228C8—21no—
229C8—42no—
230C8—42no—
231C8—41no—
232C8—84no—
233C8—81no—
234C8#35312yescomplemented
235C22—21no—
236C22—21no—
237C22—24no—
238C22—44no—
239C22—44no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—82no—
244C22—82no—
245C22—81no—
246C22—81no—
247C22#175211yescharacteristic
248C22C4×#9911yescharacteristic
249C22#175711yescharacteristic
250C4—24no—
251C4—21no—
252C4—21no—
253C4—21no—
254C4—42no—
255C4—42no—
256C4—41no—
257C4—81no—
258C4—164no—
259C4C2×#35311yescharacteristic
260C4#212611yescharacteristic
261C2—21no—
262C2—21no—
263C2—44no—
264C2—82no—
265C2#816011yescharacteristic
266C2C4×#35311yescharacteristic
267C2#1263311yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
socleC22C4×#99
prime radicalC2×#127C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8160C2semidirect
5. #7205C2semidirect
7. C2×C8×D4C4semidirect
12. #1954C4semidirect
14. #1842C4semidirect
17. C42×C8C4semidirect
20. #1325C4semidirect
21. #1318C4semidirect
126. C2×D4#97semidirect
182. C42#97semidirect
234. C8#353semidirect

Maximal subgroups

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

subgroupindex
C8×#1272
C4×#3532
#114752
#81602
#72052