Finite Group Databasebeta

← #129401

Subgroups — #129401

Order 512.

584 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 122 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×#2961C211yesmaximal, characteristic
2#12850C212yesmaximal
3C2×#1328C212yesmaximal
4C2×#1327C212yesmaximal
5C2×#533C411yescharacteristic
6#2961—24no—
7#2961—24no—
8C2×#446—22no—
9C2×#446—22no—
10C2×#424C411yescharacteristic
11C2×#424C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×#405C411yescharacteristic
14C2×#405C2211yescharacteristic
15C22×#97C411yescharacteristic
16C2×#349C2214yes—
17#1376—22no—
18#1375—22no—
19#1328—24no—
20#1327—24no—
21C22×M16C2×C411yescharacteristic
22C22×M16C2311yescharacteristic
23C23×C8C2×C411yescharacteristic
24C23×C8C2×C411yescharacteristic
25#533—22no—
26#533—22no—
27#533—22no—
28#533C814yes—
29C2×C4⋊Q8—21no—
30C2×#125—21no—
31C2×C22⋊Q8—22no—
32C2×C4×Q8—22no—
33C2×#117C2×C411yescharacteristic
34C2×#117C2×C411yescharacteristic
35C22×C42C2×C411yescharacteristic
36C2×M32—24no—
37#446—216no—
38#446—216no—
39#424—22no—
40#424—22no—
41#424—22no—
42#424D414yes—
43#423—22no—
44#423D414yes—
45C2×C4⋊C8—22no—
46C2×C4⋊C8—22no—
47C2×C4⋊C8—22no—
48#405—22no—
49#405—22no—
50#405—22no—
51#405—22no—
52C2×#97—21no—
53C2×#97—22no—
54C2×#97—21no—
55C2×#97—21no—
56C2×#97—22no—
57C2×#97D412yes—
58C2×C4×C8—21no—
59C2×C4×C8D412yes—
60#349—24no—
61#349—24no—
62C22×Q8—22no—
63C23×C4—21no—
64C23×C4C22×C411yescharacteristic
65C2×M16—21no—
66C2×M16—41no—
67C2×M16—41no—
68C2×M16—41no—
69C2×M16—41no—
70C2×M16C22⋊C412yes—
71C2×M16C2×D412yes—
72C2×M16C2×D412yes—
73C22×C8—21no—
74C22×C8—21no—
75C22×C8—22no—
76C22×C8—22no—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—21no—
80C22×C8—41no—
81C22×C8—41no—
82C22×C8C22⋊C411yescharacteristic
83C22×C8C22⋊C412yes—
84C22×C8C22⋊C411yescharacteristic
85C22×C8C2×D412yes—
86C4⋊Q8—24no—
87C4⋊Q8—24no—
88#125—24no—
89#125—24no—
90C22⋊Q8—28no—
91C22⋊Q8—28no—
92C4×Q8—28no—
93C4×Q8—28no—
94#117—21no—
95#117—21no—
96#117—21no—
97#117—21no—
98#117C22⋊C412yes—
99#117C2×C812yes—
100#117C2×C812yes—
101#117M1612yes—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4M1612yes—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42C22⋊C412yes—
113C2×C42C22⋊C412yes—
114C2×C42C2×C812yes—
115C2×C42M1611yescharacteristic
116C2×C42M1611yescharacteristic
117M32—44no—
118M32—44no—
119C2×C16—44no—
120C4⋊C8—28no—
121C4⋊C8—28no—
122C4⋊C8—28no—
123#97—24no—
124#97—24no—
125#97—24no—
126#97—24no—
127C4×C8—24no—
128C4×C8—24no—
129C24C2×M1611yescharacteristic
130C2×Q8—28no—
131C2×Q8—84no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—82no—
143C22×C4C2×C22⋊C411yescharacteristic
144C22×C4C2×C22⋊C411yescharacteristic
145C22×C4C2×C22⋊C411yescharacteristic
146C22×C4C22×C811yescharacteristic
147C22×C4C2×M1611yescharacteristic
148C22×C4C2×M1611yescharacteristic
149M16—42no—
150M16—42no—
151M16—42no—
152M16—42no—
153C2×C8—24no—
154C2×C8—24no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—22no—
162C2×C8—22no—
163C2×C8—42no—
164C2×C8—41no—
165C2×C8—41no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8—42no—
169C2×C8#10112yes—
170C2×C8#10112yes—
171C2×C8#12014yes—
172C4⋊C4—216no—
173C4⋊C4—28no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C22⋊C4—22no—
179C22⋊C4—22no—
180C22⋊C4—22no—
181C22⋊C4C22⋊C814yes—
182C42—22no—
183C42—22no—
184C42—22no—
185C42—22no—
186C42—22no—
187C42—21no—
188C42—21no—
189C42C22⋊C812yes—
190C42C22⋊C812yes—
191C42C22⋊C814yes—
192C16—84no—
193C23—21no—
194C23—21no—
195C23—42no—
196C23C2×C22⋊C811yescharacteristic
197C23#40711yescharacteristic
198C23#40711yescharacteristic
199Q8—88no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—24no—
213C2×C4—24no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4#37911yescharacteristic
221C2×C4C2×C22⋊C811yescharacteristic
222C2×C4C2×C22⋊C811yescharacteristic
223C2×C4#40711yescharacteristic
224C2×C4#41011yescharacteristic
225C2×C4#41011yescharacteristic
226C2×C4C2×#10111yescharacteristic
227C2×C4C2×#10111yescharacteristic
228C8—24no—
229C8—22no—
230C8—22no—
231C8—42no—
232C8—42no—
233C22—22no—
234C22—21no—
235C22—21no—
236C22—41no—
237C22—41no—
238C22—42no—
239C22—42no—
240C22#146711yescharacteristic
241C22#146711yescharacteristic
242C22#177911yescharacteristic
243C4—21no—
244C4—21no—
245C4—42no—
246C4—42no—
247C4—44no—
248C4—44no—
249C4#146111yescharacteristic
250C4#146111yescharacteristic
251C4#178211yescharacteristic
252C4#178411yescharacteristic
253C2—21no—
254C2—21no—
255C2—41no—
256C2—41no—
257C2#830311yescharacteristic
258C2#1154012yes—
subgroupG/H
centerC2×C4#410
commutatorC22×C4C22×C8
Frattini, cosocleC22×M16C23
socleC22#1467
prime radicalC24C2×M16
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#29612
#128502
C2×#13282
C2×#13272