Finite Group Databasebeta

← #124394

Subgroups — #124394

Order 512.

346 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 94 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#21597C211yesmaximal, characteristic
2#12844C211yesmaximal, characteristic
3#12599C211yesmaximal, characteristic
4#7128C212yesmaximal, complemented
5#7118C212yesmaximal, complemented
6C2×#446C411yescharacteristic
7C22×C4×C8C2211yescharacteristic
8C4×C22⋊Q8C411yescharacteristic
9C2×C22⋊C16—21no—
10C2×C22⋊C16—21no—
11C2×#346—21no—
12#1984C2211yescharacteristic
13#1939C2211yescharacteristic
14#1916—21no—
15#1850—21no—
16#1797C411yescharacteristic
17#1771—21no—
18#1770C411yescharacteristic
19#1752—21no—
20#1374—21no—
21#1374C2212yes—
22#1374C2212yes—
23C23×C8D411yescharacteristic
24C23×C8D411yescharacteristic
25C2×C22⋊Q8C811yescharacteristic
26C2×C4×Q8C2×C411yescharacteristic
27C22×C42C2×C411yescharacteristic
28C22×C16—22no—
29C22×C16—22no—
30#446—42no—
31C2×C4⋊C8—21no—
32C2×C4⋊C8—21no—
33C2×C4⋊C8C2×C411yescharacteristic
34C2×C4⋊C8C2×C411yescharacteristic
35C2×C22⋊C8—21no—
36C2×#97—21no—
37C2×#97—21no—
38C2×C4×C8—42no—
39C2×C4×C8C2×C411yescharacteristic
40C2×C4×C8D411yescharacteristic
41C2×C4×C8D411yescharacteristic
42C2×C4×C8C2311yescharacteristic
43#391C811yescharacteristic
44#389C811yescharacteristic
45#387—21no—
46#385—21no—
47#380C811yescharacteristic
48C4×C4⋊C4—21no—
49C4×C4⋊C4C2×C411yescharacteristic
50C4×C22⋊C4—21no—
51C22⋊C16—24no—
52C22⋊C16—42no—
53#346—42no—
54#336—21no—
55#336—21no—
56#336—21no—
57#336—21no—
58#336—21no—
59C22×Q8C2×C811yescharacteristic
60C23×C4C22⋊C411yescharacteristic
61C23×C4C2×C811yescharacteristic
62C23×C4M1611yescharacteristic
63C22×C8—22no—
64C22×C8—22no—
65C22×C8—22no—
66C22×C8—22no—
67C22×C8—21no—
68C22×C8—21no—
69C22×C8—41no—
70C22×C8—41no—
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22⋊C411yescharacteristic
73C22×C8C2×D411yescharacteristic
74C22×C8C2×D411yescharacteristic
75C22⋊Q8—42no—
76C4×Q8—41no—
77C4×Q8—41no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4C2×C811yescharacteristic
82C2×C4⋊C4C2×C811yescharacteristic
83C2×C4⋊C4C2×C811yescharacteristic
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C42—21no—
87C2×C42—21no—
88C2×C42—41no—
89C2×C42—41no—
90C2×C42C22⋊C411yescharacteristic
91C2×C42C2×C811yescharacteristic
92C2×C42M1611yescharacteristic
93C2×C42C22×C411yescharacteristic
94C2×C16—24no—
95C2×C16—42no—
96C2×C16—42no—
97C2×C16—42no—
98C2×C16—42no—
99C2×C16—42no—
100C4⋊C8—41no—
101C4⋊C8—41no—
102C4⋊C8—42no—
103C4⋊C8—42no—
104C22⋊C8—42no—
105#97—42no—
106#97—42no—
107C4×C8—22no—
108C4×C8—22no—
109C4×C8—41no—
110C4×C8—41no—
111#95—21no—
112#95—21no—
113#95—21no—
114#95—21no—
115#95—21no—
116C24C22⋊C811yescharacteristic
117C2×Q8—22no—
118C2×Q8—41no—
119C2×Q8—41no—
120C22×C4—22no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4C22⋊C811yescharacteristic
137C22×C4C22⋊C811yescharacteristic
138C22×C4C22⋊C811yescharacteristic
139C22×C4C2×C22⋊C411yescharacteristic
140C22×C4C22×C811yescharacteristic
141C22×C4C2×M1611yescharacteristic
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—44no—
151C2×C8—44no—
152C2×C8—41no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8#9912yes—
159C2×C8#10412yes—
160C2×C8#13512yes—
161C2×C8#13712yes—
162C4⋊C4—22no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C22⋊C4—42no—
169C22⋊C4—42no—
170C42—21no—
171C42—21no—
172C42—21no—
173C42—21no—
174C42—41no—
175C42—41no—
176C42—41no—
177C42—41no—
178C42—41no—
179C42—41no—
180C16—44no—
181C16—82no—
182C23—21no—
183C23—21no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—21no—
188C23#34912yes—
189C23C2×C22⋊C811yescharacteristic
190Q8—82no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4C2×#9911yescharacteristic
224C2×C4#41311yescharacteristic
225C2×C4#41611yescharacteristic
226C2×C4#41911yescharacteristic
227C2×C4C2×#10411yescharacteristic
228C2×C4#42111yescharacteristic
229C2×C4#50412yes—
230C8—22no—
231C8—22no—
232C8—22no—
233C8—22no—
234C8—81no—
235C8—81no—
236C22—22no—
237C22—22no—
238C22—22no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—41no—
243C22#145911yescharacteristic
244C22#147611yescharacteristic
245C22#147911yescharacteristic
246C22#211311yescharacteristic
247C22C2×#34911yescharacteristic
248C22#211711yescharacteristic
249C22#211711yescharacteristic
250C4—22no—
251C4—41no—
252C4—41no—
253C4—41no—
254C4—41no—
255C4—81no—
256C4—81no—
257C4—81no—
258C4—81no—
259C4#150811yescharacteristic
260C4#151611yescharacteristic
261C4#152911yescharacteristic
262C4#155411yescharacteristic
263C2—42no—
264C2#758311yescharacteristic
265C2#1146711yescharacteristic
266C2#1146911yescharacteristic
267C2#1161411yescharacteristic
268C2#1161911yescharacteristic
269C2#1164011yescharacteristic
270C2#1164411yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC23C2×C22⋊C8
prime radicalC24C22⋊C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7128C2semidirect
5. #7118C2semidirect

Maximal subgroups

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

subgroupindex
#215972
#128442
#125992
#71282
#71182