Finite Group Databasebeta

← #124403

Subgroups — #124403

Order 512.

309 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 309 conjugacy classes of subgroups — too many to draw. Only its 82 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#21604C211yesmaximal, characteristic
2#12847C211yesmaximal, characteristic
3#12602C211yesmaximal, characteristic
4#7133C212yesmaximal, complemented
5#7124C212yesmaximal, complemented
6C2×#446C411yescharacteristic
7C2×#405C2211yescharacteristic
8#2396C411yescharacteristic
9C2×#349—21no—
10#2112—21no—
11C4×M32—21no—
12#1985C2211yescharacteristic
13#1940C2211yescharacteristic
14#1916—21no—
15#1850—21no—
16#1798C411yescharacteristic
17#1770—21no—
18C8×C4⋊C4C411yescharacteristic
19#1753—21no—
20#1377—21no—
21#1377C2212yes—
22#1377C2212yes—
23C22×M16D411yescharacteristic
24C23×C8D411yescharacteristic
25C2×C4×Q8C2×C411yescharacteristic
26C2×#117C2×C411yescharacteristic
27C2×M32—22no—
28C2×M32—22no—
29#446—41no—
30#446—41no—
31C2×C4⋊C8—21no—
32C2×C4⋊C8—21no—
33C2×C4⋊C8C2×C411yescharacteristic
34C2×C4⋊C8C2×C411yescharacteristic
35C2×C22⋊C8—21no—
36#405—41no—
37#405—41no—
38C2×#97—21no—
39C2×#97D411yescharacteristic
40C2×#97C2311yescharacteristic
41C2×C4×C8—21no—
42C2×C4×C8C2×C411yescharacteristic
43C2×C4×C8D411yescharacteristic
44#400C811yescharacteristic
45#399C811yescharacteristic
46#398—21no—
47#396—21no—
48#395C811yescharacteristic
49#393C811yescharacteristic
50C4×C4⋊C4—21no—
51C4×C4⋊C4C2×C411yescharacteristic
52C4×C22⋊C4—21no—
53#349—24no—
54C22⋊C16—41no—
55C22⋊C16—41no—
56#346—41no—
57C4×C16—41no—
58#336—21no—
59#336—21no—
60#336—21no—
61#336—21no—
62#336—21no—
63C22×Q8C2×C811yescharacteristic
64C23×C4C22⋊C411yescharacteristic
65C2×M16—22no—
66C2×M16—22no—
67C2×M16—41no—
68C22×C8—21no—
69C22×C8—21no—
70C22×C8—22no—
71C22×C8—21no—
72C22×C8—21no—
73C22×C8—41no—
74C22×C8C22⋊C411yescharacteristic
75C22×C8C22⋊C411yescharacteristic
76C22×C8C2×D411yescharacteristic
77C22×C8C2×D411yescharacteristic
78C4×Q8—41no—
79C4×Q8—41no—
80#117—41no—
81#117—41no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4C2×C811yescharacteristic
87C2×C4⋊C4C2×C811yescharacteristic
88C2×C4⋊C4M1611yescharacteristic
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4C2×C811yescharacteristic
92C2×C22⋊C4M1611yescharacteristic
93C2×C42—21no—
94C2×C42—21no—
95C2×C42C22⋊C411yescharacteristic
96C2×C42C22×C411yescharacteristic
97M32—24no—
98M32—42no—
99M32—42no—
100M32—42no—
101C2×C16—42no—
102C2×C16—42no—
103C4⋊C8—41no—
104C4⋊C8—41no—
105C4⋊C8—41no—
106C4⋊C8—41no—
107C4⋊C8—42no—
108C22⋊C8—42no—
109#97—41no—
110#97—41no—
111#97—42no—
112C4×C8—21no—
113C4×C8—21no—
114C4×C8—21no—
115C4×C8—21no—
116C4×C8—42no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95C2×C811yescharacteristic
122#95C2×C811yescharacteristic
123C24C22⋊C811yescharacteristic
124C2×Q8—41no—
125C2×Q8—81no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4C22⋊C811yescharacteristic
137C22×C4C22⋊C811yescharacteristic
138C22×C4C22⋊C811yescharacteristic
139C22×C4C2×C22⋊C411yescharacteristic
140C22×C4C22×C811yescharacteristic
141C22×C4C2×M1611yescharacteristic
142M16—44no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—42no—
152C2×C8—42no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8#9911yescharacteristic
160C2×C8#9911yescharacteristic
161C2×C8#10412yes—
162C2×C8#13512yes—
163C2×C8#13712yes—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—81no—
171C22⋊C4—41no—
172C22⋊C4—41no—
173C22⋊C4—42no—
174C22⋊C4—81no—
175C42—21no—
176C42—21no—
177C42—21no—
178C42—21no—
179C42—41no—
180C42—41no—
181C42—41no—
182C42—41no—
183C16—44no—
184C16—82no—
185C23—21no—
186C23—21no—
187C23—21no—
188C23—41no—
189C23—41no—
190C23C2×C22⋊C811yescharacteristic
191Q8—82no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
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×C4C2×#9911yescharacteristic
220C2×C4#41311yescharacteristic
221C2×C4#41611yescharacteristic
222C2×C4#41911yescharacteristic
223C2×C4C2×#10411yescharacteristic
224C2×C4#42111yescharacteristic
225C8—22no—
226C8—22no—
227C8—22no—
228C8—21no—
229C8—21no—
230C8—81no—
231C8—81no—
232C22—21no—
233C22—21no—
234C22—42no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—81no—
239C22#145911yescharacteristic
240C22#147611yescharacteristic
241C22#147911yescharacteristic
242C4—42no—
243C4—41no—
244C4—41no—
245C4—41no—
246C4—41no—
247C4—41no—
248C4—81no—
249C4—81no—
250C4—81no—
251C4#150811yescharacteristic
252C4#151611yescharacteristic
253C4#152911yescharacteristic
254C4#155411yescharacteristic
255C2—21no—
256C2—21no—
257C2—81no—
258C2#758311yescharacteristic
259C2#1148411yescharacteristic
260C2#1148511yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#97C23
socleC22#1459
prime radicalC24C22⋊C8
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7133C2semidirect
5. #7124C2semidirect

Maximal subgroups

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

subgroupindex
#216042
#128472
#126022
#71332
#71242