Finite Group Databasebeta

← #134388

Subgroups — #134388

Order 512.

404 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 118 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#20485C211yesmaximal, characteristic
2#12883C211yesmaximal, characteristic, complemented
3C4×#364C211yesmaximal, characteristic, complemented
4#11587C212yesmaximal
5#7168C212yesmaximal, complemented
6C2×#436C2211yescharacteristic
7C2×C8×D4C411yescharacteristic
8C2×#405—21no—
9#2293C411yescharacteristic
10C2×#364—21no—
11#2150C2212yes—
12C4×M32C2212yes—
13#1770C2211yescharacteristic
14C8×C4⋊C4C411yescharacteristic
15#1767C411yescharacteristic
16#1753—21no—
17C8×C22⋊C4—21no—
18#1750—21no—
19C42×C8C2211yescharacteristic
20#1376—21no—
21#1376—21no—
22#1376—21no—
23#1330—24no—
24C22×M16—21no—
25C23×C8—21no—
26C2×C4×D4C2×C411yescharacteristic
27C2×#117—21no—
28C2×M32—22no—
29C2×M32—22no—
30#436—21no—
31#436—21no—
32#436D412yes—
33#436Q812yes—
34C8×D4—22no—
35C8×D4C814yes—
36C2×C4⋊C8C2×C411yescharacteristic
37C2×C4⋊C8C2×C411yescharacteristic
38C2×C22⋊C8—21no—
39C2×C22⋊C8—21no—
40#405—44no—
41C2×#97—21no—
42C2×#97—21no—
43C2×C4×C8—21no—
44C2×C4×C8C2×C411yescharacteristic
45C2×C4×C8D411yescharacteristic
46C2×C4×C8D411yescharacteristic
47C2×C4×C8D411yescharacteristic
48C2×C4×C8D411yescharacteristic
49C2×C4×C8C2311yescharacteristic
50#390—21no—
51#389—21no—
52#388—22no—
53C4×C4⋊C4C2×C411yescharacteristic
54C4×C22⋊C4—21no—
55C43C2×C411yescharacteristic
56#364—24no—
57C4⋊C16—22no—
58C4⋊C16—22no—
59#346—22no—
60C4×C16—22no—
61#336—21no—
62#336—21no—
63#336—21no—
64C22×D4M1611yescharacteristic
65C23×C4—21no—
66C2×M16—22no—
67C2×M16—22no—
68C2×M16—42no—
69C2×M16—42no—
70C22×C8—21no—
71C22×C8—21no—
72C22×C8—24no—
73C22×C8—42no—
74C22×C8—42no—
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22⋊C411yescharacteristic
77C22×C8C2×D411yescharacteristic
78C22×C8C2×D411yescharacteristic
79C4×D4—21no—
80C4×D4—21no—
81C4×D4C4⋊C412yes—
82C4×D4C2×C812yes—
83#117—42no—
84#117—42no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4M1611yescharacteristic
88C2×C22⋊C4—21no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—22no—
91C2×C42—21no—
92C2×C42—21no—
93C2×C42—21no—
94C2×C42C22⋊C411yescharacteristic
95C2×C42C22⋊C411yescharacteristic
96C2×C42C22×C411yescharacteristic
97M32—24no—
98M32—42no—
99M32—42no—
100M32—42no—
101C2×C16—42no—
102C2×C16—42no—
103C4⋊C8—21no—
104C4⋊C8—21no—
105C4⋊C8C4⋊C412yes—
106C4⋊C8C2×C812yes—
107C22⋊C8—24no—
108C22⋊C8—22no—
109C22⋊C8—22no—
110#97—42no—
111#97—42no—
112C4×C8—22no—
113C4×C8—22no—
114C4×C8—22no—
115C4×C8—41no—
116C4×C8—42no—
117C4×C8C2×C812yes—
118C4×C8M1612yes—
119C4×C8C2×D411yescharacteristic
120C4×C8C2×Q811yescharacteristic
121C4×C8#4111yescharacteristic
122C4×C8#4111yescharacteristic
123#95—22no—
124#95M1612yes—
125C24—21no—
126C2×D4—21no—
127C2×D4—42no—
128C2×D4C4⋊C812yes—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4C2×C22⋊C411yescharacteristic
140C22×C4C2×M1611yescharacteristic
141C22×C4C2×M1611yescharacteristic
142M16—44no—
143M16—44no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—48no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8C22⋊C812yes—
158C2×C8C22⋊C812yes—
159C2×C8#12111yescharacteristic
160C2×C8#12111yescharacteristic
161C2×C8C22⋊Q811yescharacteristic
162C2×C8C22⋊Q811yescharacteristic
163C2×C8#13512yes—
164C2×C8AGL(1,ℤ/8)12yes—
165C4⋊C4—21no—
166C4⋊C4—42no—
167C4⋊C4—44no—
168C4⋊C4C4⋊C812yes—
169C22⋊C4—22no—
170C22⋊C4—22no—
171C22⋊C4—44no—
172C22⋊C4—44no—
173C42—21no—
174C42—21no—
175C42—21no—
176C42—42no—
177C42—42no—
178C42—41no—
179C42—41no—
180C42#10412yes—
181C42C2×C4⋊C411yescharacteristic
182C42#11711yescharacteristic
183C42C22×C811yescharacteristic
184C42C2×M1611yescharacteristic
185C16—44no—
186C16—82no—
187C23—22no—
188C23—42no—
189C23—42no—
190C23—41no—
191C23—41no—
192C23#40711yescharacteristic
193D4—42no—
194D4—42no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4#38011yescharacteristic
214C2×C4C2×C22⋊C811yescharacteristic
215C2×C4#41611yescharacteristic
216C2×C4#41811yescharacteristic
217C2×C4C2×#10411yescharacteristic
218C2×C4#42111yescharacteristic
219C2×C4C2×C4⋊C811yescharacteristic
220C2×C4#42311yescharacteristic
221C2×C4#43111yescharacteristic
222C2×C4#43211yescharacteristic
223C8—21no—
224C8—21no—
225C8—22no—
226C8—22no—
227C8—42no—
228C8—42no—
229C8#46312yes—
230C8#47812yes—
231C22—21no—
232C22—21no—
233C22—42no—
234C22—42no—
235C22—42no—
236C22—82no—
237C22—82no—
238C22#148811yescharacteristic
239C22#149011yescharacteristic
240C22#179711yescharacteristic
241C4—21no—
242C4—21no—
243C4—21no—
244C4—42no—
245C4—42no—
246C4—42no—
247C4—82no—
248C4#147611yescharacteristic
249C4#147811yescharacteristic
250C4#180411yescharacteristic
251C4#180611yescharacteristic
252C4#216612yes—
253C2—21no—
254C2—21no—
255C2—82no—
256C2#856511yescharacteristic
257C2#1279011yescharacteristic
258C2#1279211yescharacteristic
subgroupG/H
centerC2×C4#380
commutatorC42C22×C8
Frattini, cosocleC2×C4×C8C23
socleC22#1797
prime radicalC22×D4M16
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×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
2. #12883C2semidirect
3. C4×#364C2semidirect
5. #7168C2semidirect

Maximal subgroups

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

subgroupindex
#204852
#128832
C4×#3642
#115872
#71682