Finite Group Databasebeta

← #141497

Subgroups — #141497

Order 512.

393 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 393 conjugacy classes of subgroups — too many to draw. Only its 78 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#27661C211yesmaximal, characteristic
2#13220C212yesmaximal
3#11881C212yesmaximal
4#8614C211yesmaximal, characteristic
5#8160C211yesmaximal, characteristic
6C2×C8×D4C2211yescharacteristic
7C22×#107C411yescharacteristic, complemented
8#2486C2212yes—
9C2×#352—24no—
10#2086—21no—
11#2075—21no—
12#1942—21no—
13#1823C412yescomplemented
14#1808—21no—
15#1808C411yescharacteristic, complemented
16#1800C2212yes—
17C4×#99C2211yescharacteristic
18#1752—21no—
19#1566—21no—
20#1459C2211yescharacteristic
21#1346—24no—
22#1292—21no—
23#1282—21no—
24C23×C8C2×C411yescharacteristic
25C23×C8D411yescharacteristic
26C2×#123—22no—
27C2×C4×D4C2311yescharacteristic
28C22×C4⋊C4C2×C412yes—
29C8×D4—41no—
30C8×D4—41no—
31C2×#107—21no—
32C2×#107—21no—
33C2×#107—21no—
34C2×#107—41no—
35C2×#107—41no—
36#423—22no—
37C2×C4⋊C8D411yescharacteristic
38C2×#100—22no—
39C2×#99D412yes—
40C2×C22⋊C8—21no—
41C2×C22⋊C8C2×C411yescharacteristic
42C2×C22⋊C8D411yescharacteristic
43C2×#97—41no—
44C2×C4×C8D411yescharacteristic
45#398—22no—
46#397—22no—
47#389—22no—
48#381—22no—
49#380—22no—
50#380C2×C412yes—
51C4×C22⋊C4—21no—
52#352—48no—
53#343—22no—
54#342—21no—
55#340—22no—
56#340—21no—
57#340—22no—
58#336—41no—
59#334—41no—
60#334—41no—
61#324D412yes—
62#323—21no—
63C22×D4C2×D411yescharacteristic
64C23×C4—22no—
65C23×C4C22×C411yescharacteristic
66C23×C4C2×D411yescharacteristic
67C2×M16—42no—
68C22×C8—21no—
69C22×C8—21no—
70C22×C8—21no—
71C22×C8—21no—
72C22×C8—42no—
73C22×C8—42no—
74C22×C8—41no—
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22⋊C411yescharacteristic
77C22×C8C22⋊C411yescharacteristic
78C22×C8C22⋊C411yescharacteristic
79#123—44no—
80#123—44no—
81C4×D4—41no—
82C4×D4—41no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4C2×D411yescharacteristic
90C2×C22⋊C4—22no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4#4111yescharacteristic
94C2×C22⋊C4#4111yescharacteristic
95C2×C42—41no—
96C2×C42C2×D411yescharacteristic
97#107—42no—
98#107—42no—
99#107—41no—
100#107—41no—
101C4⋊C8—41no—
102C4⋊C8—42no—
103C4⋊C8—42no—
104#100—44no—
105#99—24no—
106C22⋊C8—22no—
107C22⋊C8—41no—
108C22⋊C8—41no—
109C22⋊C8—41no—
110#97—82no—
111C4×C8—21no—
112C4×C8—21no—
113#95—22no—
114#95—41no—
115#95—42no—
116#95—42no—
117C24C4×D411yescharacteristic
118C24#12111yescharacteristic
119C2×D4—41no—
120C2×D4—41no—
121C2×D4AGL(1,ℤ/8)14yes—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—44no—
134C22×C4—82no—
135C22×C4—82no—
136C22×C4C2×C22⋊C411yescharacteristic
137C22×C4C4×D411yescharacteristic
138C22×C4#12111yescharacteristic
139C22×C4#12411yescharacteristic
140C22×C4#12711yescharacteristic
141M16—82no—
142M16—82no—
143C2×C8—22no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—42no—
148C2×C8—42no—
149C2×C8—42no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—42no—
154C2×C8—81no—
155C2×C8—81no—
156C2×C8—81no—
157C2×C8—82no—
158C2×C8#9911yescharacteristic
159C2×C8#9911yescharacteristic
160C4⋊C4—41no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—82no—
166C4⋊C4—82no—
167C4⋊C4—82no—
168C22⋊C4—41no—
169C22⋊C4—41no—
170C22⋊C4—42no—
171C22⋊C4—44no—
172C22⋊C4—42no—
173C22⋊C4—84no—
174C42—21no—
175C42—21no—
176C42—81no—
177C42—81no—
178C23—21no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—41no—
183C23—41no—
184C23—41no—
185C23—41no—
186C23#39011yescharacteristic
187C23#44212yes—
188C23#46912yes—
189D4—44no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—81no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—164no—
216C2×C4C2×#9911yescharacteristic
217C2×C4#41011yescharacteristic
218C2×C4#49612yes—
219C8—21no—
220C8—41no—
221C8—41no—
222C8—41no—
223C8—82no—
224C8—162no—
225C8#35212yescomplemented
226C22—21no—
227C22—21no—
228C22—22no—
229C22—22no—
230C22—42no—
231C22—42no—
232C22—42no—
233C22—41no—
234C22—41no—
235C22—41no—
236C22—41no—
237C22—81no—
238C22#196411yescharacteristic
239C22#197211yescharacteristic
240C22#197611yescharacteristic
241C4—22no—
242C4—21no—
243C4—41no—
244C4—41no—
245C4—41no—
246C4—162no—
247C4—162no—
248C4—162no—
249C4C2×#35211yescharacteristic
250C4#212311yescharacteristic
251C2—21no—
252C2—21no—
253C2—42no—
254C2—42no—
255C2#978511yescharacteristic
256C2#1322011yescharacteristic
257C2#1323311yescharacteristic
subgroupG/H
center, socleC22#1964
commutatorC23×C4C22×C4
Frattini, cosocleC2×C4×D4C23
prime radicalC22×D4C2×D4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ 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
7. C22×#107C4semidirect
13. #1823C4semidirect
15. #1808C4semidirect
225. C8#352semidirect

Maximal subgroups

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

subgroupindex
#276612
#132202
#118812
#86142
#81602