Finite Group Databasebeta

← #126305

Subgroups — #126305

Order 512.

296 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 64 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#32458C211yesmaximal, characteristic
2#13537C211yesmaximal, characteristic, complemented
3#13505C211yesmaximal, characteristic, complemented
4#10615C211yesmaximal, characteristic
5#7120C211yesmaximal, characteristic, complemented
6#7120C211yesmaximal, characteristic, complemented
7#7102C211yesmaximal, characteristic
8#2486—21no—
9C2×#366—21no—
10#2086—21no—
11#2075—21no—
12#2075—21no—
13#1954C2211yescharacteristic
14#1942C2211yescharacteristic
15#1942C2211yescharacteristic
16#1852—21no—
17#1852—21no—
18#1845C2211yescharacteristic
19#1802—21no—
20#1380—21no—
21#1380—21no—
22#1380—21no—
23#1374C2211yescharacteristic
24#1374C2211yescharacteristic
25#1296—21no—
26#1277C2211yescharacteristic
27C2×#123—41no—
28C2×C4×D4D411yescharacteristic
29C22×C4⋊C4—21no—
30C22×C16—21no—
31C22×C16—21no—
32C2×#107—21no—
33C2×#107—21no—
34C2×#107—21no—
35C2×#107D411yescharacteristic
36C2×#107D411yescharacteristic
37C2×#107D411yescharacteristic
38C2×#107D411yescharacteristic
39C2×C4⋊C8—21no—
40C2×C4⋊C8D411yescharacteristic
41C2×#102—41no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—21no—
44C2×C4×C8C2311yescharacteristic
45#398—41no—
46#397—41no—
47#389—21no—
48#389—21no—
49#384—21no—
50#381—41no—
51#380—21no—
52#380—21no—
53#380—21no—
54#366—42no—
55#340—21no—
56#340—21no—
57#340—21no—
58#340—21no—
59#340—21no—
60#340—41no—
61C22×D4D811yescharacteristic
62C23×C4—21no—
63C23×C4—41no—
64C22×C8—21no—
65C22×C8—41no—
66C22×C8C2×D411yescharacteristic
67C22×C8C2×D411yescharacteristic
68#123—82no—
69#123—82no—
70C4×D4—41no—
71C4×D4—41no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—81no—
83C2×C4⋊C4—81no—
84C2×C4⋊C4D811yescharacteristic
85C2×C22⋊C4—21no—
86C2×C22⋊C4—41no—
87C2×C22⋊C4—41no—
88C2×C22⋊C4—41no—
89C2×C42C2×D411yescharacteristic
90C2×C16—22no—
91C2×C16—41no—
92C2×C16—41no—
93C2×C16—41no—
94C2×C16—41no—
95C2×C16—41no—
96#107—21no—
97#107—21no—
98#107—21no—
99#107—21no—
100#107—41no—
101#107—41no—
102#107—42no—
103#107—41no—
104#107—41no—
105#107—81no—
106C4⋊C8—41no—
107C4⋊C8—41no—
108C4⋊C8—41no—
109#102—82no—
110C22⋊C8—41no—
111C22⋊C8—41no—
112C22⋊C8—41no—
113C22⋊C8—41no—
114C4×C8—41no—
115#95—41no—
116#95—41no—
117#95—41no—
118#95—41no—
119#95—41no—
120C24—21no—
121C2×D4—21no—
122C2×D4—41no—
123C2×D4—41no—
124C2×D4D1612yes—
125C22×C4—21no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—82no—
137C22×C4—82no—
138C22×C4—82no—
139C22×C4—81no—
140C22×C4#12011yescharacteristic
141C22×C4C2×D811yescharacteristic
142C22×C4AGL(1,ℤ/8)11yescharacteristic
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—41no—
148C2×C8—41no—
149C2×C8—41no—
150C2×C8—81no—
151C2×C8—81no—
152C2×C8—81no—
153C2×C8#13511yescharacteristic
154C2×C8#13511yescharacteristic
155C2×C8AGL(1,ℤ/8)11yescharacteristic
156C2×C8#13711yescharacteristic
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—82no—
167C4⋊C4—81no—
168C4⋊C4—81no—
169C4⋊C4—81no—
170C4⋊C4—81no—
171C4⋊C4—81no—
172C4⋊C4—81no—
173C4⋊C4—81no—
174C4⋊C4—81no—
175C4⋊C4—161no—
176C22⋊C4—81no—
177C22⋊C4—82no—
178C22⋊C4—82no—
179C22⋊C4—82no—
180C42—21no—
181C42—21no—
182C16—42no—
183C16—81no—
184C23—22no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23—41no—
190C23—41no—
191C23#44711yescharacteristic
192D4—42no—
193D4—81no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4—81no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—164no—
219C2×C4—161no—
220C2×C4—161no—
221C2×C4—161no—
222C2×C4#44911yescharacteristic
223C2×C4#45211yescharacteristic
224C2×C4#45411yescharacteristic
225C2×C4#45511yescharacteristic
226C2×C4#45611yescharacteristic
227C2×C4#45811yescharacteristic
228C2×C4C2×D1611yescharacteristic
229C2×C4#50911yescharacteristic
230C8—21no—
231C8—21no—
232C8—21no—
233C8—21no—
234C8—81no—
235C8—161no—
236C22—42no—
237C22—42no—
238C22—42no—
239C22—81no—
240C22—81no—
241C22—81no—
242C22—81no—
243C22#160711yescharacteristic
244C22#162511yescharacteristic
245C22#164711yescharacteristic
246C22#218411yescharacteristic
247C22#219111yescharacteristic
248C22#219311yescharacteristic
249C22#219511yescharacteristic
250C4—21no—
251C4—21no—
252C4—41no—
253C4—81no—
254C4—161no—
255C4—161no—
256C4—161no—
257C4—161no—
258C4—321no—
259C4#160911yescharacteristic
260C4#161511yescharacteristic
261C4#164011yescharacteristic
262C4#166411yescharacteristic
263C2—82no—
264C2#778811yescharacteristic
265C2#1227811yescharacteristic
266C2#1228611yescharacteristic
267C2#1234011yescharacteristic
268C2#1235111yescharacteristic
269C2#1239211yescharacteristic
270C2#1240111yescharacteristic
subgroupG/H
center, socleC23#447
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC22×D4D8
LCSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1954 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13537C2semidirect
3. #13505C2semidirect
5. #7120C2semidirect
6. #7120C2semidirect

Maximal subgroups

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

subgroupindex
#324582
#135372
#135052
#106152
#71202
#71022