Finite Group Databasebeta

← #126308

Subgroups — #126308

Order 512.

275 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 275 conjugacy classes of subgroups — too many to draw. Only its 62 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#32462C211yesmaximal, characteristic
2#13537C211yesmaximal, characteristic, complemented
3#13506C211yesmaximal, characteristic, complemented
4#10615C211yesmaximal, characteristic
5#7121C211yesmaximal, characteristic, complemented
6#7120C211yesmaximal, characteristic, complemented
7#7102C211yesmaximal, characteristic
8#2485—21no—
9C2×#367—21no—
10#2088—21no—
11#2076—21no—
12#2075—21no—
13#1954C2211yescharacteristic
14#1942C2211yescharacteristic
15#1942C2211yescharacteristic
16#1854—21no—
17#1854—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×C4×D4D411yescharacteristic
28C2×#117—21no—
29C22×C16—21no—
30C22×C16—21no—
31C2×#107—21no—
32C2×#107—21no—
33C2×#107D411yescharacteristic
34C2×#107D411yescharacteristic
35C2×#107D411yescharacteristic
36C2×#107D411yescharacteristic
37C2×#106—21no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8D411yescharacteristic
40C2×#102—41no—
41C2×C22⋊C8—21no—
42C2×C22⋊C8—21no—
43C2×C4×C8C2311yescharacteristic
44#397—41no—
45#394—41no—
46#389—21no—
47#389—21no—
48#388—41no—
49#387—41no—
50#384—21no—
51#380—21no—
52#380—21no—
53#380—21no—
54#367—42no—
55#340—21no—
56#340—21no—
57#340—21no—
58#340—21no—
59#340—21no—
60#340—41no—
61C22×D4D811yescharacteristic
62C23×C4—21no—
63C22×C8—21no—
64C22×C8—41no—
65C22×C8C2×D411yescharacteristic
66C22×C8C2×D411yescharacteristic
67C4×D4—41no—
68C4×D4—41no—
69#117—81no—
70#117—81no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4D811yescharacteristic
81C2×C22⋊C4—21no—
82C2×C22⋊C4—41no—
83C2×C22⋊C4—41no—
84C2×C22⋊C4—41no—
85C2×C22⋊C4—41no—
86C2×C42—41no—
87C2×C42C2×D411yescharacteristic
88C2×C16—22no—
89C2×C16—41no—
90C2×C16—41no—
91C2×C16—41no—
92C2×C16—41no—
93C2×C16—41no—
94#107—21no—
95#107—21no—
96#107—21no—
97#107—21no—
98#107—41no—
99#107—41no—
100#107—42no—
101#107—41no—
102#107—41no—
103#106—81no—
104C4⋊C8—41no—
105C4⋊C8—41no—
106C4⋊C8—41no—
107#102—82no—
108C22⋊C8—41no—
109C22⋊C8—41no—
110C22⋊C8—41no—
111C22⋊C8—41no—
112C4×C8—41no—
113#95—41no—
114#95—41no—
115#95—41no—
116#95—41no—
117#95—41no—
118C24—21no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4—41no—
122C2×D4—41no—
123C22×C4—21no—
124C22×C4—41no—
125C22×C4—41no—
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—81no—
135C22×C4#12011yescharacteristic
136C22×C4C2×D811yescharacteristic
137C22×C4AGL(1,ℤ/8)11yescharacteristic
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—41no—
145C2×C8—81no—
146C2×C8—81no—
147C2×C8—81no—
148C2×C8#13511yescharacteristic
149C2×C8#13511yescharacteristic
150C2×C8AGL(1,ℤ/8)11yescharacteristic
151C2×C8#13711yescharacteristic
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—81no—
162C4⋊C4—81no—
163C4⋊C4—81no—
164C4⋊C4—81no—
165C4⋊C4—81no—
166C4⋊C4—81no—
167C4⋊C4—161no—
168C4⋊C4—161no—
169C22⋊C4—81no—
170C22⋊C4—82no—
171C22⋊C4—82no—
172C22⋊C4—161no—
173C22⋊C4—161no—
174C42—21no—
175C42—21no—
176C42—82no—
177C16—42no—
178C16—81no—
179C23—41no—
180C23—41no—
181C23—41no—
182C23—41no—
183C23—41no—
184C23—41no—
185C23—41no—
186C23#44711yescharacteristic
187D4—42no—
188D4—81no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4—81no—
201C2×C4—81no—
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—161no—
214C2×C4—161no—
215C2×C4—161no—
216C2×C4#44911yescharacteristic
217C2×C4#45211yescharacteristic
218C2×C4#45411yescharacteristic
219C2×C4#45511yescharacteristic
220C2×C4#45611yescharacteristic
221C2×C4#45811yescharacteristic
222C2×C4#50811yescharacteristic
223C2×C4#50911yescharacteristic
224C8—21no—
225C8—21no—
226C8—21no—
227C8—21no—
228C8—81no—
229C8—161no—
230C22—81no—
231C22—81no—
232C22—81no—
233C22—81no—
234C22—81no—
235C22—81no—
236C22—81no—
237C22#160711yescharacteristic
238C22#162511yescharacteristic
239C22#164711yescharacteristic
240C22#218611yescharacteristic
241C22#219111yescharacteristic
242C22#219311yescharacteristic
243C22#219411yescharacteristic
244C4—21no—
245C4—21no—
246C4—41no—
247C4—81no—
248C4—161no—
249C4—161no—
250C4—161no—
251C4—161no—
252C4—321no—
253C4#160911yescharacteristic
254C4#161511yescharacteristic
255C4#164011yescharacteristic
256C4#166411yescharacteristic
257C2—161no—
258C2#778811yescharacteristic
259C2#1228011yescharacteristic
260C2#1228511yescharacteristic
261C2#1234111yescharacteristic
262C2#1235011yescharacteristic
263C2#1239511yescharacteristic
264C2#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. #13506C2semidirect
5. #7121C2semidirect
6. #7120C2semidirect

Maximal subgroups

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

subgroupindex
#324622
#135372
#135062
#106152
#71212
#71202
#71022