Finite Group Databasebeta

← #126306

Subgroups — #126306

Order 512.

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

normal subgroups only

This group has 286 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#32460C211yesmaximal, characteristic
2#13537C211yesmaximal, characteristic, complemented
3#13506C211yesmaximal, characteristic, complemented
4#10615C211yesmaximal, characteristic
5#7121C211yesmaximal, characteristic, complemented
6#7121C211yesmaximal, characteristic, complemented
7#7102C211yesmaximal, characteristic
8#2487—21no—
9C2×#367—21no—
10#2087—21no—
11#2076—21no—
12#2076—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×C4×D4D411yescharacteristic
28C22×C4⋊C4—21no—
29C22×C16—21no—
30C22×C16—21no—
31C2×#107—21no—
32C2×#107—21no—
33C2×#107—21no—
34C2×#107D411yescharacteristic
35C2×#107D411yescharacteristic
36C2×#107D411yescharacteristic
37C2×#107D411yescharacteristic
38C2×C4⋊C8—21no—
39C2×C4⋊C8D411yescharacteristic
40C2×#102—41no—
41C2×C22⋊C8—21no—
42C2×C22⋊C8—21no—
43C2×C4×C8C2311yescharacteristic
44#400—41no—
45#399—41no—
46#389—21no—
47#389—21no—
48#386—41no—
49#385—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—
63C23×C4—41no—
64C22×C8—21no—
65C22×C8—41no—
66C22×C8C2×D411yescharacteristic
67C22×C8C2×D411yescharacteristic
68C4×D4—41no—
69C4×D4—41no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—81no—
81C2×C4⋊C4—81no—
82C2×C4⋊C4D811yescharacteristic
83C2×C22⋊C4—21no—
84C2×C22⋊C4—41no—
85C2×C22⋊C4—41no—
86C2×C22⋊C4—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#107—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—41no—
121C2×D4—41no—
122C2×D4SD3212yes—
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—82no—
135C22×C4—81no—
136C22×C4—161no—
137C22×C4—161no—
138C22×C4#12011yescharacteristic
139C22×C4C2×D811yescharacteristic
140C22×C4AGL(1,ℤ/8)11yescharacteristic
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—41no—
146C2×C8—41no—
147C2×C8—41no—
148C2×C8—81no—
149C2×C8—81no—
150C2×C8—81no—
151C2×C8#13511yescharacteristic
152C2×C8#13511yescharacteristic
153C2×C8AGL(1,ℤ/8)11yescharacteristic
154C2×C8#13711yescharacteristic
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—82no—
165C4⋊C4—81no—
166C4⋊C4—81no—
167C4⋊C4—81no—
168C4⋊C4—81no—
169C4⋊C4—161no—
170C4⋊C4—161no—
171C4⋊C4—161no—
172C22⋊C4—81no—
173C22⋊C4—82no—
174C22⋊C4—161no—
175C22⋊C4—161no—
176C42—21no—
177C42—21no—
178C16—42no—
179C16—81no—
180C23—22no—
181C23—41no—
182C23—41no—
183C23—41no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23#44711yescharacteristic
188D4—42no—
189D4—81no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
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—81no—
214C2×C4—164no—
215C2×C4—161no—
216C2×C4—161no—
217C2×C4—161no—
218C2×C4#44911yescharacteristic
219C2×C4#45211yescharacteristic
220C2×C4#45411yescharacteristic
221C2×C4#45511yescharacteristic
222C2×C4#45611yescharacteristic
223C2×C4#45811yescharacteristic
224C2×C4C2×SD3211yescharacteristic
225C2×C4#51011yescharacteristic
226C8—21no—
227C8—21no—
228C8—21no—
229C8—21no—
230C8—81no—
231C8—161no—
232C22—42no—
233C22—42no—
234C22—42no—
235C22—81no—
236C22—81no—
237C22—81no—
238C22—81no—
239C22#160711yescharacteristic
240C22#162511yescharacteristic
241C22#164711yescharacteristic
242C22#218511yescharacteristic
243C22#219111yescharacteristic
244C22#219211yescharacteristic
245C22#219411yescharacteristic
246C4—21no—
247C4—21no—
248C4—41no—
249C4—81no—
250C4—161no—
251C4—161no—
252C4—161no—
253C4—161no—
254C4—321no—
255C4#160911yescharacteristic
256C4#161511yescharacteristic
257C4#164011yescharacteristic
258C4#166411yescharacteristic
259C2—82no—
260C2#778811yescharacteristic
261C2#1227911yescharacteristic
262C2#1228411yescharacteristic
263C2#1234211yescharacteristic
264C2#1235011yescharacteristic
265C2#1239411yescharacteristic
266C2#1240011yescharacteristic
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. #7121C2semidirect

Maximal subgroups

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

subgroupindex
#324602
#135372
#135062
#106152
#71212
#71022