Finite Group Databasebeta

← #126423

Subgroups — #126423

Order 512.

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

normal subgroups only

This group has 272 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#32529C211yesmaximal, characteristic
2#13537C211yesmaximal, characteristic, complemented
3#13480C211yesmaximal, characteristic
4#10649C211yesmaximal, characteristic, complemented
5#7121C211yesmaximal, characteristic, complemented
6#7118C211yesmaximal, characteristic, complemented
7#7112C211yesmaximal, characteristic
8#2513—21no—
9C2×#357—21no—
10#2092—21no—
11#2065—21no—
12#2054—21no—
13#1970C2211yescharacteristic
14#1949—21no—
15#1947—21no—
16#1942C2211yescharacteristic
17#1939C2211yescharacteristic
18#1925—21no—
19#1845C2211yescharacteristic
20#1380—21no—
21#1380—21no—
22#1380—21no—
23#1374C2211yescharacteristic
24#1374C2211yescharacteristic
25#1296—21no—
26#1277C2211yescharacteristic
27C22×D8D411yescharacteristic
28C22×C16—21no—
29C22×C16—21no—
30C2×#107—21no—
31C2×#107—21no—
32C2×#107—41no—
33C2×#107D411yescharacteristic
34C2×#107D411yescharacteristic
35C2×#107D411yescharacteristic
36C2×#106—21no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8D411yescharacteristic
40C2×#102—21no—
41C2×#102—21no—
42C2×#102—21no—
43C2×C4×C8C2311yescharacteristic
44#397—41no—
45#397—41no—
46#390—21no—
47#390—21no—
48#390—21no—
49#389—21no—
50#388—41no—
51#388—41no—
52#384—21no—
53#382—21no—
54C4×C4⋊C4—21no—
55C4×C4⋊C4D411yescharacteristic
56#357—42no—
57#340—21no—
58#340—21no—
59#340—41no—
60#336—21no—
61C22×D4—21no—
62C2×D8—21no—
63C2×D8—21no—
64C2×D8—41no—
65C2×D8—41no—
66C22×C8—21no—
67C22×C8—41no—
68C22×C8C2×D411yescharacteristic
69C22×C8C2×D411yescharacteristic
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4D811yescharacteristic
78C2×C4⋊C4D811yescharacteristic
79C2×C22⋊C4—41no—
80C2×C22⋊C4—41no—
81C2×C22⋊C4—41no—
82C2×C22⋊C4—41no—
83C2×C22⋊C4—41no—
84C2×C22⋊C4—41no—
85C2×C42—21no—
86C2×C42—41no—
87C2×C42C2×D411yescharacteristic
88C2×C16—41no—
89C2×C16—41no—
90C2×C16—41no—
91C2×C16—41no—
92C2×C16—41no—
93C2×C16—41no—
94#107—21no—
95#107—21no—
96#107—41no—
97#107—41no—
98#107—42no—
99#107—41no—
100#107—41no—
101#107—82no—
102#106—81no—
103C4⋊C8—22no—
104C4⋊C8—41no—
105C4⋊C8—41no—
106C4⋊C8—41no—
107C4⋊C8—41no—
108#102—41no—
109#102—41no—
110#102—81no—
111#102—81no—
112C4×C8—41no—
113#95—21no—
114#95—41no—
115#95—41no—
116#95—41no—
117#95—41no—
118C24—41no—
119C2×D4—41no—
120C2×D4—41no—
121C2×D4—41no—
122C2×D4—41no—
123C2×D4—41no—
124C2×D4—41no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—81no—
131C22×C4#12011yescharacteristic
132C22×C4C2×D811yescharacteristic
133C22×C4AGL(1,ℤ/8)11yescharacteristic
134D8—42no—
135D8—81no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—42no—
141C2×C8—41no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—81no—
145C2×C8—81no—
146C2×C8#13511yescharacteristic
147C2×C8#13511yescharacteristic
148C2×C8AGL(1,ℤ/8)11yescharacteristic
149C2×C8#13711yescharacteristic
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—81no—
160C4⋊C4—81no—
161C4⋊C4—81no—
162C4⋊C4—81no—
163C4⋊C4—161no—
164C4⋊C4—161no—
165C22⋊C4—82no—
166C22⋊C4—81no—
167C22⋊C4—81no—
168C22⋊C4—161no—
169C22⋊C4—161no—
170C22⋊C4—161no—
171C22⋊C4—161no—
172C42—21no—
173C42—21no—
174C42—41no—
175C42—41no—
176C42—161no—
177C16—81no—
178C16—81no—
179C23—81no—
180C23—81no—
181C23—81no—
182C23—81no—
183C23—81no—
184C23—81no—
185C23—81no—
186C23#44711yescharacteristic
187D4—81no—
188D4—81no—
189D4—81no—
190D4—81no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
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—161no—
212C2×C4—161no—
213C2×C4—161no—
214C2×C4#44911yescharacteristic
215C2×C4#45211yescharacteristic
216C2×C4#45411yescharacteristic
217C2×C4#45511yescharacteristic
218C2×C4#45611yescharacteristic
219C2×C4#45811yescharacteristic
220C2×C4#50811yescharacteristic
221C2×C4#50911yescharacteristic
222C8—21no—
223C8—21no—
224C8—21no—
225C8—21no—
226C8—82no—
227C8—81no—
228C22—161no—
229C22—161no—
230C22—161no—
231C22—161no—
232C22—161no—
233C22—161no—
234C22—161no—
235C22#161011yescharacteristic
236C22#162511yescharacteristic
237C22#164111yescharacteristic
238C22#218611yescharacteristic
239C22#219111yescharacteristic
240C22#219211yescharacteristic
241C22#219411yescharacteristic
242C4—21no—
243C4—21no—
244C4—41no—
245C4—81no—
246C4—81no—
247C4—161no—
248C4—161no—
249C4—161no—
250C4—321no—
251C4#161211yescharacteristic
252C4#161511yescharacteristic
253C4#165011yescharacteristic
254C4#166411yescharacteristic
255C2—321no—
256C2#779911yescharacteristic
257C2#1229011yescharacteristic
258C2#1229411yescharacteristic
259C2#1234311yescharacteristic
260C2#1235011yescharacteristic
261C2#1238111yescharacteristic
262C2#1238811yescharacteristic
subgroupG/H
center, socleC23#447
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC22×D8D4
LCSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1939 ◁ 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
4. #10649C2semidirect
5. #7121C2semidirect
6. #7118C2semidirect

Maximal subgroups

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

subgroupindex
#325292
#135372
#134802
#106492
#71212
#71182
#71122