Finite Group Databasebeta

← #126421

Subgroups — #126421

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#32524C211yesmaximal, characteristic
2#13537C211yesmaximal, characteristic, complemented
3#13480C211yesmaximal, characteristic
4#10649C211yesmaximal, characteristic, complemented
5#7120C211yesmaximal, characteristic, complemented
6#7118C211yesmaximal, characteristic, complemented
7#7112C211yesmaximal, characteristic
8#2512—21no—
9C2×#357—21no—
10#2093—21no—
11#2067—21no—
12#2054—21no—
13#1970C2211yescharacteristic
14#1947—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
28C2×#126—41no—
29C22×C16—21no—
30C22×C16—21no—
31C2×#107—21no—
32C2×#107—21no—
33C2×#107—21no—
34C2×#107—41no—
35C2×#107D411yescharacteristic
36C2×#107D411yescharacteristic
37C2×#107D411yescharacteristic
38C2×C4⋊C8—21no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8D411yescharacteristic
41C2×#102—21no—
42C2×#102—21no—
43C2×#102—21no—
44C2×C4×C8C2311yescharacteristic
45#397—41no—
46#396—41no—
47#390—21no—
48#390—21no—
49#390—21no—
50#389—21no—
51#388—41no—
52#384—21no—
53#384—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
70#126—82no—
71#126—82no—
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⋊C4—41no—
81C2×C4⋊C4D811yescharacteristic
82C2×C4⋊C4D811yescharacteristic
83C2×C22⋊C4—41no—
84C2×C22⋊C4—41no—
85C2×C22⋊C4—41no—
86C2×C22⋊C4—41no—
87C2×C22⋊C4—41no—
88C2×C22⋊C4—41no—
89C2×C42—21no—
90C2×C42—41no—
91C2×C42C2×D411yescharacteristic
92C2×C16—41no—
93C2×C16—41no—
94C2×C16—41no—
95C2×C16—41no—
96C2×C16—41no—
97C2×C16—41no—
98#107—21no—
99#107—21no—
100#107—41no—
101#107—41no—
102#107—42no—
103#107—41no—
104#107—41no—
105#107—81no—
106#107—82no—
107C4⋊C8—22no—
108C4⋊C8—41no—
109C4⋊C8—41no—
110C4⋊C8—41no—
111C4⋊C8—41no—
112#102—41no—
113#102—41no—
114#102—81no—
115#102—81no—
116C4×C8—41no—
117#95—21no—
118#95—41no—
119#95—41no—
120C24—41no—
121C2×D4—41no—
122C2×D4—41no—
123C2×D4—41no—
124C2×D4—41no—
125C2×D4—41no—
126C2×D4—41no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—81no—
133C22×C4#12011yescharacteristic
134C22×C4C2×D811yescharacteristic
135C22×C4AGL(1,ℤ/8)11yescharacteristic
136D8—42no—
137D8—81no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—42no—
143C2×C8—41no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—81no—
147C2×C8—81no—
148C2×C8#13511yescharacteristic
149C2×C8#13511yescharacteristic
150C2×C8AGL(1,ℤ/8)11yescharacteristic
151C2×C8#13711yescharacteristic
152C4⋊C4—21no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—82no—
161C4⋊C4—81no—
162C4⋊C4—81no—
163C4⋊C4—81no—
164C4⋊C4—81no—
165C4⋊C4—81no—
166C4⋊C4—81no—
167C4⋊C4—161no—
168C4⋊C4—161no—
169C4⋊C4D1612yes—
170C22⋊C4—82no—
171C22⋊C4—82no—
172C22⋊C4—82no—
173C22⋊C4—81no—
174C22⋊C4—81no—
175C22⋊C4—161no—
176C22⋊C4—161no—
177C42—21no—
178C42—21no—
179C42—41no—
180C42—41no—
181C42—82no—
182C16—81no—
183C16—81no—
184C23—81no—
185C23—81no—
186C23—81no—
187C23—81no—
188C23—81no—
189C23—81no—
190C23—81no—
191C23#44711yescharacteristic
192D4—81no—
193D4—81no—
194D4—81no—
195D4—81no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
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—161no—
217C2×C4—161no—
218C2×C4—161no—
219C2×C4#44911yescharacteristic
220C2×C4#45211yescharacteristic
221C2×C4#45411yescharacteristic
222C2×C4#45511yescharacteristic
223C2×C4#45611yescharacteristic
224C2×C4#45811yescharacteristic
225C2×C4C2×D1611yescharacteristic
226C2×C4#50911yescharacteristic
227C8—21no—
228C8—21no—
229C8—21no—
230C8—21no—
231C8—82no—
232C8—81no—
233C22—161no—
234C22—161no—
235C22—161no—
236C22—161no—
237C22—161no—
238C22—161no—
239C22—161no—
240C22#161011yescharacteristic
241C22#162511yescharacteristic
242C22#164111yescharacteristic
243C22#218411yescharacteristic
244C22#219111yescharacteristic
245C22#219311yescharacteristic
246C22#219411yescharacteristic
247C4—21no—
248C4—21no—
249C4—41no—
250C4—42no—
251C4—81no—
252C4—161no—
253C4—161no—
254C4—161no—
255C4—321no—
256C4#161211yescharacteristic
257C4#161511yescharacteristic
258C4#165011yescharacteristic
259C4#166411yescharacteristic
260C2—321no—
261C2#779911yescharacteristic
262C2#1228811yescharacteristic
263C2#1229511yescharacteristic
264C2#1234011yescharacteristic
265C2#1235011yescharacteristic
266C2#1238011yescharacteristic
267C2#1238911yescharacteristic
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. #7120C2semidirect
6. #7118C2semidirect

Maximal subgroups

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

subgroupindex
#325242
#135372
#134802
#106492
#71202
#71182
#71122