Finite Group Databasebeta

← #125524

Subgroups — #125524

Order 512.

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

normal subgroups only

This group has 258 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#33284C211yesmaximal, characteristic
2#12897C211yesmaximal, characteristic, complemented
3#12884C211yesmaximal, characteristic, complemented
4#7135C212yesmaximal, complemented
5#7107C212yesmaximal
6#2647C411yescharacteristic
7#2641C411yescharacteristic
8#2150—21no—
9#1992C411yescharacteristic
10#1991C2211yescharacteristic
11#1961C411yescharacteristic
12#1957C2211yescharacteristic
13#1936—21no—
14#1936—21no—
15#1934—21no—
16#1934—21no—
17#1726C2211yescharacteristic
18#1377—21no—
19#1376—21no—
20#1376C2212yes—
21#1376C2212yes—
22#1375—21no—
23C2×#127C2×C411yescharacteristic
24C2×#125C2×C411yescharacteristic
25C2×M32—22no—
26C2×M32—22no—
27C2×#102—21no—
28C2×#102—21no—
29C2×#102—21no—
30C2×#102—21no—
31C2×#97C2×C411yescharacteristic
32C2×#97D411yescharacteristic
33C2×#97D411yescharacteristic
34C2×C4×C8D411yescharacteristic
35C2×C4×C8D411yescharacteristic
36C2×C4×C8C2311yescharacteristic
37#400—41no—
38#398—41no—
39#394—41no—
40#394—41no—
41#390—21no—
42#390—21no—
43#389—21no—
44#389—21no—
45#384C2×C411yescharacteristic
46#382C2×C411yescharacteristic
47#376C2×C411yescharacteristic
48C4⋊C16—42no—
49#340—21no—
50#340—21no—
51#340—21no—
52#340—21no—
53C22×D4—21no—
54C22×D4—41no—
55C22×C8C22⋊C411yescharacteristic
56C22×C8C22⋊C411yescharacteristic
57C22×C8C2×D411yescharacteristic
58C22×C8C2×D411yescharacteristic
59#127—22no—
60#127—41no—
61#125—22no—
62#125—41no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—41no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—41no—
70C2×C22⋊C4—41no—
71C2×C22⋊C4—41no—
72C2×C22⋊C4—41no—
73C2×C22⋊C4—41no—
74C2×C42C22⋊C411yescharacteristic
75C2×C42C22⋊C411yescharacteristic
76C2×C42C22×C411yescharacteristic
77M32—42no—
78M32—42no—
79M32—42no—
80M32—42no—
81C2×C16—42no—
82C2×C16—42no—
83#102—42no—
84#102—42no—
85#102—42no—
86#102—41no—
87#102—41no—
88#97—22no—
89#97—22no—
90#97—41no—
91C4×C8—21no—
92C4×C8—21no—
93C4×C8—41no—
94C4×C8—41no—
95#95—21no—
96#95—21no—
97#95—41no—
98#95—41no—
99#95—41no—
100C24—41no—
101C2×D4—41no—
102C2×D4—41no—
103C2×D4—41no—
104C2×D4—41no—
105C2×D4—41no—
106C2×D4—41no—
107C2×D4—82no—
108C2×D4—161no—
109C2×D4—161no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—41no—
113C22×C4—41no—
114C22×C4—41no—
115C22×C4#10011yescharacteristic
116C22×C4#10011yescharacteristic
117C22×C4C2×C22⋊C411yescharacteristic
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8#13512yes—
129C2×C8AGL(1,ℤ/8)11yescharacteristic
130C2×C8#13711yescharacteristic
131C4⋊C4—41no—
132C4⋊C4—41no—
133C4⋊C4—41no—
134C4⋊C4—41no—
135C4⋊C4—41no—
136C4⋊C4—41no—
137C4⋊C4—82no—
138C4⋊C4—161no—
139C4⋊C4—161no—
140C4⋊C4—161no—
141C22⋊C4—161no—
142C22⋊C4—161no—
143C22⋊C4—161no—
144C22⋊C4—161no—
145C42—21no—
146C42—21no—
147C42—21no—
148C42—21no—
149C42—21no—
150C42#13112yes—
151C16—82no—
152C16—82no—
153C23—81no—
154C23—81no—
155C23—81no—
156C23—81no—
157C23—81no—
158C23—81no—
159C23—81no—
160C23C2×#10011yescharacteristic
161D4—81no—
162D4—81no—
163D4—81no—
164D4—81no—
165D4—164no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—81no—
177C2×C4—81no—
178C2×C4—81no—
179C2×C4—81no—
180C2×C4—81no—
181C2×C4—81no—
182C2×C4—81no—
183C2×C4—81no—
184C2×C4—81no—
185C2×C4#40811yescharacteristic
186C2×C4#41011yescharacteristic
187C2×C4#41611yescharacteristic
188C2×C4#41711yescharacteristic
189C2×C4#42111yescharacteristic
190C2×C4#42111yescharacteristic
191C8—21no—
192C8—21no—
193C8—22no—
194C8—41no—
195C8—41no—
196C22—21no—
197C22—21no—
198C22—161no—
199C22—161no—
200C22—161no—
201C22—161no—
202C22—161no—
203C22—161no—
204C22—161no—
205C22#146911yescharacteristic
206C22#153611yescharacteristic
207C22#155111yescharacteristic
208C4—42no—
209C4—41no—
210C4—41no—
211C4—81no—
212C4—161no—
213C4—161no—
214C4—161no—
215C4#148211yescharacteristic
216C4#149511yescharacteristic
217C4#151211yescharacteristic
218C4#152111yescharacteristic
219C2—21no—
220C2—21no—
221C2—321no—
222C2#771611yescharacteristic
223C2#1198911yescharacteristic
224C2#1199111yescharacteristic
subgroupG/H
center, socleC22#1551
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC2×#127C2×C4
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12897C2semidirect
3. #12884C2semidirect
4. #7135C2semidirect

Maximal subgroups

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

subgroupindex
#332842
#128972
#128842
#71352
#71072