Finite Group Databasebeta

← #125409

Subgroups — #125409

Order 512.

271 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 271 conjugacy classes of subgroups — too many to draw. Only its 66 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#12881C211yesmaximal, characteristic, complemented
3#12657C211yesmaximal, characteristic, complemented
4#7130C212yesmaximal, complemented
5#7116C212yesmaximal
6#2647C411yescharacteristic
7#2641C411yescharacteristic
8C2×#364—21no—
9#2150—21no—
10C2×#347—21no—
11#1992C2211yescharacteristic
12#1991C411yescharacteristic
13#1961C2211yescharacteristic
14#1957C411yescharacteristic
15#1936—21no—
16#1936—21no—
17#1934—21no—
18#1934—21no—
19#1726C2211yescharacteristic
20#1377—21no—
21#1375C2212yes—
22#1375C2212yes—
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×#97D411yescharacteristic
32C2×#97D411yescharacteristic
33C2×#97C2311yescharacteristic
34C2×C4×C8C2×C411yescharacteristic
35C2×C4×C8D411yescharacteristic
36C2×C4×C8D411yescharacteristic
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
48#364—42no—
49C4⋊C16—24no—
50#347—42no—
51#340—21no—
52#340—21no—
53#340—21no—
54#340—21no—
55C22×D4—21no—
56C22×D4—41no—
57C22×C8C22⋊C411yescharacteristic
58C22×C8C22⋊C411yescharacteristic
59C22×C8C2×D411yescharacteristic
60C22×C8C2×D411yescharacteristic
61#127—22no—
62#127—41no—
63#125—22no—
64#125—41no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4—41no—
72C2×C22⋊C4—41no—
73C2×C22⋊C4—41no—
74C2×C22⋊C4—41no—
75C2×C22⋊C4—41no—
76C2×C42C22⋊C411yescharacteristic
77C2×C42C22⋊C411yescharacteristic
78C2×C42C22×C411yescharacteristic
79M32—42no—
80M32—42no—
81M32—42no—
82M32—42no—
83C2×C16—24no—
84C2×C16—42no—
85#102—42no—
86#102—42no—
87#102—41no—
88#102—41no—
89#102—41no—
90#102—41no—
91#97—21no—
92#97—21no—
93#97—41no—
94#97—41no—
95C4×C8—22no—
96C4×C8—21no—
97C4×C8—21no—
98C4×C8—22no—
99#95—21no—
100#95—21no—
101#95—41no—
102#95—41no—
103#95—41no—
104C24—41no—
105C2×D4—41no—
106C2×D4—41no—
107C2×D4—41no—
108C2×D4—41no—
109C2×D4—41no—
110C2×D4—41no—
111C2×D4—82no—
112C2×D4—161no—
113C2×D4—161no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—41no—
117C22×C4—41no—
118C22×C4—41no—
119C22×C4#10011yescharacteristic
120C22×C4#10011yescharacteristic
121C22×C4C2×C22⋊C411yescharacteristic
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8#10412yes—
132C2×C8AGL(1,ℤ/8)11yescharacteristic
133C2×C8AGL(1,ℤ/8)11yescharacteristic
134C2×C8#13711yescharacteristic
135C2×C8#13711yescharacteristic
136C4⋊C4—41no—
137C4⋊C4—41no—
138C4⋊C4—41no—
139C4⋊C4—41no—
140C4⋊C4—41no—
141C4⋊C4—41no—
142C4⋊C4—82no—
143C4⋊C4—161no—
144C4⋊C4—161no—
145C4⋊C4—161no—
146C22⋊C4—161no—
147C22⋊C4—161no—
148C22⋊C4—161no—
149C22⋊C4—161no—
150C42—21no—
151C42—21no—
152C42—21no—
153C42—21no—
154C42—21no—
155C42#13112yes—
156C16—44no—
157C16—82no—
158C23—81no—
159C23—81no—
160C23—81no—
161C23—81no—
162C23—81no—
163C23—81no—
164C23—81no—
165C23C2×#10011yescharacteristic
166D4—81no—
167D4—81no—
168D4—81no—
169D4—81no—
170D4—164no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—81no—
182C2×C4—81no—
183C2×C4—81no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—81no—
187C2×C4—81no—
188C2×C4—81no—
189C2×C4—81no—
190C2×C4#40811yescharacteristic
191C2×C4#41011yescharacteristic
192C2×C4#41711yescharacteristic
193C2×C4#41711yescharacteristic
194C2×C4C2×#10411yescharacteristic
195C2×C4#42111yescharacteristic
196C8—22no—
197C8—21no—
198C8—21no—
199C8—21no—
200C8—21no—
201C8—41no—
202C22—21no—
203C22—21no—
204C22—161no—
205C22—161no—
206C22—161no—
207C22—161no—
208C22—161no—
209C22—161no—
210C22—161no—
211C22#146911yescharacteristic
212C22#153011yescharacteristic
213C22#156011yescharacteristic
214C4—42no—
215C4—41no—
216C4—41no—
217C4—81no—
218C4—161no—
219C4—161no—
220C4—161no—
221C4#148311yescharacteristic
222C4#149511yescharacteristic
223C4#150911yescharacteristic
224C4#151211yescharacteristic
225C2—21no—
226C2—21no—
227C2—321no—
228C2#769611yescharacteristic
229C2#1191211yescharacteristic
230C2#1191311yescharacteristic
subgroupG/H
center, socleC22#1530
commutatorC2×C42C22×C4
Frattini, cosocleC2×#97C23
prime radicalC2×#127C2×C4
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12881C2semidirect
3. #12657C2semidirect
4. #7130C2semidirect

Maximal subgroups

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

subgroupindex
#332842
#128812
#126572
#71302
#71162