Finite Group Databasebeta

← #149982

Subgroups — #149982

Order 512.

317 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 317 conjugacy classes of subgroups — too many to draw. Only its 56 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#28722C211yesmaximal, characteristic
2#11529C211yesmaximal, characteristic
3#11475C211yesmaximal, characteristic
4#7358C214yesmaximal, complemented
5C2×#490C412yes—
6#2859C411yescharacteristic
7C4×#127C2211yescharacteristic
8#2071—21no—
9#2071—21no—
10#1992C2212yes—
11#1804—21no—
12#1804—21no—
13#1767C411yescharacteristic
14#1465—21no—
15#1459—21no—
16#1325C2212yes—
17#1325C2212yes—
18#1318—21no—
19#1318—21no—
20C2×#127C2×C411yescharacteristic
21C2×#125C2×C412yes—
22C2×C4×D4—41no—
23C2×C4×D4D411yescharacteristic
24C2×C4×D4D411yescharacteristic
25#490—44no—
26C2×C4⋊C8—21no—
27C2×#102—22no—
28C2×#102—22no—
29#407—41no—
30C2×C22⋊C8—41no—
31C2×#97C2×C412yes—
32#390—41no—
33#390D411yescharacteristic
34#390D411yescharacteristic
35#389—41no—
36#389—41no—
37#384—21no—
38#384C2311yescharacteristic
39#382—21no—
40C43C2×C411yescharacteristic
41#340—22no—
42#340—22no—
43#336—21no—
44#324—22no—
45#324—22no—
46#323—41no—
47#323—41no—
48C22×D4—41no—
49C22×D4C22⋊C411yescharacteristic
50C22×D4C22⋊C411yescharacteristic
51C23×C4—21no—
52C23×C4—21no—
53C2×M16—81no—
54C22×C8—22no—
55C22×C8—81no—
56#127—22no—
57#127—22no—
58#125—24no—
59#125—42no—
60C4×D4—41no—
61C4×D4—41no—
62C4×D4—41no—
63C4×D4—41no—
64C4×D4—84no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—41no—
67C2×C4⋊C4—42no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4C2×D411yescharacteristic
71C2×C4⋊C4C2×D411yescharacteristic
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—41no—
75C2×C22⋊C4—41no—
76C2×C42—21no—
77C2×C42—21no—
78C2×C42C22⋊C411yescharacteristic
79C2×C42C22⋊C411yescharacteristic
80C2×C42C22×C411yescharacteristic
81C4⋊C8—41no—
82C4⋊C8—41no—
83#102—44no—
84#102—44no—
85C22⋊C8—42no—
86C22⋊C8—42no—
87C22⋊C8—81no—
88C22⋊C8—81no—
89#97—42no—
90#95—41no—
91#95—41no—
92C24—21no—
93C24—21no—
94C2×D4—21no—
95C2×D4—21no—
96C2×D4—21no—
97C2×D4—41no—
98C2×D4—41no—
99C2×D4—41no—
100C2×D4—41no—
101C2×D4—44no—
102C2×D4—82no—
103C2×D4—82no—
104C2×D4#10012yes—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—41no—
110C22×C4—41no—
111C22×C4—41no—
112C22×C4—41no—
113C22×C4—42no—
114C22×C4—82no—
115C22×C4—82no—
116C22×C4#9911yescharacteristic
117C22×C4#9911yescharacteristic
118C22×C4C2×C22⋊C411yescharacteristic
119M16—162no—
120C2×C8—42no—
121C2×C8—42no—
122C2×C8—42no—
123C2×C8—82no—
124C2×C8—82no—
125C2×C8—162no—
126C4⋊C4—41no—
127C4⋊C4—41no—
128C4⋊C4—42no—
129C4⋊C4—42no—
130C4⋊C4—84no—
131C4⋊C4—82no—
132C4⋊C4—82no—
133C4⋊C4—82no—
134C22⋊C4—81no—
135C22⋊C4—81no—
136C22⋊C4—82no—
137C22⋊C4—82no—
138C42—21no—
139C42—21no—
140C42—21no—
141C42—41no—
142C42—42no—
143C42—81no—
144C42#13112yes—
145C23—22no—
146C23—41no—
147C23—41no—
148C23—41no—
149C23—41no—
150C23—41no—
151C23—81no—
152C23—81no—
153C23—81no—
154C23—81no—
155C23C2×#9911yescharacteristic
156D4—42no—
157D4—42no—
158D4—81no—
159D4—81no—
160D4—88no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—22no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—82no—
175C2×C4—82no—
176C2×C4—81no—
177C2×C4—81no—
178C2×C4—82no—
179C2×C4—82no—
180C2×C4—82no—
181C2×C4—161no—
182C2×C4—161no—
183C2×C4#35411yescharacteristic
184C2×C4#35411yescharacteristic
185C2×C4#40811yescharacteristic
186C2×C4C2×#10011yescharacteristic
187C8—82no—
188C8—162no—
189C8—162no—
190C22—21no—
191C22—21no—
192C22—42no—
193C22—82no—
194C22—82no—
195C22—81no—
196C22—81no—
197C22—81no—
198C22—81no—
199C22—81no—
200C22—161no—
201C22—161no—
202C22#147111yescharacteristic
203C22C2×#35411yescharacteristic
204C22#213511yescharacteristic
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—42no—
210C4—42no—
211C4—41no—
212C4—81no—
213C4—81no—
214C4—162no—
215C2—21no—
216C2—21no—
217C2—82no—
218C2—161no—
219C2#1156411yescharacteristic
220C2#1359511yescharacteristic
221C2#1359811yescharacteristic
subgroupG/H
center, socleC22C2×#354
commutatorC2×C42C22×C4
Frattini, cosocle#384C23
prime radicalC2×#127C2×C4
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#127 ◁ G
LECSG ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7358C2semidirect

Maximal subgroups

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

subgroupindex
#287222
#115292
#114752
#73582