Finite Group Databasebeta

← #139684

Subgroups — #139684

Order 512.

364 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 88 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#33497C211yesmaximal, characteristic, complemented
2#28185C211yesmaximal, characteristic
3#11657C212yesmaximal
4#8617C211yesmaximal, characteristic, complemented
5#7217C212yesmaximal, complemented
6C2×C8×Q8C2211yescharacteristic
7C2×C4×D8C2211yescharacteristic
8#2538—21no—
9#2183—24no—
10#2163—22no—
11#2162C2212yes—
12#2086C412yes—
13#2054C412yes—
14#1934—22no—
15#1834C2211yescharacteristic
16#1802—21no—
17#1801—21no—
18#1567C2212yes—
19#1331—24no—
20#1292—21no—
21C22×D8C2×C411yescharacteristic
22C2×C22⋊Q8—41no—
23C2×C4×Q8D411yescharacteristic
24C2×C4×D4—21no—
25C8×Q8—22no—
26C8×Q8—22no—
27C4×D8—22no—
28C4×D8—22no—
29C2×#107C2×C411yescharacteristic
30#423—22no—
31C2×C4⋊C8C2×C412yes—
32C2×C4⋊C8D411yescharacteristic
33C2×#102—22no—
34C2×#102—21no—
35C2×#101—41no—
36C2×C4×C8D412yes—
37C2×C4×C8C2311yescharacteristic
38#396—42no—
39#390—22no—
40#389—22no—
41#384—21no—
42#380—41no—
43C4⋊C16—24no—
44C4⋊C16—22no—
45C22⋊C16—22no—
46#346—24no—
47C4×C16D414yes—
48#343—41no—
49#340—22no—
50#340—21no—
51#340C2×C412yes—
52#335—22no—
53#335—22no—
54C22×Q8D811yescharacteristic
55C22×D4—21no—
56C23×C4—41no—
57C2×D8—22no—
58C2×D8—42no—
59C2×M16—42no—
60C22×C8C22⋊C412yes—
61C22×C8C22×C411yescharacteristic
62C22×C8C2×D411yescharacteristic
63C22⋊Q8—84no—
64C4×Q8—22no—
65C4×Q8—22no—
66C4×D4—42no—
67C4×D4—42no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—42no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4D811yescharacteristic
74C2×C4⋊C4SD1612yes—
75C2×C22⋊C4—42no—
76C2×C22⋊C4—41no—
77C2×C22⋊C4—42no—
78C2×C42C22⋊C412yes—
79C2×C42C2×D411yescharacteristic
80C2×C16—24no—
81#107—22no—
82C4⋊C8—24no—
83C4⋊C8—22no—
84C4⋊C8—42no—
85C4⋊C8—42no—
86#102—44no—
87#102—41no—
88#102—41no—
89#101—82no—
90C4×C8—22no—
91C4×C8—22no—
92C4×C8C2×D412yes—
93C4×C8#4112yes—
94#95—42no—
95C24—41no—
96C2×Q8—22no—
97C2×Q8—21no—
98C2×Q8—22no—
99C2×Q8—21no—
100C2×D4—22no—
101C2×D4—41no—
102C2×D4—41no—
103C2×D4—41no—
104C2×D4—41no—
105C2×D4—41no—
106C22×C4—41no—
107C22×C4—42no—
108C22×C4—82no—
109C22×C4—82no—
110C22×C4—81no—
111C22×C4—81no—
112C22×C4#10212yes—
113C22×C4#10212yes—
114C22×C4C2×C22⋊C411yescharacteristic
115C22×C4C2×D811yescharacteristic
116C22×C4C2×SD1611yescharacteristic
117D8—24no—
118D8—42no—
119M16—82no—
120M16—82no—
121C2×C8—22no—
122C2×C8—41no—
123C2×C8—41no—
124C2×C8—42no—
125C2×C8—82no—
126C2×C8C4×D412yes—
127C2×C8#12011yescharacteristic
128C2×C8#12311yescharacteristic
129C4⋊C4—24no—
130C4⋊C4—22no—
131C4⋊C4—42no—
132C4⋊C4—42no—
133C4⋊C4—41no—
134C4⋊C4—41no—
135C4⋊C4—84no—
136C4⋊C4—82no—
137C4⋊C4—161no—
138C22⋊C4—84no—
139C22⋊C4—82no—
140C22⋊C4—162no—
141C42—22no—
142C42—22no—
143C42#12112yes—
144C42#13512yes—
145C16—44no—
146C23—42no—
147C23—81no—
148C23—81no—
149C23—81no—
150C23—81no—
151C23—81no—
152C23—81no—
153C23C2×#10211yescharacteristic
154Q8—42no—
155Q8—42no—
156D4—42no—
157D4—42no—
158D4—81no—
159D4—81no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—82no—
169C2×C4—82no—
170C2×C4—81no—
171C2×C4—82no—
172C2×C4—81no—
173C2×C4—81no—
174C2×C4—162no—
175C2×C4—162no—
176C2×C4#38611yescharacteristic
177C2×C4#41611yescharacteristic
178C2×C4#46512yes—
179C2×C4#46612yes—
180C2×C4#50911yescharacteristic
181C2×C4#50912yes—
182C2×C4#51011yescharacteristic
183C8—22no—
184C8—21no—
185C8—21no—
186C8—42no—
187C8—162no—
188C22—21no—
189C22—21no—
190C22—82no—
191C22—82no—
192C22—82no—
193C22—161no—
194C22—161no—
195C22—161no—
196C22—161no—
197C22#189311yescharacteristic
198C22#213911yescharacteristic
199C22#214011yescharacteristic
200C4—21no—
201C4—22no—
202C4—42no—
203C4—42no—
204C4—162no—
205C4—161no—
206C4#189111yescharacteristic
207C4#191211yescharacteristic
208C4#221812yes—
209C4#222012yes—
210C2—21no—
211C2—21no—
212C2—162no—
213C2#948611yescharacteristic
214C2#1306111yescharacteristic
215C2#1307611yescharacteristic
subgroupG/H
center, socleC22#1893
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC22×D8C2×C4
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33497C2semidirect
4. #8617C2semidirect
5. #7217C2semidirect

Maximal subgroups

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

subgroupindex
#334972
#281852
#116572
#86172
#72172