Finite Group Databasebeta

← #149597

Subgroups — #149597

Order 512.

338 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 72 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#21858C211yesmaximal, characteristic
2#11456C212yesmaximal, complemented
3#7361C212yesmaximal, complemented
4#7356C212yesmaximal, complemented
5#3455C411yescharacteristic
6#2905C2211yescharacteristic
7#2298C411yescharacteristic
8C2×#349—22no—
9#1995C411yescharacteristic
10#1994C2211yescharacteristic
11#1963C2211yescharacteristic
12#1962C411yescharacteristic
13#1871—22no—
14#1806—22no—
15#1327—22no—
16#1327C2214yes—
17C22×M16D412yes—
18#542—22no—
19#541C812yes—
20#540C812yes—
21#539—41no—
22#538—41no—
23#533C2×C411yescharacteristic
24#532C2×C411yescharacteristic
25#529—42no—
26#525—21no—
27C4×#41—21no—
28C2×#117C2×C411yescharacteristic
29C2×M32—42no—
30#424—42no—
31#423C2×C411yescharacteristic
32#423D412yes—
33#423C2311yescharacteristic
34C2×#104—22no—
35C2×#104—22no—
36#388—41no—
37#387—41no—
38C4×C4⋊C4—21no—
39C4×C22⋊C4—21no—
40#376C2×C411yescharacteristic
41C43C2×C411yescharacteristic
42#349—24no—
43#349—42no—
44#339—22no—
45#339—22no—
46C2×#41—21no—
47C23×C4C22⋊C411yescharacteristic
48C2×M16—22no—
49C2×M16—82no—
50C2×M16C22⋊C412yes—
51C2×M16C2×D412yes—
52C22×C8—22no—
53C4⋊Q8—21no—
54C4⋊Q8C2×C811yescharacteristic
55#127C2×C811yescharacteristic
56#126—42no—
57#126—42no—
58#125—42no—
59#125C2×C812yes—
60#124—21no—
61#124—42no—
62#123—42no—
63#123—41no—
64#123—42no—
65C22⋊Q8—22no—
66C22⋊Q8—42no—
67C22⋊Q8—41no—
68#121—22no—
69#121—42no—
70#121—41no—
71#120—41no—
72C4×Q8—22no—
73C4×Q8—41no—
74C4×D4—22no—
75C4×D4—41no—
76C4×D4—42no—
77C4×D4—42no—
78#117—22no—
79#117—21no—
80#117—41no—
81#117C2×C812yes—
82#117M1612yes—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—41no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—41no—
87C2×C42—21no—
88C2×C42—21no—
89C2×C42—41no—
90C2×C42—41no—
91C2×C42C22⋊C411yescharacteristic
92C2×C42C22×C411yescharacteristic
93M32—44no—
94M32—82no—
95C2×C16—82no—
96C4⋊C8—41no—
97C4⋊C8—42no—
98C4⋊C8—41no—
99#104—82no—
100#104—82no—
101#95—41no—
102#95—41no—
103#95—41no—
104C24#10011yescharacteristic
105#41—42no—
106#41—42no—
107C2×Q8—21no—
108C2×Q8—41no—
109C2×D4—21no—
110C2×D4—41no—
111C2×D4—41no—
112C2×D4—42no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—41no—
116C22×C4—41no—
117C22×C4—42no—
118C22×C4—41no—
119C22×C4—41no—
120C22×C4—41no—
121C22×C4C22⋊C812yes—
122C22×C4#9912yes—
123C22×C4#10011yescharacteristic
124C22×C4C2×C22⋊C411yescharacteristic
125M16—42no—
126M16—42no—
127M16—82no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—42no—
131C2×C8—42no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—41no—
136C4⋊C4—42no—
137C4⋊C4—42no—
138C4⋊C4—41no—
139C4⋊C4—42no—
140C4⋊C4—82no—
141C22⋊C4—22no—
142C22⋊C4—22no—
143C22⋊C4—42no—
144C22⋊C4—42no—
145C22⋊C4—42no—
146C22⋊C4—41no—
147C22⋊C4—42no—
148C22⋊C4—82no—
149C42—21no—
150C42—21no—
151C42—42no—
152C42—41no—
153C42—41no—
154C42—41no—
155C42—41no—
156C42—41no—
157C42—82no—
158C42C22⋊C812yes—
159C42C22×C811yescharacteristic
160C42C2×M1611yescharacteristic
161C16—84no—
162C23—41no—
163C23—41no—
164C23—81no—
165C23#32312yes—
166C23C2×#10011yescharacteristic
167Q8—42no—
168Q8—161no—
169D4—42no—
170D4—82no—
171D4—161no—
172D4—162no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—81no—
186C2×C4—81no—
187C2×C4—81no—
188C2×C4—81no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4—81no—
194C2×C4—162no—
195C2×C4#32312yes—
196C2×C4C2×C22⋊C811yescharacteristic
197C2×C4C2×#9911yescharacteristic
198C8—42no—
199C8—42no—
200C22—21no—
201C22—22no—
202C22—41no—
203C22—42no—
204C22—82no—
205C22—81no—
206C22—161no—
207C22C2×#32311yescharacteristic
208C4—22no—
209C4—21no—
210C4—42no—
211C4—41no—
212C4—41no—
213C4—82no—
214C4—82no—
215C4—81no—
216C4—161no—
217C4#212711yescharacteristic
218C4#213311yescharacteristic
219C2—21no—
220C2—41no—
221C2—42no—
222C2—161no—
223C2#1143211yescharacteristic
subgroupG/H
centerC4#2127
commutatorC42C22×C8
Frattini, cosocle#423C23
socleC2#11432
prime radical#3455C4
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C2×C4 ◁ C23×C4 ◁ #2905 ◁ G
LECSG ▷ #423 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11456C2semidirect
3. #7361C2semidirect
4. #7356C2semidirect

Maximal subgroups

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

subgroupindex
#218582
#114562
#73612
#73562