Finite Group Databasebeta

← #124669

Subgroups — #124669

Order 512.

364 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 84 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#21165C211yesmaximal, characteristic
2#12849C211yesmaximal, characteristic
3#7850C211yesmaximal, characteristic
4#7119C214yesmaximal, complemented
5C2×#431—21no—
6C2×#430C411yescharacteristic
7#2911C411yescharacteristic
8#2905—21no—
9C2×C4×M16C2211yescharacteristic
10C2×#349—21no—
11#1946—21no—
12#1946C412yes—
13#1945—21no—
14#1945C2212yes—
15C2×#339—21no—
16#1376C2212yes—
17#1375—21no—
18#1273—21no—
19#1273C2212yes—
20C22×M16—21no—
21C22×M16C2×C411yescharacteristic
22C22×M16D411yescharacteristic
23C2×#117—21no—
24C22×C42D411yescharacteristic
25C2×M32—22no—
26#431—42no—
27#431—42no—
28#430—42no—
29C2×#108—21no—
30C2×#108C2×C412yes—
31#428—24no—
32#427—21no—
33#427C2×C412yes—
34#424—22no—
35#424—22no—
36#424—42no—
37#423—42no—
38C2×C4⋊C8—22no—
39C2×C22⋊C8—22no—
40C4×M16—42no—
41C2×#97C2×C411yescharacteristic
42C2×#97D411yescharacteristic
43C2×C4×C8—22no—
44C2×C4×C8D411yescharacteristic
45C2×C4×C8C2311yescharacteristic
46#349—42no—
47#339—22no—
48#339—22no—
49C23×C4—21no—
50C23×C4C22⋊C411yescharacteristic
51C2×M16—22no—
52C2×M16—22no—
53C2×M16—22no—
54C2×M16—22no—
55C2×M16—22no—
56C2×M16—22no—
57C2×M16—41no—
58C2×M16—41no—
59C2×M16—42no—
60C2×M16—42no—
61C22×C8—24no—
62C22×C8—41no—
63C22×C8C22⋊C411yescharacteristic
64C22×C8C22⋊C411yescharacteristic
65C22×C8C22×C411yescharacteristic
66C22×C8C2×D411yescharacteristic
67#117—22no—
68#117—22no—
69#117—41no—
70#117—41no—
71C2×C4⋊C4—22no—
72C2×C22⋊C4—41no—
73C2×C42—22no—
74C2×C42—21no—
75C2×C42—41no—
76C2×C42C22⋊C411yescharacteristic
77C2×C42SD1612yes—
78C2×C42Q1612yes—
79C2×C42C2×D411yescharacteristic
80M32—42no—
81M32—42no—
82C2×C16—42no—
83#108—42no—
84#108—42no—
85#107—22no—
86#107M1614yes—
87#106—22no—
88#106—22no—
89C4⋊C8—44no—
90C4⋊C8—44no—
91C4⋊C8—44no—
92C22⋊C8—44no—
93#97—41no—
94#97—41no—
95C4×C8—41no—
96C4×C8—41no—
97C4×C8—42no—
98C4×C8—42no—
99C24#10411yescharacteristic
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—21no—
108C22×C4—44no—
109C22×C4—42no—
110C22×C4#9912yes—
111C22×C4#10312yes—
112C22×C4#10411yescharacteristic
113C22×C4C2×C22⋊C411yescharacteristic
114M16—28no—
115M16—44no—
116M16—42no—
117M16—42no—
118M16—82no—
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—44no—
129C2×C8—48no—
130C2×C8—42no—
131C2×C8—82no—
132C2×C8#10112yes—
133C2×C8C2×M1612yes—
134C2×C8AGL(1,ℤ/8)12yes—
135C4⋊C4—24no—
136C4⋊C4—42no—
137C22⋊C4—42no—
138C22⋊C4—42no—
139C42—22no—
140C42—22no—
141C42—21no—
142C42—21no—
143C42—42no—
144C42—42no—
145C42#10312yes—
146C42C2×SD1611yescharacteristic
147C42C2×Q1611yescharacteristic
148C16—82no—
149C23—21no—
150C23—21no—
151C23—42no—
152C23#32812yes—
153C23C2×#10411yescharacteristic
154C2×C4—24no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—44no—
165C2×C4—44no—
166C2×C4—42no—
167C2×C4—44no—
168C2×C4—162no—
169C2×C4#32812yes—
170C2×C4#40711yescharacteristic
171C2×C4C2×#9911yescharacteristic
172C2×C4C2×#10111yescharacteristic
173C2×C4C2×#10311yescharacteristic
174C2×C4#41811yescharacteristic
175C2×C4#42111yescharacteristic
176C8—22no—
177C8—22no—
178C8—22no—
179C8—41no—
180C8—84no—
181C8—82no—
182C22—22no—
183C22—22no—
184C22—42no—
185C22—42no—
186C22—81no—
187C22#149011yescharacteristic
188C22C2×#32811yescharacteristic
189C22#152811yescharacteristic
190C4—22no—
191C4—22no—
192C4—22no—
193C4—42no—
194C4—84no—
195C4#146311yescharacteristic
196C4#148711yescharacteristic
197C4#151011yescharacteristic
198C4#154911yescharacteristic
199C2—21no—
200C2—21no—
201C2—42no—
202C2#760711yescharacteristic
203C2#1167012yes—
subgroupG/H
centerC2×C4#418
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
socleC22#1490
prime radicalC24#104
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×M16 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7119C2semidirect

Maximal subgroups

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

subgroupindex
#211652
#128492
#78502
#71192