Finite Group Databasebeta

← #124668

Subgroups — #124668

Order 512.

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

normal subgroups only

This group has 396 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#7852C211yesmaximal, characteristic
4#7119C214yesmaximal, complemented
5C2×#430C411yescharacteristic
6#2911C411yescharacteristic
7#2905—21no—
8C2×C4×M16—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—22no—
21C22×M16—21no—
22C22×M16C2×C411yescharacteristic
23C22×M16D411yescharacteristic
24C2×#117—21no—
25C22×C42D411yescharacteristic
26C2×M32—22no—
27#430—42no—
28C2×#108—21no—
29C2×#108C2×C412yes—
30#428—24no—
31#427—21no—
32#427C2×C412yes—
33#424—22no—
34#424—22no—
35#424—42no—
36#423—42no—
37C4×M16—42no—
38C4×M16—42no—
39C4×M16—42no—
40C2×#97—22no—
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—48no—
59C2×M16—44no—
60C2×M16—41no—
61C2×M16—42no—
62C2×M16—42no—
63C22×C8—24no—
64C22×C8—41no—
65C22×C8C22⋊C411yescharacteristic
66C22×C8C22⋊C411yescharacteristic
67C22×C8C22×C411yescharacteristic
68C22×C8C2×D411yescharacteristic
69#117—22no—
70#117—22no—
71#117—41no—
72#117—41no—
73C2×C4⋊C4—22no—
74C2×C22⋊C4—41no—
75C2×C42—22no—
76C2×C42—21no—
77C2×C42—41no—
78C2×C42C22⋊C411yescharacteristic
79C2×C42SD1612yes—
80C2×C42Q1612yes—
81C2×C42C2×D411yescharacteristic
82M32—42no—
83M32—42no—
84C2×C16—42no—
85#108—42no—
86#108—42no—
87#107—22no—
88#107M1614yescomplemented
89#106—22no—
90#106—22no—
91C4⋊C8—44no—
92C4⋊C8—44no—
93#97—41no—
94#97—41no—
95#97—44no—
96C4×C8—41no—
97C4×C8—41no—
98C4×C8—42no—
99C4×C8—42no—
100C24#10411yescharacteristic
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—21no—
109C22×C4—44no—
110C22×C4—42no—
111C22×C4#9912yes—
112C22×C4#10312yes—
113C22×C4#10411yescharacteristic
114C22×C4C2×C22⋊C411yescharacteristic
115M16—28no—
116M16—44no—
117M16—416no—
118M16—42no—
119M16—42no—
120M16—82no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—28no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—44no—
132C2×C8—44no—
133C2×C8—42no—
134C2×C8—82no—
135C2×C8#10112yes—
136C2×C8C2×M1612yes—
137C2×C8AGL(1,ℤ/8)12yes—
138C4⋊C4—24no—
139C4⋊C4—42no—
140C22⋊C4—42no—
141C22⋊C4—42no—
142C42—22no—
143C42—22no—
144C42—21no—
145C42—21no—
146C42—42no—
147C42—42no—
148C42#10312yes—
149C42C2×SD1611yescharacteristic
150C42C2×Q1611yescharacteristic
151C16—82no—
152C23—21no—
153C23—21no—
154C23—42no—
155C23#32812yes—
156C23C2×#10411yescharacteristic
157C2×C4—24no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—44no—
168C2×C4—44no—
169C2×C4—42no—
170C2×C4—44no—
171C2×C4—162no—
172C2×C4#32812yes—
173C2×C4#40711yescharacteristic
174C2×C4C2×#9911yescharacteristic
175C2×C4C2×#10111yescharacteristic
176C2×C4C2×#10311yescharacteristic
177C2×C4#41811yescharacteristic
178C2×C4#42111yescharacteristic
179C8—22no—
180C8—22no—
181C8—22no—
182C8—48no—
183C8—41no—
184C8—82no—
185C22—22no—
186C22—22no—
187C22—42no—
188C22—42no—
189C22—81no—
190C22#149011yescharacteristic
191C22C2×#32811yescharacteristic
192C22#152811yescharacteristic
193C4—22no—
194C4—22no—
195C4—22no—
196C4—42no—
197C4—84no—
198C4#146311yescharacteristic
199C4#148711yescharacteristic
200C4#151011yescharacteristic
201C4#154911yescharacteristic
202C2—21no—
203C2—21no—
204C2—42no—
205C2#760711yescharacteristic
206C2#1166912yes—
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
88. #107M16semidirect

Maximal subgroups

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

subgroupindex
#211652
#128492
#78522
#71192