Finite Group Databasebeta

← #124802

Subgroups — #124802

Order 512.

284 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 78 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#21736C211yesmaximal, characteristic
2#12896C211yesmaximal, characteristic
3#8263C211yesmaximal, characteristic
4#7119C212yesmaximal
5#7119C212yesmaximal
6#3162C411yescharacteristic
7#2961—21no—
8#1994—21no—
9#1946—21no—
10#1945C2212yes—
11#1849C411yescharacteristic
12#1773C412yes—
13#1772—21no—
14C4×C4⋊C8—21no—
15C4×#97C2211yescharacteristic
16C2×#322—21no—
17#1377—21no—
18#1376C2212yes—
19#1375—21no—
20#1273C2211yescharacteristic
21#1273C2211yescharacteristic
22#533—21no—
23C2×M32—22no—
24#501—22no—
25#501—22no—
26#446—42no—
27#446—42no—
28C2×#108—21no—
29C2×#108C2×C412yes—
30#427—21no—
31#427C2×C412yes—
32#424—22no—
33#423—21no—
34C2×C4⋊C8—22no—
35#405—22no—
36C2×#97C2×C411yescharacteristic
37C2×#97C2×C411yescharacteristic
38C2×#97D411yescharacteristic
39C2×#97D411yescharacteristic
40C2×C4×C8—21no—
41C2×C4×C8—21no—
42C2×C4×C8D411yescharacteristic
43C2×C4×C8C2311yescharacteristic
44C43D411yescharacteristic
45#339—22no—
46#339—22no—
47#336—22no—
48#322—42no—
49C2×M16—22no—
50C22×C8—22no—
51C22×C8—22no—
52C22×C8C22⋊C411yescharacteristic
53C22×C8C22⋊C411yescharacteristic
54C22×C8C22×C411yescharacteristic
55C22×C8C2×D411yescharacteristic
56C4⋊Q8—21no—
57C4⋊Q8—21no—
58#125—22no—
59C22⋊Q8—41no—
60C22⋊Q8—41no—
61C4×Q8—41no—
62C4×Q8—41no—
63#117—22no—
64C2×C42—21no—
65C2×C42—21no—
66C2×C42C22⋊C411yescharacteristic
67C2×C42C22⋊C411yescharacteristic
68C2×C42C2×D411yescharacteristic
69M32—42no—
70M32—42no—
71C2×C16—42no—
72#108—42no—
73#108—42no—
74#107—21no—
75#107—21no—
76#107M1612yes—
77#107M1612yes—
78#106—22no—
79#106—22no—
80C4⋊C8—41no—
81C4⋊C8—41no—
82C4⋊C8—44no—
83C4⋊C8—44no—
84#97—22no—
85#97—22no—
86#97—22no—
87#97—41no—
88#97—42no—
89C4×C8—41no—
90C4×C8—41no—
91C4×C8—41no—
92C4×C8—42no—
93C4×C8—41no—
94C4×C8—41no—
95C4×C8SD1612yes—
96C4×C8Q1612yes—
97C2×Q8—41no—
98C2×Q8—41no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4#10411yescharacteristic
102C22×C4#10411yescharacteristic
103C22×C4C2×C22⋊C411yescharacteristic
104M16—42no—
105M16—42no—
106C2×C8—21no—
107C2×C8—21no—
108C2×C8—21no—
109C2×C8—21no—
110C2×C8—21no—
111C2×C8—21no—
112C2×C8—21no—
113C2×C8—21no—
114C2×C8—24no—
115C2×C8—42no—
116C2×C8—42no—
117C2×C8—42no—
118C2×C8—42no—
119C2×C8—42no—
120C2×C8—42no—
121C2×C8#10312yes—
122C2×C8#10312yes—
123C2×C8C2×M1612yes—
124C2×C8AGL(1,ℤ/8)12yes—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—42no—
128C4⋊C4—42no—
129C4⋊C4—41no—
130C4⋊C4—41no—
131C22⋊C4—42no—
132C42—21no—
133C42—21no—
134C42—21no—
135C42—21no—
136C42—42no—
137C42—42no—
138C42—42no—
139C42#10112yes—
140C42C2×SD1611yescharacteristic
141C42C2×Q1611yescharacteristic
142C16—82no—
143C23C2×#10411yescharacteristic
144Q8—161no—
145Q8—161no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—22no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4#40711yescharacteristic
160C2×C4#41011yescharacteristic
161C2×C4C2×#10111yescharacteristic
162C2×C4C2×#10311yescharacteristic
163C2×C4#41811yescharacteristic
164C2×C4#42111yescharacteristic
165C8—22no—
166C8—22no—
167C8—22no—
168C8—22no—
169C8—44no—
170C8—82no—
171C8—82no—
172C22—21no—
173C22—21no—
174C22#149011yescharacteristic
175C22#150311yescharacteristic
176C22#153111yescharacteristic
177C4—22no—
178C4—22no—
179C4—42no—
180C4—41no—
181C4—41no—
182C4—82no—
183C4—82no—
184C4#146711yescharacteristic
185C4#148711yescharacteristic
186C4#151411yescharacteristic
187C4#154911yescharacteristic
188C2—21no—
189C2—21no—
190C2#762211yescharacteristic
191C2#1166911yescharacteristic
192C2#1167011yescharacteristic
subgroupG/H
centerC2×C4#418
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
socleC22#1490
prime radicalC23C2×#104
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C4×#97 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#217362
#128962
#82632
#71192