Finite Group Databasebeta

← #140999

Subgroups — #140999

Order 512.

322 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 76 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#33249C211yesmaximal, characteristic
2#12657C211yesmaximal, characteristic
3#12596C211yesmaximal, characteristic
4C2×#1356C214yesmaximal
5C2×#499C411yescharacteristic
6C2×#494C411yescharacteristic
7C2×#491C2212yes—
8#2642C412yes—
9#2150—21no—
10C2×#347C2212yes—
11C2×#347C2212yes—
12#1971—21no—
13#1971—21no—
14#1969—21no—
15#1969—21no—
16#1726C2211yescharacteristic
17#1377—21no—
18#1376—21no—
19#1375—21no—
20#1356—28no—
21C22×Q16—21no—
22C22×Q16—21no—
23C2×C4⋊Q8C2×C412yes—
24C2×#125C2×C412yes—
25C2×M32—22no—
26C2×M32—22no—
27#499—42no—
28#494—42no—
29#491—24no—
30#491—24no—
31C2×#107—21no—
32C2×#106—21no—
33C2×#103—22no—
34C2×#103—22no—
35C2×#97D411yescharacteristic
36C2×#97D411yescharacteristic
37C2×#97C2311yescharacteristic
38C2×C4×C8C2×C411yescharacteristic
39C2×C4×C8D411yescharacteristic
40C2×C4×C8D411yescharacteristic
41#400—44no—
42#391—22no—
43#391—22no—
44#376C2×C411yescharacteristic
45C4⋊C16—24no—
46#347—24no—
47#347—22no—
48#347D414yes—
49C22×Q8—22no—
50C2×Q16—42no—
51C2×Q16—42no—
52C2×Q16—42no—
53C2×Q16—42no—
54C2×Q16—42no—
55C2×Q16—42no—
56C22×C8C22⋊C411yescharacteristic
57C22×C8C22⋊C411yescharacteristic
58C22×C8C2×D411yescharacteristic
59C22×C8C2×D411yescharacteristic
60C4⋊Q8—24no—
61C4⋊Q8—42no—
62#125—24no—
63#125—42no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—42no—
66C2×C4⋊C4—42no—
67C2×C42C22⋊C411yescharacteristic
68C2×C42C22⋊C411yescharacteristic
69C2×C42C22×C411yescharacteristic
70M32—24no—
71M32—42no—
72M32—42no—
73M32—42no—
74C2×C16—24no—
75C2×C16—42no—
76#107—42no—
77#106—42no—
78#103—44no—
79#103—42no—
80#103—42no—
81#97—41no—
82#97—41no—
83#97C2×D412yes—
84#97#4112yes—
85C4×C8—22no—
86C4×C8—22no—
87C4×C8—41no—
88#95—22no—
89#95—44no—
90#95—44no—
91C2×Q8—42no—
92C2×Q8—42no—
93C2×Q8—42no—
94C2×Q8—42no—
95C2×Q8—42no—
96C2×Q8—42no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—42no—
100C22×C4—42no—
101C22×C4#10012yes—
102C22×C4C2×C22⋊C411yescharacteristic
103Q16—84no—
104Q16—84no—
105C2×C8—21no—
106C2×C8—21no—
107C2×C8—21no—
108C2×C8—21no—
109C2×C8—21no—
110C2×C8—21no—
111C2×C8—21no—
112C2×C8—21no—
113C2×C8#10012yes—
114C2×C8#10412yes—
115C2×C8#12112yes—
116C2×C8#12411yescharacteristic
117C2×C8#12711yescharacteristic
118C4⋊C4—42no—
119C4⋊C4—42no—
120C4⋊C4—84no—
121C4⋊C4—84no—
122C42—21no—
123C42—21no—
124C42—21no—
125C42—21no—
126C42—21no—
127C42C4×D412yes—
128C16—44no—
129C16—44no—
130C23C2×#10011yescharacteristic
131Q8—82no—
132Q8—82no—
133Q8—82no—
134Q8—82no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4—41no—
141C2×C4—84no—
142C2×C4—82no—
143C2×C4—82no—
144C2×C4—82no—
145C2×C4—82no—
146C2×C4#39011yescharacteristic
147C2×C4C2×#10011yescharacteristic
148C2×C4C2×#10411yescharacteristic
149C2×C4#42111yescharacteristic
150C2×C4#42111yescharacteristic
151C2×C4#42111yescharacteristic
152C8—22no—
153C8—21no—
154C8—21no—
155C8—22no—
156C8—22no—
157C22—22no—
158C22#153011yescharacteristic
159C22#153611yescharacteristic
160C22#196511yescharacteristic
161C4—42no—
162C4—42no—
163C4—81no—
164C4—162no—
165C4—162no—
166C4#152711yescharacteristic
167C4#153311yescharacteristic
168C4#196311yescharacteristic
169C4#198011yescharacteristic
170C2—21no—
171C2—21no—
172C2#968911yescharacteristic
173C2#1191412yes—
subgroupG/H
center, socleC22#1530
commutatorC2×C42C22×C4
Frattini, cosocleC2×#97C23
prime radicalC23C2×#100
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G
LECSG ▷ C2×#97 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#332492
#126572
#125962
C2×#13562