Finite Group Databasebeta

← #146201

Subgroups — #146201

Order 512.

187 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 187 conjugacy classes of subgroups — too many to draw. Only its 56 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#12034C211yesmaximal, characteristic
2#12031C211yesmaximal, characteristic
3#11653C211yesmaximal, characteristic
4#11650C211yesmaximal, characteristic
5#9765C211yesmaximal, characteristic
6#8782C211yesmaximal, characteristic
7#7144C211yesmaximal, characteristic
8#3170C2211yescharacteristic
9#2163C2211yescharacteristic
10#2162C2211yescharacteristic
11#1746C2211yescharacteristic
12#1568C2211yescharacteristic
13#1564C2211yescharacteristic
14#1564C2211yescharacteristic
15#1531—21no—
16#1506—21no—
17#1372—21no—
18#1371—21no—
19#1368—22no—
20#1334—22no—
21#1332—21no—
22#1332—21no—
23#1281—21no—
24#1274—21no—
25#533—21no—
26#500Q811yescharacteristic
27#494Q811yescharacteristic
28#487—21no—
29#467—21no—
30#467—21no—
31C4×Q16—21no—
32C4×Q16—21no—
33#432—21no—
34#431—21no—
35#431—21no—
36C2×#97D411yescharacteristic
37C2×C4×C8C2311yescharacteristic
38#376D411yescharacteristic
39C4⋊C16—21no—
40C4⋊C16—21no—
41C22⋊C16—21no—
42C22⋊C16—21no—
43#346D412yes—
44C4×C16D411yescharacteristic
45C4×C16D411yescharacteristic
46#336—21no—
47#336—21no—
48#335—21no—
49#335—21no—
50#335—21no—
51#334—21no—
52#334—21no—
53#334—21no—
54#330—42no—
55#328—42no—
56C2×Q16—21no—
57C2×Q16—21no—
58C22×C8—21no—
59C22×C8#4111yescharacteristic
60C22×C8#4111yescharacteristic
61C4⋊Q8—21no—
62C4⋊Q8—21no—
63#125—41no—
64C22⋊Q8—41no—
65C22⋊Q8—41no—
66C4×Q8—41no—
67C4×Q8—41no—
68#117—41no—
69C2×C42—21no—
70C2×C42C2×D411yescharacteristic
71C2×C16—21no—
72C2×C16—21no—
73C2×C16—22no—
74#107—21no—
75#107—21no—
76C4⋊C8—41no—
77C4⋊C8—41no—
78C4⋊C8—41no—
79#103—41no—
80#103—41no—
81C22⋊C8—41no—
82C22⋊C8—41no—
83C22⋊C8—41no—
84#97—21no—
85#97—21no—
86#97—41no—
87C4×C8—41no—
88C4×C8—41no—
89C4×C8C2×D411yescharacteristic
90C4×C8C2×D411yescharacteristic
91C4×C8C2×Q811yescharacteristic
92C4×C8#4111yescharacteristic
93#95—21no—
94#95—21no—
95C2×Q8—41no—
96C2×Q8—41no—
97C22×C4—41no—
98C22×C4#12311yescharacteristic
99C22×C4#13511yescharacteristic
100C22×C4AGL(1,ℤ/8)11yescharacteristic
101Q16—42no—
102Q16—42no—
103C2×C8—41no—
104C2×C8—41no—
105C2×C8—41no—
106C2×C8—41no—
107C2×C8—41no—
108C2×C8—81no—
109C2×C8—81no—
110C2×C8—81no—
111C2×C8C22⋊Q811yescharacteristic
112C2×C8C22⋊Q811yescharacteristic
113C2×C8#12311yescharacteristic
114C2×C8#12311yescharacteristic
115C4⋊C4—41no—
116C4⋊C4—41no—
117C4⋊C4—41no—
118C4⋊C4—41no—
119C4⋊C4—81no—
120C4⋊C4—81no—
121C22⋊C4—81no—
122C42—41no—
123C42—41no—
124C42—81no—
125C42C22⋊Q811yescharacteristic
126C42#12711yescharacteristic
127C42#13511yescharacteristic
128C42AGL(1,ℤ/8)11yescharacteristic
129C16—41no—
130C16—41no—
131C16—42no—
132C23#48211yescharacteristic
133Q8—81no—
134Q8—81no—
135Q8—161no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—81no—
141C2×C4—81no—
142C2×C4—81no—
143C2×C4—81no—
144C2×C4—81no—
145C2×C4#39911yescharacteristic
146C2×C4#47811yescharacteristic
147C2×C4#47811yescharacteristic
148C2×C4#48211yescharacteristic
149C2×C4#49511yescharacteristic
150C2×C4#49611yescharacteristic
151C8—21no—
152C8—21no—
153C8—21no—
154C8—21no—
155C8—81no—
156C8—161no—
157C8—161no—
158C8—161no—
159C22—21no—
160C22—21no—
161C22—21no—
162C22#209211yescharacteristic
163C4—21no—
164C4—21no—
165C4—21no—
166C4—21no—
167C4—21no—
168C4—161no—
169C4—161no—
170C4—161no—
171C4#209811yescharacteristic
172C4#209811yescharacteristic
173C2—41no—
174C2#1066511yescharacteristic
175C2#1352411yescharacteristic
176C2#1352511yescharacteristic
subgroupG/H
center, socleC22#2092
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC23#482
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1746 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#120342
#120312
#116532
#116502
#97652
#87822
#71442