Finite Group Databasebeta

← #144580

Subgroups — #144580

Order 512.

311 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 311 conjugacy classes of subgroups — too many to draw. Only its 74 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#23835C211yesmaximal, characteristic
2#11743C212yesmaximal
3#9848C211yesmaximal, characteristic
4#7296C212yesmaximal
5#7142C211yesmaximal, characteristic
6#3170C2211yescharacteristic
7C4×C4⋊Q8—21no—
8#2162C2212yes—
9#2083—22no—
10#2057—22no—
11#1994—21no—
12C4×#107—21no—
13C4×#107C2212yes—
14#1744C2211yescharacteristic
15#1567C2211yescharacteristic
16#1493—21no—
17#1348—22no—
18#1348—22no—
19#533—21no—
20C2×C4⋊Q8—21no—
21C2×C4×Q8—41no—
22#500Q811yescharacteristic, complemented
23#494Q811yescharacteristic, complemented
24#487—21no—
25#467—22no—
26C4×Q16—22no—
27C2×#107—22no—
28C2×#107—22no—
29#423—21no—
30C2×C4×C8D411yescharacteristic
31C2×C4×C8C2311yescharacteristic
32#391—41no—
33#389—22no—
34#389—41no—
35#384—21no—
36C4×C4⋊C4—22no—
37C4×C4⋊C4—41no—
38C43D411yescharacteristic
39C4⋊C16—22no—
40C22⋊C16—22no—
41C4×C16—22no—
42#340—21no—
43#340—21no—
44#340D412yes—
45#340D412yes—
46#339—41no—
47#339—41no—
48#336—22no—
49#335—22no—
50Q8⋊C8—42no—
51C22×Q8—41no—
52C2×Q16—22no—
53C2×M16—41no—
54C22×C8—21no—
55C22×C8C2×D412yes—
56C4⋊Q8—21no—
57C4⋊Q8—21no—
58C4⋊Q8—44no—
59#125—41no—
60C22⋊Q8—41no—
61C22⋊Q8—41no—
62C4×Q8—41no—
63C4×Q8—41no—
64C4×Q8—82no—
65C4×Q8—82no—
66#117—41no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4—42no—
72C2×C4⋊C4—41no—
73C2×C42—21no—
74C2×C42—21no—
75C2×C42—42no—
76C2×C42D811yescharacteristic
77C2×C42D811yescharacteristic
78C2×C42C2×D411yescharacteristic
79C2×C16—42no—
80#107—22no—
81#107—44no—
82#107—44no—
83C4⋊C8—41no—
84C4⋊C8—41no—
85#103—42no—
86C4×C8—21no—
87C4×C8SD1612yes—
88C4×C8C2×Q811yescharacteristic
89C4×C8#4111yescharacteristic
90C4×C8#4112yes—
91#95—42no—
92#95—42no—
93C2×Q8—44no—
94C2×Q8—41no—
95C2×Q8—41no—
96C2×Q8—82no—
97C2×Q8—82no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—42no—
101C22×C4—42no—
102C22×C4#12011yescharacteristic
103C22×C4C2×D811yescharacteristic
104C22×C4AGL(1,ℤ/8)11yescharacteristic
105Q16—82no—
106M16—81no—
107M16—81no—
108C2×C8—22no—
109C2×C8—42no—
110C2×C8—42no—
111C2×C8—81no—
112C2×C8C22⋊Q812yes—
113C2×C8#12412yes—
114C4⋊C4—24no—
115C4⋊C4—42no—
116C4⋊C4—42no—
117C4⋊C4—42no—
118C4⋊C4—44no—
119C4⋊C4—41no—
120C4⋊C4—41no—
121C4⋊C4—41no—
122C4⋊C4—41no—
123C4⋊C4—84no—
124C4⋊C4—82no—
125C4⋊C4—82no—
126C4⋊C4—81no—
127C4⋊C4—81no—
128C22⋊C4—81no—
129C42—21no—
130C42—21no—
131C42—42no—
132C42—81no—
133C42—81no—
134C42—82no—
135C42—82no—
136C42SD3212yes—
137C42Q3212yes—
138C42C22⋊Q811yescharacteristic
139C42#12411yescharacteristic
140C42C2×SD1611yescharacteristic
141C42#13711yescharacteristic
142C16—82no—
143C23#44711yescharacteristic
144Q8—82no—
145Q8—88no—
146Q8—82no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—44no—
156C2×C4—82no—
157C2×C4—82no—
158C2×C4—82no—
159C2×C4—82no—
160C2×C4—82no—
161C2×C4—81no—
162C2×C4—81no—
163C2×C4#39311yescharacteristic
164C2×C4#44811yescharacteristic
165C2×C4#47411yescharacteristic
166C2×C4#47611yescharacteristic
167C2×C4#48611yescharacteristic
168C2×C4#48911yescharacteristic
169C2×C4C2×SD3211yescharacteristic
170C2×C4C2×Q3211yescharacteristic
171C2×C4#50911yescharacteristic
172C2×C4#51011yescharacteristic
173C8—22no—
174C8—22no—
175C8—42no—
176C8—161no—
177C22—21no—
178C22—21no—
179C22#202311yescharacteristic
180C22#218711yescharacteristic
181C22#218811yescharacteristic
182C4—21no—
183C4—21no—
184C4—21no—
185C4—42no—
186C4—42no—
187C4—84no—
188C4—162no—
189C4—161no—
190C4—161no—
191C4#203112yes—
192C4#222511yescharacteristic
193C4#222711yescharacteristic
194C4#224111yescharacteristic
195C4#224311yescharacteristic
196C2—21no—
197C2—21no—
198C2#1037011yescharacteristic
199C2#1336411yescharacteristic
200C2#1337311yescharacteristic
subgroupG/H
centerC2×C4#486
commutator, Frattini, cosocleC2×C4×C8C23
socleC22#2023
prime radicalC23#447
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1744 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
22. #500Q8semidirect
23. #494Q8semidirect

Maximal subgroups

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

subgroupindex
#238352
#117432
#98482
#72962
#71422