Finite Group Databasebeta

← #144246

Subgroups — #144246

Order 512.

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

normal subgroups only

This group has 449 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#33755C211yesmaximal, characteristic
2#33130C212yesmaximal
3#12844C211yesmaximal, characteristic
4#11887C212yesmaximal
5#7230C211yesmaximal, characteristic
6C2×#467C2212yes—
7C22×C4×C8C2211yescharacteristic
8#2855—22no—
9#2236—22no—
10#2216—22no—
11#2188—22no—
12C2×C22⋊C16C2211yescharacteristic
13C2×C22⋊C16C2211yescharacteristic
14#1969—22no—
15#1942—22no—
16#1933—24no—
17#1892—22no—
18#1808C2212yes—
19#1374—22no—
20#1348—24no—
21C22×Q16D412yes—
22C23×C8D411yescharacteristic
23C23×C8C2311yescharacteristic
24C2×C22⋊Q8—22no—
25C22×C42D411yescharacteristic
26C2×Q32—22no—
27C22×C16—21no—
28C22×C16D412yes—
29#467—22no—
30#467—22no—
31#467—22no—
32#467—22no—
33C2×#107—22no—
34C2×#107D412yes—
35C2×#103—24no—
36C2×#103—22no—
37C2×C4×C8—21no—
38C2×C4×C8—22no—
39C2×C4×C8—22no—
40C2×C4×C8—21no—
41#391—24no—
42#389—24no—
43#387—44no—
44#385—42no—
45#381—42no—
46#380—22no—
47#366—22no—
48#358—24no—
49C22⋊C16—22no—
50C22⋊C16—22no—
51#340—24no—
52#340D414yes—
53C22×Q8—22no—
54C23×C4D811yescharacteristic
55C23×C4D811yescharacteristic
56C23×C4C2×D411yescharacteristic
57C2×Q16—22no—
58C2×Q16—22no—
59C2×Q16—44no—
60C22×C8—22no—
61C22×C8—21no—
62C22×C8—42no—
63C22×C8—42no—
64C22×C8C2×D412yes—
65C22×C8C2×D411yescharacteristic
66C22×C8C2×D412yes—
67C22×C8#4111yescharacteristic
68C22⋊Q8—42no—
69C22⋊Q8—42no—
70C22⋊Q8—42no—
71C22⋊Q8—42no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—44no—
75C2×C4⋊C4—42no—
76C2×C22⋊C4—42no—
77C2×C22⋊C4—42no—
78C2×C42—22no—
79C2×C42—21no—
80C2×C42—21no—
81C2×C42—22no—
82Q32—82no—
83C2×C16—22no—
84C2×C16—42no—
85C2×C16—42no—
86C2×C16—41no—
87#107—22no—
88#107—22no—
89#107—44no—
90#103—48no—
91#103—42no—
92#103—42no—
93C4×C8—41no—
94C4×C8—42no—
95C4×C8—41no—
96#95—44no—
97#95—44no—
98#95—44no—
99#95—42no—
100C24C2×D811yescharacteristic
101C2×Q8—44no—
102C2×Q8—44no—
103C2×Q8—42no—
104C2×Q8—42no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—41no—
112C22×C4—41no—
113C22×C4—42no—
114C22×C4—44no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4Q3212yescomplemented
118C22×C4#12012yes—
119C22×C4#12111yescharacteristic
120C22×C4#13511yescharacteristic
121C22×C4AGL(1,ℤ/8)12yes—
122Q16—44no—
123Q16—82no—
124C2×C8—24no—
125C2×C8—22no—
126C2×C8—24no—
127C2×C8—22no—
128C2×C8—42no—
129C2×C8—41no—
130C2×C8—42no—
131C2×C8—41no—
132C2×C8—41no—
133C2×C8#12011yescharacteristic
134C2×C8#12112yes—
135C2×C8#12711yescharacteristic
136C4⋊C4—44no—
137C4⋊C4—44no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4—88no—
141C4⋊C4—82no—
142C4⋊C4—82no—
143C22⋊C4—82no—
144C22⋊C4—82no—
145C22⋊C4—82no—
146C22⋊C4—82no—
147C42—41no—
148C42—42no—
149C42—41no—
150C16—42no—
151C16—81no—
152C23—41no—
153C23—41no—
154C23#44712yes—
155C23#46611yescharacteristic
156C23C2×Q3211yescharacteristic
157C23#50811yescharacteristic
158C23#51012yes—
159Q8—84no—
160Q8—82no—
161Q8—82no—
162C2×C4—22no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—21no—
166C2×C4—24no—
167C2×C4—22no—
168C2×C4—24no—
169C2×C4—22no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—42no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—84no—
179C2×C4—84no—
180C2×C4—84no—
181C2×C4—84no—
182C2×C4—84no—
183C2×C4—82no—
184C2×C4—82no—
185C2×C4#39211yescharacteristic
186C2×C4#44912yes—
187C2×C4#46911yescharacteristic
188C8—21no—
189C8—22no—
190C8—42no—
191C8#45812yes—
192C22—22no—
193C22—21no—
194C22—22no—
195C22—21no—
196C22—41no—
197C22—41no—
198C22—42no—
199C22#199911yescharacteristic
200C22#218812yes—
201C22#218912yes—
202C22#221611yescharacteristic
203C22#221911yescharacteristic
204C4—22no—
205C4—21no—
206C4—42no—
207C4—42no—
208C4—164no—
209C4—162no—
210C4—162no—
211C4#201011yescharacteristic
212C4#202211yescharacteristic
213C2—22no—
214C2—22no—
215C2—21no—
216C2—21no—
217C2#1032111yescharacteristic
218C2#1330611yescharacteristic
219C2#1330911yescharacteristic
subgroupG/H
center, socleC22#1999
commutator, Frattini, cosocleC23×C8C23
prime radicalC24C2×D8
LCSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
117. C22×C4Q32semidirect

Maximal subgroups

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

subgroupindex
#337552
#331302
#128442
#118872
#72302