Finite Group Databasebeta

← #150474

Subgroups — #150474

Order 512.

273 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 273 conjugacy classes of subgroups — too many to draw. Only its 54 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#27672C211yesmaximal, characteristic
2#12623C211yesmaximal, characteristic
3#11537C211yesmaximal, characteristic
4#7411C212yesmaximal, complemented
5#7410C212yesmaximal, complemented
6#3092—21no—
7#2929C2211yescharacteristic
8C22×#106C411yescharacteristic, complemented
9#2166—21no—
10#2120—21no—
11#2090C412yescomplemented
12#2079C2212yes—
13#1951—21no—
14#1821—21no—
15#1815—21no—
16#1810C411yescharacteristic, complemented
17#1757—21no—
18#1392C2211yescharacteristic
19#1391C2211yescharacteristic
20#1321C2212yes—
21#1320—21no—
22C2×#131D411yescharacteristic
23C23×C8C2×C411yescharacteristic
24#525D411yescharacteristic
25C22×C4⋊C4—21no—
26#484—41no—
27#483—41no—
28#481—41no—
29#480—41no—
30#435—41no—
31C8×D4—41no—
32C2×#108—21no—
33#428—22no—
34C2×#106—21no—
35C2×#106—41no—
36C2×#106—41no—
37C2×#106C2×C412yes—
38#424C2×C411yescharacteristic
39#417—22no—
40#413—21no—
41#410—21no—
42#408C2311yescharacteristic
43#407—21no—
44#407D411yescharacteristic
45#405—41no—
46#405D411yescharacteristic
47#364—41no—
48#352—22no—
49AΣL(1,𝔽16)—22no—
50#349—22no—
51#347—41no—
52#344—41no—
53#343—22no—
54#343—22no—
55#341C2×C412yes—
56C2×#41C2×D411yescharacteristic
57C23×C4—41no—
58C23×C4C22⋊C411yescharacteristic
59#131—41no—
60#131—41no—
61C2×M16—21no—
62C2×M16—22no—
63C2×M16—41no—
64C2×M16C22⋊C411yescharacteristic
65C2×M16C2×D411yescharacteristic
66C22×C8—21no—
67C22×C8—21no—
68C22×C8—21no—
69C22×C8—41no—
70C22×C8—41no—
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22×C411yescharacteristic
73#123—21no—
74C22⋊Q8—21no—
75#121—21no—
76#120—21no—
77C4×D4—41no—
78#117—41no—
79#117C22⋊C411yescharacteristic
80C2×C4⋊C4—22no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86M32—42no—
87#108—41no—
88#108—41no—
89#107—42no—
90#106—42no—
91#106—41no—
92#106—41no—
93#106—42no—
94C4⋊C8—41no—
95#103—42no—
96#102—42no—
97#101—41no—
98#100—22no—
99#99—22no—
100#99—41no—
101C22⋊C8—41no—
102C22⋊C8—41no—
103C22⋊C8—41no—
104C22⋊C8—41no—
105#97—21no—
106#97—81no—
107C4×C8—21no—
108C4×C8—81no—
109C24#10411yescharacteristic
110#41—41no—
111#41—41no—
112C2×Q8AGL(1,ℤ/8)11yescharacteristic
113C2×D4—21no—
114C2×D4—41no—
115C2×D4AGL(1,ℤ/8)11yescharacteristic
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—41no—
120C22×C4—41no—
121C22×C4—42no—
122C22×C4—81no—
123C22×C4—81no—
124C22×C4—162no—
125C22×C4#10411yescharacteristic
126C22×C4C2×C22⋊C411yescharacteristic
127M16—41no—
128M16—41no—
129M16—42no—
130M16—42no—
131M16—42no—
132M16—81no—
133M16—81no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—42no—
140C2×C8—41no—
141C2×C8—41no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—41no—
145C2×C8—42no—
146C2×C8—81no—
147C2×C8#9911yescharacteristic
148C2×C8#9911yescharacteristic
149C4⋊C4—21no—
150C4⋊C4—41no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—81no—
154C4⋊C4—81no—
155C4⋊C4—81no—
156C22⋊C4—21no—
157C22⋊C4—41no—
158C22⋊C4—41no—
159C22⋊C4—42no—
160C42—21no—
161C42—81no—
162C16—82no—
163C23—21no—
164C23—21no—
165C23—41no—
166C23—42no—
167C23C2×#10411yescharacteristic
168Q8—41no—
169D4—41no—
170D4—81no—
171D4—82no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—42no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—82no—
184C2×C4—82no—
185C2×C4—162no—
186C2×C4—162no—
187C2×C4—162no—
188C2×C4C2×#9911yescharacteristic
189C2×C4#41811yescharacteristic
190C8—21no—
191C8—41no—
192C8—41no—
193C8—41no—
194C8—82no—
195C8—82no—
196C8#35211yescharacteristic
197C8#35211yescharacteristic
198C22—22no—
199C22—21no—
200C22—41no—
201C22—41no—
202C22—42no—
203C22—42no—
204C22—41no—
205C22—41no—
206C22—81no—
207C22#150711yescharacteristic
208C4—21no—
209C4—41no—
210C4—41no—
211C4—81no—
212C4—162no—
213C4—162no—
214C4C2×#35211yescharacteristic
215C4#212911yescharacteristic
216C2—21no—
217C2—42no—
218C2—41no—
219C2—81no—
220C2#1177311yescharacteristic
subgroupG/H
center, socleC2#11773
commutatorC22×C8C22×C4
Frattini, cosocle#408C23
prime radical#525D4
LCSG ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ #2929 ◁ G
LECSG ▷ #408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7411C2semidirect
5. #7410C2semidirect
8. C22×#106C4semidirect
11. #2090C4semidirect
16. #1810C4semidirect

Maximal subgroups

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

subgroupindex
#276722
#126232
#115372
#74112
#74102