Finite Group Databasebeta

← #141176

Subgroups — #141176

Order 512.

420 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 88 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#33180C211yesmaximal, characteristic
2#12786C211yesmaximal, characteristic
3C2×#1340C214yesmaximal
4#10814C211yesmaximal, characteristic
5C2×#494C411yescharacteristic, complemented
6C2×#470C2212yes—
7C2×#467—21no—
8C2×#405C2211yescharacteristic
9#2621—21no—
10#2150—21no—
11C2×#349C2212yes—
12#1971C412yescomplemented
13#1969—21no—
14#1960—21no—
15#1846C411yescharacteristic, complemented
16#1740—21no—
17#1735—21no—
18C2×H(3,ℤ/4)C2212yes—
19#1377—21no—
20#1340—28no—
21C22×Q16—21no—
22C22×Q16C2×C412yes—
23C22×M16C2311yescharacteristic
24C23×C8D411yescharacteristic
25C2×C4⋊Q8—21no—
26C2×C22⋊Q8—22no—
27C2×#117—22no—
28C2×#117D411yescharacteristic
29C2×M32—22no—
30#494—42no—
31#470—24no—
32#470—24no—
33#467—44no—
34C2×#107—21no—
35C2×#106C2×C412yes—
36C2×#103—22no—
37C2×#103—22no—
38#405—42no—
39C2×#97C2×C411yescharacteristic
40C2×#97D411yescharacteristic
41C2×C4×C8C2×C411yescharacteristic
42C2×C4×C8D411yescharacteristic
43#398—42no—
44#397—42no—
45#391—22no—
46#382—21no—
47C4⋊C16—42no—
48#349—24no—
49#342—42no—
50#340—42no—
51H(3,ℤ/4)—22no—
52H(3,ℤ/4)D414yes—
53C22×Q8—22no—
54C23×C4C2×D411yescharacteristic
55C2×Q16—24no—
56C2×Q16—42no—
57C2×Q16—42no—
58C2×Q16—42no—
59C2×Q16—42no—
60C2×Q16—42no—
61C2×M16—41no—
62C2×M16—41no—
63C2×M16C2×D412yes—
64C2×M16#4112yes—
65C22×C8—22no—
66C22×C8—22no—
67C22×C8—41no—
68C22×C8C22⋊C411yescharacteristic
69C22×C8C22⋊C411yescharacteristic
70C22×C8C22×C411yescharacteristic
71C22×C8C2×D411yescharacteristic
72C4⋊Q8—42no—
73C4⋊Q8—42no—
74C22⋊Q8—44no—
75C22⋊Q8—44no—
76#117—24no—
77#117—41no—
78#117—41no—
79#117—42no—
80#117—42no—
81#117—42no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4SD1612yes—
87C2×C22⋊C4—42no—
88C2×C22⋊C4—42no—
89C2×C22⋊C4D812yes—
90C2×C42—42no—
91C2×C42C22⋊C411yescharacteristic
92C2×C42C22⋊C411yescharacteristic
93M32—42no—
94M32—42no—
95C2×C16—42no—
96#107—42no—
97#106—24no—
98#103—44no—
99#103—42no—
100#103—42no—
101#97—41no—
102#97—41no—
103C4×C8—22no—
104C4×C8—22no—
105#95—42no—
106#95—44no—
107C24C2×D811yescharacteristic
108C2×Q8—42no—
109C2×Q8—42no—
110C2×Q8—42no—
111C2×Q8—42no—
112C2×Q8—42no—
113C2×Q8—42no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4—82no—
121C22×C4#10212yes—
122C22×C4#10212yes—
123C22×C4C2×C22⋊C411yescharacteristic
124C22×C4#12112yes—
125C22×C4C2×SD1611yescharacteristic
126C22×C4AGL(1,ℤ/8)12yes—
127Q16—84no—
128Q16—84no—
129M16—42no—
130M16—42no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—44no—
141C2×C8#9912yes—
142C2×C8C4×D412yes—
143C2×C8#12411yescharacteristic
144C2×C8#12711yescharacteristic
145C4⋊C4—42no—
146C4⋊C4—44no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—82no—
150C4⋊C4—84no—
151C4⋊C4—84no—
152C22⋊C4—42no—
153C22⋊C4—44no—
154C22⋊C4—82no—
155C22⋊C4—84no—
156C42—21no—
157C42—21no—
158C42—21no—
159C42—21no—
160C42—44no—
161C42—82no—
162C16—82no—
163C23—21no—
164C23—21no—
165C23—42no—
166C23C2×#10211yescharacteristic
167C23#45912yes—
168Q8—82no—
169Q8—82no—
170Q8—82no—
171Q8—82no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—84no—
183C2×C4—84no—
184C2×C4—82no—
185C2×C4—82no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—164no—
189C2×C4#39011yescharacteristic
190C2×C4C2×#9911yescharacteristic
191C2×C4#41011yescharacteristic
192C2×C4#41811yescharacteristic
193C2×C4#42111yescharacteristic
194C2×C4#42111yescharacteristic
195C2×C4#46012yes—
196C8—21no—
197C8—22no—
198C8—21no—
199C8—22no—
200C8—41no—
201C22—22no—
202C22—22no—
203C22—42no—
204C22—42no—
205C22—81no—
206C22#151711yescharacteristic
207C22#151911yescharacteristic
208C22#198211yescharacteristic
209C4—22no—
210C4—42no—
211C4—42no—
212C4—162no—
213C4—164no—
214C4—162no—
215C4#151011yescharacteristic
216C4#151411yescharacteristic
217C4#196311yescharacteristic
218C4#196411yescharacteristic
219C2—21no—
220C2—21no—
221C2—42no—
222C2#971511yescharacteristic
223C2#1186912yes—
subgroupG/H
center, socleC22#1519
commutatorC22×C8C22×C4
Frattini, cosocleC22×M16C23
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C2×#494C4semidirect
12. #1971C4semidirect
15. #1846C4semidirect

Maximal subgroups

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

subgroupindex
#331802
#127862
C2×#13402
#108142