Finite Group Databasebeta

← #141171

Subgroups — #141171

Order 512.

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

normal subgroups only

This group has 412 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×#1342C214yesmaximal
4#10815C211yesmaximal, characteristic
5C2×#494C411yescharacteristic
6C2×#470C2212yes—
7C2×#467—21no—
8C2×#405C2211yescharacteristic
9#2621—21no—
10#2150—21no—
11C2×#349C2212yes—
12#1971C412yes—
13#1969—21no—
14#1960—21no—
15#1846C411yescharacteristic
16#1742—21no—
17C2×#341—21no—
18C2×#338C2212yes—
19#1377—21no—
20#1342—28no—
21C22×Q16—21no—
22C22×Q16C2×C412yes—
23C22×M16—22no—
24C22×M16C2311yescharacteristic
25C23×C8D411yescharacteristic
26C2×C4⋊Q8—21no—
27C2×C22⋊Q8—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#343—42no—
50#341—42no—
51#338—22no—
52#338D414yes—
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—24no—
62C2×M16—41no—
63C2×M16—41no—
64C2×M16—42no—
65C2×M16—42no—
66C2×M16—42no—
67C2×M16—42no—
68C2×M16—42no—
69C2×M16C2×D412yes—
70C2×M16#4112yes—
71C22×C8—22no—
72C22×C8—22no—
73C22×C8—41no—
74C22×C8—42no—
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22⋊C411yescharacteristic
77C22×C8C22×C411yescharacteristic
78C22×C8C2×D411yescharacteristic
79C4⋊Q8—42no—
80C4⋊Q8—42no—
81C22⋊Q8—44no—
82C22⋊Q8—44no—
83#117—41no—
84#117—41no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4D812yes—
89C2×C22⋊C4—42no—
90C2×C22⋊C4SD1612yes—
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×SD1611yescharacteristic
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#10212yes—
121C22×C4#10212yes—
122C22×C4C2×C22⋊C411yescharacteristic
123C22×C4#12112yes—
124C22×C4C2×D811yescharacteristic
125C22×C4AGL(1,ℤ/8)12yes—
126Q16—84no—
127Q16—84no—
128M16—42no—
129M16—42no—
130M16—44no—
131M16—44no—
132M16—82no—
133M16—82no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—44no—
144C2×C8—44no—
145C2×C8—82no—
146C2×C8—82no—
147C2×C8#9912yes—
148C2×C8C4×D412yes—
149C2×C8#12411yescharacteristic
150C2×C8#12711yescharacteristic
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—84no—
155C4⋊C4—84no—
156C22⋊C4—42no—
157C22⋊C4—84no—
158C42—21no—
159C42—21no—
160C42—21no—
161C42—21no—
162C16—82no—
163C23—21no—
164C23—21no—
165C23—42no—
166C23C2×#10211yescharacteristic
167C23#46012yes—
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—82no—
184C2×C4—82no—
185C2×C4—82no—
186C2×C4—82no—
187C2×C4#39011yescharacteristic
188C2×C4C2×#9911yescharacteristic
189C2×C4#41011yescharacteristic
190C2×C4#41811yescharacteristic
191C2×C4#42111yescharacteristic
192C2×C4#42111yescharacteristic
193C2×C4#45912yes—
194C8—21no—
195C8—22no—
196C8—21no—
197C8—22no—
198C8—41no—
199C8—84no—
200C22—22no—
201C22—22no—
202C22—42no—
203C22—42no—
204C22—81no—
205C22#151711yescharacteristic
206C22#151911yescharacteristic
207C22#198211yescharacteristic
208C4—22no—
209C4—42no—
210C4—42no—
211C4—162no—
212C4—162no—
213C4#151011yescharacteristic
214C4#151411yescharacteristic
215C4#196311yescharacteristic
216C4#196411yescharacteristic
217C2—21no—
218C2—21no—
219C2—42no—
220C2#971511yescharacteristic
221C2#1187112yes—
subgroupG/H
center, socleC22#1519
commutatorC22×C8C22×C4
Frattini, cosocleC22×M16C23
prime radicalC24C2×SD16
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#331802
#127862
C2×#13422
#108152