Finite Group Databasebeta

← #136069

Subgroups — #136069

Order 512.

348 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 86 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#21968C211yesmaximal, characteristic
2#12931C212yesmaximal
3C2×#1328C211yesmaximal, characteristic
4#11237C211yesmaximal, characteristic
5#7255C212yesmaximal
6C2×#430C2211yescharacteristic
7C2×#424C2211yescharacteristic
8#2314—21no—
9C2×#365—22no—
10C2×#349C2212yes—
11#1949—22no—
12#1947—22no—
13#1809C2212yes—
14#1808—21no—
15C2×#344—21no—
16C2×#341—21no—
17C2×#338C2211yescharacteristic
18#1387—22no—
19#1328C414yescomplemented
20C22×M16—21no—
21C22×M16—21no—
22C22×M16C2311yescharacteristic
23C23×C8D411yescharacteristic
24C2×#117D411yescharacteristic
25C2×M32—22no—
26#430—42no—
27C2×#108—22no—
28C2×#107—21no—
29C2×#107—21no—
30C2×#107D412yes—
31C2×#106D412yes—
32#424—22no—
33#424C2×C412yes—
34#424Q812yes—
35C2×C4⋊C8—22no—
36#388—42no—
37#387—42no—
38#384—21no—
39#382—21no—
40#380—22no—
41C4×C4⋊C4—22no—
42#365—44no—
43#349—22no—
44#349C2×C414yes—
45#344—42no—
46#341—42no—
47#340—22no—
48#340—22no—
49#338—22no—
50C23×C4C2×D411yescharacteristic
51C2×M16—22no—
52C2×M16—22no—
53C2×M16—41no—
54C2×M16—41no—
55C2×M16—41no—
56C2×M16—41no—
57C2×M16—41no—
58C2×M16—41no—
59C2×M16—41no—
60C2×M16—41no—
61C2×M16—41no—
62C2×M16—41no—
63C2×M16—41no—
64C2×M16—41no—
65C2×M16C22×C411yescharacteristic
66C2×M16C2×Q811yescharacteristic
67C2×M16#4112yes—
68C22×C8—21no—
69C22×C8—21no—
70C22×C8—41no—
71C22×C8—41no—
72C22×C8—41no—
73C22×C8—41no—
74C22×C8C4⋊C412yes—
75C22×C8C2×D412yes—
76#117—21no—
77#117—21no—
78#117C4⋊C412yes—
79#117SD1612yes—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—42no—
84C2×C22⋊C4—42no—
85C2×C22⋊C4SD1612yes—
86C2×C42—21no—
87C2×C42—42no—
88M32—24no—
89M32—42no—
90C2×C16—42no—
91#108—24no—
92#108—42no—
93#107—42no—
94#107—81no—
95#107—81no—
96#106—42no—
97C4⋊C8—24no—
98C4⋊C8—24no—
99#95—42no—
100#95—42no—
101#95—42no—
102C24C2×SD1611yescharacteristic
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4C2×C4⋊C411yescharacteristic
111C22×C4#12011yescharacteristic
112C22×C4#12311yescharacteristic
113C22×C4C2×SD1611yescharacteristic
114C22×C4AGL(1,ℤ/8)11yescharacteristic
115C22×C4#13711yescharacteristic
116M16—41no—
117M16—41no—
118M16—42no—
119M16—42no—
120M16—42no—
121M16—42no—
122M16—42no—
123M16—81no—
124M16—81no—
125M16—81no—
126M16—81no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—41no—
131C2×C8—41no—
132C2×C8—42no—
133C2×C8—42no—
134C2×C8—42no—
135C2×C8—81no—
136C2×C8—81no—
137C2×C8—81no—
138C2×C8—81no—
139C2×C8—81no—
140C2×C8C4×D412yes—
141C2×C8C22⋊Q812yes—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—162no—
149C22⋊C4—22no—
150C22⋊C4—162no—
151C22⋊C4#10614yes—
152C42—22no—
153C42—22no—
154C42—162no—
155C16—44no—
156C23—21no—
157C23—21no—
158C23—41no—
159C23—41no—
160C23C2×#10611yescharacteristic
161C23#45011yescharacteristic
162C23#48311yescharacteristic
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—24no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—82no—
177C2×C4—82no—
178C2×C4—82no—
179C2×C4—82no—
180C2×C4—82no—
181C2×C4—82no—
182C2×C4#38511yescharacteristic
183C2×C4#42811yescharacteristic
184C2×C4#44811yescharacteristic
185C2×C4#45411yescharacteristic
186C2×C4#45611yescharacteristic
187C2×C4B(4,𝔽2)11yescharacteristic
188C2×C4#45811yescharacteristic
189C2×C4#48111yescharacteristic
190C8—22no—
191C8—22no—
192C8—42no—
193C8—82no—
194C8—82no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—41no—
200C22—41no—
201C22—41no—
202C22—41no—
203C22—81no—
204C22#160411yescharacteristic
205C22#161611yescharacteristic
206C22#185311yescharacteristic
207C4—21no—
208C4—21no—
209C4—44no—
210C4—162no—
211C4—162no—
212C4#159911yescharacteristic
213C4#161111yescharacteristic
214C4#185711yescharacteristic
215C4#186711yescharacteristic
216C2—21no—
217C2—21no—
218C2—41no—
219C2—41no—
220C2#885711yescharacteristic
221C2#1230411yescharacteristic
222C2#1230511yescharacteristic
subgroupG/H
center, socleC22#1616
commutatorC2×M16C22×C4
Frattini, cosocleC22×M16C23
prime radicalC24C2×SD16
LCSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
19. #1328C4semidirect

Maximal subgroups

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

subgroupindex
#219682
#129312
C2×#13282
#112372
#72552