Finite Group Databasebeta

← #93545

Subgroups — #93545

Order 512.

288 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 58 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#12607C211yesmaximal, characteristic
2#10825C211yesmaximal, characteristic
3#8147C211yesmaximal, characteristic
4C2×C4×M16C2211yescharacteristic
5C2×#349—21no—
6C2×#347C412yescomplemented
7C4×C22⋊C8—21no—
8C2×#344—21no—
9C2×#339—21no—
10C2×#338C411yescharacteristic
11C2×H(3,ℤ/4)C411yescharacteristic
12C2×#322—21no—
13#1376—21no—
14#1273C411yescharacteristic, complemented
15#1273C411yescharacteristic, complemented
16C22×M16—21no—
17C22×M16C2×C411yescharacteristic
18C22×M16D411yescharacteristic
19C2×#117—21no—
20C22×C42D411yescharacteristic
21C2×M32—22no—
22C2×C22⋊C8—22no—
23C4×M16—42no—
24C2×#97C2×C411yescharacteristic
25C2×#97Q811yescharacteristic
26C2×C4×C8—21no—
27C2×C4×C8—21no—
28C2×C4×C8C2×C411yescharacteristic
29C2×C4×C8D411yescharacteristic
30#349—22no—
31#349—22no—
32#347—42no—
33#344—42no—
34#339—22no—
35#339—22no—
36#338—41no—
37H(3,ℤ/4)—41no—
38#336—22no—
39#322—42no—
40C23×C4—21no—
41C23×C4C22⋊C411yescharacteristic
42C2×M16—21no—
43C2×M16—21no—
44C2×M16—21no—
45C2×M16—21no—
46C2×M16—22no—
47C2×M16—22no—
48C2×M16—41no—
49C2×M16—41no—
50C2×M16—41no—
51C2×M16—41no—
52C2×M16—41no—
53C2×M16—41no—
54C2×M16—41no—
55C2×M16—41no—
56C22×C8—22no—
57C22×C8—22no—
58C22×C8—41no—
59C22×C8C4211yescharacteristic
60C22×C8C22⋊C411yescharacteristic
61C22×C8C4⋊C411yescharacteristic
62C22×C8C4⋊C411yescharacteristic
63#117—22no—
64#117—41no—
65#117—41no—
66#117—41no—
67C2×C4⋊C4—41no—
68C2×C22⋊C4—41no—
69C2×C42—21no—
70C2×C42—21no—
71C2×C42—21no—
72C2×C42—41no—
73C2×C42—41no—
74C2×C42C22⋊C411yescharacteristic
75C2×C42C4⋊C411yescharacteristic
76M32—24no—
77M32—42no—
78C2×C16—42no—
79C22⋊C8—44no—
80#97—22no—
81#97—41no—
82C4×C8—22no—
83C4×C8—41no—
84C4×C8—41no—
85C4×C8—41no—
86C4×C8—41no—
87C4×C8—41no—
88C24#10411yescharacteristic
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—41no—
96C22×C4—41no—
97C22×C4—42no—
98C22×C4—42no—
99C22×C4—81no—
100C22×C4#9511yescharacteristic
101C22×C4#10011yescharacteristic
102C22×C4#10111yescharacteristic
103C22×C4#10411yescharacteristic
104C22×C4#10411yescharacteristic
105C22×C4#10411yescharacteristic
106M16—42no—
107M16—42no—
108M16—42no—
109M16—42no—
110M16—42no—
111M16—42no—
112M16—81no—
113M16—81no—
114C2×C8—21no—
115C2×C8—21no—
116C2×C8—24no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—42no—
127C2×C8—42no—
128C2×C8—42no—
129C2×C8—42no—
130C2×C8—42no—
131C2×C8—42no—
132C2×C8—81no—
133C2×C8—81no—
134C2×C8#9912yes—
135C4⋊C4—42no—
136C4⋊C4—81no—
137C22⋊C4—42no—
138C22⋊C4—81no—
139C42—21no—
140C42—21no—
141C42—21no—
142C42—41no—
143C42—41no—
144C42—42no—
145C42—81no—
146C42#10411yescharacteristic
147C42#10411yescharacteristic
148C16—44no—
149C23—21no—
150C23—21no—
151C23—41no—
152C23—41no—
153C23#32911yescharacteristic
154C23#33011yescharacteristic
155C23#33911yescharacteristic
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—82no—
174C2×C4—82no—
175C2×C4—82no—
176C2×C4—161no—
177C2×C4—161no—
178C2×C4#32911yescharacteristic
179C2×C4#33011yescharacteristic
180C2×C4H(3,ℤ/4)11yescharacteristic
181C2×C4#33911yescharacteristic
182C2×C4#33911yescharacteristic
183C2×C4#34211yescharacteristic
184C2×C4#34311yescharacteristic
185C2×C4#34411yescharacteristic
186C8—22no—
187C8—44no—
188C8—41no—
189C8—41no—
190C8—41no—
191C8—82no—
192C8—82no—
193C22—21no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—41no—
198C22—41no—
199C22—41no—
200C22—41no—
201C22—81no—
202C22#128811yescharacteristic
203C22#129511yescharacteristic
204C22#129711yescharacteristic
205C4—22no—
206C4—22no—
207C4—21no—
208C4—21no—
209C4—81no—
210C4—162no—
211C4#128411yescharacteristic
212C4#128511yescharacteristic
213C4#130011yescharacteristic
214C4#130511yescharacteristic
215C2—21no—
216C2—21no—
217C2—41no—
218C2—41no—
219C2#703611yescharacteristic
220C2#722511yescharacteristic
221C2#722611yescharacteristic
subgroupG/H
centerC2×C4H(3,ℤ/4)
commutatorC22×C8C42
Frattini, cosocleC2×C4×M16C22
socleC22#1288
prime radicalC24#104
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ G
LECSG ▷ C2×C4×M16 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C2×#347C4semidirect
14. #1273C4semidirect
15. #1273C4semidirect

Maximal subgroups

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

subgroupindex
#126072
#108252
#81472