Finite Group Databasebeta

← #141996

Subgroups — #141996

Order 512.

377 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 377 conjugacy classes of subgroups — too many to draw. Only its 102 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#33214C211yesmaximal, characteristic
2#12602C211yesmaximal, characteristic
3#10866C211yesmaximal, characteristic
4#7254C214yesmaximal
5C2×#499C411yescharacteristic, complemented
6C2×#427—21no—
7C2×#405C2211yescharacteristic
8#2629—21no—
9C2×#349C2212yes—
10C4×M32—21no—
11#1992—21no—
12#1944C412yescomplemented
13#1941—21no—
14#1844C411yescharacteristic, complemented
15#1809C2212yes—
16#1807—21no—
17#1740—21no—
18C2×H(3,ℤ/4)C2212yes—
19#1377—21no—
20C22×M16C2311yescharacteristic
21C23×C8D411yescharacteristic
22C2×#125—21no—
23C2×#117—22no—
24C2×#117D411yescharacteristic
25C2×M32—22no—
26#499—41no—
27#499—41no—
28#427—24no—
29#427—42no—
30C2×#107—41no—
31C2×#107C2×C412yes—
32C2×#106—21no—
33C2×#106—21no—
34C2×#106—41no—
35C2×#106C2×C412yes—
36#405—41no—
37#405—41no—
38C2×#97C2×C411yescharacteristic
39C2×#97D411yescharacteristic
40C2×C4×C8C2×C411yescharacteristic
41C2×C4×C8D411yescharacteristic
42#398—42no—
43#396—42no—
44#389—22no—
45#384—21no—
46#380—22no—
47#349—22no—
48#349Q814yes—
49#346—41no—
50C4×C16—41no—
51#342—42no—
52#340—22no—
53#340—22no—
54H(3,ℤ/4)—24no—
55C23×C4C2×D411yescharacteristic
56C2×M16—41no—
57C2×M16—41no—
58C2×M16C2×Q812yes—
59C2×M16#4112yes—
60C22×C8—21no—
61C22×C8—21no—
62C22×C8—22no—
63C22×C8C22⋊C411yescharacteristic
64C22×C8C22⋊C411yescharacteristic
65C22×C8SD1612yes—
66C22×C8Q1612yes—
67C22×C8C22×C411yescharacteristic
68C22×C8C2×D411yescharacteristic
69#125—42no—
70#125—42no—
71#117—24no—
72#117—41no—
73#117—41no—
74#117—42no—
75#117—42no—
76#117—42no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4SD1612yes—
84C2×C22⋊C4—42no—
85C2×C22⋊C4—42no—
86C2×C22⋊C4D812yes—
87C2×C42—42no—
88C2×C42C22⋊C411yescharacteristic
89C2×C42C22⋊C411yescharacteristic
90M32—24no—
91M32—42no—
92C2×C16—42no—
93#107—44no—
94#107—42no—
95#106—44no—
96#106—41no—
97#106—41no—
98#106—42no—
99#106—42no—
100#97—41no—
101#97—41no—
102C4×C8—21no—
103C4×C8—21no—
104C4×C8—21no—
105C4×C8—21no—
106#95—42no—
107#95—42no—
108C24C2×D811yescharacteristic
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—82no—
116C22×C4#10212yes—
117C22×C4#10212yes—
118C22×C4C2×C22⋊C411yescharacteristic
119C22×C4#12411yescharacteristic
120C22×C4C4⋊Q811yescharacteristic
121C22×C4C2×SD1611yescharacteristic
122C22×C4C2×SD1611yescharacteristic
123C22×C4C2×Q1611yescharacteristic
124M16—42no—
125M16—42no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8#9911yescharacteristic
137C2×C8#9911yescharacteristic
138C2×C8#10312yes—
139C2×C8#10312yes—
140C2×C8#10412yes—
141C2×C8C4×Q812yes—
142C2×C8C22⋊Q812yes—
143C4⋊C4—42no—
144C4⋊C4—44no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—82no—
150C4⋊C4—84no—
151C4⋊C4—162no—
152C4⋊C4—162no—
153C22⋊C4—42no—
154C22⋊C4—44no—
155C22⋊C4—82no—
156C22⋊C4—162no—
157C42—21no—
158C42—21no—
159C42—21no—
160C42—21no—
161C42—44no—
162C42—82no—
163C16—44no—
164C23—21no—
165C23—21no—
166C23—42no—
167C23C2×#10211yescharacteristic
168C23#48611yescharacteristic
169C23#50011yescharacteristic
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—84no—
185C2×C4—82no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—164no—
192C2×C4#39111yescharacteristic
193C2×C4C2×#9911yescharacteristic
194C2×C4#41011yescharacteristic
195C2×C4C2×#10311yescharacteristic
196C2×C4C2×#10411yescharacteristic
197C2×C4#42111yescharacteristic
198C2×C4#48711yescharacteristic
199C2×C4#49811yescharacteristic
200C8—22no—
201C8—22no—
202C8—22no—
203C8#32812yes—
204C8#32812yes—
205C22—22no—
206C22—21no—
207C22—21no—
208C22—41no—
209C22—41no—
210C22—42no—
211C22—81no—
212C22#150211yescharacteristic
213C22#151911yescharacteristic
214C22#199011yescharacteristic
215C4—21no—
216C4—21no—
217C4—42no—
218C4—42no—
219C4—162no—
220C4—164no—
221C4—162no—
222C4C2×#32811yescharacteristic
223C4#151811yescharacteristic
224C4#198611yescharacteristic
225C4#199611yescharacteristic
226C2—21no—
227C2—21no—
228C2—41no—
229C2—41no—
230C2#988411yescharacteristic
231C2#1186712yes—
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×#499C4semidirect
12. #1944C4semidirect
14. #1844C4semidirect

Maximal subgroups

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

subgroupindex
#332142
#126022
#108662
#72542