Finite Group Databasebeta

← #137237

Subgroups — #137237

Order 512.

476 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 110 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
1C2×#2086C211yesmaximal, characteristic
2#21961C211yesmaximal, characteristic
3#12866C212yesmaximal
4C2×#1322C211yesmaximal, characteristic
5#7258C212yesmaximal
6C22×#107C2211yescharacteristic
7C22×C4⋊C8C2211yescharacteristic
8C2×#389—21no—
9#2285—21no—
10C2×C22⋊C16C2212yes—
11#2086—22no—
12#2086C414yescomplemented
13#1947—24no—
14#1808—21no—
15#1808C2212yes—
16C2×#340—21no—
17C2×#340C2211yescharacteristic
18#1380—24no—
19#1322—22no—
20C23×C8D411yescharacteristic
21C23×C8C2311yescharacteristic
22C2×#117—41no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4—41no—
26C22×C42D411yescharacteristic
27C22×C16—22no—
28C2×#107—22no—
29C2×#107—22no—
30C2×#107—21no—
31C2×#107—22no—
32C2×#107C2×C412yes—
33C2×#107D414yes—
34C2×C4⋊C8—22no—
35C2×C4⋊C8—22no—
36C2×C4⋊C8C2×C412yes—
37C2×C4⋊C8D412yes—
38#389—24no—
39#389—42no—
40#384—22no—
41#382—41no—
42#380—22no—
43#380—41no—
44C4×C4⋊C4—22no—
45C4×C4⋊C4—41no—
46C2×#95—41no—
47C22⋊C16—22no—
48C22⋊C16—22no—
49#340—22no—
50#340—24no—
51#340—22no—
52#340—21no—
53#340C2×C412yes—
54C23×C4—41no—
55C23×C4—41no—
56C23×C4D811yescharacteristic
57C23×C4D811yescharacteristic
58C23×C4C2×D411yescharacteristic
59C22×C8—21no—
60C22×C8—21no—
61C22×C8—42no—
62C22×C8—42no—
63C22×C8C22⋊C412yes—
64C22×C8C22×C411yescharacteristic
65C22×C8C2×D412yes—
66C22×C8C2×D411yescharacteristic
67C22×C8#4112yes—
68#117—82no—
69#117—82no—
70C2×C4⋊C4—24no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—44no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—82no—
85C2×C4⋊C4—82no—
86C2×C22⋊C4—42no—
87C2×C42—21no—
88C2×C42—21no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42—42no—
92C2×C42C22⋊C412yes—
93C2×C42SD1612yes—
94C2×C16—44no—
95C2×C16—42no—
96#107—24no—
97#107—42no—
98#107—44no—
99#107—82no—
100C4⋊C8—24no—
101C4⋊C8—24no—
102#95—42no—
103#95—44no—
104#95—42no—
105C24C2×D811yescharacteristic
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—42no—
115C22×C4—44no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—44no—
119C22×C4—82no—
120C22×C4—81no—
121C22×C4—81no—
122C22×C4—81no—
123C22×C4—81no—
124C22×C4—81no—
125C22×C4—81no—
126C22×C4—81no—
127C22×C4—81no—
128C22×C4#10212yes—
129C22×C4#10212yes—
130C22×C4D1612yes—
131C22×C4SD3212yes—
132C22×C4C2×C22⋊C411yescharacteristic
133C22×C4#12011yescharacteristic
134C22×C4#12311yescharacteristic
135C22×C4C2×SD1611yescharacteristic
136C22×C4AGL(1,ℤ/8)11yescharacteristic
137C22×C4#13711yescharacteristic
138C2×C8—24no—
139C2×C8—22no—
140C2×C8—41no—
141C2×C8—41no—
142C2×C8—42no—
143C2×C8—42no—
144C2×C8—42no—
145C2×C8—81no—
146C2×C8C4×D412yes—
147C2×C8#12112yes—
148C4⋊C4—24no—
149C4⋊C4—44no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—44no—
154C4⋊C4—44no—
155C4⋊C4—84no—
156C4⋊C4—84no—
157C4⋊C4—88no—
158C4⋊C4—81no—
159C4⋊C4—81no—
160C22⋊C4—82no—
161C22⋊C4—82no—
162C42—22no—
163C42—22no—
164C42—22no—
165C42—22no—
166C42—84no—
167C16—82no—
168C23—22no—
169C23—22no—
170C23C2×#10211yescharacteristic
171C23#44711yescharacteristic
172C23#48011yescharacteristic
173C23C2×D1611yescharacteristic
174C23C2×SD3211yescharacteristic
175C23#50911yescharacteristic
176C23#51011yescharacteristic
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—44no—
194C2×C4—81no—
195C2×C4—84no—
196C2×C4—84no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—84no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4—84no—
204C2×C4—84no—
205C2×C4—161no—
206C2×C4—161no—
207C2×C4—161no—
208C2×C4—161no—
209C2×C4#35714yes—
210C2×C4#38611yescharacteristic
211C2×C4#41711yescharacteristic
212C2×C4#44811yescharacteristic
213C2×C4#45612yes—
214C2×C4#45812yes—
215C2×C4#48111yescharacteristic
216C8—22no—
217C8—22no—
218C8—42no—
219C22—22no—
220C22—22no—
221C22—22no—
222C22—22no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22#161012yes—
228C22#187411yescharacteristic
229C22C2×#35711yescharacteristic
230C22#213911yescharacteristic
231C22#218411yescharacteristic
232C22#218511yescharacteristic
233C22#219112yes—
234C22#223211yescharacteristic
235C22#223311yescharacteristic
236C4—21no—
237C4—21no—
238C4—24no—
239C4—42no—
240C4—84no—
241C4—164no—
242C4—162no—
243C4#161112yes—
244C4#188611yescharacteristic
245C4#189011yescharacteristic
246C2—21no—
247C2—21no—
248C2—21no—
249C2—21no—
250C2#902711yescharacteristic
251C2#1228812yes—
252C2#1228912yes—
253C2#1299111yescharacteristic
254C2#1300011yescharacteristic
subgroupG/H
center, socleC23#447
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #2086C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20862
#219612
#128662
C2×#13222
#72582