Finite Group Databasebeta

← #136554

Subgroups — #136554

Order 512.

291 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 291 conjugacy classes of subgroups — too many to draw. Only its 90 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#23159C211yesmaximal, characteristic
2#13289C211yesmaximal, characteristic
3#13185C211yesmaximal, characteristic
4#11935C211yesmaximal, characteristic
5#9627C211yesmaximal, characteristic
6#7174C211yesmaximal, characteristic
7#7169C211yesmaximal, characteristic
8C2×#499C2211yescharacteristic
9#2370—21no—
10C2×#365—21no—
11#2150C2211yescharacteristic
12C4×M32C2211yescharacteristic
13#1989—21no—
14#1941—21no—
15#1921—21no—
16#1918—21no—
17#1848C2211yescharacteristic
18C4×#106—21no—
19C4×#106C2211yescharacteristic
20C8×C4⋊C4C2211yescharacteristic
21#1571—21no—
22#1564C2211yescharacteristic
23#1387—21no—
24#1387—21no—
25#1383—21no—
26#1363C414yescomplemented
27#1303—21no—
28C2×#125—21no—
29C2×M32—21no—
30C2×M32—21no—
31#499—21no—
32#499—21no—
33#499C2×C412yes—
34#499Q812yes—
35#431—21no—
36C2×#108—21no—
37C2×#108—21no—
38C2×#107D411yescharacteristic
39C2×#107D411yescharacteristic
40C2×#106—21no—
41C2×#106D411yescharacteristic
42C2×#106D411yescharacteristic
43#423—21no—
44C2×C4⋊C8D411yescharacteristic
45C2×C4×C8—21no—
46C2×C4×C8C2311yescharacteristic
47#389—21no—
48#389—21no—
49#389—41no—
50#387—41no—
51#387—41no—
52C4×C4⋊C4—21no—
53C4×C4⋊C4—21no—
54C4×C4⋊C4—21no—
55C4×C4⋊C4—41no—
56C4×C4⋊C4D411yescharacteristic
57#365—42no—
58C4⋊C16—21no—
59C4⋊C16C2×C412yes—
60#346—21no—
61C4×C16—21no—
62#340—21no—
63#340—21no—
64#340—21no—
65#340—21no—
66#340—21no—
67#336—21no—
68#335—21no—
69#335—42no—
70#334C2×C412yes—
71C2×M16—41no—
72C22×C8—21no—
73C22×C8C2×D411yescharacteristic
74C22×C8C2×D411yescharacteristic
75#125—22no—
76#125—22no—
77#125—41no—
78#125—41no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C42—21no—
89C2×C42—41no—
90C2×C42—41no—
91C2×C42—41no—
92C2×C42C2×D411yescharacteristic
93M32—22no—
94M32—41no—
95M32—41no—
96M32—41no—
97C2×C16—22no—
98C2×C16—41no—
99#108—42no—
100#108—41no—
101#108—41no—
102#107—21no—
103#107—21no—
104#107C4⋊C412yes—
105#107C4⋊C412yes—
106#106—22no—
107#106—22no—
108#106—21no—
109#106—21no—
110#106—21no—
111#106Q1612yes—
112C4⋊C8—22no—
113C4⋊C8—21no—
114C4⋊C8—21no—
115C4⋊C8—41no—
116C4⋊C8—41no—
117C22⋊C8—41no—
118C4×C8—41no—
119C4×C8—41no—
120C4×C8—41no—
121C4×C8C22×C411yescharacteristic
122C4×C8C2×Q811yescharacteristic
123C4×C8#4111yescharacteristic
124C4×C8#4111yescharacteristic
125#95—41no—
126#95—41no—
127#95—41no—
128#95—41no—
129#95—41no—
130#95SD1611yescharacteristic
131#95SD1611yescharacteristic
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4#12011yescharacteristic
139C22×C4C2×SD1611yescharacteristic
140C22×C4AGL(1,ℤ/8)11yescharacteristic
141M16—81no—
142M16—81no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—42no—
146C2×C8—41no—
147C2×C8—41no—
148C2×C8—41no—
149C2×C8—81no—
150C2×C8C2×C4⋊C411yescharacteristic
151C2×C8C4×D411yescharacteristic
152C2×C8C22⋊Q811yescharacteristic
153C2×C8#12311yescharacteristic
154C2×C8C2×Q1611yescharacteristic
155C2×C8#13511yescharacteristic
156C2×C8AGL(1,ℤ/8)11yescharacteristic
157C2×C8#13711yescharacteristic
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—82no—
172C4⋊C4—161no—
173C42—41no—
174C42—41no—
175C42—82no—
176C42—161no—
177C42—161no—
178C42C4×D411yescharacteristic
179C42C22⋊Q811yescharacteristic
180C42#13511yescharacteristic
181C42AGL(1,ℤ/8)11yescharacteristic
182C16—42no—
183C16—42no—
184C23#45011yescharacteristic
185C2×C4—22no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4—81no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4#38511yescharacteristic
206C2×C4C4×SD1611yescharacteristic
207C2×C4#44211yescharacteristic
208C2×C4#44911yescharacteristic
209C2×C4#44911yescharacteristic
210C2×C4#45111yescharacteristic
211C2×C4#45411yescharacteristic
212C2×C4#45611yescharacteristic
213C2×C4#47611yescharacteristic
214C2×C4#47811yescharacteristic
215C8—21no—
216C8—21no—
217C8—82no—
218C8—81no—
219C8—161no—
220C8C4×Q1611yescharacteristic
221C8#44011yescharacteristic
222C8#47711yescharacteristic
223C8#47911yescharacteristic
224C22—21no—
225C22—21no—
226C22#163711yescharacteristic
227C22#165411yescharacteristic
228C22#186411yescharacteristic
229C4—21no—
230C4—21no—
231C4—21no—
232C4—21no—
233C4—42no—
234C4—81no—
235C4—82no—
236C4—82no—
237C4—82no—
238C4—161no—
239C4#162811yescharacteristic
240C4#164811yescharacteristic
241C4#185711yescharacteristic
242C4#186311yescharacteristic
243C2—21no—
244C2—21no—
245C2#892611yescharacteristic
246C2#1294911yescharacteristic
247C2#1295111yescharacteristic
subgroupG/H
center, socleC22#1864
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC23#450
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
26. #1363C4semidirect

Maximal subgroups

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

subgroupindex
#231592
#132892
#131852
#119352
#96272
#71742
#71692