Finite Group Databasebeta

← #136381

Subgroups — #136381

Order 512.

303 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 303 conjugacy classes of subgroups — too many to draw. Only its 96 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#22730C211yesmaximal, characteristic
2#13185C211yesmaximal, characteristic
3#12904C211yesmaximal, characteristic
4#11611C211yesmaximal, characteristic
5#8663C211yesmaximal, characteristic
6#7279C211yesmaximal, characteristic
7#7175C211yesmaximal, characteristic
8C2×#446C2211yescharacteristic
9#2355—21no—
10C2×#365—21no—
11C2×#365—21no—
12C2×#365—21no—
13#2150C2211yescharacteristic
14C4×M32C2211yescharacteristic
15#1949—21no—
16#1949—21no—
17#1948—21no—
18#1848C2211yescharacteristic
19#1847—21no—
20#1846C2211yescharacteristic
21#1776—21no—
22C4×#106C2211yescharacteristic
23#1564C2211yescharacteristic
24#1487—21no—
25#1383—21no—
26#1338C414yescomplemented
27#1301—21no—
28C2×C4×Q8D411yescharacteristic
29C2×M32—21no—
30C2×M32—21no—
31#446—21no—
32#446—21no—
33#446C2×C412yes—
34#446Q812yes—
35#431—21no—
36C2×#108—21no—
37C2×#108—21no—
38C2×#107—21no—
39C2×#107—21no—
40C2×#106—21no—
41C2×#106D411yescharacteristic
42C2×#106D411yescharacteristic
43C2×#106D411yescharacteristic
44C2×#106D411yescharacteristic
45#423—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8D411yescharacteristic
48C2×#97—21no—
49C2×C4×C8C2311yescharacteristic
50#391—41no—
51#387—41no—
52#387—41no—
53#384—21no—
54#382—21no—
55#382—21no—
56C4×C4⋊C4—21no—
57C4×C4⋊C4—21no—
58C4×C4⋊C4—21no—
59C4×C4⋊C4—41no—
60#365—42no—
61#365—42no—
62#365—42no—
63C4⋊C16—21no—
64C4⋊C16C2×C412yes—
65#346C2×C412yes—
66C4×C16—21no—
67#340—21no—
68#340—21no—
69#340—21no—
70#335—21no—
71#334—21no—
72Q8⋊C8—42no—
73C22×Q8SD1611yescharacteristic
74C2×M16—41no—
75C22×C8—21no—
76C22×C8C2×D411yescharacteristic
77C22×C8C2×D411yescharacteristic
78C4×Q8—21no—
79C4×Q8—21no—
80C4×Q8C4⋊C412yes—
81C4×Q8SD1612yes—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4SD1611yescharacteristic
90C2×C42—21no—
91C2×C42—41no—
92C2×C42—41no—
93C2×C42—41no—
94C2×C42C2×D411yescharacteristic
95M32—22no—
96M32—22no—
97M32—41no—
98M32—41no—
99C2×C16—22no—
100C2×C16—22no—
101#108—41no—
102#108—41no—
103#108—41no—
104#108—41no—
105#107—41no—
106#107—41no—
107#107—81no—
108#106—21no—
109#106—21no—
110#106—21no—
111#106—41no—
112#106—41no—
113#106—81no—
114#106SD1612yes—
115C4⋊C8—22no—
116C4⋊C8—22no—
117C4⋊C8—21no—
118C4⋊C8—41no—
119C4⋊C8—41no—
120C4⋊C8—41no—
121C4⋊C8C4⋊C412yes—
122C22⋊C8—41no—
123#97—22no—
124#97—22no—
125C4×C8—41no—
126C4×C8C22×C411yescharacteristic
127C4×C8C2×Q811yescharacteristic
128C4×C8#4111yescharacteristic
129C4×C8#4111yescharacteristic
130#95—41no—
131#95—41no—
132#95—41no—
133#95—41no—
134#95—41no—
135#95—41no—
136#95—41no—
137C2×Q8—21no—
138C2×Q8—81no—
139C2×Q8#10612yes—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4#12011yescharacteristic
147C22×C4C2×SD1611yescharacteristic
148C22×C4AGL(1,ℤ/8)11yescharacteristic
149M16—81no—
150M16—81no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—22no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—81no—
157C2×C8—81no—
158C2×C8C4×D411yescharacteristic
159C2×C8C4×D411yescharacteristic
160C2×C8C22⋊Q811yescharacteristic
161C2×C8C22⋊Q811yescharacteristic
162C2×C8C2×SD1611yescharacteristic
163C2×C8#13511yescharacteristic
164C2×C8#13711yescharacteristic
165C2×C8#13711yescharacteristic
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—21no—
169C4⋊C4—22no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—82no—
178C4⋊C4—161no—
179C4⋊C4#10612yes—
180C42—22no—
181C42—22no—
182C42—82no—
183C42—161no—
184C42—161no—
185C42C2×C4⋊C411yescharacteristic
186C42#12311yescharacteristic
187C42C2×SD1611yescharacteristic
188C42#13711yescharacteristic
189C16—24no—
190C16—42no—
191C23#45011yescharacteristic
192Q8—42no—
193Q8—81no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4—81no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4#38511yescharacteristic
215C2×C4C2×#10611yescharacteristic
216C2×C4#42811yescharacteristic
217C2×C4#44811yescharacteristic
218C2×C4#44811yescharacteristic
219C2×C4#45211yescharacteristic
220C2×C4#45311yescharacteristic
221C2×C4#45611yescharacteristic
222C2×C4#48111yescharacteristic
223C2×C4#48311yescharacteristic
224C8—21no—
225C8—21no—
226C8—42no—
227C8—161no—
228C8—161no—
229C8C4×SD1611yescharacteristic
230C8#44111yescharacteristic
231C8#47511yescharacteristic
232C8#47911yescharacteristic
233C22—21no—
234C22—21no—
235C22#163111yescharacteristic
236C22#165111yescharacteristic
237C22#185311yescharacteristic
238C4—21no—
239C4—21no—
240C4—21no—
241C4—21no—
242C4—42no—
243C4—42no—
244C4—82no—
245C4—161no—
246C4—161no—
247C4—161no—
248C4#162311yescharacteristic
249C4#165311yescharacteristic
250C4#185911yescharacteristic
251C4#186511yescharacteristic
252C2—21no—
253C2—21no—
254C2#888811yescharacteristic
255C2#1293211yescharacteristic
256C2#1293411yescharacteristic
subgroupG/H
center, socleC22#1853
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC23#450
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#446 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
26. #1338C4semidirect

Maximal subgroups

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

subgroupindex
#227302
#131852
#129042
#116112
#86632
#72792
#71752