Finite Group Databasebeta

← #136441

Subgroups — #136441

Order 512.

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

normal subgroups only

This group has 295 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#12887C211yesmaximal, characteristic
4#11611C211yesmaximal, characteristic
5#8663C211yesmaximal, characteristic
6#7279C211yesmaximal, characteristic
7#7175C211yesmaximal, characteristic
8C2×#446C2211yescharacteristic
9#2355—21no—
10C2×#365—21no—
11#2150C2211yescharacteristic
12C4×M32C2211yescharacteristic
13#1949—21no—
14#1949—21no—
15#1948—21no—
16#1848C2211yescharacteristic
17#1847—21no—
18#1846C2211yescharacteristic
19#1776—21no—
20C4×#106C2211yescharacteristic
21#1564C2211yescharacteristic
22#1487—21no—
23#1383—21no—
24#1383—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—
61C4⋊C16—21no—
62C4⋊C16C2×C412yes—
63#346C2×C412yes—
64C4×C16—21no—
65#340—21no—
66#340—21no—
67#340—21no—
68#335—21no—
69#334—21no—
70Q8⋊C8—42no—
71C22×Q8SD1611yescharacteristic
72C2×M16—41no—
73C22×C8—21no—
74C22×C8C2×D411yescharacteristic
75C22×C8C2×D411yescharacteristic
76C4×Q8—21no—
77C4×Q8—21no—
78C4×Q8C4⋊C412yes—
79C4×Q8SD1612yes—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4SD1611yescharacteristic
88C2×C42—21no—
89C2×C42—41no—
90C2×C42—41no—
91C2×C42—41no—
92C2×C42C2×D411yescharacteristic
93M32—41no—
94M32—41no—
95M32—41no—
96M32—41no—
97C2×C16—22no—
98C2×C16—22no—
99#108—42no—
100#108—41no—
101#108—41no—
102#107—41no—
103#107—41no—
104#107—81no—
105#106—21no—
106#106—21no—
107#106—21no—
108#106—41no—
109#106—41no—
110#106—81no—
111#106Q1612yes—
112C4⋊C8—22no—
113C4⋊C8—22no—
114C4⋊C8—21no—
115C4⋊C8—41no—
116C4⋊C8—41no—
117C4⋊C8—41no—
118C4⋊C8C4⋊C412yes—
119C22⋊C8—41no—
120#97—22no—
121#97—22no—
122C4×C8—41no—
123C4×C8C22×C411yescharacteristic
124C4×C8C2×Q811yescharacteristic
125C4×C8#4111yescharacteristic
126C4×C8#4111yescharacteristic
127#95—41no—
128#95—41no—
129#95—41no—
130#95—41no—
131#95—41no—
132#95—41no—
133#95—41no—
134C2×Q8—21no—
135C2×Q8—81no—
136C2×Q8#10612yes—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4#12011yescharacteristic
144C22×C4C2×SD1611yescharacteristic
145C22×C4AGL(1,ℤ/8)11yescharacteristic
146M16—81no—
147M16—81no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—22no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—81no—
154C2×C8—81no—
155C2×C8C4×D411yescharacteristic
156C2×C8C4×D411yescharacteristic
157C2×C8C22⋊Q811yescharacteristic
158C2×C8C22⋊Q811yescharacteristic
159C2×C8C2×Q1611yescharacteristic
160C2×C8#13511yescharacteristic
161C2×C8AGL(1,ℤ/8)11yescharacteristic
162C2×C8#13711yescharacteristic
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—21no—
166C4⋊C4—22no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—82no—
175C4⋊C4—161no—
176C4⋊C4#10612yes—
177C42—22no—
178C42—22no—
179C42—82no—
180C42—161no—
181C42—161no—
182C42C2×C4⋊C411yescharacteristic
183C42#12311yescharacteristic
184C42C2×SD1611yescharacteristic
185C42#13711yescharacteristic
186C16—42no—
187C16—42no—
188C23#45011yescharacteristic
189Q8—42no—
190Q8—81no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—81no—
201C2×C4—81no—
202C2×C4—81no—
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#38511yescharacteristic
212C2×C4C2×#10611yescharacteristic
213C2×C4#42811yescharacteristic
214C2×C4#44811yescharacteristic
215C2×C4#44911yescharacteristic
216C2×C4#45111yescharacteristic
217C2×C4#45411yescharacteristic
218C2×C4#45511yescharacteristic
219C2×C4#48111yescharacteristic
220C2×C4#48311yescharacteristic
221C8—21no—
222C8—21no—
223C8—42no—
224C8—161no—
225C8—161no—
226C8C4×Q1611yescharacteristic
227C8#44011yescharacteristic
228C8#47711yescharacteristic
229C8#47911yescharacteristic
230C22—21no—
231C22—21no—
232C22#163711yescharacteristic
233C22#164211yescharacteristic
234C22#185311yescharacteristic
235C4—21no—
236C4—21no—
237C4—21no—
238C4—21no—
239C4—42no—
240C4—42no—
241C4—82no—
242C4—161no—
243C4—161no—
244C4—161no—
245C4#162611yescharacteristic
246C4#164511yescharacteristic
247C4#186011yescharacteristic
248C4#186311yescharacteristic
249C2—21no—
250C2—21no—
251C2#889611yescharacteristic
252C2#1293211yescharacteristic
253C2#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
#128872
#116112
#86632
#72792
#71752