Finite Group Databasebeta

← #143391

Subgroups — #143391

Order 512.

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

normal subgroups only

This group has 366 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
1C2×#2255C211yesmaximal, characteristic
2#28680C211yesmaximal, characteristic
3C2×#1696C211yesmaximal, characteristic
4#13275C211yesmaximal, characteristic
5C2×#1363C211yesmaximal, characteristic
6C2×#1338C211yesmaximal, characteristic
7#7178C211yesmaximal, characteristic
8C2×#500C2211yescharacteristic
9C2×#499C2211yescharacteristic
10C2×#477—21no—
11C2×#446C2211yescharacteristic
12#2584—21no—
13#2255—22no—
14#2255—22no—
15C2×#367—21no—
16C2×#366—21no—
17C2×C4⋊C16C2211yescharacteristic
18C2×#346C2211yescharacteristic
19#2087—21no—
20#2086—21no—
21#1989—21no—
22#1943—22no—
23C4×#107C2211yescharacteristic
24#1696—22no—
25#1696—22no—
26C2×#334C2211yescharacteristic
27C2×#333—21no—
28#1380—22no—
29#1363—22no—
30#1338—22no—
31C2×C4⋊Q8—21no—
32C2×#125—21no—
33C2×C4×Q8D411yescharacteristic
34C22×C16—21no—
35C22×C16—21no—
36#500—21no—
37#500—21no—
38#500—21no—
39#500D412yes—
40#499—21no—
41#499—21no—
42#499—21no—
43#499—21no—
44#477—24no—
45#477—42no—
46#446—21no—
47#446—21no—
48#446—21no—
49#446—21no—
50C2×#107—21no—
51C2×#107—21no—
52C2×#107—41no—
53C2×#107D411yescharacteristic
54C2×#107D411yescharacteristic
55C2×#107D412yes—
56C2×#106—21no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8D411yescharacteristic
60C2×#103—41no—
61C2×#97—21no—
62C2×C4×C8C2311yescharacteristic
63#398—41no—
64#395—41no—
65#389—22no—
66#389—22no—
67#389—42no—
68C4×C4⋊C4—21no—
69#367—24no—
70#366—24no—
71C4⋊C16—21no—
72C4⋊C16—21no—
73#346—21no—
74#346D412yes—
75#340—22no—
76#340—22no—
77#340—21no—
78#334—21no—
79#334D412yes—
80#333—24no—
81C22×Q8—41no—
82C22×Q8D811yescharacteristic
83C22×C8—21no—
84C22×C8—41no—
85C22×C8C2×D411yescharacteristic
86C22×C8C2×D411yescharacteristic
87C4⋊Q8—22no—
88C4⋊Q8—41no—
89C4⋊Q8—41no—
90C4⋊Q8—41no—
91#125—41no—
92#125—41no—
93#125—41no—
94#125—41no—
95C4×Q8—21no—
96C4×Q8—21no—
97C4×Q8—21no—
98C4×Q8—21no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4D811yescharacteristic
110C2×C42—21no—
111C2×C42—41no—
112C2×C42C2×D411yescharacteristic
113C2×C16—24no—
114C2×C16—22no—
115C2×C16—41no—
116C2×C16—42no—
117#107—22no—
118#107—21no—
119#107—22no—
120#107—21no—
121#107—22no—
122#107—21no—
123#107—21no—
124#107—44no—
125#107—41no—
126#107—41no—
127#107Q1614yes—
128#106—41no—
129#106—41no—
130C4⋊C8—21no—
131C4⋊C8—22no—
132C4⋊C8—21no—
133C4⋊C8—41no—
134C4⋊C8—41no—
135C4⋊C8—41no—
136#103—44no—
137#97—41no—
138#97—41no—
139C4×C8C2×D411yescharacteristic
140C4×C8C2×D411yescharacteristic
141C4×C8C2×D411yescharacteristic
142C4×C8#4111yescharacteristic
143#95—42no—
144#95—42no—
145#95—41no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—42no—
149C2×Q8—42no—
150C2×Q8—81no—
151C2×Q8—161no—
152C2×Q8—161no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—42no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—81no—
159C22×C4#12011yescharacteristic
160C22×C4C2×D811yescharacteristic
161C22×C4AGL(1,ℤ/8)11yescharacteristic
162C2×C8—22no—
163C2×C8—41no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—82no—
167C2×C8#12011yescharacteristic
168C2×C8#12111yescharacteristic
169C2×C8#12111yescharacteristic
170C2×C8#12711yescharacteristic
171C2×C8C2×Q1612yes—
172C2×C8#13712yes—
173C4⋊C4—21no—
174C4⋊C4—24no—
175C4⋊C4—21no—
176C4⋊C4—41no—
177C4⋊C4—42no—
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—84no—
186C4⋊C4—81no—
187C4⋊C4—81no—
188C4⋊C4—81no—
189C4⋊C4—81no—
190C4⋊C4—81no—
191C4⋊C4—81no—
192C4⋊C4—162no—
193C42—41no—
194C42—41no—
195C42—161no—
196C42#12011yescharacteristic
197C42#12111yescharacteristic
198C42#13511yescharacteristic
199C42AGL(1,ℤ/8)11yescharacteristic
200C16—24no—
201C16—81no—
202C23#44711yescharacteristic
203Q8—42no—
204Q8—81no—
205Q8—162no—
206Q8—162no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—44no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—82no—
224C2×C4—161no—
225C2×C4—161no—
226C2×C4#39211yescharacteristic
227C2×C4#44711yescharacteristic
228C2×C4#44911yescharacteristic
229C2×C4#44911yescharacteristic
230C2×C4#45112yes—
231C2×C4#45612yes—
232C2×C4#46611yescharacteristic
233C2×C4#46911yescharacteristic
234C8—21no—
235C8—21no—
236C8—82no—
237C8—81no—
238C8#46212yes—
239C8#49411yescharacteristic
240C8#49711yescharacteristic
241C22#164912yes—
242C22#199911yescharacteristic
243C22#219112yes—
244C22#219211yescharacteristic
245C22#219311yescharacteristic
246C4—21no—
247C4—21no—
248C4—21no—
249C4—21no—
250C4—81no—
251C4—84no—
252C4—81no—
253C4—161no—
254C4—161no—
255C4—162no—
256C4#164812yes—
257C4#200511yescharacteristic
258C4#201611yescharacteristic
259C2#1006711yescharacteristic
260C2#1241212yes—
261C2#1241312yes—
262C2#1330711yescharacteristic
263C2#1331711yescharacteristic
subgroupG/H
center, socle, prime radicalC23#447
commutator, Frattini, cosocleC2×C4×C8C23
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#446 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#22552
#286802
C2×#16962
#132752
C2×#13632
C2×#13382
#71782