Finite Group Databasebeta

← #120578

Subgroups — #120578

Order 512.

532 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 136 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#21355C211yesmaximal, characteristic
2C2×#1778C211yesmaximal, characteristic
3#8068C212yesmaximal
4#7993C212yesmaximal
5#7861C211yesmaximal, characteristic
6#2920C2212yes—
7C2×#432C2211yescharacteristic
8C22×#108—21no—
9C2×C4×M16C2212yes—
10C22×#97C2211yescharacteristic
11C4×#117C2211yescharacteristic
12#1940C412yes—
13#1940C412yes—
14#1850—21no—
15#1850—21no—
16#1778—24no—
17#1767—22no—
18C2×#341—21no—
19#1300—24no—
20#1291—24no—
21#1275—22no—
22C22×M16—22no—
23C23×C8D411yescharacteristic
24C23×C8D411yescharacteristic
25C2×#117—21no—
26C22×C42C2311yescharacteristic
27#432—22no—
28#432—22no—
29#432—22no—
30#432—22no—
31#431—22no—
32#431Q814yes—
33C2×#108—28no—
34C2×#108—24no—
35#423—22no—
36C2×C4⋊C8—22no—
37C2×C4⋊C8—22no—
38C2×C4⋊C8C2×C411yescharacteristic
39C2×C4⋊C8C2×C411yescharacteristic
40#407—22no—
41C2×C22⋊C8—21no—
42C4×M16—22no—
43C4×M16—22no—
44C4×M16—22no—
45C4×M16—22no—
46C4×M16D414yes—
47C2×#97—22no—
48C2×#97—22no—
49C2×#97—22no—
50C2×#97—21no—
51C2×#97C2×C412yes—
52C2×#97D412yes—
53C2×C4×C8—22no—
54C4×C4⋊C4—21no—
55C4×C4⋊C4C2×C412yes—
56C4×C22⋊C4—21no—
57C4×C22⋊C4Q811yescharacteristic
58C4×C22⋊C4Q811yescharacteristic
59#376D412yes—
60C43—21no—
61#341—24no—
62#336—22no—
63#336—22no—
64C23×C4C2×D411yescharacteristic
65C23×C4C2×Q811yescharacteristic
66C23×C4#4111yescharacteristic
67C2×M16—28no—
68C2×M16—24no—
69C2×M16—44no—
70C2×M16—42no—
71C2×M16—42no—
72C22×C8—21no—
73C22×C8—21no—
74C22×C8—24no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—42no—
80C22×C8—42no—
81C22×C8C22⋊C412yes—
82C22×C8C22⋊C412yes—
83#117—44no—
84C2×C4⋊C4—22no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C42—22no—
88C2×C42—22no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42C22×C411yescharacteristic
92C2×C42C2×D412yes—
93C2×C42C2×D411yescharacteristic
94C2×C42C2×D411yescharacteristic
95C2×C42C2×Q812yes—
96C2×C42#4112yes—
97C2×C42#4112yes—
98C2×C42#4111yescharacteristic
99#108—216no—
100C4⋊C8—22no—
101C4⋊C8—22no—
102C4⋊C8—24no—
103C4⋊C8—44no—
104C4⋊C8—44no—
105C4⋊C8—42no—
106C22⋊C8—24no—
107C22⋊C8—41no—
108C22⋊C8—42no—
109C22⋊C8—41no—
110#97—24no—
111#97—24no—
112#97—24no—
113#97—42no—
114#97—42no—
115#97—42no—
116C4×C8—24no—
117C4×C8—24no—
118C4×C8—42no—
119#95—22no—
120#95C4⋊C412yes—
121#95C4⋊C412yes—
122C24C22⋊Q811yescharacteristic
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—42no—
128C22×C4—81no—
129C22×C4C2×C22⋊C411yescharacteristic
130C22×C4C2×C4⋊C411yescharacteristic
131C22×C4#11711yescharacteristic
132C22×C4#12011yescharacteristic
133C22×C4#12111yescharacteristic
134C22×C4#12111yescharacteristic
135C22×C4C22⋊Q812yes—
136C22×C4C22⋊Q811yescharacteristic
137C22×C4#12311yescharacteristic
138C22×C4#12411yescharacteristic
139C22×C4#12512yes—
140C22×C4#12612yes—
141C22×C4C4⋊Q811yescharacteristic
142C22×C4C4⋊Q812yes—
143M16—28no—
144M16—48no—
145M16—44no—
146M16—84no—
147C2×C8—24no—
148C2×C8—24no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—42no—
155C2×C8—42no—
156C2×C8—48no—
157C2×C8—44no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—44no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8#10414yes—
164C4⋊C4—44no—
165C22⋊C4—24no—
166C22⋊C4—42no—
167C42—44no—
168C42—42no—
169C42—42no—
170C42—81no—
171C42—81no—
172C42C4×D412yes—
173C42C4×Q812yes—
174C42#12112yes—
175C42#12112yes—
176C42C22⋊Q812yes—
177C42C22⋊Q812yes—
178C42#12312yes—
179C42#12512yes—
180C23—22no—
181C23—22no—
182C23—41no—
183C23#38011yescharacteristic
184C23#39511yescharacteristic
185C23#40011yescharacteristic
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4#38611yescharacteristic
208C2×C4#38712yes—
209C2×C4#38912yes—
210C2×C4#39111yescharacteristic
211C2×C4#39411yescharacteristic
212C2×C4#39612yes—
213C2×C4#39812yes—
214C2×C4#39811yescharacteristic
215C2×C4C2×#10412yes—
216C2×C4#42112yes—
217C8—24no—
218C8—48no—
219C8—42no—
220C8—84no—
221C8—81no—
222C8—81no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—21no—
230C22—44no—
231C22#144111yescharacteristic
232C22#180612yes—
233C22#210212yes—
234C4—21no—
235C4—21no—
236C4—22no—
237C4—22no—
238C4—22no—
239C4—22no—
240C4—44no—
241C4—82no—
242C4#189611yescharacteristic
243C4#189711yescharacteristic
244C4#198612yes—
245C2—22no—
246C2—22no—
247C2—41no—
248C2#1122611yescharacteristic
249C2#1123511yescharacteristic
250C2#1124411yescharacteristic
subgroupG/H
centerC2×C4#391
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1441
prime radicalC24C22⋊Q8
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#213552
C2×#17782
#80682
#79932
#78612