Finite Group Databasebeta

← #139378

Subgroups — #139378

Order 512.

348 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 84 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#33487C211yesmaximal, characteristic
2#28705C211yesmaximal, characteristic
3#11974C212yesmaximal, complemented
4#8622C211yesmaximal, characteristic
5#7301C212yesmaximal, complemented
6C2×#500C2211yescharacteristic
7C2×C8×D4C2211yescharacteristic
8#2563—21no—
9#2168—24no—
10#2163C2212yes—
11#2112—21no—
12C2×C22⋊C16—21no—
13#2076C412yes—
14#2076C412yes—
15#1987—21no—
16#1808—21no—
17#1808—21no—
18C4×#107C2211yescharacteristic
19#1568C2212yes—
20#1466—21no—
21#1364—24no—
22#1290—21no—
23C23×C8D411yescharacteristic
24C23×C8D411yescharacteristic
25C2×C4⋊Q8—21no—
26C2×C4×D4D411yescharacteristic
27C2×M32—22no—
28C22×C16—22no—
29#500—22no—
30#500—22no—
31C8×D4—22no—
32C8×D4—22no—
33#432—22no—
34C2×#107—21no—
35C2×#107—21no—
36C2×#107C2×C412yes—
37C2×C4⋊C8D411yescharacteristic
38C2×C22⋊C8C2×C411yescharacteristic
39C2×C22⋊C8C2×C411yescharacteristic
40C2×C4×C8C2311yescharacteristic
41#393—41no—
42#393—41no—
43#390—41no—
44#382—41no—
45#380—22no—
46#380—22no—
47C4×C4⋊C4—21no—
48C4⋊C16D414yes—
49C22⋊C16—22no—
50C22⋊C16—22no—
51C22⋊C16—22no—
52#346—22no—
53#340—22no—
54#340—22no—
55#340C2×C412yes—
56#335—22no—
57#334—22no—
58#323—41no—
59#323—41no—
60C22×Q8—41no—
61C22×D4SD1611yescharacteristic
62C23×C4C22⋊C411yescharacteristic
63C23×C4C22⋊C411yescharacteristic
64C22×C8—21no—
65C22×C8—21no—
66C22×C8—21no—
67C22×C8—21no—
68C22×C8—42no—
69C22×C8—42no—
70C22×C8C22⋊C411yescharacteristic
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22×C411yescharacteristic
73C22×C8C2×D411yescharacteristic
74C4⋊Q8—42no—
75C4⋊Q8—42no—
76C4×D4—22no—
77C4×D4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4SD1611yescharacteristic
82C2×C22⋊C4—42no—
83C2×C22⋊C4—42no—
84C2×C22⋊C4D811yescharacteristic
85C2×C22⋊C4D811yescharacteristic
86C2×C42—41no—
87C2×C42C2×D411yescharacteristic
88M32—82no—
89C2×C16—24no—
90C2×C16—82no—
91#107—24no—
92#107—42no—
93#107—41no—
94#107—41no—
95C4⋊C8—22no—
96C4⋊C8—42no—
97C22⋊C8—22no—
98C22⋊C8—22no—
99C22⋊C8—42no—
100#97—42no—
101C4×C8C2×D412yes—
102C4×C8#4112yes—
103#95—42no—
104#95—42no—
105#95—41no—
106C24#10211yescharacteristic
107C24#10211yescharacteristic
108C2×Q8—82no—
109C2×Q8—162no—
110C2×D4—22no—
111C2×D4—22no—
112C2×D4—41no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—41no—
118C22×C4—41no—
119C22×C4—41no—
120C22×C4—41no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4#10211yescharacteristic
124C22×C4#10211yescharacteristic
125C22×C4C2×C22⋊C411yescharacteristic
126C22×C4C2×D811yescharacteristic
127C22×C4C2×SD1611yescharacteristic
128C2×C8—22no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—41no—
132C2×C8—42no—
133C2×C8—42no—
134C2×C8—41no—
135C2×C8—41no—
136C2×C8—81no—
137C2×C8—81no—
138C2×C8—82no—
139C2×C8C4×D412yes—
140C2×C8#12011yescharacteristic
141C2×C8#12311yescharacteristic
142C4⋊C4—22no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—82no—
148C22⋊C4—22no—
149C22⋊C4—22no—
150C22⋊C4—162no—
151C22⋊C4—162no—
152C42—82no—
153C42#12112yes—
154C42#13512yes—
155C16—44no—
156C23—21no—
157C23—21no—
158C23—21no—
159C23—21no—
160C23—21no—
161C23—41no—
162C23—41no—
163C23—41no—
164C23—41no—
165C23#36212yes—
166C23C2×#10211yescharacteristic
167Q8—164no—
168D4—44no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—81no—
183C2×C4—81no—
184C2×C4—82no—
185C2×C4—82no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4#38611yescharacteristic
191C2×C4#41611yescharacteristic
192C2×C4#46512yes—
193C2×C4#46612yes—
194C8—21no—
195C8—22no—
196C8—41no—
197C8—41no—
198C8—162no—
199C8#45512yes—
200C22—21no—
201C22—21no—
202C22—22no—
203C22—42no—
204C22—42no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—41no—
209C22—41no—
210C22—81no—
211C22—81no—
212C22#189311yescharacteristic
213C22C2×#36211yescharacteristic
214C22#214811yescharacteristic
215C4—22no—
216C4—22no—
217C4—21no—
218C4—41no—
219C4—41no—
220C4—162no—
221C4—162no—
222C4#188511yescharacteristic
223C4#191511yescharacteristic
224C2—21no—
225C2—21no—
226C2—42no—
227C2—81no—
228C2#939311yescharacteristic
229C2#1306611yescharacteristic
230C2#1307411yescharacteristic
subgroupG/H
center, socleC22#1893
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC22×D4SD16
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11974C2semidirect
5. #7301C2semidirect

Maximal subgroups

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

subgroupindex
#334872
#287052
#119742
#86222
#73012