Finite Group Databasebeta

← #99716

Subgroups — #99716

Order 512.

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

normal subgroups only

This group has 385 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
1#23744C211yesmaximal, characteristic
2#21921C211yesmaximal, characteristic
3#8710C212yesmaximal
4#7564C211yesmaximal, characteristic
5#7029C212yesmaximal
6#2296C2211yescharacteristic
7#2285C2211yescharacteristic
8#2101—22no—
9#2100C412yes—
10#2099C412yes—
11#1917—22no—
12#1916—22no—
13#1818—21no—
14#1797C2212yes—
15#1750—21no—
16C4×C22⋊C8C2212yes—
17#1735—21no—
18#1735—21no—
19C2×#336C2211yescharacteristic
20#1279—22no—
21#1279—22no—
22#1271—22no—
23C23×C8—21no—
24C2×#117—21no—
25C2×#117D411yescharacteristic
26C2×#117D411yescharacteristic
27C22×C42C2311yescharacteristic
28C2×C4⋊C8D412yes—
29C2×C4⋊C8D412yes—
30C2×C22⋊C8—22no—
31C2×C22⋊C8—22no—
32C2×#97—21no—
33C2×C4×C8—22no—
34C2×C4×C8—21no—
35#389C2×C412yes—
36#389D412yes—
37#385—22no—
38#384C2×C411yescharacteristic
39#384C2×C411yescharacteristic
40#382—21no—
41#380—22no—
42C4×C4⋊C4—22no—
43C4×C4⋊C4—22no—
44#340—22no—
45#340—22no—
46#340—22no—
47#340—22no—
48#336—21no—
49#336—22no—
50#336—21no—
51#336—22no—
52#336—22no—
53#336—21no—
54#336—22no—
55#336C2×C412yes—
56C23×C4C2×D411yescharacteristic
57C23×C4#4112yes—
58C22×C8—22no—
59C22×C8—22no—
60C22×C8—22no—
61C22×C8—42no—
62C22×C8—41no—
63C22×C8—41no—
64C22×C8—42no—
65C22×C8—42no—
66#117—22no—
67#117—22no—
68#117—41no—
69#117—41no—
70#117—44no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4C22⋊C412yes—
78C2×C4⋊C4C22⋊C412yes—
79C2×C22⋊C4—21no—
80C2×C22⋊C4—21no—
81C2×C22⋊C4—41no—
82C2×C42—21no—
83C2×C42—21no—
84C2×C42—41no—
85C2×C42—41no—
86C2×C42—41no—
87C2×C42C22×C411yescharacteristic
88C2×C42C2×D412yes—
89C2×C42C2×D411yescharacteristic
90C4⋊C8—42no—
91C4⋊C8—42no—
92C22⋊C8—24no—
93C22⋊C8—42no—
94C22⋊C8—42no—
95C22⋊C8—42no—
96#97—81no—
97C4×C8—42no—
98C4×C8—42no—
99C4×C8—81no—
100#95—22no—
101#95—22no—
102#95—42no—
103C24#12311yescharacteristic
104C22×C4—22no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—41no—
114C22×C4—41no—
115C22×C4C2×C22⋊C411yescharacteristic
116C22×C4C4×D412yes—
117C22×C4#12011yescharacteristic
118C22×C4#12112yes—
119C22×C4#13512yes—
120C22×C4#13712yes—
121C2×C8—44no—
122C2×C8—42no—
123C2×C8—42no—
124C2×C8—42no—
125C2×C8—42no—
126C2×C8—42no—
127C2×C8—42no—
128C2×C8—42no—
129C2×C8—42no—
130C2×C8—42no—
131C2×C8—82no—
132C2×C8—82no—
133C2×C8—82no—
134C2×C8—82no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—42no—
138C4⋊C4—44no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—82no—
146C22⋊C4—42no—
147C22⋊C4—81no—
148C22⋊C4—82no—
149C42—22no—
150C42—22no—
151C42—41no—
152C42—41no—
153C42—42no—
154C42—41no—
155C42—41no—
156C42—82no—
157C23—22no—
158C23—22no—
159C23—21no—
160C23—21no—
161C23#38611yescharacteristic
162C23#48512yes—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—22no—
166C2×C4—21no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—82no—
181C2×C4—82no—
182C2×C4—82no—
183C2×C4—82no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4#41611yescharacteristic
187C2×C4#41911yescharacteristic
188C2×C4#44312yes—
189C2×C4#44412yes—
190C2×C4#45212yes—
191C2×C4#45412yes—
192C2×C4#45811yescharacteristic
193C2×C4#45811yescharacteristic
194C8—82no—
195C8—84no—
196C8—82no—
197C8—82no—
198C22—24no—
199C22—22no—
200C22—41no—
201C22—42no—
202C22—41no—
203C22#187311yescharacteristic
204C22#187612yes—
205C22#188112yes—
206C22#188811yescharacteristic
207C22#189011yescharacteristic
208C4—21no—
209C4—21no—
210C4—22no—
211C4—22no—
212C4—42no—
213C4—82no—
214C4—82no—
215C4—161no—
216C4—161no—
217C4#161512yes—
218C4#161512yes—
219C2—42no—
220C2#899412yes—
221C2#900811yescharacteristic
222C2#902211yescharacteristic
223C2#903812yes—
224C2#907011yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC23#386
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#237442
#219212
#87102
#75642
#70292