Finite Group Databasebeta

← #139393

Subgroups — #139393

Order 512.

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

normal subgroups only

This group has 316 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#33497C211yesmaximal, characteristic
2#28678C211yesmaximal, characteristic
3#11969C212yesmaximal
4#8615C211yesmaximal, characteristic
5#7302C212yesmaximal
6C2×#499C2211yescharacteristic
7C2×C8×Q8C2211yescharacteristic
8#2584—21no—
9#2183—24no—
10#2163—21no—
11#2163C2212yes—
12#2162—21no—
13#2087C411yescharacteristic
14#2087C411yescharacteristic
15#2086C411yescharacteristic
16#2086C411yescharacteristic
17#1989—21no—
18#1941—21no—
19#1941—21no—
20#1801—21no—
21C4×#106C2211yescharacteristic
22#1566C2212yes—
23#1363—24no—
24#1292—21no—
25C2×#125—21no—
26C2×C4×Q8D411yescharacteristic
27#499—22no—
28#499—22no—
29C8×Q8—22no—
30C8×Q8—22no—
31C2×#107C2×C411yescharacteristic
32C2×#107C2×C411yescharacteristic
33C2×#106—21no—
34C2×#106—21no—
35#423—22no—
36C2×C4⋊C8C2×C411yescharacteristic
37C2×C4⋊C8C2×C411yescharacteristic
38C2×C4⋊C8D411yescharacteristic
39C2×#101—41no—
40C2×C4×C8D411yescharacteristic
41C2×C4×C8D411yescharacteristic
42C2×C4×C8C2311yescharacteristic
43#398—41no—
44#395—41no—
45#389—21no—
46#389—21no—
47#389—21no—
48#389—21no—
49#389—41no—
50#389—41no—
51C4×C4⋊C4—21no—
52C4⋊C16—22no—
53C4⋊C16—22no—
54C4⋊C16D414yes—
55C22⋊C16—22no—
56#346—22no—
57#346—22no—
58C4×C16—22no—
59#343—41no—
60#340—21no—
61#340—21no—
62#340—21no—
63#340—21no—
64#340C2×C411yescharacteristic
65#340C2×C411yescharacteristic
66#334—22no—
67#334—22no—
68C22×Q8D811yescharacteristic
69C2×M16—42no—
70C22×C8C22⋊C411yescharacteristic
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22×C411yescharacteristic
73C22×C8C2×D411yescharacteristic
74#125—42no—
75#125—42no—
76C4×Q8—22no—
77C4×Q8—22no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4D811yescharacteristic
89C2×C4⋊C4SD1611yescharacteristic
90C2×C4⋊C4SD1611yescharacteristic
91C2×C42—41no—
92C2×C42C22⋊C411yescharacteristic
93C2×C42C22⋊C411yescharacteristic
94C2×C42C2×D411yescharacteristic
95C2×C16—24no—
96#107—22no—
97#107—22no—
98#106—24no—
99#106—41no—
100#106—41no—
101C4⋊C8—22no—
102C4⋊C8—22no—
103C4⋊C8—22no—
104C4⋊C8—42no—
105C4⋊C8—42no—
106#101—82no—
107C4×C8—21no—
108C4×C8—21no—
109C4×C8—21no—
110C4×C8—21no—
111C4×C8C2×D412yes—
112C4×C8#4112yes—
113#95—41no—
114#95—41no—
115#95—41no—
116#95—41no—
117#95—41no—
118C2×Q8—22no—
119C2×Q8—22no—
120C2×Q8—41no—
121C22×C4—41no—
122C22×C4—41no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4#10211yescharacteristic
126C22×C4#10211yescharacteristic
127C22×C4#10211yescharacteristic
128C22×C4#10211yescharacteristic
129C22×C4C2×C22⋊C411yescharacteristic
130C22×C4C2×D811yescharacteristic
131C22×C4C2×SD1611yescharacteristic
132M16—82no—
133M16—82no—
134C2×C8—22no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—41no—
138C2×C8—41no—
139C2×C8—41no—
140C2×C8—82no—
141C2×C8C4×D412yes—
142C2×C8#12011yescharacteristic
143C2×C8#12311yescharacteristic
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4—41no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—82no—
156C4⋊C4—82no—
157C4⋊C4—82no—
158C4⋊C4—82no—
159C4⋊C4—161no—
160C4⋊C4—161no—
161C42—21no—
162C42—21no—
163C42—21no—
164C42—21no—
165C42—161no—
166C42#12112yes—
167C42#13512yes—
168C16—44no—
169C23C2×#10211yescharacteristic
170Q8—44no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—81no—
188C2×C4—81no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4—81no—
194C2×C4—81no—
195C2×C4—81no—
196C2×C4—81no—
197C2×C4#36012yes—
198C2×C4#38611yescharacteristic
199C2×C4#41611yescharacteristic
200C2×C4#46512yes—
201C2×C4#46612yes—
202C8—21no—
203C8—22no—
204C8—41no—
205C8—41no—
206C8—162no—
207C8#45612yes—
208C22—21no—
209C22—21no—
210C22#189311yescharacteristic
211C22C2×#36011yescharacteristic
212C22#214511yescharacteristic
213C4—22no—
214C4—22no—
215C4—22no—
216C4—21no—
217C4—41no—
218C4—41no—
219C4—41no—
220C4—82no—
221C4—82no—
222C4—161no—
223C4—161no—
224C4#188611yescharacteristic
225C4#191511yescharacteristic
226C2—21no—
227C2—21no—
228C2#939411yescharacteristic
229C2#1306511yescharacteristic
230C2#1307611yescharacteristic
subgroupG/H
center, socleC22#1893
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC23C2×#102
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ 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
#334972
#286782
#119692
#86152
#73022