Finite Group Databasebeta

← #144479

Subgroups — #144479

Order 512.

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

normal subgroups only

This group has 377 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#30292C211yesmaximal, characteristic
2#12867C212yesmaximal
3#11377C211yesmaximal, characteristic
4#7258C212yesmaximal
5#7183C211yesmaximal, characteristic
6C22×#107C2211yescharacteristic
7#2851—21no—
8C2×C22⋊C16C2212yes—
9#2085—22no—
10#2059—22no—
11#1808C2212yes—
12#1807—21no—
13C2×#340—21no—
14C2×#340C2211yescharacteristic
15C2×#336C2211yescharacteristic
16C4⋊#95—21no—
17#1380—22no—
18#1380—22no—
19C23×C8D411yescharacteristic
20C23×C8C2311yescharacteristic
21C2×#125—21no—
22C22×C4⋊C4—21no—
23C22×C4⋊C4—21no—
24C22×C42D411yescharacteristic
25C22×C16—22no—
26C2×#107—22no—
27C2×#107—22no—
28C2×#107—21no—
29C2×#107D412yes—
30C2×#107D412yes—
31C2×#107Q812yes—
32C2×#106—21no—
33C2×#106—21no—
34#389—22no—
35#387—42no—
36#385—42no—
37#382—21no—
38#380—22no—
39C2×#95—42no—
40C22⋊C16—22no—
41C22⋊C16—22no—
42#340—22no—
43#340—22no—
44#340—22no—
45#340—42no—
46#336—22no—
47C23×C4—41no—
48C23×C4—41no—
49C23×C4D812yes—
50C23×C4C2×D411yescharacteristic
51C22×C8—21no—
52C22×C8—21no—
53C22×C8—41no—
54C22×C8—41no—
55C22×C8—41no—
56C22×C8—41no—
57C22×C8SD1612yes—
58C22×C8C2×D412yes—
59C22×C8C2×Q811yescharacteristic
60C22×C8#4112yes—
61C22×C8#4111yescharacteristic
62#125—81no—
63#125—81no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—41no—
80C2×C22⋊C4—42no—
81C2×C42—21no—
82C2×C42—21no—
83C2×C42—41no—
84C2×C42—41no—
85C2×C16—42no—
86C2×C16—42no—
87C2×C16—42no—
88#107—24no—
89#107—42no—
90#107—42no—
91#107—42no—
92#106—81no—
93#106—81no—
94#95—42no—
95#95—42no—
96#95—42no—
97#95—84no—
98C24C2×D811yescharacteristic
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4—42no—
112C22×C4—82no—
113C22×C4—82no—
114C22×C4—81no—
115C22×C4—82no—
116C22×C4—81no—
117C22×C4—81no—
118C22×C4—81no—
119C22×C4—82no—
120C22×C4#12011yescharacteristic
121C22×C4C22⋊Q811yescharacteristic
122C22×C4#12411yescharacteristic
123C22×C4C2×SD1611yescharacteristic
124C22×C4AGL(1,ℤ/8)11yescharacteristic
125C22×C4#13711yescharacteristic
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—41no—
130C2×C8—41no—
131C2×C8—42no—
132C2×C8—41no—
133C2×C8—41no—
134C2×C8—42no—
135C2×C8—81no—
136C2×C8C22⋊Q812yes—
137C2×C8#12412yes—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—81no—
147C4⋊C4—81no—
148C4⋊C4—84no—
149C4⋊C4—84no—
150C4⋊C4—81no—
151C4⋊C4—81no—
152C22⋊C4—82no—
153C22⋊C4—82no—
154C42—41no—
155C42—41no—
156C42—81no—
157C16—82no—
158C23—21no—
159C23—21no—
160C23—22no—
161C23#44711yescharacteristic
162C23#47411yescharacteristic
163C23#48611yescharacteristic
164C23#50812yes—
165C23#51012yes—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—44no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—82no—
181C2×C4—82no—
182C2×C4—84no—
183C2×C4—84no—
184C2×C4—82no—
185C2×C4—82no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—161no—
192C2×C4—162no—
193C2×C4—161no—
194C2×C4—161no—
195C2×C4—162no—
196C2×C4—161no—
197C2×C4#39311yescharacteristic
198C2×C4#44811yescharacteristic
199C2×C4#45511yescharacteristic
200C2×C4#45611yescharacteristic
201C2×C4#45811yescharacteristic
202C2×C4#45811yescharacteristic
203C2×C4#47611yescharacteristic
204C2×C4#48911yescharacteristic
205C8—22no—
206C8—22no—
207C8—42no—
208C22—22no—
209C22—22no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—22no—
215C22—22no—
216C22#160711yescharacteristic
217C22#161011yescharacteristic
218C22#202311yescharacteristic
219C22#218912yes—
220C22#219012yes—
221C22#222912yes—
222C22#224211yescharacteristic
223C22#224511yescharacteristic
224C4—21no—
225C4—21no—
226C4—82no—
227C4—162no—
228C4—162no—
229C4—162no—
230C4—162no—
231C4#160811yescharacteristic
232C4#161111yescharacteristic
233C4#202811yescharacteristic
234C4#203011yescharacteristic
235C2—21no—
236C2—22no—
237C2—21no—
238C2#1034211yescharacteristic
239C2#1228312yes—
240C2#1229312yes—
241C2#1336611yescharacteristic
242C2#1336711yescharacteristic
subgroupG/H
center, socleC23#447
commutator, Frattini, cosocleC23×C8C23
prime radicalC24C2×D8
LCSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#336 ◁ G
LECSG ▷ C23×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
#302922
#128672
#113772
#72582
#71832