Finite Group Databasebeta

← #99255

Subgroups — #99255

Order 512.

546 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 82 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#13953C211yesmaximal, characteristic
2#8152C212yesmaximal
3#7036C214yesmaximal, complemented
4C2×#381C411yescharacteristic
5C2×C4×C22⋊C4C411yescharacteristic
6C2×C4×C22⋊C4C2211yescharacteristic
7#1749C2214yes—
8C2×#323—22no—
9#1439—22no—
10#1436—22no—
11C4⋊#95C412yes—
12#1434C2212yes—
13#1280—22no—
14C24×C4C2×C411yescharacteristic
15C22×C4⋊C4—22no—
16C22×C22⋊C4—21no—
17C22×C22⋊C4—21no—
18C22×C22⋊C4D412yes—
19C22×C42—21no—
20C22×C42C2×C411yescharacteristic
21C22×C42C2311yescharacteristic
22C2×C22⋊C8—24no—
23C2×#97—24no—
24#381—42no—
25C4×C22⋊C4—24no—
26C4×C22⋊C4—42no—
27C2×#95—22no—
28C2×#95—21no—
29C2×#95—22no—
30C2×#95—22no—
31C2×#95C2×C412yes—
32C2×#95C2×C412yes—
33C2×#95D412yes—
34#336—24no—
35#323—28no—
36C25C22⋊C411yescharacteristic
37C23×C4—22no—
38C23×C4—22no—
39C23×C4—22no—
40C23×C4—22no—
41C23×C4—22no—
42C23×C4—81no—
43C23×C4—81no—
44C23×C4C22⋊C411yescharacteristic
45C23×C4C22⋊C412yes—
46C23×C4C22×C411yescharacteristic
47C23×C4C2×D412yes—
48C22×C8—44no—
49C2×C4⋊C4—44no—
50C2×C4⋊C4—44no—
51C2×C4⋊C4—82no—
52C2×C22⋊C4—24no—
53C2×C22⋊C4—24no—
54C2×C22⋊C4—42no—
55C2×C22⋊C4—44no—
56C2×C22⋊C4—42no—
57C2×C22⋊C4—42no—
58C2×C22⋊C4—81no—
59C2×C22⋊C4—81no—
60C2×C42—22no—
61C2×C42—42no—
62C2×C42—81no—
63C2×C42—81no—
64C2×C42—81no—
65C2×C42—81no—
66C2×C42#4114yes—
67C22⋊C8—28no—
68C22⋊C8—44no—
69#97—84no—
70#95—24no—
71#95—42no—
72#95—42no—
73#95—44no—
74#95—42no—
75#95—44no—
76#95—82no—
77C24—21no—
78C24—21no—
79C24—22no—
80C24—22no—
81C24—42no—
82C24—41no—
83C24—41no—
84C24#10012yes—
85C24C2×C22⋊C411yescharacteristic
86C22×C4—21no—
87C22×C4—22no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—22no—
92C22×C4—28no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—24no—
96C22×C4—41no—
97C22×C4—41no—
98C22×C4—42no—
99C22×C4—42no—
100C22×C4—42no—
101C22×C4—41no—
102C22×C4—41no—
103C22×C4—42no—
104C22×C4—42no—
105C22×C4—42no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4—82no—
118C22×C4—82no—
119C22×C4—84no—
120C22×C4—84no—
121C22×C4#10112yes—
122C22×C4#10414yes—
123C22×C4#11712yes—
124C22×C4#12314yes—
125C2×C8—48no—
126C2×C8—84no—
127C2×C8—84no—
128C4⋊C4—88no—
129C22⋊C4—48no—
130C22⋊C4—48no—
131C22⋊C4—84no—
132C42—44no—
133C42—82no—
134C42—84no—
135C23—22no—
136C23—21no—
137C23—21no—
138C23—22no—
139C23—24no—
140C23—42no—
141C23—42no—
142C23—42no—
143C23—42no—
144C23—41no—
145C23—42no—
146C23—42no—
147C23—41no—
148C23—41no—
149C23—41no—
150C23—82no—
151C23—82no—
152C23—82no—
153C23—82no—
154C23#38111yescharacteristic
155C23C2×#10011yescharacteristic
156C23C2×#10111yescharacteristic
157C23C2×#10412yes—
158C23#42112yes—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—24no—
164C2×C4—24no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—42no—
168C2×C4—44no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—44no—
177C2×C4—44no—
178C2×C4—44no—
179C2×C4—44no—
180C2×C4—82no—
181C2×C4—82no—
182C2×C4—88no—
183C2×C4—82no—
184C2×C4—82no—
185C2×C4—82no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—82no—
193C2×C4—82no—
194C8—88no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—22no—
200C22—22no—
201C22—22no—
202C22—22no—
203C22—42no—
204C22—42no—
205C22—42no—
206C22—84no—
207C22—82no—
208C22—82no—
209C22—82no—
210C22—82no—
211C22#152611yescharacteristic
212C22#153012yes—
213C22#153711yescharacteristic
214C22#182111yescharacteristic
215C22#182612yes—
216C22#182712yes—
217C22#182811yescharacteristic
218C22#182911yescharacteristic
219C4—42no—
220C4—42no—
221C4—44no—
222C4—44no—
223C4—84no—
224C4—162no—
225C2—22no—
226C2—21no—
227C2—21no—
228C2—82no—
229C2#872211yescharacteristic
230C2#874211yescharacteristic
231C2#875611yescharacteristic
232C2#877312yes—
233C2#877812yes—
subgroupG/H
center, socleC23#381
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7036C2semidirect

Maximal subgroups

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

subgroupindex
#139532
#81522
#70362