Finite Group Databasebeta

← #135478

Subgroups — #135478

Order 512.

548 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 66 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#14166C211yesmaximal, characteristic
2#8556C211yesmaximal, characteristic
3#8545C211yesmaximal, characteristic
4#7213C214yesmaximal
5C22×#126C2211yescharacteristic
6C2×C4×C4⋊C4C411yescharacteristic, complemented
7C2×#376C411yescharacteristic, complemented
8#1821—21no—
9#1819—21no—
10#1819—21no—
11#1819C2212yes—
12#1463—21no—
13#1462C2212yes—
14#1441—21no—
15#1439—21no—
16#1437—21no—
17#1437C2212yes—
18#1436—21no—
19#1436C412yescomplemented
20C2×#126—24no—
21C2×#126—42no—
22C2×#126—42no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4C2×C411yescharacteristic
25C22×C4⋊C4D411yescharacteristic
26C22×C4⋊C4D411yescharacteristic
27C22×C22⋊C4—22no—
28C22×C22⋊C4D411yescharacteristic
29C22×C22⋊C4D411yescharacteristic
30C22×C22⋊C4C2311yescharacteristic
31C22×C42—21no—
32C22×C42C2×C411yescharacteristic
33C22×C42C2×C412yes—
34#407—22no—
35#379—22no—
36C4×C4⋊C4—42no—
37#376—42no—
38C2×#95—22no—
39C2×#95—21no—
40C2×#95—21no—
41C2×#95—22no—
42C2×#95—22no—
43C2×#95C2×C412yes—
44#343—42no—
45#342—24no—
46#342—42no—
47#342—42no—
48#324—42no—
49#323—24no—
50C25C2×D411yescharacteristic
51C23×C4—22no—
52C23×C4—22no—
53C23×C4C22⋊C411yescharacteristic
54C23×C4C22⋊C411yescharacteristic
55C23×C4C22⋊C411yescharacteristic
56C23×C4C22⋊C411yescharacteristic
57C23×C4C22×C411yescharacteristic
58C23×C4C2×D411yescharacteristic
59C2×M16—42no—
60#126—416no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—41no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—44no—
71C2×C4⋊C4—42no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—41no—
77C2×C22⋊C4—41no—
78C2×C22⋊C4—41no—
79C2×C22⋊C4—41no—
80C2×C22⋊C4—44no—
81C2×C22⋊C4—44no—
82C2×C22⋊C4—42no—
83C2×C22⋊C4—42no—
84C2×C22⋊C4—42no—
85C2×C22⋊C4—82no—
86C2×C22⋊C4#4114yes—
87C2×C42—22no—
88C2×C42—22no—
89C2×C42—22no—
90C2×C42—44no—
91C2×C42—42no—
92C2×C42—42no—
93C2×C42—82no—
94C22⋊C8—42no—
95C22⋊C8—42no—
96#95—42no—
97#95—44no—
98#95—44no—
99#95—42no—
100#95—42no—
101#95—82no—
102C24—21no—
103C24—21no—
104C24—42no—
105C24—42no—
106C24—41no—
107C24—41no—
108C24C2×C22⋊C411yescharacteristic
109C24#12312yes—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—24no—
119C22×C4—24no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—42no—
127C22×C4—41no—
128C22×C4—42no—
129C22×C4—41no—
130C22×C4—42no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—82no—
147C22×C4#9912yes—
148C22×C4#9912yes—
149C22×C4#11712yes—
150C22×C4#12312yes—
151M16—84no—
152C2×C8—82no—
153C4⋊C4—44no—
154C4⋊C4—44no—
155C4⋊C4—44no—
156C4⋊C4—48no—
157C22⋊C4—44no—
158C22⋊C4—44no—
159C22⋊C4—44no—
160C22⋊C4—88no—
161C22⋊C4—84no—
162C22⋊C4—82no—
163C22⋊C4—82no—
164C22⋊C4—84no—
165C42—28no—
166C42—48no—
167C42—84no—
168C23—22no—
169C23—21no—
170C23—21no—
171C23—22no—
172C23—22no—
173C23—24no—
174C23—42no—
175C23—42no—
176C23—41no—
177C23—41no—
178C23—42no—
179C23—81no—
180C23—81no—
181C23—81no—
182C23—81no—
183C23—82no—
184C23—82no—
185C23—82no—
186C23—82no—
187C23#38111yescharacteristic
188C23C2×#9911yescharacteristic
189C23C2×#9911yescharacteristic
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—44no—
209C2×C4—44no—
210C2×C4—44no—
211C2×C4—44no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—81no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4—162no—
231C2×C4—162no—
232C2×C4—162no—
233C8—162no—
234C22—22no—
235C22—22no—
236C22—22no—
237C22—22no—
238C22—21no—
239C22—21no—
240C22—21no—
241C22—21no—
242C22—21no—
243C22—21no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—84no—
248C22—161no—
249C22—162no—
250C22—161no—
251C22#181911yescharacteristic
252C22#182712yes—
253C22#212211yescharacteristic
254C22#212311yescharacteristic
255C22#212612yes—
256C4—42no—
257C4—42no—
258C4—42no—
259C4—42no—
260C4—84no—
261C4—84no—
262C4—81no—
263C4—81no—
264C4—322no—
265C2—21no—
266C2—22no—
267C2—22no—
268C2—21no—
269C2—82no—
270C2#873811yescharacteristic
271C2#1285511yescharacteristic
272C2#1285711yescharacteristic
subgroupG/H
center, socleC22#1819
commutatorC23×C4C22×C4
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#126 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C2×C4×C4⋊C4C4semidirect
7. C2×#376C4semidirect
19. #1436C4semidirect

Maximal subgroups

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

subgroupindex
#141662
#85562
#85452
#72132