Finite Group Databasebeta

← #122435

Subgroups — #122435

Order 512.

288 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 72 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#22209C211yesmaximal, characteristic
2#11494C211yesmaximal, characteristic, complemented
3#11493C211yesmaximal, characteristic, complemented
4#7066C212yesmaximal, complemented
5#7060C212yesmaximal
6C2×C8×Q8C2211yescharacteristic
7C2×C8×D4C411yescharacteristic
8#2398C411yescharacteristic
9#2163—21no—
10#2162—21no—
11#1985C411yescharacteristic
12#1955C2211yescharacteristic
13#1923—21no—
14#1923—21no—
15#1850—21no—
16#1850—21no—
17#1843C2211yescharacteristic
18#1746C411yescharacteristic
19#1322—21no—
20#1322—21no—
21#1322C2212yes—
22#1322C2212yes—
23C23×C8—21no—
24C2×C4×Q8C2×C411yescharacteristic
25C2×C4×D4C2×C411yescharacteristic
26C8×Q8—41no—
27C8×Q8—41no—
28C8×D4—41no—
29C8×D4—41no—
30C2×C4⋊C8—21no—
31C2×C4⋊C8—21no—
32C2×C4⋊C8C2×C411yescharacteristic
33C2×C4⋊C8D411yescharacteristic
34C2×C4⋊C8D411yescharacteristic
35C2×C4⋊C8C2311yescharacteristic
36C2×C22⋊C8—21no—
37C2×C22⋊C8—21no—
38C2×C22⋊C8—21no—
39C2×#97C2×C411yescharacteristic
40C2×C4×C8C2×C411yescharacteristic
41C2×C4×C8D411yescharacteristic
42C2×C4×C8D411yescharacteristic
43#400C811yescharacteristic
44#400C811yescharacteristic
45#397—21no—
46#394C811yescharacteristic
47#394C811yescharacteristic
48#393—21no—
49C4×C22⋊C4—21no—
50C4×C22⋊C4—21no—
51#376C2×C411yescharacteristic
52C4⋊C16—41no—
53C4⋊C16—41no—
54C22⋊C16—22no—
55C22⋊C16—22no—
56#346—41no—
57C4×C16—41no—
58#336—21no—
59#336—21no—
60#336—21no—
61#336—21no—
62C22×Q8M1611yescharacteristic
63C22×D4C2×C811yescharacteristic
64C23×C4—21no—
65C22×C8—21no—
66C22×C8—21no—
67C22×C8—42no—
68C22×C8—81no—
69C22×C8—81no—
70C22×C8C22⋊C411yescharacteristic
71C22×C8C22⋊C411yescharacteristic
72C22×C8C2×D411yescharacteristic
73C22×C8C2×D411yescharacteristic
74C4×Q8—41no—
75C4×Q8—41no—
76C4×D4—41no—
77C4×D4—41no—
78C2×C4⋊C4C2×C811yescharacteristic
79C2×C4⋊C4C2×C811yescharacteristic
80C2×C4⋊C4C2×C811yescharacteristic
81C2×C4⋊C4M1611yescharacteristic
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C42—21no—
88C2×C42C22⋊C411yescharacteristic
89C2×C42C22⋊C411yescharacteristic
90C2×C42C22×C411yescharacteristic
91C2×C16—42no—
92C2×C16—42no—
93C4⋊C8—41no—
94C4⋊C8—41no—
95C4⋊C8—41no—
96C4⋊C8—41no—
97C4⋊C8—81no—
98C4⋊C8—81no—
99C22⋊C8—41no—
100C22⋊C8—41no—
101C22⋊C8—81no—
102C22⋊C8—81no—
103#97—41no—
104C4×C8—21no—
105C4×C8—21no—
106C4×C8—21no—
107C4×C8—21no—
108C4×C8—21no—
109C4×C8—21no—
110#95—21no—
111#95—21no—
112#95—21no—
113#95—21no—
114#95—21no—
115#95C2×C811yescharacteristic
116#95C2×C811yescharacteristic
117C24—21no—
118C2×Q8—22no—
119C2×Q8—41no—
120C2×Q8—41no—
121C2×D4—41no—
122C2×D4—42no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4C22⋊C811yescharacteristic
133C22×C4C22⋊C811yescharacteristic
134C22×C4C22⋊C811yescharacteristic
135C22×C4C22⋊C811yescharacteristic
136C22×C4C2×C22⋊C411yescharacteristic
137C22×C4C22×C811yescharacteristic
138C22×C4C2×M1611yescharacteristic
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—41no—
147C2×C8—41no—
148C2×C8—41no—
149C2×C8—41no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8—84no—
153C2×C8#9911yescharacteristic
154C2×C8#9911yescharacteristic
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C22⋊C4—42no—
160C22⋊C4—42no—
161C22⋊C4—42no—
162C22⋊C4—42no—
163C22⋊C4—42no—
164C42—21no—
165C42—21no—
166C42—21no—
167C42—21no—
168C42—21no—
169C42—21no—
170C42—41no—
171C42—41no—
172C16—82no—
173C16—82no—
174C23—22no—
175C23—41no—
176C23—41no—
177C23—41no—
178C23—41no—
179C23—41no—
180C23—41no—
181C23C2×C22⋊C811yescharacteristic
182Q8—44no—
183D4—42no—
184D4—42no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4#35012yes—
215C2×C4C2×#9911yescharacteristic
216C2×C4#41311yescharacteristic
217C8—21no—
218C8—21no—
219C8—41no—
220C8—41no—
221C8—41no—
222C8—81no—
223C8—81no—
224C22—21no—
225C22—21no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—81no—
230C22—81no—
231C22—81no—
232C22—81no—
233C22#145911yescharacteristic
234C22C2×#35011yescharacteristic
235C22#211711yescharacteristic
236C4—22no—
237C4—21no—
238C4—41no—
239C4—41no—
240C4—41no—
241C4—41no—
242C4—41no—
243C4—81no—
244C4—81no—
245C4—81no—
246C4#212811yescharacteristic
247C4#213311yescharacteristic
248C2—21no—
249C2—21no—
250C2—82no—
251C2#1147011yescharacteristic
252C2#1148011yescharacteristic
253C2#1148311yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4⋊C8C23
socleC22#1459
prime radicalC22×D4C2×C8
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11494C2semidirect
3. #11493C2semidirect
4. #7066C2semidirect

Maximal subgroups

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

subgroupindex
#222092
#114942
#114932
#70662
#70602