Finite Group Databasebeta

← #128975

Subgroups — #128975

Order 512.

483 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 483 conjugacy classes of subgroups — too many to draw. Only its 92 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#21683C211yesmaximal, characteristic
2#11726C212yesmaximal
3#8160C212yesmaximal, complemented
4#7287C211yesmaximal, characteristic, complemented
5#7283C211yesmaximal, characteristic, complemented
6C2×C8×D4—22no—
7C2×C8×D4C411yescharacteristic, complemented
8C2×C8×D4C2211yescharacteristic
9#2397C2211yescharacteristic
10#1955—22no—
11#1955C411yescharacteristic, complemented
12#1955C2211yescharacteristic
13#1843C411yescharacteristic, complemented
14C4×#99C2212yes—
15#1752—22no—
16#1726C411yescharacteristic, complemented
17#1459C2212yes—
18#1352C414yescomplemented
19#1343C414yescomplemented
20#1282—22no—
21C23×C8—21no—
22C23×C8D412yes—
23C2×C4×D4—22no—
24C2×C4×D4C2×C411yescharacteristic
25C2×C4×D4C2311yescharacteristic
26C8×D4—41no—
27C8×D4—41no—
28C8×D4—41no—
29C8×D4—41no—
30C8×D4—48no—
31C2×C4⋊C8—22no—
32C2×C4⋊C8C2×C411yescharacteristic
33C2×C4⋊C8C2×C411yescharacteristic
34C2×#99—22no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—21no—
39C2×C22⋊C8D412yes—
40C2×#97—21no—
41C2×#97—42no—
42C2×#97C2×C411yescharacteristic
43C2×C4×C8—21no—
44C2×C4×C8C2×C411yescharacteristic
45#399—21no—
46#399C2×C412yes—
47#392—21no—
48#392C2×C412yes—
49C4×C22⋊C4—22no—
50#376C2×C411yescharacteristic
51#342C2×C414yes—
52#336—42no—
53#324—22no—
54#323C2×C414yes—
55C22×D4—22no—
56C22×D4C4211yescharacteristic
57C22×D4#4111yescharacteristic
58C23×C4—21no—
59C23×C4C2×D412yes—
60C22×C8—21no—
61C22×C8—22no—
62C22×C8—22no—
63C22×C8—21no—
64C22×C8—44no—
65C22×C8—44no—
66C22×C8—42no—
67C22×C8—41no—
68C22×C8—41no—
69C22×C8C22⋊C412yes—
70C22×C8C22⋊C411yescharacteristic
71C22×C8C22⋊C411yescharacteristic
72C4×D4—41no—
73C4×D4—41no—
74C4×D4—44no—
75C4×D4—44no—
76C4×D4—41no—
77C4×D4—41no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4C4211yescharacteristic
80C2×C4⋊C4#4111yescharacteristic
81C2×C22⋊C4—22no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—24no—
85C2×C22⋊C4C22×C412yes—
86C2×C42—21no—
87C2×C42—42no—
88C2×C42C22×C411yescharacteristic
89C4⋊C8—41no—
90C4⋊C8—41no—
91C4⋊C8—44no—
92#99—44no—
93C22⋊C8—24no—
94C22⋊C8—42no—
95C22⋊C8—44no—
96C22⋊C8—44no—
97C22⋊C8—42no—
98C22⋊C8—42no—
99#97—41no—
100#97—42no—
101#97—84no—
102C4×C8—24no—
103C4×C8—21no—
104C4×C8—21no—
105#95—21no—
106#95—42no—
107#95C4212yes—
108C24—21no—
109C24C4×D412yes—
110C2×D4—24no—
111C2×D4—41no—
112C2×D4—41no—
113C2×D4—44no—
114C2×D4—44no—
115C2×D4—44no—
116C2×D4—42no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—21no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4—44no—
128C22×C4C2×C4211yescharacteristic
129C22×C4C2×C22⋊C411yescharacteristic
130C22×C4#11711yescharacteristic
131C22×C4C4×D412yes—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—21no—
135C2×C8—22no—
136C2×C8—41no—
137C2×C8—41no—
138C2×C8—44no—
139C2×C8—41no—
140C2×C8—42no—
141C2×C8—44no—
142C2×C8—42no—
143C2×C8—41no—
144C2×C8—82no—
145C2×C8—82no—
146C2×C8—82no—
147C2×C8—82no—
148C2×C8—82no—
149C2×C8#9911yescharacteristic
150C2×C8#9911yescharacteristic
151C4⋊C4—41no—
152C4⋊C4—41no—
153C4⋊C4—44no—
154C22⋊C4—42no—
155C22⋊C4—44no—
156C22⋊C4—44no—
157C22⋊C4—42no—
158C22⋊C4—44no—
159C22⋊C4—44no—
160C42—22no—
161C42—22no—
162C42—21no—
163C42—21no—
164C42—82no—
165C42—82no—
166C23—22no—
167C23—22no—
168C23—22no—
169C23—42no—
170C23—42no—
171C23—42no—
172C23—41no—
173C23—41no—
174C23—42no—
175C23C4×C22⋊C411yescharacteristic
176C23#44414yes—
177D4—44no—
178D4—42no—
179D4—416no—
180D4—42no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—24no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4—84no—
201C2×C4—84no—
202C2×C4—84no—
203C2×C4C2×#9911yescharacteristic
204C2×C4#41011yescharacteristic
205C8—21no—
206C8—42no—
207C8—42no—
208C8—41no—
209C8—84no—
210C8—81no—
211C8#35312yes—
212C22—21no—
213C22—21no—
214C22—24no—
215C22—44no—
216C22—44no—
217C22—42no—
218C22—42no—
219C22—42no—
220C22—82no—
221C22—82no—
222C22—81no—
223C22—81no—
224C22C4×#9911yescharacteristic
225C22#176512yes—
226C4—21no—
227C4—44no—
228C4—41no—
229C4—41no—
230C4—42no—
231C4—81no—
232C4—164no—
233C4C2×#35311yescharacteristic
234C4#212611yescharacteristic
235C2—22no—
236C2—44no—
237C2—82no—
238C2#823911yescharacteristic
239C2#1262311yescharacteristic
240C2C4×#35311yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
socleC22C4×#99
prime radical#2397C22
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8160C2semidirect
4. #7287C2semidirect
5. #7283C2semidirect
7. C2×C8×D4C4semidirect
11. #1955C4semidirect
13. #1843C4semidirect
16. #1726C4semidirect
18. #1352C4semidirect
19. #1343C4semidirect

Maximal subgroups

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

subgroupindex
#216832
#117262
#81602
#72872
#72832