Finite Group Databasebeta

← #134811

Subgroups — #134811

Order 512.

454 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 78 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
1C8×#127C211yesmaximal, characteristic
2#11529C212yesmaximal
3#8612C212yesmaximal, complemented
4#7249C212yesmaximal, complemented
5C2×C8×D4—22no—
6C2×C8×D4C411yescharacteristic
7C2×C8×D4C2211yescharacteristic
8C4×#127C2211yescharacteristic
9#1954—22no—
10#1954C411yescharacteristic
11#1954C2211yescharacteristic
12#1842C411yescharacteristic
13#1810—22no—
14C42×C8C411yescharacteristic
15#1465C2214yes—
16#1325—22no—
17#1318—22no—
18#1293—22no—
19C23×C8—21no—
20C23×C8D412yes—
21C2×#127Q811yescharacteristic
22C2×C4×D4—22no—
23C2×C4×D4C2×C411yescharacteristic
24C2×C4×D4C2311yescharacteristic
25C8×D4—41no—
26C8×D4—41no—
27C8×D4—41no—
28C8×D4—41no—
29C8×D4—48no—
30C2×#108—42no—
31C2×C4⋊C8—22no—
32C2×C4⋊C8C2×C411yescharacteristic
33C2×C4⋊C8C2×C411yescharacteristic
34#407—24no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—21no—
38C2×C22⋊C8D412yes—
39C2×C4×C8—21no—
40C2×C4×C8—21no—
41C2×C4×C8C2×C411yescharacteristic
42C2×C4×C8C2×C411yescharacteristic
43#390—22no—
44#390D411yescharacteristic
45#390Q811yescharacteristic
46#384D411yescharacteristic
47C43C2×C411yescharacteristic
48#341—42no—
49#324—24no—
50#323—24no—
51C22×D4—22no—
52C22×D4C4⋊C411yescharacteristic
53C22×D4C2×Q811yescharacteristic
54C23×C4—21no—
55C23×C4C2×D412yes—
56C2×M16—44no—
57C22×C8—21no—
58C22×C8—22no—
59C22×C8—22no—
60C22×C8—21no—
61C22×C8—44no—
62C22×C8—44no—
63C22×C8—41no—
64C22×C8—41no—
65C22×C8C22⋊C411yescharacteristic
66C22×C8C22⋊C412yes—
67C22×C8C22⋊C411yescharacteristic
68#127—22no—
69#127—22no—
70C4×D4—41no—
71C4×D4—41no—
72C4×D4—44no—
73C4×D4—44no—
74C4×D4—41no—
75C4×D4—41no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4C4⋊C411yescharacteristic
78C2×C4⋊C4C2×D411yescharacteristic
79C2×C22⋊C4—22no—
80C2×C22⋊C4—21no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4#4112yes—
83C2×C42—21no—
84C2×C42—21no—
85C2×C42C4⋊C411yescharacteristic
86C2×C42C4⋊C411yescharacteristic
87C2×C42C22×C411yescharacteristic
88#108—84no—
89C4⋊C8—41no—
90C4⋊C8—41no—
91C4⋊C8—44no—
92C22⋊C8—42no—
93C22⋊C8—44no—
94C22⋊C8—44no—
95C22⋊C8—42no—
96C22⋊C8—44no—
97C22⋊C8—44no—
98C4×C8—24no—
99C4×C8—21no—
100C4×C8—21no—
101C4×C8—41no—
102C4×C8—42no—
103C24—21no—
104C24C22⋊Q812yes—
105C2×D4—28no—
106C2×D4—21no—
107C2×D4—21no—
108C2×D4—21no—
109C2×D4—41no—
110C2×D4—41no—
111C2×D4—41no—
112C2×D4—41no—
113C2×D4—44no—
114C2×D4—44no—
115C2×D4#10812yescomplemented
116C22×C4—21no—
117C22×C4—21no—
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×C4C2×C22⋊C411yescharacteristic
128C22×C4C2×C4⋊C411yescharacteristic
129C22×C4#11711yescharacteristic
130C22×C4#12112yes—
131M16—84no—
132M16—84no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—21no—
136C2×C8—22no—
137C2×C8—41no—
138C2×C8—41no—
139C2×C8—44no—
140C2×C8—41no—
141C2×C8—42no—
142C2×C8—42no—
143C2×C8—41no—
144C2×C8—82no—
145C2×C8—82no—
146C2×C8—82no—
147C2×C8—84no—
148C2×C8#9911yescharacteristic
149C2×C8#9911yescharacteristic
150C4⋊C4—41no—
151C4⋊C4—41no—
152C4⋊C4—44no—
153C22⋊C4—42no—
154C22⋊C4—44no—
155C22⋊C4—44no—
156C22⋊C4—42no—
157C42—22no—
158C42—22no—
159C42—21no—
160C42—21no—
161C42—21no—
162C42—41no—
163C42—42no—
164C42#10812yescomplemented
165C23—22no—
166C23—22no—
167C23—22no—
168C23—22no—
169C23—42no—
170C23—42no—
171C23—42no—
172C23—42no—
173C23—41no—
174C23—41no—
175C23#38011yescharacteristic
176D4—42no—
177D4—416no—
178D4—42no—
179D4—42no—
180D4—42no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4C2×#9911yescharacteristic
204C2×C4#41011yescharacteristic
205C2×C4C2×#10811yescharacteristic
206C2×C4C2×#10811yescharacteristic
207C8—21no—
208C8—42no—
209C8—42no—
210C8—41no—
211C8—81no—
212C8—164no—
213C8#35312yes—
214C22—21no—
215C22—21no—
216C22—44no—
217C22—44no—
218C22—42no—
219C22—42no—
220C22—42no—
221C22—42no—
222C22—82no—
223C22—82no—
224C22—81no—
225C22—81no—
226C22#179911yescharacteristic
227C22#181011yescharacteristic
228C22#181311yescharacteristic
229C4—24no—
230C4—21no—
231C4—21no—
232C4—21no—
233C4—42no—
234C4—41no—
235C4—42no—
236C4—81no—
237C4C2×#35311yescharacteristic
238C4#212611yescharacteristic
239C2—21no—
240C2—21no—
241C2—82no—
242C2—82no—
243C2#861211yescharacteristic
244C2#1279711yescharacteristic
245C2#1280911yescharacteristic
subgroupG/H
center, socleC22#1799
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×D4C23
prime radicalC2×#127Q8
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ 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. #8612C2semidirect
4. #7249C2semidirect
115. C2×D4#108semidirect
164. C42#108semidirect

Maximal subgroups

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

subgroupindex
C8×#1272
#115292
#86122
#72492