Finite Group Databasebeta

← #134813

Subgroups — #134813

Order 512.

393 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 393 conjugacy classes of subgroups — too many to draw. Only its 76 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#22260C211yesmaximal, characteristic
2#11529C212yesmaximal
3#8612C212yesmaximal, complemented
4#7249C212yesmaximal, complemented
5C2×C8×D4C411yescharacteristic
6C2×C8×D4C2211yescharacteristic
7C4×#127C2211yescharacteristic
8#1954C411yescharacteristic
9#1954C2211yescharacteristic
10#1869—22no—
11#1869—22no—
12#1831C411yescharacteristic
13#1810—22no—
14#1744C411yescharacteristic
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—42no—
26C8×D4—42no—
27C2×#108—42no—
28C2×C4⋊C8C2×C411yescharacteristic
29C2×C4⋊C8C2×C411yescharacteristic
30#407—24no—
31C2×C22⋊C8—21no—
32C2×C22⋊C8—22no—
33C2×C22⋊C8—22no—
34C2×C22⋊C8D412yes—
35C2×C4×C8C2×C411yescharacteristic
36C2×C4×C8C2×C411yescharacteristic
37#390—22no—
38#390D411yescharacteristic
39#390Q811yescharacteristic
40#384D411yescharacteristic
41C43C2×C411yescharacteristic
42#341—42no—
43#336—22no—
44#336—21no—
45#336—21no—
46#324—24no—
47#323—24no—
48C22×D4—22no—
49C22×D4C4⋊C411yescharacteristic
50C22×D4C2×Q811yescharacteristic
51C23×C4—21no—
52C23×C4C2×D412yes—
53C2×M16—44no—
54C22×C8—22no—
55C22×C8—22no—
56C22×C8—21no—
57C22×C8—21no—
58C22×C8—42no—
59C22×C8—82no—
60C22×C8—82no—
61C22×C8C22⋊C411yescharacteristic
62C22×C8C22⋊C412yes—
63C22×C8C22⋊C411yescharacteristic
64#127—22no—
65#127—22no—
66C4×D4—41no—
67C4×D4—41no—
68C4×D4—41no—
69C4×D4—41no—
70C4×D4—82no—
71C4×D4—82no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4C4⋊C411yescharacteristic
74C2×C4⋊C4C2×D411yescharacteristic
75C2×C22⋊C4—21no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4#4112yes—
79C2×C42—21no—
80C2×C42—21no—
81C2×C42C4⋊C411yescharacteristic
82C2×C42C4⋊C411yescharacteristic
83C2×C42C22×C411yescharacteristic
84#108—84no—
85C4⋊C8—41no—
86C4⋊C8—41no—
87C22⋊C8—42no—
88C22⋊C8—42no—
89C22⋊C8—44no—
90C22⋊C8—44no—
91C22⋊C8—82no—
92C22⋊C8—82no—
93C4×C8—22no—
94C4×C8—41no—
95C24—21no—
96C24C22⋊Q812yes—
97C2×D4—28no—
98C2×D4—21no—
99C2×D4—21no—
100C2×D4—21no—
101C2×D4—41no—
102C2×D4—41no—
103C2×D4—41no—
104C2×D4—41no—
105C2×D4—82no—
106C2×D4—82no—
107C2×D4#10812yescomplemented
108C22×C4—21no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4—41no—
118C22×C4—41no—
119C22×C4C2×C22⋊C411yescharacteristic
120C22×C4C2×C4⋊C411yescharacteristic
121C22×C4#11711yescharacteristic
122C22×C4#12112yes—
123M16—84no—
124M16—84no—
125C2×C8—21no—
126C2×C8—22no—
127C2×C8—41no—
128C2×C8—44no—
129C2×C8—42no—
130C2×C8—41no—
131C2×C8—41no—
132C2×C8—41no—
133C2×C8—42no—
134C2×C8—41no—
135C2×C8—41no—
136C2×C8—82no—
137C2×C8—84no—
138C2×C8—84no—
139C2×C8#9912yes—
140C4⋊C4—41no—
141C4⋊C4—41no—
142C4⋊C4—82no—
143C22⋊C4—42no—
144C22⋊C4—42no—
145C22⋊C4—82no—
146C22⋊C4—82no—
147C42—21no—
148C42—21no—
149C42—21no—
150C42—41no—
151C42—41no—
152C42—41no—
153C42—81no—
154C42#10812yescomplemented
155C23—22no—
156C23—22no—
157C23—22no—
158C23—22no—
159C23—42no—
160C23—42no—
161C23—42no—
162C23—42no—
163C23—41no—
164C23—41no—
165C23#38011yescharacteristic
166D4—42no—
167D4—42no—
168D4—42no—
169D4—42no—
170D4—88no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—41no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4C2×#9911yescharacteristic
194C2×C4#41011yescharacteristic
195C2×C4C2×#10811yescharacteristic
196C2×C4C2×#10811yescharacteristic
197C8—22no—
198C8—42no—
199C8—41no—
200C8—81no—
201C8—81no—
202C8—164no—
203C22—21no—
204C22—21no—
205C22—44no—
206C22—44no—
207C22—42no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—82no—
212C22—82no—
213C22—81no—
214C22—81no—
215C22#179911yescharacteristic
216C22#181011yescharacteristic
217C22#181311yescharacteristic
218C4—21no—
219C4—21no—
220C4—21no—
221C4—42no—
222C4—41no—
223C4—42no—
224C4—81no—
225C4—81no—
226C4#212711yescharacteristic
227C4#212811yescharacteristic
228C2—21no—
229C2—21no—
230C2—82no—
231C2—82no—
232C2#861211yescharacteristic
233C2#1279911yescharacteristic
234C2#1281011yescharacteristic
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
107. C2×D4#108semidirect
154. C42#108semidirect

Maximal subgroups

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

subgroupindex
#222602
#115292
#86122
#72492