Finite Group Databasebeta

← #134812

Subgroups — #134812

Order 512.

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

normal subgroups only

This group has 360 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×#124C211yesmaximal, characteristic
2#11531C212yesmaximal
3#8612C212yesmaximal
4#7249C212yesmaximal
5C2×C8×Q8C411yescharacteristic
6C2×C8×D4C2211yescharacteristic
7C4×#124C2211yescharacteristic
8#1984C411yescharacteristic
9#1954C2211yescharacteristic
10#1923—22no—
11#1831C411yescharacteristic
12#1810—22no—
13C8×C22⋊C4—22no—
14C42×C8C411yescharacteristic
15#1465C2214yes—
16#1324—22no—
17#1319—22no—
18#1293—22no—
19C23×C8D412yes—
20C2×#124Q811yescharacteristic
21C2×C4×Q8C2×C411yescharacteristic
22C2×C4×D4C2311yescharacteristic
23C8×Q8—41no—
24C8×Q8—41no—
25C8×D4—41no—
26C8×D4—41no—
27C2×#108—42no—
28C2×C4⋊C8—21no—
29C2×C4⋊C8C2×C411yescharacteristic
30C2×C4⋊C8C2×C411yescharacteristic
31#407—24no—
32C2×C22⋊C8—22no—
33C2×C22⋊C8D412yes—
34C2×C4×C8—21no—
35C2×C4×C8—21no—
36C2×C4×C8—21no—
37C2×C4×C8C2×C411yescharacteristic
38C2×C4×C8C2×C411yescharacteristic
39#391D411yescharacteristic
40#390Q811yescharacteristic
41#388—22no—
42#382D411yescharacteristic
43C4×C22⋊C4—22no—
44C43C2×C411yescharacteristic
45#341—42no—
46#336—22no—
47#336—22no—
48#324—24no—
49#323—24no—
50C22×Q8C4⋊C411yescharacteristic
51C22×D4C2×Q811yescharacteristic
52C23×C4C2×D412yes—
53C2×M16—44no—
54C22×C8—21no—
55C22×C8—22no—
56C22×C8—22no—
57C22×C8—21no—
58C22×C8—44no—
59C22×C8C22⋊C411yescharacteristic
60C22×C8C22⋊C412yes—
61C22×C8C22⋊C411yescharacteristic
62#124—22no—
63#124—22no—
64C4×Q8—41no—
65C4×Q8—41no—
66C4×D4—41no—
67C4×D4—41no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4C4⋊C411yescharacteristic
70C2×C4⋊C4C2×D411yescharacteristic
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4#4112yes—
74C2×C42—21no—
75C2×C42—21no—
76C2×C42—21no—
77C2×C42C4⋊C411yescharacteristic
78C2×C42C4⋊C411yescharacteristic
79C2×C42C22×C411yescharacteristic
80#108—84no—
81C4⋊C8—41no—
82C4⋊C8—41no—
83C4⋊C8—42no—
84C22⋊C8—42no—
85C22⋊C8—44no—
86C22⋊C8—44no—
87C22⋊C8—44no—
88C4×C8—21no—
89C4×C8—21no—
90C4×C8—41no—
91C4×C8—42no—
92C4×C8—42no—
93C4×C8—41no—
94C4×C8—41no—
95#95—22no—
96#95—22no—
97C24C22⋊Q812yes—
98C2×Q8—21no—
99C2×Q8—41no—
100C2×Q8—41no—
101C2×Q8#10812yescomplemented
102C2×D4—21no—
103C2×D4—21no—
104C2×D4—41no—
105C2×D4—41no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—21no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4C2×C22⋊C411yescharacteristic
115C22×C4C2×C4⋊C411yescharacteristic
116C22×C4#11711yescharacteristic
117C22×C4#12112yes—
118M16—84no—
119M16—84no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—21no—
123C2×C8—22no—
124C2×C8—41no—
125C2×C8—41no—
126C2×C8—44no—
127C2×C8—41no—
128C2×C8—42no—
129C2×C8—42no—
130C2×C8—41no—
131C2×C8—82no—
132C2×C8—84no—
133C2×C8#9911yescharacteristic
134C2×C8#9911yescharacteristic
135C4⋊C4—41no—
136C4⋊C4—41no—
137C4⋊C4—42no—
138C22⋊C4—28no—
139C22⋊C4—42no—
140C22⋊C4—44no—
141C42—21no—
142C42—21no—
143C42—21no—
144C42—41no—
145C42—41no—
146C42—41no—
147C42—42no—
148C42—41no—
149C42—41no—
150C42#10812yescomplemented
151C23—22no—
152C23—22no—
153C23—22no—
154C23—42no—
155C23—42no—
156C23#38011yescharacteristic
157Q8—42no—
158Q8—42no—
159D4—42no—
160D4—42no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—22no—
166C2×C4—21no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—82no—
183C2×C4C2×#9911yescharacteristic
184C2×C4#41011yescharacteristic
185C2×C4C2×#10811yescharacteristic
186C2×C4C2×#10811yescharacteristic
187C8—21no—
188C8—42no—
189C8—42no—
190C8—41no—
191C8—81no—
192C8—164no—
193C8#35412yes—
194C22—21no—
195C22—21no—
196C22—44no—
197C22—42no—
198C22—42no—
199C22—42no—
200C22—82no—
201C22#179911yescharacteristic
202C22#181011yescharacteristic
203C22#181311yescharacteristic
204C4—21no—
205C4—21no—
206C4—21no—
207C4—42no—
208C4—42no—
209C4—41no—
210C4—42no—
211C4—42no—
212C4—81no—
213C4C2×#35411yescharacteristic
214C4#212611yescharacteristic
215C2—21no—
216C2—21no—
217C2—82no—
218C2#861211yescharacteristic
219C2#1279811yescharacteristic
220C2#1280911yescharacteristic
subgroupG/H
center, socleC22#1799
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×D4C23
prime radicalC22×D4C2×Q8
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
101. C2×Q8#108semidirect
150. C42#108semidirect

Maximal subgroups

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

subgroupindex
C8×#1242
#115312
#86122
#72492