Finite Group Databasebeta

← #134815

Subgroups — #134815

Order 512.

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

normal subgroups only

This group has 342 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#11530C212yesmaximal
3#8623C212yesmaximal, complemented
4#7248C212yesmaximal, complemented
5C2×C8×Q8C2211yescharacteristic
6C2×C8×D4C411yescharacteristic
7C4×#124C2211yescharacteristic
8#1984C2211yescharacteristic
9#1954C411yescharacteristic
10#1923—22no—
11#1831C411yescharacteristic
12C8×C22⋊C4—22no—
13C42×C8C411yescharacteristic
14#1568—22no—
15#1467C2214yes—
16#1324—22no—
17#1318—22no—
18#1290—22no—
19C23×C8—21no—
20C2×#124Q811yescharacteristic
21C2×C4×Q8C2311yescharacteristic
22C2×C4×D4C2×C411yescharacteristic
23C8×Q8—41no—
24C8×Q8—41no—
25C8×D4—41no—
26C8×D4—41no—
27#432—24no—
28C2×C4⋊C8C2×C411yescharacteristic
29C2×C4⋊C8C2×C411yescharacteristic
30C2×C4⋊C8D412yes—
31C2×C22⋊C8—21no—
32C2×C22⋊C8—22no—
33C2×C4×C8—21no—
34C2×C4×C8—21no—
35C2×C4×C8C2×C411yescharacteristic
36C2×C4×C8C2×C411yescharacteristic
37C2×C4×C8D412yes—
38#391Q811yescharacteristic
39#390D411yescharacteristic
40#388—22no—
41#382D411yescharacteristic
42C4×C22⋊C4—22no—
43C43C2×C411yescharacteristic
44#336—22no—
45#336—22no—
46#335—42no—
47#334—42no—
48#324—24no—
49#324—24no—
50C22×Q8C2×Q811yescharacteristic
51C22×D4C4⋊C411yescharacteristic
52C23×C4—21no—
53C22×C8—21no—
54C22×C8—21no—
55C22×C8—44no—
56C22×C8—41no—
57C22×C8—41no—
58C22×C8C22⋊C411yescharacteristic
59C22×C8C22⋊C412yes—
60C22×C8C22⋊C411yescharacteristic
61#124—22no—
62#124—22no—
63C4×Q8—41no—
64C4×Q8—41no—
65C4×D4—41no—
66C4×D4—41no—
67C2×C4⋊C4C4⋊C411yescharacteristic
68C2×C4⋊C4C2×D411yescharacteristic
69C2×C4⋊C4#4112yes—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—22no—
73C2×C42—21no—
74C2×C42—21no—
75C2×C42C4⋊C411yescharacteristic
76C2×C42C4⋊C411yescharacteristic
77C2×C42C22×C411yescharacteristic
78C2×C42C2×D412yes—
79C4⋊C8—41no—
80C4⋊C8—42no—
81C4⋊C8—41no—
82C4⋊C8—44no—
83C22⋊C8—44no—
84C22⋊C8—42no—
85C22⋊C8—44no—
86#97—44no—
87C4×C8—22no—
88C4×C8—22no—
89C4×C8—21no—
90C4×C8—21no—
91C4×C8—41no—
92C4×C8—42no—
93C4×C8—42no—
94#95—22no—
95#95—22no—
96C24—21no—
97C2×Q8—21no—
98C2×Q8—21no—
99C2×Q8—41no—
100C2×Q8—41no—
101C2×D4—21no—
102C2×D4—41no—
103C2×D4—41no—
104C2×D4#10812yes—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—41no—
111C22×C4—41no—
112C22×C4C2×C22⋊C411yescharacteristic
113C22×C4C2×C4⋊C411yescharacteristic
114C22×C4#11711yescharacteristic
115C22×C4#12112yes—
116C22×C4C22⋊Q812yes—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—21no—
120C2×C8—22no—
121C2×C8—41no—
122C2×C8—41no—
123C2×C8—41no—
124C2×C8—42no—
125C2×C8—42no—
126C2×C8—41no—
127C2×C8—82no—
128C2×C8—82no—
129C2×C8—84no—
130C2×C8#9911yescharacteristic
131C2×C8#9911yescharacteristic
132C4⋊C4—41no—
133C4⋊C4—42no—
134C4⋊C4—41no—
135C22⋊C4—28no—
136C22⋊C4—44no—
137C22⋊C4—42no—
138C42—22no—
139C42—21no—
140C42—21no—
141C42—22no—
142C42—21no—
143C42—41no—
144C42—41no—
145C42—41no—
146C42—42no—
147C42#10812yes—
148C23—22no—
149C23—42no—
150C23—42no—
151C23—41no—
152C23—41no—
153C23#38011yescharacteristic
154Q8—42no—
155Q8—42no—
156D4—42no—
157D4—42no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—22no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—82no—
178C2×C4—81no—
179C2×C4—81no—
180C2×C4C2×#9911yescharacteristic
181C2×C4#41011yescharacteristic
182C2×C4C2×#10811yescharacteristic
183C2×C4C2×#10811yescharacteristic
184C8—21no—
185C8—42no—
186C8—41no—
187C8—42no—
188C8—81no—
189C8—164no—
190C8#35512yes—
191C22—21no—
192C22—21no—
193C22—44no—
194C22—42no—
195C22—82no—
196C22—81no—
197C22—81no—
198C22#179911yescharacteristic
199C22#181011yescharacteristic
200C22#181311yescharacteristic
201C4—21no—
202C4—21no—
203C4—21no—
204C4—42no—
205C4—42no—
206C4—42no—
207C4—41no—
208C4—42no—
209C4—81no—
210C4C2×#35511yescharacteristic
211C4#213211yescharacteristic
212C2—21no—
213C2—21no—
214C2—82no—
215C2#861211yescharacteristic
216C2#1280111yescharacteristic
217C2#1280711yescharacteristic
subgroupG/H
center, socleC22#1799
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×Q8C23
prime radicalC22×D4C4⋊C4
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8623C2semidirect
4. #7248C2semidirect

Maximal subgroups

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

subgroupindex
C8×#1242
#115302
#86232
#72482