Finite Group Databasebeta

← #134758

Subgroups — #134758

Order 512.

387 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 387 conjugacy classes of subgroups — too many to draw. Only its 88 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#22208C211yesmaximal, characteristic
2#12798C212yesmaximal
3#8609C212yesmaximal
4#7237C212yesmaximal
5C2×C8×Q8C411yescharacteristic, complemented
6C2×C8×D4C2211yescharacteristic
7C4×#124C2211yescharacteristic
8C2×#354—22no—
9#1984C411yescharacteristic, complemented
10#1954C2211yescharacteristic
11#1923—22no—
12#1850—22no—
13#1842C411yescharacteristic, complemented
14#1808—22no—
15#1799C2214yes—
16#1744C411yescharacteristic, complemented
17#1393—22no—
18#1289—22no—
19C23×C8D412yes—
20C2×#124D411yescharacteristic
21C2×C4×Q8C2×C411yescharacteristic
22C2×C4×D4C2311yescharacteristic
23C8×Q8—42no—
24C8×D4—42no—
25C2×#107—42no—
26C2×C4⋊C8—21no—
27C2×C4⋊C8—22no—
28C2×C4⋊C8C2×C411yescharacteristic
29C2×C4⋊C8C2×C411yescharacteristic
30C2×#99—24no—
31C2×C22⋊C8—22no—
32C2×C22⋊C8D412yes—
33C2×C4×C8—21no—
34C2×C4×C8C2×C411yescharacteristic
35C2×C4×C8C2×C411yescharacteristic
36#391Q811yescharacteristic
37#390D411yescharacteristic
38#388—22no—
39#382Q811yescharacteristic
40#380—24no—
41C4×C22⋊C4—22no—
42C43C2×C411yescharacteristic
43#354—44no—
44#342—24no—
45#340—42no—
46#336—21no—
47#336—22no—
48#336—21no—
49C22×Q8C4⋊C411yescharacteristic
50C22×D4C2×D411yescharacteristic
51C23×C4C2×D412yes—
52C22×C8—22no—
53C22×C8—22no—
54C22×C8—21no—
55C22×C8—21no—
56C22×C8—82no—
57C22×C8C22⋊C412yes—
58C22×C8C22⋊C411yescharacteristic
59C22×C8C22⋊C411yescharacteristic
60#124—22no—
61#124D812yes—
62#124Q1612yes—
63C4×Q8—41no—
64C4×Q8—41no—
65C4×D4—41no—
66C4×D4—41no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—44no—
69C2×C4⋊C4C4⋊C411yescharacteristic
70C2×C4⋊C4C2×Q811yescharacteristic
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—44no—
74C2×C22⋊C4#4112yes—
75C2×C42—21no—
76C2×C42—21no—
77C2×C42—21no—
78C2×C42C4⋊C411yescharacteristic
79C2×C42C4⋊C411yescharacteristic
80C2×C42C22×C411yescharacteristic
81#107—84no—
82C4⋊C8—41no—
83C4⋊C8—41no—
84C4⋊C8—42no—
85C4⋊C8—82no—
86#99—44no—
87#99—44no—
88C22⋊C8—42no—
89C22⋊C8—82no—
90C4×C8—22no—
91C4×C8—41no—
92C4×C8—42no—
93#95—22no—
94#95—22no—
95#95—44no—
96C24#12112yes—
97C2×Q8—21no—
98C2×Q8—41no—
99C2×Q8—41no—
100C2×Q8#10712yescomplemented
101C2×D4—41no—
102C2×D4—41no—
103C2×D4C2×D811yescharacteristic
104C2×D4C2×Q1611yescharacteristic
105C2×D4#13512yes—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4—84no—
115C22×C4C2×C22⋊C411yescharacteristic
116C22×C4C2×C4⋊C411yescharacteristic
117C22×C4#11711yescharacteristic
118C22×C4C22⋊Q812yes—
119C2×C8—21no—
120C2×C8—22no—
121C2×C8—41no—
122C2×C8—44no—
123C2×C8—42no—
124C2×C8—41no—
125C2×C8—41no—
126C2×C8—42no—
127C2×C8—41no—
128C2×C8—41no—
129C2×C8—41no—
130C2×C8—82no—
131C2×C8#9912yes—
132C4⋊C4—41no—
133C4⋊C4—41no—
134C4⋊C4—42no—
135C4⋊C4—84no—
136C4⋊C4—84no—
137C22⋊C4—24no—
138C22⋊C4—24no—
139C22⋊C4—42no—
140C22⋊C4—82no—
141C22⋊C4—84no—
142C22⋊C4—84no—
143C42—21no—
144C42—21no—
145C42—21no—
146C42—41no—
147C42—41no—
148C42—41no—
149C42—41no—
150C42—41no—
151C42—81no—
152C42#10712yescomplemented
153C23—22no—
154C23—22no—
155C23—42no—
156C23—42no—
157C23#38011yescharacteristic
158C23#46612yes—
159C23#46712yes—
160Q8—42no—
161Q8—42no—
162D4—41no—
163D4—41no—
164D4—42no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—82no—
187C2×C4—164no—
188C2×C4—164no—
189C2×C4—164no—
190C2×C4C2×#9911yescharacteristic
191C2×C4#41011yescharacteristic
192C2×C4C2×#10711yescharacteristic
193C2×C4#42711yescharacteristic
194C8—22no—
195C8—42no—
196C8—41no—
197C8—81no—
198C8—81no—
199C22—21no—
200C22—21no—
201C22—22no—
202C22—22no—
203C22—42no—
204C22—42no—
205C22—42no—
206C22—42no—
207C22—82no—
208C22#179911yescharacteristic
209C22#180811yescharacteristic
210C22#181211yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—42no—
215C4—42no—
216C4—41no—
217C4—42no—
218C4—81no—
219C4—81no—
220C4—324no—
221C4#212711yescharacteristic
222C4#212911yescharacteristic
223C2—21no—
224C2—21no—
225C2—42no—
226C2—42no—
227C2#860911yescharacteristic
228C2#1280011yescharacteristic
229C2#1280811yescharacteristic
subgroupG/H
center, socleC22#1799
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×D4C23
prime radicalC22×D4C2×D4
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
5. C2×C8×Q8C4semidirect
9. #1984C4semidirect
13. #1842C4semidirect
16. #1744C4semidirect
100. C2×Q8#107semidirect
152. C42#107semidirect

Maximal subgroups

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

subgroupindex
#222082
#127982
#86092
#72372