Finite Group Databasebeta

← #120658

Subgroups — #120658

Order 512.

504 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 138 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#21959C211yesmaximal, characteristic
2#10833C211yesmaximal, characteristic
3#8044C212yesmaximal
4#8022C212yesmaximal
5C2×#1290C211yesmaximal, characteristic
6C2×#432C2212yes—
7C22×C4⋊C8C2211yescharacteristic
8C22×C4⋊C8C2211yescharacteristic
9#2285C2212yes—
10C2×#376C2211yescharacteristic
11#1948—22no—
12#1948—22no—
13#1807—22no—
14C2×#336—22no—
15#1296—22no—
16#1296—22no—
17#1294—24no—
18#1290C414yescomplemented
19C23×C8—21no—
20C23×C8D412yes—
21C2×#117—22no—
22C22×C42—21no—
23C22×C42C2311yescharacteristic
24#432—22no—
25#432—22no—
26#432—22no—
27#432C2×C414yes—
28C2×#106—22no—
29C2×#106—22no—
30C2×C4⋊C8—22no—
31C2×C4⋊C8—22no—
32C2×C4⋊C8—22no—
33C2×C4⋊C8—22no—
34C2×C4⋊C8—22no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8C2×C412yes—
37C2×C4⋊C8Q812yes—
38C2×C4⋊C8Q812yes—
39C2×C22⋊C8—22no—
40C2×#97—22no—
41#384D412yes—
42#384D412yes—
43#382—22no—
44#380—22no—
45#380—22no—
46C4×C4⋊C4—22no—
47C4×C4⋊C4—22no—
48#376—22no—
49#376—22no—
50C2×#95—21no—
51C2×#95D412yes—
52#340—24no—
53#336—44no—
54C23×C4—22no—
55C23×C4C2×D411yescharacteristic
56C23×C4C2×D411yescharacteristic
57C23×C4#4111yescharacteristic
58C22×C8—22no—
59C22×C8—22no—
60C22×C8—22no—
61C22×C8—42no—
62C22×C8—42no—
63C22×C8—42no—
64C22×C8—42no—
65C22×C8—41no—
66C22×C8—42no—
67C22×C8—41no—
68C22×C8C4⋊C414yes—
69#117—84no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C22⋊C4—42no—
77C2×C42—22no—
78C2×C42—22no—
79C2×C42—41no—
80C2×C42—41no—
81C2×C42—41no—
82C2×C42—41no—
83C2×C42—42no—
84C2×C42C22×C411yescharacteristic
85C2×C42C2×D412yes—
86C2×C42C2×Q811yescharacteristic
87C2×C42C2×Q811yescharacteristic
88C2×C42#4112yes—
89C2×C42#4112yes—
90C2×C42#4112yes—
91C2×C42#4111yescharacteristic
92#106—82no—
93#106—82no—
94C4⋊C8—24no—
95C4⋊C8—24no—
96C4⋊C8—24no—
97C4⋊C8—24no—
98C4⋊C8—42no—
99C4⋊C8—42no—
100C22⋊C8—24no—
101C22⋊C8—42no—
102#97—24no—
103#97—42no—
104#95—24no—
105#95—42no—
106#95—42no—
107#95—42no—
108C24#12111yescharacteristic
109C22×C4—24no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—24no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4—44no—
118C22×C4—42no—
119C22×C4—44no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4C2×C4⋊C411yescharacteristic
123C22×C4C4×D411yescharacteristic
124C22×C4C4×Q811yescharacteristic
125C22×C4#12011yescharacteristic
126C22×C4#12112yes—
127C22×C4C22⋊Q811yescharacteristic
128C22×C4#12311yescharacteristic
129C22×C4#12312yes—
130C22×C4#12312yes—
131C22×C4#12411yescharacteristic
132C22×C4#12411yescharacteristic
133C22×C4#12511yescharacteristic
134C22×C4#12612yes—
135C22×C4C4⋊Q811yescharacteristic
136C2×C8—24no—
137C2×C8—42no—
138C2×C8—42no—
139C2×C8—44no—
140C2×C8—44no—
141C2×C8—44no—
142C2×C8—42no—
143C2×C8—82no—
144C2×C8—82no—
145C2×C8—82no—
146C2×C8—82no—
147C2×C8—81no—
148C2×C8—81no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—82no—
154C4⋊C4—82no—
155C4⋊C4—82no—
156C4⋊C4—82no—
157C22⋊C4—84no—
158C42—42no—
159C42—42no—
160C42—42no—
161C42—42no—
162C42—84no—
163C42#11712yes—
164C42C4×D412yes—
165C42C22⋊Q812yes—
166C42C22⋊Q812yes—
167C42#12412yes—
168C42#12512yes—
169C42#12512yes—
170C42#12612yes—
171C23—22no—
172C23—21no—
173C23—21no—
174C23#38911yescharacteristic
175C23#39411yescharacteristic
176C23#39711yescharacteristic
177C23#47114yes—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—24no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—44no—
199C2×C4—48no—
200C2×C4—84no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4#38211yescharacteristic
210C2×C4#38511yescharacteristic
211C2×C4#38712yes—
212C2×C4#38812yes—
213C2×C4#39311yescharacteristic
214C2×C4#39812yes—
215C2×C4#40012yes—
216C2×C4#40011yescharacteristic
217C2×C4#45311yescharacteristic
218C2×C4#45411yescharacteristic
219C2×C4#45511yescharacteristic
220C2×C4#45611yescharacteristic
221C8—44no—
222C8—84no—
223C8—82no—
224C22—24no—
225C22—24no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—21no—
230C22#144211yescharacteristic
231C22#195112yes—
232C22#203911yescharacteristic
233C22#204011yescharacteristic
234C22#204612yes—
235C22#205312yes—
236C22#207912yes—
237C4—22no—
238C4—22no—
239C4—22no—
240C4—22no—
241C4—21no—
242C4—21no—
243C4—84no—
244C4—162no—
245C4—162no—
246C4#185611yescharacteristic
247C4#185711yescharacteristic
248C4#202711yescharacteristic
249C4#202811yescharacteristic
250C2—22no—
251C2—22no—
252C2#1051412yes—
253C2#1051512yes—
254C2#1127111yescharacteristic
255C2#1128511yescharacteristic
256C2#1129611yescharacteristic
subgroupG/H
center, socleC23#394
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
18. #1290C4semidirect

Maximal subgroups

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

subgroupindex
#219592
#108332
#80442
#80222
C2×#12902