Finite Group Databasebeta

← #120547

Subgroups — #120547

Order 512.

453 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 453 conjugacy classes of subgroups — too many to draw. Only its 126 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#21973C211yesmaximal, characteristic
2#21960C211yesmaximal, characteristic
3#8096C212yesmaximal
4#7988C211yesmaximal, characteristic
5#7944C212yesmaximal
6#2920C2212yes—
7C2×#431C2211yescharacteristic
8C22×C4⋊C8C2211yescharacteristic
9#2283C2212yes—
10#2283C2211yescharacteristic
11#1949—24no—
12#1939—22no—
13#1850—21no—
14#1850—21no—
15#1843C412yescomplemented
16#1832C412yescomplemented
17#1808—21no—
18#1807—21no—
19#1303—24no—
20#1290—22no—
21#1284—24no—
22C23×C8D411yescharacteristic
23C23×C8D411yescharacteristic
24C22×C42C2311yescharacteristic
25#432—22no—
26#432Q814yes—
27#431—22no—
28#431—22no—
29#431—22no—
30#431—22no—
31C2×#107—22no—
32C2×#106—22no—
33#423—22no—
34C2×C4⋊C8—22no—
35C2×C4⋊C8—22no—
36C2×C4⋊C8—22no—
37C2×C4⋊C8—22no—
38C2×C4⋊C8C2×C411yescharacteristic
39C2×C4⋊C8C2×C411yescharacteristic
40C2×C4⋊C8C2×C412yes—
41C2×C4⋊C8D412yes—
42#407—22no—
43C2×C22⋊C8—21no—
44C4×M16—22no—
45C2×C4×C8—21no—
46#383—21no—
47#383—22no—
48#382—22no—
49#382Q812yes—
50#381—22no—
51#381D412yes—
52#380—21no—
53#380—22no—
54C4×C4⋊C4—21no—
55C4×C4⋊C4D414yes—
56C4×C22⋊C4—21no—
57C4×C22⋊C4—22no—
58#376—22no—
59#376C2×C412yes—
60#340—22no—
61#340—22no—
62#336—22no—
63#336—22no—
64C23×C4C2×D411yescharacteristic
65C23×C4C2×D411yescharacteristic
66C23×C4#4111yescharacteristic
67C2×M16—42no—
68C22×C8—21no—
69C22×C8—21no—
70C22×C8—21no—
71C22×C8—21no—
72C22×C8—21no—
73C22×C8—21no—
74C22×C8—81no—
75C22×C8—81no—
76C22×C8C22⋊C412yes—
77C22×C8C22⋊C412yes—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—24no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—42no—
82C2×C42—22no—
83C2×C42—24no—
84C2×C42C22×C411yescharacteristic
85C2×C42C2×D412yes—
86C2×C42C2×D411yescharacteristic
87C2×C42C2×Q812yes—
88C2×C42C2×Q811yescharacteristic
89C2×C42#4112yes—
90C2×C42#4112yes—
91C2×C42#4111yescharacteristic
92#107—82no—
93#106—82no—
94C4⋊C8—24no—
95C4⋊C8—22no—
96C4⋊C8—22no—
97C4⋊C8—24no—
98C4⋊C8—24no—
99C4⋊C8—42no—
100C4⋊C8—82no—
101C4⋊C8—82no—
102C22⋊C8—24no—
103C22⋊C8—41no—
104C22⋊C8—42no—
105C22⋊C8—41no—
106#97—24no—
107C4×C8—42no—
108C4×C8—42no—
109#95—22no—
110#95—22no—
111#95—22no—
112#95—24no—
113#95—42no—
114#95—42no—
115#95—42no—
116#95C4⋊C414yes—
117C24#12111yescharacteristic
118C22×C4—24no—
119C22×C4—41no—
120C22×C4—41no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4—44no—
124C22×C4C2×C22⋊C411yescharacteristic
125C22×C4C2×C4⋊C411yescharacteristic
126C22×C4#11711yescharacteristic
127C22×C4#12011yescharacteristic
128C22×C4#12112yes—
129C22×C4#12111yescharacteristic
130C22×C4C22⋊Q811yescharacteristic
131C22×C4C22⋊Q812yes—
132C22×C4C22⋊Q811yescharacteristic
133C22×C4#12311yescharacteristic
134C22×C4#12312yes—
135C22×C4#12411yescharacteristic
136C22×C4#12512yes—
137C22×C4C4⋊Q811yescharacteristic
138M16—84no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—42no—
142C2×C8—42no—
143C2×C8—42no—
144C2×C8—42no—
145C2×C8—42no—
146C2×C8—42no—
147C2×C8—41no—
148C2×C8—41no—
149C2×C8—44no—
150C2×C8—81no—
151C2×C8—81no—
152C4⋊C4—44no—
153C4⋊C4—44no—
154C4⋊C4—44no—
155C22⋊C4—44no—
156C22⋊C4—84no—
157C42—42no—
158C42—42no—
159C42—84no—
160C42C4×D412yes—
161C42C4×Q812yes—
162C42#12112yes—
163C42C22⋊Q812yes—
164C42C22⋊Q812yes—
165C42#12512yes—
166C42#12612yes—
167C42C4⋊Q812yes—
168C23—22no—
169C23—22no—
170C23—41no—
171C23#38011yescharacteristic
172C23#39411yescharacteristic
173C23#39811yescharacteristic
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—44no—
192C2×C4—44no—
193C2×C4—44no—
194C2×C4—84no—
195C2×C4—84no—
196C2×C4—84no—
197C2×C4#38611yescharacteristic
198C2×C4#38712yes—
199C2×C4#38912yes—
200C2×C4#39111yescharacteristic
201C2×C4#39511yescharacteristic
202C2×C4#39612yes—
203C2×C4#39812yes—
204C2×C4#40011yescharacteristic
205C8—42no—
206C8—42no—
207C8—84no—
208C8—81no—
209C8—81no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—44no—
218C22#144111yescharacteristic
219C22#181412yes—
220C22#205312yes—
221C4—21no—
222C4—21no—
223C4—21no—
224C4—22no—
225C4—22no—
226C4—22no—
227C4—22no—
228C4—84no—
229C4—164no—
230C4#191111yescharacteristic
231C4#191211yescharacteristic
232C2—22no—
233C2—22no—
234C2—41no—
235C2#1122411yescharacteristic
236C2#1123611yescharacteristic
237C2#1124111yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #1843C4semidirect
16. #1832C4semidirect

Maximal subgroups

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

subgroupindex
#219732
#219602
#80962
#79882
#79442