Finite Group Databasebeta

← #120653

Subgroups — #120653

Order 512.

504 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 142 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#21955C211yesmaximal, characteristic
2#10835C211yesmaximal, characteristic
3C2×#1302C212yesmaximal
4#7908C211yesmaximal, characteristic
5#7879C212yesmaximal
6C2×#432C2212yes—
7C2×#431C2212yes—
8C22×C4⋊C8C2211yescharacteristic
9C22×#97C2211yescharacteristic
10C2×C4×C4⋊C4C2211yescharacteristic
11#1940—22no—
12#1940—22no—
13#1797—22no—
14C2×#336—22no—
15#1302—24no—
16#1280C412yes—
17#1280C412yes—
18#1277—22no—
19#1277—22no—
20C23×C8—21no—
21C23×C8—21no—
22C22×C4⋊C4Q812yes—
23C22×C42D412yes—
24C22×C42C2311yescharacteristic
25#432—22no—
26#432—22no—
27#432—22no—
28#432—22no—
29#431—22no—
30#431—22no—
31#431—22no—
32#431—22no—
33C2×C4⋊C8—22no—
34C2×C4⋊C8—22no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8—22no—
37C2×C4⋊C8—22no—
38C2×C4⋊C8—22no—
39C2×C4⋊C8D412yes—
40C2×C4⋊C8D412yes—
41C2×C4⋊C8Q812yes—
42C2×C22⋊C8—22no—
43C2×C22⋊C8C2×C414yes—
44C2×#97—22no—
45C2×#97—22no—
46C2×#97—22no—
47C2×#97—21no—
48C2×#97D412yes—
49C2×C4×C8—22no—
50C4×C4⋊C4—22no—
51C4×C4⋊C4—22no—
52C2×#95C2×C411yescharacteristic
53C2×#95C2×C411yescharacteristic
54#336—24no—
55#336—44no—
56C23×C4C4⋊C412yes—
57C23×C4C4⋊C412yes—
58C23×C4C22×C411yescharacteristic
59C23×C4C2×D411yescharacteristic
60C23×C4C2×Q811yescharacteristic
61C22×C8—22no—
62C22×C8—22no—
63C22×C8—22no—
64C22×C8—24no—
65C22×C8—22no—
66C22×C8—22no—
67C22×C8—22no—
68C22×C8—41no—
69C22×C8—41no—
70C22×C8—41no—
71C22×C8—41no—
72C22×C8—42no—
73C22×C8—42no—
74C22×C8—41no—
75C22×C8—41no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4—42no—
79C2×C42—22no—
80C2×C42—22no—
81C2×C42—42no—
82C2×C42—42no—
83C2×C42C2×D411yescharacteristic
84C2×C42C2×D412yes—
85C2×C42C2×Q811yescharacteristic
86C2×C42#4112yes—
87C2×C42#4112yes—
88C2×C42#4111yescharacteristic
89C2×C42#4112yes—
90C2×C42#4111yescharacteristic
91C4⋊C8—22no—
92C4⋊C8—22no—
93C4⋊C8—24no—
94C4⋊C8—22no—
95C4⋊C8—22no—
96C4⋊C8—42no—
97C4⋊C8—42no—
98C4⋊C8—42no—
99C4⋊C8—82no—
100C4⋊C8—82no—
101C22⋊C8—24no—
102C22⋊C8—24no—
103C22⋊C8—42no—
104C22⋊C8—42no—
105#97—24no—
106#97—42no—
107#97—42no—
108#97—42no—
109C4×C8—42no—
110C4×C8—42no—
111#95—22no—
112#95—22no—
113C24C2×C4⋊C411yescharacteristic
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4#10814yes—
131C22×C4#11712yes—
132C22×C4C4×D412yes—
133C22×C4C4×D411yescharacteristic
134C22×C4C4×Q811yescharacteristic
135C22×C4#12011yescharacteristic
136C22×C4#12111yescharacteristic
137C22×C4C22⋊Q811yescharacteristic
138C22×C4C22⋊Q812yes—
139C22×C4#12311yescharacteristic
140C22×C4#12411yescharacteristic
141C22×C4#12411yescharacteristic
142C22×C4#12512yes—
143C22×C4#12511yescharacteristic
144C22×C4C4⋊Q811yescharacteristic
145C2×C8—24no—
146C2×C8—42no—
147C2×C8—42no—
148C2×C8—42no—
149C2×C8—42no—
150C2×C8—42no—
151C2×C8—42no—
152C2×C8—48no—
153C2×C8—44no—
154C2×C8—44no—
155C2×C8—42no—
156C2×C8—42no—
157C2×C8—42no—
158C2×C8—42no—
159C2×C8—42no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8—82no—
163C2×C8—81no—
164C2×C8—81no—
165C4⋊C4—48no—
166C42—42no—
167C42—42no—
168C42—82no—
169C42#12112yes—
170C42C22⋊Q812yes—
171C42#12312yes—
172C42#12312yes—
173C42#12412yes—
174C42#12512yes—
175C42#12612yes—
176C42#12612yes—
177C23—21no—
178C23—21no—
179C23—22no—
180C23#38211yescharacteristic
181C23#38511yescharacteristic
182C23#38911yescharacteristic
183C23C2×#10812yes—
184C23#43012yes—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—24no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—24no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—44no—
208C2×C4—44no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4#38712yes—
215C2×C4#38812yes—
216C2×C4#39311yescharacteristic
217C2×C4#39411yescharacteristic
218C2×C4#39711yescharacteristic
219C2×C4#39812yes—
220C2×C4#40012yes—
221C2×C4#40011yescharacteristic
222C2×C4#45311yescharacteristic
223C2×C4#45411yescharacteristic
224C2×C4#45511yescharacteristic
225C2×C4#45611yescharacteristic
226C8—44no—
227C8—84no—
228C8—82no—
229C8—82no—
230C8—82no—
231C22—21no—
232C22—21no—
233C22—21no—
234C22—21no—
235C22—22no—
236C22—22no—
237C22—22no—
238C22—22no—
239C22#144211yescharacteristic
240C22#183711yescharacteristic
241C22#183811yescharacteristic
242C22#185512yes—
243C22#185611yescharacteristic
244C22#185711yescharacteristic
245C22#186812yes—
246C22#195012yes—
247C4—22no—
248C4—22no—
249C4—22no—
250C4—22no—
251C4—21no—
252C4—21no—
253C4—44no—
254C4—82no—
255C4#202711yescharacteristic
256C4#202811yescharacteristic
257C4#203911yescharacteristic
258C4#204011yescharacteristic
259C2—21no—
260C2—21no—
261C2—22no—
262C2#886412yes—
263C2#886512yes—
264C2#1126911yescharacteristic
265C2#1128811yescharacteristic
266C2#1129611yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
socleC23#385
prime radicalC24C2×C4⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#219552
#108352
C2×#13022
#79082
#78792