Finite Group Databasebeta

← #120559

Subgroups — #120559

Order 512.

590 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 150 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#21351C211yesmaximal, characteristic
2C2×C4×#106C211yesmaximal, characteristic
3#8021C212yesmaximal
4C2×#1285C212yesmaximal
5#7875C211yesmaximal, characteristic
6C2×#432C2212yes—
7C2×#432C2211yescharacteristic
8C22×#106—21no—
9C22×C4×C8C2211yescharacteristic
10#2283C2211yescharacteristic
11C2×C4×C4⋊C4C2212yes—
12#1939C412yescomplemented
13#1939C412yescomplemented
14#1850—21no—
15#1850—21no—
16C4×#106—24no—
17#1768—22no—
18C2×#340—21no—
19#1294—22no—
20#1294—22no—
21#1285—24no—
22#1277—22no—
23C23×C8D411yescharacteristic
24C23×C8D411yescharacteristic
25C22×C4⋊C4—22no—
26C22×C42—22no—
27C22×C42C2311yescharacteristic
28#432—22no—
29#432—22no—
30#432—22no—
31#432—22no—
32#432—22no—
33#432Q814yes—
34C2×#106—28no—
35C2×#106—24no—
36C2×C4⋊C8—22no—
37C2×C4⋊C8—22no—
38C2×C4⋊C8—22no—
39C2×C4⋊C8C2×C411yescharacteristic
40C2×C4⋊C8C2×C411yescharacteristic
41C2×C22⋊C8—22no—
42C2×C22⋊C8—21no—
43C2×#97—22no—
44C2×#97—21no—
45C2×C4×C8—22no—
46C2×C4×C8—22no—
47C2×C4×C8C2×C412yes—
48C2×C4×C8D412yes—
49#383—21no—
50#382Q812yes—
51#381—21no—
52#380D411yescharacteristic
53#380D411yescharacteristic
54C4×C4⋊C4—22no—
55C4×C4⋊C4—22no—
56C4×C4⋊C4—22no—
57C4×C4⋊C4C2×C412yes—
58C4×C4⋊C4D414yes—
59C4×C22⋊C4—21no—
60#376—21no—
61C2×#95—22no—
62#340—24no—
63#336—22no—
64#336—22no—
65C23×C4—42no—
66C23×C4C2×D411yescharacteristic
67C23×C4C2×D411yescharacteristic
68C23×C4#4111yescharacteristic
69C22×C8—21no—
70C22×C8—21no—
71C22×C8—24no—
72C22×C8—21no—
73C22×C8—21no—
74C22×C8—21no—
75C22×C8—21no—
76C22×C8—42no—
77C22×C8—42no—
78C22×C8C22⋊C412yes—
79C22×C8C22⋊C412yes—
80C2×C4⋊C4—28no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—44no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4C4⋊C412yes—
85C2×C4⋊C4C4⋊C412yes—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C42—22no—
89C2×C42—24no—
90C2×C42—42no—
91C2×C42—82no—
92C2×C42—82no—
93C2×C42C22×C411yescharacteristic
94C2×C42C2×D412yes—
95C2×C42C2×D411yescharacteristic
96C2×C42C2×Q812yes—
97C2×C42C2×Q811yescharacteristic
98C2×C42#4112yes—
99C2×C42#4112yes—
100C2×C42#4111yescharacteristic
101#106—216no—
102C4⋊C8—22no—
103C4⋊C8—22no—
104C4⋊C8—24no—
105C4⋊C8—44no—
106C4⋊C8—44no—
107C4⋊C8—42no—
108C22⋊C8—24no—
109C22⋊C8—41no—
110C22⋊C8—42no—
111C22⋊C8—41no—
112#97—24no—
113#97—42no—
114#97—42no—
115C4×C8—24no—
116C4×C8—24no—
117#95—22no—
118#95—21no—
119#95—21no—
120#95—22no—
121#95—22no—
122#95—24no—
123#95—42no—
124C24#12111yescharacteristic
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—42no—
132C22×C4—48no—
133C22×C4—44no—
134C22×C4—84no—
135C22×C4—82no—
136C22×C4—82no—
137C22×C4C2×C22⋊C411yescharacteristic
138C22×C4C2×C4⋊C411yescharacteristic
139C22×C4#11711yescharacteristic
140C22×C4#12011yescharacteristic
141C22×C4#12111yescharacteristic
142C22×C4#12112yes—
143C22×C4C22⋊Q811yescharacteristic
144C22×C4C22⋊Q811yescharacteristic
145C22×C4C22⋊Q812yes—
146C22×C4#12311yescharacteristic
147C22×C4#12312yes—
148C22×C4#12411yescharacteristic
149C22×C4#12512yes—
150C22×C4C4⋊Q811yescharacteristic
151C2×C8—24no—
152C2×C8—24no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—22no—
158C2×C8—42no—
159C2×C8—42no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—44no—
163C2×C8—42no—
164C2×C8—42no—
165C2×C8—84no—
166C2×C8#10414yes—
167C4⋊C4—28no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—44no—
171C4⋊C4—48no—
172C22⋊C4—42no—
173C22⋊C4—42no—
174C42—42no—
175C42—42no—
176C42—84no—
177C42—84no—
178C42C4×D412yes—
179C42C4×Q812yes—
180C42#12112yes—
181C42C22⋊Q812yes—
182C42C22⋊Q812yes—
183C42#12512yes—
184C42#12612yes—
185C42C4⋊Q812yes—
186C23—22no—
187C23—22no—
188C23#38011yescharacteristic
189C23#39411yescharacteristic
190C23#39811yescharacteristic
191C23#47112yes—
192C23#47112yes—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—24no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—48no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—84no—
219C2×C4—88no—
220C2×C4—88no—
221C2×C4—84no—
222C2×C4#38611yescharacteristic
223C2×C4#38712yes—
224C2×C4#38912yes—
225C2×C4#39111yescharacteristic
226C2×C4#39511yescharacteristic
227C2×C4#39612yes—
228C2×C4#39812yes—
229C2×C4#40011yescharacteristic
230C2×C4C2×#10412yes—
231C2×C4#42112yes—
232C8—24no—
233C8—42no—
234C8—84no—
235C8—81no—
236C8—81no—
237C22—21no—
238C22—21no—
239C22—21no—
240C22—21no—
241C22—24no—
242C22—24no—
243C22#144111yescharacteristic
244C22#180612yes—
245C22#181212yes—
246C22#181511yescharacteristic
247C22#181511yescharacteristic
248C22#204612yes—
249C22#209012yes—
250C4—21no—
251C4—21no—
252C4—22no—
253C4—22no—
254C4—22no—
255C4—22no—
256C4—88no—
257C4—82no—
258C4—82no—
259C4#189611yescharacteristic
260C4#189711yescharacteristic
261C4#198612yes—
262C2—22no—
263C2—22no—
264C2#869412yes—
265C2#869612yes—
266C2#1122611yescharacteristic
267C2#1122911yescharacteristic
268C2#1123811yescharacteristic
subgroupG/H
center, socleC23#380
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
12. #1939C4semidirect
13. #1939C4semidirect

Maximal subgroups

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

subgroupindex
#213512
C2×C4×#1062
#80212
C2×#12852
#78752