Finite Group Databasebeta

← #149913

Subgroups — #149913

Order 512.

340 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 56 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#27905C211yesmaximal, characteristic
2#11512C211yesmaximal, characteristic
3C2×#1325C211yesmaximal, characteristic
4#7371C212yesmaximal, complemented
5#7355C212yesmaximal, complemented
6C2×#483—21no—
7C2×#483—21no—
8#2857C411yescharacteristic
9C2×#384C2211yescharacteristic
10#2096—21no—
11#2096—21no—
12#1956C411yescharacteristic
13#1956C2211yescharacteristic
14#1844C411yescharacteristic
15#1844C2211yescharacteristic
16C4×C22⋊C8C411yescharacteristic
17#1463—21no—
18C2×#324—21no—
19#1325—21no—
20#1325—21no—
21#1325C2212yes—
22#1325C2212yes—
23C2×C4⋊Q8C2×C411yescharacteristic
24C2×C22⋊Q8—21no—
25C22×C4⋊C4—41no—
26C22×C4⋊C4D411yescharacteristic
27C22×C4⋊C4D411yescharacteristic
28C22×C42C2×C411yescharacteristic
29#483—42no—
30#483—42no—
31#483—42no—
32#483—42no—
33C2×#106—21no—
34C2×#106—21no—
35C2×#106—21no—
36C2×#106—21no—
37C2×#103—21no—
38C2×#103—21no—
39C2×#103—21no—
40C2×#103—21no—
41#407—41no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—41no—
44C2×C4×C8C2×C411yescharacteristic
45C2×C4×C8C2×C411yescharacteristic
46#391—21no—
47#389—41no—
48#384—41no—
49#384C2×C411yescharacteristic
50#384C2×C411yescharacteristic
51#384D411yescharacteristic
52#384D411yescharacteristic
53#384C2311yescharacteristic
54#380—41no—
55#336—21no—
56#324—22no—
57#324—22no—
58#324—41no—
59#324—41no—
60C22×Q8—21no—
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4C22⋊C411yescharacteristic
65C2×M16—81no—
66C22×C8—21no—
67C22×C8—21no—
68C22×C8—81no—
69C4⋊Q8—22no—
70C4⋊Q8—41no—
71C22⋊Q8—41no—
72C22⋊Q8—41no—
73C22⋊Q8—41no—
74C22⋊Q8—41no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—84no—
90C2×C4⋊C4C22⋊C411yescharacteristic
91C2×C4⋊C4C22⋊C411yescharacteristic
92C2×C4⋊C4C2×D411yescharacteristic
93C2×C4⋊C4C2×D411yescharacteristic
94C2×C22⋊C4—41no—
95C2×C42—22no—
96C2×C42—21no—
97C2×C42—41no—
98C2×C42C22⋊C411yescharacteristic
99C2×C42C22×C411yescharacteristic
100#106—42no—
101#106—42no—
102#106—42no—
103#106—42no—
104#103—42no—
105#103—42no—
106#103—42no—
107#103—42no—
108C22⋊C8—22no—
109C22⋊C8—22no—
110C22⋊C8—42no—
111C22⋊C8—42no—
112C22⋊C8—81no—
113C22⋊C8—81no—
114C4×C8—22no—
115C4×C8—41no—
116#95—42no—
117C24#9911yescharacteristic
118C2×Q8—41no—
119C2×Q8—41no—
120C2×Q8—41no—
121C2×Q8—41no—
122C2×Q8—41no—
123C2×Q8—41no—
124C22×C4—21no—
125C22×C4—24no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—41no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—82no—
145C22×C4—82no—
146C22×C4#9911yescharacteristic
147C22×C4C2×C22⋊C411yescharacteristic
148M16—162no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—82no—
156C2×C8—82no—
157C2×C8—162no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—81no—
165C4⋊C4—81no—
166C4⋊C4—82no—
167C4⋊C4—82no—
168C4⋊C4—82no—
169C4⋊C4—82no—
170C4⋊C4—82no—
171C4⋊C4—82no—
172C4⋊C4—81no—
173C4⋊C4—81no—
174C22⋊C4—81no—
175C22⋊C4—81no—
176C42—21no—
177C42—21no—
178C42—21no—
179C42—42no—
180C42#13112yes—
181C23—21no—
182C23—22no—
183C23—21no—
184C23—41no—
185C23#35311yescharacteristic
186C23#35311yescharacteristic
187C23C2×#9911yescharacteristic
188Q8—81no—
189Q8—81no—
190Q8—81no—
191Q8—81no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—48no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—82no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—161no—
224C2×C4—161no—
225C2×C4#40811yescharacteristic
226C2×C4#41311yescharacteristic
227C8—42no—
228C8—42no—
229C8—162no—
230C8—162no—
231C22—21no—
232C22—21no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—42no—
239C22—42no—
240C22#140912yes—
241C22#146811yescharacteristic
242C22C2×#35311yescharacteristic
243C22#213311yescharacteristic
244C4—24no—
245C4—21no—
246C4—21no—
247C4—42no—
248C4—41no—
249C4—81no—
250C4—81no—
251C4—81no—
252C4—81no—
253C4—161no—
254C4—161no—
255C2—21no—
256C2—21no—
257C2—22no—
258C2—41no—
259C2#1154911yescharacteristic
260C2C2×#140911yescharacteristic
261C2#1359011yescharacteristic
subgroupG/H
center, socleC22C2×#353
commutatorC2×C42C22×C4
Frattini, cosocle#384C23
prime radicalC24#99
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#384 ◁ G
LECSG ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7371C2semidirect
5. #7355C2semidirect

Maximal subgroups

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

subgroupindex
#279052
#115122
C2×#13252
#73712
#73552