Finite Group Databasebeta

← #150448

Subgroups — #150448

Order 512.

354 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 58 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#29106C211yesmaximal, characteristic
2C2×#1394C211yesmaximal, characteristic
3#11506C211yesmaximal, characteristic
4#7429C212yesmaximal, complemented
5#7413C212yesmaximal, complemented
6C22×#101—21no—
7#2851C411yescharacteristic
8C2×#391C2211yescharacteristic
9#2074C411yescharacteristic
10#2074C2211yescharacteristic
11#2026C411yescharacteristic
12#2026C2211yescharacteristic
13#1933—21no—
14#1933—21no—
15#1823—21no—
16#1787—21no—
17C4×C22⋊C8C411yescharacteristic
18#1463—21no—
19#1394—21no—
20#1394C2212yes—
21#1319—21no—
22#1319C2212yes—
23C23×Q8D411yescharacteristic
24C22×M16—41no—
25C2×#125C2×C411yescharacteristic
26C22×C4⋊C4D411yescharacteristic
27C22×C42C2×C411yescharacteristic
28C2×#103—21no—
29C2×#103—21no—
30C2×#103—21no—
31C2×#103—21no—
32C2×#101—24no—
33C2×#101—22no—
34C2×#101—41no—
35C2×#101—42no—
36#407—41no—
37C2×C22⋊C8—21no—
38C2×C4×C8—21no—
39#391—41no—
40#391C2×C411yescharacteristic
41#391D411yescharacteristic
42#391D411yescharacteristic
43#391C2311yescharacteristic
44#389—21no—
45#387—41no—
46#385—41no—
47#382C2×C411yescharacteristic
48#380—21no—
49C2×#95—41no—
50#340—21no—
51#340—21no—
52#340—21no—
53#340—21no—
54#336C2×C411yescharacteristic
55#336C2×C411yescharacteristic
56#324—22no—
57#324—41no—
58C22×Q8—22no—
59C22×Q8—21no—
60C22×Q8—21no—
61C22×Q8—21no—
62C22×Q8—41no—
63C22×Q8—41no—
64C22×Q8—81no—
65C22×Q8C22⋊C411yescharacteristic
66C22×Q8C2×D411yescharacteristic
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4C22⋊C411yescharacteristic
71C2×M16—44no—
72C2×M16—42no—
73C2×M16—82no—
74C2×M16—81no—
75C2×M16—81no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—81no—
79#125—81no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4C22⋊C411yescharacteristic
88C2×C4⋊C4C2×D411yescharacteristic
89C2×C22⋊C4—41no—
90C2×C42—21no—
91C2×C42—81no—
92C2×C42C22⋊C411yescharacteristic
93C2×C42C22×C411yescharacteristic
94#103—42no—
95#103—41no—
96#103—41no—
97#103—81no—
98#103—81no—
99#101—24no—
100#101—44no—
101#101—42no—
102C22⋊C8—41no—
103C22⋊C8—41no—
104C22⋊C8—42no—
105C22⋊C8—81no—
106C4×C8—81no—
107#95—41no—
108#95—41no—
109#95—41no—
110#95—41no—
111#95—41no—
112#95—41no—
113#95—41no—
114C24#9911yescharacteristic
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8—21no—
118C2×Q8—21no—
119C2×Q8—41no—
120C2×Q8—41no—
121C2×Q8—41no—
122C2×Q8—41no—
123C2×Q8—81no—
124C2×Q8—81no—
125C2×Q8—81no—
126C2×Q8—82no—
127C2×Q8—81no—
128C2×Q8—82no—
129C2×Q8—81no—
130C2×Q8—81no—
131C2×Q8#10412yes—
132C2×Q8AGL(1,ℤ/8)11yescharacteristic
133C2×Q8#13711yescharacteristic
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—41no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—42no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—81no—
157C22×C4—82no—
158C22×C4#9911yescharacteristic
159C22×C4C2×C22⋊C411yescharacteristic
160M16—44no—
161M16—84no—
162M16—82no—
163M16—162no—
164C2×C8—41no—
165C2×C8—41no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—84no—
171C2×C8—82no—
172C2×C8—82no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—81no—
180C4⋊C4—82no—
181C4⋊C4—82no—
182C22⋊C4—82no—
183C42—41no—
184C42—41no—
185C42—81no—
186C23—21no—
187C23—22no—
188C23—21no—
189C23—41no—
190C23#35411yescharacteristic
191C23#35411yescharacteristic
192C23C2×#9911yescharacteristic
193Q8—42no—
194Q8—41no—
195Q8—41no—
196Q8—81no—
197Q8—81no—
198Q8—82no—
199Q8—82no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—84no—
229C2×C4—82no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4—161no—
239C2×C4#41711yescharacteristic
240C2×C4C2×#10411yescharacteristic
241C8—81no—
242C8—81no—
243C8—84no—
244C8—162no—
245C22—21no—
246C22—21no—
247C22—21no—
248C22—21no—
249C22—21no—
250C22—21no—
251C22—21no—
252C22—42no—
253C22—42no—
254C22#141412yes—
255C22#150411yescharacteristic
256C22#212111yescharacteristic
257C22C2×#35411yescharacteristic
258C4—21no—
259C4—21no—
260C4—42no—
261C4—41no—
262C4—41no—
263C4—42no—
264C4—41no—
265C4—81no—
266C4—81no—
267C4—81no—
268C4—161no—
269C4—161no—
270C2—21no—
271C2—21no—
272C2—22no—
273C2—41no—
274C2#1175911yescharacteristic
275C2#1359911yescharacteristic
276C2C2×#141411yescharacteristic
subgroupG/H
center, socleC22C2×#354
commutatorC2×C42C22×C4
Frattini, cosocle#391C23
prime radicalC24#99
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#391 ◁ G
LECSG ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7429C2semidirect
5. #7413C2semidirect

Maximal subgroups

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

subgroupindex
#291062
C2×#13942
#115062
#74292
#74132