Finite Group Databasebeta

← #577219

Subgroups — #577219

Order 512.

2894 conjugacy classes of subgroups; only the 846 normal subgroups are shown.

normal subgroups only

This group has 2894 conjugacy classes of subgroups — too many to draw. Only its 846 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
1C2×#2840C211yesmaximal, characteristic
2#14553C2112yesmaximal, complemented
3#14220C2112yesmaximal, complemented
4#14127C213yesmaximal
5#13775C213yesmaximal
6C23×C22⋊C4C2213yes—
7#2840C4116yescomplemented
8C2×#393C22112yes—
9C2×#391C22112yescomplemented
10C2×#385C2216yes—
11C2×#385C2216yes—
12C2×#379C2213yes—
13C2×#379C2211yescharacteristic
14C2×C4×C22⋊C4C2216yes—
15#1446C2214yescomplemented
16#1443C2214yescomplemented
17#1443C2214yescomplemented
18#1442C22112yescomplemented
19#1439C22112yes—
20#1437C22112yes—
21#1437C22112yes—
22#1436C22112yescomplemented
23#1436C22112yescomplemented
24C22⋊#95C2216yes—
25C22⋊#95C2216yes—
26C22⋊#95C2216yes—
27C4×#95C2214yescomplemented
28C26C2311yescharacteristic
29C23×Q8C2314yes—
30C24×C4C2313yes—
31C24×C4C2313yes—
32C22×C4⋊C4C23112yes—
33C22×C22⋊C4—212no—
34C22×C22⋊C4—412no—
35C22×C22⋊C4C2316yes—
36C22×C22⋊C4C2316yes—
37C22×C22⋊C4C2316yes—
38C22×C22⋊C4C2316yes—
39C22×C22⋊C4C2316yes—
40C22×C22⋊C4C2316yes—
41C22×C42C23112yes—
42#393C2×C4196yescomplemented
43#391—424no—
44#385—412no—
45#385—412no—
46#379C2×C4124yes—
47#379Q818yescomplemented
48C4×C22⋊C4—412no—
49C2×#95C23112yes—
50C2×#95C23112yes—
51C2×#95C23112yes—
52C2×#95C23112yes—
53C2×#95C23112yes—
54C2×#95C23112yes—
55C2×#95C2314yes—
56C2×#95C2314yes—
57C2×#95C2314yes—
58C25—46no—
59C25—46no—
60C25C22×C414yes—
61C25C2×Q814yes—
62C25#4114yes—
63C25C2413yes—
64C22×Q8—412no—
65C22×Q8—412no—
66C22×Q8C22×C4116yes—
67C23×C4—26no—
68C23×C4—212no—
69C23×C4—26no—
70C23×C4—46no—
71C23×C4—46no—
72C23×C4—46no—
73C23×C4—46no—
74C23×C4—46no—
75C23×C4C2416yes—
76C23×C4C24112yes—
77C23×C4C2414yes—
78C23×C4C2416yes—
79C2×C4⋊C4—224no—
80C2×C4⋊C4—412no—
81C2×C4⋊C4—412no—
82C2×C22⋊C4—212no—
83C2×C22⋊C4—212no—
84C2×C22⋊C4—212no—
85C2×C22⋊C4—212no—
86C2×C22⋊C4—46no—
87C2×C22⋊C4—46no—
88C2×C22⋊C4—46no—
89C2×C22⋊C4—46no—
90C2×C22⋊C4—46no—
91C2×C22⋊C4—46no—
92C2×C22⋊C4—46no—
93C2×C22⋊C4—46no—
94C2×C22⋊C4—46no—
95C2×C22⋊C4—46no—
96C2×C22⋊C4—448no—
97C2×C22⋊C4C22×C4124yes—
98C2×C22⋊C4C22×C4124yes—
99C2×C22⋊C4C22×C4124yes—
100C2×C22⋊C4C2×Q8124yes—
101C2×C42—412no—
102C2×C42—412no—
103C2×C42—412no—
104#95—224no—
105#95—224no—
106#95—412no—
107#95—412no—
108#95—412no—
109#95—412no—
110#95C22×C4116yes—
111#95C22×C4116yes—
112#95C22×C4116yes—
113C24—26no—
114C24—26no—
115C24—26no—
116C24—26no—
117C24—26no—
118C24—26no—
119C24—412no—
120C24—424no—
121C24—424no—
122C24—412no—
123C24—824no—
124C24—88no—
125C24C4×Q8116yes—
126C24C23×C416yes—
127C24C22×Q816yes—
128C24C2×#4116yes—
129C24C2511yescharacteristic
130C2×Q8—448no—
131C2×Q8—824no—
132C2×Q8—88no—
133C22×C4—26no—
134C22×C4—212no—
135C22×C4—212no—
136C22×C4—212no—
137C22×C4—212no—
138C22×C4—212no—
139C22×C4—26no—
140C22×C4—212no—
141C22×C4—212no—
142C22×C4—212no—
143C22×C4—212no—
144C22×C4—212no—
145C22×C4—224no—
146C22×C4—26no—
147C22×C4—212no—
148C22×C4—26no—
149C22×C4—26no—
150C22×C4—212no—
151C22×C4—412no—
152C22×C4—824no—
153C22×C4—86no—
154C22×C4—86no—
155C22×C4—86no—
156C22×C4—86no—
157C22×C4—824no—
158C22×C4C23×C4124yes—
159C22×C4C22×Q818yes—
160C22×C4H(5,𝔽2)112yes—
161C4⋊C4—824no—
162C22⋊C4—424no—
163C22⋊C4—424no—
164C22⋊C4—424no—
165C22⋊C4—424no—
166C22⋊C4—424no—
167C22⋊C4—424no—
168C42—824no—
169C23—212no—
170C23—212no—
171C23—212no—
172C23—26no—
173C23—26no—
174C23—26no—
175C23—26no—
176C23—26no—
177C23—26no—
178C23—212no—
179C23—212no—
180C23—212no—
181C23—448no—
182C23—412no—
183C23—412no—
184C23—412no—
185C23—412no—
186C23—824no—
187C23—88no—
188C23—832no—
189C23—832no—
190C23C2×C4×Q8112yes—
191C23C24×C411yescharacteristic
192C23C23×Q811yescharacteristic
193C23C22×#4111yescharacteristic
194C23C2×H(5,𝔽2)16yes—
195C23#58516yes—
196Q8—832no—
197C2×C4—224no—
198C2×C4—224no—
199C2×C4—224no—
200C2×C4—212no—
201C2×C4—224no—
202C2×C4—224no—
203C2×C4—46no—
204C2×C4—46no—
205C2×C4—412no—
206C2×C4—412no—
207C2×C4—412no—
208C2×C4—412no—
209C2×C4—412no—
210C2×C4—44no—
211C2×C4—412no—
212C2×C4—46no—
213C2×C4—46no—
214C2×C4—46no—
215C2×C4—412no—
216C2×C4—412no—
217C2×C4—824no—
218C2×C4—824no—
219C22—212no—
220C22—212no—
221C22—212no—
222C22—46no—
223C22—46no—
224C22—46no—
225C22—46no—
226C22—424no—
227C22—424no—
228C22—832no—
229C22C22×C4×Q811yescharacteristic
230C22C4×H(5,𝔽2)16yes—
231C22#349013yes—
232C22#349113yes—
233C22#350116yes—
234C22#3558112yes—
235C22#356814yes—
236C4—412no—
237C4—48no—
238C4—424no—
239C4—86no—
240C2—412no—
241C2#3427013yes—
242C2#3452914yes—
243C2#4638513yes—
244C2#4643214yes—
245C2#4719611yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC26C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14553C2semidirect
3. #14220C2semidirect
7. #2840C4semidirect
9. C2×#391C22semidirect
15. #1446C22semidirect
16. #1443C22semidirect
17. #1443C22semidirect
18. #1442C22semidirect
22. #1436C22semidirect
23. #1436C22semidirect
27. C4×#95C22semidirect
42. #393C2×C4semidirect
47. #379Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28402
#145532
#142202
#141272
#137752