Finite Group Databasebeta

← #6132636

Subgroups — #6132636

Order 512.

2124 conjugacy classes of subgroups; only the 546 normal subgroups are shown.

normal subgroups only

This group has 2124 conjugacy classes of subgroups — too many to draw. Only its 546 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×#2866C211yesmaximal, characteristic, complemented
2C2×#2865C211yesmaximal, characteristic, complemented
3#16563C217yesmaximal, complemented
4#16515C217yesmaximal, complemented
5#16515C217yesmaximal, complemented
6#16514C217yesmaximal
7#14692C211yesmaximal, characteristic, complemented
8#2866—28no—
9#2865—28no—
10C2×#398C2217yes—
11C2×#396C2217yes—
12C2×#396C2217yes—
13C2×#394C2217yes—
14C2×#388C2217yes—
15C2×#388C2217yes—
16C2×#387C2217yes—
17C2×#387C2217yes—
18C2×#384C2217yes—
19C2×#384C2217yes—
20C2×C43C2211yescharacteristic
21#1441C2217yes—
22#1441C2217yes—
23#1441C2217yes—
24#1441C2217yes—
25#1441C2217yes—
26#1441C2217yes—
27#1441C2217yes—
28#1441C2217yes—
29#1439C2217yes—
30#1439C2217yes—
31#1439C2217yes—
32#1439C2217yes—
33C23×D4C2311yescharacteristic
34C22×C4⋊C4C2317yes—
35C22×C4⋊C4C2317yes—
36C22×C4⋊C4C2317yes—
37C22×C4⋊C4C2317yes—
38C22×C4⋊C4C2317yes—
39C22×C4⋊C4C2317yes—
40C22×C4⋊C4C2317yes—
41C22×C22⋊C4C2317yes—
42C22×C22⋊C4C2317yes—
43C22×C22⋊C4C2317yes—
44C22×C22⋊C4C2317yes—
45C22×C42C2317yes—
46#398—414no—
47#396—414no—
48#396—414no—
49#394—414no—
50#388—228no—
51#388—228no—
52#387—228no—
53#387—228no—
54#384—414no—
55#384—414no—
56C2×#95C2317yes—
57C2×#95C2317yes—
58C2×#95C2317yes—
59C2×#95C2317yes—
60C2×#95C2317yes—
61C2×#95C2317yes—
62C2×#95C2317yes—
63C2×#95C2317yes—
64C2×#95C2317yes—
65C2×#95C2317yes—
66C43D418yescomplemented
67C25C2411yescharacteristic
68C25C2411yescharacteristic
69C22×D4—47no—
70C23×C4C2417yes—
71C23×C4C2417yes—
72C23×C4C2417yes—
73C23×C4C2411yescharacteristic
74C23×C4C2417yes—
75C2×C4⋊C4—214no—
76C2×C4⋊C4—214no—
77C2×C4⋊C4—214no—
78C2×C4⋊C4—214no—
79C2×C4⋊C4—214no—
80C2×C4⋊C4—214no—
81C2×C4⋊C4—214no—
82C2×C4⋊C4—214no—
83C2×C4⋊C4—214no—
84C2×C4⋊C4—214no—
85C2×C4⋊C4—47no—
86C2×C4⋊C4—47no—
87C2×C4⋊C4—47no—
88C2×C4⋊C4—47no—
89C2×C4⋊C4—47no—
90C2×C4⋊C4—47no—
91C2×C4⋊C4—47no—
92C2×C4⋊C4—47no—
93C2×C4⋊C4—47no—
94C2×C4⋊C4—47no—
95C2×C4⋊C4—47no—
96C2×C22⋊C4—214no—
97C2×C22⋊C4—214no—
98C2×C22⋊C4—214no—
99C2×C22⋊C4—214no—
100C2×C22⋊C4—214no—
101C2×C22⋊C4—214no—
102C2×C22⋊C4—47no—
103C2×C22⋊C4—47no—
104C2×C22⋊C4—47no—
105C2×C22⋊C4—47no—
106C2×C22⋊C4—47no—
107C2×C22⋊C4—47no—
108C2×C42—214no—
109C2×C42—214no—
110C2×C42C2×D4128yes—
111#95—214no—
112#95—214no—
113#95—214no—
114#95—214no—
115#95—47no—
116#95—47no—
117#95—47no—
118#95—47no—
119#95—47no—
120#95—47no—
121C24—21no—
122C24—21no—
123C24—27no—
124C24—27no—
125C24—27no—
126C24—27no—
127C24C2511yescharacteristic
128C2×D4—814no—
129C22×C4—27no—
130C22×C4—27no—
131C22×C4—27no—
132C22×C4—27no—
133C22×C4—27no—
134C22×C4—27no—
135C22×C4—27no—
136C22×C4—27no—
137C22×C4—27no—
138C22×C4—27no—
139C22×C4—27no—
140C22×C4—27no—
141C22×C4—27no—
142C22×C4—27no—
143C22×C4—27no—
144C22×C4—27no—
145C22×C4—27no—
146C22×C4—27no—
147C22×C4—27no—
148C22×C4—27no—
149C22×C4—27no—
150C22×C4—27no—
151C22×C4—27no—
152C22×C4—27no—
153C22×C4—27no—
154C22×C4—27no—
155C22×C4C22×D4114yes—
156C22×C4H(5,𝔽2)114yes—
157C22×C42−1+4114yes—
158C4⋊C4—428no—
159C4⋊C4—428no—
160C4⋊C4—814no—
161C4⋊C4—814no—
162C4⋊C4—814no—
163C4⋊C4—814no—
164C4⋊C4—814no—
165C22⋊C4—428no—
166C22⋊C4—428no—
167C22⋊C4—814no—
168C22⋊C4—814no—
169C42—256no—
170C23—47no—
171C23—47no—
172C23—47no—
173C23—47no—
174C23—47no—
175C23—47no—
176C23—47no—
177C23—47no—
178C23—47no—
179C23—47no—
180C23C23×D411yescharacteristic
181C23C2×H(5,𝔽2)17yes—
182C23C2×2−1+417yes—
183D4—88no—
184C2×C4—214no—
185C2×C4—214no—
186C2×C4—214no—
187C2×C4—214no—
188C2×C4—214no—
189C2×C4—214no—
190C2×C4—47no—
191C2×C4—47no—
192C2×C4—47no—
193C2×C4—47no—
194C2×C4—47no—
195C2×C4—47no—
196C2×C4—47no—
197C2×C4—47no—
198C2×C4—47no—
199C2×C4—47no—
200C2×C4—47no—
201C2×C4—47no—
202C2×C4—47no—
203C2×C4—47no—
204C2×C4—47no—
205C2×C4—47no—
206C2×C4—47no—
207C2×C4—47no—
208C2×C4—47no—
209C2×C4—47no—
210C2×C4—47no—
211C2×C4—47no—
212C2×C4#537128yes—
213C22—81no—
214C22—81no—
215C22—87no—
216C22—87no—
217C22—87no—
218C22—87no—
219C22C2×#53717yes—
220C22#354317yes—
221C22#354417yes—
222C22#355217yes—
223C22#355217yes—
224C4—228no—
225C4—87no—
226C4—81no—
227C4—87no—
228C4—87no—
229C2—161no—
230C2—161no—
231C2#3762017yes—
232C2#5338611yescharacteristic
233C2#5338717yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2866C2semidirect
2. C2×#2865C2semidirect
3. #16563C2semidirect
4. #16515C2semidirect
5. #16515C2semidirect
7. #14692C2semidirect
66. C43D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28662
C2×#28652
#165632
#165152
#165142
#146922