Finite Group Databasebeta

← #318404

Subgroups — #318404

Order 512.

688 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 162 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#28279C212yesmaximal
2#22270C211yesmaximal, characteristic, complemented
3#22264C211yesmaximal, characteristic
4C2×#1689C212yesmaximal, complemented
5#10484C214yesmaximal, complemented
6#8809C211yesmaximal, characteristic, complemented
7#8656C212yesmaximal, complemented
8#8591C212yesmaximal, complemented
9C2×#494C2211yescharacteristic
10C2×#474C2214yes—
11C2×#436C2212yes—
12C4×C4⋊Q8C2212yes—
13C4×#127C2211yescharacteristic
14#2067—24no—
15#1982—22no—
16#1956C2212yes—
17#1954C2212yes—
18#1947—22no—
19#1878—24no—
20#1870—24no—
21#1842C2212yes—
22#1834C2211yescharacteristic
23#1831C2211yescharacteristic
24#1802C2214yes—
25#1744C2211yescharacteristic
26#1689—28no—
27C2×#334C2212yes—
28C2×#334C2212yes—
29C2×#332C2214yes—
30#1296C2214yes—
31C22×Q16—22no—
32C22×M16—22no—
33C2×C4⋊Q8C2312yes—
34C2×#127D411yescharacteristic
35C2×C4×Q8—22no—
36C2×C4×D4—22no—
37C2×C4×D4C2312yes—
38#494—22no—
39#494D414yescomplemented
40#474—28no—
41#474—44no—
42#436—24no—
43#436—24no—
44C2×#107—24no—
45C2×C4⋊C8—22no—
46C2×C4⋊C8—22no—
47C2×C4⋊C8C2312yes—
48C2×C4⋊C8C2314yes—
49C2×#103—24no—
50C2×#102—24no—
51C2×#102—24no—
52C2×C22⋊C8—22no—
53C2×C22⋊C8—24no—
54C2×C4×C8C2311yescharacteristic
55C2×C4×C8C2311yescharacteristic
56#391—22no—
57#390—22no—
58#390D412yes—
59#389—22no—
60#389—22no—
61#384D411yescharacteristic
62#384C2312yes—
63C4×C4⋊C4—22no—
64C4×C4⋊C4—22no—
65C43C2311yescharacteristic
66#340—22no—
67#340—24no—
68#336—22no—
69#336D414yes—
70#334—24no—
71#334—42no—
72#332—28no—
73C22×Q8—22no—
74C22×D4—22no—
75C22×D4C2×D412yes—
76C23×C4—22no—
77C2×Q16—28no—
78C2×Q16—84no—
79C2×M16—44no—
80C2×M16—84no—
81C22×C8—22no—
82C22×C8—24no—
83C22×C8C2×D412yes—
84C22×C8C2×D412yes—
85C4⋊Q8—22no—
86C4⋊Q8—22no—
87C4⋊Q8—22no—
88C4⋊Q8C2×D414yes—
89#127—24no—
90C4×Q8—84no—
91C4×D4—22no—
92C4×D4—22no—
93C4×D4—22no—
94C4×D4—22no—
95C4×D4—84no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4C2×D412yes—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—24no—
106C2×C42—42no—
107C2×C42C2×D412yes—
108C2×C42C2×D411yescharacteristic
109C2×C42C2×D411yescharacteristic
110C2×C42#4112yes—
111C2×C42C2411yescharacteristic
112#107—48no—
113C4⋊C8—22no—
114C4⋊C8—24no—
115C4⋊C8—22no—
116C4⋊C8—24no—
117C4⋊C8—44no—
118C4⋊C8—82no—
119#103—84no—
120#102—48no—
121#102—84no—
122C22⋊C8—44no—
123C22⋊C8—84no—
124C4×C8—21no—
125C4×C8—21no—
126C4×C8C2×D412yes—
127C4×C8#4112yes—
128#95—42no—
129#95—42no—
130C24—22no—
131C2×Q8—24no—
132C2×Q8—42no—
133C2×Q8—44no—
134C2×Q8—44no—
135C2×D4—22no—
136C2×D4—22no—
137C2×D4—24no—
138C2×D4—24no—
139C2×D4—44no—
140C2×D4—82no—
141C2×D4—82no—
142C22×C4—22no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—44no—
147C22×C4—44no—
148C22×C4—42no—
149C22×C4#12112yes—
150C22×C4#12112yes—
151C22×C4C22×D411yescharacteristic
152C22×C4C22×D411yescharacteristic
153C22×C4C2×#4111yescharacteristic
154Q16—88no—
155M16—88no—
156C2×C8—24no—
157C2×C8—22no—
158C2×C8—24no—
159C2×C8—42no—
160C2×C8—44no—
161C2×C8—44no—
162C2×C8—44no—
163C2×C8—44no—
164C2×C8#12114yes—
165C4⋊C4—24no—
166C4⋊C4—22no—
167C4⋊C4—24no—
168C4⋊C4—22no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—44no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—82no—
175C4⋊C4—82no—
176C4⋊C4—84no—
177C22⋊C4—42no—
178C22⋊C4—42no—
179C22⋊C4—84no—
180C42—24no—
181C42—22no—
182C42—42no—
183C42—84no—
184C42#13714yes—
185C42C22×D411yescharacteristic
186C42C2×#4111yescharacteristic
187C42H(5,𝔽2)11yescharacteristic
188C422−1+411yescharacteristic
189C23—44no—
190C23—42no—
191C23—42no—
192C23—42no—
193C23—44no—
194C23C2×#12111yescharacteristic
195Q8—48no—
196Q8—84no—
197D4—44no—
198D4—44no—
199D4—88no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—44no—
207C2×C4—42no—
208C2×C4—44no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—84no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4#47014yes—
218C2×C4C2×#12111yescharacteristic
219C2×C4#53711yescharacteristic
220C2×C4D4211yescharacteristic
221C2×C4#54711yescharacteristic
222C2×C4#54711yescharacteristic
223C2×C4#55511yescharacteristic
224C2×C4C2×#13712yes—
225C2×C4#57614yes—
226C2×C4#57712yes—
227C8—24no—
228C8—42no—
229C8—84no—
230C8—82no—
231C22—84no—
232C22—82no—
233C22—82no—
234C22—82no—
235C22—84no—
236C22#253711yescharacteristic
237C22#304112yes—
238C22C2×#47011yescharacteristic
239C22#306011yescharacteristic
240C22#306912yes—
241C4—21no—
242C4—21no—
243C4—24no—
244C4—42no—
245C4—82no—
246C4—84no—
247C4—162no—
248C4#304112yes—
249C4#325912yes—
250C4#327812yes—
251C4#335612yes—
252C2—162no—
253C2#1942312yes—
254C2#1950812yes—
255C2#2823211yescharacteristic
256C2#2824411yescharacteristic
257C2#2825911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22270C2semidirect
4. C2×#1689C2semidirect
5. #10484C2semidirect
6. #8809C2semidirect
7. #8656C2semidirect
8. #8591C2semidirect
39. #494D4semidirect

Maximal subgroups

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

subgroupindex
#282792
#222702
#222642
C2×#16892
#104842
#88092
#86562
#85912