Finite Group Databasebeta

← #318346

Subgroups — #318346

Order 512.

710 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 170 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#22260C211yesmaximal, characteristic
4C2×#1669C212yesmaximal, complemented
5#10485C214yesmaximal, complemented
6#8810C211yesmaximal, characteristic, complemented
7#8648C212yesmaximal, complemented
8#8591C212yesmaximal, complemented
9C2×#494C2211yescharacteristic
10C2×#474C2214yes—
11C2×C8×D4C2212yes—
12C4×C4⋊Q8C2212yes—
13C4×#127C2211yescharacteristic
14#2067—24no—
15#1982—22no—
16#1956C2212yes—
17#1954C2212yes—
18#1947—22no—
19#1878—24no—
20#1869—24no—
21#1842C2212yes—
22#1834C2211yescharacteristic
23#1831C2211yescharacteristic
24#1802C2214yes—
25#1744C2211yescharacteristic
26#1669—28no—
27C2×#335C2212yes—
28C2×#335C2212yes—
29C2×#332C2214yes—
30#1296C2214yes—
31C22×Q16—22no—
32C23×C8—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—
42C8×D4—24no—
43C8×D4—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#335—24no—
71#335—42no—
72#332—28no—
73C22×Q8—22no—
74C22×D4—22no—
75C22×D4C2×D412yes—
76C23×C4—22no—
77C2×Q16—28no—
78C2×Q16—84no—
79C22×C8—22no—
80C22×C8—24no—
81C22×C8—44no—
82C22×C8—84no—
83C22×C8C2×D412yes—
84C22×C8C2×D412yes—
85C4⋊Q8—22no—
86C4⋊Q8—22no—
87C4⋊Q8—22no—
88C4⋊Q8C2×D414yes—
89#127—22no—
90#127Q1614yescomplemented
91C4×Q8—84no—
92C4×D4—22no—
93C4×D4—22no—
94C4×D4—22no—
95C4×D4—22no—
96C4×D4—84no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4C2×D412yes—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—24no—
107C2×C42—42no—
108C2×C42C2×D412yes—
109C2×C42C2×D411yescharacteristic
110C2×C42C2×D411yescharacteristic
111C2×C42#4112yes—
112C2×C42C2411yescharacteristic
113#107—48no—
114C4⋊C8—22no—
115C4⋊C8—24no—
116C4⋊C8—22no—
117C4⋊C8—24no—
118C4⋊C8—44no—
119C4⋊C8—82no—
120#103—84no—
121#102—48no—
122#102—84no—
123C22⋊C8—44no—
124C22⋊C8—84no—
125C4×C8—21no—
126C4×C8—21no—
127C4×C8C2×D412yes—
128C4×C8#4112yes—
129#95—42no—
130#95—42no—
131C24—22no—
132C2×Q8—24no—
133C2×Q8—42no—
134C2×Q8—44no—
135C2×Q8—44no—
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—22no—
140C2×D4—22no—
141C2×D4—28no—
142C2×D4—82no—
143C2×D4—82no—
144C2×D4C2×Q1614yes—
145C22×C4—22no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—44no—
150C22×C4—44no—
151C22×C4—42no—
152C22×C4#12112yes—
153C22×C4#12112yes—
154C22×C4C22×D411yescharacteristic
155C22×C4C22×D411yescharacteristic
156C22×C4C2×#4111yescharacteristic
157Q16—88no—
158C2×C8—24no—
159C2×C8—22no—
160C2×C8—24no—
161C2×C8—42no—
162C2×C8—44no—
163C2×C8—44no—
164C2×C8—44no—
165C2×C8—44no—
166C2×C8—88no—
167C2×C8#12114yes—
168C4⋊C4—24no—
169C4⋊C4—22no—
170C4⋊C4—24no—
171C4⋊C4—22no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—44no—
176C4⋊C4—42no—
177C4⋊C4—82no—
178C4⋊C4—82no—
179C4⋊C4—84no—
180C22⋊C4—42no—
181C22⋊C4—42no—
182C22⋊C4—84no—
183C42—21no—
184C42—24no—
185C42—21no—
186C42—42no—
187C42—84no—
188C42C2×Q1612yes—
189C42#13512yes—
190C42C22×D411yescharacteristic
191C42C2×#4111yescharacteristic
192C42H(5,𝔽2)11yescharacteristic
193C422−1+411yescharacteristic
194C23—24no—
195C23—44no—
196C23—42no—
197C23—42no—
198C23—44no—
199C23C2×#12111yescharacteristic
200Q8—48no—
201Q8—84no—
202D4—24no—
203D4—24no—
204D4—42no—
205D4—42no—
206D4—88no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—44no—
214C2×C4—42no—
215C2×C4—44no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—84no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4#46714yes—
225C2×C4C2×#12111yescharacteristic
226C2×C4#53711yescharacteristic
227C2×C4D4211yescharacteristic
228C2×C4#54711yescharacteristic
229C2×C4#54711yescharacteristic
230C2×C4#55511yescharacteristic
231C2×C4C22×Q1611yescharacteristic
232C2×C4C2×#13511yescharacteristic
233C2×C4#57611yescharacteristic
234C2×C4#57614yes—
235C2×C4#57811yescharacteristic
236C8—24no—
237C8—42no—
238C8—84no—
239C8—82no—
240C22—44no—
241C22—48no—
242C22—84no—
243C22—82no—
244C22—82no—
245C22#253711yescharacteristic
246C22#304112yes—
247C22C2×#46711yescharacteristic
248C22#306111yescharacteristic
249C22#306912yes—
250C4—21no—
251C4—21no—
252C4—24no—
253C4—42no—
254C4—82no—
255C4—84no—
256C4—162no—
257C4#304112yes—
258C4#325912yes—
259C4D4×Q1611yescharacteristic
260C4#328711yescharacteristic
261C4#333811yescharacteristic
262C4#334211yescharacteristic
263C2—84no—
264C2#1938712yes—
265C2#1951112yes—
266C2#2822811yescharacteristic
267C2#2824411yescharacteristic
268C2#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×#1669C2semidirect
5. #10485C2semidirect
6. #8810C2semidirect
7. #8648C2semidirect
8. #8591C2semidirect
39. #494D4semidirect
90. #127Q16semidirect

Maximal subgroups

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

subgroupindex
#282792
#222702
#222602
C2×#16692
#104852
#88102
#86482
#85912