Finite Group Databasebeta

← #315889

Subgroups — #315889

Order 512.

714 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 714 conjugacy classes of subgroups — too many to draw. Only its 178 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#33210C211yesmaximal, characteristic
2#27910C212yesmaximal
3#27875C212yesmaximal
4#20764C211yesmaximal, characteristic
5#10679C212yesmaximal, complemented
6#10675C212yesmaximal, complemented
7#9828C212yesmaximal, complemented
8#8820C212yesmaximal, complemented
9#8147C211yesmaximal, characteristic
10C2×#498C2211yescharacteristic
11C2×#484—24no—
12C2×C4×M16C2211yescharacteristic
13#2857C2212yes—
14#2285C2211yescharacteristic
15C2×#384C2211yescharacteristic
16#2095—24no—
17#2077—24no—
18#1990C2212yes—
19#1990C2212yes—
20#1990C2212yes—
21#1956C2212yes—
22#1956C2212yes—
23#1943—22no—
24#1920—24no—
25#1845C2212yes—
26#1845C2212yes—
27#1845C2211yescharacteristic
28#1844C2212yes—
29#1836—22no—
30#1834C2212yes—
31#1834C2212yes—
32#1834C2212yes—
33#1809—22no—
34C4×C22⋊C8C2212yes—
35#1736—22no—
36C2×#322C2212yes—
37#1273C2211yescharacteristic
38#1273C2211yescharacteristic
39C22×M16D412yes—
40C2×C4⋊Q8C2312yes—
41C2×C22⋊Q8—22no—
42C2×#117—21no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—21no—
45C22×C42C2311yescharacteristic
46#498—24no—
47#484—416no—
48C2×#107—22no—
49C2×#107—24no—
50C2×#107—24no—
51C2×#106—22no—
52C2×#106—24no—
53C2×#103—24no—
54C2×#103—24no—
55C2×C22⋊C8D414yes—
56C4×M16—24no—
57C2×#97D412yes—
58C2×C4×C8C2311yescharacteristic
59C2×C4×C8C2312yes—
60C2×C4×C8C2312yes—
61C2×C4×C8C2311yescharacteristic
62#391—22no—
63#389—22no—
64#389—22no—
65#384—21no—
66#384—22no—
67#384C2312yes—
68#384C2312yes—
69#384C2312yes—
70#382—21no—
71#380—22no—
72#380—22no—
73#380—22no—
74C4×C4⋊C4—22no—
75#340—24no—
76#340—24no—
77#340—24no—
78#340—24no—
79#340—24no—
80#340—24no—
81#336D414yes—
82#322—24no—
83C22×Q8—22no—
84C23×C4—41no—
85C23×C4C2×D412yes—
86C23×C4C2×D411yescharacteristic
87C2×M16—24no—
88C2×M16—24no—
89C2×M16—42no—
90C22×C8C2×D412yes—
91C22×C8C2×D414yes—
92C22×C8C2×D412yes—
93C22×C8C2×D414yes—
94C4⋊Q8—22no—
95C4⋊Q8—22no—
96C4⋊Q8—22no—
97C4⋊Q8—22no—
98C22⋊Q8—48no—
99#117—44no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—44no—
112C2×C4⋊C4—48no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—42no—
116C2×C22⋊C4—41no—
117C2×C22⋊C4—42no—
118C2×C42—22no—
119C2×C42—21no—
120C2×C42—21no—
121C2×C42—41no—
122C2×C42C2×D412yes—
123C2×C42C2×D411yescharacteristic
124C2×C42C2411yescharacteristic
125#107—44no—
126#107—48no—
127#107—48no—
128#106—44no—
129#106—48no—
130#103—48no—
131#103—48no—
132C22⋊C8—28no—
133#97—24no—
134C4×C8—22no—
135C4×C8—22no—
136C4×C8—22no—
137C4×C8—22no—
138C4×C8—22no—
139C4×C8#4114yes—
140#95—42no—
141#95—42no—
142#95—42no—
143C24#12711yescharacteristic
144C2×Q8—42no—
145C2×Q8—42no—
146C2×Q8—44no—
147C2×Q8—44no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—24no—
154C22×C4—24no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—84no—
160C22×C4—82no—
161C22×C4#12712yes—
162C22×C4#12711yescharacteristic
163C22×C4#13714yes—
164C22×C4C22×D411yescharacteristic
165C22×C4C22×D412yes—
166M16—48no—
167C2×C8—24no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—24no—
171C2×C8—22no—
172C2×C8—24no—
173C2×C8—24no—
174C2×C8—22no—
175C2×C8—22no—
176C2×C8—24no—
177C2×C8#12314yes—
178C2×C8AGL(1,ℤ/8)18yes—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—44no—
184C4⋊C4—44no—
185C4⋊C4—44no—
186C4⋊C4—44no—
187C4⋊C4—44no—
188C4⋊C4—44no—
189C4⋊C4—44no—
190C4⋊C4—44no—
191C4⋊C4—44no—
192C4⋊C4—82no—
193C4⋊C4—84no—
194C4⋊C4—84no—
195C22⋊C4—84no—
196C22⋊C4—82no—
197C42—21no—
198C42—21no—
199C42—22no—
200C42—22no—
201C42—82no—
202C42C2×#4112yes—
203C422−1+411yescharacteristic
204C422−1+411yescharacteristic
205C23—22no—
206C23—21no—
207C23—22no—
208C23—21no—
209C23#49712yes—
210C23C2×#12711yescharacteristic
211Q8—84no—
212Q8—84no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—24no—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—164no—
235C2×C4#49712yes—
236C2×C4C2×#12311yescharacteristic
237C2×C4#53611yescharacteristic
238C2×C4#54712yes—
239C2×C4#54712yes—
240C2×C4C2×AGL(1,ℤ/8)14yes—
241C2×C4C2×#13712yes—
242C2×C4#57512yes—
243C2×C4#57814yes—
244C8—28no—
245C8—24no—
246C8—42no—
247C8—44no—
248C22—22no—
249C22—24no—
250C22—41no—
251C22—41no—
252C22—42no—
253C22#250111yescharacteristic
254C22C2×#49711yescharacteristic
255C22#315112yes—
256C22#315311yescharacteristic
257C22#315612yes—
258C4—22no—
259C4—42no—
260C4—162no—
261C4—162no—
262C4—162no—
263C4—162no—
264C4#309512yes—
265C4#323612yes—
266C4#334314yes—
267C4#335314yes—
268C2—42no—
269C2#1968412yes—
270C2#1973211yescharacteristic
271C2#1974711yescharacteristic
272C2#2791911yescharacteristic
273C2#2794412yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#127
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10679C2semidirect
6. #10675C2semidirect
7. #9828C2semidirect
8. #8820C2semidirect

Maximal subgroups

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

subgroupindex
#332102
#279102
#278752
#207642
#106792
#106752
#98282
#88202
#81472