Finite Group Databasebeta

← #345601

Subgroups — #345601

Order 512.

572 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 150 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#26466C212yesmaximal
2#26440C212yesmaximal
3#10590C212yesmaximal
4#10588C212yesmaximal
5#10573C211yesmaximal, characteristic
6#10573C211yesmaximal, characteristic
7#9841C211yesmaximal, characteristic
8#9599C212yesmaximal
9#9059C212yesmaximal
10C2×#498C2211yescharacteristic
11C2×#477—22no—
12#2551C2211yescharacteristic
13#2551C2211yescharacteristic
14#2365C2212yes—
15#2086—22no—
16#2068—22no—
17#2057—22no—
18#2024—22no—
19#1990C2212yes—
20#1988—22no—
21#1988C2212yes—
22#1981C2212yes—
23#1956C2212yes—
24#1947C2212yes—
25#1947C2212yes—
26#1939C2212yes—
27#1939C2212yes—
28#1918—22no—
29#1863—22no—
30#1845C2211yescharacteristic
31C4×#107C2212yes—
32C2×#334C2212yes—
33C2×#332C2212yes—
34C2×#322C2211yescharacteristic
35#1296C2211yescharacteristic
36#1296C2211yescharacteristic
37#1279C2212yes—
38#1279C2212yes—
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8—22no—
41C2×C4⋊Q8C2312yes—
42C2×C4×Q8—22no—
43#498—41no—
44#498Q814yescomplemented
45#477—48no—
46C2×#107—22no—
47C2×#107—22no—
48C2×#107—41no—
49C2×#107D412yes—
50C2×#107D412yes—
51C2×#106—41no—
52C2×C4⋊C8—22no—
53C2×C4⋊C8—22no—
54C2×C4⋊C8C2312yes—
55C2×C4⋊C8C2312yes—
56C2×#103—22no—
57C2×#103—22no—
58C2×#103—22no—
59C2×#103—22no—
60C2×C4×C8C2312yes—
61C2×C4×C8C2311yescharacteristic
62#395—41no—
63#395—41no—
64#391—22no—
65#389—22no—
66#389—22no—
67#389—22no—
68#387—42no—
69#384C2311yescharacteristic
70#384C2311yescharacteristic
71C4×C4⋊C4—22no—
72C4×C4⋊C4C2312yes—
73C4×C4⋊C4C2312yes—
74#340—22no—
75#340—22no—
76#340—22no—
77#340—22no—
78#340—22no—
79#340—22no—
80#340D412yes—
81#340D412yes—
82#336—22no—
83#336—22no—
84#334—42no—
85#332—42no—
86#322—41no—
87C22×Q8—22no—
88C22×Q8—41no—
89C22×Q8—41no—
90C22×C8—21no—
91C22×C8—22no—
92C22×C8—22no—
93C22×C8C2×D412yes—
94C22×C8C2×D412yes—
95C4⋊Q8—22no—
96C4⋊Q8—22no—
97C4⋊Q8—22no—
98C4⋊Q8—44no—
99C4⋊Q8—44no—
100C4⋊Q8—82no—
101C4⋊Q8—82no—
102C4⋊Q8C2×Q814yes—
103C4×Q8—44no—
104C4×Q8—82no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4C2×D412yes—
115C2×C4⋊C4C2×D412yes—
116C2×C4⋊C4C2×D412yes—
117C2×C4⋊C4C2×D412yes—
118C2×C42—22no—
119C2×C42—22no—
120C2×C42—42no—
121C2×C42C2411yescharacteristic
122#107—24no—
123#107—24no—
124#107—44no—
125#107—82no—
126#107—82no—
127#106—44no—
128C4⋊C8—22no—
129C4⋊C8—22no—
130C4⋊C8—22no—
131C4⋊C8—22no—
132C4⋊C8—44no—
133C4⋊C8—82no—
134#103—44no—
135#103—44no—
136#103—44no—
137#103—82no—
138C4×C8—22no—
139C4×C8—21no—
140C4×C8—22no—
141C4×C8C2×Q812yes—
142#95—22no—
143#95—22no—
144#95—42no—
145#95—42no—
146#95—41no—
147#95—41no—
148C2×Q8—24no—
149C2×Q8—42no—
150C2×Q8—42no—
151C2×Q8—42no—
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—84no—
155C2×Q8—84no—
156C2×Q8—161no—
157C2×Q8—161no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—42no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4C22×D412yes—
166C22×C4H(5,𝔽2)11yescharacteristic
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—22no—
176C2×C8—42no—
177C2×C8—42no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8—42no—
181C2×C8—42no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—24no—
185C4⋊C4—24no—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—84no—
197C4⋊C4—84no—
198C4⋊C4—81no—
199C4⋊C4—81no—
200C4⋊C4—84no—
201C4⋊C4AGL(1,ℤ/8)14yes—
202C4⋊C4#13714yes—
203C42—42no—
204C42—42no—
205C42—42no—
206C42—42no—
207C42—84no—
208C42C22×Q811yescharacteristic
209C42H(5,𝔽2)11yescharacteristic
210C422−1+412yes—
211C23D4211yescharacteristic
212Q8—44no—
213Q8—44no—
214Q8—82no—
215Q8—82no—
216Q8—164no—
217Q8—164no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—24no—
221C2×C4—24no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—44no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4—82no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4#53712yes—
240C2×C4#54311yescharacteristic
241C2×C4D4×Q812yes—
242C2×C4#56311yescharacteristic
243C2×C4C2×AGL(1,ℤ/8)12yes—
244C2×C4C2×#13712yes—
245C2×C4#57612yes—
246C2×C4#57812yes—
247C8—24no—
248C8—42no—
249C8—42no—
250C8—82no—
251C8—82no—
252C22#268311yescharacteristic
253C22#326412yes—
254C22#327012yes—
255C22#329111yescharacteristic
256C22#329311yescharacteristic
257C4—21no—
258C4—21no—
259C4—44no—
260C4—44no—
261C4—84no—
262C4—82no—
263C4—82no—
264C4—161no—
265C4—161no—
266C4#325412yes—
267C4#326212yes—
268C4#338512yes—
269C4#338612yes—
270C2#2002311yescharacteristic
271C2#2006212yes—
272C2#2009711yescharacteristic
273C2#3125212yes—
274C2#3125611yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
44. #498Q8semidirect

Maximal subgroups

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

subgroupindex
#264662
#264402
#105902
#105882
#105732
#98412
#95992
#90592