Finite Group Databasebeta

← #345670

Subgroups — #345670

Order 512.

508 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 142 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#26527C212yesmaximal
2#26513C212yesmaximal
3#10598C212yesmaximal
4#10596C212yesmaximal
5#10576C211yesmaximal, characteristic
6#10576C211yesmaximal, characteristic
7#9841C211yesmaximal, characteristic
8#9599C212yesmaximal
9#9059C212yesmaximal
10C2×#498C2211yescharacteristic
11#2571C2211yescharacteristic
12#2571C2211yescharacteristic
13#2365C2212yes—
14#2101—22no—
15#2097—22no—
16#2087—22no—
17#2066—22no—
18#1990C2212yes—
19#1988C2212yes—
20#1981C2212yes—
21#1956C2212yes—
22#1949—22no—
23#1948C2212yes—
24#1948C2212yes—
25#1939C2212yes—
26#1939C2212yes—
27#1919—22no—
28#1863—22no—
29#1845C2211yescharacteristic
30#1805—22no—
31C4×#107C2212yes—
32C2×#334C2212yes—
33C2×#332C2212yes—
34C2×#322C2211yescharacteristic
35#1296C2211yescharacteristic
36#1296C2211yescharacteristic
37#1279C2212yes—
38#1279C2212yes—
39C2×C4⋊Q8C2312yes—
40C2×C4×Q8—22no—
41#498—41no—
42#498Q814yes—
43C2×#107—41no—
44C2×#107D412yes—
45C2×#107D412yes—
46C2×#106—22no—
47C2×#106—22no—
48C2×#106—41no—
49C2×C4⋊C8—22no—
50C2×C4⋊C8—22no—
51C2×C4⋊C8C2312yes—
52C2×C4⋊C8C2312yes—
53C2×#103—22no—
54C2×#103—22no—
55C2×#103—22no—
56C2×#103—22no—
57C2×C4×C8C2312yes—
58C2×C4×C8C2311yescharacteristic
59#400—41no—
60#400—41no—
61#391—22no—
62#389—22no—
63#389—22no—
64#389—22no—
65#387—42no—
66#384C2311yescharacteristic
67#384C2311yescharacteristic
68#382—22no—
69#382—22no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4C2312yes—
72C4×C4⋊C4C2312yes—
73#340—22no—
74#340—22no—
75#340—22no—
76#340—22no—
77#340—22no—
78#340—22no—
79#340D412yes—
80#340D412yes—
81#336—22no—
82#336—22no—
83#334—42no—
84#332—42no—
85#322—41no—
86C22×Q8—22no—
87C22×C8—21no—
88C22×C8—22no—
89C22×C8—22no—
90C22×C8C2×D412yes—
91C22×C8C2×D412yes—
92C4⋊Q8—22no—
93C4⋊Q8—22no—
94C4⋊Q8—22no—
95C4⋊Q8C2×Q814yes—
96C4×Q8—44no—
97C4×Q8—82no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4C2×D412yes—
108C2×C4⋊C4C2×D412yes—
109C2×C4⋊C4C2×D412yes—
110C2×C4⋊C4C2×D412yes—
111C2×C42—22no—
112C2×C42—22no—
113C2×C42—42no—
114C2×C42C2411yescharacteristic
115#107—42no—
116#107—42no—
117#107—82no—
118#106—44no—
119#106—44no—
120#106—82no—
121C4⋊C8—22no—
122C4⋊C8—22no—
123C4⋊C8—22no—
124C4⋊C8—22no—
125C4⋊C8—44no—
126C4⋊C8—82no—
127#103—44no—
128#103—82no—
129#103—82no—
130#103—82no—
131C4×C8—22no—
132C4×C8—21no—
133C4×C8—22no—
134C4×C8C2×Q812yes—
135#95—22no—
136#95—22no—
137#95—42no—
138#95—41no—
139#95—41no—
140#95—42no—
141#95—42no—
142#95—42no—
143C2×Q8—24no—
144C2×Q8—42no—
145C2×Q8—42no—
146C2×Q8—42no—
147C2×Q8—42no—
148C2×Q8—42no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—42no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4C22×D412yes—
157C22×C4H(5,𝔽2)11yescharacteristic
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—22no—
163C2×C8—22no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—42no—
168C2×C8—42no—
169C2×C8—42no—
170C2×C8—42no—
171C2×C8—42no—
172C2×C8—42no—
173C4⋊C4—24no—
174C4⋊C4—24no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—81no—
190C4⋊C4—81no—
191C4⋊C4—84no—
192C4⋊C4—161no—
193C4⋊C4—161no—
194C42—42no—
195C42—42no—
196C42—42no—
197C42—42no—
198C42—84no—
199C42C22×Q811yescharacteristic
200C42H(5,𝔽2)11yescharacteristic
201C422−1+412yes—
202C23D4211yescharacteristic
203Q8—44no—
204Q8—44no—
205Q8—82no—
206Q8—82no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—44no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4#53712yes—
229C2×C4#54311yescharacteristic
230C2×C4D4×Q812yes—
231C2×C4#56311yescharacteristic
232C2×C4#57512yes—
233C2×C4#57512yes—
234C2×C4#57712yes—
235C2×C4#57712yes—
236C8—24no—
237C8—42no—
238C8—42no—
239C8—82no—
240C8—82no—
241C22#268311yescharacteristic
242C22#326712yes—
243C22#327112yes—
244C22#329111yescharacteristic
245C22#329311yescharacteristic
246C4—21no—
247C4—21no—
248C4—84no—
249C4—82no—
250C4—82no—
251C4—82no—
252C4—82no—
253C4—161no—
254C4—161no—
255C4#325612yes—
256C4#326012yes—
257C4#338812yes—
258C4#338812yes—
259C2#2013512yes—
260C2#2014311yescharacteristic
261C2#2015611yescharacteristic
262C2#3125112yes—
263C2#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

Maximal subgroups

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

subgroupindex
#265272
#265132
#105982
#105962
#105762
#98412
#95992
#90592