Finite Group Databasebeta

← #317614

Subgroups — #317614

Order 512.

538 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 158 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#29288C212yesmaximal
2#22299C211yesmaximal, characteristic
3#22270C211yesmaximal, characteristic
4#10656C212yesmaximal
5#10538C212yesmaximal
6#9562C212yesmaximal
7#9561C212yesmaximal
8#8787C211yesmaximal, characteristic
9#8586C212yesmaximal
10C2×#494C2211yescharacteristic
11C4×C4⋊Q8C2212yes—
12C4×#125C2211yescharacteristic
13#2088—22no—
14#2087—22no—
15#1989—22no—
16#1988C2212yes—
17#1988C2212yes—
18#1956C2212yes—
19#1949—22no—
20#1948C2212yes—
21#1947C2212yes—
22#1939—22no—
23#1939C2212yes—
24#1916—22no—
25#1878—22no—
26#1878—22no—
27#1842C2211yescharacteristic
28#1842C2212yes—
29#1834C2211yescharacteristic
30C8×C4⋊C4C2212yes—
31#1744C2211yescharacteristic
32C2×#332C2212yes—
33C2×#332C2212yes—
34#1296C2212yes—
35#1296C2212yes—
36#1277C2212yes—
37#1277C2211yescharacteristic
38#1277C2211yescharacteristic
39C22×Q16—21no—
40C22×Q16—21no—
41C2×C4⋊Q8C2312yes—
42C2×#125D411yescharacteristic
43C2×C4×Q8—22no—
44#494—24no—
45C2×#107—22no—
46C2×#106—22no—
47C2×C4⋊C8—22no—
48C2×C4⋊C8—22no—
49C2×C4⋊C8—22no—
50C2×C4⋊C8—22no—
51C2×C4⋊C8C2312yes—
52C2×C4⋊C8C2312yes—
53C2×C4⋊C8C2312yes—
54C2×#103—22no—
55C2×#103—22no—
56C2×C4×C8—22no—
57C2×C4×C8C2311yescharacteristic
58C2×C4×C8C2311yescharacteristic
59#391—22no—
60#389—22no—
61#389—22no—
62#389D412yes—
63#387—22no—
64#384C2312yes—
65#382D411yescharacteristic
66C4×C4⋊C4—22no—
67C4×C4⋊C4—22no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4C2312yes—
70C43C2311yescharacteristic
71#340—21no—
72#340—21no—
73#340—22no—
74#340—22no—
75#340—22no—
76#340—22no—
77#340—22no—
78#340—22no—
79#336—22no—
80#336—22no—
81#336D412yes—
82#336D412yes—
83#332—24no—
84#332—24no—
85C22×Q8—22no—
86C2×Q16—24no—
87C2×Q16—24no—
88C2×Q16—82no—
89C2×Q16—82no—
90C22×C8—22no—
91C22×C8—22no—
92C22×C8—22no—
93C22×C8C2×D411yescharacteristic
94C22×C8C2×D411yescharacteristic
95C22×C8C2×D412yes—
96C4⋊Q8—22no—
97C4⋊Q8—22no—
98C4⋊Q8—22no—
99C4⋊Q8#4114yes—
100#125—22no—
101#125Q1614yescomplemented
102C4×Q8—82no—
103C4×Q8—82no—
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—42no—
112C2×C4⋊C4C2×D412yes—
113C2×C4⋊C4C2×D412yes—
114C2×C42—22no—
115C2×C42—42no—
116C2×C42C2×D412yes—
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D411yescharacteristic
119C2×C42#4112yes—
120C2×C42C2411yescharacteristic
121#107—82no—
122#106—82no—
123C4⋊C8—22no—
124C4⋊C8—22no—
125C4⋊C8—22no—
126C4⋊C8—22no—
127C4⋊C8—22no—
128C4⋊C8—22no—
129C4⋊C8—44no—
130C4⋊C8—82no—
131C4⋊C8—82no—
132C4⋊C8—82no—
133#103—82no—
134#103—82no—
135C4×C8—22no—
136C4×C8—22no—
137C4×C8—82no—
138#95—22no—
139#95—22no—
140#95—22no—
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—42no—
152C22×C4—42no—
153C22×C4#12112yes—
154C22×C4#12112yes—
155C22×C4C22×D411yescharacteristic
156C22×C4C22×D411yescharacteristic
157C22×C4C2×#4111yescharacteristic
158Q16—84no—
159Q16—84no—
160C2×C8—21no—
161C2×C8—21no—
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—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—42no—
176C4⋊C4—22no—
177C4⋊C4—28no—
178C4⋊C4—24no—
179C4⋊C4—24no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—82no—
189C4⋊C4—82no—
190C4⋊C4—84no—
191C4⋊C4C2×Q1614yes—
192C42—22no—
193C42—22no—
194C42—22no—
195C42—42no—
196C42—42no—
197C42—42no—
198C42—84no—
199C42C2×Q1612yes—
200C42#13512yes—
201C42C2×#4112yes—
202C422−1+412yes—
203C23C2×#12111yescharacteristic
204Q8—44no—
205Q8—44no—
206Q8—82no—
207Q8—82no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—24no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—44no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4#46714yes—
227C2×C4#52511yescharacteristic
228C2×C4#53611yescharacteristic
229C2×C4#54812yes—
230C2×C4#55212yes—
231C2×C4C22×Q1611yescharacteristic
232C2×C4C2×#13511yescharacteristic
233C2×C4#57612yes—
234C2×C4#57712yes—
235C2×C4#57712yes—
236C8—41no—
237C8—41no—
238C8—42no—
239C8—82no—
240C8—82no—
241C8—82no—
242C22#253611yescharacteristic
243C22#304512yes—
244C22C2×#46711yescharacteristic
245C22#306011yescharacteristic
246C22#306912yes—
247C4—22no—
248C4—24no—
249C4—42no—
250C4—44no—
251C4—84no—
252C4—82no—
253C4—162no—
254C4#312211yescharacteristic
255C4#312311yescharacteristic
256C4#324112yes—
257C4#336312yes—
258C4#336512yes—
259C2#1941112yes—
260C2#1954112yes—
261C2#2818711yescharacteristic
262C2#2821411yescharacteristic
263C2#2822211yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
commutator, Frattini, cosocleC2×C42C24
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
101. #125Q16semidirect

Maximal subgroups

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

subgroupindex
#292882
#222992
#222702
#106562
#105382
#95622
#95612
#87872
#85862