Finite Group Databasebeta

← #13147855

Subgroups — #13147855

Order 1728.

1123 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1123 conjugacy classes of subgroups — too many to draw. Only its 187 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
1C2×#88651C212yesmaximal
2C42×#272C211yesmaximal, characteristic
3#64078⋊C3C212yesmaximal, complemented
4#11721152C214yesmaximal
5#11721117C214yesmaximal, complemented
6#11721099C212yesmaximal, complemented
7D3×#869—32nomaximal
8C2×C4×#272C2211yescharacteristic
9C2×C4×#272C2212yes—
10C2×#634⋊C3C2214yes—
11C2×#6199C2214yes—
12C2×#6196C2214yes—
13#88651—28no—
14C42×H(3,𝔽3)C2211yescharacteristic
15C4×#1084C2211yescharacteristic
16C4×#1084C2212yes—
17C4⋊((C3⋊Dic3)⋊C3)C2218yes—
18#88449C2218yes—
19C2×D3×Dic6—34no—
20#64078D312yescomplemented
21C3×C42×D3—32no—
22C3×C42×D3—61no—
23C3×#869—32no—
24#63918—38no—
25D3×C4⋊Dic3—38no—
26#63866—34no—
27C22×#272C2311yescharacteristic
28C2×C4×H(3,𝔽3)C2311yescharacteristic
29C2×C4×H(3,𝔽3)C2312yes—
30C2×#1084C2311yescharacteristic
31C2×#1084C2312yes—
32C4×#272D414yes—
33C4×#272Q818yescomplemented
34C2×(C3⋊Dic3)⋊C3C2318yes—
35#634⋊C3—28no—
36#6199—28no—
37#6196—28no—
38D3×C4⋊Q8—92no—
39C2×#869—91no—
40C2×#634C2×D314yes—
41C2×C3×C4×D3—32no—
42C2×C3×C4×D3—34no—
43C2×C3×C4×D3—61no—
44C2×C3×C4×D3—62no—
45C2×C3×Dic6—34no—
46C2×#627—38no—
47C2×D3×Dic3—38no—
48D3×Dic6—616no—
49C32×C42—21no—
50C32×C42C2×D312yes—
51C4⋊(C3⋊Dic3)C2×D318yes—
52C3×C4⋊Dic3—38no—
53C3×C4×Dic3—32no—
54C3×C4×Dic3—34no—
55C3×C4×Dic3—61no—
56C3×C4×Dic3—62no—
57Dic3⋊Dic3—316no—
58C22×H(3,𝔽3)C2411yescharacteristic
59C2×#272C2×D412yes—
60C2×#272C2×Q814yes—
61C4×H(3,𝔽3)C2×D412yes—
62C4×H(3,𝔽3)C2×Q814yescomplemented
63#1084C2×D412yes—
64#1084C2×Q814yescomplemented
65(C3⋊Dic3)⋊C3—28no—
66C2×D3×Q8—94no—
67C22×Dic6—92no—
68C3×C4⋊Q8—92no—
69C2×C3×C42—91no—
70#944—94no—
71C2×C4⋊Dic3—94no—
72D3×C4⋊C4—98no—
73#888—98no—
74C42×D3—32no—
75C42×D3—61no—
76#869—32no—
77#869—92no—
78#869—94no—
79#869—92no—
80C22×C3×D3—32no—
81C22×C3×D3—61no—
82C2×C32×C4—22no—
83C2×C32×C4—21no—
84C2×C32×C4C22×D312yes—
85C2×C32×C4C22×D314yes—
86C2×C3⋊Dic3C22×D318yes—
87#634—28no—
88C2×C3×Dic3—34no—
89C2×C3×Dic3—32no—
90C2×C3×Dic3—38no—
91C2×C3×Dic3—62no—
92C2×C3×Dic3—61no—
93C3×C4×D3—38no—
94C3×C4×D3—316no—
95C3×C4×D3—64no—
96C3×C4×D3—68no—
97C3×Dic6—68no—
98#627—616no—
99D3×Dic3—616no—
100C2×C4⋊Q8—271noSylow
101C2×H(3,𝔽3)C22×D411yescharacteristic
102C2×H(3,𝔽3)C22×Q812yes—
103#272C4⋊Q814yescomplemented
104C2×C3×Q8—94no—
105C22×C3×C4—92no—
106C22×C3×C4—91no—
107C22×Dic3—94no—
108D3×Q8—1816no—
109C2×C4×D3—34no—
110C2×C4×D3—32no—
111C2×C4×D3—62no—
112C2×C4×D3—61no—
113C2×C4×D3—98no—
114C2×Dic6—34no—
115C2×Dic6—94no—
116C2×Dic6—94no—
117C2×Dic6—98no—
118C2×Dic6—188no—
119C3×C4⋊C4—98no—
120C3×C42—32no—
121C3×C42—61no—
122C3×C42—91no—
123C3×C42—92no—
124C3×C42D3211yescharacteristic
125C4⋊Dic3—38no—
126C4⋊Dic3—98no—
127C4⋊Dic3—98no—
128Dic3⋊C4—916no—
129C4×Dic3—34no—
130C4×Dic3—32no—
131C4×Dic3—61no—
132C4×Dic3—62no—
133C22×C32—21no—
134C22×C32C23×D312yes—
135C2×C3×D3—34no—
136C2×C3×D3—38no—
137C2×C3×D3—62no—
138C2×C3×D3—64no—
139C32×C4—24no—
140C32×C4—22no—
141C32×C4D3×D414yes—
142C32×C4D3×Q818yescomplemented
143C3⋊Dic3—28no—
144C3×Dic3—34no—
145C3×Dic3—38no—
146C3×Dic3—62no—
147C3×Dic3—64no—
148C3×Dic3—68no—
149C22×Q8—272no—
150C4⋊Q8—92no—
151C4⋊Q8—274no—
152C4⋊Q8—272no—
153C2×C4⋊C4—274no—
154C2×C42—91no—
155H(3,𝔽3)C2×C4⋊Q811yescharacteristic, complemented, Sylow
156C23×C3—91no—
157C22×D3—32no—
158C22×D3—61no—
159C3×Q8—188no—
160C2×C3×C4—34no—
161C2×C3×C4—32no—
162C2×C3×C4—62no—
163C2×C3×C4—61no—
164C2×C3×C4—92no—
165C2×C3×C4—91no—
166C2×C3×C4—98no—
167C2×C3×C4—94no—
168C2×C3×C4—98no—
169C2×C3×C4C2×D3212yes—
170C2×C3×C4C2×D3211yescharacteristic
171C2×Dic3—34no—
172C2×Dic3—32no—
173C2×Dic3—38no—
174C2×Dic3—62no—
175C2×Dic3—61no—
176C2×Dic3—98no—
177C2×Dic3—188no—
178C4×D3—316no—
179C4×D3—38no—
180C4×D3—68no—
181C4×D3—64no—
182C4×D3—1816no—
183Dic6—68no—
184Dic6—188no—
185Dic6—1816no—
186Dic6—188no—
187C2×C32—22no—
188C2×C32—21no—
189C2×C32C2×D3×D412yes—
190C2×C32C2×D3×Q814yes—
191C3×D3—38no—
192C3×D3—64no—
193C2×Q8—94no—
194C2×Q8—274no—
195C2×Q8—548no—
196C22×C4—92no—
197C22×C4—91no—
198C22×C4—274no—
199C4⋊C4—98no—
200C4⋊C4—278no—
201C42—91no—
202C42—92no—
203C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
204C22×C3—32no—
205C22×C3—61no—
206C22×C3—94no—
207C22×C3—92no—
208C22×C3C22×D3211yescharacteristic
209C2×D3—34no—
210C2×D3—38no—
211C2×D3—64no—
212C2×D3—62no—
213C3×C4—38no—
214C3×C4—34no—
215C3×C4—64no—
216C3×C4—62no—
217C3×C4—94no—
218C3×C4—92no—
219C3×C4—188no—
220C3×C4#379714yes—
221C3×C4#380212yes—
222Dic3—34no—
223Dic3—38no—
224Dic3—64no—
225Dic3—62no—
226Dic3—68no—
227Dic3—188no—
228C32—21no—
229C32D3×C4⋊Q812yescomplemented
230C23—91no—
231Q8—188no—
232Q8—548no—
233C2×C4—92no—
234C2×C4—91no—
235C2×C4—98no—
236C2×C4—94no—
237C2×C4—98no—
238C2×C4—548no—
239C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
240C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
241C2×C3—34no—
242C2×C3—32no—
243C2×C3—62no—
244C2×C3—61no—
245C2×C3—94no—
246C2×C3C2×#379712yes—
247C2×C3C2×#380211yescharacteristic
248D3—38no—
249D3—64no—
250C22—94no—
251C22—92no—
252C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
253C4—94no—
254C4—92no—
255C4—188no—
256C4#8865114yescomplemented
257C4#8865512yes—
258C3—32no—
259C3—61no—
260C3#1059390911yescharacteristic
261C2—94no—
262C2C2×#8865112yes—
263C2C2×#8865511yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC22×#272C23
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×C4⋊Q8
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
3. #64078⋊C3C2semidirect
5. #11721117C2semidirect
6. #11721099C2semidirect
20. #64078D3semidirect
33. C4×#272Q8semidirect
62. C4×H(3,𝔽3)C2×Q8semidirect
64. #1084C2×Q8semidirect
103. #272C4⋊Q8semidirect
142. C32×C4D3×Q8semidirect
155. H(3,𝔽3)C2×C4⋊Q8semidirect
229. C32D3×C4⋊Q8semidirect
256. C4#88651semidirect

Maximal subgroups

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

subgroupindex
C2×#886512
C42×#2722
#64078⋊C32
#117211522
#117211172
#117210992
D3×#8693