Finite Group Databasebeta

← #13147867

Subgroups — #13147867

Order 1728.

899 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 899 conjugacy classes of subgroups — too many to draw. Only its 155 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
1C42×#272C211yesmaximal, characteristic
2#64079⋊C3C212yesmaximal, complemented
3#11721152C212yesmaximal
4#11721147C214yesmaximal
5#11721098C214yesmaximal, complemented
6#11721095C212yesmaximal, complemented
7D3×#870—32nomaximal
8C2×C4×#272C2212yes—
9C2×C4×#272C2211yescharacteristic
10C2×#6199C2214yes—
11C42×H(3,𝔽3)C2211yescharacteristic
12C4×#1084C2211yescharacteristic
13C4×#1084C2212yes—
14C4⋊((C3⋊Dic3)⋊C3)C2214yes—
15#3750⋊C3C2218yes—
16#88449C2214yes—
17#88448C2218yes—
18#64079D312yescomplemented
19C3×C42×D3—32no—
20C3×C42×D3—61no—
21C3×#870—32no—
22D3×C4⋊Dic3—34no—
23D3×Dic3⋊C4—38no—
24#63869—34no—
25#63867—38no—
26C22×#272C2311yescharacteristic
27C2×C4×H(3,𝔽3)C2312yes—
28C2×C4×H(3,𝔽3)C2311yescharacteristic
29C2×#1084C2312yes—
30C2×#1084C2311yescharacteristic
31C4×#272—24no—
32C4×#272Q814yes—
33C2×(C3⋊Dic3)⋊C3C2318yes—
34#6199—28no—
35D3×#125—92no—
36C2×#870—91no—
37C2×C3×C4×D3—32no—
38C2×C3×C4×D3—34no—
39C2×C3×C4×D3—61no—
40C2×C3×C4×D3—62no—
41C2×D3×Dic3—38no—
42C32×C42—21no—
43C32×C42C2×D312yes—
44C4⋊(C3⋊Dic3)C2×D314yes—
45#3750C2×D318yes—
46C3×C4⋊Dic3—34no—
47C3×Dic3⋊C4—38no—
48C3×C4×Dic3—34no—
49C3×C4×Dic3—32no—
50C3×C4×Dic3—61no—
51C3×C4×Dic3—62no—
52#3724—38no—
53Dic3⋊Dic3—38no—
54Dic3⋊Dic3—38no—
55C22×H(3,𝔽3)C2411yescharacteristic
56C2×#272C2×Q812yes—
57C2×#272#4114yes—
58C4×H(3,𝔽3)—22no—
59C4×H(3,𝔽3)C2×Q812yes—
60#1084—22no—
61#1084C2×Q812yes—
62(C3⋊Dic3)⋊C3—28no—
63C3×#125—92no—
64C2×C3×C42—91no—
65C2×C4⋊Dic3—92no—
66C2×Dic3⋊C4—94no—
67D3×C4⋊C4—98no—
68D3×C4⋊C4—94no—
69#890—94no—
70#889—98no—
71C42×D3—32no—
72C42×D3—61no—
73#870—32no—
74#870—92no—
75#870—94no—
76#870—92no—
77C22×C3×D3—32no—
78C22×C3×D3—61no—
79C2×C32×C4—22no—
80C2×C32×C4—21no—
81C2×C32×C4C22×D314yes—
82C2×C32×C4C22×D312yes—
83C2×C3⋊Dic3C22×D318yes—
84C2×C3×Dic3—32no—
85C2×C3×Dic3—38no—
86C2×C3×Dic3—34no—
87C2×C3×Dic3—61no—
88C2×C3×Dic3—62no—
89C3×C4×D3—38no—
90C3×C4×D3—64no—
91C3×C4×D3—68no—
92C3×C4×D3—68no—
93D3×Dic3—616no—
94C2×#125—271noSylow
95C2×H(3,𝔽3)C22×Q811yescharacteristic
96C2×H(3,𝔽3)C2×#4112yes—
97#272#12514yescomplemented
98C22×C3×C4—91no—
99C22×C3×C4—92no—
100C22×Dic3—94no—
101C2×C4×D3—32no—
102C2×C4×D3—34no—
103C2×C4×D3—61no—
104C2×C4×D3—62no—
105C2×C4×D3—98no—
106C3×C4⋊C4—98no—
107C3×C4⋊C4—94no—
108C3×C42—32no—
109C3×C42—61no—
110C3×C42—92no—
111C3×C42—91no—
112C3×C42D3211yescharacteristic
113C4⋊Dic3—34no—
114C4⋊Dic3—94no—
115C4⋊Dic3—94no—
116C4⋊Dic3—98no—
117Dic3⋊C4—38no—
118Dic3⋊C4—98no—
119Dic3⋊C4—98no—
120Dic3⋊C4—98no—
121Dic3⋊C4—98no—
122C4×Dic3—34no—
123C4×Dic3—32no—
124C4×Dic3—62no—
125C4×Dic3—61no—
126C22×C32—21no—
127C22×C32C23×D312yes—
128C2×C3×D3—34no—
129C2×C3×D3—38no—
130C2×C3×D3—62no—
131C2×C3×D3—64no—
132C32×C4—24no—
133C32×C4—24no—
134C32×C4—22no—
135C32×C4D3×Q814yes—
136C3⋊Dic3—28no—
137C3×Dic3—34no—
138C3×Dic3—62no—
139C3×Dic3—64no—
140C3×Dic3—68no—
141C3×Dic3—64no—
142#125—92no—
143#125—274no—
144#125—272no—
145C2×C4⋊C4—274no—
146C2×C4⋊C4—272no—
147C2×C42—91no—
148H(3,𝔽3)C2×#12511yescharacteristic, complemented, Sylow
149C23×C3—91no—
150C22×D3—32no—
151C22×D3—61no—
152C2×C3×C4—32no—
153C2×C3×C4—34no—
154C2×C3×C4—62no—
155C2×C3×C4—61no—
156C2×C3×C4—91no—
157C2×C3×C4—94no—
158C2×C3×C4—98no—
159C2×C3×C4—92no—
160C2×C3×C4—184no—
161C2×C3×C4C2×D3212yes—
162C2×C3×C4C2×D3211yescharacteristic
163C2×Dic3—32no—
164C2×Dic3—34no—
165C2×Dic3—38no—
166C2×Dic3—61no—
167C2×Dic3—62no—
168C2×Dic3—98no—
169C2×Dic3—188no—
170C4×D3—38no—
171C4×D3—64no—
172C4×D3—68no—
173C4×D3—68no—
174C4×D3—1816no—
175C2×C32—21no—
176C2×C32—22no—
177C2×C32C2×D3×Q812yes—
178C2×C32D3×#4114yes—
179C3×D3—38no—
180C3×D3—64no—
181C22×C4—91no—
182C22×C4—92no—
183C22×C4—274no—
184C4⋊C4—98no—
185C4⋊C4—94no—
186C4⋊C4—274no—
187C4⋊C4—278no—
188C42—92no—
189C42—91no—
190C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
191C22×C3—32no—
192C22×C3—61no—
193C22×C3—92no—
194C22×C3—94no—
195C22×C3C22×D3211yescharacteristic
196C2×D3—34no—
197C2×D3—38no—
198C2×D3—62no—
199C2×D3—64no—
200C3×C4—22no—
201C3×C4—34no—
202C3×C4—64no—
203C3×C4—64no—
204C3×C4—62no—
205C3×C4—92no—
206C3×C4—188no—
207C3×C4—182no—
208C3×C4#379712yes—
209Dic3—34no—
210Dic3—62no—
211Dic3—64no—
212Dic3—64no—
213Dic3—68no—
214Dic3—188no—
215C32—21no—
216C32D3×#12512yescomplemented
217C23—91no—
218C2×C4—91no—
219C2×C4—94no—
220C2×C4—92no—
221C2×C4—98no—
222C2×C4—184no—
223C2×C4—548no—
224C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
225C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
226C2×C3—32no—
227C2×C3—34no—
228C2×C3—61no—
229C2×C3—62no—
230C2×C3—94no—
231C2×C3C2×#379711yescharacteristic
232C2×C3#6430612yes—
233D3—38no—
234D3—64no—
235C22—92no—
236C22—94no—
237C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
238C4—22no—
239C4—92no—
240C4—182no—
241C4—188no—
242C4#8865112yes—
243C3—32no—
244C3—61no—
245C3#1059394811yescharacteristic
246C2—94no—
247C2C2×#8865111yescharacteristic
248C2#1172249112yes—
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×#125
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
2. #64079⋊C3C2semidirect
5. #11721098C2semidirect
6. #11721095C2semidirect
18. #64079D3semidirect
97. #272#125semidirect
148. H(3,𝔽3)C2×#125semidirect
216. C32D3×#125semidirect

Maximal subgroups

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

subgroupindex
C42×#2722
#64079⋊C32
#117211522
#117211472
#117210982
#117210952
D3×#8703