Finite Group Databasebeta

← #13140045

Subgroups — #13140045

Order 1728.

475 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 475 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C22×#6180C211yesmaximal, characteristic
2#64133⋊C3C212yesmaximal
3C2×#88433C211yesmaximal, characteristic, complemented
4#11720315C211yesmaximal, characteristic, complemented
5#11720313C212yesmaximal, complemented
6#10591187—32nomaximal
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C2×#634⋊C3C2212yes—
9C2×#6180—22no—
10C2×#6180C2211yescharacteristic
11C2×#6180C2211yescharacteristic
12#3757⋊C3—22no—
13C4⋊((C3⋊Dic3)⋊C3)C2212yes—
14#3750⋊C3—22no—
15#88433—22no—
16#64133D312yes—
17C3×#924—32no—
18C22×C3×C3⋊C8—32no—
19C22×C3×C3⋊C8—61no—
20C2×#3719—32no—
21#63575—32no—
22#63567—32no—
23#63566—32no—
24C23×H(3,𝔽3)D411yescharacteristic
25C2×C4×H(3,𝔽3)—21no—
26C2×C4×H(3,𝔽3)—21no—
27C2×C4×H(3,𝔽3)D411yescharacteristic
28C2×C4×H(3,𝔽3)C2311yescharacteristic
29C2×(C3⋊Dic3)⋊C3—22no—
30C2×(C3⋊Dic3)⋊C3—22no—
31#634⋊C3—22no—
32#634⋊C3—22no—
33#6180—21no—
34#6180D412yes—
35#5116—91no—
36#5051—92no—
37C22×C32×C4—21no—
38C22×C32×C4C2×D312yes—
39C2×#634C2×D312yes—
40C2×C3×Dic6—32no—
41#3757—22no—
42C4⋊(C3⋊Dic3)C2×D312yes—
43#3750—22no—
44C3×#217—62no—
45C3×C4⋊Dic3—32no—
46C3×Dic3⋊C4—62no—
47C2×C3×C3⋊C8—32no—
48C2×C3×C3⋊C8—32no—
49C2×C3×C3⋊C8—64no—
50C2×C3×C3⋊C8—61no—
51C2×C3×C3⋊C8—61no—
52C2×C3×C3⋊C8—64no—
53#3719—64no—
54C22×H(3,𝔽3)—21no—
55C22×H(3,𝔽3)—21no—
56C22×H(3,𝔽3)Q1612yescomplemented
57C22×H(3,𝔽3)C2×D411yescharacteristic
58C4×H(3,𝔽3)—21no—
59C4×H(3,𝔽3)C2×D411yescharacteristic
60C4×H(3,𝔽3)#4111yescharacteristic
61(C3⋊Dic3)⋊C3—42no—
62(C3⋊Dic3)⋊C3—42no—
63C22×C3×C8—91no—
64C3×C22⋊Q8—92no—
65C2×#216—92no—
66#924—32no—
67#924—92no—
68C22×C3⋊C8—32no—
69C22×C3⋊C8—61no—
70C2×Dic12—91no—
71#835—92no—
72#818—91no—
73#816—92no—
74#810—92no—
75#807—92no—
76C23×C32—21no—
77C23×C32#6712yes—
78C2×C32×C4—22no—
79C2×C32×C4—22no—
80C2×C32×C4—22no—
81C2×C32×C4—22no—
82C2×C32×C4—21no—
83C2×C32×C4—21no—
84C2×C32×C4#6712yes—
85C2×C32×C4C22×D312yes—
86C2×C3⋊Dic3—22no—
87C2×C3⋊Dic3—22no—
88#634—22no—
89#634—22no—
90C2×C3×Dic3—62no—
91C2×C3×Dic3—62no—
92C3×Dic6—62no—
93C3×Dic6—62no—
94C3×C3⋊C8—34no—
95C3×C3⋊C8—62no—
96C3×C3⋊C8—62no—
97C3×C3⋊C8—62no—
98#467—271noSylow
99C2×H(3,𝔽3)—22no—
100C2×H(3,𝔽3)#12111yescharacteristic
101C2×H(3,𝔽3)C2×Q1611yescharacteristic
102C2×H(3,𝔽3)#13511yescharacteristic
103C2×C3×Q8—92no—
104C22×C3×C4—32no—
105C22×C3×C4—61no—
106C22×C3×C4D3211yescharacteristic
107C2×Dic6—32no—
108C2×Dic6—92no—
109C2×C3×C8—91no—
110C2×C3×C8—91no—
111C2×C3×C8—182no—
112C3×C4⋊C4—92no—
113C3×C4⋊C4—182no—
114C3×C22⋊C4—182no—
115#217—62no—
116#217—182no—
117#216—184no—
118C4⋊Dic3—32no—
119C4⋊Dic3—92no—
120Dic3⋊C4—62no—
121Dic3⋊C4—182no—
122C2×C3⋊C8—32no—
123C2×C3⋊C8—32no—
124C2×C3⋊C8—61no—
125C2×C3⋊C8—61no—
126C2×C3⋊C8—64no—
127C2×C3⋊C8—64no—
128Dic12—182no—
129C22×C32—22no—
130C22×C32—22no—
131C22×C32—22no—
132C22×C32—22no—
133C22×C32—22no—
134C22×C32—21no—
135C22×C32#21614yescomplemented
136C22×C32C2×#6712yes—
137C32×C4—22no—
138C32×C4—22no—
139C32×C4—21no—
140C32×C4—21no—
141C32×C4D3×D412yes—
142C32×C4#23712yes—
143C3⋊Dic3—42no—
144C3⋊Dic3—42no—
145C3×Dic3—122no—
146C3×Dic3—122no—
147C2×Q16—271no—
148C22×C8—91no—
149C22⋊Q8—92no—
150#107—271no—
151#103—272no—
152H(3,𝔽3)#46711yescharacteristic, complemented, Sylow
153C23×C3—32no—
154C23×C3—61no—
155C23×C3#62511yescharacteristic
156C3×Q8—182no—
157C3×Q8—182no—
158C2×C3×C4—21no—
159C2×C3×C4—21no—
160C2×C3×C4—32no—
161C2×C3×C4—32no—
162C2×C3×C4—62no—
163C2×C3×C4—62no—
164C2×C3×C4—62no—
165C2×C3×C4—62no—
166C2×C3×C4—61no—
167C2×C3×C4—61no—
168C2×C3×C4—182no—
169C2×C3×C4—182no—
170C2×C3×C4#62511yescharacteristic
171C2×C3×C4C2×D3211yescharacteristic
172C2×Dic3—62no—
173C2×Dic3—62no—
174C2×Dic3—182no—
175C2×Dic3—182no—
176Dic6—62no—
177Dic6—62no—
178Dic6—182no—
179Dic6—182no—
180C3×C8—92no—
181C3×C8—181no—
182C3⋊C8—34no—
183C3⋊C8—62no—
184C3⋊C8—62no—
185C3⋊C8—62no—
186C2×C32—24no—
187C2×C32—24no—
188C2×C32—21no—
189C2×C32—21no—
190C2×C32—21no—
191C2×C32#93912yes—
192C2×C32C2×#21612yes—
193C2×C32#95012yes—
194C2×Q8—92no—
195C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
196Q16—542no—
197C2×C8—91no—
198C2×C8—91no—
199C2×C8—182no—
200C4⋊C4—92no—
201C4⋊C4—182no—
202C22⋊C4—182no—
203C22×C3—21no—
204C22×C3—21no—
205C22×C3—32no—
206C22×C3—34no—
207C22×C3—62no—
208C22×C3—62no—
209C22×C3—62no—
210C22×C3—62no—
211C22×C3—61no—
212C22×C3—62no—
213C22×C3#371812yes—
214C22×C3C2×#62511yescharacteristic
215C3×C4—21no—
216C3×C4—32no—
217C3×C4—32no—
218C3×C4—62no—
219C3×C4—62no—
220C3×C4—61no—
221C3×C4—61no—
222C3×C4—362no—
223C3×C4—362no—
224C3×C4#380511yescharacteristic
225C3×C4#381111yescharacteristic
226Dic3—122no—
227Dic3—122no—
228Dic3—362no—
229Dic3—362no—
230C32—21no—
231C32#505112yescomplemented
232C23#619811yescharacteristic
233Q8—182no—
234Q8—182no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—182no—
238C2×C4—182no—
239C2×C4#619811yescharacteristic
240C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
241C8—92no—
242C8—181no—
243C2×C3—22no—
244C2×C3—32no—
245C2×C3—32no—
246C2×C3—32no—
247C2×C3—64no—
248C2×C3—64no—
249C2×C3—61no—
250C2×C3—61no—
251C2×C3—61no—
252C2×C3#6382211yescharacteristic
253C2×C3C2×#371811yescharacteristic
254C2×C3#6398011yescharacteristic
255C22—21no—
256C22—21no—
257C22#8843312yescomplemented
258C22C2×#619811yescharacteristic
259C4—21no—
260C4—362no—
261C4—362no—
262C4#8867211yescharacteristic
263C4#8867711yescharacteristic
264C3—32no—
265C3—61no—
266C3#1059118811yescharacteristic
267C2—22no—
268C2#1172106011yescharacteristic
269C2C2×#8843311yescharacteristic
270C2#1172131811yescharacteristic
subgroupG/H
centerC22C2×#6198
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#625
prime radicalC23×H(3,𝔽3)D4
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#467
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. C2×#88433C2semidirect
4. #11720315C2semidirect
5. #11720313C2semidirect
56. C22×H(3,𝔽3)Q16semidirect
135. C22×C32#216semidirect
152. H(3,𝔽3)#467semidirect
231. C32#5051semidirect
257. C22#88433semidirect

Maximal subgroups

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

subgroupindex
C22×#61802
#64133⋊C32
C2×#884332
#117203152
#117203132
#105911873