Finite Group Databasebeta

← #13140046

Subgroups — #13140046

Order 1728.

453 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 453 conjugacy classes of subgroups — too many to draw. Only its 77 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×#88538C211yesmaximal, characteristic
2#64133⋊C3C212yesmaximal
3C2×#88433C211yesmaximal, characteristic, complemented
4#11720314C211yesmaximal, characteristic, complemented
5#11720313C212yesmaximal, complemented
6#10591121—32nomaximal
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C2×#634⋊C3C2212yes—
9#88538—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—
18C2×C3×#208—32no—
19C2×C3×#208—61no—
20C2×#3719—32no—
21#63573—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#5137—91no—
36#5052—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—
47C3×#208—64no—
48C3×#208—64no—
49C2×C3×C3⋊C8—32no—
50C2×C3×C3⋊C8—32no—
51C2×C3×C3⋊C8—61no—
52C2×C3×C3⋊C8—61no—
53#3719—64no—
54C22×H(3,𝔽3)—21no—
55C22×H(3,𝔽3)—22no—
56C22×H(3,𝔽3)C2×D411yescharacteristic
57C4×H(3,𝔽3)—21no—
58C4×H(3,𝔽3)C2×D411yescharacteristic
59C4×H(3,𝔽3)#4111yescharacteristic
60(C3⋊Dic3)⋊C3—42no—
61(C3⋊Dic3)⋊C3—42no—
62C2×C3×M16—91no—
63C3×C22⋊Q8—92no—
64C2×#216—92no—
65#924—32no—
66#924—92no—
67C2×#208—32no—
68C2×#208—61no—
69C2×Dic12—91no—
70#835—92no—
71#817—91no—
72#816—92no—
73#810—92no—
74#808—92no—
75C23×C32—21no—
76C23×C32#6712yes—
77C2×C32×C4—22no—
78C2×C32×C4—22no—
79C2×C32×C4—22no—
80C2×C32×C4—22no—
81C2×C32×C4—21no—
82C2×C32×C4—21no—
83C2×C32×C4#6712yes—
84C2×C32×C4C22×D312yes—
85C2×C3⋊Dic3—22no—
86C2×C3⋊Dic3—22no—
87#634—22no—
88#634—22no—
89C2×C3×Dic3—62no—
90C2×C3×Dic3—62no—
91C3×Dic6—62no—
92C3×Dic6—62no—
93C3×C3⋊C8—34no—
94C3×C3⋊C8—62no—
95C3×C3⋊C8—62no—
96C3×C3⋊C8—62no—
97#470—271noSylow
98C2×H(3,𝔽3)—41no—
99C2×H(3,𝔽3)#12111yescharacteristic
100C2×H(3,𝔽3)#13712yes—
101C2×C3×Q8—92no—
102C22×C3×C4—32no—
103C22×C3×C4—61no—
104C22×C3×C4D3211yescharacteristic
105C2×Dic6—32no—
106C2×Dic6—92no—
107C3×M16—182no—
108C2×C3×C8—91no—
109C2×C3×C8—91no—
110C3×C4⋊C4—92no—
111C3×C4⋊C4—182no—
112C3×C22⋊C4—182no—
113#217—62no—
114#217—182no—
115#216—184no—
116C4⋊Dic3—32no—
117C4⋊Dic3—92no—
118Dic3⋊C4—62no—
119Dic3⋊C4—182no—
120#208—64no—
121#208—64no—
122C2×C3⋊C8—32no—
123C2×C3⋊C8—32no—
124C2×C3⋊C8—61no—
125C2×C3⋊C8—61no—
126Dic12—182no—
127C22×C32—22no—
128C22×C32—22no—
129C22×C32—22no—
130C22×C32—22no—
131C22×C32—21no—
132C22×C32—42no—
133C22×C32C2×#6712yes—
134C32×C4—22no—
135C32×C4—22no—
136C32×C4—21no—
137C32×C4—21no—
138C32×C4D3×D412yes—
139C32×C4#23712yes—
140C3⋊Dic3—42no—
141C3⋊Dic3—42no—
142C3×Dic3—122no—
143C3×Dic3—122no—
144C2×Q16—271no—
145C2×M16—91no—
146C22⋊Q8—92no—
147#106—271no—
148#103—272no—
149H(3,𝔽3)#47011yescharacteristic, complemented, Sylow
150C23×C3—32no—
151C23×C3—61no—
152C23×C3#62511yescharacteristic
153C3×Q8—182no—
154C3×Q8—182no—
155C2×C3×C4—21no—
156C2×C3×C4—21no—
157C2×C3×C4—32no—
158C2×C3×C4—32no—
159C2×C3×C4—62no—
160C2×C3×C4—62no—
161C2×C3×C4—62no—
162C2×C3×C4—62no—
163C2×C3×C4—61no—
164C2×C3×C4—61no—
165C2×C3×C4—182no—
166C2×C3×C4—182no—
167C2×C3×C4#62511yescharacteristic
168C2×C3×C4C2×D3211yescharacteristic
169C2×Dic3—62no—
170C2×Dic3—62no—
171C2×Dic3—182no—
172C2×Dic3—182no—
173Dic6—62no—
174Dic6—62no—
175Dic6—182no—
176Dic6—182no—
177C3×C8—92no—
178C3×C8—181no—
179C3⋊C8—34no—
180C3⋊C8—62no—
181C3⋊C8—62no—
182C3⋊C8—62no—
183C2×C32—22no—
184C2×C32—21no—
185C2×C32—42no—
186C2×C32—42no—
187C2×C32#93912yes—
188C2×C32#94212yes—
189C2×C32#95112yes—
190C2×Q8—92no—
191C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
192Q16—542no—
193M16—182no—
194C2×C8—91no—
195C2×C8—91no—
196C4⋊C4—92no—
197C4⋊C4—182no—
198C22⋊C4—182no—
199C22×C3—22no—
200C22×C3—21no—
201C22×C3—32no—
202C22×C3—62no—
203C22×C3—62no—
204C22×C3—62no—
205C22×C3—62no—
206C22×C3—61no—
207C22×C3—122no—
208C22×C3C2×#62511yescharacteristic
209C3×C4—21no—
210C3×C4—32no—
211C3×C4—32no—
212C3×C4—62no—
213C3×C4—62no—
214C3×C4—61no—
215C3×C4—61no—
216C3×C4—362no—
217C3×C4—362no—
218C3×C4#380511yescharacteristic
219C3×C4#381111yescharacteristic
220Dic3—122no—
221Dic3—122no—
222Dic3—362no—
223Dic3—362no—
224C32—21no—
225C32#505212yescomplemented
226C23#619811yescharacteristic
227Q8—182no—
228Q8—182no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—182no—
232C2×C4—182no—
233C2×C4#619811yescharacteristic
234C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
235C8—92no—
236C8—181no—
237C2×C3—32no—
238C2×C3—32no—
239C2×C3—32no—
240C2×C3—41no—
241C2×C3—62no—
242C2×C3—61no—
243C2×C3—122no—
244C2×C3—122no—
245C2×C3#6382712yes—
246C2×C3#6398011yescharacteristic
247C22—22no—
248C22—21no—
249C22C2×#619811yescharacteristic
250C4—21no—
251C4—362no—
252C4—362no—
253C4#8867211yescharacteristic
254C4#8867711yescharacteristic
255C3—32no—
256C3—61no—
257C3#1059118911yescharacteristic
258C2—41no—
259C2#1172105812yes—
260C2#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)#470
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. #11720314C2semidirect
5. #11720313C2semidirect
149. H(3,𝔽3)#470semidirect
225. C32#5052semidirect

Maximal subgroups

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

subgroupindex
C2×#885382
#64133⋊C32
C2×#884332
#117203142
#117203132
#105911213