Finite Group Databasebeta

← #13140970

Subgroups — #13140970

Order 1728.

565 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 565 conjugacy classes of subgroups — too many to draw. Only its 79 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#11721861C211yesmaximal, characteristic, complemented
2C2×#3755⋊C3C212yesmaximal, complemented
3#11721144C211yesmaximal, characteristic
4C2×#88438C212yesmaximal, complemented
5#11720302C211yesmaximal, characteristic, complemented
6#10591618—32nomaximal
7C2×Q8×H(3,𝔽3)C2211yescharacteristic
8C2×#6229C2211yescharacteristic
9C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
10C2×#6198—22no—
11#88540—22no—
12C4×#1084C2211yescharacteristic
13#3755⋊C3—24no—
14C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
15#88438—24no—
16C2×#3755D312yescomplemented
17C3×#944—32no—
18C3×#944—61no—
19C2×C3×#215—32no—
20#63911—32no—
21C2×#3717—32no—
22C2×#3715—32no—
23Dic3×C3⋊C8—32no—
24C22×(C3⋊D3)⋊C3—22no—
25Q8×H(3,𝔽3)—22no—
26C2×C4×H(3,𝔽3)—21no—
27C2×C4×H(3,𝔽3)C2311yescharacteristic
28C2×#1084—21no—
29C2×#1084D412yes—
30#6229—22no—
31C4⋊((C3⋊D3)⋊C3)—22no—
32C4⋊((C3⋊D3)⋊C3)—22no—
33#6198—44no—
34(C3⋊(C3⋊C8))⋊C3D414yescomplemented
35#5175—92no—
36#5085—91no—
37C2×C32×Q8—21no—
38C2×C32×Q8C2×D312yes—
39C2×C4⋊(C3⋊D3)C2×D312yes—
40C2×C3×D12—32no—
41C2×C3×Dic6—32no—
42C2×C3×Dic6—61no—
43C2×#626—64no—
44#3755—24no—
45C2×C3⋊(C3⋊C8)C2×D312yes—
46C3×#215—64no—
47C3×Dic3⋊C4—64no—
48C3×Dic3⋊C4—64no—
49C3×C4×Dic3—32no—
50C3×C4×Dic3—61no—
51C2×C3×C3⋊C8—32no—
52#3717—64no—
53#3715—64no—
54C22×H(3,𝔽3)C2×D411yescharacteristic
55C2×(C3⋊D3)⋊C3—42no—
56C2×(C3⋊D3)⋊C3—44no—
57C4×H(3,𝔽3)—41no—
58C4×H(3,𝔽3)C2×D412yes—
59#1084—41no—
60#1084SD1614yescomplemented
61C2×C3×SD16—92no—
62C3×C4⋊Q8—91no—
63#944—32no—
64#944—61no—
65C2×#215—32no—
66C2×#215—92no—
67C2×#215—92no—
68#940—92no—
69C2×#214—92no—
70C2×#204—92no—
71#874—91no—
72C8×Dic3—92no—
73C4×C3⋊C8—91no—
74C22×C3⋊D3—22no—
75C22×C3×D3—62no—
76C32×Q8—22no—
77C32×Q8—22no—
78C32×Q8—42no—
79C2×C32×C4—22no—
80C2×C32×C4—22no—
81C2×C32×C4—21no—
82C2×C32×C4C22×D312yes—
83C4⋊(C3⋊D3)—22no—
84C4⋊(C3⋊D3)—22no—
85C2×C3×Dic3—34no—
86C2×C3×Dic3—62no—
87C2×C3×Dic3—62no—
88C2×C3×Dic3—62no—
89C3×D12—62no—
90C3×D12—62no—
91C3×Dic6—62no—
92C3×Dic6—62no—
93C3×Dic6—122no—
94#626—128no—
95C3⋊(C3⋊C8)#6714yescomplemented
96C3×C3⋊C8—34no—
97#492—271noSylow
98C2×H(3,𝔽3)#12711yescharacteristic
99C2×H(3,𝔽3)C2×SD1612yes—
100(C3⋊D3)⋊C3—82no—
101C2×C3×Q8—32no—
102C2×C3×Q8—61no—
103C2×C3×Q8—91no—
104C2×C3×Q8D3211yescharacteristic
105C2×C3×D4—92no—
106C2×#67—184no—
107C2×D12—32no—
108C2×D12—92no—
109C2×D12—182no—
110C2×Dic6—32no—
111C2×Dic6—61no—
112C3×SD16—184no—
113C2×C3×C8—92no—
114C3×C4⋊C4—182no—
115C3×C42—91no—
116#215—64no—
117#215—184no—
118#215—184no—
119#214—184no—
120Dic3⋊C4—64no—
121Dic3⋊C4—64no—
122C4×Dic3—32no—
123C4×Dic3—61no—
124C2×C3⋊C8—32no—
125C2×C3⋊C8—92no—
126#204—184no—
127C22×C32—21no—
128C22×C32D3×D412yes—
129C2×C3⋊D3—44no—
130C2×C3⋊D3—42no—
131C2×C3×D3—124no—
132C2×C3×D3—122no—
133C32×C4—22no—
134C32×C4—42no—
135C32×C4—42no—
136C32×C4D3×D412yes—
137C32×C4C2×#6712yes—
138C3×Dic3—38no—
139C3×Dic3—64no—
140C3×Dic3—122no—
141C3×Dic3—122no—
142C2×SD16—92no—
143C2×SD16—272no—
144C4⋊Q8—91no—
145#127—271no—
146C4×C8—271no—
147H(3,𝔽3)#49211yescharacteristic, complemented, Sylow
148C23×C3—182no—
149C22×D3—62no—
150C22×D3—182no—
151C3×Q8—22no—
152C3×Q8—62no—
153C3×Q8—62no—
154C3×Q8—122no—
155C3×Q8—182no—
156C3×D4—182no—
157C3×D4—182no—
158C2×C3×C4—21no—
159C2×C3×C4—32no—
160C2×C3×C4—62no—
161C2×C3×C4—62no—
162C2×C3×C4—61no—
163C2×C3×C4—92no—
164C2×C3×C4—181no—
165C2×C3×C4C2×D3211yescharacteristic
166#67—368no—
167C2×Dic3—34no—
168C2×Dic3—62no—
169C2×Dic3—62no—
170C2×Dic3—62no—
171D12—62no—
172D12—62no—
173D12—182no—
174D12—182no—
175D12—364no—
176Dic6—62no—
177Dic6—62no—
178Dic6—122no—
179C3×C8—94no—
180C3⋊C8—34no—
181C3⋊C8—94no—
182C2×C32—22no—
183C2×C32—21no—
184C2×C32D3×SD1614yes—
185C2×C32#94012yes—
186C3⋊D3—82no—
187C3×D3—242no—
188C2×Q8—91no—
189C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
190C2×D4—92no—
191C2×D4—542no—
192SD16—184no—
193SD16—544no—
194C2×C8—92no—
195C4⋊C4—182no—
196C42—91no—
197C22×C3—32no—
198C22×C3—61no—
199C22×C3—364no—
200C22×C3—362no—
201C22×C3#381111yescharacteristic
202C2×D3—124no—
203C2×D3—122no—
204C2×D3—362no—
205C2×D3—364no—
206C3×C4—32no—
207C3×C4—32no—
208C3×C4—41no—
209C3×C4—62no—
210C3×C4—94no—
211C3×C4—122no—
212C3×C4—122no—
213C3×C4—361no—
214C3×C4D3×#6712yes—
215Dic3—38no—
216Dic3—64no—
217Dic3—122no—
218Dic3—122no—
219C32—21no—
220C32#517512yescomplemented
221C23—182no—
222Q8—22no—
223Q8—182no—
224D4—182no—
225D4—182no—
226D4—1084no—
227C2×C4—21no—
228C2×C4—92no—
229C2×C4—181no—
230C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
231C8—94no—
232C2×C3—34no—
233C2×C3—32no—
234C2×C3—62no—
235C2×C3—61no—
236C2×C3—722no—
237C2×C3#6394012yes—
238C2×C3#6397911yescharacteristic
239D3—242no—
240D3—722no—
241C22—364no—
242C22—362no—
243C22#8867711yescharacteristic
244C4—41no—
245C4—94no—
246C4—361no—
247C4#8867612yes—
248C3—32no—
249C3—61no—
250C3#1059162011yescharacteristic
251C2—722no—
252C2#1172122812yes—
253C2#1172131711yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3#3811
prime radical#11721144C2
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#492
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. #11721861C2semidirect
2. C2×#3755⋊C3C2semidirect
4. C2×#88438C2semidirect
5. #11720302C2semidirect
16. C2×#3755D3semidirect
34. (C3⋊(C3⋊C8))⋊C3D4semidirect
60. #1084SD16semidirect
95. C3⋊(C3⋊C8)#67semidirect
147. H(3,𝔽3)#492semidirect
220. C32#5175semidirect

Maximal subgroups

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

subgroupindex
#117218612
C2×#3755⋊C32
#117211442
C2×#884382
#117203022
#105916183