Finite Group Databasebeta

← #13138800

Subgroups — #13138800

Order 1728.

541 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 541 conjugacy classes of subgroups — too many to draw. Only its 83 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×#3744⋊C3C212yesmaximal, complemented
2#11721144C211yesmaximal, characteristic
3#11721099C211yesmaximal, characteristic, complemented
4C2×#88431C212yesmaximal, complemented
5C8×#1084C211yesmaximal, characteristic, complemented
6#10590877—32nomaximal
7C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
8C2×#634⋊C3C2212yes—
9C2×#6198—22no—
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11C4×#1084C2211yescharacteristic
12C2×#6180C2211yescharacteristic
13#3744⋊C3—24no—
14#88449—22no—
15#88431—24no—
16C2×#3744D312yescomplemented
17C2×C3×#204—32no—
18#63911—32no—
19#63866—32no—
20C2×#3717—32no—
21C2×#3716—32no—
22C3×C8×Dic3—32no—
23C3×C8×Dic3—61no—
24C22×(C3⋊D3)⋊C3—22no—
25C2×C4×H(3,𝔽3)C2311yescharacteristic
26C2×#1084D412yes—
27C2×(C3⋊Dic3)⋊C3—22no—
28C4⋊((C3⋊D3)⋊C3)—22no—
29C4⋊((C3⋊D3)⋊C3)—22no—
30#634⋊C3—22no—
31#634⋊C3—22no—
32#6198—44no—
33C8×H(3,𝔽3)D412yescomplemented
34#6180D412yescomplemented
35#5175—92no—
36#4693—91no—
37C2×C4⋊(C3⋊D3)C2×D312yes—
38C2×#634C2×D312yes—
39C2×C3×D12—32no—
40C2×C3×Dic6—32no—
41C2×#626—64no—
42C2×C32×C8—21no—
43C2×C32×C8C2×D312yes—
44#3744—24no—
45C3×C4×Dic3—32no—
46C3×C4×Dic3—61no—
47C2×C3×C3⋊C8—32no—
48C2×C3×C3⋊C8—61no—
49C3×#204—64no—
50Dic3⋊Dic3—64no—
51#3717—64no—
52#3716—64no—
53C22×H(3,𝔽3)C2×D411yescharacteristic
54C2×(C3⋊D3)⋊C3—42no—
55C2×(C3⋊D3)⋊C3—44no—
56C4×H(3,𝔽3)C2×D411yescharacteristic
57C4×H(3,𝔽3)C2×D411yescharacteristic
58#1084SD1614yescomplemented
59(C3⋊Dic3)⋊C3—42no—
60C2×C3×SD16—92no—
61#944—92no—
62C2×#215—92no—
63#940—92no—
64C2×#214—92no—
65C2×#204—32no—
66C2×#204—92no—
67C2×#204—92no—
68#874—91no—
69#869—91no—
70C3×C4×C8—91no—
71C8×Dic3—32no—
72C8×Dic3—61no—
73C22×C3⋊D3—22no—
74C22×C3×D3—62no—
75C2×C32×C4—21no—
76C2×C32×C4C22×D312yes—
77C2×C3⋊Dic3—22no—
78C4⋊(C3⋊D3)—22no—
79C4⋊(C3⋊D3)—22no—
80#634—22no—
81#634—22no—
82C2×C3×Dic3—34no—
83C2×C3×Dic3—62no—
84C2×C3×Dic3—62no—
85C3×D12—62no—
86C3×D12—62no—
87C3×Dic6—62no—
88C3×Dic6—62no—
89#626—128no—
90C32×C8—22no—
91C32×C8#6714yescomplemented
92C3×C3⋊C8—34no—
93C3×C3⋊C8—62no—
94#492—271noSylow
95C2×H(3,𝔽3)#12711yescharacteristic
96C2×H(3,𝔽3)C2×SD1612yes—
97(C3⋊D3)⋊C3—82no—
98C2×C3×Q8—92no—
99C2×C3×D4—92no—
100C2×#67—184no—
101C2×D12—32no—
102C2×D12—92no—
103C2×D12—182no—
104C2×Dic6—32no—
105C2×Dic6—92no—
106C3×SD16—184no—
107C2×C3×C8—32no—
108C2×C3×C8—61no—
109C2×C3×C8—91no—
110C2×C3×C8D3211yescharacteristic
111C3×C42—91no—
112#215—184no—
113#214—184no—
114C4⋊Dic3—182no—
115Dic3⋊C4—184no—
116C4×Dic3—32no—
117C4×Dic3—61no—
118C2×C3⋊C8—32no—
119C2×C3⋊C8—61no—
120#204—64no—
121#204—184no—
122#204—184no—
123C22×C32—21no—
124C22×C32D3×D412yes—
125C2×C3⋊D3—44no—
126C2×C3⋊D3—42no—
127C2×C3×D3—124no—
128C2×C3×D3—122no—
129C32×C4—21no—
130C32×C4—21no—
131C32×C4D3×D412yes—
132C32×C4C2×#6712yes—
133C3⋊Dic3—42no—
134C3×Dic3—38no—
135C3×Dic3—64no—
136C3×Dic3—122no—
137C2×SD16—92no—
138C2×SD16—272no—
139C4⋊Q8—271no—
140#127—271no—
141C4×C8—91no—
142H(3,𝔽3)#49211yescharacteristic, complemented, Sylow
143C23×C3—182no—
144C22×D3—62no—
145C22×D3—182no—
146C3×Q8—182no—
147C3×Q8—182no—
148C3×D4—182no—
149C3×D4—182no—
150C2×C3×C4—32no—
151C2×C3×C4—61no—
152C2×C3×C4—92no—
153C2×C3×C4—182no—
154C2×C3×C4C2×D3211yescharacteristic
155#67—368no—
156C2×Dic3—34no—
157C2×Dic3—62no—
158C2×Dic3—62no—
159C2×Dic3—182no—
160D12—62no—
161D12—62no—
162D12—182no—
163D12—182no—
164D12—364no—
165Dic6—62no—
166Dic6—62no—
167Dic6—182no—
168Dic6—182no—
169C3×C8—34no—
170C3×C8—62no—
171C3×C8—92no—
172C3×C8#62512yes—
173C3⋊C8—34no—
174C3⋊C8—62no—
175C2×C32—22no—
176C2×C32—21no—
177C2×C32D3×SD1614yes—
178C2×C32#94012yes—
179C3⋊D3—82no—
180C3×D3—242no—
181C2×Q8—92no—
182C2×D4—92no—
183C2×D4—542no—
184SD16—184no—
185SD16—544no—
186C2×C8—91no—
187C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
188C4⋊C4—542no—
189C42—91no—
190C22×C3—32no—
191C22×C3—61no—
192C22×C3—364no—
193C22×C3—362no—
194C22×C3#380211yescharacteristic
195C2×D3—124no—
196C2×D3—122no—
197C2×D3—362no—
198C2×D3—364no—
199C3×C4—32no—
200C3×C4—32no—
201C3×C4—61no—
202C3×C4—61no—
203C3×C4—94no—
204C3×C4—362no—
205C3×C4#380211yescharacteristic
206C3×C4C2×#62511yescharacteristic
207Dic3—38no—
208Dic3—64no—
209Dic3—122no—
210Dic3—362no—
211C32—21no—
212C32#517512yescomplemented
213C23—182no—
214Q8—182no—
215Q8—182no—
216D4—182no—
217D4—182no—
218D4—1084no—
219C2×C4—92no—
220C2×C4—182no—
221C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
222C8—92no—
223C8#619812yescomplemented
224C2×C3—34no—
225C2×C3—32no—
226C2×C3—62no—
227C2×C3—61no—
228C2×C3—722no—
229C2×C3#6379612yes—
230C2×C3#6391411yescharacteristic
231D3—242no—
232D3—722no—
233C22—364no—
234C22—362no—
235C22#8865511yescharacteristic
236C4—94no—
237C4—362no—
238C4#8865511yescharacteristic
239C4C2×#619811yescharacteristic
240C3—32no—
241C3—61no—
242C3#1059088411yescharacteristic
243C2—722no—
244C2#1172083712yes—
245C2#1172114411yescharacteristic
subgroupG/H
centerC22#88655
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radical#11721144C2
hypercenterC2×C8((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×C8

Complements

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

NH ≅ G/Ntype
1. C2×#3744⋊C3C2semidirect
3. #11721099C2semidirect
4. C2×#88431C2semidirect
5. C8×#1084C2semidirect
16. C2×#3744D3semidirect
33. C8×H(3,𝔽3)D4semidirect
34. #6180D4semidirect
58. #1084SD16semidirect
91. C32×C8#67semidirect
142. H(3,𝔽3)#492semidirect
212. C32#5175semidirect
223. C8#6198semidirect

Maximal subgroups

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

subgroupindex
C2×#3744⋊C32
#117211442
#117210992
C2×#884312
C8×#10842
#105908773