Finite Group Databasebeta

← #13145685

Subgroups — #13145685

Order 1728.

536 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 98 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
1C3×#64156C211yesmaximal, characteristic
2#11722206C211yesmaximal, characteristic, complemented
3C32×#836C211yesmaximal, characteristic, complemented
4#10589343—31nomaximal
5#10589182—34nomaximal
6C2×C33×Q8C2211yescharacteristic
7C2×C3×C4⋊(C3⋊D3)C411yescharacteristic
8#88855—21no—
9C3×#3752—21no—
10C3×#3752—21no—
11C3×C4×C3⋊Dic3C411yescharacteristic
12C32×#208—21no—
13#64156D311yescharacteristic
14C3×#947—34no—
15C32×#101—31no—
16#63663—31no—
17#63663—34no—
18#63663—64no—
19C3×#836D314yescomplemented
20C22×C3×C3⋊D3—21no—
21C33×Q8—41no—
22C2×C33×C4C2×C411yescharacteristic
23C2×C33×C4D411yescharacteristic
24C2×C33×C4D411yescharacteristic
25C2×C3×C3⋊Dic3—21no—
26C3×C4⋊(C3⋊D3)—41no—
27C3⋊(C3⋊(C3⋊C8))—41no—
28C32×C3⋊C8—41no—
29#4542—91no—
30#4477—94no—
31C2×C32×Q8—24no—
32C2×C32×Q8C2×D314yes—
33C2×C32×Q8C2×D311yescharacteristic
34C2×C4⋊(C3⋊D3)Dic311yescharacteristic
35C2×C3×D12—34no—
36C32×M16—61no—
37#3752—21no—
38#3752—21no—
39C4×C3⋊Dic3Dic311yescharacteristic
40#3748—68no—
41#3748—64no—
42#3748—61no—
43C3×(C2×D3)⋊C4—64no—
44C3×(C2×D3)⋊C4—64no—
45C3×C4×Dic3—34no—
46C3×#208—24no—
47C22×C33C22⋊C411yescharacteristic
48C2×C3×C3⋊D3—42no—
49C2×C3×C3⋊D3—41no—
50C33×C4—21no—
51C33×C4—21no—
52C33×C4—21no—
53C3×C3⋊Dic3—41no—
54C3×#124—91no—
55#947—34no—
56C3×#101—34no—
57#836—94no—
58#836—91no—
59#836—184no—
60#836C3⋊D311yescharacteristic, complemented
61C22×C3⋊D3—21no—
62C22×C3×D3—64no—
63C32×Q8—41no—
64C32×Q8—44no—
65C32×Q8—48no—
66C2×C32×C4—24no—
67C2×C32×C4—24no—
68C2×C32×C4—24no—
69C2×C32×C4C4×D314yes—
70C2×C32×C4D1214yes—
71C2×C32×C4C2×Dic311yescharacteristic
72C2×C32×C4#6714yes—
73C2×C32×C4#6711yescharacteristic
74C2×C32×C4#6711yescharacteristic
75C2×C3⋊Dic3—21no—
76C4⋊(C3⋊D3)—41no—
77C2×C3×Dic3—64no—
78C3×D12—124no—
79C32×C8—121no—
80C3⋊(C3⋊C8)—128no—
81C3⋊(C3⋊C8)—124no—
82C3⋊(C3⋊C8)—121no—
83C3×C3⋊C8—44no—
84#355—271noSylow
85C2×C33—21no—
86C2×C33#9911yescharacteristic
87C3×C3⋊D3—81no—
88C2×C3×Q8—24no—
89C2×C3×Q8D3214yes—
90C2×C3×Q8C2×C3⋊D311yescharacteristic
91C2×C3×D4—91no—
92C2×D12—34no—
93C3×M16—64no—
94C3×C22⋊C4—181no—
95C3×C22⋊C4—181no—
96C3×C42—91no—
97(C2×D3)⋊C4—64no—
98(C2×D3)⋊C4—64no—
99C4×Dic3—34no—
100#208—21no—
101#208—188no—
102#208—184no—
103#208—181no—
104C22×C32—24no—
105C22×C32(C2×D3)⋊C414yes—
106C22×C32#21711yescharacteristic
107C2×C3⋊D3—42no—
108C2×C3⋊D3—41no—
109C2×C3×D3—128no—
110C2×C3×D3—124no—
111C32×C4—28no—
112C32×C4—24no—
113C32×C4—21no—
114C32×C4—21no—
115C32×C4—21no—
116C32×C4—24no—
117C32×C4—24no—
118C32×C4—44no—
119C32×C4—44no—
120C3⋊Dic3—41no—
121C3×Dic3—124no—
122#124—91no—
123#101—31no—
124#101—271no—
125C33#35511yescharacteristic, complemented, Sylow
126C23×C3—181no—
127C22×D3—64no—
128C3×Q8—44no—
129C3×Q8—41no—
130C3×Q8—48no—
131C3×D4—361no—
132C2×C3×C4—24no—
133C2×C3×C4—24no—
134C2×C3×C4—24no—
135C2×C3×C4—181no—
136C2×C3×C4D3×Dic314yes—
137C2×C3×C4#62514yes—
138C2×C3×C4#62614yes—
139C2×C3×C4C4×C3⋊D311yescharacteristic
140C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
141C2×C3×C4#63811yescharacteristic
142C2×Dic3—64no—
143D12—124no—
144C3×C8—124no—
145C3⋊C8—41no—
146C3⋊C8—368no—
147C3⋊C8—364no—
148C3⋊C8—361no—
149C2×C32—21no—
150C2×C32—24no—
151C2×C32—24no—
152C2×C32—44no—
153C2×C32#80614yes—
154C2×C32#83411yescharacteristic
155C3⋊D3—81no—
156C3×D3—244no—
157C2×Q8D3×C3⋊D311yescharacteristic
158C2×D4—91no—
159M16—61no—
160M16—541no—
161C22⋊C4—181no—
162C22⋊C4—181no—
163C42—91no—
164C22×C3—24no—
165C22×C3—362no—
166C22×C3—361no—
167C22×C3(C2×D3)⋊Dic314yes—
168C22×C3#375211yescharacteristic
169C2×D3—128no—
170C2×D3—124no—
171C3×C4—28no—
172C3×C4—24no—
173C3×C4—24no—
174C3×C4—24no—
175C3×C4—21no—
176C3×C4—21no—
177C3×C4—21no—
178C3×C4—44no—
179C3×C4—44no—
180C3×C4—361no—
181Dic3—124no—
182C32—24no—
183C32#447714yescomplemented
184C32#454211yescharacteristic, complemented
185C23—181no—
186Q8—41no—
187D4—361no—
188C2×C4—181no—
189C2×C4Dic3×C3⋊D311yescharacteristic
190C2×C4#629011yescharacteristic
191C2×C4#629211yescharacteristic
192C8—121no—
193C8—1081no—
194C2×C3—21no—
195C2×C3—24no—
196C2×C3—24no—
197C2×C3—44no—
198C2×C3—721no—
199C2×C3#6358014yes—
200C2×C3#6363311yescharacteristic
201D3—244no—
202C22—362no—
203C22—361no—
204C22#8880311yescharacteristic
205C4—21no—
206C4—21no—
207C4—21no—
208C4—361no—
209C3—24no—
210C3#1058918214yescomplemented
211C3#1058934311yescharacteristic, complemented
212C2—21no—
213C2—721no—
214C2#1172206311yescharacteristic
subgroupG/H
centerC2#11722063
commutatorC2×C33×C4C2×C4
Frattini, hypercenterC2×Q8D3×C3⋊D3
Fitting, cosocleC2×C33×Q8C22
socleC2×C33#99
prime radicalC2×C3×C4⋊(C3⋊D3)C4
hypocenterC33#355
LCSG ▷ C2×C33×C4 ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. #11722206C2semidirect
3. C32×#836C2semidirect
19. C3×#836D3semidirect
60. #836C3⋊D3semidirect
125. C33#355semidirect
183. C32#4477semidirect
184. C32#4542semidirect
210. C3#10589182semidirect
211. C3#10589343semidirect

Maximal subgroups

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

subgroupindex
C3×#641562
#117222062
C32×#8362
#105893433
#105891823