Finite Group Databasebeta

← #13146358

Subgroups — #13146358

Order 1728.

236 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 37 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×#63691C211yesmaximal, characteristic
2C3⋊(#131⋊C9)C211yesmaximal, characteristic, complemented
3C3⋊#63422C211yesmaximal, characteristic, complemented
4#10591674—31nomaximal
5#10589507—31nomaximal
6C3×#41⋊C9C2211yescharacteristic
7C3×#3689C411yescharacteristic, complemented
8C3×#3688C411yescharacteristic, complemented
9#88413—41nomaximal
10C3×#950—31no—
11C3×#948D311yescharacteristic
12#63814—31no—
13#63691D311yescharacteristic
14#63664—31no—
15#63577—31no—
16#63568—31no—
17#63555—31no—
18#131⋊C9—31no—
19#63422—31no—
20#63422—61no—
21C3×C4×D9—41no—
22C3×Q8⋊C9C2×C411yescharacteristic
23C3⋊(C9⋊C8)—41no—
24C9×C3⋊C8—41no—
25#5197—91no—
26#5145—91no—
27C32×#41C2×D311yescharacteristic
28C3×#235—31no—
29C4×C3⋊Dic3—31no—
30#3748—31no—
31C3×#216—31no—
32C3×#215—31no—
33C3×#214—31no—
34C3×#213—31no—
35C3×#208—31no—
36C2×C3×C3⋊C8—31no—
37C2×C3×C3⋊C8—31no—
38#3711—61no—
39D3×C3⋊C8—61no—
40#41⋊C9—21no—
41#41⋊C9C2×D311yescharacteristic
42#3689Dic311yescharacteristic, complemented
43#3688Dic311yescharacteristic, complemented
44#3663—121no—
45C2×C3×D9—41no—
46C3×C4×C9—41no—
47C3×Dic9—41no—
48C3×#135—91no—
49C3×#131—31no—
50#950—31no—
51#948—91no—
52#948D911yescharacteristic, complemented
53#901—91no—
54#837—91no—
55#837—91no—
56#837—91no—
57#837—181no—
58#825—91no—
59#822—91no—
60#819—91no—
61#815—91no—
62#815—91no—
63C32×Q8C4×D311yescharacteristic
64C32×D4—31no—
65C2×C32×C4—31no—
66C2×C3⋊Dic3—61no—
67C3×#67—61no—
68C3×D12—31no—
69C3×C4×D3—61no—
70C3×Dic6—31no—
71C3⋊(C3⋊C8)—61no—
72C3×C3⋊C8—31no—
73C3×C3⋊C8—31no—
74C3×C3⋊C8—31no—
75C3×C3⋊C8S411yescharacteristic
76C4×D9—41no—
77Q8⋊C9—21no—
78Q8⋊C9C2×Dic311yescharacteristic
79C8×C9—121no—
80C9⋊C8—121no—
81C9⋊C8—241no—
82#444—271noSylow
83C2×C3×C9—41no—
84C3×D9—81no—
85C3×#41—21no—
86C3×#41—91no—
87C3×#41C2×D911yescharacteristic
88C3×#41D3211yescharacteristic
89#235—31no—
90C3×Q16—91no—
91C3×SD16—91no—
92C3×SD16—91no—
93C3×D8—91no—
94C3×M16—91no—
95C2×C3×C8—91no—
96C2×C3×C8—91no—
97#216—31no—
98#215—31no—
99#214—31no—
100#213—31no—
101C4×Dic3—91no—
102C4×Dic3—91no—
103C4×Dic3—181no—
104#208—31no—
105#208—91no—
106#208—91no—
107#208—181no—
108#208—181no—
109C2×C3⋊C8—31no—
110C2×C3⋊C8—31no—
111C2×C3⋊C8—181no—
112ΓL(1,𝔽25)—181no—
113D3×C8—181no—
114C22×C32—31no—
115C2×C3×D3—61no—
116C32×C4—31no—
117C32×C4C2×S411yescharacteristic
118C3⋊Dic3—121no—
119C3×Dic3—61no—
120C2×D9—41no—
121C4×C9—41no—
122C4×C9—81no—
123Dic9—41no—
124#135—91no—
125#131—31no—
126#131—271no—
127#108—271no—
128#104—271no—
129#104—271no—
130#97—271no—
131C3×C9—41noSylow
132C3×Q8—21no—
133C3×Q8—91no—
134C3×Q8C4×D911yescharacteristic
135C3×Q8D3×Dic311yescharacteristic
136C3×D4—31no—
137C3×D4—31no—
138C3×D4—61no—
139C3×D4—91no—
140C3×D4—181no—
141C2×C3×C4—31no—
142C2×C3×C4—31no—
143C2×C3×C4—61no—
144C2×C3×C4—181no—
145#67—61no—
146C2×Dic3—181no—
147C2×Dic3—181no—
148C2×Dic3—361no—
149D12—31no—
150C4×D3—61no—
151Dic6—31no—
152C3×C8—31no—
153C3×C8—91no—
154C3×C8—91no—
155C3×C8—91no—
156C3⋊C8—31no—
157C3⋊C8—31no—
158C3⋊C8—31no—
159C3⋊C8—181no—
160C3⋊C8—181no—
161C3⋊C8—361no—
162C3⋊C8(C22⋊C9)⋊C211yescharacteristic
163C2×C32—61no—
164C2×C32C4×S411yescharacteristic
165C3×D3—121no—
166C2×C9—41no—
167C2×C9—81no—
168D9—81no—
169#41—91no—
170#41D3×D911yescharacteristic
171Q16—91no—
172SD16—91no—
173SD16—91no—
174D8—91no—
175M16—91no—
176M16—271no—
177M16—541no—
178C2×C8—91no—
179C2×C8—91no—
180C2×C8—541no—
181C42—271no—
182C22×C3—31no—
183C22×C3—31no—
184C22×C3—61no—
185C22×C3—181no—
186C2×D3—61no—
187C3×C4—21no—
188C3×C4—31no—
189C3×C4—31no—
190C3×C4—61no—
191C3×C4—181no—
192C3×C4C2×(C22⋊C9)⋊C211yescharacteristic
193C3×C4D3×S411yescharacteristic
194Dic3—61no—
195Dic3—361no—
196Dic3—361no—
197Dic3—362no—
198C32#540511yescharacteristic
199C9—41no—
200C9—81no—
201Q8—91no—
202Q8Dic3×D911yescharacteristic
203D4—31no—
204D4—91no—
205D4—181no—
206C2×C4—31no—
207C2×C4—181no—
208C2×C4—541no—
209C8—31no—
210C8—91no—
211C8—91no—
212C8—91no—
213C8—541no—
214C2×C3—21no—
215C2×C3—61no—
216C2×C3—61no—
217C2×C3—121no—
218C2×C3—361no—
219C2×C3C4×(C22⋊C9)⋊C211yescharacteristic
220C2×C3Dic3×S411yescharacteristic
221D3—121no—
222C22—31no—
223C22—181no—
224C4—31no—
225C4—181no—
226C4—1081no—
227C4D3×(C22⋊C9)⋊C211yescharacteristic
228C3—21no—
229C3#1058950711yescharacteristic, complemented
230C3#1059322111yescharacteristic
231C2—61no—
232C2—361no—
233C2Dic3×(C22⋊C9)⋊C211yescharacteristic
subgroupG/H
center, hypercenterC4D3×(C22⋊C9)⋊C2
commutator, hypocenterC3×Q8⋊C9C2×C4
FrattiniC3×C4D3×S4
FittingC32×#41C2×D3
socleC2×C32C4×S4
prime radicalC3×#63691C2
cosocleC3×#41⋊C9C22
LCSG ▷ C3×Q8⋊C9
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
2. C3⋊(#131⋊C9)C2semidirect
3. C3⋊#63422C2semidirect
7. C3×#3689C4semidirect
8. C3×#3688C4semidirect
42. #3689Dic3semidirect
43. #3688Dic3semidirect
52. #948D9semidirect
229. C3#10589507semidirect

Maximal subgroups

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

subgroupindex
C3×#636912
C3⋊(#131⋊C9)2
C3⋊#634222
#105916743
#105895073
#884134