Finite Group Databasebeta

← #10594008⋊C3 #13166120

Subgroups — #10594008⋊C3 #13166120

Order 1728.

490 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 35 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⋊D3×C42⋊C3C211yesmaximal, characteristic, complemented
2C3×C3⋊(#127⋊C3)C212yesmaximal, complemented
3#10594008C311yesmaximal, characteristic, complemented
4D3×#127⋊C3—32nomaximal
5D32×A4—41nomaximal
6C32×C42⋊C3C2211yescharacteristic, complemented
7C2×#3802—31no—
8C2×#3802—31no—
9C42×C3⋊D3C2×C311yescharacteristic, complemented
10C3×#874C2×C312yescomplemented
11C3×#127⋊C3—32no—
12#63914—32no—
13D3×C42⋊C3—32no—
14D3×C42⋊C3—61no—
15C3⋊(#127⋊C3)—22no—
16C3⋊(#127⋊C3)D312yescomplemented
17A4×C3⋊D3—41no—
18C3×D3×A4—42no—
19D3×#127—32no—
20C2×#127⋊C3—91no—
21C22×D32—31no—
22C22×D32A411yescharacteristic
23C2×D3×A4—122no—
24C2×C4×C3⋊D3—31no—
25C2×C3×D12—32no—
26C2×C3×D12—32no—
27C2×#625—32no—
28C2×#625—32no—
29#3802—62no—
30#3802—62no—
31#3802—62no—
32#3802—62no—
33C32×C42C22×C311yescharacteristic, complemented
34C4×C3⋊Dic3—31no—
35C3×C42⋊C3—22no—
36C3×C42⋊C3—21no—
37C3×C42⋊C3—41no—
38C3×C42⋊C3C2×D312yescomplemented
39C32×A4—41no—
40C3×D32—161no—
41C2×D3×D4—92no—
42C2×D3×D4—92no—
43C3×#127—32no—
44#940—92no—
45#940—92no—
46#874C3×D312yescomplemented
47C42×D3—32no—
48C42×D3—61no—
49#127⋊C3—32no—
50#127⋊C3—62no—
51C2×C42⋊C3—91no—
52C22×C3⋊D3C2×A411yescharacteristic
53C22×C3×D3—32no—
54C22×C3×D3C2×A412yes—
55C2×C3×A4—122no—
56C2×D32—61no—
57C2×D32—62no—
58C2×D32—61no—
59C2×D32—61no—
60C2×D32—62no—
61C2×D32—61no—
62D3×A4—42no—
63D3×A4—82no—
64D3×A4—122no—
65D3×A4—241no—
66C2×C32×C4—31no—
67C2×C3⋊Dic3—31no—
68C4×C3⋊D3—32no—
69C4×C3⋊D3—32no—
70C3×D12—64no—
71C3×D12—64no—
72#625—64no—
73#625—64no—
74C2×#127—91noSylow
75C3×C3⋊D3—161no—
76C32×D3—162no—
77C23×D3—32no—
78C23×D3—92no—
79C22×A4—361no—
80C2×C3×D4—92no—
81C2×C3×D4—92no—
82C2×#67—92no—
83C2×#67—94no—
84C2×#67—92no—
85D3×D4—184no—
86D3×D4—184no—
87D3×D4—184no—
88D3×D4—184no—
89C2×D12—32no—
90C2×D12—32no—
91C2×C4×D3—92no—
92C2×C4×D3—181no—
93C3×C42—21no—
94C3×C42C2×C3×D312yescomplemented
95C4×Dic3—92no—
96C4×Dic3—181no—
97C42⋊C3—22no—
98C42⋊C3—41no—
99C42⋊C3D3211yescharacteristic, complemented
100C22×C32C22×A411yescharacteristic
101C2×C3⋊D3—31no—
102C2×C3⋊D3—31no—
103C2×C3×D3—62no—
104C2×C3×D3—62no—
105C2×C3×D3—64no—
106C2×C3×D3—482no—
107C3×A4—42no—
108C3×A4—81no—
109C3×A4—82no—
110C3×A4—161no—
111D32—41no—
112D32—121no—
113D32—121no—
114D32—122no—
115D32—121no—
116C32×C4—32no—
117C3⋊Dic3—32no—
118C22×D4—271no—
119C22×D4—271no—
120#127—32no—
121#127—272no—
122C2×C42—91no—
123C33—161noSylow
124C23×C3—32no—
125C23×C3—92no—
126C22×D3—32no—
127C22×D3—32no—
128C22×D3—61no—
129C22×D3—182no—
130C22×D3—182no—
131C22×D3—184no—
132C22×D3—184no—
133C22×D3—182no—
134C22×D3—182no—
135C22×D3D3×A412yes—
136C2×A4—122no—
137C2×A4—242no—
138C2×A4—361no—
139C3×D4—184no—
140C3×D4—184no—
141C2×C3×C4—32no—
142C2×C3×C4—61no—
143#67—184no—
144#67—184no—
145#67—188no—
146C2×Dic3—92no—
147C2×Dic3—181no—
148D12—64no—
149D12—64no—
150C4×D3—94no—
151C4×D3—94no—
152C4×D3—182no—
153C4×D3—182no—
154C2×C32—31no—
155C2×C32—482no—
156C3⋊D3—31no—
157C3⋊D3#127⋊C311yescharacteristic, complemented
158C3×D3—42no—
159C3×D3—122no—
160C3×D3—162no—
161C3×D3—322no—
162C3×D3—482no—
163C3×D3—961no—
164C24—91no—
165C24—271no—
166C2×D4—92no—
167C2×D4—92no—
168C2×D4—272no—
169C2×D4—272no—
170C2×D4—542no—
171C2×D4—542no—
172C2×D4—542no—
173C2×D4—542no—
174C22×C4—271no—
175C42—271no—
176C42C3×D3211yescharacteristic, complemented
177C22×C3—21no—
178C22×C3—182no—
179C22×C3—182no—
180C22×C3—184no—
181C22×C3—1441no—
182C22×C3C2×D3×A412yes—
183C2×D3—62no—
184C2×D3—62no—
185C2×D3—64no—
186C2×D3—92no—
187C2×D3—92no—
188C2×D3—122no—
189C2×D3—181no—
190C2×D3—181no—
191C2×D3—362no—
192C2×D3—362no—
193C2×D3—362no—
194C2×D3—364no—
195A4—41no—
196A4—82no—
197A4—161no—
198C3×C4—34no—
199C3×C4—62no—
200Dic3—94no—
201Dic3—182no—
202C32—162no—
203C32—321no—
204C32—322no—
205C32—641no—
206C32C2×#127⋊C311yescharacteristic, complemented
207C23—32no—
208C23—92no—
209C23—91no—
210C23—541no—
211C23—541no—
212C23—541no—
213C23—542no—
214C23—541no—
215C23—542no—
216D4—184no—
217D4—184no—
218D4—544no—
219D4—544no—
220C2×C4—31no—
221C2×C4—272no—
222C2×C4—272no—
223C2×C4—271no—
224C2×C3—32no—
225C2×C3—61no—
226C2×C3—122no—
227C2×C3—362no—
228C2×C3—482no—
229C2×C3—962no—
230C2×C3—1441no—
231D3—32no—
232D3—42no—
233D3—61no—
234D3—92no—
235D3—122no—
236D3—181no—
237C22—182no—
238C22—182no—
239C22—184no—
240C22—271no—
241C22—271no—
242C22—361no—
243C22—1081no—
244C22—1081no—
245C22—1081no—
246C22—1082no—
247C22D32×A411yescharacteristic
248C4—32no—
249C4—272no—
250C3—21no—
251C3—161no—
252C3—322no—
253C3—641no—
254C3D3×#127⋊C312yescomplemented
255C2—31no—
256C2—91no—
257C2—122no—
258C2—271no—
259C2—362no—
subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC32×C42C22×C3
FrattiniC22D32×A4
socleC22×C32C22×A4
LCSG ▷ C32×C42
UCSC1

Complements

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

NH ≅ G/Ntype
1. C3⋊D3×C42⋊C3C2semidirect
2. C3×C3⋊(#127⋊C3)C2semidirect
3. #10594008C3semidirect
6. C32×C42⋊C3C22semidirect
9. C42×C3⋊D3C2×C3semidirect
10. C3×#874C2×C3semidirect
16. C3⋊(#127⋊C3)D3semidirect
33. C32×C42C22×C3semidirect
38. C3×C42⋊C3C2×D3semidirect
46. #874C3×D3semidirect
94. C3×C42C2×C3×D3semidirect
99. C42⋊C3D32semidirect
157. C3⋊D3#127⋊C3semidirect
176. C42C3×D32semidirect
206. C32C2×#127⋊C3semidirect
254. C3D3×#127⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3⋊D3×C42⋊C32
C3×C3⋊(#127⋊C3)2
#105940083
D3×#127⋊C33
D32×A44