Finite Group Databasebeta

← #13048658

Subgroups — #13048658

Order 1296.

505 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 505 conjugacy classes of subgroups — too many to draw. Only its 74 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#10619988C211yesmaximal, characteristic, complemented
2C4⋊(C3⋊((C3⋊D3)⋊C3))C211yesmaximal, characteristic, complemented
3C3×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
4#10619884C212yesmaximal, complemented
5#10619883C212yesmaximal, complemented
6#89041—31nomaximal
7#89022—31nomaximal
8#88648—33nomaximal
9C2×C3⋊((C3⋊D3)⋊C3)C2212yes—
10C2×C3×(C3⋊D3)⋊C3C2211yescharacteristic
11C3×C4×H(3,𝔽3)C2211yescharacteristic
12C3⋊#1084C2212yes—
13C3×(C3⋊Dic3)⋊C3C2211yescharacteristic
14C4⋊(C3⋊(C3⋊D3))D311yescharacteristic, complemented
15C3×C4⋊(C3⋊D3)—31no—
16C3×C4×C3⋊D3D311yescharacteristic, complemented
17C3×#634—31no—
18C3×#634—31no—
19C3×#634—61no—
20C32×C4×D3—31no—
21#6295—32no—
22#6294—32no—
23#6291—32no—
24#6289—32no—
25#6229—33no—
26#6229—63no—
27C4⋊((C3⋊D3)⋊C3)—33no—
28C4×(C3⋊D3)⋊C3D313yescomplemented
29#6198—36no—
30((C3⋊D3)⋊C3)⋊C4—36no—
31C2×C3×H(3,𝔽3)C2311yescharacteristic
32C3⋊((C3⋊D3)⋊C3)—22no—
33C3×(C3⋊D3)⋊C3—21no—
34#3828—91no—
35#3799—93no—
36#3799—91no—
37#3796—91no—
38#3795—91no—
39C2×C3⋊(C3⋊D3)C2×D312yes—
40C2×C3×C3⋊D3—32no—
41C2×C3×C3⋊D3C2×D311yescharacteristic
42C2×C32×D3—31no—
43C33×C4—21no—
44C33×C4C2×D311yescharacteristic
45C33×C4C2×D311yescharacteristic
46C3×C3⋊Dic3—32no—
47C3×C3⋊Dic3—32no—
48C3×C3⋊Dic3—62no—
49C3×C3⋊Dic3C2×D311yescharacteristic
50C32×Dic3—31no—
51C2×(C3⋊D3)⋊C3—36no—
52C2×(C3⋊D3)⋊C3C2×D313yes—
53C4×H(3,𝔽3)—23no—
54C4×H(3,𝔽3)C2×D313yes—
55#1084—36no—
56#1084—66no—
57(C3⋊Dic3)⋊C3C2×D313yes—
58C3×H(3,𝔽3)#4111yescharacteristic, complemented, Sylow
59C2×C32×C4—91no—
60#638—92no—
61C4⋊(C3⋊D3)—31no—
62C4⋊(C3⋊D3)—33no—
63C4⋊(C3⋊D3)—91no—
64C4⋊(C3⋊D3)—181no—
65C4×C3⋊D3—92no—
66C4×C3⋊D3C3⋊D311yescharacteristic, complemented
67#634—31no—
68#634—31no—
69#634—61no—
70C3×D12—91no—
71C3×D12—91no—
72C3×D12—181no—
73C3×C4×D3—33no—
74C3×C4×D3—31no—
75C3×C4×D3—91no—
76C3×Dic6—93no—
77C3×Dic6—91no—
78C3×Dic6—91no—
79C3×Dic6—183no—
80C3×Dic6—181no—
81#626—96no—
82#626—92no—
83#626—92no—
84#625—92no—
85#624—96no—
86#624—92no—
87D3×Dic3—92no—
88D3×Dic3—92no—
89C2×C33—21no—
90C2×C33C22×D311yescharacteristic
91C2×C33C22×D311yescharacteristic
92C3⋊(C3⋊D3)—22no—
93C3×C3⋊D3—21no—
94C3×C3⋊D3—62no—
95C32×D3—61no—
96C2×H(3,𝔽3)—23no—
97C2×H(3,𝔽3)C22×D313yes—
98(C3⋊D3)⋊C3—23no—
99(C3⋊D3)⋊C3—66no—
100#239—271no—
101#237—271no—
102#235—271no—
103#235—273no—
104C22×C32—91no—
105C2×C3⋊D3—32no—
106C2×C3⋊D3—36no—
107C2×C3⋊D3—92no—
108C2×C3⋊D3—182no—
109C2×C3⋊D3C2×C3⋊D311yescharacteristic
110C2×C3×D3—33no—
111C2×C3×D3—31no—
112C2×C3×D3—91no—
113C2×C3×D3—92no—
114C2×C3×D3—92no—
115C2×C3×D3—182no—
116C32×C4—21no—
117C32×C4—23no—
118C32×C4—31no—
119C32×C4—31no—
120C32×C4—61no—
121C32×C4—61no—
122C32×C4—61no—
123C32×C4—91no—
124C32×C4—121no—
125C32×C4D3211yescharacteristic
126C32×C4D3213yes—
127C32×C4C2×C3⋊D311yescharacteristic
128C3⋊Dic3—32no—
129C3⋊Dic3—32no—
130C3⋊Dic3—62no—
131C3⋊Dic3C2×C3⋊D311yescharacteristic
132C3×Dic3—33no—
133C3×Dic3—31no—
134C3×Dic3—91no—
135C3×Dic3—96no—
136C3×Dic3—92no—
137C3×Dic3—92no—
138C3×Dic3—186no—
139C3×Dic3—182no—
140C33—21no—
141C33#23511yescharacteristic, complemented
142C33#23711yescharacteristic, complemented
143H(3,𝔽3)—23no—
144H(3,𝔽3)#23513yescomplemented
145C3×Q8—271no—
146C3×D4—271no—
147C2×C3×C4—93no—
148C2×C3×C4—91no—
149#67—272no—
150#67—272no—
151#67—276no—
152C2×Dic3—272no—
153D12—91no—
154D12—91no—
155D12—181no—
156D12—273no—
157D12—272no—
158C4×D3—31no—
159C4×D3—91no—
160C4×D3—272no—
161C4×D3—272no—
162C4×D3—276no—
163Dic6—93no—
164Dic6—91no—
165Dic6—91no—
166Dic6—181no—
167Dic6—183no—
168C2×C32—21no—
169C2×C32—23no—
170C2×C32—31no—
171C2×C32—31no—
172C2×C32—61no—
173C2×C32—61no—
174C2×C32—61no—
175C2×C32—121no—
176C2×C32—181no—
177C2×C32C2×D3211yescharacteristic
178C2×C32C2×D3213yes—
179C2×C32C22×C3⋊D311yescharacteristic
180C3⋊D3—21no—
181C3⋊D3—62no—
182C3⋊D3—66no—
183C3⋊D3—182no—
184C3⋊D3—362no—
185C3×D3—61no—
186C3×D3—63no—
187C3×D3—181no—
188C3×D3—182no—
189C3×D3—182no—
190C3×D3—362no—
191#41—811noSylow
192C22×C3—93no—
193C22×C3—91no—
194C22×C3—272no—
195C2×D3—31no—
196C2×D3—91no—
197C2×D3—92no—
198C2×D3—92no—
199C2×D3—182no—
200C2×D3—276no—
201C3×C4—21no—
202C3×C4—31no—
203C3×C4—33no—
204C3×C4—61no—
205C3×C4—63no—
206C3×C4—93no—
207C3×C4—91no—
208C3×C4—272no—
209C3×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
210C3×C4D3×C3⋊D311yescharacteristic
211Dic3—31no—
212Dic3—91no—
213Dic3—96no—
214Dic3—92no—
215Dic3—92no—
216Dic3—182no—
217Dic3—186no—
218C32—21no—
219C32—23no—
220C32—31no—
221C32—31no—
222C32—61no—
223C32—61no—
224C32—61no—
225C32—121no—
226C32#379511yescharacteristic
227C32#379513yescomplemented
228C32#382811yescharacteristic, complemented
229Q8—271no—
230D4—271no—
231D4—812no—
232C2×C4—91no—
233C2×C4—812no—
234C2×C3—21no—
235C2×C3—31no—
236C2×C3—33no—
237C2×C3—61no—
238C2×C3—63no—
239C2×C3—183no—
240C2×C3—181no—
241C2×C3—542no—
242C2×C3C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
243C2×C3C2×D3×C3⋊D311yescharacteristic
244D3—61no—
245D3—181no—
246D3—182no—
247D3—182no—
248D3—362no—
249D3—546no—
250C22—91no—
251C22—272no—
252C4—91no—
253C4—272no—
254C4((C3⋊D3)⋊C3)⋊D311yescharacteristic
255C3—21no—
256C3—31no—
257C3—33no—
258C3—61no—
259C3—63no—
260C3#8864811yescharacteristic, complemented
261C3#8902111yescharacteristic
262C2—181no—
263C2—542no—
264C2C2×((C3⋊D3)⋊C3)⋊D311yescharacteristic
subgroupG/H
centerC2C2×((C3⋊D3)⋊C3)⋊D3
commutator, cosocleC2×C3×H(3,𝔽3)C23
FrattiniC2×C3C2×D3×C3⋊D3
FittingC3×C4×H(3,𝔽3)C22
socleC2×C32C2×D32
hypercenterC4((C3⋊D3)⋊C3)⋊D3
hypocenterC3×H(3,𝔽3)#41
LCSG ▷ C2×C3×H(3,𝔽3) ▷ C3×H(3,𝔽3)
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. #10619988C2semidirect
2. C4⋊(C3⋊((C3⋊D3)⋊C3))C2semidirect
3. C3×C4×(C3⋊D3)⋊C3C2semidirect
4. #10619884C2semidirect
5. #10619883C2semidirect
14. C4⋊(C3⋊(C3⋊D3))D3semidirect
16. C3×C4×C3⋊D3D3semidirect
28. C4×(C3⋊D3)⋊C3D3semidirect
58. C3×H(3,𝔽3)#41semidirect
66. C4×C3⋊D3C3⋊D3semidirect
141. C33#235semidirect
142. C33#237semidirect
144. H(3,𝔽3)#235semidirect
227. C32#3795semidirect
228. C32#3828semidirect
260. C3#88648semidirect

Maximal subgroups

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

subgroupindex
#106199882
C4⋊(C3⋊((C3⋊D3)⋊C3))2
C3×C4×(C3⋊D3)⋊C32
#106198842
#106198832
#890413
#890223
#886483