Finite Group Databasebeta

← #13048663

Subgroups — #13048663

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)⋊C3)C211yesmaximal, characteristic, complemented
4#10619884C212yesmaximal, complemented
5((C3⋊D3)⋊C3)⋊Dic3C212yesmaximal, complemented
6#89041—31nomaximal
7#89015—31nomaximal
8#88648—33nomaximal
9C2×C3⋊((C3⋊D3)⋊C3)C2211yescharacteristic
10C2×C3×(C3⋊D3)⋊C3C2212yes—
11C3×C4×H(3,𝔽3)C2211yescharacteristic
12C3⋊#1084C2212yes—
13C3⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
14C4×C3⋊(C3⋊D3)D311yescharacteristic, complemented
15C3×C4⋊(C3⋊D3)D311yescharacteristic, complemented
16C3×C4×C3⋊D3—31no—
17C3×#634—31no—
18C3×#634—31no—
19C3×#634—61no—
20C32×D12—31no—
21#6295—32no—
22#6294—32no—
23#6291—32no—
24D3×C3⋊Dic3—32no—
25#6229—33no—
26#6229—63no—
27C4⋊((C3⋊D3)⋊C3)D313yescomplemented
28C4×(C3⋊D3)⋊C3—33no—
29#6198—36no—
30((C3⋊D3)⋊C3)⋊C4—36no—
31C2×C3×H(3,𝔽3)C2311yescharacteristic
32C3⋊((C3⋊D3)⋊C3)—21no—
33C3×(C3⋊D3)⋊C3—22no—
34#3830—91no—
35#3799—91no—
36#3796—93no—
37#3796—91no—
38#3795—91no—
39C2×C3⋊(C3⋊D3)C2×D311yescharacteristic
40C2×C3×C3⋊D3—31no—
41C2×C3×C3⋊D3C2×D312yes—
42C2×C32×D3—32no—
43C33×C4—21no—
44C33×C4C2×D311yescharacteristic
45C33×C4C2×D311yescharacteristic
46C3⋊(C3⋊Dic3)C2×D311yescharacteristic
47C3×C3⋊Dic3—32no—
48C3×C3⋊Dic3—31no—
49C3×C3⋊Dic3—32no—
50C3×C3⋊Dic3—62no—
51C2×(C3⋊D3)⋊C3—33no—
52C2×(C3⋊D3)⋊C3C2×D316yes—
53C4×H(3,𝔽3)—23no—
54C4×H(3,𝔽3)C2×D313yes—
55#1084—36no—
56#1084—66no—
57(C3⋊Dic3)⋊C3—33no—
58C3×H(3,𝔽3)#4111yescharacteristic, complemented, Sylow
59C32×D4—91no—
60#638—92no—
61C2×C3⋊Dic3—92no—
62C4⋊(C3⋊D3)C3⋊D311yescharacteristic, complemented
63C4×C3⋊D3—31no—
64C4×C3⋊D3—33no—
65C4×C3⋊D3—91no—
66C4×C3⋊D3—181no—
67#634—31no—
68#634—31no—
69#634—61no—
70C3×D12—33no—
71C3×D12—31no—
72C3×D12—91no—
73C3×C4×D3—91no—
74C3×C4×D3—91no—
75C3×C4×D3—181no—
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—92no—
86D3×Dic3—96no—
87D3×Dic3—92no—
88D3×Dic3—92no—
89C2×C33—21no—
90C2×C33C22×D311yescharacteristic
91C2×C33C22×D311yescharacteristic
92C3⋊(C3⋊D3)—21no—
93C3×C3⋊D3—22no—
94C3×C3⋊D3—61no—
95C32×D3—62no—
96C2×H(3,𝔽3)—23no—
97C2×H(3,𝔽3)C22×D313yes—
98(C3⋊D3)⋊C3—26no—
99(C3⋊D3)⋊C3—63no—
100#239—271no—
101#237—271no—
102#237—273no—
103#235—271no—
104C22×C32—92no—
105C2×C3⋊D3—31no—
106C2×C3⋊D3—33no—
107C2×C3⋊D3—91no—
108C2×C3⋊D3—181no—
109C2×C3⋊D3C2×C3⋊D312yes—
110C2×C3×D3—36no—
111C2×C3×D3—32no—
112C2×C3×D3—92no—
113C2×C3×D3—91no—
114C2×C3×D3—91no—
115C2×C3×D3—181no—
116C32×C4—21no—
117C32×C4—23no—
118C32×C4—31no—
119C32×C4—31no—
120C32×C4—61no—
121C32×C4—61no—
122C32×C4—61no—
123C32×C4—121no—
124C32×C4D3211yescharacteristic
125C32×C4D3213yes—
126C32×C4C2×C3⋊D311yescharacteristic
127C3⋊Dic3—32no—
128C3⋊Dic3—32no—
129C3⋊Dic3—31no—
130C3⋊Dic3—33no—
131C3⋊Dic3—62no—
132C3⋊Dic3—91no—
133C3⋊Dic3—181no—
134C3×Dic3—91no—
135C3×Dic3—96no—
136C3×Dic3—92no—
137C3×Dic3—91no—
138C3×Dic3—92no—
139C3×Dic3—186no—
140C3×Dic3—182no—
141C3×Dic3—181no—
142C33—21no—
143C33#23511yescharacteristic, complemented
144C33#23711yescharacteristic, complemented
145H(3,𝔽3)—23no—
146H(3,𝔽3)#23713yescomplemented
147C3×Q8—271no—
148C3×D4—93no—
149C3×D4—91no—
150C2×C3×C4—271no—
151#67—272no—
152#67—272no—
153#67—276no—
154C2×Dic3—272no—
155C2×Dic3—276no—
156D12—31no—
157D12—91no—
158D12—272no—
159C4×D3—91no—
160C4×D3—91no—
161C4×D3—181no—
162C4×D3—273no—
163C4×D3—272no—
164C4×D3—272no—
165Dic6—91no—
166Dic6—93no—
167Dic6—91no—
168Dic6—181no—
169Dic6—183no—
170C2×C32—21no—
171C2×C32—23no—
172C2×C32—31no—
173C2×C32—31no—
174C2×C32—61no—
175C2×C32—61no—
176C2×C32—61no—
177C2×C32—121no—
178C2×C32—182no—
179C2×C32C2×D3211yescharacteristic
180C2×C32C2×D3213yes—
181C2×C32C22×C3⋊D311yescharacteristic
182C3⋊D3—22no—
183C3⋊D3—61no—
184C3⋊D3—63no—
185C3⋊D3—181no—
186C3⋊D3—182no—
187C3×D3—62no—
188C3×D3—66no—
189C3×D3—182no—
190C3×D3—181no—
191C3×D3—182no—
192C3×D3—181no—
193#41—811noSylow
194C22×C3—96no—
195C22×C3—92no—
196C22×C3—271no—
197C2×D3—32no—
198C2×D3—92no—
199C2×D3—91no—
200C2×D3—91no—
201C2×D3—181no—
202C2×D3—273no—
203C3×C4—21no—
204C3×C4—33no—
205C3×C4—31no—
206C3×C4—61no—
207C3×C4—63no—
208C3×C4—271no—
209C3×C4—272no—
210C3×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
211C3×C4D3×C3⋊D311yescharacteristic
212Dic3—92no—
213Dic3—96no—
214Dic3—92no—
215Dic3—91no—
216Dic3—91no—
217Dic3—181no—
218Dic3—182no—
219Dic3—186no—
220Dic3—273no—
221C32—21no—
222C32—23no—
223C32—31no—
224C32—31no—
225C32—61no—
226C32—61no—
227C32—61no—
228C32—121no—
229C32#379511yescharacteristic
230C32#379513yescomplemented
231C32#383011yescharacteristic, complemented
232Q8—271no—
233D4—91no—
234D4—812no—
235C2×C4—271no—
236C2×C4—812no—
237C2×C3—21no—
238C2×C3—33no—
239C2×C3—31no—
240C2×C3—61no—
241C2×C3—63no—
242C2×C3—186no—
243C2×C3—182no—
244C2×C3—541no—
245C2×C3C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
246C2×C3C2×D3×C3⋊D311yescharacteristic
247D3—62no—
248D3—182no—
249D3—182no—
250D3—181no—
251D3—181no—
252D3—543no—
253C22—92no—
254C22—271no—
255C4—271no—
256C4—272no—
257C4((C3⋊D3)⋊C3)⋊D311yescharacteristic
258C3—21no—
259C3—31no—
260C3—33no—
261C3—61no—
262C3—63no—
263C3#8864811yescharacteristic, complemented
264C3#8901411yescharacteristic
265C2—182no—
266C2—541no—
267C2C2×((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)⋊C3)C2semidirect
4. #10619884C2semidirect
5. ((C3⋊D3)⋊C3)⋊Dic3C2semidirect
14. C4×C3⋊(C3⋊D3)D3semidirect
15. C3×C4⋊(C3⋊D3)D3semidirect
27. C4⋊((C3⋊D3)⋊C3)D3semidirect
58. C3×H(3,𝔽3)#41semidirect
62. C4⋊(C3⋊D3)C3⋊D3semidirect
143. C33#235semidirect
144. C33#237semidirect
146. H(3,𝔽3)#237semidirect
230. C32#3795semidirect
231. C32#3830semidirect
263. 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)⋊C3)2
#106198842
((C3⋊D3)⋊C3)⋊Dic32
#890413
#890153
#886483