Finite Group Databasebeta

← #13048660

Subgroups — #13048660

Order 1296.

479 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 479 conjugacy classes of subgroups — too many to draw. Only its 86 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
2#10619956C211yesmaximal, characteristic, complemented
3C3×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
4((C3⋊D3)⋊C3)⋊Dic3C212yesmaximal
5#10619881C212yesmaximal, complemented
6#89040—31nomaximal
7D3×#634—31nomaximal
8((C3⋊D3)⋊C3)⋊Q8—33nomaximal
9C2×C3×(C3⋊D3)⋊C3C2211yescharacteristic
10C3×C4×H(3,𝔽3)C2211yescharacteristic
11C3⋊#1084C2212yes—
12C3⋊((C3⋊Dic3)⋊C3)C2212yes—
13C3×(C3⋊Dic3)⋊C3C2211yescharacteristic
14#6308D311yescharacteristic, complemented
15C3×C4×C3⋊D3D311yescharacteristic
16C3×#634—31no—
17C3×#634—31no—
18C3×#634—31no—
19C3×#634—61no—
20C32×C4×D3—31no—
21#6296—32no—
22#6294—32no—
23#6293—32no—
24D3×C3⋊Dic3—32no—
25#6229—33no—
26#6229—63no—
27C4×(C3⋊D3)⋊C3D313yes—
28#634⋊C3—33no—
29((C3⋊D3)⋊C3)⋊C4—36no—
30#6196—36no—
31C2×C3×H(3,𝔽3)C2311yescharacteristic
32C3×(C3⋊D3)⋊C3Q812yescomplemented
33C2×#634—91no—
34#3797—91no—
35D3×Dic6—93no—
36D3×Dic6—91no—
37D3×Dic6—91no—
38C2×C3×C3⋊D3C2×D311yescharacteristic
39C2×C32×D3—31no—
40C33×C4—21no—
41C33×C4C2×D311yescharacteristic
42C33×C4C2×D311yescharacteristic
43C3⋊(C3⋊Dic3)C2×D312yes—
44C3×C3⋊Dic3—32no—
45C3×C3⋊Dic3—32no—
46C3×C3⋊Dic3—32no—
47C3×C3⋊Dic3—62no—
48C3×C3⋊Dic3C2×D311yescharacteristic
49C32×Dic3—31no—
50C2×(C3⋊D3)⋊C3C2×D313yes—
51C4×H(3,𝔽3)—23no—
52C4×H(3,𝔽3)C2×D313yes—
53#1084—36no—
54#1084—66no—
55(C3⋊Dic3)⋊C3—36no—
56(C3⋊Dic3)⋊C3C2×D313yes—
57C3×H(3,𝔽3)C2×Q811yescharacteristic, complemented, Sylow
58C2×C32×C4—91no—
59C2×C3⋊Dic3—92no—
60C4×C3⋊D3C3⋊D311yescharacteristic
61#634—31no—
62#634—31no—
63#634—31no—
64#634—33no—
65#634—61no—
66#634—91no—
67#634—92no—
68#634—181no—
69C3×C4×D3—33no—
70C3×C4×D3—31no—
71C3×C4×D3—91no—
72C3×Dic6—91no—
73C3×Dic6—93no—
74C3×Dic6—91no—
75C3×Dic6—91no—
76C3×Dic6—91no—
77C3×Dic6—183no—
78C3×Dic6—181no—
79C3×Dic6—181no—
80#627—96no—
81#627—92no—
82#627—92no—
83#627—92no—
84#624—92no—
85D3×Dic3—96no—
86D3×Dic3—92no—
87D3×Dic3—92no—
88C2×C33—21no—
89C2×C33C22×D311yescharacteristic
90C2×C33C22×D311yescharacteristic
91C3×C3⋊D3Dic612yescomplemented
92C32×D3—32no—
93C2×H(3,𝔽3)—23no—
94C2×H(3,𝔽3)C22×D313yes—
95(C3⋊D3)⋊C3Dic616yescomplemented
96D3×Q8—271no—
97D3×Q8—271no—
98C2×Dic6—271no—
99C2×Dic6—273no—
100C22×C32—91no—
101C2×C3⋊D3C2×C3⋊D311yescharacteristic
102C2×C3×D3—33no—
103C2×C3×D3—31no—
104C2×C3×D3—91no—
105C32×C4—21no—
106C32×C4—23no—
107C32×C4—31no—
108C32×C4—31no—
109C32×C4—61no—
110C32×C4—61no—
111C32×C4—61no—
112C32×C4—91no—
113C32×C4—121no—
114C32×C4D3211yescharacteristic
115C32×C4D3213yes—
116C32×C4C2×C3⋊D311yescharacteristic
117C3⋊Dic3—32no—
118C3⋊Dic3—32no—
119C3⋊Dic3—32no—
120C3⋊Dic3—36no—
121C3⋊Dic3—62no—
122C3⋊Dic3—92no—
123C3⋊Dic3—182no—
124C3⋊Dic3C2×C3⋊D311yescharacteristic
125C3×Dic3—33no—
126C3×Dic3—31no—
127C3×Dic3—91no—
128C3×Dic3—92no—
129C3×Dic3—96no—
130C3×Dic3—92no—
131C3×Dic3—92no—
132C3×Dic3—92no—
133C3×Dic3—186no—
134C3×Dic3—182no—
135C3×Dic3—182no—
136C33—21no—
137C33C2×Dic611yescharacteristic, complemented
138C33D3×Q811yescharacteristic, complemented
139H(3,𝔽3)—23no—
140H(3,𝔽3)C2×Dic613yescomplemented
141C3×Q8—271no—
142C3×Q8—271no—
143C2×C3×C4—93no—
144C2×C3×C4—91no—
145C2×Dic3—272no—
146C2×Dic3—276no—
147C4×D3—31no—
148C4×D3—91no—
149C4×D3—272no—
150C4×D3—272no—
151Dic6—93no—
152Dic6—91no—
153Dic6—91no—
154Dic6—91no—
155Dic6—91no—
156Dic6—181no—
157Dic6—181no—
158Dic6—183no—
159Dic6—273no—
160Dic6—272no—
161Dic6—272no—
162Dic6—272no—
163Dic6—276no—
164C2×C32—21no—
165C2×C32—23no—
166C2×C32—31no—
167C2×C32—31no—
168C2×C32—61no—
169C2×C32—61no—
170C2×C32—61no—
171C2×C32—92no—
172C2×C32—121no—
173C2×C32C2×D3211yescharacteristic
174C2×C32C2×D3213yes—
175C2×C32C22×C3⋊D311yescharacteristic
176C3⋊D3#63412yescomplemented
177C3×D3—32no—
178C3×D3—36no—
179C3×D3—92no—
180C2×Q8—811noSylow
181C22×C3—93no—
182C22×C3—91no—
183C2×D3—31no—
184C2×D3—91no—
185C3×C4—21no—
186C3×C4—33no—
187C3×C4—31no—
188C3×C4—61no—
189C3×C4—63no—
190C3×C4—93no—
191C3×C4—91no—
192C3×C4—272no—
193C3×C4—272no—
194C3×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
195C3×C4D3×C3⋊D311yescharacteristic
196Dic3—31no—
197Dic3—91no—
198Dic3—92no—
199Dic3—96no—
200Dic3—92no—
201Dic3—92no—
202Dic3—92no—
203Dic3—182no—
204Dic3—182no—
205Dic3—186no—
206Dic3—276no—
207C32—21no—
208C32—23no—
209C32—31no—
210C32—31no—
211C32—61no—
212C32—61no—
213C32—61no—
214C32—121no—
215C32D3×Dic611yescharacteristic
216C32D3×Dic613yescomplemented
217C32C2×#63411yescharacteristic, complemented
218Q8—271no—
219Q8—271no—
220Q8—812no—
221C2×C4—91no—
222C2×C4—812no—
223C2×C3—21no—
224C2×C3—31no—
225C2×C3—33no—
226C2×C3—61no—
227C2×C3—63no—
228C2×C3—96no—
229C2×C3—92no—
230C2×C3C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
231C2×C3C2×D3×C3⋊D311yescharacteristic
232D3—32no—
233D3—92no—
234C22—91no—
235C4—91no—
236C4—272no—
237C4—272no—
238C4((C3⋊D3)⋊C3)⋊D311yescharacteristic
239C3—21no—
240C3—31no—
241C3—33no—
242C3—61no—
243C3—63no—
244C3((C3⋊D3)⋊C3)⋊Q811yescharacteristic, complemented
245C3D3×#63411yescharacteristic
246C2—92no—
247C2C2×((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
prime radicalC2×C3×(C3⋊D3)⋊C3C22
hypercenterC4((C3⋊D3)⋊C3)⋊D3
hypocenterC3×H(3,𝔽3)C2×Q8
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. #10619956C2semidirect
5. #10619881C2semidirect
14. #6308D3semidirect
32. C3×(C3⋊D3)⋊C3Q8semidirect
57. C3×H(3,𝔽3)C2×Q8semidirect
91. C3×C3⋊D3Dic6semidirect
95. (C3⋊D3)⋊C3Dic6semidirect
137. C33C2×Dic6semidirect
138. C33D3×Q8semidirect
140. H(3,𝔽3)C2×Dic6semidirect
176. C3⋊D3#634semidirect
216. C32D3×Dic6semidirect
217. C32C2×#634semidirect
244. C3((C3⋊D3)⋊C3)⋊Q8semidirect

Maximal subgroups

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

subgroupindex
#106199882
#106199562
C3×C4×(C3⋊D3)⋊C32
((C3⋊D3)⋊C3)⋊Dic32
#106198812
#890403
D3×#6343
((C3⋊D3)⋊C3)⋊Q83