Finite Group Databasebeta

← #13048746

Subgroups — #13048746

Order 1296.

467 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 467 conjugacy classes of subgroups — too many to draw. Only its 63 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
1C4×C3⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
2C3×#6229C211yesmaximal, characteristic, complemented
3#10619956C211yesmaximal, characteristic, complemented
4#10619901C212yesmaximal
5#10619900C212yesmaximal, complemented
6#89019—31nomaximal
7C3⋊D3×Dic6—31nomaximal
8((C3⋊D3)⋊C3)⋊Q8—31nomaximal
9C2×C3⋊((C3⋊D3)⋊C3)C2211yescharacteristic
10C3×C4×H(3,𝔽3)C2211yescharacteristic
11C3×#1084C2212yes—
12C3⋊((C3⋊Dic3)⋊C3)C2212yes—
13C3⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
14C4×C3⋊(C3⋊D3)D311yescharacteristic
15#6308D311yescharacteristic, complemented
16C3×C4×C3⋊D3—31no—
17C3×#634—31no—
18C32×Dic6—31no—
19C32×Dic6—31no—
20C32×Dic6—61no—
21#6293—32no—
22#6293—32no—
23#6289—32no—
24Dic3×C3⋊D3—32no—
25#6229D311yescharacteristic, complemented
26C4×(C3⋊D3)⋊C3—31no—
27C4×(C3⋊D3)⋊C3—61no—
28#634⋊C3—31no—
29#634⋊C3—61no—
30((C3⋊D3)⋊C3)⋊C4—32no—
31#6196—32no—
32C2×C3×H(3,𝔽3)C2311yescharacteristic
33C3⋊((C3⋊D3)⋊C3)Q812yescomplemented
34Q8×C3⋊D3—91no—
35#3797—91no—
36#3797—93no—
37D3×Dic6—91no—
38D3×Dic6—93no—
39C2×C3⋊(C3⋊D3)C2×D311yescharacteristic
40C2×C3×C3⋊D3—31no—
41C33×C4—21no—
42C33×C4C2×D311yescharacteristic
43C33×C4C2×D311yescharacteristic
44C3⋊(C3⋊Dic3)C2×D312yes—
45C3⋊(C3⋊Dic3)C2×D311yescharacteristic
46C3×C3⋊Dic3—32no—
47C3×C3⋊Dic3—31no—
48C32×Dic3—32no—
49C32×Dic3—32no—
50C32×Dic3—62no—
51C2×(C3⋊D3)⋊C3—31no—
52C2×(C3⋊D3)⋊C3—61no—
53C4×H(3,𝔽3)—21no—
54C4×H(3,𝔽3)—21no—
55C4×H(3,𝔽3)—41no—
56C4×H(3,𝔽3)C2×D311yescharacteristic
57#1084C2×D312yes—
58(C3⋊Dic3)⋊C3—31no—
59(C3⋊Dic3)⋊C3—32no—
60(C3⋊Dic3)⋊C3—62no—
61(C3⋊Dic3)⋊C3—61no—
62C3×H(3,𝔽3)C2×Q811yescharacteristic, complemented, Sylow
63C32×Q8—91no—
64C4×C3⋊D3—31no—
65C4×C3⋊D3—31no—
66C4×C3⋊D3—61no—
67C4×C3⋊D3—93no—
68C4×C3⋊D3—92no—
69#634—31no—
70#634—31no—
71#634—61no—
72#634—93no—
73#634—92no—
74C3×C4×D3—93no—
75C3×C4×D3—91no—
76C3×Dic6—33no—
77C3×Dic6—33no—
78C3×Dic6—31no—
79C3×Dic6—31no—
80C3×Dic6—61no—
81C3×Dic6—63no—
82C3×Dic6—93no—
83C3×Dic6—91no—
84#627—92no—
85#627—92no—
86#627—96no—
87#627—96no—
88#624—92no—
89#624—96no—
90D3×Dic3—92no—
91D3×Dic3—96no—
92C2×C33—21no—
93C2×C33C22×D311yescharacteristic
94C2×C33C22×D311yescharacteristic
95C3⋊(C3⋊D3)Dic612yescomplemented
96C3×C3⋊D3—32no—
97C2×H(3,𝔽3)—21no—
98C2×H(3,𝔽3)—21no—
99C2×H(3,𝔽3)—41no—
100C2×H(3,𝔽3)C22×D311yescharacteristic
101(C3⋊D3)⋊C3—32no—
102(C3⋊D3)⋊C3—62no—
103D3×Q8—271no—
104D3×Q8—273no—
105D3×Q8—271no—
106C2×Dic6—271no—
107C2×C3⋊D3—31no—
108C2×C3⋊D3—31no—
109C2×C3⋊D3—61no—
110C2×C3⋊D3—93no—
111C2×C3×D3—93no—
112C2×C3×D3—91no—
113C32×C4—21no—
114C32×C4—21no—
115C32×C4—21no—
116C32×C4—33no—
117C32×C4—33no—
118C32×C4—41no—
119C32×C4—63no—
120C32×C4—92no—
121C32×C4D3211yescharacteristic
122C32×C4D3211yescharacteristic
123C32×C4D3211yescharacteristic
124C3⋊Dic3—32no—
125C3⋊Dic3—31no—
126C3⋊Dic3—32no—
127C3⋊Dic3—31no—
128C3⋊Dic3—62no—
129C3⋊Dic3—61no—
130C3⋊Dic3—96no—
131C3⋊Dic3—93no—
132C3×Dic3—36no—
133C3×Dic3—36no—
134C3×Dic3—32no—
135C3×Dic3—32no—
136C3×Dic3—62no—
137C3×Dic3—66no—
138C3×Dic3—96no—
139C3×Dic3—93no—
140C3×Dic3—92no—
141C3×Dic3—91no—
142C33—21no—
143C33C2×Dic611yescharacteristic, complemented
144C33D3×Q811yescharacteristic, complemented
145H(3,𝔽3)—21no—
146H(3,𝔽3)—21no—
147H(3,𝔽3)—41no—
148H(3,𝔽3)D3×Q811yescharacteristic, complemented
149C3×Q8—93no—
150C3×Q8—91no—
151C3×Q8—271no—
152C2×C3×C4—271no—
153C2×Dic3—272no—
154C4×D3—93no—
155C4×D3—91no—
156C4×D3—271no—
157C4×D3—272no—
158C4×D3—276no—
159C4×D3—272no—
160C4×D3—541no—
161Dic6—31no—
162Dic6—31no—
163Dic6—61no—
164Dic6—93no—
165Dic6—91no—
166Dic6—271no—
167Dic6—272no—
168Dic6—272no—
169Dic6—276no—
170Dic6—272no—
171Dic6—541no—
172C2×C32—21no—
173C2×C32—21no—
174C2×C32—21no—
175C2×C32—33no—
176C2×C32—33no—
177C2×C32—41no—
178C2×C32—63no—
179C2×C32C2×D3211yescharacteristic
180C2×C32C2×D3211yescharacteristic
181C2×C32C2×D3211yescharacteristic
182C3⋊D3—32no—
183C3⋊D3—32no—
184C3⋊D3—62no—
185C3⋊D3—96no—
186C3×D3—96no—
187C3×D3—92no—
188C2×Q8—811noSylow
189C22×C3—271no—
190C2×D3—93no—
191C2×D3—91no—
192C2×D3—271no—
193C2×D3—541no—
194C3×C4—31no—
195C3×C4—31no—
196C3×C4—61no—
197C3×C4—61no—
198C3×C4—61no—
199C3×C4—96no—
200C3×C4—92no—
201C3×C4—121no—
202C3×C4—271no—
203C3×C4—272no—
204C3×C4((C3⋊D3)⋊C3)⋊C213yes—
205C3×C4#111311yescharacteristic
206Dic3—32no—
207Dic3—32no—
208Dic3—62no—
209Dic3—96no—
210Dic3—93no—
211Dic3—92no—
212Dic3—91no—
213Dic3—271no—
214Dic3—272no—
215Dic3—542no—
216Dic3—541no—
217C32—21no—
218C32—21no—
219C32—21no—
220C32—33no—
221C32—33no—
222C32—41no—
223C32—63no—
224C32D3×Dic611yescharacteristic
225C32D3×Dic611yescharacteristic, complemented
226C32#379711yescharacteristic, complemented
227Q8—91no—
228Q8—271no—
229Q8—812no—
230C2×C4—271no—
231C2×C4—812no—
232C2×C3—31no—
233C2×C3—31no—
234C2×C3—61no—
235C2×C3—61no—
236C2×C3—61no—
237C2×C3—121no—
238C2×C3—272no—
239C2×C3C2×((C3⋊D3)⋊C3)⋊C213yes—
240C2×C3C2×#111311yescharacteristic
241D3—96no—
242D3—92no—
243D3—272no—
244D3—542no—
245C22—271no—
246C4—92no—
247C4—271no—
248C4—272no—
249C4#6643211yescharacteristic
250C3—31no—
251C3—31no—
252C3—61no—
253C3—61no—
254C3—61no—
255C3—121no—
256C3((C3⋊D3)⋊C3)⋊Q813yescomplemented
257C3#8904011yescharacteristic
258C2—272no—
259C2C2×#6643211yescharacteristic
subgroupG/H
centerC2C2×#66432
commutator, cosocleC2×C3×H(3,𝔽3)C23
FrattiniC2×C3C2×#1113
FittingC3×C4×H(3,𝔽3)C22
socleC2×C32C2×D32
prime radicalC2×C3⋊((C3⋊D3)⋊C3)C22
hypercenterC4#66432
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
2. C3×#6229C2semidirect
3. #10619956C2semidirect
5. #10619900C2semidirect
15. #6308D3semidirect
25. #6229D3semidirect
33. C3⋊((C3⋊D3)⋊C3)Q8semidirect
62. C3×H(3,𝔽3)C2×Q8semidirect
95. C3⋊(C3⋊D3)Dic6semidirect
143. C33C2×Dic6semidirect
144. C33D3×Q8semidirect
148. H(3,𝔽3)D3×Q8semidirect
225. C32D3×Dic6semidirect
226. C32#3797semidirect
256. 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
C4×C3⋊((C3⋊D3)⋊C3)2
C3×#62292
#106199562
#106199012
#106199002
#890193
C3⋊D3×Dic63
((C3⋊D3)⋊C3)⋊Q83