Finite Group Databasebeta

← #13048657

Subgroups — #13048657

Order 1296.

485 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 485 conjugacy classes of subgroups — too many to draw. Only its 78 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
3C3×#634⋊C3C211yesmaximal, characteristic, complemented
4#10619883C212yesmaximal
5#10619881C212yesmaximal, complemented
6#89040—31nomaximal
7#89019—31nomaximal
8((C3⋊D3)⋊C3)⋊Q8—33nomaximal
9C2×C3⋊((C3⋊D3)⋊C3)C2211yescharacteristic
10C3×C4×H(3,𝔽3)C2211yescharacteristic
11C3⋊#1084C2212yes—
12C3⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
13C3×(C3⋊Dic3)⋊C3C2212yes—
14C4×C3⋊(C3⋊D3)D311yescharacteristic
15C3×C4×C3⋊D3—31no—
16C3×#634—31no—
17C3×#634—31no—
18C3×#634—61no—
19C3×#634D311yescharacteristic, complemented
20C32×Dic6—31no—
21#6296—32no—
22#6294—32no—
23#6293—32no—
24#6289—32no—
25#6229—33no—
26#6229—63no—
27C4×(C3⋊D3)⋊C3—33no—
28#634⋊C3D313yescomplemented
29((C3⋊D3)⋊C3)⋊C4—36no—
30#6196—36no—
31C2×C3×H(3,𝔽3)C2311yescharacteristic
32C3⋊((C3⋊D3)⋊C3)Q812yescomplemented
33Q8×C3⋊D3—91no—
34#3797—93no—
35#3797—91no—
36D3×Dic6—91no—
37D3×Dic6—91no—
38C2×C3⋊(C3⋊D3)C2×D311yescharacteristic
39C2×C3×C3⋊D3—31no—
40C33×C4—21no—
41C33×C4C2×D311yescharacteristic
42C33×C4C2×D311yescharacteristic
43C3⋊(C3⋊Dic3)C2×D311yescharacteristic
44C3×C3⋊Dic3—32no—
45C3×C3⋊Dic3—31no—
46C3×C3⋊Dic3—32no—
47C3×C3⋊Dic3—62no—
48C3×C3⋊Dic3C2×D312yes—
49C32×Dic3—32no—
50C2×(C3⋊D3)⋊C3—33no—
51C4×H(3,𝔽3)—23no—
52C4×H(3,𝔽3)C2×D313yes—
53#1084—36no—
54#1084—66no—
55(C3⋊Dic3)⋊C3—33no—
56(C3⋊Dic3)⋊C3C2×D316yes—
57C3×H(3,𝔽3)C2×Q811yescharacteristic, complemented, Sylow
58C32×Q8—91no—
59C4×C3⋊D3—31no—
60C4×C3⋊D3—33no—
61C4×C3⋊D3—91no—
62C4×C3⋊D3—92no—
63C4×C3⋊D3—181no—
64#634—31no—
65#634—31no—
66#634—61no—
67#634—92no—
68#634C3⋊D311yescharacteristic, complemented
69C3×C4×D3—91no—
70C3×C4×D3—91no—
71C3×C4×D3—181no—
72C3×Dic6—33no—
73C3×Dic6—31no—
74C3×Dic6—91no—
75C3×Dic6—93no—
76C3×Dic6—91no—
77C3×Dic6—91no—
78C3×Dic6—183no—
79C3×Dic6—181no—
80#627—96no—
81#627—92no—
82#627—92no—
83#627—92no—
84#624—96no—
85#624—92no—
86D3×Dic3—92no—
87D3×Dic3—92no—
88C2×C33—21no—
89C2×C33C22×D311yescharacteristic
90C2×C33C22×D311yescharacteristic
91C3⋊(C3⋊D3)Dic612yescomplemented
92C3×C3⋊D3—32no—
93C2×H(3,𝔽3)—23no—
94C2×H(3,𝔽3)C22×D313yes—
95(C3⋊D3)⋊C3—36no—
96D3×Q8—271no—
97D3×Q8—271no—
98D3×Q8—273no—
99C2×Dic6—271no—
100C2×C3⋊D3—31no—
101C2×C3⋊D3—33no—
102C2×C3⋊D3—91no—
103C2×C3⋊D3—181no—
104C2×C3×D3—91no—
105C2×C3×D3—91no—
106C2×C3×D3—181no—
107C32×C4—21no—
108C32×C4—23no—
109C32×C4—31no—
110C32×C4—31no—
111C32×C4—61no—
112C32×C4—61no—
113C32×C4—61no—
114C32×C4—92no—
115C32×C4—121no—
116C32×C4D3211yescharacteristic
117C32×C4D3213yes—
118C32×C4C2×C3⋊D311yescharacteristic
119C3⋊Dic3—32no—
120C3⋊Dic3—32no—
121C3⋊Dic3—31no—
122C3⋊Dic3—33no—
123C3⋊Dic3—62no—
124C3⋊Dic3—91no—
125C3⋊Dic3—181no—
126C3⋊Dic3C2×C3⋊D312yes—
127C3×Dic3—36no—
128C3×Dic3—32no—
129C3×Dic3—92no—
130C3×Dic3—91no—
131C3×Dic3—96no—
132C3×Dic3—92no—
133C3×Dic3—91no—
134C3×Dic3—92no—
135C3×Dic3—186no—
136C3×Dic3—182no—
137C3×Dic3—181no—
138C33—21no—
139C33C2×Dic611yescharacteristic, complemented
140C33D3×Q811yescharacteristic, complemented
141H(3,𝔽3)—23no—
142H(3,𝔽3)D3×Q813yescomplemented
143C3×Q8—93no—
144C3×Q8—91no—
145C3×Q8—271no—
146C2×C3×C4—271no—
147C2×Dic3—272no—
148C4×D3—91no—
149C4×D3—91no—
150C4×D3—181no—
151C4×D3—273no—
152C4×D3—272no—
153C4×D3—272no—
154C4×D3—276no—
155Dic6—31no—
156Dic6—91no—
157Dic6—91no—
158Dic6—93no—
159Dic6—91no—
160Dic6—181no—
161Dic6—183no—
162Dic6—272no—
163Dic6—272no—
164Dic6—272no—
165Dic6—276no—
166C2×C32—21no—
167C2×C32—23no—
168C2×C32—31no—
169C2×C32—31no—
170C2×C32—61no—
171C2×C32—61no—
172C2×C32—61no—
173C2×C32—121no—
174C2×C32C2×D3211yescharacteristic
175C2×C32C2×D3213yes—
176C2×C32C22×C3⋊D311yescharacteristic
177C3⋊D3—32no—
178C3⋊D3—36no—
179C3⋊D3—92no—
180C3⋊D3—182no—
181C3×D3—92no—
182C3×D3—92no—
183C3×D3—182no—
184C2×Q8—811noSylow
185C22×C3—271no—
186C2×D3—91no—
187C2×D3—91no—
188C2×D3—181no—
189C2×D3—273no—
190C3×C4—21no—
191C3×C4—31no—
192C3×C4—33no—
193C3×C4—61no—
194C3×C4—63no—
195C3×C4—96no—
196C3×C4—92no—
197C3×C4—271no—
198C3×C4—272no—
199C3×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
200C3×C4D3×C3⋊D311yescharacteristic
201Dic3—32no—
202Dic3—92no—
203Dic3—96no—
204Dic3—92no—
205Dic3—92no—
206Dic3—91no—
207Dic3—91no—
208Dic3—181no—
209Dic3—182no—
210Dic3—186no—
211Dic3—273no—
212C32—21no—
213C32—23no—
214C32—31no—
215C32—31no—
216C32—61no—
217C32—61no—
218C32—61no—
219C32—121no—
220C32D3×Dic611yescharacteristic
221C32D3×Dic613yescomplemented
222C32Q8×C3⋊D311yescharacteristic, complemented
223Q8—91no—
224Q8—271no—
225Q8—812no—
226C2×C4—271no—
227C2×C4—812no—
228C2×C3—21no—
229C2×C3—31no—
230C2×C3—33no—
231C2×C3—61no—
232C2×C3—63no—
233C2×C3—272no—
234C2×C3C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
235C2×C3C2×D3×C3⋊D311yescharacteristic
236D3—92no—
237D3—92no—
238D3—182no—
239D3—276no—
240C22—271no—
241C4—92no—
242C4—271no—
243C4—272no—
244C4((C3⋊D3)⋊C3)⋊D311yescharacteristic
245C3—21no—
246C3—33no—
247C3—31no—
248C3—61no—
249C3—63no—
250C3((C3⋊D3)⋊C3)⋊Q811yescharacteristic, complemented
251C3C3⋊D3×Dic611yescharacteristic
252C2—272no—
253C2C2×((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)⋊C3)C22
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
3. C3×#634⋊C3C2semidirect
5. #10619881C2semidirect
19. C3×#634D3semidirect
28. #634⋊C3D3semidirect
32. C3⋊((C3⋊D3)⋊C3)Q8semidirect
57. C3×H(3,𝔽3)C2×Q8semidirect
68. #634C3⋊D3semidirect
91. C3⋊(C3⋊D3)Dic6semidirect
139. C33C2×Dic6semidirect
140. C33D3×Q8semidirect
142. H(3,𝔽3)D3×Q8semidirect
221. C32D3×Dic6semidirect
222. C32Q8×C3⋊D3semidirect
250. 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
C4×C3⋊((C3⋊D3)⋊C3)2
C3×#634⋊C32
#106198832
#106198812
#890403
#890193
((C3⋊D3)⋊C3)⋊Q83