Finite Group Databasebeta

← #13048659

Subgroups — #13048659

Order 1296.

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

normal subgroups only

This group has 531 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
1C4×C3⋊#272C211yesmaximal, characteristic, complemented
2C4⋊(C3⋊((C3⋊D3)⋊C3))C211yesmaximal, characteristic, complemented
3C3×#634⋊C3C211yesmaximal, characteristic, complemented
4#10619887C212yesmaximal, complemented
5#10619883C212yesmaximal, complemented
6#89041—31nomaximal
7#89018—31nomaximal
8#88649—33nomaximal
9C2×C3⋊#272C2211yescharacteristic
10C2×C3⋊((C3⋊D3)⋊C3)C2212yes—
11C3×C4×H(3,𝔽3)C2211yescharacteristic
12C3⋊#1084C2211yescharacteristic
13C3×(C3⋊Dic3)⋊C3C2212yes—
14C4⋊(C3⋊(C3⋊D3))D311yescharacteristic, complemented
15C3×C4⋊(C3⋊D3)—31no—
16C3×C4×C3⋊D3—31no—
17C3×C4×C3⋊D3—31no—
18C3×C4×C3⋊D3—61no—
19C3×#634D311yescharacteristic, complemented
20C32×Dic6—31no—
21#6295—32no—
22#6294—32no—
23#6292—32no—
24#6289—32no—
25C4×#272—33no—
26C4×#272—63no—
27C4⋊((C3⋊D3)⋊C3)—33no—
28#634⋊C3D313yescomplemented
29#6200—36no—
30((C3⋊D3)⋊C3)⋊C4—36no—
31C2×C3×H(3,𝔽3)C2311yescharacteristic
32C3⋊#272—21no—
33C3⋊((C3⋊D3)⋊C3)—22no—
34#3832—91no—
35#3799—93no—
36#3799—91no—
37#3796—91no—
38#3795—91no—
39C2×C3⋊(C3⋊D3)C2×D312yes—
40C2×C3×C3⋊D3—31no—
41C2×C3×C3⋊D3—32no—
42C2×C3×C3⋊D3—31no—
43C2×C3×C3⋊D3—61no—
44C33×C4—21no—
45C33×C4C2×D311yescharacteristic
46C33×C4C2×D311yescharacteristic
47C3×C3⋊Dic3—31no—
48C3×C3⋊Dic3—31no—
49C3×C3⋊Dic3—61no—
50C3×C3⋊Dic3C2×D312yes—
51C32×Dic3—32no—
52C2×#272—33no—
53C2×#272—63no—
54C2×(C3⋊D3)⋊C3—36no—
55C4×H(3,𝔽3)—23no—
56C4×H(3,𝔽3)C2×D313yes—
57#1084—33no—
58#1084—63no—
59(C3⋊Dic3)⋊C3C2×D316yes—
60C3×H(3,𝔽3)#4111yescharacteristic, complemented, Sylow
61C32×Q8—91no—
62C4⋊(C3⋊D3)—31no—
63C4⋊(C3⋊D3)—33no—
64C4⋊(C3⋊D3)—91no—
65C4⋊(C3⋊D3)—92no—
66C4⋊(C3⋊D3)—181no—
67C4×C3⋊D3—31no—
68C4×C3⋊D3—31no—
69C4×C3⋊D3—61no—
70C4×C3⋊D3—92no—
71#634C3⋊D311yescharacteristic, complemented
72C3×D12—91no—
73C3×D12—91no—
74C3×D12—181no—
75C3×C4×D3—93no—
76C3×C4×D3—91no—
77C3×C4×D3—91no—
78C3×C4×D3—183no—
79C3×C4×D3—181no—
80C3×Dic6—33no—
81C3×Dic6—31no—
82C3×Dic6—91no—
83#626—96no—
84#626—92no—
85#626—92no—
86#625—92no—
87#624—96no—
88#624—92no—
89D3×Dic3—92no—
90D3×Dic3—92no—
91C2×C33—21no—
92C2×C33C22×D311yescharacteristic
93C2×C33C22×D311yescharacteristic
94C3⋊(C3⋊D3)—22no—
95C3×C3⋊D3—62no—
96C3×C3⋊D3—61no—
97C3×C3⋊D3—61no—
98C3×C3⋊D3—62no—
99C2×H(3,𝔽3)—23no—
100C2×H(3,𝔽3)C22×D313yes—
101#272—66no—
102#272—63no—
103(C3⋊D3)⋊C3—66no—
104#239—271no—
105#239—273no—
106#237—271no—
107#235—271no—
108C2×C3⋊D3—31no—
109C2×C3⋊D3—31no—
110C2×C3⋊D3—32no—
111C2×C3⋊D3—36no—
112C2×C3⋊D3—61no—
113C2×C3⋊D3—92no—
114C2×C3⋊D3—182no—
115C2×C3×D3—92no—
116C2×C3×D3—93no—
117C2×C3×D3—91no—
118C2×C3×D3—92no—
119C2×C3×D3—91no—
120C2×C3×D3—183no—
121C2×C3×D3—181no—
122C2×C3×D3—182no—
123C32×C4—21no—
124C32×C4—23no—
125C32×C4—31no—
126C32×C4—31no—
127C32×C4—61no—
128C32×C4—61no—
129C32×C4—61no—
130C32×C4—92no—
131C32×C4—121no—
132C32×C4D3211yescharacteristic
133C32×C4D3213yes—
134C32×C4C2×C3⋊D311yescharacteristic
135C3⋊Dic3—31no—
136C3⋊Dic3—31no—
137C3⋊Dic3—61no—
138C3⋊Dic3C2×C3⋊D312yes—
139C3×Dic3—36no—
140C3×Dic3—32no—
141C3×Dic3—92no—
142C3×Dic3—93no—
143C3×Dic3—91no—
144C3×Dic3—91no—
145C3×Dic3—183no—
146C3×Dic3—181no—
147C33—21no—
148C33#23711yescharacteristic, complemented
149C33#23911yescharacteristic, complemented
150H(3,𝔽3)—23no—
151H(3,𝔽3)#23913yescomplemented
152C3×Q8—93no—
153C3×Q8—91no—
154C3×D4—271no—
155C2×C3×C4—271no—
156#67—272no—
157#67—272no—
158C2×Dic3—272no—
159D12—91no—
160D12—91no—
161D12—181no—
162D12—273no—
163D12—272no—
164D12—276no—
165C4×D3—91no—
166C4×D3—93no—
167C4×D3—91no—
168C4×D3—181no—
169C4×D3—183no—
170C4×D3—272no—
171C4×D3—272no—
172C4×D3—276no—
173Dic6—31no—
174Dic6—91no—
175C2×C32—21no—
176C2×C32—23no—
177C2×C32—31no—
178C2×C32—31no—
179C2×C32—61no—
180C2×C32—61no—
181C2×C32—61no—
182C2×C32—121no—
183C2×C32C2×D3211yescharacteristic
184C2×C32C2×D3213yes—
185C2×C32C22×C3⋊D311yescharacteristic
186C3⋊D3—61no—
187C3⋊D3—61no—
188C3⋊D3—62no—
189C3⋊D3—62no—
190C3⋊D3—66no—
191C3⋊D3—182no—
192C3⋊D3—362no—
193C3×D3—182no—
194C3×D3—186no—
195C3×D3—182no—
196C3×D3—181no—
197C3×D3—183no—
198C3×D3—182no—
199C3×D3—181no—
200C3×D3—362no—
201#41—811noSylow
202C22×C3—272no—
203C22×C3—271no—
204C2×D3—93no—
205C2×D3—91no—
206C2×D3—91no—
207C2×D3—92no—
208C2×D3—92no—
209C2×D3—181no—
210C2×D3—183no—
211C2×D3—182no—
212C2×D3—276no—
213C3×C4—21no—
214C3×C4—31no—
215C3×C4—33no—
216C3×C4—61no—
217C3×C4—63no—
218C3×C4—96no—
219C3×C4—92no—
220C3×C4—271no—
221C3×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
222C3×C4D3×C3⋊D311yescharacteristic
223Dic3—32no—
224Dic3—92no—
225Dic3—93no—
226Dic3—91no—
227Dic3—91no—
228Dic3—181no—
229Dic3—183no—
230C32—21no—
231C32—23no—
232C32—31no—
233C32—31no—
234C32—61no—
235C32—61no—
236C32—61no—
237C32—121no—
238C32#379611yescharacteristic
239C32#379613yescomplemented
240C32#383211yescharacteristic, complemented
241Q8—91no—
242D4—271no—
243D4—812no—
244C2×C4—271no—
245C2×C4—812no—
246C2×C3—21no—
247C2×C3—31no—
248C2×C3—33no—
249C2×C3—61no—
250C2×C3—63no—
251C2×C3—542no—
252C2×C3—541no—
253C2×C3C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
254C2×C3C2×D3×C3⋊D311yescharacteristic
255D3—183no—
256D3—181no—
257D3—182no—
258D3—186no—
259D3—181no—
260D3—182no—
261D3—182no—
262D3—362no—
263D3—546no—
264C22—271no—
265C22—272no—
266C4—92no—
267C4—271no—
268C4((C3⋊D3)⋊C3)⋊D311yescharacteristic
269C3—21no—
270C3—33no—
271C3—31no—
272C3—61no—
273C3—63no—
274C3#8864911yescharacteristic, complemented
275C3#8901711yescharacteristic
276C2—541no—
277C2—542no—
278C2C2×((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. C4×C3⋊#272C2semidirect
2. C4⋊(C3⋊((C3⋊D3)⋊C3))C2semidirect
3. C3×#634⋊C3C2semidirect
4. #10619887C2semidirect
5. #10619883C2semidirect
14. C4⋊(C3⋊(C3⋊D3))D3semidirect
19. C3×#634D3semidirect
28. #634⋊C3D3semidirect
60. C3×H(3,𝔽3)#41semidirect
71. #634C3⋊D3semidirect
148. C33#237semidirect
149. C33#239semidirect
151. H(3,𝔽3)#239semidirect
239. C32#3796semidirect
240. C32#3832semidirect
274. C3#88649semidirect

Maximal subgroups

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

subgroupindex
C4×C3⋊#2722
C4⋊(C3⋊((C3⋊D3)⋊C3))2
C3×#634⋊C32
#106198872
#106198832
#890413
#890183
#886493