Finite Group Databasebeta

← #13048664

Subgroups — #13048664

Order 1296.

505 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 505 conjugacy classes of subgroups — too many to draw. Only its 76 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
2#10619956C211yesmaximal, characteristic, complemented
3C3×#634⋊C3C211yesmaximal, characteristic, complemented
4#10619885C212yesmaximal
5#10619881C212yesmaximal, complemented
6#89040—31nomaximal
7C3⋊D3×Dic6—31nomaximal
8#88651—33nomaximal
9C2×C3⋊#272C2211yescharacteristic
10C3×C4×H(3,𝔽3)C2211yescharacteristic
11C3⋊#1084C2211yescharacteristic
12C3⋊((C3⋊Dic3)⋊C3)C2212yes—
13C3×(C3⋊Dic3)⋊C3C2212yes—
14#6308D311yescharacteristic, complemented
15C3×C4×C3⋊D3—31no—
16C3×C4×C3⋊D3—31no—
17C3×C4×C3⋊D3—61no—
18C3×#634—31no—
19C3×#634D311yescharacteristic, complemented
20C32×Dic6—31no—
21#6296—32no—
22#6294—32no—
23#6293—32no—
24Dic3×C3⋊D3—32no—
25C4×#272—33no—
26C4×#272—63no—
27#634⋊C3—33no—
28#634⋊C3D313yescomplemented
29#6199—36no—
30#6196—36no—
31C2×C3×H(3,𝔽3)C2311yescharacteristic
32C3⋊#272Q812yescomplemented
33Q8×C3⋊D3—91no—
34#3797—91no—
35D3×Dic6—93no—
36D3×Dic6—91no—
37D3×Dic6—91no—
38C2×C3×C3⋊D3—31no—
39C2×C3×C3⋊D3—31no—
40C2×C3×C3⋊D3—61no—
41C33×C4—21no—
42C33×C4C2×D311yescharacteristic
43C33×C4C2×D311yescharacteristic
44C3⋊(C3⋊Dic3)C2×D312yes—
45C3×C3⋊Dic3—31no—
46C3×C3⋊Dic3—32no—
47C3×C3⋊Dic3—31no—
48C3×C3⋊Dic3—61no—
49C3×C3⋊Dic3C2×D312yes—
50C32×Dic3—32no—
51C2×#272—33no—
52C2×#272—63no—
53C4×H(3,𝔽3)—23no—
54C4×H(3,𝔽3)C2×D313yes—
55#1084—33no—
56#1084—63no—
57(C3⋊Dic3)⋊C3—36no—
58(C3⋊Dic3)⋊C3C2×D316yes—
59C3×H(3,𝔽3)C2×Q811yescharacteristic, complemented, Sylow
60C32×Q8—91no—
61C4×C3⋊D3—31no—
62C4×C3⋊D3—31no—
63C4×C3⋊D3—61no—
64C4×C3⋊D3—92no—
65#634—31no—
66#634—33no—
67#634—91no—
68#634—92no—
69#634—181no—
70#634C3⋊D311yescharacteristic, complemented
71C3×C4×D3—93no—
72C3×C4×D3—91no—
73C3×C4×D3—91no—
74C3×C4×D3—183no—
75C3×C4×D3—181no—
76C3×Dic6—33no—
77C3×Dic6—31no—
78C3×Dic6—91no—
79C3×Dic6—91no—
80C3×Dic6—91no—
81C3×Dic6—181no—
82#627—96no—
83#627—92no—
84#627—92no—
85#627—92no—
86#624—92no—
87D3×Dic3—96no—
88D3×Dic3—92no—
89D3×Dic3—92no—
90C2×C33—21no—
91C2×C33C22×D311yescharacteristic
92C2×C33C22×D311yescharacteristic
93C3×C3⋊D3—32no—
94C3×C3⋊D3—32no—
95C3×C3⋊D3—62no—
96C2×H(3,𝔽3)—23no—
97C2×H(3,𝔽3)C22×D313yes—
98#272—36no—
99#272—66no—
100D3×Q8—271no—
101D3×Q8—271no—
102D3×Q8—273no—
103C2×Dic6—271no—
104C2×C3⋊D3—31no—
105C2×C3⋊D3—31no—
106C2×C3⋊D3—61no—
107C2×C3×D3—93no—
108C2×C3×D3—91no—
109C2×C3×D3—91no—
110C2×C3×D3—183no—
111C2×C3×D3—181no—
112C32×C4—21no—
113C32×C4—23no—
114C32×C4—31no—
115C32×C4—31no—
116C32×C4—61no—
117C32×C4—61no—
118C32×C4—61no—
119C32×C4—92no—
120C32×C4—121no—
121C32×C4D3211yescharacteristic
122C32×C4D3213yes—
123C32×C4C2×C3⋊D311yescharacteristic
124C3⋊Dic3—31no—
125C3⋊Dic3—31no—
126C3⋊Dic3—32no—
127C3⋊Dic3—36no—
128C3⋊Dic3—61no—
129C3⋊Dic3—92no—
130C3⋊Dic3—182no—
131C3⋊Dic3C2×C3⋊D312yes—
132C3×Dic3—36no—
133C3×Dic3—32no—
134C3×Dic3—92no—
135C3×Dic3—92no—
136C3×Dic3—93no—
137C3×Dic3—91no—
138C3×Dic3—92no—
139C3×Dic3—91no—
140C3×Dic3—183no—
141C3×Dic3—181no—
142C3×Dic3—182no—
143C33—21no—
144C33D3×Q811yescharacteristic, complemented
145C33D3×Q811yescharacteristic, complemented
146H(3,𝔽3)—23no—
147H(3,𝔽3)D3×Q813yescomplemented
148C3×Q8—93no—
149C3×Q8—91no—
150C3×Q8—271no—
151C2×C3×C4—271no—
152C2×Dic3—272no—
153C4×D3—91no—
154C4×D3—93no—
155C4×D3—91no—
156C4×D3—181no—
157C4×D3—183no—
158C4×D3—272no—
159C4×D3—272no—
160C4×D3—276no—
161Dic6—31no—
162Dic6—91no—
163Dic6—91no—
164Dic6—91no—
165Dic6—181no—
166Dic6—273no—
167Dic6—272no—
168Dic6—272no—
169Dic6—272no—
170Dic6—276no—
171C2×C32—21no—
172C2×C32—23no—
173C2×C32—31no—
174C2×C32—31no—
175C2×C32—61no—
176C2×C32—61no—
177C2×C32—61no—
178C2×C32—121no—
179C2×C32C2×D3211yescharacteristic
180C2×C32C2×D3213yes—
181C2×C32C22×C3⋊D311yescharacteristic
182C3⋊D3—32no—
183C3⋊D3—32no—
184C3⋊D3—62no—
185C3×D3—92no—
186C3×D3—96no—
187C3×D3—92no—
188C3×D3—186no—
189C3×D3—182no—
190C2×Q8—811noSylow
191C22×C3—271no—
192C2×D3—91no—
193C2×D3—93no—
194C2×D3—91no—
195C2×D3—181no—
196C2×D3—183no—
197C3×C4—21no—
198C3×C4—31no—
199C3×C4—33no—
200C3×C4—61no—
201C3×C4—63no—
202C3×C4—96no—
203C3×C4—92no—
204C3×C4—272no—
205C3×C4—271no—
206C3×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
207C3×C4D3×C3⋊D311yescharacteristic
208Dic3—32no—
209Dic3—92no—
210Dic3—91no—
211Dic3—93no—
212Dic3—92no—
213Dic3—91no—
214Dic3—92no—
215Dic3—182no—
216Dic3—181no—
217Dic3—183no—
218Dic3—276no—
219C32—21no—
220C32—23no—
221C32—31no—
222C32—31no—
223C32—61no—
224C32—61no—
225C32—61no—
226C32—121no—
227C32#379711yescharacteristic
228C32#379713yescomplemented
229C32Q8×C3⋊D311yescharacteristic, complemented
230Q8—91no—
231Q8—271no—
232Q8—812no—
233C2×C4—271no—
234C2×C4—812no—
235C2×C3—21no—
236C2×C3—31no—
237C2×C3—33no—
238C2×C3—61no—
239C2×C3—63no—
240C2×C3—272no—
241C2×C3C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
242C2×C3C2×D3×C3⋊D311yescharacteristic
243D3—96no—
244D3—92no—
245D3—92no—
246D3—182no—
247D3—186no—
248C22—271no—
249C4—92no—
250C4—271no—
251C4—272no—
252C4((C3⋊D3)⋊C3)⋊D311yescharacteristic
253C3—21no—
254C3—33no—
255C3—31no—
256C3—61no—
257C3—63no—
258C3#8865111yescharacteristic, complemented
259C3#8901911yescharacteristic
260C2—272no—
261C2C2×((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⋊#272C22
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
2. #10619956C2semidirect
3. C3×#634⋊C3C2semidirect
5. #10619881C2semidirect
14. #6308D3semidirect
19. C3×#634D3semidirect
28. #634⋊C3D3semidirect
32. C3⋊#272Q8semidirect
59. C3×H(3,𝔽3)C2×Q8semidirect
70. #634C3⋊D3semidirect
144. C33D3×Q8semidirect
145. C33D3×Q8semidirect
147. H(3,𝔽3)D3×Q8semidirect
228. C32#3797semidirect
229. C32Q8×C3⋊D3semidirect
258. C3#88651semidirect

Maximal subgroups

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

subgroupindex
C4×C3⋊#2722
#106199562
C3×#634⋊C32
#106198852
#106198812
#890403
C3⋊D3×Dic63
#886513