Finite Group Databasebeta

← D3×#6235 #13048735

Subgroups — D3×#6235 #13048735

Order 1296.

730 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 730 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C22×D3×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×D3×#272C211yesmaximal, characteristic, complemented
3#10619994C211yesmaximal, characteristic, complemented
4C3×#6235C211yesmaximal, characteristic, complemented
5#10619892C211yesmaximal, characteristic, complemented
6#10619891C211yesmaximal, characteristic, complemented
7D3×#1084C211yesmaximal, characteristic, complemented
8C3×D3×#67—34nomaximal
9C2×#6235—31nomaximal
10C22×C3×H(3,𝔽3)C2211yescharacteristic, complemented
11C2×C3⋊#272C2211yescharacteristic, complemented
12C2×C3×#272C2211yescharacteristic, complemented
13C2×D3×H(3,𝔽3)—22no—
14C2×D3×H(3,𝔽3)C2211yescharacteristic
15C2×D3×H(3,𝔽3)C2211yescharacteristic, complemented
16D3×#272—22no—
17C3⋊#1084C2211yescharacteristic, complemented
18C3×#1084C2211yescharacteristic, complemented
19C22×C32×D3D314yescomplemented
20C2×C3×D32—34no—
21C3×#638—34no—
22C32×#67—34no—
23C3×#626—34no—
24C3×#625—34no—
25C3×D3×Dic3—34no—
26C23×H(3,𝔽3)—31no—
27C22×#272—31no—
28#6235—31no—
29#6235—31no—
30#6235—31no—
31#6235—64no—
32#6235D311yescharacteristic, direct product factor
33C2×#1084—31no—
34C2×C3×H(3,𝔽3)—21no—
35C2×C3×H(3,𝔽3)C2311yescharacteristic
36C3⋊#272—21no—
37C3×#272—21no—
38D3×H(3,𝔽3)—21no—
39D3×H(3,𝔽3)D412yescomplemented
40C2×C3×#67—94no—
41C3×D3×D4—91no—
42D3×#67—34no—
43C22×C33C2×D314yescomplemented
44C2×C3×C3⋊D3—34no—
45C2×C32×D3—28no—
46C2×C32×D3—34no—
47C2×C32×D3C2×D314yes—
48C2×C32×D3C2×D314yescomplemented
49C3×D32—68no—
50C3×C3⋊Dic3—34no—
51C32×Dic3—34no—
52C22×H(3,𝔽3)—24no—
53C22×H(3,𝔽3)—31no—
54C22×H(3,𝔽3)—31no—
55C22×H(3,𝔽3)—62no—
56C22×H(3,𝔽3)C2×D311yescharacteristic, complemented
57C2×#272—31no—
58C2×#272—62no—
59C2×#272—64no—
60C2×#272C2×D311yescharacteristic, complemented
61#1084—31no—
62#1084—64no—
63#1084C2×D311yescharacteristic, complemented
64C3×H(3,𝔽3)C2×D411yescharacteristic, complemented, Sylow
65C23×C32—34no—
66C22×C3×D3—34no—
67C22×C3×D3—94no—
68C22×C3×D3—91no—
69C22×C3×D3C3⋊D311yescharacteristic
70C2×D32—34no—
71C32×D4—91no—
72#638—34no—
73C3×#67—34no—
74C3×#67—34no—
75C3×#67—64no—
76C3×#67—94no—
77C3×#67—94no—
78C3×#67—94no—
79C3×#67—91no—
80C3×#67—91no—
81C3×#67—184no—
82C2×C3×Dic3—94no—
83C3×D12—91no—
84C3×C4×D3—91no—
85#626—34no—
86#625—34no—
87D3×Dic3—34no—
88C2×C33—24no—
89C2×C33C22×D314yes—
90C3×C3⋊D3—64no—
91C32×D3—24no—
92C32×D3—64no—
93C32×D3#6718yescomplemented
94C2×H(3,𝔽3)—21no—
95C2×H(3,𝔽3)—24no—
96C2×H(3,𝔽3)—32no—
97C2×H(3,𝔽3)—44no—
98C2×H(3,𝔽3)—61no—
99C2×H(3,𝔽3)C22×D311yescharacteristic
100#272—21no—
101#272—61no—
102#272—124no—
103C2×C3×D4—271no—
104C2×#67—94no—
105D3×D4—91no—
106C22×C32—24no—
107C22×C32—34no—
108C22×C32—34no—
109C22×C32—34no—
110C22×C32—64no—
111C22×C32—68no—
112C22×C32—91no—
113C22×C32D3214yescomplemented
114C22×C32C2×C3⋊D311yescharacteristic
115C2×C3⋊D3—34no—
116C2×C3×D3—22no—
117C2×C3×D3—34no—
118C2×C3×D3—34no—
119C2×C3×D3—34no—
120C2×C3×D3—34no—
121C2×C3×D3—68no—
122C2×C3×D3—64no—
123C2×C3×D3—94no—
124C2×C3×D3—91no—
125C2×C3×D3—184no—
126C2×C3×D3—188no—
127C2×C3×D3—182no—
128C2×C3×D3C2×C3⋊D311yescharacteristic
129C2×C3×D3C2×C3⋊D311yescharacteristic
130D32—68no—
131C32×C4—91no—
132C3⋊Dic3—34no—
133C3×Dic3—34no—
134C3×Dic3—34no—
135C3×Dic3—64no—
136C3×Dic3—94no—
137C3×Dic3—91no—
138C3×Dic3—184no—
139C33C2×#6714yescomplemented
140H(3,𝔽3)—24no—
141H(3,𝔽3)D3×D411yescharacteristic, complemented
142C23×C3—31no—
143C23×C3—94no—
144C23×C3—271no—
145C22×D3—94no—
146C22×D3—91no—
147C22×D3#27211yescharacteristic, complemented
148C3×D4—91no—
149C3×D4—91no—
150C3×D4—181no—
151C3×D4—271no—
152C3×D4—271no—
153C3×D4—271no—
154C2×C3×C4—271no—
155#67—34no—
156#67—94no—
157#67—94no—
158#67—94no—
159#67—91no—
160#67—91no—
161#67—184no—
162C2×Dic3—94no—
163D12—91no—
164C4×D3—91no—
165C2×C32—21no—
166C2×C32—24no—
167C2×C32—24no—
168C2×C32—34no—
169C2×C32—38no—
170C2×C32—44no—
171C2×C32—64no—
172C2×C32—64no—
173C2×C32—64no—
174C2×C32—124no—
175C2×C32—181no—
176C2×C32C2×D3214yes—
177C2×C32C22×C3⋊D311yescharacteristic
178C3⋊D3—64no—
179C3×D3—21no—
180C3×D3—38no—
181C3×D3—64no—
182C3×D3—64no—
183C3×D3—64no—
184C3×D3—124no—
185C3×D3—184no—
186C3×D3—181no—
187C3×D3—364no—
188C3×D3#63812yes—
189C2×D4—271noSylow
190C22×C3—21no—
191C22×C3—31no—
192C22×C3—31no—
193C22×C3—34no—
194C22×C3—62no—
195C22×C3—64no—
196C22×C3—94no—
197C22×C3—94no—
198C22×C3—91no—
199C22×C3—91no—
200C22×C3—188no—
201C22×C3—181no—
202C22×C3—271no—
203C22×C3—542no—
204C22×C3C2×#27211yescharacteristic, complemented
205C22×C3D3×C3⋊D311yescharacteristic
206C2×D3—22no—
207C2×D3—34no—
208C2×D3—94no—
209C2×D3—91no—
210C2×D3—184no—
211C2×D3—188no—
212C2×D3—182no—
213C2×D3C2×#27211yescharacteristic
214C2×D3C2×#27211yescharacteristic, complemented
215C3×C4—91no—
216C3×C4—91no—
217C3×C4—181no—
218C3×C4—271no—
219Dic3—34no—
220Dic3—94no—
221Dic3—91no—
222Dic3—184no—
223C32—24no—
224C32—34no—
225C32—64no—
226C32D3×#6714yescomplemented
227C32C2×#63811yescharacteristic
228C23—31no—
229C23—271no—
230D4—91no—
231D4—271no—
232D4—271no—
233D4—271no—
234C2×C4—271no—
235C2×C3—21no—
236C2×C3—21no—
237C2×C3—21no—
238C2×C3—32no—
239C2×C3—34no—
240C2×C3—41no—
241C2×C3—64no—
242C2×C3—61no—
243C2×C3—64no—
244C2×C3—98no—
245C2×C3—124no—
246C2×C3—184no—
247C2×C3—181no—
248C2×C3—181no—
249C2×C3—361no—
250C2×C3—541no—
251C2×C3C22×#27211yescharacteristic
252C2×C3C2×D3×C3⋊D311yescharacteristic
253D3—21no—
254D3—64no—
255D3—184no—
256D3—181no—
257D3—364no—
258D3#623512yesdirect product factor
259C22—31no—
260C22—31no—
261C22—62no—
262C22—91no—
263C22—271no—
264C22—542no—
265C22D3×#27211yescharacteristic, complemented
266C4—91no—
267C4—271no—
268C3—21no—
269C3—34no—
270C3—64no—
271C3C2×#623511yescharacteristic, complemented
272C3D3×#63811yescharacteristic
273C2—21no—
274C2—32no—
275C2—61no—
276C2—181no—
277C2—541no—
278C2C2×D3×#27211yescharacteristic
subgroupG/H
center, FrattiniC2×C3C2×D3×C3⋊D3
commutator, cosocleC2×C3×H(3,𝔽3)C23
FittingC22×C3×H(3,𝔽3)C22
socleC2×C32C22×C3⋊D3
hypercenterC22×C3D3×C3⋊D3
hypocenterC3×H(3,𝔽3)C2×D4
LCSG ▷ C2×C3×H(3,𝔽3) ▷ C3×H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C22×C3

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C22×D3×H(3,𝔽3)C2semidirect
2. C2×D3×#272C2semidirect
3. #10619994C2semidirect
4. C3×#6235C2semidirect
5. #10619892C2semidirect
6. #10619891C2semidirect
7. D3×#1084C2semidirect
10. C22×C3×H(3,𝔽3)C22semidirect
11. C2×C3⋊#272C22semidirect
12. C2×C3×#272C22semidirect
15. C2×D3×H(3,𝔽3)C22semidirect
17. C3⋊#1084C22semidirect
18. C3×#1084C22semidirect
19. C22×C32×D3D3semidirect
32. #6235D3direct (N × H)
39. D3×H(3,𝔽3)D4semidirect
43. C22×C33C2×D3semidirect
48. C2×C32×D3C2×D3semidirect
56. C22×H(3,𝔽3)C2×D3semidirect
60. C2×#272C2×D3semidirect
63. #1084C2×D3semidirect
64. C3×H(3,𝔽3)C2×D4semidirect
93. C32×D3#67semidirect
113. C22×C32D32semidirect
139. C33C2×#67semidirect
141. H(3,𝔽3)D3×D4semidirect
147. C22×D3#272semidirect
204. C22×C3C2×#272semidirect
214. C2×D3C2×#272semidirect
226. C32D3×#67semidirect
258. D3#6235direct (N × H)
265. C22D3×#272semidirect
271. C3C2×#6235semidirect

Maximal subgroups

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

subgroupindex
C22×D3×H(3,𝔽3)2
C2×D3×#2722
#106199942
C3×#62352
#106198922
#106198912
D3×#10842
C3×D3×#673
C2×#62353