Subgroups — #13144415

Order 1728.

682 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₄×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₂×#88538	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#6180	C₂	1	1	yes	maximal, characteristic, complemented
4	#11720573	C₂	1	1	yes	maximal, characteristic, complemented
5	#11720573	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄⋊#6180	C₂	1	1	yes	maximal, characteristic, complemented
7	#11720520	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#5011	—	3	4	no	maximal
9	C₂×D₄×H(3,𝔽₃)	C₄	1	1	yes	characteristic
10	C₂²×C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
11	C₂²×C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×H(3,𝔽₃)	C₄	1	1	yes	characteristic
13	C₂²⋊C₄×H(3,𝔽₃)	C₄	1	1	yes	characteristic
14	C₂²⋊C₄×H(3,𝔽₃)	C₄	1	1	yes	characteristic
15	C₄²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
16	#88538	—	2	2	no	—
17	C₂×#6180	—	2	2	no	—
18	C₂×#6180	C₂²	1	1	yes	characteristic
19	C₂×#6180	C₂²	1	1	yes	characteristic
20	C₂×#6180	C₂²	1	1	yes	characteristic
21	C₂×#6180	C₂²	1	1	yes	characteristic
22	C₃²×C₄×D₄	D₃	1	4	yes	—
23	C₂×C₃×#208	—	3	4	no	—
24	C₂²×C₃×C₃⋊C₈	—	3	4	no	—
25	C₃×#830	—	3	4	no	—
26	C₃×#830	—	3	4	no	—
27	C₃×C₄⋊(C₃⋊C₈)	—	3	4	no	—
28	C₃×#803	—	3	4	no	—
29	C₂³×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
30	C₂³×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
31	D₄×H(3,𝔽₃)	—	2	2	no	—
32	C₂×C₄×H(3,𝔽₃)	—	2	1	no	—
33	C₂×C₄×H(3,𝔽₃)	—	2	1	no	—
34	C₂×C₄×H(3,𝔽₃)	—	2	1	no	—
35	C₂×C₄×H(3,𝔽₃)	—	2	1	no	—
36	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
41	#6180	—	2	1	no	—
42	#6180	—	2	1	no	—
43	#6180	—	2	1	no	—
44	#6180	D₄	1	2	yes	complemented
45	C₃×#435	—	9	1	no	—
46	#5011	—	3	4	no	—
47	C₂×C₃²×D₄	Dic₃	1	4	yes	—
48	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
49	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
50	C₃²×C₄⋊C₄	Dic₃	1	4	yes	—
51	C₃²×C₂²⋊C₄	Dic₃	1	4	yes	—
52	C₃²×C₂²⋊C₄	Dic₃	1	4	yes	—
53	C₃²×C₄²	C₂×D₃	1	4	yes	—
54	C₃×#208	—	6	8	no	—
55	C₂×C₃×C₃⋊C₈	—	3	4	no	—
56	C₂×C₃×C₃⋊C₈	—	3	4	no	—
57	C₂×C₃×C₃⋊C₈	—	3	4	no	—
58	C₂×C₃×C₃⋊C₈	—	3	4	no	—
59	C₂×C₃×C₃⋊C₈	—	6	8	no	—
60	C₂²×H(3,𝔽₃)	—	2	1	no	—
61	C₂²×H(3,𝔽₃)	—	2	1	no	—
62	C₂²×H(3,𝔽₃)	—	2	1	no	—
63	C₂²×H(3,𝔽₃)	—	2	1	no	—
64	C₂²×H(3,𝔽₃)	—	2	1	no	—
65	C₂²×H(3,𝔽₃)	M₁₆	1	2	yes	complemented
66	C₂²×H(3,𝔽₃)	C₂²×C₄	1	1	yes	characteristic
67	C₄×H(3,𝔽₃)	—	2	1	no	—
68	C₄×H(3,𝔽₃)	—	2	1	no	—
69	C₄×H(3,𝔽₃)	—	2	1	no	—
70	C₄×H(3,𝔽₃)	—	2	1	no	—
71	C₄×H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic
72	C₄×H(3,𝔽₃)	#41	1	1	yes	characteristic
73	C₂×C₃×M₁₆	—	9	1	no	—
74	C₂²×C₃×C₈	—	9	1	no	—
75	C₃×C₄×D₄	—	3	4	no	—
76	C₃×C₄×D₄	C₃⋊D₃	1	1	yes	characteristic
77	C₂×#208	—	3	4	no	—
78	C₂²×C₃⋊C₈	—	3	4	no	—
79	C₃×C₄⋊C₈	—	9	1	no	—
80	C₃×C₂²⋊C₈	—	9	1	no	—
81	C₃×C₂²⋊C₈	—	9	1	no	—
82	C₃×#97	—	9	1	no	—
83	#830	—	3	4	no	—
84	#830	—	3	4	no	—
85	C₄⋊(C₃⋊C₈)	—	3	4	no	—
86	#803	—	3	4	no	—
87	C₂³×C₃²	C₂×Dic₃	1	4	yes	—
88	C₂³×C₃²	C₂×Dic₃	1	4	yes	—
89	C₃²×D₄	—	2	8	no	—
90	C₂×C₃²×C₄	—	2	4	no	—
91	C₂×C₃²×C₄	—	2	4	no	—
92	C₂×C₃²×C₄	—	2	4	no	—
93	C₂×C₃²×C₄	—	2	4	no	—
94	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
95	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
96	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
97	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
98	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
99	C₃×C₃⋊C₈	—	3	8	no	—
100	C₃×C₃⋊C₈	—	6	4	no	—
101	C₃×C₃⋊C₈	—	6	4	no	—
102	C₃×C₃⋊C₈	—	6	4	no	—
103	#435	—	9	1	no	Sylow
104	C₂×H(3,𝔽₃)	—	2	2	no	—
105	C₂×H(3,𝔽₃)	—	4	1	no	—
106	C₂×H(3,𝔽₃)	C₄×D₄	1	1	yes	characteristic
107	C₂×H(3,𝔽₃)	C₂×M₁₆	1	1	yes	characteristic
108	C₂×H(3,𝔽₃)	#131	1	1	yes	characteristic
109	C₂×C₃×D₄	—	3	4	no	—
110	C₂×C₃×D₄	C₃⋊Dic₃	1	1	yes	characteristic
111	C₂²×C₃×C₄	—	3	4	no	—
112	C₂²×C₃×C₄	—	3	4	no	—
113	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
114	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
115	C₃×M₁₆	—	18	2	no	—
116	C₂×C₃×C₈	—	9	1	no	—
117	C₂×C₃×C₈	—	9	1	no	—
118	C₂×C₃×C₈	—	9	1	no	—
119	C₂×C₃×C₈	—	9	1	no	—
120	C₂×C₃×C₈	—	18	2	no	—
121	C₃×C₄⋊C₄	—	3	4	no	—
122	C₃×C₄⋊C₄	C₃⋊Dic₃	1	1	yes	characteristic
123	C₃×C₂²⋊C₄	—	3	4	no	—
124	C₃×C₂²⋊C₄	—	3	4	no	—
125	C₃×C₂²⋊C₄	C₃⋊Dic₃	1	1	yes	characteristic
126	C₃×C₂²⋊C₄	C₃⋊Dic₃	1	1	yes	characteristic
127	C₃×C₄²	—	3	4	no	—
128	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
129	#208	—	6	8	no	—
130	C₂×C₃⋊C₈	—	3	4	no	—
131	C₂×C₃⋊C₈	—	3	4	no	—
132	C₂×C₃⋊C₈	—	3	4	no	—
133	C₂×C₃⋊C₈	—	3	4	no	—
134	C₂×C₃⋊C₈	—	6	8	no	—
135	C₂²×C₃²	—	2	4	no	—
136	C₂²×C₃²	—	2	4	no	—
137	C₂²×C₃²	—	2	4	no	—
138	C₂²×C₃²	—	2	4	no	—
139	C₂²×C₃²	—	2	4	no	—
140	C₂²×C₃²	#208	1	8	yes	complemented
141	C₂²×C₃²	C₂²×Dic₃	1	4	yes	—
142	C₃²×C₄	—	2	4	no	—
143	C₃²×C₄	—	2	4	no	—
144	C₃²×C₄	—	2	4	no	—
145	C₃²×C₄	—	2	4	no	—
146	C₃²×C₄	D₃×D₄	1	4	yes	—
147	C₃²×C₄	#237	1	4	yes	—
148	C₂×M₁₆	—	9	1	no	—
149	C₂²×C₈	—	9	1	no	—
150	C₄×D₄	#272	1	1	yes	characteristic
151	C₄⋊C₈	—	9	1	no	—
152	C₂²⋊C₈	—	9	1	no	—
153	C₂²⋊C₈	—	9	1	no	—
154	#97	—	9	1	no	—
155	H(3,𝔽₃)	#435	1	1	yes	characteristic, complemented, Sylow
156	C₂³×C₃	—	3	4	no	—
157	C₂³×C₃	—	3	4	no	—
158	C₂³×C₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
159	C₂³×C₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
160	C₃×D₄	—	2	2	no	—
161	C₃×D₄	—	6	8	no	—
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	2	1	no	—
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	3	4	no	—
167	C₂×C₃×C₄	—	3	4	no	—
168	C₂×C₃×C₄	—	3	4	no	—
169	C₂×C₃×C₄	—	3	4	no	—
170	C₂×C₃×C₄	—	3	4	no	—
171	C₂×C₃×C₄	—	6	4	no	—
172	C₂×C₃×C₄	—	6	4	no	—
173	C₂×C₃×C₄	—	6	4	no	—
174	C₂×C₃×C₄	—	6	4	no	—
175	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
180	C₃×C₈	—	9	2	no	—
181	C₃×C₈	—	18	1	no	—
182	C₃×C₈	—	18	1	no	—
183	C₃×C₈	—	18	1	no	—
184	C₃⋊C₈	—	3	8	no	—
185	C₃⋊C₈	—	6	4	no	—
186	C₃⋊C₈	—	6	4	no	—
187	C₃⋊C₈	—	6	4	no	—
188	C₂×C₃²	—	2	8	no	—
189	C₂×C₃²	—	4	4	no	—
190	C₂×C₃²	C₂×#208	1	4	yes	—
191	C₂×C₃²	D₄×Dic₃	1	4	yes	—
192	C₂×C₃²	#948	1	4	yes	—
193	C₂×D₄	#1084	1	1	yes	characteristic
194	C₂²×C₄	C₂×#272	1	1	yes	characteristic
195	C₂²×C₄	C₂×#272	1	1	yes	characteristic
196	M₁₆	—	18	2	no	—
197	C₂×C₈	—	9	1	no	—
198	C₂×C₈	—	9	1	no	—
199	C₂×C₈	—	9	1	no	—
200	C₂×C₈	—	9	1	no	—
201	C₂×C₈	—	18	2	no	—
202	C₄⋊C₄	#1084	1	1	yes	characteristic
203	C₂²⋊C₄	#1084	1	1	yes	characteristic
204	C₂²⋊C₄	#1084	1	1	yes	characteristic
205	C₄²	C₂×#272	1	1	yes	characteristic
206	C₂²×C₃	—	2	1	no	—
207	C₂²×C₃	—	2	1	no	—
208	C₂²×C₃	—	2	1	no	—
209	C₂²×C₃	—	2	1	no	—
210	C₂²×C₃	—	2	1	no	—
211	C₂²×C₃	—	3	4	no	—
212	C₂²×C₃	—	3	8	no	—
213	C₂²×C₃	—	6	4	no	—
214	C₂²×C₃	—	6	4	no	—
215	C₂²×C₃	—	6	4	no	—
216	C₂²×C₃	—	6	4	no	—
217	C₂²×C₃	—	6	4	no	—
218	C₂²×C₃	#3748	1	2	yes	—
219	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
220	C₃×C₄	—	2	1	no	—
221	C₃×C₄	—	2	1	no	—
222	C₃×C₄	—	2	1	no	—
223	C₃×C₄	—	2	1	no	—
224	C₃×C₄	—	3	4	no	—
225	C₃×C₄	—	3	4	no	—
226	C₃×C₄	—	6	4	no	—
227	C₃×C₄	—	6	4	no	—
228	C₃×C₄	—	6	4	no	—
229	C₃×C₄	—	6	4	no	—
230	C₃×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
231	C₃×C₄	#3830	1	1	yes	characteristic
232	C₃²	#5011	1	4	yes	complemented
233	C₂³	C₂×#1084	1	1	yes	characteristic
234	C₂³	C₂×#1084	1	1	yes	characteristic
235	D₄	—	2	2	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	C₂×#1084	1	1	yes	characteristic
241	C₂×C₄	C₂×#1084	1	1	yes	characteristic
242	C₂×C₄	C₂×#1084	1	1	yes	characteristic
243	C₂×C₄	C₂×#1084	1	1	yes	characteristic
244	C₂×C₄	C₂²×#272	1	1	yes	characteristic
245	C₈	—	9	2	no	—
246	C₈	—	18	1	no	—
247	C₈	—	18	1	no	—
248	C₈	—	18	1	no	—
249	C₂×C₃	—	2	2	no	—
250	C₂×C₃	—	3	4	no	—
251	C₂×C₃	—	3	4	no	—
252	C₂×C₃	—	3	4	no	—
253	C₂×C₃	—	4	1	no	—
254	C₂×C₃	—	6	8	no	—
255	C₂×C₃	—	12	4	no	—
256	C₂×C₃	C₂×#3748	1	1	yes	characteristic
257	C₂×C₃	D₄×C₃⋊Dic₃	1	1	yes	characteristic
258	C₂×C₃	#64157	1	1	yes	characteristic
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	#88538	1	2	yes	complemented
265	C₂²	C₂²×#1084	1	1	yes	characteristic
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	D₄×#272	1	1	yes	characteristic
271	C₄	#88744	1	1	yes	characteristic
272	C₃	—	3	4	no	—
273	C₃	#10592710	1	1	yes	characteristic
274	C₂	—	2	2	no	—
275	C₂	—	4	1	no	—
276	C₂	C₂×#88538	1	1	yes	characteristic
277	C₂	D₄×#1084	1	1	yes	characteristic
278	C₂	#11721869	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
commutator	C₂²×H(3,𝔽₃)	C₂²×C₄
Fitting	C₄×D₄×H(3,𝔽₃)	C₂
socle	C₂²×C₃	C₂²×C₃⋊Dic₃
prime radical	C₂×D₄×H(3,𝔽₃)	C₄
cosocle	C₂×C₄×H(3,𝔽₃)	C₂³
hypercenter	C₃×C₄×D₄	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#435
LCS	G ▷ C₂²×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×#88538	C₂	semidirect
3. C₂²×#6180	C₂	semidirect
4. #11720573	C₂	semidirect
5. #11720573	C₂	semidirect
6. C₄⋊#6180	C₂	semidirect
7. #11720520	C₂	semidirect
44. #6180	D₄	semidirect
65. C₂²×H(3,𝔽₃)	M₁₆	semidirect
140. C₂²×C₃²	#208	semidirect
155. H(3,𝔽₃)	#435	semidirect
232. C₃²	#5011	semidirect
264. C₂²	#88538	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₄×H(3,𝔽₃)	2
C₂×#88538	2
C₂²×#6180	2
#11720573	2
C₄⋊#6180	2
#11720520	2
C₃×#5011	3