Subgroups — #13179693

Order 1728.

2968 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	C₃³×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#11723988	C₂	1	1	yes	maximal, characteristic
3	#11723985	C₂	1	1	yes	maximal, characteristic, complemented
4	#11723939	C₂	1	2	yes	maximal, complemented
5	#11723937	C₂	1	2	yes	maximal, complemented
6	#11723934	C₂	1	1	yes	maximal, characteristic
7	#11723928	C₂	1	1	yes	maximal, characteristic, complemented
8	#11723917	C₂	1	2	yes	maximal, complemented
9	#11723916	C₂	1	1	yes	maximal, characteristic
10	#11723913	C₂	1	2	yes	maximal, complemented
11	C₄×#6308	C₂	1	1	yes	maximal, characteristic, complemented
12	#10596039	—	3	13	no	maximal
13	C₂×C₃³×Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₃⋊(C₃⋊D₃)	C₂²	1	2	yes	—
16	C₂×#6308	C₂²	1	1	yes	characteristic
17	C₂×#6308	C₂²	1	1	yes	characteristic
18	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₃³×C₄⋊C₄	C₂²	1	2	yes	—
20	C₃³×C₄²	C₂²	1	1	yes	characteristic
21	C₃³×C₄²	C₂²	1	2	yes	—
22	#88859	C₂²	1	2	yes	—
23	#88859	C₂²	1	2	yes	—
24	#88859	C₂²	1	2	yes	—
25	#88859	C₂²	1	2	yes	—
26	#88859	C₂²	1	2	yes	—
27	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	1	yes	characteristic
28	#88857	C₂²	1	1	yes	characteristic
29	#88857	C₂²	1	1	yes	characteristic
30	#88857	C₂²	1	2	yes	—
31	#88857	C₂²	1	2	yes	—
32	#88857	C₂²	1	2	yes	—
33	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
34	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
35	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
36	C₃²×C₄×Q₈	D₃	1	13	yes	complemented
37	#64156	—	3	13	no	—
38	#64153	—	3	13	no	—
39	#64107	—	3	26	no	—
40	#64105	—	3	26	no	—
41	#64102	—	3	13	no	—
42	#64096	—	3	13	no	—
43	#64085	—	3	26	no	—
44	#64084	—	3	13	no	—
45	#64081	—	3	26	no	—
46	C₄×#634	—	3	13	no	—
47	C₂²×C₃⋊(C₃⋊D₃)	C₂³	1	2	yes	—
48	C₃³×Q₈	—	2	2	no	—
49	C₂×C₃³×C₄	C₂³	1	2	yes	—
50	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
51	C₂×C₃³×C₄	C₂³	1	2	yes	—
52	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
53	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
54	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	1	yes	characteristic
55	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	1	yes	characteristic
56	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	1	yes	characteristic
57	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	2	yes	—
58	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	1	yes	characteristic
59	C₄⋊(C₃⋊(C₃⋊D₃))	—	2	2	no	—
60	C₄×C₃⋊(C₃⋊D₃)	—	2	4	no	—
61	#6308	—	2	2	no	—
62	#6308	—	2	2	no	—
63	#5568	—	9	13	no	—
64	C₂×C₃²×Q₈	C₂×D₃	1	13	yes	—
65	C₂×C₄⋊(C₃⋊D₃)	—	3	13	no	—
66	C₂×C₄×C₃⋊D₃	—	3	26	no	—
67	C₂×#634	—	3	13	no	—
68	C₂×#634	—	3	13	no	—
69	C₃²×C₄⋊C₄	C₂×D₃	1	13	yes	—
70	C₃²×C₄⋊C₄	C₂×D₃	1	26	yes	—
71	C₃²×C₄²	C₂×D₃	1	13	yes	—
72	C₃²×C₄²	C₂×D₃	1	26	yes	—
73	#3752	—	3	26	no	—
74	#3752	—	3	26	no	—
75	#3752	—	3	26	no	—
76	#3752	—	3	26	no	—
77	#3752	—	3	26	no	—
78	C₄⋊(C₃⋊Dic₃)	—	3	13	no	—
79	#3750	—	3	13	no	—
80	#3750	—	3	13	no	—
81	#3750	—	3	26	no	—
82	#3750	—	3	26	no	—
83	#3750	—	3	26	no	—
84	C₄×C₃⋊Dic₃	—	3	13	no	—
85	C₄×C₃⋊Dic₃	—	3	13	no	—
86	C₄×C₃⋊Dic₃	—	3	26	no	—
87	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
88	C₂×C₃⋊(C₃⋊D₃)	—	2	2	no	—
89	C₂×C₃⋊(C₃⋊D₃)	—	2	2	no	—
90	C₂×C₃⋊(C₃⋊D₃)	—	2	2	no	—
91	C₃³×C₄	—	2	1	no	—
92	C₃³×C₄	—	2	2	no	—
93	C₃³×C₄	—	2	2	no	—
94	C₃³×C₄	—	2	1	no	—
95	C₃³×C₄	#41	1	2	yes	—
96	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
97	C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
98	C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
99	C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
100	C₃⋊(C₃⋊Dic₃)	#41	1	2	yes	—
101	C₃×C₄×Q₈	C₃⋊D₃	1	13	yes	complemented
102	#947	—	9	13	no	—
103	#944	—	9	13	no	—
104	#898	—	9	26	no	—
105	#896	—	9	26	no	—
106	#893	—	9	13	no	—
107	#887	—	9	13	no	—
108	#876	—	9	26	no	—
109	#875	—	9	13	no	—
110	#872	—	9	26	no	—
111	C₄×Dic₆	—	9	13	no	—
112	C₂²×C₃⋊D₃	—	3	26	no	—
113	C₃²×Q₈	—	2	26	no	—
114	C₂×C₃²×C₄	C₂²×D₃	1	26	yes	—
115	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
116	C₂×C₃²×C₄	C₂²×D₃	1	26	yes	—
117	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
118	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
119	C₂×C₃⋊Dic₃	—	3	13	no	—
120	C₂×C₃⋊Dic₃	—	3	13	no	—
121	C₂×C₃⋊Dic₃	—	3	13	no	—
122	C₂×C₃⋊Dic₃	—	3	26	no	—
123	C₂×C₃⋊Dic₃	—	3	13	no	—
124	C₄⋊(C₃⋊D₃)	—	6	26	no	—
125	C₄×C₃⋊D₃	—	6	52	no	—
126	#634	—	6	26	no	—
127	#634	—	6	26	no	—
128	#556	—	27	1	no	Sylow
129	C₂×C₃³	C₂×#41	1	1	yes	characteristic
130	C₂×C₃³	C₂×#41	1	1	yes	characteristic
131	C₂×C₃³	2₋¹⁺⁴	1	1	yes	characteristic
132	C₃⋊(C₃⋊D₃)	—	4	2	no	—
133	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	13	yes	—
134	C₂×D₁₂	—	9	13	no	—
135	C₂×C₄×D₃	—	9	26	no	—
136	C₂×Dic₆	—	9	13	no	—
137	C₂×Dic₆	—	9	13	no	—
138	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	13	yes	—
139	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	26	yes	—
140	C₃×C₄²	C₂×C₃⋊D₃	1	13	yes	—
141	C₃×C₄²	C₂×C₃⋊D₃	1	26	yes	—
142	(C₂×D₃)⋊C₄	—	9	26	no	—
143	(C₂×D₃)⋊C₄	—	9	26	no	—
144	(C₂×D₃)⋊C₄	—	9	26	no	—
145	(C₂×D₃)⋊C₄	—	9	26	no	—
146	(C₂×D₃)⋊C₄	—	9	26	no	—
147	C₄⋊Dic₃	—	9	13	no	—
148	Dic₃⋊C₄	—	9	13	no	—
149	Dic₃⋊C₄	—	9	13	no	—
150	Dic₃⋊C₄	—	9	26	no	—
151	Dic₃⋊C₄	—	9	26	no	—
152	Dic₃⋊C₄	—	9	26	no	—
153	C₄×Dic₃	—	9	13	no	—
154	C₄×Dic₃	—	9	13	no	—
155	C₄×Dic₃	—	9	26	no	—
156	C₂²×C₃²	C₂³×D₃	1	13	yes	—
157	C₂×C₃⋊D₃	—	6	26	no	—
158	C₂×C₃⋊D₃	—	6	26	no	—
159	C₂×C₃⋊D₃	—	6	26	no	—
160	C₃²×C₄	—	2	26	no	—
161	C₃²×C₄	—	2	13	no	—
162	C₃²×C₄	—	2	13	no	—
163	C₃²×C₄	—	2	26	no	—
164	C₃²×C₄	#235	1	26	yes	—
165	C₃⋊Dic₃	—	3	26	no	—
166	C₃⋊Dic₃	—	6	26	no	—
167	C₃⋊Dic₃	—	6	13	no	—
168	C₃⋊Dic₃	—	6	13	no	—
169	C₃⋊Dic₃	—	6	13	no	—
170	C₄⋊Q₈	—	27	1	no	—
171	#126	—	27	2	no	—
172	#126	—	27	2	no	—
173	#124	—	27	1	no	—
174	#124	—	27	1	no	—
175	C₂²⋊Q₈	—	27	2	no	—
176	C₄×Q₈	—	27	1	no	—
177	C₄×Q₈	—	27	1	no	—
178	C₄×Q₈	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
179	C₄×D₄	—	27	1	no	—
180	#117	—	27	2	no	—
181	C₃³	#556	1	1	yes	characteristic, complemented, Sylow
182	C₂²×D₃	—	9	26	no	—
183	C₃×Q₈	—	2	26	no	—
184	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	26	yes	—
185	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
186	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	26	yes	—
187	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
188	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
189	C₂×Dic₃	—	9	13	no	—
190	C₂×Dic₃	—	9	13	no	—
191	C₂×Dic₃	—	9	13	no	—
192	C₂×Dic₃	—	9	26	no	—
193	C₂×Dic₃	—	9	13	no	—
194	D₁₂	—	18	26	no	—
195	C₄×D₃	—	18	52	no	—
196	Dic₆	—	18	26	no	—
197	Dic₆	—	18	26	no	—
198	C₂×C₃²	C₂×#235	1	13	yes	—
199	C₂×C₃²	#1007	1	13	yes	—
200	C₂×C₃²	D₃×#41	1	13	yes	—
201	C₃⋊D₃	—	12	26	no	—
202	C₂×Q₈	—	27	1	no	—
203	C₂×Q₈	—	27	1	no	—
204	C₂×Q₈	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
205	C₂×D₄	—	27	1	no	—
206	C₂²×C₄	—	27	2	no	—
207	C₄⋊C₄	—	27	1	no	—
208	C₄⋊C₄	—	27	2	no	—
209	C₄⋊C₄	—	27	2	no	—
210	C₄⋊C₄	—	27	1	no	—
211	C₄⋊C₄	—	27	1	no	—
212	C₄⋊C₄	—	27	2	no	—
213	C₄⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
214	C₄⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
215	C₂²⋊C₄	—	27	2	no	—
216	C₂²⋊C₄	—	27	2	no	—
217	C₂²⋊C₄	—	27	2	no	—
218	C₂²⋊C₄	—	27	2	no	—
219	C₂²⋊C₄	—	27	2	no	—
220	C₄²	—	27	2	no	—
221	C₄²	—	27	1	no	—
222	C₄²	—	27	1	no	—
223	C₄²	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
224	C₄²	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
225	C₂²×C₃	C₂³×C₃⋊D₃	1	13	yes	—
226	C₂×D₃	—	18	26	no	—
227	C₂×D₃	—	18	26	no	—
228	C₂×D₃	—	18	26	no	—
229	C₃×C₄	—	2	26	no	—
230	C₃×C₄	—	2	13	no	—
231	C₃×C₄	—	2	13	no	—
232	C₃×C₄	—	2	26	no	—
233	C₃×C₄	#3828	1	26	yes	—
234	Dic₃	—	9	26	no	—
235	Dic₃	—	18	26	no	—
236	Dic₃	—	18	13	no	—
237	Dic₃	—	18	13	no	—
238	Dic₃	—	18	13	no	—
239	C₃²	#5568	1	13	yes	complemented
240	C₂³	—	27	2	no	—
241	Q₈	—	2	2	no	—
242	Q₈	—	54	2	no	—
243	Q₈	—	54	2	no	—
244	D₄	—	54	2	no	—
245	C₂×C₄	—	27	1	no	—
246	C₂×C₄	—	27	1	no	—
247	C₂×C₄	—	27	1	no	—
248	C₂×C₄	—	27	2	no	—
249	C₂×C₄	—	27	1	no	—
250	C₂×C₄	—	54	4	no	—
251	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	2	yes	—
252	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
253	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	2	yes	—
254	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
255	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
256	C₂×C₃	C₂×#3828	1	13	yes	—
257	C₂×C₃	#64364	1	13	yes	—
258	C₂×C₃	#41×C₃⋊D₃	1	13	yes	—
259	D₃	—	36	26	no	—
260	C₂²	—	54	2	no	—
261	C₂²	—	54	2	no	—
262	C₂²	—	54	2	no	—
263	C₂²	C₂³×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	2	no	—
267	C₄	—	2	1	no	—
268	C₄	—	27	2	no	—
269	C₄	—	54	2	no	—
270	C₄	—	54	1	no	—
271	C₄	—	54	1	no	—
272	C₄	—	54	1	no	—
273	C₄	#89076	1	2	yes	—
274	C₃	#10596039	1	13	yes	complemented
275	C₂	—	108	2	no	—
276	C₂	C₂×#89076	1	1	yes	characteristic
277	C₂	#11724682	1	1	yes	characteristic
278	C₂	#41×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₃⋊(C₃⋊D₃)
commutator, socle, cosocle	C₂²×C₃³	C₂⁴
Fitting	C₃³×C₄×Q₈	C₂
prime radical	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	C₂²
hypercenter	C₄×Q₈	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#556
LCS	G ▷ C₂²×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₃³×C₄×Q₈	C₂	semidirect
3. #11723985	C₂	semidirect
4. #11723939	C₂	semidirect
5. #11723937	C₂	semidirect
7. #11723928	C₂	semidirect
8. #11723917	C₂	semidirect
10. #11723913	C₂	semidirect
11. C₄×#6308	C₂	semidirect
36. C₃²×C₄×Q₈	D₃	semidirect
101. C₃×C₄×Q₈	C₃⋊D₃	semidirect
178. C₄×Q₈	C₃⋊(C₃⋊D₃)	semidirect
181. C₃³	#556	semidirect
239. C₃²	#5568	semidirect
274. C₃	#10596039	semidirect

Maximal subgroups

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

subgroup	index
C₃³×C₄×Q₈	2
#11723988	2
#11723985	2
#11723939	2
#11723937	2
#11723934	2
#11723928	2
#11723917	2
#11723916	2
#11723913	2
C₄×#6308	2
#10596039	3