Subgroups — #13179708

Order 1728.

2688 conjugacy classes of subgroups.

The subgroup lattice has 272 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	#11723987	C₂	1	1	yes	maximal, characteristic
3	#11723985	C₂	1	1	yes	maximal, characteristic, complemented
4	#11723939	C₂	1	2	yes	maximal
5	#11723938	C₂	1	1	yes	maximal, characteristic
6	#11723931	C₂	1	1	yes	maximal, characteristic, complemented
7	#11723930	C₂	1	2	yes	maximal, complemented
8	#11723917	C₂	1	2	yes	maximal
9	#11723913	C₂	1	1	yes	maximal, characteristic
10	#11723911	C₂	1	2	yes	maximal, complemented
11	C₄×#6308	C₂	1	1	yes	maximal, characteristic, complemented
12	#10596054	—	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×#6308	C₂²	1	1	yes	characteristic
16	C₃³×C₄⋊C₄	C₂²	1	2	yes	—
17	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₃³×C₄²	C₂²	1	1	yes	characteristic
19	C₃³×C₄²	C₂²	1	2	yes	—
20	#88859	C₂²	1	1	yes	characteristic
21	#88859	C₂²	1	1	yes	characteristic
22	#88859	C₂²	1	2	yes	—
23	#88859	C₂²	1	2	yes	—
24	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	1	yes	characteristic
25	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	2	yes	—
26	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	2	yes	—
27	#88857	C₂²	1	2	yes	—
28	#88857	C₂²	1	1	yes	characteristic
29	#88857	C₂²	1	2	yes	—
30	#88857	C₂²	1	2	yes	—
31	#88857	C₂²	1	1	yes	characteristic
32	#88857	C₂²	1	2	yes	—
33	#88857	C₂²	1	2	yes	—
34	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
35	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
36	C₃²×C₄×Q₈	D₃	1	13	yes	complemented
37	#64155	—	3	13	no	—
38	#64153	—	3	13	no	—
39	#64107	—	3	26	no	—
40	#64106	—	3	13	no	—
41	#64099	—	3	13	no	—
42	#64098	—	3	26	no	—
43	#64085	—	3	26	no	—
44	#64081	—	3	13	no	—
45	#64079	—	3	26	no	—
46	C₄×#634	—	3	13	no	—
47	C₂²×C₃⋊(C₃⋊D₃)	C₂³	1	1	yes	characteristic
48	C₃³×Q₈	—	2	2	no	—
49	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
50	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
51	C₂×C₃³×C₄	C₂³	1	2	yes	—
52	C₂×C₃³×C₄	C₂³	1	2	yes	—
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	2	yes	—
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	#6308	—	2	2	no	—
61	#5583	—	9	13	no	—
62	C₂×C₃²×Q₈	C₂×D₃	1	13	yes	—
63	C₂×C₄×C₃⋊D₃	—	3	13	no	—
64	C₂×#634	—	3	13	no	—
65	C₃²×C₄⋊C₄	C₂×D₃	1	26	yes	—
66	C₃²×C₄⋊C₄	C₂×D₃	1	13	yes	—
67	C₃²×C₄²	C₂×D₃	1	13	yes	—
68	C₃²×C₄²	C₂×D₃	1	26	yes	—
69	#3752	—	3	13	no	—
70	#3752	—	3	13	no	—
71	#3752	—	3	26	no	—
72	#3752	—	3	26	no	—
73	C₄⋊(C₃⋊Dic₃)	—	3	13	no	—
74	C₄⋊(C₃⋊Dic₃)	—	3	26	no	—
75	C₄⋊(C₃⋊Dic₃)	—	3	26	no	—
76	#3750	—	3	26	no	—
77	#3750	—	3	13	no	—
78	#3750	—	3	26	no	—
79	#3750	—	3	26	no	—
80	#3750	—	3	13	no	—
81	#3750	—	3	26	no	—
82	#3750	—	3	26	no	—
83	C₄×C₃⋊Dic₃	—	3	26	no	—
84	C₄×C₃⋊Dic₃	—	3	13	no	—
85	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
86	C₂×C₃⋊(C₃⋊D₃)	—	2	1	no	—
87	C₂×C₃⋊(C₃⋊D₃)	—	2	1	no	—
88	C₂×C₃⋊(C₃⋊D₃)	—	2	1	no	—
89	C₃³×C₄	—	2	2	no	—
90	C₃³×C₄	—	2	1	no	—
91	C₃³×C₄	—	2	1	no	—
92	C₃³×C₄	—	2	2	no	—
93	C₃³×C₄	#41	1	2	yes	—
94	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
95	C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
96	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
97	C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
98	C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
99	C₃×C₄×Q₈	C₃⋊D₃	1	13	yes	complemented
100	#946	—	9	13	no	—
101	#944	—	9	13	no	—
102	#898	—	9	26	no	—
103	#897	—	9	13	no	—
104	#890	—	9	13	no	—
105	#889	—	9	26	no	—
106	#876	—	9	26	no	—
107	#872	—	9	13	no	—
108	#870	—	9	26	no	—
109	C₄×Dic₆	—	9	13	no	—
110	C₂²×C₃⋊D₃	—	3	13	no	—
111	C₃²×Q₈	—	2	26	no	—
112	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
113	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
114	C₂×C₃²×C₄	C₂²×D₃	1	26	yes	—
115	C₂×C₃²×C₄	C₂²×D₃	1	26	yes	—
116	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
117	C₂×C₃⋊Dic₃	—	3	13	no	—
118	C₂×C₃⋊Dic₃	—	3	26	no	—
119	C₂×C₃⋊Dic₃	—	3	13	no	—
120	C₂×C₃⋊Dic₃	—	3	26	no	—
121	C₂×C₃⋊Dic₃	—	3	13	no	—
122	C₄×C₃⋊D₃	—	6	26	no	—
123	#634	—	6	26	no	—
124	#541	—	27	1	no	Sylow
125	C₂×C₃³	C₂×#41	1	1	yes	characteristic
126	C₂×C₃³	2₋¹⁺⁴	1	1	yes	characteristic
127	C₂×C₃³	2₋¹⁺⁴	1	1	yes	characteristic
128	C₃⋊(C₃⋊D₃)	—	4	1	no	—
129	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	13	yes	—
130	C₂×C₄×D₃	—	9	13	no	—
131	C₂×Dic₆	—	9	13	no	—
132	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	26	yes	—
133	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	13	yes	—
134	C₃×C₄²	C₂×C₃⋊D₃	1	13	yes	—
135	C₃×C₄²	C₂×C₃⋊D₃	1	26	yes	—
136	(C₂×D₃)⋊C₄	—	9	13	no	—
137	(C₂×D₃)⋊C₄	—	9	13	no	—
138	(C₂×D₃)⋊C₄	—	9	26	no	—
139	(C₂×D₃)⋊C₄	—	9	26	no	—
140	C₄⋊Dic₃	—	9	13	no	—
141	C₄⋊Dic₃	—	9	26	no	—
142	C₄⋊Dic₃	—	9	26	no	—
143	Dic₃⋊C₄	—	9	26	no	—
144	Dic₃⋊C₄	—	9	13	no	—
145	Dic₃⋊C₄	—	9	26	no	—
146	Dic₃⋊C₄	—	9	26	no	—
147	Dic₃⋊C₄	—	9	13	no	—
148	Dic₃⋊C₄	—	9	26	no	—
149	Dic₃⋊C₄	—	9	26	no	—
150	C₄×Dic₃	—	9	26	no	—
151	C₄×Dic₃	—	9	13	no	—
152	C₂²×C₃²	C₂³×D₃	1	13	yes	—
153	C₂×C₃⋊D₃	—	6	13	no	—
154	C₂×C₃⋊D₃	—	6	13	no	—
155	C₂×C₃⋊D₃	—	6	13	no	—
156	C₃²×C₄	—	2	26	no	—
157	C₃²×C₄	—	2	13	no	—
158	C₃²×C₄	—	2	13	no	—
159	C₃²×C₄	—	2	26	no	—
160	C₃²×C₄	#235	1	26	yes	—
161	C₃⋊Dic₃	—	6	26	no	—
162	C₃⋊Dic₃	—	6	13	no	—
163	C₃⋊Dic₃	—	6	26	no	—
164	C₃⋊Dic₃	—	6	13	no	—
165	C₃⋊Dic₃	—	6	13	no	—
166	C₄⋊Q₈	—	27	1	no	—
167	#126	—	27	2	no	—
168	#126	—	27	2	no	—
169	#125	—	27	2	no	—
170	#125	—	27	1	no	—
171	#125	—	27	2	no	—
172	C₂²⋊Q₈	—	27	1	no	—
173	C₂²⋊Q₈	—	27	1	no	—
174	C₄×Q₈	—	27	1	no	—
175	C₄×Q₈	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
176	#117	—	27	1	no	—
177	C₃³	#541	1	1	yes	characteristic, complemented, Sylow
178	C₂²×D₃	—	9	13	no	—
179	C₃×Q₈	—	2	26	no	—
180	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
181	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
182	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	26	yes	—
183	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	26	yes	—
184	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
185	C₂×Dic₃	—	9	13	no	—
186	C₂×Dic₃	—	9	26	no	—
187	C₂×Dic₃	—	9	13	no	—
188	C₂×Dic₃	—	9	26	no	—
189	C₂×Dic₃	—	9	13	no	—
190	C₄×D₃	—	18	26	no	—
191	Dic₆	—	18	26	no	—
192	C₂×C₃²	C₂×#235	1	13	yes	—
193	C₂×C₃²	#1007	1	13	yes	—
194	C₂×C₃²	#1010	1	13	yes	—
195	C₃⋊D₃	—	12	13	no	—
196	C₂×Q₈	—	27	1	no	—
197	C₂×Q₈	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
198	C₂²×C₄	—	27	1	no	—
199	C₄⋊C₄	—	27	1	no	—
200	C₄⋊C₄	—	27	1	no	—
201	C₄⋊C₄	—	27	1	no	—
202	C₄⋊C₄	—	27	2	no	—
203	C₄⋊C₄	—	27	2	no	—
204	C₄⋊C₄	—	27	2	no	—
205	C₄⋊C₄	—	27	2	no	—
206	C₄⋊C₄	—	27	2	no	—
207	C₄⋊C₄	—	27	2	no	—
208	C₄⋊C₄	—	27	2	no	—
209	C₄⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
210	C₄⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
211	C₂²⋊C₄	—	27	1	no	—
212	C₂²⋊C₄	—	27	1	no	—
213	C₂²⋊C₄	—	27	2	no	—
214	C₂²⋊C₄	—	27	2	no	—
215	C₄²	—	27	1	no	—
216	C₄²	—	27	2	no	—
217	C₄²	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
218	C₄²	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
219	C₂²×C₃	C₂³×C₃⋊D₃	1	13	yes	—
220	C₂×D₃	—	18	13	no	—
221	C₂×D₃	—	18	13	no	—
222	C₂×D₃	—	18	13	no	—
223	C₃×C₄	—	2	26	no	—
224	C₃×C₄	—	2	13	no	—
225	C₃×C₄	—	2	13	no	—
226	C₃×C₄	—	2	26	no	—
227	C₃×C₄	#3828	1	26	yes	—
228	Dic₃	—	18	26	no	—
229	Dic₃	—	18	13	no	—
230	Dic₃	—	18	26	no	—
231	Dic₃	—	18	13	no	—
232	Dic₃	—	18	13	no	—
233	C₃²	#5583	1	13	yes	complemented
234	C₂³	—	27	1	no	—
235	Q₈	—	2	2	no	—
236	Q₈	—	54	2	no	—
237	C₂×C₄	—	27	1	no	—
238	C₂×C₄	—	27	2	no	—
239	C₂×C₄	—	27	1	no	—
240	C₂×C₄	—	27	2	no	—
241	C₂×C₄	—	27	1	no	—
242	C₂×C₄	—	54	2	no	—
243	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
244	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
245	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	2	yes	—
246	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	2	yes	—
247	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
248	C₂×C₃	C₂×#3828	1	13	yes	—
249	C₂×C₃	#64364	1	13	yes	—
250	C₂×C₃	#64367	1	13	yes	—
251	D₃	—	36	13	no	—
252	C₂²	—	54	1	no	—
253	C₂²	—	54	1	no	—
254	C₂²	—	54	1	no	—
255	C₂²	C₂³×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
256	C₄	—	2	2	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	2	no	—
260	C₄	—	54	1	no	—
261	C₄	—	54	2	no	—
262	C₄	—	54	1	no	—
263	C₄	—	54	2	no	—
264	C₄	—	54	1	no	—
265	C₄	#89076	1	2	yes	—
266	C₃	#10596054	1	13	yes	complemented
267	C₂	—	108	1	no	—
268	C₂	C₂×#89076	1	1	yes	characteristic
269	C₂	#11724682	1	1	yes	characteristic
270	C₂	#11724685	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	C₂²×C₃⋊(C₃⋊D₃)	C₂³
hypercenter	C₄×Q₈	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#541
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
6. #11723931	C₂	semidirect
7. #11723930	C₂	semidirect
10. #11723911	C₂	semidirect
11. C₄×#6308	C₂	semidirect
36. C₃²×C₄×Q₈	D₃	semidirect
99. C₃×C₄×Q₈	C₃⋊D₃	semidirect
175. C₄×Q₈	C₃⋊(C₃⋊D₃)	semidirect
177. C₃³	#541	semidirect
233. C₃²	#5583	semidirect
266. C₃	#10596054	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
#11723987	2
#11723985	2
#11723939	2
#11723938	2
#11723931	2
#11723930	2
#11723917	2
#11723913	2
#11723911	2
C₄×#6308	2
#10596054	3