Subgroups — #11797331

Order 1152.

337 conjugacy classes of subgroups.

The subgroup lattice has 271 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic
2	#10589415	C₂	1	1	yes	maximal, characteristic
3	C₃×#4549	C₂	1	1	yes	maximal, characteristic
4	Dic₃⋊(C₃⋊C₁₆)	C₂	1	2	yes	maximal, complemented
5	#10589103	C₂	1	2	yes	maximal, complemented
6	#69745	—	3	1	no	maximal
7	#69093	—	3	1	no	maximal
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄×Dic₃	C₄	1	1	yes	characteristic
10	C₂²×C₃×C₃⋊C₈	C₄	1	1	yes	characteristic
11	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
12	C₃×C₈×Dic₃	C₄	1	1	yes	characteristic
13	C₃×C₈×Dic₃	C₄	1	1	yes	characteristic
14	C₃×C₈×Dic₃	C₂²	1	1	yes	characteristic
15	C₃×C₈×Dic₃	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₃⋊C₁₆	C₂²	1	2	yes	—
17	C₂×C₃×C₄×C₈	—	3	1	no	—
18	C₂×C₈×Dic₃	D₃	1	1	yes	characteristic
19	C₃×C₂²⋊C₁₆	—	3	1	no	—
20	#4549	—	3	1	no	—
21	#4549	—	3	1	no	—
22	#4549	—	6	1	no	—
23	#4549	D₃	1	1	yes	characteristic
24	#4502	—	3	2	no	—
25	Dic₃⋊C₁₆	—	3	2	no	—
26	C₄⋊(C₃⋊C₁₆)	—	3	2	no	—
27	#4461	—	3	2	no	—
28	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×Dic₃	C₈	1	1	yes	characteristic
30	C₂²×C₃×Dic₃	C₈	1	1	yes	characteristic
31	C₂×C₃²×C₈	—	2	2	no	—
32	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
34	C₃×C₄×Dic₃	C₈	1	1	yes	characteristic
35	C₃×C₄×Dic₃	C₈	1	1	yes	characteristic
36	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×C₃⋊C₈	—	2	2	no	—
39	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
41	C₃⋊(C₃⋊C₁₆)	—	2	2	no	—
42	C₃×C₃⋊C₁₆	—	2	2	no	—
43	#2163	—	9	1	no	Sylow
44	C₂²×C₃×C₈	—	2	1	no	—
45	C₂²×C₃×C₈	—	3	1	no	—
46	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
47	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
48	C₂×C₃×C₄²	—	3	1	no	—
49	C₂×C₄×Dic₃	Dic₃	1	1	yes	characteristic
50	C₂²×C₃⋊C₈	Dic₃	1	1	yes	characteristic
51	C₂×C₃×C₁₆	—	3	2	no	—
52	C₃×C₄×C₈	—	3	1	no	—
53	C₃×C₄×C₈	—	3	1	no	—
54	C₃×C₄×C₈	—	3	1	no	—
55	C₃×C₄×C₈	—	3	1	no	—
56	C₈×Dic₃	Dic₃	1	1	yes	characteristic
57	C₈×Dic₃	Dic₃	1	1	yes	characteristic
58	C₈×Dic₃	C₂×D₃	1	1	yes	characteristic
59	C₈×Dic₃	C₂×D₃	1	1	yes	characteristic
60	C₂×C₃⋊C₁₆	—	3	2	no	—
61	C₂×C₃⋊C₁₆	—	3	2	no	—
62	C₂×C₃⋊C₁₆	—	6	2	no	—
63	C₂×C₃⋊C₁₆	C₂×D₃	1	2	yes	—
64	C₂³×C₃²	C₂×C₈	1	1	yes	characteristic
65	C₂×C₃²×C₄	—	2	1	no	—
66	C₂×C₃²×C₄	—	2	1	no	—
67	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
68	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₃×Dic₃	—	2	2	no	—
70	C₂×C₃×Dic₃	—	2	2	no	—
71	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
72	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
73	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
74	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
75	C₃²×C₈	—	2	1	no	—
76	C₃²×C₈	#41	1	2	yes	—
77	C₃×C₃⋊C₈	—	2	1	no	—
78	C₃×C₃⋊C₈	M₁₆	1	2	yes	—
79	C₂×C₄×C₈	—	3	1	no	—
80	C₄⋊C₁₆	—	9	2	no	—
81	C₂²⋊C₁₆	—	3	1	no	—
82	C₂²⋊C₁₆	—	9	1	no	—
83	#346	—	9	2	no	—
84	C₂²×C₃×C₄	—	2	1	no	—
85	C₂²×C₃×C₄	—	3	1	no	—
86	C₂²×C₃×C₄	—	3	1	no	—
87	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
88	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
89	C₂²×Dic₃	C₃⋊C₈	1	1	yes	characteristic
90	C₂²×Dic₃	C₃⋊C₈	1	1	yes	characteristic
91	C₂×C₃×C₈	—	2	2	no	—
92	C₂×C₃×C₈	—	2	2	no	—
93	C₂×C₃×C₈	—	2	1	no	—
94	C₂×C₃×C₈	—	2	1	no	—
95	C₂×C₃×C₈	—	3	1	no	—
96	C₂×C₃×C₈	—	3	1	no	—
97	C₂×C₃×C₈	—	4	2	no	—
98	C₂×C₃×C₈	—	6	2	no	—
99	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
100	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
101	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
102	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
103	C₃×C₄²	—	3	1	no	—
104	C₃×C₄²	—	3	1	no	—
105	C₃×C₄²	—	3	1	no	—
106	C₃×C₄²	—	3	1	no	—
107	C₄×Dic₃	C₃⋊C₈	1	1	yes	characteristic
108	C₄×Dic₃	C₃⋊C₈	1	1	yes	characteristic
109	C₄×Dic₃	C₂×Dic₃	1	1	yes	characteristic
110	C₄×Dic₃	C₂×Dic₃	1	1	yes	characteristic
111	C₂×C₃⋊C₈	—	2	2	no	—
112	C₂×C₃⋊C₈	C₂×Dic₃	1	1	yes	characteristic
113	C₂×C₃⋊C₈	C₂×Dic₃	1	1	yes	characteristic
114	C₃×C₁₆	—	6	2	no	—
115	C₃⋊C₁₆	—	2	2	no	—
116	C₃⋊C₁₆	—	6	2	no	—
117	C₃⋊C₁₆	—	6	2	no	—
118	C₃⋊C₁₆	—	12	2	no	—
119	C₂²×C₃²	—	2	1	no	—
120	C₂²×C₃²	—	2	1	no	—
121	C₂²×C₃²	M₃₂	1	2	yes	—
122	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
123	C₃²×C₄	—	2	1	no	—
124	C₃²×C₄	#117	1	1	yes	characteristic
125	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
126	C₃×Dic₃	—	2	1	no	—
127	C₃×Dic₃	—	2	1	no	—
128	C₃×Dic₃	—	2	1	no	—
129	C₃×Dic₃	M₃₂	1	2	yes	—
130	C₂²×C₈	—	3	1	no	—
131	C₂²×C₈	D₃²	1	1	yes	characteristic
132	C₂×C₄²	—	3	1	no	—
133	C₂×C₁₆	—	3	2	no	—
134	C₂×C₁₆	—	9	2	no	—
135	C₄×C₈	—	3	1	no	—
136	C₄×C₈	—	3	1	no	—
137	C₄×C₈	—	3	1	no	—
138	C₄×C₈	—	3	1	no	—
139	C₂³×C₃	—	2	1	no	—
140	C₂³×C₃	D₃×C₈	1	1	yes	characteristic
141	C₂³×C₃	C₂×C₃⋊C₈	1	1	yes	characteristic
142	C₂×C₃×C₄	—	2	1	no	—
143	C₂×C₃×C₄	—	2	1	no	—
144	C₂×C₃×C₄	—	2	1	no	—
145	C₂×C₃×C₄	—	2	1	no	—
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	—	2	1	no	—
148	C₂×C₃×C₄	—	3	1	no	—
149	C₂×C₃×C₄	—	3	1	no	—
150	C₂×C₃×C₄	—	3	1	no	—
151	C₂×C₃×C₄	—	3	1	no	—
152	C₂×C₃×C₄	—	4	1	no	—
153	C₂×C₃×C₄	—	4	1	no	—
154	C₂×C₃×C₄	—	6	2	no	—
155	C₂×C₃×C₄	—	6	2	no	—
156	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
157	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
159	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
160	C₂×Dic₃	—	2	2	no	—
161	C₂×Dic₃	—	2	2	no	—
162	C₂×Dic₃	C₂×C₃⋊C₈	1	1	yes	characteristic
163	C₂×Dic₃	C₂×C₃⋊C₈	1	1	yes	characteristic
164	C₂×Dic₃	C₂×C₃⋊C₈	1	1	yes	characteristic
165	C₂×Dic₃	C₂×C₃⋊C₈	1	1	yes	characteristic
166	C₃×C₈	—	2	1	no	—
167	C₃×C₈	—	2	1	no	—
168	C₃×C₈	—	2	2	no	—
169	C₃×C₈	—	3	2	no	—
170	C₃×C₈	—	4	1	no	—
171	C₃×C₈	—	6	1	no	—
172	C₃×C₈	#235	1	2	yes	—
173	C₃×C₈	#237	1	2	yes	—
174	C₃⋊C₈	—	2	1	no	—
175	C₃⋊C₈	#208	1	2	yes	—
176	C₂×C₃²	—	2	2	no	—
177	C₂×C₃²	#431	1	1	yes	characteristic
178	C₂×C₃²	C₂×M₃₂	1	1	yes	characteristic
179	C₂×C₃²	C₂×M₃₂	1	1	yes	characteristic
180	C₂²×C₄	—	3	1	no	—
181	C₂²×C₄	—	3	1	no	—
182	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	3	1	no	—
185	C₂×C₈	—	3	1	no	—
186	C₂×C₈	—	6	2	no	—
187	C₂×C₈	D₃×Dic₃	1	1	yes	characteristic
188	C₂×C₈	C₂×D₃²	1	1	yes	characteristic
189	C₄²	—	3	1	no	—
190	C₄²	—	3	1	no	—
191	C₄²	—	3	1	no	—
192	C₄²	—	3	1	no	—
193	C₁₆	—	6	2	no	—
194	C₁₆	—	18	2	no	—
195	C₂²×C₃	—	2	1	no	—
196	C₂²×C₃	—	2	1	no	—
197	C₂²×C₃	—	2	1	no	—
198	C₂²×C₃	—	2	1	no	—
199	C₂²×C₃	—	2	2	no	—
200	C₂²×C₃	—	2	1	no	—
201	C₂²×C₃	—	4	1	no	—
202	C₂²×C₃	—	4	1	no	—
203	C₂²×C₃	#798	1	2	yes	—
204	C₂²×C₃	#812	1	2	yes	—
205	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
206	C₂²×C₃	C₂²×C₃⋊C₈	1	1	yes	characteristic
207	C₃×C₄	—	2	1	no	—
208	C₃×C₄	—	2	1	no	—
209	C₃×C₄	—	2	1	no	—
210	C₃×C₄	—	2	1	no	—
211	C₃×C₄	—	3	2	no	—
212	C₃×C₄	—	4	1	no	—
213	C₃×C₄	—	6	1	no	—
214	C₃×C₄	—	6	1	no	—
215	C₃×C₄	—	6	1	no	—
216	C₃×C₄	#877	1	1	yes	characteristic
217	C₃×C₄	D₃×M₁₆	1	1	yes	characteristic
218	C₃×C₄	C₂×#208	1	1	yes	characteristic
219	C₃×C₄	#926	1	1	yes	characteristic
220	Dic₃	—	2	1	no	—
221	Dic₃	—	2	1	no	—
222	Dic₃	—	2	1	no	—
223	Dic₃	#812	1	2	yes	—
224	C₃²	#2163	1	1	yes	characteristic, complemented, Sylow
225	C₂³	D₃×C₃⋊C₈	1	1	yes	characteristic
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	3	1	no	—
229	C₂×C₄	—	3	1	no	—
230	C₂×C₄	—	3	1	no	—
231	C₂×C₄	—	3	1	no	—
232	C₂×C₄	—	6	2	no	—
233	C₂×C₄	—	6	2	no	—
234	C₂×C₄	D₃×C₃⋊C₈	1	1	yes	characteristic
235	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	3	2	no	—
238	C₈	—	6	1	no	—
239	C₈	#3804	1	2	yes	—
240	C₂×C₃	—	2	2	no	—
241	C₂×C₃	—	2	2	no	—
242	C₂×C₃	—	2	1	no	—
243	C₂×C₃	—	2	1	no	—
244	C₂×C₃	—	2	1	no	—
245	C₂×C₃	—	4	2	no	—
246	C₂×C₃	#4718	1	1	yes	characteristic
247	C₂×C₃	C₂×#798	1	1	yes	characteristic
248	C₂×C₃	D₃×M₃₂	1	1	yes	characteristic
249	C₂×C₃	#4925	1	1	yes	characteristic
250	C₂×C₃	C₂×#812	1	1	yes	characteristic
251	C₂×C₃	C₂×#812	1	1	yes	characteristic
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	#63543	1	2	yes	—
255	C₂²	C₂×D₃×C₃⋊C₈	1	1	yes	characteristic
256	C₄	—	2	1	no	—
257	C₄	—	3	2	no	—
258	C₄	—	6	1	no	—
259	C₄	—	6	1	no	—
260	C₄	—	6	1	no	—
261	C₄	D₃×#208	1	1	yes	characteristic
262	C₄	#63955	1	1	yes	characteristic
263	C₃	—	2	1	no	—
264	C₃	#69093	1	1	yes	characteristic, complemented
265	C₃	#69745	1	1	yes	characteristic, complemented
266	C₂	—	2	2	no	—
267	C₂	D₃×#812	1	1	yes	characteristic
268	C₂	C₂×#63543	1	1	yes	characteristic
269	C₂	#10591053	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×D₃²
commutator, socle	C₂²×C₃²	C₂²×C₈
Fitting	C₂²×C₃²×C₈	C₂²
prime radical	C₂³×C₃²	C₂×C₈
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	C₂²×C₈	D₃²
hypocenter	C₃²	#2163
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
4. Dic₃⋊(C₃⋊C₁₆)	C₂	semidirect
5. #10589103	C₂	semidirect
224. C₃²	#2163	semidirect
264. C₃	#69093	semidirect
265. C₃	#69745	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₈×Dic₃	2
#10589415	2
C₃×#4549	2
Dic₃⋊(C₃⋊C₁₆)	2
#10589103	2
#69745	3
#69093	3