Subgroups — #11795892

Order 1152.

397 conjugacy classes of subgroups.

The subgroup lattice has 265 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₃×C₁₆	C₂	1	1	yes	maximal, characteristic
2	#10590692	C₂	1	1	yes	maximal, characteristic
3	C₂×#63543	C₂	1	1	yes	maximal, characteristic
4	C₁₆×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×Dic₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#10589104	C₂	1	1	yes	maximal, characteristic, complemented
7	#10589103	C₂	1	1	yes	maximal, characteristic, complemented
8	#71399	—	3	1	no	maximal
9	#69051	—	3	1	no	maximal
10	C₂×C₃×D₃×C₈	C₂²	1	1	yes	characteristic
11	#63855	C₄	1	1	yes	characteristic
12	C₂×#3711	C₄	1	1	yes	characteristic
13	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
14	C₈×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
15	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
16	C₃×C₈×Dic₃	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
19	C₃×D₃×C₁₆	C₄	1	4	yes	complemented
20	#63555	C₄	1	1	yes	characteristic
21	#63554	C₄	1	1	yes	characteristic
22	#63543	C₄	1	4	yes	complemented
23	C₂²×C₃×C₁₆	—	3	1	no	—
24	#5101	—	3	1	no	—
25	C₂×#812	—	3	1	no	—
26	C₂×#798	—	3	1	no	—
27	C₂×D₃×C₁₆	D₃	1	1	yes	characteristic
28	#4689	—	3	1	no	—
29	C₃×C₄×C₁₆	—	3	1	no	—
30	#4502	—	3	1	no	—
31	#4502	—	3	1	no	—
32	#4502	—	3	1	no	—
33	Dic₃×C₁₆	—	3	1	no	—
34	Dic₃×C₁₆	—	3	1	no	—
35	Dic₃×C₁₆	—	6	1	no	—
36	Dic₃×C₁₆	D₃	1	1	yes	characteristic, complemented
37	#4461	—	3	1	no	—
38	C₂×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
40	C₄×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
42	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₃×D₃×C₈	C₂×C₄	1	2	yes	—
46	C₃×D₃×C₈	C₂×C₄	1	2	yes	—
47	Dic₃⋊Dic₃	C₈	1	2	yes	—
48	(C₂×D₃)⋊Dic₃	C₈	1	2	yes	—
49	#3711	C₈	1	4	yes	—
50	C₃²×C₁₆	C₂×C₄	1	2	yes	—
51	C₃⋊(C₃⋊C₁₆)	C₂×C₄	1	2	yes	—
52	C₃×C₃⋊C₁₆	C₂×C₄	1	2	yes	—
53	C₃×C₃⋊C₁₆	C₂×C₄	1	2	yes	—
54	#2108	—	9	1	no	Sylow
55	C₂²×C₃×C₈	—	3	1	no	—
56	#926	—	3	1	no	—
57	C₂×#208	—	3	1	no	—
58	C₂×ΓL(1,𝔽₂₅)	—	3	1	no	—
59	C₂×D₃×C₈	C₂×D₃	1	1	yes	characteristic
60	#872	—	3	1	no	—
61	C₂×C₃×C₁₆	—	2	1	no	—
62	C₂×C₃×C₁₆	—	3	1	no	—
63	C₂×C₃×C₁₆	—	3	1	no	—
64	C₂×C₃×C₁₆	—	3	4	no	—
65	C₂×C₃×C₁₆	C₂×D₃	1	1	yes	characteristic
66	C₂×C₃×C₁₆	C₂×D₃	1	1	yes	characteristic
67	C₃×C₄×C₈	—	3	1	no	—
68	#815	—	3	1	no	—
69	#815	—	3	1	no	—
70	#815	—	3	1	no	—
71	C₈×Dic₃	—	3	1	no	—
72	C₈×Dic₃	—	3	1	no	—
73	C₈×Dic₃	—	6	1	no	—
74	C₈×Dic₃	C₂×D₃	1	1	yes	characteristic
75	#812	—	3	4	no	—
76	C₂×C₃⋊C₁₆	—	3	1	no	—
77	C₂×C₃⋊C₁₆	—	3	1	no	—
78	C₂×C₃⋊C₁₆	—	6	1	no	—
79	C₂×C₃⋊C₁₆	C₂×D₃	1	1	yes	characteristic
80	C₂×C₃⋊C₁₆	C₂×D₃	1	1	yes	characteristic
81	#803	—	3	1	no	—
82	#798	—	3	4	no	—
83	D₃×C₁₆	Dic₃	1	4	yes	complemented
84	C₂²×C₃×D₃	C₂×C₈	1	1	yes	characteristic
85	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₃⋊Dic₃	C₂×C₈	1	2	yes	—
87	C₂×C₃×Dic₃	C₂×C₈	1	2	yes	—
88	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
89	C₃×C₄×D₃	C₄²	1	2	yes	—
90	C₃×C₄×D₃	C₂×C₈	1	2	yes	—
91	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
93	C₃⋊(C₃⋊C₈)	C₂×C₈	1	2	yes	—
94	C₃×C₃⋊C₈	C₄²	1	2	yes	—
95	C₃×C₃⋊C₈	C₂×C₈	1	2	yes	—
96	C₂×M₃₂	—	9	1	no	—
97	C₂²×C₁₆	—	3	1	no	—
98	#405	—	9	1	no	—
99	#346	—	9	1	no	—
100	#346	—	9	1	no	—
101	C₄×C₁₆	—	3	1	no	—
102	C₄×C₁₆	—	9	1	no	—
103	C₂²×C₃×C₄	—	3	1	no	—
104	C₂×C₄×D₃	C₄×D₃	1	1	yes	characteristic
105	C₂×C₃×C₈	—	2	1	no	—
106	C₂×C₃×C₈	—	3	1	no	—
107	C₂×C₃×C₈	—	3	1	no	—
108	C₂×C₃×C₈	—	3	2	no	—
109	C₂×C₃×C₈	—	3	2	no	—
110	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
111	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
112	C₃×C₄²	—	3	1	no	—
113	#217	—	3	2	no	—
114	(C₂×D₃)⋊C₄	—	3	2	no	—
115	C₄⋊Dic₃	—	3	2	no	—
116	Dic₃⋊C₄	—	3	2	no	—
117	C₄×Dic₃	—	3	1	no	—
118	C₄×Dic₃	—	3	1	no	—
119	C₄×Dic₃	—	6	1	no	—
120	C₄×Dic₃	C₄×D₃	1	1	yes	characteristic
121	#208	—	3	4	no	—
122	C₂×C₃⋊C₈	—	3	1	no	—
123	C₂×C₃⋊C₈	—	3	1	no	—
124	C₂×C₃⋊C₈	—	6	1	no	—
125	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
126	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
127	ΓL(1,𝔽₂₅)	—	3	4	no	—
128	D₃×C₈	C₄×D₃	1	2	yes	—
129	D₃×C₈	C₂×Dic₃	1	2	yes	—
130	C₃×C₁₆	—	2	2	no	—
131	C₃×C₁₆	—	3	2	no	—
132	C₃×C₁₆	—	3	2	no	—
133	C₃×C₁₆	C₄×D₃	1	2	yes	—
134	C₃×C₁₆	C₂×Dic₃	1	2	yes	—
135	C₃⋊C₁₆	—	3	2	no	—
136	C₃⋊C₁₆	—	3	2	no	—
137	C₃⋊C₁₆	—	6	2	no	—
138	C₃⋊C₁₆	C₄×D₃	1	2	yes	—
139	C₃⋊C₁₆	C₂×Dic₃	1	2	yes	—
140	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
141	C₂×C₃×D₃	—	2	2	no	—
142	C₂×C₃×D₃	C₄×C₈	1	2	yes	—
143	C₃²×C₄	C₂×C₄²	1	1	yes	characteristic
144	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
145	C₃⋊Dic₃	—	2	2	no	—
146	C₃×Dic₃	—	2	2	no	—
147	C₃×Dic₃	C₄×C₈	1	2	yes	—
148	C₂×M₁₆	—	9	1	no	—
149	C₂²×C₈	—	3	1	no	—
150	#117	—	9	1	no	—
151	M₃₂	—	9	4	no	—
152	C₂×C₁₆	—	3	1	no	—
153	C₂×C₁₆	—	3	1	no	—
154	C₂×C₁₆	—	3	4	no	—
155	C₂×C₁₆	—	9	1	no	—
156	C₂×C₁₆	D₃²	1	1	yes	characteristic
157	#97	—	9	1	no	—
158	#97	—	9	1	no	—
159	C₄×C₈	—	3	1	no	—
160	C₄×C₈	—	9	1	no	—
161	C₂³×C₃	—	3	1	no	—
162	C₂²×D₃	D₃×C₈	1	1	yes	characteristic
163	C₂×C₃×C₄	—	2	1	no	—
164	C₂×C₃×C₄	—	3	1	no	—
165	C₂×C₃×C₄	—	3	2	no	—
166	C₂×C₃×C₄	—	3	2	no	—
167	C₂×C₃×C₄	—	3	2	no	—
168	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
170	C₂×Dic₃	—	3	2	no	—
171	C₂×Dic₃	—	3	2	no	—
172	C₂×Dic₃	—	6	2	no	—
173	C₂×Dic₃	D₃×C₈	1	2	yes	—
174	C₂×Dic₃	D₃×C₈	1	1	yes	characteristic
175	C₄×D₃	D₃×C₈	1	2	yes	—
176	C₄×D₃	C₄×Dic₃	1	2	yes	—
177	C₃×C₈	—	2	1	no	—
178	C₃×C₈	—	2	1	no	—
179	C₃×C₈	—	3	2	no	—
180	C₃×C₈	—	3	2	no	—
181	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
182	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
183	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
184	C₃×C₈	C₂²×Dic₃	1	1	yes	characteristic
185	C₃⋊C₈	—	3	2	no	—
186	C₃⋊C₈	—	3	2	no	—
187	C₃⋊C₈	—	6	2	no	—
188	C₃⋊C₈	D₃×C₈	1	2	yes	—
189	C₃⋊C₈	C₄×Dic₃	1	2	yes	—
190	C₂×C₃²	C₂×C₄×C₈	1	1	yes	characteristic
191	C₂×C₃²	#504	1	2	yes	—
192	C₃×D₃	—	2	2	no	—
193	C₂²×C₄	—	3	1	no	—
194	M₁₆	—	9	4	no	—
195	C₂×C₈	—	3	1	no	—
196	C₂×C₈	—	3	1	no	—
197	C₂×C₈	—	3	2	no	—
198	C₂×C₈	—	3	2	no	—
199	C₂×C₈	—	9	1	no	—
200	C₂×C₈	C₂×D₃²	1	1	yes	characteristic
201	C₄⋊C₄	—	9	2	no	—
202	C₂²⋊C₄	—	9	2	no	—
203	C₄²	—	3	1	no	—
204	C₄²	—	9	1	no	—
205	C₁₆	—	3	2	no	—
206	C₁₆	—	3	2	no	—
207	C₁₆	—	9	2	no	—
208	C₁₆	D₃×Dic₃	1	2	yes	—
209	C₂²×C₃	—	2	1	no	—
210	C₂²×C₃	—	3	2	no	—
211	C₂²×C₃	—	6	2	no	—
212	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
213	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
214	C₂×D₃	—	2	2	no	—
215	C₂×D₃	C₈×Dic₃	1	2	yes	—
216	C₃×C₄	—	2	1	no	—
217	C₃×C₄	—	2	1	no	—
218	C₃×C₄	—	3	2	no	—
219	C₃×C₄	—	6	2	no	—
220	C₃×C₄	C₄²×D₃	1	1	yes	characteristic
221	C₃×C₄	C₂×D₃×C₈	1	1	yes	characteristic
222	C₃×C₄	C₂×D₃×C₈	1	1	yes	characteristic
223	C₃×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
224	Dic₃	—	2	2	no	—
225	Dic₃	—	6	4	no	—
226	Dic₃	—	6	2	no	—
227	Dic₃	—	6	2	no	—
228	Dic₃	C₈×Dic₃	1	2	yes	—
229	C₃²	#2108	1	1	yes	characteristic, complemented, Sylow
230	C₂³	—	3	1	no	—
231	C₂×C₄	—	3	1	no	—
232	C₂×C₄	—	3	2	no	—
233	C₂×C₄	—	3	2	no	—
234	C₂×C₄	—	3	2	no	—
235	C₂×C₄	—	9	2	no	—
236	C₂×C₄	C₄×D₃²	1	1	yes	characteristic
237	C₈	—	3	2	no	—
238	C₈	—	3	2	no	—
239	C₈	—	9	2	no	—
240	C₈	C₄×D₃²	1	1	yes	characteristic
241	C₈	C₂×D₃×Dic₃	1	1	yes	characteristic
242	C₂×C₃	—	2	2	no	—
243	C₂×C₃	—	2	1	no	—
244	C₂×C₃	—	6	2	no	—
245	C₂×C₃	C₄×D₃×C₈	1	1	yes	characteristic
246	C₂×C₃	#4901	1	2	yes	—
247	C₂×C₃	#4901	1	2	yes	—
248	C₂×C₃	C₂×C₈×Dic₃	1	1	yes	characteristic
249	D₃	—	2	2	no	—
250	C₂²	—	3	2	no	—
251	C₂²	—	6	2	no	—
252	C₂²	D₃²×C₈	1	1	yes	characteristic
253	C₄	—	3	2	no	—
254	C₄	—	6	2	no	—
255	C₄	—	18	2	no	—
256	C₄	D₃²×C₈	1	1	yes	characteristic
257	C₄	C₄×D₃×Dic₃	1	1	yes	characteristic
258	C₃	—	2	1	no	—
259	C₃	#69051	1	1	yes	characteristic, complemented
260	C₃	#71399	1	1	yes	characteristic, complemented
261	C₂	—	6	2	no	—
262	C₂	#10590227	1	2	yes	—
263	C₂	D₃×C₈×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₁₆	D₃²
commutator	C₂×C₃²	C₂×C₄×C₈
Frattini	C₂×C₈	C₂×D₃²
Fitting	C₂×C₃²×C₁₆	C₂²
socle	C₂²×C₃²	C₂²×C₈
prime radical	C₂²×C₃×D₃	C₂×C₈
cosocle	C₂×C₃²×C₈	C₂³
hypocenter	C₃²	#2108
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
4. C₁₆×C₃⋊Dic₃	C₂	semidirect
5. C₃×Dic₃×C₁₆	C₂	semidirect
6. #10589104	C₂	semidirect
7. #10589103	C₂	semidirect
19. C₃×D₃×C₁₆	C₄	semidirect
22. #63543	C₄	semidirect
36. Dic₃×C₁₆	D₃	semidirect
83. D₃×C₁₆	Dic₃	semidirect
229. C₃²	#2108	semidirect
259. C₃	#69051	semidirect
260. C₃	#71399	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×D₃×C₁₆	2
#10590692	2
C₂×#63543	2
C₁₆×C₃⋊Dic₃	2
C₃×Dic₃×C₁₆	2
#10589104	2
#10589103	2
#71399	3
#69051	3