Subgroups — #11795897

Order 1152.

417 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₃×#5101	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#812	C₂	1	1	yes	maximal, characteristic
4	#10589103	C₂	1	2	yes	maximal, complemented
5	Dic₃×C₃⋊C₁₆	C₂	1	2	yes	maximal, complemented
6	#71419	—	3	1	no	maximal
7	#69881	—	3	1	no	maximal
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₃×#926	C₄	1	1	yes	characteristic
10	C₂×C₃×#208	C₄	1	1	yes	characteristic
11	C₂×C₃⋊(C₃⋊C₁₆)	C₄	1	4	yes	complemented
12	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
13	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
14	C₃×#815	C₄	1	2	yes	—
15	C₃×C₈×Dic₃	C₂²	1	2	yes	—
16	C₃×#812	C₄	1	4	yes	complemented
17	C₂×C₃×C₃⋊C₁₆	C₂²	1	2	yes	—
18	C₂×C₃×M₃₂	—	3	1	no	—
19	C₃×#405	—	3	1	no	—
20	#5101	D₃	1	1	yes	characteristic
21	C₂×#812	D₃	1	1	yes	characteristic
22	C₂²×C₃⋊C₁₆	—	3	1	no	—
23	C₂²×C₃⋊C₁₆	—	3	1	no	—
24	C₂²×C₃⋊C₁₆	—	6	1	no	—
25	#4502	—	3	2	no	—
26	Dic₃×C₁₆	—	3	2	no	—
27	#4461	—	3	2	no	—
28	C₄×C₃⋊C₁₆	—	3	2	no	—
29	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
32	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
33	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
34	C₃×#217	C₈	1	2	yes	—
35	C₃×C₄⋊Dic₃	C₈	1	2	yes	—
36	C₃×C₄×Dic₃	C₂×C₄	1	2	yes	—
37	C₃×#208	C₈	1	4	yes	—
38	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
39	C₃⋊(C₃⋊C₁₆)	C₂×C₄	1	2	yes	—
40	C₃⋊(C₃⋊C₁₆)	C₂×C₄	1	2	yes	—
41	C₃×C₃⋊C₁₆	C₂×C₄	1	4	yes	—
42	#2108	—	9	1	no	Sylow
43	C₂×C₃×M₁₆	—	3	1	no	—
44	C₂²×C₃×C₈	—	2	1	no	—
45	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
46	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
47	C₃×#117	—	3	1	no	—
48	#926	Dic₃	1	1	yes	characteristic
49	C₂×#208	Dic₃	1	1	yes	characteristic
50	C₃×M₃₂	—	3	4	no	—
51	C₂×C₃×C₁₆	—	3	2	no	—
52	C₃×#97	—	3	2	no	—
53	C₃×C₄×C₈	—	3	2	no	—
54	#815	Dic₃	1	2	yes	—
55	C₈×Dic₃	C₂×D₃	1	2	yes	—
56	#812	Dic₃	1	4	yes	complemented
57	C₂×C₃⋊C₁₆	—	3	4	no	—
58	C₂×C₃⋊C₁₆	—	3	4	no	—
59	C₂×C₃⋊C₁₆	—	3	1	no	—
60	C₂×C₃⋊C₁₆	—	3	1	no	—
61	C₂×C₃⋊C₁₆	—	3	1	no	—
62	C₂×C₃⋊C₁₆	—	3	1	no	—
63	C₂×C₃⋊C₁₆	—	6	4	no	—
64	C₂×C₃⋊C₁₆	—	6	1	no	—
65	C₂×C₃⋊C₁₆	—	6	1	no	—
66	C₂×C₃⋊C₁₆	C₂×D₃	1	2	yes	—
67	C₂³×C₃²	C₂×C₈	1	1	yes	characteristic
68	C₂×C₃²×C₄	C₄²	1	2	yes	—
69	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
70	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
71	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₃×Dic₃	C₂×C₈	1	4	yes	—
73	C₃²×C₈	C₄²	1	2	yes	—
74	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
75	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
76	C₃×C₃⋊C₈	C₂×C₈	1	4	yes	—
77	C₂×M₃₂	—	3	1	no	—
78	C₂²×C₁₆	—	9	1	no	—
79	#405	—	3	1	no	—
80	#346	—	9	2	no	—
81	C₄×C₁₆	—	9	2	no	—
82	C₂²×C₃×C₄	—	2	1	no	—
83	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
84	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
85	C₃×M₁₆	—	3	4	no	—
86	C₂×C₃×C₈	—	2	1	no	—
87	C₂×C₃×C₈	—	2	2	no	—
88	C₂×C₃×C₈	—	2	2	no	—
89	C₂×C₃×C₈	—	2	1	no	—
90	C₂×C₃×C₈	—	3	2	no	—
91	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
92	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
93	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
94	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
95	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
96	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
97	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
98	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
99	C₃×C₄⋊C₄	—	3	2	no	—
100	C₃×C₂²⋊C₄	—	3	2	no	—
101	C₃×C₄²	—	3	2	no	—
102	#217	C₃⋊C₈	1	2	yes	—
103	C₄⋊Dic₃	C₃⋊C₈	1	2	yes	—
104	C₄×Dic₃	C₂×Dic₃	1	2	yes	—
105	#208	C₃⋊C₈	1	4	yes	—
106	C₂×C₃⋊C₈	C₂×Dic₃	1	2	yes	—
107	C₃×C₁₆	—	3	4	no	—
108	C₃⋊C₁₆	—	3	2	no	—
109	C₃⋊C₁₆	—	3	2	no	—
110	C₃⋊C₁₆	—	3	2	no	—
111	C₃⋊C₁₆	—	3	2	no	—
112	C₃⋊C₁₆	—	6	2	no	—
113	C₃⋊C₁₆	—	6	2	no	—
114	C₃⋊C₁₆	C₂×Dic₃	1	4	yes	—
115	C₂²×C₃²	—	2	2	no	—
116	C₂²×C₃²	C₄×C₈	1	2	yes	—
117	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
118	C₃²×C₄	C₄×C₈	1	2	yes	—
119	C₃²×C₄	C₂×C₄²	1	1	yes	characteristic
120	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
121	C₃×Dic₃	—	2	4	no	—
122	C₂×M₁₆	—	3	1	no	—
123	C₂²×C₈	D₃²	1	1	yes	characteristic
124	#117	—	3	1	no	—
125	M₃₂	—	3	4	no	—
126	C₂×C₁₆	—	3	2	no	—
127	C₂×C₁₆	—	9	4	no	—
128	C₂×C₁₆	—	9	1	no	—
129	C₂×C₁₆	—	9	1	no	—
130	#97	—	3	2	no	—
131	C₄×C₈	—	3	2	no	—
132	C₂³×C₃	—	2	1	no	—
133	C₂³×C₃	D₃×C₈	1	1	yes	characteristic
134	C₂³×C₃	C₂×C₃⋊C₈	1	1	yes	characteristic
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	2	2	no	—
137	C₂×C₃×C₄	—	2	2	no	—
138	C₂×C₃×C₄	—	2	1	no	—
139	C₂×C₃×C₄	—	3	4	no	—
140	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
141	C₂×C₃×C₄	D₃×C₈	1	2	yes	—
142	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂×C₃⋊C₈	1	2	yes	—
144	C₂×C₃×C₄	C₄×Dic₃	1	2	yes	—
145	C₂×C₃×C₄	C₄×Dic₃	1	2	yes	—
146	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
147	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
148	C₂×Dic₃	C₂×C₃⋊C₈	1	4	yes	—
149	C₃×C₈	—	2	2	no	—
150	C₃×C₈	—	2	1	no	—
151	C₃×C₈	—	2	1	no	—
152	C₃×C₈	—	3	4	no	—
153	C₃×C₈	C₄×Dic₃	1	2	yes	—
154	C₃×C₈	C₄×Dic₃	1	2	yes	—
155	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
156	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
157	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
158	C₃×C₈	C₂²×Dic₃	1	1	yes	characteristic
159	C₃⋊C₈	C₂×C₃⋊C₈	1	4	yes	—
160	C₂×C₃²	—	2	2	no	—
161	C₂×C₃²	C₂×C₄×C₈	1	1	yes	characteristic
162	C₂×C₃²	#504	1	2	yes	—
163	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
164	M₁₆	—	3	4	no	—
165	C₂×C₈	—	3	2	no	—
166	C₂×C₈	D₃×Dic₃	1	1	yes	characteristic
167	C₂×C₈	D₃×Dic₃	1	2	yes	—
168	C₂×C₈	#624	1	2	yes	—
169	C₂×C₈	C₂×D₃²	1	1	yes	characteristic
170	C₄⋊C₄	—	3	2	no	—
171	C₂²⋊C₄	—	3	2	no	—
172	C₄²	—	3	2	no	—
173	C₁₆	—	3	4	no	—
174	C₁₆	—	9	2	no	—
175	C₁₆	—	9	2	no	—
176	C₂²×C₃	—	2	4	no	—
177	C₂²×C₃	—	2	2	no	—
178	C₂²×C₃	—	2	2	no	—
179	C₂²×C₃	—	2	2	no	—
180	C₂²×C₃	—	2	1	no	—
181	C₂²×C₃	C₄×C₃⋊C₈	1	2	yes	—
182	C₂²×C₃	C₈×Dic₃	1	2	yes	—
183	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
184	C₂²×C₃	C₂²×C₃⋊C₈	1	1	yes	characteristic
185	C₃×C₄	—	2	2	no	—
186	C₃×C₄	—	2	1	no	—
187	C₃×C₄	—	2	1	no	—
188	C₃×C₄	—	6	4	no	—
189	C₃×C₄	C₄×C₃⋊C₈	1	2	yes	—
190	C₃×C₄	C₈×Dic₃	1	2	yes	—
191	C₃×C₄	C₂×D₃×C₈	1	1	yes	characteristic
192	C₃×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
193	C₃×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
194	C₃×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
195	Dic₃	—	2	4	no	—
196	C₃²	#2108	1	1	yes	characteristic, complemented, Sylow
197	C₂³	D₃×C₃⋊C₈	1	1	yes	characteristic
198	C₂×C₄	—	3	4	no	—
199	C₂×C₄	D₃×C₃⋊C₈	1	1	yes	characteristic
200	C₂×C₄	D₃×C₃⋊C₈	1	2	yes	—
201	C₂×C₄	Dic₃²	1	2	yes	—
202	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
203	C₈	—	3	4	no	—
204	C₈	Dic₃²	1	2	yes	—
205	C₈	C₂×D₃×Dic₃	1	1	yes	characteristic
206	C₈	C₂×#624	1	1	yes	characteristic
207	C₂×C₃	—	2	4	no	—
208	C₂×C₃	—	2	2	no	—
209	C₂×C₃	—	2	2	no	—
210	C₂×C₃	—	2	1	no	—
211	C₂×C₃	—	2	2	no	—
212	C₂×C₃	#4907	1	2	yes	—
213	C₂×C₃	C₂×C₄×C₃⋊C₈	1	1	yes	characteristic
214	C₂×C₃	C₂×C₈×Dic₃	1	1	yes	characteristic
215	C₂×C₃	#5140	1	2	yes	—
216	C₂²	—	2	2	no	—
217	C₂²	Dic₃×C₃⋊C₈	1	2	yes	—
218	C₂²	C₂×D₃×C₃⋊C₈	1	1	yes	characteristic
219	C₄	—	6	4	no	—
220	C₄	Dic₃×C₃⋊C₈	1	2	yes	—
221	C₄	C₂×D₃×C₃⋊C₈	1	1	yes	characteristic
222	C₄	C₂×Dic₃²	1	1	yes	characteristic
223	C₃	—	2	1	no	—
224	C₃	#69881	1	1	yes	characteristic, complemented
225	C₃	#71419	1	1	yes	characteristic, complemented
226	C₂	—	2	2	no	—
227	C₂	#10590622	1	2	yes	—
228	C₂	C₂×Dic₃×C₃⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×D₃²
commutator	C₂×C₃²	C₂×C₄×C₈
Fitting	C₂²×C₃²×C₈	C₂²
socle	C₂²×C₃²	C₂²×C₈
prime radical	C₂³×C₃²	C₂×C₈
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	C₂²×C₈	D₃²
hypocenter	C₃²	#2108
LCS	G ▷ C₂×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. #10589103	C₂	semidirect
5. Dic₃×C₃⋊C₁₆	C₂	semidirect
11. C₂×C₃⋊(C₃⋊C₁₆)	C₄	semidirect
16. C₃×#812	C₄	semidirect
56. #812	Dic₃	semidirect
196. C₃²	#2108	semidirect
224. C₃	#69881	semidirect
225. C₃	#71419	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃⋊(C₃⋊C₁₆)	2
C₃×#5101	2
C₂×C₃×#812	2
#10589103	2
Dic₃×C₃⋊C₁₆	2
#71419	3
#69881	3