Subgroups — #11798003

Order 1152.

297 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₈×Dic₆	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	Dic₃⋊(C₃⋊C₁₆)	C₂	1	2	yes	maximal
5	Dic₃×C₃⋊C₁₆	C₂	1	2	yes	maximal
6	Q₈×C₃⋊C₁₆	—	3	1	no	maximal
7	#69280	—	3	1	no	maximal
8	C₃×C₄×Dic₆	C₄	1	1	yes	characteristic
9	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
11	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
12	C₃×Dic₃⋊C₈	C₄	1	2	yes	—
13	C₃×C₈×Dic₃	C₂²	1	2	yes	—
14	C₂×C₃×C₃⋊C₁₆	C₂²	1	2	yes	—
15	C₃×C₄⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
16	C₃×C₈×Q₈	—	3	1	no	—
17	C₈×Dic₆	D₃	1	1	yes	characteristic
18	C₃×C₄⋊C₁₆	—	3	1	no	—
19	Dic₃⋊C₁₆	—	3	2	no	—
20	Dic₃×C₁₆	—	3	2	no	—
21	C₄⋊(C₃⋊C₁₆)	—	3	2	no	—
22	C₄⋊(C₃⋊C₁₆)	D₃	1	1	yes	characteristic
23	C₄×C₃⋊C₁₆	—	3	1	no	—
24	C₄×C₃⋊C₁₆	—	3	1	no	—
25	C₄×C₃⋊C₁₆	—	3	2	no	—
26	C₄×C₃⋊C₁₆	—	6	1	no	—
27	C₂×C₃×Dic₆	C₈	1	1	yes	characteristic
28	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
30	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
31	C₃×C₄⋊Dic₃	C₈	1	1	yes	characteristic
32	C₃×Dic₃⋊C₄	C₈	1	2	yes	—
33	C₃×C₄×Dic₃	C₂×C₄	1	2	yes	—
34	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
35	C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
36	C₃⋊(C₃⋊C₁₆)	Q₈	1	2	yes	complemented
37	C₃×C₃⋊C₁₆	—	2	2	no	—
38	Q₈×C₁₆	—	9	1	no	Sylow
39	C₃×C₄×Q₈	—	3	1	no	—
40	C₄×Dic₆	Dic₃	1	1	yes	characteristic
41	C₂×C₃×C₁₆	—	3	2	no	—
42	C₃×C₄⋊C₈	—	3	2	no	—
43	C₃×C₄⋊C₈	—	3	1	no	—
44	C₃×C₄×C₈	—	2	1	no	—
45	C₃×C₄×C₈	—	3	2	no	—
46	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
47	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
48	Dic₃⋊C₈	Dic₃	1	2	yes	—
49	C₈×Dic₃	C₂×D₃	1	2	yes	—
50	C₂×C₃⋊C₁₆	—	3	1	no	—
51	C₂×C₃⋊C₁₆	—	3	1	no	—
52	C₂×C₃⋊C₁₆	—	3	1	no	—
53	C₂×C₃⋊C₁₆	—	3	1	no	—
54	C₂×C₃⋊C₁₆	—	6	1	no	—
55	C₂×C₃⋊C₁₆	—	6	1	no	—
56	C₂×C₃⋊C₁₆	C₂×D₃	1	2	yes	—
57	C₄⋊(C₃⋊C₈)	Dic₃	1	1	yes	characteristic
58	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
59	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
60	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₃×Dic₃	C₂×C₈	1	2	yes	—
62	C₂×C₃×Dic₃	C₂×C₈	1	2	yes	—
63	C₃×Dic₆	C₁₆	1	4	yes	complemented
64	C₃²×C₈	—	2	1	no	—
65	C₃²×C₈	C₂×Q₈	1	1	yes	characteristic
66	C₃²×C₈	#41	1	1	yes	characteristic
67	C₃×C₃⋊C₈	—	2	2	no	—
68	C₈×Q₈	—	3	1	no	—
69	C₄⋊C₁₆	—	3	1	no	—
70	C₄⋊C₁₆	—	9	2	no	—
71	C₄×C₁₆	—	9	2	no	—
72	C₄×C₁₆	—	9	1	no	—
73	C₂×C₃×Q₈	—	3	1	no	—
74	C₂×Dic₆	C₃⋊C₈	1	1	yes	characteristic
75	C₂×C₃×C₈	—	2	1	no	—
76	C₂×C₃×C₈	—	2	1	no	—
77	C₂×C₃×C₈	—	3	2	no	—
78	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
79	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
80	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
81	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
82	C₃×C₄⋊C₄	—	3	2	no	—
83	C₃×C₄⋊C₄	—	3	1	no	—
84	C₃×C₄²	—	2	1	no	—
85	C₃×C₄²	—	3	2	no	—
86	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
87	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
88	C₄⋊Dic₃	C₃⋊C₈	1	1	yes	characteristic
89	Dic₃⋊C₄	C₃⋊C₈	1	2	yes	—
90	C₄×Dic₃	C₂×Dic₃	1	2	yes	—
91	C₂×C₃⋊C₈	C₂×Dic₃	1	2	yes	—
92	C₃×C₁₆	—	6	2	no	—
93	C₃⋊C₁₆	—	2	2	no	—
94	C₃⋊C₁₆	—	3	2	no	—
95	C₃⋊C₁₆	—	3	2	no	—
96	C₃⋊C₁₆	—	6	2	no	—
97	C₃⋊C₁₆	—	6	2	no	—
98	C₃⋊C₁₆	—	6	1	no	—
99	C₃⋊C₁₆	—	6	1	no	—
100	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
101	C₃²×C₄	—	2	1	no	—
102	C₃²×C₄	C₂×C₁₆	1	2	yes	—
103	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
104	C₃²×C₄	#131	1	1	yes	characteristic
105	C₃×Dic₃	—	2	2	no	—
106	C₃×Dic₃	C₂×C₁₆	1	4	yes	—
107	C₄×Q₈	—	3	1	no	—
108	C₂×C₁₆	—	3	2	no	—
109	C₂×C₁₆	—	9	1	no	—
110	C₂×C₁₆	—	9	1	no	—
111	C₄⋊C₈	—	3	2	no	—
112	C₄⋊C₈	—	3	1	no	—
113	C₄×C₈	—	3	2	no	—
114	C₄×C₈	D₃²	1	1	yes	characteristic
115	C₃×Q₈	—	3	4	no	—
116	C₂×C₃×C₄	—	2	1	no	—
117	C₂×C₃×C₄	—	2	1	no	—
118	C₂×C₃×C₄	—	2	1	no	—
119	C₂×C₃×C₄	—	3	2	no	—
120	C₂×C₃×C₄	—	3	2	no	—
121	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
122	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
123	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
124	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
125	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
126	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
127	C₂×Dic₃	C₂×C₃⋊C₈	1	2	yes	—
128	C₂×Dic₃	C₂×C₃⋊C₈	1	2	yes	—
129	Dic₆	C₃⋊C₁₆	1	4	yes	complemented
130	C₃×C₈	—	2	2	no	—
131	C₃×C₈	—	2	1	no	—
132	C₃×C₈	—	2	1	no	—
133	C₃×C₈	—	2	1	no	—
134	C₃×C₈	—	2	1	no	—
135	C₃×C₈	—	6	2	no	—
136	C₃×C₈	#237	1	1	yes	characteristic
137	C₃×C₈	D₃×Q₈	1	1	yes	characteristic
138	C₃×C₈	D₃×Q₈	1	1	yes	characteristic
139	C₃×C₈	#239	1	1	yes	characteristic
140	C₃⋊C₈	—	2	2	no	—
141	C₂×C₃²	C₈×Q₈	1	1	yes	characteristic
142	C₂×C₃²	C₂²×C₁₆	1	1	yes	characteristic
143	C₂×C₃²	#504	1	1	yes	characteristic
144	C₂×Q₈	—	3	1	no	—
145	C₂×C₈	—	3	2	no	—
146	C₂×C₈	D₃×Dic₃	1	1	yes	characteristic
147	C₂×C₈	C₂×D₃²	1	1	yes	characteristic
148	C₄⋊C₄	—	3	2	no	—
149	C₄⋊C₄	—	3	1	no	—
150	C₄²	—	3	2	no	—
151	C₄²	D₃×Dic₃	1	1	yes	characteristic
152	C₁₆	—	6	2	no	—
153	C₁₆	—	9	2	no	—
154	C₁₆	—	18	1	no	—
155	C₂²×C₃	—	2	1	no	—
156	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
157	C₂²×C₃	C₂²×C₃⋊C₈	1	1	yes	characteristic
158	C₃×C₄	—	2	2	no	—
159	C₃×C₄	—	2	1	no	—
160	C₃×C₄	—	2	1	no	—
161	C₃×C₄	—	2	2	no	—
162	C₃×C₄	—	2	1	no	—
163	C₃×C₄	—	2	1	no	—
164	C₃×C₄	—	3	4	no	—
165	C₃×C₄	—	6	2	no	—
166	C₃×C₄	D₃×C₁₆	1	2	yes	—
167	C₃×C₄	C₂×C₃⋊C₁₆	1	2	yes	—
168	C₃×C₄	#887	1	1	yes	characteristic
169	C₃×C₄	#907	1	1	yes	characteristic
170	C₃×C₄	Q₈×Dic₃	1	1	yes	characteristic
171	C₃×C₄	#948	1	1	yes	characteristic
172	Dic₃	—	2	2	no	—
173	Dic₃	C₂×C₃⋊C₁₆	1	4	yes	—
174	C₃²	Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
175	Q₈	—	3	4	no	—
176	C₂×C₄	—	3	2	no	—
177	C₂×C₄	—	3	2	no	—
178	C₂×C₄	D₃×C₃⋊C₈	1	1	yes	characteristic
179	C₂×C₄	D₃×C₃⋊C₈	1	1	yes	characteristic
180	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
181	C₈	—	2	1	no	—
182	C₈	—	6	2	no	—
183	C₈	#3796	1	1	yes	characteristic
184	C₈	#3797	1	1	yes	characteristic
185	C₂×C₃	—	2	1	no	—
186	C₂×C₃	—	2	1	no	—
187	C₂×C₃	—	2	1	no	—
188	C₂×C₃	#4830	1	1	yes	characteristic
189	C₂×C₃	C₂×D₃×C₁₆	1	1	yes	characteristic
190	C₂×C₃	#4907	1	1	yes	characteristic
191	C₂×C₃	Q₈×C₃⋊C₈	1	1	yes	characteristic
192	C₂×C₃	C₂²×C₃⋊C₁₆	1	1	yes	characteristic
193	C₂×C₃	#5140	1	1	yes	characteristic
194	C₂²	C₂×D₃×C₃⋊C₈	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	3	4	no	—
197	C₄	—	6	2	no	—
198	C₄	D₃×C₃⋊C₁₆	1	2	yes	—
199	C₄	#63815	1	1	yes	characteristic
200	C₄	#63843	1	1	yes	characteristic
201	C₃	—	2	1	no	—
202	C₃	#69280	1	1	yes	characteristic, complemented
203	C₃	Q₈×C₃⋊C₁₆	1	1	yes	characteristic, complemented
204	C₂	#10590250	1	1	yes	characteristic
205	C₂	C₂×D₃×C₃⋊C₁₆	1	1	yes	characteristic
206	C₂	#10590622	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×D₃²
commutator	C₂×C₃²	C₂²×C₁₆
Fitting	C₃²×C₄×C₈	C₂²
socle, prime radical	C₂²×C₃²	C₂²×C₈
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	C₄×C₈	D₃²
hypocenter	C₃²	Q₈×C₁₆
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
36. C₃⋊(C₃⋊C₁₆)	Q₈	semidirect
63. C₃×Dic₆	C₁₆	semidirect
129. Dic₆	C₃⋊C₁₆	semidirect
174. C₃²	Q₈×C₁₆	semidirect
202. C₃	#69280	semidirect
203. C₃	Q₈×C₃⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₈×Dic₆	2
C₄×C₃⋊(C₃⋊C₁₆)	2
C₃×C₄⋊(C₃⋊C₁₆)	2
Dic₃⋊(C₃⋊C₁₆)	2
Dic₃×C₃⋊C₁₆	2
Q₈×C₃⋊C₁₆	3
#69280	3