Subgroups — #11770913

Order 1152.

468 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#431	C₂	1	1	yes	maximal, characteristic
2	#10592635	C₂	1	1	yes	maximal, characteristic
3	#10592623	C₂	1	1	yes	maximal, characteristic
4	#67738	—	3	4	no	maximal
5	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
6	C₂×C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
7	C₂×#3748	—	2	1	no	—
8	C₃²×C₄⋊C₈	—	2	1	no	—
9	C₃²×C₂²⋊C₈	—	2	1	no	—
10	C₃²×C₄×C₈	C₄	1	2	yes	complemented
11	#63658	—	2	1	no	—
12	#63658	C₄	1	2	yes	—
13	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
14	C₄⋊(C₃⋊(C₃⋊C₈))	C₄	1	2	yes	complemented
15	C₃×#431	D₃	1	4	yes	—
16	#4936	—	3	4	no	—
17	#4924	—	3	4	no	—
18	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
20	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
21	C₂²×C₃⋊Dic₃	—	2	2	no	—
22	C₂×C₃²×C₈	—	2	2	no	—
23	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
24	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
25	C₃²×C₄²	D₄	1	1	yes	characteristic
26	C₃²×C₄²	Q₈	1	1	yes	characteristic
27	C₄⋊(C₃⋊Dic₃)	—	4	1	no	—
28	C₄⋊(C₃⋊Dic₃)	—	4	1	no	—
29	#3748	—	4	1	no	—
30	#3748	—	4	1	no	—
31	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
32	#1286	—	9	1	no	Sylow
33	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
34	C₂×C₄⋊Dic₃	—	6	4	no	—
35	C₂×#208	—	6	4	no	—
36	C₃×C₄⋊C₈	—	2	4	no	—
37	C₃×C₂²⋊C₈	—	2	4	no	—
38	C₃×C₄×C₈	Dic₃	1	8	yes	complemented
39	C₃⋊#95	—	3	8	no	—
40	C₃⋊#95	—	6	4	no	—
41	C₄⋊(C₃⋊C₈)	—	3	8	no	—
42	C₄⋊(C₃⋊C₈)	—	6	4	no	—
43	C₂³×C₃²	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₃²×C₄	—	4	1	no	—
45	C₂×C₃²×C₄	—	4	1	no	—
46	C₂×C₃²×C₄	C₄²	1	1	yes	characteristic
47	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
53	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
54	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
55	C₂×C₃²×C₄	Q₁₆	1	1	yes	characteristic
56	C₂×C₃⋊Dic₃	—	4	2	no	—
57	C₂×C₃⋊Dic₃	—	4	2	no	—
58	C₂×C₃⋊Dic₃	—	4	2	no	—
59	C₃²×C₈	—	4	2	no	—
60	C₃⋊(C₃⋊C₈)	—	4	2	no	—
61	#431	C₃⋊D₃	1	1	yes	characteristic
62	#423	—	9	1	no	—
63	#382	—	9	1	no	—
64	C₂²×C₃×C₄	C₄×D₃	1	4	yes	—
65	C₂²×C₃×C₄	D₁₂	1	4	yes	—
66	C₂²×C₃×C₄	#67	1	4	yes	—
67	C₂²×Dic₃	—	6	8	no	—
68	C₂×C₃×C₈	—	2	8	no	—
69	C₃×C₄²	Dic₆	1	4	yes	—
70	C₃×C₄²	C₄×D₃	1	4	yes	—
71	C₃×C₄²	C₂×Dic₃	1	4	yes	—
72	C₃×C₄²	#67	1	4	yes	—
73	C₄⋊Dic₃	—	12	4	no	—
74	C₄⋊Dic₃	—	12	4	no	—
75	#208	—	12	4	no	—
76	#208	—	12	4	no	—
77	C₂×C₃⋊C₈	—	6	8	no	—
78	C₂²×C₃²	—	2	1	no	—
79	C₂²×C₃²	—	2	1	no	—
80	C₂²×C₃²	#95	1	1	yes	characteristic
81	C₂²×C₃²	#102	1	1	yes	characteristic
82	C₂²×C₃²	#103	1	1	yes	characteristic
83	C₃²×C₄	—	2	1	no	—
84	C₃²×C₄	—	2	1	no	—
85	C₃²×C₄	—	2	1	no	—
86	C₃²×C₄	—	2	1	no	—
87	C₃²×C₄	#102	1	1	yes	characteristic
88	C₃²×C₄	#103	1	1	yes	characteristic
89	C₃²×C₄	#106	1	1	yes	characteristic
90	C₃²×C₄	#107	1	1	yes	characteristic
91	C₃⋊Dic₃	—	8	2	no	—
92	C₂×M₁₆	—	18	1	no	—
93	C₂×C₄⋊C₄	—	18	1	no	—
94	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
95	C₄⋊C₈	—	2	1	no	—
96	C₄⋊C₈	—	9	2	no	—
97	C₄⋊C₈	—	18	1	no	—
98	C₂²⋊C₈	—	2	1	no	—
99	C₄×C₈	C₃⋊Dic₃	1	2	yes	complemented
100	#95	—	9	2	no	—
101	#95	—	18	1	no	—
102	C₂³×C₃	(C₂×D₃)⋊C₄	1	4	yes	—
103	C₂×C₃×C₄	—	4	4	no	—
104	C₂×C₃×C₄	—	4	4	no	—
105	C₂×C₃×C₄	#204	1	4	yes	—
106	C₂×C₃×C₄	#204	1	4	yes	—
107	C₂×C₃×C₄	D₂₄	1	4	yes	—
108	C₂×C₃×C₄	Dic₁₂	1	4	yes	—
109	C₂×C₃×C₄	C₄×Dic₃	1	4	yes	—
110	C₂×C₃×C₄	Dic₃⋊C₄	1	4	yes	—
111	C₂×C₃×C₄	Dic₃⋊C₄	1	4	yes	—
112	C₂×C₃×C₄	C₄⋊Dic₃	1	4	yes	—
113	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
114	C₂×C₃×C₄	#217	1	4	yes	—
115	C₂×Dic₃	—	12	8	no	—
116	C₂×Dic₃	—	12	8	no	—
117	C₂×Dic₃	—	12	8	no	—
118	C₃×C₈	—	4	8	no	—
119	C₃⋊C₈	—	12	8	no	—
120	C₂×C₃²	—	2	1	no	—
121	C₂×C₃²	—	2	1	no	—
122	C₂×C₃²	H(3,ℤ/4)	1	1	yes	characteristic
123	C₂×C₃²	#340	1	1	yes	characteristic
124	C₂×C₃²	#344	1	1	yes	characteristic
125	C₂²×C₄	—	18	2	no	—
126	C₂²×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
127	C₂²×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
128	C₂²×C₄	#638	1	1	yes	characteristic
129	M₁₆	—	36	1	no	—
130	M₁₆	—	36	1	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	18	2	no	—
133	C₄⋊C₄	—	36	1	no	—
134	C₄⋊C₄	—	36	1	no	—
135	C₄²	#634	1	1	yes	characteristic
136	C₄²	C₄×C₃⋊D₃	1	1	yes	characteristic
137	C₄²	C₂×C₃⋊Dic₃	1	1	yes	characteristic
138	C₄²	#638	1	1	yes	characteristic
139	C₂²×C₃	—	2	4	no	—
140	C₂²×C₃	—	2	4	no	—
141	C₂²×C₃	#816	1	4	yes	—
142	C₂²×C₃	#821	1	4	yes	—
143	C₂²×C₃	C₃⋊#95	1	4	yes	—
144	C₃×C₄	—	2	4	no	—
145	C₃×C₄	—	2	4	no	—
146	C₃×C₄	—	2	4	no	—
147	C₃×C₄	—	2	4	no	—
148	C₃×C₄	#816	1	4	yes	—
149	C₃×C₄	#817	1	4	yes	—
150	C₃×C₄	#818	1	4	yes	—
151	C₃×C₄	#821	1	4	yes	—
152	Dic₃	—	24	8	no	—
153	C₃²	#1286	1	1	yes	characteristic, complemented, Sylow
154	C₂³	#3752	1	1	yes	characteristic
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	36	2	no	—
158	C₂×C₄	—	36	2	no	—
159	C₂×C₄	—	36	2	no	—
160	C₂×C₄	#3744	1	1	yes	characteristic
161	C₂×C₄	#3744	1	1	yes	characteristic
162	C₂×C₄	#3745	1	1	yes	characteristic
163	C₂×C₄	#3746	1	1	yes	characteristic
164	C₂×C₄	C₄×C₃⋊Dic₃	1	1	yes	characteristic
165	C₂×C₄	#3750	1	1	yes	characteristic
166	C₂×C₄	#3750	1	1	yes	characteristic
167	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
168	C₂×C₄	#3752	1	1	yes	characteristic
169	C₂×C₄	#3757	1	1	yes	characteristic
170	C₈	—	4	2	no	—
171	C₈	—	36	2	no	—
172	C₂×C₃	—	2	4	no	—
173	C₂×C₃	—	2	4	no	—
174	C₂×C₃	#4531	1	4	yes	—
175	C₂×C₃	#4551	1	4	yes	—
176	C₂×C₃	#4556	1	4	yes	—
177	C₂²	—	2	1	no	—
178	C₂²	—	2	1	no	—
179	C₂²	#63643	1	1	yes	characteristic
180	C₂²	#63648	1	1	yes	characteristic
181	C₂²	#63658	1	1	yes	characteristic
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	72	2	no	—
187	C₄	#63643	1	1	yes	characteristic
188	C₄	#63644	1	1	yes	characteristic
189	C₄	#63645	1	1	yes	characteristic
190	C₄	#63648	1	1	yes	characteristic
191	C₃	#67738	1	4	yes	complemented
192	C₂	—	2	1	no	—
193	C₂	—	2	1	no	—
194	C₂	#10589397	1	1	yes	characteristic
195	C₂	#10589417	1	1	yes	characteristic
196	C₂	#10589422	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#63658
commutator	C₂×C₃²×C₄	C₄²
Frattini	C₂×C₄²	C₂×C₃⋊D₃
Fitting	C₃²×#431	C₂
socle	C₂²×C₃²	#95
prime radical	C₂³×C₃²	C₂²⋊C₄
cosocle	C₂×C₃²×C₄²	C₂²
hypercenter	#431	C₃⋊D₃
hypocenter	C₃²	#1286
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

Complements

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

N	H ≅ G/N	type
10. C₃²×C₄×C₈	C₄	semidirect
14. C₄⋊(C₃⋊(C₃⋊C₈))	C₄	semidirect
38. C₃×C₄×C₈	Dic₃	semidirect
99. C₄×C₈	C₃⋊Dic₃	semidirect
153. C₃²	#1286	semidirect
191. C₃	#67738	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#431	2
#10592635	2
#10592623	2
#67738	3