Subgroups — #11771155

Order 1152.

444 conjugacy classes of subgroups.

The subgroup lattice has 189 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#423	C₂	1	1	yes	maximal, characteristic
2	#10592624	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic
4	#67714	—	3	4	no	maximal
5	C₂×C₃²×M₁₆	—	2	1	no	—
6	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
7	C₂²×C₃⋊(C₃⋊C₈)	—	2	1	no	—
8	C₃²×C₄⋊C₈	—	2	1	no	—
9	C₃²×C₄⋊C₈	C₄	1	2	yes	—
10	#63657	C₄	1	2	yes	—
11	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
12	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
13	C₄⋊(C₃⋊(C₃⋊C₈))	C₄	1	2	yes	—
14	C₄×C₃⋊(C₃⋊C₈)	—	2	1	no	—
15	C₃×#423	D₃	1	4	yes	—
16	#4925	—	3	4	no	—
17	C₂×C₄⋊(C₃⋊C₈)	—	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	Q₈	1	1	yes	characteristic
21	C₃²×M₁₆	—	4	1	no	—
22	C₃²×M₁₆	—	4	1	no	—
23	C₂×C₃²×C₈	—	2	2	no	—
24	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
25	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
26	C₃²×C₄²	D₄	1	1	yes	characteristic
27	C₃²×C₄²	D₄	1	1	yes	characteristic
28	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
29	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
30	C₂×C₃⋊(C₃⋊C₈)	—	4	1	no	—
31	C₂×C₃⋊(C₃⋊C₈)	—	4	1	no	—
32	#1301	—	9	1	no	Sylow
33	C₂×C₃×M₁₆	—	2	4	no	—
34	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
35	C₂²×C₃⋊C₈	—	6	4	no	—
36	C₃×C₄⋊C₈	—	2	4	no	—
37	C₃×C₄⋊C₈	Dic₃	1	8	yes	—
38	#830	—	3	8	no	—
39	C₄⋊(C₃⋊C₈)	—	3	8	no	—
40	C₄⋊(C₃⋊C₈)	—	6	4	no	—
41	C₄⋊(C₃⋊C₈)	—	6	4	no	—
42	C₄×C₃⋊C₈	—	6	4	no	—
43	C₂³×C₃²	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₃²×C₄	—	2	2	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	C₂²⋊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	C₃²×C₈	—	4	2	no	—
57	C₃⋊(C₃⋊C₈)	—	4	2	no	—
58	C₃⋊(C₃⋊C₈)	—	4	2	no	—
59	#431	—	9	1	no	—
60	#423	C₃⋊D₃	1	1	yes	characteristic
61	C₂×C₄⋊C₈	—	9	1	no	—
62	C₂²×C₃×C₄	Dic₆	1	4	yes	—
63	C₂²×C₃×C₄	C₄×D₃	1	4	yes	—
64	C₂²×C₃×C₄	#67	1	4	yes	—
65	C₃×M₁₆	—	4	4	no	—
66	C₃×M₁₆	—	4	4	no	—
67	C₂×C₃×C₈	—	2	8	no	—
68	C₃×C₄²	C₄×D₃	1	4	yes	—
69	C₃×C₄²	D₁₂	1	4	yes	—
70	C₃×C₄²	C₂×Dic₃	1	4	yes	—
71	C₃×C₄²	#67	1	4	yes	—
72	C₂×C₃⋊C₈	—	6	8	no	—
73	C₂×C₃⋊C₈	—	6	8	no	—
74	C₂×C₃⋊C₈	—	12	4	no	—
75	C₂×C₃⋊C₈	—	12	4	no	—
76	C₂²×C₃²	—	2	1	no	—
77	C₂²×C₃²	—	2	1	no	—
78	C₂²×C₃²	#95	1	1	yes	characteristic
79	C₂²×C₃²	#106	1	1	yes	characteristic
80	C₂²×C₃²	#107	1	1	yes	characteristic
81	C₃²×C₄	—	2	1	no	—
82	C₃²×C₄	—	2	1	no	—
83	C₃²×C₄	—	2	1	no	—
84	C₃²×C₄	#102	1	1	yes	characteristic
85	C₃²×C₄	#102	1	1	yes	characteristic
86	C₃²×C₄	#103	1	1	yes	characteristic
87	C₃²×C₄	#103	1	1	yes	characteristic
88	C₃²×C₄	#108	1	2	yes	—
89	C₂×M₁₆	—	2	1	no	—
90	C₂²×C₈	—	18	1	no	—
91	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
92	C₄⋊C₈	—	2	1	no	—
93	C₄⋊C₈	—	9	2	no	—
94	C₄⋊C₈	—	18	1	no	—
95	C₄⋊C₈	—	18	1	no	—
96	C₄⋊C₈	C₃⋊Dic₃	1	2	yes	—
97	C₂²⋊C₈	—	9	2	no	—
98	C₄×C₈	—	18	1	no	—
99	C₂³×C₃	Dic₃⋊C₄	1	4	yes	—
100	C₂×C₃×C₄	—	2	8	no	—
101	C₂×C₃×C₄	—	4	4	no	—
102	C₂×C₃×C₄	C₄×Dic₃	1	4	yes	—
103	C₂×C₃×C₄	Dic₃⋊C₄	1	4	yes	—
104	C₂×C₃×C₄	C₄⋊Dic₃	1	4	yes	—
105	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
106	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
107	C₂×C₃×C₄	#213	1	4	yes	—
108	C₂×C₃×C₄	#214	1	4	yes	—
109	C₂×C₃×C₄	#215	1	4	yes	—
110	C₂×C₃×C₄	#216	1	4	yes	—
111	C₂×C₃×C₄	#217	1	4	yes	—
112	C₃×C₈	—	4	8	no	—
113	C₃⋊C₈	—	12	8	no	—
114	C₃⋊C₈	—	12	8	no	—
115	C₂×C₃²	—	2	1	no	—
116	C₂×C₃²	—	2	1	no	—
117	C₂×C₃²	#340	1	1	yes	characteristic
118	C₂×C₃²	#341	1	1	yes	characteristic
119	C₂×C₃²	#344	1	1	yes	characteristic
120	C₂²×C₄	#634	1	1	yes	characteristic
121	C₂²×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
122	C₂²×C₄	#638	1	1	yes	characteristic
123	M₁₆	—	4	1	no	—
124	M₁₆	—	4	1	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	18	2	no	—
127	C₂×C₈	—	18	2	no	—
128	C₂×C₈	—	36	1	no	—
129	C₂×C₈	—	36	1	no	—
130	C₄²	C₄×C₃⋊D₃	1	1	yes	characteristic
131	C₄²	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
132	C₄²	C₂×C₃⋊Dic₃	1	1	yes	characteristic
133	C₄²	#638	1	1	yes	characteristic
134	C₂²×C₃	—	2	4	no	—
135	C₂²×C₃	—	2	4	no	—
136	C₂²×C₃	#807	1	4	yes	—
137	C₂²×C₃	#808	1	4	yes	—
138	C₂²×C₃	C₃⋊#95	1	4	yes	—
139	C₃×C₄	—	2	4	no	—
140	C₃×C₄	—	2	4	no	—
141	C₃×C₄	—	2	4	no	—
142	C₃×C₄	#809	1	4	yes	—
143	C₃×C₄	#810	1	4	yes	—
144	C₃×C₄	#822	1	8	yes	—
145	C₃×C₄	#832	1	4	yes	—
146	C₃×C₄	#835	1	4	yes	—
147	C₃²	#1301	1	1	yes	characteristic, complemented, Sylow
148	C₂³	#3750	1	1	yes	characteristic
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	C₄×C₃⋊Dic₃	1	1	yes	characteristic
152	C₂×C₄	#3750	1	1	yes	characteristic
153	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
154	C₂×C₄	#3752	1	1	yes	characteristic
155	C₂×C₄	#3752	1	1	yes	characteristic
156	C₂×C₄	#3753	1	1	yes	characteristic
157	C₂×C₄	#3754	1	1	yes	characteristic
158	C₂×C₄	#3755	1	1	yes	characteristic
159	C₂×C₄	#3756	1	1	yes	characteristic
160	C₂×C₄	#3757	1	1	yes	characteristic
161	C₈	—	4	2	no	—
162	C₈	—	36	2	no	—
163	C₈	—	36	2	no	—
164	C₂×C₃	—	2	4	no	—
165	C₂×C₃	—	2	4	no	—
166	C₂×C₃	#4529	1	4	yes	—
167	C₂×C₃	#4556	1	4	yes	—
168	C₂×C₃	#4558	1	4	yes	—
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	#63634	1	1	yes	characteristic
172	C₂²	#63635	1	1	yes	characteristic
173	C₂²	#63658	1	1	yes	characteristic
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	—	2	1	no	—
177	C₄	#63636	1	1	yes	characteristic
178	C₄	#63637	1	1	yes	characteristic
179	C₄	#63649	1	2	yes	—
180	C₄	#63659	1	1	yes	characteristic
181	C₄	#63662	1	1	yes	characteristic
182	C₃	#67714	1	4	yes	complemented
183	C₂	—	2	1	no	—
184	C₂	—	2	1	no	—
185	C₂	#10589395	1	1	yes	characteristic
186	C₂	#10589422	1	1	yes	characteristic
187	C₂	#10589424	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#3757
commutator	C₂×C₃²×C₄	C₄²
Frattini	C₂×C₄²	C₂×C₃⋊D₃
Fitting	C₃²×#423	C₂
socle	C₂²×C₃²	#95
prime radical	C₂³×C₃²	C₄⋊C₄
cosocle	C₂×C₃²×C₄²	C₂²
hypercenter	#423	C₃⋊D₃
hypocenter	C₃²	#1301
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

Complements

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

N	H ≅ G/N	type
147. C₃²	#1301	semidirect
182. C₃	#67714	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#423	2
#10592624	2
C₂×C₄⋊(C₃⋊(C₃⋊C₈))	2
#67714	3