Subgroups — #11772491

Order 1152.

408 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#108	C₂	1	1	yes	maximal, characteristic
2	C₂×#63645	C₂	1	1	yes	maximal, characteristic
3	C₂×#63639	C₂	1	1	yes	maximal, characteristic
4	#67732	—	3	4	no	maximal
5	C₂×C₃²×M₁₆	—	2	1	no	—
6	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
8	C₃²×#108	—	2	1	no	—
9	C₃²×#108	C₄	1	2	yes	complemented
10	#63645	—	2	1	no	—
11	#63645	C₄	1	2	yes	—
12	#63639	—	2	1	no	—
13	#63639	C₄	1	2	yes	complemented
14	C₂×C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
15	C₂×C₃×#108	D₃	1	4	yes	—
16	C₂×#818	—	3	4	no	—
17	C₂×#812	—	3	4	no	—
18	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
19	C₂²×C₃⋊Dic₃	—	4	1	no	—
20	C₃²×M₁₆	—	2	2	no	—
21	C₃²×M₁₆	—	4	1	no	—
22	C₂×C₃²×C₈	—	4	1	no	—
23	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
27	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
28	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
29	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
30	C₄⋊(C₃⋊Dic₃)	—	4	1	no	—
31	C₃⋊(C₃⋊C₁₆)	—	2	2	no	—
32	#1387	—	9	1	no	Sylow
33	C₂×C₃×M₁₆	—	2	4	no	—
34	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
35	C₂×C₄⋊Dic₃	—	6	4	no	—
36	C₃×#108	—	2	4	no	—
37	C₃×#108	Dic₃	1	8	yes	complemented
38	#818	—	3	8	no	—
39	#818	—	6	4	no	—
40	#812	—	3	8	no	—
41	#812	—	6	4	no	—
42	C₂×C₃⋊C₁₆	—	6	4	no	—
43	C₂³×C₃²	SD₁₆	1	1	yes	characteristic
44	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
48	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
49	C₂×C₃²×C₄	Q₁₆	1	1	yes	characteristic
50	C₂×C₃⋊Dic₃	—	4	2	no	—
51	C₂×C₃⋊Dic₃	—	8	2	no	—
52	C₃²×C₈	—	4	2	no	—
53	C₃²×C₈	C₄²	1	1	yes	characteristic
54	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
55	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
56	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
57	C₂×M₃₂	—	9	1	no	—
58	C₂×#108	C₃⋊D₃	1	1	yes	characteristic
59	C₂×#107	—	9	1	no	—
60	C₂²×C₃×C₄	#67	1	4	yes	—
61	C₂²×Dic₃	—	12	4	no	—
62	C₃×M₁₆	—	2	8	no	—
63	C₃×M₁₆	—	4	4	no	—
64	C₂×C₃×C₈	—	4	4	no	—
65	C₂×C₃×C₈	Dic₆	1	4	yes	—
66	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
67	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
68	C₂×C₃×C₈	D₁₂	1	4	yes	—
69	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
70	C₂×C₃×C₈	#67	1	4	yes	—
71	C₄⋊Dic₃	—	6	8	no	—
72	C₄⋊Dic₃	—	12	4	no	—
73	C₃⋊C₁₆	—	6	8	no	—
74	C₂²×C₃²	—	2	1	no	—
75	C₂²×C₃²	—	2	1	no	—
76	C₂²×C₃²	#102	1	1	yes	characteristic
77	C₂²×C₃²	#103	1	1	yes	characteristic
78	C₂²×C₃²	#106	1	1	yes	characteristic
79	C₃²×C₄	#95	1	1	yes	characteristic
80	C₃²×C₄	#102	1	1	yes	characteristic
81	C₃²×C₄	#103	1	1	yes	characteristic
82	C₃²×C₄	#107	1	1	yes	characteristic
83	C₃⋊Dic₃	—	8	2	no	—
84	C₂×M₁₆	—	2	1	no	—
85	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
86	C₂×C₄⋊C₄	—	18	1	no	—
87	M₃₂	—	9	2	no	—
88	M₃₂	—	18	1	no	—
89	C₂×C₁₆	—	18	1	no	—
90	#108	—	2	1	no	—
91	#108	C₃⋊Dic₃	1	2	yes	complemented
92	#107	—	9	2	no	—
93	#107	—	18	1	no	—
94	C₂³×C₃	#215	1	4	yes	—
95	C₂×C₃×C₄	Dic₃⋊C₄	1	4	yes	—
96	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
97	C₂×C₃×C₄	#213	1	4	yes	—
98	C₂×C₃×C₄	#214	1	4	yes	—
99	C₂×C₃×C₄	#216	1	4	yes	—
100	C₂×C₃×C₄	#217	1	4	yes	—
101	C₂×Dic₃	—	12	8	no	—
102	C₂×Dic₃	—	24	8	no	—
103	C₃×C₈	—	4	8	no	—
104	C₃×C₈	C₄×Dic₃	1	4	yes	—
105	C₃×C₈	Dic₃⋊C₄	1	4	yes	—
106	C₃×C₈	C₄⋊Dic₃	1	4	yes	—
107	C₃×C₈	(C₂×D₃)⋊C₄	1	4	yes	—
108	C₂×C₃²	—	2	1	no	—
109	C₂×C₃²	—	2	1	no	—
110	C₂×C₃²	#340	1	1	yes	characteristic
111	C₂×C₃²	#361	1	1	yes	characteristic
112	C₂×C₃²	#362	1	1	yes	characteristic
113	C₂²×C₄	—	36	1	no	—
114	C₂²×C₄	#638	1	1	yes	characteristic
115	M₁₆	—	2	2	no	—
116	M₁₆	—	4	1	no	—
117	C₂×C₈	—	4	1	no	—
118	C₂×C₈	#634	1	1	yes	characteristic
119	C₂×C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
120	C₂×C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
121	C₂×C₈	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
122	C₂×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
123	C₂×C₈	#638	1	1	yes	characteristic
124	C₄⋊C₄	—	18	2	no	—
125	C₄⋊C₄	—	36	1	no	—
126	C₁₆	—	18	2	no	—
127	C₂²×C₃	—	2	4	no	—
128	C₂²×C₃	—	2	4	no	—
129	C₂²×C₃	#808	1	4	yes	—
130	C₂²×C₃	#809	1	4	yes	—
131	C₂²×C₃	#835	1	4	yes	—
132	C₃×C₄	#807	1	4	yes	—
133	C₃×C₄	#810	1	4	yes	—
134	C₃×C₄	C₃⋊#95	1	4	yes	—
135	C₃×C₄	#832	1	4	yes	—
136	Dic₃	—	24	8	no	—
137	C₃²	#1387	1	1	yes	characteristic, complemented, Sylow
138	C₂³	#3755	1	1	yes	characteristic
139	C₂×C₄	—	36	2	no	—
140	C₂×C₄	—	72	2	no	—
141	C₂×C₄	#3750	1	1	yes	characteristic
142	C₂×C₄	#3752	1	1	yes	characteristic
143	C₂×C₄	#3753	1	1	yes	characteristic
144	C₂×C₄	#3754	1	1	yes	characteristic
145	C₂×C₄	#3756	1	1	yes	characteristic
146	C₂×C₄	#3757	1	1	yes	characteristic
147	C₈	—	4	2	no	—
148	C₈	C₄×C₃⋊Dic₃	1	1	yes	characteristic
149	C₈	#3750	1	1	yes	characteristic
150	C₈	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
151	C₈	#3752	1	1	yes	characteristic
152	C₂×C₃	—	2	4	no	—
153	C₂×C₃	—	2	4	no	—
154	C₂×C₃	#4495	1	4	yes	—
155	C₂×C₃	#4496	1	4	yes	—
156	C₂×C₃	#4529	1	4	yes	—
157	C₂²	—	2	1	no	—
158	C₂²	—	2	1	no	—
159	C₂²	#63635	1	1	yes	characteristic
160	C₂²	#63636	1	1	yes	characteristic
161	C₂²	#63662	1	1	yes	characteristic
162	C₄	—	72	2	no	—
163	C₄	#63634	1	1	yes	characteristic
164	C₄	#63637	1	1	yes	characteristic
165	C₄	#63658	1	1	yes	characteristic
166	C₄	#63659	1	1	yes	characteristic
167	C₃	#67732	1	4	yes	complemented
168	C₂	—	2	1	no	—
169	C₂	—	2	1	no	—
170	C₂	#10589361	1	1	yes	characteristic
171	C₂	#10589362	1	1	yes	characteristic
172	C₂	#10589395	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#63636
commutator	C₃²×C₈	C₄²
Frattini	C₂²×C₈	C₂×C₃⋊D₃
Fitting	C₂×C₃²×#108	C₂
socle	C₂²×C₃²	#102
prime radical	C₂³×C₃²	SD₁₆
cosocle	C₂²×C₃²×C₈	C₂²
hypercenter	C₂×#108	C₃⋊D₃
hypocenter	C₃²	#1387
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
9. C₃²×#108	C₄	semidirect
13. #63639	C₄	semidirect
37. C₃×#108	Dic₃	semidirect
91. #108	C₃⋊Dic₃	semidirect
137. C₃²	#1387	semidirect
167. C₃	#67732	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×#108	2
C₂×#63645	2
C₂×#63639	2
#67732	3