Subgroups — #11858504

Order 1152.

1476 conjugacy classes of subgroups.

The subgroup lattice has 201 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#64135	C₂	1	2	yes	maximal
3	#10592820	C₂	1	8	yes	maximal, complemented
4	#10592805	C₂	1	4	yes	maximal, complemented
5	#83752	—	3	4	no	maximal
6	C₂³×C₃²×C₄	C₂²	1	1	yes	characteristic
7	C₂×C₃²×M₁₆	C₄	1	8	yes	complemented
8	C₂×C₃²×M₁₆	C₂²	1	4	yes	—
9	C₂²×C₃²×C₈	C₂²	1	2	yes	—
10	C₂×#3757	—	2	2	no	—
11	#64135	—	2	2	no	—
12	#64135	—	2	2	no	—
13	#64135	C₂²	1	4	yes	—
14	#64135	C₂²	1	4	yes	—
15	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	4	yes	—
16	C₂×C₄×C₃⋊Dic₃	—	2	2	no	—
17	#63645	C₂²	1	8	yes	complemented
18	#63644	C₂²	1	8	yes	complemented
19	C₂²×C₃×M₁₆	D₃	1	4	yes	—
20	C₂×#926	—	3	8	no	—
21	#5121	—	3	32	no	—
22	#5106	—	3	16	no	—
23	C₂⁴×C₃²	D₄	1	1	yes	characteristic
24	C₂²×C₃²×C₄	C₂×C₄	1	4	yes	—
25	C₂²×C₃²×C₄	D₄	1	2	yes	—
26	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
27	C₂²×C₃²×C₄	Q₈	1	4	yes	—
28	C₂²×C₃²×C₄	C₂³	1	2	yes	—
29	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
30	C₂²×C₃⋊Dic₃	—	2	4	no	—
31	C₃²×M₁₆	C₂×C₄	1	16	yes	complemented
32	C₂×C₃²×C₈	C₂×C₄	1	8	yes	complemented
33	C₂×C₃²×C₈	C₂³	1	4	yes	—
34	#3757	—	2	4	no	—
35	#3757	—	2	4	no	—
36	C₄⋊(C₃⋊Dic₃)	—	2	4	no	—
37	C₄⋊(C₃⋊Dic₃)	C₂³	1	8	yes	—
38	C₄×C₃⋊Dic₃	—	2	4	no	—
39	C₄×C₃⋊Dic₃	—	2	4	no	—
40	#2911	—	9	1	no	Sylow
41	C₂³×C₃×C₄	C₂×D₃	1	4	yes	—
42	C₂×C₃×M₁₆	Dic₃	1	32	yes	complemented
43	C₂×C₃×M₁₆	C₂×D₃	1	16	yes	—
44	C₂²×C₃×C₈	C₂×D₃	1	8	yes	—
45	C₂×#217	—	6	8	no	—
46	#926	—	3	16	no	—
47	#926	—	3	16	no	—
48	#926	—	6	8	no	—
49	#926	—	6	8	no	—
50	C₂×C₄⋊Dic₃	—	3	16	no	—
51	C₂×C₄×Dic₃	—	6	8	no	—
52	#818	—	3	32	no	—
53	#817	—	3	32	no	—
54	C₂³×C₃²	—	4	2	no	—
55	C₂³×C₃²	C₄⋊C₄	1	4	yes	—
56	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
57	C₂³×C₃²	C₂×D₄	1	2	yes	—
58	C₂×C₃²×C₄	C₄⋊C₄	1	8	yes	—
59	C₂×C₃²×C₄	C₄⋊C₄	1	4	yes	—
60	C₂×C₃²×C₄	C₂²×C₄	1	2	yes	—
61	C₂×C₃²×C₄	C₂²×C₄	1	4	yes	—
62	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
63	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
64	C₂×C₃²×C₄	C₂×Q₈	1	4	yes	—
65	C₂×C₃²×C₄	C₂×Q₈	1	2	yes	—
66	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
67	C₂×C₃⋊Dic₃	—	2	8	no	—
68	C₂×C₃⋊Dic₃	—	4	8	no	—
69	C₃²×C₈	C₂²×C₄	1	8	yes	—
70	C₂²×M₁₆	C₃⋊D₃	1	1	yes	characteristic
71	C₂×#117	—	9	2	no	—
72	#428	—	9	8	no	—
73	#427	—	9	4	no	—
74	C₂⁴×C₃	D₁₂	1	4	yes	—
75	C₂²×C₃×C₄	Dic₆	1	16	yes	—
76	C₂²×C₃×C₄	D₁₂	1	8	yes	—
77	C₂²×C₃×C₄	D₁₂	1	4	yes	—
78	C₂²×C₃×C₄	C₂×Dic₃	1	16	yes	—
79	C₂²×C₃×C₄	C₂²×D₃	1	8	yes	—
80	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
81	C₂²×Dic₃	—	6	16	no	—
82	C₃×M₁₆	C₂×Dic₃	1	64	yes	complemented
83	C₂×C₃×C₈	C₂×Dic₃	1	32	yes	complemented
84	C₂×C₃×C₈	C₂²×D₃	1	16	yes	—
85	#217	—	6	16	no	—
86	#217	—	6	16	no	—
87	C₄⋊Dic₃	—	3	32	no	—
88	C₄⋊Dic₃	—	6	16	no	—
89	C₄×Dic₃	—	6	16	no	—
90	C₄×Dic₃	—	6	16	no	—
91	C₂²×C₃²	—	2	4	no	—
92	C₂²×C₃²	—	2	2	no	—
93	C₂²×C₃²	—	4	4	no	—
94	C₂²×C₃²	C₂×C₄⋊C₄	1	2	yes	—
95	C₂²×C₃²	C₂×C₄⋊C₄	1	4	yes	—
96	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
97	C₃²×C₄	C₂×C₄⋊C₄	1	2	yes	—
98	C₃²×C₄	C₂×C₄⋊C₄	1	4	yes	—
99	C₃²×C₄	C₂³×C₄	1	1	yes	characteristic
100	C₃²×C₄	C₂²×Q₈	1	1	yes	characteristic
101	C₃⋊Dic₃	—	4	8	no	—
102	C₂³×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
103	C₂×M₁₆	C₃⋊Dic₃	1	8	yes	complemented
104	C₂×M₁₆	C₂×C₃⋊D₃	1	4	yes	—
105	C₂²×C₈	C₂×C₃⋊D₃	1	2	yes	—
106	#117	—	9	4	no	—
107	#117	—	9	4	no	—
108	#117	—	18	2	no	—
109	#117	—	18	2	no	—
110	C₂×C₄⋊C₄	—	9	4	no	—
111	C₂×C₂²⋊C₄	—	18	2	no	—
112	C₂×C₄²	—	18	2	no	—
113	#107	—	9	8	no	—
114	#106	—	9	8	no	—
115	C₂³×C₃	—	4	8	no	—
116	C₂³×C₃	C₄⋊Dic₃	1	16	yes	—
117	C₂³×C₃	C₂×D₁₂	1	4	yes	—
118	C₂³×C₃	C₂×D₁₂	1	8	yes	—
119	C₂×C₃×C₄	C₄⋊Dic₃	1	32	yes	—
120	C₂×C₃×C₄	C₄⋊Dic₃	1	16	yes	—
121	C₂×C₃×C₄	C₂×Dic₆	1	16	yes	—
122	C₂×C₃×C₄	C₂×Dic₆	1	8	yes	—
123	C₂×C₃×C₄	C₂×D₁₂	1	8	yes	—
124	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
125	C₂×C₃×C₄	C₂²×Dic₃	1	8	yes	—
126	C₂×C₃×C₄	C₂²×Dic₃	1	16	yes	—
127	C₂×C₃×C₄	C₂³×D₃	1	4	yes	—
128	C₂×Dic₃	—	6	32	no	—
129	C₂×Dic₃	—	12	32	no	—
130	C₃×C₈	C₂²×Dic₃	1	32	yes	—
131	C₂×C₃²	—	2	4	no	—
132	C₂×C₃²	—	2	2	no	—
133	C₂×C₃²	C₂²×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₃²	#575	1	2	yes	—
135	C₂⁴	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
136	C₂²×C₄	—	18	4	no	—
137	C₂²×C₄	#634	1	4	yes	—
138	C₂²×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
139	C₂²×C₄	C₄⋊(C₃⋊D₃)	1	2	yes	—
140	C₂²×C₄	C₂×C₃⋊Dic₃	1	4	yes	—
141	C₂²×C₄	C₂²×C₃⋊D₃	1	2	yes	—
142	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
143	M₁₆	C₂×C₃⋊Dic₃	1	16	yes	complemented
144	C₂×C₈	C₂×C₃⋊Dic₃	1	8	yes	complemented
145	C₂×C₈	C₂²×C₃⋊D₃	1	4	yes	—
146	C₄⋊C₄	—	9	8	no	—
147	C₄⋊C₄	—	18	4	no	—
148	C₂²⋊C₄	—	18	4	no	—
149	C₂²⋊C₄	—	18	4	no	—
150	C₄²	—	18	4	no	—
151	C₄²	—	18	4	no	—
152	C₂²×C₃	—	2	16	no	—
153	C₂²×C₃	—	2	8	no	—
154	C₂²×C₃	—	4	16	no	—
155	C₂²×C₃	C₂×C₄⋊Dic₃	1	8	yes	—
156	C₂²×C₃	C₂×C₄⋊Dic₃	1	16	yes	—
157	C₂²×C₃	C₂²×D₁₂	1	4	yes	—
158	C₃×C₄	C₂×C₄⋊Dic₃	1	16	yes	—
159	C₃×C₄	C₂×C₄⋊Dic₃	1	8	yes	—
160	C₃×C₄	C₂²×Dic₆	1	4	yes	—
161	C₃×C₄	C₂³×Dic₃	1	4	yes	—
162	Dic₃	—	12	32	no	—
163	C₃²	#2911	1	1	yes	characteristic, complemented, Sylow
164	C₂³	—	4	2	no	—
165	C₂³	C₄⋊(C₃⋊Dic₃)	1	4	yes	—
166	C₂³	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
167	C₂³	C₂×C₄⋊(C₃⋊D₃)	1	2	yes	—
168	C₂×C₄	—	18	8	no	—
169	C₂×C₄	—	36	8	no	—
170	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	4	yes	—
171	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	8	yes	—
172	C₂×C₄	C₂×#634	1	4	yes	—
173	C₂×C₄	C₂×#634	1	2	yes	—
174	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	2	yes	—
175	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
176	C₂×C₄	C₂²×C₃⋊Dic₃	1	2	yes	—
177	C₂×C₄	C₂²×C₃⋊Dic₃	1	4	yes	—
178	C₂×C₄	C₂³×C₃⋊D₃	1	1	yes	characteristic
179	C₈	C₂²×C₃⋊Dic₃	1	8	yes	—
180	C₂×C₃	—	2	16	no	—
181	C₂×C₃	—	2	8	no	—
182	C₂×C₃	#5748	1	8	yes	—
183	C₂×C₃	C₂²×C₄⋊Dic₃	1	4	yes	—
184	C₂²	—	2	4	no	—
185	C₂²	—	2	2	no	—
186	C₂²	—	4	4	no	—
187	C₂²	C₂×C₄⋊(C₃⋊Dic₃)	1	2	yes	—
188	C₂²	C₂×C₄⋊(C₃⋊Dic₃)	1	4	yes	—
189	C₂²	C₂²×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
190	C₄	—	36	8	no	—
191	C₄	C₂×C₄⋊(C₃⋊Dic₃)	1	4	yes	—
192	C₄	C₂×C₄⋊(C₃⋊Dic₃)	1	2	yes	—
193	C₄	C₂²×#634	1	1	yes	characteristic
194	C₄	C₂³×C₃⋊Dic₃	1	1	yes	characteristic
195	C₃	#83752	1	4	yes	complemented
196	C₂	—	2	4	no	—
197	C₂	—	2	2	no	—
198	C₂	#10596219	1	2	yes	—
199	C₂	C₂²×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)
commutator	C₃²×C₄	C₂³×C₄
Frattini	C₂×C₄	C₂³×C₃⋊D₃
Fitting	C₂²×C₃²×M₁₆	C₂
socle	C₂²×C₃²	C₂²×D₄
prime radical	C₂⁴×C₃²	D₄
cosocle	C₂×C₃²×C₄	C₂⁴
hypercenter	C₂²×M₁₆	C₃⋊D₃
hypocenter	C₃²	#2911
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #10592820	C₂	semidirect
4. #10592805	C₂	semidirect
7. C₂×C₃²×M₁₆	C₄	semidirect
17. #63645	C₂²	semidirect
18. #63644	C₂²	semidirect
31. C₃²×M₁₆	C₂×C₄	semidirect
32. C₂×C₃²×C₈	C₂×C₄	semidirect
42. C₂×C₃×M₁₆	Dic₃	semidirect
82. C₃×M₁₆	C₂×Dic₃	semidirect
83. C₂×C₃×C₈	C₂×Dic₃	semidirect
103. C₂×M₁₆	C₃⋊Dic₃	semidirect
143. M₁₆	C₂×C₃⋊Dic₃	semidirect
144. C₂×C₈	C₂×C₃⋊Dic₃	semidirect
163. C₃²	#2911	semidirect
195. C₃	#83752	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃²×M₁₆	2
C₂×#64135	2
#10592820	2
#10592805	2
#83752	3