Subgroups — C₂³×#3711 #11918447

Order 1152.

3609 conjugacy classes of subgroups.

The subgroup lattice has 203 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄×D₃	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×C₃×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#3711	C₂	1	28	yes	maximal, direct product factor
5	C₂³×#208	—	3	1	no	maximal
6	C₂³×ΓL(1,𝔽₂₅)	—	3	1	no	maximal
7	C₂⁴×C₃×D₃	C₄	1	1	yes	characteristic
8	C₂³×C₃²×C₄	C₂²	1	1	yes	characteristic
9	C₂³×C₃×Dic₃	C₄	1	1	yes	characteristic
10	C₂²×C₃×C₄×D₃	C₄	1	14	yes	—
11	C₂²×C₃×C₄×D₃	C₂²	1	14	yes	—
12	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	14	yes	complemented
13	C₂²×C₃×C₃⋊C₈	C₂²	1	14	yes	complemented
14	C₂×#3711	C₂²	1	112	yes	direct product factor
15	C₂⁴×C₃×C₄	—	3	1	no	—
16	C₂³×C₄×D₃	D₃	1	1	yes	characteristic
17	C₂³×C₃×C₈	—	3	1	no	—
18	C₂²×#208	—	3	28	no	—
19	C₂³×C₃⋊C₈	—	3	1	no	—
20	C₂³×C₃⋊C₈	—	3	1	no	—
21	C₂³×C₃⋊C₈	—	6	1	no	—
22	C₂³×C₃⋊C₈	D₃	1	1	yes	characteristic, complemented
23	C₂²×ΓL(1,𝔽₂₅)	—	3	28	no	—
24	C₂⁴×C₃²	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₃×D₃	—	2	8	no	—
26	C₂³×C₃×D₃	C₂×C₄	1	14	yes	—
27	C₂²×C₃²×C₄	C₂×C₄	1	7	yes	—
28	C₂²×C₃²×C₄	C₂³	1	7	yes	—
29	C₂²×C₃×Dic₃	C₂×C₄	1	14	yes	—
30	C₂×C₃×C₄×D₃	C₂×C₄	1	84	yes	—
31	C₂×C₃×C₄×D₃	C₂³	1	28	yes	—
32	C₂×C₃⋊(C₃⋊C₈)	C₂³	1	28	yes	complemented
33	C₂×C₃×C₃⋊C₈	C₂³	1	28	yes	complemented
34	#3711	C₂³	1	64	yes	direct product factor
35	C₂³×M₁₆	—	9	1	no	Sylow
36	C₂⁵×C₃	—	3	1	no	—
37	C₂⁴×D₃	Dic₃	1	1	yes	characteristic
38	C₂³×C₃×C₄	—	2	1	no	—
39	C₂³×C₃×C₄	—	3	1	no	—
40	C₂³×C₃×C₄	—	3	14	no	—
41	C₂³×C₃×C₄	—	3	14	no	—
42	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
43	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
44	C₂³×Dic₃	Dic₃	1	1	yes	characteristic
45	C₂²×C₄×D₃	Dic₃	1	14	yes	—
46	C₂²×C₄×D₃	C₂×D₃	1	14	yes	—
47	C₂²×C₃×C₈	—	3	14	no	—
48	C₂×#208	—	3	112	no	—
49	C₂²×C₃⋊C₈	—	3	14	no	—
50	C₂²×C₃⋊C₈	—	3	14	no	—
51	C₂²×C₃⋊C₈	—	6	14	no	—
52	C₂²×C₃⋊C₈	C₂×D₃	1	14	yes	complemented
53	C₂×ΓL(1,𝔽₂₅)	—	3	112	no	—
54	C₂³×C₃²	M₁₆	1	8	yes	complemented
55	C₂³×C₃²	C₂²×C₄	1	7	yes	—
56	C₂²×C₃×D₃	—	2	56	no	—
57	C₂²×C₃×D₃	C₂²×C₄	1	28	yes	—
58	C₂×C₃²×C₄	C₂²×C₄	1	21	yes	—
59	C₂×C₃²×C₄	C₂⁴	1	7	yes	—
60	C₂×C₃×Dic₃	C₂²×C₄	1	28	yes	—
61	C₃×C₄×D₃	C₂²×C₄	1	56	yes	—
62	C₃×C₄×D₃	C₂⁴	1	8	yes	—
63	C₃⋊(C₃⋊C₈)	C₂⁴	1	8	yes	complemented
64	C₃×C₃⋊C₈	C₂⁴	1	8	yes	complemented
65	C₂⁴×C₄	—	3	1	no	—
66	C₂²×M₁₆	—	9	28	no	—
67	C₂³×C₈	—	3	1	no	—
68	C₂³×C₈	—	9	1	no	—
69	C₂⁴×C₃	—	2	1	no	—
70	C₂⁴×C₃	—	3	14	no	—
71	C₂⁴×C₃	—	6	8	no	—
72	C₂⁴×C₃	C₄×D₃	1	1	yes	characteristic
73	C₂⁴×C₃	C₂×Dic₃	1	1	yes	characteristic
74	C₂³×D₃	—	2	8	no	—
75	C₂³×D₃	C₂×Dic₃	1	14	yes	—
76	C₂²×C₃×C₄	—	2	7	no	—
77	C₂²×C₃×C₄	—	2	7	no	—
78	C₂²×C₃×C₄	—	3	14	no	—
79	C₂²×C₃×C₄	—	3	84	no	—
80	C₂²×C₃×C₄	—	3	28	no	—
81	C₂²×C₃×C₄	C₄×D₃	1	7	yes	—
82	C₂²×C₃×C₄	C₂×Dic₃	1	7	yes	—
83	C₂²×C₃×C₄	C₂²×D₃	1	7	yes	—
84	C₂²×C₃×C₄	C₂²×D₃	1	7	yes	—
85	C₂²×Dic₃	C₂×Dic₃	1	14	yes	—
86	C₂×C₄×D₃	C₂×Dic₃	1	84	yes	—
87	C₂×C₄×D₃	C₂²×D₃	1	28	yes	—
88	C₂×C₃×C₈	—	3	28	no	—
89	#208	—	3	64	no	—
90	C₂×C₃⋊C₈	—	3	28	no	—
91	C₂×C₃⋊C₈	—	3	28	no	—
92	C₂×C₃⋊C₈	—	6	28	no	—
93	C₂×C₃⋊C₈	C₂²×D₃	1	28	yes	complemented
94	ΓL(1,𝔽₂₅)	—	3	64	no	—
95	C₂²×C₃²	C₂×M₁₆	1	28	yes	complemented
96	C₂²×C₃²	C₂³×C₄	1	7	yes	—
97	C₂×C₃×D₃	—	2	56	no	—
98	C₂×C₃×D₃	C₂³×C₄	1	8	yes	—
99	C₃²×C₄	C₂³×C₄	1	7	yes	—
100	C₃²×C₄	C₂⁵	1	1	yes	characteristic
101	C₃×Dic₃	C₂³×C₄	1	8	yes	—
102	C₂⁵	—	3	1	no	—
103	C₂³×C₄	—	3	1	no	—
104	C₂³×C₄	—	3	14	no	—
105	C₂³×C₄	—	3	14	no	—
106	C₂³×C₄	D₃²	1	1	yes	characteristic
107	C₂×M₁₆	—	9	112	no	—
108	C₂²×C₈	—	3	14	no	—
109	C₂²×C₈	—	9	14	no	—
110	C₂³×C₃	—	2	8	no	—
111	C₂³×C₃	—	2	7	no	—
112	C₂³×C₃	—	3	28	no	—
113	C₂³×C₃	—	6	56	no	—
114	C₂³×C₃	ΓL(1,𝔽₂₅)	1	8	yes	complemented
115	C₂³×C₃	#208	1	8	yes	complemented
116	C₂³×C₃	C₂×C₄×D₃	1	7	yes	—
117	C₂³×C₃	C₂²×Dic₃	1	7	yes	—
118	C₂²×D₃	—	2	56	no	—
119	C₂²×D₃	C₂²×Dic₃	1	28	yes	—
120	C₂×C₃×C₄	—	2	7	no	—
121	C₂×C₃×C₄	—	2	21	no	—
122	C₂×C₃×C₄	—	3	28	no	—
123	C₂×C₃×C₄	—	3	56	no	—
124	C₂×C₃×C₄	—	3	8	no	—
125	C₂×C₃×C₄	C₂×C₄×D₃	1	21	yes	—
126	C₂×C₃×C₄	C₂²×Dic₃	1	21	yes	—
127	C₂×C₃×C₄	C₂³×D₃	1	7	yes	—
128	C₂×C₃×C₄	C₂³×D₃	1	7	yes	—
129	C₂×Dic₃	C₂²×Dic₃	1	28	yes	—
130	C₄×D₃	C₂²×Dic₃	1	56	yes	—
131	C₄×D₃	C₂³×D₃	1	8	yes	—
132	C₃×C₈	—	3	8	no	—
133	C₃⋊C₈	—	3	8	no	—
134	C₃⋊C₈	—	3	8	no	—
135	C₃⋊C₈	—	6	8	no	—
136	C₃⋊C₈	C₂³×D₃	1	8	yes	complemented
137	C₂×C₃²	C₂²×M₁₆	1	14	yes	complemented
138	C₂×C₃²	C₂⁴×C₄	1	1	yes	characteristic
139	C₃×D₃	—	2	8	no	—
140	C₂⁴	—	3	14	no	—
141	C₂⁴	—	6	8	no	—
142	C₂⁴	D₃×Dic₃	1	1	yes	characteristic
143	C₂²×C₄	—	3	14	no	—
144	C₂²×C₄	—	3	28	no	—
145	C₂²×C₄	—	3	84	no	—
146	C₂²×C₄	D₃×Dic₃	1	7	yes	—
147	C₂²×C₄	C₂×D₃²	1	7	yes	—
148	M₁₆	—	9	64	no	—
149	C₂×C₈	—	3	28	no	—
150	C₂×C₈	—	9	28	no	—
151	C₂²×C₃	—	2	7	no	—
152	C₂²×C₃	—	2	28	no	—
153	C₂²×C₃	—	3	8	no	—
154	C₂²×C₃	—	6	56	no	—
155	C₂²×C₃	C₂×ΓL(1,𝔽₂₅)	1	28	yes	complemented
156	C₂²×C₃	C₂×#208	1	28	yes	complemented
157	C₂²×C₃	C₂²×C₄×D₃	1	7	yes	—
158	C₂²×C₃	C₂³×Dic₃	1	7	yes	—
159	C₂×D₃	—	2	56	no	—
160	C₂×D₃	C₂³×Dic₃	1	8	yes	—
161	C₃×C₄	—	2	7	no	—
162	C₃×C₄	—	2	1	no	—
163	C₃×C₄	—	3	8	no	—
164	C₃×C₄	C₂²×C₄×D₃	1	7	yes	—
165	C₃×C₄	C₂³×Dic₃	1	7	yes	—
166	C₃×C₄	C₂⁴×D₃	1	1	yes	characteristic
167	C₃×C₄	C₂⁴×D₃	1	1	yes	characteristic
168	Dic₃	C₂³×Dic₃	1	8	yes	—
169	C₃²	C₂³×M₁₆	1	1	yes	characteristic, complemented, Sylow
170	C₂³	—	3	28	no	—
171	C₂³	—	6	56	no	—
172	C₂³	#3711	1	8	yes	direct product factor
173	C₂³	C₂×D₃×Dic₃	1	7	yes	—
174	C₂×C₄	—	3	28	no	—
175	C₂×C₄	—	3	56	no	—
176	C₂×C₄	—	3	8	no	—
177	C₂×C₄	C₂×D₃×Dic₃	1	21	yes	—
178	C₂×C₄	C₂²×D₃²	1	7	yes	—
179	C₈	—	3	8	no	—
180	C₈	—	9	8	no	—
181	C₂×C₃	—	2	1	no	—
182	C₂×C₃	—	2	14	no	—
183	C₂×C₃	—	6	8	no	—
184	C₂×C₃	C₂²×ΓL(1,𝔽₂₅)	1	14	yes	complemented
185	C₂×C₃	C₂²×#208	1	14	yes	complemented
186	C₂×C₃	C₂³×C₄×D₃	1	1	yes	characteristic
187	C₂×C₃	C₂⁴×Dic₃	1	1	yes	characteristic
188	D₃	—	2	8	no	—
189	C₂²	—	3	8	no	—
190	C₂²	—	6	56	no	—
191	C₂²	C₂×#3711	1	28	yes	direct product factor
192	C₂²	C₂²×D₃×Dic₃	1	7	yes	—
193	C₄	—	3	8	no	—
194	C₄	C₂²×D₃×Dic₃	1	7	yes	—
195	C₄	C₂³×D₃²	1	1	yes	characteristic
196	C₃	—	2	1	no	—
197	C₃	C₂³×ΓL(1,𝔽₂₅)	1	1	yes	characteristic, complemented
198	C₃	C₂³×#208	1	1	yes	characteristic, complemented
199	C₂	—	6	8	no	—
200	C₂	C₂²×#3711	1	14	yes	direct product factor
201	C₂	C₂³×D₃×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₄	D₃²
commutator	C₂×C₃²	C₂⁴×C₄
Frattini	C₄	C₂³×D₃²
Fitting	C₂³×C₃²×C₄	C₂²
socle	C₂⁴×C₃²	C₂×C₄
prime radical	C₂⁴×C₃×D₃	C₄
cosocle	C₃²×C₄	C₂⁵
hypocenter	C₃²	C₂³×M₁₆
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
2. C₂³×C₃⋊(C₃⋊C₈)	C₂	semidirect
3. C₂³×C₃×C₃⋊C₈	C₂	semidirect
4. C₂²×#3711	C₂	direct (N × H)
12. C₂²×C₃⋊(C₃⋊C₈)	C₂²	semidirect
13. C₂²×C₃×C₃⋊C₈	C₂²	semidirect
14. C₂×#3711	C₂²	direct (N × H)
22. C₂³×C₃⋊C₈	D₃	semidirect
32. C₂×C₃⋊(C₃⋊C₈)	C₂³	semidirect
33. C₂×C₃×C₃⋊C₈	C₂³	semidirect
34. #3711	C₂³	direct (N × H)
52. C₂²×C₃⋊C₈	C₂×D₃	semidirect
54. C₂³×C₃²	M₁₆	semidirect
63. C₃⋊(C₃⋊C₈)	C₂⁴	semidirect
64. C₃×C₃⋊C₈	C₂⁴	semidirect
93. C₂×C₃⋊C₈	C₂²×D₃	semidirect
95. C₂²×C₃²	C₂×M₁₆	semidirect
114. C₂³×C₃	ΓL(1,𝔽₂₅)	semidirect
115. C₂³×C₃	#208	semidirect
136. C₃⋊C₈	C₂³×D₃	semidirect
137. C₂×C₃²	C₂²×M₁₆	semidirect
155. C₂²×C₃	C₂×ΓL(1,𝔽₂₅)	semidirect
156. C₂²×C₃	C₂×#208	semidirect
169. C₃²	C₂³×M₁₆	semidirect
172. C₂³	#3711	direct (N × H)
184. C₂×C₃	C₂²×ΓL(1,𝔽₂₅)	semidirect
185. C₂×C₃	C₂²×#208	semidirect
191. C₂²	C₂×#3711	direct (N × H)
197. C₃	C₂³×ΓL(1,𝔽₂₅)	semidirect
198. C₃	C₂³×#208	semidirect
200. C₂	C₂²×#3711	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₄×D₃	2
C₂³×C₃⋊(C₃⋊C₈)	2
C₂³×C₃×C₃⋊C₈	2
C₂²×#3711	2
C₂³×#208	3
C₂³×ΓL(1,𝔽₂₅)	3

Subgroups — C23×#3711 #11918447

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#3711 #11918447