Subgroups — #41×C₃⋊(C₃⋊C₈) #11861545

Order 1152.

1368 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×#41	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₃⋊(C₃⋊C₈)	C₂	1	2	yes	maximal, complemented
3	D₄×C₃⋊(C₃⋊C₈)	C₂	1	6	yes	maximal, complemented
4	#10592624	C₂	1	3	yes	maximal, complemented
5	C₂×C₄×C₃⋊(C₃⋊C₈)	C₂	1	3	yes	maximal, complemented
6	#41×C₃⋊C₈	—	3	4	no	maximal
7	C₂×C₃²×#41	C₄	1	1	yes	characteristic
8	C₃²×C₄×Q₈	C₄	1	1	yes	characteristic
9	C₃²×C₄×Q₈	C₂²	1	1	yes	characteristic
10	C₃²×C₄×D₄	C₄	1	3	yes	—
11	C₃²×C₄×D₄	C₂²	1	3	yes	—
12	C₃²×#117	C₄	1	3	yes	—
13	C₂×C₃²×C₄²	C₂²	1	3	yes	—
14	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	6	yes	—
15	#63657	C₂²	1	6	yes	—
16	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	6	yes	—
17	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
18	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	3	yes	—
19	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	6	yes	—
20	C₃×C₄×#41	D₃	1	4	yes	—
21	Q₈×C₃⋊C₈	—	3	8	no	—
22	D₄×C₃⋊C₈	—	3	24	no	—
23	#4925	—	3	12	no	—
24	C₂×C₄×C₃⋊C₈	—	3	12	no	—
25	C₃²×#41	C₈	1	8	yes	complemented
26	C₂×C₃²×Q₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃²×D₄	C₂×C₄	1	3	yes	—
28	C₂²×C₃²×C₄	C₂×C₄	1	3	yes	—
29	C₂²×C₃²×C₄	C₂×C₄	1	3	yes	—
30	C₂²×C₃²×C₄	C₂³	1	3	yes	—
31	C₃²×C₄⋊C₄	C₂×C₄	1	3	yes	—
32	C₃²×C₄⋊C₄	C₂×C₄	1	3	yes	—
33	C₃²×C₂²⋊C₄	C₂×C₄	1	6	yes	—
34	C₃²×C₄²	C₂×C₄	1	3	yes	—
35	C₃²×C₄²	C₂×C₄	1	3	yes	—
36	C₃²×C₄²	C₂³	1	1	yes	characteristic
37	C₃²×C₄²	C₂³	1	3	yes	—
38	C₂×C₃⋊(C₃⋊C₈)	—	2	12	no	—
39	C₂×C₃⋊(C₃⋊C₈)	C₂³	1	6	yes	—
40	C₂×C₃⋊(C₃⋊C₈)	C₂³	1	2	yes	—
41	C₈×#41	—	9	1	no	Sylow
42	C₂×C₃×#41	Dic₃	1	4	yes	—
43	C₃×C₄×Q₈	Dic₃	1	4	yes	—
44	C₃×C₄×Q₈	C₂×D₃	1	4	yes	—
45	C₃×C₄×D₄	Dic₃	1	12	yes	—
46	C₃×C₄×D₄	C₂×D₃	1	12	yes	—
47	C₃×#117	Dic₃	1	12	yes	—
48	C₂×C₃×C₄²	C₂×D₃	1	12	yes	—
49	C₂²×C₃⋊C₈	—	3	24	no	—
50	#830	—	3	24	no	—
51	C₄⋊(C₃⋊C₈)	—	3	24	no	—
52	C₄×C₃⋊C₈	—	3	4	no	—
53	C₄×C₃⋊C₈	—	3	12	no	—
54	C₄×C₃⋊C₈	—	3	24	no	—
55	C₂³×C₃²	C₂²×C₄	1	3	yes	—
56	C₃²×Q₈	C₂×C₈	1	4	yes	complemented
57	C₃²×D₄	C₂×C₈	1	12	yes	complemented
58	C₂×C₃²×C₄	—	2	6	no	—
59	C₂×C₃²×C₄	—	2	6	no	—
60	C₂×C₃²×C₄	C₂×C₈	1	12	yes	complemented
61	C₂×C₃²×C₄	C₂²×C₄	1	3	yes	—
62	C₂×C₃²×C₄	C₂²×C₄	1	3	yes	—
63	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₃²×C₄	C₂²×C₄	1	3	yes	—
65	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
67	C₃⋊(C₃⋊C₈)	—	2	6	no	—
68	C₃⋊(C₃⋊C₈)	#41	1	4	yes	direct product factor
69	C₄×#41	C₃⋊D₃	1	1	yes	characteristic
70	C₈×Q₈	—	9	2	no	—
71	C₈×D₄	—	9	6	no	—
72	#431	—	9	3	no	—
73	C₂×C₄×C₈	—	9	3	no	—
74	C₃×#41	C₃⋊C₈	1	32	yes	complemented
75	C₂×C₃×Q₈	C₂×Dic₃	1	4	yes	—
76	C₂×C₃×D₄	C₂×Dic₃	1	12	yes	—
77	C₂²×C₃×C₄	C₂×Dic₃	1	12	yes	—
78	C₂²×C₃×C₄	C₂×Dic₃	1	12	yes	—
79	C₂²×C₃×C₄	C₂²×D₃	1	12	yes	—
80	C₃×C₄⋊C₄	C₂×Dic₃	1	12	yes	—
81	C₃×C₄⋊C₄	C₂×Dic₃	1	12	yes	—
82	C₃×C₂²⋊C₄	C₂×Dic₃	1	24	yes	—
83	C₃×C₄²	C₂×Dic₃	1	12	yes	—
84	C₃×C₄²	C₂×Dic₃	1	12	yes	—
85	C₃×C₄²	C₂²×D₃	1	12	yes	—
86	C₃×C₄²	C₂²×D₃	1	4	yes	—
87	C₂×C₃⋊C₈	—	3	8	no	—
88	C₂×C₃⋊C₈	—	3	24	no	—
89	C₂×C₃⋊C₈	—	6	48	no	—
90	C₂²×C₃²	—	2	3	no	—
91	C₂²×C₃²	—	2	3	no	—
92	C₂²×C₃²	C₂²×C₈	1	6	yes	—
93	C₂²×C₃²	C₂³×C₄	1	1	yes	characteristic
94	C₃²×C₄	—	2	3	no	—
95	C₃²×C₄	—	2	3	no	—
96	C₃²×C₄	C₂²×C₈	1	2	yes	—
97	C₃²×C₄	C₂²×C₈	1	6	yes	—
98	C₃²×C₄	#131	1	2	yes	—
99	C₃²×C₄	C₂×#41	1	2	yes	—
100	C₂×#41	C₃⋊Dic₃	1	1	yes	characteristic
101	C₂²×C₈	—	9	6	no	—
102	C₄×Q₈	C₃⋊Dic₃	1	1	yes	characteristic
103	C₄×Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic
104	C₄×D₄	C₃⋊Dic₃	1	3	yes	—
105	C₄×D₄	C₂×C₃⋊D₃	1	3	yes	—
106	#117	C₃⋊Dic₃	1	3	yes	—
107	C₂×C₄²	C₂×C₃⋊D₃	1	3	yes	—
108	C₄⋊C₈	—	9	6	no	—
109	C₂²⋊C₈	—	9	6	no	—
110	C₄×C₈	—	9	6	no	—
111	C₄×C₈	—	9	1	no	—
112	C₄×C₈	—	9	3	no	—
113	C₂³×C₃	C₂²×Dic₃	1	12	yes	—
114	C₃×Q₈	C₂×C₃⋊C₈	1	16	yes	complemented
115	C₃×D₄	C₂×C₃⋊C₈	1	48	yes	complemented
116	C₂×C₃×C₄	—	2	24	no	—
117	C₂×C₃×C₄	—	2	24	no	—
118	C₂×C₃×C₄	C₂×C₃⋊C₈	1	48	yes	complemented
119	C₂×C₃×C₄	C₂²×Dic₃	1	12	yes	—
120	C₂×C₃×C₄	C₂²×Dic₃	1	12	yes	—
121	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
122	C₂×C₃×C₄	C₂²×Dic₃	1	12	yes	—
123	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
124	C₂×C₃×C₄	C₂³×D₃	1	4	yes	—
125	C₃⋊C₈	—	3	16	no	—
126	C₃⋊C₈	—	6	24	no	—
127	C₂×C₃²	—	2	6	no	—
128	C₂×C₃²	C₄×#41	1	1	yes	characteristic
129	C₂×C₃²	C₂³×C₈	1	1	yes	characteristic
130	C₂×C₃²	C₂×#131	1	1	yes	characteristic
131	#41	C₃⋊(C₃⋊C₈)	1	8	yes	direct product factor
132	C₂×Q₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
133	C₂×D₄	C₂×C₃⋊Dic₃	1	3	yes	—
134	C₂²×C₄	C₂×C₃⋊Dic₃	1	3	yes	—
135	C₂²×C₄	C₂×C₃⋊Dic₃	1	3	yes	—
136	C₂²×C₄	C₂²×C₃⋊D₃	1	3	yes	—
137	C₂×C₈	—	9	2	no	—
138	C₂×C₈	—	9	6	no	—
139	C₂×C₈	—	18	12	no	—
140	C₄⋊C₄	C₂×C₃⋊Dic₃	1	3	yes	—
141	C₄⋊C₄	C₂×C₃⋊Dic₃	1	3	yes	—
142	C₂²⋊C₄	C₂×C₃⋊Dic₃	1	6	yes	—
143	C₄²	C₂×C₃⋊Dic₃	1	3	yes	—
144	C₄²	C₂×C₃⋊Dic₃	1	3	yes	—
145	C₄²	C₂²×C₃⋊D₃	1	3	yes	—
146	C₄²	C₂²×C₃⋊D₃	1	1	yes	characteristic
147	C₂²×C₃	—	2	12	no	—
148	C₂²×C₃	—	2	12	no	—
149	C₂²×C₃	C₂²×C₃⋊C₈	1	24	yes	—
150	C₂²×C₃	C₂³×Dic₃	1	4	yes	—
151	C₃×C₄	—	2	12	no	—
152	C₃×C₄	—	2	12	no	—
153	C₃×C₄	C₂²×C₃⋊C₈	1	8	yes	—
154	C₃×C₄	C₂²×C₃⋊C₈	1	24	yes	—
155	C₃×C₄	#948	1	8	yes	—
156	C₃×C₄	D₃×#41	1	8	yes	—
157	C₃²	C₈×#41	1	1	yes	characteristic, complemented, Sylow
158	C₂³	C₂²×C₃⋊Dic₃	1	3	yes	—
159	Q₈	C₂×C₃⋊(C₃⋊C₈)	1	4	yes	complemented
160	D₄	C₂×C₃⋊(C₃⋊C₈)	1	12	yes	complemented
161	C₂×C₄	—	2	6	no	—
162	C₂×C₄	—	2	6	no	—
163	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	12	yes	complemented
164	C₂×C₄	C₂²×C₃⋊Dic₃	1	3	yes	—
165	C₂×C₄	C₂²×C₃⋊Dic₃	1	3	yes	—
166	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
167	C₂×C₄	C₂²×C₃⋊Dic₃	1	3	yes	—
168	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
169	C₂×C₄	C₂³×C₃⋊D₃	1	1	yes	characteristic
170	C₈	—	9	4	no	—
171	C₈	—	18	6	no	—
172	C₂×C₃	—	2	24	no	—
173	C₂×C₃	C₂³×C₃⋊C₈	1	4	yes	—
174	C₂×C₃	C₂×#948	1	4	yes	—
175	C₂×C₃	Dic₃×#41	1	4	yes	—
176	C₂²	—	2	3	no	—
177	C₂²	—	2	3	no	—
178	C₂²	C₂²×C₃⋊(C₃⋊C₈)	1	6	yes	—
179	C₂²	C₂³×C₃⋊Dic₃	1	1	yes	characteristic
180	C₄	—	2	3	no	—
181	C₄	—	2	3	no	—
182	C₄	C₂²×C₃⋊(C₃⋊C₈)	1	2	yes	—
183	C₄	C₂²×C₃⋊(C₃⋊C₈)	1	6	yes	—
184	C₄	#64157	1	2	yes	—
185	C₄	#41×C₃⋊D₃	1	2	yes	—
186	C₃	#41×C₃⋊C₈	1	4	yes	complemented
187	C₂	—	2	6	no	—
188	C₂	C₂³×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
189	C₂	C₂×#64157	1	1	yes	characteristic
190	C₂	#41×C₃⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂²×C₃⋊D₃
commutator	C₂×C₃²	C₂³×C₈
Frattini	C₂×C₄	C₂³×C₃⋊D₃
Fitting	C₃²×C₄×#41	C₂
socle	C₂²×C₃²	C₂³×C₄
prime radical	C₂×C₃²×#41	C₄
cosocle	C₂×C₃²×C₄	C₂⁴
hypercenter	C₄×#41	C₃⋊D₃
hypocenter	C₃²	C₈×#41
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
2. Q₈×C₃⋊(C₃⋊C₈)	C₂	semidirect
3. D₄×C₃⋊(C₃⋊C₈)	C₂	semidirect
4. #10592624	C₂	semidirect
5. C₂×C₄×C₃⋊(C₃⋊C₈)	C₂	semidirect
25. C₃²×#41	C₈	semidirect
56. C₃²×Q₈	C₂×C₈	semidirect
57. C₃²×D₄	C₂×C₈	semidirect
60. C₂×C₃²×C₄	C₂×C₈	semidirect
68. C₃⋊(C₃⋊C₈)	#41	direct (N × H)
74. C₃×#41	C₃⋊C₈	semidirect
114. C₃×Q₈	C₂×C₃⋊C₈	semidirect
115. C₃×D₄	C₂×C₃⋊C₈	semidirect
118. C₂×C₃×C₄	C₂×C₃⋊C₈	semidirect
131. #41	C₃⋊(C₃⋊C₈)	direct (N × H)
157. C₃²	C₈×#41	semidirect
159. Q₈	C₂×C₃⋊(C₃⋊C₈)	semidirect
160. D₄	C₂×C₃⋊(C₃⋊C₈)	semidirect
163. C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	semidirect
186. C₃	#41×C₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄×#41	2
Q₈×C₃⋊(C₃⋊C₈)	2
D₄×C₃⋊(C₃⋊C₈)	2
#10592624	2
C₂×C₄×C₃⋊(C₃⋊C₈)	2
#41×C₃⋊C₈	3

Subgroups — #41×C3⋊(C3⋊C8) #11861545

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #41×C₃⋊(C₃⋊C₈) #11861545