Subgroups — #11795999

Order 1152.

684 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	C₂×C₃²×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×#63639	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
4	#10589415	C₂	1	4	yes	maximal, complemented
5	#72004	—	3	4	no	maximal
6	C₂×C₃²×#41	C₄	1	1	yes	characteristic
7	C₃²×#131	—	2	2	no	—
8	C₃²×#131	—	2	2	no	—
9	C₂×C₃²×M₁₆	C₄	1	1	yes	characteristic
10	C₂×C₃²×M₁₆	C₄	1	2	yes	—
11	C₂²×C₃²×C₈	C₂²	1	2	yes	—
12	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
13	#63639	—	2	2	no	—
14	C₂×C₃⋊(C₃⋊C₁₆)	—	2	2	no	—
15	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
16	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
17	C₂×C₃×#131	D₃	1	4	yes	—
18	C₂×#812	—	3	4	no	—
19	C₂²×C₃⋊C₁₆	—	3	4	no	—
20	#4549	—	3	16	no	—
21	C₃²×#41	—	2	2	no	—
22	C₃²×#41	—	2	2	no	—
23	C₂×C₃²×Q₈	C₈	1	1	yes	characteristic
24	C₂×C₃²×D₄	C₈	1	2	yes	—
25	C₂×C₃²×D₄	C₈	1	1	yes	characteristic
26	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
27	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
28	C₃²×M₁₆	—	2	2	no	—
29	C₃²×M₁₆	—	2	4	no	—
30	C₂×C₃²×C₈	—	2	2	no	—
31	C₂×C₃²×C₈	—	2	2	no	—
32	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
33	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃²×C₈	D₄	1	2	yes	—
35	C₂×C₃²×C₈	D₄	1	2	yes	—
36	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
37	C₃⋊(C₃⋊C₁₆)	—	2	2	no	—
38	C₃⋊(C₃⋊C₁₆)	—	2	2	no	—
39	#2113	—	9	1	no	Sylow
40	C₂×C₃×#41	Dic₃	1	4	yes	—
41	C₃×#131	—	2	8	no	—
42	C₃×#131	—	2	8	no	—
43	C₂×C₃×M₁₆	Dic₃	1	4	yes	—
44	C₂×C₃×M₁₆	Dic₃	1	8	yes	—
45	C₂²×C₃×C₈	C₂×D₃	1	8	yes	—
46	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
47	#812	—	6	8	no	—
48	C₂×C₃⋊C₁₆	—	3	8	no	—
49	C₂×C₃⋊C₁₆	—	3	8	no	—
50	C₂×C₃⋊C₁₆	—	6	8	no	—
51	C₂³×C₃²	C₂×C₈	1	2	yes	—
52	C₂³×C₃²	C₂×C₈	1	1	yes	characteristic
53	C₃²×Q₈	—	2	2	no	—
54	C₃²×D₄	—	2	4	no	—
55	C₃²×D₄	—	2	2	no	—
56	C₂×C₃²×C₄	—	2	2	no	—
57	C₂×C₃²×C₄	—	2	2	no	—
58	C₂×C₃²×C₄	C₂²⋊C₄	1	2	yes	—
59	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
60	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
61	C₂×C₃²×C₄	M₁₆	1	2	yes	—
62	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
63	C₃²×C₈	—	2	2	no	—
64	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
65	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
66	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
67	C₂×#131	C₃⋊D₃	1	1	yes	characteristic
68	C₂×M₃₂	—	9	1	no	—
69	C₂²×C₁₆	—	9	1	no	—
70	C₂²⋊C₁₆	—	9	4	no	—
71	C₃×#41	—	2	8	no	—
72	C₃×#41	—	2	8	no	—
73	C₂×C₃×Q₈	C₃⋊C₈	1	4	yes	—
74	C₂×C₃×D₄	C₃⋊C₈	1	8	yes	—
75	C₂×C₃×D₄	C₃⋊C₈	1	4	yes	—
76	C₂²×C₃×C₄	C₂×Dic₃	1	8	yes	—
77	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
78	C₃×M₁₆	—	2	8	no	—
79	C₃×M₁₆	—	2	16	no	—
80	C₂×C₃×C₈	—	2	8	no	—
81	C₂×C₃×C₈	—	2	8	no	—
82	C₂×C₃×C₈	C₂×Dic₃	1	8	yes	—
83	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
84	C₂×C₃×C₈	#67	1	8	yes	—
85	C₂×C₃×C₈	#67	1	8	yes	—
86	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
87	C₃⋊C₁₆	—	6	8	no	—
88	C₃⋊C₁₆	—	6	8	no	—
89	C₂²×C₃²	—	2	2	no	—
90	C₂²×C₃²	—	2	2	no	—
91	C₂²×C₃²	—	2	4	no	—
92	C₂²×C₃²	C₂²⋊C₈	1	2	yes	—
93	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
94	C₃²×C₄	—	2	2	no	—
95	C₃²×C₄	C₂²⋊C₈	1	2	yes	—
96	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
98	C₂×#41	C₃⋊Dic₃	1	1	yes	characteristic
99	#131	—	2	2	no	—
100	#131	—	2	2	no	—
101	C₂×M₁₆	C₃⋊Dic₃	1	1	yes	characteristic
102	C₂×M₁₆	C₃⋊Dic₃	1	2	yes	—
103	C₂²×C₈	C₂×C₃⋊D₃	1	2	yes	—
104	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
105	M₃₂	—	18	2	no	—
106	C₂×C₁₆	—	9	2	no	—
107	C₂×C₁₆	—	9	2	no	—
108	C₂×C₁₆	—	18	2	no	—
109	C₂³×C₃	C₂×C₃⋊C₈	1	8	yes	—
110	C₂³×C₃	C₂×C₃⋊C₈	1	4	yes	—
111	C₃×Q₈	—	2	8	no	—
112	C₃×D₄	—	2	16	no	—
113	C₃×D₄	—	2	8	no	—
114	C₂×C₃×C₄	—	2	8	no	—
115	C₂×C₃×C₄	—	2	8	no	—
116	C₂×C₃×C₄	C₂×C₃⋊C₈	1	8	yes	—
117	C₂×C₃×C₄	C₂×C₃⋊C₈	1	4	yes	—
118	C₂×C₃×C₄	#208	1	8	yes	—
119	C₂×C₃×C₄	#217	1	8	yes	—
120	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
121	C₃×C₈	—	2	8	no	—
122	C₃×C₈	#217	1	8	yes	—
123	C₃×C₈	C₂×#67	1	4	yes	—
124	C₃×C₈	C₂×#67	1	4	yes	—
125	C₂×C₃²	—	2	2	no	—
126	C₂×C₃²	—	4	2	no	—
127	C₂×C₃²	C₂×C₂²⋊C₈	1	1	yes	characteristic
128	C₂×C₃²	#504	1	2	yes	—
129	#41	—	2	2	no	—
130	#41	—	2	2	no	—
131	C₂×Q₈	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
132	C₂×D₄	C₃⋊(C₃⋊C₈)	1	2	yes	—
133	C₂×D₄	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
134	C₂²×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
135	C₂²×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
136	M₁₆	—	2	2	no	—
137	M₁₆	—	2	4	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	C₂×C₃⋊Dic₃	1	2	yes	—
141	C₂×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
142	C₂×C₈	#638	1	2	yes	—
143	C₂×C₈	#638	1	2	yes	—
144	C₂×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
145	C₁₆	—	18	2	no	—
146	C₁₆	—	18	2	no	—
147	C₂²×C₃	—	2	8	no	—
148	C₂²×C₃	—	2	16	no	—
149	C₂²×C₃	—	2	8	no	—
150	C₂²×C₃	#830	1	8	yes	—
151	C₂²×C₃	C₂²×C₃⋊C₈	1	4	yes	—
152	C₃×C₄	—	2	8	no	—
153	C₃×C₄	#830	1	8	yes	—
154	C₃×C₄	C₂×#208	1	4	yes	—
155	C₃×C₄	C₂×#217	1	4	yes	—
156	C₃²	#2113	1	1	yes	characteristic, complemented, Sylow
157	C₂³	C₂×C₃⋊(C₃⋊C₈)	1	2	yes	—
158	C₂³	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
159	Q₈	—	2	2	no	—
160	D₄	—	2	4	no	—
161	D₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	2	yes	—
165	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
166	C₂×C₄	#3748	1	2	yes	—
167	C₂×C₄	#3757	1	2	yes	—
168	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
169	C₈	—	2	2	no	—
170	C₈	#3757	1	2	yes	—
171	C₈	C₂×#638	1	1	yes	characteristic
172	C₈	C₂×#638	1	1	yes	characteristic
173	C₂×C₃	—	2	8	no	—
174	C₂×C₃	—	4	8	no	—
175	C₂×C₃	#5140	1	8	yes	—
176	C₂×C₃	C₂×#830	1	4	yes	—
177	C₂²	—	2	2	no	—
178	C₂²	—	2	4	no	—
179	C₂²	—	2	2	no	—
180	C₂²	#63657	1	2	yes	—
181	C₂²	C₂²×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
182	C₄	—	2	2	no	—
183	C₄	#63657	1	2	yes	—
184	C₄	C₂×#3748	1	1	yes	characteristic
185	C₄	C₂×#3757	1	1	yes	characteristic
186	C₃	#72004	1	4	yes	complemented
187	C₂	—	2	2	no	—
188	C₂	—	4	2	no	—
189	C₂	#10592839	1	2	yes	—
190	C₂	C₂×#63657	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×C₃⋊D₃
commutator, socle	C₂²×C₃²	C₂²×C₈
Fitting	C₂×C₃²×#131	C₂
prime radical	C₂×C₃²×#41	C₄
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	C₂×#131	C₃⋊D₃
hypocenter	C₃²	#2113
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. C₂×#63639	C₂	semidirect
3. C₂²×C₃⋊(C₃⋊C₁₆)	C₂	semidirect
4. #10589415	C₂	semidirect
156. C₃²	#2113	semidirect
186. C₃	#72004	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×#131	2
C₂×#63639	2
C₂²×C₃⋊(C₃⋊C₁₆)	2
#10589415	2
#72004	3