Subgroups — #11776865

Order 1152.

552 conjugacy classes of subgroups.

The subgroup lattice has 183 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#423	C₂	1	1	yes	maximal, characteristic
2	#10592623	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#10589346	C₂	1	4	yes	maximal, complemented
5	#70490	—	3	4	no	maximal
6	C₂×C₃²×M₁₆	—	2	1	no	—
7	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
8	C₂×#3748	—	2	1	no	—
9	C₂²×C₃⋊(C₃⋊C₈)	C₄	1	2	yes	—
10	C₃²×C₄⋊C₈	—	2	1	no	—
11	C₃²×C₄⋊C₈	C₂²	1	2	yes	—
12	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
13	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	2	yes	—
14	C₄×C₃⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
15	C₄×C₃⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
16	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
17	C₃×#423	D₃	1	4	yes	—
18	#4924	—	3	4	no	—
19	C₂×C₄×C₃⋊C₈	—	3	4	no	—
20	#4480	—	3	16	no	—
21	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
23	C₂²×C₃²×C₄	Q₈	1	1	yes	characteristic
24	C₃²×M₁₆	—	4	2	no	—
25	C₂×C₃²×C₈	—	2	2	no	—
26	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
27	C₃²×C₄²	D₄	1	1	yes	characteristic
28	C₃²×C₄²	Q₈	1	1	yes	characteristic
29	C₃²×C₄²	C₂³	1	1	yes	characteristic
30	#3748	—	4	2	no	—
31	C₂×C₃⋊(C₃⋊C₈)	—	2	4	no	—
32	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
33	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
34	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
35	#1566	—	9	1	no	Sylow
36	C₂×C₃×M₁₆	—	2	4	no	—
37	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
38	C₂×#208	—	6	4	no	—
39	C₂²×C₃⋊C₈	—	3	8	no	—
40	C₃×C₄⋊C₈	—	2	4	no	—
41	C₃×C₄⋊C₈	C₂×D₃	1	8	yes	—
42	C₄⋊(C₃⋊C₈)	—	3	8	no	—
43	C₄⋊(C₃⋊C₈)	—	6	4	no	—
44	C₄×C₃⋊C₈	—	3	4	no	—
45	C₄×C₃⋊C₈	—	3	4	no	—
46	C₄×C₃⋊C₈	—	3	8	no	—
47	C₂³×C₃²	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₃²×C₄	—	2	2	no	—
49	C₂×C₃²×C₄	—	2	2	no	—
50	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₃²×C₄	SD₁₆	1	2	yes	—
54	C₂×C₃²×C₄	SD₁₆	1	2	yes	—
55	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
57	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
58	C₃²×C₈	—	4	2	no	—
59	C₃⋊(C₃⋊C₈)	—	2	2	no	—
60	C₃⋊(C₃⋊C₈)	—	4	2	no	—
61	C₃⋊(C₃⋊C₈)	M₁₆	1	4	yes	complemented
62	#423	—	9	1	no	—
63	#423	C₃⋊D₃	1	1	yes	characteristic
64	C₂×C₄×C₈	—	9	1	no	—
65	#334	—	9	4	no	—
66	C₂²×C₃×C₄	Dic₆	1	4	yes	—
67	C₂²×C₃×C₄	C₄×D₃	1	4	yes	—
68	C₂²×C₃×C₄	#67	1	4	yes	—
69	C₃×M₁₆	—	4	8	no	—
70	C₂×C₃×C₈	—	2	8	no	—
71	C₃×C₄²	Dic₆	1	4	yes	—
72	C₃×C₄²	C₄×D₃	1	4	yes	—
73	C₃×C₄²	#67	1	4	yes	—
74	C₃×C₄²	C₂²×D₃	1	4	yes	—
75	#208	—	12	8	no	—
76	C₂×C₃⋊C₈	—	3	8	no	—
77	C₂×C₃⋊C₈	—	3	8	no	—
78	C₂×C₃⋊C₈	—	6	16	no	—
79	C₂×C₃⋊C₈	—	6	8	no	—
80	C₂²×C₃²	—	2	1	no	—
81	C₂²×C₃²	—	2	1	no	—
82	C₂²×C₃²	#106	1	2	yes	—
83	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₃²×C₄	—	2	1	no	—
85	C₃²×C₄	—	2	1	no	—
86	C₃²×C₄	#106	1	2	yes	—
87	C₃²×C₄	#108	1	2	yes	—
88	C₃²×C₄	C₂×M₁₆	1	2	yes	—
89	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
90	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
91	C₂×M₁₆	—	2	1	no	—
92	C₂×M₁₆	—	18	1	no	—
93	C₂²×C₈	—	9	2	no	—
94	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
95	C₄⋊C₈	—	2	1	no	—
96	C₄⋊C₈	—	9	2	no	—
97	C₄⋊C₈	—	18	1	no	—
98	C₄⋊C₈	C₂×C₃⋊D₃	1	2	yes	—
99	C₄×C₈	—	9	1	no	—
100	C₄×C₈	—	9	1	no	—
101	C₄×C₈	—	9	2	no	—
102	C₂³×C₃	Dic₃⋊C₄	1	4	yes	—
103	C₂×C₃×C₄	—	2	8	no	—
104	C₂×C₃×C₄	—	2	8	no	—
105	C₂×C₃×C₄	Dic₃⋊C₄	1	4	yes	—
106	C₂×C₃×C₄	Dic₃⋊C₄	1	4	yes	—
107	C₂×C₃×C₄	Dic₃⋊C₄	1	4	yes	—
108	C₂×C₃×C₄	#214	1	8	yes	—
109	C₂×C₃×C₄	#215	1	8	yes	—
110	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
111	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
112	C₂×C₃×C₄	C₂×#67	1	4	yes	—
113	C₃×C₈	—	4	8	no	—
114	C₃⋊C₈	—	3	16	no	—
115	C₃⋊C₈	—	6	8	no	—
116	C₃⋊C₈	—	12	8	no	—
117	C₂×C₃²	—	2	2	no	—
118	C₂×C₃²	#423	1	1	yes	characteristic
119	C₂×C₃²	C₂×#106	1	1	yes	characteristic
120	C₂×C₃²	C₂×#108	1	1	yes	characteristic
121	C₂²×C₄	#634	1	1	yes	characteristic
122	C₂²×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
123	C₂²×C₄	#638	1	1	yes	characteristic
124	M₁₆	—	4	2	no	—
125	M₁₆	—	36	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	9	2	no	—
128	C₂×C₈	—	9	2	no	—
129	C₂×C₈	—	18	4	no	—
130	C₂×C₈	—	18	2	no	—
131	C₄²	#634	1	1	yes	characteristic
132	C₄²	C₄×C₃⋊D₃	1	1	yes	characteristic
133	C₄²	#638	1	1	yes	characteristic
134	C₄²	C₂²×C₃⋊D₃	1	1	yes	characteristic
135	C₂²×C₃	—	2	4	no	—
136	C₂²×C₃	—	2	4	no	—
137	C₂²×C₃	#808	1	8	yes	—
138	C₂²×C₃	C₂×Dic₃⋊C₄	1	4	yes	—
139	C₃×C₄	—	2	4	no	—
140	C₃×C₄	—	2	4	no	—
141	C₃×C₄	#808	1	8	yes	—
142	C₃×C₄	#822	1	8	yes	—
143	C₃×C₄	D₃×M₁₆	1	8	yes	—
144	C₃×C₄	C₂×#214	1	4	yes	—
145	C₃×C₄	C₂×#215	1	4	yes	—
146	C₃²	#1566	1	1	yes	characteristic, complemented, Sylow
147	C₂³	#3750	1	1	yes	characteristic
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	#3750	1	1	yes	characteristic
151	C₂×C₄	#3750	1	1	yes	characteristic
152	C₂×C₄	#3750	1	1	yes	characteristic
153	C₂×C₄	#3754	1	2	yes	—
154	C₂×C₄	#3755	1	2	yes	—
155	C₂×C₄	C₂×#634	1	1	yes	characteristic
156	C₂×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
157	C₂×C₄	C₂×#638	1	1	yes	characteristic
158	C₈	—	4	2	no	—
159	C₈	—	9	4	no	—
160	C₈	—	18	2	no	—
161	C₈	—	36	2	no	—
162	C₂×C₃	—	2	8	no	—
163	C₂×C₃	C₂×#808	1	4	yes	—
164	C₂×C₃	#5118	1	4	yes	—
165	C₂×C₃	C₂×#822	1	4	yes	—
166	C₂²	—	2	1	no	—
167	C₂²	—	2	1	no	—
168	C₂²	#63635	1	2	yes	—
169	C₂²	C₂×#3750	1	1	yes	characteristic
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	#63635	1	2	yes	—
173	C₄	#63649	1	2	yes	—
174	C₄	M₁₆×C₃⋊D₃	1	2	yes	—
175	C₄	C₂×#3754	1	1	yes	characteristic
176	C₄	C₂×#3755	1	1	yes	characteristic
177	C₃	#70490	1	4	yes	complemented
178	C₂	—	2	2	no	—
179	C₂	C₂×#63635	1	1	yes	characteristic
180	C₂	#10592817	1	1	yes	characteristic
181	C₂	C₂×#63649	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#638
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₄²	C₂²×C₃⋊D₃
Fitting	C₃²×#423	C₂
socle	C₂²×C₃²	C₂×C₄⋊C₄
prime radical	C₂³×C₃²	C₄⋊C₄
cosocle	C₃²×C₄²	C₂³
hypercenter	#423	C₃⋊D₃
hypocenter	C₃²	#1566
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

Complements

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

N	H ≅ G/N	type
4. #10589346	C₂	semidirect
61. C₃⋊(C₃⋊C₈)	M₁₆	semidirect
146. C₃²	#1566	semidirect
177. C₃	#70490	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#423	2
#10592623	2
C₂×C₄×C₃⋊(C₃⋊C₈)	2
#10589346	2
#70490	3