Subgroups — #11786002

Order 1152.

516 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#108	C₂	1	1	yes	maximal, characteristic
2	#10592805	C₂	1	1	yes	maximal, characteristic
3	#10592800	C₂	1	1	yes	maximal, characteristic
4	#10589425	C₂	1	2	yes	maximal
5	#10589423	C₂	1	2	yes	maximal
6	#70698	—	3	4	no	maximal
7	C₂×C₃²×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	#64135	C₂²	1	2	yes	—
10	#64135	C₂²	1	1	yes	characteristic
11	C₂×#3748	C₂²	1	1	yes	characteristic
12	C₃²×#108	—	2	1	no	—
13	C₃²×#108	—	2	1	no	—
14	#63645	—	2	1	no	—
15	#63644	—	2	1	no	—
16	#63642	C₄	1	2	yes	complemented
17	C₈×C₃⋊Dic₃	C₄	1	2	yes	complemented
18	C₂×C₃×#108	D₃	1	4	yes	—
19	#5106	—	3	4	no	—
20	#5101	—	3	4	no	—
21	#4559	—	3	8	no	—
22	#4557	—	3	8	no	—
23	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
24	C₃²×M₁₆	—	2	2	no	—
25	C₃²×M₁₆	—	2	2	no	—
26	C₂×C₃²×C₈	—	2	2	no	—
27	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
30	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
31	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
32	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
33	#3757	—	2	1	no	—
34	#3757	—	2	2	no	—
35	C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
36	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
37	C₄×C₃⋊Dic₃	—	2	2	no	—
38	C₄×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
39	#3748	—	2	1	no	—
40	#3748	—	2	1	no	—
41	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
42	#1833	—	9	1	no	Sylow
43	C₂×C₃×M₁₆	C₂×D₃	1	8	yes	—
44	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
45	#926	—	3	4	no	—
46	#926	—	3	8	no	—
47	C₂×#208	—	3	4	no	—
48	C₃×#108	—	2	4	no	—
49	C₃×#108	—	2	4	no	—
50	#818	—	6	4	no	—
51	#817	—	6	4	no	—
52	#815	—	3	8	no	—
53	C₈×Dic₃	—	3	8	no	—
54	C₂³×C₃²	#41	1	1	yes	characteristic
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	C₂×C₃²×C₄	#41	1	1	yes	characteristic
59	C₂×C₃²×C₄	#41	1	1	yes	characteristic
60	C₂×C₃²×C₄	#41	1	1	yes	characteristic
61	C₂×C₃⋊Dic₃	—	2	2	no	—
62	C₂×C₃⋊Dic₃	—	4	2	no	—
63	C₃²×C₈	—	4	2	no	—
64	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
65	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
66	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
67	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
68	C₃⋊(C₃⋊C₈)	—	2	2	no	—
69	C₂×#108	C₃⋊D₃	1	1	yes	characteristic
70	#427	—	9	1	no	—
71	#405	—	9	1	no	—
72	#344	—	9	2	no	—
73	H(3,ℤ/4)	—	9	2	no	—
74	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
75	C₃×M₁₆	—	2	8	no	—
76	C₃×M₁₆	—	2	8	no	—
77	C₂×C₃×C₈	—	2	8	no	—
78	C₂×C₃×C₈	Dic₆	1	4	yes	—
79	C₂×C₃×C₈	Dic₆	1	4	yes	—
80	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
81	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
82	C₂×C₃×C₈	#67	1	4	yes	—
83	C₂×C₃×C₈	#67	1	4	yes	—
84	#217	—	6	4	no	—
85	#217	—	6	8	no	—
86	C₄⋊Dic₃	—	6	4	no	—
87	C₄⋊Dic₃	—	6	8	no	—
88	C₄×Dic₃	—	3	8	no	—
89	C₄×Dic₃	—	6	8	no	—
90	#208	—	6	4	no	—
91	#208	—	6	4	no	—
92	C₂×C₃⋊C₈	—	3	8	no	—
93	C₂²×C₃²	—	4	1	no	—
94	C₂²×C₃²	#117	1	1	yes	characteristic
95	C₂²×C₃²	#124	1	1	yes	characteristic
96	C₂²×C₃²	#125	1	1	yes	characteristic
97	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₃²×C₄	#117	1	1	yes	characteristic
99	C₃²×C₄	#124	1	1	yes	characteristic
100	C₃²×C₄	#125	1	1	yes	characteristic
101	C₃⋊Dic₃	—	4	2	no	—
102	C₃⋊Dic₃	—	8	2	no	—
103	C₂×M₁₆	—	9	1	no	—
104	C₂×M₁₆	C₂×C₃⋊D₃	1	2	yes	—
105	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
106	#117	—	9	1	no	—
107	#117	—	9	2	no	—
108	#108	—	2	1	no	—
109	#108	—	2	1	no	—
110	#107	—	18	1	no	—
111	#106	—	18	1	no	—
112	#97	—	9	2	no	—
113	C₄×C₈	—	9	2	no	—
114	C₂³×C₃	#239	1	4	yes	—
115	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
116	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
117	C₂×C₃×C₄	#237	1	4	yes	—
118	C₂×C₃×C₄	#237	1	4	yes	—
119	C₂×C₃×C₄	#239	1	4	yes	—
120	C₂×C₃×C₄	C₂×#67	1	4	yes	—
121	C₂×Dic₃	—	6	8	no	—
122	C₂×Dic₃	—	12	8	no	—
123	C₃×C₈	—	4	8	no	—
124	C₃×C₈	Dic₃⋊C₄	1	4	yes	—
125	C₃×C₈	Dic₃⋊C₄	1	4	yes	—
126	C₃×C₈	Dic₃⋊C₄	1	4	yes	—
127	C₃×C₈	Dic₃⋊C₄	1	4	yes	—
128	C₃⋊C₈	—	6	8	no	—
129	C₂×C₃²	—	2	1	no	—
130	C₂×C₃²	—	2	1	no	—
131	C₂×C₃²	—	2	1	no	—
132	C₂×C₃²	#382	1	1	yes	characteristic
133	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
134	M₁₆	—	2	2	no	—
135	M₁₆	—	2	2	no	—
136	M₁₆	—	18	1	no	—
137	M₁₆	—	18	1	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	9	2	no	—
140	C₂×C₈	#634	1	1	yes	characteristic
141	C₂×C₈	#634	1	1	yes	characteristic
142	C₂×C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
143	C₂×C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
144	C₂×C₈	#638	1	1	yes	characteristic
145	C₂×C₈	#638	1	1	yes	characteristic
146	C₄⋊C₄	—	18	1	no	—
147	C₄⋊C₄	—	18	2	no	—
148	C₂²⋊C₄	—	18	1	no	—
149	C₂²⋊C₄	—	18	2	no	—
150	C₄²	—	9	2	no	—
151	C₄²	—	18	2	no	—
152	C₂²×C₃	—	4	4	no	—
153	C₂²×C₃	#890	1	4	yes	—
154	C₂²×C₃	#892	1	4	yes	—
155	C₂²×C₃	#947	1	4	yes	—
156	C₃×C₄	#890	1	4	yes	—
157	C₃×C₄	#892	1	4	yes	—
158	C₃×C₄	C₂×Dic₃⋊C₄	1	4	yes	—
159	C₃×C₄	#936	1	4	yes	—
160	Dic₃	—	12	8	no	—
161	Dic₃	—	24	8	no	—
162	C₃²	#1833	1	1	yes	characteristic, complemented, Sylow
163	C₂³	#3832	1	1	yes	characteristic
164	C₂×C₄	—	18	2	no	—
165	C₂×C₄	—	36	2	no	—
166	C₂×C₄	C₂×#634	1	1	yes	characteristic
167	C₂×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
168	C₂×C₄	#3830	1	1	yes	characteristic
169	C₂×C₄	#3830	1	1	yes	characteristic
170	C₂×C₄	#3832	1	1	yes	characteristic
171	C₂×C₄	C₂×#638	1	1	yes	characteristic
172	C₈	—	4	2	no	—
173	C₈	—	18	2	no	—
174	C₈	#3750	1	1	yes	characteristic
175	C₈	#3750	1	1	yes	characteristic
176	C₈	#3750	1	1	yes	characteristic
177	C₈	#3750	1	1	yes	characteristic
178	C₂×C₃	—	2	4	no	—
179	C₂×C₃	—	2	4	no	—
180	C₂×C₃	—	2	4	no	—
181	C₂×C₃	#4977	1	4	yes	—
182	C₂²	—	4	1	no	—
183	C₂²	#64099	1	1	yes	characteristic
184	C₂²	#64101	1	1	yes	characteristic
185	C₂²	#64156	1	1	yes	characteristic
186	C₄	—	36	2	no	—
187	C₄	—	72	2	no	—
188	C₄	#64099	1	1	yes	characteristic
189	C₄	#64101	1	1	yes	characteristic
190	C₄	C₂×#3750	1	1	yes	characteristic
191	C₄	#64145	1	1	yes	characteristic
192	C₃	#70698	1	4	yes	complemented
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	—	2	1	no	—
196	C₂	#10592676	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#64101
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃²×#108	C₂
socle	C₂×C₃²	#382
prime radical	C₂³×C₃²	#41
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂×#108	C₃⋊D₃
hypocenter	C₃²	#1833
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
16. #63642	C₄	semidirect
17. C₈×C₃⋊Dic₃	C₄	semidirect
162. C₃²	#1833	semidirect
192. C₃	#70698	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×#108	2
#10592805	2
#10592800	2
#10589425	2
#10589423	2
#70698	3