Subgroups — #11784406

Order 1152.

672 conjugacy classes of subgroups.

The subgroup lattice has 189 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	#10592821	C₂	1	2	yes	maximal
3	#10592805	C₂	1	1	yes	maximal, characteristic
4	C₂×#63642	C₂	1	1	yes	maximal, characteristic
5	#10589421	C₂	1	2	yes	maximal
6	#70703	—	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	C₂×C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
11	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
12	C₃²×#108	C₄	1	4	yes	complemented
13	#63645	C₄	1	2	yes	complemented
14	#63644	C₄	1	2	yes	complemented
15	#63642	—	2	2	no	—
16	#63642	C₄	1	4	yes	complemented
17	C₈×C₃⋊Dic₃	—	2	2	no	—
18	C₂×C₃×#108	D₃	1	4	yes	—
19	#5122	—	3	8	no	—
20	#5106	—	3	4	no	—
21	C₂×#815	—	3	4	no	—
22	#4555	—	3	8	no	—
23	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
24	C₂²×C₃⋊Dic₃	—	2	1	no	—
25	C₃²×M₁₆	—	2	2	no	—
26	C₃²×M₁₆	C₂×C₄	1	4	yes	—
27	C₂×C₃²×C₈	—	2	2	no	—
28	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
29	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
32	#3757	—	2	2	no	—
33	C₄⋊(C₃⋊Dic₃)	C₂×C₄	1	4	yes	—
34	C₄×C₃⋊Dic₃	—	2	2	no	—
35	C₄×C₃⋊Dic₃	—	2	1	no	—
36	C₄×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
37	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
38	C₂×C₃⋊(C₃⋊C₈)	D₄	1	2	yes	—
39	C₂×C₃⋊(C₃⋊C₈)	Q₈	1	2	yes	—
40	#1773	—	9	1	no	Sylow
41	C₂×C₃×M₁₆	C₂×D₃	1	8	yes	—
42	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
43	#926	—	3	8	no	—
44	C₂×C₄×Dic₃	—	3	4	no	—
45	C₂²×C₃⋊C₈	—	3	4	no	—
46	C₃×#108	Dic₃	1	16	yes	complemented
47	#818	—	3	8	no	—
48	#817	—	3	8	no	—
49	#815	—	3	16	no	—
50	#815	—	6	8	no	—
51	C₈×Dic₃	—	6	8	no	—
52	C₂³×C₃²	#41	1	1	yes	characteristic
53	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₃²×C₄	C₂²×C₄	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×C₃⋊Dic₃	—	2	2	no	—
60	C₂×C₃⋊Dic₃	—	2	4	no	—
61	C₂×C₃⋊Dic₃	—	4	2	no	—
62	C₃²×C₈	—	2	4	no	—
63	C₃²×C₈	C₄²	1	2	yes	—
64	C₃²×C₈	C₄²	1	2	yes	—
65	C₃⋊(C₃⋊C₈)	C₄⋊C₄	1	4	yes	—
66	C₂×#108	C₃⋊D₃	1	1	yes	characteristic
67	#427	—	9	1	no	—
68	#405	—	9	2	no	—
69	C₂×#97	—	9	1	no	—
70	#339	—	9	2	no	—
71	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
72	C₂²×Dic₃	—	6	4	no	—
73	C₃×M₁₆	—	2	8	no	—
74	C₃×M₁₆	C₂×Dic₃	1	16	yes	—
75	C₂×C₃×C₈	—	2	8	no	—
76	C₂×C₃×C₈	C₄×D₃	1	8	yes	—
77	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
78	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
79	C₂×C₃×C₈	C₂×Dic₃	1	8	yes	—
80	#217	—	6	8	no	—
81	C₄⋊Dic₃	—	3	16	no	—
82	C₄×Dic₃	—	3	8	no	—
83	C₄×Dic₃	—	6	8	no	—
84	C₄×Dic₃	—	6	4	no	—
85	C₂×C₃⋊C₈	—	3	8	no	—
86	C₂×C₃⋊C₈	—	3	8	no	—
87	C₂×C₃⋊C₈	—	3	8	no	—
88	C₂²×C₃²	—	2	2	no	—
89	C₂²×C₃²	#117	1	1	yes	characteristic
90	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
91	C₂²×C₃²	C₄×Q₈	1	1	yes	characteristic
92	C₃²×C₄	C₂×C₄²	1	1	yes	characteristic
93	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
95	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
96	C₃⋊Dic₃	—	4	2	no	—
97	C₃⋊Dic₃	—	4	4	no	—
98	C₂×M₁₆	C₂×C₃⋊D₃	1	2	yes	—
99	C₂²×C₈	—	9	1	no	—
100	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
101	#117	—	9	2	no	—
102	C₂×C₄²	—	9	1	no	—
103	#108	C₃⋊Dic₃	1	4	yes	complemented
104	#107	—	9	2	no	—
105	#106	—	9	2	no	—
106	#97	—	9	4	no	—
107	#97	—	18	2	no	—
108	C₄×C₈	—	18	2	no	—
109	C₂³×C₃	#239	1	4	yes	—
110	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
111	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
112	C₂×C₃×C₄	D₃×D₄	1	4	yes	—
113	C₂×C₃×C₄	#237	1	4	yes	—
114	C₂×C₃×C₄	D₃×Q₈	1	4	yes	—
115	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
116	C₂×Dic₃	—	6	8	no	—
117	C₂×Dic₃	—	6	16	no	—
118	C₂×Dic₃	—	12	8	no	—
119	C₃×C₈	—	2	16	no	—
120	C₃×C₈	C₄×Dic₃	1	8	yes	—
121	C₃×C₈	C₄×Dic₃	1	8	yes	—
122	C₃⋊C₈	—	3	16	no	—
123	C₂×C₃²	—	2	1	no	—
124	C₂×C₃²	—	2	1	no	—
125	C₂×C₃²	C₄×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₃²	#444	1	2	yes	—
127	C₂²×C₄	—	18	1	no	—
128	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
129	M₁₆	—	2	2	no	—
130	M₁₆	C₂×C₃⋊Dic₃	1	4	yes	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	9	2	no	—
133	C₂×C₈	—	9	2	no	—
134	C₂×C₈	—	9	2	no	—
135	C₂×C₈	C₄×C₃⋊D₃	1	2	yes	—
136	C₂×C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
137	C₂×C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
138	C₂×C₈	C₂×C₃⋊Dic₃	1	2	yes	—
139	C₄⋊C₄	—	9	4	no	—
140	C₂²⋊C₄	—	18	2	no	—
141	C₄²	—	9	2	no	—
142	C₄²	—	18	2	no	—
143	C₄²	—	18	1	no	—
144	C₂²×C₃	—	2	8	no	—
145	C₂²×C₃	#892	1	4	yes	—
146	C₂²×C₃	#893	1	4	yes	—
147	C₂²×C₃	Q₈×Dic₃	1	4	yes	—
148	C₃×C₄	#887	1	4	yes	—
149	C₃×C₄	D₃×C₄⋊C₄	1	4	yes	—
150	C₃×C₄	C₂×C₄×Dic₃	1	4	yes	—
151	C₃×C₄	D₄×Dic₃	1	4	yes	—
152	Dic₃	—	12	8	no	—
153	Dic₃	—	12	16	no	—
154	C₃²	#1773	1	1	yes	characteristic, complemented, Sylow
155	C₂³	#3832	1	1	yes	characteristic
156	C₂×C₄	—	18	2	no	—
157	C₂×C₄	—	18	4	no	—
158	C₂×C₄	—	36	2	no	—
159	C₂×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
160	C₂×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
161	C₂×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
162	C₂×C₄	#3830	1	1	yes	characteristic
163	C₂×C₄	Q₈×C₃⋊D₃	1	1	yes	characteristic
164	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
165	C₈	—	2	4	no	—
166	C₈	—	9	4	no	—
167	C₈	C₄×C₃⋊Dic₃	1	2	yes	—
168	C₈	C₄×C₃⋊Dic₃	1	2	yes	—
169	C₂×C₃	—	2	4	no	—
170	C₂×C₃	—	2	4	no	—
171	C₂×C₃	#4894	1	8	yes	—
172	C₂×C₃	Dic₃×C₄⋊C₄	1	4	yes	—
173	C₂²	—	2	2	no	—
174	C₂²	#64101	1	1	yes	characteristic
175	C₂²	#64102	1	1	yes	characteristic
176	C₂²	Q₈×C₃⋊Dic₃	1	1	yes	characteristic
177	C₄	—	36	2	no	—
178	C₄	—	36	4	no	—
179	C₄	#64096	1	1	yes	characteristic
180	C₄	C₄⋊C₄×C₃⋊D₃	1	1	yes	characteristic
181	C₄	C₂×C₄×C₃⋊Dic₃	1	1	yes	characteristic
182	C₄	D₄×C₃⋊Dic₃	1	1	yes	characteristic
183	C₃	#70703	1	4	yes	complemented
184	C₂	—	2	1	no	—
185	C₂	—	2	1	no	—
186	C₂	#10592593	1	2	yes	—
187	C₂	C₄⋊C₄×C₃⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×C₃⋊D₃
commutator	C₃²×C₄	C₂×C₄²
Frattini	C₂²×C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃²×#108	C₂
socle	C₂²×C₃²	C₄×D₄
prime radical	C₂³×C₃²	#41
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂×#108	C₃⋊D₃
hypocenter	C₃²	#1773
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
12. C₃²×#108	C₄	semidirect
13. #63645	C₄	semidirect
14. #63644	C₄	semidirect
16. #63642	C₄	semidirect
46. C₃×#108	Dic₃	semidirect
103. #108	C₃⋊Dic₃	semidirect
154. C₃²	#1773	semidirect
183. C₃	#70703	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
#10592821	2
#10592805	2
C₂×#63642	2
#10589421	2
#70703	3