Subgroups — #11790130

Order 1152.

744 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#63645	C₂	1	1	yes	maximal, characteristic
3	C₂×#63645	C₂	1	1	yes	maximal, characteristic
4	#10592627	C₂	1	2	yes	maximal
5	#10589417	C₂	1	2	yes	maximal
6	#69770	—	3	4	no	maximal
7	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
11	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
12	C₂×#3750	—	2	2	no	—
13	C₃²×C₄×C₈	—	2	2	no	—
14	#63658	—	2	2	no	—
15	#63645	—	2	2	no	—
16	#63645	C₄	1	4	yes	complemented
17	C₂×C₃×C₄×C₈	D₃	1	4	yes	—
18	C₂×#818	—	3	4	no	—
19	C₂×#818	—	3	4	no	—
20	#4928	—	3	8	no	—
21	#4551	—	3	8	no	—
22	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
23	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
24	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
25	C₂²×C₃⋊Dic₃	—	2	2	no	—
26	C₂²×C₃⋊Dic₃	—	2	2	no	—
27	C₂×C₃²×C₈	—	2	1	no	—
28	C₂×C₃²×C₈	—	2	1	no	—
29	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
30	C₂×C₃²×C₈	D₄	1	2	yes	—
31	C₂×C₃²×C₈	Q₈	1	2	yes	—
32	C₂×C₃²×C₈	Q₈	1	2	yes	—
33	C₃²×C₄²	—	2	1	no	—
34	C₃²×C₄²	—	2	1	no	—
35	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
36	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
37	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
38	C₄⋊(C₃⋊Dic₃)	C₂×C₄	1	4	yes	—
39	#3750	—	4	4	no	—
40	#1942	—	9	1	no	Sylow
41	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
42	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
43	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
44	C₂×C₄⋊Dic₃	—	3	8	no	—
45	C₂×C₄⋊Dic₃	—	3	8	no	—
46	C₂×Dic₃⋊C₄	—	6	8	no	—
47	C₃×C₄×C₈	—	2	8	no	—
48	C₃⋊#95	—	6	8	no	—
49	#818	—	3	16	no	—
50	#818	—	6	8	no	—
51	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
52	C₂×C₃²×C₄	—	2	1	no	—
53	C₂×C₃²×C₄	—	2	1	no	—
54	C₂×C₃²×C₄	—	2	1	no	—
55	C₂×C₃²×C₄	—	2	1	no	—
56	C₂×C₃²×C₄	D₈	1	2	yes	—
57	C₂×C₃²×C₄	Q₁₆	1	2	yes	—
58	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₃²×C₄	#41	1	1	yes	characteristic
63	C₂×C₃²×C₄	#41	1	1	yes	characteristic
64	C₂×C₃⋊Dic₃	—	2	4	no	—
65	C₂×C₃⋊Dic₃	—	4	2	no	—
66	C₂×C₃⋊Dic₃	—	4	2	no	—
67	C₂×C₃⋊Dic₃	—	4	2	no	—
68	C₂×C₃⋊Dic₃	—	4	2	no	—
69	C₂×C₃⋊Dic₃	—	4	2	no	—
70	C₃²×C₈	—	2	2	no	—
71	C₃²×C₈	C₄⋊C₄	1	4	yes	complemented
72	C₂×#107	—	9	1	no	—
73	C₂×#107	—	9	1	no	—
74	C₂×C₄×C₈	C₃⋊D₃	1	1	yes	characteristic
75	#389	—	9	2	no	—
76	#340	—	9	2	no	—
77	C₂²×C₃×C₄	D₁₂	1	4	yes	—
78	C₂²×C₃×C₄	D₁₂	1	4	yes	—
79	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
80	C₂²×Dic₃	—	6	8	no	—
81	C₂²×Dic₃	—	6	8	no	—
82	C₂×C₃×C₈	—	2	4	no	—
83	C₂×C₃×C₈	—	2	4	no	—
84	C₂×C₃×C₈	Dic₆	1	8	yes	—
85	C₂×C₃×C₈	Dic₆	1	8	yes	—
86	C₂×C₃×C₈	C₄×D₃	1	8	yes	—
87	C₂×C₃×C₈	#67	1	8	yes	—
88	C₃×C₄²	—	2	4	no	—
89	C₃×C₄²	—	2	4	no	—
90	C₄⋊Dic₃	—	3	16	no	—
91	C₄⋊Dic₃	—	6	8	no	—
92	C₄⋊Dic₃	—	6	8	no	—
93	C₄⋊Dic₃	—	6	8	no	—
94	Dic₃⋊C₄	—	12	16	no	—
95	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
96	C₂²×C₃²	#121	1	1	yes	characteristic
97	C₂²×C₃²	C₄⋊Q₈	1	1	yes	characteristic
98	C₂²×C₃²	C₂×D₈	1	1	yes	characteristic
99	C₂²×C₃²	C₂×Q₁₆	1	1	yes	characteristic
100	C₂²×C₃²	#135	1	1	yes	characteristic
101	C₂²×C₃²	#135	1	1	yes	characteristic
102	C₃²×C₄	—	2	2	no	—
103	C₃²×C₄	—	2	2	no	—
104	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
106	C₃²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
107	C₃²×C₄	#125	1	1	yes	characteristic
108	C₃⋊Dic₃	—	4	4	no	—
109	C₃⋊Dic₃	—	8	2	no	—
110	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
111	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
112	C₂×C₄⋊C₄	—	9	2	no	—
113	C₂×C₄⋊C₄	—	9	2	no	—
114	C₂×C₄⋊C₄	—	18	2	no	—
115	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
116	#107	—	9	4	no	—
117	#107	—	18	2	no	—
118	C₄×C₈	—	2	2	no	—
119	#95	—	18	2	no	—
120	C₂³×C₃	C₂×D₁₂	1	4	yes	—
121	C₂×C₃×C₄	—	2	4	no	—
122	C₂×C₃×C₄	—	2	4	no	—
123	C₂×C₃×C₄	—	2	4	no	—
124	C₂×C₃×C₄	—	2	4	no	—
125	C₂×C₃×C₄	D₂₄	1	8	yes	—
126	C₂×C₃×C₄	Dic₁₂	1	8	yes	—
127	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
128	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
129	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
130	C₂×C₃×C₄	#235	1	4	yes	—
131	C₂×C₃×C₄	#235	1	4	yes	—
132	C₂×C₃×C₄	C₂×#67	1	4	yes	—
133	C₂×Dic₃	—	6	16	no	—
134	C₂×Dic₃	—	12	8	no	—
135	C₂×Dic₃	—	12	8	no	—
136	C₂×Dic₃	—	12	8	no	—
137	C₂×Dic₃	—	12	8	no	—
138	C₂×Dic₃	—	12	8	no	—
139	C₃×C₈	—	2	8	no	—
140	C₃×C₈	Dic₃⋊C₄	1	16	yes	complemented
141	C₂×C₃²	#389	1	1	yes	characteristic
142	C₂×C₃²	C₄×D₈	1	1	yes	characteristic
143	C₂×C₃²	C₄×Q₁₆	1	1	yes	characteristic
144	C₂×C₃²	#466	1	1	yes	characteristic
145	C₂×C₃²	#467	1	1	yes	characteristic
146	C₂×C₃²	#499	1	1	yes	characteristic
147	C₂×C₃²	#500	1	1	yes	characteristic
148	C₂²×C₄	—	18	2	no	—
149	C₂²×C₄	—	18	2	no	—
150	C₂²×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
151	C₂²×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
152	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	#634	1	2	yes	—
156	C₂×C₈	#634	1	2	yes	—
157	C₂×C₈	C₄×C₃⋊D₃	1	2	yes	—
158	C₂×C₈	#638	1	2	yes	—
159	C₄⋊C₄	—	9	4	no	—
160	C₄⋊C₄	—	18	2	no	—
161	C₄⋊C₄	—	18	2	no	—
162	C₄⋊C₄	—	18	2	no	—
163	C₄⋊C₄	—	36	4	no	—
164	C₄²	—	2	1	no	—
165	C₄²	—	2	1	no	—
166	C₂²×C₃	#869	1	4	yes	—
167	C₂²×C₃	C₄×D₁₂	1	4	yes	—
168	C₂²×C₃	C₂×D₂₄	1	4	yes	—
169	C₂²×C₃	#904	1	4	yes	—
170	C₂²×C₃	#904	1	4	yes	—
171	C₂²×C₃	C₂×Dic₁₂	1	4	yes	—
172	C₂²×C₃	#930	1	4	yes	—
173	C₃×C₄	—	2	8	no	—
174	C₃×C₄	—	2	8	no	—
175	C₃×C₄	C₄×Dic₆	1	4	yes	—
176	C₃×C₄	#870	1	4	yes	—
177	C₃×C₄	C₂×Dic₃⋊C₄	1	4	yes	—
178	C₃×C₄	#924	1	4	yes	—
179	Dic₃	—	12	16	no	—
180	Dic₃	—	24	8	no	—
181	C₃²	#1942	1	1	yes	characteristic, complemented, Sylow
182	C₂³	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	18	4	no	—
188	C₂×C₄	—	36	2	no	—
189	C₂×C₄	—	36	2	no	—
190	C₂×C₄	—	36	2	no	—
191	C₂×C₄	—	36	2	no	—
192	C₂×C₄	—	36	2	no	—
193	C₂×C₄	#3745	1	2	yes	—
194	C₂×C₄	#3746	1	2	yes	—
195	C₂×C₄	C₂×#634	1	1	yes	characteristic
196	C₂×C₄	C₂×#634	1	1	yes	characteristic
197	C₂×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
198	C₂×C₄	#3828	1	1	yes	characteristic
199	C₂×C₄	#3828	1	1	yes	characteristic
200	C₂×C₄	C₂×#638	1	1	yes	characteristic
201	C₈	—	2	2	no	—
202	C₈	#3750	1	4	yes	complemented
203	C₂×C₃	#4682	1	4	yes	—
204	C₂×C₃	#4683	1	4	yes	—
205	C₂×C₃	C₄×D₂₄	1	4	yes	—
206	C₂×C₃	C₄×Dic₁₂	1	4	yes	—
207	C₂×C₃	#4928	1	4	yes	—
208	C₂×C₃	#5115	1	4	yes	—
209	C₂×C₃	#5116	1	4	yes	—
210	C₂²	#64078	1	1	yes	characteristic
211	C₂²	C₄×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
212	C₂²	C₂×#3745	1	1	yes	characteristic
213	C₂²	#64113	1	1	yes	characteristic
214	C₂²	#64113	1	1	yes	characteristic
215	C₂²	C₂×#3746	1	1	yes	characteristic
216	C₂²	#64139	1	1	yes	characteristic
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	36	4	no	—
220	C₄	—	72	2	no	—
221	C₄	C₄×#634	1	1	yes	characteristic
222	C₄	#64079	1	1	yes	characteristic
223	C₄	C₂×#3750	1	1	yes	characteristic
224	C₄	#64133	1	1	yes	characteristic
225	C₃	#69770	1	4	yes	complemented
226	C₂	#10592381	1	1	yes	characteristic
227	C₂	#10592382	1	1	yes	characteristic
228	C₂	C₄×#3745	1	1	yes	characteristic
229	C₂	C₄×#3746	1	1	yes	characteristic
230	C₂	#10592627	1	1	yes	characteristic
231	C₂	#10592814	1	1	yes	characteristic
232	C₂	#10592815	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊(C₃⋊D₃)
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃²×C₄×C₈	C₂
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂×C₄×C₈	C₃⋊D₃
hypocenter	C₃²	#1942
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
16. #63645	C₄	semidirect
71. C₃²×C₈	C₄⋊C₄	semidirect
140. C₃×C₈	Dic₃⋊C₄	semidirect
181. C₃²	#1942	semidirect
202. C₈	#3750	semidirect
225. C₃	#69770	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₄×C₈	2
C₂×#63645	2
#10592627	2
#10589417	2
#69770	3