Subgroups — #11770782

Order 1152.

624 conjugacy classes of subgroups.

The subgroup lattice has 213 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	C₄×C₄⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#67704	—	3	4	no	maximal
5	C₂²×C₃²×C₈	—	2	1	no	—
6	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊(C₃⋊Dic₃)	C₄	1	2	yes	—
8	C₂×C₄×C₃⋊Dic₃	—	2	1	no	—
9	C₂²×C₃⋊(C₃⋊C₈)	—	2	1	no	—
10	C₃²×C₄⋊C₈	—	2	1	no	—
11	C₃²×C₄⋊C₈	C₄	1	2	yes	complemented
12	#63658	—	2	1	no	—
13	C₄×C₃⋊(C₃⋊C₈)	—	2	1	no	—
14	C₄×C₃⋊(C₃⋊C₈)	C₄	1	2	yes	complemented
15	C₂×C₃×C₄⋊C₈	D₃	1	4	yes	—
16	C₄×C₄⋊Dic₃	—	3	4	no	—
17	C₂×C₄×C₃⋊C₈	—	3	4	no	—
18	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
20	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
21	C₂²×C₃⋊Dic₃	—	2	2	no	—
22	C₂×C₃²×C₈	—	2	2	no	—
23	C₂×C₃²×C₈	—	4	1	no	—
24	C₂×C₃²×C₈	—	4	1	no	—
25	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
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	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
30	C₄⋊(C₃⋊Dic₃)	C₈	1	4	yes	complemented
31	C₄×C₃⋊Dic₃	—	4	1	no	—
32	C₄×C₃⋊Dic₃	—	4	1	no	—
33	C₂×C₃⋊(C₃⋊C₈)	—	2	4	no	—
34	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
35	#1279	—	9	1	no	Sylow
36	C₂²×C₃×C₈	—	2	4	no	—
37	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
38	C₂×C₄⋊Dic₃	—	3	8	no	—
39	C₂×C₄×Dic₃	—	6	4	no	—
40	C₂²×C₃⋊C₈	—	6	4	no	—
41	C₃×C₄⋊C₈	—	2	4	no	—
42	C₃×C₄⋊C₈	Dic₃	1	8	yes	complemented
43	C₃⋊#95	—	6	4	no	—
44	C₄×C₃⋊C₈	—	3	8	no	—
45	C₄×C₃⋊C₈	—	6	4	no	—
46	C₂³×C₃²	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₃²×C₄	—	2	2	no	—
48	C₂×C₃²×C₄	C₄²	1	1	yes	characteristic
49	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₃²×C₄	C₂²⋊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	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
55	C₂×C₃²×C₄	M₁₆	1	2	yes	—
56	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
57	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
58	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
59	C₂×C₃²×C₄	Q₁₆	1	1	yes	characteristic
60	C₂×C₃⋊Dic₃	—	2	4	no	—
61	C₂×C₃⋊Dic₃	—	4	2	no	—
62	C₂×C₃⋊Dic₃	—	4	2	no	—
63	C₃²×C₈	—	4	2	no	—
64	C₃⋊(C₃⋊C₈)	—	2	4	no	—
65	C₂×C₄⋊C₈	C₃⋊D₃	1	1	yes	characteristic
66	C₂×C₄×C₈	—	9	1	no	—
67	C₄×C₄⋊C₄	—	9	1	no	—
68	C₂²×C₃×C₄	C₄×D₃	1	4	yes	—
69	C₂²×C₃×C₄	D₁₂	1	4	yes	—
70	C₂²×C₃×C₄	#67	1	4	yes	—
71	C₂²×Dic₃	—	6	8	no	—
72	C₂×C₃×C₈	—	2	8	no	—
73	C₂×C₃×C₈	—	4	4	no	—
74	C₂×C₃×C₈	—	4	4	no	—
75	C₃×C₄²	Dic₆	1	4	yes	—
76	C₃×C₄²	C₄×D₃	1	4	yes	—
77	C₃×C₄²	C₂×Dic₃	1	4	yes	—
78	C₃×C₄²	#67	1	4	yes	—
79	C₄⋊Dic₃	—	3	16	no	—
80	C₄⋊Dic₃	—	6	8	no	—
81	C₄×Dic₃	—	12	4	no	—
82	C₄×Dic₃	—	12	4	no	—
83	C₂×C₃⋊C₈	—	6	16	no	—
84	C₂×C₃⋊C₈	—	6	8	no	—
85	C₂²×C₃²	#95	1	1	yes	characteristic
86	C₂²×C₃²	C₂²⋊C₈	1	2	yes	—
87	C₂²×C₃²	#102	1	1	yes	characteristic
88	C₂²×C₃²	#103	1	1	yes	characteristic
89	C₂²×C₃²	#104	1	2	yes	—
90	C₃²×C₄	—	2	1	no	—
91	C₃²×C₄	—	2	1	no	—
92	C₃²×C₄	C₄×C₈	1	1	yes	characteristic
93	C₃²×C₄	#97	1	1	yes	characteristic
94	C₃²×C₄	#102	1	1	yes	characteristic
95	C₃²×C₄	#103	1	1	yes	characteristic
96	C₃²×C₄	C₄⋊C₈	1	2	yes	—
97	C₃²×C₄	#106	1	1	yes	characteristic
98	C₃²×C₄	#107	1	1	yes	characteristic
99	C₃⋊Dic₃	—	4	4	no	—
100	C₂²×C₈	—	2	1	no	—
101	C₂²×C₈	—	18	1	no	—
102	C₂×C₄⋊C₄	—	9	2	no	—
103	C₂×C₄²	—	18	1	no	—
104	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
105	C₄⋊C₈	—	2	1	no	—
106	C₄⋊C₈	C₃⋊Dic₃	1	2	yes	complemented
107	C₄×C₈	—	9	2	no	—
108	C₄×C₈	—	18	1	no	—
109	#95	—	18	1	no	—
110	C₂³×C₃	(C₂×D₃)⋊C₄	1	4	yes	—
111	C₂×C₃×C₄	—	2	8	no	—
112	C₂×C₃×C₄	D₃×C₈	1	8	yes	—
113	C₂×C₃×C₄	ΓL(1,𝔽₂₅)	1	8	yes	—
114	C₂×C₃×C₄	C₄×Dic₃	1	4	yes	—
115	C₂×C₃×C₄	Dic₃⋊C₄	1	4	yes	—
116	C₂×C₃×C₄	Dic₃⋊C₄	1	4	yes	—
117	C₂×C₃×C₄	C₄⋊Dic₃	1	4	yes	—
118	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
119	C₂×C₃×C₄	#213	1	4	yes	—
120	C₂×C₃×C₄	#214	1	4	yes	—
121	C₂×C₃×C₄	#215	1	4	yes	—
122	C₂×C₃×C₄	#216	1	4	yes	—
123	C₂×C₃×C₄	#217	1	4	yes	—
124	C₂×Dic₃	—	6	16	no	—
125	C₂×Dic₃	—	12	8	no	—
126	C₂×Dic₃	—	12	8	no	—
127	C₃×C₈	—	4	8	no	—
128	C₃⋊C₈	—	6	16	no	—
129	C₂×C₃²	D₄⋊C₈	1	2	yes	—
130	C₂×C₃²	Q₈⋊C₈	1	2	yes	—
131	C₂×C₃²	#336	1	1	yes	characteristic
132	C₂×C₃²	#339	1	1	yes	characteristic
133	C₂×C₃²	#340	1	1	yes	characteristic
134	C₂²×C₄	—	18	2	no	—
135	C₂²×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
136	C₂²×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
137	C₂²×C₄	#638	1	1	yes	characteristic
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	4	1	no	—
140	C₂×C₈	—	4	1	no	—
141	C₂×C₈	—	18	4	no	—
142	C₂×C₈	—	18	2	no	—
143	C₄⋊C₄	—	9	4	no	—
144	C₄⋊C₄	—	18	2	no	—
145	C₄²	—	36	1	no	—
146	C₄²	—	36	1	no	—
147	C₄²	#634	1	1	yes	characteristic
148	C₄²	C₄×C₃⋊D₃	1	1	yes	characteristic
149	C₄²	C₂×C₃⋊Dic₃	1	1	yes	characteristic
150	C₄²	#638	1	1	yes	characteristic
151	C₂²×C₃	#809	1	4	yes	—
152	C₂²×C₃	#810	1	4	yes	—
153	C₂²×C₃	(C₂×D₃)⋊C₈	1	8	yes	—
154	C₂²×C₃	#825	1	8	yes	—
155	C₂²×C₃	C₃⋊#95	1	4	yes	—
156	C₃×C₄	—	2	4	no	—
157	C₃×C₄	—	2	4	no	—
158	C₃×C₄	#807	1	4	yes	—
159	C₃×C₄	#808	1	4	yes	—
160	C₃×C₄	C₈×Dic₃	1	4	yes	—
161	C₃×C₄	Dic₃⋊C₈	1	8	yes	—
162	C₃×C₄	#815	1	4	yes	—
163	C₃×C₄	#832	1	4	yes	—
164	C₃×C₄	#835	1	4	yes	—
165	Dic₃	—	12	16	no	—
166	C₃²	#1279	1	1	yes	characteristic, complemented, Sylow
167	C₂³	#3752	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	18	4	no	—
170	C₂×C₄	—	36	2	no	—
171	C₂×C₄	—	36	2	no	—
172	C₂×C₄	C₈×C₃⋊D₃	1	2	yes	—
173	C₂×C₄	#3743	1	2	yes	—
174	C₂×C₄	C₄×C₃⋊Dic₃	1	1	yes	characteristic
175	C₂×C₄	#3750	1	1	yes	characteristic
176	C₂×C₄	#3750	1	1	yes	characteristic
177	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
178	C₂×C₄	#3752	1	1	yes	characteristic
179	C₂×C₄	#3753	1	1	yes	characteristic
180	C₂×C₄	#3754	1	1	yes	characteristic
181	C₂×C₄	#3755	1	1	yes	characteristic
182	C₂×C₄	#3756	1	1	yes	characteristic
183	C₂×C₄	#3757	1	1	yes	characteristic
184	C₈	—	4	2	no	—
185	C₈	—	18	4	no	—
186	C₂×C₃	#4483	1	8	yes	—
187	C₂×C₃	#4484	1	8	yes	—
188	C₂×C₃	#4529	1	4	yes	—
189	C₂×C₃	#4550	1	4	yes	—
190	C₂×C₃	#4555	1	4	yes	—
191	C₂²	#63636	1	1	yes	characteristic
192	C₂²	#63637	1	1	yes	characteristic
193	C₂²	#63647	1	2	yes	—
194	C₂²	#63652	1	2	yes	—
195	C₂²	#63658	1	1	yes	characteristic
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	36	4	no	—
199	C₄	#63634	1	1	yes	characteristic
200	C₄	#63635	1	1	yes	characteristic
201	C₄	C₈×C₃⋊Dic₃	1	1	yes	characteristic
202	C₄	#63641	1	2	yes	—
203	C₄	#63642	1	1	yes	characteristic
204	C₄	#63659	1	1	yes	characteristic
205	C₄	#63662	1	1	yes	characteristic
206	C₃	#67704	1	4	yes	complemented
207	C₂	#10589349	1	2	yes	—
208	C₂	#10589350	1	2	yes	—
209	C₂	#10589395	1	1	yes	characteristic
210	C₂	#10589416	1	1	yes	characteristic
211	C₂	#10589421	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#638
commutator	C₃²×C₄	C₄×C₈
Frattini	C₂×C₄²	C₂×C₃⋊D₃
Fitting	C₂×C₃²×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₃²	C₂²⋊C₄
cosocle	C₂×C₃²×C₄²	C₂²
hypercenter	C₂×C₄⋊C₈	C₃⋊D₃
hypocenter	C₃²	#1279
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
11. C₃²×C₄⋊C₈	C₄	semidirect
14. C₄×C₃⋊(C₃⋊C₈)	C₄	semidirect
30. C₄⋊(C₃⋊Dic₃)	C₈	semidirect
42. C₃×C₄⋊C₈	Dic₃	semidirect
106. C₄⋊C₈	C₃⋊Dic₃	semidirect
166. C₃²	#1279	semidirect
206. C₃	#67704	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
C₄×C₄⋊(C₃⋊Dic₃)	2
C₂×C₄×C₃⋊(C₃⋊C₈)	2
#67704	3