Subgroups — #11771201

Order 1152.

528 conjugacy classes of subgroups.

The subgroup lattice has 201 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	#10592624	C₂	1	1	yes	maximal, characteristic
3	#10592623	C₂	1	1	yes	maximal, characteristic
4	#67620	—	3	4	no	maximal
5	C₂×C₃²×C₄⋊C₄	—	2	1	no	—
6	C₂×C₃²×C₄²	—	2	1	no	—
7	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
8	C₂×#3748	—	2	1	no	—
9	C₃²×#95	C₄	1	2	yes	—
10	#63657	—	2	1	no	—
11	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
12	C₄⋊(C₃⋊(C₃⋊C₈))	C₄	1	2	yes	complemented
13	C₄⋊(C₃⋊(C₃⋊C₈))	C₄	1	2	yes	complemented
14	C₄×C₃⋊(C₃⋊C₈)	—	2	1	no	—
15	C₃×C₄×C₄⋊C₄	D₃	1	4	yes	—
16	#4925	—	3	4	no	—
17	#4924	—	3	4	no	—
18	C₂²×C₃²×C₄	—	2	2	no	—
19	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
21	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
22	C₃²×C₄⋊C₄	—	2	4	no	—
23	C₃²×C₄²	—	4	1	no	—
24	C₃²×C₄²	—	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	#3748	—	4	1	no	—
30	#3748	—	4	1	no	—
31	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
32	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
33	#1303	—	9	1	no	Sylow
34	C₂×C₃×C₄⋊C₄	—	2	4	no	—
35	C₂×C₃×C₄²	—	2	4	no	—
36	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
37	C₂×#208	—	6	4	no	—
38	C₃×#95	Dic₃	1	8	yes	—
39	#830	—	6	4	no	—
40	C₄⋊(C₃⋊C₈)	—	3	8	no	—
41	C₄⋊(C₃⋊C₈)	—	3	8	no	—
42	C₄⋊(C₃⋊C₈)	—	6	4	no	—
43	C₄×C₃⋊C₈	—	6	4	no	—
44	C₂³×C₃²	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₃²×C₄	—	2	2	no	—
46	C₂×C₃²×C₄	—	2	4	no	—
47	C₂×C₃²×C₄	—	4	1	no	—
48	C₂×C₃²×C₄	—	4	2	no	—
49	C₂×C₃²×C₄	—	4	2	no	—
50	C₂×C₃²×C₄	C₄²	1	1	yes	characteristic
51	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
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	C₃⋊(C₃⋊C₈)	—	4	2	no	—
61	C₃⋊(C₃⋊C₈)	—	4	2	no	—
62	#431	—	9	1	no	—
63	#423	—	9	1	no	—
64	C₄×C₄⋊C₄	C₃⋊D₃	1	1	yes	characteristic
65	C₂²×C₃×C₄	—	2	8	no	—
66	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
67	C₂²×C₃×C₄	#67	1	4	yes	—
68	C₂²×C₃×C₄	#67	1	4	yes	—
69	C₃×C₄⋊C₄	—	2	16	no	—
70	C₃×C₄²	—	4	4	no	—
71	C₃×C₄²	—	4	4	no	—
72	C₃×C₄²	Dic₆	1	4	yes	—
73	C₃×C₄²	C₄×D₃	1	4	yes	—
74	C₃×C₄²	C₄×D₃	1	4	yes	—
75	C₃×C₄²	D₁₂	1	4	yes	—
76	#208	—	12	4	no	—
77	#208	—	12	4	no	—
78	C₂×C₃⋊C₈	—	6	8	no	—
79	C₂×C₃⋊C₈	—	6	8	no	—
80	C₂²×C₃²	—	2	1	no	—
81	C₂²×C₃²	—	2	1	no	—
82	C₂²×C₃²	#95	1	1	yes	characteristic
83	C₂²×C₃²	#102	1	1	yes	characteristic
84	C₂²×C₃²	#103	1	1	yes	characteristic
85	C₃²×C₄	—	2	1	no	—
86	C₃²×C₄	—	2	1	no	—
87	C₃²×C₄	—	2	1	no	—
88	C₃²×C₄	—	4	4	no	—
89	C₃²×C₄	#102	1	1	yes	characteristic
90	C₃²×C₄	#103	1	1	yes	characteristic
91	C₃²×C₄	#104	1	2	yes	—
92	C₃²×C₄	#106	1	1	yes	characteristic
93	C₃²×C₄	#107	1	1	yes	characteristic
94	C₂×M₁₆	—	18	1	no	—
95	C₂×C₄⋊C₄	—	2	1	no	—
96	C₂×C₄²	—	2	1	no	—
97	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
98	C₄⋊C₈	—	9	2	no	—
99	C₄⋊C₈	—	9	2	no	—
100	C₄⋊C₈	—	18	1	no	—
101	C₂²⋊C₈	—	18	1	no	—
102	C₄×C₈	—	18	1	no	—
103	#95	C₃⋊Dic₃	1	2	yes	—
104	C₂³×C₃	#217	1	4	yes	—
105	C₂×C₃×C₄	—	2	8	no	—
106	C₂×C₃×C₄	—	2	16	no	—
107	C₂×C₃×C₄	—	4	4	no	—
108	C₂×C₃×C₄	—	4	8	no	—
109	C₂×C₃×C₄	—	4	8	no	—
110	C₂×C₃×C₄	C₄×Dic₃	1	4	yes	—
111	C₂×C₃×C₄	Dic₃⋊C₄	1	4	yes	—
112	C₂×C₃×C₄	Dic₃⋊C₄	1	4	yes	—
113	C₂×C₃×C₄	C₄⋊Dic₃	1	4	yes	—
114	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
115	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
116	C₂×C₃×C₄	#213	1	4	yes	—
117	C₂×C₃×C₄	#214	1	4	yes	—
118	C₂×C₃×C₄	#215	1	4	yes	—
119	C₂×C₃×C₄	#216	1	4	yes	—
120	C₃⋊C₈	—	12	8	no	—
121	C₃⋊C₈	—	12	8	no	—
122	C₂×C₃²	—	2	1	no	—
123	C₂×C₃²	—	2	1	no	—
124	C₂×C₃²	#339	1	1	yes	characteristic
125	C₂×C₃²	#340	1	1	yes	characteristic
126	C₂×C₃²	#344	1	1	yes	characteristic
127	C₂²×C₄	—	2	2	no	—
128	C₂²×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
129	C₂²×C₄	#638	1	1	yes	characteristic
130	C₂²×C₄	#638	1	1	yes	characteristic
131	M₁₆	—	36	1	no	—
132	M₁₆	—	36	1	no	—
133	C₂×C₈	—	18	2	no	—
134	C₂×C₈	—	18	2	no	—
135	C₄⋊C₄	—	2	4	no	—
136	C₄²	—	4	1	no	—
137	C₄²	—	4	1	no	—
138	C₄²	#634	1	1	yes	characteristic
139	C₄²	C₄×C₃⋊D₃	1	1	yes	characteristic
140	C₄²	C₄×C₃⋊D₃	1	1	yes	characteristic
141	C₄²	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
142	C₂²×C₃	—	2	4	no	—
143	C₂²×C₃	—	2	4	no	—
144	C₂²×C₃	C₃⋊#95	1	4	yes	—
145	C₂²×C₃	#832	1	4	yes	—
146	C₂²×C₃	#835	1	4	yes	—
147	C₃×C₄	—	2	4	no	—
148	C₃×C₄	—	2	4	no	—
149	C₃×C₄	—	2	4	no	—
150	C₃×C₄	—	4	16	no	—
151	C₃×C₄	#805	1	8	yes	—
152	C₃×C₄	#807	1	4	yes	—
153	C₃×C₄	#808	1	4	yes	—
154	C₃×C₄	#809	1	4	yes	—
155	C₃×C₄	#810	1	4	yes	—
156	C₃²	#1303	1	1	yes	characteristic, complemented, Sylow
157	C₂³	#3757	1	1	yes	characteristic
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	C₄×C₃⋊Dic₃	1	1	yes	characteristic
164	C₂×C₄	#3750	1	1	yes	characteristic
165	C₂×C₄	#3750	1	1	yes	characteristic
166	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
167	C₂×C₄	#3752	1	1	yes	characteristic
168	C₂×C₄	#3752	1	1	yes	characteristic
169	C₂×C₄	#3753	1	1	yes	characteristic
170	C₂×C₄	#3754	1	1	yes	characteristic
171	C₂×C₄	#3755	1	1	yes	characteristic
172	C₂×C₄	#3756	1	1	yes	characteristic
173	C₈	—	36	2	no	—
174	C₈	—	36	2	no	—
175	C₂×C₃	—	2	4	no	—
176	C₂×C₃	—	2	4	no	—
177	C₂×C₃	#4523	1	4	yes	—
178	C₂×C₃	#4529	1	4	yes	—
179	C₂×C₃	#4533	1	4	yes	—
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	#63658	1	1	yes	characteristic
183	C₂²	#63659	1	1	yes	characteristic
184	C₂²	#63662	1	1	yes	characteristic
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	—	4	4	no	—
189	C₄	#63632	1	2	yes	—
190	C₄	#63634	1	1	yes	characteristic
191	C₄	#63635	1	1	yes	characteristic
192	C₄	#63636	1	1	yes	characteristic
193	C₄	#63637	1	1	yes	characteristic
194	C₃	#67620	1	4	yes	complemented
195	C₂	—	2	1	no	—
196	C₂	—	2	1	no	—
197	C₂	#10589389	1	1	yes	characteristic
198	C₂	#10589395	1	1	yes	characteristic
199	C₂	#10589399	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#3752
commutator	C₂×C₃²×C₄	C₄²
Frattini	C₂×C₄²	C₂×C₃⋊D₃
Fitting	C₃²×C₄×C₄⋊C₄	C₂
socle	C₂²×C₃²	#95
prime radical	C₂³×C₃²	C₂²⋊C₄
cosocle	C₂×C₃²×C₄²	C₂²
hypercenter	C₄×C₄⋊C₄	C₃⋊D₃
hypocenter	C₃²	#1303
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
12. C₄⋊(C₃⋊(C₃⋊C₈))	C₄	semidirect
13. C₄⋊(C₃⋊(C₃⋊C₈))	C₄	semidirect
156. C₃²	#1303	semidirect
194. C₃	#67620	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄×C₄⋊C₄	2
#10592624	2
#10592623	2
#67620	3