Subgroups — #11790488

Order 1152.

588 conjugacy classes of subgroups.

The subgroup lattice has 207 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#405	C₂	1	1	yes	maximal, characteristic
2	C₂×#63646	C₂	1	1	yes	maximal, characteristic
3	C₂×#63645	C₂	1	1	yes	maximal, characteristic
4	#10592694	C₂	1	2	yes	maximal
5	#10589420	C₂	1	2	yes	maximal
6	#69894	—	3	4	no	maximal
7	C₂×C₃²×M₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₃²×#117	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
11	C₂×#3748	C₂²	1	2	yes	—
12	C₃²×#97	—	2	1	no	—
13	C₃²×C₄×C₈	—	2	1	no	—
14	#63646	C₄	1	4	yes	complemented
15	#63645	—	2	1	no	—
16	#63645	—	2	1	no	—
17	#63635	—	2	2	no	—
18	#63634	—	2	2	no	—
19	C₃×#405	D₃	1	4	yes	—
20	C₂×#819	—	3	4	no	—
21	C₂×#818	—	3	4	no	—
22	#4995	—	3	8	no	—
23	#4554	—	3	8	no	—
24	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
25	C₂²×C₃⋊Dic₃	—	2	2	no	—
26	C₃²×M₁₆	—	2	1	no	—
27	C₃²×M₁₆	Q₈	1	2	yes	—
28	C₂×C₃²×C₈	—	2	1	no	—
29	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
30	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
31	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
32	C₂×C₃²×C₈	Q₈	1	2	yes	—
33	C₃²×C₄⋊C₄	D₄	1	1	yes	characteristic
34	C₃²×C₄⋊C₄	D₄	1	1	yes	characteristic
35	C₃²×C₂²⋊C₄	—	2	1	no	—
36	C₃²×C₄²	—	2	1	no	—
37	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
38	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
39	#3748	—	2	2	no	—
40	#3748	C₂×C₄	1	4	yes	—
41	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
42	#1952	—	9	1	no	Sylow
43	C₂×C₃×M₁₆	C₂×D₃	1	4	yes	—
44	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
45	C₃×#117	C₂×D₃	1	4	yes	—
46	C₂×C₄⋊Dic₃	—	3	8	no	—
47	C₂×#208	—	3	8	no	—
48	C₃×#97	—	2	4	no	—
49	C₃×C₄×C₈	—	2	4	no	—
50	#819	—	3	16	no	—
51	#818	—	6	4	no	—
52	#818	—	6	4	no	—
53	#808	—	6	8	no	—
54	#807	—	6	8	no	—
55	C₂³×C₃²	#41	1	1	yes	characteristic
56	C₂×C₃²×C₄	—	2	1	no	—
57	C₂×C₃²×C₄	—	2	1	no	—
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×D₄	1	1	yes	characteristic
61	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₃²×C₄	#41	1	1	yes	characteristic
64	C₂×C₃⋊Dic₃	—	4	2	no	—
65	C₂×C₃⋊Dic₃	—	8	2	no	—
66	C₃²×C₈	—	2	2	no	—
67	C₃²×C₈	C₄⋊C₄	1	2	yes	—
68	C₃²×C₈	C₄⋊C₄	1	2	yes	—
69	C₃⋊(C₃⋊C₈)	—	2	4	no	—
70	C₂×#108	—	9	1	no	—
71	#428	—	9	2	no	—
72	C₂×#107	—	9	1	no	—
73	#405	C₃⋊D₃	1	1	yes	characteristic
74	#343	—	9	2	no	—
75	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
76	C₂²×Dic₃	—	6	8	no	—
77	C₃×M₁₆	—	2	4	no	—
78	C₃×M₁₆	Dic₆	1	8	yes	—
79	C₂×C₃×C₈	—	2	4	no	—
80	C₂×C₃×C₈	Dic₆	1	8	yes	—
81	C₂×C₃×C₈	C₄×D₃	1	8	yes	—
82	C₂×C₃×C₈	#67	1	4	yes	—
83	C₂×C₃×C₈	#67	1	4	yes	—
84	C₃×C₄⋊C₄	D₁₂	1	4	yes	—
85	C₃×C₄⋊C₄	D₁₂	1	4	yes	—
86	C₃×C₂²⋊C₄	—	2	4	no	—
87	C₃×C₄²	—	2	4	no	—
88	C₄⋊Dic₃	—	6	8	no	—
89	C₄⋊Dic₃	—	6	8	no	—
90	#208	—	3	16	no	—
91	#208	—	6	8	no	—
92	C₂×C₃⋊C₈	—	6	8	no	—
93	C₂²×C₃²	—	2	1	no	—
94	C₂²×C₃²	—	2	1	no	—
95	C₂²×C₃²	C₄×Q₈	1	1	yes	characteristic
96	C₂²×C₃²	#121	1	1	yes	characteristic
97	C₂²×C₃²	#125	1	1	yes	characteristic
98	C₃²×C₄	—	4	1	no	—
99	C₃²×C₄	—	4	1	no	—
100	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
102	C₃²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
103	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
104	C₃⋊Dic₃	—	8	2	no	—
105	C₂×M₁₆	—	9	2	no	—
106	C₂×M₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic
107	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
108	#117	C₂×C₃⋊D₃	1	1	yes	characteristic
109	C₂×C₄⋊C₄	—	9	2	no	—
110	#108	—	9	4	no	—
111	#107	—	18	1	no	—
112	#107	—	18	1	no	—
113	#107	—	18	2	no	—
114	#106	—	18	2	no	—
115	#97	—	2	1	no	—
116	C₄×C₈	—	2	1	no	—
117	C₂³×C₃	#235	1	4	yes	—
118	C₂×C₃×C₄	—	2	4	no	—
119	C₂×C₃×C₄	—	2	4	no	—
120	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
121	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
122	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
123	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
124	C₂×C₃×C₄	#235	1	4	yes	—
125	C₂×C₃×C₄	C₂×#67	1	4	yes	—
126	C₂×Dic₃	—	12	8	no	—
127	C₂×Dic₃	—	24	8	no	—
128	C₃×C₈	—	2	8	no	—
129	C₃×C₈	Dic₃⋊C₄	1	8	yes	—
130	C₃×C₈	Dic₃⋊C₄	1	8	yes	—
131	C₃⋊C₈	—	6	16	no	—
132	C₂×C₃²	—	2	1	no	—
133	C₂×C₃²	—	2	1	no	—
134	C₂×C₃²	#389	1	1	yes	characteristic
135	C₂×C₃²	#472	1	1	yes	characteristic
136	C₂×C₃²	#473	1	1	yes	characteristic
137	C₂²×C₄	—	18	2	no	—
138	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
139	M₁₆	—	2	1	no	—
140	M₁₆	—	9	4	no	—
141	M₁₆	—	18	2	no	—
142	M₁₆	#634	1	2	yes	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	18	2	no	—
145	C₂×C₈	#634	1	2	yes	—
146	C₂×C₈	C₄×C₃⋊D₃	1	2	yes	—
147	C₂×C₈	#638	1	1	yes	characteristic
148	C₂×C₈	#638	1	1	yes	characteristic
149	C₄⋊C₄	—	18	2	no	—
150	C₄⋊C₄	—	18	2	no	—
151	C₄⋊C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
152	C₄⋊C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
153	C₂²⋊C₄	—	2	1	no	—
154	C₄²	—	2	1	no	—
155	C₂²×C₃	—	2	4	no	—
156	C₂²×C₃	—	2	4	no	—
157	C₂²×C₃	C₄×Dic₆	1	4	yes	—
158	C₂²×C₃	#870	1	4	yes	—
159	C₂²×C₃	#930	1	4	yes	—
160	C₃×C₄	—	4	4	no	—
161	C₃×C₄	—	4	4	no	—
162	C₃×C₄	#869	1	4	yes	—
163	C₃×C₄	C₄×D₁₂	1	4	yes	—
164	C₃×C₄	C₂×Dic₃⋊C₄	1	4	yes	—
165	C₃×C₄	#924	1	4	yes	—
166	Dic₃	—	24	8	no	—
167	C₃²	#1952	1	1	yes	characteristic, complemented, Sylow
168	C₂³	#3828	1	1	yes	characteristic
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	36	2	no	—
172	C₂×C₄	—	72	2	no	—
173	C₂×C₄	C₂×#634	1	1	yes	characteristic
174	C₂×C₄	C₂×#634	1	1	yes	characteristic
175	C₂×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
176	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
177	C₂×C₄	#3828	1	1	yes	characteristic
178	C₂×C₄	C₂×#638	1	1	yes	characteristic
179	C₈	—	2	2	no	—
180	C₈	—	18	4	no	—
181	C₈	#3750	1	2	yes	—
182	C₈	#3750	1	2	yes	—
183	C₂×C₃	—	2	4	no	—
184	C₂×C₃	—	2	4	no	—
185	C₂×C₃	#4928	1	4	yes	—
186	C₂×C₃	#5142	1	4	yes	—
187	C₂×C₃	#5143	1	4	yes	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	C₄×#634	1	1	yes	characteristic
191	C₂²	#64079	1	1	yes	characteristic
192	C₂²	#64139	1	1	yes	characteristic
193	C₄	—	4	1	no	—
194	C₄	—	4	1	no	—
195	C₄	—	72	2	no	—
196	C₄	#64078	1	1	yes	characteristic
197	C₄	C₄×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
198	C₄	C₂×#3750	1	1	yes	characteristic
199	C₄	#64133	1	1	yes	characteristic
200	C₃	#69894	1	4	yes	complemented
201	C₂	—	2	1	no	—
202	C₂	—	2	1	no	—
203	C₂	#10592627	1	1	yes	characteristic
204	C₂	#10592841	1	1	yes	characteristic
205	C₂	#10592842	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#64139
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×C₃⋊D₃
Fitting	C₃²×#405	C₂
socle	C₂²×C₃²	#121
prime radical	C₂³×C₃²	#41
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	#405	C₃⋊D₃
hypocenter	C₃²	#1952
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
14. #63646	C₄	semidirect
167. C₃²	#1952	semidirect
200. C₃	#69894	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#405	2
C₂×#63646	2
C₂×#63645	2
#10592694	2
#10589420	2
#69894	3