Subgroups — #83786

Order 384.

500 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#131	C₂	1	1	yes	maximal, characteristic, complemented
2	#5832	C₂	1	1	yes	maximal, characteristic
3	C₂×#901	C₂	1	1	yes	maximal, characteristic, complemented
4	#5234	C₂	1	1	yes	maximal, characteristic, complemented
5	#5129	C₂	1	1	yes	maximal, characteristic, complemented
6	#5128	C₂	1	4	yes	maximal, complemented
7	#5118	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×#67	C₂	1	4	yes	maximal, complemented
9	C₂×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#2931	—	3	1	no	maximal, Sylow
11	C₂×C₃×#41	C₂²	1	1	yes	characteristic
12	C₂×#235	C₂²	1	1	yes	characteristic
13	C₃×#131	—	2	4	no	—
14	C₂×C₃×M₁₆	C₂²	1	2	yes	—
15	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
16	C₂²×C₃×C₈	C₂²	1	2	yes	—
17	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
18	#947	C₄	1	1	yes	characteristic
19	#944	C₄	1	1	yes	characteristic
20	#940	C₄	1	1	yes	characteristic
21	#939	C₄	1	4	yes	—
22	#936	C₄	1	1	yes	characteristic
23	C₄×#67	C₂²	1	4	yes	—
24	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
25	C₂×#208	C₂²	1	1	yes	characteristic
26	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
27	#901	—	2	4	no	—
28	C₂×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	—
29	C₂×D₃×C₈	C₂²	1	2	yes	—
30	#830	C₂²	1	4	yes	—
31	(C₂×D₃)⋊C₈	C₂²	1	4	yes	—
32	Dic₃⋊C₈	C₂²	1	4	yes	—
33	C₈×Dic₃	C₂²	1	2	yes	—
34	C₈×Dic₃	C₂²	1	2	yes	—
35	C₂×#131	—	3	1	no	—
36	C₂×#131	D₃	1	1	yes	characteristic, complemented
37	#532	—	3	1	no	—
38	#436	—	3	4	no	—
39	C₈×D₄	—	3	4	no	—
40	#423	—	3	1	no	—
41	#408	—	3	1	no	—
42	#408	—	3	1	no	—
43	C₂×C₄×C₈	—	3	1	no	—
44	C₃×#41	—	2	2	no	—
45	C₃×#41	—	2	2	no	—
46	C₂×C₃×Q₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₃×D₄	C₂×C₄	1	1	yes	characteristic
48	C₂×C₃×D₄	C₂×C₄	1	2	yes	—
49	C₂²×C₃×C₄	C₂³	1	2	yes	—
50	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
51	C₂×#67	C₂×C₄	1	4	yes	—
52	C₂×#67	C₂×C₄	1	2	yes	—
53	C₂²×Dic₃	C₂×C₄	1	2	yes	—
54	#235	—	2	2	no	—
55	#235	—	2	2	no	—
56	C₂×D₁₂	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄×D₃	C₂³	1	2	yes	—
58	C₂×Dic₆	C₂×C₄	1	1	yes	characteristic
59	C₃×M₁₆	—	2	2	no	—
60	C₃×M₁₆	—	2	4	no	—
61	C₂×C₃×C₈	—	2	4	no	—
62	C₂×C₃×C₈	D₄	1	4	yes	complemented
63	C₂×C₃×C₈	C₂³	1	2	yes	—
64	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
65	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
66	#217	C₂×C₄	1	4	yes	—
67	(C₂×D₃)⋊C₄	C₂×C₄	1	4	yes	—
68	Dic₃⋊C₄	C₂×C₄	1	4	yes	—
69	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₃	C₂³	1	2	yes	—
72	#208	—	2	2	no	—
73	C₂×C₃⋊C₈	—	2	2	no	—
74	C₂×C₃⋊C₈	C₂³	1	2	yes	—
75	C₂×C₃⋊C₈	C₂³	1	2	yes	—
76	ΓL(1,𝔽₂₅)	—	2	4	no	—
77	D₃×C₈	—	2	4	no	—
78	C₂×#41	—	3	1	no	—
79	C₂×#41	C₂×D₃	1	1	yes	characteristic
80	#131	—	2	4	no	—
81	#131	—	6	4	no	—
82	C₂×M₁₆	—	3	1	no	—
83	C₂×M₁₆	—	3	2	no	—
84	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
85	C₂×M₁₆	C₂×D₃	1	2	yes	—
86	C₂²×C₈	—	3	1	no	—
87	C₂²×C₈	—	3	2	no	—
88	C₂²×C₈	C₂×D₃	1	2	yes	—
89	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
90	C₄⋊Q₈	—	3	1	no	—
91	#127	—	3	1	no	—
92	#124	—	3	1	no	—
93	#124	—	3	1	no	—
94	#121	—	3	4	no	—
95	C₄×D₄	—	3	4	no	—
96	C₂×C₄²	—	3	1	no	—
97	C₄⋊C₈	—	3	4	no	—
98	C₂²⋊C₈	—	3	4	no	—
99	C₂²⋊C₈	—	3	4	no	—
100	C₄×C₈	—	3	2	no	—
101	C₄×C₈	—	3	2	no	—
102	C₂³×C₃	C₂²×C₄	1	2	yes	—
103	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
104	C₂²×D₃	C₂²×C₄	1	2	yes	—
105	C₃×Q₈	—	2	2	no	—
106	C₃×D₄	—	2	4	no	—
107	C₃×D₄	—	2	2	no	—
108	C₂×C₃×C₄	—	2	2	no	—
109	C₂×C₃×C₄	—	2	2	no	—
110	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
111	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
113	C₂×C₃×C₄	#41	1	2	yes	—
114	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
115	#67	—	2	4	no	—
116	#67	—	2	8	no	—
117	C₂×Dic₃	—	2	4	no	—
118	C₂×Dic₃	C₂²×C₄	1	2	yes	—
119	C₂×Dic₃	C₂²×C₄	1	2	yes	—
120	C₂×Dic₃	C₂²×C₄	1	2	yes	—
121	D₁₂	—	2	2	no	—
122	C₄×D₃	—	2	2	no	—
123	C₄×D₃	—	2	2	no	—
124	Dic₆	—	2	2	no	—
125	C₃×C₈	—	2	2	no	—
126	C₃×C₈	C₂×D₄	1	2	yes	—
127	C₃×C₈	C₂×D₄	1	2	yes	—
128	C₃⋊C₈	—	2	2	no	—
129	C₃⋊C₈	—	2	2	no	—
130	#41	—	2	2	no	—
131	#41	—	2	2	no	—
132	#41	—	6	2	no	—
133	#41	—	6	2	no	—
134	C₂×Q₈	—	3	1	no	—
135	C₂×Q₈	C₄×D₃	1	1	yes	characteristic
136	C₂×D₄	—	3	4	no	—
137	C₂×D₄	—	3	2	no	—
138	C₂×D₄	—	3	1	no	—
139	C₂×D₄	C₄×D₃	1	2	yes	—
140	C₂×D₄	C₄×D₃	1	1	yes	characteristic
141	C₂²×C₄	—	3	2	no	—
142	C₂²×C₄	—	3	2	no	—
143	C₂²×C₄	C₂²×D₃	1	2	yes	—
144	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
145	M₁₆	—	2	2	no	—
146	M₁₆	—	2	4	no	—
147	M₁₆	—	6	2	no	—
148	M₁₆	—	6	4	no	—
149	C₂×C₈	—	2	4	no	—
150	C₂×C₈	—	3	2	no	—
151	C₂×C₈	—	3	2	no	—
152	C₂×C₈	—	6	2	no	—
153	C₂×C₈	—	6	4	no	—
154	C₂×C₈	#67	1	4	yes	complemented
155	C₂×C₈	C₂²×D₃	1	2	yes	—
156	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
157	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
158	C₄⋊C₄	—	3	4	no	—
159	C₂²⋊C₄	—	3	4	no	—
160	C₂²⋊C₄	—	3	4	no	—
161	C₄²	—	3	1	no	—
162	C₄²	—	3	1	no	—
163	C₄²	—	3	2	no	—
164	C₂²×C₃	—	2	2	no	—
165	C₂²×C₃	—	2	2	no	—
166	C₂²×C₃	—	2	4	no	—
167	C₂²×C₃	C₄×D₄	1	2	yes	—
168	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
169	C₂×D₃	—	2	2	no	—
170	C₂×D₃	—	2	4	no	—
171	C₃×C₄	—	2	2	no	—
172	C₃×C₄	C₄×D₄	1	2	yes	—
173	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
174	C₃×C₄	C₂×#41	1	1	yes	characteristic
175	Dic₃	—	2	2	no	—
176	Dic₃	—	2	2	no	—
177	Dic₃	#131	1	4	yes	complemented
178	C₂³	—	3	2	no	—
179	C₂³	C₂×C₄×D₃	1	2	yes	—
180	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
181	Q₈	—	2	2	no	—
182	Q₈	—	6	2	no	—
183	D₄	—	2	4	no	—
184	D₄	—	2	2	no	—
185	D₄	—	6	8	no	—
186	D₄	—	6	4	no	—
187	D₄	—	6	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	3	2	no	—
191	C₂×C₄	—	3	2	no	—
192	C₂×C₄	—	3	2	no	—
193	C₂×C₄	—	6	4	no	—
194	C₂×C₄	—	6	2	no	—
195	C₂×C₄	—	6	2	no	—
196	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
197	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
198	C₂×C₄	#235	1	2	yes	—
199	C₂×C₄	C₂×#67	1	2	yes	—
200	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
201	C₈	—	2	2	no	—
202	C₈	—	6	2	no	—
203	C₈	—	6	2	no	—
204	C₈	C₂×#67	1	2	yes	—
205	C₈	C₂×#67	1	2	yes	—
206	C₂×C₃	—	2	2	no	—
207	C₂×C₃	—	4	2	no	—
208	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
209	C₂×C₃	C₂×#131	1	2	yes	—
210	D₃	—	4	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	4	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	6	4	no	—
215	C₂²	—	6	2	no	—
216	C₂²	C₄×#67	1	2	yes	—
217	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	3	4	no	—
220	C₄	—	6	2	no	—
221	C₄	—	6	2	no	—
222	C₄	C₄×#67	1	2	yes	—
223	C₄	C₂×#235	1	1	yes	characteristic
224	C₄	C₂²×#67	1	1	yes	characteristic
225	C₃	#2931	1	1	yes	characteristic, complemented, Sylow
226	C₂	—	2	2	no	—
227	C₂	—	4	2	no	—
228	C₂	—	12	2	no	—
229	C₂	D₃×#131	1	2	yes	—
230	C₂	C₂×C₄×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂×C₃×#131	C₂
prime radical	#5832	C₂
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₂×#131	D₃
hypocenter	C₃	#2931
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#131	C₂	semidirect
3. C₂×#901	C₂	semidirect
4. #5234	C₂	semidirect
5. #5129	C₂	semidirect
6. #5128	C₂	semidirect
7. #5118	C₂	semidirect
8. C₈×#67	C₂	semidirect
9. C₂×C₈×Dic₃	C₂	semidirect
36. C₂×#131	D₃	semidirect
62. C₂×C₃×C₈	D₄	semidirect
154. C₂×C₈	#67	semidirect
177. Dic₃	#131	semidirect
225. C₃	#2931	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#131	2
#5832	2
C₂×#901	2
#5234	2
#5129	2
#5128	2
#5118	2
C₈×#67	2
C₂×C₈×Dic₃	2
#2931	3