Subgroups — #82547

Order 384.

470 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#519	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#942	C₂	1	1	yes	maximal, characteristic
3	#5523	C₂	1	1	yes	maximal, characteristic
4	C₂×#835	C₂	1	1	yes	maximal, characteristic, complemented
5	#5030	C₂	1	2	yes	maximal, complemented
6	C₄×#216	C₂	1	2	yes	maximal, complemented
7	#5026	C₂	1	2	yes	maximal, complemented
8	C₄×#215	C₂	1	2	yes	maximal, complemented
9	#5002	C₂	1	1	yes	maximal, characteristic
10	#4997	C₂	1	1	yes	maximal, characteristic, complemented
11	#4995	C₂	1	1	yes	maximal, characteristic, complemented
12	#2947	—	3	1	no	maximal, Sylow
13	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
14	C₂×#235	C₂²	1	1	yes	characteristic
15	C₃×C₄×Q₈	—	2	1	no	—
16	C₃×C₄×Q₈	—	2	1	no	—
17	C₃×C₄×Q₈	C₂²	1	2	yes	—
18	C₃×C₄×Q₈	C₂²	1	2	yes	—
19	C₃×#117	—	2	2	no	—
20	C₃×#117	C₂²	1	1	yes	characteristic
21	C₃×#117	C₂²	1	1	yes	characteristic
22	C₂×#216	C₂²	1	2	yes	—
23	#942	C₄	1	8	yes	complemented
24	C₂×#215	C₂²	1	2	yes	—
25	C₂×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
26	C₂×#208	C₂²	1	1	yes	characteristic
27	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
28	#892	—	2	1	no	—
29	D₃×C₄⋊C₄	—	2	1	no	—
30	#67⋊C₄	—	2	2	no	—
31	C₄×D₁₂	C₂²	1	2	yes	—
32	C₄×Dic₆	C₂²	1	2	yes	—
33	#835	C₂²	1	2	yes	—
34	#835	C₂²	1	2	yes	—
35	#810	C₂²	1	1	yes	characteristic
36	#810	C₂²	1	1	yes	characteristic
37	#809	C₂²	1	1	yes	characteristic
38	#809	C₂²	1	1	yes	characteristic
39	#808	C₂²	1	2	yes	—
40	#807	C₂²	1	2	yes	—
41	#803	C₂²	1	2	yes	—
42	C₄×C₃⋊C₈	C₂²	1	2	yes	—
43	C₂×#137	—	3	1	no	—
44	#520	—	3	1	no	—
45	#519	D₃	1	1	yes	characteristic, complemented
46	#441	—	3	2	no	—
47	#440	—	3	2	no	—
48	C₄×Q₁₆	—	3	2	no	—
49	C₄×SD₁₆	—	3	2	no	—
50	#428	—	3	1	no	—
51	#416	—	3	1	no	—
52	C₂×#103	—	3	1	no	—
53	#405	—	3	1	no	—
54	C₂×C₃×Q₈	—	2	1	no	—
55	C₂×C₃×Q₈	—	2	2	no	—
56	C₂×C₃×Q₈	C₂×C₄	1	4	yes	complemented
57	C₂×C₃×Q₈	C₂³	1	2	yes	—
58	C₂²×C₃×C₄	—	2	1	no	—
59	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
60	C₂×#67	—	2	1	no	—
61	C₂²×Dic₃	—	2	1	no	—
62	#235	—	2	2	no	—
63	#235	C₂×C₄	1	4	yes	—
64	C₂×D₁₂	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₃	—	2	1	no	—
66	C₂×C₄×D₃	—	2	2	no	—
67	C₂×Dic₆	C₂³	1	1	yes	characteristic
68	C₃×C₄⋊C₄	—	2	2	no	—
69	C₃×C₄⋊C₄	—	2	2	no	—
70	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
71	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
72	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
73	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
74	C₃×C₂²⋊C₄	—	2	1	no	—
75	C₃×C₂²⋊C₄	D₄	1	2	yes	—
76	C₃×C₄²	—	2	2	no	—
77	C₃×C₄²	—	2	2	no	—
78	C₃×C₄²	—	2	1	no	—
79	C₃×C₄²	C₂³	1	2	yes	—
80	#216	C₂×C₄	1	8	yes	complemented
81	#215	C₂×C₄	1	8	yes	complemented
82	(C₂×D₃)⋊C₄	—	2	2	no	—
83	C₄⋊Dic₃	—	2	1	no	—
84	C₄⋊Dic₃	C₂³	1	2	yes	—
85	Dic₃⋊C₄	—	2	2	no	—
86	C₄×Dic₃	—	2	2	no	—
87	#208	C₂×C₄	1	4	yes	complemented
88	C₂×C₃⋊C₈	—	2	2	no	—
89	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
90	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
91	C₂×C₃⋊C₈	C₂³	1	2	yes	—
92	C₂×#41	—	3	1	no	—
93	C₂²×Q₈	C₂×D₃	1	1	yes	characteristic
94	#137	—	3	8	no	—
95	C₂×Q₁₆	—	3	2	no	—
96	C₂×SD₁₆	—	3	2	no	—
97	C₂×M₁₆	—	3	1	no	—
98	C₂²×C₈	—	3	1	no	—
99	C₄×Q₈	—	2	1	no	—
100	C₄×Q₈	—	2	1	no	—
101	C₄×Q₈	—	3	2	no	—
102	C₄×Q₈	C₂×D₃	1	2	yes	—
103	C₄×Q₈	C₂×D₃	1	2	yes	—
104	C₄×D₄	—	3	2	no	—
105	C₄×D₄	—	6	2	no	—
106	#117	—	2	2	no	—
107	#117	—	6	1	no	—
108	#117	C₂×D₃	1	1	yes	characteristic
109	#117	C₂×D₃	1	1	yes	characteristic
110	C₂×C₄⋊C₄	—	3	1	no	—
111	C₂×C₄⋊C₄	—	6	1	no	—
112	#107	—	3	2	no	—
113	#106	—	3	2	no	—
114	#103	—	3	2	no	—
115	#103	—	3	2	no	—
116	#103	—	3	1	no	—
117	#103	—	3	1	no	—
118	#102	—	3	1	no	—
119	#102	—	3	1	no	—
120	#97	—	3	2	no	—
121	C₄×C₈	—	3	2	no	—
122	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
123	C₂²×D₃	—	2	1	no	—
124	C₃×Q₈	—	2	4	no	—
125	C₃×Q₈	—	2	2	no	—
126	C₃×Q₈	C₂²×C₄	1	4	yes	—
127	C₂×C₃×C₄	—	2	4	no	—
128	C₂×C₃×C₄	—	2	2	no	—
129	C₂×C₃×C₄	—	2	1	no	—
130	C₂×C₃×C₄	—	2	1	no	—
131	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
132	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
133	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
134	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
135	C₂×C₃×C₄	#41	1	2	yes	—
136	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
137	#67	—	2	4	no	—
138	C₂×Dic₃	—	2	1	no	—
139	C₂×Dic₃	—	2	2	no	—
140	C₂×Dic₃	—	8	1	no	—
141	D₁₂	—	2	1	no	—
142	D₁₂	C₂²×C₄	1	2	yes	—
143	C₄×D₃	—	2	4	no	—
144	C₄×D₃	—	8	2	no	—
145	Dic₆	—	2	1	no	—
146	Dic₆	C₂²×C₄	1	2	yes	—
147	C₃⋊C₈	—	2	1	no	—
148	C₃⋊C₈	—	2	1	no	—
149	C₃⋊C₈	C₂²×C₄	1	4	yes	—
150	#41	—	3	4	no	—
151	#41	—	6	2	no	—
152	C₂×Q₈	—	2	2	no	—
153	C₂×Q₈	—	2	1	no	—
154	C₂×Q₈	—	3	1	no	—
155	C₂×Q₈	C₄×D₃	1	4	yes	complemented
156	C₂×Q₈	C₂²×D₃	1	2	yes	—
157	C₂×D₄	—	3	1	no	—
158	C₂×D₄	—	6	1	no	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	6	1	no	—
161	C₂²×C₄	—	6	1	no	—
162	C₂²×C₄	—	6	2	no	—
163	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
164	Q₁₆	—	3	8	no	—
165	SD₁₆	—	3	8	no	—
166	M₁₆	—	3	4	no	—
167	C₂×C₈	—	3	2	no	—
168	C₂×C₈	—	3	1	no	—
169	C₂×C₈	—	3	1	no	—
170	C₂×C₈	—	6	2	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	3	2	no	—
174	C₄⋊C₄	—	6	1	no	—
175	C₄⋊C₄	—	6	2	no	—
176	C₄⋊C₄	#67	1	1	yes	characteristic
177	C₄⋊C₄	#67	1	1	yes	characteristic
178	C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
179	C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
180	C₂²⋊C₄	—	2	1	no	—
181	C₂²⋊C₄	—	6	2	no	—
182	C₂²⋊C₄	#67	1	2	yes	—
183	C₄²	—	2	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	1	no	—
186	C₄²	—	6	2	no	—
187	C₄²	C₂²×D₃	1	2	yes	—
188	C₂²×C₃	—	2	1	no	—
189	C₂²×C₃	—	2	1	no	—
190	C₂²×C₃	C₄×D₄	1	2	yes	—
191	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
192	C₂×D₃	—	2	2	no	—
193	C₂×D₃	—	4	1	no	—
194	C₂×D₃	—	4	1	no	—
195	C₃×C₄	—	2	4	no	—
196	C₃×C₄	—	2	2	no	—
197	C₃×C₄	—	2	2	no	—
198	C₃×C₄	—	4	1	no	—
199	C₃×C₄	—	4	1	no	—
200	C₃×C₄	C₄×D₄	1	2	yes	—
201	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
202	C₃×C₄	C₂×#41	1	1	yes	characteristic
203	Dic₃	—	2	2	no	—
204	Dic₃	—	4	2	no	—
205	C₂³	—	6	1	no	—
206	C₂³	C₂×#67	1	1	yes	characteristic
207	Q₈	—	2	4	no	—
208	Q₈	—	2	2	no	—
209	Q₈	—	3	2	no	—
210	Q₈	—	6	1	no	—
211	Q₈	C₂×C₄×D₃	1	4	yes	—
212	D₄	—	3	2	no	—
213	D₄	—	6	4	no	—
214	D₄	—	6	1	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	6	1	no	—
220	C₂×C₄	—	6	4	no	—
221	C₂×C₄	—	6	2	no	—
222	C₂×C₄	—	24	2	no	—
223	C₂×C₄	—	24	1	no	—
224	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
225	C₂×C₄	#235	1	2	yes	—
226	C₂×C₄	C₂×#67	1	2	yes	—
227	C₂×C₄	C₂×#67	1	2	yes	—
228	C₂×C₄	C₂×#67	1	1	yes	characteristic
229	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
230	C₈	—	3	4	no	—
231	C₈	—	6	1	no	—
232	C₈	—	6	1	no	—
233	C₂×C₃	—	2	2	no	—
234	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
235	C₂×C₃	#577	1	1	yes	characteristic
236	C₂×C₃	#578	1	1	yes	characteristic
237	D₃	—	4	2	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	6	2	no	—
241	C₂²	—	12	1	no	—
242	C₂²	—	12	1	no	—
243	C₂²	C₄×#67	1	2	yes	—
244	C₂²	C₂²×#67	1	1	yes	characteristic
245	C₄	—	2	4	no	—
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	4	1	no	—
249	C₄	—	4	1	no	—
250	C₄	—	6	2	no	—
251	C₄	—	12	2	no	—
252	C₄	C₄×#67	1	2	yes	—
253	C₄	C₂²×C₄×D₃	1	1	yes	characteristic
254	C₄	C₂×#235	1	1	yes	characteristic
255	C₃	#2947	1	1	yes	characteristic, complemented, Sylow
256	C₂	—	2	2	no	—
257	C₂	—	12	2	no	—
258	C₂	C₂×C₄×#67	1	1	yes	characteristic
259	C₂	#5837	1	1	yes	characteristic
260	C₂	#5838	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#67
commutator	C₃×C₄	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#519	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂×#235	C₂²
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	#519	D₃
hypocenter	C₃	#2947
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #519

Complements

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

N	H ≅ G/N	type
1. C₃×#519	C₂	semidirect
4. C₂×#835	C₂	semidirect
5. #5030	C₂	semidirect
6. C₄×#216	C₂	semidirect
7. #5026	C₂	semidirect
8. C₄×#215	C₂	semidirect
10. #4997	C₂	semidirect
11. #4995	C₂	semidirect
23. #942	C₄	semidirect
45. #519	D₃	semidirect
56. C₂×C₃×Q₈	C₂×C₄	semidirect
80. #216	C₂×C₄	semidirect
81. #215	C₂×C₄	semidirect
87. #208	C₂×C₄	semidirect
155. C₂×Q₈	C₄×D₃	semidirect
255. C₃	#2947	semidirect

Maximal subgroups

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

subgroup	index
C₃×#519	2
C₂×#942	2
#5523	2
C₂×#835	2
#5030	2
C₄×#216	2
#5026	2
C₄×#215	2
#5002	2
#4997	2
#4995	2
#2947	3