Subgroups — #79540

Order 384.

544 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃×M₁₆	C₂	1	2	yes	maximal, complemented
2	C₄×D₃×D₄	C₂	1	1	yes	maximal, characteristic
3	C₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
4	#5127	C₂	1	2	yes	maximal, complemented
5	#5012	C₂	1	1	yes	maximal, characteristic, complemented
6	#4837	C₂	1	1	yes	maximal, characteristic, complemented
7	#4834	C₂	1	1	yes	maximal, characteristic, complemented
8	#4728	C₂	1	2	yes	maximal, complemented
9	#4726	C₂	1	2	yes	maximal, complemented
10	C₄×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
12	D₄×M₁₆	—	3	1	no	maximal, Sylow
13	C₂×D₃×D₄	C₄	1	1	yes	characteristic
14	C₂²×C₄×D₃	C₂²	1	2	yes	—
15	C₂×C₃×M₁₆	C₂²	1	2	yes	complemented
16	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
17	D₄×Dic₃	C₄	1	1	yes	characteristic
18	C₄×#67	C₂²	1	2	yes	—
19	C₂×#208	C₂²	1	2	yes	complemented
20	D₃×M₁₆	—	2	8	no	—
21	C₂×ΓL(1,𝔽₂₅)	C₂²	1	1	yes	characteristic
22	C₂×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	—
23	C₂×ΓL(1,𝔽₂₅)	C₂²	1	1	yes	characteristic
24	C₂×D₃×C₈	C₂²	1	2	yes	complemented
25	C₂×D₃×C₈	C₂²	1	2	yes	complemented
26	#893	C₄	1	1	yes	characteristic
27	D₃×C₄⋊C₄	C₄	1	1	yes	characteristic
28	#67⋊C₄	C₄	1	2	yes	—
29	D₃×C₂²⋊C₄	C₄	1	2	yes	—
30	C₄×D₁₂	C₂²	1	1	yes	characteristic
31	C₄²×D₃	C₂²	1	1	yes	characteristic
32	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
33	C₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
34	C₃×C₄×C₈	C₂²	1	1	yes	characteristic, complemented
35	#830	C₂²	1	2	yes	complemented
36	(C₂×D₃)⋊C₈	C₂²	1	2	yes	complemented
37	(C₂×D₃)⋊C₈	C₂²	1	2	yes	complemented
38	#815	C₂²	1	1	yes	characteristic, complemented
39	#815	C₂²	1	1	yes	characteristic, complemented
40	Dic₃⋊C₈	C₂²	1	2	yes	complemented
41	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic, complemented
42	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
43	C₂²×M₁₆	—	3	2	no	—
44	C₂×C₄×D₄	—	3	1	no	—
45	#436	—	3	1	no	—
46	#436	D₃	1	1	yes	characteristic, complemented
47	#435	—	3	2	no	—
48	#435	—	3	2	no	—
49	C₈×D₄	—	3	1	no	—
50	C₈×D₄	—	3	1	no	—
51	#423	—	3	1	no	—
52	#407	—	3	2	no	—
53	C₄×M₁₆	—	3	1	no	—
54	C₂³×D₃	C₂×C₄	1	2	yes	—
55	C₂×C₃×D₄	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₃×C₄	C₂³	1	2	yes	—
57	C₂×#67	C₂×C₄	1	2	yes	—
58	C₂²×Dic₃	C₂×C₄	1	2	yes	—
59	D₃×D₄	—	2	4	no	—
60	C₂×D₁₂	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×D₃	—	2	4	no	—
62	C₂×C₄×D₃	—	2	4	no	—
63	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×D₃	C₂×C₄	1	2	yes	—
65	C₂×C₄×D₃	C₂×C₄	1	2	yes	—
66	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄×D₃	C₂³	1	2	yes	—
68	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
69	C₃×M₁₆	—	2	4	no	—
70	C₂×C₃×C₈	C₂³	1	2	yes	—
71	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
72	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
73	C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
74	C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
75	C₃×C₄²	C₂³	1	1	yes	characteristic
76	#217	C₂×C₄	1	2	yes	—
77	(C₂×D₃)⋊C₄	C₂×C₄	1	2	yes	—
78	(C₂×D₃)⋊C₄	C₂×C₄	1	2	yes	—
79	C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
80	Dic₃⋊C₄	C₂×C₄	1	2	yes	—
81	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
82	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
83	C₄×Dic₃	C₂³	1	1	yes	characteristic
84	#208	—	2	4	no	—
85	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
86	C₂×C₃⋊C₈	C₂³	1	2	yes	—
87	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
88	ΓL(1,𝔽₂₅)	—	2	4	no	—
89	ΓL(1,𝔽₂₅)	—	2	2	no	—
90	ΓL(1,𝔽₂₅)	D₄	1	4	yes	complemented
91	D₃×C₈	—	2	4	no	—
92	D₃×C₈	—	2	4	no	—
93	C₂²×D₄	—	3	1	no	—
94	C₂³×C₄	—	3	2	no	—
95	C₂×M₁₆	—	3	2	no	—
96	C₂×M₁₆	—	3	1	no	—
97	C₂×M₁₆	—	3	2	no	—
98	C₂×M₁₆	—	3	1	no	—
99	C₂×M₁₆	—	6	8	no	—
100	C₂×M₁₆	C₂×D₃	1	2	yes	complemented
101	C₂²×C₈	—	3	2	no	—
102	C₂²×C₈	—	3	2	no	—
103	C₄×D₄	—	3	1	no	—
104	C₄×D₄	—	3	2	no	—
105	C₄×D₄	—	3	1	no	—
106	C₄×D₄	—	3	2	no	—
107	C₄×D₄	—	3	1	no	—
108	C₄×D₄	C₂×D₃	1	1	yes	characteristic
109	C₂×C₄⋊C₄	—	3	1	no	—
110	C₂×C₂²⋊C₄	—	3	2	no	—
111	C₂×C₄²	—	3	1	no	—
112	C₄⋊C₈	—	3	2	no	—
113	C₄⋊C₈	—	3	1	no	—
114	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic, complemented
115	C₂²⋊C₈	—	3	2	no	—
116	C₂²⋊C₈	—	3	2	no	—
117	C₂²⋊C₈	—	3	2	no	—
118	C₂²⋊C₈	C₂×D₃	1	2	yes	complemented
119	#97	—	3	1	no	—
120	#97	—	3	1	no	—
121	C₄×C₈	—	3	1	no	—
122	C₄×C₈	C₂×D₃	1	1	yes	characteristic, complemented
123	C₂³×C₃	C₂²×C₄	1	2	yes	—
124	C₂²×D₃	—	2	4	no	—
125	C₂²×D₃	—	2	4	no	—
126	C₂²×D₃	—	4	2	no	—
127	C₂²×D₃	C₂²×C₄	1	1	yes	characteristic
128	C₂²×D₃	C₂²×C₄	1	2	yes	—
129	C₃×D₄	—	2	2	no	—
130	C₂×C₃×C₄	—	2	2	no	—
131	C₂×C₃×C₄	—	2	2	no	—
132	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
134	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
136	#67	—	2	4	no	—
137	C₂×Dic₃	—	2	4	no	—
138	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
139	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
140	C₂×Dic₃	C₂²×C₄	1	2	yes	—
141	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
142	D₁₂	M₁₆	1	4	yes	complemented
143	C₄×D₃	—	2	4	no	—
144	C₄×D₃	—	2	2	no	—
145	C₄×D₃	—	2	2	no	—
146	C₄×D₃	—	2	2	no	—
147	C₄×D₃	—	2	4	no	—
148	C₄×D₃	—	4	1	no	—
149	C₄×D₃	C₂×D₄	1	2	yes	—
150	C₄×D₃	#41	1	2	yes	—
151	C₃×C₈	—	2	2	no	—
152	C₃×C₈	—	2	1	no	—
153	C₃×C₈	C₂×D₄	1	2	yes	complemented
154	C₃⋊C₈	—	2	2	no	—
155	C₃⋊C₈	—	2	1	no	—
156	C₃⋊C₈	C₂×D₄	1	2	yes	complemented
157	C₂⁴	—	3	2	no	—
158	C₂×D₄	—	3	2	no	—
159	C₂×D₄	—	3	1	no	—
160	C₂×D₄	—	6	4	no	—
161	C₂×D₄	C₄×D₃	1	1	yes	characteristic
162	C₂²×C₄	—	3	1	no	—
163	C₂²×C₄	—	3	2	no	—
164	C₂²×C₄	—	3	1	no	—
165	C₂²×C₄	—	3	2	no	—
166	C₂²×C₄	—	3	2	no	—
167	C₂²×C₄	—	3	2	no	—
168	C₂²×C₄	—	3	1	no	—
169	C₂²×C₄	—	6	4	no	—
170	C₂²×C₄	—	6	4	no	—
171	C₂²×C₄	C₂²×D₃	1	2	yes	—
172	M₁₆	—	2	4	no	—
173	M₁₆	—	3	4	no	—
174	M₁₆	—	6	4	no	—
175	M₁₆	—	6	2	no	—
176	M₁₆	—	6	4	no	—
177	C₂×C₈	—	3	1	no	—
178	C₂×C₈	—	3	1	no	—
179	C₂×C₈	—	3	2	no	—
180	C₂×C₈	—	6	4	no	—
181	C₂×C₈	—	6	4	no	—
182	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
183	C₂×C₈	C₂²×D₃	1	2	yes	—
184	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
185	C₄⋊C₄	—	3	2	no	—
186	C₄⋊C₄	—	3	1	no	—
187	C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
188	C₂²⋊C₄	—	3	2	no	—
189	C₂²⋊C₄	—	3	2	no	—
190	C₂²⋊C₄	—	3	2	no	—
191	C₂²⋊C₄	C₄×D₃	1	2	yes	—
192	C₄²	—	3	1	no	—
193	C₄²	—	3	1	no	—
194	C₄²	—	3	1	no	—
195	C₄²	C₂²×D₃	1	1	yes	characteristic
196	C₂²×C₃	—	2	2	no	—
197	C₂²×C₃	—	2	2	no	—
198	C₂²×C₃	—	2	2	no	—
199	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
200	C₂×D₃	—	2	2	no	—
201	C₂×D₃	—	2	1	no	—
202	C₂×D₃	—	2	1	no	—
203	C₂×D₃	—	2	2	no	—
204	C₂×D₃	—	4	8	no	—
205	C₂×D₃	C₄×D₄	1	2	yes	—
206	C₂×D₃	C₂×M₁₆	1	4	yes	complemented
207	C₃×C₄	—	2	2	no	—
208	C₃×C₄	—	2	1	no	—
209	C₃×C₄	C₂×M₁₆	1	2	yes	complemented
210	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
211	C₃×C₄	C₂×#41	1	1	yes	characteristic
212	Dic₃	—	2	2	no	—
213	Dic₃	—	2	1	no	—
214	Dic₃	—	2	1	no	—
215	Dic₃	C₄×D₄	1	2	yes	—
216	C₂³	—	3	2	no	—
217	C₂³	—	3	1	no	—
218	C₂³	—	6	4	no	—
219	C₂³	—	6	4	no	—
220	C₂³	—	12	2	no	—
221	C₂³	C₂×C₄×D₃	1	2	yes	—
222	D₄	—	2	2	no	—
223	D₄	—	3	4	no	—
224	D₄	—	6	4	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	3	1	no	—
228	C₂×C₄	—	3	2	no	—
229	C₂×C₄	—	3	2	no	—
230	C₂×C₄	—	3	1	no	—
231	C₂×C₄	—	3	1	no	—
232	C₂×C₄	—	3	2	no	—
233	C₂×C₄	—	6	4	no	—
234	C₂×C₄	—	6	4	no	—
235	C₂×C₄	—	6	4	no	—
236	C₂×C₄	—	6	2	no	—
237	C₂×C₄	—	6	2	no	—
238	C₂×C₄	—	6	2	no	—
239	C₂×C₄	—	12	1	no	—
240	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
241	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
242	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
243	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	—	2	1	no	—
246	C₈	—	3	2	no	—
247	C₈	—	6	2	no	—
248	C₈	—	6	1	no	—
249	C₈	D₃×D₄	1	2	yes	complemented
250	C₂×C₃	—	4	2	no	—
251	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
252	C₂×C₃	C₂²×M₁₆	1	1	yes	characteristic
253	C₂×C₃	#568	1	1	yes	characteristic
254	D₃	—	2	2	no	—
255	D₃	—	2	4	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	3	4	no	—
260	C₂²	—	3	2	no	—
261	C₂²	—	6	1	no	—
262	C₂²	—	6	1	no	—
263	C₂²	—	6	2	no	—
264	C₂²	—	6	2	no	—
265	C₂²	—	12	8	no	—
266	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
267	C₄	—	2	1	no	—
268	C₄	—	2	2	no	—
269	C₄	—	3	2	no	—
270	C₄	—	6	1	no	—
271	C₄	—	6	1	no	—
272	C₄	—	6	2	no	—
273	C₄	D₃×M₁₆	1	2	yes	complemented
274	C₄	C₂×D₃×D₄	1	1	yes	characteristic
275	C₄	D₃×#41	1	1	yes	characteristic
276	C₃	D₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
277	C₂	—	4	2	no	—
278	C₂	—	6	2	no	—
279	C₂	—	6	4	no	—
280	C₂	C₄×D₃×D₄	1	1	yes	characteristic
281	C₂	C₂×D₃×M₁₆	1	1	yes	characteristic
282	C₂	#5750	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₃×#436	C₂
prime radical	C₂×D₃×D₄	C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	#436	D₃
hypocenter	C₃	D₄×M₁₆
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
1. C₂×D₃×M₁₆	C₂	semidirect
3. C₃×#436	C₂	semidirect
4. #5127	C₂	semidirect
5. #5012	C₂	semidirect
6. #4837	C₂	semidirect
7. #4834	C₂	semidirect
8. #4728	C₂	semidirect
9. #4726	C₂	semidirect
10. C₄×ΓL(1,𝔽₂₅)	C₂	semidirect
11. C₈×D₁₂	C₂	semidirect
15. C₂×C₃×M₁₆	C₂²	semidirect
19. C₂×#208	C₂²	semidirect
24. C₂×D₃×C₈	C₂²	semidirect
25. C₂×D₃×C₈	C₂²	semidirect
32. C₃×C₄⋊C₈	C₂²	semidirect
33. C₃×C₂²⋊C₈	C₂²	semidirect
34. C₃×C₄×C₈	C₂²	semidirect
35. #830	C₂²	semidirect
36. (C₂×D₃)⋊C₈	C₂²	semidirect
37. (C₂×D₃)⋊C₈	C₂²	semidirect
38. #815	C₂²	semidirect
39. #815	C₂²	semidirect
40. Dic₃⋊C₈	C₂²	semidirect
41. C₄⋊(C₃⋊C₈)	C₂²	semidirect
42. C₄×C₃⋊C₈	C₂²	semidirect
46. #436	D₃	semidirect
90. ΓL(1,𝔽₂₅)	D₄	semidirect
100. C₂×M₁₆	C₂×D₃	semidirect
114. C₄⋊C₈	C₂×D₃	semidirect
118. C₂²⋊C₈	C₂×D₃	semidirect
122. C₄×C₈	C₂×D₃	semidirect
142. D₁₂	M₁₆	semidirect
153. C₃×C₈	C₂×D₄	semidirect
156. C₃⋊C₈	C₂×D₄	semidirect
206. C₂×D₃	C₂×M₁₆	semidirect
209. C₃×C₄	C₂×M₁₆	semidirect
249. C₈	D₃×D₄	semidirect
273. C₄	D₃×M₁₆	semidirect
276. C₃	D₄×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃×M₁₆	2
C₄×D₃×D₄	2
C₃×#436	2
#5127	2
#5012	2
#4837	2
#4834	2
#4728	2
#4726	2
C₄×ΓL(1,𝔽₂₅)	2
C₈×D₁₂	2
D₄×M₁₆	3