Subgroups — #79720

Order 384.

456 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	#5572	C₂	1	2	yes	maximal
3	C₃×C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	Dic₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#5030	C₂	1	2	yes	maximal, complemented
6	#4880	C₂	1	1	yes	maximal, characteristic
7	#4873	C₂	1	1	yes	maximal, characteristic, complemented
8	#4802	C₂	1	2	yes	maximal
9	#4789	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₁₂	C₂	1	1	yes	maximal, characteristic, complemented
11	#4689	C₂	1	1	yes	maximal, characteristic
12	#2950	—	3	1	no	maximal, Sylow
13	C₂×D₃×Q₈	C₂²	1	2	yes	—
14	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic
15	C₃×C₄×Q₈	C₂²	1	2	yes	—
16	Q₈×Dic₃	C₂²	1	2	yes	—
17	C₂×#216	C₂²	1	2	yes	—
18	D₃×Q₁₆	C₄	1	8	yes	complemented
19	C₂×Dic₁₂	C₂²	1	1	yes	characteristic
20	C₂×ΓL(1,𝔽₂₅)	C₂²	1	1	yes	characteristic
21	C₂×D₃×C₈	C₂²	1	1	yes	characteristic
22	#892	—	2	2	no	—
23	#892	C₂²	1	2	yes	—
24	#887	—	2	2	no	—
25	#887	C₂²	1	2	yes	—
26	#872	—	2	2	no	—
27	#872	C₂²	1	1	yes	characteristic
28	C₄×Dic₆	—	2	2	no	—
29	C₄×Dic₆	C₂²	1	2	yes	—
30	C₃×#107	C₂²	1	1	yes	characteristic
31	C₃×#103	C₂²	1	2	yes	—
32	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
33	#835	C₂²	1	2	yes	—
34	#818	C₂²	1	1	yes	characteristic
35	#816	C₂²	1	2	yes	—
36	#815	C₂²	1	1	yes	characteristic
37	C₈×Dic₃	C₂²	1	1	yes	characteristic
38	#810	C₂²	1	2	yes	—
39	#808	C₂²	1	2	yes	—
40	#803	C₂²	1	1	yes	characteristic
41	C₂²×Q₁₆	—	3	1	no	—
42	#519	—	3	2	no	—
43	#441	—	3	2	no	—
44	#441	—	3	2	no	—
45	C₄×Q₁₆	—	3	1	no	—
46	C₄×Q₁₆	—	3	1	no	—
47	C₄×Q₁₆	—	3	1	no	—
48	C₄×Q₁₆	D₃	1	1	yes	characteristic, complemented
49	#427	—	3	1	no	—
50	#419	—	3	2	no	—
51	#405	—	3	1	no	—
52	C₂×C₃×Q₈	C₂³	1	2	yes	—
53	D₃×Q₈	—	2	4	no	—
54	D₃×Q₈	C₂×C₄	1	8	yes	—
55	C₂×C₄×D₃	—	2	2	no	—
56	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
57	C₂×Dic₆	—	2	2	no	—
58	C₂×Dic₆	C₂³	1	2	yes	—
59	C₃×Q₁₆	C₂×C₄	1	4	yes	—
60	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
61	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
62	C₃×C₄⋊C₄	—	2	2	no	—
63	C₃×C₄⋊C₄	C₂³	1	2	yes	—
64	C₃×C₄²	—	2	2	no	—
65	C₃×C₄²	C₂³	1	1	yes	characteristic
66	#216	C₂×C₄	1	8	yes	—
67	(C₂×D₃)⋊C₄	—	2	2	no	—
68	(C₂×D₃)⋊C₄	D₄	1	1	yes	characteristic
69	(C₂×D₃)⋊C₄	D₄	1	1	yes	characteristic
70	C₄⋊Dic₃	—	2	2	no	—
71	C₄⋊Dic₃	C₂³	1	2	yes	—
72	Dic₃⋊C₄	—	2	2	no	—
73	Dic₃⋊C₄	—	2	2	no	—
74	Dic₃⋊C₄	D₄	1	1	yes	characteristic
75	Dic₃⋊C₄	D₄	1	1	yes	characteristic
76	C₄×Dic₃	—	2	2	no	—
77	C₄×Dic₃	—	2	2	no	—
78	C₄×Dic₃	—	2	2	no	—
79	C₄×Dic₃	—	2	2	no	—
80	C₄×Dic₃	C₂³	1	1	yes	characteristic
81	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
82	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
83	Dic₁₂	C₂×C₄	1	4	yes	—
84	ΓL(1,𝔽₂₅)	—	2	2	no	—
85	D₃×C₈	C₂×C₄	1	4	yes	—
86	C₂²×Q₈	—	3	2	no	—
87	C₂×Q₁₆	—	3	2	no	—
88	C₂×Q₁₆	—	3	8	no	—
89	C₂×Q₁₆	—	3	1	no	—
90	C₂×Q₁₆	C₂×D₃	1	1	yes	characteristic
91	C₂×M₁₆	—	3	1	no	—
92	C₂²×C₈	—	3	1	no	—
93	C₄×Q₈	—	3	2	no	—
94	C₄×Q₈	—	3	2	no	—
95	C₄×Q₈	—	3	2	no	—
96	C₄×Q₈	—	6	2	no	—
97	C₄×Q₈	—	6	2	no	—
98	C₄×Q₈	C₂×D₃	1	2	yes	—
99	#117	—	3	1	no	—
100	#117	—	3	2	no	—
101	#117	—	6	2	no	—
102	#117	—	6	2	no	—
103	#107	—	3	1	no	—
104	#107	C₂×D₃	1	1	yes	characteristic
105	#106	—	3	2	no	—
106	#103	—	3	2	no	—
107	#103	—	3	2	no	—
108	#103	—	3	2	no	—
109	#103	C₂×D₃	1	2	yes	—
110	#97	—	3	1	no	—
111	#97	—	3	1	no	—
112	C₄×C₈	—	3	1	no	—
113	C₄×C₈	C₂×D₃	1	1	yes	characteristic
114	C₂²×D₃	C₂×D₄	1	1	yes	characteristic
115	C₃×Q₈	—	2	2	no	—
116	C₃×Q₈	C₂²×C₄	1	4	yes	—
117	C₂×C₃×C₄	—	2	2	no	—
118	C₂×C₃×C₄	—	2	2	no	—
119	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
120	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
121	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
122	C₂×Dic₃	—	2	2	no	—
123	C₂×Dic₃	—	2	2	no	—
124	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
127	C₄×D₃	—	2	8	no	—
128	C₄×D₃	C₂²×C₄	1	2	yes	—
129	C₄×D₃	#41	1	2	yes	—
130	Dic₆	—	2	2	no	—
131	Dic₆	—	2	8	no	—
132	Dic₆	C₂²×C₄	1	4	yes	—
133	C₃×C₈	—	2	1	no	—
134	C₃×C₈	C₂²×C₄	1	2	yes	—
135	C₃⋊C₈	—	2	1	no	—
136	C₃⋊C₈	C₂²×C₄	1	2	yes	—
137	C₂×Q₈	—	3	8	no	—
138	C₂×Q₈	—	3	2	no	—
139	C₂×Q₈	—	6	2	no	—
140	C₂×Q₈	—	6	4	no	—
141	C₂×Q₈	C₂²×D₃	1	2	yes	—
142	C₂²×C₄	—	3	1	no	—
143	C₂²×C₄	—	6	2	no	—
144	Q₁₆	—	3	8	no	—
145	Q₁₆	—	3	4	no	—
146	Q₁₆	C₄×D₃	1	4	yes	—
147	M₁₆	—	6	2	no	—
148	C₂×C₈	—	3	1	no	—
149	C₂×C₈	—	3	4	no	—
150	C₂×C₈	—	3	1	no	—
151	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
152	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	3	1	no	—
155	C₄⋊C₄	—	3	1	no	—
156	C₄⋊C₄	—	3	2	no	—
157	C₄⋊C₄	—	6	2	no	—
158	C₄⋊C₄	—	6	2	no	—
159	C₄⋊C₄	—	6	2	no	—
160	C₄⋊C₄	C₂²×D₃	1	2	yes	—
161	C₂²⋊C₄	—	3	1	no	—
162	C₂²⋊C₄	—	3	1	no	—
163	C₂²⋊C₄	—	6	2	no	—
164	C₄²	—	2	2	no	—
165	C₄²	—	3	1	no	—
166	C₄²	—	6	2	no	—
167	C₄²	—	6	2	no	—
168	C₄²	—	6	2	no	—
169	C₄²	—	6	2	no	—
170	C₄²	C₂²×D₃	1	1	yes	characteristic
171	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
172	C₂×D₃	—	2	1	no	—
173	C₂×D₃	—	2	1	no	—
174	C₂×D₃	C₄×D₄	1	2	yes	—
175	C₃×C₄	—	2	4	no	—
176	C₃×C₄	—	2	1	no	—
177	C₃×C₄	—	2	1	no	—
178	C₃×C₄	—	4	2	no	—
179	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
180	C₃×C₄	C₂×#41	1	1	yes	characteristic
181	Dic₃	—	2	4	no	—
182	Dic₃	—	2	1	no	—
183	Dic₃	—	2	1	no	—
184	Dic₃	—	4	2	no	—
185	Dic₃	C₄×D₄	1	2	yes	—
186	C₂³	—	3	1	no	—
187	Q₈	—	2	2	no	—
188	Q₈	—	3	4	no	—
189	Q₈	—	6	8	no	—
190	Q₈	—	6	2	no	—
191	Q₈	C₂×C₄×D₃	1	4	yes	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	3	1	no	—
195	C₂×C₄	—	3	2	no	—
196	C₂×C₄	—	3	2	no	—
197	C₂×C₄	—	3	1	no	—
198	C₂×C₄	—	3	1	no	—
199	C₂×C₄	—	6	8	no	—
200	C₂×C₄	—	6	2	no	—
201	C₂×C₄	—	6	2	no	—
202	C₂×C₄	D₃×D₄	1	1	yes	characteristic
203	C₂×C₄	D₃×D₄	1	1	yes	characteristic
204	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
205	C₈	—	2	1	no	—
206	C₈	—	3	2	no	—
207	C₈	—	6	1	no	—
208	C₈	C₂×C₄×D₃	1	2	yes	—
209	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
210	C₂×C₃	#578	1	1	yes	characteristic
211	C₂×C₃	#578	1	1	yes	characteristic
212	D₃	—	2	2	no	—
213	C₂²	—	3	2	no	—
214	C₂²	—	6	1	no	—
215	C₂²	—	6	1	no	—
216	C₂²	C₂×D₃×D₄	1	1	yes	characteristic
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	4	no	—
220	C₄	—	3	2	no	—
221	C₄	—	4	2	no	—
222	C₄	—	6	1	no	—
223	C₄	—	6	1	no	—
224	C₄	—	6	4	no	—
225	C₄	—	12	2	no	—
226	C₄	C₂²×C₄×D₃	1	1	yes	characteristic
227	C₄	D₃×#41	1	1	yes	characteristic
228	C₃	#2950	1	1	yes	characteristic, complemented, Sylow
229	C₂	—	6	2	no	—
230	C₂	C₄×D₃×D₄	1	1	yes	characteristic
231	C₂	#5766	1	1	yes	characteristic
232	C₂	#5771	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₃×D₄
commutator	C₃×C₄	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×C₄×Q₁₆	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂²×D₃	C₂×D₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₄×Q₁₆	D₃
hypocenter	C₃	#2950
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
3. C₃×C₄×Q₁₆	C₂	semidirect
4. Dic₃×Q₁₆	C₂	semidirect
5. #5030	C₂	semidirect
7. #4873	C₂	semidirect
9. #4789	C₂	semidirect
10. C₄×Dic₁₂	C₂	semidirect
18. D₃×Q₁₆	C₄	semidirect
48. C₄×Q₁₆	D₃	semidirect
228. C₃	#2950	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃×Q₁₆	2
#5572	2
C₃×C₄×Q₁₆	2
Dic₃×Q₁₆	2
#5030	2
#4880	2
#4873	2
#4802	2
#4789	2
C₄×Dic₁₂	2
#4689	2
#2950	3