Subgroups — #79721

Order 384.

484 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#919	C₂	1	1	yes	maximal, characteristic
2	#5574	C₂	1	2	yes	maximal, complemented
3	C₃×C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	Dic₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#5026	C₂	1	2	yes	maximal, complemented
6	D₃×#107	C₂	1	1	yes	maximal, characteristic, complemented
7	#4872	C₂	1	1	yes	maximal, characteristic, complemented
8	#4817	C₂	1	2	yes	maximal, complemented
9	#4803	C₂	1	2	yes	maximal, complemented
10	C₄×D₂₄	C₂	1	1	yes	maximal, characteristic, complemented
11	#4689	C₂	1	1	yes	maximal, characteristic, complemented
12	#2951	—	3	1	no	maximal, Sylow
13	C₂×#239	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×#215	C₂²	1	2	yes	—
18	#919	C₄	1	8	yes	complemented
19	C₂×D₂₄	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	#893	—	2	2	no	—
23	#893	C₂²	1	2	yes	—
24	D₃×C₄⋊C₄	—	2	2	no	—
25	D₃×C₄⋊C₄	C₂²	1	2	yes	—
26	C₄×D₁₂	—	2	2	no	—
27	C₄×D₁₂	C₂²	1	2	yes	—
28	#872	—	2	2	no	—
29	#872	C₂²	1	1	yes	characteristic
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	#821	C₂²	1	2	yes	—
35	#818	C₂²	1	1	yes	characteristic
36	#815	C₂²	1	1	yes	characteristic
37	C₈×Dic₃	C₂²	1	1	yes	characteristic
38	#809	C₂²	1	2	yes	—
39	#807	C₂²	1	2	yes	—
40	#803	C₂²	1	1	yes	characteristic
41	C₂×#135	—	3	1	no	—
42	#520	—	3	2	no	—
43	#440	—	3	2	no	—
44	#440	—	3	2	no	—
45	C₄×Q₁₆	—	3	1	no	—
46	C₄×Q₁₆	D₃	1	1	yes	characteristic, complemented
47	C₄×D₈	—	3	1	no	—
48	C₄×D₈	—	3	1	no	—
49	C₂×#107	—	3	1	no	—
50	#417	—	3	2	no	—
51	#405	—	3	1	no	—
52	C₂×C₃×Q₈	C₂³	1	2	yes	—
53	#239	—	2	4	no	—
54	#239	C₂×C₄	1	8	yes	complemented
55	C₂×D₁₂	—	2	2	no	—
56	C₂×D₁₂	C₂³	1	2	yes	—
57	C₂×C₄×D₃	—	2	2	no	—
58	C₂×C₄×D₃	—	2	2	no	—
59	C₂×C₄×D₃	—	2	2	no	—
60	C₂×C₄×D₃	—	2	2	no	—
61	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
62	C₃×Q₁₆	C₂×C₄	1	4	yes	complemented
63	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
64	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
65	C₃×C₄⋊C₄	—	2	2	no	—
66	C₃×C₄⋊C₄	C₂³	1	2	yes	—
67	C₃×C₄²	—	2	2	no	—
68	C₃×C₄²	C₂³	1	1	yes	characteristic
69	#215	C₂×C₄	1	8	yes	complemented
70	(C₂×D₃)⋊C₄	—	2	2	no	—
71	(C₂×D₃)⋊C₄	—	2	2	no	—
72	(C₂×D₃)⋊C₄	D₄	1	1	yes	characteristic
73	(C₂×D₃)⋊C₄	D₄	1	1	yes	characteristic
74	C₄⋊Dic₃	—	2	2	no	—
75	C₄⋊Dic₃	C₂³	1	2	yes	—
76	Dic₃⋊C₄	—	2	2	no	—
77	Dic₃⋊C₄	D₄	1	1	yes	characteristic
78	Dic₃⋊C₄	D₄	1	1	yes	characteristic
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	D₂₄	C₂×C₄	1	4	yes	complemented
84	ΓL(1,𝔽₂₅)	—	2	2	no	—
85	D₃×C₈	C₂×C₄	1	4	yes	complemented
86	C₂×#41	—	3	2	no	—
87	#135	—	3	8	no	—
88	C₂×Q₁₆	C₂×D₃	1	1	yes	characteristic
89	C₂×SD₁₆	—	3	2	no	—
90	C₂×D₈	—	3	1	no	—
91	C₂×M₁₆	—	3	1	no	—
92	C₂²×C₈	—	3	1	no	—
93	C₄×Q₈	—	3	2	no	—
94	C₄×Q₈	C₂×D₃	1	2	yes	—
95	C₄×D₄	—	3	2	no	—
96	C₄×D₄	—	3	2	no	—
97	C₄×D₄	—	6	2	no	—
98	C₄×D₄	—	6	2	no	—
99	#117	—	3	1	no	—
100	#117	—	6	2	no	—
101	C₂×C₄⋊C₄	—	3	2	no	—
102	C₂×C₄⋊C₄	—	6	2	no	—
103	#107	—	3	1	no	—
104	#107	—	3	2	no	—
105	#107	C₂×D₃	1	1	yes	characteristic
106	#103	—	3	2	no	—
107	#103	C₂×D₃	1	2	yes	—
108	#102	—	3	2	no	—
109	#102	—	3	2	no	—
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	—	2	2	no	—
115	C₂²×D₃	C₂×D₄	1	1	yes	characteristic
116	C₃×Q₈	—	2	2	no	—
117	C₃×Q₈	C₂²×C₄	1	4	yes	—
118	C₂×C₃×C₄	—	2	2	no	—
119	C₂×C₃×C₄	—	2	2	no	—
120	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
121	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
122	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
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	D₁₂	—	2	8	no	—
128	D₁₂	—	2	2	no	—
129	D₁₂	C₂²×C₄	1	4	yes	—
130	C₄×D₃	—	2	8	no	—
131	C₄×D₃	—	8	2	no	—
132	C₄×D₃	—	8	2	no	—
133	C₄×D₃	—	8	2	no	—
134	C₄×D₃	C₂²×C₄	1	2	yes	—
135	C₄×D₃	#41	1	2	yes	—
136	C₃×C₈	—	2	1	no	—
137	C₃×C₈	C₂²×C₄	1	2	yes	—
138	C₃⋊C₈	—	2	1	no	—
139	C₃⋊C₈	C₂²×C₄	1	2	yes	—
140	#41	—	3	8	no	—
141	#41	—	6	4	no	—
142	C₂×Q₈	C₂²×D₃	1	2	yes	—
143	C₂×D₄	—	3	2	no	—
144	C₂×D₄	—	6	2	no	—
145	C₂²×C₄	—	3	1	no	—
146	C₂²×C₄	—	6	2	no	—
147	C₂²×C₄	—	6	2	no	—
148	C₂²×C₄	—	6	2	no	—
149	C₂²×C₄	—	6	2	no	—
150	Q₁₆	C₄×D₃	1	4	yes	complemented
151	SD₁₆	—	3	8	no	—
152	D₈	—	3	4	no	—
153	M₁₆	—	6	2	no	—
154	C₂×C₈	—	3	1	no	—
155	C₂×C₈	—	3	4	no	—
156	C₂×C₈	—	3	1	no	—
157	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
158	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	3	1	no	—
161	C₄⋊C₄	—	3	1	no	—
162	C₄⋊C₄	—	3	2	no	—
163	C₄⋊C₄	—	6	2	no	—
164	C₄⋊C₄	—	6	2	no	—
165	C₄⋊C₄	C₂²×D₃	1	2	yes	—
166	C₂²⋊C₄	—	3	1	no	—
167	C₂²⋊C₄	—	3	1	no	—
168	C₂²⋊C₄	—	6	2	no	—
169	C₂²⋊C₄	—	6	2	no	—
170	C₄²	—	2	2	no	—
171	C₄²	—	3	1	no	—
172	C₄²	—	6	2	no	—
173	C₄²	C₂²×D₃	1	1	yes	characteristic
174	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
175	C₂×D₃	—	2	1	no	—
176	C₂×D₃	—	2	1	no	—
177	C₂×D₃	—	2	4	no	—
178	C₂×D₃	—	4	2	no	—
179	C₂×D₃	—	4	2	no	—
180	C₂×D₃	C₄×D₄	1	2	yes	—
181	C₃×C₄	—	2	4	no	—
182	C₃×C₄	—	2	1	no	—
183	C₃×C₄	—	2	1	no	—
184	C₃×C₄	—	4	2	no	—
185	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
186	C₃×C₄	C₂×#41	1	1	yes	characteristic
187	Dic₃	—	2	1	no	—
188	Dic₃	—	2	1	no	—
189	Dic₃	—	4	2	no	—
190	Dic₃	C₄×D₄	1	2	yes	—
191	C₂³	—	3	1	no	—
192	C₂³	—	6	2	no	—
193	Q₈	—	2	2	no	—
194	Q₈	C₂×C₄×D₃	1	4	yes	—
195	D₄	—	3	4	no	—
196	D₄	—	6	8	no	—
197	D₄	—	6	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	3	1	no	—
201	C₂×C₄	—	3	2	no	—
202	C₂×C₄	—	3	2	no	—
203	C₂×C₄	—	3	1	no	—
204	C₂×C₄	—	3	1	no	—
205	C₂×C₄	—	6	8	no	—
206	C₂×C₄	—	6	2	no	—
207	C₂×C₄	—	24	2	no	—
208	C₂×C₄	—	24	2	no	—
209	C₂×C₄	—	24	2	no	—
210	C₂×C₄	D₃×D₄	1	1	yes	characteristic
211	C₂×C₄	D₃×D₄	1	1	yes	characteristic
212	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
213	C₈	—	2	1	no	—
214	C₈	—	3	2	no	—
215	C₈	—	6	1	no	—
216	C₈	C₂×C₄×D₃	1	2	yes	—
217	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
218	C₂×C₃	#576	1	1	yes	characteristic
219	C₂×C₃	#578	1	1	yes	characteristic
220	D₃	—	2	2	no	—
221	D₃	—	4	4	no	—
222	C₂²	—	3	2	no	—
223	C₂²	—	6	1	no	—
224	C₂²	—	6	1	no	—
225	C₂²	—	6	4	no	—
226	C₂²	—	12	2	no	—
227	C₂²	—	12	2	no	—
228	C₂²	C₂×D₃×D₄	1	1	yes	characteristic
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	4	no	—
232	C₄	—	3	2	no	—
233	C₄	—	4	2	no	—
234	C₄	—	6	1	no	—
235	C₄	—	6	1	no	—
236	C₄	—	12	2	no	—
237	C₄	C₂²×C₄×D₃	1	1	yes	characteristic
238	C₄	D₃×#41	1	1	yes	characteristic
239	C₃	#2951	1	1	yes	characteristic, complemented, Sylow
240	C₂	—	6	2	no	—
241	C₂	—	12	4	no	—
242	C₂	C₄×D₃×D₄	1	1	yes	characteristic
243	C₂	#5766	1	1	yes	characteristic
244	C₂	#5769	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	C₂×#919	C₂
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₄×Q₁₆	D₃
hypocenter	C₃	#2951
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
2. #5574	C₂	semidirect
3. C₃×C₄×Q₁₆	C₂	semidirect
4. Dic₃×Q₁₆	C₂	semidirect
5. #5026	C₂	semidirect
6. D₃×#107	C₂	semidirect
7. #4872	C₂	semidirect
8. #4817	C₂	semidirect
9. #4803	C₂	semidirect
10. C₄×D₂₄	C₂	semidirect
11. #4689	C₂	semidirect
18. #919	C₄	semidirect
46. C₄×Q₁₆	D₃	semidirect
54. #239	C₂×C₄	semidirect
62. C₃×Q₁₆	C₂×C₄	semidirect
69. #215	C₂×C₄	semidirect
83. D₂₄	C₂×C₄	semidirect
85. D₃×C₈	C₂×C₄	semidirect
150. Q₁₆	C₄×D₃	semidirect
239. C₃	#2951	semidirect

Maximal subgroups

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

subgroup	index
C₂×#919	2
#5574	2
C₃×C₄×Q₁₆	2
Dic₃×Q₁₆	2
#5026	2
D₃×#107	2
#4872	2
#4817	2
#4803	2
C₄×D₂₄	2
#4689	2
#2951	3