Subgroups — #79017

Order 384.

364 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5479	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#5111	C₂	1	2	yes	maximal, complemented
4	C₈×#67	C₂	1	2	yes	maximal, complemented
5	#4925	C₂	1	1	yes	maximal, characteristic, complemented
6	#4840	C₂	1	2	yes	maximal, complemented
7	#4839	C₂	1	2	yes	maximal, complemented
8	#4838	C₂	1	1	yes	maximal, characteristic, complemented
9	#4834	C₂	1	1	yes	maximal, characteristic, complemented
10	#4831	C₂	1	1	yes	maximal, characteristic, complemented
11	#4830	C₂	1	1	yes	maximal, characteristic, complemented
12	#2994	—	3	1	no	maximal, Sylow
13	C₂²×C₃×C₈	C₂²	1	2	yes	—
14	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
15	#930	C₄	1	1	yes	characteristic
16	#929	C₄	1	2	yes	—
17	C₄×#67	C₂²	1	2	yes	—
18	#924	C₄	1	1	yes	characteristic
19	C₂×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	—
20	C₂×D₃×C₈	C₂²	1	2	yes	—
21	#876	C₄	1	2	yes	—
22	#875	C₄	1	1	yes	characteristic
23	C₄×D₁₂	C₂²	1	1	yes	characteristic
24	#872	C₂²	1	2	yes	—
25	#870	C₄	1	1	yes	characteristic
26	C₄×Dic₆	C₂²	1	1	yes	characteristic
27	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₃×C₄⋊C₈	C₂²	1	2	yes	—
30	#830	C₂²	1	1	yes	characteristic
31	#830	C₂²	1	1	yes	characteristic
32	(C₂×D₃)⋊C₈	C₂²	1	2	yes	—
33	(C₂×D₃)⋊C₈	C₂²	1	2	yes	—
34	#815	C₂²	1	2	yes	—
35	Dic₃⋊C₈	C₂²	1	2	yes	—
36	Dic₃⋊C₈	C₂²	1	2	yes	—
37	C₈×Dic₃	C₂²	1	2	yes	—
38	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
39	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
40	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
41	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
42	#529	—	3	1	no	—
43	#446	—	3	1	no	—
44	C₈×Q₈	—	3	1	no	—
45	#436	—	3	1	no	—
46	#435	—	3	2	no	—
47	C₈×D₄	—	3	2	no	—
48	C₈×D₄	—	3	1	no	—
49	#433	—	3	2	no	—
50	#431	—	3	1	no	—
51	#424	—	3	2	no	—
52	C₂×C₄⋊C₈	D₃	1	1	yes	characteristic, complemented
53	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
56	C₂×#67	C₂×C₄	1	2	yes	—
57	C₂×D₁₂	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×D₃	C₂³	1	2	yes	—
59	C₂×Dic₆	C₂×C₄	1	1	yes	characteristic
60	C₂×C₃×C₈	—	4	2	no	—
61	C₂×C₃×C₈	C₂³	1	2	yes	—
62	C₂×C₃×C₈	C₂³	1	2	yes	—
63	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
64	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
65	C₃×C₄²	C₂³	1	1	yes	characteristic
66	C₃×C₄²	C₂³	1	1	yes	characteristic
67	#217	C₂×C₄	1	2	yes	—
68	(C₂×D₃)⋊C₄	C₂×C₄	1	2	yes	—
69	(C₂×D₃)⋊C₄	C₂×C₄	1	2	yes	—
70	(C₂×D₃)⋊C₄	C₂×C₄	1	2	yes	—
71	(C₂×D₃)⋊C₄	C₂×C₄	1	2	yes	—
72	C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
73	C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
74	Dic₃⋊C₄	C₂×C₄	1	2	yes	—
75	Dic₃⋊C₄	C₂×C₄	1	2	yes	—
76	Dic₃⋊C₄	C₂×C₄	1	2	yes	—
77	Dic₃⋊C₄	C₂×C₄	1	2	yes	—
78	C₄×Dic₃	C₂³	1	2	yes	—
79	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
80	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
81	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
82	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
83	ΓL(1,𝔽₂₅)	—	4	2	no	—
84	D₃×C₈	—	4	2	no	—
85	C₂×M₁₆	—	3	2	no	—
86	C₂²×C₈	—	3	2	no	—
87	C₂²×C₈	C₂×D₃	1	2	yes	—
88	#126	—	3	2	no	—
89	#125	—	3	1	no	—
90	#124	—	3	1	no	—
91	#123	—	3	2	no	—
92	C₂²⋊Q₈	—	3	1	no	—
93	#121	—	3	1	no	—
94	C₄×Q₈	—	3	1	no	—
95	C₄×D₄	—	3	2	no	—
96	C₄×D₄	—	3	1	no	—
97	#117	—	3	2	no	—
98	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
99	C₄⋊C₈	—	3	2	no	—
100	C₄⋊C₈	—	3	2	no	—
101	C₄⋊C₈	—	3	1	no	—
102	C₄⋊C₈	—	3	1	no	—
103	C₄⋊C₈	C₂×D₃	1	2	yes	—
104	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
105	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
106	C₂²⋊C₈	—	3	1	no	—
107	C₂²⋊C₈	—	3	1	no	—
108	C₂²⋊C₈	—	3	2	no	—
109	C₂²⋊C₈	—	3	2	no	—
110	#97	—	3	2	no	—
111	C₄×C₈	—	3	2	no	—
112	C₄×C₈	—	3	1	no	—
113	C₄×C₈	—	3	1	no	—
114	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
115	C₂²×D₃	C₂²×C₄	1	2	yes	—
116	C₂×C₃×C₄	—	2	2	no	—
117	C₂×C₃×C₄	—	2	2	no	—
118	C₂×C₃×C₄	—	2	1	no	—
119	C₂×C₃×C₄	—	2	1	no	—
120	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
126	#67	—	2	4	no	—
127	C₂×Dic₃	C₂²×C₄	1	2	yes	—
128	C₂×Dic₃	C₂²×C₄	1	2	yes	—
129	C₂×Dic₃	C₂²×C₄	1	2	yes	—
130	D₁₂	—	2	2	no	—
131	C₄×D₃	—	2	2	no	—
132	C₄×D₃	—	2	2	no	—
133	Dic₆	—	2	2	no	—
134	C₃×C₈	—	2	2	no	—
135	C₃×C₈	—	2	2	no	—
136	C₃⋊C₈	—	2	1	no	—
137	C₃⋊C₈	—	2	1	no	—
138	C₃⋊C₈	—	2	1	no	—
139	C₃⋊C₈	—	2	1	no	—
140	C₂×Q₈	—	3	1	no	—
141	C₂×D₄	—	3	2	no	—
142	C₂×D₄	—	3	1	no	—
143	C₂²×C₄	—	3	2	no	—
144	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
145	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
146	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
147	M₁₆	—	12	2	no	—
148	C₂×C₈	—	3	1	no	—
149	C₂×C₈	—	3	1	no	—
150	C₂×C₈	—	3	1	no	—
151	C₂×C₈	—	3	1	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	12	2	no	—
154	C₂×C₈	C₂²×D₃	1	2	yes	—
155	C₂×C₈	C₂²×D₃	1	2	yes	—
156	C₄⋊C₄	—	3	2	no	—
157	C₄⋊C₄	—	3	2	no	—
158	C₄⋊C₄	—	3	1	no	—
159	C₄⋊C₄	—	3	2	no	—
160	C₄⋊C₄	—	3	1	no	—
161	C₄⋊C₄	—	3	2	no	—
162	C₂²⋊C₄	—	3	2	no	—
163	C₂²⋊C₄	—	3	2	no	—
164	C₂²⋊C₄	—	3	2	no	—
165	C₂²⋊C₄	—	3	2	no	—
166	C₂²⋊C₄	—	3	2	no	—
167	C₄²	—	3	2	no	—
168	C₄²	C₄×D₃	1	1	yes	characteristic
169	C₄²	C₄×D₃	1	1	yes	characteristic
170	C₄²	C₂²×D₃	1	1	yes	characteristic
171	C₄²	C₂²×D₃	1	1	yes	characteristic
172	C₂²×C₃	—	2	1	no	—
173	C₂²×C₃	—	2	1	no	—
174	C₂²×C₃	#131	1	2	yes	—
175	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
176	C₂×D₃	—	2	2	no	—
177	C₂×D₃	—	2	2	no	—
178	C₂×D₃	—	2	2	no	—
179	C₃×C₄	—	2	1	no	—
180	C₃×C₄	—	2	1	no	—
181	C₃×C₄	—	2	1	no	—
182	C₃×C₄	—	2	1	no	—
183	C₃×C₄	#131	1	2	yes	—
184	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
185	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
186	Dic₃	—	2	2	no	—
187	Dic₃	—	2	2	no	—
188	Dic₃	—	2	2	no	—
189	C₂³	—	3	2	no	—
190	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
191	Q₈	—	6	2	no	—
192	D₄	—	6	4	no	—
193	D₄	—	6	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	3	2	no	—
199	C₂×C₄	—	3	2	no	—
200	C₂×C₄	—	3	2	no	—
201	C₂×C₄	—	6	2	no	—
202	C₂×C₄	—	6	2	no	—
203	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
204	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
205	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
206	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
207	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
208	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
209	C₈	—	2	2	no	—
210	C₈	—	2	2	no	—
211	C₈	—	6	1	no	—
212	C₈	—	6	1	no	—
213	C₈	—	6	1	no	—
214	C₈	—	6	1	no	—
215	C₂×C₃	—	2	2	no	—
216	C₂×C₃	#520	1	1	yes	characteristic
217	C₂×C₃	C₂×#131	1	1	yes	characteristic
218	C₂×C₃	C₂×#131	1	1	yes	characteristic
219	D₃	—	4	2	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	6	2	no	—
223	C₂²	—	6	2	no	—
224	C₂²	—	6	2	no	—
225	C₂²	#907	1	2	yes	—
226	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	6	2	no	—
232	C₄	—	6	2	no	—
233	C₄	—	6	2	no	—
234	C₄	#901	1	2	yes	—
235	C₄	#1004	1	1	yes	characteristic
236	C₄	#1007	1	1	yes	characteristic
237	C₃	#2994	1	1	yes	characteristic, complemented, Sylow
238	C₂	—	2	2	no	—
239	C₂	—	12	2	no	—
240	C₂	#5504	1	1	yes	characteristic
241	C₂	C₂×#901	1	1	yes	characteristic
242	C₂	C₂×#907	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₂×C₃×C₄⋊C₈	C₂
prime radical	#930	C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₂×C₄⋊C₈	D₃
hypocenter	C₃	#2994
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₄⋊C₈	C₂	semidirect
3. #5111	C₂	semidirect
4. C₈×#67	C₂	semidirect
5. #4925	C₂	semidirect
6. #4840	C₂	semidirect
7. #4839	C₂	semidirect
8. #4838	C₂	semidirect
9. #4834	C₂	semidirect
10. #4831	C₂	semidirect
11. #4830	C₂	semidirect
52. C₂×C₄⋊C₈	D₃	semidirect
237. C₃	#2994	semidirect

Maximal subgroups

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

subgroup	index
#5479	2
C₂×C₃×C₄⋊C₈	2
#5111	2
C₈×#67	2
#4925	2
#4840	2
#4839	2
#4838	2
#4834	2
#4831	2
#4830	2
#2994	3