Subgroups — #79526

Order 384.

354 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	#5557	C₂	1	1	yes	maximal, characteristic
2	C₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
3	#5128	C₂	1	2	yes	maximal, complemented
4	#5119	C₂	1	2	yes	maximal, complemented
5	D₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#4837	C₂	1	1	yes	maximal, characteristic, complemented
7	#4830	C₂	1	1	yes	maximal, characteristic, complemented
8	#4730	C₂	1	2	yes	maximal, complemented
9	#4720	C₂	1	2	yes	maximal, complemented
10	#4685	C₂	1	1	yes	maximal, characteristic, complemented
11	#4679	C₂	1	1	yes	maximal, characteristic, complemented
12	#2993	—	3	1	no	maximal, Sylow
13	C₂×C₃×M₁₆	C₂²	1	2	yes	—
14	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
15	#938	C₄	1	1	yes	characteristic
16	#936	C₄	1	1	yes	characteristic
17	C₄×#67	C₂²	1	2	yes	—
18	#926	C₂²	1	2	yes	—
19	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
20	C₂×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	—
21	#897	C₄	1	1	yes	characteristic
22	#890	C₄	1	1	yes	characteristic
23	#883	C₄	1	2	yes	—
24	#879	C₄	1	2	yes	—
25	C₄²×D₃	C₂²	1	1	yes	characteristic
26	C₄×Dic₆	C₂²	1	1	yes	characteristic
27	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
29	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
30	#830	C₂²	1	2	yes	—
31	(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
32	(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
33	#815	C₂²	1	2	yes	—
34	Dic₃⋊C₈	C₂²	1	2	yes	—
35	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
36	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
37	Dic₃⋊C₈	C₂²	1	2	yes	—
38	Dic₃⋊C₈	C₂²	1	2	yes	—
39	C₈×Dic₃	C₂²	1	2	yes	—
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	2	no	—
46	#436	D₃	1	1	yes	characteristic, complemented
47	#435	—	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	#423	—	3	1	no	—
53	C₂×C₃×D₄	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₃×C₄	C₂³	1	2	yes	—
55	C₂×#67	C₂×C₄	1	2	yes	—
56	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
59	C₂×Dic₆	C₂×C₄	1	1	yes	characteristic
60	C₃×M₁₆	—	4	2	no	—
61	C₂×C₃×C₈	C₂³	1	2	yes	—
62	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
63	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
64	C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
65	C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
66	C₃×C₄²	C₂³	1	1	yes	characteristic
67	#217	C₂×C₄	1	2	yes	—
68	#217	C₂×C₄	1	2	yes	—
69	#217	C₂×C₄	1	2	yes	—
70	(C₂×D₃)⋊C₄	C₂×C₄	1	2	yes	—
71	C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
72	C₄⋊Dic₃	C₂×C₄	1	2	yes	—
73	C₄⋊Dic₃	C₂×C₄	1	2	yes	—
74	Dic₃⋊C₄	C₂×C₄	1	2	yes	—
75	Dic₃⋊C₄	C₂×C₄	1	2	yes	—
76	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
77	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
78	C₄×Dic₃	C₂³	1	2	yes	—
79	C₄×Dic₃	C₂³	1	1	yes	characteristic
80	C₂×C₃⋊C₈	—	4	2	no	—
81	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
82	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
83	C₂×C₃⋊C₈	C₂³	1	2	yes	—
84	ΓL(1,𝔽₂₅)	—	4	2	no	—
85	C₂×M₁₆	—	3	2	no	—
86	C₂×M₁₆	C₂×D₃	1	2	yes	—
87	C₂²×C₈	—	3	2	no	—
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	C₂×D₃	1	1	yes	characteristic
97	#117	—	3	2	no	—
98	C₂×C₄²	—	3	1	no	—
99	C₄⋊C₈	—	3	2	no	—
100	C₄⋊C₈	—	3	1	no	—
101	C₄⋊C₈	—	3	2	no	—
102	C₄⋊C₈	—	3	2	no	—
103	C₄⋊C₈	—	3	1	no	—
104	C₄⋊C₈	—	3	1	no	—
105	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
106	C₂²⋊C₈	—	3	2	no	—
107	C₂²⋊C₈	—	3	1	no	—
108	C₂²⋊C₈	—	3	1	no	—
109	C₂²⋊C₈	C₂×D₃	1	2	yes	—
110	#97	—	3	2	no	—
111	C₄×C₈	—	3	2	no	—
112	C₄×C₈	—	3	1	no	—
113	C₄×C₈	C₂×D₃	1	1	yes	characteristic
114	C₂³×C₃	C₂²×C₄	1	2	yes	—
115	C₂²×D₃	C₂²×C₄	1	1	yes	characteristic
116	C₃×D₄	—	2	2	no	—
117	C₂×C₃×C₄	—	2	2	no	—
118	C₂×C₃×C₄	—	2	2	no	—
119	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
121	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
123	#67	—	4	2	no	—
124	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
125	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
126	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
127	C₂×Dic₃	C₂²×C₄	1	2	yes	—
128	C₂×Dic₃	C₂²×C₄	1	2	yes	—
129	C₄×D₃	—	2	2	no	—
130	C₄×D₃	—	2	1	no	—
131	C₄×D₃	—	2	1	no	—
132	C₄×D₃	—	4	1	no	—
133	Dic₆	—	2	2	no	—
134	C₃×C₈	—	2	2	no	—
135	C₃×C₈	—	2	1	no	—
136	C₃×C₈	—	2	1	no	—
137	C₃⋊C₈	—	2	2	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	C₄×D₃	1	1	yes	characteristic
143	C₂²×C₄	—	3	1	no	—
144	C₂²×C₄	—	3	1	no	—
145	C₂²×C₄	—	3	1	no	—
146	C₂²×C₄	C₂²×D₃	1	2	yes	—
147	M₁₆	—	4	2	no	—
148	M₁₆	—	12	2	no	—
149	C₂×C₈	—	3	1	no	—
150	C₂×C₈	—	3	1	no	—
151	C₂×C₈	—	3	2	no	—
152	C₂×C₈	—	12	2	no	—
153	C₂×C₈	C₂²×D₃	1	2	yes	—
154	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
155	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
156	C₄⋊C₄	—	3	2	no	—
157	C₄⋊C₄	—	3	2	no	—
158	C₄⋊C₄	—	3	2	no	—
159	C₄⋊C₄	—	3	1	no	—
160	C₄⋊C₄	—	3	2	no	—
161	C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
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	C₄×D₃	1	2	yes	—
167	C₄²	—	3	1	no	—
168	C₄²	—	3	1	no	—
169	C₄²	—	3	2	no	—
170	C₄²	—	3	1	no	—
171	C₄²	C₂²×D₃	1	1	yes	characteristic
172	C₂²×C₃	—	2	2	no	—
173	C₂²×C₃	—	2	2	no	—
174	C₂²×C₃	—	2	2	no	—
175	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
176	C₂×D₃	—	2	1	no	—
177	C₂×D₃	—	2	1	no	—
178	C₂×D₃	—	2	1	no	—
179	C₃×C₄	—	2	2	no	—
180	C₃×C₄	—	2	1	no	—
181	C₃×C₄	#131	1	2	yes	—
182	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
183	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
184	Dic₃	—	2	1	no	—
185	Dic₃	—	2	2	no	—
186	Dic₃	—	2	1	no	—
187	Dic₃	—	2	1	no	—
188	Dic₃	—	2	2	no	—
189	C₂³	—	3	1	no	—
190	C₂³	C₂×C₄×D₃	1	2	yes	—
191	Q₈	—	6	2	no	—
192	D₄	—	2	2	no	—
193	D₄	—	12	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	3	1	no	—
197	C₂×C₄	—	3	1	no	—
198	C₂×C₄	—	3	1	no	—
199	C₂×C₄	—	3	2	no	—
200	C₂×C₄	—	3	2	no	—
201	C₂×C₄	—	6	2	no	—
202	C₂×C₄	—	6	1	no	—
203	C₂×C₄	—	6	1	no	—
204	C₂×C₄	—	12	1	no	—
205	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
206	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
207	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
208	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
209	C₈	—	2	1	no	—
210	C₈	—	2	2	no	—
211	C₈	—	2	1	no	—
212	C₈	—	6	1	no	—
213	C₈	—	6	2	no	—
214	C₈	—	6	1	no	—
215	C₂×C₃	—	4	2	no	—
216	C₂×C₃	#520	1	1	yes	characteristic
217	C₂×C₃	C₂×#131	1	1	yes	characteristic
218	C₂×C₃	#568	1	1	yes	characteristic
219	D₃	—	4	1	no	—
220	C₂²	—	2	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	6	1	no	—
224	C₂²	—	6	1	no	—
225	C₂²	—	6	1	no	—
226	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
227	C₄	—	2	1	no	—
228	C₄	—	2	2	no	—
229	C₄	—	6	1	no	—
230	C₄	—	6	2	no	—
231	C₄	—	6	1	no	—
232	C₄	—	6	2	no	—
233	C₄	—	6	1	no	—
234	C₄	#907	1	2	yes	—
235	C₄	#1004	1	1	yes	characteristic
236	C₄	#1010	1	1	yes	characteristic
237	C₃	#2993	1	1	yes	characteristic, complemented, Sylow
238	C₂	—	4	2	no	—
239	C₂	—	12	1	no	—
240	C₂	#5546	1	1	yes	characteristic
241	C₂	C₂×#907	1	1	yes	characteristic
242	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	#938	C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	#436	D₃
hypocenter	C₃	#2993
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
2. C₃×#436	C₂	semidirect
3. #5128	C₂	semidirect
4. #5119	C₂	semidirect
5. D₄×C₃⋊C₈	C₂	semidirect
6. #4837	C₂	semidirect
7. #4830	C₂	semidirect
8. #4730	C₂	semidirect
9. #4720	C₂	semidirect
10. #4685	C₂	semidirect
11. #4679	C₂	semidirect
46. #436	D₃	semidirect
237. C₃	#2993	semidirect

Maximal subgroups

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

subgroup	index
#5557	2
C₃×#436	2
#5128	2
#5119	2
D₄×C₃⋊C₈	2
#4837	2
#4830	2
#4730	2
#4720	2
#4685	2
#4679	2
#2993	3