Subgroups — #78511

Order 384.

370 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	#5479	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#5128	C₂	1	2	yes	maximal, complemented
4	#5127	C₂	1	2	yes	maximal, complemented
5	#4925	C₂	1	1	yes	maximal, characteristic, complemented
6	#4709	C₂	1	2	yes	maximal, complemented
7	#4708	C₂	1	2	yes	maximal, complemented
8	#4688	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
10	#4679	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#2969	—	3	1	no	maximal, Sylow
13	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
16	#930	C₄	1	1	yes	characteristic
17	#929	C₄	1	2	yes	—
18	C₄×#67	C₂²	1	2	yes	—
19	#924	C₄	1	1	yes	characteristic
20	C₂×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	—
21	C₂×D₃×C₈	C₂²	1	2	yes	—
22	#876	C₄	1	2	yes	—
23	#875	C₄	1	1	yes	characteristic
24	C₄×D₁₂	C₂²	1	1	yes	characteristic
25	#872	C₂²	1	2	yes	—
26	#870	C₄	1	1	yes	characteristic
27	C₄×Dic₆	C₂²	1	1	yes	characteristic
28	C₃×#97	C₂²	1	1	yes	characteristic
29	C₃×#97	C₂²	1	1	yes	characteristic
30	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
31	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
32	#830	C₂²	1	2	yes	—
33	(C₂×D₃)⋊C₈	C₂²	1	2	yes	—
34	(C₂×D₃)⋊C₈	C₂²	1	2	yes	—
35	#815	C₂²	1	2	yes	—
36	Dic₃⋊C₈	C₂²	1	2	yes	—
37	Dic₃⋊C₈	C₂²	1	2	yes	—
38	C₈×Dic₃	C₂²	1	2	yes	—
39	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
40	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
41	C₄×C₃⋊C₈	C₂²	1	2	yes	—
42	#529	—	3	1	no	—
43	#446	—	3	1	no	—
44	C₈×Q₈	—	3	1	no	—
45	#436	—	3	2	no	—
46	#436	—	3	1	no	—
47	#435	—	3	2	no	—
48	C₈×D₄	—	3	1	no	—
49	#433	—	3	2	no	—
50	#431	—	3	1	no	—
51	#405	—	3	2	no	—
52	C₄×M₁₆	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	C₃×M₁₆	—	2	2	no	—
61	C₃×M₁₆	—	2	2	no	—
62	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
63	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
64	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
65	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
66	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
67	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
68	C₃×C₄²	C₂³	1	1	yes	characteristic
69	C₃×C₄²	C₂³	1	1	yes	characteristic
70	#217	C₂×C₄	1	2	yes	—
71	(C₂×D₃)⋊C₄	C₂×C₄	1	2	yes	—
72	(C₂×D₃)⋊C₄	C₂×C₄	1	2	yes	—
73	(C₂×D₃)⋊C₄	C₂×C₄	1	2	yes	—
74	(C₂×D₃)⋊C₄	C₂×C₄	1	2	yes	—
75	C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
76	C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
77	Dic₃⋊C₄	C₂×C₄	1	2	yes	—
78	Dic₃⋊C₄	C₂×C₄	1	2	yes	—
79	Dic₃⋊C₄	C₂×C₄	1	2	yes	—
80	Dic₃⋊C₄	C₂×C₄	1	2	yes	—
81	C₄×Dic₃	C₂³	1	2	yes	—
82	C₂×C₃⋊C₈	C₂³	1	2	yes	—
83	C₂×C₃⋊C₈	C₂³	1	2	yes	—
84	ΓL(1,𝔽₂₅)	—	2	4	no	—
85	D₃×C₈	—	2	4	no	—
86	C₂×M₁₆	—	3	2	no	—
87	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
88	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
89	C₂²×C₈	—	3	2	no	—
90	#126	—	3	2	no	—
91	#125	—	3	1	no	—
92	#124	—	3	1	no	—
93	#123	—	3	2	no	—
94	C₂²⋊Q₈	—	3	1	no	—
95	#121	—	3	1	no	—
96	C₄×Q₈	—	3	1	no	—
97	C₄×D₄	—	3	2	no	—
98	C₄×D₄	—	3	1	no	—
99	#117	—	3	2	no	—
100	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
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	C₂²⋊C₈	—	3	2	no	—
106	C₂²⋊C₈	—	3	2	no	—
107	C₂²⋊C₈	—	3	2	no	—
108	#97	—	3	2	no	—
109	#97	C₂×D₃	1	1	yes	characteristic
110	#97	C₂×D₃	1	1	yes	characteristic
111	C₄×C₈	—	3	2	no	—
112	C₄×C₈	—	3	2	no	—
113	C₄×C₈	C₂×D₃	1	1	yes	characteristic
114	C₄×C₈	C₂×D₃	1	1	yes	characteristic
115	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
116	C₂²×D₃	C₂²×C₄	1	2	yes	—
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	—	4	1	no	—
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	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
127	#67	—	4	2	no	—
128	C₂×Dic₃	C₂²×C₄	1	2	yes	—
129	C₂×Dic₃	C₂²×C₄	1	2	yes	—
130	C₂×Dic₃	C₂²×C₄	1	2	yes	—
131	D₁₂	—	2	2	no	—
132	C₄×D₃	—	2	2	no	—
133	C₄×D₃	—	2	2	no	—
134	Dic₆	—	2	2	no	—
135	C₃×C₈	—	2	1	no	—
136	C₃×C₈	—	2	1	no	—
137	C₃×C₈	#41	1	2	yes	—
138	C₃×C₈	#41	1	2	yes	—
139	C₃⋊C₈	—	2	2	no	—
140	C₃⋊C₈	—	2	2	no	—
141	C₂×Q₈	—	3	1	no	—
142	C₂×D₄	—	3	2	no	—
143	C₂×D₄	—	3	1	no	—
144	C₂²×C₄	—	3	2	no	—
145	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
146	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
147	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
148	M₁₆	—	2	2	no	—
149	M₁₆	—	2	2	no	—
150	M₁₆	—	6	4	no	—
151	C₂×C₈	—	3	2	no	—
152	C₂×C₈	—	3	2	no	—
153	C₂×C₈	—	6	4	no	—
154	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
155	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
156	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
157	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
158	C₄⋊C₄	—	3	2	no	—
159	C₄⋊C₄	—	3	2	no	—
160	C₄⋊C₄	—	3	1	no	—
161	C₄⋊C₄	—	3	2	no	—
162	C₄⋊C₄	—	3	1	no	—
163	C₄⋊C₄	—	3	2	no	—
164	C₂²⋊C₄	—	3	2	no	—
165	C₂²⋊C₄	—	3	2	no	—
166	C₂²⋊C₄	—	3	2	no	—
167	C₂²⋊C₄	—	3	2	no	—
168	C₂²⋊C₄	—	3	2	no	—
169	C₄²	—	3	2	no	—
170	C₄²	C₄×D₃	1	1	yes	characteristic
171	C₄²	C₄×D₃	1	1	yes	characteristic
172	C₄²	C₂²×D₃	1	1	yes	characteristic
173	C₄²	C₂²×D₃	1	1	yes	characteristic
174	C₂²×C₃	—	2	1	no	—
175	C₂²×C₃	—	2	1	no	—
176	C₂²×C₃	—	2	1	no	—
177	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
178	C₂×D₃	—	2	2	no	—
179	C₂×D₃	—	2	2	no	—
180	C₂×D₃	—	2	2	no	—
181	C₃×C₄	—	2	1	no	—
182	C₃×C₄	—	2	1	no	—
183	C₃×C₄	—	2	1	no	—
184	C₃×C₄	—	2	1	no	—
185	C₃×C₄	#131	1	2	yes	—
186	C₃×C₄	C₂×#41	1	1	yes	characteristic
187	C₃×C₄	C₂×#41	1	1	yes	characteristic
188	Dic₃	—	2	2	no	—
189	Dic₃	—	2	2	no	—
190	Dic₃	—	2	2	no	—
191	C₂³	—	3	2	no	—
192	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
193	Q₈	—	6	2	no	—
194	D₄	—	6	2	no	—
195	D₄	—	12	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	3	2	no	—
200	C₂×C₄	—	3	2	no	—
201	C₂×C₄	—	3	2	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	6	2	no	—
204	C₂×C₄	—	6	2	no	—
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	C₂×C₄×D₃	1	1	yes	characteristic
209	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
210	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
211	C₈	—	2	1	no	—
212	C₈	—	2	1	no	—
213	C₈	—	6	2	no	—
214	C₈	—	6	2	no	—
215	C₈	#235	1	2	yes	—
216	C₈	#235	1	2	yes	—
217	C₂×C₃	—	4	1	no	—
218	C₂×C₃	C₄×#41	1	1	yes	characteristic
219	C₂×C₃	C₂×#131	1	1	yes	characteristic
220	C₂×C₃	#568	1	1	yes	characteristic
221	D₃	—	4	2	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	6	2	no	—
226	C₂²	—	6	2	no	—
227	C₂²	—	6	2	no	—
228	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	6	2	no	—
234	C₄	—	6	2	no	—
235	C₄	—	6	2	no	—
236	C₄	#907	1	2	yes	—
237	C₄	C₂×#235	1	1	yes	characteristic
238	C₄	C₂×#235	1	1	yes	characteristic
239	C₃	#2969	1	1	yes	characteristic, complemented, Sylow
240	C₂	—	4	1	no	—
241	C₂	—	12	2	no	—
242	C₂	C₄×#235	1	1	yes	characteristic
243	C₂	C₂×#907	1	1	yes	characteristic
244	C₂	#5745	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₃×C₄×M₁₆	C₂
prime radical	#930	C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₄×M₁₆	D₃
hypocenter	C₃	#2969
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×M₁₆	C₂	semidirect
3. #5128	C₂	semidirect
4. #5127	C₂	semidirect
5. #4925	C₂	semidirect
6. #4709	C₂	semidirect
7. #4708	C₂	semidirect
8. #4688	C₂	semidirect
9. C₈×D₁₂	C₂	semidirect
10. #4679	C₂	semidirect
11. C₈×Dic₆	C₂	semidirect
52. C₄×M₁₆	D₃	semidirect
239. C₃	#2969	semidirect

Maximal subgroups

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

subgroup	index
#5479	2
C₃×C₄×M₁₆	2
#5128	2
#5127	2
#4925	2
#4709	2
#4708	2
#4688	2
C₈×D₁₂	2
#4679	2
C₈×Dic₆	2
#2969	3