Subgroups — #10700992

Order 768.

842 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
2	#82062	C₂	1	2	yes	maximal
3	C₂×#4557	C₂	1	4	yes	maximal
4	#10837	—	3	1	no	maximal, Sylow
5	C₂²×C₃×M₁₆	C₂²	1	2	yes	—
6	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
7	C₂×#926	C₂²	1	4	yes	—
8	C₃×C₄×M₁₆	—	2	2	no	—
9	C₃×C₄×M₁₆	—	2	2	no	—
10	C₂×C₃×#97	C₄	1	2	yes	complemented
11	C₂×C₃×C₄×C₈	C₄	1	2	yes	complemented
12	#5212	—	2	4	no	—
13	#4936	—	2	2	no	—
14	#4935	—	2	2	no	—
15	C₄×C₄⋊Dic₃	C₄	1	8	yes	complemented
16	#4557	—	2	8	no	—
17	C₂×C₄×M₁₆	D₃	1	1	yes	characteristic
18	#2285	—	3	2	no	—
19	C₂×H(3,ℤ/4)	—	3	4	no	—
20	C₂³×C₃×C₄	D₄	1	2	yes	—
21	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
22	C₂×C₃×M₁₆	—	2	4	no	—
23	C₂×C₃×M₁₆	—	2	4	no	—
24	C₂×C₃×M₁₆	D₄	1	4	yes	—
25	C₂×C₃×M₁₆	Q₈	1	4	yes	—
26	C₂²×C₃×C₈	C₂×C₄	1	4	yes	—
27	C₂×C₃×C₄²	C₂×C₄	1	4	yes	—
28	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₄²	D₄	1	2	yes	—
31	C₂×C₃×C₄²	D₄	1	2	yes	—
32	C₂×C₃×C₄²	Q₈	1	2	yes	—
33	C₂×#217	—	2	4	no	—
34	#926	—	2	8	no	—
35	#926	—	2	8	no	—
36	C₂×C₄⋊Dic₃	—	2	8	no	—
37	C₂×C₄⋊Dic₃	—	2	4	no	—
38	C₂×C₄×Dic₃	C₂×C₄	1	8	yes	—
39	C₃×#97	—	2	4	no	—
40	C₃×C₄×C₈	—	2	2	no	—
41	C₃×C₄×C₈	—	2	2	no	—
42	C₃⋊#95	—	2	8	no	—
43	C₂²×M₁₆	C₂×D₃	1	2	yes	—
44	C₂×#117	—	3	4	no	—
45	C₂²×C₄²	C₂×D₃	1	1	yes	characteristic
46	C₄×M₁₆	—	2	2	no	—
47	C₄×M₁₆	—	2	2	no	—
48	C₂×#97	Dic₃	1	2	yes	complemented
49	C₂×C₄×C₈	Dic₃	1	2	yes	complemented
50	#384	—	6	2	no	—
51	#382	—	6	2	no	—
52	#380	—	6	4	no	—
53	C₄×C₄⋊C₄	—	3	8	no	—
54	H(3,ℤ/4)	—	6	8	no	—
55	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₃×C₄	—	4	4	no	—
57	C₂²×C₃×C₄	C₂²⋊C₄	1	4	yes	—
58	C₂²×C₃×C₄	C₂²⋊C₄	1	4	yes	—
59	C₂²×C₃×C₄	C₄⋊C₄	1	2	yes	—
60	C₂²×C₃×C₄	C₄⋊C₄	1	2	yes	—
61	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
62	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
64	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
65	C₂²×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₃×C₄	C₂×Q₈	1	2	yes	—
67	C₂²×C₃×C₄	#41	1	4	yes	—
68	C₂²×Dic₃	—	2	8	no	—
69	C₃×M₁₆	—	2	8	no	—
70	C₃×M₁₆	—	2	8	no	—
71	C₂×C₃×C₈	—	2	4	no	—
72	C₂×C₃×C₈	—	2	4	no	—
73	C₂×C₃×C₈	C₄⋊C₄	1	8	yes	complemented
74	C₃×C₄²	C₄²	1	2	yes	—
75	C₃×C₄²	C₄²	1	2	yes	—
76	C₃×C₄²	C₂²⋊C₄	1	1	yes	characteristic
77	C₃×C₄²	C₂²⋊C₄	1	1	yes	characteristic
78	C₃×C₄²	C₂²⋊C₄	1	2	yes	—
79	C₃×C₄²	C₄⋊C₄	1	4	yes	—
80	C₃×C₄²	C₄⋊C₄	1	4	yes	—
81	#217	—	4	4	no	—
82	#217	—	4	4	no	—
83	C₄⋊Dic₃	—	4	16	no	—
84	C₄⋊Dic₃	—	4	8	no	—
85	C₄×Dic₃	—	2	16	no	—
86	C₂³×C₄	D₁₂	1	2	yes	—
87	C₂³×C₄	C₂²×D₃	1	1	yes	characteristic
88	C₂×M₁₆	—	2	4	no	—
89	C₂×M₁₆	—	2	4	no	—
90	C₂×M₁₆	Dic₆	1	4	yes	—
91	C₂×M₁₆	D₁₂	1	4	yes	—
92	C₂²×C₈	C₂×Dic₃	1	4	yes	—
93	#117	—	6	8	no	—
94	#117	—	6	8	no	—
95	C₂×C₄⋊C₄	—	6	8	no	—
96	C₂×C₄⋊C₄	—	6	4	no	—
97	C₂×C₂²⋊C₄	—	6	4	no	—
98	C₂×C₄²	—	3	8	no	—
99	C₂×C₄²	Dic₆	1	2	yes	—
100	C₂×C₄²	C₄×D₃	1	4	yes	—
101	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
102	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
103	C₂×C₄²	#67	1	2	yes	—
104	C₂×C₄²	#67	1	2	yes	—
105	#97	—	2	4	no	—
106	C₄×C₈	—	2	2	no	—
107	C₄×C₈	—	2	2	no	—
108	#95	—	6	8	no	—
109	C₂³×C₃	—	4	2	no	—
110	C₂³×C₃	C₂×C₂²⋊C₄	1	2	yes	—
111	C₂³×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂³×C₃	#124	1	2	yes	—
113	C₂³×C₃	#127	1	1	yes	characteristic
114	C₂³×C₃	C₄⋊Q₈	1	1	yes	characteristic
115	C₂×C₃×C₄	—	2	8	no	—
116	C₂×C₃×C₄	—	2	4	no	—
117	C₂×C₃×C₄	—	4	4	no	—
118	C₂×C₃×C₄	—	4	2	no	—
119	C₂×C₃×C₄	#95	1	2	yes	—
120	C₂×C₃×C₄	#95	1	4	yes	—
121	C₂×C₃×C₄	#95	1	2	yes	—
122	C₂×C₃×C₄	C₂×C₄²	1	1	yes	characteristic
123	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₃×C₄	#117	1	2	yes	—
127	C₂×C₃×C₄	C₄×D₄	1	4	yes	—
128	C₂×C₃×C₄	C₄×Q₈	1	4	yes	—
129	C₂×C₃×C₄	#121	1	4	yes	—
130	C₂×C₃×C₄	C₂²⋊Q₈	1	4	yes	—
131	C₂×C₃×C₄	#125	1	2	yes	—
132	C₂×C₃×C₄	C₄⋊Q₈	1	2	yes	—
133	C₂×Dic₃	—	4	16	no	—
134	C₂×Dic₃	—	4	8	no	—
135	C₃×C₈	—	2	8	no	—
136	C₂⁴	C₂×D₁₂	1	1	yes	characteristic
137	C₂²×C₄	—	4	4	no	—
138	C₂²×C₄	—	6	8	no	—
139	C₂²×C₄	C₄⋊Dic₃	1	2	yes	—
140	C₂²×C₄	C₄⋊Dic₃	1	2	yes	—
141	C₂²×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
142	C₂²×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
143	C₂²×C₄	C₂×Dic₆	1	2	yes	—
144	C₂²×C₄	C₂×Dic₆	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₄×D₃	1	2	yes	—
146	C₂²×C₄	C₂×D₁₂	1	2	yes	—
147	C₂²×C₄	#235	1	4	yes	—
148	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
149	C₂²×C₄	C₂×#67	1	2	yes	—
150	M₁₆	—	2	8	no	—
151	M₁₆	—	2	8	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	C₄⋊Dic₃	1	8	yes	complemented
155	C₄⋊C₄	—	12	16	no	—
156	C₄⋊C₄	—	12	8	no	—
157	C₂²⋊C₄	—	12	4	no	—
158	C₂²⋊C₄	—	12	4	no	—
159	C₄²	—	6	16	no	—
160	C₄²	C₄×Dic₃	1	2	yes	—
161	C₄²	C₄×Dic₃	1	2	yes	—
162	C₄²	Dic₃⋊C₄	1	4	yes	—
163	C₄²	Dic₃⋊C₄	1	4	yes	—
164	C₄²	#217	1	1	yes	characteristic
165	C₄²	#217	1	1	yes	characteristic
166	C₄²	#217	1	2	yes	—
167	C₂²×C₃	—	2	4	no	—
168	C₂²×C₃	—	2	2	no	—
169	C₂²×C₃	—	4	2	no	—
170	C₂²×C₃	—	4	1	no	—
171	C₂²×C₃	—	4	1	no	—
172	C₂²×C₃	C₂×#95	1	1	yes	characteristic
173	C₂²×C₃	#382	1	1	yes	characteristic
174	C₂²×C₃	#384	1	1	yes	characteristic
175	C₂²×C₃	#390	1	2	yes	—
176	C₂²×C₃	#391	1	2	yes	—
177	C₃×C₄	—	2	4	no	—
178	C₃×C₄	—	2	2	no	—
179	C₃×C₄	H(3,ℤ/4)	1	4	yes	complemented
180	C₃×C₄	C₄×C₄⋊C₄	1	2	yes	—
181	C₃×C₄	#380	1	2	yes	—
182	C₃×C₄	#389	1	4	yes	—
183	Dic₃	—	8	8	no	—
184	C₂³	—	4	2	no	—
185	C₂³	#869	1	1	yes	characteristic
186	C₂³	#874	1	1	yes	characteristic
187	C₂³	#875	1	2	yes	—
188	C₂³	C₂×C₄⋊Dic₃	1	1	yes	characteristic
189	C₂³	C₂×(C₂×D₃)⋊C₄	1	2	yes	—
190	C₂×C₄	—	2	8	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	4	no	—
194	C₂×C₄	—	12	16	no	—
195	C₂×C₄	—	12	8	no	—
196	C₂×C₄	C₃⋊#95	1	2	yes	—
197	C₂×C₄	C₃⋊#95	1	4	yes	—
198	C₂×C₄	C₃⋊#95	1	2	yes	—
199	C₂×C₄	C₄×Dic₆	1	4	yes	—
200	C₂×C₄	#869	1	2	yes	—
201	C₂×C₄	#870	1	2	yes	—
202	C₂×C₄	C₄×D₁₂	1	4	yes	—
203	C₂×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
204	C₂×C₄	C₂×Dic₃⋊C₄	1	2	yes	—
205	C₂×C₄	#924	1	4	yes	—
206	C₂×C₄	C₂×C₄⋊Dic₃	1	2	yes	—
207	C₂×C₄	#926	1	2	yes	—
208	C₂×C₄	#930	1	4	yes	—
209	C₂×C₄	C₂×#217	1	1	yes	characteristic
210	C₈	—	2	8	no	—
211	C₂×C₃	—	2	2	no	—
212	C₂×C₃	—	2	1	no	—
213	C₂×C₃	—	2	1	no	—
214	C₂×C₃	—	2	2	no	—
215	C₂×C₃	C₄⋊#95	1	1	yes	characteristic
216	C₂×C₃	C₂×H(3,ℤ/4)	1	2	yes	—
217	C₂²	—	2	4	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	4	1	no	—
220	C₂²	—	4	1	no	—
221	C₂²	—	4	2	no	—
222	C₂²	#4929	1	2	yes	—
223	C₂²	#4935	1	1	yes	characteristic
224	C₂²	#4936	1	1	yes	characteristic
225	C₂²	#4939	1	2	yes	—
226	C₂²	C₂×C₃⋊#95	1	1	yes	characteristic
227	C₄	—	2	4	no	—
228	C₄	—	2	2	no	—
229	C₄	—	24	8	no	—
230	C₄	#4557	1	4	yes	complemented
231	C₄	#4928	1	4	yes	—
232	C₄	C₄×C₄⋊Dic₃	1	2	yes	—
233	C₄	#5212	1	2	yes	—
234	C₃	#10837	1	1	yes	characteristic, complemented, Sylow
235	C₂	—	2	2	no	—
236	C₂	—	2	2	no	—
237	C₂	—	2	1	no	—
238	C₂	—	2	1	no	—
239	C₂	#71535	1	1	yes	characteristic
240	C₂	C₂×#4557	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₃
commutator	C₂×C₃×C₄	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₃
Fitting	C₂×C₃×C₄×M₁₆	C₂
socle	C₂²×C₃	C₂×#95
prime radical	C₂⁴×C₃	C₂×D₄
cosocle	C₂³×C₃×C₄	C₂³
hypercenter	C₂×C₄×M₁₆	D₃
hypocenter	C₃	#10837
LCS	G ▷ C₂×C₃×C₄ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
10. C₂×C₃×#97	C₄	semidirect
11. C₂×C₃×C₄×C₈	C₄	semidirect
15. C₄×C₄⋊Dic₃	C₄	semidirect
48. C₂×#97	Dic₃	semidirect
49. C₂×C₄×C₈	Dic₃	semidirect
73. C₂×C₃×C₈	C₄⋊C₄	semidirect
154. C₂×C₈	C₄⋊Dic₃	semidirect
179. C₃×C₄	H(3,ℤ/4)	semidirect
230. C₄	#4557	semidirect
234. C₃	#10837	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×M₁₆	2
#82062	2
C₂×#4557	2
#10837	3