Subgroups — C₂³×#67⋊C₄ #11701749

Order 768.

19998 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×#67	C₂	1	1	yes	maximal, characteristic
2	C₂⁵×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂⁴×C₄×D₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×C₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×(C₂×D₃)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×Dic₃⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₄×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#67⋊C₄	C₂	1	56	yes	maximal, direct product factor
9	C₂³×C₄×D₄	—	3	1	no	maximal, Sylow
10	C₂⁶×C₃	C₂²	1	1	yes	characteristic
11	C₂⁵×D₃	C₂²	1	1	yes	characteristic
12	C₂⁴×C₃×C₄	C₂²	1	1	yes	characteristic
13	C₂⁴×C₃×C₄	C₂²	1	1	yes	characteristic
14	C₂³×#67	C₄	1	32	yes	complemented
15	C₂³×#67	C₂²	1	28	yes	—
16	C₂⁴×Dic₃	—	2	16	no	—
17	C₂⁴×Dic₃	C₂²	1	1	yes	characteristic
18	C₂⁴×Dic₃	C₂²	1	1	yes	characteristic
19	C₂⁴×Dic₃	C₂²	1	1	yes	characteristic
20	C₂⁴×Dic₃	C₂²	1	28	yes	complemented
21	C₂³×C₄×D₃	—	2	16	no	—
22	C₂³×C₄×D₃	C₂²	1	28	yes	complemented
23	C₂²×C₃×C₂²⋊C₄	C₂²	1	28	yes	complemented
24	C₂²×(C₂×D₃)⋊C₄	C₂²	1	28	yes	complemented
25	C₂²×Dic₃⋊C₄	C₂²	1	28	yes	complemented
26	C₂²×C₄×Dic₃	C₂²	1	28	yes	complemented
27	C₂×#67⋊C₄	C₂²	1	448	yes	direct product factor
28	C₂⁴×D₄	—	3	1	no	—
29	C₂⁵×C₄	—	3	1	no	—
30	C₂⁵×C₄	—	3	1	no	—
31	C₂²×C₄×D₄	—	3	56	no	—
32	C₂³×C₄⋊C₄	—	3	1	no	—
33	C₂³×C₂²⋊C₄	—	3	1	no	—
34	C₂³×C₂²⋊C₄	D₃	1	1	yes	characteristic, complemented
35	C₂³×C₄²	—	3	1	no	—
36	C₂⁵×C₃	—	2	8	no	—
37	C₂⁵×C₃	—	2	8	no	—
38	C₂⁵×C₃	C₂×C₄	1	16	yes	complemented
39	C₂⁵×C₃	C₂³	1	14	yes	—
40	C₂⁵×C₃	C₂³	1	1	yes	characteristic
41	C₂⁴×D₃	—	2	8	no	—
42	C₂⁴×D₃	—	2	8	no	—
43	C₂⁴×D₃	C₂×C₄	1	16	yes	complemented
44	C₂⁴×D₃	C₂³	1	14	yes	—
45	C₂³×C₃×C₄	—	2	8	no	—
46	C₂³×C₃×C₄	—	2	8	no	—
47	C₂³×C₃×C₄	C₂³	1	14	yes	—
48	C₂³×C₃×C₄	C₂³	1	14	yes	—
49	C₂²×#67	C₂×C₄	1	448	yes	complemented
50	C₂²×#67	C₂³	1	112	yes	—
51	C₂³×Dic₃	—	2	8	no	—
52	C₂³×Dic₃	—	2	224	no	—
53	C₂³×Dic₃	C₂×C₄	1	16	yes	complemented
54	C₂³×Dic₃	D₄	1	16	yes	complemented
55	C₂³×Dic₃	C₂³	1	14	yes	—
56	C₂³×Dic₃	C₂³	1	14	yes	—
57	C₂³×Dic₃	C₂³	1	14	yes	—
58	C₂³×Dic₃	C₂³	1	112	yes	complemented
59	C₂²×C₄×D₃	—	2	224	no	—
60	C₂²×C₄×D₃	C₂³	1	112	yes	complemented
61	C₂×C₃×C₂²⋊C₄	C₂³	1	112	yes	complemented
62	C₂×(C₂×D₃)⋊C₄	C₂³	1	112	yes	complemented
63	C₂×Dic₃⋊C₄	C₂³	1	112	yes	complemented
64	C₂×C₄×Dic₃	C₂³	1	112	yes	complemented
65	#67⋊C₄	C₂³	1	512	yes	direct product factor
66	C₂⁶	—	3	1	no	—
67	C₂⁶	C₂×D₃	1	1	yes	characteristic
68	C₂³×D₄	—	3	32	no	—
69	C₂³×D₄	—	3	28	no	—
70	C₂⁴×C₄	—	3	1	no	—
71	C₂⁴×C₄	—	3	1	no	—
72	C₂⁴×C₄	—	3	28	no	—
73	C₂⁴×C₄	—	3	1	no	—
74	C₂⁴×C₄	—	3	28	no	—
75	C₂⁴×C₄	—	6	16	no	—
76	C₂⁴×C₄	—	6	16	no	—
77	C₂⁴×C₄	C₂×D₃	1	1	yes	characteristic
78	C₂⁴×C₄	C₂×D₃	1	1	yes	characteristic
79	C₂×C₄×D₄	—	3	448	no	—
80	C₂²×C₄⋊C₄	—	3	28	no	—
81	C₂²×C₂²⋊C₄	—	3	28	no	—
82	C₂²×C₂²⋊C₄	C₂×D₃	1	28	yes	complemented
83	C₂²×C₄²	—	3	28	no	—
84	C₂⁴×C₃	—	2	128	no	—
85	C₂⁴×C₃	—	2	56	no	—
86	C₂⁴×C₃	—	2	56	no	—
87	C₂⁴×C₃	C₂²×C₄	1	8	yes	—
88	C₂⁴×C₃	C₂²×C₄	1	112	yes	complemented
89	C₂⁴×C₃	C₂×D₄	1	8	yes	—
90	C₂⁴×C₃	#41	1	8	yes	—
91	C₂⁴×C₃	C₂⁴	1	28	yes	—
92	C₂⁴×C₃	C₂⁴	1	7	yes	—
93	C₂³×D₃	—	2	128	no	—
94	C₂³×D₃	—	2	56	no	—
95	C₂³×D₃	—	2	56	no	—
96	C₂³×D₃	C₂²×C₄	1	112	yes	complemented
97	C₂³×D₃	C₂⁴	1	28	yes	—
98	C₂²×C₃×C₄	—	2	56	no	—
99	C₂²×C₃×C₄	—	2	56	no	—
100	C₂²×C₃×C₄	C₂⁴	1	28	yes	—
101	C₂²×C₃×C₄	C₂⁴	1	28	yes	—
102	C₂×#67	C₂²×C₄	1	896	yes	complemented
103	C₂×#67	C₂⁴	1	64	yes	—
104	C₂²×Dic₃	—	2	56	no	—
105	C₂²×Dic₃	—	2	448	no	—
106	C₂²×Dic₃	C₂²×C₄	1	112	yes	complemented
107	C₂²×Dic₃	C₂×D₄	1	112	yes	complemented
108	C₂²×Dic₃	C₂⁴	1	28	yes	—
109	C₂²×Dic₃	C₂⁴	1	28	yes	—
110	C₂²×Dic₃	C₂⁴	1	28	yes	—
111	C₂²×Dic₃	C₂⁴	1	64	yes	complemented
112	C₂×C₄×D₃	—	2	448	no	—
113	C₂×C₄×D₃	C₂⁴	1	64	yes	complemented
114	C₃×C₂²⋊C₄	C₂⁴	1	64	yes	complemented
115	(C₂×D₃)⋊C₄	C₂⁴	1	64	yes	complemented
116	Dic₃⋊C₄	C₂⁴	1	64	yes	complemented
117	C₄×Dic₃	C₂⁴	1	64	yes	complemented
118	C₂⁵	—	2	8	no	—
119	C₂⁵	—	2	8	no	—
120	C₂⁵	—	3	16	no	—
121	C₂⁵	—	3	14	no	—
122	C₂⁵	—	6	8	no	—
123	C₂⁵	—	6	8	no	—
124	C₂⁵	C₄×D₃	1	16	yes	complemented
125	C₂⁵	C₂²×D₃	1	14	yes	—
126	C₂⁵	C₂²×D₃	1	1	yes	characteristic
127	C₂²×D₄	—	3	448	no	—
128	C₂²×D₄	—	3	112	no	—
129	C₂³×C₄	—	2	8	no	—
130	C₂³×C₄	—	2	8	no	—
131	C₂³×C₄	—	3	16	no	—
132	C₂³×C₄	—	3	16	no	—
133	C₂³×C₄	—	3	14	no	—
134	C₂³×C₄	—	3	14	no	—
135	C₂³×C₄	—	3	14	no	—
136	C₂³×C₄	—	3	112	no	—
137	C₂³×C₄	—	3	112	no	—
138	C₂³×C₄	—	6	224	no	—
139	C₂³×C₄	—	6	224	no	—
140	C₂³×C₄	—	6	8	no	—
141	C₂³×C₄	C₂²×D₃	1	14	yes	—
142	C₂³×C₄	C₂²×D₃	1	14	yes	—
143	C₄×D₄	—	3	512	no	—
144	C₂×C₄⋊C₄	—	3	112	no	—
145	C₂×C₂²⋊C₄	—	3	112	no	—
146	C₂×C₂²⋊C₄	C₂²×D₃	1	112	yes	complemented
147	C₂×C₄²	—	3	112	no	—
148	C₂³×C₃	—	2	448	no	—
149	C₂³×C₃	—	2	56	no	—
150	C₂³×C₃	—	2	56	no	—
151	C₂³×C₃	C₄×D₄	1	64	yes	complemented
152	C₂³×C₃	C₂³×C₄	1	28	yes	—
153	C₂³×C₃	C₂³×C₄	1	112	yes	complemented
154	C₂³×C₃	C₂²×D₄	1	28	yes	—
155	C₂³×C₃	C₂×#41	1	28	yes	—
156	C₂³×C₃	C₂⁵	1	8	yes	—
157	C₂³×C₃	C₂⁵	1	7	yes	—
158	C₂²×D₃	—	2	448	no	—
159	C₂²×D₃	—	2	56	no	—
160	C₂²×D₃	—	2	56	no	—
161	C₂²×D₃	C₂³×C₄	1	112	yes	complemented
162	C₂²×D₃	C₂⁵	1	8	yes	—
163	C₂×C₃×C₄	—	2	56	no	—
164	C₂×C₃×C₄	—	2	56	no	—
165	C₂×C₃×C₄	C₂⁵	1	8	yes	—
166	C₂×C₃×C₄	C₂⁵	1	8	yes	—
167	#67	C₂³×C₄	1	256	yes	complemented
168	C₂×Dic₃	—	2	56	no	—
169	C₂×Dic₃	—	2	128	no	—
170	C₂×Dic₃	C₂³×C₄	1	112	yes	complemented
171	C₂×Dic₃	C₂²×D₄	1	112	yes	complemented
172	C₂×Dic₃	C₂⁵	1	8	yes	—
173	C₂×Dic₃	C₂⁵	1	8	yes	—
174	C₂×Dic₃	C₂⁵	1	8	yes	—
175	C₄×D₃	—	2	128	no	—
176	C₂⁴	—	2	128	no	—
177	C₂⁴	—	2	56	no	—
178	C₂⁴	—	2	56	no	—
179	C₂⁴	—	3	112	no	—
180	C₂⁴	—	3	28	no	—
181	C₂⁴	—	6	128	no	—
182	C₂⁴	—	6	56	no	—
183	C₂⁴	—	6	56	no	—
184	C₂⁴	C₂×C₄×D₃	1	8	yes	—
185	C₂⁴	C₂×C₄×D₃	1	112	yes	complemented
186	C₂⁴	D₃×D₄	1	8	yes	—
187	C₂⁴	#237	1	8	yes	—
188	C₂⁴	C₂³×D₃	1	28	yes	—
189	C₂⁴	C₂³×D₃	1	7	yes	—
190	C₂×D₄	—	3	896	no	—
191	C₂×D₄	—	3	64	no	—
192	C₂²×C₄	—	2	56	no	—
193	C₂²×C₄	—	2	56	no	—
194	C₂²×C₄	—	3	112	no	—
195	C₂²×C₄	—	3	112	no	—
196	C₂²×C₄	—	3	28	no	—
197	C₂²×C₄	—	3	28	no	—
198	C₂²×C₄	—	3	28	no	—
199	C₂²×C₄	—	3	64	no	—
200	C₂²×C₄	—	3	64	no	—
201	C₂²×C₄	—	6	448	no	—
202	C₂²×C₄	—	6	448	no	—
203	C₂²×C₄	—	6	56	no	—
204	C₂²×C₄	C₂³×D₃	1	28	yes	—
205	C₂²×C₄	C₂³×D₃	1	28	yes	—
206	C₄⋊C₄	—	3	64	no	—
207	C₂²⋊C₄	—	3	64	no	—
208	C₂²⋊C₄	C₂³×D₃	1	64	yes	complemented
209	C₄²	—	3	64	no	—
210	C₂²×C₃	—	2	8	no	—
211	C₂²×C₃	—	2	8	no	—
212	C₂²×C₃	—	2	224	no	—
213	C₂²×C₃	C₂×C₄×D₄	1	112	yes	complemented
214	C₂²×C₃	C₂⁴×C₄	1	14	yes	—
215	C₂²×C₃	C₂⁴×C₄	1	16	yes	complemented
216	C₂²×C₃	C₂³×D₄	1	14	yes	—
217	C₂²×C₃	C₂²×#41	1	14	yes	—
218	C₂²×C₃	C₂⁶	1	1	yes	characteristic
219	C₂×D₃	—	2	8	no	—
220	C₂×D₃	—	2	8	no	—
221	C₂×D₃	—	2	224	no	—
222	C₂×D₃	C₂⁴×C₄	1	16	yes	complemented
223	C₃×C₄	—	2	8	no	—
224	C₃×C₄	—	2	8	no	—
225	Dic₃	—	2	8	no	—
226	Dic₃	C₂⁴×C₄	1	16	yes	complemented
227	Dic₃	C₂³×D₄	1	16	yes	complemented
228	C₂³	—	2	56	no	—
229	C₂³	—	2	56	no	—
230	C₂³	—	2	448	no	—
231	C₂³	—	3	112	no	—
232	C₂³	—	3	8	no	—
233	C₂³	—	6	56	no	—
234	C₂³	—	6	56	no	—
235	C₂³	—	6	448	no	—
236	C₂³	#67⋊C₄	1	64	yes	direct product factor
237	C₂³	C₂²×C₄×D₃	1	28	yes	—
238	C₂³	C₂²×C₄×D₃	1	112	yes	complemented
239	C₂³	C₂×D₃×D₄	1	28	yes	—
240	C₂³	C₂×#237	1	28	yes	—
241	C₂³	C₂⁴×D₃	1	8	yes	—
242	C₂³	C₂⁴×D₃	1	7	yes	—
243	D₄	—	3	256	no	—
244	C₂×C₄	—	2	56	no	—
245	C₂×C₄	—	2	56	no	—
246	C₂×C₄	—	3	112	no	—
247	C₂×C₄	—	3	112	no	—
248	C₂×C₄	—	3	8	no	—
249	C₂×C₄	—	3	8	no	—
250	C₂×C₄	—	3	8	no	—
251	C₂×C₄	—	6	128	no	—
252	C₂×C₄	—	6	128	no	—
253	C₂×C₄	—	6	56	no	—
254	C₂×C₄	C₂⁴×D₃	1	8	yes	—
255	C₂×C₄	C₂⁴×D₃	1	8	yes	—
256	C₂×C₃	—	2	16	no	—
257	C₂×C₃	C₂²×C₄×D₄	1	28	yes	complemented
258	C₂×C₃	C₂⁵×C₄	1	1	yes	characteristic
259	C₂×C₃	C₂⁴×D₄	1	1	yes	characteristic
260	C₂×C₃	C₂³×#41	1	1	yes	characteristic
261	D₃	—	2	16	no	—
262	C₂²	—	2	8	no	—
263	C₂²	—	2	8	no	—
264	C₂²	—	2	224	no	—
265	C₂²	—	3	16	no	—
266	C₂²	—	6	8	no	—
267	C₂²	—	6	8	no	—
268	C₂²	—	6	224	no	—
269	C₂²	C₂×#67⋊C₄	1	112	yes	direct product factor
270	C₂²	C₂³×C₄×D₃	1	14	yes	—
271	C₂²	C₂³×C₄×D₃	1	16	yes	complemented
272	C₂²	C₂²×D₃×D₄	1	14	yes	—
273	C₂²	C₂²×#237	1	14	yes	—
274	C₂²	C₂⁵×D₃	1	1	yes	characteristic
275	C₄	—	2	8	no	—
276	C₄	—	2	8	no	—
277	C₄	—	3	16	no	—
278	C₄	—	3	16	no	—
279	C₄	—	6	8	no	—
280	C₃	C₂³×C₄×D₄	1	1	yes	characteristic, complemented, Sylow
281	C₂	—	2	16	no	—
282	C₂	—	6	16	no	—
283	C₂	C₂²×#67⋊C₄	1	28	yes	direct product factor
284	C₂	C₂⁴×C₄×D₃	1	1	yes	characteristic
285	C₂	C₂³×#237	1	1	yes	characteristic
286	C₂	C₂³×D₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₃
commutator	C₂×C₃	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₃
Fitting	C₂³×C₃×C₂²⋊C₄	C₂
socle	C₂⁵×C₃	C₂³
prime radical	C₂⁴×#67	C₂
cosocle	C₂²×C₃	C₂⁶
hypercenter	C₂³×C₂²⋊C₄	D₃
hypocenter	C₃	C₂³×C₄×D₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂⁵×Dic₃	C₂	semidirect
3. C₂⁴×C₄×D₃	C₂	semidirect
4. C₂³×C₃×C₂²⋊C₄	C₂	semidirect
5. C₂³×(C₂×D₃)⋊C₄	C₂	semidirect
6. C₂³×Dic₃⋊C₄	C₂	semidirect
7. C₂³×C₄×Dic₃	C₂	semidirect
8. C₂²×#67⋊C₄	C₂	direct (N × H)
14. C₂³×#67	C₄	semidirect
20. C₂⁴×Dic₃	C₂²	semidirect
22. C₂³×C₄×D₃	C₂²	semidirect
23. C₂²×C₃×C₂²⋊C₄	C₂²	semidirect
24. C₂²×(C₂×D₃)⋊C₄	C₂²	semidirect
25. C₂²×Dic₃⋊C₄	C₂²	semidirect
26. C₂²×C₄×Dic₃	C₂²	semidirect
27. C₂×#67⋊C₄	C₂²	direct (N × H)
34. C₂³×C₂²⋊C₄	D₃	semidirect
38. C₂⁵×C₃	C₂×C₄	semidirect
43. C₂⁴×D₃	C₂×C₄	semidirect
49. C₂²×#67	C₂×C₄	semidirect
53. C₂³×Dic₃	C₂×C₄	semidirect
54. C₂³×Dic₃	D₄	semidirect
58. C₂³×Dic₃	C₂³	semidirect
60. C₂²×C₄×D₃	C₂³	semidirect
61. C₂×C₃×C₂²⋊C₄	C₂³	semidirect
62. C₂×(C₂×D₃)⋊C₄	C₂³	semidirect
63. C₂×Dic₃⋊C₄	C₂³	semidirect
64. C₂×C₄×Dic₃	C₂³	semidirect
65. #67⋊C₄	C₂³	direct (N × H)
82. C₂²×C₂²⋊C₄	C₂×D₃	semidirect
88. C₂⁴×C₃	C₂²×C₄	semidirect
96. C₂³×D₃	C₂²×C₄	semidirect
102. C₂×#67	C₂²×C₄	semidirect
106. C₂²×Dic₃	C₂²×C₄	semidirect
107. C₂²×Dic₃	C₂×D₄	semidirect
111. C₂²×Dic₃	C₂⁴	semidirect
113. C₂×C₄×D₃	C₂⁴	semidirect
114. C₃×C₂²⋊C₄	C₂⁴	semidirect
115. (C₂×D₃)⋊C₄	C₂⁴	semidirect
116. Dic₃⋊C₄	C₂⁴	semidirect
117. C₄×Dic₃	C₂⁴	semidirect
124. C₂⁵	C₄×D₃	semidirect
146. C₂×C₂²⋊C₄	C₂²×D₃	semidirect
151. C₂³×C₃	C₄×D₄	semidirect
153. C₂³×C₃	C₂³×C₄	semidirect
161. C₂²×D₃	C₂³×C₄	semidirect
167. #67	C₂³×C₄	semidirect
170. C₂×Dic₃	C₂³×C₄	semidirect
171. C₂×Dic₃	C₂²×D₄	semidirect
185. C₂⁴	C₂×C₄×D₃	semidirect
208. C₂²⋊C₄	C₂³×D₃	semidirect
213. C₂²×C₃	C₂×C₄×D₄	semidirect
215. C₂²×C₃	C₂⁴×C₄	semidirect
222. C₂×D₃	C₂⁴×C₄	semidirect
226. Dic₃	C₂⁴×C₄	semidirect
227. Dic₃	C₂³×D₄	semidirect
236. C₂³	#67⋊C₄	direct (N × H)
238. C₂³	C₂²×C₄×D₃	semidirect
257. C₂×C₃	C₂²×C₄×D₄	semidirect
269. C₂²	C₂×#67⋊C₄	direct (N × H)
271. C₂²	C₂³×C₄×D₃	semidirect
280. C₃	C₂³×C₄×D₄	semidirect
283. C₂	C₂²×#67⋊C₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×#67	2
C₂⁵×Dic₃	2
C₂⁴×C₄×D₃	2
C₂³×C₃×C₂²⋊C₄	2
C₂³×(C₂×D₃)⋊C₄	2
C₂³×Dic₃⋊C₄	2
C₂³×C₄×Dic₃	2
C₂²×#67⋊C₄	2
C₂³×C₄×D₄	3

Subgroups — C23×#67⋊C4 #11701749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#67⋊C₄ #11701749