Subgroups — C₂³×#885 #11701848

Order 768.

14334 conjugacy classes of subgroups.

The subgroup lattice has 280 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	C₂⁴×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×C₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×#217	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×(C₂×D₃)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×(C₂×D₃)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₄×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#885	C₂	1	56	yes	maximal, direct product factor
9	C₂³×#124	—	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	—	2	16	no	—
15	C₂³×#67	C₂²	1	28	yes	—
16	C₂⁴×Dic₃	C₂²	1	1	yes	characteristic
17	C₂⁴×Dic₃	C₂²	1	1	yes	characteristic
18	C₂⁴×Dic₃	C₂²	1	1	yes	characteristic
19	C₂³×Dic₆	—	2	16	no	—
20	C₂³×Dic₆	C₂²	1	28	yes	complemented
21	C₂²×C₃×C₂²⋊C₄	C₂²	1	28	yes	complemented
22	C₂²×#217	C₂²	1	28	yes	complemented
23	C₂²×(C₂×D₃)⋊C₄	C₂²	1	28	yes	complemented
24	C₂²×(C₂×D₃)⋊C₄	C₂²	1	28	yes	complemented
25	C₂²×C₄×Dic₃	C₂²	1	28	yes	complemented
26	C₂×#885	C₂²	1	448	yes	direct product factor
27	C₂⁴×Q₈	—	3	1	no	—
28	C₂⁴×D₄	—	3	1	no	—
29	C₂²×#124	—	3	56	no	—
30	C₂³×C₂²⋊C₄	—	3	1	no	—
31	C₂³×C₂²⋊C₄	—	3	1	no	—
32	C₂³×C₂²⋊C₄	—	3	1	no	—
33	C₂³×C₂²⋊C₄	D₃	1	1	yes	characteristic, complemented
34	C₂³×C₄²	—	3	1	no	—
35	C₂⁵×C₃	—	2	8	no	—
36	C₂⁵×C₃	—	2	8	no	—
37	C₂⁵×C₃	—	2	8	no	—
38	C₂⁵×C₃	C₂³	1	14	yes	—
39	C₂⁵×C₃	C₂³	1	1	yes	characteristic
40	C₂⁴×D₃	—	2	8	no	—
41	C₂⁴×D₃	—	2	8	no	—
42	C₂⁴×D₃	—	2	8	no	—
43	C₂⁴×D₃	C₂³	1	14	yes	—
44	C₂³×C₃×C₄	—	2	8	no	—
45	C₂³×C₃×C₄	—	2	8	no	—
46	C₂³×C₃×C₄	C₂³	1	14	yes	—
47	C₂³×C₃×C₄	C₂³	1	14	yes	—
48	C₂²×#67	—	2	224	no	—
49	C₂²×#67	C₂³	1	112	yes	—
50	C₂³×Dic₃	—	2	8	no	—
51	C₂³×Dic₃	—	2	8	no	—
52	C₂³×Dic₃	D₄	1	16	yes	—
53	C₂³×Dic₃	C₂³	1	14	yes	—
54	C₂³×Dic₃	C₂³	1	14	yes	—
55	C₂³×Dic₃	C₂³	1	14	yes	—
56	C₂²×Dic₆	—	2	224	no	—
57	C₂²×Dic₆	C₂³	1	112	yes	complemented
58	C₂×C₃×C₂²⋊C₄	C₂³	1	112	yes	complemented
59	C₂×#217	C₂³	1	112	yes	complemented
60	C₂×(C₂×D₃)⋊C₄	C₂³	1	112	yes	complemented
61	C₂×(C₂×D₃)⋊C₄	C₂³	1	112	yes	complemented
62	C₂×C₄×Dic₃	C₂³	1	112	yes	complemented
63	#885	C₂³	1	512	yes	direct product factor
64	C₂⁶	—	3	1	no	—
65	C₂⁶	C₂×D₃	1	1	yes	characteristic
66	C₂³×Q₈	—	3	28	no	—
67	C₂³×Q₈	—	6	16	no	—
68	C₂³×D₄	—	3	28	no	—
69	C₂³×D₄	—	6	16	no	—
70	C₂⁴×C₄	—	3	1	no	—
71	C₂⁴×C₄	—	3	1	no	—
72	C₂⁴×C₄	—	3	1	no	—
73	C₂⁴×C₄	C₂×D₃	1	1	yes	characteristic
74	C₂⁴×C₄	C₂×D₃	1	1	yes	characteristic
75	C₂×#124	—	3	448	no	—
76	C₂²×C₂²⋊C₄	—	3	28	no	—
77	C₂²×C₂²⋊C₄	—	3	28	no	—
78	C₂²×C₂²⋊C₄	—	3	28	no	—
79	C₂²×C₂²⋊C₄	C₂×D₃	1	28	yes	complemented
80	C₂²×C₄²	—	3	28	no	—
81	C₂⁴×C₃	—	2	56	no	—
82	C₂⁴×C₃	—	2	56	no	—
83	C₂⁴×C₃	—	2	56	no	—
84	C₂⁴×C₃	—	4	64	no	—
85	C₂⁴×C₃	C₂×D₄	1	8	yes	—
86	C₂⁴×C₃	#41	1	8	yes	—
87	C₂⁴×C₃	#41	1	8	yes	—
88	C₂⁴×C₃	C₂⁴	1	28	yes	—
89	C₂⁴×C₃	C₂⁴	1	7	yes	—
90	C₂³×D₃	—	2	56	no	—
91	C₂³×D₃	—	2	56	no	—
92	C₂³×D₃	—	2	56	no	—
93	C₂³×D₃	—	4	64	no	—
94	C₂³×D₃	C₂⁴	1	28	yes	—
95	C₂²×C₃×C₄	—	2	56	no	—
96	C₂²×C₃×C₄	—	2	56	no	—
97	C₂²×C₃×C₄	C₂⁴	1	28	yes	—
98	C₂²×C₃×C₄	C₂⁴	1	28	yes	—
99	C₂×#67	—	2	448	no	—
100	C₂×#67	C₂⁴	1	64	yes	—
101	C₂²×Dic₃	—	2	56	no	—
102	C₂²×Dic₃	—	2	56	no	—
103	C₂²×Dic₃	C₂×D₄	1	112	yes	—
104	C₂²×Dic₃	C₂⁴	1	28	yes	—
105	C₂²×Dic₃	C₂⁴	1	28	yes	—
106	C₂²×Dic₃	C₂⁴	1	28	yes	—
107	C₂×Dic₆	—	2	448	no	—
108	C₂×Dic₆	C₂⁴	1	64	yes	complemented
109	C₃×C₂²⋊C₄	C₂⁴	1	64	yes	complemented
110	#217	C₂⁴	1	64	yes	complemented
111	(C₂×D₃)⋊C₄	C₂⁴	1	64	yes	complemented
112	(C₂×D₃)⋊C₄	C₂⁴	1	64	yes	complemented
113	C₄×Dic₃	C₂⁴	1	64	yes	complemented
114	C₂⁵	—	2	8	no	—
115	C₂⁵	—	2	8	no	—
116	C₂⁵	—	2	8	no	—
117	C₂⁵	—	3	14	no	—
118	C₂⁵	—	6	8	no	—
119	C₂⁵	—	6	8	no	—
120	C₂⁵	—	6	8	no	—
121	C₂⁵	C₂²×D₃	1	14	yes	—
122	C₂⁵	C₂²×D₃	1	1	yes	characteristic
123	C₂²×Q₈	—	3	112	no	—
124	C₂²×Q₈	—	6	224	no	—
125	C₂²×D₄	—	3	112	no	—
126	C₂²×D₄	—	6	224	no	—
127	C₂³×C₄	—	2	8	no	—
128	C₂³×C₄	—	2	8	no	—
129	C₂³×C₄	—	3	16	no	—
130	C₂³×C₄	—	3	14	no	—
131	C₂³×C₄	—	3	14	no	—
132	C₂³×C₄	—	3	14	no	—
133	C₂³×C₄	—	6	8	no	—
134	C₂³×C₄	—	6	8	no	—
135	C₂³×C₄	C₂²×D₃	1	14	yes	—
136	C₂³×C₄	C₂²×D₃	1	14	yes	—
137	#124	—	3	512	no	—
138	C₂×C₂²⋊C₄	—	3	112	no	—
139	C₂×C₂²⋊C₄	—	3	112	no	—
140	C₂×C₂²⋊C₄	—	3	112	no	—
141	C₂×C₂²⋊C₄	C₂²×D₃	1	112	yes	complemented
142	C₂×C₄²	—	3	112	no	—
143	C₂³×C₃	—	2	56	no	—
144	C₂³×C₃	—	2	56	no	—
145	C₂³×C₃	—	2	56	no	—
146	C₂³×C₃	—	4	224	no	—
147	C₂³×C₃	#124	1	64	yes	complemented
148	C₂³×C₃	C₂²×D₄	1	28	yes	—
149	C₂³×C₃	C₂×#41	1	28	yes	—
150	C₂³×C₃	C₂×#41	1	28	yes	—
151	C₂³×C₃	C₂⁵	1	8	yes	—
152	C₂³×C₃	C₂⁵	1	7	yes	—
153	C₂²×D₃	—	2	56	no	—
154	C₂²×D₃	—	2	56	no	—
155	C₂²×D₃	—	2	56	no	—
156	C₂²×D₃	—	4	224	no	—
157	C₂²×D₃	C₂⁵	1	8	yes	—
158	C₂×C₃×C₄	—	2	56	no	—
159	C₂×C₃×C₄	—	2	56	no	—
160	C₂×C₃×C₄	C₂⁵	1	8	yes	—
161	C₂×C₃×C₄	C₂⁵	1	8	yes	—
162	#67	—	2	128	no	—
163	C₂×Dic₃	—	2	56	no	—
164	C₂×Dic₃	—	2	56	no	—
165	C₂×Dic₃	C₂²×D₄	1	112	yes	—
166	C₂×Dic₃	C₂⁵	1	8	yes	—
167	C₂×Dic₃	C₂⁵	1	8	yes	—
168	C₂×Dic₃	C₂⁵	1	8	yes	—
169	Dic₆	—	2	128	no	—
170	C₂⁴	—	2	56	no	—
171	C₂⁴	—	2	56	no	—
172	C₂⁴	—	2	56	no	—
173	C₂⁴	—	3	28	no	—
174	C₂⁴	—	4	64	no	—
175	C₂⁴	—	6	56	no	—
176	C₂⁴	—	6	56	no	—
177	C₂⁴	—	6	56	no	—
178	C₂⁴	—	12	64	no	—
179	C₂⁴	#235	1	8	yes	—
180	C₂⁴	D₃×D₄	1	8	yes	—
181	C₂⁴	#237	1	8	yes	—
182	C₂⁴	C₂³×D₃	1	28	yes	—
183	C₂⁴	C₂³×D₃	1	7	yes	—
184	C₂×Q₈	—	3	64	no	—
185	C₂×Q₈	—	6	448	no	—
186	C₂×D₄	—	3	64	no	—
187	C₂×D₄	—	6	448	no	—
188	C₂²×C₄	—	2	56	no	—
189	C₂²×C₄	—	2	56	no	—
190	C₂²×C₄	—	3	112	no	—
191	C₂²×C₄	—	3	28	no	—
192	C₂²×C₄	—	3	28	no	—
193	C₂²×C₄	—	3	28	no	—
194	C₂²×C₄	—	6	56	no	—
195	C₂²×C₄	—	6	56	no	—
196	C₂²×C₄	C₂³×D₃	1	28	yes	—
197	C₂²×C₄	C₂³×D₃	1	28	yes	—
198	C₂²⋊C₄	—	3	64	no	—
199	C₂²⋊C₄	—	3	64	no	—
200	C₂²⋊C₄	—	3	64	no	—
201	C₂²⋊C₄	C₂³×D₃	1	64	yes	complemented
202	C₄²	—	3	64	no	—
203	C₂²×C₃	—	2	8	no	—
204	C₂²×C₃	—	2	8	no	—
205	C₂²×C₃	—	2	8	no	—
206	C₂²×C₃	—	4	112	no	—
207	C₂²×C₃	C₂×#124	1	112	yes	complemented
208	C₂²×C₃	C₂³×D₄	1	14	yes	—
209	C₂²×C₃	C₂²×#41	1	14	yes	—
210	C₂²×C₃	C₂²×#41	1	14	yes	—
211	C₂²×C₃	C₂⁶	1	1	yes	characteristic
212	C₂×D₃	—	2	8	no	—
213	C₂×D₃	—	2	8	no	—
214	C₂×D₃	—	2	8	no	—
215	C₂×D₃	—	4	112	no	—
216	C₃×C₄	—	2	8	no	—
217	C₃×C₄	—	2	8	no	—
218	Dic₃	—	2	8	no	—
219	Dic₃	—	2	8	no	—
220	Dic₃	C₂³×D₄	1	16	yes	—
221	C₂³	—	2	56	no	—
222	C₂³	—	2	56	no	—
223	C₂³	—	2	56	no	—
224	C₂³	—	3	8	no	—
225	C₂³	—	4	224	no	—
226	C₂³	—	6	56	no	—
227	C₂³	—	6	56	no	—
228	C₂³	—	6	56	no	—
229	C₂³	—	12	224	no	—
230	C₂³	#885	1	64	yes	direct product factor
231	C₂³	C₂×#235	1	28	yes	—
232	C₂³	C₂×D₃×D₄	1	28	yes	—
233	C₂³	C₂×#237	1	28	yes	—
234	C₂³	C₂⁴×D₃	1	8	yes	—
235	C₂³	C₂⁴×D₃	1	7	yes	—
236	Q₈	—	6	128	no	—
237	D₄	—	6	128	no	—
238	C₂×C₄	—	2	56	no	—
239	C₂×C₄	—	2	56	no	—
240	C₂×C₄	—	3	112	no	—
241	C₂×C₄	—	3	8	no	—
242	C₂×C₄	—	3	8	no	—
243	C₂×C₄	—	3	8	no	—
244	C₂×C₄	—	6	56	no	—
245	C₂×C₄	—	6	56	no	—
246	C₂×C₄	C₂⁴×D₃	1	8	yes	—
247	C₂×C₄	C₂⁴×D₃	1	8	yes	—
248	C₂×C₃	—	4	8	no	—
249	C₂×C₃	C₂²×#124	1	28	yes	complemented
250	C₂×C₃	C₂⁴×D₄	1	1	yes	characteristic
251	C₂×C₃	C₂³×#41	1	1	yes	characteristic
252	C₂×C₃	C₂³×#41	1	1	yes	characteristic
253	D₃	—	4	8	no	—
254	C₂²	—	2	8	no	—
255	C₂²	—	2	8	no	—
256	C₂²	—	2	8	no	—
257	C₂²	—	4	112	no	—
258	C₂²	—	6	8	no	—
259	C₂²	—	6	8	no	—
260	C₂²	—	6	8	no	—
261	C₂²	—	12	112	no	—
262	C₂²	C₂×#885	1	112	yes	direct product factor
263	C₂²	C₂²×#235	1	14	yes	—
264	C₂²	C₂²×D₃×D₄	1	14	yes	—
265	C₂²	C₂²×#237	1	14	yes	—
266	C₂²	C₂⁵×D₃	1	1	yes	characteristic
267	C₄	—	2	8	no	—
268	C₄	—	2	8	no	—
269	C₄	—	3	16	no	—
270	C₄	—	6	8	no	—
271	C₄	—	6	8	no	—
272	C₃	C₂³×#124	1	1	yes	characteristic, complemented, Sylow
273	C₂	—	4	8	no	—
274	C₂	—	12	8	no	—
275	C₂	C₂²×#885	1	28	yes	direct product factor
276	C₂	C₂³×#235	1	1	yes	characteristic
277	C₂	C₂³×#237	1	1	yes	characteristic
278	C₂	C₂³×D₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂⁶
Frattini	C₂²	C₂⁵×D₃
Fitting	C₂³×C₃×C₂²⋊C₄	C₂
socle	C₂⁵×C₃	C₂³
prime radical	C₂⁴×#67	C₂
hypercenter	C₂³×C₂²⋊C₄	D₃
hypocenter	C₃	C₂³×#124
LCS	G ▷ C₂²×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. C₂⁴×Dic₆	C₂	semidirect
3. C₂³×C₃×C₂²⋊C₄	C₂	semidirect
4. C₂³×#217	C₂	semidirect
5. C₂³×(C₂×D₃)⋊C₄	C₂	semidirect
6. C₂³×(C₂×D₃)⋊C₄	C₂	semidirect
7. C₂³×C₄×Dic₃	C₂	semidirect
8. C₂²×#885	C₂	direct (N × H)
20. C₂³×Dic₆	C₂²	semidirect
21. C₂²×C₃×C₂²⋊C₄	C₂²	semidirect
22. C₂²×#217	C₂²	semidirect
23. C₂²×(C₂×D₃)⋊C₄	C₂²	semidirect
24. C₂²×(C₂×D₃)⋊C₄	C₂²	semidirect
25. C₂²×C₄×Dic₃	C₂²	semidirect
26. C₂×#885	C₂²	direct (N × H)
33. C₂³×C₂²⋊C₄	D₃	semidirect
57. C₂²×Dic₆	C₂³	semidirect
58. C₂×C₃×C₂²⋊C₄	C₂³	semidirect
59. C₂×#217	C₂³	semidirect
60. C₂×(C₂×D₃)⋊C₄	C₂³	semidirect
61. C₂×(C₂×D₃)⋊C₄	C₂³	semidirect
62. C₂×C₄×Dic₃	C₂³	semidirect
63. #885	C₂³	direct (N × H)
79. C₂²×C₂²⋊C₄	C₂×D₃	semidirect
108. C₂×Dic₆	C₂⁴	semidirect
109. C₃×C₂²⋊C₄	C₂⁴	semidirect
110. #217	C₂⁴	semidirect
111. (C₂×D₃)⋊C₄	C₂⁴	semidirect
112. (C₂×D₃)⋊C₄	C₂⁴	semidirect
113. C₄×Dic₃	C₂⁴	semidirect
141. C₂×C₂²⋊C₄	C₂²×D₃	semidirect
147. C₂³×C₃	#124	semidirect
201. C₂²⋊C₄	C₂³×D₃	semidirect
207. C₂²×C₃	C₂×#124	semidirect
230. C₂³	#885	direct (N × H)
249. C₂×C₃	C₂²×#124	semidirect
262. C₂²	C₂×#885	direct (N × H)
272. C₃	C₂³×#124	semidirect
275. C₂	C₂²×#885	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
C₂⁴×Dic₆	2
C₂³×C₃×C₂²⋊C₄	2
C₂³×#217	2
C₂³×(C₂×D₃)⋊C₄	2
C₂³×C₄×Dic₃	2
C₂²×#885	2
C₂³×#124	3

Subgroups — C23×#885 #11701848

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#885 #11701848