Subgroups — C₂³×#918 #11708986

Order 768.

11166 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	C₂³×D₃×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×#239	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×#216	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×#215	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#918	C₂	1	56	yes	maximal, direct product factor
7	C₂³×#204	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×#137	—	3	1	no	maximal, Sylow
10	C₂³×C₃×Q₈	C₂²	1	1	yes	characteristic
11	C₂³×C₃×Q₈	C₂²	1	1	yes	characteristic
12	C₂²×#239	—	2	14	no	—
13	C₂²×#239	C₂²	1	28	yes	—
14	C₂²×D₃×Q₈	—	2	14	no	—
15	C₂²×D₃×Q₈	C₂²	1	28	yes	complemented
16	C₂³×D₁₂	—	2	1	no	—
17	C₂³×D₁₂	C₂²	1	1	yes	characteristic
18	C₂³×C₄×D₃	—	2	1	no	—
19	C₂³×C₄×D₃	—	2	1	no	—
20	C₂³×C₄×D₃	C₂²	1	1	yes	characteristic
21	C₂³×Dic₆	—	2	1	no	—
22	C₂³×Dic₆	C₂²	1	1	yes	characteristic
23	C₂²×C₃×Q₁₆	C₂²	1	28	yes	complemented
24	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
25	C₂²×#216	C₂²	1	28	yes	complemented
26	C₂²×#215	C₂²	1	28	yes	complemented
27	C₂³×C₃⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×#918	C₂²	1	448	yes	direct product factor
29	C₂²×#204	C₂²	1	28	yes	complemented
30	C₂²×ΓL(1,𝔽₂₅)	C₂²	1	28	yes	complemented
31	C₂³×#41	—	3	1	no	—
32	C₂⁴×Q₈	—	3	1	no	—
33	C₂²×#137	—	3	56	no	—
34	C₂³×Q₁₆	—	3	1	no	—
35	C₂³×Q₁₆	D₃	1	1	yes	characteristic, complemented
36	C₂³×SD₁₆	—	3	1	no	—
37	C₂³×SD₁₆	—	3	1	no	—
38	C₂³×M₁₆	—	3	1	no	—
39	C₂⁴×D₃	—	2	1	no	—
40	C₂⁴×D₃	D₄	1	1	yes	characteristic
41	C₂²×C₃×Q₈	—	2	7	no	—
42	C₂²×C₃×Q₈	—	2	7	no	—
43	C₂²×C₃×Q₈	C₂³	1	14	yes	—
44	C₂²×C₃×Q₈	C₂³	1	14	yes	—
45	C₂³×C₃×C₄	—	2	1	no	—
46	C₂³×C₃×C₄	—	2	1	no	—
47	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
48	C₂³×Dic₃	—	2	1	no	—
49	C₂³×Dic₃	D₄	1	1	yes	characteristic
50	C₂×#239	—	2	168	no	—
51	C₂×#239	C₂³	1	112	yes	—
52	C₂×D₃×Q₈	—	2	168	no	—
53	C₂×D₃×Q₈	C₂³	1	112	yes	complemented
54	C₂²×D₁₂	—	2	28	no	—
55	C₂²×D₁₂	—	2	7	no	—
56	C₂²×D₁₂	C₂³	1	14	yes	—
57	C₂²×C₄×D₃	—	2	28	no	—
58	C₂²×C₄×D₃	—	2	28	no	—
59	C₂²×C₄×D₃	D₄	1	14	yes	—
60	C₂²×C₄×D₃	C₂³	1	14	yes	—
61	C₂²×Dic₆	—	2	7	no	—
62	C₂²×Dic₆	—	2	28	no	—
63	C₂²×Dic₆	C₂³	1	14	yes	—
64	C₂×C₃×Q₁₆	C₂³	1	112	yes	complemented
65	C₂²×C₃×C₈	C₂³	1	14	yes	—
66	C₂×#216	C₂³	1	112	yes	complemented
67	C₂×#215	C₂³	1	112	yes	complemented
68	C₂²×C₃⋊C₈	C₂³	1	14	yes	—
69	#918	C₂³	1	512	yes	direct product factor
70	C₂×#204	C₂³	1	112	yes	complemented
71	C₂×ΓL(1,𝔽₂₅)	C₂³	1	112	yes	complemented
72	C₂²×#41	—	3	28	no	—
73	C₂²×#41	—	6	14	no	—
74	C₂³×Q₈	—	3	28	no	—
75	C₂³×Q₈	—	3	1	no	—
76	C₂³×Q₈	—	6	1	no	—
77	C₂³×Q₈	—	6	14	no	—
78	C₂³×Q₈	C₂×D₃	1	1	yes	characteristic
79	C₂³×Q₈	C₂×D₃	1	1	yes	characteristic
80	C₂³×D₄	—	3	1	no	—
81	C₂³×D₄	—	6	1	no	—
82	C₂⁴×C₄	—	3	1	no	—
83	C₂⁴×C₄	—	6	1	no	—
84	C₂⁴×C₄	—	6	1	no	—
85	C₂×#137	—	3	448	no	—
86	C₂²×Q₁₆	—	3	28	no	—
87	C₂²×Q₁₆	C₂×D₃	1	28	yes	complemented
88	C₂²×SD₁₆	—	3	28	no	—
89	C₂²×SD₁₆	—	3	28	no	—
90	C₂²×M₁₆	—	3	28	no	—
91	C₂³×C₈	—	3	1	no	—
92	C₂³×C₈	C₂×D₃	1	1	yes	characteristic
93	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
94	C₂³×D₃	—	2	8	no	—
95	C₂³×D₃	—	2	14	no	—
96	C₂³×D₃	—	4	8	no	—
97	C₂³×D₃	C₂×D₄	1	14	yes	—
98	C₂×C₃×Q₈	—	2	42	no	—
99	C₂×C₃×Q₈	—	2	42	no	—
100	C₂×C₃×Q₈	C₂⁴	1	28	yes	—
101	C₂×C₃×Q₈	C₂⁴	1	28	yes	—
102	C₂²×C₃×C₄	—	2	14	no	—
103	C₂²×C₃×C₄	—	2	14	no	—
104	C₂²×C₃×C₄	C₂×D₄	1	7	yes	—
105	C₂²×C₃×C₄	C₂⁴	1	7	yes	—
106	C₂²×Dic₃	—	2	14	no	—
107	C₂²×Dic₃	C₂×D₄	1	14	yes	—
108	#239	—	2	224	no	—
109	#239	C₂⁴	1	64	yes	—
110	D₃×Q₈	—	2	224	no	—
111	D₃×Q₈	C₂⁴	1	64	yes	complemented
112	C₂×D₁₂	—	2	112	no	—
113	C₂×D₁₂	—	2	42	no	—
114	C₂×D₁₂	C₂⁴	1	28	yes	—
115	C₂×C₄×D₃	—	2	112	no	—
116	C₂×C₄×D₃	—	2	112	no	—
117	C₂×C₄×D₃	C₂×D₄	1	84	yes	—
118	C₂×C₄×D₃	C₂⁴	1	28	yes	—
119	C₂×Dic₆	—	2	42	no	—
120	C₂×Dic₆	—	2	112	no	—
121	C₂×Dic₆	C₂⁴	1	28	yes	—
122	C₃×Q₁₆	C₂⁴	1	64	yes	complemented
123	C₂×C₃×C₈	C₂⁴	1	28	yes	—
124	#216	C₂⁴	1	64	yes	complemented
125	#215	C₂⁴	1	64	yes	complemented
126	C₂×C₃⋊C₈	C₂⁴	1	28	yes	—
127	#204	C₂⁴	1	64	yes	complemented
128	ΓL(1,𝔽₂₅)	C₂⁴	1	64	yes	complemented
129	C₂⁵	—	3	1	no	—
130	C₂⁵	—	6	1	no	—
131	C₂×#41	—	3	112	no	—
132	C₂×#41	—	6	168	no	—
133	C₂²×Q₈	—	2	7	no	—
134	C₂²×Q₈	—	2	7	no	—
135	C₂²×Q₈	—	3	112	no	—
136	C₂²×Q₈	—	3	14	no	—
137	C₂²×Q₈	—	6	28	no	—
138	C₂²×Q₈	—	6	168	no	—
139	C₂²×Q₈	—	6	7	no	—
140	C₂²×Q₈	C₂²×D₃	1	14	yes	—
141	C₂²×Q₈	C₂²×D₃	1	14	yes	—
142	C₂²×D₄	—	3	14	no	—
143	C₂²×D₄	—	6	28	no	—
144	C₂²×D₄	—	6	7	no	—
145	C₂³×C₄	—	2	1	no	—
146	C₂³×C₄	—	2	1	no	—
147	C₂³×C₄	—	3	1	no	—
148	C₂³×C₄	—	3	14	no	—
149	C₂³×C₄	—	3	14	no	—
150	C₂³×C₄	—	6	28	no	—
151	C₂³×C₄	—	6	28	no	—
152	C₂³×C₄	—	6	1	no	—
153	C₂³×C₄	C₂²×D₃	1	1	yes	characteristic
154	#137	—	3	512	no	—
155	C₂×Q₁₆	—	3	112	no	—
156	C₂×Q₁₆	C₂²×D₃	1	112	yes	complemented
157	C₂×SD₁₆	—	3	112	no	—
158	C₂×SD₁₆	—	3	112	no	—
159	C₂×M₁₆	—	3	112	no	—
160	C₂²×C₈	—	3	14	no	—
161	C₂²×C₈	C₂²×D₃	1	14	yes	—
162	C₂³×C₃	#137	1	8	yes	complemented
163	C₂³×C₃	C₂²×D₄	1	7	yes	—
164	C₂²×D₃	—	2	56	no	—
165	C₂²×D₃	—	2	28	no	—
166	C₂²×D₃	—	4	56	no	—
167	C₂²×D₃	C₂²×D₄	1	28	yes	—
168	C₃×Q₈	—	2	28	no	—
169	C₃×Q₈	—	2	28	no	—
170	C₃×Q₈	C₂⁵	1	8	yes	—
171	C₃×Q₈	C₂⁵	1	8	yes	—
172	C₂×C₃×C₄	—	2	28	no	—
173	C₂×C₃×C₄	—	2	28	no	—
174	C₂×C₃×C₄	C₂²×D₄	1	21	yes	—
175	C₂×C₃×C₄	C₂⁵	1	7	yes	—
176	C₂×Dic₃	—	2	28	no	—
177	C₂×Dic₃	C₂²×D₄	1	28	yes	—
178	D₁₂	—	2	64	no	—
179	D₁₂	—	2	28	no	—
180	D₁₂	C₂⁵	1	8	yes	—
181	C₄×D₃	—	2	64	no	—
182	C₄×D₃	—	2	64	no	—
183	C₄×D₃	C₂²×D₄	1	56	yes	—
184	C₄×D₃	C₂⁵	1	8	yes	—
185	Dic₆	—	2	28	no	—
186	Dic₆	—	2	64	no	—
187	Dic₆	C₂⁵	1	8	yes	—
188	C₃×C₈	C₂⁵	1	8	yes	—
189	C₃⋊C₈	C₂⁵	1	8	yes	—
190	C₂⁴	—	3	14	no	—
191	C₂⁴	—	6	8	no	—
192	C₂⁴	—	6	14	no	—
193	C₂⁴	—	12	8	no	—
194	C₂⁴	D₃×D₄	1	1	yes	characteristic
195	#41	—	3	64	no	—
196	#41	—	6	224	no	—
197	C₂×Q₈	—	2	42	no	—
198	C₂×Q₈	—	2	42	no	—
199	C₂×Q₈	—	3	64	no	—
200	C₂×Q₈	—	3	28	no	—
201	C₂×Q₈	—	6	112	no	—
202	C₂×Q₈	—	6	224	no	—
203	C₂×Q₈	—	6	42	no	—
204	C₂×Q₈	C₂³×D₃	1	28	yes	—
205	C₂×Q₈	C₂³×D₃	1	28	yes	—
206	C₂×D₄	—	3	28	no	—
207	C₂×D₄	—	6	112	no	—
208	C₂×D₄	—	6	42	no	—
209	C₂²×C₄	—	2	14	no	—
210	C₂²×C₄	—	2	14	no	—
211	C₂²×C₄	—	3	14	no	—
212	C₂²×C₄	—	3	84	no	—
213	C₂²×C₄	—	3	28	no	—
214	C₂²×C₄	—	6	112	no	—
215	C₂²×C₄	—	6	112	no	—
216	C₂²×C₄	—	6	14	no	—
217	C₂²×C₄	D₃×D₄	1	7	yes	—
218	C₂²×C₄	C₂³×D₃	1	7	yes	—
219	Q₁₆	—	3	64	no	—
220	Q₁₆	C₂³×D₃	1	64	yes	complemented
221	SD₁₆	—	3	64	no	—
222	SD₁₆	—	3	64	no	—
223	M₁₆	—	3	64	no	—
224	C₂×C₈	—	3	28	no	—
225	C₂×C₈	C₂³×D₃	1	28	yes	—
226	C₂²×C₃	C₂×#137	1	28	yes	complemented
227	C₂²×C₃	C₂³×D₄	1	7	yes	—
228	C₂×D₃	—	2	56	no	—
229	C₂×D₃	—	2	8	no	—
230	C₂×D₃	—	4	56	no	—
231	C₂×D₃	C₂³×D₄	1	8	yes	—
232	C₃×C₄	—	2	8	no	—
233	C₃×C₄	—	2	8	no	—
234	C₃×C₄	C₂³×D₄	1	7	yes	—
235	C₃×C₄	C₂⁶	1	1	yes	characteristic
236	Dic₃	—	2	8	no	—
237	Dic₃	C₂³×D₄	1	8	yes	—
238	C₂³	—	3	28	no	—
239	C₂³	—	6	56	no	—
240	C₂³	—	6	28	no	—
241	C₂³	—	12	56	no	—
242	C₂³	#918	1	8	yes	direct product factor
243	C₂³	C₂×D₃×D₄	1	7	yes	—
244	Q₈	—	2	28	no	—
245	Q₈	—	2	28	no	—
246	Q₈	—	3	8	no	—
247	Q₈	—	6	64	no	—
248	Q₈	—	6	28	no	—
249	Q₈	C₂⁴×D₃	1	8	yes	—
250	Q₈	C₂⁴×D₃	1	8	yes	—
251	D₄	—	3	8	no	—
252	D₄	—	6	64	no	—
253	D₄	—	6	28	no	—
254	C₂×C₄	—	2	28	no	—
255	C₂×C₄	—	2	28	no	—
256	C₂×C₄	—	3	28	no	—
257	C₂×C₄	—	3	56	no	—
258	C₂×C₄	—	3	8	no	—
259	C₂×C₄	—	6	64	no	—
260	C₂×C₄	—	6	64	no	—
261	C₂×C₄	—	6	28	no	—
262	C₂×C₄	C₂×D₃×D₄	1	21	yes	—
263	C₂×C₄	C₂⁴×D₃	1	7	yes	—
264	C₈	—	3	8	no	—
265	C₈	C₂⁴×D₃	1	8	yes	—
266	C₂×C₃	C₂²×#137	1	14	yes	complemented
267	C₂×C₃	C₂⁴×D₄	1	1	yes	characteristic
268	D₃	—	2	8	no	—
269	D₃	—	4	8	no	—
270	C₂²	—	3	8	no	—
271	C₂²	—	6	56	no	—
272	C₂²	—	6	8	no	—
273	C₂²	—	12	56	no	—
274	C₂²	C₂×#918	1	28	yes	direct product factor
275	C₂²	C₂²×D₃×D₄	1	7	yes	—
276	C₄	—	2	8	no	—
277	C₄	—	2	8	no	—
278	C₄	—	3	8	no	—
279	C₄	—	6	8	no	—
280	C₄	C₂²×D₃×D₄	1	7	yes	—
281	C₄	C₂⁵×D₃	1	1	yes	characteristic
282	C₃	C₂³×#137	1	1	yes	characteristic, complemented, Sylow
283	C₂	—	6	8	no	—
284	C₂	—	12	8	no	—
285	C₂	C₂²×#918	1	14	yes	direct product factor
286	C₂	C₂³×D₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₃×D₄
commutator, cosocle	C₃×C₄	C₂⁶
Frattini	C₄	C₂⁵×D₃
Fitting	C₂³×C₃×Q₁₆	C₂
socle	C₂⁴×C₃	C₂×D₄
prime radical	C₂³×#239	C₂
hypercenter	C₂³×Q₁₆	D₃
hypocenter	C₃	C₂³×#137
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×Q₁₆

Complements

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

N	H ≅ G/N	type
1. C₂³×D₃×Q₈	C₂	semidirect
3. C₂³×C₃×Q₁₆	C₂	semidirect
4. C₂³×#216	C₂	semidirect
5. C₂³×#215	C₂	semidirect
6. C₂²×#918	C₂	direct (N × H)
7. C₂³×#204	C₂	semidirect
8. C₂³×ΓL(1,𝔽₂₅)	C₂	semidirect
15. C₂²×D₃×Q₈	C₂²	semidirect
23. C₂²×C₃×Q₁₆	C₂²	semidirect
25. C₂²×#216	C₂²	semidirect
26. C₂²×#215	C₂²	semidirect
28. C₂×#918	C₂²	direct (N × H)
29. C₂²×#204	C₂²	semidirect
30. C₂²×ΓL(1,𝔽₂₅)	C₂²	semidirect
35. C₂³×Q₁₆	D₃	semidirect
53. C₂×D₃×Q₈	C₂³	semidirect
64. C₂×C₃×Q₁₆	C₂³	semidirect
66. C₂×#216	C₂³	semidirect
67. C₂×#215	C₂³	semidirect
69. #918	C₂³	direct (N × H)
70. C₂×#204	C₂³	semidirect
71. C₂×ΓL(1,𝔽₂₅)	C₂³	semidirect
87. C₂²×Q₁₆	C₂×D₃	semidirect
111. D₃×Q₈	C₂⁴	semidirect
122. C₃×Q₁₆	C₂⁴	semidirect
124. #216	C₂⁴	semidirect
125. #215	C₂⁴	semidirect
127. #204	C₂⁴	semidirect
128. ΓL(1,𝔽₂₅)	C₂⁴	semidirect
156. C₂×Q₁₆	C₂²×D₃	semidirect
162. C₂³×C₃	#137	semidirect
220. Q₁₆	C₂³×D₃	semidirect
226. C₂²×C₃	C₂×#137	semidirect
242. C₂³	#918	direct (N × H)
266. C₂×C₃	C₂²×#137	semidirect
274. C₂²	C₂×#918	direct (N × H)
282. C₃	C₂³×#137	semidirect
285. C₂	C₂²×#918	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₃×Q₈	2
C₂³×#239	2
C₂³×C₃×Q₁₆	2
C₂³×#216	2
C₂³×#215	2
C₂²×#918	2
C₂³×#204	2
C₂³×ΓL(1,𝔽₂₅)	2
C₂³×#137	3

Subgroups — C23×#918 #11708986

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#918 #11708986