Subgroups — D₄×Q₈×D₂₃ #15686867

Order 2944.

1568 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×D₂₃	C₂	1	2	yes	maximal, complemented
2	Q₈×D₉₂	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×Dic₄₆	C₂	1	3	yes	maximal, complemented
4	#13093082	C₂	1	3	yes	maximal, complemented
5	Q₈×C₂²⋊D₂₃	C₂	1	2	yes	maximal, complemented
6	#13093080	C₂	1	6	yes	maximal, complemented
7	C₄⋊Q₈×D₂₃	C₂	1	3	yes	maximal, complemented
8	C₂²⋊Q₈×D₂₃	C₂	1	6	yes	maximal, complemented
9	C₄×Q₈×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₄×D₂₃	C₂	1	3	yes	maximal
11	D₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×C₂₃	C₂²	1	2	yes	complemented
13	C₂×Q₈×D₂₃	—	2	16	no	—
14	C₂×Q₈×D₂₃	C₂²	1	6	yes	—
15	C₂×Q₈×D₂₃	C₂²	1	1	yes	characteristic
16	C₂×Q₈×D₂₃	C₂²	1	6	yes	complemented
17	C₂×Q₈×D₂₃	C₂²	1	2	yes	complemented
18	C₂×D₄×D₂₃	C₂²	1	1	yes	characteristic
19	C₂²×C₄×D₂₃	C₂²	1	6	yes	—
20	C₂²×Dic₄₆	C₂²	1	6	yes	complemented
21	C₂₃×C₄⋊Q₈	C₂²	1	3	yes	complemented
22	C₂₃×C₂²⋊Q₈	C₂²	1	6	yes	complemented
23	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
24	C₄×D₄×C₂₃	C₂²	1	3	yes	—
25	#10625117	C₂²	1	6	yes	complemented
26	Q₈×Dic₂₃	C₂²	1	1	yes	characteristic, complemented
27	#10625115	C₂²	1	6	yes	complemented
28	D₄×Dic₂₃	C₂²	1	3	yes	—
29	C₄×C₂²⋊D₂₃	C₂²	1	6	yes	—
30	#10625095	C₂²	1	6	yes	complemented
31	#10625068	C₂²	1	6	yes	complemented
32	#10625067	C₂²	1	12	yes	complemented
33	#10625064	C₂²	1	3	yes	—
34	C₄⋊C₄×D₂₃	C₂²	1	3	yes	—
35	C₄⋊C₄×D₂₃	C₂²	1	3	yes	—
36	C₄⋊C₄×D₂₃	C₂²	1	6	yes	—
37	#10625059	C₂²	1	6	yes	complemented
38	#10625059	C₂²	1	6	yes	complemented
39	#10625058	C₂²	1	3	yes	complemented
40	(C₂²⋊D₂₃)⋊C₄	C₂²	1	6	yes	—
41	C₂²⋊C₄×D₂₃	C₂²	1	6	yes	—
42	#10625049	C₂²	1	12	yes	complemented
43	C₄×D₉₂	C₂²	1	3	yes	—
44	C₄²×D₂₃	C₂²	1	3	yes	—
45	#10625040	C₂²	1	3	yes	complemented
46	C₄×Dic₄₆	C₂²	1	3	yes	complemented
47	C₂³×D₂₃	C₂³	1	2	yes	—
48	C₂×Q₈×C₂₃	—	2	8	no	—
49	C₂×Q₈×C₂₃	C₂³	1	6	yes	—
50	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
51	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₂₃	C₂³	1	6	yes	—
53	C₂×C₂²⋊D₂₃	C₂³	1	2	yes	—
54	C₂²×Dic₂₃	C₂³	1	6	yes	—
55	Q₈×D₂₃	—	2	8	no	—
56	Q₈×D₂₃	—	2	24	no	—
57	Q₈×D₂₃	—	2	24	no	—
58	Q₈×D₂₃	D₄	1	8	yes	direct product factor
59	D₄×D₂₃	Q₈	1	8	yes	direct product factor
60	C₂×D₉₂	C₂³	1	1	yes	characteristic
61	C₂×C₄×D₂₃	—	2	24	no	—
62	C₂×C₄×D₂₃	C₂³	1	3	yes	—
63	C₂×C₄×D₂₃	C₂³	1	6	yes	—
64	C₂×C₄×D₂₃	C₂³	1	6	yes	—
65	C₂×C₄×D₂₃	C₂³	1	6	yes	—
66	C₂×C₄×D₂₃	C₂³	1	3	yes	—
67	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
68	C₂×Dic₄₆	—	2	24	no	—
69	C₂×Dic₄₆	C₂³	1	6	yes	—
70	C₂×Dic₄₆	C₂³	1	12	yes	—
71	C₂×Dic₄₆	C₂³	1	3	yes	—
72	C₄⋊C₄×C₂₃	C₂³	1	3	yes	—
73	C₄⋊C₄×C₂₃	C₂³	1	3	yes	—
74	C₄⋊C₄×C₂₃	C₂³	1	6	yes	—
75	C₂²⋊C₄×C₂₃	C₂³	1	6	yes	—
76	C₄²×C₂₃	C₂³	1	3	yes	—
77	C₂²⋊Dic₂₃	C₂³	1	6	yes	—
78	(C₂×D₂₃)⋊C₄	C₂³	1	6	yes	—
79	(C₂×D₂₃)⋊C₄	C₂³	1	6	yes	—
80	C₄⋊Dic₂₃	C₂³	1	3	yes	—
81	C₄⋊Dic₂₃	C₂³	1	3	yes	—
82	C₄⋊Dic₂₃	C₂³	1	6	yes	—
83	Dic₂₃⋊C₄	C₂³	1	6	yes	—
84	Dic₂₃⋊C₄	C₂³	1	12	yes	—
85	Dic₂₃⋊C₄	C₂³	1	6	yes	—
86	C₄×Dic₂₃	C₂³	1	3	yes	—
87	C₄×Dic₂₃	C₂³	1	3	yes	—
88	C₄×Dic₂₃	C₂³	1	3	yes	—
89	C₂³×C₂₃	C₂⁴	1	2	yes	—
90	C₂²×D₂₃	—	2	4	no	—
91	C₂²×D₂₃	—	2	4	no	—
92	C₂²×D₂₃	C₂×Q₈	1	8	yes	complemented
93	C₂²×D₂₃	C₂⁴	1	2	yes	—
94	C₂²×D₂₃	C₂⁴	1	1	yes	characteristic
95	Q₈×C₂₃	—	2	12	no	—
96	Q₈×C₂₃	C₂×D₄	1	4	yes	complemented
97	D₄×C₂₃	C₂×Q₈	1	4	yes	complemented
98	C₂×C₄×C₂₃	—	2	12	no	—
99	C₂×C₄×C₂₃	C₂⁴	1	3	yes	—
100	C₂×C₄×C₂₃	C₂⁴	1	6	yes	—
101	C₂×C₄×C₂₃	C₂⁴	1	3	yes	—
102	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
103	C₂²⋊D₂₃	C₂×Q₈	1	8	yes	complemented
104	C₂×Dic₂₃	—	2	12	no	—
105	C₂×Dic₂₃	C₂⁴	1	3	yes	—
106	C₂×Dic₂₃	C₂⁴	1	6	yes	—
107	C₂×Dic₂₃	C₂⁴	1	3	yes	—
108	C₂×Dic₂₃	C₂⁴	1	1	yes	characteristic
109	D₉₂	C₂×Q₈	1	4	yes	complemented
110	C₄×D₂₃	—	2	12	no	—
111	C₄×D₂₃	—	2	6	no	—
112	C₄×D₂₃	—	2	12	no	—
113	C₄×D₂₃	—	2	12	no	—
114	C₄×D₂₃	C₂×D₄	1	12	yes	—
115	C₄×D₂₃	C₂×Q₈	1	4	yes	complemented
116	Dic₄₆	—	2	12	no	—
117	Dic₄₆	—	2	24	no	—
118	Dic₄₆	C₂×D₄	1	12	yes	complemented
119	C₂×D₄×Q₈	—	23	1	no	maximal, Sylow
120	C₂²×C₂₃	—	2	2	no	—
121	C₂²×C₂₃	—	2	2	no	—
122	C₂²×C₂₃	C₂²×Q₈	1	4	yes	complemented
123	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
124	C₂×D₂₃	—	2	2	no	—
125	C₂×D₂₃	—	2	2	no	—
126	C₂×D₂₃	—	2	16	no	—
127	C₂×D₂₃	C₂²×D₄	1	2	yes	—
128	C₂×D₂₃	C₂²×Q₈	1	2	yes	—
129	C₂×D₂₃	C₂²×Q₈	1	4	yes	complemented
130	C₂×D₂₃	2₋¹⁺⁴	1	2	yes	—
131	C₄×C₂₃	—	2	6	no	—
132	C₄×C₂₃	—	2	3	no	—
133	C₄×C₂₃	C₂²×D₄	1	6	yes	—
134	C₄×C₂₃	C₂²×Q₈	1	2	yes	complemented
135	Dic₂₃	—	2	6	no	—
136	Dic₂₃	—	2	3	no	—
137	Dic₂₃	C₂²×D₄	1	6	yes	—
138	Dic₂₃	C₂²×Q₈	1	2	yes	complemented
139	C₂³×Q₈	—	23	2	no	—
140	D₄×Q₈	—	23	2	no	—
141	D₄×Q₈	—	23	3	no	—
142	D₄×Q₈	—	23	6	no	—
143	D₄×Q₈	—	23	1	no	—
144	D₄×Q₈	—	23	3	no	—
145	D₄×Q₈	D₂₃	1	1	yes	characteristic, direct product factor
146	C₂×C₄⋊Q₈	—	23	3	no	—
147	C₂×C₂²⋊Q₈	—	23	6	no	—
148	C₂×C₄×Q₈	—	23	1	no	—
149	C₂×C₄×D₄	—	23	3	no	—
150	C₂×C₂₃	—	2	4	no	—
151	C₂×C₂₃	C₂³×D₄	1	1	yes	characteristic
152	C₂×C₂₃	C₂³×Q₈	1	1	yes	characteristic
153	C₂×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
154	D₂₃	—	2	4	no	—
155	D₂₃	D₄×Q₈	1	4	yes	direct product factor
156	C₂²×Q₈	—	23	1	no	—
157	C₂²×Q₈	—	23	6	no	—
158	C₂²×Q₈	—	23	6	no	—
159	C₂²×Q₈	—	23	2	no	—
160	C₂²×Q₈	—	23	6	no	—
161	C₂²×Q₈	—	46	16	no	—
162	C₂²×Q₈	C₂×D₂₃	1	2	yes	complemented
163	C₂²×D₄	—	23	1	no	—
164	C₂³×C₄	—	23	6	no	—
165	C₄⋊Q₈	—	23	6	no	—
166	C₄⋊Q₈	—	23	6	no	—
167	C₄⋊Q₈	—	23	6	no	—
168	C₄⋊Q₈	—	23	3	no	—
169	C₄⋊Q₈	C₂×D₂₃	1	3	yes	complemented
170	C₂²⋊Q₈	—	23	6	no	—
171	C₂²⋊Q₈	—	23	6	no	—
172	C₂²⋊Q₈	—	23	12	no	—
173	C₂²⋊Q₈	—	23	6	no	—
174	C₂²⋊Q₈	—	23	12	no	—
175	C₂²⋊Q₈	C₂×D₂₃	1	6	yes	complemented
176	C₄×Q₈	—	23	1	no	—
177	C₄×Q₈	—	23	3	no	—
178	C₄×Q₈	—	23	3	no	—
179	C₄×Q₈	C₂×D₂₃	1	1	yes	characteristic, complemented
180	C₄×D₄	—	23	3	no	—
181	C₄×D₄	—	23	6	no	—
182	C₄×D₄	—	23	3	no	—
183	C₄×D₄	—	23	6	no	—
184	C₄×D₄	—	23	3	no	—
185	C₄×D₄	C₂×D₂₃	1	3	yes	—
186	C₂×C₄⋊C₄	—	23	3	no	—
187	C₂×C₄⋊C₄	—	23	3	no	—
188	C₂×C₄⋊C₄	—	23	6	no	—
189	C₂×C₂²⋊C₄	—	23	6	no	—
190	C₂×C₄²	—	23	3	no	—
191	C₂₃	C₂×D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
192	C₂⁴	—	23	2	no	—
193	C₂×Q₈	—	2	8	no	—
194	C₂×Q₈	—	23	8	no	—
195	C₂×Q₈	—	23	3	no	—
196	C₂×Q₈	—	23	12	no	—
197	C₂×Q₈	—	23	6	no	—
198	C₂×Q₈	—	46	24	no	—
199	C₂×Q₈	—	46	24	no	—
200	C₂×Q₈	—	46	24	no	—
201	C₂×Q₈	—	46	8	no	—
202	C₂×Q₈	C₂²×D₂₃	1	6	yes	—
203	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
204	C₂×D₄	—	23	2	no	—
205	C₂×D₄	—	23	8	no	—
206	C₂×D₄	—	23	1	no	—
207	C₂×D₄	C₂²×D₂₃	1	1	yes	characteristic
208	C₂²×C₄	—	23	1	no	—
209	C₂²×C₄	—	23	3	no	—
210	C₂²×C₄	—	23	6	no	—
211	C₂²×C₄	—	23	3	no	—
212	C₂²×C₄	—	23	6	no	—
213	C₂²×C₄	—	23	6	no	—
214	C₂²×C₄	—	23	6	no	—
215	C₂²×C₄	—	46	24	no	—
216	C₂²×C₄	C₂²×D₂₃	1	6	yes	—
217	C₄⋊C₄	—	23	6	no	—
218	C₄⋊C₄	—	23	3	no	—
219	C₄⋊C₄	—	23	6	no	—
220	C₄⋊C₄	—	23	3	no	—
221	C₄⋊C₄	—	23	12	no	—
222	C₄⋊C₄	—	23	6	no	—
223	C₄⋊C₄	C₂²×D₂₃	1	3	yes	—
224	C₄⋊C₄	C₂²×D₂₃	1	3	yes	—
225	C₄⋊C₄	C₂²×D₂₃	1	6	yes	—
226	C₂²⋊C₄	—	23	6	no	—
227	C₂²⋊C₄	—	23	6	no	—
228	C₂²⋊C₄	—	23	6	no	—
229	C₂²⋊C₄	C₂²×D₂₃	1	6	yes	—
230	C₄²	—	23	3	no	—
231	C₄²	—	23	3	no	—
232	C₄²	—	23	3	no	—
233	C₄²	C₂²×D₂₃	1	3	yes	—
234	C₂³	—	23	1	no	—
235	C₂³	—	23	8	no	—
236	C₂³	—	23	2	no	—
237	C₂³	—	46	4	no	—
238	C₂³	—	46	4	no	—
239	C₂³	C₂³×D₂₃	1	2	yes	—
240	Q₈	—	2	12	no	—
241	Q₈	—	23	12	no	—
242	Q₈	—	46	24	no	—
243	Q₈	—	46	12	no	—
244	Q₈	D₄×D₂₃	1	4	yes	direct product factor
245	D₄	—	23	8	no	—
246	D₄	—	23	4	no	—
247	D₄	Q₈×D₂₃	1	4	yes	direct product factor
248	C₂×C₄	—	2	12	no	—
249	C₂×C₄	—	23	1	no	—
250	C₂×C₄	—	23	3	no	—
251	C₂×C₄	—	23	12	no	—
252	C₂×C₄	—	23	4	no	—
253	C₂×C₄	—	23	3	no	—
254	C₂×C₄	—	23	6	no	—
255	C₂×C₄	—	46	12	no	—
256	C₂×C₄	—	46	12	no	—
257	C₂×C₄	—	46	6	no	—
258	C₂×C₄	—	46	12	no	—
259	C₂×C₄	—	46	12	no	—
260	C₂×C₄	C₂³×D₂₃	1	3	yes	—
261	C₂×C₄	C₂³×D₂₃	1	6	yes	—
262	C₂×C₄	C₂³×D₂₃	1	3	yes	—
263	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	23	2	no	—
267	C₂²	—	23	2	no	—
268	C₂²	—	23	2	no	—
269	C₂²	—	23	4	no	—
270	C₂²	—	46	16	no	—
271	C₂²	—	46	2	no	—
272	C₂²	—	46	2	no	—
273	C₂²	C₂×Q₈×D₂₃	1	4	yes	complemented
274	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
275	C₄	—	2	6	no	—
276	C₄	—	2	3	no	—
277	C₄	—	23	6	no	—
278	C₄	—	23	2	no	—
279	C₄	—	46	6	no	—
280	C₄	—	46	3	no	—
281	C₄	C₂×D₄×D₂₃	1	6	yes	—
282	C₄	C₂×Q₈×D₂₃	1	2	yes	complemented
283	C₂	—	2	4	no	—
284	C₂	—	23	4	no	—
285	C₂	—	46	4	no	—
286	C₂	C₂²×D₄×D₂₃	1	1	yes	characteristic
287	C₂	C₂²×Q₈×D₂₃	1	1	yes	characteristic
288	C₂	2₋¹⁺⁴×D₂₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁵
Fitting	D₄×Q₈×C₂₃	C₂
prime radical	C₂×D₄×D₂₃	C₂²
hypercenter	D₄×Q₈	D₂₃
hypocenter	C₂₃	C₂×D₄×Q₈
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ D₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×Q₈×D₂₃	C₂	semidirect
2. Q₈×D₉₂	C₂	semidirect
3. D₄×Dic₄₆	C₂	semidirect
4. #13093082	C₂	semidirect
5. Q₈×C₂²⋊D₂₃	C₂	semidirect
6. #13093080	C₂	semidirect
7. C₄⋊Q₈×D₂₃	C₂	semidirect
8. C₂²⋊Q₈×D₂₃	C₂	semidirect
9. C₄×Q₈×D₂₃	C₂	semidirect
11. D₄×Q₈×C₂₃	C₂	semidirect
12. C₂²×Q₈×C₂₃	C₂²	semidirect
16. C₂×Q₈×D₂₃	C₂²	semidirect
17. C₂×Q₈×D₂₃	C₂²	semidirect
20. C₂²×Dic₄₆	C₂²	semidirect
21. C₂₃×C₄⋊Q₈	C₂²	semidirect
22. C₂₃×C₂²⋊Q₈	C₂²	semidirect
23. C₄×Q₈×C₂₃	C₂²	semidirect
25. #10625117	C₂²	semidirect
26. Q₈×Dic₂₃	C₂²	semidirect
27. #10625115	C₂²	semidirect
30. #10625095	C₂²	semidirect
31. #10625068	C₂²	semidirect
32. #10625067	C₂²	semidirect
37. #10625059	C₂²	semidirect
38. #10625059	C₂²	semidirect
39. #10625058	C₂²	semidirect
42. #10625049	C₂²	semidirect
45. #10625040	C₂²	semidirect
46. C₄×Dic₄₆	C₂²	semidirect
58. Q₈×D₂₃	D₄	direct (N × H)
59. D₄×D₂₃	Q₈	direct (N × H)
92. C₂²×D₂₃	C₂×Q₈	semidirect
96. Q₈×C₂₃	C₂×D₄	semidirect
97. D₄×C₂₃	C₂×Q₈	semidirect
103. C₂²⋊D₂₃	C₂×Q₈	semidirect
109. D₉₂	C₂×Q₈	semidirect
115. C₄×D₂₃	C₂×Q₈	semidirect
118. Dic₄₆	C₂×D₄	semidirect
122. C₂²×C₂₃	C₂²×Q₈	semidirect
129. C₂×D₂₃	C₂²×Q₈	semidirect
134. C₄×C₂₃	C₂²×Q₈	semidirect
138. Dic₂₃	C₂²×Q₈	semidirect
145. D₄×Q₈	D₂₃	direct (N × H)
155. D₂₃	D₄×Q₈	direct (N × H)
162. C₂²×Q₈	C₂×D₂₃	semidirect
169. C₄⋊Q₈	C₂×D₂₃	semidirect
175. C₂²⋊Q₈	C₂×D₂₃	semidirect
179. C₄×Q₈	C₂×D₂₃	semidirect
191. C₂₃	C₂×D₄×Q₈	semidirect
244. Q₈	D₄×D₂₃	direct (N × H)
247. D₄	Q₈×D₂₃	direct (N × H)
273. C₂²	C₂×Q₈×D₂₃	semidirect
282. C₄	C₂×Q₈×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Q₈×D₂₃	2
Q₈×D₉₂	2
D₄×Dic₄₆	2
#13093082	2
Q₈×C₂²⋊D₂₃	2
#13093080	2
C₄⋊Q₈×D₂₃	2
C₂²⋊Q₈×D₂₃	2
C₄×Q₈×D₂₃	2
C₄×D₄×D₂₃	2
D₄×Q₈×C₂₃	2
C₂×D₄×Q₈	23

Subgroups — D4×Q8×D23 #15686867

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×Q₈×D₂₃ #15686867