Subgroups — D₄×#67267 #15682134

Order 2944.

544 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#67267	C₂	1	2	yes	maximal, complemented
2	#13092495	C₂	1	2	yes	maximal, complemented
3	#13092489	C₂	1	4	yes	maximal, complemented
4	D₄×C₂₃⋊C₈	C₂	1	2	yes	maximal, complemented
5	#13092441	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092381	C₂	1	2	yes	maximal, complemented
7	C₄×#67267	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×D₄×C₂₃	C₄	1	1	yes	characteristic
10	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
11	C₄×D₄×C₂₃	C₄	1	2	yes	—
12	C₄×D₄×C₂₃	C₄	1	1	yes	characteristic
13	C₄×D₄×C₂₃	C₄	1	1	yes	characteristic
14	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₄×D₄×C₂₃	C₂²	1	2	yes	—
16	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₂₃	C₄	1	1	yes	characteristic
18	C₂×C₂²⋊C₄×C₂₃	C₄	1	2	yes	—
19	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×#67267	—	2	8	no	—
21	C₂×#67267	C₂²	1	1	yes	characteristic
22	C₂×#67267	C₂²	1	2	yes	—
23	C₂×#67267	C₂²	1	1	yes	characteristic
24	C₂×#67267	C₂²	1	4	yes	complemented
25	C₂²×C₂₃⋊C₈	C₂²	1	4	yes	complemented
26	C₂²⋊(C₂₃⋊C₈)	C₂²	1	4	yes	complemented
27	C₂²⋊(C₂₃⋊C₈)	C₂²	1	4	yes	complemented
28	C₄⋊(C₂₃⋊C₈)	C₂²	1	2	yes	complemented
29	C₄⋊(C₂₃⋊C₈)	C₂²	1	2	yes	complemented
30	#10624985	C₂²	1	2	yes	complemented
31	C₄×C₂₃⋊C₈	C₂²	1	2	yes	complemented
32	C₂⁴×C₂₃	C₂×C₄	1	2	yes	—
33	C₂×D₄×C₂₃	—	2	4	no	—
34	C₂×D₄×C₂₃	C₂×C₄	1	2	yes	—
35	C₂×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
36	C₂×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₂₃	—	2	4	no	—
38	C₂²×C₄×C₂₃	—	2	4	no	—
39	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
41	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
42	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
48	C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
49	C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
50	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
51	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
52	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
53	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
54	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
55	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
56	C₄²×C₂₃	C₂³	1	1	yes	characteristic
57	C₄²×C₂₃	C₂³	1	1	yes	characteristic
58	#67267	—	2	4	no	—
59	#67267	—	2	2	no	—
60	#67267	—	2	8	no	—
61	#67267	D₄	1	4	yes	direct product factor
62	C₂×C₂₃⋊C₈	—	2	8	no	—
63	C₂×C₂₃⋊C₈	C₂³	1	4	yes	—
64	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
65	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
66	C₂³×C₂₃	—	2	4	no	—
67	C₂³×C₂₃	—	2	4	no	—
68	C₂³×C₂₃	—	4	2	no	—
69	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
70	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
71	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
72	D₄×C₂₃	—	2	4	no	—
73	D₄×C₂₃	—	2	2	no	—
74	D₄×C₂₃	M₁₆	1	4	yes	complemented
75	C₂×C₄×C₂₃	—	2	4	no	—
76	C₂×C₄×C₂₃	—	2	4	no	—
77	C₂×C₄×C₂₃	—	2	4	no	—
78	C₂×C₄×C₂₃	—	2	2	no	—
79	C₂×C₄×C₂₃	—	2	2	no	—
80	C₂×C₄×C₂₃	—	2	2	no	—
81	C₂×C₄×C₂₃	—	2	2	no	—
82	C₂×C₄×C₂₃	—	2	2	no	—
83	C₂×C₄×C₂₃	—	4	1	no	—
84	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₂₃	#41	1	2	yes	—
93	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
94	C₂₃⋊C₈	—	2	4	no	—
95	C₂₃⋊C₈	—	2	2	no	—
96	C₂₃⋊C₈	C₂×D₄	1	4	yes	complemented
97	D₄×M₁₆	—	23	1	no	maximal, Sylow
98	C₂²×C₂₃	—	2	2	no	—
99	C₂²×C₂₃	—	2	1	no	—
100	C₂²×C₂₃	—	2	1	no	—
101	C₂²×C₂₃	—	2	2	no	—
102	C₂²×C₂₃	—	2	2	no	—
103	C₂²×C₂₃	—	2	2	no	—
104	C₂²×C₂₃	—	2	2	no	—
105	C₂²×C₂₃	—	4	8	no	—
106	C₂²×C₂₃	C₄×D₄	1	2	yes	—
107	C₂²×C₂₃	C₂×M₁₆	1	4	yes	complemented
108	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
109	C₄×C₂₃	—	2	2	no	—
110	C₄×C₂₃	—	2	2	no	—
111	C₄×C₂₃	—	2	1	no	—
112	C₄×C₂₃	—	2	1	no	—
113	C₄×C₂₃	—	2	1	no	—
114	C₄×C₂₃	C₄×D₄	1	2	yes	—
115	C₄×C₂₃	C₂×M₁₆	1	2	yes	complemented
116	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
117	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
118	C₂²×M₁₆	—	23	2	no	—
119	C₂×C₄×D₄	D₂₃	1	1	yes	characteristic
120	#436	—	23	2	no	—
121	#435	—	23	4	no	—
122	C₈×D₄	—	23	2	no	—
123	#423	—	23	1	no	—
124	#407	—	23	2	no	—
125	C₄×M₁₆	—	23	1	no	—
126	C₂×C₂₃	—	2	2	no	—
127	C₂×C₂₃	—	2	4	no	—
128	C₂×C₂₃	—	4	2	no	—
129	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
130	C₂×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
131	C₂×C₂₃	#568	1	1	yes	characteristic
132	C₂²×D₄	Dic₂₃	1	1	yes	characteristic
133	C₂³×C₄	C₂×D₂₃	1	2	yes	—
134	C₂×M₁₆	—	23	1	no	—
135	C₂×M₁₆	—	23	2	no	—
136	C₂×M₁₆	—	23	4	no	—
137	C₂×M₁₆	—	23	1	no	—
138	C₂×M₁₆	—	46	8	no	—
139	C₂²×C₈	—	23	4	no	—
140	C₄×D₄	Dic₂₃	1	2	yes	—
141	C₄×D₄	Dic₂₃	1	1	yes	characteristic
142	C₄×D₄	Dic₂₃	1	1	yes	characteristic
143	C₄×D₄	C₂×D₂₃	1	1	yes	characteristic
144	C₄×D₄	C₂×D₂₃	1	2	yes	—
145	C₄×D₄	C₂×D₂₃	1	1	yes	characteristic
146	C₂×C₄⋊C₄	Dic₂₃	1	1	yes	characteristic
147	C₂×C₂²⋊C₄	Dic₂₃	1	2	yes	—
148	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
149	C₄⋊C₈	—	23	2	no	—
150	C₄⋊C₈	—	23	2	no	—
151	C₂²⋊C₈	—	23	4	no	—
152	C₂²⋊C₈	—	23	4	no	—
153	#97	—	23	2	no	—
154	C₄×C₈	—	23	2	no	—
155	C₂₃	D₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
156	C₂⁴	C₂×Dic₂₃	1	2	yes	—
157	C₂×D₄	—	2	4	no	—
158	C₂×D₄	C₂×Dic₂₃	1	2	yes	—
159	C₂×D₄	C₂×Dic₂₃	1	1	yes	characteristic
160	C₂×D₄	C₂×Dic₂₃	1	1	yes	characteristic
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
164	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
165	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
166	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
167	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
168	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
169	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
170	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
171	M₁₆	—	23	4	no	—
172	M₁₆	—	46	8	no	—
173	M₁₆	—	46	4	no	—
174	M₁₆	—	46	2	no	—
175	C₂×C₈	—	23	2	no	—
176	C₂×C₈	—	23	2	no	—
177	C₂×C₈	—	23	4	no	—
178	C₂×C₈	—	46	8	no	—
179	C₄⋊C₄	C₂×Dic₂₃	1	2	yes	—
180	C₄⋊C₄	C₂×Dic₂₃	1	1	yes	characteristic
181	C₄⋊C₄	C₂×Dic₂₃	1	1	yes	characteristic
182	C₂²⋊C₄	C₂×Dic₂₃	1	2	yes	—
183	C₂²⋊C₄	C₂×Dic₂₃	1	2	yes	—
184	C₂²⋊C₄	C₂×Dic₂₃	1	2	yes	—
185	C₂²⋊C₄	C₂×Dic₂₃	1	2	yes	—
186	C₄²	C₂×Dic₂₃	1	1	yes	characteristic
187	C₄²	C₂×Dic₂₃	1	1	yes	characteristic
188	C₄²	C₂²×D₂₃	1	1	yes	characteristic
189	C₄²	C₂²×D₂₃	1	1	yes	characteristic
190	C₂³	—	2	4	no	—
191	C₂³	—	2	4	no	—
192	C₂³	—	4	2	no	—
193	C₂³	C₂²×Dic₂₃	1	2	yes	—
194	C₂³	C₂²×Dic₂₃	1	2	yes	—
195	C₂³	C₂²×Dic₂₃	1	1	yes	characteristic
196	D₄	—	2	4	no	—
197	D₄	—	2	2	no	—
198	D₄	#67267	1	4	yes	direct product factor
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	D₄×D₂₃	1	2	yes	—
209	C₂×C₄	#67290	1	2	yes	—
210	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
211	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
212	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
213	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
214	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
215	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
216	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
217	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
218	C₈	—	23	4	no	—
219	C₈	—	46	4	no	—
220	C₈	—	46	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	4	8	no	—
229	C₂²	C₂×#67267	1	4	yes	complemented
230	C₂²	D₄×Dic₂₃	1	2	yes	—
231	C₂²	C₂³×Dic₂₃	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	C₂×#67267	1	2	yes	complemented
238	C₄	D₄×Dic₂₃	1	2	yes	—
239	C₄	C₂×D₄×D₂₃	1	1	yes	characteristic
240	C₄	C₂×#67290	1	1	yes	characteristic
241	C₂	—	2	2	no	—
242	C₂	—	2	4	no	—
243	C₂	—	4	2	no	—
244	C₂	C₂×D₄×Dic₂₃	1	1	yes	characteristic
245	C₂	C₂²×#67267	1	1	yes	characteristic
246	C₂	#13093184	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₃
commutator, socle	C₂²×C₂₃	C₂³×C₄
Fitting	C₂×C₄×D₄×C₂₃	C₂
prime radical	C₂²×D₄×C₂₃	C₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₂×C₄×D₄	D₂₃
hypocenter	C₂₃	D₄×M₁₆
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#67267	C₂	semidirect
2. #13092495	C₂	semidirect
3. #13092489	C₂	semidirect
4. D₄×C₂₃⋊C₈	C₂	semidirect
5. #13092441	C₂	semidirect
6. #13092381	C₂	semidirect
7. C₄×#67267	C₂	semidirect
24. C₂×#67267	C₂²	semidirect
25. C₂²×C₂₃⋊C₈	C₂²	semidirect
26. C₂²⋊(C₂₃⋊C₈)	C₂²	semidirect
27. C₂²⋊(C₂₃⋊C₈)	C₂²	semidirect
28. C₄⋊(C₂₃⋊C₈)	C₂²	semidirect
29. C₄⋊(C₂₃⋊C₈)	C₂²	semidirect
30. #10624985	C₂²	semidirect
31. C₄×C₂₃⋊C₈	C₂²	semidirect
61. #67267	D₄	direct (N × H)
74. D₄×C₂₃	M₁₆	semidirect
96. C₂₃⋊C₈	C₂×D₄	semidirect
107. C₂²×C₂₃	C₂×M₁₆	semidirect
115. C₄×C₂₃	C₂×M₁₆	semidirect
155. C₂₃	D₄×M₁₆	semidirect
198. D₄	#67267	direct (N × H)
229. C₂²	C₂×#67267	semidirect
237. C₄	C₂×#67267	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#67267	2
#13092495	2
#13092489	2
D₄×C₂₃⋊C₈	2
#13092441	2
#13092381	2
C₄×#67267	2
C₂×C₄×D₄×C₂₃	2
D₄×M₁₆	23

Subgroups — D4×#67267 #15682134

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#67267 #15682134