Subgroups — C₄×Q₈⋊(C₄₃⋊C₈) #24987194

Order 11008.

654 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊(C₄₃⋊C₈)	C₂	1	2	yes	maximal
4	#19132436	C₂	1	2	yes	maximal
5	C₄²×Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₄₃⋊C₈	—	2	2	no	—
8	C₂×C₄×C₄₃⋊C₈	—	2	1	no	—
9	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
10	#14582654	—	2	1	no	—
11	Q₈⋊(C₄₃⋊C₈)	C₄	1	8	yes	direct product factor
12	C₂×C₄×Q₈×C₄₃	C₄	1	2	yes	—
13	C₂×C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
14	C₄×C₄⋊C₄×C₄₃	—	2	1	no	—
15	C₄×C₄⋊C₄×C₄₃	—	2	2	no	—
16	C₄×C₄⋊C₄×C₄₃	C₄	1	2	yes	—
17	C₄×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
18	C₄³×C₄₃	—	2	1	no	—
19	C₄³×C₄₃	C₂²	1	1	yes	characteristic
20	C₂²×C₄₃⋊C₈	—	2	2	no	—
21	C₂²×C₄₃⋊C₈	—	2	2	no	—
22	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
23	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	4	yes	—
24	C₄×C₄₃⋊C₈	—	2	8	no	—
25	C₄×C₄₃⋊C₈	—	2	2	no	—
26	C₄×C₄₃⋊C₈	—	4	2	no	—
27	C₄×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
28	C₂²×Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
29	C₄×Q₈×C₄₃	—	2	4	no	—
30	C₄×Q₈×C₄₃	—	2	2	no	—
31	C₄×Q₈×C₄₃	C₈	1	8	yes	complemented
32	C₄×Q₈×C₄₃	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
35	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₄₃	—	2	2	no	—
38	C₂×C₄²×C₄₃	—	2	1	no	—
39	C₂×C₄²×C₄₃	—	2	2	no	—
40	C₂×C₄²×C₄₃	—	2	1	no	—
41	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₄₃	D₄	1	2	yes	—
43	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
46	#95×C₄₃	—	2	2	no	—
47	#95×C₄₃	—	2	1	no	—
48	C₂×Q₈×C₄₃	—	2	1	no	—
49	C₂×Q₈×C₄₃	—	2	2	no	—
50	C₂×Q₈×C₄₃	C₄²	1	2	yes	—
51	C₂×Q₈×C₄₃	C₂×C₈	1	4	yes	—
52	C₂²×C₄×C₄₃	—	2	1	no	—
53	C₂²×C₄×C₄₃	—	2	1	no	—
54	C₂²×C₄×C₄₃	—	2	2	no	—
55	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
60	C₄⋊C₄×C₄₃	—	2	8	no	—
61	C₄⋊C₄×C₄₃	—	2	1	no	—
62	C₄⋊C₄×C₄₃	—	2	4	no	—
63	C₄⋊C₄×C₄₃	—	2	2	no	—
64	C₄⋊C₄×C₄₃	C₄²	1	2	yes	—
65	C₄⋊C₄×C₄₃	C₂×C₈	1	4	yes	—
66	C₄²×C₄₃	—	2	8	no	—
67	C₄²×C₄₃	—	2	4	no	—
68	C₄²×C₄₃	—	2	4	no	—
69	C₄²×C₄₃	—	2	2	no	—
70	C₄²×C₄₃	—	4	4	no	—
71	C₄²×C₄₃	—	4	2	no	—
72	C₄²×C₄₃	C₂×C₈	1	4	yes	—
73	C₄²×C₄₃	M₁₆	1	4	yes	—
74	C₄²×C₄₃	SD₁₆	1	2	yes	—
75	C₄²×C₄₃	Q₁₆	1	2	yes	—
76	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
77	C₄²×C₄₃	#41	1	2	yes	—
78	C₂×C₄₃⋊C₈	—	2	8	no	—
79	C₂×C₄₃⋊C₈	—	2	4	no	—
80	C₂×C₄₃⋊C₈	—	2	4	no	—
81	C₂×C₄₃⋊C₈	—	4	2	no	—
82	C₂×C₄₃⋊C₈	—	4	2	no	—
83	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	Q₈×C₄₃	—	2	2	no	—
85	Q₈×C₄₃	—	2	4	no	—
86	Q₈×C₄₃	C₄×C₈	1	4	yes	complemented
87	C₂×C₄×C₄₃	—	2	2	no	—
88	C₂×C₄×C₄₃	—	2	4	no	—
89	C₂×C₄×C₄₃	—	2	2	no	—
90	C₂×C₄×C₄₃	—	2	2	no	—
91	C₂×C₄×C₄₃	—	2	2	no	—
92	C₂×C₄×C₄₃	—	2	2	no	—
93	C₂×C₄×C₄₃	—	2	4	no	—
94	C₂×C₄×C₄₃	—	4	2	no	—
95	C₂×C₄×C₄₃	—	4	2	no	—
96	C₂×C₄×C₄₃	—	4	2	no	—
97	C₂×C₄×C₄₃	C₂²⋊C₈	1	4	yes	—
98	C₂×C₄×C₄₃	C₂²⋊C₈	1	4	yes	—
99	C₂×C₄×C₄₃	#103	1	4	yes	—
100	C₂×C₄×C₄₃	#104	1	4	yes	—
101	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
103	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
107	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
109	C₂×C₄×C₄₃	#135	1	2	yes	—
110	C₄₃⋊C₈	—	2	8	no	—
111	C₄₃⋊C₈	—	4	4	no	—
112	C₄×Q₈⋊C₈	—	43	1	no	maximal, Sylow
113	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
114	C₂²×C₄₃	C₂×C₂²⋊C₈	1	2	yes	—
115	C₂²×C₄₃	C₂×#103	1	1	yes	characteristic
116	C₂²×C₄₃	#416	1	1	yes	characteristic
117	C₂²×C₄₃	C₂×#104	1	2	yes	—
118	C₄×C₄₃	—	2	4	no	—
119	C₄×C₄₃	—	2	4	no	—
120	C₄×C₄₃	—	2	2	no	—
121	C₄×C₄₃	—	4	4	no	—
122	C₄×C₄₃	—	4	2	no	—
123	C₄×C₄₃	Q₈⋊C₈	1	8	yes	complemented
124	C₄×C₄₃	C₂×C₄×C₈	1	1	yes	characteristic
125	C₄×C₄₃	C₄×M₁₆	1	1	yes	characteristic
126	C₄×C₄₃	#431	1	2	yes	—
127	C₄×C₄₃	C₄×SD₁₆	1	2	yes	—
128	C₄×C₄₃	C₄×Q₁₆	1	2	yes	—
129	C₄²×Q₈	D₄₃	1	1	yes	characteristic
130	C₄×C₄⋊C₈	—	43	1	no	—
131	C₄²×C₈	—	43	1	no	—
132	C₂×Q₈⋊C₈	—	43	2	no	—
133	#1279	—	43	2	no	—
134	C₂×C₄₃	C₂×Q₈⋊C₈	1	2	yes	—
135	C₂×C₄₃	#1476	1	2	yes	—
136	C₂×C₄₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
137	C₂×C₄₃	C₄×#104	1	1	yes	characteristic
138	C₂×C₄₃	C₄×#103	1	1	yes	characteristic
139	C₂×C₄×Q₈	Dic₄₃	1	2	yes	—
140	C₂×C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
141	C₂×C₄⋊C₈	—	43	2	no	—
142	C₂×C₄×C₈	—	43	2	no	—
143	C₂×C₄×C₈	—	86	2	no	—
144	C₂×C₄×C₈	—	86	1	no	—
145	C₄×C₄⋊C₄	—	2	1	no	—
146	C₄×C₄⋊C₄	—	2	2	no	—
147	C₄×C₄⋊C₄	Dic₄₃	1	2	yes	—
148	C₄×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
149	C₄³	—	2	1	no	—
150	C₄³	C₂×D₄₃	1	1	yes	characteristic
151	#336	—	86	1	no	—
152	Q₈⋊C₈	—	43	8	no	—
153	C₄₃	C₄×Q₈⋊C₈	1	1	yes	characteristic, complemented, Sylow
154	C₂²×Q₈	C₂×Dic₄₃	1	1	yes	characteristic
155	C₂²×C₈	—	86	2	no	—
156	C₂²×C₈	—	86	2	no	—
157	C₄×Q₈	—	2	4	no	—
158	C₄×Q₈	—	2	2	no	—
159	C₄×Q₈	C₄₃⋊C₈	1	8	yes	complemented
160	C₄×Q₈	C₄×D₄₃	1	4	yes	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	—	2	1	no	—
163	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
164	C₂×C₄⋊C₄	C₂×Dic₄₃	1	1	yes	characteristic
165	C₂×C₄²	—	2	2	no	—
166	C₂×C₄²	—	2	1	no	—
167	C₂×C₄²	—	2	2	no	—
168	C₂×C₄²	—	2	1	no	—
169	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
170	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
171	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
172	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
173	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
174	C₄⋊C₈	—	43	4	no	—
175	C₄⋊C₈	—	86	2	no	—
176	C₄×C₈	—	43	4	no	—
177	C₄×C₈	—	86	8	no	—
178	C₄×C₈	—	86	2	no	—
179	C₄×C₈	—	172	2	no	—
180	#95	—	2	2	no	—
181	#95	—	2	1	no	—
182	C₂×Q₈	—	2	1	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂×Q₈	C₂×C₄₃⋊C₈	1	4	yes	—
185	C₂×Q₈	C₄×Dic₄₃	1	2	yes	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	C₂²⋊Dic₄₃	1	2	yes	—
190	C₂²×C₄	C₂²⋊Dic₄₃	1	2	yes	—
191	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
194	C₂×C₈	—	86	8	no	—
195	C₂×C₈	—	86	4	no	—
196	C₂×C₈	—	86	4	no	—
197	C₂×C₈	—	172	2	no	—
198	C₂×C₈	—	172	2	no	—
199	C₄⋊C₄	—	2	8	no	—
200	C₄⋊C₄	—	2	1	no	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	C₂×C₄₃⋊C₈	1	4	yes	—
204	C₄⋊C₄	C₄×Dic₄₃	1	2	yes	—
205	C₄²	—	2	8	no	—
206	C₄²	—	2	4	no	—
207	C₄²	—	2	4	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	4	4	no	—
210	C₄²	—	4	2	no	—
211	C₄²	C₂×C₄₃⋊C₈	1	4	yes	—
212	C₄²	#10621877	1	4	yes	—
213	C₄²	#10621884	1	2	yes	—
214	C₄²	#10621885	1	2	yes	—
215	C₄²	C₂×C₄×D₄₃	1	2	yes	—
216	C₄²	#10621898	1	2	yes	—
217	C₂³	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
218	Q₈	—	2	2	no	—
219	Q₈	—	2	4	no	—
220	Q₈	C₄×C₄₃⋊C₈	1	4	yes	complemented
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	C₂²⋊(C₄₃⋊C₈)	1	4	yes	—
232	C₂×C₄	C₂²⋊(C₄₃⋊C₈)	1	4	yes	—
233	C₂×C₄	#13063249	1	4	yes	—
234	C₂×C₄	#13063252	1	4	yes	—
235	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
236	C₂×C₄	#13063284	1	1	yes	characteristic
237	C₂×C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
238	C₂×C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
239	C₂×C₄	C₂²×C₄₃⋊C₈	1	2	yes	—
240	C₂×C₄	C₂×#10621877	1	2	yes	—
241	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
242	C₂×C₄	C₂×#10621885	1	1	yes	characteristic
243	C₂×C₄	#13063345	1	2	yes	—
244	C₈	—	86	8	no	—
245	C₈	—	172	4	no	—
246	C₂²	C₄×C₂²⋊Dic₄₃	1	1	yes	characteristic
247	C₂²	C₂×C₂²⋊(C₄₃⋊C₈)	1	2	yes	—
248	C₂²	C₂×#13063249	1	1	yes	characteristic
249	C₂²	#14582895	1	1	yes	characteristic
250	C₂²	C₂×#13063252	1	2	yes	—
251	C₄	—	2	4	no	—
252	C₄	—	2	4	no	—
253	C₄	—	2	2	no	—
254	C₄	—	4	4	no	—
255	C₄	—	4	2	no	—
256	C₄	Q₈⋊(C₄₃⋊C₈)	1	8	yes	direct product factor
257	C₄	C₂×C₄×C₄₃⋊C₈	1	1	yes	characteristic
258	C₄	C₄×#10621877	1	1	yes	characteristic
259	C₄	#14582952	1	2	yes	—
260	C₄	C₄×#10621884	1	2	yes	—
261	C₄	C₄×#10621885	1	2	yes	—
262	C₂	C₂×Q₈⋊(C₄₃⋊C₈)	1	2	yes	—
263	C₂	#19132977	1	2	yes	—
264	C₂	C₄×C₂²⋊(C₄₃⋊C₈)	1	1	yes	characteristic
265	C₂	C₄×#13063252	1	1	yes	characteristic
266	C₂	C₄×#13063249	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂²⋊D₄₃
commutator	C₄×C₄₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄²×Q₈×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₄²×Q₈	D₄₃
hypocenter	C₄₃	C₄×Q₈⋊C₈
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄³ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
11. Q₈⋊(C₄₃⋊C₈)	C₄	direct (N × H)
31. C₄×Q₈×C₄₃	C₈	semidirect
86. Q₈×C₄₃	C₄×C₈	semidirect
123. C₄×C₄₃	Q₈⋊C₈	semidirect
153. C₄₃	C₄×Q₈⋊C₈	semidirect
159. C₄×Q₈	C₄₃⋊C₈	semidirect
220. Q₈	C₄×C₄₃⋊C₈	semidirect
256. C₄	Q₈⋊(C₄₃⋊C₈)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₄⋊(C₄₃⋊C₈)	2
C₄²×C₄₃⋊C₈	2
C₂×Q₈⋊(C₄₃⋊C₈)	2
#19132436	2
C₄²×Q₈×C₄₃	2
C₄×Q₈⋊C₈	43

Subgroups — C4×Q8⋊(C43⋊C8) #24987194

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×Q₈⋊(C₄₃⋊C₈) #24987194