Subgroups — C₃⋊C₈×#208 #11773550

Order 1152.

507 conjugacy classes of subgroups.

The subgroup lattice has 275 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10592624	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×#208	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#10589078	C₂	1	2	yes	maximal, complemented
5	C₃⋊C₈²	C₂	1	2	yes	maximal, complemented
6	M₁₆×C₃⋊C₈	—	3	1	no	maximal
7	C₈×#208	—	3	1	no	maximal
8	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
9	C₂×C₃×#208	C₄	1	2	yes	—
10	C₂²×C₃×C₃⋊C₈	C₄	1	2	yes	—
11	#63657	C₄	1	2	yes	—
12	C₄⋊(C₃⋊(C₃⋊C₈))	C₄	1	2	yes	—
13	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
14	C₃×#803	C₄	1	2	yes	—
15	C₃×C₄×C₃⋊C₈	C₄	1	2	yes	—
16	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₃×C₄×C₃⋊C₈	C₂²	1	2	yes	—
19	C₃×C₄×M₁₆	—	3	1	no	—
20	C₂×C₃×C₄×C₈	—	3	1	no	—
21	#4925	—	3	1	no	—
22	#4925	—	3	1	no	—
23	#4925	—	6	1	no	—
24	C₄×#208	D₃	1	1	yes	characteristic
25	C₂×C₄×C₃⋊C₈	D₃	1	1	yes	characteristic
26	#4455	—	3	2	no	—
27	#4454	—	3	2	no	—
28	C₈×C₃⋊C₈	—	3	2	no	—
29	C₈×C₃⋊C₈	—	3	2	no	—
30	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
33	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
34	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
35	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
36	C₃²×C₄²	C₂³	1	1	yes	characteristic
37	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	4	yes	—
38	C₃×#208	C₈	1	8	yes	complemented
39	C₂×C₃×C₃⋊C₈	—	2	4	no	—
40	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
41	C₂×C₃×C₃⋊C₈	C₂×C₄	1	4	yes	—
42	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
43	C₈×M₁₆	—	9	1	no	Sylow
44	C₂×C₃×M₁₆	—	3	2	no	—
45	C₂²×C₃×C₈	—	3	2	no	—
46	C₂×C₃×C₄²	—	2	1	no	—
47	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
48	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
49	C₂×#208	Dic₃	1	2	yes	—
50	C₂²×C₃⋊C₈	Dic₃	1	2	yes	—
51	C₃×#97	—	3	2	no	—
52	C₃×C₄×C₈	—	3	2	no	—
53	C₃×C₄×C₈	—	3	1	no	—
54	C₃×C₄×C₈	—	3	2	no	—
55	C₃×C₄×C₈	—	3	1	no	—
56	#830	—	3	2	no	—
57	#830	—	3	2	no	—
58	#830	—	6	2	no	—
59	C₄⋊(C₃⋊C₈)	—	3	2	no	—
60	C₄⋊(C₃⋊C₈)	—	3	2	no	—
61	C₄⋊(C₃⋊C₈)	—	6	2	no	—
62	#803	Dic₃	1	2	yes	—
63	C₄×C₃⋊C₈	—	3	2	no	—
64	C₄×C₃⋊C₈	—	3	2	no	—
65	C₄×C₃⋊C₈	—	6	2	no	—
66	C₄×C₃⋊C₈	Dic₃	1	2	yes	—
67	C₄×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
68	C₄×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
69	C₄×C₃⋊C₈	C₂×D₃	1	2	yes	—
70	C₂³×C₃²	C₄²	1	1	yes	characteristic
71	C₂×C₃²×C₄	—	2	2	no	—
72	C₂×C₃²×C₄	—	2	2	no	—
73	C₂×C₃²×C₄	C₄²	1	1	yes	characteristic
74	C₂×C₃²×C₄	C₄²	1	1	yes	characteristic
75	C₂×C₃²×C₄	C₄²	1	1	yes	characteristic
76	C₂×C₃²×C₄	C₂×C₈	1	4	yes	—
77	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
80	C₃⋊(C₃⋊C₈)	—	2	4	no	—
81	C₃×C₃⋊C₈	—	2	2	no	—
82	C₃×C₃⋊C₈	C₂×C₈	1	8	yes	complemented
83	C₃×C₃⋊C₈	M₁₆	1	4	yes	complemented
84	#431	—	9	1	no	—
85	C₄×M₁₆	—	3	1	no	—
86	C₂×C₄×C₈	—	3	1	no	—
87	#322	—	9	2	no	—
88	C₈²	—	9	2	no	—
89	C₂²×C₃×C₄	—	2	1	no	—
90	C₂²×C₃×C₄	—	2	1	no	—
91	C₂²×C₃×C₄	—	2	1	no	—
92	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
93	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
94	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
95	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
96	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
97	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
98	C₃×M₁₆	—	3	8	no	—
99	C₂×C₃×C₈	—	3	2	no	—
100	C₂×C₃×C₈	—	3	4	no	—
101	C₂×C₃×C₈	—	3	2	no	—
102	C₂×C₃×C₈	—	6	4	no	—
103	C₃×C₄²	—	2	1	no	—
104	C₃×C₄²	—	2	1	no	—
105	C₃×C₄²	—	2	1	no	—
106	C₃×C₄²	—	2	1	no	—
107	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
108	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
109	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
110	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
111	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
112	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
113	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
114	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
115	#208	C₃⋊C₈	1	8	yes	direct product factor
116	C₂×C₃⋊C₈	—	2	4	no	—
117	C₂×C₃⋊C₈	—	3	4	no	—
118	C₂×C₃⋊C₈	—	3	4	no	—
119	C₂×C₃⋊C₈	—	6	4	no	—
120	C₂×C₃⋊C₈	C₂×Dic₃	1	2	yes	—
121	C₂×C₃⋊C₈	C₂×Dic₃	1	4	yes	—
122	C₂×C₃⋊C₈	C₂×Dic₃	1	2	yes	—
123	C₂²×C₃²	—	2	1	no	—
124	C₂²×C₃²	—	2	1	no	—
125	C₂²×C₃²	C₄×C₈	1	2	yes	—
126	C₂²×C₃²	C₂×C₄²	1	1	yes	characteristic
127	C₃²×C₄	—	2	1	no	—
128	C₃²×C₄	—	2	1	no	—
129	C₃²×C₄	C₄×C₈	1	2	yes	—
130	C₃²×C₄	C₂²×C₈	1	2	yes	—
131	C₃²×C₄	C₂×M₁₆	1	2	yes	—
132	C₃²×C₄	#131	1	2	yes	—
133	C₂×M₁₆	—	3	2	no	—
134	C₂²×C₈	—	3	2	no	—
135	C₂×C₄²	D₃²	1	1	yes	characteristic
136	C₄⋊C₈	—	9	2	no	—
137	C₂²⋊C₈	—	9	2	no	—
138	#97	—	3	2	no	—
139	C₄×C₈	—	3	2	no	—
140	C₄×C₈	—	3	1	no	—
141	C₄×C₈	—	3	2	no	—
142	C₄×C₈	—	3	1	no	—
143	C₄×C₈	—	9	2	no	—
144	C₂³×C₃	—	2	1	no	—
145	C₂³×C₃	C₄×Dic₃	1	1	yes	characteristic
146	C₂³×C₃	C₄×Dic₃	1	1	yes	characteristic
147	C₂×C₃×C₄	—	2	2	no	—
148	C₂×C₃×C₄	—	2	2	no	—
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	2	4	no	—
152	C₂×C₃×C₄	—	2	1	no	—
153	C₂×C₃×C₄	—	2	1	no	—
154	C₂×C₃×C₄	—	2	1	no	—
155	C₂×C₃×C₄	—	2	1	no	—
156	C₂×C₃×C₄	—	2	1	no	—
157	C₂×C₃×C₄	—	2	1	no	—
158	C₂×C₃×C₄	—	4	2	no	—
159	C₂×C₃×C₄	—	4	2	no	—
160	C₂×C₃×C₄	D₃×C₈	1	4	yes	—
161	C₂×C₃×C₄	C₂×C₃⋊C₈	1	4	yes	—
162	C₂×C₃×C₄	C₄×Dic₃	1	1	yes	characteristic
163	C₂×C₃×C₄	C₄×Dic₃	1	1	yes	characteristic
164	C₂×C₃×C₄	C₄×Dic₃	1	1	yes	characteristic
165	C₂×C₃×C₄	C₄×Dic₃	1	1	yes	characteristic
166	C₂×C₃×C₄	C₄×Dic₃	1	1	yes	characteristic
167	C₂×C₃×C₄	C₄×Dic₃	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
174	C₃×C₈	—	3	8	no	—
175	C₃×C₈	—	3	4	no	—
176	C₃×C₈	—	6	2	no	—
177	C₃⋊C₈	—	2	2	no	—
178	C₃⋊C₈	—	6	4	no	—
179	C₃⋊C₈	—	6	4	no	—
180	C₃⋊C₈	—	12	4	no	—
181	C₃⋊C₈	C₂×C₃⋊C₈	1	8	yes	complemented
182	C₃⋊C₈	#208	1	4	yes	direct product factor
183	C₂×C₃²	—	2	2	no	—
184	C₂×C₃²	C₂×C₄×C₈	1	1	yes	characteristic
185	C₂×C₃²	C₄×M₁₆	1	1	yes	characteristic
186	C₂×C₃²	#405	1	1	yes	characteristic
187	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
188	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
189	C₂²×C₄	#624	1	1	yes	characteristic
190	M₁₆	—	3	8	no	—
191	C₂×C₈	—	3	2	no	—
192	C₂×C₈	—	3	4	no	—
193	C₂×C₈	—	3	2	no	—
194	C₂×C₈	—	6	4	no	—
195	C₂×C₈	—	9	4	no	—
196	C₄²	D₃×Dic₃	1	1	yes	characteristic
197	C₄²	D₃×Dic₃	1	1	yes	characteristic
198	C₄²	#624	1	1	yes	characteristic
199	C₄²	C₂×D₃²	1	1	yes	characteristic
200	C₂²×C₃	—	2	1	no	—
201	C₂²×C₃	—	2	1	no	—
202	C₂²×C₃	—	2	1	no	—
203	C₂²×C₃	—	2	1	no	—
204	C₂²×C₃	—	2	2	no	—
205	C₂²×C₃	—	2	1	no	—
206	C₂²×C₃	—	4	1	no	—
207	C₂²×C₃	—	4	1	no	—
208	C₂²×C₃	C₄×C₃⋊C₈	1	2	yes	—
209	C₂²×C₃	C₈×Dic₃	1	2	yes	—
210	C₂²×C₃	C₂×C₄×Dic₃	1	1	yes	characteristic
211	C₂²×C₃	C₂×C₄×Dic₃	1	1	yes	characteristic
212	C₃×C₄	—	2	1	no	—
213	C₃×C₄	—	2	1	no	—
214	C₃×C₄	—	2	1	no	—
215	C₃×C₄	—	2	1	no	—
216	C₃×C₄	—	2	2	no	—
217	C₃×C₄	—	2	2	no	—
218	C₃×C₄	—	2	2	no	—
219	C₃×C₄	—	2	2	no	—
220	C₃×C₄	—	4	1	no	—
221	C₃×C₄	—	4	1	no	—
222	C₃×C₄	C₄×C₃⋊C₈	1	2	yes	—
223	C₃×C₄	C₈×Dic₃	1	2	yes	—
224	C₃×C₄	C₂×D₃×C₈	1	2	yes	—
225	C₃×C₄	#901	1	2	yes	—
226	C₃×C₄	D₃×M₁₆	1	2	yes	—
227	C₃×C₄	#907	1	2	yes	—
228	C₃×C₄	C₂²×C₃⋊C₈	1	2	yes	—
229	C₃×C₄	C₂×#208	1	2	yes	—
230	C₃²	C₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
231	C₂³	Dic₃²	1	1	yes	characteristic
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	D₃×C₃⋊C₈	1	4	yes	—
235	C₂×C₄	Dic₃²	1	1	yes	characteristic
236	C₂×C₄	Dic₃²	1	1	yes	characteristic
237	C₂×C₄	Dic₃²	1	1	yes	characteristic
238	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
239	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
240	C₂×C₄	C₂×#624	1	1	yes	characteristic
241	C₈	—	3	8	no	—
242	C₈	—	3	4	no	—
243	C₈	—	6	2	no	—
244	C₈	—	18	4	no	—
245	C₂×C₃	—	2	2	no	—
246	C₂×C₃	—	2	2	no	—
247	C₂×C₃	—	2	1	no	—
248	C₂×C₃	—	2	1	no	—
249	C₂×C₃	—	2	1	no	—
250	C₂×C₃	—	4	2	no	—
251	C₂×C₃	C₂×C₄×C₃⋊C₈	1	1	yes	characteristic
252	C₂×C₃	C₄×#208	1	1	yes	characteristic
253	C₂×C₃	C₂×C₈×Dic₃	1	1	yes	characteristic
254	C₂×C₃	#5101	1	1	yes	characteristic
255	C₂×C₃	Dic₃×M₁₆	1	1	yes	characteristic
256	C₂×C₃	#5122	1	1	yes	characteristic
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	Dic₃×C₃⋊C₈	1	2	yes	—
260	C₂²	C₂×Dic₃²	1	1	yes	characteristic
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	Dic₃×C₃⋊C₈	1	2	yes	—
264	C₄	C₂×D₃×C₃⋊C₈	1	2	yes	—
265	C₄	D₃×#208	1	2	yes	—
266	C₄	#63817	1	2	yes	—
267	C₃	—	2	1	no	—
268	C₃	C₈×#208	1	1	yes	characteristic, complemented
269	C₃	M₁₆×C₃⋊C₈	1	1	yes	characteristic, complemented
270	C₂	—	2	2	no	—
271	C₂	C₂×Dic₃×C₃⋊C₈	1	1	yes	characteristic
272	C₂	Dic₃×#208	1	1	yes	characteristic
273	C₂	#10591059	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂×D₃²
commutator	C₂×C₃²	C₂×C₄×C₈
Fitting	C₂×C₃²×C₄²	C₂²
socle	C₂²×C₃²	C₂×C₄²
prime radical	C₂³×C₃²	C₄²
cosocle	C₃²×C₄²	C₂³
hypercenter	C₂×C₄²	D₃²
hypocenter	C₃²	C₈×M₁₆
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄² ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
4. #10589078	C₂	semidirect
5. C₃⋊C₈²	C₂	semidirect
38. C₃×#208	C₈	semidirect
82. C₃×C₃⋊C₈	C₂×C₈	semidirect
83. C₃×C₃⋊C₈	M₁₆	semidirect
115. #208	C₃⋊C₈	direct (N × H)
181. C₃⋊C₈	C₂×C₃⋊C₈	semidirect
182. C₃⋊C₈	#208	direct (N × H)
230. C₃²	C₈×M₁₆	semidirect
268. C₃	C₈×#208	semidirect
269. C₃	M₁₆×C₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#10592624	2
C₃×C₄×#208	2
C₂×C₃×C₄×C₃⋊C₈	2
#10589078	2
C₃⋊C₈²	2
M₁₆×C₃⋊C₈	3
C₈×#208	3

Subgroups — C3⋊C8×#208 #11773550

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃⋊C₈×#208 #11773550