Subgroups — C₂×Dic₃⋊(C₃⋊C₁₆) #11796981

Order 1152.

417 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	Dic₃⋊(C₃⋊C₁₆)	C₂	1	4	yes	maximal, direct product factor
5	C₂×Dic₃⋊C₁₆	—	3	1	no	maximal
6	C₂×C₄⋊(C₃⋊C₁₆)	—	3	1	no	maximal
7	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄×Dic₃	C₄	1	1	yes	characteristic
9	C₂²×C₃×C₃⋊C₈	C₄	1	1	yes	characteristic
10	C₂×C₃⋊(C₃⋊C₁₆)	—	2	2	no	—
11	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
12	C₃×C₈×Dic₃	C₄	1	2	yes	—
13	C₃×C₈×Dic₃	C₂²	1	2	yes	—
14	C₂×C₃×C₃⋊C₁₆	—	2	2	no	—
15	C₂×C₃×C₃⋊C₁₆	C₂²	1	2	yes	—
16	C₂²×C₃×C₁₆	—	3	1	no	—
17	C₂×C₃×C₄×C₈	—	3	1	no	—
18	C₂×C₈×Dic₃	D₃	1	1	yes	characteristic
19	C₂²×C₃⋊C₁₆	—	3	1	no	—
20	C₂²×C₃⋊C₁₆	—	3	1	no	—
21	C₂²×C₃⋊C₁₆	—	6	1	no	—
22	C₂²×C₃⋊C₁₆	D₃	1	1	yes	characteristic
23	Dic₃⋊C₁₆	—	3	4	no	—
24	C₄⋊(C₃⋊C₁₆)	—	3	4	no	—
25	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×Dic₃	C₈	1	1	yes	characteristic
27	C₂²×C₃×Dic₃	C₈	1	1	yes	characteristic
28	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃²×C₈	D₄	1	2	yes	—
30	C₂×C₃²×C₈	Q₈	1	2	yes	—
31	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
32	C₃×C₄×Dic₃	C₈	1	2	yes	—
33	C₃×C₄×Dic₃	C₂×C₄	1	2	yes	—
34	C₂×C₃×C₃⋊C₈	—	2	2	no	—
35	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
36	C₃⋊(C₃⋊C₁₆)	—	2	2	no	—
37	C₃×C₃⋊C₁₆	—	2	2	no	—
38	C₂×C₄⋊C₁₆	—	9	1	no	Sylow
39	C₂²×C₃×C₈	—	2	1	no	—
40	C₂²×C₃×C₈	—	3	1	no	—
41	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
42	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
43	C₂×C₃×C₄²	—	3	1	no	—
44	C₂×C₄×Dic₃	Dic₃	1	1	yes	characteristic
45	C₂²×C₃⋊C₈	Dic₃	1	1	yes	characteristic
46	C₂×C₃×C₁₆	—	3	2	no	—
47	C₂×C₃×C₁₆	—	6	2	no	—
48	C₃×C₄×C₈	—	3	2	no	—
49	C₃×C₄×C₈	—	3	2	no	—
50	C₈×Dic₃	Dic₃	1	2	yes	—
51	C₈×Dic₃	C₂×D₃	1	2	yes	—
52	C₂×C₃⋊C₁₆	—	2	2	no	—
53	C₂×C₃⋊C₁₆	—	3	2	no	—
54	C₂×C₃⋊C₁₆	—	3	2	no	—
55	C₂×C₃⋊C₁₆	—	6	2	no	—
56	C₂×C₃⋊C₁₆	—	6	4	no	—
57	C₂×C₃⋊C₁₆	—	6	2	no	—
58	C₂×C₃⋊C₁₆	—	6	2	no	—
59	C₂×C₃⋊C₁₆	C₂×D₃	1	2	yes	—
60	C₂³×C₃²	C₂×C₈	1	1	yes	characteristic
61	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
62	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
63	C₂×C₃²×C₄	M₁₆	1	2	yes	—
64	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₃×Dic₃	—	2	2	no	—
66	C₂×C₃×Dic₃	C₁₆	1	4	yes	complemented
67	C₂×C₃×Dic₃	C₂×C₈	1	2	yes	—
68	C₂×C₃×Dic₃	C₂×C₈	1	2	yes	—
69	C₃²×C₈	C₄⋊C₄	1	2	yes	—
70	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
71	C₃²×C₈	C₂×Q₈	1	1	yes	characteristic
72	C₃×C₃⋊C₈	—	2	2	no	—
73	C₂²×C₁₆	—	3	1	no	—
74	C₂²×C₁₆	—	9	1	no	—
75	C₂×C₄×C₈	—	3	1	no	—
76	C₄⋊C₁₆	—	9	4	no	—
77	C₂²×C₃×C₄	—	2	1	no	—
78	C₂²×C₃×C₄	—	3	1	no	—
79	C₂²×C₃×C₄	—	3	1	no	—
80	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
81	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
82	C₂²×Dic₃	C₃⋊C₈	1	1	yes	characteristic
83	C₂²×Dic₃	C₃⋊C₈	1	1	yes	characteristic
84	C₂×C₃×C₈	—	2	1	no	—
85	C₂×C₃×C₈	—	2	2	no	—
86	C₂×C₃×C₈	—	2	2	no	—
87	C₂×C₃×C₈	—	2	1	no	—
88	C₂×C₃×C₈	—	3	2	no	—
89	C₂×C₃×C₈	—	6	2	no	—
90	C₂×C₃×C₈	Dic₆	1	2	yes	—
91	C₂×C₃×C₈	Dic₆	1	2	yes	—
92	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
93	C₂×C₃×C₈	D₁₂	1	2	yes	—
94	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
95	C₂×C₃×C₈	#67	1	2	yes	—
96	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
97	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
98	C₃×C₄²	—	3	2	no	—
99	C₃×C₄²	—	3	2	no	—
100	C₄×Dic₃	C₃⋊C₈	1	2	yes	—
101	C₄×Dic₃	C₂×Dic₃	1	2	yes	—
102	C₂×C₃⋊C₈	—	2	2	no	—
103	C₂×C₃⋊C₈	C₂×Dic₃	1	2	yes	—
104	C₃×C₁₆	—	6	2	no	—
105	C₃⋊C₁₆	—	2	2	no	—
106	C₃⋊C₁₆	—	6	4	no	—
107	C₃⋊C₁₆	—	6	2	no	—
108	C₃⋊C₁₆	—	6	2	no	—
109	C₂²×C₃²	C₄⋊C₈	1	2	yes	—
110	C₂²×C₃²	C₂×C₁₆	1	2	yes	—
111	C₂²×C₃²	M₃₂	1	2	yes	—
112	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
113	C₃²×C₄	C₄⋊C₈	1	2	yes	—
114	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
116	C₃×Dic₃	—	2	2	no	—
117	C₃×Dic₃	C₂×C₁₆	1	4	yes	complemented
118	C₂²×C₈	—	3	1	no	—
119	C₂²×C₈	D₃²	1	1	yes	characteristic
120	C₂×C₄²	—	3	1	no	—
121	C₂×C₁₆	—	3	2	no	—
122	C₂×C₁₆	—	6	2	no	—
123	C₂×C₁₆	—	9	2	no	—
124	C₂×C₁₆	—	18	2	no	—
125	C₄×C₈	—	3	2	no	—
126	C₄×C₈	—	3	2	no	—
127	C₂³×C₃	—	2	1	no	—
128	C₂³×C₃	D₃×C₈	1	1	yes	characteristic
129	C₂³×C₃	C₂×C₃⋊C₈	1	1	yes	characteristic
130	C₂×C₃×C₄	—	2	1	no	—
131	C₂×C₃×C₄	—	2	2	no	—
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	—	2	1	no	—
134	C₂×C₃×C₄	—	3	4	no	—
135	C₂×C₃×C₄	—	3	2	no	—
136	C₂×C₃×C₄	—	3	2	no	—
137	C₂×C₃×C₄	—	6	2	no	—
138	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
139	C₂×C₃×C₄	ΓL(1,𝔽₂₅)	1	2	yes	—
140	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
141	C₂×C₃×C₄	#208	1	2	yes	—
142	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
143	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
144	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
145	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
146	C₂×Dic₃	—	2	2	no	—
147	C₂×Dic₃	C₃⋊C₁₆	1	4	yes	complemented
148	C₂×Dic₃	C₂×C₃⋊C₈	1	2	yes	—
149	C₂×Dic₃	C₂×C₃⋊C₈	1	2	yes	—
150	C₃×C₈	—	2	2	no	—
151	C₃×C₈	—	2	1	no	—
152	C₃×C₈	—	2	1	no	—
153	C₃×C₈	—	6	2	no	—
154	C₃×C₈	Dic₃⋊C₄	1	2	yes	—
155	C₃×C₈	C₄⋊Dic₃	1	2	yes	—
156	C₃×C₈	C₂×Dic₆	1	1	yes	characteristic
157	C₃×C₈	C₂×Dic₆	1	1	yes	characteristic
158	C₃×C₈	C₂×D₁₂	1	1	yes	characteristic
159	C₃×C₈	C₂×#67	1	1	yes	characteristic
160	C₃⋊C₈	—	2	2	no	—
161	C₂×C₃²	C₄⋊C₁₆	1	4	yes	complemented
162	C₂×C₃²	C₂×C₄⋊C₈	1	1	yes	characteristic
163	C₂×C₃²	C₂²×C₁₆	1	1	yes	characteristic
164	C₂×C₃²	C₂×M₃₂	1	1	yes	characteristic
165	C₂²×C₄	—	3	1	no	—
166	C₂²×C₄	—	3	1	no	—
167	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
168	C₂×C₈	—	3	2	no	—
169	C₂×C₈	—	6	2	no	—
170	C₂×C₈	D₃×Dic₃	1	1	yes	characteristic
171	C₂×C₈	#626	1	2	yes	—
172	C₂×C₈	#627	1	2	yes	—
173	C₂×C₈	C₂×D₃²	1	1	yes	characteristic
174	C₄²	—	3	2	no	—
175	C₄²	—	3	2	no	—
176	C₁₆	—	6	2	no	—
177	C₁₆	—	18	2	no	—
178	C₂²×C₃	—	2	2	no	—
179	C₂²×C₃	—	2	2	no	—
180	C₂²×C₃	—	2	1	no	—
181	C₂²×C₃	—	2	2	no	—
182	C₂²×C₃	D₃×C₁₆	1	2	yes	—
183	C₂²×C₃	#798	1	2	yes	—
184	C₂²×C₃	C₄⋊(C₃⋊C₈)	1	2	yes	—
185	C₂²×C₃	C₂×C₃⋊C₁₆	1	2	yes	—
186	C₂²×C₃	#812	1	2	yes	—
187	C₂²×C₃	Dic₃⋊C₈	1	2	yes	—
188	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
189	C₂²×C₃	C₂²×C₃⋊C₈	1	1	yes	characteristic
190	C₃×C₄	—	2	1	no	—
191	C₃×C₄	—	2	2	no	—
192	C₃×C₄	—	2	1	no	—
193	C₃×C₄	—	3	4	no	—
194	C₃×C₄	—	6	2	no	—
195	C₃×C₄	C₄⋊(C₃⋊C₈)	1	2	yes	—
196	C₃×C₄	Dic₃⋊C₈	1	2	yes	—
197	C₃×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
198	C₃×C₄	C₂×#208	1	1	yes	characteristic
199	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
200	C₃×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
201	Dic₃	—	2	2	no	—
202	Dic₃	C₂×C₃⋊C₁₆	1	4	yes	complemented
203	C₃²	C₂×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
204	C₂³	D₃×C₃⋊C₈	1	1	yes	characteristic
205	C₂×C₄	—	3	4	no	—
206	C₂×C₄	—	3	2	no	—
207	C₂×C₄	—	3	2	no	—
208	C₂×C₄	—	6	2	no	—
209	C₂×C₄	D₃×C₃⋊C₈	1	1	yes	characteristic
210	C₂×C₄	#3711	1	2	yes	—
211	C₂×C₄	Dic₃⋊Dic₃	1	2	yes	—
212	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
213	C₈	—	6	2	no	—
214	C₈	Dic₃⋊Dic₃	1	2	yes	—
215	C₈	C₂×#626	1	1	yes	characteristic
216	C₈	C₂×#627	1	1	yes	characteristic
217	C₂×C₃	—	2	1	no	—
218	C₂×C₃	—	2	1	no	—
219	C₂×C₃	—	2	1	no	—
220	C₂×C₃	—	2	4	no	—
221	C₂×C₃	C₄⋊(C₃⋊C₁₆)	1	4	yes	complemented
222	C₂×C₃	Dic₃⋊C₁₆	1	4	yes	complemented
223	C₂×C₃	C₂×D₃×C₁₆	1	1	yes	characteristic
224	C₂×C₃	C₂×#798	1	1	yes	characteristic
225	C₂×C₃	C₂×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
226	C₂×C₃	C₂²×C₃⋊C₁₆	1	1	yes	characteristic
227	C₂×C₃	C₂×#812	1	1	yes	characteristic
228	C₂×C₃	C₂×Dic₃⋊C₈	1	1	yes	characteristic
229	C₂²	D₃×C₃⋊C₁₆	1	2	yes	—
230	C₂²	#63543	1	2	yes	—
231	C₂²	Dic₃⋊(C₃⋊C₈)	1	2	yes	—
232	C₂²	C₂×D₃×C₃⋊C₈	1	1	yes	characteristic
233	C₄	—	3	4	no	—
234	C₄	—	6	2	no	—
235	C₄	Dic₃⋊(C₃⋊C₈)	1	2	yes	—
236	C₄	C₂×#3711	1	1	yes	characteristic
237	C₄	C₂×Dic₃⋊Dic₃	1	1	yes	characteristic
238	C₃	—	2	1	no	—
239	C₃	C₂×C₄⋊(C₃⋊C₁₆)	1	1	yes	characteristic, complemented
240	C₃	C₂×Dic₃⋊C₁₆	1	1	yes	characteristic, complemented
241	C₂	Dic₃⋊(C₃⋊C₁₆)	1	4	yes	direct product factor
242	C₂	C₂×D₃×C₃⋊C₁₆	1	1	yes	characteristic
243	C₂	C₂×#63543	1	1	yes	characteristic
244	C₂	C₂×Dic₃⋊(C₃⋊C₈)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₈	D₃²
commutator	C₂×C₃²	C₂²×C₁₆
Frattini	C₂×C₈	C₂×D₃²
Fitting	C₂²×C₃²×C₈	C₂²
socle, prime radical	C₂³×C₃²	C₂×C₈
cosocle	C₂×C₃²×C₈	C₂³
hypocenter	C₃²	C₂×C₄⋊C₁₆
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
4. Dic₃⋊(C₃⋊C₁₆)	C₂	direct (N × H)
66. C₂×C₃×Dic₃	C₁₆	semidirect
117. C₃×Dic₃	C₂×C₁₆	semidirect
147. C₂×Dic₃	C₃⋊C₁₆	semidirect
161. C₂×C₃²	C₄⋊C₁₆	semidirect
202. Dic₃	C₂×C₃⋊C₁₆	semidirect
203. C₃²	C₂×C₄⋊C₁₆	semidirect
221. C₂×C₃	C₄⋊(C₃⋊C₁₆)	semidirect
222. C₂×C₃	Dic₃⋊C₁₆	semidirect
239. C₃	C₂×C₄⋊(C₃⋊C₁₆)	semidirect
240. C₃	C₂×Dic₃⋊C₁₆	semidirect
241. C₂	Dic₃⋊(C₃⋊C₁₆)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃⋊(C₃⋊C₁₆)	2
C₂×C₃×C₈×Dic₃	2
C₂²×C₃×C₃⋊C₁₆	2
Dic₃⋊(C₃⋊C₁₆)	2
C₂×Dic₃⋊C₁₆	3
C₂×C₄⋊(C₃⋊C₁₆)	3

Subgroups — C2×Dic3⋊(C3⋊C16) #11796981

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Dic₃⋊(C₃⋊C₁₆) #11796981