Subgroups — C₃×#68052 #11777763

Order 1152.

397 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×D₂₄	C₂	1	1	yes	maximal, characteristic
2	C₃×#4688	C₂	1	2	yes	maximal, complemented
3	C₃²×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4459	C₂	1	2	yes	maximal, complemented
5	C₃×#4458	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#1589	—	3	1	no	maximal
7	#68052	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×D₂₄	C₄	1	1	yes	characteristic
9	C₂×C₃×ΓL(1,𝔽₂₅)	—	2	2	no	—
10	C₃×C₄×D₁₂	C₂²	1	2	yes	—
11	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₃×#821	C₄	1	2	yes	—
15	C₃×(C₂×D₃)⋊C₈	—	2	2	no	—
16	C₃×#818	C₄	1	1	yes	characteristic
17	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
18	C₃×C₄×D₈	—	3	1	no	—
19	C₃×#436	—	3	2	no	—
20	C₄×D₂₄	C₂×C₃	1	1	yes	characteristic
21	#4688	C₂×C₃	1	2	yes	complemented
22	C₃×#335	—	3	1	no	—
23	C₃×D₄⋊C₈	—	3	2	no	—
24	C₃×C₈²	—	2	1	no	—
25	C₃×C₈²	D₃	1	1	yes	characteristic, complemented
26	C₃×C₈²	C₂×C₃	1	1	yes	characteristic, complemented
27	#4459	C₂×C₃	1	2	yes	complemented
28	#4458	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂×C₃×D₁₂	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₄×D₃	—	2	2	no	—
31	C₂×C₃²×C₈	—	2	1	no	—
32	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
35	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
36	C₃²×C₄²	C₂³	1	1	yes	characteristic
37	C₃×(C₂×D₃)⋊C₄	—	2	2	no	—
38	C₃×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
39	C₂×C₃×C₃⋊C₈	—	2	2	no	—
40	C₃×D₂₄	—	2	2	no	—
41	C₃×ΓL(1,𝔽₂₅)	—	4	4	no	—
42	#1589	—	3	1	no	Sylow
43	C₂×C₃×D₈	—	3	1	no	—
44	C₂×C₃×M₁₆	—	6	2	no	—
45	C₃×C₄×D₄	—	3	2	no	—
46	C₂×D₂₄	C₃×C₄	1	1	yes	characteristic
47	C₂×ΓL(1,𝔽₂₅)	—	2	2	no	—
48	C₄×D₁₂	C₂²×C₃	1	2	yes	—
49	C₃×#107	—	3	1	no	—
50	C₃×C₄⋊C₈	—	3	2	no	—
51	C₃×#102	—	3	2	no	—
52	C₃×C₂²⋊C₈	—	6	2	no	—
53	C₃×C₄×C₈	—	2	1	no	—
54	C₃×C₄×C₈	—	2	1	no	—
55	C₃×C₄×C₈	—	2	1	no	—
56	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
57	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
58	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
59	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
60	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
61	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
62	#821	C₃×C₄	1	2	yes	—
63	(C₂×D₃)⋊C₈	—	2	2	no	—
64	#818	C₃×C₄	1	1	yes	characteristic
65	C₄⋊(C₃⋊C₈)	C₂²×C₃	1	2	yes	—
66	C₂²×C₃×D₃	—	2	2	no	—
67	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
69	C₂×C₃²×C₄	#41	1	1	yes	characteristic
70	C₂×C₃×Dic₃	—	2	2	no	—
71	C₃×D₁₂	—	2	2	no	—
72	C₃×D₁₂	—	2	2	no	—
73	C₃×C₄×D₃	—	4	2	no	—
74	C₃×C₄×D₃	—	4	2	no	—
75	C₃²×C₈	—	2	2	no	—
76	C₃²×C₈	—	2	1	no	—
77	C₃²×C₈	—	2	1	no	—
78	C₃²×C₈	M₁₆	1	2	yes	complemented
79	C₃²×C₈	D₈	1	2	yes	complemented
80	C₃×C₃⋊C₈	—	4	2	no	—
81	C₄×D₈	—	3	1	no	—
82	#436	—	3	2	no	—
83	#335	—	3	1	no	—
84	D₄⋊C₈	—	3	2	no	—
85	C₈²	C₃×D₃	1	1	yes	characteristic, complemented
86	C₂×C₃×D₄	—	3	2	no	—
87	C₂²×C₃×C₄	—	6	2	no	—
88	C₂×D₁₂	C₂×C₃×C₄	1	2	yes	—
89	C₂×C₄×D₃	—	2	2	no	—
90	C₃×D₈	—	6	2	no	—
91	C₃×M₁₆	—	12	4	no	—
92	C₂×C₃×C₈	—	2	2	no	—
93	C₂×C₃×C₈	—	2	1	no	—
94	C₂×C₃×C₈	—	2	1	no	—
95	C₂×C₃×C₈	—	2	1	no	—
96	C₂×C₃×C₈	—	2	1	no	—
97	C₂×C₃×C₈	—	2	1	no	—
98	C₂×C₃×C₈	—	2	1	no	—
99	C₂×C₃×C₈	—	6	2	no	—
100	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
101	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
102	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
103	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
104	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
105	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
106	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
107	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
108	C₃×C₄⋊C₄	—	3	2	no	—
109	C₃×C₂²⋊C₄	—	6	2	no	—
110	C₃×C₄²	—	2	1	no	—
111	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
112	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
113	(C₂×D₃)⋊C₄	—	2	2	no	—
114	C₄⋊Dic₃	C₂×C₃×C₄	1	2	yes	—
115	C₂×C₃⋊C₈	—	2	2	no	—
116	D₂₄	—	2	2	no	—
117	ΓL(1,𝔽₂₅)	—	4	4	no	—
118	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
119	C₂×C₃×D₃	—	4	2	no	—
120	C₂×C₃×D₃	—	4	2	no	—
121	C₂×C₃×D₃	—	4	2	no	—
122	C₃²×C₄	—	2	1	no	—
123	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
124	C₃²×C₄	#131	1	1	yes	characteristic
125	C₃²×C₄	C₂×D₈	1	1	yes	characteristic
126	C₃²×C₄	#135	1	1	yes	characteristic
127	C₃×Dic₃	—	4	2	no	—
128	C₂×D₈	—	3	1	no	—
129	C₂×M₁₆	—	6	2	no	—
130	C₄×D₄	—	3	2	no	—
131	#107	—	3	1	no	—
132	C₄⋊C₈	—	3	2	no	—
133	#102	—	3	2	no	—
134	C₂²⋊C₈	—	6	2	no	—
135	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
136	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
137	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
138	C₂³×C₃	—	6	2	no	—
139	C₂²×D₃	—	2	2	no	—
140	C₃×D₄	—	6	2	no	—
141	C₃×D₄	—	6	2	no	—
142	C₂×C₃×C₄	—	2	1	no	—
143	C₂×C₃×C₄	—	2	1	no	—
144	C₂×C₃×C₄	—	2	1	no	—
145	C₂×C₃×C₄	—	6	2	no	—
146	C₂×C₃×C₄	—	12	2	no	—
147	C₂×C₃×C₄	—	12	2	no	—
148	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
150	C₂×C₃×C₄	#235	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
152	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
153	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
154	C₂×Dic₃	—	2	2	no	—
155	D₁₂	—	2	2	no	—
156	D₁₂	—	2	2	no	—
157	C₄×D₃	—	4	2	no	—
158	C₄×D₃	—	4	2	no	—
159	C₃×C₈	—	2	4	no	—
160	C₃×C₈	—	2	2	no	—
161	C₃×C₈	—	2	2	no	—
162	C₃×C₈	—	2	2	no	—
163	C₃×C₈	—	2	2	no	—
164	C₃×C₈	—	2	1	no	—
165	C₃×C₈	—	2	1	no	—
166	C₃×C₈	—	2	1	no	—
167	C₃×C₈	—	2	1	no	—
168	C₃×C₈	—	2	2	no	—
169	C₃×C₈	—	2	2	no	—
170	C₃×C₈	—	12	2	no	—
171	C₃×C₈	ΓL(1,𝔽₂₅)	1	2	yes	complemented
172	C₃×C₈	D₂₄	1	2	yes	complemented
173	C₃×C₈	C₃×M₁₆	1	2	yes	complemented
174	C₃×C₈	C₃×D₈	1	2	yes	complemented
175	C₃⋊C₈	—	4	2	no	—
176	C₂×C₃²	#436	1	1	yes	characteristic
177	C₂×C₃²	C₄×D₈	1	1	yes	characteristic
178	C₂×C₃²	#443	1	1	yes	characteristic
179	C₃×D₃	—	8	2	no	—
180	C₂×D₄	—	3	2	no	—
181	C₂²×C₄	—	6	2	no	—
182	D₈	—	6	2	no	—
183	M₁₆	—	12	4	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	6	2	no	—
186	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
187	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
188	C₂×C₈	C₃×D₁₂	1	1	yes	characteristic
189	C₂×C₈	C₃×D₁₂	1	1	yes	characteristic
190	C₄⋊C₄	—	3	2	no	—
191	C₂²⋊C₄	—	6	2	no	—
192	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
193	C₂²×C₃	—	2	1	no	—
194	C₂²×C₃	—	12	2	no	—
195	C₂²×C₃	—	12	2	no	—
196	C₂²×C₃	—	12	2	no	—
197	C₂²×C₃	C₄×D₁₂	1	1	yes	characteristic
198	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
199	C₂×D₃	—	4	2	no	—
200	C₂×D₃	—	4	2	no	—
201	C₂×D₃	—	4	2	no	—
202	C₃×C₄	—	2	2	no	—
203	C₃×C₄	—	2	1	no	—
204	C₃×C₄	—	2	1	no	—
205	C₃×C₄	—	2	1	no	—
206	C₃×C₄	—	2	1	no	—
207	C₃×C₄	—	2	1	no	—
208	C₃×C₄	—	2	1	no	—
209	C₃×C₄	—	12	2	no	—
210	C₃×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
211	C₃×C₄	#901	1	1	yes	characteristic
212	C₃×C₄	C₂×D₂₄	1	1	yes	characteristic
213	C₃×C₄	#904	1	1	yes	characteristic
214	C₃×C₄	C₂×C₃×M₁₆	1	1	yes	characteristic
215	C₃×C₄	C₃×#131	1	1	yes	characteristic
216	C₃×C₄	C₂×C₃×D₈	1	1	yes	characteristic
217	C₃×C₄	C₃×#135	1	1	yes	characteristic
218	Dic₃	—	4	2	no	—
219	C₃²	#1589	1	1	yes	characteristic, complemented, Sylow
220	C₂³	—	6	2	no	—
221	D₄	—	6	2	no	—
222	D₄	—	6	2	no	—
223	C₂×C₄	—	6	2	no	—
224	C₂×C₄	—	12	2	no	—
225	C₂×C₄	—	12	2	no	—
226	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
227	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
228	C₂×C₄	C₃×#235	1	1	yes	characteristic
229	C₈	—	2	2	no	—
230	C₈	—	2	1	no	—
231	C₈	—	2	1	no	—
232	C₈	—	12	2	no	—
233	C₈	C₃×ΓL(1,𝔽₂₅)	1	2	yes	complemented
234	C₈	C₃×D₂₄	1	2	yes	complemented
235	C₂×C₃	—	2	1	no	—
236	C₂×C₃	—	2	1	no	—
237	C₂×C₃	—	2	1	no	—
238	C₂×C₃	—	24	2	no	—
239	C₂×C₃	#4688	1	1	yes	characteristic
240	C₂×C₃	C₄×D₂₄	1	1	yes	characteristic
241	C₂×C₃	#4700	1	1	yes	characteristic
242	C₂×C₃	C₃×#436	1	1	yes	characteristic
243	C₂×C₃	C₃×C₄×D₈	1	1	yes	characteristic
244	C₂×C₃	C₃×#443	1	1	yes	characteristic
245	D₃	—	8	2	no	—
246	C₂²	—	12	2	no	—
247	C₂²	—	12	2	no	—
248	C₂²	—	12	2	no	—
249	C₂²	C₃×C₄×D₁₂	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	12	2	no	—
252	C₄	C₂×C₃×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
253	C₄	C₃×#901	1	1	yes	characteristic
254	C₄	C₂×C₃×D₂₄	1	1	yes	characteristic
255	C₄	C₃×#904	1	1	yes	characteristic
256	C₃	—	2	1	no	—
257	C₃	#68052	1	1	yes	characteristic, direct product factor
258	C₃	C₃×#1589	1	1	yes	characteristic, complemented
259	C₂	—	24	2	no	—
260	C₂	C₃×#4688	1	1	yes	characteristic
261	C₂	C₃×C₄×D₂₄	1	1	yes	characteristic
262	C₂	C₃×#4700	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₄²	C₂²×C₃×D₃
Fitting	C₃²×C₈²	C₂
socle	C₂²×C₃²	C₄×D₄
prime radical	C₂×C₃×D₂₄	C₄
cosocle	C₃×C₄²	C₂³×C₃
hypercenter	C₃×C₈²	D₃
hypocenter	C₃	C₃×#1589
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄×C₈ ◁ C₃×C₈²

Complements

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

N	H ≅ G/N	type
2. C₃×#4688	C₂	semidirect
3. C₃²×C₈²	C₂	semidirect
4. C₃×#4459	C₂	semidirect
5. C₃×#4458	C₂	semidirect
7. #68052	C₃	direct (N × H)
21. #4688	C₂×C₃	semidirect
25. C₃×C₈²	D₃	semidirect
26. C₃×C₈²	C₂×C₃	semidirect
27. #4459	C₂×C₃	semidirect
28. #4458	C₂×C₃	semidirect
78. C₃²×C₈	M₁₆	semidirect
79. C₃²×C₈	D₈	semidirect
85. C₈²	C₃×D₃	semidirect
171. C₃×C₈	ΓL(1,𝔽₂₅)	semidirect
172. C₃×C₈	D₂₄	semidirect
173. C₃×C₈	C₃×M₁₆	semidirect
174. C₃×C₈	C₃×D₈	semidirect
219. C₃²	#1589	semidirect
233. C₈	C₃×ΓL(1,𝔽₂₅)	semidirect
234. C₈	C₃×D₂₄	semidirect
257. C₃	#68052	direct (N × H)
258. C₃	C₃×#1589	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×D₂₄	2
C₃×#4688	2
C₃²×C₈²	2
C₃×#4459	2
C₃×#4458	2
C₃×#1589	3
#68052	3

Subgroups — C3×#68052 #11777763

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#68052 #11777763