Subgroups — C₃×#78442 #11874981

Order 1152.

793 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#5470	C₂	1	2	yes	maximal
2	C₂×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×#5106	C₂	1	4	yes	maximal
4	C₃×#4683	C₂	1	4	yes	maximal, complemented
5	C₃×#4682	C₂	1	2	yes	maximal, complemented
6	C₃×#4680	C₂	1	2	yes	maximal, complemented
7	C₃×#3160	—	3	1	no	maximal
8	#78442	C₃	1	1	yes	maximal, characteristic, direct product factor
9	C₂²×C₃²×C₈	C₂²	1	2	yes	—
10	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
11	C₃×#926	C₂²	1	4	yes	—
12	C₃×#924	—	2	4	no	—
13	C₃×#870	C₂²	1	4	yes	—
14	C₃×#869	C₂²	1	4	yes	—
15	C₃×C₄×Dic₆	—	2	4	no	—
16	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
17	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
18	C₃²×C₄×C₈	C₂²	1	2	yes	—
19	C₃×#818	C₂²	1	8	yes	—
20	C₃×#817	C₂²	1	8	yes	—
21	C₃×#533	—	3	2	no	—
22	#5470	C₂×C₃	1	2	yes	—
23	C₃×#500	—	3	2	no	—
24	C₃×#499	—	3	4	no	—
25	C₃×#498	—	3	2	no	—
26	C₃×#427	—	3	4	no	—
27	C₂×C₃×C₄×C₈	—	2	1	no	—
28	C₂×C₃×C₄×C₈	D₃	1	1	yes	characteristic
29	C₂×C₃×C₄×C₈	C₂×C₃	1	1	yes	characteristic
30	#5106	C₂×C₃	1	4	yes	—
31	#4683	C₂×C₃	1	4	yes	complemented
32	#4682	C₂×C₃	1	2	yes	complemented
33	#4680	C₂×C₃	1	2	yes	complemented
34	C₂²×C₃²×C₄	D₄	1	2	yes	—
35	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
36	C₂×C₃×Dic₆	—	2	4	no	—
37	C₂×C₃²×C₈	Q₈	1	8	yes	complemented
38	C₂×C₃²×C₈	C₂³	1	2	yes	—
39	C₂×C₃²×C₈	C₂³	1	2	yes	—
40	C₃²×C₄²	D₄	1	2	yes	—
41	C₃²×C₄²	C₂³	1	1	yes	characteristic
42	C₃²×C₄²	C₂³	1	1	yes	characteristic
43	C₃×#217	—	2	4	no	—
44	C₃×C₄⋊Dic₃	—	2	4	no	—
45	C₃×C₄⋊Dic₃	C₂³	1	8	yes	—
46	C₃×Dic₃⋊C₄	—	2	8	no	—
47	C₃×C₄×Dic₃	—	2	4	no	—
48	#3160	—	3	1	no	Sylow
49	C₂²×C₃×C₈	—	2	2	no	—
50	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
51	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
52	C₃×C₄⋊Q₈	—	3	4	no	—
53	C₃×#125	—	3	4	no	—
54	C₃×C₂²⋊Q₈	—	6	4	no	—
55	C₃×C₄×Q₈	—	6	4	no	—
56	C₃×#117	—	3	4	no	—
57	C₂×C₃×C₄²	—	2	1	no	—
58	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
59	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
60	#926	C₂²×C₃	1	4	yes	—
61	#924	—	2	4	no	—
62	#870	C₂²×C₃	1	4	yes	—
63	#869	C₂²×C₃	1	4	yes	—
64	C₄×Dic₆	—	2	4	no	—
65	C₃×#107	—	3	8	no	—
66	C₃×#106	—	3	8	no	—
67	C₃×C₄×C₈	—	2	1	no	—
68	C₃×C₄×C₈	—	2	2	no	—
69	C₃×C₄×C₈	—	2	1	no	—
70	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
71	C₃×C₄×C₈	C₂×D₃	1	2	yes	—
72	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
73	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
74	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
75	C₃×C₄×C₈	C₂²×C₃	1	2	yes	—
76	#818	C₂²×C₃	1	8	yes	—
77	#817	C₂²×C₃	1	8	yes	—
78	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃²×C₄	—	2	4	no	—
80	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
81	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
82	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃²×C₄	C₂×Q₈	1	4	yes	—
84	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
85	C₂×C₃×Dic₃	—	2	8	no	—
86	C₃×Dic₆	—	4	8	no	—
87	C₃²×C₈	C₂×Q₈	1	4	yes	—
88	C₃²×C₈	C₂×Q₈	1	4	yes	—
89	#533	—	3	2	no	—
90	#500	—	3	2	no	—
91	#499	—	3	4	no	—
92	#498	—	3	2	no	—
93	#427	—	3	4	no	—
94	C₂×C₄×C₈	C₃×D₃	1	1	yes	characteristic
95	C₂×C₃×Q₈	—	6	4	no	—
96	C₂²×C₃×C₄	—	2	2	no	—
97	C₂²×C₃×C₄	—	2	1	no	—
98	C₂²×C₃×C₄	D₁₂	1	2	yes	—
99	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
100	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
101	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
102	C₂×Dic₆	—	2	4	no	—
103	C₂×C₃×C₈	—	2	8	no	—
104	C₂×C₃×C₈	—	2	2	no	—
105	C₂×C₃×C₈	—	2	2	no	—
106	C₂×C₃×C₈	Dic₆	1	8	yes	complemented
107	C₂×C₃×C₈	C₃×Q₈	1	8	yes	complemented
108	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
109	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
110	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
111	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
112	C₃×C₄⋊C₄	—	3	8	no	—
113	C₃×C₄⋊C₄	—	6	8	no	—
114	C₃×C₄⋊C₄	—	6	4	no	—
115	C₃×C₂²⋊C₄	—	6	4	no	—
116	C₃×C₄²	—	2	2	no	—
117	C₃×C₄²	—	2	1	no	—
118	C₃×C₄²	—	2	1	no	—
119	C₃×C₄²	—	6	4	no	—
120	C₃×C₄²	D₁₂	1	2	yes	—
121	C₃×C₄²	C₃×D₄	1	2	yes	—
122	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
123	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
124	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
125	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
126	#217	—	2	4	no	—
127	C₄⋊Dic₃	—	2	4	no	—
128	C₄⋊Dic₃	C₂³×C₃	1	8	yes	—
129	Dic₃⋊C₄	—	2	8	no	—
130	C₄×Dic₃	—	2	4	no	—
131	C₂²×C₃²	—	2	2	no	—
132	C₂²×C₃²	C₄⋊Q₈	1	2	yes	—
133	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
134	C₃²×C₄	—	2	2	no	—
135	C₃²×C₄	C₄⋊Q₈	1	2	yes	—
136	C₃²×C₄	#135	1	4	yes	—
137	C₃²×C₄	C₂²×Q₈	1	2	yes	—
138	C₃×Dic₃	—	4	8	no	—
139	C₂²×C₈	C₂×C₃×D₃	1	2	yes	—
140	C₄⋊Q₈	—	3	4	no	—
141	#125	—	3	4	no	—
142	C₂²⋊Q₈	—	6	4	no	—
143	C₄×Q₈	—	6	4	no	—
144	#117	—	3	4	no	—
145	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
146	#107	—	3	8	no	—
147	#106	—	3	8	no	—
148	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
149	C₄×C₈	C₂×C₃×D₃	1	2	yes	—
150	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
151	C₂³×C₃	—	2	1	no	—
152	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
153	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
154	C₃×Q₈	—	12	8	no	—
155	C₂×C₃×C₄	—	2	8	no	—
156	C₂×C₃×C₄	—	2	4	no	—
157	C₂×C₃×C₄	—	2	4	no	—
158	C₂×C₃×C₄	—	2	1	no	—
159	C₂×C₃×C₄	—	2	4	no	—
160	C₂×C₃×C₄	—	2	2	no	—
161	C₂×C₃×C₄	—	2	2	no	—
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	6	8	no	—
164	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
165	C₂×C₃×C₄	C₂×D₁₂	1	2	yes	—
166	C₂×C₃×C₄	C₂×D₁₂	1	2	yes	—
167	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
169	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
171	C₂×C₃×C₄	C₂×C₃×Q₈	1	4	yes	—
172	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
174	C₂×Dic₃	—	2	8	no	—
175	Dic₆	—	4	8	no	—
176	C₃×C₈	—	2	4	no	—
177	C₃×C₈	—	2	4	no	—
178	C₃×C₈	C₂×Dic₆	1	4	yes	—
179	C₃×C₈	C₂×Dic₆	1	4	yes	—
180	C₃×C₈	C₂×C₃×Q₈	1	4	yes	—
181	C₃×C₈	C₂×C₃×Q₈	1	4	yes	—
182	C₂×C₃²	—	2	2	no	—
183	C₂×C₃²	C₂×C₄⋊Q₈	1	1	yes	characteristic
184	C₂×C₃²	C₂×#135	1	2	yes	—
185	C₂×Q₈	—	6	4	no	—
186	C₂²×C₄	C₃×D₁₂	1	2	yes	—
187	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
188	C₂×C₈	C₃×Dic₆	1	8	yes	complemented
189	C₂×C₈	C₂²×C₃×D₃	1	2	yes	—
190	C₂×C₈	C₂²×C₃×D₃	1	2	yes	—
191	C₄⋊C₄	—	3	8	no	—
192	C₄⋊C₄	—	6	8	no	—
193	C₄⋊C₄	—	6	4	no	—
194	C₂²⋊C₄	—	6	4	no	—
195	C₄²	—	6	4	no	—
196	C₄²	C₃×D₁₂	1	2	yes	—
197	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
198	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
199	C₂²×C₃	—	2	4	no	—
200	C₂²×C₃	—	2	2	no	—
201	C₂²×C₃	—	2	2	no	—
202	C₂²×C₃	—	2	1	no	—
203	C₂²×C₃	—	2	2	no	—
204	C₂²×C₃	#869	1	2	yes	—
205	C₂²×C₃	C₃×C₄⋊Q₈	1	2	yes	—
206	C₂²×C₃	C₂²×D₁₂	1	1	yes	characteristic
207	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
208	C₃×C₄	—	2	4	no	—
209	C₃×C₄	—	2	2	no	—
210	C₃×C₄	—	2	2	no	—
211	C₃×C₄	—	2	4	no	—
212	C₃×C₄	—	2	2	no	—
213	C₃×C₄	—	2	2	no	—
214	C₃×C₄	—	12	8	no	—
215	C₃×C₄	#869	1	2	yes	—
216	C₃×C₄	#904	1	4	yes	—
217	C₃×C₄	C₃×C₄⋊Q₈	1	2	yes	—
218	C₃×C₄	C₃×#135	1	4	yes	—
219	C₃×C₄	C₂²×Dic₆	1	2	yes	—
220	C₃×C₄	C₂²×C₃×Q₈	1	2	yes	—
221	Dic₃	—	4	8	no	—
222	C₃²	#3160	1	1	yes	characteristic, complemented, Sylow
223	C₂³	C₂×C₃×D₁₂	1	1	yes	characteristic
224	Q₈	—	12	8	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	6	8	no	—
227	C₂×C₄	C₂×C₃×Dic₆	1	4	yes	—
228	C₂×C₄	C₂×C₃×D₁₂	1	2	yes	—
229	C₂×C₄	C₂×C₃×D₁₂	1	2	yes	—
230	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
231	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
232	C₈	C₂×C₃×Dic₆	1	4	yes	—
233	C₈	C₂×C₃×Dic₆	1	4	yes	—
234	C₂×C₃	—	2	4	no	—
235	C₂×C₃	—	2	2	no	—
236	C₂×C₃	—	2	2	no	—
237	C₂×C₃	—	2	1	no	—
238	C₂×C₃	—	2	2	no	—
239	C₂×C₃	C₂×#869	1	1	yes	characteristic
240	C₂×C₃	C₂×#904	1	2	yes	—
241	C₂×C₃	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
242	C₂×C₃	C₂×C₃×#135	1	2	yes	—
243	C₂²	—	2	2	no	—
244	C₂²	C₃×#869	1	2	yes	—
245	C₂²	C₂²×C₃×D₁₂	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	12	8	no	—
248	C₄	C₃×#869	1	2	yes	—
249	C₄	C₃×#904	1	4	yes	—
250	C₄	C₂²×C₃×Dic₆	1	2	yes	—
251	C₃	—	2	1	no	—
252	C₃	#78442	1	1	yes	characteristic, direct product factor
253	C₃	C₃×#3160	1	1	yes	characteristic, complemented
254	C₂	—	2	2	no	—
255	C₂	C₂×C₃×#869	1	1	yes	characteristic
256	C₂	C₂×C₃×#904	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₃
Fitting	C₂×C₃²×C₄×C₈	C₂
socle	C₂²×C₃²	C₂²×D₄
prime radical	C₂³×C₃²	C₂×D₄
hypercenter	C₂×C₃×C₄×C₈	D₃
hypocenter	C₃	C₃×#3160
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. C₃×#4683	C₂	semidirect
5. C₃×#4682	C₂	semidirect
6. C₃×#4680	C₂	semidirect
8. #78442	C₃	direct (N × H)
31. #4683	C₂×C₃	semidirect
32. #4682	C₂×C₃	semidirect
33. #4680	C₂×C₃	semidirect
37. C₂×C₃²×C₈	Q₈	semidirect
106. C₂×C₃×C₈	Dic₆	semidirect
107. C₂×C₃×C₈	C₃×Q₈	semidirect
188. C₂×C₈	C₃×Dic₆	semidirect
222. C₃²	#3160	semidirect
252. C₃	#78442	direct (N × H)
253. C₃	C₃×#3160	semidirect

Maximal subgroups

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

subgroup	index
C₃×#5470	2
C₂×C₃²×C₄×C₈	2
C₃×#5106	2
C₃×#4683	2
C₃×#4682	2
C₃×#4680	2
C₃×#3160	3
#78442	3

Subgroups — C3×#78442 #11874981

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#78442 #11874981