Subgroups — C₃×#81888 #11876979

Order 1152.

941 conjugacy classes of subgroups.

The subgroup lattice has 329 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#5725	C₂	1	2	yes	maximal, complemented
2	C₃²×#498	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#5088	C₂	1	2	yes	maximal, complemented
4	C₃×#4869	C₂	1	4	yes	maximal, complemented
5	C₃×#4865	C₂	1	4	yes	maximal, complemented
6	C₃×#4695	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₄×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#3184	—	3	1	no	maximal
9	#81888	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃×#239	—	2	2	no	—
11	C₃²×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₃×#947	—	2	2	no	—
13	C₂×C₃×D₂₄	C₂²	1	4	yes	—
14	C₂×C₃×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	—
15	C₃×#895	—	2	2	no	—
16	C₃×#895	C₂²	1	4	yes	—
17	C₃×#893	—	2	2	no	—
18	C₃×#893	C₂²	1	4	yes	—
19	C₃×#874	C₂²	1	2	yes	—
20	C₃×C₄²×D₃	C₂²	1	1	yes	characteristic
21	C₃²×#106	C₂²	1	4	yes	—
22	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
23	C₃×#815	C₂²	1	2	yes	—
24	C₃×#809	C₂²	1	8	yes	—
25	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
26	C₃×#532	—	3	2	no	—
27	#5725	C₂×C₃	1	2	yes	complemented
28	C₃×#498	—	2	1	no	—
29	C₃×#498	D₃	1	1	yes	characteristic, complemented
30	C₃×#498	C₂×C₃	1	1	yes	characteristic, complemented
31	C₃×#493	—	3	1	no	—
32	C₃×#486	—	3	2	no	—
33	C₃×#469	—	3	4	no	—
34	C₃×#442	—	3	4	no	—
35	C₃×C₄×M₁₆	—	3	1	no	—
36	#5088	C₂×C₃	1	2	yes	complemented
37	#4869	C₂×C₃	1	4	yes	complemented
38	#4865	C₂×C₃	1	4	yes	complemented
39	#4695	C₂×C₃	1	1	yes	characteristic, complemented
40	C₄×ΓL(1,𝔽₂₅)	C₂×C₃	1	1	yes	characteristic, complemented
41	C₂×C₃²×Q₈	—	2	2	no	—
42	C₃×#239	—	4	8	no	—
43	C₂×C₃×D₁₂	—	2	4	no	—
44	C₂×C₃×D₁₂	—	2	2	no	—
45	C₂×C₃×D₁₂	—	2	2	no	—
46	C₂×C₃×D₁₂	C₂³	1	4	yes	—
47	C₂×C₃×C₄×D₃	—	2	4	no	—
48	C₂×C₃×C₄×D₃	D₄	1	2	yes	—
49	C₂×C₃×C₄×D₃	C₂³	1	1	yes	characteristic
50	C₂×C₃²×C₈	C₂³	1	2	yes	—
51	C₃²×C₄⋊C₄	—	2	2	no	—
52	C₃²×C₄⋊C₄	C₂³	1	4	yes	—
53	C₃²×C₄²	C₂³	1	1	yes	characteristic
54	C₃×(C₂×D₃)⋊C₄	—	2	4	no	—
55	C₃×(C₂×D₃)⋊C₄	—	2	4	no	—
56	C₃×C₄×Dic₃	D₄	1	2	yes	—
57	C₃×C₄×Dic₃	C₂³	1	1	yes	characteristic
58	C₂×C₃×C₃⋊C₈	C₂³	1	2	yes	—
59	C₃×D₂₄	—	2	8	no	—
60	C₃×ΓL(1,𝔽₂₅)	—	2	4	no	—
61	#3184	—	3	1	no	Sylow
62	C₂×C₃×#41	—	6	2	no	—
63	C₂×#239	—	2	2	no	—
64	C₂×C₃×D₈	—	3	4	no	—
65	C₂×C₃×M₁₆	—	3	2	no	—
66	C₃×C₄⋊Q₈	—	2	2	no	—
67	C₃×C₄⋊Q₈	C₂×D₃	1	2	yes	—
68	C₃×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
69	C₃×#127	—	3	2	no	—
70	C₃×#124	—	6	2	no	—
71	C₃×#121	—	3	4	no	—
72	C₃×#121	—	6	2	no	—
73	C₃×C₄×D₄	—	3	4	no	—
74	C₃×C₄×D₄	—	6	2	no	—
75	C₂×C₃×C₄²	—	3	1	no	—
76	#947	—	2	2	no	—
77	C₂×D₂₄	C₂²×C₃	1	4	yes	—
78	C₂×ΓL(1,𝔽₂₅)	C₂²×C₃	1	2	yes	—
79	#895	—	2	2	no	—
80	#895	C₂²×C₃	1	4	yes	—
81	#893	—	2	2	no	—
82	#893	C₂²×C₃	1	4	yes	—
83	#874	C₂²×C₃	1	2	yes	—
84	C₄²×D₃	C₂²×C₃	1	1	yes	characteristic
85	C₃×#106	—	2	4	no	—
86	C₃×#106	C₂×D₃	1	4	yes	—
87	C₃×#106	C₂²×C₃	1	4	yes	—
88	C₃×#102	—	3	8	no	—
89	C₃×#97	—	3	2	no	—
90	C₃×C₄×C₈	—	2	1	no	—
91	C₃×C₄×C₈	—	3	1	no	—
92	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
93	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
94	#815	C₂²×C₃	1	2	yes	—
95	#809	C₂²×C₃	1	8	yes	—
96	C₄×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
97	C₂²×C₃×D₃	—	2	4	no	—
98	C₂²×C₃×D₃	C₂×D₄	1	1	yes	characteristic
99	C₃²×Q₈	—	4	4	no	—
100	C₂×C₃²×C₄	—	2	4	no	—
101	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
102	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
103	C₂×C₃×Dic₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₃×Dic₃	C₂×D₄	1	2	yes	—
105	C₃×D₁₂	—	2	8	no	—
106	C₃×D₁₂	—	4	8	no	—
107	C₃×D₁₂	—	4	4	no	—
108	C₃×D₁₂	—	4	4	no	—
109	C₃×C₄×D₃	—	2	4	no	—
110	C₃×C₄×D₃	—	2	2	no	—
111	C₃×C₄×D₃	—	4	8	no	—
112	C₃²×C₈	#41	1	4	yes	complemented
113	C₃×C₃⋊C₈	—	2	2	no	—
114	#532	—	3	2	no	—
115	#498	C₃×D₃	1	1	yes	characteristic, complemented
116	#493	—	3	1	no	—
117	#486	—	3	2	no	—
118	#469	—	3	4	no	—
119	#442	—	3	4	no	—
120	C₄×M₁₆	—	3	1	no	—
121	C₃×#41	—	12	8	no	—
122	C₂×C₃×Q₈	—	2	2	no	—
123	C₂×C₃×Q₈	—	2	2	no	—
124	C₂×C₃×Q₈	—	4	2	no	—
125	C₂×C₃×D₄	—	3	4	no	—
126	C₂×C₃×D₄	—	6	4	no	—
127	C₂×C₃×D₄	—	6	2	no	—
128	C₂×C₃×D₄	—	6	2	no	—
129	C₂²×C₃×C₄	—	3	2	no	—
130	C₂²×C₃×C₄	—	3	1	no	—
131	C₂²×C₃×C₄	—	6	4	no	—
132	#239	—	4	8	no	—
133	C₂×D₁₂	—	2	4	no	—
134	C₂×D₁₂	—	2	2	no	—
135	C₂×D₁₂	—	2	2	no	—
136	C₂×D₁₂	C₂³×C₃	1	4	yes	—
137	C₂×C₄×D₃	—	2	4	no	—
138	C₂×C₄×D₃	C₃×D₄	1	2	yes	—
139	C₂×C₄×D₃	C₂³×C₃	1	1	yes	characteristic
140	C₃×D₈	—	6	8	no	—
141	C₃×M₁₆	—	6	4	no	—
142	C₂×C₃×C₈	—	2	2	no	—
143	C₂×C₃×C₈	—	3	2	no	—
144	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
145	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
146	C₃×C₄⋊C₄	—	2	2	no	—
147	C₃×C₄⋊C₄	—	2	2	no	—
148	C₃×C₄⋊C₄	—	2	4	no	—
149	C₃×C₄⋊C₄	—	4	2	no	—
150	C₃×C₄⋊C₄	C₂²×D₃	1	4	yes	—
151	C₃×C₄⋊C₄	C₂³×C₃	1	4	yes	—
152	C₃×C₂²⋊C₄	—	6	4	no	—
153	C₃×C₂²⋊C₄	—	6	4	no	—
154	C₃×C₄²	—	2	1	no	—
155	C₃×C₄²	—	3	2	no	—
156	C₃×C₄²	—	3	1	no	—
157	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
158	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
159	(C₂×D₃)⋊C₄	—	2	4	no	—
160	(C₂×D₃)⋊C₄	—	2	4	no	—
161	C₄×Dic₃	C₃×D₄	1	2	yes	—
162	C₄×Dic₃	C₂³×C₃	1	1	yes	characteristic
163	C₂×C₃⋊C₈	C₂³×C₃	1	2	yes	—
164	D₂₄	—	2	8	no	—
165	ΓL(1,𝔽₂₅)	—	2	4	no	—
166	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
167	C₂×C₃×D₃	—	2	1	no	—
168	C₂×C₃×D₃	—	2	2	no	—
169	C₂×C₃×D₃	—	4	4	no	—
170	C₂×C₃×D₃	—	4	4	no	—
171	C₂×C₃×D₃	—	4	4	no	—
172	C₃²×C₄	—	4	4	no	—
173	C₃²×C₄	AGL(1,ℤ/8)	1	4	yes	—
174	C₃²×C₄	C₂×#41	1	2	yes	—
175	C₃×Dic₃	—	2	1	no	—
176	C₃×Dic₃	—	2	2	no	—
177	C₂×#41	—	6	2	no	—
178	C₂×D₈	—	3	4	no	—
179	C₂×M₁₆	—	3	2	no	—
180	C₄⋊Q₈	C₂×C₃×D₃	1	2	yes	—
181	#127	—	3	2	no	—
182	#124	—	6	2	no	—
183	#121	—	3	4	no	—
184	#121	—	6	2	no	—
185	C₄×D₄	—	3	4	no	—
186	C₄×D₄	—	6	2	no	—
187	C₂×C₄²	—	3	1	no	—
188	#106	C₂×C₃×D₃	1	4	yes	—
189	#102	—	3	8	no	—
190	#97	—	3	2	no	—
191	C₄×C₈	—	3	1	no	—
192	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
193	C₂³×C₃	—	3	1	no	—
194	C₂³×C₃	—	6	4	no	—
195	C₂²×D₃	—	2	4	no	—
196	C₂²×D₃	C₂×C₃×D₄	1	1	yes	characteristic
197	C₃×Q₈	—	4	4	no	—
198	C₃×Q₈	—	4	4	no	—
199	C₃×Q₈	—	8	4	no	—
200	C₃×D₄	—	6	8	no	—
201	C₃×D₄	—	12	8	no	—
202	C₃×D₄	—	12	4	no	—
203	C₃×D₄	—	12	4	no	—
204	C₂×C₃×C₄	—	2	4	no	—
205	C₂×C₃×C₄	—	2	4	no	—
206	C₂×C₃×C₄	—	2	1	no	—
207	C₂×C₃×C₄	—	2	2	no	—
208	C₂×C₃×C₄	—	3	1	no	—
209	C₂×C₃×C₄	—	3	2	no	—
210	C₂×C₃×C₄	—	4	4	no	—
211	C₂×C₃×C₄	—	6	4	no	—
212	C₂×C₃×C₄	—	6	2	no	—
213	C₂×C₃×C₄	—	12	8	no	—
214	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
215	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
216	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
217	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
218	C₂×Dic₃	C₂×C₃×D₄	1	1	yes	characteristic
219	C₂×Dic₃	C₂×C₃×D₄	1	2	yes	—
220	D₁₂	—	2	8	no	—
221	D₁₂	—	4	8	no	—
222	D₁₂	—	4	4	no	—
223	D₁₂	—	4	4	no	—
224	C₄×D₃	—	2	4	no	—
225	C₄×D₃	—	2	2	no	—
226	C₄×D₃	—	4	8	no	—
227	C₃×C₈	—	2	4	no	—
228	C₃×C₈	—	6	2	no	—
229	C₃×C₈	#239	1	4	yes	complemented
230	C₃×C₈	C₃×#41	1	4	yes	complemented
231	C₃⋊C₈	—	2	2	no	—
232	C₂×C₃²	#532	1	1	yes	characteristic
233	C₂×C₃²	C₂×AGL(1,ℤ/8)	1	2	yes	—
234	C₃×D₃	—	4	1	no	—
235	C₃×D₃	—	8	4	no	—
236	#41	—	12	8	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×D₄	—	3	4	no	—
239	C₂×D₄	—	6	4	no	—
240	C₂×D₄	—	6	2	no	—
241	C₂×D₄	—	6	2	no	—
242	C₂²×C₄	—	3	1	no	—
243	C₂²×C₄	—	3	2	no	—
244	C₂²×C₄	—	6	4	no	—
245	D₈	—	6	8	no	—
246	M₁₆	—	6	4	no	—
247	C₂×C₈	—	3	2	no	—
248	C₂×C₈	C₂²×C₃×D₃	1	2	yes	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	C₂²×C₃×D₃	1	4	yes	—
251	C₂²⋊C₄	—	6	4	no	—
252	C₂²⋊C₄	—	6	4	no	—
253	C₄²	—	3	1	no	—
254	C₄²	—	3	2	no	—
255	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
256	C₂²×C₃	—	2	1	no	—
257	C₂²×C₃	—	6	1	no	—
258	C₂²×C₃	—	6	2	no	—
259	C₂²×C₃	—	12	4	no	—
260	C₂²×C₃	—	12	4	no	—
261	C₂²×C₃	—	12	4	no	—
262	C₂²×C₃	C₂×D₃×D₄	1	1	yes	characteristic
263	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
264	C₂×D₃	—	2	2	no	—
265	C₂×D₃	—	2	1	no	—
266	C₂×D₃	—	4	4	no	—
267	C₂×D₃	—	4	4	no	—
268	C₂×D₃	—	4	4	no	—
269	C₃×C₄	—	2	4	no	—
270	C₃×C₄	—	2	2	no	—
271	C₃×C₄	—	4	4	no	—
272	C₃×C₄	—	4	4	no	—
273	C₃×C₄	—	6	1	no	—
274	C₃×C₄	—	6	2	no	—
275	C₃×C₄	—	8	4	no	—
276	C₃×C₄	#914	1	4	yes	—
277	C₃×C₄	C₃×AGL(1,ℤ/8)	1	4	yes	—
278	C₃×C₄	C₂×#239	1	2	yes	—
279	C₃×C₄	C₂×C₃×#41	1	2	yes	—
280	Dic₃	—	2	2	no	—
281	Dic₃	—	2	1	no	—
282	C₃²	#3184	1	1	yes	characteristic, complemented, Sylow
283	C₂³	—	3	1	no	—
284	C₂³	—	6	4	no	—
285	Q₈	—	4	4	no	—
286	D₄	—	6	8	no	—
287	D₄	—	12	8	no	—
288	D₄	—	12	4	no	—
289	D₄	—	12	4	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	3	1	no	—
292	C₂×C₄	—	3	2	no	—
293	C₂×C₄	—	6	4	no	—
294	C₂×C₄	—	6	2	no	—
295	C₂×C₄	—	12	8	no	—
296	C₂×C₄	C₃×D₃×D₄	1	2	yes	—
297	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
298	C₈	—	6	2	no	—
299	C₈	C₃×#239	1	4	yes	complemented
300	C₂×C₃	—	2	2	no	—
301	C₂×C₃	—	2	1	no	—
302	C₂×C₃	—	12	1	no	—
303	C₂×C₃	—	24	4	no	—
304	C₂×C₃	#5725	1	1	yes	characteristic
305	C₂×C₃	C₂×#914	1	2	yes	—
306	C₂×C₃	C₃×#532	1	1	yes	characteristic
307	C₂×C₃	C₂×C₃×AGL(1,ℤ/8)	1	2	yes	—
308	D₃	—	4	1	no	—
309	D₃	—	8	4	no	—
310	C₂²	—	6	2	no	—
311	C₂²	—	6	1	no	—
312	C₂²	—	12	4	no	—
313	C₂²	—	12	4	no	—
314	C₂²	—	12	4	no	—
315	C₂²	C₂×C₃×D₃×D₄	1	1	yes	characteristic
316	C₄	—	4	4	no	—
317	C₄	—	6	2	no	—
318	C₄	—	6	1	no	—
319	C₄	C₃×#914	1	4	yes	—
320	C₄	C₂×C₃×#239	1	2	yes	—
321	C₃	—	2	1	no	—
322	C₃	#81888	1	1	yes	characteristic, direct product factor
323	C₃	C₃×#3184	1	1	yes	characteristic, complemented
324	C₂	—	12	1	no	—
325	C₂	—	24	4	no	—
326	C₂	C₃×#5725	1	1	yes	characteristic
327	C₂	C₂×C₃×#914	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₃
Fitting	C₃²×#498	C₂
socle	C₂²×C₃²	C₂²×D₄
hypercenter	C₃×#498	D₃
hypocenter	C₃	C₃×#3184
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#498

Complements

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

N	H ≅ G/N	type
1. C₃×#5725	C₂	semidirect
2. C₃²×#498	C₂	semidirect
3. C₃×#5088	C₂	semidirect
4. C₃×#4869	C₂	semidirect
5. C₃×#4865	C₂	semidirect
6. C₃×#4695	C₂	semidirect
7. C₃×C₄×ΓL(1,𝔽₂₅)	C₂	semidirect
9. #81888	C₃	direct (N × H)
27. #5725	C₂×C₃	semidirect
29. C₃×#498	D₃	semidirect
30. C₃×#498	C₂×C₃	semidirect
36. #5088	C₂×C₃	semidirect
37. #4869	C₂×C₃	semidirect
38. #4865	C₂×C₃	semidirect
39. #4695	C₂×C₃	semidirect
40. C₄×ΓL(1,𝔽₂₅)	C₂×C₃	semidirect
112. C₃²×C₈	#41	semidirect
115. #498	C₃×D₃	semidirect
229. C₃×C₈	#239	semidirect
230. C₃×C₈	C₃×#41	semidirect
282. C₃²	#3184	semidirect
299. C₈	C₃×#239	semidirect
322. C₃	#81888	direct (N × H)
323. C₃	C₃×#3184	semidirect

Maximal subgroups

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

subgroup	index
C₃×#5725	2
C₃²×#498	2
C₃×#5088	2
C₃×#4869	2
C₃×#4865	2
C₃×#4695	2
C₃×C₄×ΓL(1,𝔽₂₅)	2
C₃×#3184	3
#81888	3

Subgroups — C3×#81888 #11876979

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#81888 #11876979