Subgroups — C₃×#60401 #10679008

Order 768.

4854 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×H(7,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#3485	C₂	1	3	yes	maximal, complemented
3	C₃×#3028	C₂	1	4	yes	maximal, complemented
4	C₃×#3026	C₂	1	8	yes	maximal, complemented
5	C₃×#3025	C₂	1	4	yes	maximal, complemented
6	C₂×C₃×#458	C₂	1	8	yes	maximal, complemented
7	C₃×#2883	C₂	1	3	yes	maximal, complemented
8	#60401	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₃×#585	—	2	12	no	—
10	C₃×#585	C₂²	1	4	yes	—
11	C₂×C₃×H(5,𝔽₂)	—	2	12	no	—
12	C₂×C₃×H(5,𝔽₂)	C₂²	1	8	yes	—
13	C₂×C₃×H(5,𝔽₂)	C₂²	1	3	yes	—
14	C₂²×C₃×#41	C₂²	1	1	yes	characteristic, complemented
15	C₃×#547	—	2	12	no	—
16	C₃×#546	—	2	12	no	—
17	C₃×#536	C₂²	1	6	yes	complemented
18	C₃×#534	C₂²	1	12	yes	complemented
19	C₃×#525	C₂²	1	6	yes	complemented
20	C₂×C₃×#123	C₂²	1	12	yes	complemented
21	C₃×#518	C₂²	1	3	yes	complemented
22	C₃×#458	C₂²	1	16	yes	complemented
23	C₃×#458	C₂²	1	32	yes	complemented
24	C₃×B(4,𝔽₂)	C₂²	1	16	yes	complemented
25	C₃×#410	C₂²	1	12	yes	complemented
26	C₂×C₃×#99	C₂²	1	24	yes	complemented
27	H(7,𝔽₂)	C₂×C₃	1	1	yes	characteristic, complemented
28	#3485	C₂×C₃	1	3	yes	complemented
29	#3028	C₂×C₃	1	4	yes	complemented
30	#3026	C₂×C₃	1	8	yes	complemented
31	#3025	C₂×C₃	1	4	yes	complemented
32	C₂×#458	C₂×C₃	1	8	yes	complemented
33	#2883	C₂×C₃	1	3	yes	complemented
34	C₃×2₋¹⁺⁴	—	2	16	no	—
35	C₃×2₋¹⁺⁴	—	2	12	no	—
36	C₃×H(5,𝔽₂)	—	2	48	no	—
37	C₃×H(5,𝔽₂)	—	2	12	no	—
38	C₃×H(5,𝔽₂)	—	2	48	no	—
39	C₃×H(5,𝔽₂)	—	2	24	no	—
40	C₃×H(5,𝔽₂)	C₂³	1	16	yes	—
41	C₂×C₃×#41	—	2	48	no	—
42	C₂×C₃×#41	—	2	12	no	—
43	C₂×C₃×#41	—	2	24	no	—
44	C₂×C₃×#41	—	2	24	no	—
45	C₂×C₃×#41	D₄	1	12	yes	complemented
46	C₂×C₃×#41	C₂³	1	6	yes	—
47	C₂²×C₃×Q₈	C₂³	1	1	yes	characteristic, complemented
48	C₂²×C₃×D₄	—	2	16	no	—
49	C₂²×C₃×D₄	—	2	24	no	—
50	C₂²×C₃×D₄	D₄	1	12	yes	complemented
51	C₂²×C₃×D₄	C₂³	1	12	yes	—
52	C₂²×C₃×D₄	C₂³	1	1	yes	characteristic
53	C₂²×C₃×D₄	C₂³	1	2	yes	complemented
54	C₂³×C₃×C₄	C₂³	1	2	yes	complemented
55	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic, complemented
56	C₃×C₄⋊Q₈	—	2	6	no	—
57	C₃×#124	—	2	6	no	—
58	C₃×#123	—	2	24	no	—
59	C₃×#123	—	2	12	no	—
60	C₃×#123	C₂³	1	12	yes	complemented
61	C₃×#123	C₂³	1	24	yes	complemented
62	C₃×C₂²⋊Q₈	—	2	6	no	—
63	C₃×C₂²⋊Q₈	—	2	12	no	—
64	C₃×#121	—	2	6	no	—
65	C₃×#121	—	2	24	no	—
66	C₃×#120	—	2	12	no	—
67	C₃×#120	C₂³	1	12	yes	complemented
68	C₃×C₄×D₄	—	2	12	no	—
69	C₃×C₄×D₄	—	2	12	no	—
70	C₃×#117	C₂³	1	6	yes	complemented
71	C₂×C₃×C₄⋊C₄	—	2	12	no	—
72	C₂×C₃×C₂²⋊C₄	—	2	12	no	—
73	C₂×C₃×C₂²⋊C₄	C₂³	1	12	yes	complemented
74	C₃×#99	C₂³	1	48	yes	complemented
75	#585	—	2	12	no	—
76	#585	C₂²×C₃	1	4	yes	—
77	C₂×H(5,𝔽₂)	—	2	12	no	—
78	C₂×H(5,𝔽₂)	C₂²×C₃	1	8	yes	—
79	C₂×H(5,𝔽₂)	C₂²×C₃	1	3	yes	—
80	C₂²×#41	C₂²×C₃	1	1	yes	characteristic, complemented
81	#547	—	2	12	no	—
82	#546	—	2	12	no	—
83	#536	C₂²×C₃	1	6	yes	complemented
84	#534	C₂²×C₃	1	12	yes	complemented
85	#525	C₂²×C₃	1	6	yes	complemented
86	C₂×#123	C₂²×C₃	1	12	yes	complemented
87	#518	C₂²×C₃	1	3	yes	complemented
88	#458	C₂²×C₃	1	16	yes	complemented
89	#458	C₂²×C₃	1	32	yes	complemented
90	B(4,𝔽₂)	C₂²×C₃	1	16	yes	complemented
91	#410	C₂²×C₃	1	12	yes	complemented
92	C₂×#99	C₂²×C₃	1	24	yes	complemented
93	C₂⁴×C₃	—	2	8	no	—
94	C₂⁴×C₃	C₂×D₄	1	12	yes	—
95	C₂⁴×C₃	C₂⁴	1	1	yes	characteristic
96	C₂⁴×C₃	C₂⁴	1	2	yes	—
97	C₃×#41	—	2	96	no	—
98	C₃×#41	—	2	48	no	—
99	C₃×#41	—	2	16	no	—
100	C₃×#41	—	2	48	no	—
101	C₃×#41	—	2	48	no	—
102	C₃×#41	—	2	24	no	—
103	C₃×#41	—	4	16	no	—
104	C₂×C₃×Q₈	—	2	24	no	—
105	C₂×C₃×Q₈	—	2	8	no	—
106	C₂×C₃×Q₈	—	2	6	no	—
107	C₂×C₃×Q₈	C₂×D₄	1	6	yes	—
108	C₂×C₃×D₄	—	2	96	no	—
109	C₂×C₃×D₄	—	2	24	no	—
110	C₂×C₃×D₄	—	2	24	no	—
111	C₂×C₃×D₄	—	2	48	no	—
112	C₂×C₃×D₄	—	2	48	no	—
113	C₂×C₃×D₄	—	2	48	no	—
114	C₂×C₃×D₄	—	2	12	no	—
115	C₂×C₃×D₄	—	2	12	no	—
116	C₂×C₃×D₄	C₂×D₄	1	24	yes	—
117	C₂×C₃×D₄	C₂×D₄	1	6	yes	—
118	C₂×C₃×D₄	C₂×D₄	1	12	yes	—
119	C₂×C₃×D₄	C₂⁴	1	12	yes	—
120	C₂²×C₃×C₄	—	2	24	no	—
121	C₂²×C₃×C₄	—	2	16	no	—
122	C₂²×C₃×C₄	—	2	12	no	—
123	C₂²×C₃×C₄	—	2	12	no	—
124	C₂²×C₃×C₄	—	2	6	no	—
125	C₂²×C₃×C₄	C₂×D₄	1	12	yes	—
126	C₂²×C₃×C₄	C₂×D₄	1	12	yes	—
127	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
128	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
129	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
130	C₃×C₄⋊C₄	—	2	12	no	—
131	C₃×C₄⋊C₄	—	2	24	no	—
132	C₃×C₄⋊C₄	—	2	6	no	—
133	C₃×C₂²⋊C₄	—	2	12	no	—
134	C₃×C₂²⋊C₄	—	2	24	no	—
135	C₃×C₂²⋊C₄	—	2	12	no	—
136	C₃×C₂²⋊C₄	—	2	12	no	—
137	C₃×C₂²⋊C₄	C₂⁴	1	12	yes	—
138	C₃×C₄²	—	2	6	no	—
139	2₋¹⁺⁴	—	2	16	no	—
140	2₋¹⁺⁴	—	2	12	no	—
141	H(5,𝔽₂)	—	2	48	no	—
142	H(5,𝔽₂)	—	2	12	no	—
143	H(5,𝔽₂)	—	2	48	no	—
144	H(5,𝔽₂)	—	2	24	no	—
145	H(5,𝔽₂)	C₂³×C₃	1	16	yes	—
146	C₂×#41	—	2	48	no	—
147	C₂×#41	—	2	12	no	—
148	C₂×#41	—	2	24	no	—
149	C₂×#41	—	2	24	no	—
150	C₂×#41	C₃×D₄	1	12	yes	complemented
151	C₂×#41	C₂³×C₃	1	6	yes	—
152	C₂²×Q₈	C₂³×C₃	1	1	yes	characteristic, complemented
153	C₂²×D₄	—	2	16	no	—
154	C₂²×D₄	—	2	24	no	—
155	C₂²×D₄	C₃×D₄	1	12	yes	complemented
156	C₂²×D₄	C₂³×C₃	1	12	yes	—
157	C₂²×D₄	C₂³×C₃	1	1	yes	characteristic
158	C₂²×D₄	C₂³×C₃	1	2	yes	complemented
159	C₂³×C₄	C₂³×C₃	1	2	yes	complemented
160	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic, complemented
161	C₄⋊Q₈	—	2	6	no	—
162	#124	—	2	6	no	—
163	#123	—	2	24	no	—
164	#123	—	2	12	no	—
165	#123	C₂³×C₃	1	12	yes	complemented
166	#123	C₂³×C₃	1	24	yes	complemented
167	C₂²⋊Q₈	—	2	6	no	—
168	C₂²⋊Q₈	—	2	12	no	—
169	#121	—	2	6	no	—
170	#121	—	2	24	no	—
171	#120	—	2	12	no	—
172	#120	C₂³×C₃	1	12	yes	complemented
173	C₄×D₄	—	2	12	no	—
174	C₄×D₄	—	2	12	no	—
175	#117	C₂³×C₃	1	6	yes	complemented
176	C₂×C₄⋊C₄	—	2	12	no	—
177	C₂×C₂²⋊C₄	—	2	12	no	—
178	C₂×C₂²⋊C₄	C₂³×C₃	1	12	yes	complemented
179	#99	C₂³×C₃	1	48	yes	complemented
180	C₂³×C₃	—	2	16	no	—
181	C₂³×C₃	—	2	24	no	—
182	C₂³×C₃	—	2	3	no	—
183	C₂³×C₃	—	8	2	no	—
184	C₂³×C₃	—	8	1	no	—
185	C₂³×C₃	—	8	12	no	—
186	C₂³×C₃	—	16	8	no	—
187	C₂³×C₃	C₂²×D₄	1	12	yes	—
188	C₂³×C₃	C₂²×D₄	1	12	yes	—
189	C₂³×C₃	C₂⁵	1	1	yes	characteristic
190	C₃×Q₈	—	2	16	no	—
191	C₃×Q₈	—	2	12	no	—
192	C₃×Q₈	—	4	4	no	—
193	C₃×D₄	—	2	48	no	—
194	C₃×D₄	—	2	48	no	—
195	C₃×D₄	—	2	24	no	—
196	C₃×D₄	—	2	12	no	—
197	C₃×D₄	—	4	8	no	—
198	C₃×D₄	—	8	12	no	—
199	C₃×D₄	#120	1	16	yes	—
200	C₂×C₃×C₄	—	2	48	no	—
201	C₂×C₃×C₄	—	2	24	no	—
202	C₂×C₃×C₄	—	2	24	no	—
203	C₂×C₃×C₄	—	2	12	no	—
204	C₂×C₃×C₄	—	2	3	no	—
205	C₂×C₃×C₄	—	2	6	no	—
206	C₂×C₃×C₄	—	4	8	no	—
207	C₂×C₃×C₄	—	4	4	no	—
208	C₂×C₃×C₄	—	8	12	no	—
209	C₂×C₃×C₄	C₂²×D₄	1	12	yes	—
210	C₂×C₃×C₄	C₂²×D₄	1	6	yes	—
211	C₂⁴	—	2	8	no	—
212	C₂⁴	C₂×C₃×D₄	1	12	yes	—
213	C₂⁴	C₂⁴×C₃	1	1	yes	characteristic
214	C₂⁴	C₂⁴×C₃	1	2	yes	—
215	#41	—	2	96	no	—
216	#41	—	2	48	no	—
217	#41	—	2	16	no	—
218	#41	—	2	48	no	—
219	#41	—	2	48	no	—
220	#41	—	2	24	no	—
221	#41	—	4	16	no	—
222	C₂×Q₈	—	2	24	no	—
223	C₂×Q₈	—	2	8	no	—
224	C₂×Q₈	—	2	6	no	—
225	C₂×Q₈	C₂×C₃×D₄	1	6	yes	—
226	C₂×D₄	—	2	96	no	—
227	C₂×D₄	—	2	24	no	—
228	C₂×D₄	—	2	24	no	—
229	C₂×D₄	—	2	48	no	—
230	C₂×D₄	—	2	48	no	—
231	C₂×D₄	—	2	48	no	—
232	C₂×D₄	—	2	12	no	—
233	C₂×D₄	—	2	12	no	—
234	C₂×D₄	C₂×C₃×D₄	1	24	yes	—
235	C₂×D₄	C₂×C₃×D₄	1	6	yes	—
236	C₂×D₄	C₂×C₃×D₄	1	12	yes	—
237	C₂×D₄	C₂⁴×C₃	1	12	yes	—
238	C₂²×C₄	—	2	24	no	—
239	C₂²×C₄	—	2	16	no	—
240	C₂²×C₄	—	2	12	no	—
241	C₂²×C₄	—	2	12	no	—
242	C₂²×C₄	—	2	6	no	—
243	C₂²×C₄	C₂×C₃×D₄	1	12	yes	—
244	C₂²×C₄	C₂×C₃×D₄	1	12	yes	—
245	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
246	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
247	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
248	C₄⋊C₄	—	2	12	no	—
249	C₄⋊C₄	—	2	24	no	—
250	C₄⋊C₄	—	2	6	no	—
251	C₂²⋊C₄	—	2	12	no	—
252	C₂²⋊C₄	—	2	24	no	—
253	C₂²⋊C₄	—	2	12	no	—
254	C₂²⋊C₄	—	2	12	no	—
255	C₂²⋊C₄	C₂⁴×C₃	1	12	yes	—
256	C₄²	—	2	6	no	—
257	C₂²×C₃	—	2	12	no	—
258	C₂²×C₃	—	4	12	no	—
259	C₂²×C₃	—	4	1	no	—
260	C₂²×C₃	—	4	12	no	—
261	C₂²×C₃	—	4	6	no	—
262	C₂²×C₃	—	4	1	no	—
263	C₂²×C₃	—	8	24	no	—
264	C₂²×C₃	—	8	16	no	—
265	C₂²×C₃	C₂×#120	1	8	yes	—
266	C₂²×C₃	C₂³×D₄	1	3	yes	—
267	C₃×C₄	—	2	12	no	—
268	C₃×C₄	—	4	12	no	—
269	C₃×C₄	—	4	1	no	—
270	C₃×C₄	—	4	2	no	—
271	C₃×C₄	C₂×#120	1	4	yes	—
272	C₂³	—	2	16	no	—
273	C₂³	—	2	24	no	—
274	C₂³	—	2	3	no	—
275	C₂³	—	8	2	no	—
276	C₂³	—	8	1	no	—
277	C₂³	—	8	12	no	—
278	C₂³	—	16	8	no	—
279	C₂³	C₂²×C₃×D₄	1	12	yes	—
280	C₂³	C₂²×C₃×D₄	1	12	yes	—
281	C₂³	C₂⁵×C₃	1	1	yes	characteristic
282	Q₈	—	2	16	no	—
283	Q₈	—	2	12	no	—
284	Q₈	—	4	4	no	—
285	D₄	—	2	48	no	—
286	D₄	—	2	24	no	—
287	D₄	—	2	48	no	—
288	D₄	—	2	12	no	—
289	D₄	—	4	8	no	—
290	D₄	—	8	12	no	—
291	D₄	C₃×#120	1	16	yes	—
292	C₂×C₄	—	2	48	no	—
293	C₂×C₄	—	2	24	no	—
294	C₂×C₄	—	2	24	no	—
295	C₂×C₄	—	2	12	no	—
296	C₂×C₄	—	2	3	no	—
297	C₂×C₄	—	2	6	no	—
298	C₂×C₄	—	4	8	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	8	12	no	—
301	C₂×C₄	C₂²×C₃×D₄	1	12	yes	—
302	C₂×C₄	C₂²×C₃×D₄	1	6	yes	—
303	C₂×C₃	—	2	3	no	—
304	C₂×C₃	—	2	8	no	—
305	C₂×C₃	—	4	12	no	—
306	C₂×C₃	—	8	1	no	—
307	C₂×C₃	C₂²×#120	1	1	yes	characteristic
308	C₂²	—	2	12	no	—
309	C₂²	—	4	12	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	4	12	no	—
312	C₂²	—	4	6	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	8	24	no	—
315	C₂²	—	8	16	no	—
316	C₂²	C₂×C₃×#120	1	8	yes	—
317	C₂²	C₂³×C₃×D₄	1	3	yes	—
318	C₄	—	2	12	no	—
319	C₄	—	4	12	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	2	no	—
322	C₄	C₂×C₃×#120	1	4	yes	—
323	C₃	#60401	1	1	yes	characteristic, direct product factor, Sylow
324	C₂	—	2	3	no	—
325	C₂	—	2	8	no	—
326	C₂	—	4	12	no	—
327	C₂	—	8	1	no	—
328	C₂	C₂²×C₃×#120	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	C₂²×#120
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×H(7,𝔽₂)	C₂	semidirect
2. C₃×#3485	C₂	semidirect
3. C₃×#3028	C₂	semidirect
4. C₃×#3026	C₂	semidirect
5. C₃×#3025	C₂	semidirect
6. C₂×C₃×#458	C₂	semidirect
7. C₃×#2883	C₂	semidirect
8. #60401	C₃	direct (N × H)
14. C₂²×C₃×#41	C₂²	semidirect
17. C₃×#536	C₂²	semidirect
18. C₃×#534	C₂²	semidirect
19. C₃×#525	C₂²	semidirect
20. C₂×C₃×#123	C₂²	semidirect
21. C₃×#518	C₂²	semidirect
22. C₃×#458	C₂²	semidirect
23. C₃×#458	C₂²	semidirect
24. C₃×B(4,𝔽₂)	C₂²	semidirect
25. C₃×#410	C₂²	semidirect
26. C₂×C₃×#99	C₂²	semidirect
27. H(7,𝔽₂)	C₂×C₃	semidirect
28. #3485	C₂×C₃	semidirect
29. #3028	C₂×C₃	semidirect
30. #3026	C₂×C₃	semidirect
31. #3025	C₂×C₃	semidirect
32. C₂×#458	C₂×C₃	semidirect
33. #2883	C₂×C₃	semidirect
45. C₂×C₃×#41	D₄	semidirect
47. C₂²×C₃×Q₈	C₂³	semidirect
50. C₂²×C₃×D₄	D₄	semidirect
53. C₂²×C₃×D₄	C₂³	semidirect
54. C₂³×C₃×C₄	C₂³	semidirect
55. C₂³×C₃×C₄	C₂³	semidirect
60. C₃×#123	C₂³	semidirect
61. C₃×#123	C₂³	semidirect
67. C₃×#120	C₂³	semidirect
70. C₃×#117	C₂³	semidirect
73. C₂×C₃×C₂²⋊C₄	C₂³	semidirect
74. C₃×#99	C₂³	semidirect
80. C₂²×#41	C₂²×C₃	semidirect
83. #536	C₂²×C₃	semidirect
84. #534	C₂²×C₃	semidirect
85. #525	C₂²×C₃	semidirect
86. C₂×#123	C₂²×C₃	semidirect
87. #518	C₂²×C₃	semidirect
88. #458	C₂²×C₃	semidirect
89. #458	C₂²×C₃	semidirect
90. B(4,𝔽₂)	C₂²×C₃	semidirect
91. #410	C₂²×C₃	semidirect
92. C₂×#99	C₂²×C₃	semidirect
150. C₂×#41	C₃×D₄	semidirect
152. C₂²×Q₈	C₂³×C₃	semidirect
155. C₂²×D₄	C₃×D₄	semidirect
158. C₂²×D₄	C₂³×C₃	semidirect
159. C₂³×C₄	C₂³×C₃	semidirect
160. C₂³×C₄	C₂³×C₃	semidirect
165. #123	C₂³×C₃	semidirect
166. #123	C₂³×C₃	semidirect
172. #120	C₂³×C₃	semidirect
175. #117	C₂³×C₃	semidirect
178. C₂×C₂²⋊C₄	C₂³×C₃	semidirect
179. #99	C₂³×C₃	semidirect
323. C₃	#60401	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×H(7,𝔽₂)	2
C₃×#3485	2
C₃×#3028	2
C₃×#3026	2
C₃×#3025	2
C₂×C₃×#458	2
C₃×#2883	2
#60401	3

Subgroups — C3×#60401 #10679008

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#60401 #10679008