Subgroups — C₃×#35285 #10653892

Order 768.

3206 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×#127	C₂	1	1	yes	maximal, characteristic
3	C₃×#2858	C₂	1	4	yes	maximal, complemented
4	C₃×#2641	C₂	1	4	yes	maximal, complemented
5	C₃×#2548	C₂	1	8	yes	maximal, complemented
6	C₃×#2435	C₂	1	8	yes	maximal, complemented
7	C₃×#2309	C₂	1	4	yes	maximal, complemented
8	C₃×C₄×#117	C₂	1	1	yes	maximal, characteristic, complemented
9	#35285	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂³×C₃×D₄	C₂²	1	2	yes	—
11	C₂³×C₃×D₄	C₂²	1	4	yes	—
12	C₃×#533	—	2	8	no	—
13	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
14	C₂×C₃×#127	—	2	8	no	—
15	C₂×C₃×#127	C₂²	1	2	yes	—
16	C₂×C₃×#127	C₂²	1	4	yes	—
17	C₂×C₃×#127	C₂²	1	2	yes	—
18	C₂×C₃×#125	C₂²	1	2	yes	complemented
19	C₂×C₃×#124	C₂²	1	16	yes	complemented
20	C₂×C₃×C₂²⋊Q₈	C₂²	1	4	yes	—
21	C₂×C₃×C₄×Q₈	C₂²	1	4	yes	complemented
22	C₂×C₃×#117	C₂²	1	2	yes	—
23	C₂×C₃×#117	C₂²	1	2	yes	—
24	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
25	C₃×#394	C₂²	1	32	yes	complemented
26	C₃×#390	C₂²	1	16	yes	complemented
27	C₃×#390	C₂²	1	16	yes	complemented
28	C₃×#390	C₂²	1	16	yes	complemented
29	C₃×#379	C₂²	1	16	yes	—
30	C₃×C₄×C₄⋊C₄	C₂²	1	4	yes	complemented
31	C₃×C₄×C₂²⋊C₄	C₂²	1	4	yes	—
32	C₃×#376	C₂²	1	2	yes	complemented
33	C₃×C₄³	C₂²	1	2	yes	complemented
34	C₂×#533	C₂×C₃	1	1	yes	characteristic, complemented
35	C₂²×#127	C₂×C₃	1	1	yes	characteristic
36	#2858	C₂×C₃	1	4	yes	complemented
37	#2641	C₂×C₃	1	4	yes	complemented
38	#2548	C₂×C₃	1	8	yes	complemented
39	#2435	C₂×C₃	1	8	yes	complemented
40	#2309	C₂×C₃	1	4	yes	complemented
41	C₄×#117	C₂×C₃	1	1	yes	characteristic, complemented
42	C₂⁵×C₃	C₂³	1	4	yes	—
43	C₂²×C₃×Q₈	C₂³	1	4	yes	—
44	C₂²×C₃×D₄	—	2	16	no	—
45	C₂²×C₃×D₄	—	2	8	no	—
46	C₂²×C₃×D₄	—	2	16	no	—
47	C₂²×C₃×D₄	—	4	16	no	—
48	C₂²×C₃×D₄	C₂³	1	4	yes	—
49	C₂²×C₃×D₄	C₂³	1	4	yes	—
50	C₂²×C₃×D₄	C₂³	1	8	yes	—
51	C₂²×C₃×D₄	C₂³	1	8	yes	—
52	C₂³×C₃×C₄	C₂³	1	2	yes	—
53	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
54	C₃×C₄⋊Q₈	—	2	8	no	—
55	C₃×#127	—	2	16	no	—
56	C₃×#127	—	2	8	no	—
57	C₃×#127	—	2	8	no	—
58	C₃×#125	—	2	8	no	—
59	C₃×#124	—	4	32	no	—
60	C₃×C₂²⋊Q₈	—	2	16	no	—
61	C₃×C₄×Q₈	—	2	16	no	—
62	C₃×#117	—	2	8	no	—
63	C₃×#117	—	2	8	no	—
64	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
65	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
66	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
67	C₂×C₃×C₄⋊C₄	C₂³	1	8	yes	—
68	C₂×C₃×C₂²⋊C₄	C₂³	1	4	yes	—
69	C₂×C₃×C₂²⋊C₄	C₂³	1	4	yes	—
70	C₂×C₃×C₂²⋊C₄	C₂³	1	64	yes	—
71	C₂×C₃×C₄²	D₄	1	8	yes	—
72	C₂×C₃×C₄²	C₂³	1	4	yes	—
73	C₂×C₃×C₄²	C₂³	1	8	yes	—
74	C₂×C₃×C₄²	C₂³	1	4	yes	—
75	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
76	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
77	C₂×C₃×C₄²	C₂³	1	2	yes	—
78	C₃×#95	C₂³	1	8	yes	—
79	C₂³×D₄	C₂²×C₃	1	2	yes	—
80	C₂³×D₄	C₂²×C₃	1	4	yes	—
81	#533	—	2	8	no	—
82	C₂×C₄⋊Q₈	C₂²×C₃	1	2	yes	complemented
83	C₂×#127	—	2	8	no	—
84	C₂×#127	C₂²×C₃	1	2	yes	—
85	C₂×#127	C₂²×C₃	1	2	yes	—
86	C₂×#127	C₂²×C₃	1	4	yes	—
87	C₂×#125	C₂²×C₃	1	2	yes	complemented
88	C₂×#124	C₂²×C₃	1	16	yes	complemented
89	C₂×C₂²⋊Q₈	C₂²×C₃	1	4	yes	—
90	C₂×C₄×Q₈	C₂²×C₃	1	4	yes	complemented
91	C₂×#117	C₂²×C₃	1	2	yes	—
92	C₂×#117	C₂²×C₃	1	2	yes	—
93	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
94	#394	C₂²×C₃	1	32	yes	complemented
95	#390	C₂²×C₃	1	16	yes	complemented
96	#390	C₂²×C₃	1	16	yes	complemented
97	#390	C₂²×C₃	1	16	yes	complemented
98	#379	C₂²×C₃	1	16	yes	—
99	C₄×C₄⋊C₄	C₂²×C₃	1	4	yes	complemented
100	C₄×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
101	#376	C₂²×C₃	1	2	yes	complemented
102	C₄³	C₂²×C₃	1	2	yes	complemented
103	C₂⁴×C₃	—	2	8	no	—
104	C₂⁴×C₃	—	2	16	no	—
105	C₂⁴×C₃	—	4	16	no	—
106	C₂⁴×C₃	C₂⁴	1	8	yes	—
107	C₂⁴×C₃	C₂⁴	1	1	yes	characteristic
108	C₂×C₃×Q₈	—	2	8	no	—
109	C₂×C₃×Q₈	—	2	16	no	—
110	C₂×C₃×D₄	—	2	16	no	—
111	C₂×C₃×D₄	—	2	8	no	—
112	C₂×C₃×D₄	—	2	8	no	—
113	C₂×C₃×D₄	—	2	16	no	—
114	C₂×C₃×D₄	—	2	16	no	—
115	C₂×C₃×D₄	—	2	16	no	—
116	C₂×C₃×D₄	—	2	32	no	—
117	C₂×C₃×D₄	—	4	16	no	—
118	C₂×C₃×D₄	—	4	64	no	—
119	C₂×C₃×D₄	—	4	32	no	—
120	C₂²×C₃×C₄	—	2	8	no	—
121	C₂²×C₃×C₄	C₂×D₄	1	8	yes	—
122	C₂²×C₃×C₄	C₂×D₄	1	4	yes	—
123	C₂²×C₃×C₄	#41	1	8	yes	—
124	C₂²×C₃×C₄	C₂⁴	1	8	yes	—
125	C₂²×C₃×C₄	C₂⁴	1	8	yes	—
126	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
127	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
128	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
129	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
130	C₃×C₄⋊C₄	—	2	8	no	—
131	C₃×C₄⋊C₄	—	2	16	no	—
132	C₃×C₄⋊C₄	—	2	8	no	—
133	C₃×C₄⋊C₄	—	2	8	no	—
134	C₃×C₂²⋊C₄	—	2	8	no	—
135	C₃×C₂²⋊C₄	—	2	8	no	—
136	C₃×C₂²⋊C₄	—	4	64	no	—
137	C₃×C₄²	—	2	8	no	—
138	C₃×C₄²	—	2	8	no	—
139	C₃×C₄²	—	2	16	no	—
140	C₃×C₄²	C₂×D₄	1	8	yes	—
141	C₃×C₄²	C₂×D₄	1	4	yes	—
142	C₃×C₄²	C₂×D₄	1	4	yes	—
143	C₂⁵	C₂³×C₃	1	4	yes	—
144	C₂²×Q₈	C₂³×C₃	1	4	yes	—
145	C₂²×D₄	—	2	16	no	—
146	C₂²×D₄	—	2	8	no	—
147	C₂²×D₄	—	2	16	no	—
148	C₂²×D₄	—	4	16	no	—
149	C₂²×D₄	C₂³×C₃	1	8	yes	—
150	C₂²×D₄	C₂³×C₃	1	4	yes	—
151	C₂²×D₄	C₂³×C₃	1	4	yes	—
152	C₂²×D₄	C₂³×C₃	1	8	yes	—
153	C₂³×C₄	C₂³×C₃	1	2	yes	—
154	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
155	C₄⋊Q₈	—	2	8	no	—
156	#127	—	2	16	no	—
157	#127	—	2	8	no	—
158	#127	—	2	8	no	—
159	#125	—	2	8	no	—
160	#124	—	4	32	no	—
161	C₂²⋊Q₈	—	2	16	no	—
162	C₄×Q₈	—	2	16	no	—
163	#117	—	2	8	no	—
164	#117	—	2	8	no	—
165	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
166	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
167	C₂×C₄⋊C₄	C₂³×C₃	1	8	yes	—
168	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
169	C₂×C₂²⋊C₄	C₂³×C₃	1	4	yes	—
170	C₂×C₂²⋊C₄	C₂³×C₃	1	4	yes	—
171	C₂×C₂²⋊C₄	C₂³×C₃	1	64	yes	—
172	C₂×C₄²	C₃×D₄	1	8	yes	—
173	C₂×C₄²	C₂³×C₃	1	4	yes	—
174	C₂×C₄²	C₂³×C₃	1	4	yes	—
175	C₂×C₄²	C₂³×C₃	1	8	yes	—
176	C₂×C₄²	C₂³×C₃	1	2	yes	—
177	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
178	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
179	#95	C₂³×C₃	1	8	yes	—
180	C₂³×C₃	—	2	4	no	—
181	C₂³×C₃	—	2	8	no	—
182	C₂³×C₃	—	2	16	no	—
183	C₂³×C₃	—	2	32	no	—
184	C₂³×C₃	—	4	32	no	—
185	C₂³×C₃	—	4	16	no	—
186	C₂³×C₃	—	8	32	no	—
187	C₂³×C₃	C₂²×D₄	1	2	yes	—
188	C₂³×C₃	C₂×#41	1	4	yes	—
189	C₂³×C₃	C₂⁵	1	1	yes	characteristic
190	C₃×Q₈	—	4	16	no	—
191	C₃×D₄	—	4	32	no	—
192	C₃×D₄	—	4	16	no	—
193	C₃×D₄	—	4	16	no	—
194	C₃×D₄	—	4	32	no	—
195	C₂×C₃×C₄	—	2	4	no	—
196	C₂×C₃×C₄	—	2	16	no	—
197	C₂×C₃×C₄	—	2	8	no	—
198	C₂×C₃×C₄	—	2	8	no	—
199	C₂×C₃×C₄	—	2	16	no	—
200	C₂×C₃×C₄	—	2	16	no	—
201	C₂×C₃×C₄	#124	1	16	yes	complemented
202	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
203	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
204	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
205	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
206	C₂×C₃×C₄	C₂×#41	1	4	yes	—
207	C₂×C₃×C₄	C₂×#41	1	4	yes	—
208	C₂×C₃×C₄	H(5,𝔽₂)	1	8	yes	—
209	C₂⁴	—	2	8	no	—
210	C₂⁴	—	2	16	no	—
211	C₂⁴	—	4	16	no	—
212	C₂⁴	C₂⁴×C₃	1	8	yes	—
213	C₂⁴	C₂⁴×C₃	1	1	yes	characteristic
214	C₂×Q₈	—	2	8	no	—
215	C₂×Q₈	—	2	16	no	—
216	C₂×D₄	—	2	16	no	—
217	C₂×D₄	—	2	8	no	—
218	C₂×D₄	—	2	8	no	—
219	C₂×D₄	—	2	16	no	—
220	C₂×D₄	—	2	16	no	—
221	C₂×D₄	—	2	16	no	—
222	C₂×D₄	—	2	32	no	—
223	C₂×D₄	—	4	16	no	—
224	C₂×D₄	—	4	64	no	—
225	C₂×D₄	—	4	32	no	—
226	C₂²×C₄	—	2	8	no	—
227	C₂²×C₄	C₂×C₃×D₄	1	8	yes	—
228	C₂²×C₄	C₂×C₃×D₄	1	4	yes	—
229	C₂²×C₄	C₃×#41	1	8	yes	—
230	C₂²×C₄	C₂⁴×C₃	1	8	yes	—
231	C₂²×C₄	C₂⁴×C₃	1	8	yes	—
232	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
233	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
234	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
235	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
236	C₄⋊C₄	—	2	8	no	—
237	C₄⋊C₄	—	2	16	no	—
238	C₄⋊C₄	—	2	8	no	—
239	C₄⋊C₄	—	2	8	no	—
240	C₂²⋊C₄	—	2	8	no	—
241	C₂²⋊C₄	—	2	8	no	—
242	C₂²⋊C₄	—	4	64	no	—
243	C₄²	—	2	8	no	—
244	C₄²	—	2	8	no	—
245	C₄²	—	2	16	no	—
246	C₄²	C₂×C₃×D₄	1	8	yes	—
247	C₄²	C₂×C₃×D₄	1	4	yes	—
248	C₄²	C₂×C₃×D₄	1	4	yes	—
249	C₂²×C₃	—	2	4	no	—
250	C₂²×C₃	—	2	8	no	—
251	C₂²×C₃	—	4	8	no	—
252	C₂²×C₃	—	4	16	no	—
253	C₂²×C₃	—	4	32	no	—
254	C₂²×C₃	—	8	32	no	—
255	C₂²×C₃	C₂×#124	1	4	yes	—
256	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
257	C₂²×C₃	C₂²×#41	1	2	yes	—
258	C₂²×C₃	C₂×H(5,𝔽₂)	1	4	yes	—
259	C₃×C₄	—	2	4	no	—
260	C₃×C₄	—	4	8	no	—
261	C₃×C₄	—	4	8	no	—
262	C₃×C₄	C₂×#124	1	4	yes	—
263	C₃×C₄	C₂×#124	1	4	yes	—
264	C₃×C₄	#535	1	8	yes	—
265	C₂³	—	2	4	no	—
266	C₂³	—	2	8	no	—
267	C₂³	—	2	16	no	—
268	C₂³	—	2	32	no	—
269	C₂³	—	4	32	no	—
270	C₂³	—	4	16	no	—
271	C₂³	—	8	32	no	—
272	C₂³	C₂²×C₃×D₄	1	2	yes	—
273	C₂³	C₂×C₃×#41	1	4	yes	—
274	C₂³	C₂⁵×C₃	1	1	yes	characteristic
275	Q₈	—	4	16	no	—
276	D₄	—	4	32	no	—
277	D₄	—	4	16	no	—
278	D₄	—	4	16	no	—
279	D₄	—	4	32	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	16	no	—
282	C₂×C₄	—	2	8	no	—
283	C₂×C₄	—	2	8	no	—
284	C₂×C₄	—	2	16	no	—
285	C₂×C₄	—	2	16	no	—
286	C₂×C₄	C₃×#124	1	16	yes	complemented
287	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
288	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
289	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
290	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
291	C₂×C₄	C₂×C₃×#41	1	4	yes	—
292	C₂×C₄	C₂×C₃×#41	1	4	yes	—
293	C₂×C₄	C₃×H(5,𝔽₂)	1	8	yes	—
294	C₂×C₃	—	2	4	no	—
295	C₂×C₃	—	8	8	no	—
296	C₂×C₃	C₂²×#124	1	1	yes	characteristic
297	C₂×C₃	C₂×#535	1	2	yes	—
298	C₂×C₃	#3508	1	4	yes	—
299	C₂²	—	2	4	no	—
300	C₂²	—	2	8	no	—
301	C₂²	—	4	8	no	—
302	C₂²	—	4	16	no	—
303	C₂²	—	4	32	no	—
304	C₂²	—	8	32	no	—
305	C₂²	C₂×C₃×#124	1	4	yes	—
306	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
307	C₂²	C₂²×C₃×#41	1	2	yes	—
308	C₂²	C₂×C₃×H(5,𝔽₂)	1	4	yes	—
309	C₄	—	2	4	no	—
310	C₄	—	4	8	no	—
311	C₄	—	4	8	no	—
312	C₄	C₂×C₃×#124	1	4	yes	—
313	C₄	C₂×C₃×#124	1	4	yes	—
314	C₄	C₃×#535	1	8	yes	—
315	C₃	#35285	1	1	yes	characteristic, direct product factor, Sylow
316	C₂	—	2	4	no	—
317	C₂	—	8	8	no	—
318	C₂	C₂²×C₃×#124	1	1	yes	characteristic
319	C₂	C₂×C₃×#535	1	2	yes	—
320	C₂	C₃×#3508	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃
prime radical	C₂²×C₃×#127	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#533	C₂	semidirect
3. C₃×#2858	C₂	semidirect
4. C₃×#2641	C₂	semidirect
5. C₃×#2548	C₂	semidirect
6. C₃×#2435	C₂	semidirect
7. C₃×#2309	C₂	semidirect
8. C₃×C₄×#117	C₂	semidirect
9. #35285	C₃	direct (N × H)
13. C₂×C₃×C₄⋊Q₈	C₂²	semidirect
18. C₂×C₃×#125	C₂²	semidirect
19. C₂×C₃×#124	C₂²	semidirect
21. C₂×C₃×C₄×Q₈	C₂²	semidirect
25. C₃×#394	C₂²	semidirect
26. C₃×#390	C₂²	semidirect
27. C₃×#390	C₂²	semidirect
28. C₃×#390	C₂²	semidirect
30. C₃×C₄×C₄⋊C₄	C₂²	semidirect
32. C₃×#376	C₂²	semidirect
33. C₃×C₄³	C₂²	semidirect
34. C₂×#533	C₂×C₃	semidirect
36. #2858	C₂×C₃	semidirect
37. #2641	C₂×C₃	semidirect
38. #2548	C₂×C₃	semidirect
39. #2435	C₂×C₃	semidirect
40. #2309	C₂×C₃	semidirect
41. C₄×#117	C₂×C₃	semidirect
82. C₂×C₄⋊Q₈	C₂²×C₃	semidirect
87. C₂×#125	C₂²×C₃	semidirect
88. C₂×#124	C₂²×C₃	semidirect
90. C₂×C₄×Q₈	C₂²×C₃	semidirect
94. #394	C₂²×C₃	semidirect
95. #390	C₂²×C₃	semidirect
96. #390	C₂²×C₃	semidirect
97. #390	C₂²×C₃	semidirect
99. C₄×C₄⋊C₄	C₂²×C₃	semidirect
101. #376	C₂²×C₃	semidirect
102. C₄³	C₂²×C₃	semidirect
201. C₂×C₃×C₄	#124	semidirect
286. C₂×C₄	C₃×#124	semidirect
315. C₃	#35285	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#533	2
C₂²×C₃×#127	2
C₃×#2858	2
C₃×#2641	2
C₃×#2548	2
C₃×#2435	2
C₃×#2309	2
C₃×C₄×#117	2
#35285	3

Subgroups — C3×#35285 #10653892

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#35285 #10653892