Subgroups — C₂×C₃×#2155 #10652010

Order 768.

658 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×#108	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#427	C₂	1	1	yes	maximal, characteristic
4	C₃×#2155	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×#365	C₂	1	4	yes	maximal, complemented
6	C₂×#2155	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂²×C₃×M₁₆	—	2	1	no	—
8	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×#117	—	2	1	no	—
10	C₂×C₃×M₃₂	C₄	1	2	yes	complemented
11	C₂×C₃×M₃₂	C₄	1	4	yes	complemented
12	C₂×C₃×M₃₂	C₂²	1	2	yes	—
13	C₂×C₃×M₃₂	C₂²	1	4	yes	—
14	C₂²×C₃×C₁₆	C₄	1	2	yes	complemented
15	C₂×C₃×#108	—	2	1	no	—
16	C₂×C₃×#108	—	2	2	no	—
17	C₂×C₃×#108	C₂²	1	2	yes	—
18	C₂×C₃×#108	C₂²	1	4	yes	—
19	C₃×#427	—	2	2	no	—
20	C₃×#427	C₂²	1	4	yes	—
21	C₂×C₃×#107	—	2	1	no	—
22	C₂×C₃×#106	C₂²	1	2	yes	—
23	C₃×#365	C₂²	1	16	yes	complemented
24	C₂²×M₃₂	C₂×C₃	1	1	yes	characteristic
25	C₂²×#108	C₂×C₃	1	1	yes	characteristic
26	C₂×#427	C₂×C₃	1	1	yes	characteristic
27	#2155	C₂×C₃	1	8	yes	direct product factor
28	C₂×#365	C₂×C₃	1	4	yes	complemented
29	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
30	C₂×C₃×M₁₆	—	2	2	no	—
31	C₂×C₃×M₁₆	—	2	4	no	—
32	C₂×C₃×M₁₆	—	4	1	no	—
33	C₂×C₃×M₁₆	—	4	2	no	—
34	C₂²×C₃×C₈	—	4	1	no	—
35	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
38	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
39	C₂²×C₃×C₈	D₄	1	2	yes	—
40	C₂²×C₃×C₈	Q₈	1	1	yes	characteristic
41	C₂²×C₃×C₈	Q₈	1	2	yes	—
42	C₂²×C₃×C₈	Q₈	1	1	yes	characteristic
43	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
44	C₂²×C₃×C₈	C₂³	1	2	yes	—
45	C₃×#117	—	2	4	no	—
46	C₃×#117	—	4	2	no	—
47	C₂×C₃×C₄⋊C₄	—	2	2	no	—
48	C₂×C₃×C₂²⋊C₄	—	4	1	no	—
49	C₂×C₃×C₄²	—	4	1	no	—
50	C₃×M₃₂	C₂×C₄	1	4	yes	—
51	C₃×M₃₂	C₂×C₄	1	8	yes	complemented
52	C₃×M₃₂	C₂³	1	4	yes	—
53	C₂×C₃×C₁₆	C₂×C₄	1	4	yes	—
54	C₂×C₃×C₁₆	C₂×C₄	1	8	yes	complemented
55	C₃×#108	—	2	2	no	—
56	C₃×#108	—	2	4	no	—
57	C₃×#108	C₂³	1	4	yes	—
58	C₃×#107	—	2	4	no	—
59	C₃×#106	—	2	2	no	—
60	C₃×#106	C₂³	1	4	yes	—
61	C₂²×M₁₆	—	2	1	no	—
62	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
63	C₂×#117	—	2	1	no	—
64	C₂×M₃₂	C₃×C₄	1	2	yes	complemented
65	C₂×M₃₂	C₃×C₄	1	4	yes	complemented
66	C₂×M₃₂	C₂²×C₃	1	2	yes	—
67	C₂×M₃₂	C₂²×C₃	1	4	yes	—
68	C₂²×C₁₆	C₃×C₄	1	2	yes	complemented
69	C₂×#108	—	2	1	no	—
70	C₂×#108	—	2	2	no	—
71	C₂×#108	C₂²×C₃	1	2	yes	—
72	C₂×#108	C₂²×C₃	1	4	yes	—
73	#427	—	2	2	no	—
74	#427	C₂²×C₃	1	4	yes	—
75	C₂×#107	—	2	1	no	—
76	C₂×#106	C₂²×C₃	1	2	yes	—
77	#365	C₂²×C₃	1	16	yes	complemented
78	C₂⁴×C₃	SD₁₆	1	1	yes	characteristic
79	C₂²×C₃×C₄	—	4	2	no	—
80	C₂²×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₃×C₄	C₄⋊C₄	1	2	yes	—
82	C₂²×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₃×C₄	SD₁₆	1	2	yes	—
84	C₂²×C₃×C₄	SD₁₆	1	1	yes	characteristic
85	C₂²×C₃×C₄	SD₁₆	1	1	yes	characteristic
86	C₂²×C₃×C₄	SD₁₆	1	2	yes	—
87	C₂²×C₃×C₄	SD₁₆	1	1	yes	characteristic
88	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
90	C₃×M₁₆	—	2	4	no	—
91	C₃×M₁₆	—	4	2	no	—
92	C₃×M₁₆	—	4	4	no	—
93	C₂×C₃×C₈	—	4	2	no	—
94	C₂×C₃×C₈	—	4	4	no	—
95	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₃×C₈	C₄⋊C₄	1	2	yes	—
97	C₂×C₃×C₈	C₄⋊C₄	1	2	yes	—
98	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₃×C₈	C₄⋊C₄	1	2	yes	—
100	C₂×C₃×C₈	C₄⋊C₄	1	2	yes	—
101	C₂×C₃×C₈	C₄⋊C₄	1	2	yes	—
102	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
104	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
106	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
108	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
109	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
110	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
111	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
112	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
113	C₃×C₄⋊C₄	—	2	4	no	—
114	C₃×C₄⋊C₄	—	4	2	no	—
115	C₃×C₂²⋊C₄	—	4	4	no	—
116	C₃×C₄²	—	4	4	no	—
117	C₃×C₁₆	C₂²×C₄	1	8	yes	—
118	C₂³×C₄	C₃×D₄	1	1	yes	characteristic
119	C₂×M₁₆	—	2	2	no	—
120	C₂×M₁₆	—	2	4	no	—
121	C₂×M₁₆	—	4	1	no	—
122	C₂×M₁₆	—	4	2	no	—
123	C₂²×C₈	—	4	1	no	—
124	C₂²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
125	C₂²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
126	C₂²×C₈	C₂×C₃×C₄	1	2	yes	—
127	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
128	C₂²×C₈	C₃×D₄	1	2	yes	—
129	C₂²×C₈	C₃×Q₈	1	1	yes	characteristic
130	C₂²×C₈	C₃×Q₈	1	2	yes	—
131	C₂²×C₈	C₃×Q₈	1	1	yes	characteristic
132	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
133	C₂²×C₈	C₂³×C₃	1	2	yes	—
134	#117	—	2	4	no	—
135	#117	—	4	2	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₂²⋊C₄	—	4	1	no	—
138	C₂×C₄²	—	4	1	no	—
139	M₃₂	C₂×C₃×C₄	1	4	yes	—
140	M₃₂	C₂×C₃×C₄	1	8	yes	complemented
141	M₃₂	C₂³×C₃	1	4	yes	—
142	C₂×C₁₆	C₂×C₃×C₄	1	4	yes	—
143	C₂×C₁₆	C₂×C₃×C₄	1	8	yes	complemented
144	#108	—	2	2	no	—
145	#108	—	2	4	no	—
146	#108	C₂³×C₃	1	4	yes	—
147	#107	—	2	4	no	—
148	#106	—	2	2	no	—
149	#106	C₂³×C₃	1	4	yes	—
150	C₂³×C₃	—	4	2	no	—
151	C₂³×C₃	#106	1	1	yes	characteristic
152	C₂³×C₃	#106	1	1	yes	characteristic
153	C₂³×C₃	#106	1	2	yes	—
154	C₂³×C₃	C₂×SD₁₆	1	1	yes	characteristic
155	C₂³×C₃	C₂×SD₁₆	1	2	yes	—
156	C₂×C₃×C₄	—	4	4	no	—
157	C₂×C₃×C₄	—	8	4	no	—
158	C₂×C₃×C₄	#106	1	2	yes	—
159	C₂×C₃×C₄	#106	1	2	yes	—
160	C₂×C₃×C₄	#106	1	1	yes	characteristic
161	C₂×C₃×C₄	#106	1	2	yes	—
162	C₂×C₃×C₄	#106	1	2	yes	—
163	C₂×C₃×C₄	#106	1	2	yes	—
164	C₂×C₃×C₄	#106	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
166	C₂×C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
167	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
169	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂×SD₁₆	1	2	yes	—
171	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂×SD₁₆	1	2	yes	—
173	C₂×C₃×C₄	C₂×SD₁₆	1	2	yes	—
174	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
176	C₃×C₈	—	4	4	no	—
177	C₃×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
178	C₃×C₈	C₂×C₄⋊C₄	1	2	yes	—
179	C₃×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
180	C₃×C₈	C₂×C₄⋊C₄	1	2	yes	—
181	C₃×C₈	C₂³×C₄	1	1	yes	characteristic
182	C₃×C₈	C₂²×Q₈	1	1	yes	characteristic
183	C₂⁴	C₃×SD₁₆	1	1	yes	characteristic
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
186	C₂²×C₄	C₃×C₄⋊C₄	1	2	yes	—
187	C₂²×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
188	C₂²×C₄	C₃×SD₁₆	1	1	yes	characteristic
189	C₂²×C₄	C₃×SD₁₆	1	2	yes	—
190	C₂²×C₄	C₃×SD₁₆	1	1	yes	characteristic
191	C₂²×C₄	C₃×SD₁₆	1	1	yes	characteristic
192	C₂²×C₄	C₃×SD₁₆	1	2	yes	—
193	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
195	M₁₆	—	2	4	no	—
196	M₁₆	—	4	2	no	—
197	M₁₆	—	4	4	no	—
198	C₂×C₈	—	4	4	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	C₃×C₄⋊C₄	1	1	yes	characteristic
201	C₂×C₈	C₃×C₄⋊C₄	1	2	yes	—
202	C₂×C₈	C₃×C₄⋊C₄	1	2	yes	—
203	C₂×C₈	C₃×C₄⋊C₄	1	1	yes	characteristic
204	C₂×C₈	C₃×C₄⋊C₄	1	2	yes	—
205	C₂×C₈	C₃×C₄⋊C₄	1	2	yes	—
206	C₂×C₈	C₃×C₄⋊C₄	1	2	yes	—
207	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
208	C₂×C₈	C₂²×C₃×C₄	1	2	yes	—
209	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
210	C₂×C₈	C₂²×C₃×C₄	1	2	yes	—
211	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
212	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
213	C₂×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
214	C₂×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
215	C₂×C₈	C₂×C₃×Q₈	1	2	yes	—
216	C₂×C₈	C₂×C₃×Q₈	1	2	yes	—
217	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	4	2	no	—
220	C₂²⋊C₄	—	4	4	no	—
221	C₄²	—	4	4	no	—
222	C₁₆	C₂²×C₃×C₄	1	8	yes	—
223	C₂²×C₃	—	2	2	no	—
224	C₂²×C₃	—	2	2	no	—
225	C₂²×C₃	—	2	2	no	—
226	C₂²×C₃	—	4	2	no	—
227	C₂²×C₃	—	4	2	no	—
228	C₂²×C₃	C₂×#106	1	2	yes	—
229	C₂²×C₃	C₂×#106	1	2	yes	—
230	C₂²×C₃	C₂×#106	1	1	yes	characteristic
231	C₂²×C₃	C₂×#106	1	1	yes	characteristic
232	C₂²×C₃	C₂²×SD₁₆	1	1	yes	characteristic
233	C₃×C₄	—	8	4	no	—
234	C₃×C₄	C₂×#106	1	1	yes	characteristic
235	C₃×C₄	C₂×#106	1	2	yes	—
236	C₃×C₄	C₂×#106	1	1	yes	characteristic
237	C₃×C₄	C₂×#106	1	2	yes	—
238	C₃×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
239	C₃×C₄	C₂²×SD₁₆	1	1	yes	characteristic
240	C₂³	—	4	2	no	—
241	C₂³	C₃×#106	1	1	yes	characteristic
242	C₂³	C₃×#106	1	1	yes	characteristic
243	C₂³	C₃×#106	1	2	yes	—
244	C₂³	C₂×C₃×SD₁₆	1	1	yes	characteristic
245	C₂³	C₂×C₃×SD₁₆	1	2	yes	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	8	4	no	—
248	C₂×C₄	C₃×#106	1	2	yes	—
249	C₂×C₄	C₃×#106	1	2	yes	—
250	C₂×C₄	C₃×#106	1	2	yes	—
251	C₂×C₄	C₃×#106	1	2	yes	—
252	C₂×C₄	C₃×#106	1	2	yes	—
253	C₂×C₄	C₃×#106	1	1	yes	characteristic
254	C₂×C₄	C₃×#106	1	1	yes	characteristic
255	C₂×C₄	C₂×C₃×C₄⋊C₄	1	2	yes	—
256	C₂×C₄	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
257	C₂×C₄	C₂×C₃×C₄⋊C₄	1	2	yes	—
258	C₂×C₄	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
259	C₂×C₄	C₂×C₃×SD₁₆	1	2	yes	—
260	C₂×C₄	C₂×C₃×SD₁₆	1	1	yes	characteristic
261	C₂×C₄	C₂×C₃×SD₁₆	1	1	yes	characteristic
262	C₂×C₄	C₂×C₃×SD₁₆	1	2	yes	—
263	C₂×C₄	C₂×C₃×SD₁₆	1	2	yes	—
264	C₂×C₄	C₂×C₃×SD₁₆	1	1	yes	characteristic
265	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
266	C₈	—	4	4	no	—
267	C₈	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
268	C₈	C₂×C₃×C₄⋊C₄	1	2	yes	—
269	C₈	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
270	C₈	C₂×C₃×C₄⋊C₄	1	2	yes	—
271	C₈	C₂³×C₃×C₄	1	1	yes	characteristic
272	C₈	C₂²×C₃×Q₈	1	1	yes	characteristic
273	C₂×C₃	—	2	1	no	—
274	C₂×C₃	—	2	1	no	—
275	C₂×C₃	—	2	2	no	—
276	C₂×C₃	—	2	2	no	—
277	C₂×C₃	#2155	1	2	yes	direct product factor
278	C₂×C₃	C₂²×#106	1	1	yes	characteristic
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	C₂×C₃×#106	1	2	yes	—
285	C₂²	C₂×C₃×#106	1	2	yes	—
286	C₂²	C₂×C₃×#106	1	1	yes	characteristic
287	C₂²	C₂×C₃×#106	1	1	yes	characteristic
288	C₂²	C₂²×C₃×SD₁₆	1	1	yes	characteristic
289	C₄	—	8	4	no	—
290	C₄	C₂×C₃×#106	1	1	yes	characteristic
291	C₄	C₂×C₃×#106	1	2	yes	—
292	C₄	C₂×C₃×#106	1	1	yes	characteristic
293	C₄	C₂×C₃×#106	1	2	yes	—
294	C₄	C₂²×C₃×C₄⋊C₄	1	1	yes	characteristic
295	C₄	C₂²×C₃×SD₁₆	1	1	yes	characteristic
296	C₃	C₂×#2155	1	1	yes	characteristic, direct product factor, Sylow
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	2	no	—
300	C₂	—	2	2	no	—
301	C₂	C₃×#2155	1	2	yes	direct product factor
302	C₂	C₂²×C₃×#106	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	#106
commutator	C₈	C₂³×C₃×C₄
Frattini, cosocle	C₂×C₈	C₂⁴×C₃
socle	C₂²×C₃	C₂×#106
prime radical	C₂⁴×C₃	SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂³×C₃×C₄ ◁ C₂³×C₃×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃×#2155	C₂	direct (N × H)
5. C₂×C₃×#365	C₂	semidirect
6. C₂×#2155	C₃	direct (N × H)
10. C₂×C₃×M₃₂	C₄	semidirect
11. C₂×C₃×M₃₂	C₄	semidirect
14. C₂²×C₃×C₁₆	C₄	semidirect
23. C₃×#365	C₂²	semidirect
27. #2155	C₂×C₃	direct (N × H)
28. C₂×#365	C₂×C₃	semidirect
51. C₃×M₃₂	C₂×C₄	semidirect
54. C₂×C₃×C₁₆	C₂×C₄	semidirect
64. C₂×M₃₂	C₃×C₄	semidirect
65. C₂×M₃₂	C₃×C₄	semidirect
68. C₂²×C₁₆	C₃×C₄	semidirect
77. #365	C₂²×C₃	semidirect
140. M₃₂	C₂×C₃×C₄	semidirect
143. C₂×C₁₆	C₂×C₃×C₄	semidirect
277. C₂×C₃	#2155	direct (N × H)
296. C₃	C₂×#2155	direct (N × H)
301. C₂	C₃×#2155	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×M₃₂	2
C₂²×C₃×#108	2
C₂×C₃×#427	2
C₃×#2155	2
C₂×C₃×#365	2
C₂×#2155	3

Subgroups — C2×C3×#2155 #10652010

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#2155 #10652010