Subgroups — C₂×#71425 #10947844

Order 768.

738 conjugacy classes of subgroups.

The subgroup lattice has 312 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	C₂×#5101	C₂	1	1	yes	maximal, characteristic
3	C₂²×#812	C₂	1	1	yes	maximal, characteristic
4	#71425	C₂	1	8	yes	maximal, direct product factor
5	C₂×#4512	C₂	1	4	yes	maximal, complemented
6	C₂×#2110	—	3	1	no	maximal, Sylow
7	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
8	C₂×#926	C₄	1	1	yes	characteristic
9	C₂²×#208	C₄	1	1	yes	characteristic
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	#5101	C₄	1	4	yes	—
16	#5101	C₂²	1	4	yes	—
17	C₂×#815	C₂²	1	2	yes	—
18	C₂×C₈×Dic₃	C₄	1	2	yes	—
19	C₂×#812	C₄	1	2	yes	complemented
20	C₂×#812	C₄	1	4	yes	complemented
21	C₂×#812	C₂²	1	2	yes	—
22	C₂×#812	C₂²	1	4	yes	—
23	C₂²×C₃⋊C₁₆	C₄	1	2	yes	complemented
24	#4512	C₂²	1	16	yes	complemented
25	C₂²×M₃₂	—	3	1	no	—
26	C₂²×M₃₂	D₃	1	1	yes	characteristic
27	C₂×#405	—	3	1	no	—
28	#2110	—	3	8	no	—
29	C₂×#347	—	3	4	no	—
30	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
33	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
36	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
39	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
40	C₂²×C₃×C₈	C₂³	1	2	yes	—
41	C₂×#217	—	2	1	no	—
42	#926	—	2	2	no	—
43	#926	C₂×C₄	1	4	yes	—
44	C₂×C₄⋊Dic₃	—	2	1	no	—
45	C₂×C₄×Dic₃	C₂×C₄	1	2	yes	—
46	C₂×#208	—	2	1	no	—
47	C₂×#208	—	2	1	no	—
48	C₂×#208	—	2	2	no	—
49	C₂×#208	C₂×C₄	1	4	yes	—
50	C₂²×C₃⋊C₈	C₂×C₄	1	2	yes	—
51	C₃×M₃₂	C₂×C₄	1	4	yes	—
52	C₃×M₃₂	C₂×C₄	1	8	yes	complemented
53	C₃×M₃₂	C₂³	1	4	yes	—
54	C₂×C₃×C₁₆	C₂×C₄	1	4	yes	—
55	C₂×C₃×C₁₆	C₂×C₄	1	8	yes	complemented
56	#815	C₂×C₄	1	4	yes	—
57	#815	C₂³	1	4	yes	—
58	C₈×Dic₃	C₂×C₄	1	8	yes	—
59	#812	C₂×C₄	1	4	yes	—
60	#812	C₂×C₄	1	8	yes	complemented
61	#812	C₂³	1	4	yes	—
62	C₂×C₃⋊C₁₆	C₂×C₄	1	4	yes	—
63	C₂×C₃⋊C₁₆	C₂×C₄	1	8	yes	complemented
64	C₂²×M₁₆	—	3	1	no	—
65	C₂³×C₈	C₂×D₃	1	1	yes	characteristic
66	C₂×#117	—	3	1	no	—
67	C₂×M₃₂	—	3	2	no	—
68	C₂×M₃₂	—	3	4	no	—
69	C₂×M₃₂	—	3	2	no	—
70	C₂×M₃₂	—	3	4	no	—
71	C₂×M₃₂	Dic₃	1	2	yes	complemented
72	C₂×M₃₂	Dic₃	1	4	yes	complemented
73	C₂×M₃₂	C₂×D₃	1	2	yes	—
74	C₂×M₃₂	C₂×D₃	1	4	yes	—
75	C₂²×C₁₆	—	3	2	no	—
76	C₂²×C₁₆	Dic₃	1	2	yes	complemented
77	#405	—	3	4	no	—
78	#405	—	3	4	no	—
79	C₂×#97	—	3	2	no	—
80	C₂×C₄×C₈	—	3	2	no	—
81	#347	—	3	16	no	—
82	C₂⁴×C₃	M₁₆	1	1	yes	characteristic
83	C₂²×C₃×C₄	C₄²	1	1	yes	characteristic
84	C₂²×C₃×C₄	C₄²	1	1	yes	characteristic
85	C₂²×C₃×C₄	C₄²	1	2	yes	—
86	C₂²×C₃×C₄	M₁₆	1	1	yes	characteristic
87	C₂²×C₃×C₄	M₁₆	1	2	yes	—
88	C₂²×C₃×C₄	M₁₆	1	1	yes	characteristic
89	C₂²×C₃×C₄	M₁₆	1	1	yes	characteristic
90	C₂²×C₃×C₄	M₁₆	1	2	yes	—
91	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
93	C₂²×Dic₃	—	2	2	no	—
94	C₂×C₃×C₈	C₄²	1	1	yes	characteristic
95	C₂×C₃×C₈	C₄²	1	2	yes	—
96	C₂×C₃×C₈	C₄²	1	2	yes	—
97	C₂×C₃×C₈	C₄²	1	2	yes	—
98	C₂×C₃×C₈	C₄²	1	2	yes	—
99	C₂×C₃×C₈	C₄²	1	2	yes	—
100	C₂×C₃×C₈	C₄²	1	1	yes	characteristic
101	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
102	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
105	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
107	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
109	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
111	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
112	#217	—	2	4	no	—
113	C₄⋊Dic₃	—	2	4	no	—
114	C₄×Dic₃	—	2	2	no	—
115	C₄×Dic₃	C₂²×C₄	1	4	yes	—
116	#208	—	2	4	no	—
117	#208	—	2	4	no	—
118	C₂×C₃⋊C₈	—	2	2	no	—
119	C₂×C₃⋊C₈	—	2	2	no	—
120	C₂×C₃⋊C₈	C₂²×C₄	1	4	yes	—
121	C₃×C₁₆	C₂²×C₄	1	8	yes	—
122	C₃⋊C₁₆	C₂²×C₄	1	8	yes	—
123	C₂³×C₄	C₄×D₃	1	1	yes	characteristic
124	C₂×M₁₆	—	3	4	no	—
125	C₂×M₁₆	—	6	1	no	—
126	C₂×M₁₆	—	6	1	no	—
127	C₂×M₁₆	—	6	2	no	—
128	C₂²×C₈	—	3	2	no	—
129	C₂²×C₈	C₄×D₃	1	2	yes	—
130	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
131	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
132	C₂²×C₈	C₄×D₃	1	2	yes	—
133	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
134	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
135	C₂²×C₈	C₂×Dic₃	1	2	yes	—
136	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
137	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
138	C₂²×C₈	C₂²×D₃	1	2	yes	—
139	#117	—	3	4	no	—
140	#117	—	6	2	no	—
141	C₂×C₄⋊C₄	—	6	1	no	—
142	C₂×C₂²⋊C₄	—	6	1	no	—
143	C₂×C₄²	—	3	2	no	—
144	M₃₂	—	3	4	no	—
145	M₃₂	—	3	8	no	—
146	M₃₂	—	3	4	no	—
147	M₃₂	C₂×Dic₃	1	4	yes	—
148	M₃₂	C₂×Dic₃	1	8	yes	complemented
149	M₃₂	C₂²×D₃	1	4	yes	—
150	C₂×C₁₆	—	3	4	no	—
151	C₂×C₁₆	—	3	8	no	—
152	C₂×C₁₆	C₂×Dic₃	1	4	yes	—
153	C₂×C₁₆	C₂×Dic₃	1	8	yes	complemented
154	#97	—	3	4	no	—
155	#97	—	3	4	no	—
156	C₄×C₈	—	3	8	no	—
157	C₂³×C₃	—	4	2	no	—
158	C₂³×C₃	#97	1	1	yes	characteristic
159	C₂³×C₃	#97	1	1	yes	characteristic
160	C₂³×C₃	#97	1	2	yes	—
161	C₂³×C₃	C₂×M₁₆	1	1	yes	characteristic
162	C₂³×C₃	C₂×M₁₆	1	2	yes	—
163	C₂×C₃×C₄	#97	1	2	yes	—
164	C₂×C₃×C₄	#97	1	2	yes	—
165	C₂×C₃×C₄	#97	1	2	yes	—
166	C₂×C₃×C₄	#97	1	2	yes	—
167	C₂×C₃×C₄	#97	1	2	yes	—
168	C₂×C₃×C₄	#97	1	1	yes	characteristic
169	C₂×C₃×C₄	#97	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂×C₄²	1	2	yes	—
171	C₂×C₃×C₄	C₂×C₄²	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂×C₄²	1	2	yes	—
173	C₂×C₃×C₄	C₂×C₄²	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
177	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
178	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
180	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
181	C₂×Dic₃	—	2	4	no	—
182	C₂×Dic₃	—	4	4	no	—
183	C₃×C₈	C₂×C₄²	1	2	yes	—
184	C₃×C₈	C₂×C₄²	1	1	yes	characteristic
185	C₃×C₈	C₂×C₄²	1	1	yes	characteristic
186	C₃×C₈	C₂×C₄²	1	2	yes	—
187	C₃×C₈	C₂³×C₄	1	1	yes	characteristic
188	C₃×C₈	C₂³×C₄	1	1	yes	characteristic
189	C₃⋊C₈	—	2	4	no	—
190	C₂⁴	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
191	C₂²×C₄	—	6	2	no	—
192	C₂²×C₄	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
193	C₂²×C₄	ΓL(1,𝔽₂₅)	1	2	yes	—
194	C₂²×C₄	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
195	C₂²×C₄	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
196	C₂²×C₄	ΓL(1,𝔽₂₅)	1	2	yes	—
197	C₂²×C₄	C₄×Dic₃	1	2	yes	—
198	C₂²×C₄	C₄×Dic₃	1	1	yes	characteristic
199	C₂²×C₄	C₄×Dic₃	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₄×D₃	1	2	yes	—
202	M₁₆	—	6	4	no	—
203	M₁₆	—	6	4	no	—
204	C₂×C₈	—	3	4	no	—
205	C₂×C₈	—	6	2	no	—
206	C₂×C₈	—	6	2	no	—
207	C₂×C₈	C₄×Dic₃	1	2	yes	—
208	C₂×C₈	C₄×Dic₃	1	2	yes	—
209	C₂×C₈	C₄×Dic₃	1	1	yes	characteristic
210	C₂×C₈	C₄×Dic₃	1	2	yes	—
211	C₂×C₈	C₄×Dic₃	1	2	yes	—
212	C₂×C₈	C₄×Dic₃	1	2	yes	—
213	C₂×C₈	C₄×Dic₃	1	1	yes	characteristic
214	C₂×C₈	C₂×C₄×D₃	1	2	yes	—
215	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
216	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
217	C₂×C₈	C₂×C₄×D₃	1	2	yes	—
218	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
219	C₂×C₈	C₂×C₄×D₃	1	2	yes	—
220	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
221	C₂×C₈	C₂²×Dic₃	1	2	yes	—
222	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
223	C₂×C₈	C₂²×Dic₃	1	2	yes	—
224	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
225	C₄⋊C₄	—	6	4	no	—
226	C₂²⋊C₄	—	6	4	no	—
227	C₄²	—	3	4	no	—
228	C₄²	—	6	2	no	—
229	C₁₆	—	3	8	no	—
230	C₁₆	C₂²×Dic₃	1	8	yes	—
231	C₂²×C₃	—	2	2	no	—
232	C₂²×C₃	—	2	2	no	—
233	C₂²×C₃	—	2	2	no	—
234	C₂²×C₃	—	4	2	no	—
235	C₂²×C₃	—	4	2	no	—
236	C₂²×C₃	C₂×#97	1	1	yes	characteristic
237	C₂²×C₃	C₂×#97	1	2	yes	—
238	C₂²×C₃	C₂×#97	1	2	yes	—
239	C₂²×C₃	C₂×#97	1	1	yes	characteristic
240	C₂²×C₃	C₂²×M₁₆	1	1	yes	characteristic
241	C₃×C₄	C₂×#97	1	1	yes	characteristic
242	C₃×C₄	C₂×#97	1	1	yes	characteristic
243	C₃×C₄	C₂×#97	1	2	yes	—
244	C₃×C₄	C₂×#97	1	2	yes	—
245	C₃×C₄	C₂²×C₄²	1	1	yes	characteristic
246	C₃×C₄	C₂²×M₁₆	1	1	yes	characteristic
247	Dic₃	—	4	4	no	—
248	C₂³	—	4	2	no	—
249	C₂³	#815	1	1	yes	characteristic
250	C₂³	#815	1	1	yes	characteristic
251	C₂³	#815	1	2	yes	—
252	C₂³	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
253	C₂³	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
254	C₂×C₄	—	6	4	no	—
255	C₂×C₄	—	12	4	no	—
256	C₂×C₄	#815	1	2	yes	—
257	C₂×C₄	#815	1	2	yes	—
258	C₂×C₄	#815	1	2	yes	—
259	C₂×C₄	#815	1	2	yes	—
260	C₂×C₄	#815	1	2	yes	—
261	C₂×C₄	#815	1	1	yes	characteristic
262	C₂×C₄	#815	1	1	yes	characteristic
263	C₂×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
264	C₂×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
265	C₂×C₄	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
266	C₂×C₄	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
267	C₂×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
268	C₂×C₄	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
269	C₂×C₄	C₂×C₄×Dic₃	1	2	yes	—
270	C₂×C₄	C₂×C₄×Dic₃	1	2	yes	—
271	C₂×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
272	C₂×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
273	C₂×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
274	C₈	—	6	4	no	—
275	C₈	C₂×C₄×Dic₃	1	2	yes	—
276	C₈	C₂×C₄×Dic₃	1	1	yes	characteristic
277	C₈	C₂×C₄×Dic₃	1	1	yes	characteristic
278	C₈	C₂×C₄×Dic₃	1	2	yes	—
279	C₈	C₂²×C₄×D₃	1	1	yes	characteristic
280	C₈	C₂³×Dic₃	1	1	yes	characteristic
281	C₂×C₃	—	2	1	no	—
282	C₂×C₃	—	2	1	no	—
283	C₂×C₃	—	2	2	no	—
284	C₂×C₃	—	2	2	no	—
285	C₂×C₃	#2110	1	2	yes	complemented
286	C₂×C₃	C₂²×#97	1	1	yes	characteristic
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	4	2	no	—
291	C₂²	—	4	2	no	—
292	C₂²	C₂×#815	1	1	yes	characteristic
293	C₂²	C₂×#815	1	2	yes	—
294	C₂²	C₂×#815	1	2	yes	—
295	C₂²	C₂×#815	1	1	yes	characteristic
296	C₂²	C₂²×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
297	C₄	—	12	4	no	—
298	C₄	C₂×#815	1	1	yes	characteristic
299	C₄	C₂×#815	1	1	yes	characteristic
300	C₄	C₂×#815	1	2	yes	—
301	C₄	C₂×#815	1	2	yes	—
302	C₄	C₂²×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
303	C₄	C₂²×C₄×Dic₃	1	1	yes	characteristic
304	C₃	C₂×#2110	1	1	yes	characteristic, complemented, Sylow
305	C₂	—	2	1	no	—
306	C₂	—	2	1	no	—
307	C₂	—	2	2	no	—
308	C₂	—	2	2	no	—
309	C₂	#71425	1	2	yes	direct product factor
310	C₂	C₂²×#815	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄×Dic₃
commutator	C₃×C₄	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₃
Fitting	C₂²×C₃×M₃₂	C₂
socle	C₂²×C₃	C₂×#97
prime radical	C₂⁴×C₃	M₁₆
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	C₂²×M₃₂	D₃
hypocenter	C₃	C₂×#2110
LCS	G ▷ C₃×C₄ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
4. #71425	C₂	direct (N × H)
5. C₂×#4512	C₂	semidirect
10. C₂×C₃×M₃₂	C₄	semidirect
11. C₂×C₃×M₃₂	C₄	semidirect
14. C₂²×C₃×C₁₆	C₄	semidirect
19. C₂×#812	C₄	semidirect
20. C₂×#812	C₄	semidirect
23. C₂²×C₃⋊C₁₆	C₄	semidirect
24. #4512	C₂²	semidirect
52. C₃×M₃₂	C₂×C₄	semidirect
55. C₂×C₃×C₁₆	C₂×C₄	semidirect
60. #812	C₂×C₄	semidirect
63. C₂×C₃⋊C₁₆	C₂×C₄	semidirect
71. C₂×M₃₂	Dic₃	semidirect
72. C₂×M₃₂	Dic₃	semidirect
76. C₂²×C₁₆	Dic₃	semidirect
148. M₃₂	C₂×Dic₃	semidirect
153. C₂×C₁₆	C₂×Dic₃	semidirect
285. C₂×C₃	#2110	semidirect
304. C₃	C₂×#2110	semidirect
309. C₂	#71425	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
C₂×#5101	2
C₂²×#812	2
#71425	2
C₂×#4512	2
C₂×#2110	3

Subgroups — C2×#71425 #10947844

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#71425 #10947844