Subgroups — #10712942

Order 768.

332 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#4508	C₂	1	1	yes	maximal, characteristic
3	#71409	C₂	1	1	yes	maximal, characteristic
4	#71401	C₂	1	1	yes	maximal, characteristic, complemented
5	#69789	C₂	1	1	yes	maximal, characteristic
6	#67805	C₂	1	1	yes	maximal, characteristic, complemented
7	#67658	C₂	1	1	yes	maximal, characteristic, complemented
8	#13124	—	3	1	no	maximal, Sylow
9	C₂×#901	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	#5113	C₄	1	1	yes	characteristic
14	#5101	C₂²	1	1	yes	characteristic
15	C₂×#812	C₂²	1	1	yes	characteristic
16	C₂×#812	C₂²	1	1	yes	characteristic
17	C₂×#805	C₄	1	1	yes	characteristic
18	C₃×C₄×C₁₆	—	2	2	no	—
19	#4552	C₄	1	1	yes	characteristic
20	#4523	C₄	1	1	yes	characteristic
21	#4508	—	2	2	no	—
22	(C₂×D₃)⋊C₁₆	—	2	2	no	—
23	Dic₃⋊C₁₆	—	2	2	no	—
24	#2151	—	3	1	no	—
25	C₂×#350	—	3	1	no	—
26	#2113	—	3	1	no	—
27	C₂×C₄×C₁₆	D₃	1	1	yes	characteristic, complemented
28	#1764	—	3	1	no	—
29	#1379	—	3	1	no	—
30	#1376	—	3	1	no	—
31	C₂×#235	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
34	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
35	#926	C₂×C₄	1	1	yes	characteristic
36	C₂×#208	C₂×C₄	1	1	yes	characteristic
37	C₂×#208	C₂×C₄	1	1	yes	characteristic
38	#901	—	2	1	no	—
39	#901	—	2	1	no	—
40	#901	—	2	1	no	—
41	#901	—	2	1	no	—
42	C₂×ΓL(1,𝔽₂₅)	—	2	1	no	—
43	C₂×D₃×C₈	—	2	1	no	—
44	C₂×C₃×C₁₆	—	2	1	no	—
45	C₂×C₃×C₁₆	—	2	1	no	—
46	C₂×C₃×C₁₆	—	2	1	no	—
47	C₂×C₃×C₁₆	—	2	1	no	—
48	C₂×C₃×C₁₆	D₄	1	2	yes	—
49	C₂×C₃×C₁₆	D₄	1	2	yes	—
50	C₃×C₄×C₈	—	2	1	no	—
51	C₃×C₄×C₈	—	2	1	no	—
52	#821	C₈	1	2	yes	—
53	#816	C₈	1	2	yes	—
54	#815	—	2	1	no	—
55	C₈×Dic₃	—	2	1	no	—
56	#812	—	2	1	no	—
57	#812	—	2	1	no	—
58	#812	—	2	1	no	—
59	#812	—	2	1	no	—
60	C₂×C₃⋊C₁₆	—	2	1	no	—
61	C₂×C₃⋊C₁₆	—	2	1	no	—
62	#805	—	2	2	no	—
63	C₂×#131	—	3	1	no	—
64	C₂×M₃₂	—	3	1	no	—
65	C₂×M₃₂	—	3	1	no	—
66	C₂²×C₁₆	C₂×D₃	1	1	yes	characteristic
67	C₂²×C₁₆	C₂×D₃	1	1	yes	characteristic
68	C₂×#104	—	3	1	no	—
69	#416	—	3	1	no	—
70	#405	—	3	1	no	—
71	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
72	C₄⋊C₁₆	—	6	2	no	—
73	#350	—	6	2	no	—
74	C₂²⋊C₁₆	—	6	2	no	—
75	C₄×C₁₆	—	2	2	no	—
76	#341	—	3	1	no	—
77	#339	—	3	1	no	—
78	C₂²×C₃×C₄	—	2	1	no	—
79	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
80	C₂×#67	—	2	1	no	—
81	#235	—	2	1	no	—
82	#235	—	2	1	no	—
83	#235	—	2	1	no	—
84	#235	—	2	1	no	—
85	C₂×D₁₂	C₂×C₈	1	1	yes	characteristic
86	C₂×C₄×D₃	—	2	1	no	—
87	C₂×Dic₆	C₂×C₈	1	1	yes	characteristic
88	C₂×C₃×C₈	—	2	1	no	—
89	C₂×C₃×C₈	—	2	1	no	—
90	C₂×C₃×C₈	C₂×C₈	1	2	yes	—
91	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
92	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
93	C₂×C₃×C₈	#41	1	1	yes	characteristic
94	C₂×C₃×C₈	#41	1	1	yes	characteristic
95	C₂×C₃×C₈	#41	1	1	yes	characteristic
96	C₂×C₃×C₈	#41	1	1	yes	characteristic
97	C₃×C₄²	—	2	1	no	—
98	C₃×C₄²	M₁₆	1	2	yes	—
99	#217	—	2	1	no	—
100	C₄⋊Dic₃	C₂×C₈	1	2	yes	—
101	C₄×Dic₃	—	2	1	no	—
102	#208	—	2	1	no	—
103	#208	—	2	1	no	—
104	#208	—	2	1	no	—
105	#208	—	2	1	no	—
106	C₂×C₃⋊C₈	—	2	1	no	—
107	C₂×C₃⋊C₈	—	2	1	no	—
108	ΓL(1,𝔽₂₅)	—	4	1	no	—
109	ΓL(1,𝔽₂₅)	—	4	1	no	—
110	D₃×C₈	—	4	1	no	—
111	D₃×C₈	—	4	1	no	—
112	C₃×C₁₆	—	2	2	no	—
113	C₃×C₁₆	—	2	2	no	—
114	C₃⋊C₁₆	—	4	1	no	—
115	C₃⋊C₁₆	—	4	1	no	—
116	C₂×#41	—	3	1	no	—
117	#131	—	6	1	no	—
118	#131	—	6	1	no	—
119	#131	—	6	1	no	—
120	#131	—	6	1	no	—
121	C₂×M₁₆	—	3	1	no	—
122	C₂×M₁₆	—	3	1	no	—
123	C₂×M₁₆	—	6	1	no	—
124	C₂²×C₈	—	6	1	no	—
125	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
127	#117	—	3	1	no	—
128	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
129	M₃₂	—	6	1	no	—
130	M₃₂	—	6	1	no	—
131	M₃₂	—	6	1	no	—
132	M₃₂	—	6	1	no	—
133	C₂×C₁₆	—	2	1	no	—
134	C₂×C₁₆	—	2	1	no	—
135	C₂×C₁₆	—	2	1	no	—
136	C₂×C₁₆	—	2	1	no	—
137	C₂×C₁₆	—	6	1	no	—
138	C₂×C₁₆	—	6	1	no	—
139	C₂×C₁₆	D₁₂	1	2	yes	—
140	C₂×C₁₆	#67	1	2	yes	—
141	#104	—	6	2	no	—
142	#103	—	3	2	no	—
143	#102	—	3	2	no	—
144	#97	—	6	1	no	—
145	C₄×C₈	—	2	1	no	—
146	C₄×C₈	—	2	1	no	—
147	C₄×C₈	—	6	1	no	—
148	C₂³×C₃	#131	1	1	yes	characteristic
149	C₂²×D₃	—	2	1	no	—
150	C₂×C₃×C₄	—	2	4	no	—
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	#117	1	1	yes	characteristic
153	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
154	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
157	C₂×C₃×C₄	#131	1	1	yes	characteristic
158	#67	—	4	1	no	—
159	#67	—	4	1	no	—
160	C₂×Dic₃	—	2	2	no	—
161	C₂×Dic₃	—	2	1	no	—
162	D₁₂	—	2	1	no	—
163	D₁₂	—	2	1	no	—
164	C₄×D₃	—	4	1	no	—
165	C₄×D₃	—	4	1	no	—
166	Dic₆	—	2	1	no	—
167	Dic₆	—	2	1	no	—
168	C₃×C₈	—	2	2	no	—
169	C₃×C₈	#121	1	1	yes	characteristic
170	C₃×C₈	#123	1	1	yes	characteristic
171	C₃×C₈	#124	1	1	yes	characteristic
172	C₃×C₈	#126	1	1	yes	characteristic
173	C₃⋊C₈	—	4	1	no	—
174	C₃⋊C₈	—	4	1	no	—
175	#41	—	6	1	no	—
176	#41	—	6	1	no	—
177	#41	—	6	1	no	—
178	#41	—	6	1	no	—
179	C₂×Q₈	—	3	1	no	—
180	C₂×D₄	—	3	1	no	—
181	C₂×D₄	—	6	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	6	1	no	—
184	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
185	M₁₆	—	6	1	no	—
186	M₁₆	—	6	1	no	—
187	M₁₆	—	6	1	no	—
188	M₁₆	—	6	1	no	—
189	M₁₆	—	12	1	no	—
190	M₁₆	—	12	1	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	6	1	no	—
194	C₂×C₈	—	6	1	no	—
195	C₂×C₈	—	12	1	no	—
196	C₂×C₈	—	12	1	no	—
197	C₂×C₈	D₃×C₈	1	2	yes	—
198	C₂×C₈	C₂×D₁₂	1	1	yes	characteristic
199	C₂×C₈	#235	1	1	yes	characteristic
200	C₂×C₈	#235	1	1	yes	characteristic
201	C₂×C₈	#235	1	1	yes	characteristic
202	C₂×C₈	#235	1	1	yes	characteristic
203	C₂×C₈	C₂×#67	1	1	yes	characteristic
204	C₄⋊C₄	—	3	2	no	—
205	C₂²⋊C₄	—	6	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	6	1	no	—
208	C₄²	ΓL(1,𝔽₂₅)	1	2	yes	—
209	C₁₆	—	2	2	no	—
210	C₁₆	—	2	2	no	—
211	C₁₆	—	12	1	no	—
212	C₁₆	—	12	1	no	—
213	C₂²×C₃	—	2	2	no	—
214	C₂²×C₃	#433	1	1	yes	characteristic
215	C₂²×C₃	C₈×D₄	1	1	yes	characteristic
216	C₂²×C₃	#435	1	1	yes	characteristic
217	C₂×D₃	—	4	2	no	—
218	C₂×D₃	—	4	1	no	—
219	C₃×C₄	—	2	4	no	—
220	C₃×C₄	#388	1	1	yes	characteristic
221	C₃×C₄	#431	1	1	yes	characteristic
222	C₃×C₄	C₈×D₄	1	1	yes	characteristic
223	C₃×C₄	#436	1	1	yes	characteristic
224	Dic₃	—	4	2	no	—
225	Dic₃	—	4	1	no	—
226	C₂³	—	6	1	no	—
227	C₂³	#901	1	1	yes	characteristic
228	Q₈	—	6	1	no	—
229	Q₈	—	6	1	no	—
230	D₄	—	6	1	no	—
231	D₄	—	6	1	no	—
232	D₄	—	12	1	no	—
233	D₄	—	12	1	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	6	2	no	—
237	C₂×C₄	—	6	1	no	—
238	C₂×C₄	—	12	1	no	—
239	C₂×C₄	—	12	1	no	—
240	C₂×C₄	#872	1	1	yes	characteristic
241	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
242	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
243	C₂×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
244	C₂×C₄	#901	1	1	yes	characteristic
245	C₂×C₄	C₄×#67	1	1	yes	characteristic
246	C₈	—	2	2	no	—
247	C₈	—	12	1	no	—
248	C₈	—	12	1	no	—
249	C₈	#875	1	1	yes	characteristic
250	C₈	#876	1	1	yes	characteristic
251	C₈	#929	1	1	yes	characteristic
252	C₈	#930	1	1	yes	characteristic
253	C₂×C₃	—	2	1	no	—
254	C₂×C₃	—	2	1	no	—
255	C₂×C₃	#1923	1	1	yes	characteristic
256	C₂×C₃	#2170	1	2	yes	—
257	D₃	—	8	1	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	12	1	no	—
260	C₂²	—	12	2	no	—
261	C₂²	C₈×D₁₂	1	1	yes	characteristic
262	C₂²	#4690	1	1	yes	characteristic
263	C₂²	#5111	1	1	yes	characteristic
264	C₄	—	2	4	no	—
265	C₄	—	12	1	no	—
266	C₄	—	12	2	no	—
267	C₄	#4685	1	1	yes	characteristic
268	C₄	#4688	1	1	yes	characteristic
269	C₄	#4940	1	1	yes	characteristic
270	C₄	C₈×#67	1	1	yes	characteristic
271	C₃	#13124	1	1	yes	characteristic, complemented, Sylow
272	C₂	—	2	1	no	—
273	C₂	—	2	1	no	—
274	C₂	—	24	1	no	—
275	C₂	#69056	1	2	yes	—
276	C₂	#69785	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₂×C₃×C₄×C₁₆	C₂
socle	C₂²×C₃	C₈×D₄
prime radical	C₂×#235	C₂×C₄
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	C₂×C₄×C₁₆	D₃
hypocenter	C₃	#13124
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₄×C₁₆	C₂	semidirect
4. #71401	C₂	semidirect
6. #67805	C₂	semidirect
7. #67658	C₂	semidirect
27. C₂×C₄×C₁₆	D₃	semidirect
271. C₃	#13124	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₁₆	2
C₂×#4508	2
#71409	2
#71401	2
#69789	2
#67805	2
#67658	2
#13124	3