Subgroups — #10695321

Order 768.

450 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#73307	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#69394	C₂	1	1	yes	maximal, characteristic
4	#67702	C₂	1	2	yes	maximal
5	#67702	C₂	1	2	yes	maximal
6	#9900	—	3	1	no	maximal, Sylow
7	C₂×#870	—	2	1	no	—
8	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₃×C₄×C₈	—	2	1	no	—
10	C₃×C₄³	C₂²	1	1	yes	characteristic
11	#4936	—	2	1	no	—
12	#4935	C₂²	1	1	yes	characteristic
13	#4935	C₂²	1	1	yes	characteristic
14	#4928	—	2	1	no	—
15	#4928	—	2	1	no	—
16	#4928	C₄	1	2	yes	—
17	#4928	C₄	1	2	yes	—
18	C₂×C₄⋊(C₃⋊C₈)	—	2	1	no	—
19	C₂×C₄⋊(C₃⋊C₈)	—	2	1	no	—
20	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
21	C₃×#336	—	2	1	no	—
22	#2859	—	3	1	no	—
23	#1842	—	3	1	no	—
24	C₄×C₄⋊C₈	D₃	1	1	yes	characteristic
25	#1296	—	3	2	no	—
26	#1296	—	3	2	no	—
27	C₂²×C₃×C₈	—	2	2	no	—
28	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
31	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
32	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
33	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
34	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊Dic₃	—	2	1	no	—
36	C₂×C₄⋊Dic₃	—	2	1	no	—
37	C₂×C₄⋊Dic₃	—	2	1	no	—
38	C₂×C₄⋊Dic₃	—	2	1	no	—
39	C₂×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
41	C₂×Dic₃⋊C₄	—	2	2	no	—
42	C₂×Dic₃⋊C₄	—	2	2	no	—
43	C₂²×C₃⋊C₈	—	2	2	no	—
44	#870	—	4	2	no	—
45	#870	—	4	2	no	—
46	C₃×C₄⋊C₈	—	2	2	no	—
47	C₃×C₄⋊C₈	Q₈	1	4	yes	—
48	C₃×C₄×C₈	—	4	2	no	—
49	C₃⋊#95	—	2	2	no	—
50	C₃⋊#95	—	2	2	no	—
51	C₄⋊(C₃⋊C₈)	—	2	4	no	—
52	C₄⋊(C₃⋊C₈)	—	4	2	no	—
53	C₄⋊(C₃⋊C₈)	—	4	2	no	—
54	C₂×#125	—	6	1	no	—
55	C₂×C₄⋊C₈	—	3	2	no	—
56	C₂×C₄⋊C₈	—	6	1	no	—
57	C₂×C₄⋊C₈	—	6	1	no	—
58	C₂×C₄⋊C₈	C₂×D₃	1	2	yes	—
59	C₂×C₄×C₈	—	2	1	no	—
60	#389	—	3	2	no	—
61	#389	—	3	2	no	—
62	#389	—	6	1	no	—
63	#389	—	6	1	no	—
64	#384	—	3	1	no	—
65	#384	—	3	1	no	—
66	#382	—	6	1	no	—
67	C₄³	C₂×D₃	1	1	yes	characteristic
68	#336	—	2	1	no	—
69	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
76	C₂²×Dic₃	—	2	2	no	—
77	C₂²×Dic₃	—	2	2	no	—
78	C₂×C₃×C₈	—	2	4	no	—
79	C₂×C₃×C₈	—	4	2	no	—
80	C₂×C₃×C₈	—	4	2	no	—
81	C₃×C₄²	—	2	2	no	—
82	C₃×C₄²	—	2	1	no	—
83	C₃×C₄²	—	2	1	no	—
84	C₃×C₄²	—	4	1	no	—
85	C₃×C₄²	—	4	1	no	—
86	C₃×C₄²	D₈	1	2	yes	—
87	C₃×C₄²	SD₁₆	1	2	yes	—
88	C₃×C₄²	SD₁₆	1	2	yes	—
89	C₃×C₄²	Q₁₆	1	2	yes	—
90	C₃×C₄²	C₂×Q₈	1	2	yes	—
91	C₃×C₄²	#41	1	2	yes	—
92	C₄⋊Dic₃	—	4	1	no	—
93	C₄⋊Dic₃	—	4	1	no	—
94	C₄⋊Dic₃	—	4	1	no	—
95	C₄⋊Dic₃	—	4	1	no	—
96	C₄⋊Dic₃	—	4	2	no	—
97	C₄⋊Dic₃	—	4	1	no	—
98	C₄⋊Dic₃	—	4	1	no	—
99	C₄⋊Dic₃	—	4	2	no	—
100	C₄⋊Dic₃	—	4	2	no	—
101	Dic₃⋊C₄	—	4	4	no	—
102	Dic₃⋊C₄	—	4	4	no	—
103	C₂×C₃⋊C₈	—	4	2	no	—
104	C₂×C₃⋊C₈	—	4	2	no	—
105	C₂×C₃⋊C₈	—	4	2	no	—
106	C₂²×C₈	—	2	2	no	—
107	C₂²×C₈	—	6	2	no	—
108	#125	—	12	2	no	—
109	#125	—	12	2	no	—
110	C₂×C₄⋊C₄	—	3	2	no	—
111	C₂×C₄⋊C₄	—	3	2	no	—
112	C₂×C₄⋊C₄	—	6	2	no	—
113	C₂×C₄⋊C₄	—	6	2	no	—
114	C₂×C₄⋊C₄	—	6	1	no	—
115	C₂×C₄⋊C₄	—	6	1	no	—
116	C₂×C₄⋊C₄	—	6	1	no	—
117	C₂×C₄⋊C₄	—	6	1	no	—
118	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
119	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
120	C₂×C₄²	D₁₂	1	1	yes	characteristic
121	C₂×C₄²	D₁₂	1	1	yes	characteristic
122	C₂×C₄²	#67	1	1	yes	characteristic
123	C₂×C₄²	#67	1	1	yes	characteristic
124	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	6	4	no	—
127	C₄⋊C₈	—	12	2	no	—
128	C₄⋊C₈	—	12	2	no	—
129	C₄⋊C₈	Dic₆	1	4	yes	—
130	C₄×C₈	—	4	2	no	—
131	#95	—	6	2	no	—
132	#95	—	6	2	no	—
133	C₂³×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	2	1	no	—
137	C₂×C₃×C₄	—	2	1	no	—
138	C₂×C₃×C₄	—	2	1	no	—
139	C₂×C₃×C₄	—	2	1	no	—
140	C₂×C₃×C₄	—	2	1	no	—
141	C₂×C₃×C₄	—	2	1	no	—
142	C₂×C₃×C₄	—	2	1	no	—
143	C₂×C₃×C₄	—	2	1	no	—
144	C₂×C₃×C₄	#102	1	2	yes	—
145	C₂×C₃×C₄	#102	1	2	yes	—
146	C₂×C₃×C₄	#103	1	2	yes	—
147	C₂×C₃×C₄	#103	1	2	yes	—
148	C₂×C₃×C₄	C₄×Q₈	1	2	yes	—
149	C₂×C₃×C₄	C₂²⋊Q₈	1	2	yes	—
150	C₂×C₃×C₄	#124	1	1	yes	characteristic
151	C₂×C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
152	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
156	C₂×C₃×C₄	#135	1	2	yes	—
157	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
158	C₂×C₃×C₄	#137	1	1	yes	characteristic
159	C₂×Dic₃	—	4	2	no	—
160	C₂×Dic₃	—	4	2	no	—
161	C₂×Dic₃	—	4	2	no	—
162	C₂×Dic₃	—	4	2	no	—
163	C₂×Dic₃	—	4	2	no	—
164	C₂×Dic₃	—	4	2	no	—
165	C₃×C₈	—	4	4	no	—
166	C₃⋊C₈	—	8	2	no	—
167	C₂²×C₄	—	6	2	no	—
168	C₂²×C₄	—	6	2	no	—
169	C₂²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
170	C₂²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
171	C₂²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
172	C₂²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
174	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
175	C₂²×C₄	C₂×#67	1	1	yes	characteristic
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	—	4	2	no	—
179	C₂×C₈	—	12	2	no	—
180	C₂×C₈	—	12	2	no	—
181	C₂×C₈	—	12	2	no	—
182	C₄⋊C₄	—	12	2	no	—
183	C₄⋊C₄	—	12	2	no	—
184	C₄⋊C₄	—	12	4	no	—
185	C₄⋊C₄	—	12	4	no	—
186	C₄⋊C₄	—	12	1	no	—
187	C₄⋊C₄	—	12	1	no	—
188	C₄⋊C₄	—	12	1	no	—
189	C₄⋊C₄	—	12	1	no	—
190	C₄⋊C₄	—	12	1	no	—
191	C₄⋊C₄	—	12	1	no	—
192	C₄⋊C₄	—	12	2	no	—
193	C₄²	—	2	1	no	—
194	C₄²	—	2	1	no	—
195	C₄²	—	2	2	no	—
196	C₄²	—	4	1	no	—
197	C₄²	—	4	1	no	—
198	C₄²	#204	1	2	yes	—
199	C₄²	#204	1	2	yes	—
200	C₄²	D₂₄	1	2	yes	—
201	C₄²	Dic₁₂	1	2	yes	—
202	C₄²	C₂×Dic₆	1	2	yes	—
203	C₄²	#235	1	2	yes	—
204	C₂²×C₃	#391	1	1	yes	characteristic
205	C₂²×C₃	#413	1	1	yes	characteristic
206	C₂²×C₃	#413	1	1	yes	characteristic
207	C₂²×C₃	C₂×#102	1	1	yes	characteristic
208	C₂²×C₃	C₂×#103	1	1	yes	characteristic
209	C₂²×C₃	#416	1	1	yes	characteristic
210	C₂²×C₃	#417	1	1	yes	characteristic
211	C₃×C₄	—	2	2	no	—
212	C₃×C₄	—	2	2	no	—
213	C₃×C₄	—	2	2	no	—
214	C₃×C₄	—	2	2	no	—
215	C₃×C₄	—	2	2	no	—
216	C₃×C₄	#478	1	2	yes	—
217	C₃×C₄	#479	1	2	yes	—
218	C₃×C₄	#486	1	1	yes	characteristic
219	C₃×C₄	#487	1	1	yes	characteristic
220	C₃×C₄	#498	1	1	yes	characteristic
221	C₃×C₄	#500	1	1	yes	characteristic
222	Dic₃	—	8	2	no	—
223	Dic₃	—	8	2	no	—
224	C₂³	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	12	2	no	—
236	C₂×C₄	—	12	2	no	—
237	C₂×C₄	—	12	2	no	—
238	C₂×C₄	—	12	2	no	—
239	C₂×C₄	—	12	2	no	—
240	C₂×C₄	—	12	2	no	—
241	C₂×C₄	#816	1	2	yes	—
242	C₂×C₄	#816	1	2	yes	—
243	C₂×C₄	#821	1	2	yes	—
244	C₂×C₄	#821	1	2	yes	—
245	C₂×C₄	C₄×Dic₆	1	2	yes	—
246	C₂×C₄	#869	1	1	yes	characteristic
247	C₂×C₄	#875	1	1	yes	characteristic
248	C₂×C₄	C₂×#204	1	1	yes	characteristic
249	C₂×C₄	C₂×#204	1	1	yes	characteristic
250	C₂×C₄	C₂×D₂₄	1	1	yes	characteristic
251	C₂×C₄	C₂×Dic₁₂	1	1	yes	characteristic
252	C₂×C₄	#924	1	2	yes	—
253	C₂×C₄	#932	1	1	yes	characteristic
254	C₂×C₄	#942	1	1	yes	characteristic
255	C₂×C₄	#950	1	2	yes	—
256	C₈	—	4	4	no	—
257	C₈	—	24	2	no	—
258	C₂×C₃	#1540	1	1	yes	characteristic
259	C₂×C₃	#1542	1	1	yes	characteristic
260	C₂×C₃	#1547	1	1	yes	characteristic
261	C₂×C₃	#1550	1	1	yes	characteristic
262	C₂×C₃	#1989	1	1	yes	characteristic
263	C₂×C₃	#1990	1	1	yes	characteristic
264	C₂×C₃	#1998	1	1	yes	characteristic
265	C₂²	#4929	1	1	yes	characteristic
266	C₂²	#4966	1	1	yes	characteristic
267	C₂²	#5002	1	1	yes	characteristic
268	C₂²	C₂×#816	1	1	yes	characteristic
269	C₂²	C₂×#821	1	1	yes	characteristic
270	C₂²	#5132	1	1	yes	characteristic
271	C₂²	#5132	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	24	2	no	—
278	C₄	—	24	2	no	—
279	C₄	#4680	1	1	yes	characteristic
280	C₄	#4681	1	1	yes	characteristic
281	C₄	#4682	1	1	yes	characteristic
282	C₄	#4694	1	1	yes	characteristic
283	C₄	#5009	1	2	yes	—
284	C₄	#5022	1	2	yes	—
285	C₃	#9900	1	1	yes	characteristic, complemented, Sylow
286	C₂	#69491	1	1	yes	characteristic
287	C₂	#69777	1	1	yes	characteristic
288	C₂	#69880	1	1	yes	characteristic
289	C₂	#70432	1	1	yes	characteristic
290	C₂	#70438	1	1	yes	characteristic
291	C₂	#70772	1	1	yes	characteristic
292	C₂	#70776	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₃)⋊C₄
commutator	C₂²×C₃×C₄	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×C₄×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₃	C₂×C₂²⋊C₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₄×C₄⋊C₈	D₃
hypocenter	C₃	#9900
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
285. C₃	#9900	semidirect

Maximal subgroups

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

subgroup	index
#73307	2
C₃×C₄×C₄⋊C₈	2
#69394	2
#67702	2
#9900	3