Subgroups — #10694551

Order 768.

502 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	C₃×#3147	C₂	1	1	yes	maximal, characteristic
2	#71811	C₂	1	2	yes	maximal, complemented
3	#71800	C₂	1	2	yes	maximal, complemented
4	#69757	C₂	1	1	yes	maximal, characteristic, complemented
5	#69729	C₂	1	1	yes	maximal, characteristic, complemented
6	#9753	—	3	1	no	maximal, Sylow
7	C₂×C₃×#135	—	2	1	no	—
8	C₂×C₃×#135	—	2	1	no	—
9	C₃×#532	C₂²	1	2	yes	—
10	C₃×#495	—	2	1	no	—
11	C₃×#495	—	2	1	no	—
12	C₃×#494	C₄	1	1	yes	characteristic, complemented
13	C₃×#493	C₄	1	1	yes	characteristic, complemented
14	C₃×#492	C₄	1	2	yes	complemented
15	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₂×#815	—	2	1	no	—
17	C₄×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
18	#4925	C₂²	1	2	yes	—
19	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
20	#4550	—	2	1	no	—
21	#4537	—	2	4	no	—
22	D₄⋊(C₃⋊C₈)	—	2	4	no	—
23	#4458	—	2	1	no	—
24	#4457	—	2	1	no	—
25	#3147	D₃	1	1	yes	characteristic
26	#1770	—	3	1	no	—
27	#1564	—	3	1	no	—
28	#1502	—	3	2	no	—
29	#1480	—	3	2	no	—
30	C₂×C₃×#41	—	2	2	no	—
31	C₃×#135	—	4	2	no	—
32	C₃×#135	—	4	2	no	—
33	C₃×#135	—	4	2	no	—
34	C₃×#135	—	4	2	no	—
35	C₂×C₃×Q₁₆	—	2	1	no	—
36	C₂×C₃×Q₁₆	—	2	1	no	—
37	C₂×C₃×SD₁₆	—	2	2	no	—
38	C₂×C₃×SD₁₆	—	2	2	no	—
39	C₂×C₃×D₈	—	2	1	no	—
40	C₂×C₃×D₈	—	2	1	no	—
41	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
42	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
43	C₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
44	C₃×#127	C₂×C₄	1	2	yes	—
45	C₃×#124	—	2	2	no	—
46	C₃×#121	—	2	2	no	—
47	C₃×#121	—	2	2	no	—
48	C₃×C₄×D₄	—	2	2	no	—
49	C₃×C₄×D₄	—	2	2	no	—
50	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
51	C₂×C₄⋊Dic₃	—	2	1	no	—
52	C₂×C₄×Dic₃	—	2	1	no	—
53	C₂²×C₃⋊C₈	—	2	1	no	—
54	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
55	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
56	C₃×C₄×C₈	D₄	1	1	yes	characteristic
57	C₃×C₄×C₈	D₄	1	1	yes	characteristic
58	C₃⋊#95	D₄	1	2	yes	—
59	#830	—	2	2	no	—
60	#815	—	4	2	no	—
61	C₄⋊(C₃⋊C₈)	—	2	1	no	—
62	C₄⋊(C₃⋊C₈)	—	2	2	no	—
63	C₄⋊(C₃⋊C₈)	D₄	1	2	yes	—
64	C₄×C₃⋊C₈	—	2	2	no	—
65	C₂×#135	—	2	1	no	—
66	C₂×#135	—	2	1	no	—
67	#532	C₂×D₃	1	2	yes	—
68	#495	—	2	1	no	—
69	#495	—	2	1	no	—
70	#494	Dic₃	1	1	yes	characteristic, complemented
71	#493	Dic₃	1	1	yes	characteristic, complemented
72	#492	Dic₃	1	2	yes	complemented
73	#431	—	3	2	no	—
74	C₂×C₄⋊C₈	—	3	1	no	—
75	C₂×#97	—	6	1	no	—
76	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
77	C₄×C₄⋊C₄	—	3	1	no	—
78	#336	—	6	1	no	—
79	#335	—	6	1	no	—
80	#334	—	6	1	no	—
81	#327	—	6	4	no	—
82	D₄⋊C₈	—	6	4	no	—
83	C₃×#41	—	4	2	no	—
84	C₃×#41	—	4	2	no	—
85	C₃×#41	—	4	2	no	—
86	C₃×#41	—	4	2	no	—
87	C₂×C₃×Q₈	—	2	2	no	—
88	C₂×C₃×D₄	—	2	2	no	—
89	C₂×C₃×D₄	—	2	2	no	—
90	C₂×C₃×D₄	—	2	2	no	—
91	C₂×C₃×D₄	—	4	2	no	—
92	C₂²×C₃×C₄	—	4	2	no	—
93	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₃×C₄	#41	1	1	yes	characteristic
96	C₂²×Dic₃	—	2	2	no	—
97	C₃×Q₁₆	—	4	2	no	—
98	C₃×Q₁₆	—	4	2	no	—
99	C₃×SD₁₆	—	4	4	no	—
100	C₃×SD₁₆	—	4	4	no	—
101	C₃×D₈	—	4	2	no	—
102	C₃×D₈	—	4	2	no	—
103	C₂×C₃×C₈	—	2	1	no	—
104	C₂×C₃×C₈	—	2	1	no	—
105	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₃×C₈	D₈	1	2	yes	—
110	C₂×C₃×C₈	SD₁₆	1	2	yes	—
111	C₃×C₄⋊C₄	—	2	2	no	—
112	C₃×C₄⋊C₄	—	2	2	no	—
113	C₃×C₂²⋊C₄	—	4	2	no	—
114	C₃×C₂²⋊C₄	—	4	2	no	—
115	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
116	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
117	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
118	C₃×C₄²	#41	1	1	yes	characteristic
119	C₄⋊Dic₃	—	4	2	no	—
120	C₄×Dic₃	—	4	1	no	—
121	C₄×Dic₃	—	4	1	no	—
122	C₂×C₃⋊C₈	—	2	2	no	—
123	C₂×C₃⋊C₈	—	4	1	no	—
124	C₂×C₃⋊C₈	—	4	1	no	—
125	C₂×C₃⋊C₈	—	4	2	no	—
126	C₂×#41	—	2	2	no	—
127	#135	—	4	2	no	—
128	#135	—	4	2	no	—
129	#135	—	4	2	no	—
130	#135	—	4	2	no	—
131	C₂×Q₁₆	—	2	1	no	—
132	C₂×Q₁₆	—	2	1	no	—
133	C₂×SD₁₆	—	2	2	no	—
134	C₂×SD₁₆	—	2	2	no	—
135	C₂×D₈	—	2	1	no	—
136	C₂×D₈	—	2	1	no	—
137	C₂²×C₈	—	6	1	no	—
138	C₂²×C₈	#67	1	1	yes	characteristic
139	C₂²×C₈	#67	1	1	yes	characteristic
140	C₄⋊Q₈	C₂×Dic₃	1	2	yes	—
141	#127	C₂×Dic₃	1	2	yes	—
142	#124	—	2	2	no	—
143	#121	—	2	2	no	—
144	#121	—	2	2	no	—
145	C₄×D₄	—	2	2	no	—
146	C₄×D₄	—	2	2	no	—
147	C₂×C₄⋊C₄	—	6	1	no	—
148	C₂×C₄²	—	6	1	no	—
149	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
150	C₄⋊C₈	—	3	2	no	—
151	C₄⋊C₈	—	6	1	no	—
152	C₄⋊C₈	—	6	2	no	—
153	C₂²⋊C₈	—	6	2	no	—
154	#97	—	12	2	no	—
155	C₄×C₈	—	6	2	no	—
156	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
157	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
158	C₄×C₈	#67	1	1	yes	characteristic
159	C₄×C₈	#67	1	1	yes	characteristic
160	#95	—	3	2	no	—
161	C₂³×C₃	—	4	2	no	—
162	C₂³×C₃	#121	1	1	yes	characteristic
163	C₃×Q₈	—	4	2	no	—
164	C₃×Q₈	—	4	2	no	—
165	C₃×D₄	—	4	2	no	—
166	C₃×D₄	—	4	2	no	—
167	C₃×D₄	—	4	4	no	—
168	C₃×D₄	—	8	2	no	—
169	C₃×D₄	—	8	2	no	—
170	C₃×D₄	—	8	2	no	—
171	C₂×C₃×C₄	—	2	2	no	—
172	C₂×C₃×C₄	—	4	1	no	—
173	C₂×C₃×C₄	—	4	2	no	—
174	C₂×C₃×C₄	—	8	2	no	—
175	C₂×C₃×C₄	—	8	2	no	—
176	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
177	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
178	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
179	C₂×C₃×C₄	#121	1	1	yes	characteristic
180	C₂×C₃×C₄	#124	1	1	yes	characteristic
181	C₂×C₃×C₄	#127	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
184	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
185	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
186	C₂×Dic₃	—	4	2	no	—
187	C₂×Dic₃	—	4	2	no	—
188	C₂×Dic₃	—	4	2	no	—
189	C₃×C₈	—	2	1	no	—
190	C₃×C₈	—	2	1	no	—
191	C₃×C₈	#102	1	2	yes	—
192	C₃×C₈	#102	1	2	yes	—
193	C₃⋊C₈	—	4	2	no	—
194	C₃⋊C₈	—	8	2	no	—
195	#41	—	4	2	no	—
196	#41	—	4	2	no	—
197	#41	—	4	2	no	—
198	#41	—	4	2	no	—
199	C₂×Q₈	—	2	2	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	—	4	2	no	—
204	C₂²×C₄	—	4	2	no	—
205	C₂²×C₄	—	6	2	no	—
206	C₂²×C₄	D₃×D₄	1	1	yes	characteristic
207	C₂²×C₄	#237	1	1	yes	characteristic
208	C₂²×C₄	C₂×#67	1	1	yes	characteristic
209	Q₁₆	—	4	2	no	—
210	Q₁₆	—	4	2	no	—
211	SD₁₆	—	4	4	no	—
212	SD₁₆	—	4	4	no	—
213	D₈	—	4	2	no	—
214	D₈	—	4	2	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	6	2	no	—
218	C₂×C₈	—	12	1	no	—
219	C₂×C₈	—	12	1	no	—
220	C₂×C₈	—	12	2	no	—
221	C₂×C₈	#213	1	2	yes	—
222	C₂×C₈	#214	1	2	yes	—
223	C₂×C₈	#217	1	1	yes	characteristic
224	C₂×C₈	#217	1	1	yes	characteristic
225	C₂×C₈	#217	1	1	yes	characteristic
226	C₂×C₈	#217	1	1	yes	characteristic
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	12	2	no	—
230	C₂²⋊C₄	—	4	2	no	—
231	C₂²⋊C₄	—	4	2	no	—
232	C₄²	—	12	1	no	—
233	C₄²	—	12	1	no	—
234	C₄²	D₃×D₄	1	1	yes	characteristic
235	C₄²	#237	1	1	yes	characteristic
236	C₄²	C₂²×Dic₃	1	1	yes	characteristic
237	C₄²	C₂×#67	1	1	yes	characteristic
238	C₂²×C₃	—	2	1	no	—
239	C₂²×C₃	—	2	1	no	—
240	C₂²×C₃	—	8	2	no	—
241	C₂²×C₃	—	8	2	no	—
242	C₂²×C₃	—	8	2	no	—
243	C₂²×C₃	#390	1	1	yes	characteristic
244	C₂²×C₃	#459	1	1	yes	characteristic
245	C₂²×C₃	#460	1	1	yes	characteristic
246	C₃×C₄	—	2	1	no	—
247	C₃×C₄	—	2	1	no	—
248	C₃×C₄	—	2	1	no	—
249	C₃×C₄	—	8	2	no	—
250	C₃×C₄	C₂×#102	1	1	yes	characteristic
251	C₃×C₄	#418	1	1	yes	characteristic
252	C₃×C₄	#444	1	2	yes	—
253	C₃×C₄	#459	1	1	yes	characteristic
254	C₃×C₄	#460	1	1	yes	characteristic
255	Dic₃	—	8	2	no	—
256	C₂³	—	4	2	no	—
257	C₂³	#938	1	1	yes	characteristic
258	Q₈	—	4	2	no	—
259	Q₈	—	4	2	no	—
260	D₄	—	4	4	no	—
261	D₄	—	4	2	no	—
262	D₄	—	4	2	no	—
263	D₄	—	8	2	no	—
264	D₄	—	8	2	no	—
265	D₄	—	8	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	8	2	no	—
270	C₂×C₄	—	8	2	no	—
271	C₂×C₄	—	12	2	no	—
272	C₂×C₄	—	12	2	no	—
273	C₂×C₄	—	12	2	no	—
274	C₂×C₄	C₂×#213	1	1	yes	characteristic
275	C₂×C₄	#932	1	1	yes	characteristic
276	C₂×C₄	#932	1	1	yes	characteristic
277	C₂×C₄	C₂×#214	1	1	yes	characteristic
278	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
279	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
280	C₂×C₄	#936	1	1	yes	characteristic
281	C₂×C₄	#938	1	1	yes	characteristic
282	C₂×C₄	#940	1	1	yes	characteristic
283	C₂×C₄	C₂×#217	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	12	2	no	—
287	C₈	—	24	2	no	—
288	C₈	#832	1	2	yes	—
289	C₈	#832	1	2	yes	—
290	C₂×C₃	—	2	1	no	—
291	C₂×C₃	—	2	1	no	—
292	C₂×C₃	—	16	2	no	—
293	C₂×C₃	#1975	1	1	yes	characteristic
294	C₂×C₃	#1976	1	1	yes	characteristic
295	C₂×C₃	#1982	1	1	yes	characteristic
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	8	2	no	—
299	C₂²	—	8	2	no	—
300	C₂²	—	8	2	no	—
301	C₂²	#5072	1	1	yes	characteristic
302	C₂²	#5075	1	1	yes	characteristic
303	C₂²	#5221	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	8	2	no	—
308	C₄	—	24	2	no	—
309	C₄	#5072	1	1	yes	characteristic
310	C₄	#5075	1	1	yes	characteristic
311	C₄	#5197	1	2	yes	—
312	C₄	C₂×#832	1	1	yes	characteristic
313	C₄	#5215	1	1	yes	characteristic
314	C₃	#9753	1	1	yes	characteristic, complemented, Sylow
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	16	2	no	—
318	C₂	#71761	1	1	yes	characteristic
319	C₂	#72085	1	1	yes	characteristic
320	C₂	#72086	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#938
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting, prime radical	C₃×#3147	C₂
socle	C₂²×C₃	#390
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	#3147	D₃
hypocenter	C₃	#9753
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

Complements

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

N	H ≅ G/N	type
2. #71811	C₂	semidirect
3. #71800	C₂	semidirect
4. #69757	C₂	semidirect
5. #69729	C₂	semidirect
12. C₃×#494	C₄	semidirect
13. C₃×#493	C₄	semidirect
14. C₃×#492	C₄	semidirect
70. #494	Dic₃	semidirect
71. #493	Dic₃	semidirect
72. #492	Dic₃	semidirect
314. C₃	#9753	semidirect

Maximal subgroups

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

subgroup	index
C₃×#3147	2
#71811	2
#71800	2
#69757	2
#69729	2
#9753	3