Subgroups — #10693712

Order 768.

318 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#1542	C₂	1	1	yes	maximal, characteristic
2	#71841	C₂	1	1	yes	maximal, characteristic
3	#71826	C₂	1	1	yes	maximal, characteristic
4	#70741	C₂	1	1	yes	maximal, characteristic
5	#70474	C₂	1	1	yes	maximal, characteristic
6	#67746	C₂	1	1	yes	maximal, characteristic
7	#67704	C₂	1	1	yes	maximal, characteristic
8	#9591	—	3	1	no	maximal, Sylow
9	C₃×#533	C₂²	1	1	yes	characteristic
10	C₃×#431	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×#819	—	2	1	no	—
13	C₄×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
14	#4924	C₂²	1	1	yes	characteristic
15	C₄×#208	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₃×#333	C₄	1	1	yes	characteristic, complemented
18	C₃×#333	C₄	1	1	yes	characteristic, complemented
19	C₃×#332	C₄	1	1	yes	characteristic, complemented
20	C₃×#332	C₄	1	1	yes	characteristic, complemented
21	#4552	—	2	1	no	—
22	#4535	—	2	1	no	—
23	#4535	—	2	1	no	—
24	#4532	—	2	1	no	—
25	#4532	—	2	1	no	—
26	#4455	—	2	1	no	—
27	#4454	—	2	1	no	—
28	#1986	—	3	1	no	—
29	#1950	—	3	1	no	—
30	#1542	D₃	1	1	yes	characteristic
31	#1493	—	3	1	no	—
32	#1451	—	3	1	no	—
33	#1303	—	3	1	no	—
34	#1279	—	3	1	no	—
35	C₂²×C₃×C₈	—	2	1	no	—
36	C₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
38	C₃×#125	—	2	1	no	—
39	C₃×C₂²⋊Q₈	—	2	1	no	—
40	C₃×C₂²⋊Q₈	—	2	1	no	—
41	C₃×C₄×Q₈	—	2	1	no	—
42	C₃×C₄×Q₈	—	2	1	no	—
43	C₃×#117	—	2	1	no	—
44	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
45	C₂×C₄⋊Dic₃	—	2	1	no	—
46	C₂×C₄×Dic₃	—	2	1	no	—
47	C₂×#208	—	2	1	no	—
48	C₂×#208	D₄	1	1	yes	characteristic
49	C₂×#208	D₄	1	1	yes	characteristic
50	C₂²×C₃⋊C₈	—	2	1	no	—
51	C₃×C₄⋊C₈	—	2	1	no	—
52	C₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
54	C₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
55	C₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
56	C₃×C₂²⋊C₈	—	2	1	no	—
57	C₃×C₄×C₈	—	2	1	no	—
58	C₃⋊#95	—	2	1	no	—
59	#819	—	4	2	no	—
60	C₄⋊(C₃⋊C₈)	—	2	1	no	—
61	C₄⋊(C₃⋊C₈)	—	2	1	no	—
62	#803	—	2	1	no	—
63	C₄×C₃⋊C₈	—	2	1	no	—
64	C₄×C₃⋊C₈	—	2	1	no	—
65	C₄×C₃⋊C₈	D₄	1	1	yes	characteristic
66	C₄×C₃⋊C₈	D₄	1	1	yes	characteristic
67	#533	C₂×D₃	1	1	yes	characteristic
68	#431	C₂×D₃	1	1	yes	characteristic
69	C₂×#108	—	6	1	no	—
70	#423	—	3	1	no	—
71	C₂×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
72	C₄×M₁₆	—	3	1	no	—
73	C₂×C₄×C₈	—	3	1	no	—
74	C₄×C₄⋊C₄	—	3	1	no	—
75	#341	—	6	1	no	—
76	#339	—	6	1	no	—
77	#339	—	6	1	no	—
78	#333	Dic₃	1	1	yes	characteristic, complemented
79	#333	Dic₃	1	1	yes	characteristic, complemented
80	#332	Dic₃	1	1	yes	characteristic, complemented
81	#332	Dic₃	1	1	yes	characteristic, complemented
82	Q₈⋊C₈	—	6	1	no	—
83	Q₈⋊C₈	—	6	1	no	—
84	#322	—	6	1	no	—
85	#322	—	6	1	no	—
86	C₂×C₃×Q₈	—	2	1	no	—
87	C₂×C₃×Q₈	—	2	1	no	—
88	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₃×C₄	#41	1	1	yes	characteristic
90	C₂²×C₃×C₄	#41	1	1	yes	characteristic
91	C₂²×Dic₃	—	4	1	no	—
92	C₂×C₃×C₈	—	2	1	no	—
93	C₂×C₃×C₈	—	2	1	no	—
94	C₂×C₃×C₈	—	2	1	no	—
95	C₂×C₃×C₈	—	2	1	no	—
96	C₂×C₃×C₈	—	4	1	no	—
97	C₂×C₃×C₈	—	4	1	no	—
98	C₃×C₄⋊C₄	—	2	1	no	—
99	C₃×C₄⋊C₄	—	2	1	no	—
100	C₃×C₄⋊C₄	—	2	1	no	—
101	C₃×C₄⋊C₄	—	2	1	no	—
102	C₃×C₄⋊C₄	—	4	1	no	—
103	C₃×C₄⋊C₄	—	4	1	no	—
104	C₃×C₂²⋊C₄	—	4	1	no	—
105	C₃×C₄²	—	4	1	no	—
106	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
107	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
108	C₃×C₄²	#41	1	1	yes	characteristic
109	C₃×C₄²	#41	1	1	yes	characteristic
110	C₄⋊Dic₃	—	2	2	no	—
111	C₄⋊Dic₃	—	4	1	no	—
112	C₄×Dic₃	—	4	1	no	—
113	C₄×Dic₃	—	4	1	no	—
114	#208	—	2	1	no	—
115	#208	—	2	1	no	—
116	#208	—	2	1	no	—
117	#208	—	2	1	no	—
118	#208	—	4	1	no	—
119	#208	—	4	1	no	—
120	C₂×C₃⋊C₈	—	2	1	no	—
121	C₂×C₃⋊C₈	—	2	1	no	—
122	C₂×C₃⋊C₈	—	2	1	no	—
123	C₂×C₃⋊C₈	—	2	1	no	—
124	C₂×C₃⋊C₈	—	4	2	no	—
125	C₂×C₃⋊C₈	—	4	1	no	—
126	C₂×M₁₆	—	3	1	no	—
127	C₂×M₁₆	—	3	1	no	—
128	C₂×M₁₆	—	6	1	no	—
129	C₂²×C₈	—	2	1	no	—
130	C₂²×C₈	—	6	1	no	—
131	C₄⋊Q₈	C₂×Dic₃	1	1	yes	characteristic
132	C₄⋊Q₈	C₂×Dic₃	1	1	yes	characteristic
133	#125	—	2	1	no	—
134	C₂²⋊Q₈	—	2	1	no	—
135	C₂²⋊Q₈	—	2	1	no	—
136	C₄×Q₈	—	2	1	no	—
137	C₄×Q₈	—	2	1	no	—
138	#117	—	2	1	no	—
139	C₂×C₄⋊C₄	—	6	1	no	—
140	C₂×C₄²	—	6	1	no	—
141	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
142	#108	—	12	2	no	—
143	C₄⋊C₈	—	2	1	no	—
144	C₄⋊C₈	—	6	1	no	—
145	C₄⋊C₈	—	6	1	no	—
146	C₄⋊C₈	C₂×Dic₃	1	1	yes	characteristic
147	C₄⋊C₈	C₂×Dic₃	1	1	yes	characteristic
148	C₄⋊C₈	C₂×Dic₃	1	1	yes	characteristic
149	C₄⋊C₈	C₂×Dic₃	1	1	yes	characteristic
150	C₂²⋊C₈	—	2	1	no	—
151	#97	—	6	1	no	—
152	C₄×C₈	—	2	1	no	—
153	C₄×C₈	—	3	1	no	—
154	C₄×C₈	—	3	1	no	—
155	C₄×C₈	—	6	1	no	—
156	C₄×C₈	—	6	1	no	—
157	#95	—	6	1	no	—
158	C₂³×C₃	#124	1	1	yes	characteristic
159	C₃×Q₈	—	4	1	no	—
160	C₃×Q₈	—	4	1	no	—
161	C₃×Q₈	—	8	1	no	—
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	2	1	no	—
164	C₂×C₃×C₄	—	4	1	no	—
165	C₂×C₃×C₄	—	4	1	no	—
166	C₂×C₃×C₄	—	4	1	no	—
167	C₂×C₃×C₄	#117	1	1	yes	characteristic
168	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
169	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
170	C₂×C₃×C₄	#121	1	1	yes	characteristic
171	C₂×C₃×C₄	#123	1	1	yes	characteristic
172	C₂×C₃×C₄	#126	1	1	yes	characteristic
173	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
174	C₂×C₃×C₄	#137	1	1	yes	characteristic
175	C₂×C₃×C₄	#137	1	1	yes	characteristic
176	C₂×C₃×C₄	#137	1	1	yes	characteristic
177	C₂×Dic₃	—	4	2	no	—
178	C₂×Dic₃	—	8	1	no	—
179	C₂×Dic₃	—	8	1	no	—
180	C₃×C₈	—	4	1	no	—
181	C₃×C₈	—	4	1	no	—
182	C₃×C₈	—	4	1	no	—
183	C₃×C₈	—	4	1	no	—
184	C₃⋊C₈	—	2	2	no	—
185	C₃⋊C₈	—	4	1	no	—
186	C₃⋊C₈	—	4	1	no	—
187	C₃⋊C₈	—	4	1	no	—
188	C₃⋊C₈	—	8	1	no	—
189	C₂×Q₈	—	2	1	no	—
190	C₂×Q₈	—	2	1	no	—
191	C₂²×C₄	—	12	1	no	—
192	C₂²×C₄	#235	1	1	yes	characteristic
193	C₂²×C₄	D₃×D₄	1	1	yes	characteristic
194	C₂²×C₄	#237	1	1	yes	characteristic
195	M₁₆	—	6	1	no	—
196	M₁₆	—	6	1	no	—
197	M₁₆	—	6	1	no	—
198	M₁₆	—	6	1	no	—
199	M₁₆	—	12	1	no	—
200	M₁₆	—	12	1	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	—	4	1	no	—
206	C₂×C₈	—	4	1	no	—
207	C₂×C₈	—	6	1	no	—
208	C₂×C₈	—	6	1	no	—
209	C₂×C₈	—	6	1	no	—
210	C₂×C₈	—	6	1	no	—
211	C₂×C₈	—	12	2	no	—
212	C₂×C₈	—	12	1	no	—
213	C₄⋊C₄	—	2	1	no	—
214	C₄⋊C₄	—	2	1	no	—
215	C₄⋊C₄	—	2	1	no	—
216	C₄⋊C₄	—	2	1	no	—
217	C₄⋊C₄	—	4	1	no	—
218	C₄⋊C₄	—	4	1	no	—
219	C₄⋊C₄	—	6	2	no	—
220	C₄⋊C₄	—	12	1	no	—
221	C₂²⋊C₄	—	4	1	no	—
222	C₄²	—	4	1	no	—
223	C₄²	—	12	1	no	—
224	C₄²	—	12	1	no	—
225	C₄²	#235	1	1	yes	characteristic
226	C₄²	D₃×D₄	1	1	yes	characteristic
227	C₄²	#237	1	1	yes	characteristic
228	C₄²	C₂²×Dic₃	1	1	yes	characteristic
229	C₂²×C₃	—	2	1	no	—
230	C₂²×C₃	—	2	1	no	—
231	C₂²×C₃	#388	1	1	yes	characteristic
232	C₂²×C₃	#489	1	1	yes	characteristic
233	C₂²×C₃	#491	1	1	yes	characteristic
234	C₃×C₄	—	2	1	no	—
235	C₃×C₄	—	2	1	no	—
236	C₃×C₄	—	2	1	no	—
237	C₃×C₄	—	8	1	no	—
238	C₃×C₄	—	8	1	no	—
239	C₃×C₄	#440	1	1	yes	characteristic
240	C₃×C₄	#441	1	1	yes	characteristic
241	C₃×C₄	#443	1	1	yes	characteristic
242	C₃×C₄	#444	1	1	yes	characteristic
243	C₃×C₄	#468	1	1	yes	characteristic
244	C₃×C₄	#470	1	1	yes	characteristic
245	Dic₃	—	8	2	no	—
246	C₂³	#885	1	1	yes	characteristic
247	Q₈	—	4	1	no	—
248	Q₈	—	4	1	no	—
249	Q₈	—	8	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	12	2	no	—
256	C₂×C₄	—	24	1	no	—
257	C₂×C₄	—	24	1	no	—
258	C₂×C₄	#879	1	1	yes	characteristic
259	C₂×C₄	#883	1	1	yes	characteristic
260	C₂×C₄	#884	1	1	yes	characteristic
261	C₂×C₄	#914	1	1	yes	characteristic
262	C₂×C₄	#915	1	1	yes	characteristic
263	C₂×C₄	#918	1	1	yes	characteristic
264	C₂×C₄	#918	1	1	yes	characteristic
265	C₂×C₄	#926	1	1	yes	characteristic
266	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
267	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
268	C₈	—	4	1	no	—
269	C₈	—	4	1	no	—
270	C₈	—	4	1	no	—
271	C₈	—	4	1	no	—
272	C₈	—	6	2	no	—
273	C₈	—	12	1	no	—
274	C₈	—	12	1	no	—
275	C₈	—	12	1	no	—
276	C₈	—	24	1	no	—
277	C₂×C₃	—	2	1	no	—
278	C₂×C₃	—	2	1	no	—
279	C₂×C₃	#1929	1	1	yes	characteristic
280	C₂×C₃	#1932	1	1	yes	characteristic
281	C₂×C₃	#1935	1	1	yes	characteristic
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	#4794	1	1	yes	characteristic
285	C₂²	#4797	1	1	yes	characteristic
286	C₂²	#4951	1	1	yes	characteristic
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	8	1	no	—
291	C₄	—	8	1	no	—
292	C₄	—	24	2	no	—
293	C₄	#4812	1	1	yes	characteristic
294	C₄	#4816	1	1	yes	characteristic
295	C₄	#5164	1	1	yes	characteristic
296	C₄	#5184	1	1	yes	characteristic
297	C₄	#5196	1	1	yes	characteristic
298	C₄	#5197	1	1	yes	characteristic
299	C₃	#9591	1	1	yes	characteristic, complemented, Sylow
300	C₂	—	2	1	no	—
301	C₂	—	2	1	no	—
302	C₂	#70130	1	1	yes	characteristic
303	C₂	#70255	1	1	yes	characteristic
304	C₂	#70291	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#884
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#1542	C₂
socle	C₂²×C₃	#388
prime radical	C₂³×C₃	#124
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	#1542	D₃
hypocenter	C₃	#9591
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1542

Complements

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

N	H ≅ G/N	type
17. C₃×#333	C₄	semidirect
18. C₃×#333	C₄	semidirect
19. C₃×#332	C₄	semidirect
20. C₃×#332	C₄	semidirect
78. #333	Dic₃	semidirect
79. #333	Dic₃	semidirect
80. #332	Dic₃	semidirect
81. #332	Dic₃	semidirect
299. C₃	#9591	semidirect

Maximal subgroups

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

subgroup	index
C₃×#1542	2
#71841	2
#71826	2
#70741	2
#70474	2
#67746	2
#67704	2
#9591	3