Subgroups — #10804036

Order 768.

654 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#79045	C₂	1	2	yes	maximal
2	#78442	C₂	1	1	yes	maximal, characteristic
3	C₃×#1848	C₂	1	1	yes	maximal, characteristic
4	#70705	C₂	1	2	yes	maximal
5	#70704	C₂	1	2	yes	maximal
6	#70703	C₂	1	2	yes	maximal
7	#70702	C₂	1	2	yes	maximal
8	#70515	C₂	1	2	yes	maximal
9	#69749	C₂	1	1	yes	maximal, characteristic
10	#21733	—	3	1	no	maximal, Sylow
11	#5470	C₂²	1	2	yes	—
12	C₂×C₃×#108	C₂²	1	2	yes	—
13	C₂×C₃×#108	C₂²	1	2	yes	—
14	C₃×#423	C₂²	1	2	yes	—
15	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
16	#5122	C₂²	1	4	yes	—
17	#5119	C₂²	1	4	yes	—
18	#5106	C₂²	1	2	yes	—
19	#5106	C₂²	1	2	yes	—
20	C₂×#815	C₂²	1	1	yes	characteristic
21	C₂×#815	C₂²	1	1	yes	characteristic
22	C₂×C₈×Dic₃	C₂²	1	1	yes	characteristic
23	C₂×C₈×Dic₃	C₂²	1	1	yes	characteristic
24	#4932	C₂²	1	1	yes	characteristic
25	C₂×#803	C₂²	1	1	yes	characteristic
26	#4828	—	2	8	no	—
27	#4683	C₄	1	2	yes	complemented
28	#4683	C₄	1	2	yes	complemented
29	#4682	C₄	1	2	yes	complemented
30	#4680	C₄	1	2	yes	complemented
31	#4555	C₂²	1	4	yes	—
32	#4555	C₂²	1	4	yes	—
33	#3160	—	3	1	no	—
34	#2961	—	3	2	no	—
35	#1995	—	3	2	no	—
36	#1946	—	3	2	no	—
37	#1945	—	3	2	no	—
38	#1848	D₃	1	1	yes	characteristic
39	#1773	—	3	2	no	—
40	#1772	—	3	2	no	—
41	#1726	—	3	1	no	—
42	C₂×C₃×M₁₆	C₂³	1	4	yes	—
43	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
44	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
45	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
46	#926	C₂³	1	4	yes	—
47	#924	—	2	4	no	—
48	C₂×C₄×Dic₃	C₂³	1	1	yes	characteristic
49	C₂×C₄×Dic₃	C₂³	1	1	yes	characteristic
50	C₂²×C₃⋊C₈	C₂³	1	1	yes	characteristic
51	C₂²×C₃⋊C₈	C₂³	1	1	yes	characteristic
52	#870	C₂×C₄	1	4	yes	—
53	#869	C₂×C₄	1	4	yes	—
54	C₄×Dic₆	—	2	4	no	—
55	C₃×#108	—	2	4	no	—
56	C₃×#108	—	2	4	no	—
57	C₃×C₄⋊C₈	—	2	4	no	—
58	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
59	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
60	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
61	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
62	C₃⋊#95	—	2	2	no	—
63	#818	C₂×C₄	1	4	yes	—
64	#818	C₂×C₄	1	4	yes	—
65	#817	C₂×C₄	1	4	yes	—
66	#817	C₂×C₄	1	4	yes	—
67	#815	—	2	2	no	—
68	#815	—	2	4	no	—
69	#815	Q₈	1	4	yes	complemented
70	Dic₃⋊C₈	—	2	8	no	—
71	C₈×Dic₃	—	2	2	no	—
72	C₈×Dic₃	—	2	4	no	—
73	C₈×Dic₃	Q₈	1	4	yes	complemented
74	#803	D₄	1	4	yes	—
75	#533	—	3	2	no	—
76	#500	—	3	2	no	—
77	#499	—	3	2	no	—
78	#499	—	3	2	no	—
79	#498	—	3	2	no	—
80	#446	—	6	8	no	—
81	C₂×#108	C₂×D₃	1	2	yes	—
82	C₂×#108	C₂×D₃	1	2	yes	—
83	#427	—	3	2	no	—
84	#427	—	3	2	no	—
85	#424	—	3	4	no	—
86	#423	C₂×D₃	1	2	yes	—
87	#405	—	3	4	no	—
88	C₂×#97	—	3	1	no	—
89	C₂×#97	—	3	1	no	—
90	C₂×#97	—	3	1	no	—
91	C₂×C₄×C₈	—	3	1	no	—
92	C₂×C₄×C₈	—	3	1	no	—
93	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
94	#376	—	3	1	no	—
95	#339	—	3	4	no	—
96	#339	—	3	4	no	—
97	C₂²×C₃×C₄	#41	1	2	yes	—
98	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
99	C₂²×Dic₃	—	2	1	no	—
100	C₂²×Dic₃	—	2	1	no	—
101	C₂×Dic₆	—	2	4	no	—
102	C₃×M₁₆	—	2	4	no	—
103	C₃×M₁₆	—	2	4	no	—
104	C₂×C₃×C₈	—	2	4	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	C₂×Q₈	1	2	yes	—
110	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
111	C₂×C₃×C₈	#41	1	2	yes	—
112	C₂×C₃×C₈	#41	1	2	yes	—
113	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
114	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
115	C₃×C₄²	C₂×D₄	1	2	yes	—
116	#217	—	2	4	no	—
117	C₄⋊Dic₃	—	2	4	no	—
118	C₄⋊Dic₃	C₂²×C₄	1	8	yes	—
119	Dic₃⋊C₄	—	2	8	no	—
120	C₄×Dic₃	—	2	4	no	—
121	C₄×Dic₃	—	2	1	no	—
122	C₄×Dic₃	—	2	1	no	—
123	C₄×Dic₃	C₂×Q₈	1	2	yes	—
124	C₄×Dic₃	C₂×Q₈	1	2	yes	—
125	C₂×C₃⋊C₈	—	2	1	no	—
126	C₂×C₃⋊C₈	—	2	1	no	—
127	C₂×C₃⋊C₈	C₂×D₄	1	2	yes	—
128	C₂×C₃⋊C₈	C₂×D₄	1	2	yes	—
129	C₂×C₃⋊C₈	C₂×Q₈	1	2	yes	—
130	C₂×C₃⋊C₈	C₂×Q₈	1	2	yes	—
131	C₂×M₁₆	C₂²×D₃	1	4	yes	—
132	C₂²×C₈	—	3	1	no	—
133	C₂²×C₈	—	3	1	no	—
134	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
136	C₄⋊Q₈	—	3	4	no	—
137	#125	—	3	4	no	—
138	C₂²⋊Q₈	—	6	4	no	—
139	C₄×Q₈	—	6	4	no	—
140	#117	—	3	4	no	—
141	C₂×C₄²	—	3	1	no	—
142	C₂×C₄²	—	3	1	no	—
143	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
144	#108	—	2	4	no	—
145	#108	—	2	4	no	—
146	#107	—	3	4	no	—
147	#107	—	3	4	no	—
148	#106	—	3	4	no	—
149	#106	—	3	4	no	—
150	C₄⋊C₈	—	2	4	no	—
151	C₄⋊C₈	—	6	8	no	—
152	#97	—	3	4	no	—
153	#97	—	3	4	no	—
154	#97	—	6	4	no	—
155	#97	—	6	2	no	—
156	C₄×C₈	—	3	4	no	—
157	C₄×C₈	—	6	4	no	—
158	C₄×C₈	—	6	2	no	—
159	C₄×C₈	C₄×D₃	1	1	yes	characteristic
160	C₄×C₈	C₄×D₃	1	1	yes	characteristic
161	C₄×C₈	C₄×D₃	1	1	yes	characteristic
162	C₄×C₈	C₄×D₃	1	1	yes	characteristic
163	#95	—	6	2	no	—
164	C₂³×C₃	C₂×#41	1	1	yes	characteristic
165	C₂×C₃×C₄	—	4	2	no	—
166	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
167	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
168	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
174	C₂×Dic₃	—	2	8	no	—
175	C₂×Dic₃	—	2	2	no	—
176	C₂×Dic₃	—	2	2	no	—
177	C₂×Dic₃	—	4	2	no	—
178	C₂×Dic₃	—	4	2	no	—
179	Dic₆	—	8	4	no	—
180	C₃×C₈	—	4	4	no	—
181	C₃×C₈	C₄×Q₈	1	2	yes	—
182	C₃×C₈	C₄×Q₈	1	2	yes	—
183	C₃×C₈	C₄×Q₈	1	2	yes	—
184	C₃×C₈	C₄×Q₈	1	2	yes	—
185	C₃⋊C₈	—	2	2	no	—
186	C₃⋊C₈	C₄⋊Q₈	1	4	yes	—
187	C₂×Q₈	—	6	4	no	—
188	C₂²×C₄	—	6	1	no	—
189	C₂²×C₄	—	6	1	no	—
190	C₂²×C₄	#239	1	2	yes	—
191	C₂²×C₄	C₂³×D₃	1	1	yes	characteristic
192	M₁₆	—	2	4	no	—
193	M₁₆	—	2	4	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	3	2	no	—
196	C₂×C₈	—	3	2	no	—
197	C₂×C₈	—	3	2	no	—
198	C₂×C₈	—	3	2	no	—
199	C₂×C₈	—	6	1	no	—
200	C₂×C₈	—	6	1	no	—
201	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
202	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
203	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
204	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
205	C₂×C₈	#237	1	2	yes	—
206	C₂×C₈	#237	1	2	yes	—
207	C₂×C₈	D₃×Q₈	1	2	yes	—
208	C₂×C₈	D₃×Q₈	1	2	yes	—
209	C₄⋊C₄	—	3	8	no	—
210	C₄⋊C₄	—	6	8	no	—
211	C₄⋊C₄	—	6	4	no	—
212	C₂²⋊C₄	—	6	4	no	—
213	C₄²	—	3	2	no	—
214	C₄²	—	3	2	no	—
215	C₄²	—	6	4	no	—
216	C₄²	—	6	1	no	—
217	C₄²	—	6	1	no	—
218	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
219	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
220	C₄²	D₃×D₄	1	2	yes	—
221	C₂²×C₃	—	2	2	no	—
222	C₂²×C₃	C₂×C₄×D₄	1	1	yes	characteristic
223	C₂²×C₃	#533	1	1	yes	characteristic
224	C₂²×C₃	#533	1	1	yes	characteristic
225	C₃×C₄	—	2	2	no	—
226	C₃×C₄	—	2	2	no	—
227	C₃×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
228	C₃×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
229	C₃×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
230	C₃×C₄	#532	1	1	yes	characteristic
231	Dic₃	—	4	2	no	—
232	Dic₃	—	4	2	no	—
233	Dic₃	—	4	8	no	—
234	C₂³	C₂×#239	1	1	yes	characteristic
235	Q₈	—	24	4	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	6	2	no	—
238	C₂×C₄	—	6	2	no	—
239	C₂×C₄	—	6	8	no	—
240	C₂×C₄	—	12	2	no	—
241	C₂×C₄	—	12	2	no	—
242	C₂×C₄	#893	1	2	yes	—
243	C₂×C₄	#893	1	2	yes	—
244	C₂×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
245	C₂×C₄	C₂×D₃×D₄	1	1	yes	characteristic
246	C₂×C₄	C₂×#237	1	1	yes	characteristic
247	C₂×C₄	C₂×#237	1	1	yes	characteristic
248	C₂×C₄	C₂×D₃×Q₈	1	1	yes	characteristic
249	C₂×C₄	C₂×D₃×Q₈	1	1	yes	characteristic
250	C₈	—	3	4	no	—
251	C₈	—	4	4	no	—
252	C₈	—	6	2	no	—
253	C₈	#887	1	2	yes	—
254	C₈	#887	1	2	yes	—
255	C₈	#887	1	2	yes	—
256	C₈	#887	1	2	yes	—
257	C₂×C₃	—	2	1	no	—
258	C₂×C₃	—	2	1	no	—
259	C₂×C₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
260	C₂×C₃	#2957	1	2	yes	—
261	C₂²	—	2	2	no	—
262	C₂²	C₂×#893	1	1	yes	characteristic
263	C₂²	#5728	1	1	yes	characteristic
264	C₂²	#5728	1	1	yes	characteristic
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	12	2	no	—
268	C₄	—	12	2	no	—
269	C₄	—	12	8	no	—
270	C₄	C₂×#887	1	1	yes	characteristic
271	C₄	C₂×#887	1	1	yes	characteristic
272	C₄	#5716	1	1	yes	characteristic
273	C₄	D₃×C₄⋊Q₈	1	1	yes	characteristic
274	C₃	#21733	1	1	yes	characteristic, complemented, Sylow
275	C₂	—	2	1	no	—
276	C₂	—	2	1	no	—
277	C₂	#74320	1	1	yes	characteristic
278	C₂	#79249	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₃×D₄
commutator	C₂×C₃×C₄	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₃
Fitting	C₃×#1848	C₂
socle	C₂²×C₃	C₂×C₄×D₄
prime radical	C₂³×C₃	C₂×#41
cosocle	C₂²×C₃×C₄	C₂⁴
hypercenter	#1848	D₃
hypocenter	C₃	#21733
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
27. #4683	C₄	semidirect
28. #4683	C₄	semidirect
29. #4682	C₄	semidirect
30. #4680	C₄	semidirect
69. #815	Q₈	semidirect
73. C₈×Dic₃	Q₈	semidirect
274. C₃	#21733	semidirect

Maximal subgroups

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

subgroup	index
#79045	2
#78442	2
C₃×#1848	2
#70705	2
#70704	2
#70703	2
#70702	2
#70515	2
#69749	2
#21733	3