Subgroups — #10804059

Order 768.

670 conjugacy classes of subgroups.

The subgroup lattice has 256 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	#78479	C₂	1	1	yes	maximal, characteristic
3	C₃×#1849	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	#70514	C₂	1	2	yes	maximal
9	Dic₃×#97	C₂	1	1	yes	maximal, characteristic
10	#21736	—	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×#97	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	C₄²×Dic₃	C₂²	1	1	yes	characteristic
25	C₂×#803	C₂²	1	1	yes	characteristic
26	#4828	—	2	8	no	—
27	#4702	C₄	1	8	yes	complemented
28	#4555	C₂²	1	4	yes	—
29	#4555	C₂²	1	4	yes	—
30	#3162	—	3	1	no	—
31	#2961	—	3	2	no	—
32	#1994	—	3	2	no	—
33	#1946	—	3	2	no	—
34	#1945	—	3	2	no	—
35	#1849	D₃	1	1	yes	characteristic
36	#1773	—	3	2	no	—
37	#1772	—	3	2	no	—
38	C₄×#97	—	3	1	no	—
39	C₂×C₃×M₁₆	C₂³	1	4	yes	—
40	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
42	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
43	#926	C₂³	1	4	yes	—
44	#924	—	2	4	no	—
45	C₂×C₄×Dic₃	—	2	2	no	—
46	C₂×C₄×Dic₃	C₂³	1	1	yes	characteristic
47	C₂×C₄×Dic₃	C₂³	1	1	yes	characteristic
48	C₂²×C₃⋊C₈	C₂³	1	1	yes	characteristic
49	C₂²×C₃⋊C₈	C₂³	1	1	yes	characteristic
50	#870	C₂×C₄	1	4	yes	—
51	#869	C₂×C₄	1	4	yes	—
52	C₄×Dic₆	—	2	4	no	—
53	C₃×#108	—	2	4	no	—
54	C₃×#108	—	2	4	no	—
55	C₃×C₄⋊C₈	—	2	4	no	—
56	C₃×#97	C₂×C₄	1	4	yes	—
57	#818	C₂×C₄	1	4	yes	—
58	#818	C₂×C₄	1	4	yes	—
59	#817	C₂×C₄	1	4	yes	—
60	#817	C₂×C₄	1	4	yes	—
61	#815	—	2	2	no	—
62	#815	—	2	4	no	—
63	#815	Q₈	1	4	yes	complemented
64	Dic₃⋊C₈	—	2	8	no	—
65	C₈×Dic₃	—	2	2	no	—
66	C₈×Dic₃	—	2	4	no	—
67	C₈×Dic₃	Q₈	1	4	yes	complemented
68	#803	D₄	1	4	yes	—
69	#533	—	3	2	no	—
70	#501	—	3	8	no	—
71	#446	—	6	8	no	—
72	C₂×#108	C₂×D₃	1	2	yes	—
73	C₂×#108	C₂×D₃	1	2	yes	—
74	#427	—	3	2	no	—
75	#427	—	3	2	no	—
76	#424	—	3	4	no	—
77	#423	C₂×D₃	1	2	yes	—
78	#405	—	3	4	no	—
79	C₂×#97	—	3	1	no	—
80	C₂×#97	—	3	1	no	—
81	C₂×#97	—	3	1	no	—
82	C₂×#97	C₂×D₃	1	1	yes	characteristic
83	C₂×C₄×C₈	—	3	1	no	—
84	C₂×C₄×C₈	—	3	1	no	—
85	C₄³	—	3	1	no	—
86	#339	—	3	4	no	—
87	#339	—	3	4	no	—
88	C₂²×C₃×C₄	#41	1	2	yes	—
89	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
90	C₂²×Dic₃	—	2	1	no	—
91	C₂²×Dic₃	—	2	1	no	—
92	C₂×Dic₆	—	2	4	no	—
93	C₃×M₁₆	—	2	4	no	—
94	C₃×M₁₆	—	2	4	no	—
95	C₂×C₃×C₈	—	2	4	no	—
96	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
97	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
98	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
99	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
100	C₂×C₃×C₈	#41	1	2	yes	—
101	C₂×C₃×C₈	#41	1	2	yes	—
102	C₃×C₄²	C₂²×C₄	1	2	yes	—
103	C₃×C₄²	C₂×D₄	1	2	yes	—
104	#217	—	2	4	no	—
105	C₄⋊Dic₃	—	2	4	no	—
106	C₄⋊Dic₃	C₂²×C₄	1	8	yes	—
107	Dic₃⋊C₄	—	2	8	no	—
108	C₄×Dic₃	—	2	4	no	—
109	C₄×Dic₃	—	2	1	no	—
110	C₄×Dic₃	—	2	1	no	—
111	C₄×Dic₃	—	4	4	no	—
112	C₄×Dic₃	C₂×Q₈	1	2	yes	—
113	C₄×Dic₃	C₂×Q₈	1	2	yes	—
114	C₂×C₃⋊C₈	—	2	1	no	—
115	C₂×C₃⋊C₈	—	2	1	no	—
116	C₂×C₃⋊C₈	C₂×D₄	1	2	yes	—
117	C₂×C₃⋊C₈	C₂×D₄	1	2	yes	—
118	C₂×C₃⋊C₈	C₂×Q₈	1	2	yes	—
119	C₂×C₃⋊C₈	C₂×Q₈	1	2	yes	—
120	C₂×M₁₆	C₂²×D₃	1	4	yes	—
121	C₂²×C₈	—	3	1	no	—
122	C₂²×C₈	—	3	1	no	—
123	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
124	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
125	C₄⋊Q₈	—	3	4	no	—
126	#125	—	3	4	no	—
127	C₂²⋊Q₈	—	6	4	no	—
128	C₄×Q₈	—	6	4	no	—
129	#117	—	3	4	no	—
130	C₂×C₄²	—	3	1	no	—
131	C₂×C₄²	—	3	1	no	—
132	C₂×C₄²	—	6	2	no	—
133	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
134	#108	—	2	4	no	—
135	#108	—	2	4	no	—
136	#107	—	3	4	no	—
137	#107	—	3	4	no	—
138	#106	—	3	4	no	—
139	#106	—	3	4	no	—
140	C₄⋊C₈	—	2	4	no	—
141	C₄⋊C₈	—	6	8	no	—
142	#97	—	3	4	no	—
143	#97	—	3	4	no	—
144	#97	—	6	4	no	—
145	#97	—	6	2	no	—
146	#97	C₄×D₃	1	4	yes	—
147	C₄×C₈	—	3	4	no	—
148	C₄×C₈	—	6	4	no	—
149	C₄×C₈	—	6	2	no	—
150	C₂³×C₃	C₂×#41	1	1	yes	characteristic
151	C₂×C₃×C₄	—	4	2	no	—
152	C₂×C₃×C₄	C₄×D₄	1	4	yes	—
153	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
157	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
159	C₂×Dic₃	—	2	4	no	—
160	C₂×Dic₃	—	2	2	no	—
161	C₂×Dic₃	—	2	8	no	—
162	C₂×Dic₃	—	2	2	no	—
163	C₂×Dic₃	—	4	2	no	—
164	Dic₆	—	8	4	no	—
165	C₃×C₈	—	4	4	no	—
166	C₃×C₈	C₄×Q₈	1	4	yes	—
167	C₃×C₈	C₄×Q₈	1	4	yes	—
168	C₃⋊C₈	—	2	2	no	—
169	C₃⋊C₈	C₄⋊Q₈	1	4	yes	—
170	C₂×Q₈	—	6	4	no	—
171	C₂²×C₄	—	6	1	no	—
172	C₂²×C₄	—	6	1	no	—
173	C₂²×C₄	#239	1	2	yes	—
174	C₂²×C₄	C₂³×D₃	1	1	yes	characteristic
175	M₁₆	—	2	4	no	—
176	M₁₆	—	2	4	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	3	2	no	—
179	C₂×C₈	—	3	2	no	—
180	C₂×C₈	—	3	2	no	—
181	C₂×C₈	—	3	2	no	—
182	C₂×C₈	—	6	1	no	—
183	C₂×C₈	—	6	1	no	—
184	C₂×C₈	C₂×C₄×D₃	1	2	yes	—
185	C₂×C₈	C₂×C₄×D₃	1	2	yes	—
186	C₂×C₈	#237	1	2	yes	—
187	C₂×C₈	#237	1	2	yes	—
188	C₂×C₈	D₃×Q₈	1	2	yes	—
189	C₂×C₈	D₃×Q₈	1	2	yes	—
190	C₄⋊C₄	—	3	8	no	—
191	C₄⋊C₄	—	6	8	no	—
192	C₄⋊C₄	—	6	4	no	—
193	C₂²⋊C₄	—	6	4	no	—
194	C₄²	—	3	2	no	—
195	C₄²	—	3	2	no	—
196	C₄²	—	6	4	no	—
197	C₄²	—	6	1	no	—
198	C₄²	—	6	1	no	—
199	C₄²	—	12	4	no	—
200	C₄²	C₂×C₄×D₃	1	2	yes	—
201	C₄²	D₃×D₄	1	2	yes	—
202	C₂²×C₃	—	2	2	no	—
203	C₂²×C₃	C₂×C₄×D₄	1	1	yes	characteristic
204	C₂²×C₃	#533	1	1	yes	characteristic
205	C₂²×C₃	#533	1	1	yes	characteristic
206	C₃×C₄	—	2	4	no	—
207	C₃×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
208	C₃×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
209	C₃×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
210	C₃×C₄	#532	1	1	yes	characteristic
211	Dic₃	—	2	4	no	—
212	Dic₃	—	4	2	no	—
213	Dic₃	—	4	8	no	—
214	C₂³	C₂×#239	1	1	yes	characteristic
215	Q₈	—	24	4	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	6	4	no	—
218	C₂×C₄	—	6	2	no	—
219	C₂×C₄	—	6	2	no	—
220	C₂×C₄	—	6	8	no	—
221	C₂×C₄	—	12	2	no	—
222	C₂×C₄	#893	1	4	yes	—
223	C₂×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
224	C₂×C₄	C₂×D₃×D₄	1	1	yes	characteristic
225	C₂×C₄	C₂×#237	1	1	yes	characteristic
226	C₂×C₄	C₂×#237	1	1	yes	characteristic
227	C₂×C₄	C₂×D₃×Q₈	1	1	yes	characteristic
228	C₂×C₄	C₂×D₃×Q₈	1	1	yes	characteristic
229	C₈	—	3	4	no	—
230	C₈	—	4	4	no	—
231	C₈	—	6	2	no	—
232	C₈	#887	1	4	yes	—
233	C₈	#887	1	4	yes	—
234	C₂×C₃	—	2	1	no	—
235	C₂×C₃	—	2	1	no	—
236	C₂×C₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
237	C₂×C₃	#2956	1	2	yes	—
238	C₂²	—	2	2	no	—
239	C₂²	C₂×#893	1	1	yes	characteristic
240	C₂²	#5728	1	1	yes	characteristic
241	C₂²	#5728	1	1	yes	characteristic
242	C₄	—	2	4	no	—
243	C₄	—	6	4	no	—
244	C₄	—	12	2	no	—
245	C₄	—	12	8	no	—
246	C₄	C₂×#887	1	1	yes	characteristic
247	C₄	C₂×#887	1	1	yes	characteristic
248	C₄	#5716	1	1	yes	characteristic
249	C₄	D₃×C₄⋊Q₈	1	1	yes	characteristic
250	C₃	#21736	1	1	yes	characteristic, complemented, Sylow
251	C₂	—	2	1	no	—
252	C₂	—	2	1	no	—
253	C₂	#74320	1	1	yes	characteristic
254	C₂	#79258	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×#1849	C₂
socle	C₂²×C₃	C₂×C₄×D₄
prime radical	C₂³×C₃	C₂×#41
cosocle	C₂²×C₃×C₄	C₂⁴
hypercenter	#1849	D₃
hypocenter	C₃	#21736
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1849

Complements

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

N	H ≅ G/N	type
27. #4702	C₄	semidirect
63. #815	Q₈	semidirect
67. C₈×Dic₃	Q₈	semidirect
250. C₃	#21736	semidirect

Maximal subgroups

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

subgroup	index
#79045	2
#78479	2
C₃×#1849	2
#70705	2
#70704	2
#70703	2
#70702	2
#70514	2
Dic₃×#97	2
#21736	3