Subgroups — #10804067

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	C₃×#3162	C₂	1	1	yes	maximal, characteristic
2	#83329	C₂	1	2	yes	maximal
3	#71839	C₂	1	2	yes	maximal
4	#70663	C₂	1	2	yes	maximal
5	#70662	C₂	1	2	yes	maximal
6	#70660	C₂	1	2	yes	maximal
7	#70659	C₂	1	2	yes	maximal
8	#69842	C₂	1	1	yes	maximal, characteristic
9	Dic₃×#97	C₂	1	1	yes	maximal, characteristic
10	#21736	—	3	1	no	maximal, Sylow
11	C₃×#533	C₂²	1	2	yes	—
12	C₃×#501	C₄	1	8	yes	complemented
13	C₃×#427	C₂²	1	2	yes	—
14	C₃×#427	C₂²	1	2	yes	—
15	C₂×C₃×#97	C₂²	1	1	yes	characteristic
16	C₂×#819	C₂²	1	2	yes	—
17	C₂×#819	C₂²	1	2	yes	—
18	C₂×#815	C₂²	1	1	yes	characteristic
19	C₂×#815	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₃	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₃	C₂²	1	1	yes	characteristic
22	#5024	—	2	8	no	—
23	#4999	C₂²	1	4	yes	—
24	#4997	C₂²	1	4	yes	—
25	C₄²×Dic₃	C₂²	1	1	yes	characteristic
26	#4924	C₂²	1	2	yes	—
27	C₂×#803	C₂²	1	1	yes	characteristic
28	#4532	C₂²	1	4	yes	—
29	#4532	C₂²	1	4	yes	—
30	#3162	D₃	1	1	yes	characteristic
31	#2961	—	3	2	no	—
32	#1994	—	3	2	no	—
33	#1946	—	3	2	no	—
34	#1945	—	3	2	no	—
35	#1849	—	3	1	no	—
36	#1773	—	3	2	no	—
37	#1772	—	3	2	no	—
38	C₄×#97	—	3	1	no	—
39	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
40	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
41	C₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
42	C₃×#125	C₂×C₄	1	4	yes	—
43	C₃×C₂²⋊Q₈	—	2	4	no	—
44	C₃×C₄×Q₈	—	2	4	no	—
45	C₃×#117	C₂³	1	4	yes	—
46	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
47	C₂×C₄×Dic₃	—	2	2	no	—
48	C₂×C₄×Dic₃	C₂³	1	1	yes	characteristic
49	C₂×C₄×Dic₃	C₂³	1	1	yes	characteristic
50	C₂×#208	C₂³	1	4	yes	—
51	C₂²×C₃⋊C₈	C₂³	1	1	yes	characteristic
52	C₂²×C₃⋊C₈	C₂³	1	1	yes	characteristic
53	C₃×#107	C₂×C₄	1	4	yes	—
54	C₃×#107	C₂×C₄	1	4	yes	—
55	C₃×#106	C₂×C₄	1	4	yes	—
56	C₃×#106	C₂×C₄	1	4	yes	—
57	C₃×#97	C₂×C₄	1	4	yes	—
58	#819	—	2	4	no	—
59	#819	—	2	4	no	—
60	#815	—	2	2	no	—
61	#815	Q₈	1	4	yes	complemented
62	C₈×Dic₃	—	2	2	no	—
63	C₈×Dic₃	Q₈	1	4	yes	complemented
64	C₄⋊(C₃⋊C₈)	—	2	4	no	—
65	C₄⋊(C₃⋊C₈)	—	2	8	no	—
66	#803	—	2	4	no	—
67	#803	D₄	1	4	yes	—
68	C₄×C₃⋊C₈	—	2	4	no	—
69	#533	C₂×D₃	1	2	yes	—
70	#501	Dic₃	1	8	yes	complemented
71	#446	—	6	8	no	—
72	C₂×#108	—	3	2	no	—
73	C₂×#108	—	3	2	no	—
74	#427	C₂×D₃	1	2	yes	—
75	#427	C₂×D₃	1	2	yes	—
76	#424	—	3	4	no	—
77	#423	—	3	2	no	—
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	C₂×C₃×Q₈	—	2	4	no	—
89	C₂²×C₃×C₄	#41	1	2	yes	—
90	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
91	C₂²×Dic₃	—	2	1	no	—
92	C₂²×Dic₃	—	2	1	no	—
93	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
94	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
95	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
96	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
97	C₂×C₃×C₈	#41	1	2	yes	—
98	C₂×C₃×C₈	#41	1	2	yes	—
99	C₃×C₄⋊C₄	—	2	8	no	—
100	C₃×C₄⋊C₄	—	2	4	no	—
101	C₃×C₄⋊C₄	C₂²×C₄	1	8	yes	—
102	C₃×C₂²⋊C₄	—	2	4	no	—
103	C₃×C₄²	—	2	4	no	—
104	C₃×C₄²	C₂²×C₄	1	2	yes	—
105	C₃×C₄²	C₂×D₄	1	2	yes	—
106	C₄×Dic₃	—	2	1	no	—
107	C₄×Dic₃	—	2	1	no	—
108	C₄×Dic₃	—	4	4	no	—
109	C₄×Dic₃	C₂×Q₈	1	2	yes	—
110	C₄×Dic₃	C₂×Q₈	1	2	yes	—
111	#208	—	2	4	no	—
112	#208	—	2	4	no	—
113	C₂×C₃⋊C₈	—	2	1	no	—
114	C₂×C₃⋊C₈	—	2	1	no	—
115	C₂×C₃⋊C₈	—	2	4	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	—	3	4	no	—
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	C₂×Dic₃	1	4	yes	—
126	#125	C₂×Dic₃	1	4	yes	—
127	C₂²⋊Q₈	—	2	4	no	—
128	C₄×Q₈	—	2	4	no	—
129	#117	C₂²×D₃	1	4	yes	—
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	—	6	4	no	—
135	#108	—	6	4	no	—
136	#107	C₂×Dic₃	1	4	yes	—
137	#107	C₂×Dic₃	1	4	yes	—
138	#106	C₂×Dic₃	1	4	yes	—
139	#106	C₂×Dic₃	1	4	yes	—
140	C₄⋊C₈	—	6	8	no	—
141	C₄⋊C₈	—	6	4	no	—
142	#97	—	3	4	no	—
143	#97	—	3	4	no	—
144	#97	—	6	4	no	—
145	#97	—	6	2	no	—
146	#97	C₂×Dic₃	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	C₃×Q₈	—	8	4	no	—
152	C₂×C₃×C₄	—	2	8	no	—
153	C₂×C₃×C₄	—	4	2	no	—
154	C₂×C₃×C₄	C₄×D₄	1	4	yes	—
155	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
159	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
160	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
161	C₂×Dic₃	—	2	4	no	—
162	C₂×Dic₃	—	2	2	no	—
163	C₂×Dic₃	—	2	2	no	—
164	C₂×Dic₃	—	4	2	no	—
165	C₃×C₈	C₄×Q₈	1	4	yes	—
166	C₃×C₈	C₄×Q₈	1	4	yes	—
167	C₃⋊C₈	—	2	2	no	—
168	C₃⋊C₈	—	4	4	no	—
169	C₃⋊C₈	C₄⋊Q₈	1	4	yes	—
170	C₂×Q₈	—	2	4	no	—
171	C₂²×C₄	—	6	1	no	—
172	C₂²×C₄	—	6	1	no	—
173	C₂²×C₄	#237	1	2	yes	—
174	C₂²×C₄	C₂³×D₃	1	1	yes	characteristic
175	M₁₆	—	6	4	no	—
176	M₁₆	—	6	4	no	—
177	C₂×C₈	—	3	2	no	—
178	C₂×C₈	—	3	2	no	—
179	C₂×C₈	—	3	2	no	—
180	C₂×C₈	—	3	2	no	—
181	C₂×C₈	—	6	1	no	—
182	C₂×C₈	—	6	1	no	—
183	C₂×C₈	—	6	4	no	—
184	C₂×C₈	D₃×Q₈	1	2	yes	—
185	C₂×C₈	D₃×Q₈	1	2	yes	—
186	C₂×C₈	#239	1	2	yes	—
187	C₂×C₈	#239	1	2	yes	—
188	C₂×C₈	C₂²×Dic₃	1	2	yes	—
189	C₂×C₈	C₂²×Dic₃	1	2	yes	—
190	C₄⋊C₄	—	2	8	no	—
191	C₄⋊C₄	—	2	4	no	—
192	C₄⋊C₄	C₂²×Dic₃	1	8	yes	—
193	C₂²⋊C₄	—	2	4	no	—
194	C₄²	—	2	4	no	—
195	C₄²	—	3	2	no	—
196	C₄²	—	3	2	no	—
197	C₄²	—	6	1	no	—
198	C₄²	—	6	1	no	—
199	C₄²	—	12	4	no	—
200	C₄²	D₃×D₄	1	2	yes	—
201	C₄²	C₂²×Dic₃	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	—	4	8	no	—
208	C₃×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
209	C₃×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
210	C₃×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
211	C₃×C₄	#532	1	1	yes	characteristic
212	Dic₃	—	2	4	no	—
213	Dic₃	—	4	2	no	—
214	C₂³	C₂×#237	1	1	yes	characteristic
215	Q₈	—	8	4	no	—
216	C₂×C₄	—	2	8	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	6	4	no	—
219	C₂×C₄	—	6	2	no	—
220	C₂×C₄	—	6	2	no	—
221	C₂×C₄	—	12	2	no	—
222	C₂×C₄	D₄×Dic₃	1	4	yes	—
223	C₂×C₄	C₂×D₃×D₄	1	1	yes	characteristic
224	C₂×C₄	C₂×D₃×Q₈	1	1	yes	characteristic
225	C₂×C₄	C₂×D₃×Q₈	1	1	yes	characteristic
226	C₂×C₄	C₂×#239	1	1	yes	characteristic
227	C₂×C₄	C₂×#239	1	1	yes	characteristic
228	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
229	C₈	—	3	4	no	—
230	C₈	—	6	2	no	—
231	C₈	—	12	4	no	—
232	C₈	Q₈×Dic₃	1	4	yes	—
233	C₈	Q₈×Dic₃	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	#5728	1	1	yes	characteristic
240	C₂²	#5728	1	1	yes	characteristic
241	C₂²	C₂×D₄×Dic₃	1	1	yes	characteristic
242	C₄	—	2	4	no	—
243	C₄	—	4	8	no	—
244	C₄	—	6	4	no	—
245	C₄	—	12	2	no	—
246	C₄	D₃×C₄⋊Q₈	1	1	yes	characteristic
247	C₄	#5725	1	1	yes	characteristic
248	C₄	C₂×Q₈×Dic₃	1	1	yes	characteristic
249	C₄	C₂×Q₈×Dic₃	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	Dic₃×C₄⋊Q₈	1	1	yes	characteristic
254	C₂	#84000	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×#3162	C₂
socle	C₂²×C₃	C₂×C₄×D₄
prime radical	C₂³×C₃	C₂×#41
cosocle	C₂²×C₃×C₄	C₂⁴
hypercenter	#3162	D₃
hypocenter	C₃	#21736
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3162

Complements

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

N	H ≅ G/N	type
12. C₃×#501	C₄	semidirect
61. #815	Q₈	semidirect
63. C₈×Dic₃	Q₈	semidirect
70. #501	Dic₃	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
C₃×#3162	2
#83329	2
#71839	2
#70663	2
#70662	2
#70660	2
#70659	2
#69842	2
Dic₃×#97	2
#21736	3