Subgroups — #10765746

Order 768.

598 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3170	C₂	1	1	yes	maximal, characteristic
2	#83328	C₂	1	2	yes	maximal
3	#71826	C₂	1	2	yes	maximal
4	#70906	C₂	1	4	yes	maximal, complemented
5	#70902	C₂	1	4	yes	maximal, complemented
6	#69730	C₂	1	1	yes	maximal, characteristic
7	C₂×#4455	C₂	1	1	yes	maximal, characteristic
8	#17961	—	3	1	no	maximal, Sylow
9	C₃×#533	C₂²	1	2	yes	—
10	C₃×#500	C₄	1	1	yes	characteristic
11	C₃×#494	C₄	1	1	yes	characteristic
12	C₃×#487	C₄	1	2	yes	—
13	C₃×#467	C₄	1	4	yes	—
14	C₃×C₄×Q₁₆	C₂²	1	4	yes	—
15	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
16	#5024	—	2	2	no	—
17	#5024	C₂²	1	4	yes	—
18	Q₈×C₃⋊C₈	—	2	2	no	—
19	Q₈×C₃⋊C₈	C₂²	1	4	yes	—
20	#4999	—	2	2	no	—
21	#4997	—	2	2	no	—
22	#4924	C₂²	1	2	yes	—
23	C₂×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
25	#4535	C₂²	1	8	yes	—
26	#4457	C₂²	1	4	yes	—
27	#4455	C₂²	1	2	yes	—
28	#4455	C₂²	1	2	yes	—
29	#3170	D₃	1	1	yes	characteristic
30	#2960	—	3	2	no	—
31	#1593	—	3	4	no	—
32	#1579	—	3	4	no	—
33	#1566	—	3	1	no	—
34	#1493	—	3	2	no	—
35	C₂×#322	—	3	1	no	—
36	C₂×C₃×Q₁₆	C₂×C₄	1	4	yes	—
37	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
38	C₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
39	C₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
40	C₃×#125	—	2	2	no	—
41	C₃×C₂²⋊Q₈	—	2	2	no	—
42	C₃×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
43	C₃×C₄×Q₈	—	2	2	no	—
44	C₃×C₄×Q₈	C₂³	1	4	yes	—
45	C₃×#117	—	2	2	no	—
46	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
47	C₂×#208	—	2	2	no	—
48	C₂²×C₃⋊C₈	—	2	1	no	—
49	C₂²×C₃⋊C₈	D₄	1	2	yes	—
50	C₃×#107	C₂×C₄	1	4	yes	—
51	C₃×#103	C₂×C₄	1	8	yes	—
52	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
53	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
54	C₃×C₄×C₈	C₂³	1	2	yes	—
55	C₄⋊(C₃⋊C₈)	—	2	2	no	—
56	C₄⋊(C₃⋊C₈)	—	2	4	no	—
57	C₄⋊(C₃⋊C₈)	—	2	4	no	—
58	C₄⋊(C₃⋊C₈)	C₂³	1	4	yes	—
59	#803	—	2	4	no	—
60	C₄×C₃⋊C₈	—	2	1	no	—
61	C₄×C₃⋊C₈	—	2	4	no	—
62	C₄×C₃⋊C₈	D₄	1	2	yes	—
63	C₄×C₃⋊C₈	C₂³	1	2	yes	—
64	C₄×C₃⋊C₈	C₂³	1	1	yes	characteristic
65	C₄×C₃⋊C₈	C₂³	1	1	yes	characteristic
66	#533	C₂×D₃	1	2	yes	—
67	#500	Dic₃	1	1	yes	characteristic
68	#494	Dic₃	1	1	yes	characteristic
69	#487	Dic₃	1	2	yes	—
70	#467	Dic₃	1	4	yes	—
71	#446	—	3	4	no	—
72	#446	—	6	2	no	—
73	C₈×Q₈	—	3	4	no	—
74	C₈×Q₈	—	6	2	no	—
75	C₄×Q₁₆	C₂×D₃	1	4	yes	—
76	#424	—	6	2	no	—
77	#423	—	3	2	no	—
78	#405	—	6	2	no	—
79	C₂×C₄×C₈	—	3	1	no	—
80	C₂×C₄×C₈	—	3	1	no	—
81	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
82	#334	—	3	4	no	—
83	Q₈⋊C₈	—	3	8	no	—
84	#322	—	3	2	no	—
85	#322	—	3	2	no	—
86	C₂×C₃×Q₈	—	2	2	no	—
87	C₂×C₃×Q₈	C₂²×C₄	1	4	yes	—
88	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₃×C₄	#41	1	1	yes	characteristic
91	C₃×Q₁₆	—	2	8	no	—
92	C₂×C₃×C₈	—	2	4	no	—
93	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
94	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
95	C₃×C₄⋊C₄	—	2	2	no	—
96	C₃×C₄⋊C₄	—	2	2	no	—
97	C₃×C₄⋊C₄	—	2	2	no	—
98	C₃×C₄⋊C₄	—	2	4	no	—
99	C₃×C₄⋊C₄	—	2	4	no	—
100	C₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
101	C₃×C₂²⋊C₄	—	2	4	no	—
102	C₃×C₄²	—	2	4	no	—
103	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
104	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
105	C₃×C₄²	#41	1	1	yes	characteristic
106	C₃×C₄²	C₂⁴	1	1	yes	characteristic
107	#208	—	4	4	no	—
108	C₂×C₃⋊C₈	—	2	4	no	—
109	C₂×C₃⋊C₈	—	2	4	no	—
110	C₂×C₃⋊C₈	—	2	2	no	—
111	C₂×C₃⋊C₈	—	4	2	no	—
112	C₂×C₃⋊C₈	C₂×D₄	1	2	yes	—
113	C₂×C₃⋊C₈	C₂×D₄	1	2	yes	—
114	C₂×Q₁₆	C₂×Dic₃	1	4	yes	—
115	C₂×M₁₆	—	6	2	no	—
116	C₂²×C₈	—	3	2	no	—
117	C₂²×C₈	—	6	1	no	—
118	C₂²×C₈	C₂×Dic₃	1	2	yes	—
119	C₄⋊Q₈	C₂×Dic₃	1	2	yes	—
120	C₄⋊Q₈	C₂×Dic₃	1	2	yes	—
121	#125	—	2	2	no	—
122	C₂²⋊Q₈	—	2	2	no	—
123	C₂²⋊Q₈	C₂×Dic₃	1	4	yes	—
124	C₄×Q₈	—	2	2	no	—
125	C₄×Q₈	C₂²×D₃	1	4	yes	—
126	#117	—	2	2	no	—
127	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
128	#107	C₂×Dic₃	1	4	yes	—
129	C₄⋊C₈	—	3	4	no	—
130	C₄⋊C₈	—	6	4	no	—
131	C₄⋊C₈	—	6	4	no	—
132	C₄⋊C₈	—	6	2	no	—
133	#103	C₂×Dic₃	1	8	yes	—
134	#97	—	6	4	no	—
135	C₄×C₈	—	3	2	no	—
136	C₄×C₈	—	3	1	no	—
137	C₄×C₈	—	3	1	no	—
138	C₄×C₈	—	3	2	no	—
139	C₄×C₈	—	6	4	no	—
140	C₄×C₈	—	6	1	no	—
141	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
142	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
143	C₄×C₈	C₂²×D₃	1	2	yes	—
144	C₂³×C₃	C₄×D₄	1	1	yes	characteristic
145	C₃×Q₈	—	2	8	no	—
146	C₃×Q₈	—	4	4	no	—
147	C₂×C₃×C₄	—	2	2	no	—
148	C₂×C₃×C₄	—	2	2	no	—
149	C₂×C₃×C₄	—	2	4	no	—
150	C₂×C₃×C₄	—	2	4	no	—
151	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
152	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
153	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
154	C₂×C₃×C₄	#137	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	C₂×#41	1	1	yes	characteristic
158	C₃×C₈	—	2	2	no	—
159	C₃×C₈	#131	1	4	yes	—
160	C₃⋊C₈	—	2	2	no	—
161	C₃⋊C₈	—	2	2	no	—
162	C₃⋊C₈	—	4	2	no	—
163	C₃⋊C₈	—	4	4	no	—
164	C₂×Q₈	—	2	2	no	—
165	C₂×Q₈	C₂²×Dic₃	1	4	yes	—
166	C₂²×C₄	D₃×D₄	1	1	yes	characteristic
167	C₂²×C₄	#237	1	1	yes	characteristic
168	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
169	Q₁₆	—	2	8	no	—
170	M₁₆	—	12	4	no	—
171	C₂×C₈	—	2	4	no	—
172	C₂×C₈	—	3	2	no	—
173	C₂×C₈	—	3	2	no	—
174	C₂×C₈	—	6	4	no	—
175	C₂×C₈	—	6	4	no	—
176	C₂×C₈	—	6	2	no	—
177	C₂×C₈	—	12	2	no	—
178	C₂×C₈	C₂²×Dic₃	1	2	yes	—
179	C₂×C₈	C₂²×Dic₃	1	2	yes	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	2	4	no	—
184	C₄⋊C₄	—	2	4	no	—
185	C₄⋊C₄	C₂²×Dic₃	1	4	yes	—
186	C₂²⋊C₄	—	2	4	no	—
187	C₄²	—	2	4	no	—
188	C₄²	D₃×D₄	1	1	yes	characteristic
189	C₄²	#237	1	1	yes	characteristic
190	C₄²	C₂²×Dic₃	1	1	yes	characteristic
191	C₄²	C₂³×D₃	1	1	yes	characteristic
192	C₂²×C₃	—	2	1	no	—
193	C₂²×C₃	—	2	1	no	—
194	C₂²×C₃	#441	1	2	yes	—
195	C₂²×C₃	C₂×C₄×D₄	1	1	yes	characteristic
196	C₃×C₄	—	2	1	no	—
197	C₃×C₄	—	2	1	no	—
198	C₃×C₄	—	4	4	no	—
199	C₃×C₄	—	4	4	no	—
200	C₃×C₄	#441	1	2	yes	—
201	C₃×C₄	#444	1	2	yes	—
202	C₃×C₄	C₂×#131	1	2	yes	—
203	C₃×C₄	C₂×#137	1	2	yes	—
204	C₂³	D₄×Dic₃	1	1	yes	characteristic
205	Q₈	—	2	8	no	—
206	Q₈	—	4	4	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	#918	1	4	yes	—
212	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
213	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
214	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
215	C₂×C₄	C₂×D₃×D₄	1	1	yes	characteristic
216	C₂×C₄	C₂×#237	1	1	yes	characteristic
217	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
218	C₈	—	2	2	no	—
219	C₈	—	6	2	no	—
220	C₈	—	6	2	no	—
221	C₈	—	12	2	no	—
222	C₈	—	12	4	no	—
223	C₈	#948	1	4	yes	—
224	C₂×C₃	—	2	2	no	—
225	C₂×C₃	#2931	1	1	yes	characteristic
226	C₂×C₃	C₂×#441	1	1	yes	characteristic
227	C₂×C₃	C₂×#444	1	1	yes	characteristic
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	#5184	1	2	yes	—
231	C₂²	C₂×D₄×Dic₃	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	4	4	no	—
235	C₄	—	4	4	no	—
236	C₄	#5184	1	2	yes	—
237	C₄	#5197	1	2	yes	—
238	C₄	C₂×#918	1	2	yes	—
239	C₄	C₂×#948	1	2	yes	—
240	C₃	#17961	1	1	yes	characteristic, complemented, Sylow
241	C₂	—	2	2	no	—
242	C₂	#83275	1	1	yes	characteristic
243	C₂	C₂×#5184	1	1	yes	characteristic
244	C₂	C₂×#5197	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃×D₄
commutator	C₂×C₃×C₄	C₂³×C₄
Frattini	C₄²	C₂³×D₃
Fitting	C₃×#3170	C₂
socle	C₂²×C₃	C₂×C₄×D₄
prime radical	C₂³×C₃	C₄×D₄
cosocle	C₃×C₄²	C₂⁴
hypercenter	#3170	D₃
hypocenter	C₃	#17961
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
4. #70906	C₂	semidirect
5. #70902	C₂	semidirect
240. C₃	#17961	semidirect

Maximal subgroups

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

subgroup	index
C₃×#3170	2
#83328	2
#71826	2
#70906	2
#70902	2
#69730	2
C₂×#4455	2
#17961	3