Subgroups — #10705276

Order 768.

378 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3169	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#69731	C₂	1	1	yes	maximal, characteristic, complemented
4	#67763	C₂	1	4	yes	maximal, complemented
5	#11671	—	3	1	no	maximal, Sylow
6	C₃×#532	—	2	1	no	—
7	C₃×#500	C₄	1	1	yes	characteristic
8	C₃×#493	C₄	1	1	yes	characteristic
9	C₃×#486	—	2	1	no	—
10	C₃×#466	—	2	1	no	—
11	C₃×#466	C₄	1	2	yes	—
12	C₃×C₄×D₈	—	2	1	no	—
13	C₃×C₄×D₈	C₂²	1	2	yes	—
14	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂²×C₃⋊C₁₆	—	2	1	no	—
16	#4924	—	2	1	no	—
17	C₄×C₃⋊C₁₆	—	2	1	no	—
18	C₄×C₃⋊C₁₆	C₂²	1	2	yes	—
19	#4458	—	2	1	no	—
20	#4458	C₂²	1	2	yes	—
21	#3169	D₃	1	1	yes	characteristic
22	C₂×C₄×C₁₆	—	3	1	no	—
23	#1567	—	3	1	no	—
24	#1331	—	3	4	no	—
25	C₂×C₃×#41	—	4	1	no	—
26	C₂×C₃×D₈	—	2	1	no	—
27	C₂×C₃×D₈	C₂×C₄	1	2	yes	—
28	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
30	C₃×C₄⋊Q₈	—	2	1	no	—
31	C₃×#127	—	2	1	no	—
32	C₃×#124	—	4	1	no	—
33	C₃×#121	—	2	2	no	—
34	C₃×#121	—	4	1	no	—
35	C₃×C₄×D₄	—	2	2	no	—
36	C₃×C₄×D₄	—	4	1	no	—
37	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
38	C₂×#208	—	4	1	no	—
39	C₃×#107	—	2	1	no	—
40	C₃×#107	C₂×C₄	1	2	yes	—
41	C₃×#102	—	2	2	no	—
42	C₃×#102	—	2	2	no	—
43	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
44	C₃×C₄×C₈	D₄	1	1	yes	characteristic
45	C₃×C₄×C₈	D₄	1	1	yes	characteristic
46	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
47	C₂×C₃⋊C₁₆	—	2	4	no	—
48	C₂×C₃⋊C₁₆	—	2	2	no	—
49	C₄⋊(C₃⋊C₈)	—	2	2	no	—
50	C₄⋊(C₃⋊C₈)	—	4	1	no	—
51	#532	—	2	1	no	—
52	C₂²×C₁₆	—	6	1	no	—
53	#500	Dic₃	1	1	yes	characteristic
54	#493	Dic₃	1	1	yes	characteristic
55	#486	—	2	1	no	—
56	#466	—	2	1	no	—
57	#466	Dic₃	1	2	yes	—
58	C₄×D₈	—	2	1	no	—
59	C₄×D₈	C₂×D₃	1	2	yes	—
60	#423	—	6	1	no	—
61	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
62	C₄×C₁₆	—	3	2	no	—
63	C₄×C₁₆	—	6	1	no	—
64	#335	—	3	2	no	—
65	#335	—	6	1	no	—
66	C₃×#41	—	8	2	no	—
67	C₃×#41	—	8	2	no	—
68	C₂×C₃×Q₈	—	4	1	no	—
69	C₂×C₃×D₄	—	2	2	no	—
70	C₂×C₃×D₄	—	4	2	no	—
71	C₂×C₃×D₄	—	4	1	no	—
72	C₂×C₃×D₄	—	4	1	no	—
73	C₂²×C₃×C₄	—	4	2	no	—
74	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
76	C₂²×C₃×C₄	SD₁₆	1	1	yes	characteristic
77	C₃×D₈	—	2	4	no	—
78	C₃×D₈	—	4	2	no	—
79	C₂×C₃×C₈	—	2	2	no	—
80	C₂×C₃×C₈	—	2	2	no	—
81	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
85	C₃×C₄⋊C₄	—	2	2	no	—
86	C₃×C₄⋊C₄	—	4	1	no	—
87	C₃×C₂²⋊C₄	—	4	2	no	—
88	C₃×C₂²⋊C₄	—	4	2	no	—
89	C₃×C₄²	C₂²⋊C₄	1	1	yes	characteristic
90	C₃×C₄²	D₈	1	1	yes	characteristic
91	C₃×C₄²	SD₁₆	1	1	yes	characteristic
92	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
93	#208	—	8	2	no	—
94	C₂×C₃⋊C₈	—	4	2	no	—
95	C₃⋊C₁₆	—	2	4	no	—
96	C₂×#41	—	4	1	no	—
97	C₂×D₈	—	2	1	no	—
98	C₂×D₈	C₂×Dic₃	1	2	yes	—
99	C₂×M₁₆	—	12	1	no	—
100	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
101	C₂²×C₈	#67	1	1	yes	characteristic
102	C₄⋊Q₈	—	2	1	no	—
103	#127	—	2	1	no	—
104	#124	—	4	1	no	—
105	#121	—	2	2	no	—
106	#121	—	4	1	no	—
107	C₄×D₄	—	2	2	no	—
108	C₄×D₄	—	4	1	no	—
109	C₂×C₄²	#67	1	1	yes	characteristic
110	C₂×C₁₆	—	6	4	no	—
111	C₂×C₁₆	—	6	2	no	—
112	#107	—	2	1	no	—
113	#107	C₂×Dic₃	1	2	yes	—
114	C₄⋊C₈	—	6	2	no	—
115	C₄⋊C₈	—	12	1	no	—
116	#102	—	2	2	no	—
117	#102	—	2	2	no	—
118	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
119	C₄×C₈	#67	1	1	yes	characteristic
120	C₄×C₈	#67	1	1	yes	characteristic
121	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
122	C₂³×C₃	—	4	2	no	—
123	C₂³×C₃	#102	1	1	yes	characteristic
124	C₃×Q₈	—	8	2	no	—
125	C₃×D₄	—	4	2	no	—
126	C₃×D₄	—	4	2	no	—
127	C₃×D₄	—	8	4	no	—
128	C₃×D₄	—	8	2	no	—
129	C₃×D₄	—	8	2	no	—
130	C₂×C₃×C₄	—	2	2	no	—
131	C₂×C₃×C₄	—	2	1	no	—
132	C₂×C₃×C₄	—	2	1	no	—
133	C₂×C₃×C₄	—	4	2	no	—
134	C₂×C₃×C₄	—	8	2	no	—
135	C₂×C₃×C₄	—	8	2	no	—
136	C₂×C₃×C₄	#102	1	1	yes	characteristic
137	C₂×C₃×C₄	#102	1	1	yes	characteristic
138	C₂×C₃×C₄	#102	1	1	yes	characteristic
139	C₂×C₃×C₄	D₁₆	1	2	yes	—
140	C₂×C₃×C₄	SD₃₂	1	2	yes	—
141	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
144	C₃×C₈	—	2	1	no	—
145	C₃×C₈	—	2	1	no	—
146	C₃×C₈	#104	1	2	yes	—
147	C₃×C₈	#131	1	2	yes	—
148	C₃⋊C₈	—	8	2	no	—
149	#41	—	8	2	no	—
150	#41	—	8	2	no	—
151	C₂×Q₈	—	4	1	no	—
152	C₂×D₄	—	2	2	no	—
153	C₂×D₄	—	4	2	no	—
154	C₂×D₄	—	4	1	no	—
155	C₂×D₄	—	4	1	no	—
156	C₂²×C₄	—	4	2	no	—
157	C₂²×C₄	#213	1	1	yes	characteristic
158	C₂²×C₄	#214	1	1	yes	characteristic
159	C₂²×C₄	#217	1	1	yes	characteristic
160	D₈	—	2	4	no	—
161	D₈	—	4	2	no	—
162	M₁₆	—	24	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	12	2	no	—
166	C₂×C₈	#217	1	1	yes	characteristic
167	C₂×C₈	#217	1	1	yes	characteristic
168	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
169	C₂×C₈	C₂×#67	1	1	yes	characteristic
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	4	1	no	—
172	C₂²⋊C₄	—	4	2	no	—
173	C₂²⋊C₄	—	4	2	no	—
174	C₄²	#213	1	1	yes	characteristic
175	C₄²	#214	1	1	yes	characteristic
176	C₄²	#217	1	1	yes	characteristic
177	C₄²	C₂×#67	1	1	yes	characteristic
178	C₁₆	—	6	4	no	—
179	C₂²×C₃	—	2	1	no	—
180	C₂²×C₃	—	2	1	no	—
181	C₂²×C₃	—	8	2	no	—
182	C₂²×C₃	—	8	2	no	—
183	C₂²×C₃	—	8	2	no	—
184	C₂²×C₃	#357	1	2	yes	—
185	C₂²×C₃	C₂×#102	1	1	yes	characteristic
186	C₃×C₄	—	2	1	no	—
187	C₃×C₄	—	2	1	no	—
188	C₃×C₄	—	8	2	no	—
189	C₃×C₄	#357	1	2	yes	—
190	C₃×C₄	#359	1	2	yes	—
191	C₃×C₄	#408	1	1	yes	characteristic
192	C₃×C₄	C₂×#104	1	1	yes	characteristic
193	C₃×C₄	C₂×D₁₆	1	1	yes	characteristic
194	C₃×C₄	C₂×SD₃₂	1	1	yes	characteristic
195	C₂³	—	4	2	no	—
196	C₂³	#832	1	1	yes	characteristic
197	Q₈	—	8	2	no	—
198	D₄	—	4	2	no	—
199	D₄	—	4	2	no	—
200	D₄	—	8	4	no	—
201	D₄	—	8	2	no	—
202	D₄	—	8	2	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	8	2	no	—
208	C₂×C₄	—	8	2	no	—
209	C₂×C₄	#826	1	2	yes	—
210	C₂×C₄	#827	1	2	yes	—
211	C₂×C₄	#832	1	1	yes	characteristic
212	C₂×C₄	#832	1	1	yes	characteristic
213	C₂×C₄	#832	1	1	yes	characteristic
214	C₂×C₄	C₂×#213	1	1	yes	characteristic
215	C₂×C₄	C₂×#214	1	1	yes	characteristic
216	C₂×C₄	C₂×#217	1	1	yes	characteristic
217	C₈	—	2	1	no	—
218	C₈	—	2	1	no	—
219	C₈	—	24	2	no	—
220	C₈	#837	1	2	yes	—
221	C₈	#948	1	2	yes	—
222	C₂×C₃	—	2	2	no	—
223	C₂×C₃	—	16	2	no	—
224	C₂×C₃	#1492	1	1	yes	characteristic
225	C₂×C₃	C₂×#357	1	1	yes	characteristic
226	C₂×C₃	C₂×#359	1	1	yes	characteristic
227	C₂²	—	2	1	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	8	2	no	—
230	C₂²	—	8	2	no	—
231	C₂²	—	8	2	no	—
232	C₂²	#4562	1	2	yes	—
233	C₂²	C₂×#832	1	1	yes	characteristic
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	8	2	no	—
237	C₄	#4562	1	2	yes	—
238	C₄	#4567	1	2	yes	—
239	C₄	C₂×#826	1	1	yes	characteristic
240	C₄	C₂×#827	1	1	yes	characteristic
241	C₄	#5234	1	1	yes	characteristic
242	C₄	C₂×#837	1	1	yes	characteristic
243	C₃	#11671	1	1	yes	characteristic, complemented, Sylow
244	C₂	—	2	2	no	—
245	C₂	—	16	2	no	—
246	C₂	#71802	1	1	yes	characteristic
247	C₂	C₂×#4562	1	1	yes	characteristic
248	C₂	C₂×#4567	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#213
commutator	C₂×C₃×C₈	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₃
Fitting, prime radical	C₃×#3169	C₂
socle	C₂²×C₃	C₂×#102
cosocle	C₃×C₄×C₈	C₂³
hypercenter	#3169	D₃
hypocenter	C₃	#11671
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3169

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×C₃⋊C₁₆	C₂	semidirect
3. #69731	C₂	semidirect
4. #67763	C₂	semidirect
243. C₃	#11671	semidirect

Maximal subgroups

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

subgroup	index
C₃×#3169	2
C₂×C₄×C₃⋊C₁₆	2
#69731	2
#67763	2
#11671	3