Subgroups — #10693840

Order 768.

382 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	#69765	C₂	1	1	yes	maximal, characteristic
3	C₂×#4458	C₂	1	1	yes	maximal, characteristic
4	#69467	C₂	1	2	yes	maximal
5	#67702	C₂	1	2	yes	maximal
6	#9615	—	3	1	no	maximal, Sylow
7	C₂×C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄×C₈	—	2	1	no	—
9	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₃×C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₂×#818	—	2	1	no	—
12	C₂×#818	—	2	1	no	—
13	C₂×#808	—	2	2	no	—
14	#4935	C₂²	1	2	yes	—
15	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
16	C₃×#336	—	2	1	no	—
17	#4529	—	2	2	no	—
18	#4458	C₄	1	4	yes	complemented
19	#1948	—	3	2	no	—
20	#1845	—	3	1	no	—
21	C₈×C₄⋊C₄	D₃	1	1	yes	characteristic
22	C₂×#335	—	3	1	no	—
23	#1296	—	3	2	no	—
24	C₂²×C₃×C₈	—	2	1	no	—
25	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
26	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
27	C₂×C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
28	C₂×C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
29	C₂×C₃×C₄²	—	2	1	no	—
30	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊Dic₃	—	2	2	no	—
32	C₂×C₄⋊Dic₃	—	2	2	no	—
33	C₂×C₄⋊Dic₃	—	2	2	no	—
34	C₂²×C₃⋊C₈	—	2	2	no	—
35	C₃×C₄⋊C₈	—	2	1	no	—
36	C₃×C₄⋊C₈	Q₈	1	2	yes	—
37	C₃×C₄×C₈	—	4	2	no	—
38	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
39	C₃×C₄×C₈	Q₈	1	2	yes	—
40	C₃×#95	—	2	1	no	—
41	#818	—	4	2	no	—
42	#818	—	4	2	no	—
43	#808	—	4	4	no	—
44	C₄⋊(C₃⋊C₈)	—	2	2	no	—
45	C₄⋊(C₃⋊C₈)	C₂×C₄	1	4	yes	—
46	C₂×#107	—	6	1	no	—
47	C₂×#107	—	6	1	no	—
48	C₂×#106	—	6	2	no	—
49	C₂×C₄⋊C₈	—	3	2	no	—
50	C₂×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
51	C₂×C₄×C₈	—	2	1	no	—
52	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
53	#384	—	3	2	no	—
54	C₄×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
55	#340	—	6	2	no	—
56	#336	—	2	1	no	—
57	#335	—	3	4	no	—
58	C₂²×C₃×C₄	—	2	1	no	—
59	C₂²×C₃×C₄	—	2	1	no	—
60	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₃×C₄	#41	1	1	yes	characteristic
63	C₂²×Dic₃	—	4	2	no	—
64	C₂×C₃×C₈	—	2	1	no	—
65	C₂×C₃×C₈	—	2	1	no	—
66	C₂×C₃×C₈	—	2	2	no	—
67	C₂×C₃×C₈	—	4	1	no	—
68	C₂×C₃×C₈	—	4	1	no	—
69	C₂×C₃×C₈	C₄⋊C₄	1	2	yes	—
70	C₂×C₃×C₈	C₄⋊C₄	1	2	yes	—
71	C₂×C₃×C₈	SD₁₆	1	2	yes	—
72	C₂×C₃×C₈	SD₁₆	1	2	yes	—
73	C₃×C₄⋊C₄	—	2	1	no	—
74	C₃×C₄⋊C₄	—	2	1	no	—
75	C₃×C₄⋊C₄	D₈	1	2	yes	—
76	C₃×C₄⋊C₄	Q₁₆	1	2	yes	—
77	C₃×C₄²	—	4	1	no	—
78	C₃×C₄²	—	4	1	no	—
79	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
80	C₃×C₄²	C₂×Q₈	1	1	yes	characteristic
81	C₃×C₄²	C₂×Q₈	1	1	yes	characteristic
82	C₃×C₄²	#41	1	1	yes	characteristic
83	C₄⋊Dic₃	—	4	2	no	—
84	C₄⋊Dic₃	—	4	2	no	—
85	C₄⋊Dic₃	—	4	2	no	—
86	C₄⋊Dic₃	—	4	2	no	—
87	C₄⋊Dic₃	—	8	2	no	—
88	C₂×C₃⋊C₈	—	2	4	no	—
89	C₂×C₃⋊C₈	—	4	2	no	—
90	C₂×C₃⋊C₈	—	4	2	no	—
91	C₂²×C₈	—	2	1	no	—
92	C₂²×C₈	—	6	2	no	—
93	C₂²×C₈	#67	1	1	yes	characteristic
94	C₂²×C₈	#67	1	1	yes	characteristic
95	C₂×C₄⋊C₄	—	6	2	no	—
96	C₂×C₄⋊C₄	—	6	2	no	—
97	C₂×C₄⋊C₄	—	6	2	no	—
98	C₂×C₄⋊C₄	D₁₂	1	1	yes	characteristic
99	C₂×C₄⋊C₄	D₁₂	1	1	yes	characteristic
100	C₂×C₄²	—	2	1	no	—
101	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
102	#107	—	12	2	no	—
103	#107	—	12	2	no	—
104	#106	—	12	4	no	—
105	C₄⋊C₈	—	2	1	no	—
106	C₄⋊C₈	—	3	4	no	—
107	C₄⋊C₈	—	6	2	no	—
108	C₄⋊C₈	Dic₆	1	2	yes	—
109	C₄×C₈	—	4	2	no	—
110	C₄×C₈	Dic₆	1	2	yes	—
111	C₄×C₈	C₄×D₃	1	2	yes	—
112	#95	—	2	1	no	—
113	C₂³×C₃	#121	1	1	yes	characteristic
114	C₂×C₃×C₄	—	2	1	no	—
115	C₂×C₃×C₄	—	2	1	no	—
116	C₂×C₃×C₄	—	2	2	no	—
117	C₂×C₃×C₄	—	2	2	no	—
118	C₂×C₃×C₄	—	4	1	no	—
119	C₂×C₃×C₄	—	4	1	no	—
120	C₂×C₃×C₄	—	4	1	no	—
121	C₂×C₃×C₄	—	4	1	no	—
122	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
124	C₂×C₃×C₄	C₄×Q₈	1	1	yes	characteristic
125	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
126	C₂×C₃×C₄	#125	1	1	yes	characteristic
127	C₂×C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
128	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
129	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
131	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
132	C₂×C₃×C₄	#135	1	1	yes	characteristic
133	C₂×C₃×C₄	#135	1	1	yes	characteristic
134	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₂×C₃×C₄	#137	1	1	yes	characteristic
136	C₂×Dic₃	—	8	2	no	—
137	C₂×Dic₃	—	8	2	no	—
138	C₂×Dic₃	—	8	2	no	—
139	C₃×C₈	—	2	2	no	—
140	C₃×C₈	—	4	2	no	—
141	C₃×C₈	#106	1	4	yes	complemented
142	C₃⋊C₈	—	4	4	no	—
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	—	12	2	no	—
146	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
147	C₂²×C₄	#235	1	1	yes	characteristic
148	C₂²×C₄	C₂×#67	1	1	yes	characteristic
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	4	1	no	—
153	C₂×C₈	—	4	1	no	—
154	C₂×C₈	—	6	4	no	—
155	C₂×C₈	—	12	2	no	—
156	C₂×C₈	—	12	2	no	—
157	C₂×C₈	Dic₃⋊C₄	1	2	yes	—
158	C₂×C₈	Dic₃⋊C₄	1	2	yes	—
159	C₂×C₈	#214	1	2	yes	—
160	C₂×C₈	#215	1	2	yes	—
161	C₄⋊C₄	—	2	1	no	—
162	C₄⋊C₄	—	2	1	no	—
163	C₄⋊C₄	—	12	2	no	—
164	C₄⋊C₄	—	12	2	no	—
165	C₄⋊C₄	—	12	2	no	—
166	C₄⋊C₄	—	12	2	no	—
167	C₄⋊C₄	—	24	2	no	—
168	C₄⋊C₄	D₂₄	1	2	yes	—
169	C₄⋊C₄	Dic₁₂	1	2	yes	—
170	C₄²	—	4	1	no	—
171	C₄²	—	4	1	no	—
172	C₄²	C₂×Dic₆	1	1	yes	characteristic
173	C₄²	C₂×Dic₆	1	1	yes	characteristic
174	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
175	C₄²	#235	1	1	yes	characteristic
176	C₂²×C₃	#389	1	1	yes	characteristic
177	C₂²×C₃	#460	1	1	yes	characteristic
178	C₂²×C₃	#461	1	1	yes	characteristic
179	C₂²×C₃	#466	1	1	yes	characteristic
180	C₂²×C₃	#467	1	1	yes	characteristic
181	C₂²×C₃	#472	1	1	yes	characteristic
182	C₂²×C₃	#473	1	1	yes	characteristic
183	C₃×C₄	—	2	1	no	—
184	C₃×C₄	—	2	1	no	—
185	C₃×C₄	—	4	2	no	—
186	C₃×C₄	—	4	2	no	—
187	C₃×C₄	C₂×#106	1	1	yes	characteristic
188	C₃×C₄	#428	1	1	yes	characteristic
189	C₃×C₄	C₄×D₈	1	1	yes	characteristic
190	C₃×C₄	C₄×Q₁₆	1	1	yes	characteristic
191	C₃×C₄	#475	1	1	yes	characteristic
192	C₃×C₄	#476	1	1	yes	characteristic
193	C₃×C₄	#499	1	1	yes	characteristic
194	C₃×C₄	#500	1	1	yes	characteristic
195	Dic₃	—	16	2	no	—
196	C₂³	#930	1	1	yes	characteristic
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	1	no	—
205	C₂×C₄	—	24	2	no	—
206	C₂×C₄	—	24	2	no	—
207	C₂×C₄	—	24	2	no	—
208	C₂×C₄	C₄×Dic₆	1	1	yes	characteristic
209	C₂×C₄	#869	1	1	yes	characteristic
210	C₂×C₄	#870	1	1	yes	characteristic
211	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
212	C₂×C₄	C₂×D₂₄	1	1	yes	characteristic
213	C₂×C₄	#904	1	1	yes	characteristic
214	C₂×C₄	#904	1	1	yes	characteristic
215	C₂×C₄	C₂×Dic₁₂	1	1	yes	characteristic
216	C₂×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
217	C₂×C₄	#924	1	1	yes	characteristic
218	C₂×C₄	C₂×#214	1	1	yes	characteristic
219	C₂×C₄	C₂×#215	1	1	yes	characteristic
220	C₂×C₄	#949	1	1	yes	characteristic
221	C₂×C₄	#951	1	1	yes	characteristic
222	C₈	—	2	2	no	—
223	C₈	—	4	2	no	—
224	C₈	—	12	4	no	—
225	C₈	#808	1	4	yes	complemented
226	C₂×C₃	#1668	1	1	yes	characteristic
227	C₂×C₃	#1670	1	1	yes	characteristic
228	C₂×C₃	#1673	1	1	yes	characteristic
229	C₂×C₃	#1675	1	1	yes	characteristic
230	C₂×C₃	#1942	1	1	yes	characteristic
231	C₂×C₃	#1948	1	1	yes	characteristic
232	C₂×C₃	#1952	1	1	yes	characteristic
233	C₂²	#4928	1	1	yes	characteristic
234	C₂²	#5019	1	1	yes	characteristic
235	C₂²	#5027	1	1	yes	characteristic
236	C₂²	#5115	1	1	yes	characteristic
237	C₂²	#5116	1	1	yes	characteristic
238	C₂²	#5142	1	1	yes	characteristic
239	C₂²	#5143	1	1	yes	characteristic
240	C₄	—	2	1	no	—
241	C₄	—	2	1	no	—
242	C₄	—	4	2	no	—
243	C₄	—	4	2	no	—
244	C₄	—	48	2	no	—
245	C₄	#4682	1	1	yes	characteristic
246	C₄	#4683	1	1	yes	characteristic
247	C₄	C₄×D₂₄	1	1	yes	characteristic
248	C₄	C₄×Dic₁₂	1	1	yes	characteristic
249	C₄	C₂×#808	1	1	yes	characteristic
250	C₄	#4995	1	1	yes	characteristic
251	C₄	#5008	1	1	yes	characteristic
252	C₄	#5020	1	1	yes	characteristic
253	C₃	#9615	1	1	yes	characteristic, complemented, Sylow
254	C₂	#69467	1	1	yes	characteristic
255	C₂	#69770	1	1	yes	characteristic
256	C₂	#69894	1	1	yes	characteristic
257	C₂	#70852	1	1	yes	characteristic
258	C₂	#70853	1	1	yes	characteristic
259	C₂	#70939	1	1	yes	characteristic
260	C₂	#70942	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#930
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×C₈×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₃	#121
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₈×C₄⋊C₄	D₃
hypocenter	C₃	#9615
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
18. #4458	C₄	semidirect
141. C₃×C₈	#106	semidirect
225. C₈	#808	semidirect
253. C₃	#9615	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₈×C₄⋊C₄	2
#69765	2
C₂×#4458	2
#69467	2
#67702	2
#9615	3