Subgroups — #10713739

Order 768.

332 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#78442	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#364	C₂	1	1	yes	maximal, characteristic
3	#69779	C₂	1	1	yes	maximal, characteristic
4	#67822	C₂	1	2	yes	maximal
5	#67818	C₂	1	2	yes	maximal
6	#13271	—	3	1	no	maximal, Sylow
7	#5470	—	2	1	no	—
8	C₂×C₃×M₃₂	C₂²	1	2	yes	—
9	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₂×#819	C₂²	1	1	yes	characteristic
11	#5106	—	2	1	no	—
12	#5106	C₂²	1	2	yes	—
13	#5106	C₂²	1	1	yes	characteristic
14	#5101	—	2	1	no	—
15	#4683	—	2	1	no	—
16	#4683	C₄	1	2	yes	complemented
17	#4682	—	2	1	no	—
18	#4680	C₄	1	2	yes	complemented
19	C₃×#364	—	2	1	no	—
20	C₃×#364	—	2	1	no	—
21	#4523	—	2	1	no	—
22	#3160	—	3	1	no	—
23	C₂×#364	D₃	1	1	yes	characteristic
24	#1772	—	3	1	no	—
25	#1389	—	3	2	no	—
26	#1384	—	3	2	no	—
27	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
29	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
30	#926	—	2	2	no	—
31	#926	—	2	1	no	—
32	#924	—	4	2	no	—
33	C₂×#208	—	2	1	no	—
34	#870	—	2	2	no	—
35	#869	—	2	2	no	—
36	C₄×Dic₆	—	4	2	no	—
37	C₃×M₃₂	—	2	2	no	—
38	C₃×M₃₂	—	2	2	no	—
39	C₂×C₃×C₁₆	—	2	2	no	—
40	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
41	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
42	C₃×C₄×C₈	D₄	1	1	yes	characteristic
43	C₃×C₄×C₈	D₄	1	1	yes	characteristic
44	#819	—	2	1	no	—
45	#819	Q₈	1	2	yes	complemented
46	#818	—	2	2	no	—
47	#818	—	2	2	no	—
48	#818	—	2	1	no	—
49	#817	—	2	2	no	—
50	#817	C₂×C₄	1	4	yes	—
51	#817	Q₈	1	2	yes	complemented
52	#815	—	4	1	no	—
53	C₈×Dic₃	—	4	1	no	—
54	#533	—	6	1	no	—
55	C₂×M₃₂	C₂×D₃	1	2	yes	—
56	#500	—	6	1	no	—
57	#499	—	3	2	no	—
58	#499	—	6	1	no	—
59	#498	—	3	2	no	—
60	C₂×#108	—	3	1	no	—
61	#427	—	3	2	no	—
62	#427	—	3	1	no	—
63	#427	—	6	1	no	—
64	#405	—	6	1	no	—
65	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
66	#364	—	2	1	no	—
67	#364	—	2	1	no	—
68	#339	—	6	1	no	—
69	C₂²×C₃×C₄	—	2	1	no	—
70	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
71	C₂×Dic₆	—	4	2	no	—
72	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₈	SD₁₆	1	1	yes	characteristic
75	C₂×C₃×C₈	SD₁₆	1	1	yes	characteristic
76	C₂×C₃×C₈	Q₁₆	1	1	yes	characteristic
77	C₂×C₃×C₈	Q₁₆	1	1	yes	characteristic
78	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
81	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
82	C₂×C₃×C₈	#41	1	1	yes	characteristic
83	C₂×C₃×C₈	#41	1	1	yes	characteristic
84	C₃×C₄²	—	2	1	no	—
85	C₃×C₄²	C₂²⋊C₄	1	1	yes	characteristic
86	C₃×C₄²	C₂²⋊C₄	1	1	yes	characteristic
87	#217	—	4	2	no	—
88	#217	—	4	1	no	—
89	C₄⋊Dic₃	—	2	4	no	—
90	C₄⋊Dic₃	—	2	2	no	—
91	C₄⋊Dic₃	—	4	2	no	—
92	Dic₃⋊C₄	—	4	4	no	—
93	C₄×Dic₃	—	4	1	no	—
94	C₄×Dic₃	—	4	2	no	—
95	#208	—	2	2	no	—
96	#208	—	4	1	no	—
97	C₂×C₃⋊C₈	—	4	1	no	—
98	C₃×C₁₆	—	4	2	no	—
99	C₂×M₁₆	—	6	1	no	—
100	C₂²×C₈	D₁₂	1	1	yes	characteristic
101	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
102	C₄⋊Q₈	—	6	2	no	—
103	#125	—	6	2	no	—
104	C₂²⋊Q₈	—	12	2	no	—
105	C₄×Q₈	—	12	2	no	—
106	#117	—	6	2	no	—
107	#117	—	6	1	no	—
108	C₂×C₄²	D₁₂	1	1	yes	characteristic
109	M₃₂	—	2	2	no	—
110	M₃₂	—	2	2	no	—
111	C₂×C₁₆	—	2	2	no	—
112	#108	—	3	2	no	—
113	#108	—	6	1	no	—
114	#107	—	6	2	no	—
115	#107	—	6	1	no	—
116	#107	—	6	2	no	—
117	#106	—	3	4	no	—
118	#106	—	3	2	no	—
119	#106	—	6	2	no	—
120	#97	—	12	1	no	—
121	C₄×C₈	—	12	1	no	—
122	C₄×C₈	C₄×D₃	1	1	yes	characteristic
123	C₄×C₈	C₄×D₃	1	1	yes	characteristic
124	C₄×C₈	#67	1	1	yes	characteristic
125	C₄×C₈	#67	1	1	yes	characteristic
126	C₂³×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	—	2	1	no	—
129	C₂×C₃×C₄	—	4	2	no	—
130	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
135	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₂×Dic₃	—	4	4	no	—
137	C₂×Dic₃	—	4	2	no	—
138	Dic₆	—	8	4	no	—
139	C₃×C₈	#103	1	1	yes	characteristic
140	C₃×C₈	#103	1	1	yes	characteristic
141	C₃×C₈	#103	1	1	yes	characteristic
142	C₃×C₈	#103	1	1	yes	characteristic
143	C₃×C₈	C₄×Q₈	1	1	yes	characteristic
144	C₃×C₈	C₄×Q₈	1	1	yes	characteristic
145	C₃×C₈	#124	1	1	yes	characteristic
146	C₃×C₈	C₄⋊Q₈	1	1	yes	characteristic
147	C₃⋊C₈	—	4	2	no	—
148	C₂×Q₈	—	12	2	no	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
151	M₁₆	—	6	2	no	—
152	M₁₆	—	12	1	no	—
153	C₂×C₈	—	12	1	no	—
154	C₂×C₈	#204	1	1	yes	characteristic
155	C₂×C₈	#204	1	1	yes	characteristic
156	C₂×C₈	Dic₁₂	1	1	yes	characteristic
157	C₂×C₈	Dic₁₂	1	1	yes	characteristic
158	C₂×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
159	C₂×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
160	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
161	C₂×C₈	#237	1	1	yes	characteristic
162	C₂×C₈	#237	1	1	yes	characteristic
163	C₂×C₈	D₃×Q₈	1	1	yes	characteristic
164	C₂×C₈	D₃×Q₈	1	1	yes	characteristic
165	C₂×C₈	C₂×#67	1	1	yes	characteristic
166	C₄⋊C₄	—	6	2	no	—
167	C₄⋊C₄	—	6	4	no	—
168	C₄⋊C₄	—	12	4	no	—
169	C₄⋊C₄	—	12	2	no	—
170	C₂²⋊C₄	—	12	2	no	—
171	C₂²⋊C₄	—	12	1	no	—
172	C₄²	—	2	1	no	—
173	C₄²	—	12	1	no	—
174	C₄²	—	12	2	no	—
175	C₄²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
176	C₄²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
177	C₁₆	—	4	2	no	—
178	C₂²×C₃	—	4	1	no	—
179	C₂²×C₃	#418	1	1	yes	characteristic
180	C₂²×C₃	#476	1	1	yes	characteristic
181	C₂²×C₃	#477	1	1	yes	characteristic
182	C₃×C₄	—	2	2	no	—
183	C₃×C₄	—	4	1	no	—
184	C₃×C₄	#391	1	1	yes	characteristic
185	C₃×C₄	C₂×#103	1	1	yes	characteristic
186	C₃×C₄	#476	1	1	yes	characteristic
187	C₃×C₄	#477	1	1	yes	characteristic
188	Dic₃	—	8	4	no	—
189	Dic₃	—	8	2	no	—
190	C₂³	#908	1	1	yes	characteristic
191	Q₈	—	24	4	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	12	4	no	—
196	C₂×C₄	—	12	2	no	—
197	C₂×C₄	#897	1	1	yes	characteristic
198	C₂×C₄	#897	1	1	yes	characteristic
199	C₂×C₄	C₂×#204	1	1	yes	characteristic
200	C₂×C₄	C₂×Dic₁₂	1	1	yes	characteristic
201	C₂×C₄	#908	1	1	yes	characteristic
202	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
203	C₈	—	12	2	no	—
204	C₈	#816	1	1	yes	characteristic
205	C₈	#816	1	1	yes	characteristic
206	C₈	#816	1	1	yes	characteristic
207	C₈	#816	1	1	yes	characteristic
208	C₈	#887	1	1	yes	characteristic
209	C₈	#887	1	1	yes	characteristic
210	C₈	#936	1	1	yes	characteristic
211	C₈	#944	1	1	yes	characteristic
212	C₂×C₃	—	2	1	no	—
213	C₂×C₃	—	2	1	no	—
214	C₂×C₃	—	2	1	no	—
215	C₂×C₃	#1987	1	1	yes	characteristic
216	C₂²	—	4	1	no	—
217	C₂²	#4842	1	1	yes	characteristic
218	C₂²	#4850	1	1	yes	characteristic
219	C₂²	#5131	1	1	yes	characteristic
220	C₄	—	2	2	no	—
221	C₄	—	4	1	no	—
222	C₄	—	24	4	no	—
223	C₄	—	24	2	no	—
224	C₄	#4842	1	1	yes	characteristic
225	C₄	#4850	1	1	yes	characteristic
226	C₄	#4973	1	1	yes	characteristic
227	C₄	C₂×#816	1	1	yes	characteristic
228	C₃	#13271	1	1	yes	characteristic, complemented, Sylow
229	C₂	—	2	1	no	—
230	C₂	—	2	1	no	—
231	C₂	—	2	1	no	—
232	C₂	#70459	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#4842
commutator	C₂×C₃×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₂×C₃×#364	C₂
socle	C₂×C₃	#1987
prime radical	C₂³×C₃	AGL(1,ℤ/8)
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	C₂×#364	D₃
hypocenter	C₃	#13271
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
16. #4683	C₄	semidirect
18. #4680	C₄	semidirect
45. #819	Q₈	semidirect
51. #817	Q₈	semidirect
228. C₃	#13271	semidirect

Maximal subgroups

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

subgroup	index
#78442	2
C₂×C₃×#364	2
#69779	2
#67822	2
#67818	2
#13271	3