Subgroups — #10713304

Order 768.

310 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#364	C₂	1	1	yes	maximal, characteristic
2	#71428	C₂	1	2	yes	maximal
3	#69779	C₂	1	1	yes	maximal, characteristic
4	#69765	C₂	1	1	yes	maximal, characteristic
5	#67817	C₂	1	2	yes	maximal
6	#13181	—	3	1	no	maximal, Sylow
7	C₂×C₃×M₃₂	C₂²	1	2	yes	—
8	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₂×#819	C₂²	1	1	yes	characteristic
10	#5106	C₂²	1	1	yes	characteristic
11	C₂×#818	—	2	1	no	—
12	C₂×#818	C₂²	1	2	yes	—
13	#5101	—	2	1	no	—
14	#4935	—	2	1	no	—
15	C₃×#364	C₄	1	4	yes	complemented
16	#4523	—	2	1	no	—
17	#4505	—	2	2	no	—
18	#4504	—	2	2	no	—
19	#2159	—	3	2	no	—
20	C₂×#364	D₃	1	1	yes	characteristic
21	#1845	—	3	1	no	—
22	#1772	—	3	1	no	—
23	#1387	—	3	2	no	—
24	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
25	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
26	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
27	#926	—	2	1	no	—
28	C₂×C₄⋊Dic₃	—	2	2	no	—
29	C₂×C₄⋊Dic₃	—	4	2	no	—
30	C₂×#208	—	2	1	no	—
31	C₃×M₃₂	—	2	2	no	—
32	C₃×M₃₂	C₂×C₄	1	4	yes	—
33	C₂×C₃×C₁₆	—	2	2	no	—
34	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
35	C₃×C₄×C₈	Q₈	1	2	yes	—
36	#819	—	2	1	no	—
37	#819	Q₈	1	2	yes	—
38	#818	—	2	2	no	—
39	#818	—	2	2	no	—
40	#818	—	4	1	no	—
41	#818	—	4	1	no	—
42	#818	D₄	1	1	yes	characteristic
43	#818	D₄	1	1	yes	characteristic
44	#817	—	2	1	no	—
45	#815	—	4	1	no	—
46	C₈×Dic₃	—	4	1	no	—
47	C₂×M₃₂	C₂×D₃	1	2	yes	—
48	C₂×#108	—	3	1	no	—
49	#427	—	3	1	no	—
50	C₂×#107	—	3	2	no	—
51	C₂×#107	—	6	1	no	—
52	#405	—	6	1	no	—
53	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
54	#384	—	6	1	no	—
55	#367	—	6	2	no	—
56	#366	—	6	2	no	—
57	#364	Dic₃	1	4	yes	complemented
58	#339	—	6	1	no	—
59	C₂²×C₃×C₄	—	2	1	no	—
60	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
61	C₂²×Dic₃	—	4	2	no	—
62	C₂×C₃×C₈	C₄⋊C₄	1	2	yes	—
63	C₂×C₃×C₈	D₈	1	1	yes	characteristic
64	C₂×C₃×C₈	D₈	1	1	yes	characteristic
65	C₂×C₃×C₈	Q₁₆	1	2	yes	—
66	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₃×C₈	#41	1	1	yes	characteristic
71	C₂×C₃×C₈	#41	1	1	yes	characteristic
72	C₃×C₄²	—	2	1	no	—
73	C₃×C₄²	C₄⋊C₄	1	2	yes	—
74	#217	—	4	1	no	—
75	C₄⋊Dic₃	—	2	1	no	—
76	C₄⋊Dic₃	—	2	1	no	—
77	C₄⋊Dic₃	—	4	2	no	—
78	C₄⋊Dic₃	—	4	2	no	—
79	C₄⋊Dic₃	—	8	4	no	—
80	C₄×Dic₃	—	4	1	no	—
81	#208	—	2	2	no	—
82	#208	—	4	1	no	—
83	C₂×C₃⋊C₈	—	4	1	no	—
84	C₃×C₁₆	—	2	4	no	—
85	C₂×M₁₆	—	6	1	no	—
86	C₂²×C₈	D₁₂	1	1	yes	characteristic
87	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
88	#117	—	6	1	no	—
89	C₂×C₄⋊C₄	—	6	2	no	—
90	C₂×C₄⋊C₄	—	12	2	no	—
91	C₂×C₄²	D₁₂	1	1	yes	characteristic
92	M₃₂	—	2	2	no	—
93	M₃₂	C₂×Dic₃	1	4	yes	—
94	C₂×C₁₆	—	2	2	no	—
95	#108	—	3	2	no	—
96	#108	—	6	1	no	—
97	#107	—	3	1	no	—
98	#107	—	3	1	no	—
99	#107	—	6	2	no	—
100	#107	—	6	2	no	—
101	#107	—	12	1	no	—
102	#107	—	12	1	no	—
103	#106	—	6	1	no	—
104	#97	—	12	1	no	—
105	C₄×C₈	—	12	1	no	—
106	C₄×C₈	Dic₆	1	2	yes	—
107	C₄×C₈	C₂×Dic₃	1	2	yes	—
108	C₂³×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₂×C₃×C₄	—	2	2	no	—
110	C₂×C₃×C₄	—	2	2	no	—
111	C₂×C₃×C₄	—	2	2	no	—
112	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₃×C₄	#121	1	1	yes	characteristic
114	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
116	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
117	C₂×C₃×C₄	#137	1	1	yes	characteristic
118	C₂×Dic₃	—	4	1	no	—
119	C₂×Dic₃	—	4	1	no	—
120	C₂×Dic₃	—	8	4	no	—
121	C₂×Dic₃	—	8	2	no	—
122	C₃×C₈	#107	1	2	yes	—
123	C₃×C₈	#107	1	2	yes	—
124	C₃×C₈	C₄×D₄	1	1	yes	characteristic
125	C₃×C₈	C₄×Q₈	1	1	yes	characteristic
126	C₃×C₈	#125	1	1	yes	characteristic
127	C₃×C₈	C₄⋊Q₈	1	1	yes	characteristic
128	C₃⋊C₈	—	4	2	no	—
129	C₂²×C₄	—	2	1	no	—
130	C₂²×C₄	—	12	2	no	—
131	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
132	M₁₆	—	6	2	no	—
133	M₁₆	—	12	1	no	—
134	C₂×C₈	—	12	1	no	—
135	C₂×C₈	D₂₄	1	1	yes	characteristic
136	C₂×C₈	D₂₄	1	1	yes	characteristic
137	C₂×C₈	Dic₁₂	1	2	yes	—
138	C₂×C₈	C₄⋊Dic₃	1	2	yes	—
139	C₂×C₈	C₂×Dic₆	1	1	yes	characteristic
140	C₂×C₈	D₃×D₄	1	1	yes	characteristic
141	C₂×C₈	#237	1	1	yes	characteristic
142	C₂×C₈	D₃×Q₈	1	1	yes	characteristic
143	C₂×C₈	#239	1	1	yes	characteristic
144	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
145	C₄⋊C₄	—	6	1	no	—
146	C₄⋊C₄	—	6	1	no	—
147	C₄⋊C₄	—	12	2	no	—
148	C₄⋊C₄	—	12	2	no	—
149	C₄⋊C₄	—	24	4	no	—
150	C₂²⋊C₄	—	12	1	no	—
151	C₄²	—	2	1	no	—
152	C₄²	—	12	1	no	—
153	C₄²	C₄⋊Dic₃	1	2	yes	—
154	C₁₆	—	2	4	no	—
155	C₂²×C₃	—	2	1	no	—
156	C₂²×C₃	—	2	1	no	—
157	C₂²×C₃	#428	1	1	yes	characteristic
158	C₂²×C₃	#459	1	1	yes	characteristic
159	C₂²×C₃	#477	1	1	yes	characteristic
160	C₃×C₄	—	2	4	no	—
161	C₃×C₄	#389	1	1	yes	characteristic
162	C₃×C₄	C₂×#107	1	1	yes	characteristic
163	C₃×C₄	#462	1	1	yes	characteristic
164	C₃×C₄	#474	1	1	yes	characteristic
165	Dic₃	—	8	1	no	—
166	Dic₃	—	8	1	no	—
167	Dic₃	—	16	2	no	—
168	C₂³	#908	1	1	yes	characteristic
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	12	1	no	—
173	C₂×C₄	—	12	1	no	—
174	C₂×C₄	—	24	4	no	—
175	C₂×C₄	—	24	2	no	—
176	C₂×C₄	#895	1	1	yes	characteristic
177	C₂×C₄	#897	1	1	yes	characteristic
178	C₂×C₄	C₂×D₂₄	1	1	yes	characteristic
179	C₂×C₄	C₂×Dic₁₂	1	1	yes	characteristic
180	C₂×C₄	#909	1	1	yes	characteristic
181	C₂×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
182	C₈	—	12	2	no	—
183	C₈	#818	1	2	yes	—
184	C₈	#818	1	2	yes	—
185	C₈	#888	1	1	yes	characteristic
186	C₈	#890	1	1	yes	characteristic
187	C₈	D₄×Dic₃	1	1	yes	characteristic
188	C₈	Q₈×Dic₃	1	1	yes	characteristic
189	C₂×C₃	—	2	1	no	—
190	C₂×C₃	—	2	1	no	—
191	C₂×C₃	#1947	1	1	yes	characteristic
192	C₂×C₃	#2213	1	1	yes	characteristic
193	C₂×C₃	#2214	1	1	yes	characteristic
194	C₂²	—	2	1	no	—
195	C₂²	—	2	1	no	—
196	C₂²	#4843	1	1	yes	characteristic
197	C₂²	#4850	1	1	yes	characteristic
198	C₂²	#5121	1	1	yes	characteristic
199	C₄	—	2	4	no	—
200	C₄	—	24	1	no	—
201	C₄	—	24	1	no	—
202	C₄	—	48	2	no	—
203	C₄	#4846	1	1	yes	characteristic
204	C₄	#4849	1	1	yes	characteristic
205	C₄	#4984	1	1	yes	characteristic
206	C₄	C₂×#818	1	1	yes	characteristic
207	C₃	#13181	1	1	yes	characteristic, complemented, Sylow
208	C₂	—	2	1	no	—
209	C₂	—	2	1	no	—
210	C₂	#69305	1	1	yes	characteristic
211	C₂	#69306	1	1	yes	characteristic
212	C₂	#70453	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#4843
commutator	C₂×C₃×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₂×C₃×#364	C₂
socle	C₂²×C₃	#459
prime radical	C₂³×C₃	AGL(1,ℤ/8)
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	C₂×#364	D₃
hypocenter	C₃	#13181
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
15. C₃×#364	C₄	semidirect
57. #364	Dic₃	semidirect
207. C₃	#13181	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#364	2
#71428	2
#69779	2
#69765	2
#67817	2
#13181	3