Subgroups — #10694230

Order 768.

570 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#73311	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#4482	C₂	1	4	yes	maximal
4	#69394	C₂	1	1	yes	maximal, characteristic
5	#9694	—	3	1	no	maximal, Sylow
6	C₂×#869	—	2	1	no	—
7	C₂×#869	—	2	1	no	—
8	C₂×#869	C₂²	1	2	yes	—
9	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₃×C₄×C₈	—	2	1	no	—
11	C₃×C₄³	C₂²	1	1	yes	characteristic
12	#4935	—	2	2	no	—
13	#4935	C₄	1	4	yes	—
14	C₂×C₄⋊(C₃⋊C₈)	—	2	2	no	—
15	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
16	C₃×#336	—	2	1	no	—
17	#4482	—	2	8	no	—
18	#2868	—	3	1	no	—
19	#1842	—	3	1	no	—
20	C₄×C₄⋊C₈	D₃	1	1	yes	characteristic
21	C₂×#332	—	3	4	no	—
22	C₂²×Dic₆	—	2	2	no	—
23	C₂²×C₃×C₈	—	2	2	no	—
24	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
25	C₂×C₃×C₄²	D₄	1	2	yes	—
26	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
27	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
28	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊Dic₃	—	2	4	no	—
30	C₂×C₄⋊Dic₃	—	2	4	no	—
31	C₂×C₄⋊Dic₃	—	2	2	no	—
32	C₂×C₄⋊Dic₃	C₂×C₄	1	4	yes	—
33	C₂²×C₃⋊C₈	—	2	2	no	—
34	#869	—	2	4	no	—
35	#869	—	2	4	no	—
36	#869	—	4	4	no	—
37	#869	—	4	2	no	—
38	#869	—	4	2	no	—
39	C₃×C₄⋊C₈	—	2	2	no	—
40	C₃×C₄⋊C₈	D₄	1	4	yes	—
41	C₃×C₄×C₈	—	4	2	no	—
42	C₄⋊(C₃⋊C₈)	—	2	4	no	—
43	C₄⋊(C₃⋊C₈)	—	4	4	no	—
44	C₂×C₄⋊Q₈	—	3	2	no	—
45	C₂×C₄⋊Q₈	—	6	1	no	—
46	C₂×C₄⋊Q₈	—	6	1	no	—
47	C₂×C₄⋊C₈	—	3	2	no	—
48	C₂×C₄⋊C₈	—	6	2	no	—
49	C₂×C₄⋊C₈	C₂×D₃	1	2	yes	—
50	C₂×C₄×C₈	—	2	1	no	—
51	#384	—	3	4	no	—
52	#384	—	6	2	no	—
53	C₄³	C₂×D₃	1	1	yes	characteristic
54	#336	—	2	1	no	—
55	#332	—	6	8	no	—
56	C₂²×C₃×C₄	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₃×C₄	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
61	C₂²×Dic₃	—	2	4	no	—
62	C₂×Dic₆	—	4	4	no	—
63	C₂×Dic₆	—	4	2	no	—
64	C₂×Dic₆	—	4	2	no	—
65	C₂×Dic₆	—	4	4	no	—
66	C₂×C₃×C₈	—	2	4	no	—
67	C₂×C₃×C₈	—	4	2	no	—
68	C₂×C₃×C₈	—	4	2	no	—
69	C₃×C₄²	—	2	4	no	—
70	C₃×C₄²	—	2	2	no	—
71	C₃×C₄²	D₈	1	2	yes	—
72	C₃×C₄²	SD₁₆	1	4	yes	—
73	C₃×C₄²	SD₁₆	1	2	yes	—
74	C₃×C₄²	Q₁₆	1	4	yes	—
75	C₃×C₄²	C₂×D₄	1	2	yes	—
76	C₃×C₄²	#41	1	2	yes	—
77	C₄⋊Dic₃	—	2	8	no	—
78	C₄⋊Dic₃	—	4	8	no	—
79	C₄⋊Dic₃	—	4	8	no	—
80	C₄⋊Dic₃	—	4	2	no	—
81	C₄⋊Dic₃	—	4	2	no	—
82	C₂×C₃⋊C₈	—	4	2	no	—
83	C₂×C₃⋊C₈	—	4	4	no	—
84	C₂²×Q₈	—	6	2	no	—
85	C₂²×C₈	—	2	2	no	—
86	C₂²×C₈	—	6	2	no	—
87	C₄⋊Q₈	—	6	4	no	—
88	C₄⋊Q₈	—	6	4	no	—
89	C₄⋊Q₈	—	12	4	no	—
90	C₄⋊Q₈	—	12	2	no	—
91	C₄⋊Q₈	—	12	2	no	—
92	C₂×C₄⋊C₄	—	3	4	no	—
93	C₂×C₄⋊C₄	—	6	4	no	—
94	C₂×C₄⋊C₄	—	6	4	no	—
95	C₂×C₄⋊C₄	—	6	2	no	—
96	C₂×C₄²	C₄×D₃	1	2	yes	—
97	C₂×C₄²	D₁₂	1	2	yes	—
98	C₂×C₄²	#67	1	1	yes	characteristic
99	C₂×C₄²	#67	1	1	yes	characteristic
100	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
101	C₄⋊C₈	—	2	2	no	—
102	C₄⋊C₈	—	6	4	no	—
103	C₄⋊C₈	—	12	4	no	—
104	C₄⋊C₈	D₁₂	1	4	yes	—
105	C₄×C₈	—	4	2	no	—
106	C₂³×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₃×C₄	—	2	2	no	—
108	C₂×C₃×C₄	—	2	2	no	—
109	C₂×C₃×C₄	—	2	2	no	—
110	C₂×C₃×C₄	#101	1	4	yes	—
111	C₂×C₃×C₄	#102	1	4	yes	—
112	C₂×C₃×C₄	#103	1	4	yes	—
113	C₂×C₃×C₄	#103	1	4	yes	—
114	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
115	C₂×C₃×C₄	#121	1	2	yes	—
116	C₂×C₃×C₄	#124	1	1	yes	characteristic
117	C₂×C₃×C₄	#127	1	1	yes	characteristic
118	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
119	C₂×C₃×C₄	C₂×SD₁₆	1	2	yes	—
120	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×C₃×C₄	C₂×Q₁₆	1	2	yes	—
122	C₂×C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
123	C₂×Dic₃	—	4	4	no	—
124	C₂×Dic₃	—	4	4	no	—
125	C₂×Dic₃	—	4	4	no	—
126	Dic₆	—	8	4	no	—
127	Dic₆	—	8	4	no	—
128	C₃×C₈	—	4	4	no	—
129	C₃⋊C₈	—	8	2	no	—
130	C₂×Q₈	—	12	4	no	—
131	C₂×Q₈	—	12	2	no	—
132	C₂×Q₈	—	12	2	no	—
133	C₂×Q₈	—	12	4	no	—
134	C₂²×C₄	—	6	4	no	—
135	C₂²×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
136	C₂²×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
137	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
138	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
139	C₂²×C₄	C₂×#67	1	1	yes	characteristic
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	12	2	no	—
144	C₂×C₈	—	12	4	no	—
145	C₄⋊C₄	—	6	8	no	—
146	C₄⋊C₄	—	12	8	no	—
147	C₄⋊C₄	—	12	8	no	—
148	C₄⋊C₄	—	12	2	no	—
149	C₄⋊C₄	—	12	2	no	—
150	C₄²	—	2	4	no	—
151	C₄²	—	2	2	no	—
152	C₄²	#204	1	4	yes	—
153	C₄²	Dic₁₂	1	4	yes	—
154	C₄²	#213	1	2	yes	—
155	C₄²	#215	1	2	yes	—
156	C₄²	C₂×D₁₂	1	2	yes	—
157	C₄²	#235	1	2	yes	—
158	C₂²×C₃	#390	1	1	yes	characteristic
159	C₂²×C₃	C₂×#101	1	2	yes	—
160	C₂²×C₃	C₂×#102	1	1	yes	characteristic
161	C₂²×C₃	C₂×#103	1	2	yes	—
162	C₂²×C₃	#418	1	1	yes	characteristic
163	C₃×C₄	—	2	4	no	—
164	C₃×C₄	—	2	2	no	—
165	C₃×C₄	#332	1	8	yes	complemented
166	C₃×C₄	#459	1	2	yes	—
167	C₃×C₄	#460	1	2	yes	—
168	C₃×C₄	#487	1	2	yes	—
169	C₃×C₄	#492	1	1	yes	characteristic
170	C₃×C₄	#494	1	1	yes	characteristic
171	Dic₃	—	8	4	no	—
172	C₂³	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
173	Q₈	—	24	4	no	—
174	Q₈	—	24	4	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	12	4	no	—
179	C₂×C₄	—	12	4	no	—
180	C₂×C₄	—	12	4	no	—
181	C₂×C₄	#809	1	4	yes	—
182	C₂×C₄	#816	1	4	yes	—
183	C₂×C₄	#816	1	4	yes	—
184	C₂×C₄	#824	1	4	yes	—
185	C₂×C₄	C₄×D₁₂	1	2	yes	—
186	C₂×C₄	#874	1	1	yes	characteristic
187	C₂×C₄	#875	1	1	yes	characteristic
188	C₂×C₄	C₂×#204	1	2	yes	—
189	C₂×C₄	C₂×Dic₁₂	1	2	yes	—
190	C₂×C₄	#930	1	2	yes	—
191	C₂×C₄	C₂×#213	1	1	yes	characteristic
192	C₂×C₄	C₂×#215	1	1	yes	characteristic
193	C₂×C₄	#949	1	2	yes	—
194	C₈	—	4	4	no	—
195	C₈	—	24	2	no	—
196	C₂×C₃	C₂×#332	1	2	yes	—
197	C₂×C₃	#1549	1	2	yes	—
198	C₂×C₃	#1956	1	1	yes	characteristic
199	C₂×C₃	#1966	1	1	yes	characteristic
200	C₂×C₃	#1982	1	1	yes	characteristic
201	C₂²	#4939	1	1	yes	characteristic
202	C₂²	C₂×#809	1	1	yes	characteristic
203	C₂²	#5001	1	1	yes	characteristic
204	C₂²	C₂×#816	1	2	yes	—
205	C₂²	C₂×#824	1	2	yes	—
206	C₄	—	2	4	no	—
207	C₄	—	2	2	no	—
208	C₄	—	24	4	no	—
209	C₄	#4482	1	8	yes	complemented
210	C₄	#4681	1	2	yes	—
211	C₄	#4693	1	1	yes	characteristic
212	C₄	#4699	1	1	yes	characteristic
213	C₄	#5015	1	2	yes	—
214	C₄	#5027	1	2	yes	—
215	C₃	#9694	1	1	yes	characteristic, complemented, Sylow
216	C₂	#69487	1	1	yes	characteristic
217	C₂	#69771	1	1	yes	characteristic
218	C₂	#69877	1	1	yes	characteristic
219	C₂	C₂×#4482	1	2	yes	—
220	C₂	#70780	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×(C₂×D₃)⋊C₄
commutator	C₂²×C₃×C₄	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×C₄×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₃	C₂×C₂²⋊C₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₄×C₄⋊C₈	D₃
hypocenter	C₃	#9694
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
165. C₃×C₄	#332	semidirect
209. C₄	#4482	semidirect
215. C₃	#9694	semidirect

Maximal subgroups

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

subgroup	index
#73311	2
C₃×C₄×C₄⋊C₈	2
C₂×#4482	2
#69394	2
#9694	3