Subgroups — #10693759

Order 768.

378 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#322	C₂	1	1	yes	maximal, characteristic
2	#69786	C₂	1	2	yes	maximal
3	C₄×#817	C₂	1	1	yes	maximal, characteristic
4	C₂×#4458	C₂	1	1	yes	maximal, characteristic
5	#67655	C₂	1	2	yes	maximal
6	#9600	—	3	1	no	maximal, Sylow
7	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₂×#817	C₄	1	2	yes	—
11	C₂×Dic₃⋊C₈	—	2	2	no	—
12	C₄×C₄⋊Dic₃	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
14	C₃×#322	—	2	2	no	—
15	#4551	C₄	1	1	yes	characteristic
16	#4551	C₄	1	1	yes	characteristic
17	#4550	—	2	2	no	—
18	#4458	—	2	2	no	—
19	#1939	—	3	2	no	—
20	C₄×#106	—	3	1	no	—
21	C₂×#335	—	3	1	no	—
22	C₂×#322	D₃	1	1	yes	characteristic
23	#1279	—	3	2	no	—
24	C₂²×C₃×C₈	—	2	1	no	—
25	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
29	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
31	C₂×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
32	C₂×C₄×Dic₃	—	2	2	no	—
33	C₂²×C₃⋊C₈	—	2	2	no	—
34	C₃×C₄×C₈	—	2	1	no	—
35	C₃×C₄×C₈	—	2	1	no	—
36	C₃×C₄×C₈	—	2	1	no	—
37	C₃×C₄×C₈	D₄	1	2	yes	—
38	C₃×C₄×C₈	Q₈	1	2	yes	—
39	C₃×C₄×C₈	Q₈	1	2	yes	—
40	C₃⋊#95	—	2	2	no	—
41	#817	—	2	4	no	—
42	Dic₃⋊C₈	—	4	4	no	—
43	C₄⋊(C₃⋊C₈)	—	2	2	no	—
44	C₄⋊(C₃⋊C₈)	—	2	2	no	—
45	C₂×#106	—	3	2	no	—
46	C₂×C₄⋊C₈	—	3	2	no	—
47	C₂×C₄⋊C₈	—	6	2	no	—
48	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
49	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
50	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
51	C₄×C₄⋊C₄	—	3	2	no	—
52	#340	—	3	1	no	—
53	#340	—	3	1	no	—
54	#336	—	6	2	no	—
55	#335	—	6	2	no	—
56	#322	—	2	2	no	—
57	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₃×C₄	#41	1	1	yes	characteristic
60	C₂²×Dic₃	—	2	2	no	—
61	C₂²×Dic₃	—	2	2	no	—
62	C₂×C₃×C₈	—	2	2	no	—
63	C₂×C₃×C₈	—	2	2	no	—
64	C₂×C₃×C₈	—	2	2	no	—
65	C₂×C₃×C₈	—	2	2	no	—
66	C₂×C₃×C₈	—	2	2	no	—
67	C₂×C₃×C₈	—	2	1	no	—
68	C₂×C₃×C₈	—	2	1	no	—
69	C₂×C₃×C₈	—	2	2	no	—
70	C₂×C₃×C₈	—	2	2	no	—
71	C₂×C₃×C₈	C₄⋊C₄	1	2	yes	—
72	C₂×C₃×C₈	C₄⋊C₄	1	2	yes	—
73	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
74	C₃×C₄²	C₂×Q₈	1	1	yes	characteristic
75	C₃×C₄²	C₂×Q₈	1	1	yes	characteristic
76	C₃×C₄²	#41	1	1	yes	characteristic
77	C₄⋊Dic₃	—	2	2	no	—
78	C₄⋊Dic₃	—	2	2	no	—
79	C₄⋊Dic₃	—	2	2	no	—
80	C₄⋊Dic₃	M₁₆	1	4	yes	—
81	C₄×Dic₃	—	4	2	no	—
82	C₄×Dic₃	—	4	2	no	—
83	C₂×C₃⋊C₈	—	4	2	no	—
84	C₂×C₃⋊C₈	—	4	2	no	—
85	C₂×C₃⋊C₈	—	4	2	no	—
86	C₂²×C₈	—	2	1	no	—
87	C₂²×C₈	—	6	2	no	—
88	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
89	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
90	C₂²×C₈	D₁₂	1	1	yes	characteristic
91	C₂²×C₈	D₁₂	1	1	yes	characteristic
92	C₂×C₄⋊C₄	—	3	2	no	—
93	C₂×C₄⋊C₄	—	3	2	no	—
94	C₂×C₄²	—	6	2	no	—
95	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
96	#106	—	6	4	no	—
97	C₄⋊C₈	—	6	2	no	—
98	C₄⋊C₈	—	6	2	no	—
99	C₄⋊C₈	—	12	4	no	—
100	C₄×C₈	—	2	1	no	—
101	C₄×C₈	—	2	1	no	—
102	C₄×C₈	—	2	1	no	—
103	C₄×C₈	Dic₆	1	2	yes	—
104	C₄×C₈	Dic₆	1	2	yes	—
105	C₄×C₈	#67	1	2	yes	—
106	#95	—	6	2	no	—
107	C₂³×C₃	C₄×D₄	1	1	yes	characteristic
108	C₂×C₃×C₄	—	2	1	no	—
109	C₂×C₃×C₄	—	2	1	no	—
110	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₃×C₄	C₄×Q₈	1	1	yes	characteristic
112	C₂×C₃×C₄	#121	1	1	yes	characteristic
113	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂×C₃×C₄	#125	1	1	yes	characteristic
115	C₂×C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
116	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
117	C₂×C₃×C₄	#131	1	2	yes	—
118	C₂×C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
119	C₂×C₃×C₄	#137	1	2	yes	—
120	C₂×Dic₃	—	2	4	no	—
121	C₂×Dic₃	—	4	2	no	—
122	C₂×Dic₃	—	4	2	no	—
123	C₂×Dic₃	—	4	2	no	—
124	C₂×Dic₃	—	4	2	no	—
125	C₂×Dic₃	—	4	2	no	—
126	C₃×C₈	—	2	4	no	—
127	C₃×C₈	—	2	2	no	—
128	C₃×C₈	—	2	2	no	—
129	C₃×C₈	—	4	1	no	—
130	C₃×C₈	—	4	1	no	—
131	C₃⋊C₈	—	8	2	no	—
132	C₂²×C₄	—	6	2	no	—
133	C₂²×C₄	—	6	2	no	—
134	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
135	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
136	C₂²×C₄	#235	1	1	yes	characteristic
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	12	2	no	—
147	C₂×C₈	—	12	2	no	—
148	C₂×C₈	—	12	2	no	—
149	C₂×C₈	Dic₃⋊C₄	1	2	yes	—
150	C₂×C₈	Dic₃⋊C₄	1	2	yes	—
151	C₄⋊C₄	—	3	4	no	—
152	C₄⋊C₄	—	6	2	no	—
153	C₄⋊C₄	—	6	2	no	—
154	C₄⋊C₄	—	6	2	no	—
155	C₄²	—	12	2	no	—
156	C₄²	—	12	2	no	—
157	C₄²	C₂×Dic₆	1	1	yes	characteristic
158	C₄²	C₂×Dic₆	1	1	yes	characteristic
159	C₄²	#235	1	1	yes	characteristic
160	C₄²	C₂×#67	1	1	yes	characteristic
161	C₂²×C₃	#389	1	1	yes	characteristic
162	C₂²×C₃	#435	1	2	yes	—
163	C₂²×C₃	#440	1	2	yes	—
164	C₂²×C₃	#443	1	1	yes	characteristic
165	C₂²×C₃	#443	1	1	yes	characteristic
166	C₃×C₄	—	2	1	no	—
167	C₃×C₄	—	2	1	no	—
168	C₃×C₄	#423	1	1	yes	characteristic
169	C₃×C₄	#424	1	1	yes	characteristic
170	C₃×C₄	#446	1	2	yes	—
171	C₃×C₄	#469	1	1	yes	characteristic
172	C₃×C₄	#470	1	1	yes	characteristic
173	C₃×C₄	#501	1	2	yes	—
174	Dic₃	—	4	4	no	—
175	Dic₃	—	8	2	no	—
176	C₂³	C₄×D₁₂	1	1	yes	characteristic
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	6	4	no	—
180	C₂×C₄	—	12	2	no	—
181	C₂×C₄	—	12	2	no	—
182	C₂×C₄	—	12	2	no	—
183	C₂×C₄	—	12	2	no	—
184	C₂×C₄	—	12	2	no	—
185	C₂×C₄	C₄×Dic₆	1	1	yes	characteristic
186	C₂×C₄	#869	1	1	yes	characteristic
187	C₂×C₄	#870	1	1	yes	characteristic
188	C₂×C₄	D₃×M₁₆	1	2	yes	—
189	C₂×C₄	#907	1	2	yes	—
190	C₂×C₄	#908	1	2	yes	—
191	C₂×C₄	#909	1	2	yes	—
192	C₂×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
193	C₂×C₄	#924	1	1	yes	characteristic
194	C₂×C₄	#930	1	1	yes	characteristic
195	C₈	—	2	4	no	—
196	C₈	—	2	2	no	—
197	C₈	—	2	2	no	—
198	C₈	—	4	1	no	—
199	C₈	—	4	1	no	—
200	C₈	—	24	2	no	—
201	C₂×C₃	#1585	1	2	yes	—
202	C₂×C₃	#1588	1	2	yes	—
203	C₂×C₃	#1940	1	1	yes	characteristic
204	C₂×C₃	#1944	1	1	yes	characteristic
205	C₂×C₃	#1945	1	1	yes	characteristic
206	C₂²	#4700	1	1	yes	characteristic
207	C₂²	#4700	1	1	yes	characteristic
208	C₂²	#4706	1	2	yes	—
209	C₂²	#4710	1	2	yes	—
210	C₂²	#4928	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	12	4	no	—
214	C₄	—	24	2	no	—
215	C₄	#4701	1	2	yes	—
216	C₄	#4702	1	2	yes	—
217	C₄	#5118	1	1	yes	characteristic
218	C₄	#5119	1	1	yes	characteristic
219	C₄	#5135	1	1	yes	characteristic
220	C₄	#5137	1	1	yes	characteristic
221	C₃	#9600	1	1	yes	characteristic, complemented, Sylow
222	C₂	#68080	1	2	yes	—
223	C₂	#68088	1	2	yes	—
224	C₂	#69794	1	1	yes	characteristic
225	C₂	#69820	1	1	yes	characteristic
226	C₂	#69836	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₂
commutator	C₂²×C₃×C₄	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₂×C₃×#322	C₂
socle, prime radical	C₂³×C₃	C₄×D₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₂×#322	D₃
hypocenter	C₃	#9600
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
221. C₃	#9600	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#322	2
#69786	2
C₄×#817	2
C₂×#4458	2
#67655	2
#9600	3