Subgroups — #10693978

Order 768.

530 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	C₃×C₄×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#4535	C₂	1	4	yes	maximal
3	#69392	C₂	1	1	yes	maximal, characteristic
4	#69388	C₂	1	1	yes	maximal, characteristic
5	#9641	—	3	1	no	maximal, Sylow
6	C₂×C₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
7	C₂×C₃×C₄×Q₈	C₂²	1	2	yes	—
8	C₃×#391	C₄	1	2	yes	—
9	C₃×#389	—	2	2	no	—
10	C₃×#384	C₄	1	1	yes	characteristic
11	C₃×C₄×C₄⋊C₄	—	2	2	no	—
12	C₃×C₄³	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₄×C₃⋊C₈	C₂²	1	2	yes	—
15	#4535	—	2	8	no	—
16	#4524	—	2	2	no	—
17	#4524	—	2	2	no	—
18	C₄×C₄⋊Q₈	D₃	1	1	yes	characteristic
19	#1831	—	3	1	no	—
20	#1744	—	3	1	no	—
21	C₂×Q₈⋊C₈	—	3	4	no	—
22	C₂²×C₃×Q₈	C₂×C₄	1	2	yes	—
23	C₃×C₄⋊Q₈	—	2	2	no	—
24	C₃×C₄⋊Q₈	C₈	1	4	yes	complemented
25	C₃×C₄×Q₈	—	2	4	no	—
26	C₃×C₄×Q₈	—	2	4	no	—
27	C₂×C₃×C₄⋊C₄	—	2	2	no	—
28	C₂×C₃×C₄⋊C₄	—	2	2	no	—
29	C₂×C₃×C₄⋊C₄	—	2	2	no	—
30	C₂×C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₄²	—	2	2	no	—
32	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃×C₄²	D₄	1	2	yes	—
35	C₂×C₃×C₄²	D₄	1	2	yes	—
36	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
37	C₂²×C₃⋊C₈	—	2	2	no	—
38	C₂²×C₃⋊C₈	—	2	2	no	—
39	C₃×#95	—	2	4	no	—
40	C₄⋊(C₃⋊C₈)	—	2	4	no	—
41	C₄×C₃⋊C₈	—	2	2	no	—
42	C₄×C₃⋊C₈	D₄	1	4	yes	—
43	C₂×C₄⋊Q₈	Dic₃	1	1	yes	characteristic
44	C₂×C₄×Q₈	C₂×D₃	1	2	yes	—
45	C₂×C₄⋊C₈	—	3	2	no	—
46	C₂×C₄×C₈	—	3	2	no	—
47	#391	Dic₃	1	2	yes	—
48	#389	—	2	2	no	—
49	#384	Dic₃	1	1	yes	characteristic
50	C₄×C₄⋊C₄	—	2	2	no	—
51	C₄³	C₂×D₃	1	1	yes	characteristic
52	#336	—	6	2	no	—
53	#336	—	6	2	no	—
54	Q₈⋊C₈	—	6	8	no	—
55	C₂×C₃×Q₈	—	2	2	no	—
56	C₂×C₃×Q₈	—	2	4	no	—
57	C₂×C₃×Q₈	—	2	4	no	—
58	C₂×C₃×Q₈	C₂×C₈	1	4	yes	—
59	C₂²×C₃×C₄	—	2	2	no	—
60	C₂²×C₃×C₄	—	2	2	no	—
61	C₂²×C₃×C₄	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₃×C₄	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
66	C₃×C₄⋊C₄	—	2	8	no	—
67	C₃×C₄⋊C₄	—	2	4	no	—
68	C₃×C₄⋊C₄	—	4	4	no	—
69	C₃×C₄⋊C₄	—	4	4	no	—
70	C₃×C₄²	—	2	2	no	—
71	C₃×C₄²	—	2	4	no	—
72	C₃×C₄²	—	4	2	no	—
73	C₃×C₄²	—	4	2	no	—
74	C₃×C₄²	—	4	2	no	—
75	C₃×C₄²	C₂×C₈	1	2	yes	—
76	C₃×C₄²	M₁₆	1	2	yes	—
77	C₃×C₄²	C₂×D₄	1	2	yes	—
78	C₃×C₄²	#41	1	2	yes	—
79	C₂×C₃⋊C₈	—	2	4	no	—
80	C₂×C₃⋊C₈	—	4	4	no	—
81	C₂×C₃⋊C₈	—	4	2	no	—
82	C₂×C₃⋊C₈	—	4	4	no	—
83	C₂²×Q₈	C₂×Dic₃	1	2	yes	—
84	C₂²×C₈	—	6	2	no	—
85	C₂²×C₈	—	6	2	no	—
86	C₄⋊Q₈	—	2	2	no	—
87	C₄⋊Q₈	C₃⋊C₈	1	4	yes	complemented
88	C₄×Q₈	—	2	4	no	—
89	C₄×Q₈	—	2	4	no	—
90	C₂×C₄⋊C₄	—	2	2	no	—
91	C₂×C₄⋊C₄	—	2	2	no	—
92	C₂×C₄⋊C₄	—	2	2	no	—
93	C₂×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
94	C₂×C₄²	—	2	2	no	—
95	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
96	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
97	C₂×C₄²	#67	1	2	yes	—
98	C₂×C₄²	#67	1	2	yes	—
99	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
100	C₄⋊C₈	—	6	4	no	—
101	C₄×C₈	—	3	4	no	—
102	C₄×C₈	—	6	2	no	—
103	#95	—	2	4	no	—
104	C₂³×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₃×Q₈	—	2	8	no	—
106	C₃×Q₈	—	4	4	no	—
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	—	2	4	no	—
111	C₂×C₃×C₄	—	2	2	no	—
112	C₂×C₃×C₄	—	2	2	no	—
113	C₂×C₃×C₄	—	2	2	no	—
114	C₂×C₃×C₄	—	4	2	no	—
115	C₂×C₃×C₄	—	4	2	no	—
116	C₂×C₃×C₄	—	4	2	no	—
117	C₂×C₃×C₄	—	4	4	no	—
118	C₂×C₃×C₄	C₂²⋊C₈	1	4	yes	—
119	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
120	C₂×C₃×C₄	#121	1	2	yes	—
121	C₂×C₃×C₄	#124	1	1	yes	characteristic
122	C₂×C₃×C₄	#127	1	1	yes	characteristic
123	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
125	C₂×C₃×C₄	#131	1	2	yes	—
126	C₂×C₃×C₄	#137	1	4	yes	—
127	C₃⋊C₈	—	4	4	no	—
128	C₃⋊C₈	—	8	2	no	—
129	C₂×Q₈	—	2	2	no	—
130	C₂×Q₈	—	2	4	no	—
131	C₂×Q₈	—	2	4	no	—
132	C₂×Q₈	C₂×C₃⋊C₈	1	4	yes	—
133	C₂²×C₄	—	2	2	no	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	#217	1	2	yes	—
136	C₂²×C₄	#217	1	2	yes	—
137	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
138	C₂²×C₄	C₂×#67	1	1	yes	characteristic
139	C₂²×C₄	C₂×#67	1	1	yes	characteristic
140	C₂×C₈	—	6	4	no	—
141	C₂×C₈	—	12	4	no	—
142	C₂×C₈	—	12	2	no	—
143	C₂×C₈	—	12	4	no	—
144	C₄⋊C₄	—	2	8	no	—
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	4	4	no	—
147	C₄⋊C₄	—	4	4	no	—
148	C₄²	—	2	4	no	—
149	C₄²	—	2	2	no	—
150	C₄²	—	4	2	no	—
151	C₄²	—	4	2	no	—
152	C₄²	—	4	2	no	—
153	C₄²	C₂×C₃⋊C₈	1	2	yes	—
154	C₄²	#208	1	2	yes	—
155	C₄²	D₃×D₄	1	2	yes	—
156	C₄²	#237	1	2	yes	—
157	C₂²×C₃	#390	1	1	yes	characteristic
158	C₂²×C₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
159	C₂²×C₃	#408	1	1	yes	characteristic
160	C₂²×C₃	#419	1	2	yes	—
161	C₂²×C₃	#421	1	2	yes	—
162	C₃×C₄	—	2	2	no	—
163	C₃×C₄	—	2	4	no	—
164	C₃×C₄	—	4	4	no	—
165	C₃×C₄	—	4	2	no	—
166	C₃×C₄	—	8	2	no	—
167	C₃×C₄	C₈×D₄	1	2	yes	—
168	C₃×C₄	#436	1	2	yes	—
169	C₃×C₄	#491	1	2	yes	—
170	C₃×C₄	#497	1	2	yes	—
171	C₂³	C₂×#217	1	1	yes	characteristic
172	Q₈	—	2	8	no	—
173	Q₈	—	4	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	4	4	no	—
185	C₂×C₄	#830	1	4	yes	—
186	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
187	C₂×C₄	C₂×#208	1	1	yes	characteristic
188	C₂×C₄	D₄×Dic₃	1	2	yes	—
189	C₂×C₄	#936	1	1	yes	characteristic
190	C₂×C₄	#938	1	2	yes	—
191	C₂×C₄	#940	1	1	yes	characteristic
192	C₂×C₄	#942	1	4	yes	—
193	C₂×C₄	#948	1	2	yes	—
194	C₈	—	12	4	no	—
195	C₈	—	24	2	no	—
196	C₂×C₃	#1479	1	2	yes	—
197	C₂×C₃	#1497	1	2	yes	—
198	C₂×C₃	#1954	1	1	yes	characteristic
199	C₂×C₃	#1960	1	1	yes	characteristic
200	C₂×C₃	#1963	1	1	yes	characteristic
201	C₂²	C₂×#830	1	1	yes	characteristic
202	C₂²	#5221	1	1	yes	characteristic
203	C₂²	#5225	1	2	yes	—
204	C₂²	#5234	1	1	yes	characteristic
205	C₂²	#5236	1	2	yes	—
206	C₄	—	2	2	no	—
207	C₄	—	2	4	no	—
208	C₄	—	4	2	no	—
209	C₄	—	4	4	no	—
210	C₄	—	8	2	no	—
211	C₄	D₄×C₃⋊C₈	1	2	yes	—
212	C₄	#5012	1	2	yes	—
213	C₄	#5082	1	2	yes	—
214	C₄	#5086	1	2	yes	—
215	C₃	#9641	1	1	yes	characteristic, complemented, Sylow
216	C₂	#71614	1	1	yes	characteristic
217	C₂	#71621	1	2	yes	—
218	C₂	#71786	1	1	yes	characteristic
219	C₂	#71816	1	1	yes	characteristic
220	C₂	#71827	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×C₄×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₃	C₂×C₂²⋊C₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₄×C₄⋊Q₈	D₃
hypocenter	C₃	#9641
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
24. C₃×C₄⋊Q₈	C₈	semidirect
87. C₄⋊Q₈	C₃⋊C₈	semidirect
215. C₃	#9641	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₄⋊Q₈	2
C₂×#4535	2
#69392	2
#69388	2
#9641	3