Subgroups — #10693740

Order 768.

382 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×#106	C₂	1	1	yes	maximal, characteristic
2	C₂×#4458	C₂	1	1	yes	maximal, characteristic
3	C₂×#4455	C₂	1	1	yes	maximal, characteristic
4	#69484	C₂	1	2	yes	maximal
5	#67619	C₂	1	2	yes	maximal
6	#9596	—	3	1	no	maximal, Sylow
7	C₂×C₃×#106	C₄	1	1	yes	characteristic
8	C₂×C₃×#106	C₄	1	1	yes	characteristic
9	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₃×C₄×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₄⋊(C₃⋊C₈)	—	2	2	no	—
12	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₃×#340	C₄	1	2	yes	—
16	#4524	—	2	2	no	—
17	#4458	—	2	2	no	—
18	#4455	—	2	2	no	—
19	#1939	—	3	2	no	—
20	C₄×#106	D₃	1	1	yes	characteristic
21	C₂×#335	—	3	1	no	—
22	C₂×#322	—	3	1	no	—
23	#1279	—	3	2	no	—
24	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
27	C₂×C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₄²	—	2	2	no	—
29	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
30	C₂²×C₃⋊C₈	—	2	1	no	—
31	C₂²×C₃⋊C₈	—	2	2	no	—
32	C₂²×C₃⋊C₈	D₄	1	1	yes	characteristic
33	C₂²×C₃⋊C₈	D₄	1	1	yes	characteristic
34	C₃×#106	—	2	2	no	—
35	C₃×#106	C₈	1	4	yes	complemented
36	C₃×C₄×C₈	D₄	1	2	yes	—
37	C₃×C₄×C₈	Q₈	1	2	yes	—
38	C₃×#95	—	2	2	no	—
39	C₄⋊(C₃⋊C₈)	—	2	2	no	—
40	C₄⋊(C₃⋊C₈)	—	2	2	no	—
41	C₄⋊(C₃⋊C₈)	—	4	4	no	—
42	C₄×C₃⋊C₈	—	2	1	no	—
43	C₄×C₃⋊C₈	—	2	1	no	—
44	C₄×C₃⋊C₈	—	2	1	no	—
45	C₄×C₃⋊C₈	Q₈	1	2	yes	—
46	C₂×#106	Dic₃	1	1	yes	characteristic
47	C₂×#106	Dic₃	1	1	yes	characteristic
48	C₂×C₄⋊C₈	—	3	2	no	—
49	C₂×C₄⋊C₈	—	6	2	no	—
50	C₂×C₄×C₈	—	3	1	no	—
51	C₂×C₄×C₈	—	3	1	no	—
52	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
53	C₄×C₄⋊C₄	C₂×D₃	1	2	yes	—
54	#340	Dic₃	1	2	yes	—
55	#336	—	6	2	no	—
56	#335	—	6	2	no	—
57	#322	—	6	2	no	—
58	C₂²×C₃×C₄	—	2	2	no	—
59	C₂²×C₃×C₄	—	2	2	no	—
60	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₃×C₄	#41	1	1	yes	characteristic
63	C₂×C₃×C₈	—	2	1	no	—
64	C₂×C₃×C₈	—	2	1	no	—
65	C₂×C₃×C₈	C₄⋊C₄	1	2	yes	—
66	C₂×C₃×C₈	C₄⋊C₄	1	2	yes	—
67	C₂×C₃×C₈	C₂×C₈	1	2	yes	—
68	C₂×C₃×C₈	M₁₆	1	2	yes	—
69	C₃×C₄⋊C₄	—	2	2	no	—
70	C₃×C₄⋊C₄	—	2	2	no	—
71	C₃×C₄⋊C₄	—	2	2	no	—
72	C₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
73	C₃×C₄²	—	4	2	no	—
74	C₃×C₄²	—	4	2	no	—
75	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
76	C₃×C₄²	C₂×Q₈	1	1	yes	characteristic
77	C₃×C₄²	C₂×Q₈	1	1	yes	characteristic
78	C₃×C₄²	#41	1	1	yes	characteristic
79	C₂×C₃⋊C₈	—	2	1	no	—
80	C₂×C₃⋊C₈	—	2	1	no	—
81	C₂×C₃⋊C₈	—	2	2	no	—
82	C₂×C₃⋊C₈	—	2	2	no	—
83	C₂×C₃⋊C₈	—	2	2	no	—
84	C₂×C₃⋊C₈	—	4	2	no	—
85	C₂×C₃⋊C₈	—	4	2	no	—
86	C₂×C₃⋊C₈	—	4	2	no	—
87	C₂×C₃⋊C₈	—	4	2	no	—
88	C₂²×C₈	—	3	1	no	—
89	C₂²×C₈	—	3	1	no	—
90	C₂²×C₈	—	6	1	no	—
91	C₂²×C₈	—	6	2	no	—
92	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
93	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
94	C₂×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
95	C₂×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
96	C₂×C₄²	—	2	2	no	—
97	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
98	#106	—	2	2	no	—
99	#106	C₃⋊C₈	1	4	yes	complemented
100	C₄⋊C₈	—	6	2	no	—
101	C₄⋊C₈	—	6	2	no	—
102	C₄⋊C₈	—	12	4	no	—
103	C₄×C₈	—	3	2	no	—
104	C₄×C₈	—	6	1	no	—
105	C₄×C₈	—	6	1	no	—
106	C₄×C₈	—	6	1	no	—
107	C₄×C₈	Dic₆	1	2	yes	—
108	C₄×C₈	D₁₂	1	2	yes	—
109	#95	—	2	2	no	—
110	C₂³×C₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₃×C₄	—	2	2	no	—
112	C₂×C₃×C₄	—	2	4	no	—
113	C₂×C₃×C₄	—	4	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	2	no	—
118	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₃×C₄	C₄×Q₈	1	1	yes	characteristic
120	C₂×C₃×C₄	#121	1	1	yes	characteristic
121	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₃×C₄	#125	1	1	yes	characteristic
123	C₂×C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
124	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
126	C₂×C₃×C₄	#131	1	1	yes	characteristic
127	C₂×C₃×C₄	#131	1	1	yes	characteristic
128	C₂×C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
129	C₂×C₃×C₄	#137	1	2	yes	—
130	C₃×C₈	—	2	2	no	—
131	C₃×C₈	C₄⋊C₈	1	4	yes	complemented
132	C₃⋊C₈	—	4	2	no	—
133	C₃⋊C₈	—	4	1	no	—
134	C₃⋊C₈	—	4	1	no	—
135	C₃⋊C₈	—	8	2	no	—
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	—	2	2	no	—
138	C₂²×C₄	D₃×D₄	1	1	yes	characteristic
139	C₂²×C₄	#237	1	1	yes	characteristic
140	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	6	1	no	—
144	C₂×C₈	—	6	1	no	—
145	C₂×C₈	—	6	2	no	—
146	C₂×C₈	—	6	2	no	—
147	C₂×C₈	—	6	2	no	—
148	C₂×C₈	—	12	2	no	—
149	C₂×C₈	—	12	2	no	—
150	C₂×C₈	—	12	2	no	—
151	C₂×C₈	—	12	2	no	—
152	C₂×C₈	C₂×C₃⋊C₈	1	2	yes	—
153	C₂×C₈	#208	1	2	yes	—
154	C₂×C₈	C₄⋊Dic₃	1	2	yes	—
155	C₂×C₈	C₄⋊Dic₃	1	2	yes	—
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	C₂×C₃⋊C₈	1	4	yes	—
160	C₄²	—	4	2	no	—
161	C₄²	—	4	2	no	—
162	C₄²	C₂×Dic₆	1	1	yes	characteristic
163	C₄²	C₂×D₁₂	1	1	yes	characteristic
164	C₄²	D₃×Q₈	1	1	yes	characteristic
165	C₄²	#239	1	1	yes	characteristic
166	C₂²×C₃	#389	1	1	yes	characteristic
167	C₂²×C₃	C₈×D₄	1	1	yes	characteristic
168	C₂²×C₃	#436	1	1	yes	characteristic
169	C₂²×C₃	#440	1	2	yes	—
170	C₂²×C₃	#444	1	2	yes	—
171	C₃×C₄	—	2	1	no	—
172	C₃×C₄	—	2	1	no	—
173	C₃×C₄	—	4	4	no	—
174	C₃×C₄	—	8	2	no	—
175	C₃×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
176	C₃×C₄	#424	1	1	yes	characteristic
177	C₃×C₄	C₈×Q₈	1	1	yes	characteristic
178	C₃×C₄	#446	1	1	yes	characteristic
179	C₃×C₄	#469	1	1	yes	characteristic
180	C₃×C₄	#470	1	1	yes	characteristic
181	C₃×C₄	#501	1	2	yes	—
182	C₂³	D₄×Dic₃	1	1	yes	characteristic
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	#888	1	1	yes	characteristic
191	C₂×C₄	#890	1	1	yes	characteristic
192	C₂×C₄	#895	1	1	yes	characteristic
193	C₂×C₄	#897	1	1	yes	characteristic
194	C₂×C₄	#914	1	2	yes	—
195	C₂×C₄	#915	1	2	yes	—
196	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
197	C₂×C₄	C₂×#208	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
199	C₂×C₄	Q₈×Dic₃	1	1	yes	characteristic
200	C₂×C₄	#948	1	1	yes	characteristic
201	C₂×C₄	#948	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	12	2	no	—
204	C₈	—	12	1	no	—
205	C₈	—	12	1	no	—
206	C₈	—	24	2	no	—
207	C₈	C₄⋊(C₃⋊C₈)	1	4	yes	complemented
208	C₂×C₃	SD₁₆⋊C₈	1	2	yes	—
209	C₂×C₃	#1592	1	2	yes	—
210	C₂×C₃	#1939	1	1	yes	characteristic
211	C₂×C₃	#1944	1	1	yes	characteristic
212	C₂×C₃	#1946	1	1	yes	characteristic
213	C₂²	#4984	1	1	yes	characteristic
214	C₂²	D₄×C₃⋊C₈	1	1	yes	characteristic
215	C₂²	#5012	1	1	yes	characteristic
216	C₂²	#5164	1	2	yes	—
217	C₂²	#5197	1	2	yes	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	4	4	no	—
221	C₄	—	8	2	no	—
222	C₄	#4856	1	2	yes	—
223	C₄	#4865	1	1	yes	characteristic
224	C₄	#4867	1	1	yes	characteristic
225	C₄	C₂×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
226	C₄	#4999	1	1	yes	characteristic
227	C₄	Q₈×C₃⋊C₈	1	1	yes	characteristic
228	C₄	#5024	1	1	yes	characteristic
229	C₃	#9596	1	1	yes	characteristic, complemented, Sylow
230	C₂	#69484	1	1	yes	characteristic
231	C₂	#70562	1	1	yes	characteristic
232	C₂	#70663	1	1	yes	characteristic
233	C₂	SD₁₆⋊(C₃⋊C₈)	1	2	yes	—
234	C₂	#70900	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×C₄×#106	C₂
socle, prime radical	C₂³×C₃	C₄×D₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₄×#106	D₃
hypocenter	C₃	#9596
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
35. C₃×#106	C₈	semidirect
99. #106	C₃⋊C₈	semidirect
131. C₃×C₈	C₄⋊C₈	semidirect
207. C₈	C₄⋊(C₃⋊C₈)	semidirect
229. C₃	#9596	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×#106	2
C₂×#4458	2
C₂×#4455	2
#69484	2
#67619	2
#9596	3