Subgroups — #10693799

Order 768.

306 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×#108	C₂	1	1	yes	maximal, characteristic
2	#69850	C₂	1	2	yes	maximal
3	C₂×#4457	C₂	1	1	yes	maximal, characteristic
4	#69724	C₂	1	1	yes	maximal, characteristic
5	#67668	C₂	1	2	yes	maximal
6	#9607	—	3	1	no	maximal, Sylow
7	C₂×C₃×#108	C₄	1	1	yes	characteristic
8	C₂×C₃×#108	C₄	1	1	yes	characteristic
9	C₃×C₄×M₁₆	C₂²	1	2	yes	—
10	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
11	#4925	C₂²	1	1	yes	characteristic
12	#4925	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
14	C₃×#341	C₄	1	2	yes	—
15	#4458	—	2	2	no	—
16	#4457	—	2	1	no	—
17	#4457	—	2	1	no	—
18	#4457	—	2	2	no	—
19	#4455	—	2	1	no	—
20	C₈×C₃⋊C₈	—	2	1	no	—
21	C₄×#108	D₃	1	1	yes	characteristic
22	#1565	—	3	2	no	—
23	C₂×#334	—	3	1	no	—
24	#1450	—	3	1	no	—
25	#1283	—	3	2	no	—
26	C₂×C₃×M₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₃×M₁₆	C₂×C₄	1	2	yes	—
28	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
31	C₂²×C₃⋊C₈	—	2	2	no	—
32	C₃×#108	—	2	2	no	—
33	C₃×#108	C₈	1	4	yes	complemented
34	C₃×#97	—	2	2	no	—
35	C₃×C₄×C₈	—	2	2	no	—
36	C₃×C₄×C₈	D₄	1	1	yes	characteristic
37	C₃×C₄×C₈	D₄	1	1	yes	characteristic
38	C₃×C₄×C₈	Q₈	1	1	yes	characteristic
39	C₃×C₄×C₈	Q₈	1	1	yes	characteristic
40	#830	—	2	1	no	—
41	#830	Q₈	1	2	yes	—
42	C₄⋊(C₃⋊C₈)	—	2	1	no	—
43	C₄⋊(C₃⋊C₈)	—	2	2	no	—
44	C₄⋊(C₃⋊C₈)	—	2	2	no	—
45	C₄⋊(C₃⋊C₈)	D₄	1	2	yes	—
46	C₄×C₃⋊C₈	—	2	1	no	—
47	C₄×C₃⋊C₈	—	2	1	no	—
48	#431	—	3	1	no	—
49	#431	—	3	1	no	—
50	C₂×#108	Dic₃	1	1	yes	characteristic
51	C₂×#108	Dic₃	1	1	yes	characteristic
52	C₂×C₄⋊C₈	—	3	2	no	—
53	C₄×M₁₆	C₂×D₃	1	2	yes	—
54	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
55	#341	Dic₃	1	2	yes	—
56	#335	—	6	2	no	—
57	#334	—	6	2	no	—
58	#334	—	6	1	no	—
59	#334	—	6	1	no	—
60	#322	—	6	1	no	—
61	C₈²	—	6	1	no	—
62	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₃×C₄	#41	1	1	yes	characteristic
65	C₃×M₁₆	—	2	2	no	—
66	C₃×M₁₆	—	2	2	no	—
67	C₃×M₁₆	—	2	2	no	—
68	C₃×M₁₆	C₂×C₈	1	4	yes	—
69	C₂×C₃×C₈	—	2	2	no	—
70	C₂×C₃×C₈	—	2	1	no	—
71	C₂×C₃×C₈	—	2	1	no	—
72	C₂×C₃×C₈	—	2	2	no	—
73	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₃×C₈	C₂×C₈	1	2	yes	—
78	C₂×C₃×C₈	M₁₆	1	2	yes	—
79	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
80	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
81	C₃×C₄²	C₂×Q₈	1	1	yes	characteristic
82	C₃×C₄²	#41	1	1	yes	characteristic
83	C₂×C₃⋊C₈	—	2	2	no	—
84	C₂×C₃⋊C₈	—	2	2	no	—
85	C₂×C₃⋊C₈	—	4	2	no	—
86	C₂×C₃⋊C₈	—	8	2	no	—
87	C₂×M₁₆	C₂×Dic₃	1	2	yes	—
88	C₂×M₁₆	C₂×Dic₃	1	2	yes	—
89	C₂²×C₈	—	6	2	no	—
90	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
91	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
92	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
93	#108	—	2	2	no	—
94	#108	C₃⋊C₈	1	4	yes	complemented
95	C₄⋊C₈	—	3	2	no	—
96	C₄⋊C₈	—	6	2	no	—
97	C₄⋊C₈	—	6	2	no	—
98	C₄⋊C₈	—	6	1	no	—
99	C₂²⋊C₈	—	3	2	no	—
100	C₂²⋊C₈	—	6	1	no	—
101	#97	—	2	2	no	—
102	C₄×C₈	—	2	2	no	—
103	C₄×C₈	—	6	1	no	—
104	C₄×C₈	—	6	1	no	—
105	C₄×C₈	Dic₆	1	1	yes	characteristic
106	C₄×C₈	Dic₆	1	1	yes	characteristic
107	C₄×C₈	D₁₂	1	1	yes	characteristic
108	C₄×C₈	D₁₂	1	1	yes	characteristic
109	C₂³×C₃	C₄×Q₈	1	1	yes	characteristic
110	C₂×C₃×C₄	—	2	2	no	—
111	C₂×C₃×C₄	—	4	1	no	—
112	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
114	C₂×C₃×C₄	#121	1	1	yes	characteristic
115	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₃×C₄	#125	1	1	yes	characteristic
117	C₂×C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
118	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
120	C₂×C₃×C₄	#131	1	1	yes	characteristic
121	C₂×C₃×C₄	#131	1	1	yes	characteristic
122	C₃×C₈	—	2	1	no	—
123	C₃×C₈	—	2	1	no	—
124	C₃×C₈	—	2	4	no	—
125	C₃×C₈	—	4	2	no	—
126	C₃×C₈	C₄⋊C₈	1	2	yes	—
127	C₃×C₈	C₄⋊C₈	1	2	yes	—
128	C₃⋊C₈	—	4	2	no	—
129	C₃⋊C₈	—	4	2	no	—
130	C₃⋊C₈	—	8	2	no	—
131	C₂²×C₄	D₃×Q₈	1	1	yes	characteristic
132	C₂²×C₄	#239	1	1	yes	characteristic
133	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
134	M₁₆	—	2	2	no	—
135	M₁₆	—	2	2	no	—
136	M₁₆	—	2	2	no	—
137	M₁₆	C₂×C₃⋊C₈	1	4	yes	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	6	2	no	—
143	C₂×C₈	—	6	2	no	—
144	C₂×C₈	—	12	2	no	—
145	C₂×C₈	—	24	2	no	—
146	C₂×C₈	C₂×C₃⋊C₈	1	2	yes	—
147	C₂×C₈	#208	1	2	yes	—
148	C₂×C₈	C₄⋊Dic₃	1	1	yes	characteristic
149	C₂×C₈	C₄⋊Dic₃	1	1	yes	characteristic
150	C₂×C₈	C₄⋊Dic₃	1	1	yes	characteristic
151	C₂×C₈	C₄⋊Dic₃	1	1	yes	characteristic
152	C₄²	C₂×Dic₆	1	1	yes	characteristic
153	C₄²	C₂×D₁₂	1	1	yes	characteristic
154	C₄²	D₃×D₄	1	1	yes	characteristic
155	C₄²	#237	1	1	yes	characteristic
156	C₂²×C₃	—	2	1	no	—
157	C₂²×C₃	—	2	1	no	—
158	C₂²×C₃	#389	1	1	yes	characteristic
159	C₂²×C₃	C₈×Q₈	1	1	yes	characteristic
160	C₂²×C₃	#446	1	1	yes	characteristic
161	C₃×C₄	—	2	1	no	—
162	C₃×C₄	—	2	1	no	—
163	C₃×C₄	—	2	1	no	—
164	C₃×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
165	C₃×C₄	#424	1	1	yes	characteristic
166	C₃×C₄	C₈×D₄	1	1	yes	characteristic
167	C₃×C₄	#436	1	1	yes	characteristic
168	C₃×C₄	#471	1	2	yes	—
169	C₂³	Q₈×Dic₃	1	1	yes	characteristic
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	#888	1	1	yes	characteristic
173	C₂×C₄	#890	1	1	yes	characteristic
174	C₂×C₄	#895	1	1	yes	characteristic
175	C₂×C₄	#897	1	1	yes	characteristic
176	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
177	C₂×C₄	C₂×#208	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
179	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
180	C₂×C₄	#948	1	1	yes	characteristic
181	C₂×C₄	#948	1	1	yes	characteristic
182	C₈	—	2	1	no	—
183	C₈	—	2	1	no	—
184	C₈	—	2	4	no	—
185	C₈	—	4	2	no	—
186	C₈	—	12	2	no	—
187	C₈	—	12	2	no	—
188	C₈	—	24	2	no	—
189	C₈	C₄⋊(C₃⋊C₈)	1	2	yes	—
190	C₈	C₄⋊(C₃⋊C₈)	1	2	yes	—
191	C₂×C₃	—	2	1	no	—
192	C₂×C₃	—	2	1	no	—
193	C₂×C₃	#1939	1	1	yes	characteristic
194	C₂×C₃	#1951	1	1	yes	characteristic
195	C₂×C₃	#1953	1	1	yes	characteristic
196	C₂²	—	2	1	no	—
197	C₂²	—	2	1	no	—
198	C₂²	#4984	1	1	yes	characteristic
199	C₂²	Q₈×C₃⋊C₈	1	1	yes	characteristic
200	C₂²	#5024	1	1	yes	characteristic
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	#4898	1	2	yes	—
205	C₄	C₂×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
206	C₄	#4999	1	1	yes	characteristic
207	C₄	D₄×C₃⋊C₈	1	1	yes	characteristic
208	C₄	#5012	1	1	yes	characteristic
209	C₃	#9607	1	1	yes	characteristic, complemented, Sylow
210	C₂	—	2	1	no	—
211	C₂	—	2	1	no	—
212	C₂	#69484	1	1	yes	characteristic
213	C₂	#70700	1	1	yes	characteristic
214	C₂	#70707	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#895
commutator	C₂×C₃×C₄	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×C₄×#108	C₂
socle	C₂²×C₃	#389
prime radical	C₂³×C₃	C₄×Q₈
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₄×#108	D₃
hypocenter	C₃	#9607
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
33. C₃×#108	C₈	semidirect
94. #108	C₃⋊C₈	semidirect
209. C₃	#9607	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×#108	2
#69850	2
C₂×#4457	2
#69724	2
#67668	2
#9607	3