Subgroups — #10713707

Order 768.

398 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#67657	C₂	1	4	yes	maximal
5	#13263	—	3	1	no	maximal, Sylow
6	C₂×C₃×C₄×Q₈	C₄	1	1	yes	characteristic
7	C₃×C₈×Q₈	—	2	2	no	—
8	C₃×C₈×Q₈	—	2	2	no	—
9	C₂×C₃×C₄⋊C₈	C₄	1	2	yes	—
10	C₂×C₃×C₄⋊C₈	C₄	1	1	yes	characteristic
11	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
12	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂²×C₃⋊C₁₆	C₂²	1	2	yes	—
14	C₂²×C₃⋊C₁₆	C₂²	1	2	yes	—
15	C₄⋊(C₃⋊C₁₆)	—	2	2	no	—
16	C₄×C₃⋊C₁₆	—	2	2	no	—
17	C₂×C₈×Q₈	D₃	1	1	yes	characteristic
18	C₂×C₄⋊C₁₆	—	3	1	no	—
19	C₂×C₄×C₁₆	—	3	1	no	—
20	#1374	—	3	4	no	—
21	C₂²×C₃×Q₈	C₈	1	1	yes	characteristic
22	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
23	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
25	C₃×C₄×Q₈	—	2	2	no	—
26	C₃×C₄×Q₈	—	2	2	no	—
27	C₂×C₃×C₄⋊C₄	C₈	1	1	yes	characteristic
28	C₂×C₃×C₄⋊C₄	C₈	1	2	yes	—
29	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
31	C₃×C₄⋊C₈	—	2	4	no	—
32	C₃×C₄⋊C₈	—	2	2	no	—
33	C₃×C₄×C₈	—	2	2	no	—
34	C₃×C₄×C₈	—	2	2	no	—
35	C₃×C₄×C₈	D₄	1	2	yes	—
36	C₃×C₄×C₈	D₄	1	2	yes	—
37	C₂×C₃⋊C₁₆	—	2	2	no	—
38	C₂×C₃⋊C₁₆	—	2	2	no	—
39	C₂×C₃⋊C₁₆	—	2	2	no	—
40	C₂×C₃⋊C₁₆	—	2	4	no	—
41	C₂×C₃⋊C₁₆	Q₈	1	4	yes	complemented
42	C₂×C₄×Q₈	Dic₃	1	1	yes	characteristic
43	C₂²×C₁₆	—	3	2	no	—
44	C₂²×C₁₆	—	3	2	no	—
45	C₈×Q₈	—	2	2	no	—
46	C₈×Q₈	—	2	2	no	—
47	C₂×C₄⋊C₈	Dic₃	1	2	yes	—
48	C₂×C₄⋊C₈	Dic₃	1	1	yes	characteristic
49	C₂×C₄×C₈	C₂×D₃	1	2	yes	—
50	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
51	C₄⋊C₁₆	—	6	2	no	—
52	C₄×C₁₆	—	6	2	no	—
53	C₂×C₃×Q₈	—	2	2	no	—
54	C₂×C₃×Q₈	—	2	2	no	—
55	C₂×C₃×Q₈	C₁₆	1	4	yes	complemented
56	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
57	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
59	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₃×C₈	—	2	2	no	—
62	C₂×C₃×C₈	—	2	2	no	—
63	C₂×C₃×C₈	—	2	4	no	—
64	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
65	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
68	C₂×C₃×C₈	#41	1	2	yes	—
69	C₃×C₄⋊C₄	—	2	4	no	—
70	C₃×C₄⋊C₄	—	2	2	no	—
71	C₃×C₄²	—	2	2	no	—
72	C₃×C₄²	—	2	2	no	—
73	C₃×C₄²	C₂²⋊C₄	1	2	yes	—
74	C₃×C₄²	M₁₆	1	2	yes	—
75	C₃⋊C₁₆	—	2	4	no	—
76	C₃⋊C₁₆	—	4	2	no	—
77	C₂²×Q₈	C₃⋊C₈	1	1	yes	characteristic
78	C₂²×C₈	C₂×Dic₃	1	2	yes	—
79	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
80	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
81	C₄×Q₈	—	2	2	no	—
82	C₄×Q₈	—	2	2	no	—
83	C₂×C₄⋊C₄	C₃⋊C₈	1	1	yes	characteristic
84	C₂×C₄⋊C₄	C₃⋊C₈	1	2	yes	—
85	C₂×C₄²	C₂×Dic₃	1	2	yes	—
86	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
87	C₂×C₁₆	—	3	4	no	—
88	C₂×C₁₆	—	6	2	no	—
89	C₂×C₁₆	—	6	2	no	—
90	C₂×C₁₆	—	6	4	no	—
91	C₂×C₁₆	—	6	2	no	—
92	C₄⋊C₈	—	2	4	no	—
93	C₄⋊C₈	—	2	2	no	—
94	C₄×C₈	—	2	2	no	—
95	C₄×C₈	—	2	2	no	—
96	C₄×C₈	#67	1	2	yes	—
97	C₄×C₈	#67	1	2	yes	—
98	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
99	C₃×Q₈	—	2	8	no	—
100	C₂×C₃×C₄	—	2	2	no	—
101	C₂×C₃×C₄	—	2	2	no	—
102	C₂×C₃×C₄	—	2	2	no	—
103	C₂×C₃×C₄	—	2	2	no	—
104	C₂×C₃×C₄	—	2	4	no	—
105	C₂×C₃×C₄	C₂²⋊C₈	1	2	yes	—
106	C₂×C₃×C₄	C₂²⋊C₈	1	2	yes	—
107	C₂×C₃×C₄	C₂×C₁₆	1	4	yes	—
108	C₂×C₃×C₄	C₂×C₁₆	1	2	yes	—
109	C₂×C₃×C₄	M₃₂	1	2	yes	—
110	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₃×C₄	C₄×Q₈	1	2	yes	—
112	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
113	C₂×C₃×C₄	#131	1	2	yes	—
114	C₃×C₈	—	2	2	no	—
115	C₃×C₈	—	4	2	no	—
116	C₃×C₈	C₂²⋊Q₈	1	2	yes	—
117	C₃×C₈	#124	1	1	yes	characteristic
118	C₃×C₈	C₄⋊Q₈	1	1	yes	characteristic
119	C₂×Q₈	—	2	2	no	—
120	C₂×Q₈	—	2	2	no	—
121	C₂×Q₈	C₃⋊C₁₆	1	4	yes	complemented
122	C₂²×C₄	C₂×C₃⋊C₈	1	2	yes	—
123	C₂²×C₄	C₂×C₃⋊C₈	1	2	yes	—
124	C₂²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
125	C₂²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
126	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	#217	1	2	yes	—
131	C₂×C₈	D₃×Q₈	1	2	yes	—
132	C₂×C₈	#239	1	2	yes	—
133	C₂×C₈	C₂×#67	1	1	yes	characteristic
134	C₂×C₈	C₂×#67	1	1	yes	characteristic
135	C₄⋊C₄	—	2	4	no	—
136	C₄⋊C₄	—	2	2	no	—
137	C₄²	—	2	2	no	—
138	C₄²	—	2	2	no	—
139	C₄²	#208	1	2	yes	—
140	C₄²	#217	1	2	yes	—
141	C₁₆	—	6	4	no	—
142	C₁₆	—	12	2	no	—
143	C₂²×C₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
144	C₂²×C₃	C₈×Q₈	1	2	yes	—
145	C₂²×C₃	C₂²×C₁₆	1	1	yes	characteristic
146	C₂²×C₃	C₂×M₃₂	1	1	yes	characteristic
147	C₂²×C₃	#504	1	2	yes	—
148	C₃×C₄	—	2	4	no	—
149	C₃×C₄	—	2	2	no	—
150	C₃×C₄	—	4	2	no	—
151	C₃×C₄	C₂²⋊C₁₆	1	4	yes	—
152	C₃×C₄	#391	1	1	yes	characteristic
153	C₃×C₄	#408	1	1	yes	characteristic
154	C₃×C₄	#446	1	2	yes	—
155	C₂³	C₂²×C₃⋊C₈	1	1	yes	characteristic
156	Q₈	—	2	8	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	C₂×C₃⋊C₁₆	1	4	yes	—
163	C₂×C₄	C₂×C₃⋊C₁₆	1	2	yes	—
164	C₂×C₄	#812	1	2	yes	—
165	C₂×C₄	#830	1	2	yes	—
166	C₂×C₄	#830	1	2	yes	—
167	C₂×C₄	C₂×#208	1	1	yes	characteristic
168	C₂×C₄	Q₈×Dic₃	1	2	yes	—
169	C₂×C₄	#948	1	2	yes	—
170	C₂×C₄	C₂×#217	1	1	yes	characteristic
171	C₈	—	2	2	no	—
172	C₈	—	4	2	no	—
173	C₈	#944	1	1	yes	characteristic
174	C₈	#946	1	2	yes	—
175	C₈	#947	1	1	yes	characteristic
176	C₂×C₃	#1984	1	1	yes	characteristic
177	C₂×C₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
178	C₂×C₃	#2113	1	1	yes	characteristic
179	C₂×C₃	Q₈×C₁₆	1	2	yes	—
180	C₂×C₃	#2183	1	2	yes	—
181	C₂²	Q₈×C₃⋊C₈	1	2	yes	—
182	C₂²	C₂²×C₃⋊C₁₆	1	1	yes	characteristic
183	C₂²	C₂×#812	1	1	yes	characteristic
184	C₂²	#5140	1	2	yes	—
185	C₂²	C₂×#830	1	1	yes	characteristic
186	C₄	—	2	4	no	—
187	C₄	—	2	2	no	—
188	C₄	—	4	2	no	—
189	C₄	#4549	1	4	yes	—
190	C₄	#5024	1	2	yes	—
191	C₄	#5229	1	1	yes	characteristic
192	C₄	#5234	1	1	yes	characteristic
193	C₃	#13263	1	1	yes	characteristic, complemented, Sylow
194	C₂	Q₈×C₃⋊C₁₆	1	2	yes	—
195	C₂	#70949	1	2	yes	—
196	C₂	#71634	1	1	yes	characteristic
197	C₂	C₂×#4549	1	1	yes	characteristic
198	C₂	#72004	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₁₆
Fitting	C₂×C₃×C₈×Q₈	C₂
socle, prime radical	C₂³×C₃	C₂²×C₈
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	C₂×C₈×Q₈	D₃
hypocenter	C₃	#13263
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
41. C₂×C₃⋊C₁₆	Q₈	semidirect
55. C₂×C₃×Q₈	C₁₆	semidirect
121. C₂×Q₈	C₃⋊C₁₆	semidirect
193. C₃	#13263	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₈×Q₈	2
C₂×C₄⋊(C₃⋊C₁₆)	2
C₂×C₄×C₃⋊C₁₆	2
#67657	2
#13263	3