Subgroups — #10710285

Order 768.

410 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#5140	C₂	1	1	yes	maximal, characteristic
2	C₃×#1757	C₂	1	1	yes	maximal, characteristic
3	#72007	C₂	1	1	yes	maximal, characteristic
4	#69881	C₂	1	2	yes	maximal, complemented
5	#67672	C₂	1	2	yes	maximal, complemented
6	#12613	—	3	1	no	maximal, Sylow
7	C₂×C₃×#131	C₂²	1	1	yes	characteristic
8	C₃×#413	C₄	1	1	yes	characteristic
9	C₃×#410	C₄	1	1	yes	characteristic
10	C₃×#405	C₂²	1	2	yes	—
11	#5140	—	2	2	no	—
12	#5140	—	2	2	no	—
13	C₂×#812	C₄	1	2	yes	complemented
14	C₂×#812	C₂²	1	2	yes	—
15	C₂×#812	C₂²	1	1	yes	characteristic
16	C₂²×C₃⋊C₁₆	C₄	1	2	yes	complemented
17	C₂²×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₃×#344	C₄	1	2	yes	—
19	#4553	C₄	1	4	yes	complemented
20	#4461	—	2	2	no	—
21	C₄×C₃⋊C₁₆	—	2	2	no	—
22	C₂×#504	—	3	1	no	—
23	#2115	—	3	1	no	—
24	#2108	—	3	2	no	—
25	#1757	D₃	1	1	yes	characteristic
26	#1378	—	3	2	no	—
27	C₂×C₃×#41	C₂×C₄	1	1	yes	characteristic
28	C₃×#131	—	2	2	no	—
29	C₃×#131	—	2	2	no	—
30	C₂×C₃×M₁₆	C₂×C₄	1	2	yes	—
31	C₂×C₃×M₁₆	C₂×C₄	1	2	yes	—
32	C₂×C₃×M₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
34	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
35	C₃×#117	C₂×C₄	1	2	yes	—
36	C₃×#101	C₈	1	2	yes	—
37	C₃×#100	C₈	1	2	yes	—
38	C₃×#99	C₈	1	4	yes	—
39	C₃×#97	—	2	2	no	—
40	C₃×C₄×C₈	—	2	2	no	—
41	#812	—	2	2	no	—
42	#812	—	2	2	no	—
43	#812	—	2	4	no	—
44	#812	C₂×C₄	1	4	yes	—
45	C₂×C₃⋊C₁₆	—	2	2	no	—
46	C₂×C₃⋊C₁₆	—	2	2	no	—
47	C₂×C₃⋊C₁₆	—	2	2	no	—
48	C₂×C₃⋊C₁₆	C₂×C₄	1	2	yes	—
49	C₂×C₃⋊C₁₆	C₂×C₄	1	1	yes	characteristic
50	C₂×C₃⋊C₁₆	C₂×C₄	1	1	yes	characteristic
51	C₂×C₃⋊C₁₆	D₄	1	2	yes	—
52	C₂×C₃⋊C₁₆	D₄	1	2	yes	—
53	C₂×#131	C₂×D₃	1	1	yes	characteristic
54	#504	—	6	2	no	—
55	#504	—	6	2	no	—
56	C₂×M₃₂	—	3	2	no	—
57	C₂×M₃₂	—	3	1	no	—
58	C₂×M₃₂	—	3	2	no	—
59	C₂²×C₁₆	—	3	1	no	—
60	C₂²×C₁₆	—	3	2	no	—
61	#413	Dic₃	1	1	yes	characteristic
62	#410	Dic₃	1	1	yes	characteristic
63	#405	C₂×D₃	1	2	yes	—
64	#349	—	3	4	no	—
65	#346	—	6	2	no	—
66	C₄×C₁₆	—	6	2	no	—
67	#344	Dic₃	1	2	yes	—
68	C₃×#41	—	2	2	no	—
69	C₃×#41	—	2	2	no	—
70	C₂×C₃×Q₈	C₂×C₈	1	1	yes	characteristic
71	C₂×C₃×D₄	C₂×C₈	1	2	yes	—
72	C₂×C₃×D₄	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₃×C₄	C₄²	1	2	yes	—
74	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
75	C₃×M₁₆	—	2	2	no	—
76	C₃×M₁₆	—	2	2	no	—
77	C₃×M₁₆	—	2	4	no	—
78	C₃×M₁₆	C₂×C₈	1	4	yes	—
79	C₂×C₃×C₈	—	2	2	no	—
80	C₂×C₃×C₈	—	2	2	no	—
81	C₂×C₃×C₈	—	2	2	no	—
82	C₂×C₃×C₈	C₄²	1	2	yes	—
83	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
86	C₂×C₃×C₈	#41	1	2	yes	—
87	C₃×C₄⋊C₄	—	2	2	no	—
88	C₃×C₂²⋊C₄	C₂×C₈	1	4	yes	—
89	C₃×C₄²	—	2	2	no	—
90	C₃⋊C₁₆	—	2	2	no	—
91	C₃⋊C₁₆	—	2	4	no	—
92	C₃⋊C₁₆	C₂²⋊C₄	1	1	yes	characteristic
93	C₃⋊C₁₆	C₂²⋊C₄	1	1	yes	characteristic
94	C₃⋊C₁₆	C₂²⋊C₄	1	2	yes	—
95	C₂×#41	C₂×Dic₃	1	1	yes	characteristic
96	#131	—	2	2	no	—
97	#131	—	2	2	no	—
98	C₂×M₁₆	C₄×D₃	1	2	yes	—
99	C₂×M₁₆	C₂×Dic₃	1	2	yes	—
100	C₂×M₁₆	C₂×Dic₃	1	1	yes	characteristic
101	C₂²×C₈	C₄×D₃	1	2	yes	—
102	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
103	#117	C₂×Dic₃	1	2	yes	—
104	M₃₂	—	3	4	no	—
105	M₃₂	—	6	4	no	—
106	M₃₂	—	6	2	no	—
107	M₃₂	—	6	2	no	—
108	C₂×C₁₆	—	3	1	no	—
109	C₂×C₁₆	—	3	1	no	—
110	C₂×C₁₆	—	3	2	no	—
111	C₂×C₁₆	—	3	2	no	—
112	C₂×C₁₆	—	3	2	no	—
113	C₂×C₁₆	—	6	2	no	—
114	C₂×C₁₆	—	6	2	no	—
115	C₂×C₁₆	—	6	2	no	—
116	#101	C₃⋊C₈	1	2	yes	—
117	#100	C₃⋊C₈	1	2	yes	—
118	#99	C₃⋊C₈	1	4	yes	—
119	#97	—	2	2	no	—
120	C₄×C₈	—	2	2	no	—
121	C₂³×C₃	C₄×C₈	1	2	yes	—
122	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
123	C₃×Q₈	—	2	2	no	—
124	C₃×D₄	—	2	4	no	—
125	C₃×D₄	—	2	2	no	—
126	C₂×C₃×C₄	—	2	2	no	—
127	C₂×C₃×C₄	—	2	2	no	—
128	C₂×C₃×C₄	—	2	4	no	—
129	C₂×C₃×C₄	C₄×C₈	1	2	yes	—
130	C₂×C₃×C₄	C₂×C₄²	1	1	yes	characteristic
131	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
132	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₃×C₄	#131	1	2	yes	—
134	C₃×C₈	—	2	2	no	—
135	C₃×C₈	—	2	4	no	—
136	C₃×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₃×C₈	#117	1	1	yes	characteristic
138	C₃×C₈	C₄×D₄	1	2	yes	—
139	#41	—	2	2	no	—
140	#41	—	2	2	no	—
141	C₂×Q₈	C₂×C₃⋊C₈	1	1	yes	characteristic
142	C₂×D₄	C₂×C₃⋊C₈	1	2	yes	—
143	C₂×D₄	C₂×C₃⋊C₈	1	1	yes	characteristic
144	C₂²×C₄	C₄×Dic₃	1	2	yes	—
145	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
146	M₁₆	—	2	2	no	—
147	M₁₆	—	2	2	no	—
148	M₁₆	—	2	4	no	—
149	M₁₆	C₂×C₃⋊C₈	1	4	yes	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	C₄×Dic₃	1	2	yes	—
154	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
155	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
156	C₂×C₈	D₃×D₄	1	2	yes	—
157	C₂×C₈	#237	1	2	yes	—
158	C₄⋊C₄	—	2	2	no	—
159	C₂²⋊C₄	C₂×C₃⋊C₈	1	4	yes	—
160	C₄²	—	2	2	no	—
161	C₁₆	—	3	1	no	—
162	C₁₆	—	3	1	no	—
163	C₁₆	—	3	2	no	—
164	C₁₆	—	6	2	no	—
165	C₁₆	—	6	4	no	—
166	C₂²×C₃	—	2	2	no	—
167	C₂²×C₃	—	4	1	no	—
168	C₂²×C₃	—	4	2	no	—
169	C₂²×C₃	C₂×C₄×C₈	1	1	yes	characteristic
170	C₂²×C₃	C₈×D₄	1	2	yes	—
171	C₃×C₄	—	2	2	no	—
172	C₃×C₄	—	4	4	no	—
173	C₃×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
174	C₃×C₄	#405	1	1	yes	characteristic
175	C₃×C₄	C₈×D₄	1	2	yes	—
176	C₂³	C₄×C₃⋊C₈	1	2	yes	—
177	C₂³	C₂²×C₃⋊C₈	1	1	yes	characteristic
178	Q₈	—	2	2	no	—
179	D₄	—	2	4	no	—
180	D₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	C₄×C₃⋊C₈	1	2	yes	—
185	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
186	C₂×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
187	C₂×C₄	D₄×Dic₃	1	2	yes	—
188	C₂×C₄	#948	1	2	yes	—
189	C₈	—	2	2	no	—
190	C₈	—	2	4	no	—
191	C₈	#877	1	1	yes	characteristic
192	C₈	D₃×C₂²⋊C₄	1	1	yes	characteristic
193	C₈	#67⋊C₄	1	2	yes	—
194	C₂×C₃	—	2	2	no	—
195	C₂×C₃	—	2	1	no	—
196	C₂×C₃	—	4	2	no	—
197	C₂×C₃	C₈×C₂²⋊C₄	1	1	yes	characteristic
198	C₂²	—	2	2	no	—
199	C₂²	—	4	1	no	—
200	C₂²	—	4	2	no	—
201	C₂²	C₂×C₄×C₃⋊C₈	1	1	yes	characteristic
202	C₂²	D₄×C₃⋊C₈	1	2	yes	—
203	C₄	—	2	2	no	—
204	C₄	—	4	4	no	—
205	C₄	Dic₃×C₂²⋊C₄	1	1	yes	characteristic
206	C₄	#4997	1	1	yes	characteristic
207	C₄	D₄×C₃⋊C₈	1	2	yes	—
208	C₃	#12613	1	1	yes	characteristic, complemented, Sylow
209	C₂	—	2	2	no	—
210	C₂	—	2	1	no	—
211	C₂	—	4	2	no	—
212	C₂	C₂²⋊C₄×C₃⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₃×C₂²⋊C₄
commutator	C₂²×C₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₃×#1757	C₂
socle	C₂×C₃	C₈×C₂²⋊C₄
prime radical	C₂×C₃×#41	C₂×C₄
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	#1757	D₃
hypocenter	C₃	#12613
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1757

Complements

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

N	H ≅ G/N	type
4. #69881	C₂	semidirect
5. #67672	C₂	semidirect
13. C₂×#812	C₄	semidirect
16. C₂²×C₃⋊C₁₆	C₄	semidirect
19. #4553	C₄	semidirect
208. C₃	#12613	semidirect

Maximal subgroups

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

subgroup	index
C₂×#5140	2
C₃×#1757	2
#72007	2
#69881	2
#67672	2
#12613	3