Finite Group Databasebeta

Subgroups — #82031

Order 384.

370 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5769	C₂	1	2	yes	maximal, complemented
2	C₃×#508	C₂	1	1	yes	maximal, characteristic, complemented
3	#5193	C₂	1	2	yes	maximal, complemented
4	#4919	C₂	1	2	yes	maximal, complemented
5	#4914	C₂	1	4	yes	maximal, complemented
6	D₃×D₁₆	C₂	1	2	yes	maximal, complemented
7	C₂×D₄₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#4901	C₂	1	1	yes	maximal, characteristic, complemented
9	#3415	—	3	1	no	maximal, Sylow
10	#1009	—	2	2	no	—
11	C₃×#135	C₂²	1	2	yes	complemented
12	#949	—	2	2	no	—
13	#919	C₂²	1	4	yes	complemented
14	#914	—	2	4	no	—
15	D₃×D₈	C₂²	1	4	yes	complemented
16	C₂×D₂₄	C₂²	1	2	yes	complemented
17	#901	C₂²	1	1	yes	characteristic
18	C₃×Q₃₂	C₂²	1	1	yes	characteristic, complemented
19	C₃×SD₃₂	C₂²	1	2	yes	complemented
20	C₃×D₁₆	C₂²	1	1	yes	characteristic, complemented
21	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic, complemented
22	#828	C₂²	1	4	yes	complemented
23	#826	C₂²	1	4	yes	complemented
24	#812	C₂²	1	1	yes	characteristic, complemented
25	D₄₈	C₂²	1	2	yes	complemented
26	D₄₈	C₂²	1	2	yes	complemented
27	#798	C₂²	1	2	yes	complemented
28	D₃×C₁₆	C₂²	1	2	yes	complemented
29	#576	—	3	2	no	—
30	#509	—	3	2	no	—
31	#509	—	3	4	no	—
32	#508	—	3	2	no	—
33	#508	D₃	1	1	yes	characteristic, complemented
34	C₂×D₁₆	—	3	2	no	—
35	C₂×D₁₆	—	3	1	no	—
36	#504	—	3	1	no	—
37	C₃×#41	—	2	2	no	—
38	#239	—	2	4	no	—
39	D₃×D₄	—	2	4	no	—
40	D₃×D₄	—	4	4	no	—
41	#235	—	4	2	no	—
42	#235	D₄	1	1	yes	characteristic
43	C₂×D₁₂	—	2	2	no	—
44	C₂×D₁₂	—	4	2	no	—
45	C₃×Q₁₆	C₂³	1	2	yes	—
46	C₃×SD₁₆	—	2	2	no	—
47	C₃×D₈	C₂³	1	2	yes	—
48	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
49	#215	—	2	4	no	—
50	#213	—	2	4	no	—
51	#208	D₄	1	1	yes	characteristic
52	D₂₄	—	2	2	no	—
53	D₂₄	C₂³	1	4	yes	—
54	ΓL(1,𝔽₂₅)	D₄	1	2	yes	—
55	D₃×C₈	C₂³	1	2	yes	—
56	C₃×C₁₆	C₂³	1	1	yes	characteristic
57	C₃×C₁₆	C₂³	1	1	yes	characteristic
58	C₃⋊C₁₆	C₂³	1	2	yes	—
59	H(5,𝔽₂)	—	6	2	no	—
60	AGL(1,ℤ/8)	—	6	4	no	—
61	AGL(1,ℤ/8)	—	6	2	no	—
62	#135	—	3	4	no	—
63	#135	C₂×D₃	1	2	yes	complemented
64	C₂×D₈	—	3	4	no	—
65	C₂×D₈	—	3	2	no	—
66	#131	—	3	1	no	—
67	Q₃₂	C₂×D₃	1	1	yes	characteristic, complemented
68	SD₃₂	—	3	4	no	—
69	SD₃₂	C₂×D₃	1	2	yes	complemented
70	D₁₆	—	3	4	no	—
71	D₁₆	—	3	2	no	—
72	D₁₆	—	3	2	no	—
73	D₁₆	C₂×D₃	1	1	yes	characteristic, complemented
74	M₃₂	—	3	1	no	—
75	M₃₂	—	3	2	no	—
76	C₂×C₁₆	—	3	2	no	—
77	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic, complemented
78	C₂²×D₃	—	4	2	no	—
79	C₂²×D₃	—	4	4	no	—
80	C₃×Q₈	—	2	2	no	—
81	C₃×D₄	—	2	2	no	—
82	C₃×D₄	—	4	2	no	—
83	C₂×C₃×C₄	—	4	2	no	—
84	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
85	#67	—	4	4	no	—
86	#67	D₈	1	2	yes	—
87	D₁₂	—	2	4	no	—
88	D₁₂	—	4	4	no	—
89	D₁₂	—	4	2	no	—
90	D₁₂	D₈	1	1	yes	characteristic
91	C₄×D₃	—	4	4	no	—
92	C₄×D₃	C₂×D₄	1	2	yes	—
93	Dic₆	D₈	1	1	yes	characteristic
94	C₃×C₈	C₂×D₄	1	1	yes	characteristic
95	C₃×C₈	C₂⁴	1	1	yes	characteristic
96	C₃⋊C₈	C₂×D₄	1	2	yes	—
97	#41	—	2	2	no	—
98	#41	—	3	1	no	—
99	#41	—	6	4	no	—
100	#41	—	12	2	no	—
101	C₂×D₄	—	6	4	no	—
102	C₂×D₄	—	6	2	no	—
103	C₂×D₄	—	12	2	no	—
104	C₂×D₄	—	12	4	no	—
105	Q₁₆	C₂²×D₃	1	2	yes	—
106	SD₁₆	—	2	2	no	—
107	SD₁₆	—	6	4	no	—
108	D₈	—	3	4	no	—
109	D₈	—	6	4	no	—
110	D₈	—	6	2	no	—
111	D₈	C₂²×D₃	1	2	yes	—
112	M₁₆	—	3	1	no	—
113	M₁₆	—	3	2	no	—
114	C₂×C₈	—	3	2	no	—
115	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
116	C₁₆	—	3	2	no	—
117	C₁₆	C₂²×D₃	1	1	yes	characteristic
118	C₁₆	C₂²×D₃	1	1	yes	characteristic
119	C₂²×C₃	—	4	2	no	—
120	C₂²×C₃	C₂×D₈	1	1	yes	characteristic
121	C₂×D₃	—	4	4	no	—
122	C₂×D₃	—	16	4	no	—
123	C₂×D₃	—	16	2	no	—
124	C₂×D₃	C₂×D₈	1	2	yes	—
125	C₃×C₄	—	4	2	no	—
126	C₃×C₄	C₂×D₈	1	1	yes	characteristic
127	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
128	Dic₃	C₂×D₈	1	2	yes	—
129	C₂³	—	12	4	no	—
130	C₂³	—	12	2	no	—
131	Q₈	—	2	2	no	—
132	Q₈	—	3	1	no	—
133	D₄	—	2	2	no	—
134	D₄	—	3	2	no	—
135	D₄	—	3	1	no	—
136	D₄	—	4	2	no	—
137	D₄	—	6	4	no	—
138	D₄	—	12	4	no	—
139	D₄	—	12	4	no	—
140	D₄	—	12	2	no	—
141	C₂×C₄	—	3	2	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	12	4	no	—
144	C₂×C₄	D₃×D₄	1	1	yes	characteristic
145	C₈	—	3	2	no	—
146	C₈	D₃×D₄	1	1	yes	characteristic
147	C₈	C₂³×D₃	1	1	yes	characteristic
148	C₂×C₃	—	2	1	no	—
149	C₂×C₃	—	8	2	no	—
150	C₂×C₃	C₂²×D₈	1	1	yes	characteristic
151	D₃	—	2	2	no	—
152	D₃	—	8	4	no	—
153	C₂²	—	3	2	no	—
154	C₂²	—	4	2	no	—
155	C₂²	—	12	4	no	—
156	C₂²	—	48	2	no	—
157	C₂²	—	48	4	no	—
158	C₂²	D₃×D₈	1	1	yes	characteristic
159	C₄	—	3	2	no	—
160	C₄	—	4	2	no	—
161	C₄	D₃×D₈	1	1	yes	characteristic
162	C₄	C₂×D₃×D₄	1	1	yes	characteristic
163	C₃	#3415	1	1	yes	characteristic, complemented, Sylow
164	C₂	—	2	1	no	—
165	C₂	—	6	2	no	—
166	C₂	—	8	2	no	—
167	C₂	—	24	4	no	—
168	C₂	C₂×D₃×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₃×D₈
commutator, cosocle	C₃×C₈	C₂⁴
Frattini	C₈	C₂³×D₃
Fitting	C₃×#508	C₂
socle	C₂×C₃	C₂²×D₈
hypercenter	#508	D₃
hypocenter	C₃	#3415
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #508

Complements

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

N	H ≅ G/N	type
1. #5769	C₂	semidirect
2. C₃×#508	C₂	semidirect
3. #5193	C₂	semidirect
4. #4919	C₂	semidirect
5. #4914	C₂	semidirect
6. D₃×D₁₆	C₂	semidirect
7. C₂×D₄₈	C₂	semidirect
8. #4901	C₂	semidirect
11. C₃×#135	C₂²	semidirect
13. #919	C₂²	semidirect
15. D₃×D₈	C₂²	semidirect
16. C₂×D₂₄	C₂²	semidirect
18. C₃×Q₃₂	C₂²	semidirect
19. C₃×SD₃₂	C₂²	semidirect
20. C₃×D₁₆	C₂²	semidirect
21. C₂×C₃×C₁₆	C₂²	semidirect
22. #828	C₂²	semidirect
23. #826	C₂²	semidirect
24. #812	C₂²	semidirect
25. D₄₈	C₂²	semidirect
26. D₄₈	C₂²	semidirect
27. #798	C₂²	semidirect
28. D₃×C₁₆	C₂²	semidirect
33. #508	D₃	semidirect
63. #135	C₂×D₃	semidirect
67. Q₃₂	C₂×D₃	semidirect
69. SD₃₂	C₂×D₃	semidirect
73. D₁₆	C₂×D₃	semidirect
77. C₂×C₁₆	C₂×D₃	semidirect
163. C₃	#3415	semidirect

Maximal subgroups

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

subgroup	index
#5769	2
C₃×#508	2
#5193	2
#4919	2
#4914	2
D₃×D₁₆	2
C₂×D₄₈	2
#4901	2
#3415	3