Finite Group Databasebeta

Subgroups — #83817

Order 384.

738 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 249 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	#5964	C₂	1	1	yes	maximal, characteristic
2	C₃×#568	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₃×#41	C₂	1	1	yes	maximal, characteristic, complemented
4	#5139	C₂	1	6	yes	maximal, complemented
5	C₂×#825	C₂	1	6	yes	maximal, complemented
6	#2901	—	3	1	no	maximal, Sylow
7	C₂×C₃×#41	C₂²	1	1	yes	characteristic
8	#1010	C₄	1	4	yes	complemented
9	#1009	C₄	1	4	yes	complemented
10	D₃×#41	—	2	4	no	—
11	#1007	—	2	1	no	—
12	#1004	—	2	3	no	—
13	C₂×#235	C₂²	1	6	yes	—
14	C₃×#131	—	2	4	no	—
15	C₂×C₃×M₁₆	C₂²	1	6	yes	complemented
16	Q₈×Dic₃	—	2	1	no	—
17	D₄×Dic₃	—	2	3	no	—
18	#926	C₂²	1	3	yes	complemented
19	C₂×C₄×Dic₃	C₂²	1	3	yes	complemented
20	#825	C₂²	1	4	yes	complemented
21	#825	C₂²	1	12	yes	complemented
22	#585	—	3	1	no	—
23	#568	D₃	1	1	yes	characteristic, complemented
24	C₄×#41	—	3	1	no	—
25	#421	—	3	6	no	—
26	C₂×#104	—	3	6	no	—
27	C₃×#41	C₂×C₄	1	4	yes	—
28	C₃×#41	D₄	1	4	yes	—
29	C₂×C₃×Q₈	D₄	1	1	yes	characteristic
30	C₂×C₃×D₄	D₄	1	3	yes	—
31	C₂²×C₃×C₄	C₂³	1	3	yes	—
32	C₂×#67	—	2	6	no	—
33	C₂²×Dic₃	—	2	3	no	—
34	#239	—	2	4	no	—
35	D₃×Q₈	—	2	4	no	—
36	#237	—	2	12	no	—
37	D₃×D₄	—	2	12	no	—
38	#235	—	2	6	no	—
39	#235	—	2	6	no	—
40	#235	C₂×C₄	1	12	yes	complemented
41	#235	C₂³	1	4	yes	—
42	C₂×D₁₂	C₂×C₄	1	6	yes	complemented
43	C₂×C₄×D₃	—	2	6	no	—
44	C₂×Dic₆	C₂×C₄	1	6	yes	complemented
45	C₃×M₁₆	—	2	6	no	—
46	C₃×M₁₆	C₂³	1	4	yes	—
47	C₂×C₃×C₈	—	2	6	no	—
48	#217	—	2	3	no	—
49	C₄⋊Dic₃	—	2	3	no	—
50	C₄×Dic₃	—	2	3	no	—
51	C₄×Dic₃	C₂³	1	1	yes	characteristic
52	C₄×Dic₃	C₂³	1	3	yes	—
53	2₋¹⁺⁴	—	3	4	no	—
54	2₋¹⁺⁴	—	6	1	no	—
55	H(5,𝔽₂)	—	3	4	no	—
56	H(5,𝔽₂)	—	6	3	no	—
57	C₂×#41	—	3	6	no	—
58	C₂×#41	—	6	4	no	—
59	C₂×#41	C₂×D₃	1	1	yes	characteristic
60	#131	—	2	4	no	—
61	C₂×M₁₆	C₂×D₃	1	6	yes	complemented
62	C₄×Q₈	—	6	1	no	—
63	C₄×D₄	—	6	3	no	—
64	#117	—	3	3	no	—
65	C₂×C₄²	—	3	3	no	—
66	#104	—	3	4	no	—
67	#104	—	3	12	no	—
68	C₂³×C₃	C₂×D₄	1	3	yes	—
69	C₂²×D₃	—	2	6	no	—
70	C₃×Q₈	C₂²⋊C₄	1	4	yes	—
71	C₃×D₄	C₂²⋊C₄	1	12	yes	—
72	C₂×C₃×C₄	C₂²×C₄	1	6	yes	—
73	C₂×C₃×C₄	C₂×D₄	1	3	yes	—
74	C₂×C₃×C₄	C₂×D₄	1	6	yes	—
75	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
76	#67	—	2	12	no	—
77	#67	—	2	4	no	—
78	C₂×Dic₃	—	2	6	no	—
79	C₂×Dic₃	—	2	1	no	—
80	C₂×Dic₃	—	2	3	no	—
81	C₂×Dic₃	—	4	6	no	—
82	D₁₂	—	2	6	no	—
83	D₁₂	C₂²×C₄	1	4	yes	—
84	C₄×D₃	—	2	12	no	—
85	C₄×D₃	—	2	4	no	—
86	Dic₆	—	2	6	no	—
87	Dic₆	C₂²×C₄	1	4	yes	—
88	C₃×C₈	—	2	4	no	—
89	#41	—	3	12	no	—
90	#41	—	3	4	no	—
91	#41	—	6	4	no	—
92	#41	—	6	12	no	—
93	#41	—	6	6	no	—
94	#41	—	6	6	no	—
95	#41	C₄×D₃	1	4	yes	—
96	#41	#67	1	4	yes	—
97	C₂×Q₈	—	3	6	no	—
98	C₂×Q₈	—	6	4	no	—
99	C₂×Q₈	D₁₂	1	1	yes	characteristic
100	C₂×D₄	—	3	6	no	—
101	C₂×D₄	—	6	6	no	—
102	C₂×D₄	—	6	12	no	—
103	C₂×D₄	D₁₂	1	3	yes	—
104	C₂²×C₄	—	6	6	no	—
105	C₂²×C₄	—	6	3	no	—
106	C₂²×C₄	C₂²×D₃	1	3	yes	—
107	M₁₆	—	2	6	no	—
108	M₁₆	C₂²×D₃	1	4	yes	—
109	C₂×C₈	—	2	6	no	—
110	C₄⋊C₄	—	6	3	no	—
111	C₂²⋊C₄	—	6	3	no	—
112	C₄²	—	3	1	no	—
113	C₄²	—	3	3	no	—
114	C₄²	—	6	3	no	—
115	C₂²×C₃	—	4	3	no	—
116	C₂²×C₃	C₂×C₂²⋊C₄	1	6	yes	—
117	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×D₃	—	2	4	no	—
119	C₂×D₃	—	8	6	no	—
120	C₃×C₄	C₂×C₂²⋊C₄	1	6	yes	—
121	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
122	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
123	Dic₃	—	2	2	no	—
124	Dic₃	—	2	4	no	—
125	Dic₃	—	4	3	no	—
126	C₂³	—	6	6	no	—
127	C₂³	C₂×D₁₂	1	3	yes	—
128	Q₈	—	3	4	no	—
129	Q₈	—	6	6	no	—
130	Q₈	(C₂×D₃)⋊C₄	1	4	yes	—
131	D₄	—	3	4	no	—
132	D₄	—	6	12	no	—
133	D₄	—	6	4	no	—
134	D₄	—	6	6	no	—
135	D₄	(C₂×D₃)⋊C₄	1	12	yes	—
136	C₂×C₄	—	6	6	no	—
137	C₂×C₄	—	6	1	no	—
138	C₂×C₄	—	6	12	no	—
139	C₂×C₄	—	6	4	no	—
140	C₂×C₄	—	6	3	no	—
141	C₂×C₄	—	12	6	no	—
142	C₂×C₄	C₂×C₄×D₃	1	6	yes	—
143	C₂×C₄	C₂×D₁₂	1	3	yes	—
144	C₂×C₄	C₂×#67	1	6	yes	—
145	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
146	C₈	—	2	4	no	—
147	C₂×C₃	—	2	1	no	—
148	C₂×C₃	—	2	6	no	—
149	C₂×C₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
150	D₃	—	4	4	no	—
151	C₂²	—	4	3	no	—
152	C₂²	—	6	4	no	—
153	C₂²	—	24	6	no	—
154	C₂²	C₂×(C₂×D₃)⋊C₄	1	6	yes	—
155	C₂²	C₂²×D₁₂	1	1	yes	characteristic
156	C₄	—	6	2	no	—
157	C₄	—	6	4	no	—
158	C₄	—	12	3	no	—
159	C₄	C₂×(C₂×D₃)⋊C₄	1	6	yes	—
160	C₄	C₂²×C₄×D₃	1	1	yes	characteristic
161	C₄	C₂²×#67	1	1	yes	characteristic
162	C₃	#2901	1	1	yes	characteristic, complemented, Sylow
163	C₂	—	2	1	no	—
164	C₂	—	2	6	no	—
165	C₂	—	12	4	no	—
166	C₂	C₂²×(C₂×D₃)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×#67
commutator	C₃×C₄	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#568	C₂
socle	C₂×C₃	C₂²×C₂²⋊C₄
prime radical	#5964	C₂
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	#568	D₃
hypocenter	C₃	#2901
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂×#41 ◁ #568

Complements

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

N	H ≅ G/N	type
2. C₃×#568	C₂	semidirect
3. Dic₃×#41	C₂	semidirect
4. #5139	C₂	semidirect
5. C₂×#825	C₂	semidirect
8. #1010	C₄	semidirect
9. #1009	C₄	semidirect
15. C₂×C₃×M₁₆	C₂²	semidirect
18. #926	C₂²	semidirect
19. C₂×C₄×Dic₃	C₂²	semidirect
20. #825	C₂²	semidirect
21. #825	C₂²	semidirect
23. #568	D₃	semidirect
40. #235	C₂×C₄	semidirect
42. C₂×D₁₂	C₂×C₄	semidirect
44. C₂×Dic₆	C₂×C₄	semidirect
61. C₂×M₁₆	C₂×D₃	semidirect
162. C₃	#2901	semidirect

Maximal subgroups

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

subgroup	index
#5964	2
C₃×#568	2
Dic₃×#41	2
#5139	2
C₂×#825	2
#2901	3