Finite Group Databasebeta

Subgroups — #82033

Order 384.

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

normal subgroups only

This group has 350 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	#5771	C₂	1	2	yes	maximal, complemented
2	C₃×#508	C₂	1	1	yes	maximal, characteristic, complemented
3	#5195	C₂	1	2	yes	maximal, complemented
4	D₃×Q₃₂	C₂	1	2	yes	maximal, complemented
5	#4915	C₂	1	4	yes	maximal, complemented
6	#4912	C₂	1	2	yes	maximal, complemented
7	C₂×Dic₂₄	C₂	1	1	yes	maximal, characteristic, complemented
8	#4901	C₂	1	1	yes	maximal, characteristic, complemented
9	#3417	—	3	1	no	maximal, Sylow
10	#1010	—	2	2	no	—
11	C₃×#135	C₂²	1	2	yes	—
12	#951	—	2	2	no	—
13	D₃×Q₁₆	C₂²	1	4	yes	—
14	#915	—	2	4	no	—
15	#912	C₂²	1	4	yes	—
16	C₂×Dic₁₂	C₂²	1	2	yes	—
17	#901	C₂²	1	1	yes	characteristic
18	C₃×Q₃₂	C₂²	1	1	yes	characteristic
19	C₃×SD₃₂	C₂²	1	2	yes	—
20	C₃×D₁₆	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
22	#829	C₂²	1	4	yes	—
23	#827	C₂²	1	4	yes	—
24	#812	C₂²	1	1	yes	characteristic
25	Dic₂₄	C₂²	1	2	yes	—
26	Dic₂₄	C₂²	1	2	yes	—
27	#798	C₂²	1	2	yes	—
28	D₃×C₁₆	C₂²	1	2	yes	—
29	#578	—	3	2	no	—
30	#510	—	3	2	no	—
31	#510	—	3	4	no	—
32	#508	—	3	2	no	—
33	#508	D₃	1	1	yes	characteristic, complemented
34	C₂×Q₃₂	—	3	2	no	—
35	C₂×Q₃₂	—	3	1	no	—
36	#504	—	3	1	no	—
37	C₃×#41	—	2	2	no	—
38	D₃×Q₈	—	2	4	no	—
39	#237	—	2	4	no	—
40	#237	—	4	4	no	—
41	#235	—	4	2	no	—
42	#235	D₄	1	1	yes	characteristic
43	C₂×Dic₆	—	2	2	no	—
44	C₂×Dic₆	—	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	#216	—	2	4	no	—
50	#214	—	2	4	no	—
51	#208	D₄	1	1	yes	characteristic
52	Dic₁₂	—	2	2	no	—
53	Dic₁₂	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	2₋¹⁺⁴	—	6	2	no	—
60	#137	—	6	4	no	—
61	#137	—	6	2	no	—
62	#135	—	3	4	no	—
63	#135	C₂×D₃	1	2	yes	—
64	C₂×Q₁₆	—	3	4	no	—
65	C₂×Q₁₆	—	3	2	no	—
66	#131	—	3	1	no	—
67	Q₃₂	—	3	4	no	—
68	Q₃₂	—	3	2	no	—
69	Q₃₂	—	3	2	no	—
70	Q₃₂	C₂×D₃	1	1	yes	characteristic
71	SD₃₂	—	3	4	no	—
72	SD₃₂	C₂×D₃	1	2	yes	—
73	D₁₆	C₂×D₃	1	1	yes	characteristic
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
78	C₃×Q₈	—	2	2	no	—
79	C₃×D₄	—	2	2	no	—
80	C₃×D₄	—	4	2	no	—
81	C₂×C₃×C₄	—	4	2	no	—
82	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
83	#67	—	4	4	no	—
84	#67	D₈	1	2	yes	—
85	C₂×Dic₃	—	4	2	no	—
86	C₂×Dic₃	—	4	4	no	—
87	D₁₂	D₈	1	1	yes	characteristic
88	C₄×D₃	—	4	4	no	—
89	C₄×D₃	C₂×D₄	1	2	yes	—
90	Dic₆	—	2	4	no	—
91	Dic₆	—	4	2	no	—
92	Dic₆	—	4	4	no	—
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	#41	—	12	4	no	—
102	C₂×Q₈	—	6	4	no	—
103	C₂×Q₈	—	6	2	no	—
104	C₂×Q₈	—	12	2	no	—
105	Q₁₆	—	3	4	no	—
106	Q₁₆	—	6	4	no	—
107	Q₁₆	—	6	2	no	—
108	Q₁₆	C₂²×D₃	1	2	yes	—
109	SD₁₆	—	2	2	no	—
110	SD₁₆	—	6	4	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	C₂×D₈	1	2	yes	—
122	C₃×C₄	—	4	2	no	—
123	C₃×C₄	C₂×D₈	1	1	yes	characteristic
124	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
125	Dic₃	—	4	4	no	—
126	Dic₃	C₂×D₈	1	2	yes	—
127	Q₈	—	2	2	no	—
128	Q₈	—	3	1	no	—
129	Q₈	—	6	4	no	—
130	Q₈	—	12	4	no	—
131	Q₈	—	12	2	no	—
132	D₄	—	2	2	no	—
133	D₄	—	3	2	no	—
134	D₄	—	3	1	no	—
135	D₄	—	4	2	no	—
136	D₄	—	12	4	no	—
137	C₂×C₄	—	3	2	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	12	4	no	—
140	C₂×C₄	—	12	4	no	—
141	C₂×C₄	—	12	2	no	—
142	C₂×C₄	D₃×D₄	1	1	yes	characteristic
143	C₈	—	3	2	no	—
144	C₈	D₃×D₄	1	1	yes	characteristic
145	C₈	C₂³×D₃	1	1	yes	characteristic
146	C₂×C₃	—	2	1	no	—
147	C₂×C₃	—	8	2	no	—
148	C₂×C₃	C₂²×D₈	1	1	yes	characteristic
149	D₃	—	2	2	no	—
150	C₂²	—	3	2	no	—
151	C₂²	—	4	2	no	—
152	C₂²	D₃×D₈	1	1	yes	characteristic
153	C₄	—	3	2	no	—
154	C₄	—	4	2	no	—
155	C₄	—	12	4	no	—
156	C₄	D₃×D₈	1	1	yes	characteristic
157	C₄	C₂×D₃×D₄	1	1	yes	characteristic
158	C₃	#3417	1	1	yes	characteristic, complemented, Sylow
159	C₂	—	2	1	no	—
160	C₂	—	6	2	no	—
161	C₂	—	8	2	no	—
162	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	#3417
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. #5771	C₂	semidirect
2. C₃×#508	C₂	semidirect
3. #5195	C₂	semidirect
4. D₃×Q₃₂	C₂	semidirect
5. #4915	C₂	semidirect
6. #4912	C₂	semidirect
7. C₂×Dic₂₄	C₂	semidirect
8. #4901	C₂	semidirect
33. #508	D₃	semidirect
158. C₃	#3417	semidirect

Maximal subgroups

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

subgroup	index
#5771	2
C₃×#508	2
#5195	2
D₃×Q₃₂	2
#4915	2
#4912	2
C₂×Dic₂₄	2
#4901	2
#3417	3