Finite Group Databasebeta

Subgroups — #79258

Order 384.

458 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 177 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	#5748	C₂	1	1	yes	maximal, characteristic
2	#5745	C₂	1	2	yes	maximal, complemented
3	C₃×#430	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#825	C₂	1	2	yes	maximal, complemented
5	Dic₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#4895	C₂	1	4	yes	maximal, complemented
7	#4894	C₂	1	4	yes	maximal, complemented
8	#2956	—	3	1	no	maximal, Sylow
9	C₂×#235	C₂²	1	2	yes	—
10	C₂×C₃×M₁₆	C₂²	1	2	yes	—
11	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
13	C₂×#208	C₂²	1	1	yes	characteristic
14	#909	C₄	1	4	yes	complemented
15	#908	C₄	1	4	yes	complemented
16	#907	C₂²	1	8	yes	complemented
17	D₃×M₁₆	—	2	4	no	—
18	#904	C₂²	1	4	yes	—
19	#901	—	2	4	no	—
20	C₃×#108	C₂²	1	4	yes	complemented
21	#825	C₂²	1	4	yes	complemented
22	#825	C₂²	1	4	yes	complemented
23	#815	C₂²	1	2	yes	complemented
24	C₈×Dic₃	C₂²	1	2	yes	complemented
25	#575	—	3	1	no	—
26	#568	—	3	2	no	—
27	#444	—	3	4	no	—
28	#443	—	3	4	no	—
29	#430	D₃	1	1	yes	characteristic, complemented
30	C₂×#104	—	3	2	no	—
31	C₄×M₁₆	—	3	1	no	—
32	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
33	C₂×#67	—	2	2	no	—
34	C₂²×Dic₃	—	2	1	no	—
35	#235	—	2	2	no	—
36	#235	—	2	2	no	—
37	#235	C₂×C₄	1	4	yes	—
38	#235	C₂³	1	4	yes	—
39	C₂×D₁₂	C₂×C₄	1	2	yes	—
40	C₂×C₄×D₃	—	2	2	no	—
41	C₂×Dic₆	C₂×C₄	1	2	yes	—
42	C₃×M₁₆	—	2	2	no	—
43	C₃×M₁₆	C₂×C₄	1	4	yes	—
44	C₃×M₁₆	C₂³	1	4	yes	—
45	C₂×C₃×C₈	—	2	2	no	—
46	C₂×C₃×C₈	C₂³	1	2	yes	—
47	C₄×Dic₃	—	2	1	no	—
48	C₄×Dic₃	C₂³	1	1	yes	characteristic
49	C₄×Dic₃	C₂³	1	1	yes	characteristic
50	#208	D₄	1	4	yes	—
51	C₂×C₃⋊C₈	C₂³	1	2	yes	—
52	Dic₁₂	C₂×C₄	1	4	yes	complemented
53	D₂₄	C₂×C₄	1	4	yes	complemented
54	#204	C₂×C₄	1	8	yes	complemented
55	ΓL(1,𝔽₂₅)	—	2	8	no	—
56	D₃×C₈	—	2	8	no	—
57	C₂×#41	—	3	2	no	—
58	#137	—	3	4	no	—
59	AGL(1,ℤ/8)	—	3	4	no	—
60	#135	—	3	4	no	—
61	#131	—	3	8	no	—
62	#131	—	6	4	no	—
63	C₂×M₁₆	—	3	1	no	—
64	C₂×M₁₆	—	6	4	no	—
65	C₂×M₁₆	C₂×D₃	1	2	yes	—
66	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
67	C₂×C₄²	—	3	1	no	—
68	#108	C₂×D₃	1	4	yes	complemented
69	#104	—	3	4	no	—
70	#104	—	3	4	no	—
71	#97	—	3	2	no	—
72	C₄×C₈	—	3	2	no	—
73	C₂³×C₃	#41	1	1	yes	characteristic
74	C₂²×D₃	—	2	2	no	—
75	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
76	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
77	C₂×C₃×C₄	#41	1	1	yes	characteristic
78	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
79	#67	—	2	4	no	—
80	#67	—	2	4	no	—
81	C₂×Dic₃	—	2	2	no	—
82	C₂×Dic₃	—	2	1	no	—
83	C₂×Dic₃	—	2	1	no	—
84	C₂×Dic₃	—	4	2	no	—
85	D₁₂	—	2	2	no	—
86	D₁₂	C₂²×C₄	1	4	yes	—
87	C₄×D₃	—	2	4	no	—
88	C₄×D₃	—	2	4	no	—
89	Dic₆	—	2	2	no	—
90	Dic₆	C₂²×C₄	1	4	yes	—
91	C₃×C₈	—	2	4	no	—
92	C₃×C₈	C₂²×C₄	1	4	yes	—
93	C₃⋊C₈	C₂×D₄	1	4	yes	—
94	#41	—	3	4	no	—
95	#41	—	3	4	no	—
96	#41	—	6	2	no	—
97	#41	—	6	2	no	—
98	C₂×Q₈	—	3	2	no	—
99	C₂×D₄	—	3	2	no	—
100	C₂×D₄	—	6	2	no	—
101	C₂²×C₄	—	6	2	no	—
102	C₂²×C₄	—	6	1	no	—
103	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
104	Q₁₆	—	3	4	no	—
105	SD₁₆	—	3	8	no	—
106	D₈	—	3	4	no	—
107	M₁₆	—	2	2	no	—
108	M₁₆	—	3	4	no	—
109	M₁₆	—	6	8	no	—
110	M₁₆	C₄×D₃	1	4	yes	—
111	M₁₆	C₂²×D₃	1	4	yes	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	3	2	no	—
114	C₂×C₈	—	6	8	no	—
115	C₂×C₈	C₂²×D₃	1	2	yes	—
116	C₄²	—	3	1	no	—
117	C₄²	—	3	1	no	—
118	C₄²	—	6	1	no	—
119	C₂²×C₃	—	4	1	no	—
120	C₂²×C₃	C₄×D₄	1	2	yes	—
121	C₂²×C₃	C₂×#41	1	1	yes	characteristic
122	C₂×D₃	—	2	4	no	—
123	C₂×D₃	—	8	2	no	—
124	C₃×C₄	C₄×D₄	1	2	yes	—
125	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
126	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
127	Dic₃	—	2	2	no	—
128	Dic₃	—	2	4	no	—
129	Dic₃	—	4	1	no	—
130	C₂³	—	6	2	no	—
131	C₂³	#239	1	1	yes	characteristic
132	Q₈	—	3	4	no	—
133	Q₈	—	6	2	no	—
134	D₄	—	3	4	no	—
135	D₄	—	6	4	no	—
136	D₄	—	6	4	no	—
137	D₄	—	6	2	no	—
138	C₂×C₄	—	6	2	no	—
139	C₂×C₄	—	6	1	no	—
140	C₂×C₄	—	6	4	no	—
141	C₂×C₄	—	6	4	no	—
142	C₂×C₄	—	6	1	no	—
143	C₂×C₄	—	12	2	no	—
144	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
145	C₂×C₄	D₃×D₄	1	2	yes	—
146	C₂×C₄	#239	1	1	yes	characteristic
147	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
148	C₈	—	2	4	no	—
149	C₈	—	3	4	no	—
150	C₈	C₂×C₄×D₃	1	4	yes	—
151	C₂×C₃	—	2	1	no	—
152	C₂×C₃	—	2	2	no	—
153	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
154	D₃	—	4	4	no	—
155	C₂²	—	4	1	no	—
156	C₂²	—	6	4	no	—
157	C₂²	—	24	2	no	—
158	C₂²	#893	1	2	yes	—
159	C₂²	C₂×#239	1	1	yes	characteristic
160	C₄	—	6	2	no	—
161	C₄	—	6	4	no	—
162	C₄	—	12	1	no	—
163	C₄	#893	1	2	yes	—
164	C₄	C₂²×C₄×D₃	1	1	yes	characteristic
165	C₄	C₂×D₃×D₄	1	1	yes	characteristic
166	C₃	#2956	1	1	yes	characteristic, complemented, Sylow
167	C₂	—	2	1	no	—
168	C₂	—	2	2	no	—
169	C₂	—	12	4	no	—
170	C₂	C₂×#893	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₃×D₄
commutator	C₃×C₄	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#430	C₂
socle	C₂×C₃	C₂×C₄×D₄
prime radical	#5748	C₂
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	#430	D₃
hypocenter	C₃	#2956
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #430

Complements

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

N	H ≅ G/N	type
2. #5745	C₂	semidirect
3. C₃×#430	C₂	semidirect
4. C₂×#825	C₂	semidirect
5. Dic₃×M₁₆	C₂	semidirect
6. #4895	C₂	semidirect
7. #4894	C₂	semidirect
14. #909	C₄	semidirect
15. #908	C₄	semidirect
16. #907	C₂²	semidirect
20. C₃×#108	C₂²	semidirect
21. #825	C₂²	semidirect
22. #825	C₂²	semidirect
23. #815	C₂²	semidirect
24. C₈×Dic₃	C₂²	semidirect
29. #430	D₃	semidirect
52. Dic₁₂	C₂×C₄	semidirect
53. D₂₄	C₂×C₄	semidirect
54. #204	C₂×C₄	semidirect
68. #108	C₂×D₃	semidirect
166. C₃	#2956	semidirect

Maximal subgroups

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

subgroup	index
#5748	2
#5745	2
C₃×#430	2
C₂×#825	2
Dic₃×M₁₆	2
#4895	2
#4894	2
#2956	3