Finite Group Databasebeta

Subgroups — #69190

Order 384.

260 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₂×D₃×D₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#798	C₂	1	1	yes	maximal, characteristic, complemented
3	#4880	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#357	C₂	1	1	yes	maximal, characteristic, complemented
5	#4562	C₂	1	1	yes	maximal, characteristic, complemented
6	#4509	C₂	1	1	yes	maximal, characteristic, complemented
7	#4491	C₂	1	1	yes	maximal, characteristic, complemented
8	#2140	—	3	1	no	maximal, Sylow
9	C₂×D₃×D₄	—	2	1	no	—
10	C₂×C₃×D₈	C₂²	1	1	yes	characteristic
11	C₂×#213	—	2	1	no	—
12	D₃×D₈	—	2	2	no	—
13	D₃×D₈	C₄	1	4	yes	complemented
14	C₂×D₂₄	C₂²	1	1	yes	characteristic
15	C₂×D₃×C₈	C₂²	1	1	yes	characteristic
16	#892	—	2	1	no	—
17	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic, complemented
18	C₃×#107	C₂²	1	1	yes	characteristic, complemented
19	#818	C₂²	1	1	yes	characteristic, complemented
20	C₂×C₃⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
21	#808	—	2	1	no	—
22	#798	—	2	2	no	—
23	C₂²×D₈	—	3	1	no	—
24	C₂×M₃₂	—	3	1	no	—
25	#427	—	3	1	no	—
26	#357	—	3	1	no	—
27	#357	—	3	1	no	—
28	#357	—	3	1	no	—
29	#357	D₃	1	1	yes	characteristic, complemented
30	C₂³×D₃	—	4	1	no	—
31	C₂×C₃×D₄	—	2	1	no	—
32	C₂×#67	—	4	1	no	—
33	D₃×D₄	—	2	4	no	—
34	D₃×D₄	—	4	2	no	—
35	C₂×D₁₂	—	2	1	no	—
36	C₂×C₄×D₃	D₄	1	1	yes	characteristic
37	C₃×D₈	—	2	1	no	—
38	C₃×D₈	C₂×C₄	1	2	yes	—
39	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
40	C₃×C₄⋊C₄	—	2	1	no	—
41	#213	—	2	4	no	—
42	(C₂×D₃)⋊C₄	—	4	1	no	—
43	C₄⋊Dic₃	—	2	1	no	—
44	C₄×Dic₃	—	4	1	no	—
45	C₂×C₃⋊C₈	D₄	1	1	yes	characteristic
46	D₂₄	—	2	1	no	—
47	D₂₄	C₂×C₄	1	2	yes	—
48	D₃×C₈	C₂×C₄	1	2	yes	—
49	D₃×C₈	D₄	1	2	yes	—
50	C₃×C₁₆	—	2	1	no	—
51	C₃⋊C₁₆	—	2	1	no	—
52	C₂²×D₄	—	6	1	no	—
53	C₂×D₈	—	3	4	no	—
54	C₂×D₈	—	3	1	no	—
55	C₂×D₈	—	6	2	no	—
56	C₂×D₈	—	6	1	no	—
57	C₂×D₈	C₂×D₃	1	1	yes	characteristic
58	C₂²×C₈	—	3	1	no	—
59	#117	—	6	1	no	—
60	M₃₂	—	6	2	no	—
61	C₂×C₁₆	—	3	1	no	—
62	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic, complemented
63	#107	—	3	1	no	—
64	#107	C₂×D₃	1	1	yes	characteristic, complemented
65	#106	—	6	1	no	—
66	C₂³×C₃	—	4	1	no	—
67	C₂²×D₃	—	4	1	no	—
68	C₂²×D₃	—	4	4	no	—
69	C₂²×D₃	—	8	2	no	—
70	C₂²×D₃	—	8	2	no	—
71	C₂²×D₃	D₈	1	1	yes	characteristic
72	C₃×D₄	—	2	2	no	—
73	C₃×D₄	—	4	1	no	—
74	C₂×C₃×C₄	—	4	1	no	—
75	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
76	#67	—	4	4	no	—
77	C₂×Dic₃	—	4	1	no	—
78	C₂×Dic₃	D₈	1	1	yes	characteristic
79	D₁₂	—	2	2	no	—
80	D₁₂	—	4	1	no	—
81	C₄×D₃	C₂²⋊C₄	1	2	yes	—
82	C₄×D₃	SD₁₆	1	2	yes	—
83	C₃×C₈	C₂²×C₄	1	1	yes	characteristic
84	C₃×C₈	C₂×D₄	1	1	yes	characteristic
85	C₃⋊C₈	C₂²⋊C₄	1	2	yes	—
86	C₂⁴	—	12	1	no	—
87	C₂×D₄	—	2	1	no	—
88	C₂×D₄	—	6	4	no	—
89	C₂×D₄	—	6	1	no	—
90	C₂×D₄	—	12	2	no	—
91	C₂×D₄	—	12	1	no	—
92	C₂²×C₄	—	3	1	no	—
93	D₈	—	2	1	no	—
94	D₈	—	3	2	no	—
95	D₈	—	6	4	no	—
96	D₈	—	6	1	no	—
97	D₈	C₄×D₃	1	2	yes	—
98	C₂×C₈	—	3	1	no	—
99	C₂×C₈	—	3	2	no	—
100	C₂×C₈	—	3	2	no	—
101	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	6	1	no	—
104	C₂²⋊C₄	—	12	1	no	—
105	C₄²	—	12	1	no	—
106	C₁₆	—	2	1	no	—
107	C₁₆	—	6	1	no	—
108	C₂²×C₃	—	4	2	no	—
109	C₂²×C₃	—	8	1	no	—
110	C₂²×C₃	—	8	1	no	—
111	C₂²×C₃	C₂×D₈	1	1	yes	characteristic
112	C₂×D₃	—	2	1	no	—
113	C₂×D₃	—	2	1	no	—
114	C₂×D₃	—	4	2	no	—
115	C₂×D₃	—	8	8	no	—
116	C₂×D₃	—	8	1	no	—
117	C₂×D₃	—	8	1	no	—
118	C₂×D₃	#102	1	2	yes	—
119	C₃×C₄	—	8	1	no	—
120	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
122	Dic₃	—	8	1	no	—
123	Dic₃	#102	1	2	yes	—
124	C₂³	—	3	1	no	—
125	C₂³	—	4	1	no	—
126	C₂³	—	12	4	no	—
127	C₂³	—	12	1	no	—
128	C₂³	—	24	2	no	—
129	C₂³	—	24	2	no	—
130	D₄	—	2	2	no	—
131	D₄	—	4	1	no	—
132	D₄	—	6	2	no	—
133	D₄	—	12	4	no	—
134	D₄	—	12	1	no	—
135	C₂×C₄	—	3	1	no	—
136	C₂×C₄	—	3	2	no	—
137	C₂×C₄	—	3	2	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	—	12	1	no	—
140	C₂×C₄	D₃×D₄	1	1	yes	characteristic
141	C₈	—	3	2	no	—
142	C₈	C₂×C₄×D₃	1	1	yes	characteristic
143	C₈	D₃×D₄	1	1	yes	characteristic
144	C₂×C₃	—	8	2	no	—
145	C₂×C₃	C₂×#102	1	1	yes	characteristic
146	C₂×C₃	#509	1	1	yes	characteristic
147	C₂×C₃	#509	1	1	yes	characteristic
148	D₃	—	2	2	no	—
149	D₃	—	8	2	no	—
150	C₂²	—	3	2	no	—
151	C₂²	—	4	2	no	—
152	C₂²	—	6	1	no	—
153	C₂²	—	6	1	no	—
154	C₂²	—	8	1	no	—
155	C₂²	—	8	1	no	—
156	C₂²	—	12	2	no	—
157	C₂²	—	24	8	no	—
158	C₂²	—	24	1	no	—
159	C₂²	—	24	1	no	—
160	C₂²	D₃×D₈	1	1	yes	characteristic
161	C₄	—	3	2	no	—
162	C₄	—	8	1	no	—
163	C₄	—	24	1	no	—
164	C₄	D₃×C₂²⋊C₄	1	1	yes	characteristic
165	C₄	D₃×SD₁₆	1	1	yes	characteristic
166	C₃	#2140	1	1	yes	characteristic, complemented, Sylow
167	C₂	—	6	2	no	—
168	C₂	—	8	2	no	—
169	C₂	—	24	2	no	—
170	C₂	D₃×#102	1	1	yes	characteristic
171	C₂	#4911	1	1	yes	characteristic
172	C₂	#4914	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×D₈
commutator	C₃×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₃×#357	C₂
socle	C₂²×C₃	C₂×D₈
prime radical	C₂×D₃×D₈	C₂
cosocle	C₂×C₃×C₈	C₂³
hypercenter	#357	D₃
hypocenter	C₃	#2140
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

Complements

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

N	H ≅ G/N	type
2. C₂×#798	C₂	semidirect
3. #4880	C₂	semidirect
4. C₃×#357	C₂	semidirect
5. #4562	C₂	semidirect
6. #4509	C₂	semidirect
7. #4491	C₂	semidirect
13. D₃×D₈	C₄	semidirect
17. C₂×C₃×C₁₆	C₂²	semidirect
18. C₃×#107	C₂²	semidirect
19. #818	C₂²	semidirect
20. C₂×C₃⋊C₁₆	C₂²	semidirect
29. #357	D₃	semidirect
62. C₂×C₁₆	C₂×D₃	semidirect
64. #107	C₂×D₃	semidirect
166. C₃	#2140	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃×D₈	2
C₂×#798	2
#4880	2
C₃×#357	2
#4562	2
#4509	2
#4491	2
#2140	3