Finite Group Databasebeta

Subgroups — #68598

Order 384.

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

normal subgroups only

This group has 240 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	C₂×#815	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic
3	#4932	C₂	1	1	yes	maximal, characteristic
4	C₃×#336	C₂	1	1	yes	maximal, characteristic
5	#4550	C₂	1	1	yes	maximal, characteristic
6	#4550	C₂	1	1	yes	maximal, characteristic
7	#4524	C₂	1	1	yes	maximal, characteristic
8	#1746	—	3	1	no	maximal, Sylow
9	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
14	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₃⋊#95	C₄	1	2	yes	—
17	C₃⋊#95	C₄	1	2	yes	—
18	#815	C₄	1	4	yes	complemented
19	C₈×Dic₃	C₄	1	4	yes	complemented
20	C₂×#97	—	3	1	no	—
21	C₂×C₄×C₈	—	3	1	no	—
22	#376	—	3	1	no	—
23	#336	—	3	1	no	—
24	#336	—	3	1	no	—
25	#336	—	3	1	no	—
26	#336	D₃	1	1	yes	characteristic
27	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
28	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
29	C₂²×Dic₃	C₂×C₄	1	2	yes	—
30	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₈	—	2	1	no	—
33	C₂×C₃×C₈	—	2	1	no	—
34	C₂×C₃×C₈	—	2	1	no	—
35	C₂×C₃×C₈	—	2	1	no	—
36	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
37	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
38	C₃×C₄²	—	2	1	no	—
39	C₃×C₄²	—	2	1	no	—
40	C₄×Dic₃	—	2	1	no	—
41	C₄×Dic₃	—	2	1	no	—
42	C₄×Dic₃	C₂×C₄	1	2	yes	—
43	C₄×Dic₃	C₂×C₄	1	2	yes	—
44	C₂×C₃⋊C₈	—	2	1	no	—
45	C₂×C₃⋊C₈	—	2	1	no	—
46	C₂×C₃⋊C₈	—	2	1	no	—
47	C₂×C₃⋊C₈	—	2	1	no	—
48	C₂×C₃⋊C₈	C₂×C₄	1	2	yes	—
49	C₂×C₃⋊C₈	C₂×C₄	1	2	yes	—
50	C₂²×C₈	—	3	1	no	—
51	C₂²×C₈	—	3	1	no	—
52	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
53	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
54	C₂×C₄²	—	3	1	no	—
55	C₂×C₄²	—	3	1	no	—
56	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
57	#97	—	3	4	no	—
58	C₄×C₈	—	3	4	no	—
59	#95	—	3	2	no	—
60	#95	—	3	2	no	—
61	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₃×C₄	—	2	2	no	—
63	C₂×C₃×C₄	—	2	2	no	—
64	C₂×C₃×C₄	—	2	2	no	—
65	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₃×C₄	#41	1	1	yes	characteristic
68	C₂×C₃×C₄	#41	1	1	yes	characteristic
69	C₂×C₃×C₄	#41	1	1	yes	characteristic
70	C₂×C₃×C₄	#41	1	1	yes	characteristic
71	C₂×Dic₃	—	2	2	no	—
72	C₂×Dic₃	—	2	2	no	—
73	C₂×Dic₃	—	2	2	no	—
74	C₂×Dic₃	—	2	2	no	—
75	C₂×Dic₃	—	2	2	no	—
76	C₂×Dic₃	C₄²	1	2	yes	—
77	C₂×Dic₃	C₄²	1	2	yes	—
78	C₃×C₈	—	2	2	no	—
79	C₃×C₈	—	2	2	no	—
80	C₃⋊C₈	—	2	2	no	—
81	C₃⋊C₈	—	2	2	no	—
82	C₂²×C₄	—	3	1	no	—
83	C₂²×C₄	—	3	1	no	—
84	C₂²×C₄	—	3	2	no	—
85	C₂²×C₄	C₄×D₃	1	2	yes	—
86	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	3	2	no	—
92	C₂×C₈	—	3	2	no	—
93	C₂×C₈	—	6	1	no	—
94	C₂×C₈	—	6	1	no	—
95	C₂×C₈	—	6	1	no	—
96	C₂×C₈	—	6	1	no	—
97	C₂×C₈	C₄×D₃	1	2	yes	—
98	C₂×C₈	C₄×D₃	1	2	yes	—
99	C₄²	—	2	1	no	—
100	C₄²	—	2	1	no	—
101	C₄²	—	3	2	no	—
102	C₄²	—	3	2	no	—
103	C₄²	—	6	1	no	—
104	C₄²	—	6	1	no	—
105	C₂²×C₃	C₂×C₄²	1	1	yes	characteristic
106	C₂²×C₃	#117	1	1	yes	characteristic
107	C₂²×C₃	#117	1	1	yes	characteristic
108	C₂²×C₃	#131	1	2	yes	—
109	C₂²×C₃	#131	1	2	yes	—
110	C₃×C₄	—	4	2	no	—
111	C₃×C₄	#117	1	1	yes	characteristic
112	C₃×C₄	#117	1	1	yes	characteristic
113	C₃×C₄	#117	1	1	yes	characteristic
114	C₃×C₄	#117	1	1	yes	characteristic
115	Dic₃	—	2	2	no	—
116	Dic₃	—	2	2	no	—
117	Dic₃	—	4	2	no	—
118	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	3	2	no	—
123	C₂×C₄	—	3	2	no	—
124	C₂×C₄	—	6	2	no	—
125	C₂×C₄	—	6	2	no	—
126	C₂×C₄	—	6	2	no	—
127	C₂×C₄	—	6	2	no	—
128	C₂×C₄	—	6	2	no	—
129	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
130	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
131	C₂×C₄	#237	1	1	yes	characteristic
132	C₂×C₄	#237	1	1	yes	characteristic
133	C₂×C₄	#237	1	1	yes	characteristic
134	C₂×C₄	#239	1	1	yes	characteristic
135	C₈	—	2	2	no	—
136	C₈	—	2	2	no	—
137	C₈	—	6	2	no	—
138	C₈	—	6	2	no	—
139	C₂×C₃	#376	1	1	yes	characteristic
140	C₂×C₃	#405	1	1	yes	characteristic
141	C₂×C₃	#405	1	1	yes	characteristic
142	C₂×C₃	#433	1	2	yes	—
143	C₂×C₃	#433	1	2	yes	—
144	C₂²	C₄²×D₃	1	1	yes	characteristic
145	C₂²	#877	1	1	yes	characteristic
146	C₂²	#892	1	1	yes	characteristic
147	C₂²	#901	1	2	yes	—
148	C₂²	#907	1	2	yes	—
149	C₄	—	4	2	no	—
150	C₄	—	6	2	no	—
151	C₄	—	6	2	no	—
152	C₄	—	12	2	no	—
153	C₄	#877	1	1	yes	characteristic
154	C₄	#877	1	1	yes	characteristic
155	C₄	#892	1	1	yes	characteristic
156	C₄	#892	1	1	yes	characteristic
157	C₃	#1746	1	1	yes	characteristic, complemented, Sylow
158	C₂	#4648	1	1	yes	characteristic
159	C₂	#4689	1	1	yes	characteristic
160	C₂	#4708	1	1	yes	characteristic
161	C₂	#4720	1	2	yes	—
162	C₂	#4840	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₃
commutator	C₂²×C₃	C₂×C₄²
Fitting	C₃×#336	C₂
socle, prime radical	C₂³×C₃	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	#336	D₃
hypocenter	C₃	#1746
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ #336

Complements

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

N	H ≅ G/N	type
18. #815	C₄	semidirect
19. C₈×Dic₃	C₄	semidirect
157. C₃	#1746	semidirect

Maximal subgroups

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

subgroup	index
C₂×#815	2
C₂×C₈×Dic₃	2
#4932	2
C₃×#336	2
#4550	2
#4524	2
#1746	3