Finite Group Databasebeta

Subgroups — #10592904

Order 576.

426 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 87 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	#64367	C₂	1	1	yes	maximal, characteristic
2	C₃²×#137	C₂	1	1	yes	maximal, characteristic, complemented
3	#64153	C₂	1	1	yes	maximal, characteristic, complemented
4	#64151	C₂	1	1	yes	maximal, characteristic, complemented
5	#63664	C₂	1	1	yes	maximal, characteristic, complemented
6	#63652	C₂	1	1	yes	maximal, characteristic, complemented
7	#63651	C₂	1	1	yes	maximal, characteristic, complemented
8	#5205	—	3	4	no	maximal
9	C₃²×#41	C₂²	1	1	yes	characteristic
10	C₂×C₃²×Q₈	C₂²	1	1	yes	characteristic
11	Q₈×C₃⋊D₃	—	2	1	no	—
12	#3830	—	2	1	no	—
13	#3830	—	2	2	no	—
14	#3828	—	2	1	no	—
15	#3828	C₂²	1	1	yes	characteristic
16	C₂×#634	—	2	1	no	—
17	C₂×#634	C₂²	1	1	yes	characteristic
18	C₃²×Q₁₆	—	2	1	no	—
19	C₃²×SD₁₆	—	2	1	no	—
20	C₃²×M₁₆	C₂²	1	1	yes	characteristic
21	#3756	—	2	1	no	—
22	#3755	—	2	1	no	—
23	#3750	—	2	2	no	—
24	C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
25	#3748	C₂²	1	1	yes	characteristic
26	#1010	—	3	4	no	—
27	C₃×#137	D₃	1	4	yes	complemented
28	#944	—	3	4	no	—
29	#942	—	3	4	no	—
30	#837	—	3	4	no	—
31	#825	—	3	4	no	—
32	#824	—	3	4	no	—
33	C₃²×Q₈	—	2	1	no	—
34	C₃²×Q₈	—	2	1	no	—
35	C₃²×Q₈	D₄	1	1	yes	characteristic
36	C₃²×D₄	—	2	1	no	—
37	C₃²×D₄	D₄	1	1	yes	characteristic
38	C₂×C₃²×C₄	—	2	1	no	—
39	C₂×C₃²×C₄	—	2	1	no	—
40	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
41	#638	—	2	2	no	—
42	#638	—	2	1	no	—
43	C₂×C₃⋊Dic₃	—	2	2	no	—
44	C₂×C₃⋊Dic₃	—	2	1	no	—
45	C₂×C₃⋊Dic₃	D₄	1	2	yes	—
46	C₄⋊(C₃⋊D₃)	D₄	1	1	yes	characteristic
47	C₄×C₃⋊D₃	—	2	2	no	—
48	C₄×C₃⋊D₃	—	2	1	no	—
49	#634	—	2	2	no	—
50	#634	—	2	1	no	—
51	#634	—	2	1	no	—
52	#634	D₄	1	1	yes	characteristic
53	C₃²×C₈	—	2	1	no	—
54	C₃⋊(C₃⋊C₈)	—	2	1	no	—
55	#456	—	9	1	no	Sylow
56	C₃×#41	C₂×D₃	1	4	yes	—
57	C₂×C₃×Q₈	C₂×D₃	1	4	yes	—
58	D₃×Q₈	—	6	4	no	—
59	#237	—	6	8	no	—
60	#237	—	6	4	no	—
61	#235	—	3	4	no	—
62	#235	—	6	4	no	—
63	C₂×Dic₆	—	3	4	no	—
64	C₂×Dic₆	—	6	4	no	—
65	C₃×Q₁₆	—	2	4	no	—
66	C₃×SD₁₆	—	2	4	no	—
67	C₃×M₁₆	C₂×D₃	1	4	yes	—
68	#216	—	6	4	no	—
69	#215	—	6	4	no	—
70	Dic₃⋊C₄	—	6	8	no	—
71	C₄×Dic₃	—	3	4	no	—
72	#208	—	3	4	no	—
73	C₂²×C₃²	—	2	1	no	—
74	C₂²×C₃²	C₂×D₄	1	1	yes	characteristic
75	C₂×C₃⋊D₃	—	2	1	no	—
76	C₃²×C₄	—	2	1	no	—
77	C₃²×C₄	—	4	1	no	—
78	C₃²×C₄	C₂×D₄	1	1	yes	characteristic
79	C₃²×C₄	C₂×D₄	1	1	yes	characteristic
80	C₃⋊Dic₃	—	2	2	no	—
81	C₃⋊Dic₃	—	2	2	no	—
82	C₃⋊Dic₃	—	2	1	no	—
83	2₋¹⁺⁴	—	9	1	no	—
84	#137	—	9	1	no	—
85	#137	C₃⋊D₃	1	1	yes	characteristic, complemented
86	C₄⋊Q₈	—	9	1	no	—
87	#104	—	9	1	no	—
88	#104	—	9	1	no	—
89	#101	—	9	1	no	—
90	C₃×Q₈	—	2	4	no	—
91	C₃×Q₈	—	2	4	no	—
92	C₃×Q₈	#67	1	4	yes	—
93	C₃×D₄	—	2	4	no	—
94	C₃×D₄	#67	1	4	yes	—
95	C₂×C₃×C₄	—	2	4	no	—
96	C₂×C₃×C₄	—	2	4	no	—
97	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
98	#67	—	6	8	no	—
99	#67	—	6	4	no	—
100	C₂×Dic₃	—	3	8	no	—
101	C₂×Dic₃	—	6	8	no	—
102	C₂×Dic₃	—	6	4	no	—
103	D₁₂	—	3	4	no	—
104	C₄×D₃	—	6	8	no	—
105	C₄×D₃	—	6	4	no	—
106	Dic₆	—	3	4	no	—
107	Dic₆	—	6	8	no	—
108	Dic₆	—	6	4	no	—
109	Dic₆	—	6	4	no	—
110	C₃×C₈	—	2	4	no	—
111	C₃⋊C₈	—	6	4	no	—
112	C₂×C₃²	—	2	1	no	—
113	C₂×C₃²	—	4	1	no	—
114	C₂×C₃²	#120	1	1	yes	characteristic
115	C₃⋊D₃	—	4	1	no	—
116	#41	—	9	1	no	—
117	#41	—	18	1	no	—
118	#41	—	18	2	no	—
119	#41	—	18	1	no	—
120	#41	C₂×C₃⋊D₃	1	1	yes	characteristic
121	C₂×Q₈	—	9	1	no	—
122	C₂×Q₈	—	18	1	no	—
123	C₂×Q₈	—	18	1	no	—
124	C₂×Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic
125	Q₁₆	—	2	1	no	—
126	Q₁₆	—	18	1	no	—
127	SD₁₆	—	2	1	no	—
128	SD₁₆	—	18	1	no	—
129	M₁₆	—	9	1	no	—
130	M₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic
131	C₄⋊C₄	—	18	2	no	—
132	C₄²	—	9	1	no	—
133	C₂²×C₃	—	2	4	no	—
134	C₂²×C₃	D₃×D₄	1	4	yes	—
135	C₂×D₃	—	6	4	no	—
136	C₃×C₄	—	2	4	no	—
137	C₃×C₄	—	4	4	no	—
138	C₃×C₄	D₃×D₄	1	4	yes	—
139	C₃×C₄	C₂×#67	1	4	yes	—
140	Dic₃	—	6	8	no	—
141	Dic₃	—	6	4	no	—
142	Dic₃	—	6	8	no	—
143	C₃²	#456	1	1	yes	characteristic, complemented, Sylow
144	Q₈	—	2	1	no	—
145	Q₈	—	2	1	no	—
146	Q₈	—	9	1	no	—
147	Q₈	—	18	2	no	—
148	Q₈	—	18	1	no	—
149	Q₈	—	18	1	no	—
150	Q₈	#638	1	1	yes	characteristic
151	D₄	—	2	1	no	—
152	D₄	—	9	1	no	—
153	D₄	—	18	2	no	—
154	D₄	—	18	1	no	—
155	D₄	#638	1	1	yes	characteristic
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	9	2	no	—
159	C₂×C₄	—	18	2	no	—
160	C₂×C₄	—	18	2	no	—
161	C₂×C₄	—	18	1	no	—
162	C₂×C₄	—	18	1	no	—
163	C₂×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
164	C₈	—	2	1	no	—
165	C₈	—	18	1	no	—
166	C₂×C₃	—	2	4	no	—
167	C₂×C₃	—	4	4	no	—
168	C₂×C₃	#937	1	4	yes	—
169	D₃	—	12	4	no	—
170	C₂²	—	2	1	no	—
171	C₂²	—	18	1	no	—
172	C₂²	D₄×C₃⋊D₃	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	4	1	no	—
175	C₄	—	18	2	no	—
176	C₄	—	18	2	no	—
177	C₄	—	18	1	no	—
178	C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
179	C₄	C₂×#638	1	1	yes	characteristic
180	C₃	#5205	1	4	yes	complemented
181	C₂	—	2	1	no	—
182	C₂	—	4	1	no	—
183	C₂	—	36	1	no	—
184	C₂	#64146	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#64146
commutator, cosocle	C₂×C₃²×C₄	C₂³
Frattini	C₂×C₄	C₂²×C₃⋊D₃
Fitting	C₃²×#137	C₂
socle	C₂×C₃²	#120
prime radical	#64367	C₂
hypercenter	#137	C₃⋊D₃
hypocenter	C₃²	#456
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

Complements

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

N	H ≅ G/N	type
2. C₃²×#137	C₂	semidirect
3. #64153	C₂	semidirect
4. #64151	C₂	semidirect
5. #63664	C₂	semidirect
6. #63652	C₂	semidirect
7. #63651	C₂	semidirect
27. C₃×#137	D₃	semidirect
85. #137	C₃⋊D₃	semidirect
143. C₃²	#456	semidirect
180. C₃	#5205	semidirect

Maximal subgroups

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

subgroup	index
#64367	2
C₃²×#137	2
#64153	2
#64151	2
#63664	2
#63652	2
#63651	2
#5205	3