Finite Group Databasebeta

Subgroups — #382949

Order 512.

599 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 599 conjugacy classes of subgroups — too many to draw. Only its 232 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×#1747	C₂	1	1	yes	maximal, characteristic
2	#11311	C₂	1	2	yes	maximal
3	#11307	C₂	1	6	yes	maximal
4	#10872	C₂	1	3	yes	maximal
5	#10871	C₂	1	3	yes	maximal
6	C₂²×#108	C₂²	1	1	yes	characteristic
7	C₂×#427	C₂²	1	3	yes	—
8	C₂×#405	C₂²	1	3	yes	—
9	#1747	C₄	1	4	yes	complemented
10	#1747	C₄	1	4	yes	complemented
11	C₂×#344	C₂²	1	6	yes	—
12	#1740	C₂²	1	6	yes	—
13	#1733	C₂²	1	6	yes	—
14	#1733	C₂²	1	6	yes	—
15	C₂×#338	C₂²	1	1	yes	characteristic
16	C₂×H(3,ℤ/4)	C₂²	1	3	yes	—
17	C₂²×M₁₆	C₂³	1	2	yes	—
18	C₂²×M₁₆	C₂³	1	3	yes	—
19	C₂³×C₈	C₂³	1	1	yes	characteristic
20	C₂×#117	C₂³	1	6	yes	—
21	C₂×#117	C₂³	1	3	yes	—
22	C₂×#108	—	2	3	no	—
23	C₂×#108	—	2	3	no	—
24	#427	—	2	3	no	—
25	#427	—	2	3	no	—
26	#427	—	2	3	no	—
27	#427	—	2	3	no	—
28	C₂×#107	—	2	3	no	—
29	C₂×#106	—	2	3	no	—
30	#405	—	2	3	no	—
31	#405	—	2	3	no	—
32	#405	C₂×C₄	1	6	yes	—
33	#405	C₂×C₄	1	6	yes	—
34	C₂×#97	—	2	3	no	—
35	C₂×C₄×C₈	—	2	3	no	—
36	#344	—	4	6	no	—
37	#342	—	4	6	no	—
38	#339	—	4	6	no	—
39	#339	—	4	6	no	—
40	#338	C₂×C₄	1	4	yes	—
41	H(3,ℤ/4)	C₂×C₄	1	12	yes	—
42	C₂³×C₄	C₂⁴	1	1	yes	characteristic
43	C₂×M₁₆	—	2	3	no	—
44	C₂×M₁₆	—	2	3	no	—
45	C₂×M₁₆	—	2	3	no	—
46	C₂×M₁₆	—	2	3	no	—
47	C₂×M₁₆	—	2	3	no	—
48	C₂×M₁₆	—	2	6	no	—
49	C₂×M₁₆	—	2	6	no	—
50	C₂×M₁₆	C₂²×C₄	1	6	yes	—
51	C₂²×C₈	—	2	3	no	—
52	C₂²×C₈	—	2	2	no	—
53	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₈	#41	1	3	yes	—
56	C₂²×C₈	#41	1	3	yes	—
57	C₂²×C₈	#41	1	3	yes	—
58	C₂²×C₈	#41	1	3	yes	—
59	#117	—	2	3	no	—
60	#117	—	2	3	no	—
61	#117	—	2	3	no	—
62	#117	—	2	6	no	—
63	#117	—	2	6	no	—
64	#117	—	2	6	no	—
65	#117	—	2	6	no	—
66	#117	C₂²×C₄	1	6	yes	—
67	C₂×C₄⋊C₄	—	2	3	no	—
68	C₂×C₄⋊C₄	—	2	6	no	—
69	C₂×C₂²⋊C₄	—	2	3	no	—
70	C₂×C₂²⋊C₄	—	2	6	no	—
71	C₂×C₄²	—	2	3	no	—
72	C₂×C₄²	—	2	6	no	—
73	#108	—	4	1	no	—
74	#108	—	4	3	no	—
75	#107	—	4	3	no	—
76	#107	—	4	3	no	—
77	#106	—	4	3	no	—
78	#106	—	4	3	no	—
79	#97	—	2	6	no	—
80	#97	—	2	6	no	—
81	C₄×C₈	—	2	6	no	—
82	C₄×C₈	—	2	6	no	—
83	C₂⁴	H(5,𝔽₂)	1	1	yes	characteristic
84	C₂²×C₄	—	4	3	no	—
85	C₂²×C₄	—	4	6	no	—
86	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
87	C₂²×C₄	C₂×#41	1	3	yes	—
88	C₂²×C₄	C₂×#41	1	3	yes	—
89	C₂²×C₄	H(5,𝔽₂)	1	3	yes	—
90	C₂²×C₄	2₋¹⁺⁴	1	3	yes	—
91	C₂²×C₄	2₋¹⁺⁴	1	1	yes	characteristic
92	M₁₆	—	2	6	no	—
93	M₁₆	—	2	6	no	—
94	M₁₆	—	4	3	no	—
95	M₁₆	—	4	3	no	—
96	M₁₆	—	4	2	no	—
97	M₁₆	—	4	6	no	—
98	C₂×C₈	—	2	6	no	—
99	C₂×C₈	—	4	3	no	—
100	C₂×C₈	—	4	3	no	—
101	C₂×C₈	—	4	6	no	—
102	C₂×C₈	#117	1	3	yes	—
103	C₂×C₈	#117	1	3	yes	—
104	C₂×C₈	#117	1	3	yes	—
105	C₂×C₈	#117	1	3	yes	—
106	C₂×C₈	#126	1	3	yes	—
107	C₂×C₈	#126	1	3	yes	—
108	C₂×C₈	#126	1	3	yes	—
109	C₂×C₈	#126	1	3	yes	—
110	C₂×C₈	#126	1	1	yes	characteristic
111	C₂×C₈	#126	1	3	yes	—
112	C₄⋊C₄	—	2	6	no	—
113	C₄⋊C₄	—	4	3	no	—
114	C₄⋊C₄	—	4	6	no	—
115	C₄⋊C₄	—	4	6	no	—
116	C₂²⋊C₄	—	2	6	no	—
117	C₂²⋊C₄	—	4	3	no	—
118	C₂²⋊C₄	—	4	6	no	—
119	C₂²⋊C₄	—	4	6	no	—
120	C₄²	—	2	6	no	—
121	C₄²	—	4	3	no	—
122	C₄²	—	4	6	no	—
123	C₄²	—	4	6	no	—
124	C₂³	—	8	1	no	—
125	C₂³	#520	1	1	yes	characteristic
126	C₂³	#540	1	3	yes	—
127	C₂³	#542	1	3	yes	—
128	C₂×C₄	—	4	6	no	—
129	C₂×C₄	—	8	6	no	—
130	C₂×C₄	—	8	6	no	—
131	C₂×C₄	—	16	6	no	—
132	C₂×C₄	C₂×#117	1	3	yes	—
133	C₂×C₄	#520	1	3	yes	—
134	C₂×C₄	C₂×#126	1	1	yes	characteristic
135	C₂×C₄	C₂×#126	1	3	yes	—
136	C₂×C₄	#540	1	3	yes	—
137	C₂×C₄	#541	1	3	yes	—
138	C₂×C₄	#541	1	3	yes	—
139	C₂×C₄	#542	1	3	yes	—
140	C₂×C₄	#542	1	3	yes	—
141	C₂×C₄	#542	1	3	yes	—
142	C₈	—	4	6	no	—
143	C₈	—	8	2	no	—
144	C₈	#383	1	1	yes	characteristic
145	C₈	#383	1	1	yes	characteristic
146	C₈	#383	1	3	yes	—
147	C₈	#383	1	3	yes	—
148	C₂²	—	4	3	no	—
149	C₂²	—	4	3	no	—
150	C₂²	—	4	1	no	—
151	C₂²	#2314	1	3	yes	—
152	C₂²	#2644	1	3	yes	—
153	C₂²	#2645	1	1	yes	characteristic
154	C₄	—	8	6	no	—
155	C₄	—	16	6	no	—
156	C₄	C₂×#383	1	1	yes	characteristic
157	C₄	#2314	1	3	yes	—
158	C₄	#2644	1	1	yes	characteristic
159	C₄	#2645	1	3	yes	—
160	C₂	—	2	3	no	—
161	C₂	—	2	3	no	—
162	C₂	—	2	1	no	—
163	C₂	#14350	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#2644
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14350
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
9. #1747	C₄	semidirect
10. #1747	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#1747	2
#11311	2
#11307	2
#10872	2
#10871	2