Finite Group Databasebeta

Subgroups — #127409

Order 512.

292 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 124 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	#33428	C₂	1	2	yes	maximal
2	#33420	C₂	1	1	yes	maximal, characteristic
3	#7240	C₂	1	2	yes	maximal, complemented
4	#7239	C₂	1	2	yes	maximal, complemented
5	C₄×#131	C₂²	1	1	yes	characteristic
6	#2166	—	2	2	no	—
7	#2166	C₂²	1	4	yes	—
8	#2165	—	2	2	no	—
9	#2165	C₄	1	4	yes	—
10	#2164	—	2	2	no	—
11	#2162	—	2	2	no	—
12	C₂×#364	—	2	2	no	—
13	#2151	C₂²	1	1	yes	characteristic
14	#2151	C₂²	1	1	yes	characteristic
15	#2115	—	2	2	no	—
16	#2113	C₄	1	1	yes	characteristic
17	#2113	C₄	1	1	yes	characteristic
18	#2110	C₄	1	4	yes	—
19	C₄×M₃₂	C₄	1	1	yes	characteristic
20	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
21	C₂×#131	C₂×C₄	1	1	yes	characteristic
22	C₂×#131	D₄	1	1	yes	characteristic
23	C₄×#41	Q₈	1	1	yes	characteristic
24	C₂×M₃₂	—	2	4	no	—
25	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
26	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
29	#405	C₂×C₄	1	2	yes	—
30	#405	D₄	1	2	yes	—
31	#405	Q₈	1	1	yes	characteristic
32	#405	C₂³	1	1	yes	characteristic
33	C₄×M₁₆	C₂×C₄	1	2	yes	—
34	C₄×M₁₆	D₄	1	1	yes	characteristic
35	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×C₈	D₄	1	2	yes	—
37	#364	—	2	4	no	—
38	#364	—	2	4	no	—
39	C₄⋊C₁₆	—	2	2	no	—
40	C₄⋊C₁₆	—	2	2	no	—
41	#349	—	2	4	no	—
42	#349	—	2	4	no	—
43	C₂²⋊C₁₆	—	2	2	no	—
44	C₂²⋊C₁₆	—	2	2	no	—
45	#347	—	2	4	no	—
46	#347	C₂×C₄	1	8	yes	—
47	#346	—	2	1	no	—
48	C₄×C₁₆	—	2	1	no	—
49	C₄×C₁₆	—	2	1	no	—
50	C₄×C₁₆	—	2	1	no	—
51	C₂×#41	C₄⋊C₄	1	1	yes	characteristic
52	#131	—	4	1	no	—
53	#131	—	4	1	no	—
54	#131	—	4	1	no	—
55	#131	—	4	1	no	—
56	C₂×M₁₆	C₄²	1	2	yes	—
57	C₂×M₁₆	C₂²⋊C₄	1	2	yes	—
58	C₂×M₁₆	C₄⋊C₄	1	1	yes	characteristic
59	C₂×M₁₆	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
64	C₄×Q₈	—	2	1	no	—
65	C₄×D₄	—	2	2	no	—
66	C₄×D₄	—	2	1	no	—
67	#117	C₄⋊C₄	1	2	yes	—
68	#117	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄²	C₄⋊C₄	1	2	yes	—
71	M₃₂	—	2	8	no	—
72	M₃₂	—	4	4	no	—
73	M₃₂	—	4	1	no	—
74	M₃₂	—	4	1	no	—
75	C₂×C₁₆	—	2	1	no	—
76	C₂×C₁₆	—	2	1	no	—
77	C₂×C₁₆	—	2	1	no	—
78	C₂×C₁₆	—	2	1	no	—
79	C₂×C₁₆	—	4	4	no	—
80	C₂×C₁₆	—	4	1	no	—
81	C₂×C₁₆	—	4	1	no	—
82	#97	C₂²⋊C₄	1	2	yes	—
83	#97	C₄⋊C₄	1	2	yes	—
84	#97	C₂²×C₄	1	2	yes	—
85	C₄×C₈	C₄²	1	1	yes	characteristic
86	C₄×C₈	C₄²	1	1	yes	characteristic
87	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
88	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
89	C₄×C₈	C₂²⋊C₄	1	2	yes	—
90	C₄×C₈	C₄⋊C₄	1	2	yes	—
91	C₄×C₈	C₂×D₄	1	2	yes	—
92	#41	—	4	1	no	—
93	#41	—	4	1	no	—
94	C₂×Q₈	#108	1	1	yes	characteristic
95	C₂×D₄	#108	1	2	yes	—
96	C₂×D₄	#108	1	1	yes	characteristic
97	C₂²×C₄	—	2	2	no	—
98	C₂²×C₄	—	2	1	no	—
99	C₂²×C₄	#95	1	2	yes	—
100	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
101	M₁₆	—	4	2	no	—
102	M₁₆	—	4	2	no	—
103	M₁₆	—	4	1	no	—
104	M₁₆	—	4	1	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	#95	1	1	yes	characteristic
114	C₂×C₈	#95	1	2	yes	—
115	C₂×C₈	#95	1	1	yes	characteristic
116	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₈	C₂×C₂²⋊C₄	1	2	yes	—
118	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₂²⋊C₄	—	2	2	no	—
122	C₂²⋊C₄	—	2	1	no	—
123	C₄²	—	2	2	no	—
124	C₄²	—	2	1	no	—
125	C₄²	#95	1	1	yes	characteristic
126	C₄²	#95	1	1	yes	characteristic
127	C₄²	C₂×C₄⋊C₄	1	2	yes	—
128	C₁₆	—	4	8	no	—
129	C₁₆	—	4	1	no	—
130	C₁₆	—	4	1	no	—
131	C₁₆	—	4	1	no	—
132	C₁₆	—	4	1	no	—
133	C₂³	#341	1	2	yes	—
134	C₂³	C₂×#108	1	1	yes	characteristic
135	Q₈	—	4	1	no	—
136	D₄	—	4	2	no	—
137	D₄	—	4	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	4	4	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	#341	1	2	yes	—
148	C₂×C₄	C₂×#95	1	1	yes	characteristic
149	C₂×C₄	C₂×#108	1	1	yes	characteristic
150	C₈	—	2	2	no	—
151	C₈	—	2	1	no	—
152	C₈	—	2	2	no	—
153	C₈	—	2	1	no	—
154	C₈	H(3,ℤ/4)	1	1	yes	characteristic
155	C₈	H(3,ℤ/4)	1	1	yes	characteristic
156	C₈	#413	1	1	yes	characteristic
157	C₈	#413	1	1	yes	characteristic
158	C₂²	—	2	2	no	—
159	C₂²	—	2	1	no	—
160	C₂²	—	4	2	no	—
161	C₂²	—	4	1	no	—
162	C₂²	C₂×#341	1	1	yes	characteristic
163	C₄	—	2	2	no	—
164	C₄	—	2	2	no	—
165	C₄	—	2	1	no	—
166	C₄	—	4	1	no	—
167	C₄	—	4	2	no	—
168	C₄	C₂×H(3,ℤ/4)	1	1	yes	characteristic
169	C₄	#1742	1	1	yes	characteristic
170	C₂	—	2	1	no	—
171	C₂	—	4	2	no	—
172	C₂	—	4	1	no	—
173	C₂	#7986	1	1	yes	characteristic

	subgroup	G/H
center	C₈	H(3,ℤ/4)
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	#405	C₂³
socle	C₂	#7986
prime radical	C₂×#41	C₄⋊C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₄×C₈ ◁ G
LECS	G ▷ #405 ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #7240	C₂	semidirect
4. #7239	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33428	2
#33420	2
#7240	2
#7239	2