Finite Group Databasebeta

Subgroups — #142107

Order 512.

362 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 126 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	#33310	C₂	1	1	yes	maximal, characteristic
2	#12900	C₂	1	1	yes	maximal, characteristic
3	C₄×#364	C₂	1	1	yes	maximal, characteristic
4	#7268	C₂	1	4	yes	maximal
5	C₂×#498	—	2	1	no	—
6	C₂×#498	—	2	1	no	—
7	#2868	C₄	1	2	yes	—
8	C₂×#364	—	2	1	no	—
9	C₂×#364	C₂²	1	2	yes	—
10	C₂×#364	C₂²	1	2	yes	—
11	#2150	—	2	1	no	—
12	C₄×M₃₂	—	2	1	no	—
13	#1844	—	2	1	no	—
14	#1844	—	2	1	no	—
15	#1844	C₄	1	2	yes	—
16	#1844	C₂²	1	2	yes	—
17	C₄²×C₈	C₂²	1	1	yes	characteristic
18	#1376	—	2	1	no	—
19	C₂×C₄⋊Q₈	—	2	2	no	—
20	C₂×C₄⋊Q₈	—	4	2	no	—
21	C₂×M₃₂	—	2	2	no	—
22	C₂×M₃₂	—	2	2	no	—
23	#498	—	4	1	no	—
24	#498	—	4	1	no	—
25	#498	—	4	2	no	—
26	#498	—	4	2	no	—
27	#498	—	4	1	no	—
28	#498	—	4	1	no	—
29	C₂×#106	—	2	2	no	—
30	C₂×#106	—	2	2	no	—
31	C₂×#106	—	2	2	no	—
32	C₂×#106	—	2	2	no	—
33	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×C₈	D₄	1	1	yes	characteristic
35	C₂×C₄×C₈	D₄	1	1	yes	characteristic
36	C₂×C₄×C₈	D₄	1	1	yes	characteristic
37	C₂×C₄×C₈	D₄	1	1	yes	characteristic
38	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
39	#384	—	2	2	no	—
40	#384	—	4	2	no	—
41	#384	C₂×C₄	1	4	yes	—
42	C₄³	C₂×C₄	1	1	yes	characteristic
43	#364	—	2	2	no	—
44	#364	—	2	4	no	—
45	#364	—	4	2	no	—
46	#364	Q₈	1	4	yes	complemented
47	C₄⋊C₁₆	—	4	2	no	—
48	#346	—	4	1	no	—
49	C₄×C₁₆	—	4	1	no	—
50	C₂²×Q₈	—	4	2	no	—
51	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
55	C₄⋊Q₈	—	4	2	no	—
56	C₄⋊Q₈	—	4	2	no	—
57	C₄⋊Q₈	—	4	2	no	—
58	C₄⋊Q₈	—	4	2	no	—
59	C₄⋊Q₈	—	8	8	no	—
60	C₂×C₄⋊C₄	—	2	4	no	—
61	C₂×C₄⋊C₄	—	4	4	no	—
62	C₂×C₄⋊C₄	—	4	4	no	—
63	C₂×C₄⋊C₄	—	4	2	no	—
64	C₂×C₄²	—	2	2	no	—
65	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
68	M₃₂	—	2	4	no	—
69	M₃₂	—	4	2	no	—
70	M₃₂	—	4	2	no	—
71	M₃₂	—	4	2	no	—
72	C₂×C₁₆	—	4	2	no	—
73	C₂×C₁₆	—	4	2	no	—
74	#106	—	4	4	no	—
75	#106	—	4	2	no	—
76	#106	—	4	2	no	—
77	#106	—	4	2	no	—
78	#106	—	4	2	no	—
79	#106	—	4	2	no	—
80	#106	—	4	2	no	—
81	C₄×C₈	—	2	2	no	—
82	C₄×C₈	—	2	2	no	—
83	C₄×C₈	D₈	1	1	yes	characteristic
84	C₄×C₈	D₈	1	1	yes	characteristic
85	C₄×C₈	D₈	1	2	yes	—
86	C₄×C₈	SD₁₆	1	2	yes	—
87	C₄×C₈	SD₁₆	1	1	yes	characteristic
88	C₄×C₈	SD₁₆	1	1	yes	characteristic
89	C₄×C₈	SD₁₆	1	2	yes	—
90	C₄×C₈	Q₁₆	1	2	yes	—
91	C₄×C₈	C₂×Q₈	1	2	yes	—
92	C₄×C₈	#41	1	2	yes	—
93	C₂×Q₈	—	8	8	no	—
94	C₂×Q₈	—	8	2	no	—
95	C₂×Q₈	—	8	2	no	—
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	—	4	4	no	—
99	C₂²×C₄	#100	1	2	yes	—
100	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	#101	1	2	yes	—
104	C₂×C₈	#102	1	2	yes	—
105	C₂×C₈	#102	1	1	yes	characteristic
106	C₂×C₈	#102	1	1	yes	characteristic
107	C₂×C₈	#103	1	2	yes	—
108	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₈	#124	1	1	yes	characteristic
110	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
111	C₂×C₈	C₂×D₈	1	1	yes	characteristic
112	C₂×C₈	C₂×D₈	1	1	yes	characteristic
113	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
114	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
116	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
117	C₂×C₈	#137	1	2	yes	—
118	C₄⋊C₄	—	4	4	no	—
119	C₄⋊C₄	—	4	4	no	—
120	C₄⋊C₄	—	8	8	no	—
121	C₄⋊C₄	—	8	8	no	—
122	C₄⋊C₄	—	8	2	no	—
123	C₄⋊C₄	—	8	2	no	—
124	C₄²	—	2	8	no	—
125	C₄²	—	2	1	no	—
126	C₄²	—	2	1	no	—
127	C₄²	#102	1	1	yes	characteristic
128	C₄²	#102	1	2	yes	—
129	C₄²	#102	1	1	yes	characteristic
130	C₄²	#103	1	2	yes	—
131	C₄²	C₄×Q₈	1	2	yes	—
132	C₁₆	—	4	4	no	—
133	C₁₆	—	8	2	no	—
134	C₂³	C₂×#100	1	1	yes	characteristic
135	Q₈	—	16	4	no	—
136	Q₈	—	16	4	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	8	8	no	—
143	C₂×C₄	—	8	4	no	—
144	C₂×C₄	#331	1	2	yes	—
145	C₂×C₄	#331	1	2	yes	—
146	C₂×C₄	#391	1	1	yes	characteristic
147	C₂×C₄	C₂×#101	1	1	yes	characteristic
148	C₂×C₄	C₂×#102	1	1	yes	characteristic
149	C₂×C₄	C₂×#102	1	1	yes	characteristic
150	C₂×C₄	C₂×#103	1	1	yes	characteristic
151	C₂×C₄	#419	1	1	yes	characteristic
152	C₈	—	2	2	no	—
153	C₈	#332	1	2	yes	—
154	C₈	#332	1	2	yes	—
155	C₈	#474	1	2	yes	—
156	C₈	#475	1	2	yes	—
157	C₈	#486	1	1	yes	characteristic
158	C₈	#487	1	1	yes	characteristic
159	C₈	#498	1	1	yes	characteristic
160	C₈	#500	1	1	yes	characteristic
161	C₂²	—	2	2	no	—
162	C₂²	C₂×#331	1	1	yes	characteristic
163	C₂²	#1548	1	1	yes	characteristic
164	C₂²	#1997	1	1	yes	characteristic
165	C₄	—	2	4	no	—
166	C₄	—	2	4	no	—
167	C₄	—	2	2	no	—
168	C₄	—	16	4	no	—
169	C₄	#1359	1	4	yes	—
170	C₄	C₂×#332	1	1	yes	characteristic
171	C₄	#1546	1	1	yes	characteristic
172	C₄	#1988	1	1	yes	characteristic
173	C₄	#1990	1	1	yes	characteristic
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	#9898	1	1	yes	characteristic
177	C₂	C₂×#1359	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂×#331
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
46. #364	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#33310	2
#12900	2
C₄×#364	2
#7268	2