Finite Group Databasebeta

Subgroups — #139389

Order 512.

283 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 283 conjugacy classes of subgroups — too many to draw. Only its 80 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	#33501	C₂	1	1	yes	maximal, characteristic
2	#28699	C₂	1	1	yes	maximal, characteristic
3	#11975	C₂	1	1	yes	maximal, characteristic
4	#11974	C₂	1	1	yes	maximal, characteristic
5	#8633	C₂	1	1	yes	maximal, characteristic
6	#7303	C₂	1	1	yes	maximal, characteristic
7	#7303	C₂	1	1	yes	maximal, characteristic
8	#3165	—	2	1	no	—
9	C₂×#500	C₂²	1	1	yes	characteristic
10	#2960	C₂²	1	1	yes	characteristic
11	#2183	—	2	2	no	—
12	#2183	—	2	2	no	—
13	#2164	—	2	2	no	—
14	#2163	C₂²	1	1	yes	characteristic
15	#2163	C₂²	1	1	yes	characteristic
16	#2089	C₄	1	4	yes	—
17	#1997	—	2	1	no	—
18	#1952	—	2	2	no	—
19	C₄×#108	C₂²	1	1	yes	characteristic
20	#1568	C₂²	1	1	yes	characteristic
21	#1568	C₂²	1	1	yes	characteristic
22	#1537	—	2	1	no	—
23	#1365	—	2	2	no	—
24	#1364	—	2	2	no	—
25	#1307	—	2	1	no	—
26	#533	D₄	1	1	yes	characteristic
27	C₂×C₄⋊Q₈	—	2	1	no	—
28	#501	—	4	2	no	—
29	#500	—	2	1	no	—
30	#500	—	2	1	no	—
31	#500	—	2	1	no	—
32	#500	—	2	1	no	—
33	#500	—	4	1	no	—
34	#498	—	4	1	no	—
35	#446	—	2	1	no	—
36	#446	—	2	1	no	—
37	C₈×Q₈	—	2	1	no	—
38	C₈×Q₈	—	2	1	no	—
39	#432	—	2	1	no	—
40	#432	—	2	1	no	—
41	C₂×#108	—	2	1	no	—
42	#428	—	2	4	no	—
43	C₂×#107	—	2	1	no	—
44	C₂×#107	C₂×C₄	1	2	yes	—
45	#424	C₂×C₄	1	2	yes	—
46	#423	D₄	1	1	yes	characteristic
47	#405	D₄	1	2	yes	—
48	C₄×M₁₆	—	2	1	no	—
49	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
50	C₄⋊C₁₆	—	2	4	no	—
51	C₄⋊C₁₆	D₄	1	2	yes	—
52	C₄⋊C₁₆	D₄	1	2	yes	—
53	C₂²⋊C₁₆	—	2	1	no	—
54	C₂²⋊C₁₆	—	2	1	no	—
55	#346	—	2	2	no	—
56	#346	—	2	1	no	—
57	#346	—	2	1	no	—
58	C₄×C₁₆	—	2	2	no	—
59	#343	—	2	4	no	—
60	#341	C₂×C₄	1	2	yes	—
61	#335	—	2	1	no	—
62	#335	—	2	1	no	—
63	#334	—	2	1	no	—
64	#334	—	2	1	no	—
65	#330	—	4	2	no	—
66	C₂²×Q₈	—	4	1	no	—
67	C₂×M₁₆	—	2	2	no	—
68	C₂×M₁₆	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
71	C₄⋊Q₈	—	4	1	no	—
72	C₄⋊Q₈	—	4	1	no	—
73	C₄⋊Q₈	—	4	1	no	—
74	C₄⋊Q₈	—	4	1	no	—
75	C₄⋊Q₈	SD₁₆	1	1	yes	characteristic
76	C₄⋊Q₈	SD₁₆	1	1	yes	characteristic
77	#125	D₈	1	2	yes	—
78	C₂²⋊Q₈	—	2	1	no	—
79	C₂²⋊Q₈	—	2	1	no	—
80	C₄×Q₈	—	2	1	no	—
81	C₄×Q₈	—	2	1	no	—
82	#117	C₂²⋊C₄	1	2	yes	—
83	C₂×C₄⋊C₄	—	2	2	no	—
84	C₂×C₄⋊C₄	—	4	1	no	—
85	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
86	C₂×C₁₆	—	2	2	no	—
87	C₂×C₁₆	—	2	2	no	—
88	#108	—	2	4	no	—
89	#107	—	2	2	no	—
90	#107	—	2	2	no	—
91	#107	—	4	1	no	—
92	#107	—	4	1	no	—
93	#107	—	4	4	no	—
94	#106	—	4	4	no	—
95	C₄⋊C₈	—	2	2	no	—
96	C₄⋊C₈	—	2	2	no	—
97	C₄⋊C₈	—	2	1	no	—
98	C₄⋊C₈	—	2	1	no	—
99	C₄⋊C₈	—	4	1	no	—
100	C₄⋊C₈	—	4	1	no	—
101	C₂²⋊C₈	—	4	1	no	—
102	C₂²⋊C₈	—	4	1	no	—
103	#97	—	2	2	no	—
104	#97	—	4	1	no	—
105	#97	—	4	1	no	—
106	#97	—	4	1	no	—
107	C₄×C₈	—	2	2	no	—
108	C₄×C₈	—	4	1	no	—
109	C₄×C₈	C₂×D₄	1	1	yes	characteristic
110	C₄×C₈	C₂×D₄	1	1	yes	characteristic
111	C₄×C₈	#41	1	1	yes	characteristic
112	C₄×C₈	#41	1	1	yes	characteristic
113	C₂×Q₈	—	2	1	no	—
114	C₂×Q₈	—	2	1	no	—
115	C₂×Q₈	—	8	1	no	—
116	C₂×Q₈	—	8	1	no	—
117	C₂×Q₈	—	16	2	no	—
118	C₂²×C₄	—	4	2	no	—
119	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂²×C₄	#135	1	1	yes	characteristic
121	C₂²×C₄	#135	1	1	yes	characteristic
122	M₁₆	—	2	4	no	—
123	M₁₆	—	4	2	no	—
124	M₁₆	—	4	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	4	1	no	—
128	C₂×C₈	—	4	2	no	—
129	C₂×C₈	—	8	1	no	—
130	C₂×C₈	—	8	1	no	—
131	C₂×C₈	C₄×D₄	1	1	yes	characteristic
132	C₂×C₈	C₄×D₄	1	1	yes	characteristic
133	C₂×C₈	#120	1	1	yes	characteristic
134	C₂×C₈	#123	1	1	yes	characteristic
135	C₄⋊C₄	—	2	1	no	—
136	C₄⋊C₄	—	2	1	no	—
137	C₄⋊C₄	—	2	2	no	—
138	C₄⋊C₄	—	2	2	no	—
139	C₄⋊C₄	—	4	2	no	—
140	C₄⋊C₄	—	4	2	no	—
141	C₄⋊C₄	—	8	1	no	—
142	C₄⋊C₄	—	8	1	no	—
143	C₄⋊C₄	#102	1	2	yes	—
144	C₄⋊C₄	#102	1	2	yes	—
145	C₂²⋊C₄	—	2	2	no	—
146	C₄²	—	2	2	no	—
147	C₄²	#121	1	1	yes	characteristic
148	C₄²	#121	1	1	yes	characteristic
149	C₄²	C₂×D₈	1	1	yes	characteristic
150	C₄²	C₂×SD₁₆	1	1	yes	characteristic
151	C₁₆	—	4	2	no	—
152	C₁₆	—	4	2	no	—
153	C₂³	#416	1	1	yes	characteristic
154	Q₈	—	8	1	no	—
155	Q₈	—	8	1	no	—
156	Q₈	—	16	2	no	—
157	Q₈	—	16	2	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	4	1	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	—	16	2	no	—
166	C₂×C₄	#386	1	1	yes	characteristic
167	C₂×C₄	C₂×#102	1	1	yes	characteristic
168	C₂×C₄	#465	1	1	yes	characteristic
169	C₂×C₄	#465	1	1	yes	characteristic
170	C₂×C₄	#466	1	1	yes	characteristic
171	C₂×C₄	#466	1	1	yes	characteristic
172	C₈	—	2	1	no	—
173	C₈	—	2	1	no	—
174	C₈	—	2	1	no	—
175	C₈	—	4	4	no	—
176	C₈	—	4	2	no	—
177	C₈	—	16	1	no	—
178	C₈	—	16	1	no	—
179	C₈	#456	1	2	yes	—
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	—	2	1	no	—
183	C₂²	#1893	1	1	yes	characteristic
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	—	2	1	no	—
189	C₄	—	4	2	no	—
190	C₄	—	4	2	no	—
191	C₄	—	16	2	no	—
192	C₄	#1886	1	1	yes	characteristic
193	C₄	#1915	1	1	yes	characteristic
194	C₂	—	4	1	no	—
195	C₂	#9394	1	1	yes	characteristic
196	C₂	#13068	1	1	yes	characteristic
197	C₂	#13069	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#416
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
#33501	2
#28699	2
#11975	2
#11974	2
#8633	2
#7303	2