Finite Group Databasebeta

Subgroups — #135107

Order 512.

314 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 94 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	#17956	C₂	1	1	yes	maximal, characteristic
2	#12012	C₂	1	1	yes	maximal, characteristic, complemented
3	#11856	C₂	1	2	yes	maximal
4	#8815	C₂	1	1	yes	maximal, characteristic, complemented
5	#7217	C₂	1	2	yes	maximal, complemented
6	#3169	C₂²	1	1	yes	characteristic
7	#2931	—	2	1	no	—
8	#2162	C₂²	1	2	yes	—
9	#1844	—	2	1	no	—
10	#1834	—	2	1	no	—
11	#1834	C₂²	1	2	yes	—
12	#1589	—	2	1	no	—
13	#1589	C₄	1	2	yes	complemented
14	C₈×D₈	—	2	1	no	—
15	C₈×D₈	C₄	1	2	yes	complemented
16	#1567	C₂²	1	1	yes	characteristic
17	#1492	—	2	1	no	—
18	C₂×C₈²	C₂²	1	1	yes	characteristic
19	#1370	—	2	1	no	—
20	#1367	—	2	1	no	—
21	#1347	—	2	4	no	—
22	C₂×#131	—	4	1	no	—
23	#532	—	2	1	no	—
24	#500	Q₈	1	1	yes	characteristic
25	#493	Q₈	1	1	yes	characteristic
26	#486	—	2	1	no	—
27	#466	—	2	1	no	—
28	#466	D₄	1	2	yes	—
29	C₄×D₈	—	2	1	no	—
30	C₄×D₈	C₂×C₄	1	2	yes	—
31	#436	—	2	2	no	—
32	#436	—	4	1	no	—
33	C₈×D₄	—	2	2	no	—
34	C₈×D₄	—	4	1	no	—
35	C₂×#106	—	4	2	no	—
36	#423	—	2	1	no	—
37	#408	—	4	1	no	—
38	C₂×C₄×C₈	D₄	1	1	yes	characteristic
39	C₂×C₄×C₈	D₄	1	1	yes	characteristic
40	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
41	#384	—	2	2	no	—
42	C₄⋊C₁₆	—	2	2	no	—
43	C₂²⋊C₁₆	—	2	2	no	—
44	C₄×C₁₆	—	2	2	no	—
45	#340	—	2	2	no	—
46	#340	—	2	2	no	—
47	#340	—	4	1	no	—
48	#340	—	4	1	no	—
49	#335	—	2	1	no	—
50	#335	C₂×C₄	1	2	yes	—
51	D₄⋊C₈	—	2	2	no	—
52	D₄⋊C₈	—	2	2	no	—
53	C₈²	C₂×C₄	1	1	yes	characteristic
54	C₈²	C₂×C₄	1	1	yes	characteristic
55	C₈²	D₄	1	1	yes	characteristic
56	C₈²	D₄	1	1	yes	characteristic
57	C₂×#41	—	4	1	no	—
58	C₂×D₈	—	2	1	no	—
59	C₂×D₈	C₄⋊C₄	1	2	yes	—
60	#131	—	8	4	no	—
61	C₂×M₁₆	—	4	1	no	—
62	C₂×M₁₆	—	4	2	no	—
63	C₂²×C₈	—	2	1	no	—
64	C₂²×C₈	—	4	2	no	—
65	C₂²×C₈	D₈	1	1	yes	characteristic
66	C₂²×C₈	D₈	1	1	yes	characteristic
67	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₈	#41	1	1	yes	characteristic
69	C₄⋊Q₈	—	2	1	no	—
70	#127	—	2	1	no	—
71	#124	—	4	1	no	—
72	#121	—	2	2	no	—
73	#121	—	4	1	no	—
74	C₄×D₄	—	2	2	no	—
75	C₄×D₄	—	4	1	no	—
76	C₂×C₄⋊C₄	—	4	2	no	—
77	C₂×C₄⋊C₄	—	4	2	no	—
78	C₂×C₄⋊C₄	—	4	2	no	—
79	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
80	C₂×C₁₆	—	4	2	no	—
81	#107	—	2	1	no	—
82	#107	C₄⋊C₄	1	2	yes	—
83	#106	—	8	4	no	—
84	C₄⋊C₈	—	2	2	no	—
85	C₄⋊C₈	—	4	1	no	—
86	#102	—	2	2	no	—
87	#102	—	2	2	no	—
88	C₂²⋊C₈	—	4	2	no	—
89	C₂²⋊C₈	—	4	2	no	—
90	C₄×C₈	—	2	1	no	—
91	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
92	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
93	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
94	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
95	C₄×C₈	SD₁₆	1	1	yes	characteristic
96	C₄×C₈	SD₁₆	1	1	yes	characteristic
97	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
98	C₄×C₈	C₂×D₄	1	1	yes	characteristic
99	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
100	C₄×C₈	#41	1	1	yes	characteristic
101	#41	—	8	2	no	—
102	#41	—	8	2	no	—
103	C₂×Q₈	—	4	1	no	—
104	C₂×D₄	—	2	2	no	—
105	C₂×D₄	—	4	2	no	—
106	C₂×D₄	—	4	1	no	—
107	C₂×D₄	—	4	1	no	—
108	C₂²×C₄	—	4	2	no	—
109	C₂²×C₄	—	8	2	no	—
110	C₂²×C₄	#121	1	1	yes	characteristic
111	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
112	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
113	D₈	—	4	2	no	—
114	D₈	—	4	2	no	—
115	M₁₆	—	8	4	no	—
116	M₁₆	—	8	2	no	—
117	C₂×C₈	—	2	4	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	4	1	no	—
123	C₂×C₈	—	4	1	no	—
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	—	8	4	no	—
126	C₂×C₈	#102	1	1	yes	characteristic
127	C₂×C₈	#102	1	1	yes	characteristic
128	C₂×C₈	#102	1	1	yes	characteristic
129	C₂×C₈	#102	1	1	yes	characteristic
130	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₈	#117	1	1	yes	characteristic
132	C₂×C₈	#121	1	1	yes	characteristic
133	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
134	C₄⋊C₄	—	2	2	no	—
135	C₄⋊C₄	—	4	1	no	—
136	C₄⋊C₄	—	8	4	no	—
137	C₄⋊C₄	—	8	2	no	—
138	C₄⋊C₄	—	8	2	no	—
139	C₄⋊C₄	—	8	2	no	—
140	C₄⋊C₄	—	8	2	no	—
141	C₂²⋊C₄	—	4	2	no	—
142	C₂²⋊C₄	—	4	2	no	—
143	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
145	C₄²	C₂×SD₁₆	1	1	yes	characteristic
146	C₄²	#137	1	1	yes	characteristic
147	C₁₆	—	8	2	no	—
148	C₂³	—	4	2	no	—
149	C₂³	#459	1	1	yes	characteristic
150	Q₈	—	8	2	no	—
151	D₄	—	4	2	no	—
152	D₄	—	4	2	no	—
153	D₄	—	8	4	no	—
154	D₄	—	8	2	no	—
155	D₄	—	8	2	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	—	16	2	no	—
163	C₂×C₄	—	16	2	no	—
164	C₂×C₄	—	16	2	no	—
165	C₂×C₄	#380	1	1	yes	characteristic
166	C₂×C₄	C₂×#102	1	1	yes	characteristic
167	C₂×C₄	#417	1	1	yes	characteristic
168	C₂×C₄	#461	1	1	yes	characteristic
169	C₂×C₄	#474	1	1	yes	characteristic
170	C₂×C₄	#476	1	1	yes	characteristic
171	C₂×C₄	#508	1	2	yes	—
172	C₂×C₄	#509	1	2	yes	—
173	C₈	—	2	2	no	—
174	C₈	—	2	2	no	—
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	2	1	no	—
178	C₈	—	2	1	no	—
179	C₈	—	2	1	no	—
180	C₈	—	2	1	no	—
181	C₈	—	2	1	no	—
182	C₈	—	2	1	no	—
183	C₈	—	8	2	no	—
184	C₂²	—	2	1	no	—
185	C₂²	—	2	1	no	—
186	C₂²	—	8	2	no	—
187	C₂²	—	8	2	no	—
188	C₂²	—	8	2	no	—
189	C₂²	#1802	1	1	yes	characteristic
190	C₂²	#2210	1	2	yes	—
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	—	8	2	no	—
194	C₄	—	32	2	no	—
195	C₄	#1812	1	1	yes	characteristic
196	C₄	#1814	1	1	yes	characteristic
197	C₄	#2138	1	1	yes	characteristic
198	C₄	#2140	1	1	yes	characteristic
199	C₄	#2228	1	2	yes	—
200	C₄	#2230	1	2	yes	—
201	C₂	—	2	2	no	—
202	C₂	—	16	2	no	—
203	C₂	#8676	1	1	yes	characteristic
204	C₂	#12813	1	1	yes	characteristic
205	C₂	#12823	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#1802
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3169	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ 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
2. #12012	C₂	semidirect
4. #8815	C₂	semidirect
5. #7217	C₂	semidirect
13. #1589	C₄	semidirect
15. C₈×D₈	C₄	semidirect

Maximal subgroups

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

subgroup	index
#17956	2
#12012	2
#11856	2
#8815	2
#7217	2