Finite Group Databasebeta

Subgroups — #141326

Order 512.

352 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 86 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	#19603	C₂	1	1	yes	maximal, characteristic
2	#12013	C₂	1	1	yes	maximal, characteristic, complemented
3	#11751	C₂	1	2	yes	maximal, complemented
4	#11652	C₂	1	2	yes	maximal, complemented
5	C₈×#108	C₂	1	1	yes	maximal, characteristic, complemented
6	#3169	—	2	1	no	—
7	#3169	C₂²	1	2	yes	—
8	#3147	—	2	1	no	—
9	#2163	C₂²	1	2	yes	—
10	C₄×#108	C₂²	1	1	yes	characteristic
11	#1713	C₄	1	1	yes	characteristic
12	#1712	C₄	1	1	yes	characteristic
13	#1709	—	2	1	no	—
14	#1706	—	2	1	no	—
15	#1706	C₄	1	2	yes	—
16	#1564	C₂²	1	1	yes	characteristic
17	C₈×M₁₆	—	2	1	no	—
18	C₂×C₈²	C₂²	1	1	yes	characteristic
19	#1370	—	2	1	no	—
20	#1367	—	2	1	no	—
21	#1349	—	2	4	no	—
22	#1333	—	2	4	no	—
23	#1278	—	2	1	no	—
24	C₂×#135	—	4	1	no	—
25	C₂×#135	—	4	1	no	—
26	#532	—	2	2	no	—
27	#500	—	2	1	no	—
28	#500	C₂×C₄	1	2	yes	—
29	#495	—	4	1	no	—
30	#495	—	4	1	no	—
31	#494	—	2	1	no	—
32	#493	—	2	1	no	—
33	#493	—	2	1	no	—
34	#493	C₂×C₄	1	2	yes	—
35	#492	—	2	2	no	—
36	#486	—	2	2	no	—
37	#486	—	2	2	no	—
38	#466	—	2	2	no	—
39	#466	—	2	2	no	—
40	#466	—	4	2	no	—
41	C₄×D₈	—	2	2	no	—
42	C₄×D₈	—	2	2	no	—
43	C₄×D₈	—	4	2	no	—
44	#431	—	2	1	no	—
45	C₂×#108	—	2	1	no	—
46	C₄×M₁₆	—	2	1	no	—
47	C₂×C₄×C₈	D₄	1	1	yes	characteristic
48	C₂×C₄×C₈	D₄	1	1	yes	characteristic
49	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
50	C₄⋊C₁₆	—	2	2	no	—
51	C₂²⋊C₁₆	—	2	2	no	—
52	#346	—	2	2	no	—
53	#341	D₄	1	2	yes	—
54	#335	—	2	1	no	—
55	#334	D₄	1	2	yes	—
56	#322	—	4	1	no	—
57	C₈²	—	4	1	no	—
58	C₈²	C₂×C₄	1	1	yes	characteristic
59	C₈²	C₂×C₄	1	1	yes	characteristic
60	C₈²	D₄	1	1	yes	characteristic
61	C₈²	D₄	1	1	yes	characteristic
62	C₂×#41	—	4	2	no	—
63	#135	—	8	4	no	—
64	#135	—	8	4	no	—
65	C₂×Q₁₆	—	4	1	no	—
66	C₂×Q₁₆	—	4	1	no	—
67	C₂×SD₁₆	—	4	2	no	—
68	C₂×SD₁₆	—	4	2	no	—
69	C₂×D₈	—	2	2	no	—
70	C₂×D₈	—	2	2	no	—
71	C₂×D₈	—	4	2	no	—
72	C₂×D₈	—	4	1	no	—
73	C₂×D₈	—	4	1	no	—
74	C₂×M₁₆	—	2	2	no	—
75	C₂²×C₈	—	2	1	no	—
76	C₂²×C₈	SD₁₆	1	1	yes	characteristic
77	C₂²×C₈	SD₁₆	1	1	yes	characteristic
78	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₈	#41	1	1	yes	characteristic
80	C₄⋊Q₈	—	2	2	no	—
81	#127	—	2	2	no	—
82	#124	—	4	2	no	—
83	#121	—	4	2	no	—
84	#121	—	4	2	no	—
85	C₄×D₄	—	4	2	no	—
86	C₄×D₄	—	4	2	no	—
87	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
88	C₂×C₁₆	—	4	2	no	—
89	#108	—	2	4	no	—
90	#107	—	2	2	no	—
91	#107	—	2	2	no	—
92	#107	—	4	2	no	—
93	C₄⋊C₈	—	2	2	no	—
94	#102	—	4	4	no	—
95	#102	—	4	2	no	—
96	#102	—	4	2	no	—
97	C₂²⋊C₈	—	4	1	no	—
98	#97	—	4	1	no	—
99	C₄×C₈	—	2	1	no	—
100	C₄×C₈	—	4	1	no	—
101	C₄×C₈	—	4	1	no	—
102	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
103	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
104	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
105	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
106	C₄×C₈	D₈	1	1	yes	characteristic
107	C₄×C₈	D₈	1	1	yes	characteristic
108	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
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	#41	—	8	4	no	—
113	#41	—	8	4	no	—
114	C₂×Q₈	—	4	2	no	—
115	C₂×D₄	—	4	2	no	—
116	C₂×D₄	—	4	2	no	—
117	C₂×D₄	—	4	2	no	—
118	C₂×D₄	—	8	2	no	—
119	C₂²×C₄	—	8	2	no	—
120	C₂²×C₄	#121	1	1	yes	characteristic
121	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
122	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
123	Q₁₆	—	8	2	no	—
124	Q₁₆	—	8	2	no	—
125	SD₁₆	—	8	4	no	—
126	SD₁₆	—	8	4	no	—
127	D₈	—	4	4	no	—
128	D₈	—	8	2	no	—
129	D₈	—	8	4	no	—
130	D₈	—	8	2	no	—
131	D₈	—	8	2	no	—
132	M₁₆	—	2	4	no	—
133	M₁₆	—	4	2	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	4	1	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	#102	1	1	yes	characteristic
144	C₂×C₈	#102	1	1	yes	characteristic
145	C₂×C₈	#102	1	1	yes	characteristic
146	C₂×C₈	#102	1	1	yes	characteristic
147	C₂×C₈	C₄×D₄	1	1	yes	characteristic
148	C₂×C₈	C₄×D₄	1	1	yes	characteristic
149	C₂×C₈	#124	1	1	yes	characteristic
150	C₂×C₈	#127	1	1	yes	characteristic
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₂²⋊C₄	—	8	2	no	—
154	C₂²⋊C₄	—	8	2	no	—
155	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
156	C₄²	#121	1	1	yes	characteristic
157	C₄²	C₂×D₈	1	1	yes	characteristic
158	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
159	C₁₆	—	8	2	no	—
160	C₂³	—	8	2	no	—
161	C₂³	#460	1	1	yes	characteristic
162	Q₈	—	8	2	no	—
163	Q₈	—	8	2	no	—
164	D₄	—	8	4	no	—
165	D₄	—	8	2	no	—
166	D₄	—	8	2	no	—
167	D₄	—	8	2	no	—
168	D₄	—	8	2	no	—
169	D₄	—	16	4	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	4	1	no	—
175	C₂×C₄	—	8	2	no	—
176	C₂×C₄	—	16	4	no	—
177	C₂×C₄	#390	1	1	yes	characteristic
178	C₂×C₄	C₂×#102	1	1	yes	characteristic
179	C₂×C₄	#418	1	1	yes	characteristic
180	C₂×C₄	#459	1	1	yes	characteristic
181	C₂×C₄	#459	1	1	yes	characteristic
182	C₂×C₄	#460	1	1	yes	characteristic
183	C₈	—	2	2	no	—
184	C₈	—	2	1	no	—
185	C₈	—	2	1	no	—
186	C₈	—	2	1	no	—
187	C₈	—	2	1	no	—
188	C₈	—	2	1	no	—
189	C₈	—	4	1	no	—
190	C₈	—	4	4	no	—
191	C₈	—	8	2	no	—
192	C₈	#361	1	2	yes	—
193	C₈	#361	1	2	yes	—
194	C₈	#443	1	2	yes	—
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	2	1	no	—
198	C₂²	—	16	2	no	—
199	C₂²	—	16	2	no	—
200	C₂²	—	16	2	no	—
201	C₂²	#1982	1	1	yes	characteristic
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	16	2	no	—
207	C₄	#1974	1	1	yes	characteristic
208	C₄	#1976	1	1	yes	characteristic
209	C₄	C₂×#361	1	1	yes	characteristic
210	C₄	C₂×#361	1	1	yes	characteristic
211	C₂	—	4	1	no	—
212	C₂	—	32	2	no	—
213	C₂	#9750	1	1	yes	characteristic
214	C₂	#13257	1	1	yes	characteristic
215	C₂	#13259	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#1982
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#19603	C₂
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. #12013	C₂	semidirect
3. #11751	C₂	semidirect
4. #11652	C₂	semidirect
5. C₈×#108	C₂	semidirect

Maximal subgroups

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

subgroup	index
#19603	2
#12013	2
#11751	2
#11652	2
C₈×#108	2