Finite Group Databasebeta

Subgroups — #33618

Order 256.

328 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₂×#509	C₂	1	1	yes	maximal, characteristic, complemented
2	#3407	C₂	1	1	yes	maximal, characteristic, complemented
3	#3067	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#472	C₂	1	1	yes	maximal, characteristic, complemented
5	#2223	C₂	1	4	yes	maximal, complemented
6	#2221	C₂	1	4	yes	maximal, complemented
7	#2161	C₂	1	1	yes	maximal, characteristic, complemented
8	#2148	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#361	C₂	1	1	yes	maximal, characteristic, complemented
10	#575	—	2	1	no	—
11	C₂×AGL(1,ℤ/8)	—	2	1	no	—
12	C₂×#135	C₂²	1	1	yes	characteristic
13	C₂²×D₈	C₂²	1	1	yes	characteristic
14	C₂×#131	C₂²	1	1	yes	characteristic
15	#509	—	2	2	no	—
16	#509	—	2	2	no	—
17	C₂×SD₃₂	C₂²	1	2	yes	complemented
18	C₂×D₁₆	C₂²	1	2	yes	complemented
19	#504	—	2	2	no	—
20	#504	C₂²	1	4	yes	complemented
21	C₂×M₃₂	—	2	2	no	—
22	C₂×M₃₂	C₂²	1	1	yes	characteristic
23	C₂×M₃₂	C₂²	1	1	yes	characteristic
24	#473	C₂²	1	2	yes	complemented
25	#472	—	2	2	no	—
26	#472	C₂²	1	4	yes	—
27	#472	C₂²	1	2	yes	complemented
28	#471	—	2	2	no	—
29	C₂×#108	C₂²	1	1	yes	characteristic
30	C₂×#108	C₂²	1	1	yes	characteristic
31	#413	—	2	1	no	—
32	C₂×#100	—	2	1	no	—
33	#368	C₂²	1	4	yes	complemented
34	#362	C₂²	1	2	yes	complemented
35	#361	C₂²	1	4	yes	complemented
36	#361	C₂²	1	2	yes	complemented
37	C₂×#41	—	2	1	no	—
38	C₂×#41	D₄	1	1	yes	characteristic
39	C₂²×D₄	—	2	1	no	—
40	#137	—	2	2	no	—
41	#137	—	4	1	no	—
42	AGL(1,ℤ/8)	—	2	4	no	—
43	AGL(1,ℤ/8)	—	2	2	no	—
44	AGL(1,ℤ/8)	—	4	2	no	—
45	AGL(1,ℤ/8)	—	4	1	no	—
46	#135	—	2	1	no	—
47	#135	—	2	1	no	—
48	#135	—	2	1	no	—
49	#135	—	2	1	no	—
50	#135	—	4	2	no	—
51	C₂×Q₁₆	C₂³	1	1	yes	characteristic
52	C₂×SD₁₆	—	2	1	no	—
53	C₂×SD₁₆	—	4	1	no	—
54	C₂×D₈	—	2	1	no	—
55	C₂×D₈	—	2	1	no	—
56	C₂×D₈	—	2	1	no	—
57	C₂×D₈	—	2	1	no	—
58	C₂×D₈	—	2	1	no	—
59	C₂×D₈	—	4	1	no	—
60	C₂×D₈	C₂³	1	2	yes	—
61	C₂×D₈	C₂³	1	1	yes	characteristic
62	#131	—	2	1	no	—
63	#131	—	2	1	no	—
64	#131	D₄	1	2	yes	—
65	#131	C₂³	1	2	yes	—
66	C₂×M₁₆	—	2	1	no	—
67	C₂×M₁₆	—	2	1	no	—
68	C₂×M₁₆	—	2	1	no	—
69	C₂×M₁₆	D₄	1	1	yes	characteristic
70	C₂²×C₈	—	2	1	no	—
71	C₂²×C₈	C₂³	1	1	yes	characteristic
72	SD₃₂	—	2	4	no	—
73	D₁₆	—	2	4	no	—
74	M₃₂	—	2	1	no	—
75	M₃₂	—	2	4	no	—
76	M₃₂	D₄	1	2	yes	—
77	M₃₂	D₄	1	2	yes	—
78	M₃₂	C₂³	1	2	yes	—
79	C₂×C₁₆	—	2	1	no	—
80	C₂×C₁₆	—	2	4	no	—
81	C₂×C₁₆	C₂³	1	2	yes	—
82	#108	—	2	1	no	—
83	#108	—	2	1	no	—
84	#108	C₂³	1	2	yes	—
85	#108	C₂³	1	2	yes	—
86	#101	—	2	2	no	—
87	#100	—	2	4	no	—
88	#100	—	2	2	no	—
89	C₂⁴	—	4	1	no	—
90	#41	—	2	1	no	—
91	#41	—	2	1	no	—
92	#41	—	4	1	no	—
93	#41	—	4	1	no	—
94	#41	—	4	1	no	—
95	#41	—	4	1	no	—
96	#41	D₈	1	2	yes	—
97	#41	C₂×D₄	1	2	yes	—
98	C₂×Q₈	—	2	1	no	—
99	C₂×Q₈	D₈	1	1	yes	characteristic
100	C₂×D₄	—	2	1	no	—
101	C₂×D₄	—	2	2	no	—
102	C₂×D₄	—	2	1	no	—
103	C₂×D₄	—	4	1	no	—
104	C₂×D₄	—	4	1	no	—
105	C₂×D₄	—	4	1	no	—
106	C₂×D₄	—	4	1	no	—
107	C₂×D₄	—	4	1	no	—
108	C₂×D₄	—	4	1	no	—
109	C₂×D₄	D₈	1	1	yes	characteristic
110	C₂²×C₄	—	2	1	no	—
111	C₂²×C₄	—	4	1	no	—
112	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
113	Q₁₆	—	2	1	no	—
114	Q₁₆	—	2	1	no	—
115	Q₁₆	—	4	2	no	—
116	SD₁₆	—	4	4	no	—
117	SD₁₆	—	4	2	no	—
118	SD₁₆	—	4	2	no	—
119	SD₁₆	—	4	1	no	—
120	SD₁₆	—	4	1	no	—
121	D₈	—	2	1	no	—
122	D₈	—	2	2	no	—
123	D₈	—	2	2	no	—
124	D₈	—	2	1	no	—
125	D₈	—	4	4	no	—
126	D₈	—	4	2	no	—
127	D₈	—	4	1	no	—
128	D₈	—	4	1	no	—
129	M₁₆	—	2	1	no	—
130	M₁₆	—	2	2	no	—
131	M₁₆	—	2	2	no	—
132	M₁₆	—	2	2	no	—
133	M₁₆	—	2	2	no	—
134	M₁₆	—	4	1	no	—
135	M₁₆	—	4	1	no	—
136	M₁₆	C₂×D₄	1	2	yes	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	4	1	no	—
141	C₂×C₈	—	4	1	no	—
142	C₂×C₈	C₂×D₄	1	1	yes	characteristic
143	C₂×C₈	C₂×D₄	1	1	yes	characteristic
144	C₂×C₈	C₂×D₄	1	1	yes	characteristic
145	C₂×C₈	#41	1	1	yes	characteristic
146	C₂×C₈	#41	1	1	yes	characteristic
147	C₂×C₈	C₂⁴	1	1	yes	characteristic
148	C₁₆	—	2	2	no	—
149	C₁₆	C₂×D₄	1	4	yes	—
150	C₂³	—	2	1	no	—
151	C₂³	—	4	1	no	—
152	C₂³	—	4	2	no	—
153	C₂³	—	8	1	no	—
154	C₂³	—	8	1	no	—
155	C₂³	—	16	2	no	—
156	C₂³	#135	1	1	yes	characteristic
157	Q₈	—	2	1	no	—
158	Q₈	—	4	1	no	—
159	Q₈	—	4	1	no	—
160	Q₈	C₂×D₈	1	2	yes	—
161	D₄	—	2	2	no	—
162	D₄	—	2	2	no	—
163	D₄	—	2	1	no	—
164	D₄	—	4	2	no	—
165	D₄	—	4	1	no	—
166	D₄	—	4	2	no	—
167	D₄	—	4	1	no	—
168	D₄	—	8	1	no	—
169	D₄	—	8	1	no	—
170	D₄	—	8	1	no	—
171	D₄	—	8	1	no	—
172	D₄	C₂×D₈	1	2	yes	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	1	no	—
177	C₂×C₄	—	8	1	no	—
178	C₂×C₄	—	8	1	no	—
179	C₂×C₄	#121	1	1	yes	characteristic
180	C₂×C₄	#121	1	1	yes	characteristic
181	C₂×C₄	C₂×D₈	1	1	yes	characteristic
182	C₂×C₄	C₂×D₈	1	1	yes	characteristic
183	C₂×C₄	#135	1	1	yes	characteristic
184	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
185	C₈	—	2	2	no	—
186	C₈	—	4	2	no	—
187	C₈	—	4	2	no	—
188	C₈	#121	1	1	yes	characteristic
189	C₈	#121	1	1	yes	characteristic
190	C₈	C₂²×D₄	1	1	yes	characteristic
191	C₈	C₂×#41	1	1	yes	characteristic
192	C₂²	—	2	2	no	—
193	C₂²	—	4	1	no	—
194	C₂²	—	8	1	no	—
195	C₂²	—	8	1	no	—
196	C₂²	—	8	2	no	—
197	C₂²	—	16	1	no	—
198	C₂²	—	16	2	no	—
199	C₂²	—	16	2	no	—
200	C₂²	—	16	2	no	—
201	C₂²	#466	1	1	yes	characteristic
202	C₂²	#466	1	1	yes	characteristic
203	C₂²	C₂×#135	1	1	yes	characteristic
204	C₄	—	2	2	no	—
205	C₄	—	8	1	no	—
206	C₄	#466	1	1	yes	characteristic
207	C₄	#466	1	1	yes	characteristic
208	C₄	C₂×#121	1	1	yes	characteristic
209	C₄	C₂²×D₈	1	1	yes	characteristic
210	C₂	—	2	1	no	—
211	C₂	—	2	1	no	—
212	C₂	—	2	1	no	—
213	C₂	—	4	2	no	—
214	C₂	—	16	1	no	—
215	C₂	—	16	2	no	—
216	C₂	C₂×#466	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	C₂×#466
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#509	C₂	semidirect
2. #3407	C₂	semidirect
3. #3067	C₂	semidirect
4. C₂×#472	C₂	semidirect
5. #2223	C₂	semidirect
6. #2221	C₂	semidirect
7. #2161	C₂	semidirect
8. #2148	C₂	semidirect
9. C₂×#361	C₂	semidirect
17. C₂×SD₃₂	C₂²	semidirect
18. C₂×D₁₆	C₂²	semidirect
20. #504	C₂²	semidirect
24. #473	C₂²	semidirect
27. #472	C₂²	semidirect
33. #368	C₂²	semidirect
34. #362	C₂²	semidirect
35. #361	C₂²	semidirect
36. #361	C₂²	semidirect

Maximal subgroups

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

subgroup	index
C₂×#509	2
#3407	2
#3067	2
C₂×#472	2
#2223	2
#2221	2
#2161	2
#2148	2
C₂×#361	2