Finite Group Databasebeta

Subgroups — #33856

Order 256.

326 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 105 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	#3416	C₂	1	2	yes	maximal, complemented
2	#3293	C₂	1	1	yes	maximal, characteristic, complemented
3	#3292	C₂	1	1	yes	maximal, characteristic, complemented
4	#2248	C₂	1	1	yes	maximal, characteristic, complemented
5	#2223	C₂	1	2	yes	maximal, complemented
6	#2214	C₂	1	1	yes	maximal, characteristic, complemented
7	#2213	C₂	1	1	yes	maximal, characteristic, complemented
8	#2194	C₂	1	2	yes	maximal, complemented
9	#2193	C₂	1	2	yes	maximal, complemented
10	#2178	C₂	1	1	yes	maximal, characteristic, complemented
11	#2170	C₂	1	1	yes	maximal, characteristic, complemented
12	#578	—	2	1	no	—
13	#578	C₂²	1	2	yes	complemented
14	#576	—	2	1	no	—
15	#576	C₂²	1	2	yes	—
16	#510	—	2	2	no	—
17	#510	C₂²	1	2	yes	—
18	#509	—	2	2	no	—
19	#509	C₂²	1	2	yes	complemented
20	#508	—	2	2	no	—
21	#508	C₂²	1	1	yes	characteristic
22	C₂×SD₃₂	—	2	2	no	—
23	C₂×SD₃₂	C₂²	1	2	yes	complemented
24	C₂×SD₃₂	C₂²	1	2	yes	complemented
25	#504	C₂²	1	2	yes	complemented
26	#494	C₂²	1	1	yes	characteristic, complemented
27	#493	C₂²	1	1	yes	characteristic
28	#473	—	2	1	no	—
29	#473	C₂²	1	2	yes	complemented
30	#472	—	2	1	no	—
31	#472	C₂²	1	2	yes	—
32	#456	—	2	2	no	—
33	#453	—	2	2	no	—
34	#443	C₂²	1	1	yes	characteristic
35	#443	C₂²	1	1	yes	characteristic
36	#443	C₂²	1	1	yes	characteristic, complemented
37	#368	C₂²	1	1	yes	characteristic, complemented
38	#364	C₂²	1	1	yes	characteristic, complemented
39	#362	C₂²	1	2	yes	complemented
40	#361	C₂²	1	2	yes	complemented
41	#359	C₂²	1	1	yes	characteristic
42	#359	C₂²	1	1	yes	characteristic, complemented
43	#350	C₂²	1	2	yes	complemented
44	C₄×C₁₆	C₂²	1	1	yes	characteristic, complemented
45	2₋¹⁺⁴	—	2	2	no	—
46	H(5,𝔽₂)	—	2	2	no	—
47	#137	—	2	2	no	—
48	#137	—	2	2	no	—
49	#137	—	4	2	no	—
50	#137	—	4	2	no	—
51	AGL(1,ℤ/8)	—	2	2	no	—
52	AGL(1,ℤ/8)	—	2	2	no	—
53	AGL(1,ℤ/8)	—	4	2	no	—
54	AGL(1,ℤ/8)	—	4	2	no	—
55	#135	—	2	2	no	—
56	#135	—	2	2	no	—
57	#135	—	4	1	no	—
58	#135	—	4	1	no	—
59	#135	D₄	1	1	yes	characteristic
60	#135	C₂³	1	1	yes	characteristic
61	#135	C₂³	1	1	yes	characteristic
62	C₂×Q₁₆	—	2	2	no	—
63	C₂×Q₁₆	—	2	1	no	—
64	C₂×Q₁₆	—	4	1	no	—
65	C₂×Q₁₆	C₂³	1	2	yes	—
66	C₂×D₈	—	2	2	no	—
67	C₂×D₈	—	2	1	no	—
68	C₂×D₈	—	4	1	no	—
69	C₂×D₈	C₂³	1	2	yes	—
70	#131	—	2	1	no	—
71	#131	—	2	1	no	—
72	#131	C₂³	1	2	yes	—
73	C₄⋊Q₈	—	2	1	no	—
74	#127	—	2	1	no	—
75	Q₃₂	—	2	2	no	—
76	Q₃₂	D₄	1	1	yes	characteristic, complemented
77	SD₃₂	—	2	2	no	—
78	SD₃₂	—	2	2	no	—
79	SD₃₂	—	2	2	no	—
80	SD₃₂	—	2	2	no	—
81	SD₃₂	—	4	2	no	—
82	SD₃₂	D₄	1	1	yes	characteristic, complemented
83	SD₃₂	D₄	1	1	yes	characteristic, complemented
84	D₁₆	—	2	2	no	—
85	D₁₆	D₄	1	1	yes	characteristic, complemented
86	M₃₂	—	2	2	no	—
87	M₃₂	C₂³	1	2	yes	—
88	C₂×C₁₆	—	2	2	no	—
89	C₂×C₁₆	C₂³	1	1	yes	characteristic
90	C₂×C₁₆	C₂³	1	1	yes	characteristic
91	#108	D₄	1	1	yes	characteristic
92	#108	C₂³	1	1	yes	characteristic
93	#108	C₂³	1	1	yes	characteristic
94	#104	—	2	1	no	—
95	#104	—	2	1	no	—
96	#104	D₄	1	2	yes	—
97	#101	—	2	2	no	—
98	#100	—	2	2	no	—
99	C₄×C₈	C₂³	1	1	yes	characteristic
100	#41	—	2	1	no	—
101	#41	—	2	1	no	—
102	#41	—	4	2	no	—
103	#41	—	4	2	no	—
104	#41	—	4	2	no	—
105	#41	—	4	2	no	—
106	#41	—	8	2	no	—
107	#41	C₂×D₄	1	2	yes	—
108	C₂×Q₈	—	2	2	no	—
109	C₂×Q₈	—	4	2	no	—
110	C₂×Q₈	—	4	2	no	—
111	C₂×D₄	—	2	2	no	—
112	C₂×D₄	—	4	2	no	—
113	C₂×D₄	—	4	2	no	—
114	C₂×D₄	—	8	1	no	—
115	C₂×D₄	—	8	2	no	—
116	Q₁₆	—	2	2	no	—
117	Q₁₆	—	2	2	no	—
118	Q₁₆	—	4	2	no	—
119	Q₁₆	—	4	2	no	—
120	Q₁₆	—	4	1	no	—
121	Q₁₆	—	4	1	no	—
122	Q₁₆	D₈	1	1	yes	characteristic
123	Q₁₆	C₂×D₄	1	1	yes	characteristic
124	Q₁₆	C₂×D₄	1	1	yes	characteristic
125	SD₁₆	—	2	1	no	—
126	SD₁₆	—	2	1	no	—
127	SD₁₆	—	4	2	no	—
128	SD₁₆	—	4	2	no	—
129	SD₁₆	—	4	2	no	—
130	SD₁₆	—	4	2	no	—
131	SD₁₆	D₈	1	2	yes	—
132	D₈	—	2	2	no	—
133	D₈	—	2	2	no	—
134	D₈	—	4	2	no	—
135	D₈	—	4	2	no	—
136	D₈	—	4	1	no	—
137	D₈	—	4	1	no	—
138	D₈	D₈	1	1	yes	characteristic
139	D₈	C₂×D₄	1	1	yes	characteristic
140	D₈	C₂×D₄	1	1	yes	characteristic
141	M₁₆	—	2	2	no	—
142	M₁₆	—	2	1	no	—
143	M₁₆	—	2	1	no	—
144	M₁₆	—	4	1	no	—
145	M₁₆	—	4	1	no	—
146	M₁₆	C₂×D₄	1	2	yes	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	4	1	no	—
149	C₂×C₈	—	4	1	no	—
150	C₂×C₈	C₂×D₄	1	1	yes	characteristic
151	C₂×C₈	C₂⁴	1	1	yes	characteristic
152	C₄⋊C₄	—	8	1	no	—
153	C₄²	C₂×D₄	1	1	yes	characteristic
154	C₁₆	—	2	2	no	—
155	C₁₆	—	2	1	no	—
156	C₁₆	C₂×D₄	1	1	yes	characteristic
157	C₁₆	C₂×D₄	1	1	yes	characteristic
158	C₂³	—	4	2	no	—
159	C₂³	—	8	2	no	—
160	Q₈	—	2	1	no	—
161	Q₈	—	2	1	no	—
162	Q₈	—	4	2	no	—
163	Q₈	—	4	2	no	—
164	Q₈	—	8	2	no	—
165	Q₈	C₂×D₈	1	2	yes	—
166	D₄	—	2	2	no	—
167	D₄	—	2	1	no	—
168	D₄	—	2	1	no	—
169	D₄	—	4	1	no	—
170	D₄	—	4	1	no	—
171	D₄	—	4	2	no	—
172	D₄	—	4	2	no	—
173	D₄	—	8	2	no	—
174	D₄	—	8	2	no	—
175	D₄	—	8	2	no	—
176	D₄	—	16	2	no	—
177	D₄	C₂×D₈	1	2	yes	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	4	1	no	—
181	C₂×C₄	—	4	1	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	8	2	no	—
184	C₂×C₄	—	8	2	no	—
185	C₂×C₄	—	8	2	no	—
186	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
187	C₈	—	2	2	no	—
188	C₈	—	4	1	no	—
189	C₈	—	4	1	no	—
190	C₈	C₂×D₈	1	1	yes	characteristic
191	C₈	C₂×D₈	1	1	yes	characteristic
192	C₈	C₂²×D₄	1	1	yes	characteristic
193	C₈	H(5,𝔽₂)	1	1	yes	characteristic
194	C₂²	—	2	2	no	—
195	C₂²	—	4	1	no	—
196	C₂²	—	4	1	no	—
197	C₂²	—	8	2	no	—
198	C₂²	—	16	2	no	—
199	C₂²	—	32	2	no	—
200	C₂²	#576	1	1	yes	characteristic
201	C₄	—	2	2	no	—
202	C₄	—	2	2	no	—
203	C₄	—	4	1	no	—
204	C₄	—	4	1	no	—
205	C₄	—	8	2	no	—
206	C₄	D₄²	1	1	yes	characteristic
207	C₄	C₂²×D₈	1	1	yes	characteristic
208	C₂	—	2	1	no	—
209	C₂	—	4	2	no	—
210	C₂	—	8	1	no	—
211	C₂	—	8	1	no	—
212	C₂	—	16	2	no	—
213	C₂	D₄×D₈	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	D₄×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #443 ◁ 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. #3416	C₂	semidirect
2. #3293	C₂	semidirect
3. #3292	C₂	semidirect
4. #2248	C₂	semidirect
5. #2223	C₂	semidirect
6. #2214	C₂	semidirect
7. #2213	C₂	semidirect
8. #2194	C₂	semidirect
9. #2193	C₂	semidirect
10. #2178	C₂	semidirect
11. #2170	C₂	semidirect
13. #578	C₂²	semidirect
19. #509	C₂²	semidirect
23. C₂×SD₃₂	C₂²	semidirect
24. C₂×SD₃₂	C₂²	semidirect
25. #504	C₂²	semidirect
26. #494	C₂²	semidirect
29. #473	C₂²	semidirect
36. #443	C₂²	semidirect
37. #368	C₂²	semidirect
38. #364	C₂²	semidirect
39. #362	C₂²	semidirect
40. #361	C₂²	semidirect
42. #359	C₂²	semidirect
43. #350	C₂²	semidirect
44. C₄×C₁₆	C₂²	semidirect
76. Q₃₂	D₄	semidirect
82. SD₃₂	D₄	semidirect
83. SD₃₂	D₄	semidirect
85. D₁₆	D₄	semidirect

Maximal subgroups

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

subgroup	index
#3416	2
#3293	2
#3292	2
#2248	2
#2223	2
#2214	2
#2213	2
#2194	2
#2193	2
#2178	2
#2170	2