Finite Group Databasebeta

Subgroups — #29271

Order 256.

302 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 113 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	#2961	C₂	1	1	yes	maximal, characteristic, complemented
2	#2957	C₂	1	1	yes	maximal, characteristic
3	#2098	C₂	1	2	yes	maximal, complemented
4	#2031	C₂	1	2	yes	maximal, complemented
5	#1976	C₂	1	1	yes	maximal, characteristic
6	#1975	C₂	1	1	yes	maximal, characteristic
7	#1953	C₂	1	1	yes	maximal, characteristic, complemented
8	#1946	C₂	1	1	yes	maximal, characteristic, complemented
9	#1868	C₂	1	2	yes	maximal, complemented
10	#1861	C₂	1	2	yes	maximal, complemented
11	#1833	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135	C₂²	1	1	yes	characteristic
13	#568	C₂²	1	2	yes	—
14	#533	C₂²	1	1	yes	characteristic
15	#446	—	2	2	no	—
16	#446	—	2	2	no	—
17	#444	—	2	2	no	—
18	#443	—	2	2	no	—
19	#430	C₂²	1	2	yes	—
20	C₂×#108	C₂²	1	1	yes	characteristic
21	C₂×#108	C₂²	1	1	yes	characteristic
22	#427	C₂²	1	1	yes	characteristic
23	#424	C₂²	1	1	yes	characteristic
24	#424	C₂²	1	1	yes	characteristic
25	#424	C₂²	1	2	yes	—
26	#423	C₂²	1	1	yes	characteristic
27	#421	C₂²	1	2	yes	—
28	#421	C₂²	1	2	yes	—
29	C₂×#104	C₂²	1	2	yes	—
30	#410	C₂²	1	2	yes	—
31	#408	C₂²	1	2	yes	—
32	#405	C₂²	1	1	yes	characteristic
33	#405	C₂²	1	1	yes	characteristic
34	C₂×#97	C₂²	1	1	yes	characteristic
35	#344	C₂²	1	2	yes	—
36	#344	C₂²	1	2	yes	—
37	#339	C₂²	1	2	yes	—
38	H(3,ℤ/4)	C₂²	1	2	yes	—
39	C₂×#41	C₂³	1	2	yes	—
40	#135	—	2	1	no	—
41	#135	—	2	1	no	—
42	#135	—	2	1	no	—
43	#135	—	2	1	no	—
44	C₂×Q₁₆	Q₈	1	1	yes	characteristic, complemented
45	C₂×SD₁₆	Q₈	1	2	yes	complemented
46	C₂×D₈	Q₈	1	1	yes	characteristic, complemented
47	#131	—	2	4	no	—
48	#131	—	4	2	no	—
49	C₂×M₁₆	—	4	2	no	—
50	C₂×M₁₆	C₂³	1	2	yes	—
51	C₂×M₁₆	C₂³	1	2	yes	—
52	C₂×M₁₆	C₂³	1	1	yes	characteristic
53	C₂×M₁₆	C₂³	1	1	yes	characteristic
54	C₂²×C₈	C₂³	1	1	yes	characteristic
55	C₂²×C₈	C₂³	1	1	yes	characteristic
56	C₄⋊Q₈	—	2	1	no	—
57	#125	—	2	1	no	—
58	C₂²⋊Q₈	—	2	2	no	—
59	C₄×Q₈	—	2	2	no	—
60	#117	C₂³	1	2	yes	—
61	#117	C₂³	1	1	yes	characteristic
62	#117	C₂³	1	1	yes	characteristic
63	C₂×C₄²	C₂³	1	1	yes	characteristic
64	#108	—	2	1	no	—
65	#108	—	2	1	no	—
66	#108	—	2	1	no	—
67	#108	—	2	1	no	—
68	#108	—	4	2	no	—
69	#107	—	2	1	no	—
70	#106	—	2	1	no	—
71	C₄⋊C₈	—	2	2	no	—
72	C₄⋊C₈	—	2	2	no	—
73	C₄⋊C₈	—	2	2	no	—
74	C₄⋊C₈	—	4	2	no	—
75	#104	—	2	4	no	—
76	#104	—	2	2	no	—
77	#104	—	2	2	no	—
78	#104	—	4	2	no	—
79	#99	—	2	4	no	—
80	C₂²⋊C₈	—	4	2	no	—
81	#97	—	2	1	no	—
82	#97	—	2	1	no	—
83	#97	—	2	1	no	—
84	#97	D₄	1	2	yes	—
85	C₄×C₈	—	2	1	no	—
86	C₄×C₈	D₄	1	2	yes	—
87	#41	—	2	2	no	—
88	#41	—	2	2	no	—
89	#41	—	2	2	no	—
90	#41	—	2	2	no	—
91	C₂×Q₈	—	2	2	no	—
92	C₂×Q₈	C₂×Q₈	1	2	yes	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	C₂×Q₈	1	2	yes	—
95	C₂²×C₄	—	2	1	no	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	C₂⁴	1	1	yes	characteristic
98	Q₁₆	—	2	1	no	—
99	Q₁₆	—	2	1	no	—
100	SD₁₆	—	2	2	no	—
101	SD₁₆	—	2	2	no	—
102	D₈	—	2	1	no	—
103	D₈	—	2	1	no	—
104	M₁₆	—	2	1	no	—
105	M₁₆	—	2	1	no	—
106	M₁₆	—	2	2	no	—
107	M₁₆	—	2	2	no	—
108	M₁₆	—	2	2	no	—
109	M₁₆	—	2	1	no	—
110	M₁₆	—	2	1	no	—
111	M₁₆	—	2	2	no	—
112	M₁₆	—	4	4	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	4	4	no	—
120	C₂×C₈	C₂×D₄	1	2	yes	—
121	C₂×C₈	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
124	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
125	C₂×C₈	#41	1	1	yes	characteristic
126	C₂×C₈	#41	1	1	yes	characteristic
127	C₄⋊C₄	—	2	4	no	—
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	2	1	no	—
130	C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₂²⋊C₄	—	2	1	no	—
133	C₂²⋊C₄	—	2	2	no	—
134	C₂²⋊C₄	#41	1	2	yes	—
135	C₄²	—	2	1	no	—
136	C₄²	—	2	1	no	—
137	C₄²	—	2	2	no	—
138	C₄²	C₂×D₄	1	2	yes	—
139	C₄²	#41	1	1	yes	characteristic
140	C₄²	#41	1	1	yes	characteristic
141	C₂³	—	2	2	no	—
142	C₂³	C₂×#41	1	1	yes	characteristic
143	Q₈	—	2	2	no	—
144	Q₈	—	2	2	no	—
145	Q₈	—	8	2	no	—
146	D₄	—	2	2	no	—
147	D₄	—	2	2	no	—
148	D₄	—	4	2	no	—
149	D₄	—	4	2	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	8	1	no	—
159	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
161	C₂×C₄	C₂×#41	1	1	yes	characteristic
162	C₂×C₄	C₂×#41	1	1	yes	characteristic
163	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
164	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
165	C₈	—	2	2	no	—
166	C₈	—	4	1	no	—
167	C₈	—	4	2	no	—
168	C₈	—	4	1	no	—
169	C₈	—	4	2	no	—
170	C₈	C₂²⋊Q₈	1	1	yes	characteristic
171	C₈	C₂²⋊Q₈	1	1	yes	characteristic
172	C₈	C₂²⋊Q₈	1	1	yes	characteristic
173	C₈	C₂²⋊Q₈	1	1	yes	characteristic
174	C₂²	—	4	1	no	—
175	C₂²	—	4	2	no	—
176	C₂²	—	8	2	no	—
177	C₂²	#525	1	1	yes	characteristic
178	C₂²	#554	1	1	yes	characteristic
179	C₂²	#559	1	1	yes	characteristic
180	C₄	—	4	1	no	—
181	C₄	—	4	1	no	—
182	C₄	—	4	2	no	—
183	C₄	—	4	2	no	—
184	C₄	—	4	2	no	—
185	C₄	—	8	2	no	—
186	C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
187	C₄	#535	1	1	yes	characteristic
188	C₄	#554	1	1	yes	characteristic
189	C₄	#559	1	1	yes	characteristic
190	C₂	—	2	1	no	—
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	—	8	2	no	—
194	C₂	#2580	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#535
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂	#2580
prime radical	C₂×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #2961	C₂	semidirect
3. #2098	C₂	semidirect
4. #2031	C₂	semidirect
7. #1953	C₂	semidirect
8. #1946	C₂	semidirect
9. #1868	C₂	semidirect
10. #1861	C₂	semidirect
11. #1833	C₂	semidirect
44. C₂×Q₁₆	Q₈	semidirect
45. C₂×SD₁₆	Q₈	semidirect
46. C₂×D₈	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#2961	2
#2957	2
#2098	2
#2031	2
#1976	2
#1975	2
#1953	2
#1946	2
#1868	2
#1861	2
#1833	2