Finite Group Databasebeta

Subgroups — #19986

Order 256.

328 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 111 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	Q₈×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#3362	C₂	1	2	yes	maximal, complemented
3	#3287	C₂	1	1	yes	maximal, characteristic
4	#1706	C₂	1	1	yes	maximal, characteristic, complemented
5	#1678	C₂	1	2	yes	maximal, complemented
6	#1675	C₂	1	1	yes	maximal, characteristic, complemented
7	#1668	C₂	1	1	yes	maximal, characteristic, complemented
8	#1631	C₂	1	2	yes	maximal, complemented
9	#1623	C₂	1	2	yes	maximal, complemented
10	C₈×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135	—	2	1	no	—
13	Q₈²	C₂²	1	2	yes	—
14	#550	C₂²	1	2	yes	—
15	#500	—	2	1	no	—
16	#500	C₂²	1	1	yes	characteristic
17	#493	C₂²	1	1	yes	characteristic
18	#492	C₂²	1	2	yes	—
19	#486	—	2	2	no	—
20	#486	C₂²	1	2	yes	—
21	#477	—	2	2	no	—
22	#477	C₂²	1	2	yes	—
23	#476	—	2	2	no	—
24	#476	C₂²	1	2	yes	—
25	#466	—	2	1	no	—
26	#460	—	2	2	no	—
27	#460	C₂²	1	2	yes	—
28	#460	C₂²	1	2	yes	—
29	#449	—	2	2	no	—
30	C₈×Q₈	C₂²	1	2	yes	—
31	C₈×Q₈	C₂²	1	1	yes	characteristic
32	C₄×Q₁₆	—	2	1	no	—
33	C₄×Q₁₆	C₂²	1	1	yes	characteristic
34	C₄×SD₁₆	—	2	2	no	—
35	C₄×SD₁₆	C₂²	1	1	yes	characteristic
36	C₈×D₄	C₂²	1	1	yes	characteristic
37	#335	C₂²	1	1	yes	characteristic
38	#334	C₂²	1	1	yes	characteristic
39	#333	C₂²	1	2	yes	—
40	#331	C₂²	1	2	yes	—
41	Q₈⋊C₈	C₂²	1	1	yes	characteristic
42	Q₈⋊C₈	C₂²	1	2	yes	—
43	D₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₈²	C₂²	1	1	yes	characteristic
45	C₂×#41	—	2	2	no	—
46	#135	—	4	4	no	—
47	C₂×Q₁₆	D₄	1	1	yes	characteristic
48	C₂×SD₁₆	—	2	2	no	—
49	C₂×SD₁₆	—	2	2	no	—
50	C₂×SD₁₆	—	4	2	no	—
51	C₂×SD₁₆	D₄	1	1	yes	characteristic
52	C₂×D₈	—	2	1	no	—
53	C₂×D₈	—	2	1	no	—
54	C₂²×C₈	—	2	1	no	—
55	C₄⋊Q₈	—	2	2	no	—
56	C₄⋊Q₈	—	2	2	no	—
57	C₄⋊Q₈	—	4	2	no	—
58	C₄⋊Q₈	C₂³	1	2	yes	—
59	#127	—	2	2	no	—
60	#127	C₂³	1	2	yes	—
61	#121	—	2	2	no	—
62	#121	—	4	2	no	—
63	C₄×Q₈	—	2	2	no	—
64	C₄×Q₈	—	2	2	no	—
65	C₄×Q₈	C₂³	1	2	yes	—
66	C₄×Q₈	C₂³	1	1	yes	characteristic
67	C₄×D₄	—	2	2	no	—
68	C₄×D₄	C₂³	1	1	yes	characteristic
69	#107	—	2	1	no	—
70	#107	—	2	1	no	—
71	#107	—	2	2	no	—
72	#107	D₄	1	1	yes	characteristic
73	#106	—	2	2	no	—
74	#106	D₄	1	1	yes	characteristic
75	C₄⋊C₈	—	2	2	no	—
76	C₄⋊C₈	—	2	1	no	—
77	C₄⋊C₈	C₂³	1	1	yes	characteristic
78	C₄⋊C₈	C₂³	1	2	yes	—
79	C₄⋊C₈	C₂³	1	1	yes	characteristic
80	#103	—	2	2	no	—
81	#103	D₄	1	1	yes	characteristic
82	#103	D₄	1	2	yes	—
83	#102	—	2	2	no	—
84	#102	—	2	2	no	—
85	#102	—	2	2	no	—
86	#102	—	2	2	no	—
87	#102	—	2	2	no	—
88	#102	D₄	1	1	yes	characteristic
89	C₂²⋊C₈	—	2	1	no	—
90	C₄×C₈	—	2	2	no	—
91	C₄×C₈	—	2	1	no	—
92	C₄×C₈	C₂³	1	1	yes	characteristic
93	C₄×C₈	C₂³	1	1	yes	characteristic
94	C₄×C₈	C₂³	1	1	yes	characteristic
95	#41	—	4	4	no	—
96	#41	—	8	2	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	—	4	2	no	—
99	C₂×Q₈	C₂×D₄	1	2	yes	—
100	C₂×Q₈	C₂×D₄	1	1	yes	characteristic
101	C₂×D₄	—	2	2	no	—
102	C₂×D₄	—	2	2	no	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	4	2	no	—
105	C₂×D₄	—	4	2	no	—
106	C₂×D₄	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄	—	2	1	no	—
108	C₂²×C₄	—	4	2	no	—
109	Q₁₆	SD₁₆	1	4	yes	complemented
110	SD₁₆	—	2	2	no	—
111	SD₁₆	—	4	4	no	—
112	SD₁₆	—	4	4	no	—
113	SD₁₆	—	8	4	no	—
114	D₈	—	4	2	no	—
115	D₈	—	8	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	4	2	no	—
121	C₂×C₈	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈	C₂×D₄	1	1	yes	characteristic
124	C₂×C₈	C₂×D₄	1	1	yes	characteristic
125	C₄⋊C₄	—	2	2	no	—
126	C₄⋊C₄	—	2	1	no	—
127	C₄⋊C₄	—	2	2	no	—
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	4	2	no	—
130	C₄⋊C₄	—	4	2	no	—
131	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
132	C₄⋊C₄	C₂×D₄	1	2	yes	—
133	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
134	C₂²⋊C₄	—	2	1	no	—
135	C₂²⋊C₄	—	4	2	no	—
136	C₄²	—	2	2	no	—
137	C₄²	—	2	1	no	—
138	C₄²	—	4	2	no	—
139	C₄²	C₂⁴	1	1	yes	characteristic
140	C₂³	—	2	1	no	—
141	C₂³	—	4	2	no	—
142	Q₈	—	2	2	no	—
143	Q₈	—	2	1	no	—
144	Q₈	—	8	2	no	—
145	Q₈	—	16	2	no	—
146	Q₈	C₂×SD₁₆	1	4	yes	—
147	Q₈	#135	1	2	yes	complemented
148	D₄	—	2	1	no	—
149	D₄	—	2	1	no	—
150	D₄	—	4	2	no	—
151	D₄	—	4	2	no	—
152	D₄	—	4	2	no	—
153	D₄	—	4	2	no	—
154	D₄	—	8	4	no	—
155	D₄	—	8	2	no	—
156	D₄	—	16	2	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	4	1	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	8	4	no	—
166	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
167	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
169	C₈	—	2	2	no	—
170	C₈	—	2	1	no	—
171	C₈	—	2	1	no	—
172	C₈	—	2	1	no	—
173	C₈	—	4	1	no	—
174	C₈	—	4	2	no	—
175	C₈	—	4	1	no	—
176	C₈	C₂×SD₁₆	1	2	yes	—
177	C₂²	—	4	1	no	—
178	C₂²	—	4	1	no	—
179	C₂²	—	4	1	no	—
180	C₂²	—	8	2	no	—
181	C₂²	—	8	2	no	—
182	C₂²	—	8	2	no	—
183	C₂²	D₄²	1	1	yes	characteristic
184	C₄	—	2	4	no	—
185	C₄	—	2	2	no	—
186	C₄	—	2	1	no	—
187	C₄	—	4	1	no	—
188	C₄	—	4	2	no	—
189	C₄	—	4	1	no	—
190	C₄	—	8	2	no	—
191	C₄	C₂²×SD₁₆	1	1	yes	characteristic
192	C₄	C₂×#135	1	1	yes	characteristic
193	C₄	#577	1	1	yes	characteristic
194	C₄	#578	1	1	yes	characteristic
195	C₂	—	8	1	no	—
196	C₂	—	16	2	no	—
197	C₂	D₄×SD₁₆	1	1	yes	characteristic
198	C₂	#3285	1	1	yes	characteristic
199	C₂	#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	#3287	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. Q₈×Q₁₆	C₂	semidirect
2. #3362	C₂	semidirect
4. #1706	C₂	semidirect
5. #1678	C₂	semidirect
6. #1675	C₂	semidirect
7. #1668	C₂	semidirect
8. #1631	C₂	semidirect
9. #1623	C₂	semidirect
10. C₈×Q₁₆	C₂	semidirect
11. C₈×SD₁₆	C₂	semidirect
109. Q₁₆	SD₁₆	semidirect
147. Q₈	#135	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Q₁₆	2
#3362	2
#3287	2
#1706	2
#1678	2
#1675	2
#1668	2
#1631	2
#1623	2
C₈×Q₁₆	2
C₈×SD₁₆	2