Finite Group Databasebeta

Subgroups — #19930

Order 256.

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

normal subgroups only

This group has 285 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	#3385	C₂	1	2	yes	maximal
2	Q₈×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#3338	C₂	1	1	yes	maximal, characteristic
4	#1713	C₂	1	1	yes	maximal, characteristic, complemented
5	#1696	C₂	1	2	yes	maximal, complemented
6	#1676	C₂	1	1	yes	maximal, characteristic, complemented
7	#1675	C₂	1	1	yes	maximal, characteristic
8	#1633	C₂	1	2	yes	maximal, complemented
9	#1632	C₂	1	2	yes	maximal
10	#1591	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆	C₂	1	1	yes	maximal, characteristic
12	Q₈²	C₂²	1	2	yes	—
13	#554	C₂²	1	2	yes	—
14	#501	—	2	2	no	—
15	#500	—	2	1	no	—
16	#500	C₂²	1	1	yes	characteristic
17	#500	C₂²	1	1	yes	characteristic
18	#500	C₂²	1	2	yes	—
19	#494	C₂²	1	2	yes	—
20	#484	—	2	2	no	—
21	#479	—	2	2	no	—
22	#477	—	2	2	no	—
23	#477	C₂²	1	2	yes	—
24	#477	C₂²	1	2	yes	—
25	#477	C₂²	1	2	yes	—
26	#476	C₂²	1	2	yes	—
27	#474	—	2	2	no	—
28	#467	—	2	1	no	—
29	#446	C₂²	1	2	yes	—
30	C₈×Q₈	C₂²	1	1	yes	characteristic
31	#440	—	2	2	no	—
32	C₄×Q₁₆	—	2	1	no	—
33	C₄×Q₁₆	C₂²	1	1	yes	characteristic
34	C₄×SD₁₆	C₂²	1	1	yes	characteristic
35	C₈×D₄	C₂²	1	1	yes	characteristic
36	C₂×#107	—	2	1	no	—
37	#335	C₂²	1	1	yes	characteristic
38	#334	C₂²	1	1	yes	characteristic
39	#333	C₂²	1	2	yes	—
40	#333	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×Q₁₆	—	4	2	no	—
46	C₂×Q₁₆	D₄	1	1	yes	characteristic
47	C₂×SD₁₆	D₄	1	1	yes	characteristic
48	C₂²×C₈	—	2	1	no	—
49	C₄⋊Q₈	—	2	2	no	—
50	C₄⋊Q₈	—	2	2	no	—
51	C₄⋊Q₈	—	4	2	no	—
52	C₄⋊Q₈	C₂³	1	2	yes	—
53	C₄⋊Q₈	C₂³	1	2	yes	—
54	#125	—	2	2	no	—
55	C₂²⋊Q₈	—	2	2	no	—
56	C₂²⋊Q₈	—	4	2	no	—
57	C₄×Q₈	—	2	2	no	—
58	C₄×Q₈	—	2	2	no	—
59	C₄×Q₈	C₂³	1	2	yes	—
60	C₄×Q₈	C₂³	1	1	yes	characteristic
61	C₄×D₄	—	2	2	no	—
62	C₄×D₄	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄	—	2	2	no	—
64	#107	—	2	1	no	—
65	#107	—	2	1	no	—
66	#107	—	2	2	no	—
67	#107	—	2	2	no	—
68	#107	—	2	1	no	—
69	#107	—	2	1	no	—
70	#107	—	2	2	no	—
71	#107	—	2	2	no	—
72	#107	—	2	2	no	—
73	#107	D₄	1	1	yes	characteristic
74	#106	—	2	2	no	—
75	#106	D₄	1	1	yes	characteristic
76	C₄⋊C₈	—	2	1	no	—
77	C₄⋊C₈	—	2	2	no	—
78	C₄⋊C₈	C₂³	1	1	yes	characteristic
79	C₄⋊C₈	C₂³	1	1	yes	characteristic
80	C₄⋊C₈	C₂³	1	2	yes	—
81	#103	—	2	2	no	—
82	#103	—	2	2	no	—
83	#103	—	2	2	no	—
84	#103	D₄	1	1	yes	characteristic
85	#103	D₄	1	2	yes	—
86	#102	—	2	2	no	—
87	#102	D₄	1	1	yes	characteristic
88	C₂²⋊C₈	—	2	1	no	—
89	#97	—	2	2	no	—
90	C₄×C₈	—	2	1	no	—
91	C₄×C₈	C₂³	1	1	yes	characteristic
92	C₄×C₈	C₂³	1	1	yes	characteristic
93	C₄×C₈	C₂³	1	1	yes	characteristic
94	C₂×Q₈	—	2	2	no	—
95	C₂×Q₈	—	2	2	no	—
96	C₂×Q₈	—	4	2	no	—
97	C₂×Q₈	C₂×D₄	1	2	yes	—
98	C₂×Q₈	C₂×D₄	1	1	yes	characteristic
99	C₂×D₄	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄	—	2	1	no	—
101	C₂²×C₄	—	4	2	no	—
102	Q₁₆	—	2	2	no	—
103	Q₁₆	—	8	4	no	—
104	SD₁₆	Q₁₆	1	4	yes	complemented
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	4	2	no	—
110	C₂×C₈	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈	C₂×D₄	1	1	yes	characteristic
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	2	1	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	4	2	no	—
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	—	4	2	no	—
124	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
125	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
126	C₄⋊C₄	C₂×D₄	1	2	yes	—
127	C₂²⋊C₄	—	2	1	no	—
128	C₂²⋊C₄	—	4	2	no	—
129	C₄²	—	2	1	no	—
130	C₄²	—	2	2	no	—
131	C₄²	—	4	2	no	—
132	C₄²	C₂⁴	1	1	yes	characteristic
133	C₂³	—	2	1	no	—
134	Q₈	—	2	1	no	—
135	Q₈	—	2	2	no	—
136	Q₈	—	2	2	no	—
137	Q₈	—	8	2	no	—
138	Q₈	—	8	2	no	—
139	Q₈	—	16	2	no	—
140	Q₈	C₂×Q₁₆	1	2	yes	—
141	D₄	—	2	1	no	—
142	D₄	C₂×Q₁₆	1	2	yes	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	16	2	no	—
153	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
156	C₈	—	2	2	no	—
157	C₈	—	2	1	no	—
158	C₈	—	2	1	no	—
159	C₈	—	4	1	no	—
160	C₈	—	4	1	no	—
161	C₈	—	4	2	no	—
162	C₈	C₂×Q₁₆	1	2	yes	—
163	C₈	#137	1	2	yes	—
164	C₂²	—	2	2	no	—
165	C₂²	—	4	1	no	—
166	C₂²	—	4	1	no	—
167	C₂²	D₄²	1	1	yes	characteristic
168	C₄	—	2	2	no	—
169	C₄	—	2	1	no	—
170	C₄	—	4	1	no	—
171	C₄	—	4	2	no	—
172	C₄	—	4	2	no	—
173	C₄	—	4	1	no	—
174	C₄	—	8	2	no	—
175	C₄	—	8	2	no	—
176	C₄	C₂²×Q₁₆	1	1	yes	characteristic
177	C₄	C₂×#137	1	1	yes	characteristic
178	C₄	#578	1	1	yes	characteristic
179	C₄	#578	1	1	yes	characteristic
180	C₂	—	4	2	no	—
181	C₂	#3276	1	1	yes	characteristic
182	C₂	D₄×Q₁₆	1	1	yes	characteristic
183	C₂	#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂×D₄	C₂×D₄
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
2. Q₈×Q₁₆	C₂	semidirect
4. #1713	C₂	semidirect
5. #1696	C₂	semidirect
6. #1676	C₂	semidirect
8. #1633	C₂	semidirect
10. #1591	C₂	semidirect
104. SD₁₆	Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
#3385	2
Q₈×Q₁₆	2
#3338	2
#1713	2
#1696	2
#1676	2
#1675	2
#1633	2
#1632	2
#1591	2
C₈×SD₁₆	2