Finite Group Databasebeta

Subgroups — #33923

Order 256.

235 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 235 conjugacy classes of subgroups — too many to draw. Only its 101 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	#3342	C₂	1	1	yes	maximal, characteristic
2	#3338	C₂	1	1	yes	maximal, characteristic
3	#2254	C₂	1	1	yes	maximal, characteristic, complemented
4	#2237	C₂	1	2	yes	maximal, complemented
5	#2235	C₂	1	2	yes	maximal, complemented
6	#2227	C₂	1	1	yes	maximal, characteristic, complemented
7	#2225	C₂	1	1	yes	maximal, characteristic, complemented
8	#2219	C₂	1	2	yes	maximal, complemented
9	C₄×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
10	#2169	C₂	1	1	yes	maximal, characteristic
11	#2159	C₂	1	2	yes	maximal, complemented
12	#556	—	2	1	no	—
13	#554	—	2	1	no	—
14	C₂×Q₃₂	C₂²	1	1	yes	characteristic
15	C₂×M₃₂	C₂²	1	2	yes	—
16	#500	C₂²	1	1	yes	characteristic
17	#500	C₂²	1	1	yes	characteristic
18	#485	—	2	2	no	—
19	#484	—	2	2	no	—
20	#477	—	2	1	no	—
21	#475	—	2	1	no	—
22	#467	C₂²	1	2	yes	—
23	#467	C₂²	1	2	yes	—
24	C₄×Q₁₆	C₂²	1	1	yes	characteristic
25	C₄×Q₁₆	C₂²	1	1	yes	characteristic
26	C₈×D₄	C₂²	1	1	yes	characteristic
27	#427	C₂²	1	2	yes	—
28	C₂×#107	C₂²	1	2	yes	—
29	#367	C₂²	1	2	yes	—
30	#367	C₂²	1	2	yes	—
31	#367	C₂²	1	2	yes	—
32	#366	C₂²	1	1	yes	characteristic
33	#366	C₂²	1	2	yes	—
34	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
35	#358	C₂²	1	1	yes	characteristic
36	#358	C₂²	1	1	yes	characteristic
37	#358	C₂²	1	2	yes	—
38	#358	C₂²	1	2	yes	—
39	C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₄×C₁₆	C₂²	1	1	yes	characteristic
41	C₂×Q₁₆	C₂³	1	1	yes	characteristic
42	C₂×Q₁₆	C₂³	1	1	yes	characteristic
43	C₂²×C₈	C₂³	1	2	yes	—
44	C₄⋊Q₈	—	2	1	no	—
45	C₄⋊Q₈	—	2	1	no	—
46	#126	—	4	2	no	—
47	#125	—	4	1	no	—
48	#124	—	4	1	no	—
49	C₂²⋊Q₈	—	2	2	no	—
50	C₂²⋊Q₈	—	2	2	no	—
51	C₂²⋊Q₈	—	4	2	no	—
52	C₄×Q₈	—	2	1	no	—
53	C₄×Q₈	—	2	1	no	—
54	C₄×Q₈	—	4	1	no	—
55	C₄×D₄	—	4	1	no	—
56	C₄×D₄	D₄	1	1	yes	characteristic
57	#117	—	2	2	no	—
58	C₂×C₄⋊C₄	—	2	2	no	—
59	Q₃₂	—	2	2	no	—
60	M₃₂	—	2	4	no	—
61	C₂×C₁₆	C₂³	1	1	yes	characteristic
62	C₂×C₁₆	C₂³	1	2	yes	—
63	C₂×C₁₆	C₂³	1	1	yes	characteristic
64	#107	—	2	2	no	—
65	#107	—	2	1	no	—
66	#107	—	2	1	no	—
67	#107	C₂³	1	1	yes	characteristic
68	#107	C₂³	1	1	yes	characteristic
69	#107	C₂³	1	2	yes	—
70	#107	C₂³	1	2	yes	—
71	#106	—	2	2	no	—
72	C₄⋊C₈	D₄	1	1	yes	characteristic
73	#103	—	2	1	no	—
74	#103	—	2	1	no	—
75	#103	—	2	2	no	—
76	#103	—	2	2	no	—
77	C₂²⋊C₈	D₄	1	2	yes	—
78	C₄×C₈	C₂³	1	1	yes	characteristic
79	C₂×Q₈	—	2	1	no	—
80	C₂×Q₈	—	2	1	no	—
81	C₂×Q₈	—	4	1	no	—
82	C₂×Q₈	—	4	1	no	—
83	C₂×D₄	D₈	1	1	yes	characteristic
84	C₂²×C₄	—	4	2	no	—
85	C₂²×C₄	C₂×D₄	1	2	yes	—
86	Q₁₆	—	2	1	no	—
87	Q₁₆	—	2	1	no	—
88	Q₁₆	—	2	1	no	—
89	Q₁₆	—	2	1	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈	C₂×D₄	1	2	yes	—
94	C₂×C₈	C₂⁴	1	1	yes	characteristic
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	2	1	no	—
98	C₄⋊C₄	—	2	2	no	—
99	C₄⋊C₄	—	4	2	no	—
100	C₄⋊C₄	—	4	1	no	—
101	C₄⋊C₄	—	4	1	no	—
102	C₄⋊C₄	—	4	1	no	—
103	C₄⋊C₄	—	4	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	—	4	1	no	—
106	C₄⋊C₄	D₈	1	1	yes	characteristic
107	C₂²⋊C₄	—	4	2	no	—
108	C₂²⋊C₄	—	4	2	no	—
109	C₂²⋊C₄	—	4	2	no	—
110	C₂²⋊C₄	D₈	1	2	yes	—
111	C₄²	—	4	2	no	—
112	C₄²	—	4	1	no	—
113	C₄²	—	4	1	no	—
114	C₄²	C₂×D₄	1	1	yes	characteristic
115	C₁₆	—	2	2	no	—
116	C₁₆	—	2	1	no	—
117	C₁₆	#41	1	2	yes	—
118	C₂³	C₂×D₈	1	2	yes	—
119	Q₈	—	4	1	no	—
120	Q₈	—	4	1	no	—
121	Q₈	—	4	1	no	—
122	Q₈	—	4	1	no	—
123	Q₈	—	8	2	no	—
124	Q₈	—	16	1	no	—
125	D₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	16	2	no	—
135	C₂×C₄	C₂×D₈	1	2	yes	—
136	C₂×C₄	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
139	C₈	—	2	2	no	—
140	C₈	—	2	1	no	—
141	C₈	C₂×#41	1	1	yes	characteristic
142	C₈	2₋¹⁺⁴	1	1	yes	characteristic
143	C₂²	—	2	2	no	—
144	C₂²	—	2	2	no	—
145	C₂²	—	2	2	no	—
146	C₂²	C₂²×D₈	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	1	no	—
149	C₄	—	8	1	no	—
150	C₄	—	8	1	no	—
151	C₄	—	8	2	no	—
152	C₄	—	8	1	no	—
153	C₄	—	8	1	no	—
154	C₄	—	8	2	no	—
155	C₄	#510	1	2	yes	—
156	C₄	#547	1	1	yes	characteristic
157	C₄	#578	1	1	yes	characteristic
158	C₂	—	4	2	no	—
159	C₂	#3334	1	1	yes	characteristic
160	C₂	C₂×#510	1	1	yes	characteristic
161	C₂	#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂×D₄	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #2254	C₂	semidirect
4. #2237	C₂	semidirect
5. #2235	C₂	semidirect
6. #2227	C₂	semidirect
7. #2225	C₂	semidirect
8. #2219	C₂	semidirect
9. C₄×Q₃₂	C₂	semidirect
11. #2159	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3342	2
#3338	2
#2254	2
#2237	2
#2235	2
#2227	2
#2225	2
#2219	2
C₄×Q₃₂	2
#2169	2
#2159	2