Finite Group Databasebeta

Subgroups — #33739

Order 256.

254 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 103 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	#3175	C₂	1	2	yes	maximal, complemented
2	#2252	C₂	1	1	yes	maximal, characteristic, complemented
3	#2249	C₂	1	1	yes	maximal, characteristic, complemented
4	#2243	C₂	1	2	yes	maximal, complemented
5	#2220	C₂	1	2	yes	maximal, complemented
6	#2216	C₂	1	1	yes	maximal, characteristic, complemented
7	#2215	C₂	1	1	yes	maximal, characteristic
8	#2175	C₂	1	2	yes	maximal, complemented
9	C₄×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	#2108	C₂	1	1	yes	maximal, characteristic, complemented
12	#542	—	2	2	no	—
13	C₂×Q₃₂	C₂²	1	1	yes	characteristic
14	C₂×SD₃₂	C₂²	1	2	yes	—
15	C₂×D₁₆	C₂²	1	1	yes	characteristic
16	C₂×M₃₂	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆	C₂²	1	1	yes	characteristic
18	#500	C₂²	1	2	yes	—
19	#495	C₂²	1	2	yes	—
20	#489	—	2	2	no	—
21	#468	—	2	2	no	—
22	#467	C₂²	1	2	yes	—
23	#466	C₂²	1	2	yes	—
24	#440	—	2	2	no	—
25	C₄×Q₁₆	C₂²	1	2	yes	—
26	C₄×D₈	C₂²	1	2	yes	—
27	#405	C₂²	1	1	yes	characteristic
28	#366	C₂²	1	1	yes	characteristic
29	#366	C₂²	1	1	yes	characteristic
30	#366	C₂²	1	2	yes	—
31	#358	C₂²	1	2	yes	—
32	#358	C₂²	1	2	yes	—
33	#357	C₂²	1	2	yes	—
34	#357	C₂²	1	2	yes	—
35	#346	C₂²	1	1	yes	characteristic
36	#346	C₂²	1	1	yes	characteristic
37	C₄×C₁₆	C₂²	1	1	yes	characteristic
38	C₄×C₁₆	C₂²	1	1	yes	characteristic
39	C₂×Q₁₆	C₂³	1	2	yes	—
40	C₂×SD₁₆	—	2	2	no	—
41	C₂×D₈	C₂³	1	2	yes	—
42	C₂×M₁₆	D₄	1	1	yes	characteristic
43	C₂²×C₈	C₂³	1	1	yes	characteristic
44	C₄⋊Q₈	—	2	2	no	—
45	#126	—	4	2	no	—
46	#124	—	2	2	no	—
47	#124	—	4	2	no	—
48	#123	—	4	2	no	—
49	C₂²⋊Q₈	—	2	2	no	—
50	C₂²⋊Q₈	—	4	2	no	—
51	#121	—	2	2	no	—
52	C₄×Q₈	—	2	2	no	—
53	C₄×D₄	—	2	2	no	—
54	#117	D₄	1	1	yes	characteristic
55	Q₃₂	—	2	2	no	—
56	SD₃₂	—	2	4	no	—
57	D₁₆	—	2	2	no	—
58	M₃₂	—	2	2	no	—
59	C₂×C₁₆	—	2	2	no	—
60	C₂×C₁₆	C₂³	1	1	yes	characteristic
61	C₂×C₁₆	C₂³	1	1	yes	characteristic
62	C₂×C₁₆	C₂³	1	1	yes	characteristic
63	C₂×C₁₆	C₂³	1	1	yes	characteristic
64	#107	—	2	2	no	—
65	#107	C₂³	1	2	yes	—
66	#107	C₂³	1	2	yes	—
67	#103	—	2	2	no	—
68	#103	—	2	2	no	—
69	#102	—	2	2	no	—
70	#102	—	2	2	no	—
71	#97	D₄	1	1	yes	characteristic
72	#97	D₄	1	1	yes	characteristic
73	C₄×C₈	C₂³	1	1	yes	characteristic
74	C₄×C₈	C₂³	1	1	yes	characteristic
75	C₂×Q₈	—	2	2	no	—
76	C₂×Q₈	—	4	2	no	—
77	C₂×D₄	—	2	2	no	—
78	C₂×D₄	—	4	2	no	—
79	C₂²×C₄	—	4	2	no	—
80	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
81	Q₁₆	—	2	2	no	—
82	Q₁₆	—	2	2	no	—
83	SD₁₆	—	4	4	no	—
84	D₈	—	2	2	no	—
85	D₈	—	2	2	no	—
86	M₁₆	—	2	2	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	4	2	no	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	D₈	1	1	yes	characteristic
99	C₄⋊C₄	D₈	1	1	yes	characteristic
100	C₂²⋊C₄	—	4	2	no	—
101	C₂²⋊C₄	—	4	2	no	—
102	C₂²⋊C₄	—	4	2	no	—
103	C₂²⋊C₄	—	4	2	no	—
104	C₂²⋊C₄	—	4	2	no	—
105	C₂²⋊C₄	D₈	1	1	yes	characteristic
106	C₂²⋊C₄	D₈	1	1	yes	characteristic
107	C₄²	—	4	2	no	—
108	C₄²	C₂×D₄	1	1	yes	characteristic
109	C₄²	C₂×D₄	1	1	yes	characteristic
110	C₁₆	—	2	1	no	—
111	C₁₆	—	2	1	no	—
112	C₁₆	#41	1	2	yes	—
113	C₁₆	#41	1	2	yes	—
114	C₂³	—	4	2	no	—
115	C₂³	C₂×D₈	1	1	yes	characteristic
116	Q₈	—	4	2	no	—
117	Q₈	—	4	2	no	—
118	Q₈	—	16	2	no	—
119	D₄	—	4	2	no	—
120	D₄	—	4	2	no	—
121	D₄	—	16	2	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	16	2	no	—
128	C₂×C₄	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
134	C₈	—	2	1	no	—
135	C₈	—	2	1	no	—
136	C₈	#135	1	2	yes	—
137	C₈	C₂×#41	1	1	yes	characteristic
138	C₈	C₂×#41	1	1	yes	characteristic
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	8	2	no	—
143	C₂²	—	8	2	no	—
144	C₂²	—	8	2	no	—
145	C₂²	C₂²×D₈	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	8	2	no	—
152	C₄	—	8	2	no	—
153	C₄	—	8	2	no	—
154	C₄	#532	1	1	yes	characteristic
155	C₄	C₂×#135	1	1	yes	characteristic
156	C₂	—	4	1	no	—
157	C₂	—	16	2	no	—
158	C₂	#3169	1	1	yes	characteristic
159	C₂	#3415	1	1	yes	characteristic
160	C₂	#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	#2215	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ 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. #3175	C₂	semidirect
2. #2252	C₂	semidirect
3. #2249	C₂	semidirect
4. #2243	C₂	semidirect
5. #2220	C₂	semidirect
6. #2216	C₂	semidirect
8. #2175	C₂	semidirect
9. C₄×Q₃₂	C₂	semidirect
10. C₄×D₁₆	C₂	semidirect
11. #2108	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3175	2
#2252	2
#2249	2
#2243	2
#2220	2
#2216	2
#2215	2
#2175	2
C₄×Q₃₂	2
C₄×D₁₆	2
#2108	2