Finite Group Databasebeta

Subgroups — #33743

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	#3177	C₂	1	1	yes	maximal, characteristic, complemented
2	#3176	C₂	1	1	yes	maximal, characteristic
3	#2253	C₂	1	1	yes	maximal, characteristic, complemented
4	#2248	C₂	1	1	yes	maximal, characteristic, complemented
5	#2245	C₂	1	1	yes	maximal, characteristic, complemented
6	#2244	C₂	1	1	yes	maximal, characteristic, complemented
7	#2220	C₂	1	2	yes	maximal, complemented
8	#2217	C₂	1	2	yes	maximal, complemented
9	#2175	C₂	1	2	yes	maximal, complemented
10	C₄×SD₃₂	C₂	1	2	yes	maximal, complemented
11	#2108	C₂	1	1	yes	maximal, characteristic, complemented
12	#541	—	2	1	no	—
13	#540	—	2	1	no	—
14	C₂×SD₃₂	C₂²	1	2	yes	—
15	C₂×SD₃₂	C₂²	1	2	yes	—
16	C₂×M₃₂	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆	C₂²	1	1	yes	characteristic
18	#499	C₂²	1	1	yes	characteristic
19	#499	C₂²	1	1	yes	characteristic
20	#494	C₂²	1	1	yes	characteristic
21	#493	C₂²	1	1	yes	characteristic
22	#491	—	2	1	no	—
23	#490	—	2	1	no	—
24	#470	—	2	1	no	—
25	#469	—	2	1	no	—
26	#467	C₂²	1	2	yes	—
27	#466	C₂²	1	2	yes	—
28	#442	—	2	1	no	—
29	#441	—	2	1	no	—
30	C₄×Q₁₆	C₂²	1	2	yes	—
31	C₄×D₈	C₂²	1	2	yes	—
32	#405	C₂²	1	1	yes	characteristic
33	#367	C₂²	1	2	yes	—
34	#366	C₂²	1	2	yes	—
35	#358	C₂²	1	2	yes	—
36	#358	C₂²	1	2	yes	—
37	#357	C₂²	1	2	yes	—
38	#357	C₂²	1	2	yes	—
39	#346	C₂²	1	1	yes	characteristic
40	#346	C₂²	1	1	yes	characteristic
41	C₄×C₁₆	C₂²	1	1	yes	characteristic
42	C₄×C₁₆	C₂²	1	1	yes	characteristic
43	C₂×Q₁₆	—	2	1	no	—
44	C₂×Q₁₆	C₂³	1	2	yes	—
45	C₂×D₈	—	2	1	no	—
46	C₂×D₈	C₂³	1	2	yes	—
47	C₂×M₁₆	D₄	1	1	yes	characteristic
48	C₂²×C₈	C₂³	1	1	yes	characteristic
49	C₄⋊Q₈	—	2	1	no	—
50	#127	—	2	1	no	—
51	#126	—	4	2	no	—
52	#125	—	2	1	no	—
53	#125	—	2	1	no	—
54	#125	—	4	2	no	—
55	#123	—	4	2	no	—
56	C₂²⋊Q₈	—	2	2	no	—
57	#121	—	2	2	no	—
58	#121	—	4	2	no	—
59	C₄×Q₈	—	2	2	no	—
60	C₄×D₄	—	2	2	no	—
61	#117	D₄	1	1	yes	characteristic
62	SD₃₂	—	2	4	no	—
63	SD₃₂	—	2	4	no	—
64	M₃₂	—	2	2	no	—
65	C₂×C₁₆	—	2	2	no	—
66	C₂×C₁₆	C₂³	1	1	yes	characteristic
67	C₂×C₁₆	C₂³	1	1	yes	characteristic
68	C₂×C₁₆	C₂³	1	1	yes	characteristic
69	C₂×C₁₆	C₂³	1	1	yes	characteristic
70	#107	C₂³	1	2	yes	—
71	#107	C₂³	1	2	yes	—
72	#106	—	2	1	no	—
73	#106	—	2	1	no	—
74	#103	—	2	2	no	—
75	#103	—	2	2	no	—
76	#102	—	2	2	no	—
77	#102	—	2	2	no	—
78	#97	D₄	1	1	yes	characteristic
79	#97	D₄	1	1	yes	characteristic
80	C₄×C₈	C₂³	1	1	yes	characteristic
81	C₄×C₈	C₂³	1	1	yes	characteristic
82	C₂×Q₈	—	2	2	no	—
83	C₂×D₄	—	2	2	no	—
84	C₂×D₄	—	4	2	no	—
85	C₂×D₄	—	4	2	no	—
86	C₂²×C₄	—	4	2	no	—
87	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
88	Q₁₆	—	2	2	no	—
89	Q₁₆	—	2	2	no	—
90	Q₁₆	—	4	2	no	—
91	D₈	—	2	2	no	—
92	D₈	—	2	2	no	—
93	D₈	—	4	2	no	—
94	M₁₆	—	2	2	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈	C₂⁴	1	1	yes	characteristic
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	4	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	—	4	2	no	—
106	C₄⋊C₄	—	4	2	no	—
107	C₄⋊C₄	—	4	2	no	—
108	C₄⋊C₄	D₈	1	2	yes	—
109	C₂²⋊C₄	—	4	2	no	—
110	C₂²⋊C₄	—	4	2	no	—
111	C₂²⋊C₄	—	4	2	no	—
112	C₂²⋊C₄	D₈	1	2	yes	—
113	C₄²	—	4	2	no	—
114	C₄²	C₂×D₄	1	1	yes	characteristic
115	C₄²	C₂×D₄	1	1	yes	characteristic
116	C₁₆	—	2	1	no	—
117	C₁₆	—	2	1	no	—
118	C₁₆	#41	1	2	yes	—
119	C₁₆	#41	1	2	yes	—
120	C₂³	—	4	2	no	—
121	C₂³	C₂×D₈	1	1	yes	characteristic
122	Q₈	—	4	2	no	—
123	Q₈	—	4	2	no	—
124	D₄	—	4	2	no	—
125	D₄	—	4	2	no	—
126	D₄	—	16	2	no	—
127	D₄	—	16	2	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	16	2	no	—
134	C₂×C₄	C₂×D₈	1	2	yes	—
135	C₂×C₄	C₂×D₈	1	2	yes	—
136	C₂×C₄	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
138	C₈	—	2	1	no	—
139	C₈	—	2	1	no	—
140	C₈	#135	1	2	yes	—
141	C₈	C₂×#41	1	1	yes	characteristic
142	C₈	C₂×#41	1	1	yes	characteristic
143	C₂²	—	2	2	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	8	2	no	—
146	C₂²	—	8	2	no	—
147	C₂²	—	8	2	no	—
148	C₂²	C₂²×D₈	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	2	1	no	—
152	C₄	—	8	2	no	—
153	C₄	—	8	2	no	—
154	C₄	—	8	2	no	—
155	C₄	#532	1	1	yes	characteristic
156	C₄	C₂×#135	1	1	yes	characteristic
157	C₂	—	4	1	no	—
158	C₂	—	16	2	no	—
159	C₂	#3169	1	1	yes	characteristic
160	C₂	#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	#3176	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. #3177	C₂	semidirect
3. #2253	C₂	semidirect
4. #2248	C₂	semidirect
5. #2245	C₂	semidirect
6. #2244	C₂	semidirect
7. #2220	C₂	semidirect
8. #2217	C₂	semidirect
9. #2175	C₂	semidirect
10. C₄×SD₃₂	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
#3177	2
#3176	2
#2253	2
#2248	2
#2245	2
#2244	2
#2220	2
#2217	2
#2175	2
C₄×SD₃₂	2
#2108	2