Finite Group Databasebeta

Subgroups — #33631

Order 256.

278 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 119 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	#3164	C₂	1	1	yes	maximal, characteristic, complemented
2	#2948	C₂	1	1	yes	maximal, characteristic
3	#2228	C₂	1	2	yes	maximal, complemented
4	#2228	C₂	1	2	yes	maximal, complemented
5	#2226	C₂	1	2	yes	maximal, complemented
6	#2224	C₂	1	2	yes	maximal, complemented
7	#2159	C₂	1	1	yes	maximal, characteristic, complemented
8	#2158	C₂	1	1	yes	maximal, characteristic, complemented
9	#2151	C₂	1	1	yes	maximal, characteristic, complemented
10	#2140	C₂	1	1	yes	maximal, characteristic
11	C₂×#357	C₂	1	1	yes	maximal, characteristic
12	C₂²×D₈	C₂²	1	1	yes	characteristic
13	#544	—	2	1	no	—
14	#518	—	2	1	no	—
15	C₂×M₃₂	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆	C₂²	1	1	yes	characteristic
17	#501	—	2	2	no	—
18	#500	C₂²	1	2	yes	complemented
19	#499	C₂²	1	2	yes	complemented
20	#442	—	2	2	no	—
21	C₄×D₈	C₂²	1	2	yes	—
22	C₄×D₈	C₂²	1	2	yes	—
23	#428	—	2	1	no	—
24	#427	C₂²	1	1	yes	characteristic
25	#427	C₂²	1	1	yes	characteristic
26	C₂×#107	C₂²	1	1	yes	characteristic
27	#418	—	2	1	no	—
28	#405	C₂²	1	1	yes	characteristic
29	#367	C₂²	1	1	yes	characteristic, complemented
30	#367	C₂²	1	1	yes	characteristic, complemented
31	#367	C₂²	1	2	yes	complemented
32	#366	C₂²	1	1	yes	characteristic, complemented
33	#366	C₂²	1	1	yes	characteristic, complemented
34	#366	C₂²	1	2	yes	complemented
35	C₄⋊C₁₆	C₂²	1	2	yes	complemented
36	C₄⋊C₁₆	C₂²	1	2	yes	complemented
37	#357	C₂²	1	2	yes	—
38	#357	C₂²	1	2	yes	—
39	#357	C₂²	1	2	yes	—
40	#357	C₂²	1	2	yes	—
41	C₂²×D₄	—	2	1	no	—
42	C₂×D₈	—	2	1	no	—
43	C₂×D₈	—	2	2	no	—
44	C₂×D₈	Q₈	1	4	yes	complemented
45	C₂×D₈	C₂³	1	2	yes	—
46	C₂×M₁₆	D₄	1	1	yes	characteristic
47	C₂²×C₈	C₂³	1	1	yes	characteristic
48	C₄⋊Q₈	—	2	2	no	—
49	C₄⋊Q₈	—	4	1	no	—
50	#125	—	2	2	no	—
51	#125	—	4	1	no	—
52	C₂²⋊Q₈	—	4	2	no	—
53	C₄×D₄	—	2	2	no	—
54	C₄×D₄	—	2	2	no	—
55	C₄×D₄	—	4	1	no	—
56	C₄×D₄	—	4	1	no	—
57	#117	—	2	1	no	—
58	#117	—	2	1	no	—
59	#117	D₄	1	1	yes	characteristic
60	C₂×C₄⋊C₄	—	2	1	no	—
61	C₂×C₂²⋊C₄	—	4	2	no	—
62	M₃₂	—	2	2	no	—
63	C₂×C₁₆	—	2	2	no	—
64	C₂×C₁₆	C₂³	1	1	yes	characteristic
65	C₂×C₁₆	C₂³	1	2	yes	—
66	C₂×C₁₆	C₂³	1	1	yes	characteristic
67	#107	—	2	1	no	—
68	#107	—	2	2	no	—
69	#107	C₂³	1	2	yes	—
70	#107	C₂³	1	1	yes	characteristic
71	#107	C₂³	1	1	yes	characteristic
72	#107	C₂³	1	1	yes	characteristic
73	#107	C₂³	1	1	yes	characteristic
74	#106	—	2	1	no	—
75	#106	—	2	1	no	—
76	#106	—	2	2	no	—
77	#102	—	2	2	no	—
78	#102	—	2	2	no	—
79	#97	D₄	1	2	yes	—
80	C₄×C₈	C₂³	1	2	yes	—
81	C₂⁴	—	4	1	no	—
82	C₂×Q₈	—	4	2	no	—
83	C₂×D₄	—	2	4	no	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	4	1	no	—
86	C₂×D₄	—	4	2	no	—
87	C₂²×C₄	—	4	1	no	—
88	C₂²×C₄	—	4	2	no	—
89	C₂²×C₄	—	4	2	no	—
90	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
91	D₈	—	2	4	no	—
92	D₈	—	2	2	no	—
93	D₈	C₂×Q₈	1	4	yes	—
94	M₁₆	—	2	2	no	—
95	C₂×C₈	C₂×D₄	1	2	yes	—
96	C₂×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈	C₂×Q₈	1	2	yes	—
98	C₂×C₈	#41	1	2	yes	—
99	C₂×C₈	C₂⁴	1	1	yes	characteristic
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	4	1	no	—
106	C₄⋊C₄	—	4	2	no	—
107	C₄⋊C₄	—	4	2	no	—
108	C₄⋊C₄	—	4	2	no	—
109	C₄⋊C₄	D₈	1	1	yes	characteristic
110	C₄⋊C₄	D₈	1	1	yes	characteristic
111	C₂²⋊C₄	—	4	1	no	—
112	C₂²⋊C₄	—	4	1	no	—
113	C₂²⋊C₄	—	4	2	no	—
114	C₂²⋊C₄	—	4	2	no	—
115	C₂²⋊C₄	D₈	1	2	yes	—
116	C₄²	—	4	1	no	—
117	C₄²	—	4	1	no	—
118	C₄²	C₂×D₄	1	2	yes	—
119	C₁₆	—	2	1	no	—
120	C₁₆	—	2	1	no	—
121	C₁₆	—	2	2	no	—
122	C₂³	—	4	2	no	—
123	C₂³	—	4	4	no	—
124	C₂³	—	16	1	no	—
125	C₂³	—	16	1	no	—
126	C₂³	C₂×D₈	1	1	yes	characteristic
127	Q₈	—	16	2	no	—
128	D₄	—	2	4	no	—
129	D₄	—	4	2	no	—
130	D₄	—	4	4	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	—	16	2	no	—
137	C₂×C₄	—	16	2	no	—
138	C₂×C₄	—	16	1	no	—
139	C₂×C₄	C₂²⋊Q₈	1	2	yes	—
140	C₂×C₄	C₂×D₈	1	2	yes	—
141	C₂×C₄	C₂×D₈	1	2	yes	—
142	C₂×C₄	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄	#137	1	2	yes	—
144	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
145	C₈	—	2	2	no	—
146	C₈	C₂²⋊Q₈	1	2	yes	—
147	C₈	C₂²×Q₈	1	1	yes	characteristic
148	C₈	C₂×#41	1	1	yes	characteristic
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	4	4	no	—
152	C₂²	—	8	2	no	—
153	C₂²	—	8	2	no	—
154	C₂²	—	16	4	no	—
155	C₂²	#474	1	2	yes	—
156	C₂²	C₂²×D₈	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	—	8	1	no	—
160	C₄	—	8	2	no	—
161	C₄	—	8	1	no	—
162	C₄	—	8	1	no	—
163	C₄	—	8	1	no	—
164	C₄	#474	1	2	yes	—
165	C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
166	C₄	C₂×#137	1	1	yes	characteristic
167	C₂	—	2	2	no	—
168	C₂	—	8	4	no	—
169	C₂	C₂×#474	1	1	yes	characteristic
170	C₂	#3415	1	1	yes	characteristic
171	C₂	#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂²×D₈	C₂²
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. #3164	C₂	semidirect
3. #2228	C₂	semidirect
4. #2228	C₂	semidirect
5. #2226	C₂	semidirect
6. #2224	C₂	semidirect
7. #2159	C₂	semidirect
8. #2158	C₂	semidirect
9. #2151	C₂	semidirect
18. #500	C₂²	semidirect
19. #499	C₂²	semidirect
29. #367	C₂²	semidirect
30. #367	C₂²	semidirect
31. #367	C₂²	semidirect
32. #366	C₂²	semidirect
33. #366	C₂²	semidirect
34. #366	C₂²	semidirect
35. C₄⋊C₁₆	C₂²	semidirect
36. C₄⋊C₁₆	C₂²	semidirect
44. C₂×D₈	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#3164	2
#2948	2
#2228	2
#2226	2
#2224	2
#2159	2
#2158	2
#2151	2
#2140	2
C₂×#357	2