Finite Group Databasebeta

Subgroups — #33871

Order 256.

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

normal subgroups only

This group has 278 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, complemented
2	#3287	C₂	1	1	yes	maximal, characteristic
3	#2240	C₂	1	1	yes	maximal, characteristic, complemented
4	#2234	C₂	1	2	yes	maximal, complemented
5	#2225	C₂	1	1	yes	maximal, characteristic, complemented
6	#2217	C₂	1	2	yes	maximal, complemented
7	#2207	C₂	1	1	yes	maximal, characteristic, complemented
8	#2189	C₂	1	2	yes	maximal, complemented
9	C₄×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
10	D₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	#2138	C₂	1	2	yes	maximal, complemented
12	C₂×#135	C₂²	1	2	yes	—
13	#556	—	2	1	no	—
14	#550	—	2	1	no	—
15	C₂×Q₃₂	C₂²	1	1	yes	characteristic
16	C₂×SD₃₂	C₂²	1	2	yes	—
17	C₂²×C₁₆	C₂²	1	2	yes	—
18	#500	C₂²	1	1	yes	characteristic
19	#493	C₂²	1	1	yes	characteristic
20	#485	—	2	2	no	—
21	#475	—	2	1	no	—
22	#467	C₂²	1	2	yes	—
23	#466	C₂²	1	2	yes	—
24	#460	—	2	1	no	—
25	#449	—	2	2	no	—
26	C₄×Q₁₆	C₂²	1	1	yes	characteristic
27	C₄×Q₁₆	C₂²	1	1	yes	characteristic
28	C₈×D₄	C₂²	1	1	yes	characteristic
29	#427	C₂²	1	2	yes	—
30	#367	C₂²	1	2	yes	—
31	#366	C₂²	1	1	yes	characteristic
32	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
33	#358	C₂²	1	1	yes	characteristic
34	#358	C₂²	1	2	yes	—
35	#358	C₂²	1	1	yes	characteristic
36	#357	C₂²	1	2	yes	—
37	#357	C₂²	1	2	yes	—
38	#357	C₂²	1	2	yes	—
39	C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₄×C₁₆	C₂²	1	1	yes	characteristic
41	C₂×#41	—	2	2	no	—
42	#135	—	2	4	no	—
43	#135	—	4	2	no	—
44	C₂×Q₁₆	C₂³	1	1	yes	characteristic
45	C₂×Q₁₆	C₂³	1	1	yes	characteristic
46	C₂×SD₁₆	—	2	2	no	—
47	C₂×D₈	—	2	1	no	—
48	C₂×D₈	C₂³	1	2	yes	—
49	C₂²×C₈	C₂³	1	2	yes	—
50	C₄⋊Q₈	—	2	1	no	—
51	#127	—	2	1	no	—
52	#127	—	4	1	no	—
53	#126	—	4	2	no	—
54	#124	—	4	1	no	—
55	C₂²⋊Q₈	—	2	2	no	—
56	#121	—	2	2	no	—
57	#121	—	4	2	no	—
58	C₄×Q₈	—	2	1	no	—
59	C₄×Q₈	—	2	1	no	—
60	C₄×Q₈	—	4	1	no	—
61	C₄×D₄	—	4	1	no	—
62	C₄×D₄	D₄	1	1	yes	characteristic
63	#117	—	2	2	no	—
64	Q₃₂	—	2	2	no	—
65	SD₃₂	—	4	2	no	—
66	C₂×C₁₆	—	4	2	no	—
67	C₂×C₁₆	C₂³	1	1	yes	characteristic
68	C₂×C₁₆	C₂³	1	1	yes	characteristic
69	C₂×C₁₆	C₂³	1	2	yes	—
70	#107	—	2	1	no	—
71	#107	C₂³	1	1	yes	characteristic
72	#107	C₂³	1	2	yes	—
73	#107	C₂³	1	1	yes	characteristic
74	#106	—	2	2	no	—
75	C₄⋊C₈	D₄	1	1	yes	characteristic
76	#103	—	2	1	no	—
77	#103	—	2	1	no	—
78	#103	—	2	2	no	—
79	#102	—	2	2	no	—
80	C₂²⋊C₈	D₄	1	2	yes	—
81	C₄×C₈	C₂³	1	1	yes	characteristic
82	#41	—	4	4	no	—
83	#41	—	8	2	no	—
84	C₂×Q₈	—	2	1	no	—
85	C₂×Q₈	—	2	1	no	—
86	C₂×Q₈	—	4	1	no	—
87	C₂×D₄	—	2	2	no	—
88	C₂×D₄	—	4	2	no	—
89	C₂×D₄	—	4	2	no	—
90	C₂×D₄	—	4	1	no	—
91	C₂×D₄	—	4	1	no	—
92	C₂×D₄	D₈	1	1	yes	characteristic
93	C₂²×C₄	—	4	2	no	—
94	C₂²×C₄	C₂×D₄	1	2	yes	—
95	Q₁₆	—	2	1	no	—
96	Q₁₆	—	2	1	no	—
97	Q₁₆	—	2	1	no	—
98	Q₁₆	#41	1	2	yes	—
99	SD₁₆	—	4	4	no	—
100	D₈	—	2	2	no	—
101	D₈	—	2	2	no	—
102	D₈	—	8	1	no	—
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈	C₂×D₄	1	2	yes	—
107	C₂×C₈	C₂⁴	1	1	yes	characteristic
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	4	1	no	—
112	C₄⋊C₄	—	4	1	no	—
113	C₄⋊C₄	—	4	2	no	—
114	C₄⋊C₄	—	4	1	no	—
115	C₄⋊C₄	D₈	1	1	yes	characteristic
116	C₂²⋊C₄	—	4	2	no	—
117	C₂²⋊C₄	—	4	2	no	—
118	C₂²⋊C₄	—	4	2	no	—
119	C₂²⋊C₄	D₈	1	2	yes	—
120	C₄²	—	4	2	no	—
121	C₄²	—	4	1	no	—
122	C₄²	—	4	1	no	—
123	C₄²	C₂×D₄	1	1	yes	characteristic
124	C₁₆	—	2	1	no	—
125	C₁₆	—	2	1	no	—
126	C₁₆	—	2	2	no	—
127	C₂³	—	4	2	no	—
128	C₂³	C₂×D₈	1	2	yes	—
129	Q₈	—	2	2	no	—
130	Q₈	—	4	1	no	—
131	Q₈	—	4	1	no	—
132	Q₈	—	4	1	no	—
133	Q₈	—	8	2	no	—
134	D₄	—	2	2	no	—
135	D₄	—	4	2	no	—
136	D₄	—	4	2	no	—
137	D₄	—	8	4	no	—
138	D₄	—	8	2	no	—
139	D₄	—	16	1	no	—
140	D₄	—	16	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	8	4	no	—
149	C₂×C₄	C₂×D₈	1	2	yes	—
150	C₂×C₄	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
153	C₈	—	2	2	no	—
154	C₈	—	2	1	no	—
155	C₈	C₂×#41	1	1	yes	characteristic
156	C₈	H(5,𝔽₂)	1	1	yes	characteristic
157	C₂²	—	2	2	no	—
158	C₂²	—	2	2	no	—
159	C₂²	—	2	2	no	—
160	C₂²	—	8	2	no	—
161	C₂²	—	8	2	no	—
162	C₂²	—	8	2	no	—
163	C₂²	C₂²×D₈	1	1	yes	characteristic
164	C₄	—	2	1	no	—
165	C₄	—	2	2	no	—
166	C₄	—	4	2	no	—
167	C₄	—	8	2	no	—
168	C₄	—	8	1	no	—
169	C₄	—	8	1	no	—
170	C₄	—	8	1	no	—
171	C₄	#508	1	2	yes	—
172	C₄	#546	1	1	yes	characteristic
173	C₄	#577	1	1	yes	characteristic
174	C₂	—	4	2	no	—
175	C₂	—	16	2	no	—
176	C₂	#3294	1	1	yes	characteristic
177	C₂	C₂×#508	1	1	yes	characteristic
178	C₂	#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	#3287	C₂
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
1. #3342	C₂	semidirect
3. #2240	C₂	semidirect
4. #2234	C₂	semidirect
5. #2225	C₂	semidirect
6. #2217	C₂	semidirect
7. #2207	C₂	semidirect
8. #2189	C₂	semidirect
9. C₄×Q₃₂	C₂	semidirect
10. D₄×C₁₆	C₂	semidirect
11. #2138	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
#3287	2
#2240	2
#2234	2
#2225	2
#2217	2
#2207	2
#2189	2
C₄×Q₃₂	2
D₄×C₁₆	2
#2138	2