Finite Group Databasebeta

Subgroups — #33921

Order 256.

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

normal subgroups only

This group has 257 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	#3341	C₂	1	1	yes	maximal, characteristic, complemented
2	#3334	C₂	1	1	yes	maximal, characteristic, complemented
3	#2254	C₂	1	1	yes	maximal, characteristic, complemented
4	#2234	C₂	1	2	yes	maximal, complemented
5	#2233	C₂	1	2	yes	maximal, complemented
6	#2226	C₂	1	1	yes	maximal, characteristic, complemented
7	#2224	C₂	1	1	yes	maximal, characteristic, complemented
8	#2218	C₂	1	2	yes	maximal, complemented
9	C₄×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	#2169	C₂	1	1	yes	maximal, characteristic, complemented
11	#2159	C₂	1	2	yes	maximal, complemented
12	#555	—	2	1	no	—
13	#547	—	2	1	no	—
14	C₂×D₁₆	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	#482	—	2	2	no	—
19	#480	—	2	2	no	—
20	#476	—	2	1	no	—
21	#474	—	2	1	no	—
22	#466	C₂²	1	2	yes	—
23	#466	C₂²	1	2	yes	—
24	C₄×D₈	C₂²	1	1	yes	characteristic
25	C₄×D₈	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	#357	C₂²	1	1	yes	characteristic
36	#357	C₂²	1	1	yes	characteristic
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	—	4	1	no	—
42	C₂×D₈	C₂³	1	1	yes	characteristic
43	C₂×D₈	C₂³	1	1	yes	characteristic
44	C₂²×C₈	C₂³	1	2	yes	—
45	C₄⋊Q₈	—	2	1	no	—
46	C₄⋊Q₈	—	2	1	no	—
47	#124	—	4	2	no	—
48	#123	—	4	2	no	—
49	C₂²⋊Q₈	—	4	1	no	—
50	#121	—	2	2	no	—
51	#121	—	2	2	no	—
52	#121	—	4	1	no	—
53	C₄×D₄	—	2	1	no	—
54	C₄×D₄	—	2	1	no	—
55	C₄×D₄	D₄	1	1	yes	characteristic
56	#117	—	2	2	no	—
57	#117	—	4	1	no	—
58	C₂×C₄⋊C₄	—	2	2	no	—
59	D₁₆	—	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	#102	—	2	1	no	—
74	#102	—	2	1	no	—
75	#102	—	2	2	no	—
76	#102	—	2	2	no	—
77	C₂²⋊C₈	D₄	1	2	yes	—
78	C₄×C₈	C₂³	1	1	yes	characteristic
79	#41	—	8	4	no	—
80	C₂×Q₈	—	4	1	no	—
81	C₂×Q₈	—	4	1	no	—
82	C₂×D₄	—	2	1	no	—
83	C₂×D₄	—	2	1	no	—
84	C₂×D₄	—	4	2	no	—
85	C₂×D₄	—	4	2	no	—
86	C₂×D₄	D₈	1	1	yes	characteristic
87	C₂²×C₄	—	4	2	no	—
88	C₂²×C₄	—	4	1	no	—
89	C₂²×C₄	—	4	1	no	—
90	C₂²×C₄	C₂×D₄	1	2	yes	—
91	D₈	—	2	1	no	—
92	D₈	—	2	1	no	—
93	D₈	—	2	1	no	—
94	D₈	—	2	1	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈	C₂×D₄	1	2	yes	—
99	C₂×C₈	C₂⁴	1	1	yes	characteristic
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	—	4	1	no	—
106	C₄⋊C₄	—	4	1	no	—
107	C₄⋊C₄	D₈	1	1	yes	characteristic
108	C₂²⋊C₄	—	4	2	no	—
109	C₂²⋊C₄	—	4	2	no	—
110	C₂²⋊C₄	—	4	1	no	—
111	C₂²⋊C₄	—	4	1	no	—
112	C₂²⋊C₄	—	4	2	no	—
113	C₂²⋊C₄	D₈	1	2	yes	—
114	C₄²	—	4	2	no	—
115	C₄²	C₂×D₄	1	1	yes	characteristic
116	C₁₆	—	2	2	no	—
117	C₁₆	—	2	1	no	—
118	C₁₆	#41	1	2	yes	complemented
119	C₂³	—	4	1	no	—
120	C₂³	—	4	1	no	—
121	C₂³	C₂×D₈	1	2	yes	—
122	Q₈	—	8	2	no	—
123	Q₈	—	16	1	no	—
124	D₄	—	2	2	no	—
125	D₄	—	4	1	no	—
126	D₄	—	4	1	no	—
127	D₄	—	4	1	no	—
128	D₄	—	4	1	no	—
129	D₄	—	8	4	no	—
130	D₄	—	16	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	—	8	4	no	—
138	C₂×C₄	—	8	2	no	—
139	C₂×C₄	—	8	2	no	—
140	C₂×C₄	C₂×D₈	1	2	yes	—
141	C₂×C₄	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
144	C₈	—	2	2	no	—
145	C₈	—	2	1	no	—
146	C₈	C₂×#41	1	1	yes	characteristic
147	C₈	2₋¹⁺⁴	1	1	yes	characteristic
148	C₂²	—	2	2	no	—
149	C₂²	—	2	2	no	—
150	C₂²	—	2	2	no	—
151	C₂²	—	8	1	no	—
152	C₂²	—	8	1	no	—
153	C₂²	—	8	1	no	—
154	C₂²	—	8	1	no	—
155	C₂²	—	8	1	no	—
156	C₂²	—	8	1	no	—
157	C₂²	C₂²×D₈	1	1	yes	characteristic
158	C₄	—	2	2	no	—
159	C₄	—	2	1	no	—
160	C₄	—	8	1	no	—
161	C₄	—	8	1	no	—
162	C₄	—	8	2	no	—
163	C₄	—	8	2	no	—
164	C₄	#509	1	2	yes	—
165	C₄	#547	1	1	yes	characteristic
166	C₄	#578	1	1	yes	characteristic
167	C₂	—	4	2	no	—
168	C₂	—	16	1	no	—
169	C₂	—	16	1	no	—
170	C₂	#3334	1	1	yes	characteristic
171	C₂	C₂×#509	1	1	yes	characteristic
172	C₂	#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
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. #3341	C₂	semidirect
2. #3334	C₂	semidirect
3. #2254	C₂	semidirect
4. #2234	C₂	semidirect
5. #2233	C₂	semidirect
6. #2226	C₂	semidirect
7. #2224	C₂	semidirect
8. #2218	C₂	semidirect
9. C₄×D₁₆	C₂	semidirect
10. #2169	C₂	semidirect
11. #2159	C₂	semidirect
118. C₁₆	#41	semidirect

Maximal subgroups

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

subgroup	index
#3341	2
#3334	2
#2254	2
#2234	2
#2233	2
#2226	2
#2224	2
#2218	2
C₄×D₁₆	2
#2169	2
#2159	2