Finite Group Databasebeta

Subgroups — #33747

Order 256.

256 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 105 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	#3170	C₂	1	1	yes	maximal, characteristic, complemented
2	#3168	C₂	1	1	yes	maximal, characteristic
3	#2255	C₂	1	1	yes	maximal, characteristic, complemented
4	#2251	C₂	1	1	yes	maximal, characteristic, complemented
5	#2245	C₂	1	1	yes	maximal, characteristic, complemented
6	#2243	C₂	1	1	yes	maximal, characteristic, complemented
7	#2217	C₂	1	2	yes	maximal
8	#2216	C₂	1	2	yes	maximal, complemented
9	#2176	C₂	1	2	yes	maximal, complemented
10	#2175	C₂	1	2	yes	maximal, complemented
11	C₂×#346	C₂	1	1	yes	maximal, characteristic, complemented
12	#533	—	2	1	no	—
13	#529	—	2	1	no	—
14	C₂×Q₃₂	C₂²	1	2	yes	—
15	C₂×SD₃₂	C₂²	1	2	yes	—
16	C₂²×C₁₆	C₂²	1	2	yes	—
17	#500	C₂²	1	1	yes	characteristic
18	#499	C₂²	1	1	yes	characteristic
19	#495	C₂²	1	1	yes	characteristic
20	#494	C₂²	1	1	yes	characteristic
21	#488	—	2	1	no	—
22	#487	—	2	1	no	—
23	#467	—	2	1	no	—
24	#467	C₂²	1	1	yes	characteristic
25	#467	C₂²	1	2	yes	—
26	#466	C₂²	1	1	yes	characteristic
27	#465	—	2	1	no	—
28	C₄×Q₁₆	—	2	1	no	—
29	C₄×Q₁₆	C₂²	1	1	yes	characteristic
30	C₄×Q₁₆	C₂²	1	2	yes	—
31	C₄×SD₁₆	—	2	1	no	—
32	C₄×D₈	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
34	#367	C₂²	1	2	yes	—
35	#366	C₂²	1	2	yes	—
36	#358	C₂²	1	2	yes	—
37	#358	C₂²	1	2	yes	—
38	#358	C₂²	1	2	yes	—
39	#357	C₂²	1	2	yes	—
40	#346	C₂²	1	2	yes	—
41	#346	C₂²	1	1	yes	characteristic
42	#346	C₂²	1	1	yes	characteristic
43	C₂×Q₁₆	—	2	1	no	—
44	C₂×Q₁₆	C₂³	1	2	yes	—
45	C₂×Q₁₆	C₂³	1	1	yes	characteristic
46	C₂×SD₁₆	—	2	1	no	—
47	C₂×D₈	C₂³	1	1	yes	characteristic
48	C₂²×C₈	D₄	1	1	yes	characteristic
49	C₂²×C₈	C₂³	1	1	yes	characteristic
50	C₄⋊Q₈	—	2	1	no	—
51	C₄⋊Q₈	—	2	1	no	—
52	#126	—	4	1	no	—
53	#125	—	2	1	no	—
54	#125	—	4	1	no	—
55	#124	—	2	1	no	—
56	#123	—	4	1	no	—
57	C₂²⋊Q₈	—	2	1	no	—
58	C₂²⋊Q₈	—	2	2	no	—
59	C₂²⋊Q₈	—	4	1	no	—
60	#121	—	2	1	no	—
61	C₄×Q₈	—	2	1	no	—
62	C₄×Q₈	—	2	2	no	—
63	C₄×Q₈	—	4	1	no	—
64	C₄×D₄	—	2	1	no	—
65	C₄×D₄	—	4	1	no	—
66	#117	—	4	1	no	—
67	#117	—	4	1	no	—
68	C₂×C₄²	D₄	1	1	yes	characteristic
69	Q₃₂	—	2	4	no	—
70	SD₃₂	—	2	4	no	—
71	C₂×C₁₆	—	2	4	no	—
72	C₂×C₁₆	C₂³	1	2	yes	—
73	C₂×C₁₆	C₂³	1	2	yes	—
74	#107	—	2	1	no	—
75	#107	C₂³	1	2	yes	—
76	#107	C₂³	1	1	yes	characteristic
77	#107	C₂³	1	1	yes	characteristic
78	#106	—	2	1	no	—
79	#103	—	2	1	no	—
80	#103	—	2	2	no	—
81	#103	—	2	1	no	—
82	#103	—	2	2	no	—
83	#102	—	2	1	no	—
84	#102	—	2	1	no	—
85	C₄×C₈	D₄	1	1	yes	characteristic
86	C₄×C₈	D₄	1	1	yes	characteristic
87	C₄×C₈	C₂³	1	1	yes	characteristic
88	C₄×C₈	C₂³	1	1	yes	characteristic
89	C₂×Q₈	—	2	1	no	—
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	4	1	no	—
92	C₂×D₄	—	2	1	no	—
93	C₂×D₄	—	4	1	no	—
94	C₂²×C₄	—	4	1	no	—
95	C₂²×C₄	D₈	1	1	yes	characteristic
96	C₂²×C₄	D₈	1	1	yes	characteristic
97	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
98	Q₁₆	—	2	2	no	—
99	Q₁₆	—	2	2	no	—
100	Q₁₆	—	2	2	no	—
101	Q₁₆	—	4	2	no	—
102	SD₁₆	—	4	2	no	—
103	D₈	—	2	2	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈	C₂⁴	1	1	yes	characteristic
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	4	1	no	—
114	C₄⋊C₄	—	4	1	no	—
115	C₄⋊C₄	—	4	1	no	—
116	C₄⋊C₄	—	4	1	no	—
117	C₄⋊C₄	—	4	1	no	—
118	C₄⋊C₄	—	4	2	no	—
119	C₄⋊C₄	—	4	1	no	—
120	C₄⋊C₄	—	4	2	no	—
121	C₄⋊C₄	—	4	1	no	—
122	C₂²⋊C₄	—	4	2	no	—
123	C₂²⋊C₄	—	4	1	no	—
124	C₂²⋊C₄	—	4	1	no	—
125	C₂²⋊C₄	—	4	1	no	—
126	C₂²⋊C₄	—	4	1	no	—
127	C₂²⋊C₄	—	4	1	no	—
128	C₄²	—	4	1	no	—
129	C₄²	—	4	2	no	—
130	C₄²	D₈	1	1	yes	characteristic
131	C₄²	D₈	1	1	yes	characteristic
132	C₄²	C₂×D₄	1	1	yes	characteristic
133	C₄²	C₂×D₄	1	1	yes	characteristic
134	C₁₆	—	2	2	no	—
135	C₁₆	#41	1	4	yes	—
136	C₂³	—	4	1	no	—
137	C₂³	C₂×D₈	1	1	yes	characteristic
138	Q₈	—	4	2	no	—
139	Q₈	—	4	1	no	—
140	Q₈	—	4	1	no	—
141	Q₈	—	4	2	no	—
142	Q₈	—	8	2	no	—
143	D₄	—	4	1	no	—
144	D₄	—	4	1	no	—
145	D₄	—	8	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄	C₂×D₈	1	1	yes	characteristic
158	C₂×C₄	C₂×D₈	1	1	yes	characteristic
159	C₂×C₄	C₂×D₈	1	1	yes	characteristic
160	C₂×C₄	C₂×D₈	1	1	yes	characteristic
161	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
162	C₈	—	2	1	no	—
163	C₈	—	2	1	no	—
164	C₈	#135	1	2	yes	—
165	C₈	C₂×#41	1	2	yes	—
166	C₂²	—	2	1	no	—
167	C₂²	—	2	1	no	—
168	C₂²	—	8	1	no	—
169	C₂²	—	8	1	no	—
170	C₂²	—	8	1	no	—
171	C₂²	#510	1	2	yes	—
172	C₂²	C₂²×D₈	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	8	1	no	—
179	C₄	—	8	1	no	—
180	C₄	—	8	2	no	—
181	C₄	—	8	1	no	—
182	C₄	—	8	2	no	—
183	C₄	#532	1	1	yes	characteristic
184	C₄	C₂×#135	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	—	16	1	no	—
187	C₂	#3169	1	1	yes	characteristic
188	C₂	C₂×#510	1	1	yes	characteristic
189	C₂	#3414	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	#466	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ 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. #3170	C₂	semidirect
3. #2255	C₂	semidirect
4. #2251	C₂	semidirect
5. #2245	C₂	semidirect
6. #2243	C₂	semidirect
8. #2216	C₂	semidirect
9. #2176	C₂	semidirect
10. #2175	C₂	semidirect
11. C₂×#346	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3170	2
#3168	2
#2255	2
#2251	2
#2245	2
#2243	2
#2217	2
#2216	2
#2176	2
#2175	2
C₂×#346	2