Finite Group Databasebeta

Subgroups — #33703

Order 256.

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

normal subgroups only

This group has 285 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	#3156	C₂	1	1	yes	maximal, characteristic
3	#2245	C₂	1	2	yes	maximal, complemented
4	#2243	C₂	1	2	yes	maximal, complemented
5	#2242	C₂	1	2	yes	maximal, complemented
6	#2241	C₂	1	2	yes	maximal, complemented
7	#2141	C₂	1	1	yes	maximal, characteristic, complemented
8	#2139	C₂	1	1	yes	maximal, characteristic
9	#2138	C₂	1	1	yes	maximal, characteristic
10	C₂×#358	C₂	1	1	yes	maximal, characteristic, complemented
11	#2108	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#137	—	2	1	no	—
13	C₂×#135	C₂²	1	1	yes	characteristic
14	C₂²×Q₁₆	C₂²	1	1	yes	characteristic
15	#544	—	2	1	no	—
16	#536	—	2	1	no	—
17	C₂×M₃₂	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆	C₂²	1	1	yes	characteristic
19	#501	—	2	2	no	—
20	#500	C₂²	1	2	yes	—
21	#499	C₂²	1	2	yes	—
22	#497	—	2	2	no	—
23	#495	C₂²	1	2	yes	—
24	#494	C₂²	1	2	yes	—
25	#428	—	2	1	no	—
26	#427	C₂²	1	1	yes	characteristic
27	C₂×#107	C₂²	1	1	yes	characteristic
28	#405	C₂²	1	1	yes	characteristic
29	#358	C₂²	1	2	yes	—
30	#358	C₂²	1	2	yes	—
31	#358	C₂²	1	1	yes	characteristic
32	#358	C₂²	1	1	yes	characteristic
33	#358	C₂²	1	2	yes	—
34	#358	C₂²	1	2	yes	—
35	#358	C₂²	1	2	yes	—
36	#357	C₂²	1	1	yes	characteristic
37	#357	C₂²	1	1	yes	characteristic
38	#357	C₂²	1	2	yes	—
39	#346	C₂²	1	2	yes	—
40	C₄×C₁₆	C₂²	1	2	yes	—
41	C₂×#41	—	2	1	no	—
42	C₂²×Q₈	—	2	1	no	—
43	#137	—	4	4	no	—
44	#135	—	2	2	no	—
45	#135	—	2	2	no	—
46	C₂×Q₁₆	—	2	1	no	—
47	C₂×Q₁₆	—	2	2	no	—
48	C₂×Q₁₆	—	2	2	no	—
49	C₂×Q₁₆	—	2	2	no	—
50	C₂×Q₁₆	C₂³	1	2	yes	—
51	C₂×Q₁₆	C₂³	1	1	yes	characteristic
52	C₂×SD₁₆	—	2	1	no	—
53	C₂×SD₁₆	—	2	2	no	—
54	C₂×D₈	C₂³	1	1	yes	characteristic
55	C₂×M₁₆	D₄	1	1	yes	characteristic
56	C₂²×C₈	C₂³	1	1	yes	characteristic
57	C₄⋊Q₈	—	2	2	no	—
58	C₄⋊Q₈	—	2	2	no	—
59	C₄⋊Q₈	—	4	1	no	—
60	#125	—	2	2	no	—
61	#125	—	4	1	no	—
62	#124	—	2	2	no	—
63	#123	—	4	2	no	—
64	C₂²⋊Q₈	—	4	1	no	—
65	C₂²⋊Q₈	—	4	1	no	—
66	C₂²⋊Q₈	—	4	2	no	—
67	#117	—	2	1	no	—
68	#117	D₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄	—	2	1	no	—
70	M₃₂	—	2	2	no	—
71	C₂×C₁₆	—	2	2	no	—
72	C₂×C₁₆	C₂³	1	1	yes	characteristic
73	C₂×C₁₆	C₂³	1	2	yes	—
74	C₂×C₁₆	C₂³	1	1	yes	characteristic
75	#107	—	2	1	no	—
76	#107	—	2	2	no	—
77	#107	C₂³	1	2	yes	—
78	#107	C₂³	1	1	yes	characteristic
79	#107	C₂³	1	1	yes	characteristic
80	#106	—	2	1	no	—
81	#106	—	2	2	no	—
82	#97	D₄	1	2	yes	—
83	C₄×C₈	C₂³	1	2	yes	—
84	#41	—	4	2	no	—
85	#41	—	4	2	no	—
86	C₂×Q₈	—	2	2	no	—
87	C₂×Q₈	—	2	1	no	—
88	C₂×Q₈	—	4	1	no	—
89	C₂×Q₈	—	4	2	no	—
90	C₂×Q₈	—	4	2	no	—
91	C₂×Q₈	—	4	2	no	—
92	C₂×D₄	—	2	1	no	—
93	C₂×D₄	—	4	1	no	—
94	C₂²×C₄	—	4	1	no	—
95	C₂²×C₄	—	4	1	no	—
96	C₂²×C₄	—	4	1	no	—
97	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
98	Q₁₆	—	2	2	no	—
99	Q₁₆	—	2	2	no	—
100	Q₁₆	—	2	1	no	—
101	Q₁₆	—	2	1	no	—
102	Q₁₆	—	4	2	no	—
103	Q₁₆	—	4	4	no	—
104	SD₁₆	—	4	2	no	—
105	SD₁₆	—	4	4	no	—
106	D₈	—	2	1	no	—
107	D₈	—	2	1	no	—
108	M₁₆	SD₁₆	1	4	yes	—
109	C₂×C₈	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈	C₂×D₄	1	2	yes	—
111	C₂×C₈	#41	1	2	yes	—
112	C₂×C₈	#41	1	2	yes	—
113	C₂×C₈	C₂⁴	1	1	yes	characteristic
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	4	1	no	—
118	C₄⋊C₄	—	4	2	no	—
119	C₄⋊C₄	—	4	2	no	—
120	C₄⋊C₄	—	4	2	no	—
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	D₈	1	1	yes	characteristic
124	C₄⋊C₄	D₈	1	1	yes	characteristic
125	C₂²⋊C₄	—	4	1	no	—
126	C₂²⋊C₄	—	4	2	no	—
127	C₂²⋊C₄	—	4	2	no	—
128	C₂²⋊C₄	D₈	1	2	yes	—
129	C₄²	—	4	1	no	—
130	C₄²	C₂×D₄	1	2	yes	—
131	C₁₆	—	2	1	no	—
132	C₁₆	—	2	1	no	—
133	C₁₆	—	2	2	no	—
134	C₂³	—	4	1	no	—
135	C₂³	C₂×D₈	1	1	yes	characteristic
136	Q₈	—	4	1	no	—
137	Q₈	—	4	2	no	—
138	Q₈	—	4	2	no	—
139	Q₈	—	4	1	no	—
140	Q₈	—	8	2	no	—
141	Q₈	—	16	2	no	—
142	D₄	—	4	1	no	—
143	D₄	—	4	1	no	—
144	D₄	—	8	2	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	8	2	no	—
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	—	16	1	no	—
153	C₂×C₄	#124	1	2	yes	—
154	C₂×C₄	C₂×D₈	1	2	yes	—
155	C₂×C₄	C₂×D₈	1	2	yes	—
156	C₂×C₄	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄	C₂×SD₁₆	1	2	yes	—
158	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
159	C₈	#124	1	2	yes	—
160	C₈	C₂×SD₁₆	1	4	yes	—
161	C₈	C₂×#41	1	1	yes	characteristic
162	C₈	C₂×#41	1	1	yes	characteristic
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	8	1	no	—
166	C₂²	—	8	1	no	—
167	C₂²	—	8	1	no	—
168	C₂²	#486	1	2	yes	—
169	C₂²	C₂²×D₈	1	1	yes	characteristic
170	C₄	—	2	2	no	—
171	C₄	—	2	2	no	—
172	C₄	—	8	1	no	—
173	C₄	—	8	2	no	—
174	C₄	—	8	2	no	—
175	C₄	—	8	1	no	—
176	C₄	—	8	1	no	—
177	C₄	#486	1	2	yes	—
178	C₄	C₂×#124	1	1	yes	characteristic
179	C₄	C₂²×SD₁₆	1	1	yes	characteristic
180	C₂	—	2	2	no	—
181	C₂	—	16	1	no	—
182	C₂	C₂×#486	1	1	yes	characteristic
183	C₂	#3416	1	1	yes	characteristic
184	C₂	#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂×#135	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. #2245	C₂	semidirect
4. #2243	C₂	semidirect
5. #2242	C₂	semidirect
6. #2241	C₂	semidirect
7. #2141	C₂	semidirect
10. C₂×#358	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
#3164	2
#3156	2
#2245	2
#2243	2
#2242	2
#2241	2
#2141	2
#2139	2
#2138	2
C₂×#358	2
#2108	2