Finite Group Databasebeta

Subgroups — #19423

Order 256.

315 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 315 conjugacy classes of subgroups — too many to draw. Only its 113 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	#3209	C₂	1	1	yes	maximal, characteristic, complemented
2	#3077	C₂	1	1	yes	maximal, characteristic
3	C₂×#474	C₂	1	1	yes	maximal, characteristic
4	#2975	C₂	1	1	yes	maximal, characteristic
5	#1703	C₂	1	4	yes	maximal, complemented
6	#1689	C₂	1	4	yes	maximal, complemented
7	#1570	C₂	1	1	yes	maximal, characteristic, complemented
8	#1549	C₂	1	1	yes	maximal, characteristic, complemented
9	#1542	C₂	1	1	yes	maximal, characteristic, complemented
10	#533	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₄	C₂²	1	1	yes	characteristic
13	#494	C₂²	1	2	yes	complemented
14	#487	C₂²	1	2	yes	complemented
15	#478	—	2	2	no	—
16	#476	C₂²	1	2	yes	—
17	#474	—	2	2	no	—
18	#474	C₂²	1	2	yes	—
19	#474	C₂²	1	4	yes	—
20	#462	—	2	2	no	—
21	#451	—	2	2	no	—
22	#436	—	2	2	no	—
23	#436	C₂²	1	4	yes	—
24	#431	C₂²	1	1	yes	characteristic
25	#431	C₂²	1	1	yes	characteristic
26	#428	—	2	1	no	—
27	C₂×#107	—	2	1	no	—
28	#423	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	#418	—	2	1	no	—
31	C₂×#102	—	2	1	no	—
32	#407	—	2	2	no	—
33	#334	C₂²	1	4	yes	complemented
34	#333	C₂²	1	2	yes	complemented
35	#332	C₂²	1	2	yes	complemented
36	#332	C₂²	1	4	yes	complemented
37	C₂²×Q₈	—	2	1	no	—
38	C₂²×D₄	D₄	1	1	yes	characteristic
39	C₂³×C₄	—	2	1	no	—
40	C₂×Q₁₆	—	2	4	no	—
41	C₂×M₁₆	—	2	1	no	—
42	C₂×M₁₆	—	2	4	no	—
43	C₂²×C₈	—	2	1	no	—
44	C₄⋊Q₈	—	2	1	no	—
45	C₄⋊Q₈	—	2	1	no	—
46	C₄⋊Q₈	—	2	1	no	—
47	C₄⋊Q₈	C₂³	1	2	yes	—
48	C₄⋊Q₈	C₂³	1	1	yes	characteristic
49	C₄⋊Q₈	C₂³	1	1	yes	characteristic
50	#125	—	2	1	no	—
51	C₂²⋊Q₈	—	2	1	no	—
52	C₂²⋊Q₈	—	2	1	no	—
53	C₄×Q₈	—	2	1	no	—
54	C₄×Q₈	—	2	1	no	—
55	C₄×D₄	—	2	1	no	—
56	C₄×D₄	—	2	1	no	—
57	C₄×D₄	D₄	1	2	yes	—
58	C₄×D₄	C₂³	1	2	yes	—
59	#117	—	2	1	no	—
60	C₂×C₄⋊C₄	—	2	1	no	—
61	C₂×C₄⋊C₄	—	2	1	no	—
62	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
63	C₂×C₂²⋊C₄	—	2	1	no	—
64	C₂×C₄²	C₂³	1	1	yes	characteristic
65	#107	—	2	4	no	—
66	#107	—	2	2	no	—
67	#106	—	2	2	no	—
68	C₄⋊C₈	—	2	1	no	—
69	C₄⋊C₈	—	2	1	no	—
70	C₄⋊C₈	D₄	1	2	yes	—
71	C₄⋊C₈	C₂³	1	2	yes	—
72	C₄⋊C₈	C₂³	1	2	yes	—
73	C₄⋊C₈	C₂³	1	2	yes	—
74	#103	—	2	4	no	—
75	#102	—	2	4	no	—
76	#102	—	2	2	no	—
77	#102	—	2	2	no	—
78	C₂²⋊C₈	—	2	1	no	—
79	C₂²⋊C₈	—	2	4	no	—
80	C₂²⋊C₈	D₄	1	2	yes	—
81	C₄×C₈	—	2	1	no	—
82	C₄×C₈	C₂³	1	2	yes	—
83	C₂⁴	—	2	1	no	—
84	C₂×Q₈	—	2	2	no	—
85	C₂×Q₈	—	2	1	no	—
86	C₂×Q₈	—	2	1	no	—
87	C₂×Q₈	—	4	1	no	—
88	C₂×Q₈	—	8	2	no	—
89	C₂×D₄	—	2	1	no	—
90	C₂×D₄	—	2	2	no	—
91	C₂×D₄	—	2	2	no	—
92	C₂×D₄	C₂×D₄	1	2	yes	—
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	—	4	1	no	—
97	C₂²×C₄	—	8	2	no	—
98	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄	#41	1	1	yes	characteristic
101	Q₁₆	—	8	4	no	—
102	M₁₆	—	4	2	no	—
103	M₁₆	—	8	4	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	4	2	no	—
108	C₂×C₈	C₂×D₄	1	4	yes	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	2	1	no	—
116	C₄⋊C₄	—	4	1	no	—
117	C₄⋊C₄	—	4	1	no	—
118	C₄⋊C₄	—	4	1	no	—
119	C₄⋊C₄	—	4	1	no	—
120	C₄⋊C₄	C₂×D₄	1	2	yes	—
121	C₂²⋊C₄	—	2	2	no	—
122	C₂²⋊C₄	—	2	2	no	—
123	C₂²⋊C₄	—	4	1	no	—
124	C₄²	—	4	1	no	—
125	C₄²	C₂×D₄	1	1	yes	characteristic
126	C₄²	C₂×D₄	1	1	yes	characteristic
127	C₄²	#41	1	1	yes	characteristic
128	C₄²	C₂⁴	1	1	yes	characteristic
129	C₂³	—	2	2	no	—
130	C₂³	—	4	2	no	—
131	C₂³	—	8	1	no	—
132	C₂³	—	8	1	no	—
133	C₂³	#121	1	1	yes	characteristic
134	Q₈	—	4	2	no	—
135	Q₈	—	4	4	no	—
136	Q₈	—	8	1	no	—
137	Q₈	—	8	2	no	—
138	D₄	—	2	2	no	—
139	D₄	—	2	2	no	—
140	D₄	—	4	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	—	4	4	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	8	2	no	—
149	C₂×C₄	—	8	1	no	—
150	C₂×C₄	—	8	2	no	—
151	C₂×C₄	#121	1	1	yes	characteristic
152	C₂×C₄	#121	1	1	yes	characteristic
153	C₂×C₄	#121	1	1	yes	characteristic
154	C₂×C₄	#137	1	2	yes	—
155	C₂×C₄	#137	1	2	yes	—
156	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄	C₂×#41	1	1	yes	characteristic
159	C₈	—	2	4	no	—
160	C₈	—	4	2	no	—
161	C₈	—	4	2	no	—
162	C₈	—	4	2	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	4	2	no	—
166	C₂²	—	4	2	no	—
167	C₂²	—	4	2	no	—
168	C₂²	—	8	4	no	—
169	C₂²	#470	1	2	yes	—
170	C₂²	C₂×#121	1	1	yes	characteristic
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	4	2	no	—
175	C₄	—	8	2	no	—
176	C₄	—	8	1	no	—
177	C₄	—	8	1	no	—
178	C₄	#470	1	2	yes	—
179	C₄	C₂×#137	1	1	yes	characteristic
180	C₄	C₂×#137	1	1	yes	characteristic
181	C₄	#576	1	2	yes	—
182	C₂	—	2	2	no	—
183	C₂	—	8	2	no	—
184	C₂	#3041	1	1	yes	characteristic
185	C₂	C₂×#470	1	1	yes	characteristic
186	C₂	#3069	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂²×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #3209	C₂	semidirect
5. #1703	C₂	semidirect
6. #1689	C₂	semidirect
7. #1570	C₂	semidirect
8. #1549	C₂	semidirect
9. #1542	C₂	semidirect
13. #494	C₂²	semidirect
14. #487	C₂²	semidirect
33. #334	C₂²	semidirect
34. #333	C₂²	semidirect
35. #332	C₂²	semidirect
36. #332	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#3209	2
#3077	2
C₂×#474	2
#2975	2
#1703	2
#1689	2
#1570	2
#1549	2
#1542	2