Finite Group Databasebeta

Subgroups — #19441

Order 256.

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

normal subgroups only

This group has 285 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	#3230	C₂	1	1	yes	maximal, characteristic
2	#3078	C₂	1	1	yes	maximal, characteristic
3	C₂×#477	C₂	1	1	yes	maximal, characteristic
4	#2983	C₂	1	1	yes	maximal, characteristic
5	#1704	C₂	1	4	yes	maximal, complemented
6	#1696	C₂	1	4	yes	maximal, complemented
7	#1570	C₂	1	1	yes	maximal, characteristic
8	#1548	C₂	1	1	yes	maximal, characteristic
9	#1542	C₂	1	1	yes	maximal, characteristic
10	#533	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
13	#500	C₂²	1	2	yes	—
14	#487	C₂²	1	2	yes	—
15	#484	—	2	2	no	—
16	#479	—	2	2	no	—
17	#477	—	2	2	no	—
18	#477	—	2	2	no	—
19	#477	C₂²	1	4	yes	—
20	#477	C₂²	1	2	yes	—
21	#475	C₂²	1	2	yes	—
22	#446	—	2	2	no	—
23	#446	C₂²	1	4	yes	—
24	#432	—	2	2	no	—
25	#431	C₂²	1	1	yes	characteristic
26	#431	C₂²	1	1	yes	characteristic
27	#428	—	2	1	no	—
28	C₂×#107	—	2	1	no	—
29	#423	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	#419	—	2	1	no	—
32	C₂×#103	—	2	1	no	—
33	#334	C₂²	1	4	yes	—
34	#333	C₂²	1	2	yes	—
35	#333	C₂²	1	4	yes	—
36	#332	C₂²	1	2	yes	—
37	C₂²×Q₈	—	2	1	no	—
38	C₂²×Q₈	D₄	1	1	yes	characteristic
39	C₂×M₁₆	—	2	1	no	—
40	C₂²×C₈	—	2	1	no	—
41	C₄⋊Q₈	—	2	1	no	—
42	C₄⋊Q₈	—	2	1	no	—
43	C₄⋊Q₈	—	2	1	no	—
44	C₄⋊Q₈	C₂³	1	2	yes	—
45	C₄⋊Q₈	C₂³	1	1	yes	characteristic
46	C₄⋊Q₈	C₂³	1	1	yes	characteristic
47	#125	—	2	1	no	—
48	C₂²⋊Q₈	—	2	1	no	—
49	C₂²⋊Q₈	—	2	1	no	—
50	C₄×Q₈	—	2	1	no	—
51	C₄×Q₈	—	2	1	no	—
52	C₄×Q₈	—	2	1	no	—
53	C₄×Q₈	—	2	1	no	—
54	C₄×Q₈	D₄	1	2	yes	—
55	C₄×Q₈	C₂³	1	2	yes	—
56	#117	—	2	1	no	—
57	C₂×C₄⋊C₄	—	2	1	no	—
58	C₂×C₄⋊C₄	—	2	1	no	—
59	C₂×C₄⋊C₄	—	2	1	no	—
60	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
61	C₂×C₄²	—	2	1	no	—
62	C₂×C₄²	C₂³	1	1	yes	characteristic
63	#107	—	2	4	no	—
64	#107	—	2	4	no	—
65	#107	—	2	2	no	—
66	#106	—	2	2	no	—
67	C₄⋊C₈	—	2	4	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	#103	—	2	2	no	—
76	#103	—	2	2	no	—
77	#103	—	2	4	no	—
78	C₂²⋊C₈	—	2	1	no	—
79	C₂²⋊C₈	D₄	1	2	yes	—
80	#97	—	2	4	no	—
81	C₄×C₈	—	2	1	no	—
82	C₄×C₈	C₂³	1	2	yes	—
83	C₂×Q₈	—	2	1	no	—
84	C₂×Q₈	—	2	2	no	—
85	C₂×Q₈	—	2	2	no	—
86	C₂×Q₈	—	2	1	no	—
87	C₂×Q₈	—	2	1	no	—
88	C₂×Q₈	—	2	2	no	—
89	C₂×Q₈	—	4	1	no	—
90	C₂×Q₈	—	8	2	no	—
91	C₂×Q₈	C₂×D₄	1	2	yes	—
92	C₂²×C₄	—	2	1	no	—
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	—	4	1	no	—
95	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄	#41	1	1	yes	characteristic
98	M₁₆	—	4	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	4	2	no	—
103	C₂×C₈	C₂×D₄	1	4	yes	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	1	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	C₂×D₄	1	2	yes	—
118	C₂²⋊C₄	—	4	1	no	—
119	C₄²	—	2	2	no	—
120	C₄²	—	2	2	no	—
121	C₄²	—	4	1	no	—
122	C₄²	C₂×D₄	1	1	yes	characteristic
123	C₄²	C₂×D₄	1	1	yes	characteristic
124	C₄²	#41	1	1	yes	characteristic
125	C₄²	C₂⁴	1	1	yes	characteristic
126	C₂³	#121	1	1	yes	characteristic
127	Q₈	—	2	2	no	—
128	Q₈	—	2	2	no	—
129	Q₈	—	4	2	no	—
130	Q₈	—	4	2	no	—
131	Q₈	—	8	1	no	—
132	Q₈	—	8	2	no	—
133	Q₈	—	8	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	8	1	no	—
143	C₂×C₄	—	8	2	no	—
144	C₂×C₄	#121	1	1	yes	characteristic
145	C₂×C₄	#121	1	1	yes	characteristic
146	C₂×C₄	#121	1	1	yes	characteristic
147	C₂×C₄	#137	1	2	yes	—
148	C₂×C₄	#137	1	2	yes	—
149	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄	C₂×#41	1	1	yes	characteristic
152	C₈	—	2	4	no	—
153	C₈	—	4	2	no	—
154	C₈	—	4	2	no	—
155	C₈	—	4	2	no	—
156	C₂²	—	2	1	no	—
157	C₂²	—	2	1	no	—
158	C₂²	#470	1	2	yes	—
159	C₂²	C₂×#121	1	1	yes	characteristic
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	—	4	2	no	—
164	C₄	—	4	2	no	—
165	C₄	—	8	1	no	—
166	C₄	—	8	2	no	—
167	C₄	—	8	1	no	—
168	C₄	#470	1	2	yes	—
169	C₄	C₂×#137	1	1	yes	characteristic
170	C₄	C₂×#137	1	1	yes	characteristic
171	C₄	#578	1	2	yes	—
172	C₂	—	2	2	no	—
173	C₂	#3044	1	1	yes	characteristic
174	C₂	C₂×#470	1	1	yes	characteristic
175	C₂	#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂³	#121
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
5. #1704	C₂	semidirect
6. #1696	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3230	2
#3078	2
C₂×#477	2
#2983	2
#1704	2
#1696	2
#1570	2
#1548	2
#1542	2