Finite Group Databasebeta

Subgroups — #19944

Order 256.

277 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 277 conjugacy classes of subgroups — too many to draw. Only its 103 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	#3395	C₂	1	1	yes	maximal, characteristic
2	#3383	C₂	1	1	yes	maximal, characteristic
3	#3374	C₂	1	2	yes	maximal
4	#1707	C₂	1	1	yes	maximal, characteristic
5	#1700	C₂	1	2	yes	maximal
6	#1674	C₂	1	1	yes	maximal, characteristic
7	#1670	C₂	1	1	yes	maximal, characteristic
8	#1653	C₂	1	2	yes	maximal
9	#1653	C₂	1	2	yes	maximal
10	#1591	C₂	1	1	yes	maximal, characteristic
11	C₈×Q₁₆	C₂	1	1	yes	maximal, characteristic
12	#557	C₂²	1	2	yes	—
13	#557	C₂²	1	2	yes	—
14	#500	C₂²	1	1	yes	characteristic
15	#499	—	2	1	no	—
16	#498	C₂²	1	2	yes	—
17	#494	—	2	1	no	—
18	#494	C₂²	1	1	yes	characteristic
19	#491	—	2	2	no	—
20	#487	C₂²	1	2	yes	—
21	#479	—	2	2	no	—
22	#479	—	2	2	no	—
23	#475	C₂²	1	2	yes	—
24	#475	C₂²	1	2	yes	—
25	#462	—	2	2	no	—
26	#462	—	2	2	no	—
27	#462	C₂²	1	2	yes	—
28	#462	C₂²	1	2	yes	—
29	#446	C₂²	1	2	yes	—
30	C₈×Q₈	C₂²	1	1	yes	characteristic
31	C₈×Q₈	C₂²	1	1	yes	characteristic
32	#441	—	2	2	no	—
33	C₄×Q₁₆	—	2	1	no	—
34	C₄×Q₁₆	—	2	1	no	—
35	C₄×Q₁₆	C₂²	1	1	yes	characteristic
36	C₄×Q₁₆	C₂²	1	1	yes	characteristic
37	#335	C₂²	1	1	yes	characteristic
38	#335	C₂²	1	1	yes	characteristic
39	#332	C₂²	1	2	yes	—
40	#332	C₂²	1	2	yes	—
41	Q₈⋊C₈	C₂²	1	1	yes	characteristic
42	Q₈⋊C₈	C₂²	1	1	yes	characteristic
43	Q₈⋊C₈	C₂²	1	2	yes	—
44	C₈²	C₂²	1	1	yes	characteristic
45	C₂×Q₁₆	—	2	2	no	—
46	C₂×Q₁₆	—	2	2	no	—
47	C₂×Q₁₆	—	2	1	no	—
48	C₂×Q₁₆	—	2	1	no	—
49	C₂×Q₁₆	D₄	1	1	yes	characteristic
50	C₂×Q₁₆	D₄	1	1	yes	characteristic
51	C₄⋊Q₈	—	2	2	no	—
52	C₄⋊Q₈	—	2	2	no	—
53	C₄⋊Q₈	C₂³	1	2	yes	—
54	C₄⋊Q₈	C₂³	1	2	yes	—
55	#125	—	2	2	no	—
56	#125	—	2	2	no	—
57	#125	—	4	2	no	—
58	#125	—	4	2	no	—
59	C₄×Q₈	—	2	2	no	—
60	C₄×Q₈	—	2	2	no	—
61	C₄×Q₈	—	2	2	no	—
62	C₄×Q₈	—	2	2	no	—
63	C₄×Q₈	C₂³	1	1	yes	characteristic
64	C₄×Q₈	C₂³	1	1	yes	characteristic
65	C₄×Q₈	C₂³	1	2	yes	—
66	#107	—	2	1	no	—
67	#107	—	2	1	no	—
68	#107	D₄	1	1	yes	characteristic
69	#107	D₄	1	1	yes	characteristic
70	#106	—	2	2	no	—
71	#106	—	2	2	no	—
72	#106	—	4	2	no	—
73	C₄⋊C₈	—	2	1	no	—
74	C₄⋊C₈	—	2	1	no	—
75	C₄⋊C₈	—	2	2	no	—
76	C₄⋊C₈	C₂³	1	1	yes	characteristic
77	C₄⋊C₈	C₂³	1	1	yes	characteristic
78	C₄⋊C₈	C₂³	1	2	yes	—
79	#103	—	2	2	no	—
80	#103	—	2	2	no	—
81	#103	—	2	2	no	—
82	#103	—	2	2	no	—
83	#103	—	2	2	no	—
84	#103	—	2	2	no	—
85	#103	D₄	1	1	yes	characteristic
86	#103	D₄	1	2	yes	—
87	#103	D₄	1	1	yes	characteristic
88	#97	—	2	2	no	—
89	C₄×C₈	—	2	1	no	—
90	C₄×C₈	—	2	1	no	—
91	C₄×C₈	C₂³	1	1	yes	characteristic
92	C₄×C₈	C₂³	1	1	yes	characteristic
93	C₄×C₈	C₂³	1	1	yes	characteristic
94	C₂×Q₈	—	2	2	no	—
95	C₂×Q₈	—	2	2	no	—
96	C₂×Q₈	C₂×D₄	1	1	yes	characteristic
97	C₂×Q₈	C₂×D₄	1	2	yes	—
98	C₂×Q₈	C₂×D₄	1	1	yes	characteristic
99	Q₁₆	—	2	2	no	—
100	Q₁₆	—	2	2	no	—
101	Q₁₆	—	4	4	no	—
102	Q₁₆	—	4	2	no	—
103	Q₁₆	—	8	1	no	—
104	Q₁₆	—	8	2	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈	C₂×D₄	1	1	yes	characteristic
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	4	2	no	—
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	—	4	2	no	—
124	C₄⋊C₄	—	4	2	no	—
125	C₄⋊C₄	—	4	2	no	—
126	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
127	C₄⋊C₄	C₂×D₄	1	2	yes	—
128	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
129	C₄²	—	2	1	no	—
130	C₄²	—	2	1	no	—
131	C₄²	—	2	2	no	—
132	C₄²	—	4	2	no	—
133	C₄²	—	4	2	no	—
134	C₄²	C₂⁴	1	1	yes	characteristic
135	Q₈	—	2	1	no	—
136	Q₈	—	2	1	no	—
137	Q₈	—	2	2	no	—
138	Q₈	—	2	2	no	—
139	Q₈	—	2	1	no	—
140	Q₈	—	4	2	no	—
141	Q₈	—	4	2	no	—
142	Q₈	—	4	2	no	—
143	Q₈	—	4	2	no	—
144	Q₈	#135	1	2	yes	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
156	C₈	—	2	2	no	—
157	C₈	—	2	1	no	—
158	C₈	—	2	1	no	—
159	C₈	—	2	1	no	—
160	C₈	—	4	2	no	—
161	C₈	—	4	1	no	—
162	C₈	—	4	1	no	—
163	C₈	#137	1	2	yes	—
164	C₂²	D₄²	1	1	yes	characteristic
165	C₄	—	2	2	no	—
166	C₄	—	2	1	no	—
167	C₄	—	4	2	no	—
168	C₄	—	4	1	no	—
169	C₄	—	4	1	no	—
170	C₄	—	4	2	no	—
171	C₄	—	4	1	no	—
172	C₄	—	8	2	no	—
173	C₄	—	8	2	no	—
174	C₄	C₂×#135	1	1	yes	characteristic
175	C₄	C₂×#137	1	1	yes	characteristic
176	C₄	#577	1	1	yes	characteristic
177	C₄	#578	1	1	yes	characteristic
178	C₂	#3278	1	1	yes	characteristic
179	C₂	#3285	1	1	yes	characteristic
180	C₂	#3293	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Maximal subgroups

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

subgroup	index
#3395	2
#3383	2
#3374	2
#1707	2
#1700	2
#1674	2
#1670	2
#1653	2
#1591	2
C₈×Q₁₆	2