Finite Group Databasebeta

Subgroups — #19207

Order 256.

297 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 297 conjugacy classes of subgroups — too many to draw. Only its 123 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	#3209	C₂	1	1	yes	maximal, characteristic
3	C₂×#498	C₂	1	1	yes	maximal, characteristic
4	#2960	C₂	1	1	yes	maximal, characteristic
5	#1692	C₂	1	4	yes	maximal, complemented
6	#1672	C₂	1	4	yes	maximal, complemented
7	#1564	C₂	1	1	yes	maximal, characteristic
8	#1549	C₂	1	1	yes	maximal, characteristic
9	#1548	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	—	2	1	no	—
14	#498	—	2	1	no	—
15	#498	—	2	1	no	—
16	#498	C₂²	1	2	yes	—
17	#498	C₂²	1	2	yes	—
18	#494	—	2	1	no	—
19	#487	—	2	1	no	—
20	#487	—	2	1	no	—
21	#484	—	2	2	no	—
22	#477	C₂²	1	4	yes	—
23	#462	C₂²	1	4	yes	—
24	#451	—	2	2	no	—
25	#446	—	2	1	no	—
26	#446	C₂²	1	2	yes	—
27	C₈×Q₈	—	2	1	no	—
28	C₈×Q₈	C₂²	1	2	yes	—
29	#431	C₂²	1	1	yes	characteristic
30	#431	C₂²	1	1	yes	characteristic
31	C₂×#106	—	2	1	no	—
32	C₂×#106	—	2	1	no	—
33	#424	—	2	1	no	—
34	#423	C₂²	1	1	yes	characteristic
35	#405	—	2	1	no	—
36	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
37	#335	C₂²	1	2	yes	—
38	#334	C₂²	1	2	yes	—
39	#333	C₂²	1	4	yes	—
40	#332	C₂²	1	4	yes	—
41	C₂²×Q₈	—	2	1	no	—
42	C₂²×Q₈	—	2	1	no	—
43	C₂×Q₁₆	—	2	4	no	—
44	C₂×M₁₆	—	2	1	no	—
45	C₂²×C₈	D₄	1	1	yes	characteristic
46	C₂²×C₈	D₄	1	1	yes	characteristic
47	C₄⋊Q₈	—	2	1	no	—
48	C₄⋊Q₈	—	2	1	no	—
49	C₄⋊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	D₄	1	1	yes	characteristic
54	C₄⋊Q₈	D₄	1	1	yes	characteristic
55	C₄⋊Q₈	C₂³	1	2	yes	—
56	C₄⋊Q₈	C₂³	1	2	yes	—
57	#125	—	2	1	no	—
58	C₂²⋊Q₈	—	2	1	no	—
59	C₂²⋊Q₈	D₄	1	2	yes	—
60	C₄×Q₈	—	2	1	no	—
61	C₄×Q₈	C₂³	1	2	yes	—
62	#117	—	2	1	no	—
63	C₂×C₄⋊C₄	—	2	1	no	—
64	C₂×C₄⋊C₄	—	2	1	no	—
65	C₂×C₄⋊C₄	—	2	1	no	—
66	C₂×C₄⋊C₄	—	2	1	no	—
67	C₂×C₄²	C₂³	1	1	yes	characteristic
68	#107	—	2	4	no	—
69	#106	—	2	2	no	—
70	#106	—	2	2	no	—
71	#106	—	2	2	no	—
72	#106	—	2	2	no	—
73	C₄⋊C₈	—	2	2	no	—
74	C₄⋊C₈	—	2	2	no	—
75	C₄⋊C₈	—	2	1	no	—
76	C₄⋊C₈	C₂³	1	2	yes	—
77	C₄⋊C₈	C₂³	1	2	yes	—
78	C₄⋊C₈	C₂³	1	2	yes	—
79	#103	—	2	4	no	—
80	#103	—	2	4	no	—
81	C₂²⋊C₈	—	2	1	no	—
82	C₂²⋊C₈	—	2	1	no	—
83	#97	—	2	2	no	—
84	C₄×C₈	—	2	2	no	—
85	C₄×C₈	—	2	1	no	—
86	C₄×C₈	—	2	1	no	—
87	C₄×C₈	D₄	1	1	yes	characteristic
88	C₄×C₈	D₄	1	1	yes	characteristic
89	C₄×C₈	C₂³	1	1	yes	characteristic
90	C₄×C₈	C₂³	1	1	yes	characteristic
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	—	4	1	no	—
95	C₂×Q₈	—	4	1	no	—
96	C₂×Q₈	—	8	2	no	—
97	C₂×Q₈	—	8	2	no	—
98	C₂×Q₈	C₂×D₄	1	2	yes	—
99	C₂²×C₄	—	4	1	no	—
100	C₂²×C₄	—	4	1	no	—
101	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄	#41	1	1	yes	characteristic
104	Q₁₆	—	8	4	no	—
105	M₁₆	—	4	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	Q₁₆	1	4	yes	complemented
111	C₂×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈	C₂×D₄	1	1	yes	characteristic
115	C₄⋊C₄	—	2	1	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	—	4	1	no	—
125	C₄⋊C₄	—	4	1	no	—
126	C₄⋊C₄	—	4	1	no	—
127	C₄⋊C₄	—	4	1	no	—
128	C₄⋊C₄	C₂×D₄	1	2	yes	—
129	C₂²⋊C₄	—	2	2	no	—
130	C₄²	—	2	2	no	—
131	C₄²	C₂×D₄	1	1	yes	characteristic
132	C₄²	C₂×D₄	1	1	yes	characteristic
133	C₄²	#41	1	1	yes	characteristic
134	C₄²	C₂⁴	1	1	yes	characteristic
135	C₂³	#121	1	1	yes	characteristic
136	Q₈	—	2	4	no	—
137	Q₈	—	4	2	no	—
138	Q₈	—	4	4	no	—
139	Q₈	—	8	2	no	—
140	Q₈	—	8	2	no	—
141	Q₈	—	8	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	8	2	no	—
149	C₂×C₄	—	8	2	no	—
150	C₂×C₄	#121	1	1	yes	characteristic
151	C₂×C₄	#121	1	1	yes	characteristic
152	C₂×C₄	#121	1	1	yes	characteristic
153	C₂×C₄	C₂×Q₁₆	1	2	yes	—
154	C₂×C₄	#137	1	2	yes	—
155	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄	C₂×#41	1	1	yes	characteristic
158	C₈	—	2	1	no	—
159	C₈	—	2	1	no	—
160	C₈	—	4	2	no	—
161	C₈	—	4	2	no	—
162	C₈	—	4	2	no	—
163	C₈	C₂×Q₁₆	1	2	yes	—
164	C₈	C₂×Q₁₆	1	2	yes	—
165	C₂²	—	2	1	no	—
166	C₂²	—	2	1	no	—
167	C₂²	#462	1	2	yes	—
168	C₂²	C₂×#121	1	1	yes	characteristic
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	4	2	no	—
172	C₄	—	4	2	no	—
173	C₄	—	8	2	no	—
174	C₄	—	8	2	no	—
175	C₄	#462	1	2	yes	—
176	C₄	#471	1	2	yes	—
177	C₄	C₂²×Q₁₆	1	1	yes	characteristic
178	C₄	C₂×#137	1	1	yes	characteristic
179	C₄	#577	1	2	yes	—
180	C₂	—	2	2	no	—
181	C₂	C₂×#462	1	1	yes	characteristic
182	C₂	#3057	1	1	yes	characteristic
183	C₂	C₂×#471	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. #1692	C₂	semidirect
6. #1672	C₂	semidirect
110. C₂×C₈	Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
#3230	2
#3209	2
C₂×#498	2
#2960	2
#1692	2
#1672	2
#1564	2
#1549	2
#1548	2