Finite Group Databasebeta

Subgroups — #19241

Order 256.

286 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 121 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	#3249	C₂	1	1	yes	maximal, characteristic
2	#3230	C₂	1	1	yes	maximal, characteristic
3	#3165	C₂	1	1	yes	maximal, characteristic
4	#2962	C₂	1	1	yes	maximal, characteristic
5	#1696	C₂	1	2	yes	maximal, complemented
6	#1692	C₂	1	2	yes	maximal, complemented
7	#1676	C₂	1	2	yes	maximal, complemented
8	#1672	C₂	1	2	yes	maximal, complemented
9	#1565	C₂	1	1	yes	maximal, characteristic
10	#1542	C₂	1	1	yes	maximal, characteristic
11	C₂×#333	C₂	1	1	yes	maximal, characteristic
12	#533	C₂²	1	1	yes	characteristic
13	#533	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	#501	—	2	2	no	—
16	#500	—	2	1	no	—
17	#500	C₂²	1	2	yes	—
18	#498	C₂²	1	2	yes	—
19	#487	—	2	1	no	—
20	#484	—	2	2	no	—
21	#477	—	2	1	no	—
22	#477	C₂²	1	2	yes	—
23	#477	C₂²	1	2	yes	—
24	#477	C₂²	1	2	yes	—
25	#467	—	2	2	no	—
26	#462	—	2	1	no	—
27	#462	C₂²	1	2	yes	—
28	#446	—	2	2	no	—
29	#446	C₂²	1	2	yes	—
30	C₈×Q₈	C₂²	1	2	yes	—
31	#431	C₂²	1	1	yes	characteristic
32	#428	—	2	1	no	—
33	#428	—	2	1	no	—
34	#424	—	2	1	no	—
35	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
36	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	#405	—	2	1	no	—
38	C₄×M₁₆	C₂²	1	1	yes	characteristic
39	#335	C₂²	1	2	yes	—
40	#334	C₂²	1	2	yes	—
41	#333	C₂²	1	2	yes	—
42	#333	C₂²	1	2	yes	—
43	#333	C₂²	1	1	yes	characteristic
44	#333	C₂²	1	1	yes	characteristic
45	#332	C₂²	1	1	yes	characteristic
46	#332	C₂²	1	1	yes	characteristic
47	C₂²×Q₈	—	2	1	no	—
48	C₂×Q₁₆	—	2	2	no	—
49	C₂×M₁₆	D₄	1	1	yes	characteristic
50	C₂×M₁₆	D₄	1	1	yes	characteristic
51	C₂²×C₈	—	2	1	no	—
52	C₂²×C₈	—	2	1	no	—
53	C₄⋊Q₈	—	2	1	no	—
54	C₄⋊Q₈	—	2	1	no	—
55	C₄⋊Q₈	—	2	1	no	—
56	C₄⋊Q₈	D₄	1	1	yes	characteristic
57	C₄⋊Q₈	D₄	1	1	yes	characteristic
58	C₄⋊Q₈	C₂³	1	2	yes	—
59	C₄⋊Q₈	C₂³	1	1	yes	characteristic
60	C₄⋊Q₈	C₂³	1	1	yes	characteristic
61	#125	—	2	1	no	—
62	#125	—	2	1	no	—
63	C₂²⋊Q₈	—	2	1	no	—
64	C₂²⋊Q₈	—	2	1	no	—
65	C₂²⋊Q₈	—	2	1	no	—
66	C₂²⋊Q₈	D₄	1	2	yes	—
67	C₄×Q₈	—	2	1	no	—
68	C₄×Q₈	—	2	1	no	—
69	C₄×Q₈	—	2	1	no	—
70	C₄×Q₈	C₂³	1	2	yes	—
71	#117	—	2	1	no	—
72	#117	—	2	1	no	—
73	C₂×C₄⋊C₄	—	2	1	no	—
74	C₂×C₄⋊C₄	—	2	1	no	—
75	C₂×C₄²	C₂³	1	1	yes	characteristic
76	#107	—	2	2	no	—
77	#107	—	2	2	no	—
78	#107	—	2	2	no	—
79	#107	—	2	2	no	—
80	#107	—	2	2	no	—
81	#106	—	2	2	no	—
82	#106	—	2	2	no	—
83	C₄⋊C₈	—	2	2	no	—
84	C₄⋊C₈	—	2	2	no	—
85	C₄⋊C₈	—	2	1	no	—
86	C₄⋊C₈	—	2	1	no	—
87	C₄⋊C₈	C₂³	1	1	yes	characteristic
88	C₄⋊C₈	C₂³	1	1	yes	characteristic
89	C₄⋊C₈	C₂³	1	1	yes	characteristic
90	C₄⋊C₈	C₂³	1	1	yes	characteristic
91	C₄⋊C₈	C₂³	1	2	yes	—
92	#103	—	2	2	no	—
93	#103	—	2	2	no	—
94	#103	—	2	2	no	—
95	#103	—	2	2	no	—
96	C₂²⋊C₈	—	2	1	no	—
97	#97	—	2	2	no	—
98	#97	D₄	1	2	yes	—
99	C₄×C₈	—	2	2	no	—
100	C₄×C₈	—	2	1	no	—
101	C₄×C₈	C₂³	1	2	yes	—
102	C₂×Q₈	—	2	1	no	—
103	C₂×Q₈	—	2	1	no	—
104	C₂×Q₈	—	2	1	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	4	1	no	—
107	C₂×Q₈	—	8	2	no	—
108	C₂×Q₈	C₂×D₄	1	2	yes	—
109	C₂²×C₄	—	4	1	no	—
110	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄	#41	1	1	yes	characteristic
113	Q₁₆	—	8	2	no	—
114	M₁₆	—	2	2	no	—
115	M₁₆	Q₁₆	1	4	yes	complemented
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	—	8	1	no	—
123	C₂×C₈	C₂×D₄	1	2	yes	—
124	C₂×C₈	C₂×D₄	1	2	yes	—
125	C₄⋊C₄	—	2	1	no	—
126	C₄⋊C₄	—	2	1	no	—
127	C₄⋊C₄	—	2	1	no	—
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	—	2	2	no	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	—	2	2	no	—
134	C₄⋊C₄	—	2	1	no	—
135	C₄⋊C₄	—	2	1	no	—
136	C₄⋊C₄	—	4	1	no	—
137	C₄⋊C₄	—	4	1	no	—
138	C₄⋊C₄	—	4	1	no	—
139	C₄⋊C₄	—	4	1	no	—
140	C₄⋊C₄	C₂×D₄	1	2	yes	—
141	C₂²⋊C₄	—	2	2	no	—
142	C₂²⋊C₄	—	4	1	no	—
143	C₄²	—	2	2	no	—
144	C₄²	—	4	1	no	—
145	C₄²	C₂×D₄	1	1	yes	characteristic
146	C₄²	C₂×D₄	1	1	yes	characteristic
147	C₄²	#41	1	1	yes	characteristic
148	C₄²	C₂⁴	1	1	yes	characteristic
149	C₂³	#121	1	1	yes	characteristic
150	Q₈	—	2	2	no	—
151	Q₈	—	2	2	no	—
152	Q₈	—	4	1	no	—
153	Q₈	—	4	1	no	—
154	Q₈	—	8	1	no	—
155	Q₈	—	8	1	no	—
156	Q₈	—	8	2	no	—
157	Q₈	—	8	2	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	4	1	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	4	1	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	8	2	no	—
167	C₂×C₄	#121	1	1	yes	characteristic
168	C₂×C₄	#121	1	1	yes	characteristic
169	C₂×C₄	#121	1	1	yes	characteristic
170	C₂×C₄	C₂×Q₁₆	1	2	yes	—
171	C₂×C₄	#137	1	2	yes	—
172	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
174	C₂×C₄	C₂×#41	1	1	yes	characteristic
175	C₈	—	2	2	no	—
176	C₈	—	4	1	no	—
177	C₈	—	4	1	no	—
178	C₈	—	4	1	no	—
179	C₈	—	4	2	no	—
180	C₈	—	4	1	no	—
181	C₈	C₂×Q₁₆	1	4	yes	—
182	C₂²	—	2	1	no	—
183	C₂²	—	2	1	no	—
184	C₂²	#462	1	2	yes	—
185	C₂²	C₂×#121	1	1	yes	characteristic
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	—	2	1	no	—
189	C₄	—	4	2	no	—
190	C₄	—	4	2	no	—
191	C₄	—	8	1	no	—
192	C₄	—	8	2	no	—
193	C₄	—	8	1	no	—
194	C₄	#462	1	2	yes	—
195	C₄	C₂²×Q₁₆	1	1	yes	characteristic
196	C₄	C₂×#137	1	1	yes	characteristic
197	C₄	#577	1	1	yes	characteristic
198	C₄	#578	1	1	yes	characteristic
199	C₂	—	2	2	no	—
200	C₂	C₂×#462	1	1	yes	characteristic
201	C₂	#3063	1	1	yes	characteristic
202	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. #1696	C₂	semidirect
6. #1692	C₂	semidirect
7. #1676	C₂	semidirect
8. #1672	C₂	semidirect
115. M₁₆	Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
#3249	2
#3230	2
#3165	2
#2962	2
#1696	2
#1692	2
#1676	2
#1672	2
#1565	2
#1542	2
C₂×#333	2