Finite Group Databasebeta

Subgroups — #9352

Order 256.

207 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 207 conjugacy classes of subgroups — too many to draw. Only its 61 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	#3080	C₂	1	1	yes	maximal, characteristic
2	#2971	C₂	1	1	yes	maximal, characteristic
3	#1528	C₂	1	1	yes	maximal, characteristic, complemented
4	#1488	C₂	1	1	yes	maximal, characteristic
5	#1470	C₂	1	1	yes	maximal, characteristic, complemented
6	#1307	C₂	1	1	yes	maximal, characteristic, complemented
7	#1299	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#125	C₂²	1	1	yes	characteristic
9	C₄×#41	C₂²	1	1	yes	characteristic
10	#479	—	2	1	no	—
11	#479	—	2	1	no	—
12	#478	—	2	1	no	—
13	#478	—	2	1	no	—
14	C₈×Q₈	—	2	1	no	—
15	#436	—	2	1	no	—
16	#436	—	2	2	no	—
17	#432	—	2	1	no	—
18	#432	C₂²	1	1	yes	characteristic
19	#432	C₂²	1	1	yes	characteristic
20	#431	C₂²	1	1	yes	characteristic
21	#428	C₄	1	1	yes	characteristic
22	#428	C₄	1	1	yes	characteristic
23	#423	C₂²	1	1	yes	characteristic
24	#417	C₄	1	1	yes	characteristic
25	#417	C₄	1	1	yes	characteristic
26	C₄×M₁₆	—	2	1	no	—
27	C₄×M₁₆	C₂²	1	1	yes	characteristic
28	#330	—	2	1	no	—
29	#330	—	2	1	no	—
30	Q₈⋊C₈	—	2	1	no	—
31	D₄⋊C₈	—	2	1	no	—
32	#324	—	2	2	no	—
33	C₂×#41	C₂×C₄	1	1	yes	characteristic
34	C₂×M₁₆	—	2	2	no	—
35	C₂×M₁₆	—	2	1	no	—
36	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
37	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
38	#125	—	2	1	no	—
39	#125	—	2	1	no	—
40	#125	—	2	1	no	—
41	#125	—	2	1	no	—
42	C₄×Q₈	D₄	1	1	yes	characteristic
43	C₄×Q₈	D₄	1	1	yes	characteristic
44	C₄×D₄	—	2	1	no	—
45	C₄×D₄	—	2	1	no	—
46	C₄×D₄	D₄	1	1	yes	characteristic
47	C₄×D₄	D₄	1	1	yes	characteristic
48	#117	—	2	1	no	—
49	#117	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄	—	2	1	no	—
51	C₂×C₄⋊C₄	—	2	1	no	—
52	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²	—	2	1	no	—
55	C₂×C₄²	C₂³	1	1	yes	characteristic
56	#107	—	2	1	no	—
57	#107	—	2	1	no	—
58	#106	—	2	1	no	—
59	#106	—	2	1	no	—
60	C₄⋊C₈	—	2	2	no	—
61	C₄⋊C₈	—	2	1	no	—
62	C₄⋊C₈	—	2	1	no	—
63	C₄⋊C₈	—	2	1	no	—
64	C₄⋊C₈	—	2	1	no	—
65	C₄⋊C₈	—	2	1	no	—
66	#103	—	2	1	no	—
67	#103	—	2	1	no	—
68	#102	—	2	1	no	—
69	#102	—	2	1	no	—
70	C₂²⋊C₈	—	2	2	no	—
71	C₂²⋊C₈	—	2	1	no	—
72	C₂²⋊C₈	—	2	1	no	—
73	C₂²⋊C₈	D₄	1	2	yes	—
74	#97	—	2	1	no	—
75	#97	—	2	1	no	—
76	#97	D₄	1	1	yes	characteristic
77	#97	D₄	1	1	yes	characteristic
78	C₄×C₈	—	2	2	no	—
79	C₄×C₈	—	2	1	no	—
80	C₄×C₈	—	2	1	no	—
81	#41	—	2	2	no	—
82	#41	—	4	1	no	—
83	C₂×Q₈	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×D₄	—	2	1	no	—
85	C₂×D₄	C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×C₄	—	2	1	no	—
87	C₂²×C₄	—	2	1	no	—
88	C₂²×C₄	—	2	1	no	—
89	C₂²×C₄	—	2	1	no	—
90	C₂²×C₄	—	2	1	no	—
91	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄	#41	1	1	yes	characteristic
94	M₁₆	—	4	2	no	—
95	M₁₆	—	4	4	no	—
96	M₁₆	—	4	1	no	—
97	M₁₆	—	4	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	4	1	no	—
104	C₂×C₈	—	4	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	4	1	no	—
112	C₄⋊C₄	—	4	1	no	—
113	C₄⋊C₄	—	4	1	no	—
114	C₄⋊C₄	—	4	1	no	—
115	C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
116	C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
117	C₂²⋊C₄	—	2	1	no	—
118	C₂²⋊C₄	—	2	1	no	—
119	C₂²⋊C₄	—	2	1	no	—
120	C₄²	—	2	1	no	—
121	C₄²	—	2	1	no	—
122	C₄²	—	2	1	no	—
123	C₄²	C₂×D₄	1	1	yes	characteristic
124	C₄²	C₂×D₄	1	1	yes	characteristic
125	C₄²	C₂×D₄	1	1	yes	characteristic
126	C₄²	#41	1	1	yes	characteristic
127	C₂³	—	2	1	no	—
128	C₂³	C₄×D₄	1	1	yes	characteristic
129	Q₈	—	2	1	no	—
130	Q₈	#101	1	2	yes	—
131	D₄	—	2	1	no	—
132	D₄	—	2	1	no	—
133	D₄	—	4	2	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	4	2	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	8	1	no	—
147	C₂×C₄	—	8	1	no	—
148	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₄	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄	#120	1	1	yes	characteristic
151	C₂×C₄	#121	1	1	yes	characteristic
152	C₂×C₄	#121	1	1	yes	characteristic
153	C₂×C₄	#123	1	1	yes	characteristic
154	C₈	—	2	2	no	—
155	C₈	—	4	2	no	—
156	C₈	—	4	1	no	—
157	C₈	—	4	1	no	—
158	C₈	—	4	1	no	—
159	C₈	—	8	1	no	—
160	C₈	—	8	1	no	—
161	C₂²	—	2	1	no	—
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	4	1	no	—
165	C₂²	—	4	1	no	—
166	C₂²	—	4	1	no	—
167	C₂²	#386	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	2	no	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	4	1	no	—
175	C₄	—	4	1	no	—
176	C₄	—	8	1	no	—
177	C₄	—	8	1	no	—
178	C₄	C₂×#101	1	1	yes	characteristic
179	C₄	#413	1	1	yes	characteristic
180	C₂	—	4	1	no	—
181	C₂	—	8	1	no	—
182	C₂	#1881	1	1	yes	characteristic
183	C₂	#1901	1	1	yes	characteristic
184	C₂	#1906	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#386
commutator	C₂²×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #1528	C₂	semidirect
5. #1470	C₂	semidirect
6. #1307	C₂	semidirect
7. #1299	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3080	2
#2971	2
#1528	2
#1488	2
#1470	2
#1307	2
#1299	2