Finite Group Databasebeta

Subgroups — #19262

Order 256.

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

normal subgroups only

This group has 301 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	#3170	C₂	1	1	yes	maximal, characteristic
2	#3078	C₂	1	2	yes	maximal
3	C₂×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
4	#1684	C₂	1	4	yes	maximal, complemented
5	#1676	C₂	1	2	yes	maximal, complemented
6	#1670	C₂	1	2	yes	maximal, complemented
7	#1567	C₂	1	1	yes	maximal, characteristic
8	#1550	C₂	1	2	yes	maximal
9	#533	C₂²	1	2	yes	—
10	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
11	#500	C₂²	1	1	yes	characteristic
12	#494	C₂²	1	1	yes	characteristic
13	#487	C₂²	1	2	yes	—
14	#479	—	2	4	no	—
15	#477	C₂²	1	4	yes	—
16	#475	C₂²	1	4	yes	—
17	#467	—	2	1	no	—
18	#467	—	2	1	no	—
19	C₈×Q₈	—	2	1	no	—
20	C₈×Q₈	—	2	1	no	—
21	C₈×Q₈	C₂²	1	2	yes	—
22	C₈×Q₈	C₂²	1	2	yes	—
23	C₄×Q₁₆	—	2	1	no	—
24	C₄×Q₁₆	—	2	1	no	—
25	#428	—	2	2	no	—
26	#423	C₂²	1	2	yes	—
27	C₂×C₄⋊C₈	—	2	1	no	—
28	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	#419	—	2	2	no	—
30	C₂×C₄×C₈	—	2	1	no	—
31	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
32	#335	C₂²	1	2	yes	—
33	#335	C₂²	1	2	yes	—
34	#333	C₂²	1	4	yes	—
35	#332	C₂²	1	4	yes	—
36	C₂²×Q₈	D₄	1	1	yes	characteristic
37	C₂×Q₁₆	—	2	1	no	—
38	C₂×Q₁₆	—	2	1	no	—
39	C₂×M₁₆	—	2	2	no	—
40	C₂²×C₈	—	2	1	no	—
41	C₂²×C₈	D₄	1	1	yes	characteristic
42	C₂²×C₈	D₄	1	1	yes	characteristic
43	C₄⋊Q₈	C₂³	1	2	yes	—
44	C₄⋊Q₈	C₂³	1	2	yes	—
45	#125	—	2	2	no	—
46	C₂²⋊Q₈	—	2	2	no	—
47	C₂²⋊Q₈	—	2	2	no	—
48	C₄×Q₈	—	2	1	no	—
49	C₄×Q₈	—	2	1	no	—
50	C₄×Q₈	—	2	2	no	—
51	C₄×Q₈	—	2	2	no	—
52	C₄×Q₈	D₄	1	2	yes	—
53	C₄×Q₈	C₂³	1	2	yes	—
54	#117	—	2	2	no	—
55	C₂×C₄⋊C₄	—	2	1	no	—
56	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
57	C₂×C₄²	—	2	1	no	—
58	C₂×C₄²	C₂³	1	1	yes	characteristic
59	#107	—	2	1	no	—
60	#107	—	2	1	no	—
61	#107	—	2	4	no	—
62	#106	—	2	4	no	—
63	C₄⋊C₈	—	2	2	no	—
64	C₄⋊C₈	—	2	2	no	—
65	C₄⋊C₈	—	2	1	no	—
66	C₄⋊C₈	—	2	2	no	—
67	C₄⋊C₈	C₂³	1	2	yes	—
68	C₄⋊C₈	C₂³	1	4	yes	—
69	#103	—	2	4	no	—
70	#103	—	2	4	no	—
71	#103	—	2	2	no	—
72	#103	—	2	2	no	—
73	C₄×C₈	—	2	2	no	—
74	C₄×C₈	—	2	2	no	—
75	C₄×C₈	D₄	1	1	yes	characteristic
76	C₄×C₈	D₄	1	1	yes	characteristic
77	C₄×C₈	C₂³	1	1	yes	characteristic
78	C₄×C₈	C₂³	1	1	yes	characteristic
79	C₂×Q₈	—	2	1	no	—
80	C₂×Q₈	—	2	2	no	—
81	C₂×Q₈	—	2	2	no	—
82	C₂×Q₈	—	2	2	no	—
83	C₂×Q₈	Q₁₆	1	4	yes	complemented
84	C₂×Q₈	C₂×D₄	1	2	yes	—
85	C₂²×C₄	—	2	1	no	—
86	C₂²×C₄	—	2	1	no	—
87	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄	#41	1	1	yes	characteristic
90	Q₁₆	—	4	2	no	—
91	Q₁₆	—	4	2	no	—
92	M₁₆	—	4	4	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	4	no	—
96	C₂×C₈	—	4	1	no	—
97	C₂×C₈	—	4	2	no	—
98	C₂×C₈	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈	C₂×D₄	1	1	yes	characteristic
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	4	2	no	—
110	C₄⋊C₄	—	4	2	no	—
111	C₄⋊C₄	C₂×D₄	1	2	yes	—
112	C₂²⋊C₄	—	4	2	no	—
113	C₄²	—	2	2	no	—
114	C₄²	—	2	2	no	—
115	C₄²	—	4	2	no	—
116	C₄²	C₂×D₄	1	1	yes	characteristic
117	C₄²	C₂×D₄	1	1	yes	characteristic
118	C₄²	#41	1	1	yes	characteristic
119	C₄²	C₂⁴	1	1	yes	characteristic
120	C₂³	#121	1	1	yes	characteristic
121	Q₈	—	2	2	no	—
122	Q₈	—	2	4	no	—
123	Q₈	—	4	2	no	—
124	Q₈	—	4	2	no	—
125	Q₈	—	8	2	no	—
126	Q₈	C₂×Q₁₆	1	4	yes	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	#121	1	1	yes	characteristic
137	C₂×C₄	#121	1	1	yes	characteristic
138	C₂×C₄	#121	1	1	yes	characteristic
139	C₂×C₄	C₂×Q₁₆	1	2	yes	—
140	C₂×C₄	#135	1	2	yes	—
141	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄	C₂×#41	1	1	yes	characteristic
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	2	1	no	—
147	C₈	—	4	4	no	—
148	C₈	—	4	2	no	—
149	C₈	#137	1	2	yes	—
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	#467	1	2	yes	—
153	C₂²	C₂×#121	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	4	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	4	2	no	—
159	C₄	—	8	2	no	—
160	C₄	—	8	2	no	—
161	C₄	#467	1	2	yes	—
162	C₄	C₂²×Q₁₆	1	1	yes	characteristic
163	C₄	C₂×#135	1	1	yes	characteristic
164	C₄	C₂×#137	1	1	yes	characteristic
165	C₄	#575	1	1	yes	characteristic
166	C₂	—	2	2	no	—
167	C₂	#3037	1	1	yes	characteristic
168	C₂	C₂×#467	1	1	yes	characteristic
169	C₂	#3067	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
4. #1684	C₂	semidirect
5. #1676	C₂	semidirect
6. #1670	C₂	semidirect
83. C₂×Q₈	Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
#3170	2
#3078	2
C₂×C₈×Q₈	2
#1684	2
#1676	2
#1670	2
#1567	2
#1550	2