Finite Group Databasebeta

Subgroups — #23833

Order 256.

308 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 109 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	#2960	C₂	1	1	yes	maximal, characteristic, complemented
2	#2955	C₂	1	1	yes	maximal, characteristic
3	#2103	C₂	1	1	yes	maximal, characteristic, complemented
4	#2102	C₂	1	1	yes	maximal, characteristic, complemented
5	#2080	C₂	1	2	yes	maximal, complemented
6	#2053	C₂	1	1	yes	maximal, characteristic
7	#2050	C₂	1	1	yes	maximal, characteristic
8	#2031	C₂	1	2	yes	maximal, complemented
9	#1764	C₂	1	2	yes	maximal, complemented
10	#1757	C₂	1	2	yes	maximal, complemented
11	#1747	C₂	1	1	yes	maximal, characteristic, complemented
12	#575	C₂²	1	1	yes	characteristic
13	C₂×#131	C₂²	1	2	yes	—
14	#533	C₂²	1	1	yes	characteristic
15	#446	—	2	2	no	—
16	C₈×Q₈	—	2	2	no	—
17	#444	—	2	2	no	—
18	#443	—	2	2	no	—
19	#430	C₂²	1	1	yes	characteristic
20	#424	C₂²	1	1	yes	characteristic
21	#424	C₂²	1	1	yes	characteristic
22	#423	C₂²	1	1	yes	characteristic
23	#421	C₂²	1	2	yes	—
24	C₂×#104	C₂²	1	2	yes	—
25	#416	C₂²	1	2	yes	—
26	#413	C₂²	1	2	yes	—
27	#410	C₂²	1	2	yes	—
28	#405	C₂²	1	1	yes	characteristic
29	#405	C₂²	1	2	yes	—
30	#405	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
32	#344	C₂²	1	2	yes	—
33	#344	C₂²	1	2	yes	—
34	#341	C₂²	1	2	yes	—
35	#339	C₂²	1	2	yes	—
36	#338	C₂²	1	1	yes	characteristic
37	H(3,ℤ/4)	C₂²	1	1	yes	characteristic
38	H(3,ℤ/4)	C₂²	1	2	yes	—
39	C₂×#41	C₂³	1	2	yes	—
40	#137	—	2	2	no	—
41	AGL(1,ℤ/8)	—	2	2	no	—
42	#135	—	2	2	no	—
43	#131	—	2	2	no	—
44	#131	—	2	2	no	—
45	#131	—	2	2	no	—
46	#131	—	2	2	no	—
47	C₂×M₁₆	—	2	2	no	—
48	C₂×M₁₆	C₂³	1	2	yes	—
49	C₂×M₁₆	C₂³	1	2	yes	—
50	C₂×M₁₆	C₂³	1	1	yes	characteristic
51	C₂×M₁₆	C₂³	1	1	yes	characteristic
52	C₂²×C₈	—	2	2	no	—
53	C₂²×C₈	C₂³	1	1	yes	characteristic
54	C₂²×C₈	C₂³	1	1	yes	characteristic
55	C₄⋊Q₈	—	2	1	no	—
56	#125	—	2	1	no	—
57	C₂²⋊Q₈	—	2	2	no	—
58	C₄×Q₈	—	2	2	no	—
59	#117	C₂³	1	2	yes	—
60	#117	C₂³	1	1	yes	characteristic
61	#117	C₂³	1	1	yes	characteristic
62	C₂×C₄²	C₂³	1	1	yes	characteristic
63	#108	—	2	2	no	—
64	C₄⋊C₈	—	2	2	no	—
65	C₄⋊C₈	—	2	2	no	—
66	C₄⋊C₈	—	2	2	no	—
67	#104	—	2	4	no	—
68	#104	—	2	2	no	—
69	#104	—	2	2	no	—
70	#103	—	2	2	no	—
71	#102	—	2	2	no	—
72	#101	—	2	2	no	—
73	#100	—	2	2	no	—
74	#99	—	2	4	no	—
75	#97	—	2	1	no	—
76	#97	—	2	2	no	—
77	#97	D₄	1	2	yes	—
78	C₄×C₈	—	2	1	no	—
79	C₄×C₈	—	2	1	no	—
80	C₄×C₈	—	2	1	no	—
81	C₄×C₈	—	2	2	no	—
82	C₄×C₈	D₄	1	2	yes	—
83	#41	—	2	2	no	—
84	#41	—	2	2	no	—
85	#41	—	2	2	no	—
86	#41	—	2	2	no	—
87	C₂×Q₈	—	2	2	no	—
88	C₂×Q₈	#41	1	2	yes	—
89	C₂×D₄	—	2	2	no	—
90	C₂×D₄	#41	1	2	yes	—
91	C₂²×C₄	—	2	1	no	—
92	C₂²×C₄	—	2	2	no	—
93	C₂²×C₄	C₂⁴	1	1	yes	characteristic
94	Q₁₆	—	2	2	no	—
95	SD₁₆	—	2	4	no	—
96	D₈	—	2	2	no	—
97	M₁₆	—	2	2	no	—
98	M₁₆	—	2	2	no	—
99	M₁₆	—	2	2	no	—
100	M₁₆	—	2	2	no	—
101	M₁₆	—	2	2	no	—
102	M₁₆	—	2	2	no	—
103	M₁₆	—	4	2	no	—
104	M₁₆	—	4	2	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	4	2	no	—
111	C₂×C₈	—	4	2	no	—
112	C₂×C₈	C₂×D₄	1	2	yes	—
113	C₂×C₈	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈	#41	1	2	yes	—
116	C₂×C₈	#41	1	2	yes	—
117	C₄⋊C₄	—	2	4	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	2	2	no	—
122	C₂²⋊C₄	—	2	1	no	—
123	C₂²⋊C₄	—	2	2	no	—
124	C₂²⋊C₄	#41	1	2	yes	—
125	C₄²	—	2	1	no	—
126	C₄²	—	2	1	no	—
127	C₄²	—	2	2	no	—
128	C₄²	C₂×D₄	1	2	yes	—
129	C₄²	#41	1	1	yes	characteristic
130	C₄²	#41	1	1	yes	characteristic
131	C₂³	—	2	2	no	—
132	C₂³	C₂×#41	1	1	yes	characteristic
133	Q₈	—	2	2	no	—
134	Q₈	—	2	2	no	—
135	Q₈	—	4	4	no	—
136	D₄	—	2	2	no	—
137	D₄	—	2	2	no	—
138	D₄	—	4	2	no	—
139	D₄	—	4	2	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄	C₂×#41	1	2	yes	—
151	C₂×C₄	C₂×#41	1	2	yes	—
152	C₂×C₄	C₂×#41	1	1	yes	characteristic
153	C₈	—	2	2	no	—
154	C₈	—	4	1	no	—
155	C₈	—	4	1	no	—
156	C₈	—	4	2	no	—
157	C₈	—	4	2	no	—
158	C₈	#124	1	2	yes	—
159	C₈	#124	1	1	yes	characteristic
160	C₈	#124	1	1	yes	characteristic
161	C₂²	—	4	1	no	—
162	C₂²	—	4	2	no	—
163	C₂²	—	8	2	no	—
164	C₂²	#525	1	1	yes	characteristic
165	C₂²	#529	1	2	yes	—
166	C₄	—	2	2	no	—
167	C₄	—	4	1	no	—
168	C₄	—	4	2	no	—
169	C₄	—	4	2	no	—
170	C₄	—	4	2	no	—
171	C₄	—	8	2	no	—
172	C₄	#525	1	1	yes	characteristic
173	C₄	C₂×#124	1	1	yes	characteristic
174	C₄	#529	1	2	yes	—
175	C₂	—	2	1	no	—
176	C₂	—	2	2	no	—
177	C₂	—	8	2	no	—
178	C₂	#2398	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#124
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂	#2398
prime radical	#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #2960	C₂	semidirect
3. #2103	C₂	semidirect
4. #2102	C₂	semidirect
5. #2080	C₂	semidirect
8. #2031	C₂	semidirect
9. #1764	C₂	semidirect
10. #1757	C₂	semidirect
11. #1747	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2960	2
#2955	2
#2103	2
#2102	2
#2080	2
#2053	2
#2050	2
#2031	2
#1764	2
#1757	2
#1747	2