Finite Group Databasebeta

Subgroups — #13385

Order 256.

204 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 55 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	#3061	C₂	1	1	yes	maximal, characteristic
2	C₂×#367	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#359	C₂	1	1	yes	maximal, characteristic, complemented
4	#2139	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#350	C₂	1	1	yes	maximal, characteristic, complemented
6	#1816	C₂	1	1	yes	maximal, characteristic, complemented
7	#1387	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135	C₂²	1	1	yes	characteristic
9	C₂×#131	C₂²	1	1	yes	characteristic
10	#537	—	2	1	no	—
11	C₂×M₃₂	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆	C₂²	1	1	yes	characteristic
13	#468	—	2	2	no	—
14	#467	—	2	1	no	—
15	#466	—	2	1	no	—
16	C₂×#108	C₂²	1	1	yes	characteristic
17	C₂×#107	C₂²	1	1	yes	characteristic
18	C₂×#107	C₂²	1	1	yes	characteristic
19	#417	—	2	1	no	—
20	#417	—	2	1	no	—
21	#367	—	2	2	no	—
22	#359	—	2	2	no	—
23	#358	—	2	1	no	—
24	#357	—	2	1	no	—
25	#350	—	2	2	no	—
26	#343	—	2	1	no	—
27	C₂×#41	—	2	1	no	—
28	C₂×#41	D₄	1	1	yes	characteristic
29	#135	—	2	1	no	—
30	#135	—	2	1	no	—
31	#135	—	2	1	no	—
32	#135	—	2	1	no	—
33	C₂×Q₁₆	D₄	1	1	yes	characteristic
34	C₂×SD₁₆	—	2	1	no	—
35	C₂×D₈	D₄	1	1	yes	characteristic
36	#131	—	2	1	no	—
37	#131	—	2	1	no	—
38	#131	—	2	1	no	—
39	#131	—	2	1	no	—
40	C₂×M₁₆	—	2	1	no	—
41	C₂×M₁₆	—	2	1	no	—
42	C₂×M₁₆	D₄	1	1	yes	characteristic
43	C₂²×C₈	—	2	1	no	—
44	C₂²×C₈	C₂³	1	1	yes	characteristic
45	C₂²⋊Q₈	—	2	2	no	—
46	C₂²⋊Q₈	—	4	1	no	—
47	#121	—	2	2	no	—
48	#121	—	4	2	no	—
49	#121	—	4	1	no	—
50	C₂×C₄⋊C₄	—	2	1	no	—
51	C₂×C₄⋊C₄	—	2	1	no	—
52	M₃₂	—	2	1	no	—
53	M₃₂	—	2	1	no	—
54	C₂×C₁₆	—	2	1	no	—
55	C₂×C₁₆	—	2	1	no	—
56	C₂×C₁₆	—	2	1	no	—
57	C₂×C₁₆	D₄	1	2	yes	complemented
58	#108	—	2	1	no	—
59	#108	—	2	1	no	—
60	#107	—	2	1	no	—
61	#107	—	2	1	no	—
62	#107	—	2	1	no	—
63	#107	D₄	1	2	yes	—
64	#103	—	2	2	no	—
65	#103	—	4	1	no	—
66	#102	—	2	2	no	—
67	#102	—	4	1	no	—
68	#41	—	2	1	no	—
69	#41	—	2	1	no	—
70	#41	—	2	1	no	—
71	#41	—	2	1	no	—
72	#41	—	4	1	no	—
73	#41	—	4	1	no	—
74	#41	—	4	1	no	—
75	#41	—	4	1	no	—
76	C₂×Q₈	—	2	1	no	—
77	C₂×Q₈	D₈	1	1	yes	characteristic
78	C₂×D₄	—	2	1	no	—
79	C₂×D₄	—	2	1	no	—
80	C₂×D₄	—	4	1	no	—
81	C₂×D₄	—	4	2	no	—
82	C₂×D₄	D₈	1	1	yes	characteristic
83	C₂²×C₄	—	2	1	no	—
84	C₂²×C₄	—	4	1	no	—
85	C₂²×C₄	—	4	1	no	—
86	C₂²×C₄	—	4	1	no	—
87	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
88	Q₁₆	—	2	1	no	—
89	Q₁₆	—	2	1	no	—
90	SD₁₆	—	4	1	no	—
91	SD₁₆	—	4	1	no	—
92	D₈	—	2	1	no	—
93	D₈	—	2	1	no	—
94	M₁₆	—	2	1	no	—
95	M₁₆	—	2	1	no	—
96	M₁₆	—	4	1	no	—
97	M₁₆	—	4	1	no	—
98	M₁₆	—	4	1	no	—
99	M₁₆	—	4	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	4	1	no	—
102	C₂×C₈	—	4	1	no	—
103	C₂×C₈	—	4	1	no	—
104	C₂×C₈	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈	#41	1	1	yes	characteristic
108	C₂×C₈	#41	1	1	yes	characteristic
109	C₂×C₈	#41	1	1	yes	characteristic
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	4	2	no	—
112	C₄⋊C₄	—	4	1	no	—
113	C₄⋊C₄	—	4	1	no	—
114	C₄⋊C₄	—	4	1	no	—
115	C₂²⋊C₄	—	4	2	no	—
116	C₂²⋊C₄	—	4	2	no	—
117	C₁₆	—	2	2	no	—
118	C₁₆	—	4	1	no	—
119	C₂³	—	2	1	no	—
120	C₂³	—	4	1	no	—
121	C₂³	AGL(1,ℤ/8)	1	1	yes	characteristic
122	Q₈	—	2	1	no	—
123	Q₈	—	2	1	no	—
124	Q₈	—	4	1	no	—
125	Q₈	—	4	1	no	—
126	D₄	—	2	1	no	—
127	D₄	—	2	1	no	—
128	D₄	—	4	1	no	—
129	D₄	—	4	1	no	—
130	D₄	—	4	1	no	—
131	D₄	—	4	1	no	—
132	D₄	—	8	1	no	—
133	D₄	—	8	1	no	—
134	D₄	—	16	2	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	—	8	1	no	—
141	C₂×C₄	—	8	1	no	—
142	C₂×C₄	—	8	1	no	—
143	C₂×C₄	—	16	1	no	—
144	C₂×C₄	—	16	1	no	—
145	C₂×C₄	#120	1	1	yes	characteristic
146	C₂×C₄	#121	1	1	yes	characteristic
147	C₂×C₄	#123	1	1	yes	characteristic
148	C₂×C₄	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄	#135	1	1	yes	characteristic
150	C₂×C₄	#137	1	1	yes	characteristic
151	C₈	—	4	1	no	—
152	C₈	—	8	1	no	—
153	C₈	#121	1	1	yes	characteristic
154	C₈	#121	1	1	yes	characteristic
155	C₈	#123	1	1	yes	characteristic
156	C₈	#124	1	1	yes	characteristic
157	C₂²	—	2	1	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	4	1	no	—
160	C₂²	—	4	1	no	—
161	C₂²	—	4	1	no	—
162	C₂²	—	8	1	no	—
163	C₂²	—	8	1	no	—
164	C₂²	—	8	1	no	—
165	C₂²	#447	1	1	yes	characteristic
166	C₂²	#468	1	1	yes	characteristic
167	C₂²	#482	1	1	yes	characteristic
168	C₄	—	4	1	no	—
169	C₄	—	8	2	no	—
170	C₄	—	8	1	no	—
171	C₄	—	16	1	no	—
172	C₄	#394	1	1	yes	characteristic
173	C₄	#452	1	1	yes	characteristic
174	C₄	#466	1	1	yes	characteristic
175	C₄	#480	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	2	1	no	—
178	C₂	—	8	1	no	—
179	C₂	—	16	1	no	—
180	C₂	#2033	1	1	yes	characteristic
181	C₂	#2193	1	1	yes	characteristic
182	C₂	#2194	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³
prime radical	#3061	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂×#367	C₂	semidirect
3. C₂×#359	C₂	semidirect
4. #2139	C₂	semidirect
5. C₂×#350	C₂	semidirect
6. #1816	C₂	semidirect
7. #1387	C₂	semidirect
57. C₂×C₁₆	D₄	semidirect

Maximal subgroups

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

subgroup	index
#3061	2
C₂×#367	2
C₂×#359	2
#2139	2
C₂×#350	2
#1816	2
#1387	2