Finite Group Databasebeta

Subgroups — #9339

Order 256.

208 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 59 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	#3239	C₂	1	1	yes	maximal, characteristic
2	#2995	C₂	1	1	yes	maximal, characteristic
3	#1966	C₂	1	1	yes	maximal, characteristic, complemented
4	#1556	C₂	1	1	yes	maximal, characteristic, complemented
5	#1529	C₂	1	1	yes	maximal, characteristic
6	#1307	C₂	1	1	yes	maximal, characteristic, complemented
7	#1291	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	#529	C₂²	1	1	yes	characteristic
10	#501	—	2	1	no	—
11	#489	—	2	1	no	—
12	#484	C₄	1	2	yes	—
13	#481	C₄	1	2	yes	—
14	#475	—	2	1	no	—
15	#474	—	2	1	no	—
16	#446	—	2	1	no	—
17	#436	—	2	1	no	—
18	#435	—	2	1	no	—
19	#435	—	2	1	no	—
20	#433	—	2	1	no	—
21	#432	C₂²	1	1	yes	characteristic
22	#432	C₂²	1	1	yes	characteristic
23	#432	C₂²	1	1	yes	characteristic
24	#424	—	2	1	no	—
25	#423	C₂²	1	1	yes	characteristic
26	C₂×#101	—	2	1	no	—
27	C₂×#101	—	2	1	no	—
28	C₄×M₁₆	C₂²	1	1	yes	characteristic
29	#332	—	2	1	no	—
30	#332	—	2	1	no	—
31	#330	—	2	1	no	—
32	#329	—	2	1	no	—
33	C₂²×Q₈	—	2	1	no	—
34	C₂×M₁₆	—	2	1	no	—
35	C₂×M₁₆	—	2	1	no	—
36	C₂×M₁₆	D₄	1	1	yes	characteristic
37	C₂×M₁₆	D₄	1	1	yes	characteristic
38	C₂²×C₈	—	2	1	no	—
39	C₄⋊Q₈	—	2	1	no	—
40	C₄⋊Q₈	—	2	1	no	—
41	C₄⋊Q₈	—	2	1	no	—
42	C₄⋊Q₈	—	2	1	no	—
43	#126	—	2	1	no	—
44	#125	D₄	1	1	yes	characteristic
45	#124	D₄	1	1	yes	characteristic
46	#123	—	2	1	no	—
47	C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
48	#121	C₂×C₄	1	1	yes	characteristic
49	C₄×Q₈	D₄	1	1	yes	characteristic
50	C₄×D₄	—	2	1	no	—
51	C₄×D₄	D₄	1	1	yes	characteristic
52	#117	—	2	1	no	—
53	C₂×C₄⋊C₄	—	2	1	no	—
54	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
55	C₂×C₄²	C₂³	1	1	yes	characteristic
56	#107	—	2	2	no	—
57	#106	—	2	2	no	—
58	C₄⋊C₈	—	2	1	no	—
59	C₄⋊C₈	—	2	1	no	—
60	C₄⋊C₈	—	2	1	no	—
61	C₄⋊C₈	—	2	1	no	—
62	C₄⋊C₈	—	2	1	no	—
63	C₄⋊C₈	—	2	1	no	—
64	C₄⋊C₈	D₄	1	1	yes	characteristic
65	C₄⋊C₈	D₄	1	1	yes	characteristic
66	#103	—	2	2	no	—
67	#102	—	2	2	no	—
68	#101	—	4	2	no	—
69	#101	—	4	2	no	—
70	C₂²⋊C₈	—	2	1	no	—
71	C₂²⋊C₈	—	2	1	no	—
72	C₂²⋊C₈	—	2	1	no	—
73	C₂²⋊C₈	—	2	1	no	—
74	C₂²⋊C₈	C₂×C₄	1	2	yes	—
75	#97	—	2	1	no	—
76	#97	—	2	1	no	—
77	#97	—	2	1	no	—
78	#97	—	2	1	no	—
79	#97	—	2	1	no	—
80	C₄×C₈	—	2	1	no	—
81	C₄×C₈	—	2	1	no	—
82	C₂×Q₈	—	4	1	no	—
83	C₂×Q₈	—	4	1	no	—
84	C₂×Q₈	—	4	1	no	—
85	C₂×Q₈	—	4	1	no	—
86	C₂×Q₈	—	4	1	no	—
87	C₂×Q₈	—	4	1	no	—
88	C₂×Q₈	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×D₄	—	2	1	no	—
90	C₂×D₄	C₂²⋊C₄	1	1	yes	characteristic
91	C₂²×C₄	—	2	1	no	—
92	C₂²×C₄	—	2	2	no	—
93	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄	#41	1	1	yes	characteristic
96	M₁₆	—	2	2	no	—
97	M₁₆	—	4	1	no	—
98	M₁₆	—	4	1	no	—
99	M₁₆	—	4	1	no	—
100	M₁₆	—	8	1	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	4	1	no	—
107	C₂×C₈	—	4	1	no	—
108	C₂×C₈	—	8	1	no	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	4	1	no	—
116	C₄⋊C₄	—	4	1	no	—
117	C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
118	C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
119	C₂²⋊C₄	—	2	1	no	—
120	C₂²⋊C₄	—	2	1	no	—
121	C₂²⋊C₄	—	2	1	no	—
122	C₂²⋊C₄	—	2	1	no	—
123	C₂²⋊C₄	—	2	1	no	—
124	C₄²	—	2	1	no	—
125	C₄²	C₂×D₄	1	1	yes	characteristic
126	C₄²	C₂×D₄	1	1	yes	characteristic
127	C₄²	C₂×D₄	1	1	yes	characteristic
128	C₄²	#41	1	1	yes	characteristic
129	C₂³	—	2	1	no	—
130	C₂³	C₄×D₄	1	1	yes	characteristic
131	Q₈	—	4	1	no	—
132	Q₈	—	8	1	no	—
133	Q₈	—	8	1	no	—
134	Q₈	—	8	1	no	—
135	Q₈	—	8	1	no	—
136	D₄	—	4	1	no	—
137	D₄	—	8	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
151	C₂×C₄	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄	#120	1	1	yes	characteristic
153	C₂×C₄	#121	1	1	yes	characteristic
154	C₂×C₄	#121	1	1	yes	characteristic
155	C₂×C₄	#123	1	1	yes	characteristic
156	C₈	—	4	2	no	—
157	C₈	—	4	2	no	—
158	C₈	—	4	1	no	—
159	C₈	—	4	1	no	—
160	C₈	—	4	1	no	—
161	C₈	—	4	1	no	—
162	C₈	—	8	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	—	4	1	no	—
167	C₂²	—	4	1	no	—
168	C₂²	—	4	1	no	—
169	C₂²	#386	1	1	yes	characteristic
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	4	1	no	—
176	C₄	—	4	1	no	—
177	C₄	—	4	1	no	—
178	C₄	—	8	2	no	—
179	C₄	#453	1	1	yes	characteristic
180	C₄	#454	1	1	yes	characteristic
181	C₂	—	4	1	no	—
182	C₂	—	8	1	no	—
183	C₂	#1883	1	1	yes	characteristic
184	C₂	#1902	1	1	yes	characteristic
185	C₂	#1915	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#386
commutator	C₂²×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	#121	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. #1966	C₂	semidirect
4. #1556	C₂	semidirect
6. #1307	C₂	semidirect
7. #1291	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3239	2
#2995	2
#1966	2
#1556	2
#1529	2
#1307	2
#1291	2