Finite Group Databasebeta

Subgroups — #11685

Order 256.

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

normal subgroups only

This group has 224 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	#3054	C₂	1	1	yes	maximal, characteristic
2	C₂×#349	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic, complemented
4	#1340	C₂	1	2	yes	maximal, complemented
5	#1339	C₂	1	2	yes	maximal, complemented
6	C₂×#135	C₄	1	1	yes	characteristic, complemented
7	C₂²×M₁₆	C₂²	1	1	yes	characteristic
8	#525	—	2	1	no	—
9	C₂×#117	—	2	1	no	—
10	C₂×M₃₂	—	2	1	no	—
11	#470	C₄	1	1	yes	characteristic, complemented
12	#470	C₂²	1	1	yes	characteristic
13	#469	C₄	1	1	yes	characteristic, complemented
14	#469	C₂²	1	1	yes	characteristic
15	#468	—	2	1	no	—
16	#468	—	2	1	no	—
17	#427	C₄	1	1	yes	characteristic, complemented
18	#416	—	2	1	no	—
19	#349	—	2	1	no	—
20	#349	C₂²	1	2	yes	—
21	H(3,ℤ/4)	—	2	1	no	—
22	H(3,ℤ/4)	C₂²	1	2	yes	—
23	C₂×#41	—	2	1	no	—
24	C₂³×C₄	D₄	1	1	yes	characteristic
25	#135	—	4	1	no	—
26	#135	—	4	1	no	—
27	C₂×Q₁₆	C₂×C₄	1	1	yes	characteristic, complemented
28	C₂×SD₁₆	—	2	1	no	—
29	C₂×D₈	C₂×C₄	1	1	yes	characteristic, complemented
30	C₂×M₁₆	—	4	1	no	—
31	C₂×M₁₆	—	4	1	no	—
32	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂×M₁₆	D₄	1	1	yes	characteristic
34	C₂×M₁₆	D₄	1	1	yes	characteristic
35	C₂×M₁₆	C₂³	1	1	yes	characteristic
36	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈	D₄	1	1	yes	characteristic
38	#123	—	4	1	no	—
39	C₂²⋊Q₈	—	2	1	no	—
40	C₂²⋊Q₈	—	2	1	no	—
41	#121	—	2	1	no	—
42	#121	—	2	1	no	—
43	#120	—	4	1	no	—
44	C₄×D₄	—	4	1	no	—
45	C₄×D₄	—	4	1	no	—
46	#117	—	2	2	no	—
47	#117	—	2	2	no	—
48	#117	—	2	1	no	—
49	#117	—	4	1	no	—
50	#117	—	4	1	no	—
51	C₂×C₄⋊C₄	—	4	1	no	—
52	C₂×C₂²⋊C₄	—	4	1	no	—
53	C₂×C₄²	—	2	2	no	—
54	M₃₂	—	2	2	no	—
55	M₃₂	—	4	1	no	—
56	C₂×C₁₆	—	4	1	no	—
57	#107	—	2	1	no	—
58	#106	C₂×C₄	1	1	yes	characteristic, complemented
59	#106	C₂×C₄	1	1	yes	characteristic, complemented
60	#103	—	2	1	no	—
61	#103	—	2	1	no	—
62	#102	—	2	1	no	—
63	#102	—	2	1	no	—
64	C₂⁴	D₈	1	1	yes	characteristic
65	#41	—	4	1	no	—
66	#41	—	4	1	no	—
67	#41	—	4	1	no	—
68	#41	—	4	1	no	—
69	C₂×Q₈	—	2	1	no	—
70	C₂×D₄	—	2	1	no	—
71	C₂×D₄	—	4	1	no	—
72	C₂×D₄	—	4	1	no	—
73	C₂×D₄	—	4	1	no	—
74	C₂²×C₄	—	2	1	no	—
75	C₂²×C₄	—	2	1	no	—
76	C₂²×C₄	—	2	1	no	—
77	C₂²×C₄	—	2	1	no	—
78	C₂²×C₄	—	4	2	no	—
79	C₂²×C₄	—	4	1	no	—
80	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
82	C₂²×C₄	D₈	1	1	yes	characteristic
83	C₂²×C₄	SD₁₆	1	1	yes	characteristic
84	C₂²×C₄	SD₁₆	1	1	yes	characteristic
85	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
86	Q₁₆	—	4	1	no	—
87	SD₁₆	—	4	2	no	—
88	D₈	—	4	1	no	—
89	M₁₆	—	4	1	no	—
90	M₁₆	—	4	1	no	—
91	M₁₆	—	4	1	no	—
92	M₁₆	—	4	1	no	—
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈	C₂×D₄	1	1	yes	characteristic
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	4	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	—	4	1	no	—
106	C₄⋊C₄	—	4	1	no	—
107	C₂²⋊C₄	—	4	2	no	—
108	C₂²⋊C₄	—	4	1	no	—
109	C₂²⋊C₄	—	4	1	no	—
110	C₂²⋊C₄	—	4	2	no	—
111	C₂²⋊C₄	—	4	1	no	—
112	C₂²⋊C₄	—	4	1	no	—
113	C₂²⋊C₄	—	4	1	no	—
114	C₄²	—	2	4	no	—
115	C₄²	—	4	2	no	—
116	C₄²	—	4	1	no	—
117	C₁₆	—	4	2	no	—
118	C₂³	—	2	1	no	—
119	C₂³	—	2	1	no	—
120	C₂³	—	4	1	no	—
121	C₂³	—	8	1	no	—
122	C₂³	#102	1	1	yes	characteristic
123	C₂³	#102	1	1	yes	characteristic
124	C₂³	C₂×D₈	1	1	yes	characteristic
125	Q₈	—	4	1	no	—
126	Q₈	—	4	1	no	—
127	D₄	—	4	1	no	—
128	D₄	—	4	1	no	—
129	D₄	—	8	2	no	—
130	D₄	—	8	2	no	—
131	D₄	—	8	1	no	—
132	D₄	—	8	1	no	—
133	C₂×C₄	—	2	1	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	—	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	4	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	8	4	no	—
148	C₂×C₄	—	8	1	no	—
149	C₂×C₄	—	8	1	no	—
150	C₂×C₄	#99	1	1	yes	characteristic
151	C₂×C₄	#99	1	1	yes	characteristic
152	C₂×C₄	#102	1	1	yes	characteristic
153	C₂×C₄	#102	1	1	yes	characteristic
154	C₂×C₄	#104	1	1	yes	characteristic
155	C₂×C₄	#104	1	1	yes	characteristic
156	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
157	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
158	C₈	—	2	1	no	—
159	C₈	—	2	1	no	—
160	C₈	—	2	1	no	—
161	C₈	—	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	—	4	2	no	—
168	C₂²	—	8	1	no	—
169	C₂²	—	8	1	no	—
170	C₂²	—	8	1	no	—
171	C₂²	—	8	1	no	—
172	C₂²	#327	1	1	yes	characteristic
173	C₂²	#327	1	1	yes	characteristic
174	C₂²	C₂×#102	1	1	yes	characteristic
175	C₄	—	2	1	no	—
176	C₄	—	2	1	no	—
177	C₄	—	8	1	no	—
178	C₄	—	8	1	no	—
179	C₄	—	8	4	no	—
180	C₄	—	8	1	no	—
181	C₄	#327	1	1	yes	characteristic
182	C₄	#327	1	1	yes	characteristic
183	C₄	C₂×#99	1	1	yes	characteristic
184	C₄	C₂×#104	1	1	yes	characteristic
185	C₂	—	2	1	no	—
186	C₂	—	2	1	no	—
187	C₂	—	2	1	no	—
188	C₂	—	4	1	no	—
189	C₂	—	4	1	no	—
190	C₂	—	16	1	no	—
191	C₂	C₂×#327	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#327
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂×M₁₆	C₂³
socle	C₂	C₂×#327
prime radical	#3054	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂×#349	C₂	semidirect
3. C₂×H(3,ℤ/4)	C₂	semidirect
4. #1340	C₂	semidirect
5. #1339	C₂	semidirect
6. C₂×#135	C₄	semidirect
11. #470	C₄	semidirect
13. #469	C₄	semidirect
17. #427	C₄	semidirect
27. C₂×Q₁₆	C₂×C₄	semidirect
29. C₂×D₈	C₂×C₄	semidirect
58. #106	C₂×C₄	semidirect
59. #106	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
#3054	2
C₂×#349	2
C₂×H(3,ℤ/4)	2
#1340	2
#1339	2