Subgroups — #33955

Order 256.

229 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3383	C₂	1	1	yes	maximal, characteristic, complemented
2	#3380	C₂	1	1	yes	maximal, characteristic
3	#2255	C₂	1	1	yes	maximal, characteristic, complemented
4	#2255	C₂	1	1	yes	maximal, characteristic, complemented
5	#2253	C₂	1	1	yes	maximal, characteristic, complemented
6	#2229	C₂	1	1	yes	maximal, characteristic, complemented
7	#2229	C₂	1	1	yes	maximal, characteristic, complemented
8	#2228	C₂	1	1	yes	maximal, characteristic
9	#2228	C₂	1	1	yes	maximal, characteristic
10	#2226	C₂	1	1	yes	maximal, characteristic
11	#2225	C₂	1	1	yes	maximal, characteristic, complemented
12	#2183	C₂	1	1	yes	maximal, characteristic, complemented
13	#2175	C₂	1	1	yes	maximal, characteristic
14	#2175	C₂	1	1	yes	maximal, characteristic
15	C₄×SD₃₂	C₂	1	1	yes	maximal, characteristic
16	#557	—	2	1	no	—
17	#554	—	2	1	no	—
18	C₂×SD₃₂	C₂²	1	1	yes	characteristic
19	#500	C₂²	1	1	yes	characteristic
20	#500	C₂²	1	1	yes	characteristic
21	#499	C₂²	1	1	yes	characteristic
22	#499	C₂²	1	1	yes	characteristic
23	#499	C₂²	1	1	yes	characteristic
24	#499	C₂²	1	1	yes	characteristic
25	#479	—	2	1	no	—
26	#479	—	2	1	no	—
27	#478	—	2	1	no	—
28	#478	—	2	1	no	—
29	#476	—	2	1	no	—
30	#475	—	2	1	no	—
31	C₈×Q₈	C₂²	1	1	yes	characteristic
32	C₄×Q₁₆	C₂²	1	1	yes	characteristic
33	C₄×Q₁₆	C₂²	1	1	yes	characteristic
34	C₄×Q₁₆	C₂²	1	1	yes	characteristic
35	C₄×D₈	C₂²	1	1	yes	characteristic
36	C₄×D₈	C₂²	1	1	yes	characteristic
37	C₄×D₈	C₂²	1	1	yes	characteristic
38	#367	C₂²	1	1	yes	characteristic
39	#367	C₂²	1	1	yes	characteristic
40	#367	C₂²	1	1	yes	characteristic
41	#367	C₂²	1	1	yes	characteristic
42	#367	C₂²	1	1	yes	characteristic
43	#366	C₂²	1	1	yes	characteristic
44	#366	C₂²	1	1	yes	characteristic
45	#366	C₂²	1	1	yes	characteristic
46	#366	C₂²	1	1	yes	characteristic
47	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
48	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
49	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
50	#358	C₂²	1	1	yes	characteristic
51	#358	C₂²	1	1	yes	characteristic
52	#358	C₂²	1	1	yes	characteristic
53	#357	C₂²	1	1	yes	characteristic
54	#357	C₂²	1	1	yes	characteristic
55	#357	C₂²	1	1	yes	characteristic
56	#346	C₂²	1	1	yes	characteristic
57	#346	C₂²	1	1	yes	characteristic
58	C₄×C₁₆	C₂²	1	1	yes	characteristic
59	C₂×Q₁₆	C₂³	1	1	yes	characteristic
60	C₂×D₈	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈	—	2	1	no	—
62	C₄⋊Q₈	—	2	1	no	—
63	C₄⋊Q₈	—	4	1	no	—
64	#125	—	2	1	no	—
65	#125	—	2	1	no	—
66	#125	—	2	1	no	—
67	#125	—	2	1	no	—
68	#125	—	4	1	no	—
69	#125	—	4	1	no	—
70	C₂²⋊Q₈	—	4	1	no	—
71	C₂²⋊Q₈	—	4	1	no	—
72	C₂²⋊Q₈	—	4	1	no	—
73	C₄×Q₈	—	2	1	no	—
74	C₄×Q₈	—	2	1	no	—
75	C₄×Q₈	—	2	1	no	—
76	C₄×Q₈	—	4	1	no	—
77	C₄×Q₈	D₄	1	1	yes	characteristic
78	C₄×D₄	—	2	1	no	—
79	C₄×D₄	—	2	1	no	—
80	C₄×D₄	—	2	1	no	—
81	C₂×C₄⋊C₄	—	4	1	no	—
82	SD₃₂	Q₈	1	4	yes	complemented
83	C₂×C₁₆	C₂³	1	1	yes	characteristic
84	C₂×C₁₆	C₂³	1	1	yes	characteristic
85	C₂×C₁₆	C₂³	1	1	yes	characteristic
86	C₂×C₁₆	C₂³	1	1	yes	characteristic
87	#107	—	2	1	no	—
88	#107	—	2	1	no	—
89	#107	C₂³	1	1	yes	characteristic
90	#107	C₂³	1	1	yes	characteristic
91	#107	C₂³	1	1	yes	characteristic
92	#107	C₂³	1	1	yes	characteristic
93	#107	C₂³	1	1	yes	characteristic
94	#107	C₂³	1	1	yes	characteristic
95	#106	—	2	1	no	—
96	#106	—	2	1	no	—
97	#106	—	2	1	no	—
98	#106	—	2	1	no	—
99	C₄⋊C₈	D₄	1	1	yes	characteristic
100	C₄⋊C₈	D₄	1	1	yes	characteristic
101	C₄⋊C₈	D₄	1	1	yes	characteristic
102	#103	—	2	1	no	—
103	#103	—	2	1	no	—
104	#103	—	2	1	no	—
105	#102	—	2	1	no	—
106	#102	—	2	1	no	—
107	#102	—	2	1	no	—
108	C₄×C₈	C₂³	1	1	yes	characteristic
109	C₄×C₈	C₂³	1	1	yes	characteristic
110	C₄×C₈	C₂³	1	1	yes	characteristic
111	C₂×Q₈	—	2	1	no	—
112	C₂×Q₈	—	4	1	no	—
113	C₂×Q₈	—	4	1	no	—
114	C₂×Q₈	D₈	1	1	yes	characteristic
115	C₂×D₄	—	2	1	no	—
116	C₂²×C₄	—	4	1	no	—
117	C₂²×C₄	—	4	1	no	—
118	C₂²×C₄	—	4	1	no	—
119	Q₁₆	—	2	1	no	—
120	Q₁₆	C₂×Q₈	1	2	yes	—
121	D₈	—	2	1	no	—
122	D₈	C₂×Q₈	1	2	yes	—
123	C₂×C₈	C₂×D₄	1	1	yes	characteristic
124	C₂×C₈	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈	C₂⁴	1	1	yes	characteristic
127	C₄⋊C₄	—	2	1	no	—
128	C₄⋊C₄	—	2	1	no	—
129	C₄⋊C₄	—	2	1	no	—
130	C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	—	4	1	no	—
134	C₄⋊C₄	—	4	1	no	—
135	C₄⋊C₄	—	4	1	no	—
136	C₄⋊C₄	—	4	1	no	—
137	C₄⋊C₄	—	4	1	no	—
138	C₄⋊C₄	—	4	1	no	—
139	C₄⋊C₄	—	4	1	no	—
140	C₄⋊C₄	—	4	1	no	—
141	C₄⋊C₄	—	4	1	no	—
142	C₄⋊C₄	—	4	1	no	—
143	C₄⋊C₄	—	4	1	no	—
144	C₄⋊C₄	—	4	1	no	—
145	C₄⋊C₄	—	4	1	no	—
146	C₄⋊C₄	D₈	1	1	yes	characteristic
147	C₄⋊C₄	D₈	1	1	yes	characteristic
148	C₄⋊C₄	D₈	1	1	yes	characteristic
149	C₂²⋊C₄	—	4	1	no	—
150	C₂²⋊C₄	—	4	1	no	—
151	C₂²⋊C₄	—	4	1	no	—
152	C₄²	—	4	1	no	—
153	C₄²	—	4	1	no	—
154	C₄²	—	4	1	no	—
155	C₄²	C₂×D₄	1	1	yes	characteristic
156	C₄²	C₂×D₄	1	1	yes	characteristic
157	C₄²	C₂×D₄	1	1	yes	characteristic
158	C₁₆	—	2	1	no	—
159	C₁₆	—	2	1	no	—
160	C₁₆	—	2	1	no	—
161	C₁₆	C₂×Q₈	1	2	yes	—
162	C₂³	—	4	1	no	—
163	Q₈	—	2	2	no	—
164	Q₈	—	4	1	no	—
165	Q₈	—	4	1	no	—
166	Q₈	—	8	2	no	—
167	Q₈	—	16	1	no	—
168	D₄	—	2	2	no	—
169	D₄	—	4	1	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	—	4	1	no	—
175	C₂×C₄	—	4	1	no	—
176	C₂×C₄	—	4	1	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	—	16	1	no	—
179	C₂×C₄	—	16	1	no	—
180	C₂×C₄	C₂×D₈	1	1	yes	characteristic
181	C₂×C₄	C₂×D₈	1	1	yes	characteristic
182	C₂×C₄	C₂×D₈	1	1	yes	characteristic
183	C₂×C₄	C₂×D₈	1	1	yes	characteristic
184	C₂×C₄	C₂×D₈	1	1	yes	characteristic
185	C₂×C₄	C₂×D₈	1	1	yes	characteristic
186	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
187	C₈	—	2	1	no	—
188	C₈	—	2	1	no	—
189	C₈	—	2	1	no	—
190	C₈	C₂²×Q₈	1	1	yes	characteristic
191	C₈	2₋¹⁺⁴	1	1	yes	characteristic
192	C₂²	—	4	2	no	—
193	C₂²	—	8	1	no	—
194	C₂²	—	8	1	no	—
195	C₂²	C₂²×D₈	1	1	yes	characteristic
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	2	1	no	—
199	C₄	—	2	1	no	—
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	4	2	no	—
203	C₄	—	8	1	no	—
204	C₄	—	8	1	no	—
205	C₄	—	8	1	no	—
206	C₄	—	8	1	no	—
207	C₄	—	8	1	no	—
208	C₄	—	8	1	no	—
209	C₄	D₄×Q₈	1	1	yes	characteristic
210	C₄	#578	1	1	yes	characteristic
211	C₂	—	8	2	no	—
212	C₂	Q₈×D₈	1	1	yes	characteristic
213	C₂	#3414	1	1	yes	characteristic
214	C₂	#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂×D₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3383	C₂	semidirect
3. #2255	C₂	semidirect
4. #2255	C₂	semidirect
5. #2253	C₂	semidirect
6. #2229	C₂	semidirect
7. #2229	C₂	semidirect
11. #2225	C₂	semidirect
12. #2183	C₂	semidirect
82. SD₃₂	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#3383	2
#3380	2
#2255	2
#2253	2
#2229	2
#2228	2
#2226	2
#2225	2
#2183	2
#2175	2
C₄×SD₃₂	2