Subgroups — #21137

Order 256.

400 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#135	C₂	1	1	yes	maximal, characteristic
2	#2922	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#131	C₂	1	1	yes	maximal, characteristic, complemented
4	#1929	C₂	1	2	yes	maximal, complemented
5	#1922	C₂	1	2	yes	maximal, complemented
6	#1876	C₂	1	2	yes	maximal
7	#1866	C₂	1	2	yes	maximal, complemented
8	C₄×#108	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#104	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#104	C₂	1	1	yes	maximal, characteristic, complemented
11	#1744	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135	C₂²	1	1	yes	characteristic
13	C₂×#131	C₂²	1	2	yes	—
14	C₄×#41	C₂²	1	1	yes	characteristic
15	C₄×#41	C₂²	1	1	yes	characteristic
16	C₄×Q₁₆	C₄	1	1	yes	characteristic, complemented
17	C₄×Q₁₆	C₄	1	1	yes	characteristic, complemented
18	C₄×SD₁₆	C₄	1	1	yes	characteristic, complemented
19	C₄×SD₁₆	C₄	1	1	yes	characteristic, complemented
20	C₄×SD₁₆	C₄	1	1	yes	characteristic, complemented
21	C₄×SD₁₆	C₄	1	1	yes	characteristic, complemented
22	C₄×D₈	C₄	1	1	yes	characteristic, complemented
23	C₄×D₈	C₄	1	1	yes	characteristic, complemented
24	#433	—	2	2	no	—
25	#431	—	2	1	no	—
26	#431	—	2	1	no	—
27	C₂×#108	C₂²	1	2	yes	—
28	#427	C₂²	1	1	yes	characteristic
29	#424	C₂²	1	2	yes	—
30	C₂×#104	C₂²	1	2	yes	—
31	C₂×#104	C₂²	1	2	yes	—
32	#416	C₂²	1	1	yes	characteristic
33	#416	C₂²	1	1	yes	characteristic
34	#408	C₂²	1	2	yes	—
35	#405	—	2	2	no	—
36	#405	C₂²	1	2	yes	—
37	C₄×M₁₆	—	2	1	no	—
38	C₄×M₁₆	C₂²	1	1	yes	characteristic
39	C₄×M₁₆	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈	—	2	1	no	—
41	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
42	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
43	C₄³	C₂²	1	1	yes	characteristic
44	#341	C₂²	1	2	yes	—
45	#339	C₂²	1	2	yes	—
46	#339	C₂²	1	2	yes	—
47	#336	C₂²	1	2	yes	—
48	#336	C₂²	1	2	yes	—
49	C₂×#41	C₂³	1	1	yes	characteristic
50	C₂×#41	C₂³	1	1	yes	characteristic
51	#135	—	2	4	no	—
52	C₂×Q₁₆	C₂×C₄	1	1	yes	characteristic
53	C₂×SD₁₆	C₂×C₄	1	1	yes	characteristic
54	C₂×SD₁₆	C₂×C₄	1	1	yes	characteristic
55	C₂×D₈	C₂×C₄	1	1	yes	characteristic
56	#131	—	2	4	no	—
57	#131	—	2	4	no	—
58	C₂×M₁₆	—	2	2	no	—
59	C₂×M₁₆	C₂³	1	2	yes	—
60	C₂×M₁₆	C₂³	1	2	yes	—
61	C₂²×C₈	—	2	2	no	—
62	C₂²×C₈	C₂³	1	1	yes	characteristic
63	C₂²×C₈	C₂³	1	1	yes	characteristic
64	C₂²×C₈	C₂³	1	2	yes	—
65	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
66	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
67	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
68	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
69	C₄×D₄	—	2	1	no	—
70	C₄×D₄	—	2	1	no	—
71	C₄×D₄	—	2	1	no	—
72	C₄×D₄	—	2	1	no	—
73	C₄×D₄	C₂×C₄	1	1	yes	characteristic
74	C₄×D₄	C₂×C₄	1	1	yes	characteristic
75	C₄×D₄	C₂×C₄	1	1	yes	characteristic
76	C₄×D₄	C₂×C₄	1	1	yes	characteristic
77	#117	—	2	1	no	—
78	#117	—	2	1	no	—
79	#117	C₂³	1	1	yes	characteristic
80	#117	C₂³	1	1	yes	characteristic
81	C₂×C₄²	—	2	1	no	—
82	C₂×C₄²	—	2	1	no	—
83	C₂×C₄²	—	2	1	no	—
84	C₂×C₄²	—	2	1	no	—
85	C₂×C₄²	C₂³	1	2	yes	—
86	C₂×C₄²	C₂³	1	1	yes	characteristic
87	#108	—	2	4	no	—
88	#107	C₂×C₄	1	1	yes	characteristic
89	#107	C₂×C₄	1	1	yes	characteristic
90	#106	C₂×C₄	1	1	yes	characteristic
91	#106	C₂×C₄	1	1	yes	characteristic
92	C₄⋊C₈	—	2	2	no	—
93	C₄⋊C₈	—	2	2	no	—
94	#104	—	2	4	no	—
95	#104	—	2	4	no	—
96	#103	C₂×C₄	1	1	yes	characteristic
97	#103	C₂×C₄	1	1	yes	characteristic
98	#103	C₂×C₄	1	1	yes	characteristic
99	#103	C₂×C₄	1	1	yes	characteristic
100	#102	C₂×C₄	1	1	yes	characteristic
101	#102	C₂×C₄	1	1	yes	characteristic
102	#102	C₂×C₄	1	1	yes	characteristic
103	#102	C₂×C₄	1	1	yes	characteristic
104	C₂²⋊C₈	—	2	2	no	—
105	C₂²⋊C₈	—	2	2	no	—
106	#97	—	2	2	no	—
107	#97	—	2	1	no	—
108	#97	—	2	1	no	—
109	C₄×C₈	—	2	2	no	—
110	C₄×C₈	—	2	1	no	—
111	C₄×C₈	—	2	1	no	—
112	C₄×C₈	C₂×C₄	1	1	yes	characteristic
113	C₄×C₈	C₂×C₄	1	1	yes	characteristic
114	C₄×C₈	C₂×C₄	1	1	yes	characteristic
115	C₄×C₈	C₂×C₄	1	1	yes	characteristic
116	C₄×C₈	D₄	1	2	yes	—
117	C₄×C₈	D₄	1	1	yes	characteristic
118	C₄×C₈	D₄	1	1	yes	characteristic
119	#41	—	2	2	no	—
120	#41	—	2	2	no	—
121	#41	—	2	2	no	—
122	#41	#41	1	4	yes	—
123	C₂×Q₈	C₂²×C₄	1	1	yes	characteristic
124	C₂×Q₈	C₂²×C₄	1	1	yes	characteristic
125	C₂×D₄	—	2	1	no	—
126	C₂×D₄	—	2	1	no	—
127	C₂×D₄	C₂²×C₄	1	1	yes	characteristic
128	C₂×D₄	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄	—	2	1	no	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	—	2	1	no	—
133	C₂²×C₄	—	2	1	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	#41	1	1	yes	characteristic
137	C₂²×C₄	#41	1	1	yes	characteristic
138	C₂²×C₄	C₂⁴	1	1	yes	characteristic
139	Q₁₆	—	2	2	no	—
140	SD₁₆	—	2	2	no	—
141	SD₁₆	—	2	2	no	—
142	D₈	—	2	2	no	—
143	M₁₆	—	2	8	no	—
144	M₁₆	—	2	2	no	—
145	M₁₆	—	2	2	no	—
146	M₁₆	—	2	2	no	—
147	M₁₆	#41	1	4	yes	—
148	C₂×C₈	—	2	8	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
156	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
157	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
158	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
159	C₂×C₈	C₂×D₄	1	2	yes	—
160	C₂×C₈	C₂×D₄	1	2	yes	—
161	C₄⋊C₄	—	2	1	no	—
162	C₄⋊C₄	—	2	1	no	—
163	C₄⋊C₄	—	2	1	no	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
166	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
167	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
168	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
169	C₂²⋊C₄	—	2	1	no	—
170	C₂²⋊C₄	—	2	1	no	—
171	C₂²⋊C₄	—	2	1	no	—
172	C₂²⋊C₄	—	2	1	no	—
173	C₂²⋊C₄	—	2	1	no	—
174	C₂²⋊C₄	—	2	1	no	—
175	C₄²	—	2	4	no	—
176	C₄²	—	2	1	no	—
177	C₄²	—	2	1	no	—
178	C₄²	—	2	1	no	—
179	C₄²	—	2	1	no	—
180	C₄²	—	2	1	no	—
181	C₄²	—	2	1	no	—
182	C₄²	—	2	2	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	4	2	no	—
185	C₄²	C₂²×C₄	1	1	yes	characteristic
186	C₄²	C₂²×C₄	1	1	yes	characteristic
187	C₄²	C₂×D₄	1	1	yes	characteristic
188	C₄²	C₂×D₄	1	1	yes	characteristic
189	C₂³	—	2	1	no	—
190	C₂³	—	2	1	no	—
191	C₂³	C₂×#41	1	1	yes	characteristic
192	Q₈	—	2	1	no	—
193	Q₈	—	2	1	no	—
194	Q₈	—	2	1	no	—
195	Q₈	#117	1	2	yes	—
196	D₄	—	2	4	no	—
197	D₄	—	2	1	no	—
198	D₄	—	2	1	no	—
199	D₄	—	2	1	no	—
200	D₄	—	4	2	no	—
201	D₄	#117	1	2	yes	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	C₄×D₄	1	1	yes	characteristic
220	C₂×C₄	C₄×D₄	1	1	yes	characteristic
221	C₂×C₄	C₄×D₄	1	1	yes	characteristic
222	C₂×C₄	C₄×D₄	1	1	yes	characteristic
223	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
224	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₄	C₂×#41	1	2	yes	—
226	C₂×C₄	C₂×#41	1	2	yes	—
227	C₈	—	2	1	no	—
228	C₈	—	2	1	no	—
229	C₈	—	2	1	no	—
230	C₈	—	2	1	no	—
231	C₈	—	2	2	no	—
232	C₈	—	2	4	no	—
233	C₈	—	2	2	no	—
234	C₈	—	4	2	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	4	1	no	—
239	C₂²	—	4	1	no	—
240	C₂²	—	4	1	no	—
241	C₂²	—	4	1	no	—
242	C₂²	—	4	1	no	—
243	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
244	C₂²	#529	1	2	yes	—
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	2	2	no	—
248	C₄	—	4	4	no	—
249	C₄	—	4	1	no	—
250	C₄	—	4	1	no	—
251	C₄	—	4	1	no	—
252	C₄	—	4	1	no	—
253	C₄	—	4	1	no	—
254	C₄	#443	1	2	yes	—
255	C₄	#444	1	2	yes	—
256	C₄	C₂×#117	1	1	yes	characteristic
257	C₄	C₄×#41	1	1	yes	characteristic
258	C₄	#525	1	2	yes	—
259	C₂	—	2	2	no	—
260	C₂	—	4	2	no	—
261	C₂	—	8	1	no	—
262	C₂	#2290	1	1	yes	characteristic
263	C₂	C₂×#443	1	1	yes	characteristic
264	C₂	C₂×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂²	C₂×C₄×D₄
prime radical	C₂×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #2922	C₂	semidirect
3. C₄×#131	C₂	semidirect
4. #1929	C₂	semidirect
5. #1922	C₂	semidirect
7. #1866	C₂	semidirect
8. C₄×#108	C₂	semidirect
9. C₄×#104	C₂	semidirect
10. C₄×#104	C₂	semidirect
11. #1744	C₂	semidirect
16. C₄×Q₁₆	C₄	semidirect
17. C₄×Q₁₆	C₄	semidirect
18. C₄×SD₁₆	C₄	semidirect
19. C₄×SD₁₆	C₄	semidirect
20. C₄×SD₁₆	C₄	semidirect
21. C₄×SD₁₆	C₄	semidirect
22. C₄×D₈	C₄	semidirect
23. C₄×D₈	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₄×#135	2
#2922	2
C₄×#131	2
#1929	2
#1922	2
#1876	2
#1866	2
C₄×#108	2
C₄×#104	2
#1744	2