Subgroups — C₂×#13459 #499209

Order 512.

810 conjugacy classes of subgroups.

The subgroup lattice has 315 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#3162	C₂	1	1	yes	maximal, characteristic
2	C₂×#2155	C₂	1	2	yes	maximal
3	C₂×#1946	C₂	1	1	yes	maximal, characteristic
4	#13459	C₂	1	8	yes	maximal, direct product factor
5	C₂×#1384	C₂	1	2	yes	maximal
6	C₂×#1375	C₂	1	1	yes	maximal, characteristic
7	C₂×#533	—	2	1	no	—
8	C₂²×M₃₂	C₂²	1	2	yes	—
9	#3162	—	2	4	no	—
10	#3162	—	2	2	no	—
11	#3162	C₂²	1	4	yes	—
12	C₂×#501	—	2	4	no	—
13	C₂²×#108	C₂²	1	1	yes	characteristic
14	C₂×#427	—	2	1	no	—
15	C₂×#427	C₂²	1	2	yes	—
16	C₂×#427	C₂²	1	1	yes	characteristic
17	C₂×#424	—	2	1	no	—
18	C₂²×#97	C₂²	1	1	yes	characteristic
19	#2155	—	2	4	no	—
20	#2155	C₂²	1	8	yes	—
21	C₂×#365	—	2	4	no	—
22	#1946	—	2	2	no	—
23	#1946	C₂²	1	4	yes	—
24	C₂×#339	—	2	1	no	—
25	#1384	C₂²	1	8	yes	—
26	#1375	C₂²	1	4	yes	—
27	C₂²×M₁₆	—	2	1	no	—
28	C₂³×C₈	D₄	1	1	yes	characteristic
29	C₂³×C₈	C₂³	1	1	yes	characteristic
30	#533	—	2	4	no	—
31	#533	—	2	4	no	—
32	#533	—	4	2	no	—
33	#533	—	4	2	no	—
34	C₂×C₄⋊Q₈	—	2	2	no	—
35	C₂×#125	—	2	2	no	—
36	C₂×C₂²⋊Q₈	—	4	2	no	—
37	C₂×C₄×Q₈	—	4	2	no	—
38	C₂×#117	—	2	1	no	—
39	C₂×#117	—	2	2	no	—
40	C₂²×C₄²	D₄	1	1	yes	characteristic
41	C₂×M₃₂	—	2	2	no	—
42	C₂×M₃₂	—	2	2	no	—
43	C₂×M₃₂	—	2	2	no	—
44	C₂×M₃₂	—	2	2	no	—
45	C₂×M₃₂	Q₈	1	4	yes	complemented
46	C₂×M₃₂	C₂³	1	4	yes	—
47	C₂²×C₁₆	Q₈	1	4	yes	complemented
48	#501	—	2	16	no	—
49	#501	—	2	16	no	—
50	C₂×#108	—	2	2	no	—
51	C₂×#108	—	2	2	no	—
52	C₂×#108	—	2	1	no	—
53	C₂×#108	C₂³	1	2	yes	—
54	#427	—	2	4	no	—
55	#427	—	2	2	no	—
56	#427	—	2	2	no	—
57	#427	—	2	4	no	—
58	#427	—	2	1	no	—
59	#427	—	2	1	no	—
60	#427	D₄	1	4	yes	—
61	#427	Q₈	1	2	yes	complemented
62	#427	C₂³	1	4	yes	—
63	#427	C₂³	1	2	yes	—
64	C₂×#107	—	2	2	no	—
65	C₂×#107	—	2	2	no	—
66	C₂×#107	—	2	1	no	—
67	C₂×#106	—	2	2	no	—
68	C₂×#106	D₄	1	4	yes	—
69	C₂×#106	Q₈	1	2	yes	complemented
70	#424	—	2	4	no	—
71	#424	—	4	2	no	—
72	C₂×C₄⋊C₈	—	4	2	no	—
73	C₂×#97	—	2	2	no	—
74	C₂×#97	—	2	2	no	—
75	C₂×#97	D₄	1	2	yes	—
76	C₂×#97	C₂³	1	2	yes	—
77	#365	—	2	16	no	—
78	#339	—	2	4	no	—
79	C₂²×Q₈	—	4	2	no	—
80	C₂³×C₄	—	2	1	no	—
81	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
82	C₂×M₁₆	—	2	2	no	—
83	C₂×M₁₆	—	4	1	no	—
84	C₂×M₁₆	—	4	1	no	—
85	C₂×M₁₆	—	4	1	no	—
86	C₂×M₁₆	—	4	1	no	—
87	C₂×M₁₆	—	4	1	no	—
88	C₂²×C₈	—	2	1	no	—
89	C₂²×C₈	—	2	1	no	—
90	C₂²×C₈	—	2	1	no	—
91	C₂²×C₈	—	2	1	no	—
92	C₂²×C₈	—	4	1	no	—
93	C₂²×C₈	SD₁₆	1	2	yes	—
94	C₂²×C₈	SD₁₆	1	2	yes	—
95	C₂²×C₈	C₂×D₄	1	2	yes	—
96	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₈	C₂×D₄	1	2	yes	—
98	C₂²×C₈	C₂×D₄	1	2	yes	—
99	C₂²×C₈	C₂×Q₈	1	2	yes	—
100	C₂²×C₈	C₂×Q₈	1	2	yes	—
101	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
102	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
103	C₂²×C₈	#41	1	1	yes	characteristic
104	C₂²×C₈	#41	1	1	yes	characteristic
105	C₂²×C₈	C₂⁴	1	1	yes	characteristic
106	C₄⋊Q₈	—	2	8	no	—
107	C₄⋊Q₈	—	4	4	no	—
108	#125	—	2	8	no	—
109	#125	—	4	4	no	—
110	C₂²⋊Q₈	—	4	8	no	—
111	C₂²⋊Q₈	—	4	8	no	—
112	C₄×Q₈	—	4	8	no	—
113	C₄×Q₈	—	4	8	no	—
114	#117	—	2	4	no	—
115	#117	—	2	2	no	—
116	#117	—	2	4	no	—
117	#117	—	2	2	no	—
118	#117	—	4	2	no	—
119	#117	—	4	1	no	—
120	#117	—	4	1	no	—
121	#117	—	4	2	no	—
122	C₂×C₄⋊C₄	—	2	4	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	—	4	4	no	—
125	C₂×C₄⋊C₄	—	4	2	no	—
126	C₂×C₂²⋊C₄	—	4	1	no	—
127	C₂×C₂²⋊C₄	—	4	2	no	—
128	C₂×C₄²	—	2	1	no	—
129	C₂×C₄²	—	2	1	no	—
130	C₂×C₄²	—	2	1	no	—
131	C₂×C₄²	—	4	2	no	—
132	C₂×C₄²	—	4	1	no	—
133	C₂×C₄²	SD₁₆	1	2	yes	—
134	C₂×C₄²	SD₁₆	1	2	yes	—
135	C₂×C₄²	C₂×D₄	1	2	yes	—
136	M₃₂	—	2	4	no	—
137	M₃₂	—	2	4	no	—
138	M₃₂	—	2	4	no	—
139	M₃₂	—	2	4	no	—
140	C₂×C₁₆	—	2	4	no	—
141	C₂×C₁₆	—	2	4	no	—
142	C₂×C₁₆	C₂×Q₈	1	8	yes	complemented
143	#108	—	2	4	no	—
144	#108	—	4	2	no	—
145	#107	—	2	8	no	—
146	#107	—	2	4	no	—
147	#107	—	2	2	no	—
148	#107	—	2	4	no	—
149	#107	—	2	2	no	—
150	#106	—	2	8	no	—
151	#106	—	2	4	no	—
152	#106	—	2	2	no	—
153	#106	C₂×D₄	1	8	yes	—
154	#106	C₂×Q₈	1	4	yes	complemented
155	C₄⋊C₈	—	4	8	no	—
156	#97	—	2	4	no	—
157	#97	—	2	4	no	—
158	C₂⁴	AGL(1,ℤ/8)	1	1	yes	characteristic
159	C₂×Q₈	—	4	8	no	—
160	C₂×Q₈	—	8	8	no	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	—	4	4	no	—
165	C₂²×C₄	—	4	2	no	—
166	C₂²×C₄	—	8	2	no	—
167	C₂²×C₄	#120	1	1	yes	characteristic
168	C₂²×C₄	#120	1	1	yes	characteristic
169	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
170	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
171	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
172	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
173	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
174	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
175	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
176	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
177	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
178	C₂²×C₄	#137	1	1	yes	characteristic
179	C₂²×C₄	#137	1	1	yes	characteristic
180	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
181	M₁₆	—	4	2	no	—
182	M₁₆	—	4	2	no	—
183	M₁₆	—	8	1	no	—
184	M₁₆	—	8	1	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	8	1	no	—
193	C₂×C₈	—	8	1	no	—
194	C₂×C₈	#120	1	2	yes	—
195	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
196	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
197	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
198	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
199	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
200	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
201	C₂×C₈	C₄⋊Q₈	1	2	yes	—
202	C₂×C₈	C₄⋊Q₈	1	2	yes	—
203	C₂×C₈	C₄⋊Q₈	1	2	yes	—
204	C₂×C₈	C₄⋊Q₈	1	2	yes	—
205	C₂×C₈	C₄⋊Q₈	1	2	yes	—
206	C₂×C₈	C₂×SD₁₆	1	4	yes	—
207	C₂×C₈	C₂²×D₄	1	2	yes	—
208	C₂×C₈	C₂²×Q₈	1	2	yes	—
209	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
210	C₂×C₈	C₂×#41	1	1	yes	characteristic
211	C₄⋊C₄	—	2	8	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	4	16	no	—
214	C₄⋊C₄	—	4	8	no	—
215	C₄⋊C₄	—	4	4	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₂²⋊C₄	—	4	2	no	—
218	C₂²⋊C₄	—	4	4	no	—
219	C₂²⋊C₄	—	4	4	no	—
220	C₂²⋊C₄	—	4	2	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	4	2	no	—
225	C₄²	—	4	4	no	—
226	C₄²	—	4	4	no	—
227	C₄²	—	4	2	no	—
228	C₄²	C₂×SD₁₆	1	4	yes	—
229	C₁₆	—	2	8	no	—
230	C₂³	—	4	2	no	—
231	C₂³	#450	1	1	yes	characteristic
232	C₂³	#450	1	1	yes	characteristic
233	C₂³	#476	1	1	yes	characteristic
234	C₂³	#476	1	1	yes	characteristic
235	C₂³	#501	1	1	yes	characteristic
236	C₂³	#501	1	1	yes	characteristic
237	C₂³	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
238	Q₈	—	8	16	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	8	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	8	2	no	—
244	C₂×C₄	—	8	2	no	—
245	C₂×C₄	—	8	4	no	—
246	C₂×C₄	—	8	8	no	—
247	C₂×C₄	#395	1	1	yes	characteristic
248	C₂×C₄	#395	1	1	yes	characteristic
249	C₂×C₄	#395	1	1	yes	characteristic
250	C₂×C₄	#395	1	1	yes	characteristic
251	C₂×C₄	#448	1	1	yes	characteristic
252	C₂×C₄	#448	1	1	yes	characteristic
253	C₂×C₄	#448	1	1	yes	characteristic
254	C₂×C₄	#448	1	1	yes	characteristic
255	C₂×C₄	#450	1	1	yes	characteristic
256	C₂×C₄	#450	1	1	yes	characteristic
257	C₂×C₄	#475	1	1	yes	characteristic
258	C₂×C₄	#475	1	1	yes	characteristic
259	C₂×C₄	#475	1	1	yes	characteristic
260	C₂×C₄	#475	1	1	yes	characteristic
261	C₂×C₄	#476	1	1	yes	characteristic
262	C₂×C₄	#476	1	1	yes	characteristic
263	C₂×C₄	#498	1	1	yes	characteristic
264	C₂×C₄	#498	1	1	yes	characteristic
265	C₂×C₄	#498	1	1	yes	characteristic
266	C₂×C₄	#498	1	1	yes	characteristic
267	C₂×C₄	#501	1	1	yes	characteristic
268	C₂×C₄	#501	1	1	yes	characteristic
269	C₂×C₄	C₂×#120	1	1	yes	characteristic
270	C₂×C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
271	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
272	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
273	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
274	C₂×C₄	C₂×#137	1	1	yes	characteristic
275	C₈	—	2	4	no	—
276	C₈	—	8	2	no	—
277	C₈	#395	1	2	yes	—
278	C₈	#395	1	2	yes	—
279	C₈	C₂×C₂²⋊Q₈	1	2	yes	—
280	C₈	C₂×C₄⋊Q₈	1	2	yes	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	1	no	—
287	C₂²	—	4	1	no	—
288	C₂²	#2058	1	1	yes	characteristic
289	C₂²	#2058	1	1	yes	characteristic
290	C₂²	#2058	1	1	yes	characteristic
291	C₂²	#2058	1	1	yes	characteristic
292	C₂²	C₂×#450	1	1	yes	characteristic
293	C₂²	C₂×#476	1	1	yes	characteristic
294	C₂²	C₂×#501	1	1	yes	characteristic
295	C₄	—	4	4	no	—
296	C₄	—	8	4	no	—
297	C₄	—	8	8	no	—
298	C₄	#2058	1	1	yes	characteristic
299	C₄	#2058	1	1	yes	characteristic
300	C₄	#2058	1	1	yes	characteristic
301	C₄	#2058	1	1	yes	characteristic
302	C₄	C₂×#395	1	1	yes	characteristic
303	C₄	C₂×#448	1	1	yes	characteristic
304	C₄	C₂×#475	1	1	yes	characteristic
305	C₄	C₂×#498	1	1	yes	characteristic
306	C₂	—	2	1	no	—
307	C₂	—	2	1	no	—
308	C₂	—	2	1	no	—
309	C₂	—	2	1	no	—
310	C₂	—	2	1	no	—
311	C₂	—	2	1	no	—
312	C₂	#13459	1	2	yes	direct product factor
313	C₂	C₂×#2058	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#498
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	#2058
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #13459	C₂	direct (N × H)
45. C₂×M₃₂	Q₈	semidirect
47. C₂²×C₁₆	Q₈	semidirect
61. #427	Q₈	semidirect
69. C₂×#106	Q₈	semidirect
142. C₂×C₁₆	C₂×Q₈	semidirect
154. #106	C₂×Q₈	semidirect
312. C₂	#13459	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#3162	2
C₂×#2155	2
C₂×#1946	2
#13459	2
C₂×#1384	2
C₂×#1375	2

Subgroups — C2×#13459 #499209

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13459 #499209