Subgroups — C₄×#2230 #495997

Order 512.

586 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2230	C₂	1	1	yes	maximal, characteristic
2	C₄×#498	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#443	C₂	1	1	yes	maximal, characteristic
4	#13272	C₂	1	1	yes	maximal, characteristic, complemented
5	#13186	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116	C₂	1	2	yes	maximal, complemented
7	#12950	C₂	1	2	yes	maximal
8	#12823	C₂	1	1	yes	maximal, characteristic
9	C₄×#365	C₂	1	2	yes	maximal, complemented
10	C₄×#364	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#360	C₂	1	2	yes	maximal
12	C₂×#498	C₂²	1	1	yes	characteristic
13	C₂×#443	C₂²	1	1	yes	characteristic
14	C₂×#443	C₂²	1	1	yes	characteristic
15	C₄×#135	C₂²	1	1	yes	characteristic
16	C₄×#131	—	2	1	no	—
17	C₄×C₄⋊Q₈	—	2	1	no	—
18	#2230	C₄	1	8	yes	direct product factor
19	C₂×#365	C₂²	1	2	yes	—
20	C₂×#365	C₂²	1	2	yes	—
21	C₂×#364	C₂²	1	1	yes	characteristic
22	C₂×#364	C₂²	1	1	yes	characteristic
23	C₂×#360	C₂²	1	2	yes	—
24	C₂×#360	C₂²	1	2	yes	—
25	C₄×M₃₂	C₂²	1	2	yes	—
26	#1990	—	2	1	no	—
27	#1941	—	2	1	no	—
28	#1941	C₂²	1	2	yes	—
29	#1844	C₂²	1	1	yes	characteristic
30	C₄×#108	C₂²	1	1	yes	characteristic
31	C₄×#106	—	2	1	no	—
32	C₄×#106	C₂²	1	2	yes	—
33	#1772	C₂²	1	1	yes	characteristic
34	#1772	C₂²	1	1	yes	characteristic
35	#1764	—	2	1	no	—
36	#1764	—	2	1	no	—
37	C₄×#104	—	2	1	no	—
38	C₄²×C₈	C₂²	1	1	yes	characteristic
39	#1386	C₂²	1	2	yes	—
40	#1386	C₂²	1	2	yes	—
41	#1383	C₂²	1	2	yes	—
42	#1383	C₂²	1	2	yes	—
43	#1376	C₂²	1	2	yes	—
44	C₂×#135	C₂³	1	1	yes	characteristic
45	C₂×#131	—	2	1	no	—
46	C₂×#131	—	2	1	no	—
47	C₂×C₄⋊Q₈	—	2	1	no	—
48	C₄×#41	—	2	1	no	—
49	C₂×C₄×Q₈	—	4	1	no	—
50	C₂×M₃₂	C₂³	1	2	yes	—
51	C₂×M₃₂	C₂³	1	2	yes	—
52	#498	—	2	2	no	—
53	#498	C₂×C₄	1	4	yes	complemented
54	#443	—	2	2	no	—
55	#443	—	2	2	no	—
56	#443	—	2	2	no	—
57	#443	C₂×C₄	1	4	yes	—
58	C₄×Q₁₆	Q₈	1	1	yes	characteristic, complemented
59	C₄×Q₁₆	Q₈	1	1	yes	characteristic, complemented
60	C₄×SD₁₆	—	2	1	no	—
61	C₄×SD₁₆	—	2	1	no	—
62	C₄×D₈	Q₈	1	1	yes	characteristic, complemented
63	C₄×D₈	Q₈	1	1	yes	characteristic, complemented
64	C₂×#108	C₂³	1	1	yes	characteristic
65	C₂×#108	C₂³	1	1	yes	characteristic
66	#427	C₂³	1	1	yes	characteristic
67	C₂×#106	—	2	1	no	—
68	C₂×#106	—	2	1	no	—
69	C₂×#106	C₂³	1	2	yes	—
70	C₂×#106	C₂³	1	2	yes	—
71	C₂×#104	—	2	1	no	—
72	C₂×#104	—	2	1	no	—
73	#416	—	2	1	no	—
74	#405	—	2	1	no	—
75	#405	—	2	1	no	—
76	#405	—	4	1	no	—
77	#405	—	4	1	no	—
78	C₄×M₁₆	—	2	1	no	—
79	C₄×M₁₆	—	4	1	no	—
80	C₂×C₄×C₈	—	4	1	no	—
81	C₂×C₄×C₈	D₄	1	1	yes	characteristic
82	C₂×C₄×C₈	D₄	1	1	yes	characteristic
83	C₂×C₄×C₈	D₄	1	1	yes	characteristic
84	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
85	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
86	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
87	#391	—	4	1	no	—
88	#389	—	2	2	no	—
89	#389	—	4	1	no	—
90	#384	—	2	1	no	—
91	C₄×C₄⋊C₄	—	2	2	no	—
92	C₄×C₄⋊C₄	—	4	1	no	—
93	C₄³	D₄	1	1	yes	characteristic
94	#365	—	2	4	no	—
95	#365	C₂×C₄	1	8	yes	complemented
96	#364	—	2	2	no	—
97	#364	C₂×C₄	1	4	yes	complemented
98	#360	—	2	4	no	—
99	#360	C₂×C₄	1	8	yes	—
100	#346	—	2	2	no	—
101	C₄×C₁₆	—	2	2	no	—
102	#341	—	2	1	no	—
103	#340	—	2	2	no	—
104	#340	—	2	2	no	—
105	#339	—	2	1	no	—
106	#339	—	2	1	no	—
107	C₂×#41	—	2	1	no	—
108	C₂²×Q₈	—	4	1	no	—
109	#135	—	2	1	no	—
110	#135	—	2	1	no	—
111	#135	C₂²×C₄	1	2	yes	—
112	#135	#41	1	2	yes	—
113	C₂×Q₁₆	C₂×Q₈	1	1	yes	characteristic
114	C₂×SD₁₆	—	2	1	no	—
115	C₂×D₈	C₂×Q₈	1	1	yes	characteristic
116	#131	—	2	4	no	—
117	#131	—	2	4	no	—
118	#131	—	4	2	no	—
119	#131	—	4	2	no	—
120	C₂×M₁₆	—	2	1	no	—
121	C₂×M₁₆	—	2	1	no	—
122	C₂×M₁₆	—	4	1	no	—
123	C₂×M₁₆	—	4	1	no	—
124	C₂²×C₈	—	4	1	no	—
125	C₂²×C₈	—	4	1	no	—
126	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₈	C₂⁴	1	1	yes	characteristic
130	C₄⋊Q₈	—	2	4	no	—
131	C₄⋊Q₈	—	4	2	no	—
132	C₄×Q₈	—	2	1	no	—
133	C₄×Q₈	—	2	1	no	—
134	C₄×Q₈	—	8	4	no	—
135	C₄×D₄	—	2	1	no	—
136	C₄×D₄	—	2	1	no	—
137	C₄×D₄	—	4	1	no	—
138	C₄×D₄	—	4	1	no	—
139	#117	—	2	1	no	—
140	#117	—	4	1	no	—
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₄⋊C₄	—	2	2	no	—
143	C₂×C₄⋊C₄	—	4	1	no	—
144	C₂×C₄⋊C₄	—	4	1	no	—
145	C₂×C₄⋊C₄	—	4	2	no	—
146	C₂×C₄⋊C₄	—	4	1	no	—
147	C₂×C₄²	—	2	1	no	—
148	C₂×C₄²	—	2	1	no	—
149	C₂×C₄²	—	4	1	no	—
150	C₂×C₄²	—	4	2	no	—
151	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
152	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
153	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
154	M₃₂	—	2	2	no	—
155	M₃₂	—	2	2	no	—
156	M₃₂	—	2	2	no	—
157	M₃₂	C₂²×C₄	1	4	yes	—
158	C₂×C₁₆	—	2	2	no	—
159	C₂×C₁₆	—	2	2	no	—
160	#108	—	2	1	no	—
161	#108	—	2	1	no	—
162	#108	C₂²×C₄	1	2	yes	—
163	#108	#41	1	2	yes	—
164	#107	C₂×Q₈	1	1	yes	characteristic
165	#107	C₂×Q₈	1	1	yes	characteristic
166	#106	—	2	4	no	—
167	#106	—	2	2	no	—
168	#106	—	2	1	no	—
169	#106	—	2	2	no	—
170	#106	—	2	2	no	—
171	#106	—	4	2	no	—
172	#106	C₂²×C₄	1	4	yes	—
173	#104	—	2	4	no	—
174	#104	—	2	4	no	—
175	#103	—	2	1	no	—
176	#103	—	2	1	no	—
177	#102	—	2	1	no	—
178	#102	—	2	1	no	—
179	#97	—	4	1	no	—
180	#97	—	4	1	no	—
181	#97	—	4	1	no	—
182	C₄×C₈	—	2	2	no	—
183	C₄×C₈	—	2	1	no	—
184	C₄×C₈	—	2	1	no	—
185	C₄×C₈	—	4	1	no	—
186	C₄×C₈	—	4	1	no	—
187	C₄×C₈	—	4	1	no	—
188	C₄×C₈	—	4	1	no	—
189	C₄×C₈	SD₁₆	1	1	yes	characteristic
190	C₄×C₈	SD₁₆	1	1	yes	characteristic
191	C₄×C₈	SD₁₆	1	1	yes	characteristic
192	C₄×C₈	SD₁₆	1	1	yes	characteristic
193	C₄×C₈	C₂²×C₄	1	2	yes	—
194	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
195	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
196	C₄×C₈	#41	1	2	yes	—
197	C₄×C₈	#41	1	1	yes	characteristic
198	C₄×C₈	#41	1	1	yes	characteristic
199	#95	—	4	2	no	—
200	#95	—	4	2	no	—
201	#41	—	2	2	no	—
202	#41	—	2	2	no	—
203	#41	—	4	1	no	—
204	#41	—	4	1	no	—
205	C₂×Q₈	—	2	1	no	—
206	C₂×Q₈	—	4	4	no	—
207	C₂×Q₈	—	8	2	no	—
208	C₂×Q₈	—	8	2	no	—
209	C₂×D₄	—	2	1	no	—
210	C₂×D₄	—	4	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	4	1	no	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
219	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
220	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
221	Q₁₆	—	2	1	no	—
222	Q₁₆	C₄×Q₈	1	2	yes	complemented
223	SD₁₆	—	2	2	no	—
224	SD₁₆	—	2	2	no	—
225	D₈	—	2	1	no	—
226	D₈	C₄×Q₈	1	2	yes	complemented
227	M₁₆	—	2	2	no	—
228	M₁₆	—	2	2	no	—
229	M₁₆	—	4	4	no	—
230	M₁₆	—	4	1	no	—
231	M₁₆	—	4	1	no	—
232	M₁₆	—	4	4	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	4	4	no	—
238	C₂×C₈	—	4	1	no	—
239	C₂×C₈	—	4	1	no	—
240	C₂×C₈	—	4	4	no	—
241	C₂×C₈	C₄×D₄	1	2	yes	—
242	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
243	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
244	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
245	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
246	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
247	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
248	C₂×C₈	#135	1	2	yes	—
249	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
250	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
251	C₂×C₈	C₂×#41	1	1	yes	characteristic
252	C₂×C₈	C₂×#41	1	1	yes	characteristic
253	C₂×C₈	C₂×#41	1	1	yes	characteristic
254	C₂×C₈	C₂×#41	1	1	yes	characteristic
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	4	4	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	8	2	no	—
265	C₄⋊C₄	—	8	2	no	—
266	C₄⋊C₄	—	8	4	no	—
267	C₂²⋊C₄	—	4	1	no	—
268	C₂²⋊C₄	—	4	1	no	—
269	C₂²⋊C₄	—	4	1	no	—
270	C₄²	—	2	4	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	2	1	no	—
273	C₄²	—	2	1	no	—
274	C₄²	—	4	2	no	—
275	C₄²	—	4	1	no	—
276	C₄²	—	4	1	no	—
277	C₄²	—	4	1	no	—
278	C₄²	—	8	4	no	—
279	C₄²	C₄×D₄	1	2	yes	—
280	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
281	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
282	C₄²	C₂×SD₁₆	1	1	yes	characteristic
283	C₄²	C₂×SD₁₆	1	1	yes	characteristic
284	C₁₆	—	2	4	no	—
285	C₁₆	—	4	2	no	—
286	C₂³	—	4	1	no	—
287	C₂³	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
288	Q₈	—	2	2	no	—
289	Q₈	—	4	1	no	—
290	Q₈	—	8	8	no	—
291	D₄	—	2	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	2	no	—
294	D₄	—	4	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	8	2	no	—
310	C₂×C₄	—	8	2	no	—
311	C₂×C₄	—	8	2	no	—
312	C₂×C₄	—	8	2	no	—
313	C₂×C₄	—	8	2	no	—
314	C₂×C₄	—	8	2	no	—
315	C₂×C₄	—	8	2	no	—
316	C₂×C₄	#476	1	1	yes	characteristic
317	C₂×C₄	#476	1	1	yes	characteristic
318	C₂×C₄	#476	1	1	yes	characteristic
319	C₂×C₄	#476	1	1	yes	characteristic
320	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
321	C₂×C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
322	C₂×C₄	#525	1	1	yes	characteristic
323	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
324	C₂×C₄	C₂×#135	1	1	yes	characteristic
325	C₂×C₄	#575	1	1	yes	characteristic
326	C₈	—	2	1	no	—
327	C₈	—	2	1	no	—
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₈	—	4	2	no	—
331	C₈	—	4	2	no	—
332	C₈	C₄×SD₁₆	1	2	yes	—
333	C₈	C₄×SD₁₆	1	2	yes	—
334	C₈	C₂×C₄×Q₈	1	1	yes	characteristic
335	C₈	C₄×#41	1	1	yes	characteristic
336	C₈	#529	1	1	yes	characteristic
337	C₈	#533	1	1	yes	characteristic
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	4	2	no	—
341	C₂²	—	8	1	no	—
342	C₂²	—	8	1	no	—
343	C₂²	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
344	C₂²	C₂×#476	1	1	yes	characteristic
345	C₂²	#3106	1	1	yes	characteristic
346	C₄	—	2	4	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	4	2	no	—
351	C₄	—	4	2	no	—
352	C₄	—	8	4	no	—
353	C₄	—	8	1	no	—
354	C₄	—	8	1	no	—
355	C₄	—	16	2	no	—
356	C₄	#2230	1	2	yes	direct product factor
357	C₄	C₄×C₂²⋊Q₈	1	1	yes	characteristic
358	C₄	C₂×C₄×SD₁₆	1	1	yes	characteristic
359	C₄	#3076	1	1	yes	characteristic
360	C₄	#3101	1	1	yes	characteristic
361	C₂	—	2	1	no	—
362	C₂	—	2	1	no	—
363	C₂	—	8	2	no	—
364	C₂	C₄×#476	1	1	yes	characteristic
365	C₂	C₂×#2230	1	1	yes	characteristic
366	C₂	#33637	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#476
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#476
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₄×#498	C₂	semidirect
4. #13272	C₂	semidirect
5. #13186	C₂	semidirect
6. #13116	C₂	semidirect
9. C₄×#365	C₂	semidirect
10. C₄×#364	C₂	semidirect
18. #2230	C₄	direct (N × H)
53. #498	C₂×C₄	semidirect
58. C₄×Q₁₆	Q₈	semidirect
59. C₄×Q₁₆	Q₈	semidirect
62. C₄×D₈	Q₈	semidirect
63. C₄×D₈	Q₈	semidirect
95. #365	C₂×C₄	semidirect
97. #364	C₂×C₄	semidirect
222. Q₁₆	C₄×Q₈	semidirect
226. D₈	C₄×Q₈	semidirect
356. C₄	#2230	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×#2230	2
C₄×#498	2
C₄×#443	2
#13272	2
#13186	2
#13116	2
#12950	2
#12823	2
C₄×#365	2
C₄×#364	2
C₄×#360	2

Subgroups — C4×#2230 #495997

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#2230 #495997