Subgroups — C₂²×#16840504 #30586885

Order 15616.

10246 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840504	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#14519766	C₂	1	2	yes	maximal, complemented
3	C₂²×#14519765	C₂	1	2	yes	maximal, complemented
4	C₂²×#14519763	C₂	1	4	yes	maximal, complemented
5	C₂²×#14519747	C₂	1	4	yes	maximal, complemented
6	C₂²×#14519745	C₂	1	2	yes	maximal, complemented
7	C₂²×#124×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂⁵×D₆₁	C₂²	1	2	yes	complemented
9	C₂³×Dic₁₂₂	C₂²	1	2	yes	complemented
10	C₂³×D₂₄₄	C₂²	1	2	yes	—
11	C₂³×D₂₄₄	C₂²	1	2	yes	—
12	C₂³×C₂²⋊D₆₁	C₂²	1	4	yes	—
13	#16840504	C₂²	1	256	yes	direct product factor
14	C₂×#14519766	C₂²	1	48	yes	complemented
15	C₂×#14519765	C₂²	1	48	yes	complemented
16	C₂×#14519763	C₂²	1	96	yes	complemented
17	C₂×#14519747	C₂²	1	96	yes	complemented
18	C₂×#14519745	C₂²	1	48	yes	complemented
19	C₂²×C₂²⋊Dic₆₁	C₂²	1	2	yes	complemented
20	C₂²×#11758198	C₂²	1	4	yes	complemented
21	C₂²×#11758198	C₂²	1	4	yes	complemented
22	C₂²×#11758198	C₂²	1	4	yes	complemented
23	C₂²×C₄×Dic₆₁	C₂²	1	2	yes	complemented
24	C₂³×Q₈×C₆₁	C₂²	1	1	yes	characteristic, complemented
25	C₂³×D₄×C₆₁	C₂²	1	1	yes	characteristic
26	C₂×#124×C₆₁	C₂²	1	24	yes	complemented
27	C₂²×C₂²⋊C₄×C₆₁	C₂²	1	4	yes	complemented
28	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic, complemented
29	C₂⁴×D₆₁	—	4	16	no	—
30	C₂⁴×D₆₁	—	4	8	no	—
31	C₂⁴×D₆₁	C₂³	1	4	yes	—
32	C₂⁴×D₆₁	C₂³	1	24	yes	complemented
33	C₂²×Dic₁₂₂	—	4	8	no	—
34	C₂²×Dic₁₂₂	C₂³	1	24	yes	complemented
35	C₂²×D₂₄₄	—	4	8	no	—
36	C₂²×D₂₄₄	—	4	8	no	—
37	C₂²×D₂₄₄	C₂³	1	24	yes	—
38	C₂²×D₂₄₄	C₂³	1	24	yes	—
39	C₂²×C₂²⋊D₆₁	—	4	16	no	—
40	C₂²×C₂²⋊D₆₁	C₂³	1	48	yes	—
41	C₂³×Dic₆₁	C₂³	1	4	yes	—
42	#14519766	C₂³	1	128	yes	complemented
43	#14519765	C₂³	1	128	yes	complemented
44	#14519763	C₂³	1	256	yes	complemented
45	#14519747	C₂³	1	256	yes	complemented
46	#14519745	C₂³	1	128	yes	complemented
47	C₂×C₂²⋊Dic₆₁	C₂³	1	24	yes	complemented
48	C₂×#11758198	C₂³	1	48	yes	complemented
49	C₂×#11758198	C₂³	1	48	yes	complemented
50	C₂×#11758198	C₂³	1	48	yes	complemented
51	C₂×C₄×Dic₆₁	C₂³	1	24	yes	complemented
52	C₂⁵×C₆₁	C₂³	1	2	yes	—
53	C₂²×Q₈×C₆₁	—	4	4	no	—
54	C₂²×Q₈×C₆₁	C₂³	1	12	yes	complemented
55	C₂²×D₄×C₆₁	—	4	4	no	—
56	C₂²×D₄×C₆₁	C₂³	1	12	yes	—
57	C₂³×C₄×C₆₁	C₂³	1	2	yes	—
58	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
59	C₂³×C₄×C₆₁	C₂³	1	2	yes	—
60	#124×C₆₁	C₂³	1	64	yes	complemented
61	C₂×C₂²⋊C₄×C₆₁	C₂³	1	48	yes	complemented
62	C₂×C₄²×C₆₁	C₂³	1	12	yes	complemented
63	C₂⁴×C₆₁	—	2	16	no	—
64	C₂⁴×C₆₁	—	2	8	no	—
65	C₂⁴×C₆₁	C₂⁴	1	12	yes	—
66	C₂⁴×C₆₁	C₂⁴	1	1	yes	characteristic
67	C₂³×D₆₁	—	2	16	no	—
68	C₂³×D₆₁	—	2	16	no	—
69	C₂³×D₆₁	—	2	16	no	—
70	C₂³×D₆₁	—	4	48	no	—
71	C₂³×D₆₁	—	4	96	no	—
72	C₂³×D₆₁	—	4	128	no	—
73	C₂³×D₆₁	C₂⁴	1	24	yes	—
74	C₂³×D₆₁	C₂⁴	1	32	yes	complemented
75	C₂×Q₈×C₆₁	—	4	24	no	—
76	C₂×Q₈×C₆₁	C₂⁴	1	16	yes	complemented
77	C₂×D₄×C₆₁	—	4	24	no	—
78	C₂×D₄×C₆₁	C₂⁴	1	16	yes	—
79	C₂²×C₄×C₆₁	—	2	8	no	—
80	C₂²×C₄×C₆₁	—	2	8	no	—
81	C₂²×C₄×C₆₁	—	2	4	no	—
82	C₂²×C₄×C₆₁	C₂⁴	1	12	yes	—
83	C₂²×C₄×C₆₁	C₂⁴	1	12	yes	—
84	C₂²×C₄×C₆₁	C₂⁴	1	6	yes	—
85	C₂×C₂²⋊D₆₁	—	4	96	no	—
86	C₂×C₂²⋊D₆₁	C₂⁴	1	64	yes	—
87	C₂²×Dic₆₁	—	2	16	no	—
88	C₂²×Dic₆₁	C₂⁴	1	24	yes	—
89	C₂×D₂₄₄	—	4	48	no	—
90	C₂×D₂₄₄	—	4	48	no	—
91	C₂×D₂₄₄	C₂⁴	1	32	yes	—
92	C₂×D₂₄₄	C₂⁴	1	32	yes	—
93	C₂×Dic₁₂₂	—	4	48	no	—
94	C₂×Dic₁₂₂	C₂⁴	1	32	yes	complemented
95	C₂²⋊C₄×C₆₁	C₂⁴	1	64	yes	complemented
96	C₄²×C₆₁	C₂⁴	1	16	yes	complemented
97	C₂²⋊Dic₆₁	C₂⁴	1	32	yes	complemented
98	#11758198	C₂⁴	1	64	yes	complemented
99	#11758198	C₂⁴	1	64	yes	complemented
100	#11758198	C₂⁴	1	64	yes	complemented
101	C₄×Dic₆₁	C₂⁴	1	32	yes	complemented
102	C₂³×C₆₁	—	2	48	no	—
103	C₂³×C₆₁	—	2	24	no	—
104	C₂³×C₆₁	—	4	32	no	—
105	C₂³×C₆₁	H(5,𝔽₂)	1	4	yes	—
106	C₂³×C₆₁	H(5,𝔽₂)	1	8	yes	—
107	C₂³×C₆₁	C₂⁵	1	8	yes	—
108	C₂³×C₆₁	C₂⁵	1	3	yes	—
109	C₂²×D₆₁	—	2	48	no	—
110	C₂²×D₆₁	—	2	48	no	—
111	C₂²×D₆₁	—	2	48	no	—
112	C₂²×D₆₁	—	4	32	no	—
113	C₂²×D₆₁	—	4	64	no	—
114	C₂²×D₆₁	—	4	384	no	—
115	C₂²×D₆₁	—	4	64	no	—
116	C₂²×D₆₁	C₂⁵	1	16	yes	—
117	Q₈×C₆₁	—	4	16	no	—
118	D₄×C₆₁	—	4	16	no	—
119	C₂×C₄×C₆₁	—	2	24	no	—
120	C₂×C₄×C₆₁	—	2	24	no	—
121	C₂×C₄×C₆₁	—	2	12	no	—
122	C₂×C₄×C₆₁	C₂⁵	1	8	yes	—
123	C₂×C₄×C₆₁	C₂⁵	1	4	yes	—
124	C₂×C₄×C₆₁	C₂⁵	1	8	yes	—
125	C₂²⋊D₆₁	—	4	64	no	—
126	C₂×Dic₆₁	—	2	48	no	—
127	C₂×Dic₆₁	C₂⁵	1	16	yes	—
128	D₂₄₄	—	4	32	no	—
129	D₂₄₄	—	4	32	no	—
130	Dic₁₂₂	—	4	32	no	—
131	C₂²×H(3,𝔽₄)	—	61	1	no	maximal, Sylow
132	C₂²×C₆₁	—	2	16	no	—
133	C₂²×C₆₁	—	2	8	no	—
134	C₂²×C₆₁	—	4	48	no	—
135	C₂²×C₆₁	H(3,𝔽₄)	1	16	yes	complemented
136	C₂²×C₆₁	C₂×H(5,𝔽₂)	1	6	yes	—
137	C₂²×C₆₁	C₂×H(5,𝔽₂)	1	12	yes	—
138	C₂²×C₆₁	C₂⁶	1	1	yes	characteristic
139	C₂×D₆₁	—	2	16	no	—
140	C₂×D₆₁	—	2	16	no	—
141	C₂×D₆₁	—	2	16	no	—
142	C₂×D₆₁	—	4	128	no	—
143	C₂×D₆₁	—	4	96	no	—
144	C₄×C₆₁	—	2	8	no	—
145	C₄×C₆₁	—	2	8	no	—
146	C₄×C₆₁	—	2	4	no	—
147	Dic₆₁	—	2	16	no	—
148	C₂×H(3,𝔽₄)	—	61	48	no	—
149	C₂²×#124	—	61	2	no	—
150	C₂²×#124	—	61	2	no	—
151	C₂²×#124	—	61	4	no	—
152	C₂²×#124	D₆₁	1	1	yes	characteristic, complemented
153	C₂²×#120	—	61	2	no	—
154	C₂²×#120	—	61	4	no	—
155	C₂×C₆₁	—	4	8	no	—
156	C₂×C₆₁	C₂×H(3,𝔽₄)	1	12	yes	complemented
157	C₂×C₆₁	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
158	C₂×C₆₁	C₂²×H(5,𝔽₂)	1	2	yes	—
159	D₆₁	—	4	16	no	—
160	C₂⁶	—	61	2	no	—
161	C₂³×Q₈	—	61	2	no	—
162	C₂³×Q₈	C₂×D₆₁	1	1	yes	characteristic, complemented
163	C₂³×D₄	—	61	4	no	—
164	C₂³×D₄	—	61	2	no	—
165	C₂³×D₄	—	61	2	no	—
166	C₂³×D₄	C₂×D₆₁	1	1	yes	characteristic
167	H(3,𝔽₄)	—	61	256	no	—
168	C₂×#124	—	61	48	no	—
169	C₂×#124	—	61	96	no	—
170	C₂×#124	—	61	48	no	—
171	C₂×#124	C₂×D₆₁	1	24	yes	complemented
172	C₂×#120	—	61	48	no	—
173	C₂×#120	—	61	96	no	—
174	C₂²×C₂²⋊C₄	—	61	2	no	—
175	C₂²×C₂²⋊C₄	—	61	4	no	—
176	C₂²×C₂²⋊C₄	—	61	4	no	—
177	C₂²×C₂²⋊C₄	—	61	4	no	—
178	C₂²×C₂²⋊C₄	C₂×D₆₁	1	4	yes	complemented
179	C₂²×C₄²	—	61	2	no	—
180	C₂²×C₄²	C₂×D₆₁	1	1	yes	characteristic, complemented
181	C₆₁	C₂²×H(3,𝔽₄)	1	1	yes	characteristic, complemented, Sylow
182	C₂⁵	—	61	4	no	—
183	C₂⁵	—	61	24	no	—
184	C₂⁵	—	244	8	no	—
185	C₂⁵	—	244	16	no	—
186	C₂⁵	C₂²×D₆₁	1	2	yes	—
187	C₂²×Q₈	—	4	4	no	—
188	C₂²×Q₈	—	61	24	no	—
189	C₂²×Q₈	—	244	8	no	—
190	C₂²×Q₈	C₂²×D₆₁	1	12	yes	complemented
191	C₂²×D₄	—	4	4	no	—
192	C₂²×D₄	—	61	48	no	—
193	C₂²×D₄	—	61	24	no	—
194	C₂²×D₄	—	61	24	no	—
195	C₂²×D₄	—	244	8	no	—
196	C₂²×D₄	—	244	8	no	—
197	C₂²×D₄	—	244	16	no	—
198	C₂²×D₄	C₂²×D₆₁	1	12	yes	—
199	C₂³×C₄	—	61	4	no	—
200	C₂³×C₄	C₂²×D₆₁	1	1	yes	characteristic
201	C₂³×C₄	C₂²×D₆₁	1	2	yes	—
202	C₂³×C₄	C₂²×D₆₁	1	2	yes	—
203	#124	—	61	128	no	—
204	#124	—	61	256	no	—
205	#124	—	61	128	no	—
206	#124	C₂²×D₆₁	1	64	yes	complemented
207	#120	—	61	128	no	—
208	#120	—	61	256	no	—
209	C₂×C₂²⋊C₄	—	61	24	no	—
210	C₂×C₂²⋊C₄	—	61	48	no	—
211	C₂×C₂²⋊C₄	—	61	48	no	—
212	C₂×C₂²⋊C₄	—	61	48	no	—
213	C₂×C₂²⋊C₄	C₂²×D₆₁	1	48	yes	complemented
214	C₂×C₄²	—	61	24	no	—
215	C₂×C₄²	C₂²×D₆₁	1	12	yes	complemented
216	C₂⁴	—	2	8	no	—
217	C₂⁴	—	2	16	no	—
218	C₂⁴	—	61	24	no	—
219	C₂⁴	—	61	32	no	—
220	C₂⁴	—	122	16	no	—
221	C₂⁴	—	122	16	no	—
222	C₂⁴	—	122	16	no	—
223	C₂⁴	—	244	48	no	—
224	C₂⁴	—	244	96	no	—
225	C₂⁴	—	244	128	no	—
226	C₂⁴	C₂³×D₆₁	1	12	yes	—
227	C₂⁴	C₂³×D₆₁	1	1	yes	characteristic
228	C₂×Q₈	—	4	24	no	—
229	C₂×Q₈	—	61	32	no	—
230	C₂×Q₈	—	244	48	no	—
231	C₂×Q₈	C₂³×D₆₁	1	16	yes	complemented
232	C₂×D₄	—	4	24	no	—
233	C₂×D₄	—	61	64	no	—
234	C₂×D₄	—	61	32	no	—
235	C₂×D₄	—	61	32	no	—
236	C₂×D₄	—	244	48	no	—
237	C₂×D₄	—	244	48	no	—
238	C₂×D₄	—	244	96	no	—
239	C₂×D₄	C₂³×D₆₁	1	16	yes	—
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	2	8	no	—
242	C₂²×C₄	—	2	8	no	—
243	C₂²×C₄	—	61	24	no	—
244	C₂²×C₄	—	122	16	no	—
245	C₂²×C₄	C₂³×D₆₁	1	12	yes	—
246	C₂²×C₄	C₂³×D₆₁	1	6	yes	—
247	C₂²×C₄	C₂³×D₆₁	1	12	yes	—
248	C₂²⋊C₄	—	61	32	no	—
249	C₂²⋊C₄	—	61	64	no	—
250	C₂²⋊C₄	—	61	64	no	—
251	C₂²⋊C₄	—	61	64	no	—
252	C₂²⋊C₄	C₂³×D₆₁	1	64	yes	complemented
253	C₄²	—	61	32	no	—
254	C₄²	C₂³×D₆₁	1	16	yes	complemented
255	C₂³	—	2	24	no	—
256	C₂³	—	2	48	no	—
257	C₂³	—	4	32	no	—
258	C₂³	—	61	16	no	—
259	C₂³	—	122	48	no	—
260	C₂³	—	122	48	no	—
261	C₂³	—	122	48	no	—
262	C₂³	—	244	32	no	—
263	C₂³	—	244	64	no	—
264	C₂³	—	244	384	no	—
265	C₂³	—	244	64	no	—
266	C₂³	#14519822	1	4	yes	—
267	C₂³	#14519823	1	8	yes	—
268	C₂³	C₂⁴×D₆₁	1	8	yes	—
269	C₂³	C₂⁴×D₆₁	1	3	yes	—
270	Q₈	—	4	16	no	—
271	Q₈	—	244	32	no	—
272	D₄	—	4	16	no	—
273	D₄	—	244	32	no	—
274	D₄	—	244	32	no	—
275	D₄	—	244	64	no	—
276	C₂×C₄	—	2	12	no	—
277	C₂×C₄	—	2	24	no	—
278	C₂×C₄	—	2	24	no	—
279	C₂×C₄	—	61	16	no	—
280	C₂×C₄	—	122	48	no	—
281	C₂×C₄	C₂⁴×D₆₁	1	8	yes	—
282	C₂×C₄	C₂⁴×D₆₁	1	4	yes	—
283	C₂×C₄	C₂⁴×D₆₁	1	8	yes	—
284	C₂²	—	2	8	no	—
285	C₂²	—	2	16	no	—
286	C₂²	—	4	48	no	—
287	C₂²	—	122	16	no	—
288	C₂²	—	122	16	no	—
289	C₂²	—	122	16	no	—
290	C₂²	—	244	128	no	—
291	C₂²	—	244	96	no	—
292	C₂²	#16840504	1	16	yes	direct product factor
293	C₂²	C₂×#14519822	1	6	yes	—
294	C₂²	C₂×#14519823	1	12	yes	—
295	C₂²	C₂⁵×D₆₁	1	1	yes	characteristic
296	C₄	—	2	4	no	—
297	C₄	—	2	8	no	—
298	C₄	—	2	8	no	—
299	C₄	—	122	16	no	—
300	C₂	—	4	8	no	—
301	C₂	—	244	16	no	—
302	C₂	C₂×#16840504	1	12	yes	direct product factor
303	C₂	C₂²×#14519822	1	1	yes	characteristic
304	C₂	C₂²×#14519823	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₆₁
commutator, cosocle	C₂²×C₆₁	C₂⁶
Frattini	C₂²	C₂⁵×D₆₁
Fitting	C₂²×#124×C₆₁	C₂
socle	C₂⁴×C₆₁	C₂⁴
hypercenter	C₂²×#124	D₆₁
hypocenter	C₆₁	C₂²×H(3,𝔽₄)
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#124

Complements

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

N	H ≅ G/N	type
1. C₂×#16840504	C₂	direct (N × H)
2. C₂²×#14519766	C₂	semidirect
3. C₂²×#14519765	C₂	semidirect
4. C₂²×#14519763	C₂	semidirect
5. C₂²×#14519747	C₂	semidirect
6. C₂²×#14519745	C₂	semidirect
7. C₂²×#124×C₆₁	C₂	semidirect
8. C₂⁵×D₆₁	C₂²	semidirect
9. C₂³×Dic₁₂₂	C₂²	semidirect
13. #16840504	C₂²	direct (N × H)
14. C₂×#14519766	C₂²	semidirect
15. C₂×#14519765	C₂²	semidirect
16. C₂×#14519763	C₂²	semidirect
17. C₂×#14519747	C₂²	semidirect
18. C₂×#14519745	C₂²	semidirect
19. C₂²×C₂²⋊Dic₆₁	C₂²	semidirect
20. C₂²×#11758198	C₂²	semidirect
21. C₂²×#11758198	C₂²	semidirect
22. C₂²×#11758198	C₂²	semidirect
23. C₂²×C₄×Dic₆₁	C₂²	semidirect
24. C₂³×Q₈×C₆₁	C₂²	semidirect
26. C₂×#124×C₆₁	C₂²	semidirect
27. C₂²×C₂²⋊C₄×C₆₁	C₂²	semidirect
28. C₂²×C₄²×C₆₁	C₂²	semidirect
32. C₂⁴×D₆₁	C₂³	semidirect
34. C₂²×Dic₁₂₂	C₂³	semidirect
42. #14519766	C₂³	semidirect
43. #14519765	C₂³	semidirect
44. #14519763	C₂³	semidirect
45. #14519747	C₂³	semidirect
46. #14519745	C₂³	semidirect
47. C₂×C₂²⋊Dic₆₁	C₂³	semidirect
48. C₂×#11758198	C₂³	semidirect
49. C₂×#11758198	C₂³	semidirect
50. C₂×#11758198	C₂³	semidirect
51. C₂×C₄×Dic₆₁	C₂³	semidirect
54. C₂²×Q₈×C₆₁	C₂³	semidirect
60. #124×C₆₁	C₂³	semidirect
61. C₂×C₂²⋊C₄×C₆₁	C₂³	semidirect
62. C₂×C₄²×C₆₁	C₂³	semidirect
74. C₂³×D₆₁	C₂⁴	semidirect
76. C₂×Q₈×C₆₁	C₂⁴	semidirect
94. C₂×Dic₁₂₂	C₂⁴	semidirect
95. C₂²⋊C₄×C₆₁	C₂⁴	semidirect
96. C₄²×C₆₁	C₂⁴	semidirect
97. C₂²⋊Dic₆₁	C₂⁴	semidirect
98. #11758198	C₂⁴	semidirect
99. #11758198	C₂⁴	semidirect
100. #11758198	C₂⁴	semidirect
101. C₄×Dic₆₁	C₂⁴	semidirect
135. C₂²×C₆₁	H(3,𝔽₄)	semidirect
152. C₂²×#124	D₆₁	semidirect
156. C₂×C₆₁	C₂×H(3,𝔽₄)	semidirect
162. C₂³×Q₈	C₂×D₆₁	semidirect
171. C₂×#124	C₂×D₆₁	semidirect
178. C₂²×C₂²⋊C₄	C₂×D₆₁	semidirect
180. C₂²×C₄²	C₂×D₆₁	semidirect
181. C₆₁	C₂²×H(3,𝔽₄)	semidirect
190. C₂²×Q₈	C₂²×D₆₁	semidirect
206. #124	C₂²×D₆₁	semidirect
213. C₂×C₂²⋊C₄	C₂²×D₆₁	semidirect
215. C₂×C₄²	C₂²×D₆₁	semidirect
231. C₂×Q₈	C₂³×D₆₁	semidirect
252. C₂²⋊C₄	C₂³×D₆₁	semidirect
254. C₄²	C₂³×D₆₁	semidirect
292. C₂²	#16840504	direct (N × H)
302. C₂	C₂×#16840504	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×#16840504	2
C₂²×#14519766	2
C₂²×#14519765	2
C₂²×#14519763	2
C₂²×#14519747	2
C₂²×#14519745	2
C₂²×#124×C₆₁	2
C₂²×H(3,𝔽₄)	61

Subgroups — C22×#16840504 #30586885

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16840504 #30586885