Subgroups — C₂²×#5683 #11702059

Order 768.

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	C₂²×C₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#947	C₂	1	2	yes	maximal, complemented
3	C₂²×#937	C₂	1	2	yes	maximal, complemented
4	C₂×#5683	C₂	1	48	yes	maximal, direct product factor
5	C₂²×#882	C₂	1	4	yes	maximal, complemented
6	C₂²×#885	C₂	1	4	yes	maximal, complemented
7	C₂²×#875	C₂	1	2	yes	maximal, complemented
8	C₂²×H(3,𝔽₄)	—	3	1	no	maximal, Sylow
9	C₂⁵×D₃	C₂²	1	2	yes	complemented
10	C₂³×C₃×Q₈	C₂²	1	1	yes	characteristic, complemented
11	C₂³×C₃×D₄	C₂²	1	1	yes	characteristic
12	C₂³×#67	C₂²	1	4	yes	—
13	C₂³×D₁₂	C₂²	1	2	yes	—
14	C₂³×D₁₂	C₂²	1	2	yes	—
15	C₂³×Dic₆	C₂²	1	2	yes	complemented
16	C₂×C₃×#124	C₂²	1	24	yes	complemented
17	C₂²×C₃×C₂²⋊C₄	C₂²	1	4	yes	complemented
18	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic, complemented
19	C₂²×#217	C₂²	1	2	yes	complemented
20	C₂×#947	C₂²	1	48	yes	complemented
21	C₂×#937	C₂²	1	48	yes	complemented
22	C₂²×(C₂×D₃)⋊C₄	C₂²	1	4	yes	complemented
23	C₂²×(C₂×D₃)⋊C₄	C₂²	1	4	yes	complemented
24	C₂²×(C₂×D₃)⋊C₄	C₂²	1	4	yes	complemented
25	C₂²×C₄×Dic₃	C₂²	1	2	yes	complemented
26	#5683	C₂²	1	256	yes	direct product factor
27	C₂×#885	C₂²	1	96	yes	complemented
28	C₂×#882	C₂²	1	96	yes	complemented
29	C₂×#875	C₂²	1	48	yes	complemented
30	C₂×H(3,𝔽₄)	—	3	48	no	—
31	C₂²×#124	—	3	2	no	—
32	C₂²×#124	—	3	2	no	—
33	C₂²×#124	—	3	4	no	—
34	C₂²×#124	D₃	1	1	yes	characteristic, complemented
35	C₂²×#120	—	3	2	no	—
36	C₂²×#120	—	3	4	no	—
37	C₂⁵×C₃	C₂³	1	2	yes	—
38	C₂⁴×D₃	—	4	8	no	—
39	C₂⁴×D₃	—	4	16	no	—
40	C₂⁴×D₃	C₂³	1	4	yes	—
41	C₂⁴×D₃	C₂³	1	24	yes	complemented
42	C₂²×C₃×Q₈	—	4	4	no	—
43	C₂²×C₃×Q₈	C₂³	1	12	yes	complemented
44	C₂²×C₃×D₄	—	4	4	no	—
45	C₂²×C₃×D₄	C₂³	1	12	yes	—
46	C₂³×C₃×C₄	C₂³	1	2	yes	—
47	C₂³×C₃×C₄	C₂³	1	2	yes	—
48	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
49	C₂²×#67	—	4	16	no	—
50	C₂²×#67	C₂³	1	48	yes	—
51	C₂³×Dic₃	C₂³	1	4	yes	—
52	C₂²×D₁₂	—	4	8	no	—
53	C₂²×D₁₂	—	4	8	no	—
54	C₂²×D₁₂	C₂³	1	24	yes	—
55	C₂²×D₁₂	C₂³	1	24	yes	—
56	C₂²×Dic₆	—	4	8	no	—
57	C₂²×Dic₆	C₂³	1	24	yes	complemented
58	C₃×#124	C₂³	1	64	yes	complemented
59	C₂×C₃×C₂²⋊C₄	C₂³	1	48	yes	complemented
60	C₂×C₃×C₄²	C₂³	1	12	yes	complemented
61	C₂×#217	C₂³	1	24	yes	complemented
62	#947	C₂³	1	128	yes	complemented
63	#937	C₂³	1	128	yes	complemented
64	C₂×(C₂×D₃)⋊C₄	C₂³	1	48	yes	complemented
65	C₂×(C₂×D₃)⋊C₄	C₂³	1	48	yes	complemented
66	C₂×(C₂×D₃)⋊C₄	C₂³	1	48	yes	complemented
67	C₂×C₄×Dic₃	C₂³	1	24	yes	complemented
68	#885	C₂³	1	256	yes	complemented
69	#882	C₂³	1	256	yes	complemented
70	#875	C₂³	1	128	yes	complemented
71	C₂⁶	—	3	2	no	—
72	C₂³×Q₈	—	3	2	no	—
73	C₂³×Q₈	C₂×D₃	1	1	yes	characteristic, complemented
74	C₂³×D₄	—	3	4	no	—
75	C₂³×D₄	—	3	2	no	—
76	C₂³×D₄	—	3	2	no	—
77	C₂³×D₄	C₂×D₃	1	1	yes	characteristic
78	H(3,𝔽₄)	—	3	256	no	—
79	C₂×#124	—	3	48	no	—
80	C₂×#124	—	3	96	no	—
81	C₂×#124	—	3	48	no	—
82	C₂×#124	C₂×D₃	1	24	yes	complemented
83	C₂×#120	—	3	48	no	—
84	C₂×#120	—	3	96	no	—
85	C₂²×C₂²⋊C₄	—	3	2	no	—
86	C₂²×C₂²⋊C₄	—	3	4	no	—
87	C₂²×C₂²⋊C₄	—	3	4	no	—
88	C₂²×C₂²⋊C₄	—	3	4	no	—
89	C₂²×C₂²⋊C₄	C₂×D₃	1	4	yes	complemented
90	C₂²×C₄²	—	3	2	no	—
91	C₂²×C₄²	C₂×D₃	1	1	yes	characteristic, complemented
92	C₂⁴×C₃	—	2	16	no	—
93	C₂⁴×C₃	—	2	8	no	—
94	C₂⁴×C₃	C₂⁴	1	12	yes	—
95	C₂⁴×C₃	C₂⁴	1	1	yes	characteristic
96	C₂³×D₃	—	2	16	no	—
97	C₂³×D₃	—	2	16	no	—
98	C₂³×D₃	—	2	16	no	—
99	C₂³×D₃	—	4	48	no	—
100	C₂³×D₃	—	4	96	no	—
101	C₂³×D₃	—	4	128	no	—
102	C₂³×D₃	C₂⁴	1	24	yes	—
103	C₂³×D₃	C₂⁴	1	32	yes	complemented
104	C₂×C₃×Q₈	—	4	24	no	—
105	C₂×C₃×Q₈	C₂⁴	1	16	yes	complemented
106	C₂×C₃×D₄	—	4	24	no	—
107	C₂×C₃×D₄	C₂⁴	1	16	yes	—
108	C₂²×C₃×C₄	—	2	8	no	—
109	C₂²×C₃×C₄	—	2	8	no	—
110	C₂²×C₃×C₄	—	2	4	no	—
111	C₂²×C₃×C₄	C₂⁴	1	12	yes	—
112	C₂²×C₃×C₄	C₂⁴	1	6	yes	—
113	C₂²×C₃×C₄	C₂⁴	1	12	yes	—
114	C₂×#67	—	4	96	no	—
115	C₂×#67	C₂⁴	1	64	yes	—
116	C₂²×Dic₃	—	2	16	no	—
117	C₂²×Dic₃	C₂⁴	1	24	yes	—
118	C₂×D₁₂	—	4	48	no	—
119	C₂×D₁₂	—	4	48	no	—
120	C₂×D₁₂	C₂⁴	1	32	yes	—
121	C₂×D₁₂	C₂⁴	1	32	yes	—
122	C₂×Dic₆	—	4	48	no	—
123	C₂×Dic₆	C₂⁴	1	32	yes	complemented
124	C₃×C₂²⋊C₄	C₂⁴	1	64	yes	complemented
125	C₃×C₄²	C₂⁴	1	16	yes	complemented
126	#217	C₂⁴	1	32	yes	complemented
127	(C₂×D₃)⋊C₄	C₂⁴	1	64	yes	complemented
128	(C₂×D₃)⋊C₄	C₂⁴	1	64	yes	complemented
129	(C₂×D₃)⋊C₄	C₂⁴	1	64	yes	complemented
130	C₄×Dic₃	C₂⁴	1	32	yes	complemented
131	C₂⁵	—	3	4	no	—
132	C₂⁵	—	3	24	no	—
133	C₂⁵	—	12	8	no	—
134	C₂⁵	—	12	16	no	—
135	C₂⁵	C₂²×D₃	1	2	yes	—
136	C₂²×Q₈	—	3	24	no	—
137	C₂²×Q₈	—	4	4	no	—
138	C₂²×Q₈	—	12	8	no	—
139	C₂²×Q₈	C₂²×D₃	1	12	yes	complemented
140	C₂²×D₄	—	3	48	no	—
141	C₂²×D₄	—	3	24	no	—
142	C₂²×D₄	—	3	24	no	—
143	C₂²×D₄	—	4	4	no	—
144	C₂²×D₄	—	12	8	no	—
145	C₂²×D₄	—	12	8	no	—
146	C₂²×D₄	—	12	16	no	—
147	C₂²×D₄	C₂²×D₃	1	12	yes	—
148	C₂³×C₄	—	3	4	no	—
149	C₂³×C₄	C₂²×D₃	1	1	yes	characteristic
150	C₂³×C₄	C₂²×D₃	1	2	yes	—
151	C₂³×C₄	C₂²×D₃	1	2	yes	—
152	#124	—	3	128	no	—
153	#124	—	3	256	no	—
154	#124	—	3	128	no	—
155	#124	C₂²×D₃	1	64	yes	complemented
156	#120	—	3	128	no	—
157	#120	—	3	256	no	—
158	C₂×C₂²⋊C₄	—	3	24	no	—
159	C₂×C₂²⋊C₄	—	3	48	no	—
160	C₂×C₂²⋊C₄	—	3	48	no	—
161	C₂×C₂²⋊C₄	—	3	48	no	—
162	C₂×C₂²⋊C₄	C₂²×D₃	1	48	yes	complemented
163	C₂×C₄²	—	3	24	no	—
164	C₂×C₄²	C₂²×D₃	1	12	yes	complemented
165	C₂³×C₃	—	2	48	no	—
166	C₂³×C₃	—	2	24	no	—
167	C₂³×C₃	—	4	32	no	—
168	C₂³×C₃	H(5,𝔽₂)	1	4	yes	—
169	C₂³×C₃	H(5,𝔽₂)	1	8	yes	—
170	C₂³×C₃	C₂⁵	1	8	yes	—
171	C₂³×C₃	C₂⁵	1	3	yes	—
172	C₂²×D₃	—	2	48	no	—
173	C₂²×D₃	—	2	48	no	—
174	C₂²×D₃	—	2	48	no	—
175	C₂²×D₃	—	4	32	no	—
176	C₂²×D₃	—	4	64	no	—
177	C₂²×D₃	—	4	384	no	—
178	C₂²×D₃	—	4	64	no	—
179	C₂²×D₃	C₂⁵	1	16	yes	—
180	C₃×Q₈	—	4	16	no	—
181	C₃×D₄	—	4	16	no	—
182	C₂×C₃×C₄	—	2	24	no	—
183	C₂×C₃×C₄	—	2	24	no	—
184	C₂×C₃×C₄	—	2	12	no	—
185	C₂×C₃×C₄	C₂⁵	1	8	yes	—
186	C₂×C₃×C₄	C₂⁵	1	4	yes	—
187	C₂×C₃×C₄	C₂⁵	1	8	yes	—
188	#67	—	4	64	no	—
189	C₂×Dic₃	—	2	48	no	—
190	C₂×Dic₃	C₂⁵	1	16	yes	—
191	D₁₂	—	4	32	no	—
192	D₁₂	—	4	32	no	—
193	Dic₆	—	4	32	no	—
194	C₂⁴	—	2	8	no	—
195	C₂⁴	—	2	16	no	—
196	C₂⁴	—	3	24	no	—
197	C₂⁴	—	3	32	no	—
198	C₂⁴	—	6	16	no	—
199	C₂⁴	—	6	16	no	—
200	C₂⁴	—	6	16	no	—
201	C₂⁴	—	12	48	no	—
202	C₂⁴	—	12	96	no	—
203	C₂⁴	—	12	128	no	—
204	C₂⁴	C₂³×D₃	1	12	yes	—
205	C₂⁴	C₂³×D₃	1	1	yes	characteristic
206	C₂×Q₈	—	3	32	no	—
207	C₂×Q₈	—	4	24	no	—
208	C₂×Q₈	—	12	48	no	—
209	C₂×Q₈	C₂³×D₃	1	16	yes	complemented
210	C₂×D₄	—	3	64	no	—
211	C₂×D₄	—	3	32	no	—
212	C₂×D₄	—	3	32	no	—
213	C₂×D₄	—	4	24	no	—
214	C₂×D₄	—	12	48	no	—
215	C₂×D₄	—	12	48	no	—
216	C₂×D₄	—	12	96	no	—
217	C₂×D₄	C₂³×D₃	1	16	yes	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	8	no	—
220	C₂²×C₄	—	2	8	no	—
221	C₂²×C₄	—	3	24	no	—
222	C₂²×C₄	—	6	16	no	—
223	C₂²×C₄	C₂³×D₃	1	12	yes	—
224	C₂²×C₄	C₂³×D₃	1	6	yes	—
225	C₂²×C₄	C₂³×D₃	1	12	yes	—
226	C₂²⋊C₄	—	3	32	no	—
227	C₂²⋊C₄	—	3	64	no	—
228	C₂²⋊C₄	—	3	64	no	—
229	C₂²⋊C₄	—	3	64	no	—
230	C₂²⋊C₄	C₂³×D₃	1	64	yes	complemented
231	C₄²	—	3	32	no	—
232	C₄²	C₂³×D₃	1	16	yes	complemented
233	C₂²×C₃	—	2	16	no	—
234	C₂²×C₃	—	2	8	no	—
235	C₂²×C₃	—	4	48	no	—
236	C₂²×C₃	H(3,𝔽₄)	1	16	yes	complemented
237	C₂²×C₃	C₂×H(5,𝔽₂)	1	6	yes	—
238	C₂²×C₃	C₂×H(5,𝔽₂)	1	12	yes	—
239	C₂²×C₃	C₂⁶	1	1	yes	characteristic
240	C₂×D₃	—	2	16	no	—
241	C₂×D₃	—	2	16	no	—
242	C₂×D₃	—	2	16	no	—
243	C₂×D₃	—	4	128	no	—
244	C₂×D₃	—	4	96	no	—
245	C₃×C₄	—	2	8	no	—
246	C₃×C₄	—	2	8	no	—
247	C₃×C₄	—	2	4	no	—
248	Dic₃	—	2	16	no	—
249	C₂³	—	2	24	no	—
250	C₂³	—	2	48	no	—
251	C₂³	—	3	16	no	—
252	C₂³	—	4	32	no	—
253	C₂³	—	6	48	no	—
254	C₂³	—	6	48	no	—
255	C₂³	—	6	48	no	—
256	C₂³	—	12	32	no	—
257	C₂³	—	12	64	no	—
258	C₂³	—	12	384	no	—
259	C₂³	—	12	64	no	—
260	C₂³	#1004	1	4	yes	—
261	C₂³	#1009	1	8	yes	—
262	C₂³	C₂⁴×D₃	1	8	yes	—
263	C₂³	C₂⁴×D₃	1	3	yes	—
264	Q₈	—	4	16	no	—
265	Q₈	—	12	32	no	—
266	D₄	—	4	16	no	—
267	D₄	—	12	32	no	—
268	D₄	—	12	32	no	—
269	D₄	—	12	64	no	—
270	C₂×C₄	—	2	12	no	—
271	C₂×C₄	—	2	24	no	—
272	C₂×C₄	—	2	24	no	—
273	C₂×C₄	—	3	16	no	—
274	C₂×C₄	—	6	48	no	—
275	C₂×C₄	C₂⁴×D₃	1	8	yes	—
276	C₂×C₄	C₂⁴×D₃	1	4	yes	—
277	C₂×C₄	C₂⁴×D₃	1	8	yes	—
278	C₂×C₃	—	4	8	no	—
279	C₂×C₃	C₂×H(3,𝔽₄)	1	12	yes	complemented
280	C₂×C₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
281	C₂×C₃	C₂²×H(5,𝔽₂)	1	2	yes	—
282	D₃	—	4	16	no	—
283	C₂²	—	2	8	no	—
284	C₂²	—	2	16	no	—
285	C₂²	—	4	48	no	—
286	C₂²	—	6	16	no	—
287	C₂²	—	6	16	no	—
288	C₂²	—	6	16	no	—
289	C₂²	—	12	128	no	—
290	C₂²	—	12	96	no	—
291	C₂²	#5683	1	16	yes	direct product factor
292	C₂²	C₂×#1004	1	6	yes	—
293	C₂²	C₂×#1009	1	12	yes	—
294	C₂²	C₂⁵×D₃	1	1	yes	characteristic
295	C₄	—	2	4	no	—
296	C₄	—	2	8	no	—
297	C₄	—	2	8	no	—
298	C₄	—	6	16	no	—
299	C₃	C₂²×H(3,𝔽₄)	1	1	yes	characteristic, complemented, Sylow
300	C₂	—	4	8	no	—
301	C₂	—	12	16	no	—
302	C₂	C₂×#5683	1	12	yes	direct product factor
303	C₂	C₂²×#1004	1	1	yes	characteristic
304	C₂	C₂²×#1009	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁶
Frattini	C₂²	C₂⁵×D₃
Fitting	C₂²×C₃×#124	C₂
socle	C₂⁴×C₃	C₂⁴
hypercenter	C₂²×#124	D₃
hypocenter	C₃	C₂²×H(3,𝔽₄)
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#124

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×#124	C₂	semidirect
2. C₂²×#947	C₂	semidirect
3. C₂²×#937	C₂	semidirect
4. C₂×#5683	C₂	direct (N × H)
5. C₂²×#882	C₂	semidirect
6. C₂²×#885	C₂	semidirect
7. C₂²×#875	C₂	semidirect
9. C₂⁵×D₃	C₂²	semidirect
10. C₂³×C₃×Q₈	C₂²	semidirect
15. C₂³×Dic₆	C₂²	semidirect
16. C₂×C₃×#124	C₂²	semidirect
17. C₂²×C₃×C₂²⋊C₄	C₂²	semidirect
18. C₂²×C₃×C₄²	C₂²	semidirect
19. C₂²×#217	C₂²	semidirect
20. C₂×#947	C₂²	semidirect
21. C₂×#937	C₂²	semidirect
22. C₂²×(C₂×D₃)⋊C₄	C₂²	semidirect
23. C₂²×(C₂×D₃)⋊C₄	C₂²	semidirect
24. C₂²×(C₂×D₃)⋊C₄	C₂²	semidirect
25. C₂²×C₄×Dic₃	C₂²	semidirect
26. #5683	C₂²	direct (N × H)
27. C₂×#885	C₂²	semidirect
28. C₂×#882	C₂²	semidirect
29. C₂×#875	C₂²	semidirect
34. C₂²×#124	D₃	semidirect
41. C₂⁴×D₃	C₂³	semidirect
43. C₂²×C₃×Q₈	C₂³	semidirect
57. C₂²×Dic₆	C₂³	semidirect
58. C₃×#124	C₂³	semidirect
59. C₂×C₃×C₂²⋊C₄	C₂³	semidirect
60. C₂×C₃×C₄²	C₂³	semidirect
61. C₂×#217	C₂³	semidirect
62. #947	C₂³	semidirect
63. #937	C₂³	semidirect
64. C₂×(C₂×D₃)⋊C₄	C₂³	semidirect
65. C₂×(C₂×D₃)⋊C₄	C₂³	semidirect
66. C₂×(C₂×D₃)⋊C₄	C₂³	semidirect
67. C₂×C₄×Dic₃	C₂³	semidirect
68. #885	C₂³	semidirect
69. #882	C₂³	semidirect
70. #875	C₂³	semidirect
73. C₂³×Q₈	C₂×D₃	semidirect
82. C₂×#124	C₂×D₃	semidirect
89. C₂²×C₂²⋊C₄	C₂×D₃	semidirect
91. C₂²×C₄²	C₂×D₃	semidirect
103. C₂³×D₃	C₂⁴	semidirect
105. C₂×C₃×Q₈	C₂⁴	semidirect
123. C₂×Dic₆	C₂⁴	semidirect
124. C₃×C₂²⋊C₄	C₂⁴	semidirect
125. C₃×C₄²	C₂⁴	semidirect
126. #217	C₂⁴	semidirect
127. (C₂×D₃)⋊C₄	C₂⁴	semidirect
128. (C₂×D₃)⋊C₄	C₂⁴	semidirect
129. (C₂×D₃)⋊C₄	C₂⁴	semidirect
130. C₄×Dic₃	C₂⁴	semidirect
139. C₂²×Q₈	C₂²×D₃	semidirect
155. #124	C₂²×D₃	semidirect
162. C₂×C₂²⋊C₄	C₂²×D₃	semidirect
164. C₂×C₄²	C₂²×D₃	semidirect
209. C₂×Q₈	C₂³×D₃	semidirect
230. C₂²⋊C₄	C₂³×D₃	semidirect
232. C₄²	C₂³×D₃	semidirect
236. C₂²×C₃	H(3,𝔽₄)	semidirect
279. C₂×C₃	C₂×H(3,𝔽₄)	semidirect
291. C₂²	#5683	direct (N × H)
299. C₃	C₂²×H(3,𝔽₄)	semidirect
302. C₂	C₂×#5683	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×#124	2
C₂²×#947	2
C₂²×#937	2
C₂×#5683	2
C₂²×#882	2
C₂²×#885	2
C₂²×#875	2
C₂²×H(3,𝔽₄)	3

Subgroups — C22×#5683 #11702059

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#5683 #11702059