Subgroups — C₂³×#3444 #72301677

Order 1024.

544741 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#3444	C₂	1	224	yes	maximal, direct product factor
2	C₂³×#533	C₂	1	3	yes	maximal, complemented
3	C₂³×#532	C₂	1	3	yes	maximal, complemented
4	C₂³×#529	C₂	1	12	yes	maximal, complemented
5	C₂³×#525	C₂	1	6	yes	maximal, complemented
6	C₂³×C₄×#41	C₂	1	6	yes	maximal, complemented
7	C₂⁵×C₄²	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂⁴×#41	C₂²	1	6	yes	—
9	C₂⁶×C₄	C₂²	1	3	yes	—
10	C₂×#3444	C₂²	1	7168	yes	direct product factor
11	C₂²×#533	C₂²	1	336	yes	complemented
12	C₂²×#532	C₂²	1	336	yes	complemented
13	C₂³×C₄⋊Q₈	C₂²	1	3	yes	complemented
14	C₂³×#127	C₂²	1	1	yes	characteristic, complemented
15	C₂²×#529	C₂²	1	1344	yes	complemented
16	C₂³×#126	C₂²	1	8	yes	complemented
17	C₂³×#125	C₂²	1	6	yes	complemented
18	C₂³×#124	C₂²	1	6	yes	complemented
19	C₂²×#525	C₂²	1	672	yes	complemented
20	C₂³×#123	C₂²	1	12	yes	—
21	C₂³×C₂²⋊Q₈	C₂²	1	12	yes	—
22	C₂³×#121	C₂²	1	12	yes	—
23	C₂³×#120	C₂²	1	4	yes	—
24	C₂²×C₄×#41	C₂²	1	672	yes	complemented
25	C₂³×C₄×Q₈	C₂²	1	12	yes	complemented
26	C₂³×C₄×D₄	C₂²	1	24	yes	—
27	C₂³×C₄×D₄	C₂²	1	12	yes	complemented
28	C₂³×#117	C₂²	1	6	yes	—
29	C₂³×#117	C₂²	1	12	yes	complemented
30	C₂⁴×C₄²	C₂²	1	6	yes	—
31	C₂⁴×C₄²	C₂²	1	3	yes	—
32	C₂⁴×C₄²	C₂²	1	3	yes	—
33	C₂⁴×C₄²	C₂²	1	112	yes	complemented
34	C₂⁴×C₄²	C₂²	1	4	yes	complemented
35	C₂⁷	C₂³	1	1	yes	characteristic
36	C₂³×#41	—	2	192	no	—
37	C₂³×#41	C₂³	1	336	yes	—
38	C₂⁴×Q₈	C₂³	1	6	yes	—
39	C₂⁴×D₄	C₂³	1	12	yes	—
40	C₂⁴×D₄	C₂³	1	6	yes	—
41	C₂⁵×C₄	—	2	96	no	—
42	C₂⁵×C₄	C₂³	1	6	yes	—
43	C₂⁵×C₄	C₂³	1	6	yes	—
44	C₂⁵×C₄	C₂³	1	3	yes	—
45	C₂⁵×C₄	C₂³	1	168	yes	—
46	C₂⁵×C₄	C₂³	1	12	yes	—
47	C₂⁵×C₄	C₂³	1	3	yes	—
48	#3444	C₂³	1	32768	yes	direct product factor
49	C₂×#533	C₂³	1	5376	yes	complemented
50	C₂×#532	C₂³	1	5376	yes	complemented
51	C₂²×C₄⋊Q₈	C₂³	1	168	yes	complemented
52	C₂²×#127	C₂³	1	56	yes	complemented
53	C₂×#529	C₂³	1	21504	yes	complemented
54	C₂²×#126	C₂³	1	448	yes	complemented
55	C₂²×#125	C₂³	1	336	yes	complemented
56	C₂²×#124	C₂³	1	336	yes	complemented
57	C₂×#525	C₂³	1	10752	yes	complemented
58	C₂²×#123	C₂³	1	672	yes	—
59	C₂²×C₂²⋊Q₈	C₂³	1	672	yes	—
60	C₂²×#121	C₂³	1	672	yes	—
61	C₂²×#120	C₂³	1	224	yes	—
62	C₂×C₄×#41	C₂³	1	10752	yes	complemented
63	C₂²×C₄×Q₈	C₂³	1	672	yes	complemented
64	C₂²×C₄×D₄	C₂³	1	1344	yes	—
65	C₂²×C₄×D₄	C₂³	1	672	yes	complemented
66	C₂²×#117	C₂³	1	336	yes	—
67	C₂²×#117	C₂³	1	672	yes	complemented
68	C₂³×C₄⋊C₄	C₂³	1	12	yes	—
69	C₂³×C₄⋊C₄	C₂³	1	24	yes	—
70	C₂³×C₂²⋊C₄	C₂³	1	12	yes	—
71	C₂³×C₂²⋊C₄	C₂³	1	24	yes	—
72	C₂³×C₄²	C₂³	1	3	yes	—
73	C₂³×C₄²	C₂³	1	6	yes	—
74	C₂³×C₄²	C₂³	1	3	yes	—
75	C₂³×C₄²	C₂³	1	2	yes	—
76	C₂³×C₄²	C₂³	1	1	yes	characteristic
77	C₂³×C₄²	C₂³	1	168	yes	—
78	C₂³×C₄²	C₂³	1	336	yes	—
79	C₂³×C₄²	C₂³	1	168	yes	—
80	C₂³×C₄²	C₂³	1	12	yes	—
81	C₂³×C₄²	C₂³	1	1	yes	characteristic
82	C₂³×C₄²	C₂³	1	1792	yes	complemented
83	C₂³×C₄²	C₂³	1	224	yes	complemented
84	C₂⁶	—	2	48	no	—
85	C₂⁶	C₂⁴	1	28	yes	—
86	C₂⁶	C₂⁴	1	4	yes	—
87	C₂⁶	C₂⁴	1	3	yes	—
88	C₂²×#41	—	2	5376	no	—
89	C₂²×#41	C₂⁴	1	2688	yes	—
90	C₂³×Q₈	—	2	96	no	—
91	C₂³×Q₈	C₂⁴	1	168	yes	—
92	C₂³×D₄	—	2	192	no	—
93	C₂³×D₄	—	2	96	no	—
94	C₂³×D₄	C₂⁴	1	336	yes	—
95	C₂³×D₄	C₂⁴	1	168	yes	—
96	C₂⁴×C₄	—	2	48	no	—
97	C₂⁴×C₄	—	2	96	no	—
98	C₂⁴×C₄	—	2	96	no	—
99	C₂⁴×C₄	—	2	192	no	—
100	C₂⁴×C₄	—	2	2688	no	—
101	C₂⁴×C₄	#41	1	96	yes	complemented
102	C₂⁴×C₄	C₂⁴	1	6	yes	—
103	C₂⁴×C₄	C₂⁴	1	168	yes	—
104	C₂⁴×C₄	C₂⁴	1	168	yes	—
105	C₂⁴×C₄	C₂⁴	1	84	yes	—
106	C₂⁴×C₄	C₂⁴	1	1344	yes	—
107	C₂⁴×C₄	C₂⁴	1	12	yes	—
108	C₂⁴×C₄	C₂⁴	1	336	yes	—
109	C₂⁴×C₄	C₂⁴	1	3	yes	—
110	C₂⁴×C₄	C₂⁴	1	3	yes	—
111	C₂⁴×C₄	C₂⁴	1	84	yes	—
112	#533	C₂⁴	1	12288	yes	complemented
113	#532	C₂⁴	1	12288	yes	complemented
114	C₂×C₄⋊Q₈	C₂⁴	1	1344	yes	complemented
115	C₂×#127	C₂⁴	1	448	yes	complemented
116	#529	C₂⁴	1	49152	yes	complemented
117	C₂×#126	C₂⁴	1	3584	yes	complemented
118	C₂×#125	C₂⁴	1	2688	yes	complemented
119	C₂×#124	C₂⁴	1	2688	yes	complemented
120	#525	C₂⁴	1	24576	yes	complemented
121	C₂×#123	C₂⁴	1	5376	yes	—
122	C₂×C₂²⋊Q₈	C₂⁴	1	5376	yes	—
123	C₂×#121	C₂⁴	1	5376	yes	—
124	C₂×#120	C₂⁴	1	1792	yes	—
125	C₄×#41	C₂⁴	1	24576	yes	complemented
126	C₂×C₄×Q₈	C₂⁴	1	5376	yes	complemented
127	C₂×C₄×D₄	C₂⁴	1	10752	yes	—
128	C₂×C₄×D₄	C₂⁴	1	5376	yes	complemented
129	C₂×#117	C₂⁴	1	2688	yes	—
130	C₂×#117	C₂⁴	1	5376	yes	complemented
131	C₂²×C₄⋊C₄	C₂⁴	1	336	yes	—
132	C₂²×C₄⋊C₄	C₂⁴	1	672	yes	—
133	C₂²×C₂²⋊C₄	C₂⁴	1	336	yes	—
134	C₂²×C₂²⋊C₄	C₂⁴	1	672	yes	—
135	C₂²×C₄²	C₂⁴	1	84	yes	—
136	C₂²×C₄²	C₂⁴	1	168	yes	—
137	C₂²×C₄²	C₂⁴	1	84	yes	—
138	C₂²×C₄²	C₂⁴	1	56	yes	—
139	C₂²×C₄²	C₂⁴	1	28	yes	—
140	C₂²×C₄²	C₂⁴	1	28	yes	—
141	C₂²×C₄²	C₂⁴	1	1344	yes	—
142	C₂²×C₄²	C₂⁴	1	2688	yes	—
143	C₂²×C₄²	C₂⁴	1	1344	yes	—
144	C₂²×C₄²	C₂⁴	1	336	yes	—
145	C₂²×C₄²	C₂⁴	1	4096	yes	complemented
146	C₂²×C₄²	C₂⁴	1	1792	yes	complemented
147	C₂⁵	—	2	512	no	—
148	C₂⁵	—	2	672	no	—
149	C₂⁵	—	2	48	no	—
150	C₂⁵	—	2	96	no	—
151	C₂⁵	C₂×#41	1	48	yes	—
152	C₂⁵	C₂⁵	1	56	yes	—
153	C₂⁵	C₂⁵	1	112	yes	—
154	C₂⁵	C₂⁵	1	42	yes	—
155	C₂⁵	C₂⁵	1	1	yes	characteristic
156	C₂×#41	—	2	21504	no	—
157	C₂×#41	C₂⁵	1	3072	yes	—
158	C₂²×Q₈	—	2	1344	no	—
159	C₂²×Q₈	C₂⁵	1	672	yes	—
160	C₂²×D₄	—	2	2688	no	—
161	C₂²×D₄	—	2	1344	no	—
162	C₂²×D₄	C₂⁵	1	1344	yes	—
163	C₂²×D₄	C₂⁵	1	672	yes	—
164	C₂³×C₄	—	2	672	no	—
165	C₂³×C₄	—	2	48	no	—
166	C₂³×C₄	—	2	96	no	—
167	C₂³×C₄	—	2	1344	no	—
168	C₂³×C₄	—	2	1344	no	—
169	C₂³×C₄	—	2	2688	no	—
170	C₂³×C₄	—	2	10752	no	—
171	C₂³×C₄	C₂×#41	1	48	yes	—
172	C₂³×C₄	C₂×#41	1	48	yes	—
173	C₂³×C₄	C₂×#41	1	1344	yes	complemented
174	C₂³×C₄	C₂⁵	1	168	yes	—
175	C₂³×C₄	C₂⁵	1	84	yes	—
176	C₂³×C₄	C₂⁵	1	1344	yes	—
177	C₂³×C₄	C₂⁵	1	672	yes	—
178	C₂³×C₄	C₂⁵	1	672	yes	—
179	C₂³×C₄	C₂⁵	1	336	yes	—
180	C₂³×C₄	C₂⁵	1	1536	yes	—
181	C₂³×C₄	C₂⁵	1	42	yes	—
182	C₂³×C₄	C₂⁵	1	42	yes	—
183	C₂³×C₄	C₂⁵	1	336	yes	—
184	C₄⋊Q₈	C₂⁵	1	1536	yes	complemented
185	#127	C₂⁵	1	512	yes	complemented
186	#126	C₂⁵	1	4096	yes	complemented
187	#125	C₂⁵	1	3072	yes	complemented
188	#124	C₂⁵	1	3072	yes	complemented
189	#123	C₂⁵	1	6144	yes	—
190	C₂²⋊Q₈	C₂⁵	1	6144	yes	—
191	#121	C₂⁵	1	6144	yes	—
192	#120	C₂⁵	1	2048	yes	—
193	C₄×Q₈	C₂⁵	1	6144	yes	complemented
194	C₄×D₄	C₂⁵	1	12288	yes	—
195	C₄×D₄	C₂⁵	1	6144	yes	complemented
196	#117	C₂⁵	1	3072	yes	—
197	#117	C₂⁵	1	6144	yes	complemented
198	C₂×C₄⋊C₄	C₂⁵	1	1344	yes	—
199	C₂×C₄⋊C₄	C₂⁵	1	2688	yes	—
200	C₂×C₂²⋊C₄	C₂⁵	1	1344	yes	—
201	C₂×C₂²⋊C₄	C₂⁵	1	2688	yes	—
202	C₂×C₄²	C₂⁵	1	1344	yes	—
203	C₂×C₄²	C₂⁵	1	336	yes	—
204	C₂×C₄²	C₂⁵	1	672	yes	—
205	C₂×C₄²	C₂⁵	1	1536	yes	—
206	C₂×C₄²	C₂⁵	1	336	yes	—
207	C₂×C₄²	C₂⁵	1	224	yes	—
208	C₂×C₄²	C₂⁵	1	3072	yes	—
209	C₂×C₄²	C₂⁵	1	112	yes	—
210	C₂×C₄²	C₂⁵	1	1536	yes	—
211	C₂×C₄²	C₂⁵	1	112	yes	—
212	C₂×C₄²	C₂⁵	1	2048	yes	complemented
213	C₂⁴	—	2	3584	no	—
214	C₂⁴	—	2	1344	no	—
215	C₂⁴	—	2	336	no	—
216	C₂⁴	—	2	384	no	—
217	C₂⁴	—	2	672	no	—
218	C₂⁴	—	4	256	no	—
219	C₂⁴	C₂²×#41	1	336	yes	—
220	C₂⁴	C₂²×#41	1	24	yes	—
221	C₂⁴	C₂⁶	1	112	yes	—
222	C₂⁴	C₂⁶	1	64	yes	—
223	C₂⁴	C₂⁶	1	84	yes	—
224	C₂⁴	C₂⁶	1	7	yes	—
225	#41	—	2	12288	no	—
226	C₂×Q₈	—	2	2688	no	—
227	C₂×Q₈	C₂⁶	1	384	yes	—
228	C₂×D₄	—	2	5376	no	—
229	C₂×D₄	—	2	2688	no	—
230	C₂×D₄	C₂⁶	1	768	yes	—
231	C₂×D₄	C₂⁶	1	384	yes	—
232	C₂²×C₄	—	2	1344	no	—
233	C₂²×C₄	—	2	336	no	—
234	C₂²×C₄	—	2	672	no	—
235	C₂²×C₄	—	2	2688	no	—
236	C₂²×C₄	—	2	2688	no	—
237	C₂²×C₄	—	2	5376	no	—
238	C₂²×C₄	—	2	6144	no	—
239	C₂²×C₄	C₂²×#41	1	336	yes	—
240	C₂²×C₄	C₂²×#41	1	336	yes	—
241	C₂²×C₄	C₂²×#41	1	2688	yes	complemented
242	C₂²×C₄	C₂⁶	1	168	yes	—
243	C₂²×C₄	C₂⁶	1	384	yes	—
244	C₂²×C₄	C₂⁶	1	384	yes	—
245	C₂²×C₄	C₂⁶	1	192	yes	—
246	C₂²×C₄	C₂⁶	1	336	yes	—
247	C₂²×C₄	C₂⁶	1	768	yes	—
248	C₂²×C₄	C₂⁶	1	84	yes	—
249	C₂²×C₄	C₂⁶	1	84	yes	—
250	C₂²×C₄	C₂⁶	1	192	yes	—
251	C₄⋊C₄	C₂⁶	1	768	yes	—
252	C₄⋊C₄	C₂⁶	1	1536	yes	—
253	C₂²⋊C₄	C₂⁶	1	768	yes	—
254	C₂²⋊C₄	C₂⁶	1	1536	yes	—
255	C₄²	C₂⁶	1	64	yes	—
256	C₄²	C₂⁶	1	192	yes	—
257	C₄²	C₂⁶	1	384	yes	—
258	C₄²	C₂⁶	1	192	yes	—
259	C₄²	C₂⁶	1	128	yes	—
260	C₄²	C₂⁶	1	64	yes	—
261	C₄²	C₂⁶	1	768	yes	—
262	C₂³	—	2	3584	no	—
263	C₂³	—	2	384	no	—
264	C₂³	—	2	336	no	—
265	C₂³	—	2	1344	no	—
266	C₂³	—	2	672	no	—
267	C₂³	—	4	896	no	—
268	C₂³	#3444	1	64	yes	direct product factor
269	C₂³	C₂³×#41	1	336	yes	—
270	C₂³	C₂³×#41	1	84	yes	—
271	C₂³	C₂⁷	1	32	yes	—
272	C₂³	C₂⁷	1	24	yes	—
273	C₂³	C₂⁷	1	7	yes	—
274	Q₈	—	2	768	no	—
275	D₄	—	2	1536	no	—
276	D₄	—	2	768	no	—
277	C₂×C₄	—	2	384	no	—
278	C₂×C₄	—	2	336	no	—
279	C₂×C₄	—	2	672	no	—
280	C₂×C₄	—	2	768	no	—
281	C₂×C₄	—	2	768	no	—
282	C₂×C₄	—	2	1536	no	—
283	C₂×C₄	C₂³×#41	1	336	yes	—
284	C₂×C₄	C₂³×#41	1	336	yes	—
285	C₂×C₄	C₂³×#41	1	768	yes	complemented
286	C₂×C₄	C₂⁷	1	48	yes	—
287	C₂×C₄	C₂⁷	1	96	yes	—
288	C₂×C₄	C₂⁷	1	24	yes	—
289	C₂×C₄	C₂⁷	1	24	yes	—
290	C₂²	—	2	512	no	—
291	C₂²	—	2	672	no	—
292	C₂²	—	2	48	no	—
293	C₂²	—	2	96	no	—
294	C₂²	—	4	448	no	—
295	C₂²	C₂×#3444	1	112	yes	direct product factor
296	C₂²	C₂⁴×#41	1	48	yes	—
297	C₂²	C₂⁴×#41	1	42	yes	—
298	C₂²	C₂⁸	1	1	yes	characteristic
299	C₄	—	2	48	no	—
300	C₄	—	2	96	no	—
301	C₄	C₂⁴×#41	1	48	yes	—
302	C₄	C₂⁴×#41	1	48	yes	—
303	C₂	—	2	48	no	—
304	C₂	—	4	32	no	—
305	C₂	C₂²×#3444	1	28	yes	direct product factor
306	C₂	C₂⁵×#41	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₄²	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁸
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂²×#3444	C₂	direct (N × H)
2. C₂³×#533	C₂	semidirect
3. C₂³×#532	C₂	semidirect
4. C₂³×#529	C₂	semidirect
5. C₂³×#525	C₂	semidirect
6. C₂³×C₄×#41	C₂	semidirect
7. C₂⁵×C₄²	C₂	semidirect
10. C₂×#3444	C₂²	direct (N × H)
11. C₂²×#533	C₂²	semidirect
12. C₂²×#532	C₂²	semidirect
13. C₂³×C₄⋊Q₈	C₂²	semidirect
14. C₂³×#127	C₂²	semidirect
15. C₂²×#529	C₂²	semidirect
16. C₂³×#126	C₂²	semidirect
17. C₂³×#125	C₂²	semidirect
18. C₂³×#124	C₂²	semidirect
19. C₂²×#525	C₂²	semidirect
24. C₂²×C₄×#41	C₂²	semidirect
25. C₂³×C₄×Q₈	C₂²	semidirect
27. C₂³×C₄×D₄	C₂²	semidirect
29. C₂³×#117	C₂²	semidirect
33. C₂⁴×C₄²	C₂²	semidirect
34. C₂⁴×C₄²	C₂²	semidirect
48. #3444	C₂³	direct (N × H)
49. C₂×#533	C₂³	semidirect
50. C₂×#532	C₂³	semidirect
51. C₂²×C₄⋊Q₈	C₂³	semidirect
52. C₂²×#127	C₂³	semidirect
53. C₂×#529	C₂³	semidirect
54. C₂²×#126	C₂³	semidirect
55. C₂²×#125	C₂³	semidirect
56. C₂²×#124	C₂³	semidirect
57. C₂×#525	C₂³	semidirect
62. C₂×C₄×#41	C₂³	semidirect
63. C₂²×C₄×Q₈	C₂³	semidirect
65. C₂²×C₄×D₄	C₂³	semidirect
67. C₂²×#117	C₂³	semidirect
82. C₂³×C₄²	C₂³	semidirect
83. C₂³×C₄²	C₂³	semidirect
101. C₂⁴×C₄	#41	semidirect
112. #533	C₂⁴	semidirect
113. #532	C₂⁴	semidirect
114. C₂×C₄⋊Q₈	C₂⁴	semidirect
115. C₂×#127	C₂⁴	semidirect
116. #529	C₂⁴	semidirect
117. C₂×#126	C₂⁴	semidirect
118. C₂×#125	C₂⁴	semidirect
119. C₂×#124	C₂⁴	semidirect
120. #525	C₂⁴	semidirect
125. C₄×#41	C₂⁴	semidirect
126. C₂×C₄×Q₈	C₂⁴	semidirect
128. C₂×C₄×D₄	C₂⁴	semidirect
130. C₂×#117	C₂⁴	semidirect
145. C₂²×C₄²	C₂⁴	semidirect
146. C₂²×C₄²	C₂⁴	semidirect
173. C₂³×C₄	C₂×#41	semidirect
184. C₄⋊Q₈	C₂⁵	semidirect
185. #127	C₂⁵	semidirect
186. #126	C₂⁵	semidirect
187. #125	C₂⁵	semidirect
188. #124	C₂⁵	semidirect
193. C₄×Q₈	C₂⁵	semidirect
195. C₄×D₄	C₂⁵	semidirect
197. #117	C₂⁵	semidirect
212. C₂×C₄²	C₂⁵	semidirect
241. C₂²×C₄	C₂²×#41	semidirect
268. C₂³	#3444	direct (N × H)
285. C₂×C₄	C₂³×#41	semidirect
295. C₂²	C₂×#3444	direct (N × H)
305. C₂	C₂²×#3444	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×#3444	2
C₂³×#533	2
C₂³×#532	2
C₂³×#529	2
C₂³×#525	2
C₂³×C₄×#41	2
C₂⁵×C₄²	2

Subgroups — C23×#3444 #72301677

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#3444 #72301677