Subgroups — #3521

Order 128.

521 conjugacy classes of subgroups.

The subgroup lattice has 323 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#556	C₂	1	1	yes	maximal, characteristic, complemented
2	#556	C₂	1	1	yes	maximal, characteristic, complemented
3	#554	C₂	1	1	yes	maximal, characteristic, complemented
4	#553	C₂	1	1	yes	maximal, characteristic, complemented
5	#553	C₂	1	2	yes	maximal, complemented
6	#552	C₂	1	1	yes	maximal, characteristic, complemented
7	#552	C₂	1	2	yes	maximal, complemented
8	#551	C₂	1	2	yes	maximal, complemented
9	#551	C₂	1	2	yes	maximal, complemented
10	#547	C₂	1	1	yes	maximal, characteristic, complemented
11	#546	C₂	1	1	yes	maximal, characteristic, complemented
12	#546	C₂	1	1	yes	maximal, characteristic, complemented
13	#542	C₂	1	1	yes	maximal, characteristic, complemented
14	#541	C₂	1	1	yes	maximal, characteristic, complemented
15	#539	C₂	1	2	yes	maximal, complemented
16	#538	C₂	1	2	yes	maximal, complemented
17	#529	C₂	1	1	yes	maximal, characteristic, complemented
18	#529	C₂	1	2	yes	maximal, complemented
19	#529	C₂	1	1	yes	maximal, characteristic, complemented
20	C₂×#126	C₂	1	2	yes	maximal, complemented
21	#520	C₂	1	1	yes	maximal, characteristic, complemented
22	#518	C₂	1	1	yes	maximal, characteristic, complemented
23	C₄×#41	C₂	1	1	yes	maximal, characteristic, complemented
24	C₂×#41	C₂²	1	1	yes	characteristic
25	C₂×#41	C₂²	1	1	yes	characteristic
26	C₂²×D₄	C₂²	1	1	yes	characteristic
27	C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
28	C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
29	#126	C₂²	1	1	yes	characteristic
30	#126	C₂²	1	2	yes	—
31	#126	C₂²	1	2	yes	—
32	#126	C₂²	1	2	yes	—
33	#126	C₂²	1	2	yes	—
34	#126	C₂²	1	2	yes	complemented
35	#126	C₂²	1	2	yes	complemented
36	#126	C₂²	1	2	yes	complemented
37	#126	C₂²	1	2	yes	complemented
38	#126	C₂²	1	1	yes	characteristic, complemented
39	#126	C₂²	1	2	yes	complemented
40	#126	C₂²	1	2	yes	complemented
41	#125	C₂²	1	2	yes	complemented
42	#125	C₂²	1	1	yes	characteristic, complemented
43	#125	C₂²	1	1	yes	characteristic, complemented
44	#125	C₂²	1	2	yes	complemented
45	#125	C₂²	1	2	yes	complemented
46	#125	C₂²	1	1	yes	characteristic, complemented
47	#124	C₂²	1	1	yes	characteristic
48	#124	C₂²	1	1	yes	characteristic
49	#124	C₂²	1	2	yes	—
50	#124	C₂²	1	1	yes	characteristic
51	#124	C₂²	1	2	yes	complemented
52	#123	C₂²	1	2	yes	—
53	#123	C₂²	1	1	yes	characteristic
54	#123	C₂²	1	2	yes	—
55	#123	C₂²	1	2	yes	—
56	#123	C₂²	1	2	yes	complemented
57	#123	C₂²	1	2	yes	complemented
58	#123	C₂²	1	1	yes	characteristic, complemented
59	#123	C₂²	1	2	yes	complemented
60	#123	C₂²	1	2	yes	complemented
61	#123	C₂²	1	2	yes	complemented
62	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
63	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
64	C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
65	C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
66	C₂²⋊Q₈	C₂²	1	2	yes	complemented
67	C₂²⋊Q₈	C₂²	1	2	yes	complemented
68	C₂²⋊Q₈	C₂²	1	2	yes	complemented
69	C₂²⋊Q₈	C₂²	1	2	yes	complemented
70	C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
71	C₂²⋊Q₈	C₂²	1	2	yes	complemented
72	#121	C₂²	1	1	yes	characteristic
73	#121	C₂²	1	1	yes	characteristic
74	#121	C₂²	1	2	yes	—
75	#121	C₂²	1	2	yes	—
76	#121	C₂²	1	1	yes	characteristic
77	#121	C₂²	1	2	yes	complemented
78	#121	C₂²	1	2	yes	complemented
79	#120	C₂²	1	2	yes	—
80	#120	C₂²	1	2	yes	—
81	C₄×Q₈	C₂²	1	1	yes	characteristic, complemented
82	C₄×Q₈	C₂²	1	1	yes	characteristic, complemented
83	C₄×Q₈	C₂²	1	1	yes	characteristic, complemented
84	C₄×Q₈	C₂²	1	1	yes	characteristic, complemented
85	C₄×Q₈	C₂²	1	2	yes	complemented
86	C₄×Q₈	C₂²	1	1	yes	characteristic, complemented
87	C₄×D₄	C₂²	1	1	yes	characteristic
88	C₄×D₄	C₂²	1	1	yes	characteristic
89	C₄×D₄	C₂²	1	2	yes	—
90	C₄×D₄	C₂²	1	1	yes	characteristic
91	C₄×D₄	C₂²	1	1	yes	characteristic
92	C₄×D₄	C₂²	1	2	yes	—
93	C₄×D₄	C₂²	1	1	yes	characteristic
94	C₄×D₄	C₂²	1	1	yes	characteristic
95	C₄×D₄	C₂²	1	1	yes	characteristic
96	C₄×D₄	C₂²	1	1	yes	characteristic
97	C₄×D₄	C₂²	1	2	yes	complemented
98	C₄×D₄	C₂²	1	2	yes	complemented
99	C₄×D₄	C₂²	1	2	yes	complemented
100	C₄×D₄	C₂²	1	2	yes	complemented
101	C₄×D₄	C₂²	1	1	yes	characteristic, complemented
102	C₄×D₄	C₂²	1	2	yes	complemented
103	C₄×D₄	C₂²	1	2	yes	complemented
104	#117	C₂²	1	1	yes	characteristic
105	#117	C₂²	1	1	yes	characteristic
106	#117	C₂²	1	2	yes	complemented
107	#117	C₂²	1	1	yes	characteristic, complemented
108	#117	C₂²	1	1	yes	characteristic, complemented
109	#117	C₂²	1	2	yes	complemented
110	#117	C₂²	1	2	yes	complemented
111	#117	C₂²	1	2	yes	complemented
112	#117	C₂²	1	2	yes	complemented
113	C₂×C₄⋊C₄	C₂²	1	2	yes	complemented
114	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
115	C₂×C₄⋊C₄	C₂²	1	2	yes	complemented
116	C₂×C₄⋊C₄	C₂²	1	2	yes	complemented
117	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
118	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
119	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
120	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
121	C₂×C₄²	C₂²	1	2	yes	complemented
122	C₂×C₄²	C₂²	1	1	yes	characteristic, complemented
123	C₂⁴	C₂³	1	2	yes	—
124	#41	—	2	4	no	—
125	#41	—	2	4	no	—
126	C₂×Q₈	C₂³	1	1	yes	characteristic
127	C₂×Q₈	C₂³	1	1	yes	characteristic
128	C₂×Q₈	C₂³	1	1	yes	characteristic
129	C₂×Q₈	C₂³	1	2	yes	—
130	C₂×D₄	—	2	4	no	—
131	C₂×D₄	C₂³	1	2	yes	—
132	C₂×D₄	C₂³	1	1	yes	characteristic
133	C₂×D₄	C₂³	1	2	yes	—
134	C₂×D₄	C₂³	1	2	yes	—
135	C₂×D₄	C₂³	1	1	yes	characteristic
136	C₂×D₄	C₂³	1	1	yes	characteristic
137	C₂×D₄	C₂³	1	2	yes	—
138	C₂×D₄	C₂³	1	2	yes	—
139	C₂×D₄	C₂³	1	1	yes	characteristic
140	C₂×D₄	C₂³	1	1	yes	characteristic
141	C₂²×C₄	C₂³	1	2	yes	—
142	C₂²×C₄	C₂³	1	1	yes	characteristic
143	C₂²×C₄	C₂³	1	2	yes	—
144	C₂²×C₄	C₂³	1	1	yes	characteristic
145	C₂²×C₄	C₂³	1	1	yes	characteristic
146	C₂²×C₄	C₂³	1	2	yes	—
147	C₂²×C₄	C₂³	1	2	yes	—
148	C₂²×C₄	C₂³	1	1	yes	characteristic
149	C₂²×C₄	C₂³	1	1	yes	characteristic
150	C₂²×C₄	C₂³	1	2	yes	—
151	C₂²×C₄	C₂³	1	2	yes	—
152	C₂²×C₄	C₂³	1	2	yes	—
153	C₂²×C₄	C₂³	1	2	yes	—
154	C₂²×C₄	C₂³	1	2	yes	—
155	C₄⋊C₄	C₂³	1	1	yes	characteristic
156	C₄⋊C₄	C₂³	1	2	yes	—
157	C₄⋊C₄	C₂³	1	1	yes	characteristic
158	C₄⋊C₄	C₂³	1	1	yes	characteristic
159	C₄⋊C₄	C₂³	1	1	yes	characteristic
160	C₄⋊C₄	C₂³	1	2	yes	—
161	C₄⋊C₄	C₂³	1	2	yes	—
162	C₄⋊C₄	C₂³	1	1	yes	characteristic
163	C₄⋊C₄	C₂³	1	2	yes	—
164	C₄⋊C₄	C₂³	1	2	yes	—
165	C₄⋊C₄	C₂³	1	2	yes	—
166	C₄⋊C₄	C₂³	1	2	yes	—
167	C₄⋊C₄	C₂³	1	2	yes	—
168	C₄⋊C₄	C₂³	1	2	yes	—
169	C₄⋊C₄	C₂³	1	1	yes	characteristic
170	C₄⋊C₄	C₂³	1	1	yes	characteristic
171	C₄⋊C₄	C₂³	1	1	yes	characteristic
172	C₄⋊C₄	C₂³	1	1	yes	characteristic
173	C₄⋊C₄	C₂³	1	1	yes	characteristic
174	C₄⋊C₄	C₂³	1	1	yes	characteristic
175	C₄⋊C₄	C₂³	1	2	yes	—
176	C₄⋊C₄	C₂³	1	2	yes	—
177	C₄⋊C₄	C₂³	1	2	yes	—
178	C₄⋊C₄	C₂³	1	2	yes	—
179	C₄⋊C₄	C₂³	1	2	yes	—
180	C₄⋊C₄	C₂³	1	1	yes	characteristic
181	C₄⋊C₄	C₂³	1	1	yes	characteristic
182	C₄⋊C₄	C₂³	1	2	yes	—
183	C₄⋊C₄	C₂³	1	2	yes	—
184	C₄⋊C₄	C₂³	1	1	yes	characteristic
185	C₂²⋊C₄	C₂³	1	1	yes	characteristic
186	C₂²⋊C₄	C₂³	1	2	yes	—
187	C₂²⋊C₄	C₂³	1	1	yes	characteristic
188	C₂²⋊C₄	C₂³	1	1	yes	characteristic
189	C₂²⋊C₄	C₂³	1	1	yes	characteristic
190	C₂²⋊C₄	C₂³	1	2	yes	—
191	C₂²⋊C₄	C₂³	1	2	yes	—
192	C₂²⋊C₄	C₂³	1	2	yes	—
193	C₂²⋊C₄	C₂³	1	2	yes	—
194	C₂²⋊C₄	C₂³	1	2	yes	—
195	C₂²⋊C₄	C₂³	1	2	yes	—
196	C₂²⋊C₄	C₂³	1	2	yes	—
197	C₂²⋊C₄	C₂³	1	2	yes	—
198	C₂²⋊C₄	C₂³	1	2	yes	—
199	C₂²⋊C₄	C₂³	1	2	yes	—
200	C₂²⋊C₄	C₂³	1	1	yes	characteristic
201	C₂²⋊C₄	C₂³	1	2	yes	—
202	C₂²⋊C₄	C₂³	1	2	yes	—
203	C₂²⋊C₄	C₂³	1	1	yes	characteristic
204	C₂²⋊C₄	C₂³	1	2	yes	—
205	C₂²⋊C₄	C₂³	1	1	yes	characteristic
206	C₂²⋊C₄	C₂³	1	2	yes	—
207	C₂²⋊C₄	C₂³	1	2	yes	—
208	C₂²⋊C₄	C₂³	1	2	yes	—
209	C₂²⋊C₄	C₂³	1	1	yes	characteristic
210	C₂²⋊C₄	C₂³	1	1	yes	characteristic
211	C₂²⋊C₄	C₂³	1	2	yes	—
212	C₂²⋊C₄	C₂³	1	1	yes	characteristic
213	C₄²	C₂³	1	1	yes	characteristic
214	C₄²	C₂³	1	1	yes	characteristic
215	C₄²	C₂³	1	1	yes	characteristic
216	C₄²	C₂³	1	2	yes	—
217	C₄²	C₂³	1	2	yes	—
218	C₄²	C₂³	1	1	yes	characteristic
219	C₄²	C₂³	1	2	yes	—
220	C₄²	C₂³	1	2	yes	—
221	C₄²	C₂³	1	1	yes	characteristic
222	C₄²	C₂³	1	2	yes	—
223	C₄²	C₂³	1	2	yes	—
224	C₄²	C₂³	1	1	yes	characteristic
225	C₂³	—	2	4	no	—
226	C₂³	—	4	2	no	—
227	C₂³	—	4	2	no	—
228	C₂³	C₂⁴	1	1	yes	characteristic
229	C₂³	C₂⁴	1	2	yes	—
230	C₂³	C₂⁴	1	1	yes	characteristic
231	C₂³	C₂⁴	1	1	yes	characteristic
232	C₂³	C₂⁴	1	2	yes	—
233	Q₈	—	2	2	no	—
234	Q₈	—	2	2	no	—
235	Q₈	—	4	1	no	—
236	Q₈	—	4	2	no	—
237	D₄	—	2	4	no	—
238	D₄	—	2	4	no	—
239	D₄	—	2	4	no	—
240	D₄	—	2	2	no	—
241	D₄	—	4	2	no	—
242	D₄	—	4	1	no	—
243	D₄	—	4	1	no	—
244	D₄	—	4	2	no	—
245	D₄	—	4	1	no	—
246	D₄	#41	1	4	yes	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	C₂⁴	1	1	yes	characteristic
262	C₂×C₄	C₂⁴	1	1	yes	characteristic
263	C₂×C₄	C₂⁴	1	1	yes	characteristic
264	C₂×C₄	C₂⁴	1	1	yes	characteristic
265	C₂×C₄	C₂⁴	1	2	yes	—
266	C₂×C₄	C₂⁴	1	1	yes	characteristic
267	C₂×C₄	C₂⁴	1	2	yes	—
268	C₂×C₄	C₂⁴	1	2	yes	—
269	C₂×C₄	C₂⁴	1	1	yes	characteristic
270	C₂×C₄	C₂⁴	1	2	yes	—
271	C₂×C₄	C₂⁴	1	1	yes	characteristic
272	C₂×C₄	C₂⁴	1	2	yes	—
273	C₂×C₄	C₂⁴	1	1	yes	characteristic
274	C₂×C₄	C₂⁴	1	1	yes	characteristic
275	C₂×C₄	C₂⁴	1	1	yes	characteristic
276	C₂×C₄	C₂⁴	1	2	yes	—
277	C₂×C₄	C₂⁴	1	1	yes	characteristic
278	C₂×C₄	C₂⁴	1	1	yes	characteristic
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	1	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	8	4	no	—
294	C₂²	C₂×#41	1	4	yes	—
295	C₂²	C₂⁵	1	1	yes	characteristic
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	2	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	2	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	1	no	—
313	C₄	C₂×#41	1	2	yes	—
314	C₂	—	2	4	no	—
315	C₂	—	4	1	no	—
316	C₂	—	4	1	no	—
317	C₂	—	4	2	no	—
318	C₂	—	4	1	no	—
319	C₂	C₂²×#41	1	1	yes	characteristic
320	C₂	#585	1	1	yes	characteristic
321	C₂	#585	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #556	C₂	semidirect
2. #556	C₂	semidirect
3. #554	C₂	semidirect
4. #553	C₂	semidirect
5. #553	C₂	semidirect
6. #552	C₂	semidirect
7. #552	C₂	semidirect
8. #551	C₂	semidirect
9. #551	C₂	semidirect
10. #547	C₂	semidirect
11. #546	C₂	semidirect
12. #546	C₂	semidirect
13. #542	C₂	semidirect
14. #541	C₂	semidirect
15. #539	C₂	semidirect
16. #538	C₂	semidirect
17. #529	C₂	semidirect
18. #529	C₂	semidirect
19. #529	C₂	semidirect
20. C₂×#126	C₂	semidirect
21. #520	C₂	semidirect
22. #518	C₂	semidirect
23. C₄×#41	C₂	semidirect
27. C₄⋊Q₈	C₂²	semidirect
28. C₄⋊Q₈	C₂²	semidirect
34. #126	C₂²	semidirect
35. #126	C₂²	semidirect
36. #126	C₂²	semidirect
37. #126	C₂²	semidirect
38. #126	C₂²	semidirect
39. #126	C₂²	semidirect
40. #126	C₂²	semidirect
41. #125	C₂²	semidirect
42. #125	C₂²	semidirect
43. #125	C₂²	semidirect
44. #125	C₂²	semidirect
45. #125	C₂²	semidirect
46. #125	C₂²	semidirect
51. #124	C₂²	semidirect
56. #123	C₂²	semidirect
57. #123	C₂²	semidirect
58. #123	C₂²	semidirect
59. #123	C₂²	semidirect
60. #123	C₂²	semidirect
61. #123	C₂²	semidirect
64. C₂²⋊Q₈	C₂²	semidirect
65. C₂²⋊Q₈	C₂²	semidirect
66. C₂²⋊Q₈	C₂²	semidirect
67. C₂²⋊Q₈	C₂²	semidirect
68. C₂²⋊Q₈	C₂²	semidirect
69. C₂²⋊Q₈	C₂²	semidirect
70. C₂²⋊Q₈	C₂²	semidirect
71. C₂²⋊Q₈	C₂²	semidirect
77. #121	C₂²	semidirect
78. #121	C₂²	semidirect
81. C₄×Q₈	C₂²	semidirect
82. C₄×Q₈	C₂²	semidirect
83. C₄×Q₈	C₂²	semidirect
84. C₄×Q₈	C₂²	semidirect
85. C₄×Q₈	C₂²	semidirect
86. C₄×Q₈	C₂²	semidirect
97. C₄×D₄	C₂²	semidirect
98. C₄×D₄	C₂²	semidirect
99. C₄×D₄	C₂²	semidirect
100. C₄×D₄	C₂²	semidirect
101. C₄×D₄	C₂²	semidirect
102. C₄×D₄	C₂²	semidirect
103. C₄×D₄	C₂²	semidirect
106. #117	C₂²	semidirect
107. #117	C₂²	semidirect
108. #117	C₂²	semidirect
109. #117	C₂²	semidirect
110. #117	C₂²	semidirect
111. #117	C₂²	semidirect
112. #117	C₂²	semidirect
113. C₂×C₄⋊C₄	C₂²	semidirect
114. C₂×C₄⋊C₄	C₂²	semidirect
115. C₂×C₄⋊C₄	C₂²	semidirect
116. C₂×C₄⋊C₄	C₂²	semidirect
121. C₂×C₄²	C₂²	semidirect
122. C₂×C₄²	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#556	2
#554	2
#553	2
#552	2
#551	2
#547	2
#546	2
#542	2
#541	2
#539	2
#538	2
#529	2
C₂×#126	2
#520	2
#518	2
C₄×#41	2