Subgroups — #3533

Order 128.

530 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#563	C₂	1	1	yes	maximal, characteristic, complemented
2	#562	C₂	1	2	yes	maximal, complemented
3	#560	C₂	1	1	yes	maximal, characteristic, complemented
4	#559	C₂	1	1	yes	maximal, characteristic, complemented
5	#553	C₂	1	2	yes	maximal, complemented
6	#552	C₂	1	2	yes	maximal, complemented
7	#551	C₂	1	2	yes	maximal, complemented
8	#551	C₂	1	1	yes	maximal, characteristic, complemented
9	#547	C₂	1	2	yes	maximal, complemented
10	#546	C₂	1	2	yes	maximal, complemented
11	#542	C₂	1	2	yes	maximal, complemented
12	#540	C₂	1	2	yes	maximal, complemented
13	#539	C₂	1	2	yes	maximal, complemented
14	#538	C₂	1	2	yes	maximal, complemented
15	#538	C₂	1	2	yes	maximal, complemented
16	#535	C₂	1	1	yes	maximal, characteristic, complemented
17	#534	C₂	1	1	yes	maximal, characteristic, complemented
18	#525	C₂	1	2	yes	maximal, complemented
19	C₂×#123	C₂	1	1	yes	maximal, characteristic, complemented
20	C₂×#41	C₂²	1	2	yes	—
21	C₂×#41	C₂²	1	1	yes	characteristic, complemented
22	C₂²×D₄	C₂²	1	1	yes	characteristic
23	C₂²×D₄	C₂²	1	1	yes	characteristic
24	C₂³×C₄	C₂²	1	1	yes	characteristic, complemented
25	C₄⋊Q₈	C₂²	1	2	yes	complemented
26	#127	C₂²	1	1	yes	characteristic
27	#127	C₂²	1	1	yes	characteristic, complemented
28	#126	C₂²	1	2	yes	complemented
29	#126	C₂²	1	2	yes	complemented
30	#126	C₂²	1	2	yes	complemented
31	#126	C₂²	1	2	yes	complemented
32	#126	C₂²	1	2	yes	complemented
33	#126	C₂²	1	2	yes	complemented
34	#125	C₂²	1	2	yes	complemented
35	#125	C₂²	1	2	yes	complemented
36	#125	C₂²	1	2	yes	complemented
37	#124	C₂²	1	1	yes	characteristic
38	#124	C₂²	1	1	yes	characteristic, complemented
39	#124	C₂²	1	2	yes	complemented
40	#124	C₂²	1	2	yes	complemented
41	#124	C₂²	1	2	yes	complemented
42	#124	C₂²	1	2	yes	complemented
43	#123	C₂²	1	2	yes	—
44	#123	C₂²	1	1	yes	characteristic
45	#123	C₂²	1	1	yes	characteristic
46	#123	C₂²	1	2	yes	—
47	#123	C₂²	1	1	yes	characteristic
48	#123	C₂²	1	2	yes	—
49	#123	C₂²	1	1	yes	characteristic
50	#123	C₂²	1	2	yes	complemented
51	#123	C₂²	1	1	yes	characteristic, complemented
52	#123	C₂²	1	1	yes	characteristic, complemented
53	#123	C₂²	1	2	yes	complemented
54	#123	C₂²	1	2	yes	complemented
55	#123	C₂²	1	2	yes	complemented
56	#123	C₂²	1	2	yes	complemented
57	#123	C₂²	1	2	yes	complemented
58	#123	C₂²	1	2	yes	complemented
59	#123	C₂²	1	2	yes	complemented
60	#123	C₂²	1	2	yes	complemented
61	C₂²⋊Q₈	C₂²	1	2	yes	complemented
62	C₂²⋊Q₈	C₂²	1	2	yes	complemented
63	C₂²⋊Q₈	C₂²	1	2	yes	complemented
64	C₂²⋊Q₈	C₂²	1	2	yes	complemented
65	C₂²⋊Q₈	C₂²	1	2	yes	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	#121	C₂²	1	2	yes	—
70	#121	C₂²	1	2	yes	—
71	#121	C₂²	1	2	yes	—
72	#121	C₂²	1	2	yes	—
73	#121	C₂²	1	2	yes	complemented
74	#121	C₂²	1	2	yes	complemented
75	#121	C₂²	1	2	yes	complemented
76	#121	C₂²	1	2	yes	complemented
77	#121	C₂²	1	2	yes	complemented
78	#121	C₂²	1	2	yes	complemented
79	#121	C₂²	1	2	yes	complemented
80	#121	C₂²	1	2	yes	complemented
81	#120	C₂²	1	2	yes	—
82	#120	C₂²	1	1	yes	characteristic
83	#120	C₂²	1	1	yes	characteristic
84	#120	C₂²	1	2	yes	—
85	#120	C₂²	1	2	yes	complemented
86	#120	C₂²	1	2	yes	complemented
87	C₄×Q₈	C₂²	1	2	yes	complemented
88	C₄×D₄	C₂²	1	2	yes	—
89	C₄×D₄	C₂²	1	2	yes	complemented
90	C₄×D₄	C₂²	1	2	yes	complemented
91	C₄×D₄	C₂²	1	2	yes	complemented
92	C₄×D₄	C₂²	1	2	yes	complemented
93	C₄×D₄	C₂²	1	2	yes	complemented
94	C₄×D₄	C₂²	1	2	yes	complemented
95	C₄×D₄	C₂²	1	2	yes	complemented
96	C₄×D₄	C₂²	1	2	yes	complemented
97	#117	C₂²	1	2	yes	complemented
98	#117	C₂²	1	1	yes	characteristic, complemented
99	#117	C₂²	1	2	yes	complemented
100	#117	C₂²	1	2	yes	complemented
101	C₂×C₄⋊C₄	C₂²	1	2	yes	complemented
102	C₂×C₄⋊C₄	C₂²	1	2	yes	complemented
103	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
104	C₂×C₂²⋊C₄	C₂²	1	1	yes	characteristic
105	C₂×C₂²⋊C₄	C₂²	1	1	yes	characteristic, complemented
106	C₂×C₂²⋊C₄	C₂²	1	2	yes	complemented
107	C₂⁴	C₂³	1	1	yes	characteristic
108	C₂⁴	C₂³	1	1	yes	characteristic
109	C₂⁴	C₂³	1	1	yes	characteristic
110	#41	—	4	4	no	—
111	#41	—	4	2	no	—
112	C₂×Q₈	C₂³	1	2	yes	—
113	C₂×Q₈	C₂³	1	2	yes	—
114	C₂×Q₈	C₂³	1	1	yes	characteristic
115	C₂×D₄	—	4	2	no	—
116	C₂×D₄	—	4	2	no	—
117	C₂×D₄	C₂³	1	2	yes	—
118	C₂×D₄	C₂³	1	2	yes	—
119	C₂×D₄	C₂³	1	2	yes	—
120	C₂×D₄	C₂³	1	2	yes	—
121	C₂×D₄	C₂³	1	2	yes	—
122	C₂×D₄	C₂³	1	2	yes	—
123	C₂×D₄	C₂³	1	2	yes	—
124	C₂×D₄	C₂³	1	2	yes	—
125	C₂×D₄	C₂³	1	2	yes	—
126	C₂×D₄	C₂³	1	1	yes	characteristic
127	C₂×D₄	C₂³	1	1	yes	characteristic
128	C₂×D₄	C₂³	1	2	yes	—
129	C₂×D₄	C₂³	1	1	yes	characteristic
130	C₂×D₄	C₂³	1	2	yes	—
131	C₂×D₄	C₂³	1	1	yes	characteristic
132	C₂×D₄	C₂³	1	1	yes	characteristic
133	C₂²×C₄	—	4	2	no	—
134	C₂²×C₄	C₂³	1	2	yes	—
135	C₂²×C₄	C₂³	1	1	yes	characteristic
136	C₂²×C₄	C₂³	1	2	yes	—
137	C₂²×C₄	C₂³	1	2	yes	—
138	C₂²×C₄	C₂³	1	2	yes	—
139	C₂²×C₄	C₂³	1	2	yes	—
140	C₂²×C₄	C₂³	1	2	yes	—
141	C₂²×C₄	C₂³	1	2	yes	—
142	C₂²×C₄	C₂³	1	2	yes	—
143	C₂²×C₄	C₂³	1	1	yes	characteristic
144	C₂²×C₄	C₂³	1	2	yes	—
145	C₂²×C₄	C₂³	1	1	yes	characteristic
146	C₂²×C₄	C₂³	1	1	yes	characteristic
147	C₄⋊C₄	C₂³	1	2	yes	—
148	C₄⋊C₄	C₂³	1	2	yes	—
149	C₄⋊C₄	C₂³	1	2	yes	—
150	C₄⋊C₄	C₂³	1	2	yes	—
151	C₄⋊C₄	C₂³	1	2	yes	—
152	C₄⋊C₄	C₂³	1	2	yes	—
153	C₄⋊C₄	C₂³	1	2	yes	—
154	C₄⋊C₄	C₂³	1	2	yes	—
155	C₄⋊C₄	C₂³	1	2	yes	—
156	C₄⋊C₄	C₂³	1	2	yes	—
157	C₄⋊C₄	C₂³	1	2	yes	—
158	C₄⋊C₄	C₂³	1	2	yes	—
159	C₄⋊C₄	C₂³	1	2	yes	—
160	C₄⋊C₄	C₂³	1	2	yes	—
161	C₄⋊C₄	C₂³	1	2	yes	—
162	C₄⋊C₄	C₂³	1	2	yes	—
163	C₄⋊C₄	C₂³	1	2	yes	—
164	C₄⋊C₄	C₂³	1	2	yes	—
165	C₄⋊C₄	C₂³	1	2	yes	—
166	C₂²⋊C₄	C₂³	1	2	yes	—
167	C₂²⋊C₄	C₂³	1	2	yes	—
168	C₂²⋊C₄	C₂³	1	2	yes	—
169	C₂²⋊C₄	C₂³	1	1	yes	characteristic
170	C₂²⋊C₄	C₂³	1	1	yes	characteristic
171	C₂²⋊C₄	C₂³	1	2	yes	—
172	C₂²⋊C₄	C₂³	1	2	yes	—
173	C₂²⋊C₄	C₂³	1	2	yes	—
174	C₂²⋊C₄	C₂³	1	2	yes	—
175	C₂²⋊C₄	C₂³	1	2	yes	—
176	C₂²⋊C₄	C₂³	1	1	yes	characteristic
177	C₂²⋊C₄	C₂³	1	2	yes	—
178	C₂²⋊C₄	C₂³	1	1	yes	characteristic
179	C₂²⋊C₄	C₂³	1	2	yes	—
180	C₂²⋊C₄	C₂³	1	2	yes	—
181	C₂²⋊C₄	C₂³	1	2	yes	—
182	C₂²⋊C₄	C₂³	1	2	yes	—
183	C₂²⋊C₄	C₂³	1	2	yes	—
184	C₂²⋊C₄	C₂³	1	2	yes	—
185	C₂²⋊C₄	C₂³	1	2	yes	—
186	C₂²⋊C₄	C₂³	1	2	yes	—
187	C₂²⋊C₄	C₂³	1	2	yes	—
188	C₂²⋊C₄	C₂³	1	2	yes	—
189	C₂²⋊C₄	C₂³	1	2	yes	—
190	C₂²⋊C₄	C₂³	1	2	yes	—
191	C₂²⋊C₄	C₂³	1	2	yes	—
192	C₂²⋊C₄	C₂³	1	2	yes	—
193	C₄²	C₂³	1	1	yes	characteristic
194	C₄²	C₂³	1	2	yes	—
195	C₄²	C₂³	1	2	yes	—
196	C₄²	C₂³	1	2	yes	—
197	C₄²	C₂³	1	1	yes	characteristic
198	C₄²	C₂³	1	2	yes	—
199	C₂³	—	2	2	no	—
200	C₂³	—	2	2	no	—
201	C₂³	—	4	2	no	—
202	C₂³	—	4	1	no	—
203	C₂³	—	4	2	no	—
204	C₂³	—	4	2	no	—
205	C₂³	C₂⁴	1	1	yes	characteristic
206	C₂³	C₂⁴	1	1	yes	characteristic
207	C₂³	C₂⁴	1	1	yes	characteristic
208	C₂³	C₂⁴	1	2	yes	—
209	C₂³	C₂⁴	1	1	yes	characteristic
210	C₂³	C₂⁴	1	2	yes	—
211	C₂³	C₂⁴	1	1	yes	characteristic
212	Q₈	—	4	2	no	—
213	Q₈	—	4	2	no	—
214	Q₈	—	4	1	no	—
215	D₄	—	2	4	no	—
216	D₄	—	2	4	no	—
217	D₄	—	4	1	no	—
218	D₄	—	4	1	no	—
219	D₄	—	4	1	no	—
220	D₄	—	4	1	no	—
221	D₄	—	4	2	no	—
222	D₄	—	4	2	no	—
223	D₄	—	4	2	no	—
224	D₄	—	4	2	no	—
225	D₄	—	4	2	no	—
226	D₄	—	4	2	no	—
227	D₄	—	4	2	no	—
228	D₄	—	4	2	no	—
229	D₄	—	4	2	no	—
230	D₄	—	4	1	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	1	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	C₂⁴	1	2	yes	—
245	C₂×C₄	C₂⁴	1	2	yes	—
246	C₂×C₄	C₂⁴	1	2	yes	—
247	C₂×C₄	C₂⁴	1	2	yes	—
248	C₂×C₄	C₂⁴	1	1	yes	characteristic
249	C₂×C₄	C₂⁴	1	2	yes	—
250	C₂×C₄	C₂⁴	1	2	yes	—
251	C₂×C₄	C₂⁴	1	2	yes	—
252	C₂×C₄	C₂⁴	1	2	yes	—
253	C₂×C₄	C₂⁴	1	2	yes	—
254	C₂×C₄	C₂⁴	1	2	yes	—
255	C₂×C₄	C₂⁴	1	1	yes	characteristic
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	4	4	no	—
272	C₂²	—	8	2	no	—
273	C₂²	—	8	2	no	—
274	C₂²	H(5,𝔽₂)	1	2	yes	—
275	C₂²	C₂⁵	1	1	yes	characteristic
276	C₄	—	2	2	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₂	—	2	2	no	—
289	C₂	—	4	2	no	—
290	C₂	—	4	1	no	—
291	C₂	—	4	1	no	—
292	C₂	—	4	2	no	—
293	C₂	—	4	1	no	—
294	C₂	—	4	1	no	—
295	C₂	C₂×H(5,𝔽₂)	1	1	yes	characteristic
296	C₂	#585	1	2	yes	—

	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. #563	C₂	semidirect
2. #562	C₂	semidirect
3. #560	C₂	semidirect
4. #559	C₂	semidirect
5. #553	C₂	semidirect
6. #552	C₂	semidirect
7. #551	C₂	semidirect
8. #551	C₂	semidirect
9. #547	C₂	semidirect
10. #546	C₂	semidirect
11. #542	C₂	semidirect
12. #540	C₂	semidirect
13. #539	C₂	semidirect
14. #538	C₂	semidirect
15. #538	C₂	semidirect
16. #535	C₂	semidirect
17. #534	C₂	semidirect
18. #525	C₂	semidirect
19. C₂×#123	C₂	semidirect
21. C₂×#41	C₂²	semidirect
24. C₂³×C₄	C₂²	semidirect
25. C₄⋊Q₈	C₂²	semidirect
27. #127	C₂²	semidirect
28. #126	C₂²	semidirect
29. #126	C₂²	semidirect
30. #126	C₂²	semidirect
31. #126	C₂²	semidirect
32. #126	C₂²	semidirect
33. #126	C₂²	semidirect
34. #125	C₂²	semidirect
35. #125	C₂²	semidirect
36. #125	C₂²	semidirect
38. #124	C₂²	semidirect
39. #124	C₂²	semidirect
40. #124	C₂²	semidirect
41. #124	C₂²	semidirect
42. #124	C₂²	semidirect
50. #123	C₂²	semidirect
51. #123	C₂²	semidirect
52. #123	C₂²	semidirect
53. #123	C₂²	semidirect
54. #123	C₂²	semidirect
55. #123	C₂²	semidirect
56. #123	C₂²	semidirect
57. #123	C₂²	semidirect
58. #123	C₂²	semidirect
59. #123	C₂²	semidirect
60. #123	C₂²	semidirect
61. C₂²⋊Q₈	C₂²	semidirect
62. C₂²⋊Q₈	C₂²	semidirect
63. C₂²⋊Q₈	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
73. #121	C₂²	semidirect
74. #121	C₂²	semidirect
75. #121	C₂²	semidirect
76. #121	C₂²	semidirect
77. #121	C₂²	semidirect
78. #121	C₂²	semidirect
79. #121	C₂²	semidirect
80. #121	C₂²	semidirect
85. #120	C₂²	semidirect
86. #120	C₂²	semidirect
87. C₄×Q₈	C₂²	semidirect
89. C₄×D₄	C₂²	semidirect
90. C₄×D₄	C₂²	semidirect
91. C₄×D₄	C₂²	semidirect
92. C₄×D₄	C₂²	semidirect
93. C₄×D₄	C₂²	semidirect
94. C₄×D₄	C₂²	semidirect
95. C₄×D₄	C₂²	semidirect
96. C₄×D₄	C₂²	semidirect
97. #117	C₂²	semidirect
98. #117	C₂²	semidirect
99. #117	C₂²	semidirect
100. #117	C₂²	semidirect
101. C₂×C₄⋊C₄	C₂²	semidirect
102. C₂×C₄⋊C₄	C₂²	semidirect
105. C₂×C₂²⋊C₄	C₂²	semidirect
106. C₂×C₂²⋊C₄	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#563	2
#562	2
#560	2
#559	2
#553	2
#552	2
#551	2
#547	2
#546	2
#542	2
#540	2
#539	2
#538	2
#535	2
#534	2
#525	2
C₂×#123	2