Subgroups — #3541

Order 128.

516 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#563	C₂	1	2	yes	maximal, complemented
2	#562	C₂	1	2	yes	maximal, complemented
3	#556	C₂	1	1	yes	maximal, characteristic, complemented
4	#554	C₂	1	1	yes	maximal, characteristic, complemented
5	#554	C₂	1	2	yes	maximal, complemented
6	#553	C₂	1	1	yes	maximal, characteristic, complemented
7	#553	C₂	1	2	yes	maximal, complemented
8	#552	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
10	#548	C₂	1	1	yes	maximal, characteristic, complemented
11	#548	C₂	1	1	yes	maximal, characteristic, complemented
12	#547	C₂	1	1	yes	maximal, characteristic, complemented
13	#542	C₂	1	1	yes	maximal, characteristic, complemented
14	#542	C₂	1	2	yes	maximal, complemented
15	#541	C₂	1	1	yes	maximal, characteristic, complemented
16	#539	C₂	1	2	yes	maximal, complemented
17	#538	C₂	1	2	yes	maximal, complemented
18	#538	C₂	1	2	yes	maximal, complemented
19	#537	C₂	1	1	yes	maximal, characteristic, complemented
20	#536	C₂	1	1	yes	maximal, characteristic, complemented
21	#525	C₂	1	1	yes	maximal, characteristic, complemented
22	#525	C₂	1	1	yes	maximal, characteristic, complemented
23	C₂×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
24	C₂×#41	C₂²	1	1	yes	characteristic
25	C₂×#41	C₂²	1	1	yes	characteristic
26	C₂×#41	C₂²	1	1	yes	characteristic, complemented
27	C₂²×Q₈	C₂²	1	1	yes	characteristic
28	C₂²×Q₈	C₂²	1	1	yes	characteristic, complemented
29	C₂³×C₄	C₂²	1	1	yes	characteristic
30	C₄⋊Q₈	C₂²	1	2	yes	complemented
31	C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
32	C₄⋊Q₈	C₂²	1	2	yes	complemented
33	C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
34	#126	C₂²	1	2	yes	—
35	#126	C₂²	1	2	yes	—
36	#126	C₂²	1	2	yes	—
37	#126	C₂²	1	2	yes	—
38	#126	C₂²	1	2	yes	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	2	yes	complemented
43	#125	C₂²	1	1	yes	characteristic, complemented
44	#125	C₂²	1	2	yes	complemented
45	#125	C₂²	1	2	yes	complemented
46	#124	C₂²	1	2	yes	—
47	#124	C₂²	1	2	yes	—
48	#124	C₂²	1	2	yes	—
49	#124	C₂²	1	2	yes	—
50	#124	C₂²	1	1	yes	characteristic, complemented
51	#124	C₂²	1	2	yes	complemented
52	#123	C₂²	1	1	yes	characteristic
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	—
57	#123	C₂²	1	2	yes	—
58	#123	C₂²	1	2	yes	—
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	2	yes	—
65	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
66	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
67	C₂²⋊Q₈	C₂²	1	2	yes	—
68	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
69	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
70	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
71	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
72	C₂²⋊Q₈	C₂²	1	2	yes	—
73	C₂²⋊Q₈	C₂²	1	2	yes	—
74	C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
75	C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
76	C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
77	C₂²⋊Q₈	C₂²	1	2	yes	complemented
78	C₂²⋊Q₈	C₂²	1	2	yes	complemented
79	C₂²⋊Q₈	C₂²	1	2	yes	complemented
80	C₂²⋊Q₈	C₂²	1	2	yes	complemented
81	C₂²⋊Q₈	C₂²	1	2	yes	complemented
82	C₂²⋊Q₈	C₂²	1	2	yes	complemented
83	#121	C₂²	1	1	yes	characteristic
84	#121	C₂²	1	1	yes	characteristic
85	#121	C₂²	1	1	yes	characteristic
86	#121	C₂²	1	1	yes	characteristic
87	#121	C₂²	1	2	yes	—
88	#121	C₂²	1	2	yes	—
89	#121	C₂²	1	2	yes	—
90	#121	C₂²	1	2	yes	—
91	#121	C₂²	1	2	yes	—
92	#121	C₂²	1	2	yes	—
93	#121	C₂²	1	1	yes	characteristic, complemented
94	#121	C₂²	1	2	yes	complemented
95	#120	C₂²	1	1	yes	characteristic
96	#120	C₂²	1	2	yes	—
97	#120	C₂²	1	1	yes	characteristic
98	C₄×Q₈	C₂²	1	1	yes	characteristic, complemented
99	C₄×Q₈	C₂²	1	1	yes	characteristic, complemented
100	C₄×Q₈	C₂²	1	1	yes	characteristic, complemented
101	C₄×Q₈	C₂²	1	2	yes	complemented
102	C₄×D₄	C₂²	1	2	yes	—
103	C₄×D₄	C₂²	1	1	yes	characteristic
104	C₄×D₄	C₂²	1	1	yes	characteristic
105	C₄×D₄	C₂²	1	1	yes	characteristic
106	C₄×D₄	C₂²	1	1	yes	characteristic
107	C₄×D₄	C₂²	1	2	yes	—
108	C₄×D₄	C₂²	1	2	yes	—
109	C₄×D₄	C₂²	1	2	yes	—
110	C₄×D₄	C₂²	1	2	yes	—
111	C₄×D₄	C₂²	1	2	yes	—
112	C₄×D₄	C₂²	1	2	yes	complemented
113	C₄×D₄	C₂²	1	1	yes	characteristic, complemented
114	#117	C₂²	1	1	yes	characteristic
115	#117	C₂²	1	1	yes	characteristic
116	#117	C₂²	1	1	yes	characteristic, complemented
117	#117	C₂²	1	2	yes	complemented
118	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
119	C₂×C₄⋊C₄	C₂²	1	2	yes	—
120	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
121	C₂×C₄⋊C₄	C₂²	1	2	yes	complemented
122	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
123	C₂⁴	C₂³	1	1	yes	characteristic
124	#41	—	4	2	no	—
125	#41	—	4	2	no	—
126	#41	—	4	2	no	—
127	C₂×Q₈	—	4	2	no	—
128	C₂×Q₈	—	4	2	no	—
129	C₂×Q₈	C₂³	1	1	yes	characteristic
130	C₂×Q₈	C₂³	1	1	yes	characteristic
131	C₂×Q₈	C₂³	1	1	yes	characteristic
132	C₂×Q₈	C₂³	1	2	yes	—
133	C₂×Q₈	C₂³	1	2	yes	—
134	C₂×Q₈	C₂³	1	1	yes	characteristic
135	C₂×Q₈	C₂³	1	2	yes	—
136	C₂×D₄	C₂³	1	1	yes	characteristic
137	C₂×D₄	C₂³	1	1	yes	characteristic
138	C₂×D₄	C₂³	1	2	yes	—
139	C₂×D₄	C₂³	1	1	yes	characteristic
140	C₂×D₄	C₂³	1	1	yes	characteristic
141	C₂×D₄	C₂³	1	2	yes	—
142	C₂×D₄	C₂³	1	1	yes	characteristic
143	C₂×D₄	C₂³	1	2	yes	—
144	C₂×D₄	C₂³	1	1	yes	characteristic
145	C₂×D₄	C₂³	1	2	yes	—
146	C₂×D₄	C₂³	1	1	yes	characteristic
147	C₂×D₄	C₂³	1	2	yes	—
148	C₂×D₄	C₂³	1	1	yes	characteristic
149	C₂²×C₄	—	4	2	no	—
150	C₂²×C₄	C₂³	1	1	yes	characteristic
151	C₂²×C₄	C₂³	1	1	yes	characteristic
152	C₂²×C₄	C₂³	1	1	yes	characteristic
153	C₂²×C₄	C₂³	1	1	yes	characteristic
154	C₂²×C₄	C₂³	1	1	yes	characteristic
155	C₂²×C₄	C₂³	1	2	yes	—
156	C₂²×C₄	C₂³	1	2	yes	—
157	C₂²×C₄	C₂³	1	1	yes	characteristic
158	C₂²×C₄	C₂³	1	2	yes	—
159	C₂²×C₄	C₂³	1	2	yes	—
160	C₂²×C₄	C₂³	1	2	yes	—
161	C₂²×C₄	C₂³	1	2	yes	—
162	C₂²×C₄	C₂³	1	1	yes	characteristic
163	C₂²×C₄	C₂³	1	2	yes	—
164	C₂²×C₄	C₂³	1	1	yes	characteristic
165	C₄⋊C₄	C₂³	1	1	yes	characteristic
166	C₄⋊C₄	C₂³	1	2	yes	—
167	C₄⋊C₄	C₂³	1	2	yes	—
168	C₄⋊C₄	C₂³	1	1	yes	characteristic
169	C₄⋊C₄	C₂³	1	2	yes	—
170	C₄⋊C₄	C₂³	1	2	yes	—
171	C₄⋊C₄	C₂³	1	2	yes	—
172	C₄⋊C₄	C₂³	1	1	yes	characteristic
173	C₄⋊C₄	C₂³	1	1	yes	characteristic
174	C₄⋊C₄	C₂³	1	2	yes	—
175	C₄⋊C₄	C₂³	1	2	yes	—
176	C₄⋊C₄	C₂³	1	2	yes	—
177	C₄⋊C₄	C₂³	1	1	yes	characteristic
178	C₄⋊C₄	C₂³	1	1	yes	characteristic
179	C₄⋊C₄	C₂³	1	1	yes	characteristic
180	C₄⋊C₄	C₂³	1	2	yes	—
181	C₄⋊C₄	C₂³	1	1	yes	characteristic
182	C₄⋊C₄	C₂³	1	1	yes	characteristic
183	C₄⋊C₄	C₂³	1	2	yes	—
184	C₄⋊C₄	C₂³	1	2	yes	—
185	C₄⋊C₄	C₂³	1	1	yes	characteristic
186	C₄⋊C₄	C₂³	1	2	yes	—
187	C₄⋊C₄	C₂³	1	2	yes	—
188	C₄⋊C₄	C₂³	1	2	yes	—
189	C₄⋊C₄	C₂³	1	2	yes	—
190	C₄⋊C₄	C₂³	1	2	yes	—
191	C₄⋊C₄	C₂³	1	2	yes	—
192	C₄⋊C₄	C₂³	1	2	yes	—
193	C₄⋊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	1	yes	characteristic
199	C₂²⋊C₄	C₂³	1	2	yes	—
200	C₂²⋊C₄	C₂³	1	1	yes	characteristic
201	C₂²⋊C₄	C₂³	1	1	yes	characteristic
202	C₂²⋊C₄	C₂³	1	2	yes	—
203	C₂²⋊C₄	C₂³	1	2	yes	—
204	C₂²⋊C₄	C₂³	1	1	yes	characteristic
205	C₂²⋊C₄	C₂³	1	2	yes	—
206	C₂²⋊C₄	C₂³	1	1	yes	characteristic
207	C₂²⋊C₄	C₂³	1	1	yes	characteristic
208	C₂²⋊C₄	C₂³	1	1	yes	characteristic
209	C₂²⋊C₄	C₂³	1	2	yes	—
210	C₂²⋊C₄	C₂³	1	1	yes	characteristic
211	C₂²⋊C₄	C₂³	1	2	yes	—
212	C₂²⋊C₄	C₂³	1	2	yes	—
213	C₂²⋊C₄	C₂³	1	2	yes	—
214	C₂²⋊C₄	C₂³	1	1	yes	characteristic
215	C₂²⋊C₄	C₂³	1	1	yes	characteristic
216	C₂²⋊C₄	C₂³	1	2	yes	—
217	C₂²⋊C₄	C₂³	1	2	yes	—
218	C₂²⋊C₄	C₂³	1	2	yes	—
219	C₂²⋊C₄	C₂³	1	2	yes	—
220	C₂²⋊C₄	C₂³	1	2	yes	—
221	C₄²	C₂³	1	2	yes	—
222	C₄²	C₂³	1	1	yes	characteristic
223	C₄²	C₂³	1	1	yes	characteristic
224	C₄²	C₂³	1	2	yes	—
225	C₄²	C₂³	1	2	yes	—
226	C₄²	C₂³	1	2	yes	—
227	C₄²	C₂³	1	2	yes	—
228	C₂³	—	2	2	no	—
229	C₂³	—	4	1	no	—
230	C₂³	—	4	1	no	—
231	C₂³	C₂⁴	1	1	yes	characteristic
232	C₂³	C₂⁴	1	1	yes	characteristic
233	C₂³	C₂⁴	1	1	yes	characteristic
234	C₂³	C₂⁴	1	1	yes	characteristic
235	C₂³	C₂⁴	1	2	yes	—
236	C₂³	C₂⁴	1	1	yes	characteristic
237	Q₈	—	2	2	no	—
238	Q₈	—	2	2	no	—
239	Q₈	—	2	2	no	—
240	Q₈	—	4	2	no	—
241	Q₈	—	4	2	no	—
242	Q₈	—	4	2	no	—
243	Q₈	—	4	1	no	—
244	D₄	—	2	2	no	—
245	D₄	—	2	2	no	—
246	D₄	—	2	4	no	—
247	D₄	—	2	2	no	—
248	D₄	—	4	1	no	—
249	D₄	—	4	2	no	—
250	D₄	—	4	2	no	—
251	D₄	—	4	1	no	—
252	D₄	—	4	1	no	—
253	D₄	—	4	1	no	—
254	D₄	—	4	1	no	—
255	D₄	—	4	2	no	—
256	D₄	—	4	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
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	C₂×C₄	C₂⁴	1	2	yes	—
280	C₂×C₄	C₂⁴	1	2	yes	—
281	C₂×C₄	C₂⁴	1	1	yes	characteristic
282	C₂×C₄	C₂⁴	1	2	yes	—
283	C₂×C₄	C₂⁴	1	2	yes	—
284	C₂×C₄	C₂⁴	1	2	yes	—
285	C₂×C₄	C₂⁴	1	1	yes	characteristic
286	C₂×C₄	C₂⁴	1	1	yes	characteristic
287	C₂×C₄	C₂⁴	1	1	yes	characteristic
288	C₂×C₄	C₂⁴	1	1	yes	characteristic
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	4	4	no	—
307	C₂²	2₋¹⁺⁴	1	2	yes	—
308	C₂²	C₂⁵	1	1	yes	characteristic
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	2	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	2	1	no	—
325	C₄	H(5,𝔽₂)	1	2	yes	—
326	C₂	—	2	2	no	—
327	C₂	—	4	1	no	—
328	C₂	—	4	1	no	—
329	C₂	—	4	2	no	—
330	C₂	—	4	1	no	—
331	C₂	—	4	1	no	—
332	C₂	C₂×H(5,𝔽₂)	1	1	yes	characteristic
333	C₂	C₂×2₋¹⁺⁴	1	1	yes	characteristic
334	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. #563	C₂	semidirect
2. #562	C₂	semidirect
3. #556	C₂	semidirect
4. #554	C₂	semidirect
5. #554	C₂	semidirect
6. #553	C₂	semidirect
7. #553	C₂	semidirect
8. #552	C₂	semidirect
9. D₄×Q₈	C₂	semidirect
10. #548	C₂	semidirect
11. #548	C₂	semidirect
12. #547	C₂	semidirect
13. #542	C₂	semidirect
14. #542	C₂	semidirect
15. #541	C₂	semidirect
16. #539	C₂	semidirect
17. #538	C₂	semidirect
18. #538	C₂	semidirect
19. #537	C₂	semidirect
20. #536	C₂	semidirect
21. #525	C₂	semidirect
22. #525	C₂	semidirect
23. C₂×C₂²⋊Q₈	C₂	semidirect
26. C₂×#41	C₂²	semidirect
28. C₂²×Q₈	C₂²	semidirect
30. C₄⋊Q₈	C₂²	semidirect
31. C₄⋊Q₈	C₂²	semidirect
32. C₄⋊Q₈	C₂²	semidirect
33. C₄⋊Q₈	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
50. #124	C₂²	semidirect
51. #124	C₂²	semidirect
59. #123	C₂²	semidirect
60. #123	C₂²	semidirect
61. #123	C₂²	semidirect
74. C₂²⋊Q₈	C₂²	semidirect
75. C₂²⋊Q₈	C₂²	semidirect
76. C₂²⋊Q₈	C₂²	semidirect
77. C₂²⋊Q₈	C₂²	semidirect
78. C₂²⋊Q₈	C₂²	semidirect
79. C₂²⋊Q₈	C₂²	semidirect
80. C₂²⋊Q₈	C₂²	semidirect
81. C₂²⋊Q₈	C₂²	semidirect
82. C₂²⋊Q₈	C₂²	semidirect
93. #121	C₂²	semidirect
94. #121	C₂²	semidirect
98. C₄×Q₈	C₂²	semidirect
99. C₄×Q₈	C₂²	semidirect
100. C₄×Q₈	C₂²	semidirect
101. C₄×Q₈	C₂²	semidirect
112. C₄×D₄	C₂²	semidirect
113. C₄×D₄	C₂²	semidirect
116. #117	C₂²	semidirect
117. #117	C₂²	semidirect
120. C₂×C₄⋊C₄	C₂²	semidirect
121. C₂×C₄⋊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
#556	2
#554	2
#553	2
#552	2
D₄×Q₈	2
#548	2
#547	2
#542	2
#541	2
#539	2
#538	2
#537	2
#536	2
#525	2
C₂×C₂²⋊Q₈	2