Subgroups — #3552

Order 128.

502 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#563	C₂	1	4	yes	maximal, complemented
2	#557	C₂	1	1	yes	maximal, characteristic, complemented
3	#556	C₂	1	1	yes	maximal, characteristic, complemented
4	#554	C₂	1	2	yes	maximal, complemented
5	#553	C₂	1	2	yes	maximal, complemented
6	D₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×Q₈	C₂	1	2	yes	maximal, complemented
8	#548	C₂	1	2	yes	maximal, complemented
9	#547	C₂	1	1	yes	maximal, characteristic, complemented
10	#544	C₂	1	2	yes	maximal, complemented
11	#542	C₂	1	2	yes	maximal, complemented
12	#541	C₂	1	2	yes	maximal, complemented
13	#539	C₂	1	4	yes	maximal, complemented
14	#536	C₂	1	2	yes	maximal, complemented
15	C₂×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
16	#529	C₂	1	1	yes	maximal, characteristic, complemented
17	#529	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×#41	C₂²	1	2	yes	—
19	C₂²×Q₈	C₂²	1	2	yes	—
20	C₂²×Q₈	C₂²	1	2	yes	—
21	C₄⋊Q₈	C₂²	1	2	yes	—
22	C₄⋊Q₈	C₂²	1	2	yes	—
23	C₄⋊Q₈	C₂²	1	1	yes	characteristic
24	C₄⋊Q₈	C₂²	1	1	yes	characteristic
25	C₄⋊Q₈	C₂²	1	2	yes	—
26	C₄⋊Q₈	C₂²	1	2	yes	—
27	C₄⋊Q₈	C₂²	1	2	yes	—
28	C₄⋊Q₈	C₂²	1	4	yes	—
29	#126	C₂²	1	2	yes	—
30	#126	C₂²	1	2	yes	—
31	#126	C₂²	1	4	yes	—
32	#126	C₂²	1	4	yes	—
33	#126	C₂²	1	4	yes	—
34	#125	C₂²	1	1	yes	characteristic
35	#125	C₂²	1	1	yes	characteristic
36	#125	C₂²	1	4	yes	—
37	#125	C₂²	1	4	yes	—
38	#125	C₂²	1	4	yes	—
39	#125	C₂²	1	2	yes	—
40	#124	C₂²	1	1	yes	characteristic
41	#124	C₂²	1	1	yes	characteristic
42	#124	C₂²	1	2	yes	—
43	#124	C₂²	1	4	yes	—
44	#123	C₂²	1	2	yes	—
45	#123	C₂²	1	2	yes	—
46	#123	C₂²	1	4	yes	—
47	#123	C₂²	1	4	yes	—
48	#123	C₂²	1	4	yes	—
49	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
50	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
51	C₂²⋊Q₈	C₂²	1	4	yes	—
52	C₂²⋊Q₈	C₂²	1	4	yes	—
53	C₂²⋊Q₈	C₂²	1	4	yes	—
54	C₂²⋊Q₈	C₂²	1	4	yes	—
55	C₂²⋊Q₈	C₂²	1	2	yes	—
56	C₂²⋊Q₈	C₂²	1	2	yes	—
57	C₂²⋊Q₈	C₂²	1	4	yes	—
58	C₂²⋊Q₈	C₂²	1	2	yes	—
59	C₂²⋊Q₈	C₂²	1	4	yes	—
60	#121	C₂²	1	1	yes	characteristic
61	#121	C₂²	1	1	yes	characteristic
62	#121	C₂²	1	4	yes	—
63	#121	C₂²	1	2	yes	—
64	C₄×Q₈	C₂²	1	1	yes	characteristic
65	C₄×Q₈	C₂²	1	1	yes	characteristic
66	C₄×Q₈	C₂²	1	2	yes	—
67	C₄×Q₈	C₂²	1	2	yes	—
68	C₄×Q₈	C₂²	1	2	yes	—
69	C₄×D₄	C₂²	1	1	yes	characteristic
70	C₄×D₄	C₂²	1	1	yes	characteristic
71	C₄×D₄	C₂²	1	2	yes	—
72	C₄×D₄	C₂²	1	2	yes	—
73	C₄×D₄	C₂²	1	4	yes	—
74	C₄×D₄	C₂²	1	4	yes	—
75	C₄×D₄	C₂²	1	2	yes	—
76	#117	C₂²	1	2	yes	—
77	#117	C₂²	1	2	yes	—
78	#117	C₂²	1	2	yes	—
79	C₂×C₄⋊C₄	C₂²	1	2	yes	—
80	C₂×C₄⋊C₄	C₂²	1	2	yes	—
81	C₂×C₄⋊C₄	C₂²	1	2	yes	—
82	C₂×C₄²	C₂²	1	1	yes	characteristic
83	#41	—	4	4	no	—
84	C₂×Q₈	—	4	4	no	—
85	C₂×Q₈	—	4	4	no	—
86	C₂×Q₈	C₂³	1	1	yes	characteristic
87	C₂×Q₈	C₂³	1	1	yes	characteristic
88	C₂×Q₈	C₂³	1	2	yes	—
89	C₂×Q₈	C₂³	1	2	yes	—
90	C₂×Q₈	C₂³	1	4	yes	—
91	C₂×Q₈	C₂³	1	4	yes	—
92	C₂×D₄	C₂³	1	2	yes	—
93	C₂×D₄	C₂³	1	2	yes	—
94	C₂×D₄	C₂³	1	1	yes	characteristic
95	C₂×D₄	C₂³	1	1	yes	characteristic
96	C₂×D₄	C₂³	1	4	yes	—
97	C₂²×C₄	C₂³	1	1	yes	characteristic
98	C₂²×C₄	C₂³	1	1	yes	characteristic
99	C₂²×C₄	C₂³	1	1	yes	characteristic
100	C₂²×C₄	C₂³	1	2	yes	—
101	C₂²×C₄	C₂³	1	2	yes	—
102	C₂²×C₄	C₂³	1	4	yes	—
103	C₂²×C₄	C₂³	1	4	yes	—
104	C₂²×C₄	C₂³	1	4	yes	—
105	C₄⋊C₄	C₂³	1	1	yes	characteristic
106	C₄⋊C₄	C₂³	1	2	yes	—
107	C₄⋊C₄	C₂³	1	2	yes	—
108	C₄⋊C₄	C₂³	1	4	yes	—
109	C₄⋊C₄	C₂³	1	2	yes	—
110	C₄⋊C₄	C₂³	1	4	yes	—
111	C₄⋊C₄	C₂³	1	4	yes	—
112	C₄⋊C₄	C₂³	1	2	yes	—
113	C₄⋊C₄	C₂³	1	1	yes	characteristic
114	C₄⋊C₄	C₂³	1	2	yes	—
115	C₄⋊C₄	C₂³	1	2	yes	—
116	C₄⋊C₄	C₂³	1	2	yes	—
117	C₄⋊C₄	C₂³	1	2	yes	—
118	C₄⋊C₄	C₂³	1	1	yes	characteristic
119	C₄⋊C₄	C₂³	1	4	yes	—
120	C₄⋊C₄	C₂³	1	2	yes	—
121	C₄⋊C₄	C₂³	1	4	yes	—
122	C₄⋊C₄	C₂³	1	2	yes	—
123	C₄⋊C₄	C₂³	1	1	yes	characteristic
124	C₄⋊C₄	C₂³	1	4	yes	—
125	C₄⋊C₄	C₂³	1	4	yes	—
126	C₄⋊C₄	C₂³	1	4	yes	—
127	C₄⋊C₄	C₂³	1	2	yes	—
128	C₄⋊C₄	C₂³	1	2	yes	—
129	C₂²⋊C₄	C₂³	1	4	yes	—
130	C₂²⋊C₄	C₂³	1	2	yes	—
131	C₂²⋊C₄	C₂³	1	2	yes	—
132	C₂²⋊C₄	C₂³	1	2	yes	—
133	C₂²⋊C₄	C₂³	1	2	yes	—
134	C₂²⋊C₄	C₂³	1	2	yes	—
135	C₂²⋊C₄	C₂³	1	2	yes	—
136	C₂²⋊C₄	C₂³	1	4	yes	—
137	C₂²⋊C₄	C₂³	1	4	yes	—
138	C₂²⋊C₄	C₂³	1	2	yes	—
139	C₂²⋊C₄	C₂³	1	2	yes	—
140	C₂²⋊C₄	C₂³	1	4	yes	—
141	C₂²⋊C₄	C₂³	1	2	yes	—
142	C₂²⋊C₄	C₂³	1	2	yes	—
143	C₄²	C₂³	1	1	yes	characteristic
144	C₄²	C₂³	1	1	yes	characteristic
145	C₄²	C₂³	1	1	yes	characteristic
146	C₄²	C₂³	1	1	yes	characteristic
147	C₄²	C₂³	1	2	yes	—
148	C₄²	C₂³	1	2	yes	—
149	C₄²	C₂³	1	4	yes	—
150	C₄²	C₂³	1	4	yes	—
151	C₂³	C₂⁴	1	2	yes	—
152	C₂³	C₂⁴	1	2	yes	—
153	C₂³	C₂⁴	1	1	yes	characteristic
154	Q₈	—	2	4	no	—
155	Q₈	—	2	2	no	—
156	Q₈	—	2	4	no	—
157	Q₈	—	2	2	no	—
158	Q₈	—	4	4	no	—
159	Q₈	—	4	4	no	—
160	D₄	—	2	4	no	—
161	D₄	—	2	4	no	—
162	D₄	—	2	2	no	—
163	D₄	—	2	2	no	—
164	D₄	—	4	4	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	4	4	no	—
173	C₂×C₄	C₂⁴	1	2	yes	—
174	C₂×C₄	C₂⁴	1	2	yes	—
175	C₂×C₄	C₂⁴	1	2	yes	—
176	C₂×C₄	C₂⁴	1	1	yes	characteristic
177	C₂×C₄	C₂⁴	1	1	yes	characteristic
178	C₂×C₄	C₂⁴	1	1	yes	characteristic
179	C₂×C₄	C₂⁴	1	2	yes	—
180	C₂×C₄	C₂⁴	1	1	yes	characteristic
181	C₂×C₄	C₂⁴	1	2	yes	—
182	C₂×C₄	C₂⁴	1	4	yes	—
183	C₂×C₄	C₂⁴	1	4	yes	—
184	C₂×C₄	C₂⁴	1	2	yes	—
185	C₂×C₄	C₂⁴	1	1	yes	characteristic
186	C₂×C₄	C₂⁴	1	1	yes	characteristic
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	2	2	no	—
192	C₂²	—	2	2	no	—
193	C₂²	—	2	2	no	—
194	C₂²	—	2	2	no	—
195	C₂²	2₋¹⁺⁴	1	2	yes	—
196	C₂²	C₂⁵	1	1	yes	characteristic
197	C₄	—	2	2	no	—
198	C₄	—	2	1	no	—
199	C₄	—	2	4	no	—
200	C₄	—	2	4	no	—
201	C₄	—	2	2	no	—
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	2	2	no	—
206	C₄	—	2	2	no	—
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	H(5,𝔽₂)	1	2	yes	—
210	C₄	2₋¹⁺⁴	1	2	yes	—
211	C₂	—	2	2	no	—
212	C₂	—	4	2	no	—
213	C₂	—	4	2	no	—
214	C₂	C₂×H(5,𝔽₂)	1	1	yes	characteristic
215	C₂	C₂×2₋¹⁺⁴	1	1	yes	characteristic
216	C₂	C₂×2₋¹⁺⁴	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. #557	C₂	semidirect
3. #556	C₂	semidirect
4. #554	C₂	semidirect
5. #553	C₂	semidirect
6. D₄×Q₈	C₂	semidirect
7. D₄×Q₈	C₂	semidirect
8. #548	C₂	semidirect
9. #547	C₂	semidirect
10. #544	C₂	semidirect
11. #542	C₂	semidirect
12. #541	C₂	semidirect
13. #539	C₂	semidirect
14. #536	C₂	semidirect
15. C₂×C₄⋊Q₈	C₂	semidirect
16. #529	C₂	semidirect
17. #529	C₂	semidirect

Maximal subgroups

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

subgroup	index
#563	2
#557	2
#556	2
#554	2
#553	2
D₄×Q₈	2
#548	2
#547	2
#544	2
#542	2
#541	2
#539	2
#536	2
C₂×C₄⋊Q₈	2
#529	2