Finite Group Databasebeta

Subgroups — #3198

Order 128.

211 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 211 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#537	C₂	1	1	yes	maximal, characteristic, complemented
2	#535	C₂	1	1	yes	maximal, characteristic, complemented
3	#485	C₂	1	1	yes	maximal, characteristic, complemented
4	#482	C₂	1	1	yes	maximal, characteristic, complemented
5	#481	C₂	1	1	yes	maximal, characteristic, complemented
6	#480	C₂	1	1	yes	maximal, characteristic, complemented
7	#469	C₂	1	1	yes	maximal, characteristic, complemented
8	#468	C₂	1	1	yes	maximal, characteristic, complemented
9	#466	C₂	1	1	yes	maximal, characteristic, complemented
10	#465	C₂	1	1	yes	maximal, characteristic, complemented
11	#450	C₂	1	1	yes	maximal, characteristic, complemented
12	#449	C₂	1	1	yes	maximal, characteristic, complemented
13	#449	C₂	1	1	yes	maximal, characteristic, complemented
14	#447	C₂	1	1	yes	maximal, characteristic, complemented
15	#408	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#41	C₂²	1	1	yes	characteristic
17	C₂×#41	C₂²	1	1	yes	characteristic, complemented
18	C₂²×D₄	C₂²	1	1	yes	characteristic
19	C₂×SD₁₆	C₂²	1	1	yes	characteristic
20	C₂×SD₁₆	C₂²	1	1	yes	characteristic
21	C₂×D₈	C₂²	1	1	yes	characteristic
22	C₂×D₈	C₂²	1	1	yes	characteristic
23	C₂×M₁₆	C₂²	1	1	yes	characteristic, complemented
24	C₂²×C₈	C₂²	1	1	yes	characteristic, complemented
25	#127	—	2	1	no	—
26	#124	—	2	1	no	—
27	C₂²⋊Q₈	—	2	1	no	—
28	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
29	C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
30	#121	—	2	1	no	—
31	#121	—	2	1	no	—
32	#121	—	2	1	no	—
33	#121	C₂²	1	1	yes	characteristic
34	#121	C₂²	1	1	yes	characteristic
35	#121	C₂²	1	1	yes	characteristic
36	#121	C₂²	1	1	yes	characteristic
37	#121	C₂²	1	1	yes	characteristic
38	#121	C₂²	1	1	yes	characteristic, complemented
39	#120	—	2	1	no	—
40	#120	—	2	1	no	—
41	#117	C₂²	1	1	yes	characteristic
42	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
43	#107	C₂²	1	1	yes	characteristic, complemented
44	#107	C₂²	1	1	yes	characteristic, complemented
45	#106	C₂²	1	1	yes	characteristic, complemented
46	#106	C₂²	1	1	yes	characteristic, complemented
47	#103	C₂²	1	1	yes	characteristic, complemented
48	#103	C₂²	1	1	yes	characteristic, complemented
49	#102	C₂²	1	1	yes	characteristic
50	#102	C₂²	1	1	yes	characteristic
51	#102	C₂²	1	1	yes	characteristic
52	#102	C₂²	1	1	yes	characteristic
53	#102	C₂²	1	1	yes	characteristic, complemented
54	#102	C₂²	1	1	yes	characteristic, complemented
55	C₂²⋊C₈	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈	C₂²	1	1	yes	characteristic
57	C₂²⋊C₈	C₂²	1	1	yes	characteristic, complemented
58	C₂²⋊C₈	C₂²	1	1	yes	characteristic, complemented
59	C₂⁴	—	2	1	no	—
60	#41	—	4	1	no	—
61	#41	—	4	1	no	—
62	#41	—	4	1	no	—
63	#41	—	4	1	no	—
64	C₂×Q₈	D₄	1	1	yes	characteristic
65	C₂×Q₈	C₂³	1	1	yes	characteristic
66	C₂×D₄	—	2	1	no	—
67	C₂×D₄	—	2	1	no	—
68	C₂×D₄	—	2	1	no	—
69	C₂×D₄	—	2	1	no	—
70	C₂×D₄	—	2	1	no	—
71	C₂×D₄	—	2	1	no	—
72	C₂×D₄	—	2	1	no	—
73	C₂×D₄	—	2	1	no	—
74	C₂×D₄	—	4	1	no	—
75	C₂×D₄	—	4	1	no	—
76	C₂×D₄	D₄	1	1	yes	characteristic
77	C₂×D₄	D₄	1	1	yes	characteristic
78	C₂×D₄	D₄	1	1	yes	characteristic
79	C₂×D₄	C₂³	1	1	yes	characteristic
80	C₂×D₄	C₂³	1	1	yes	characteristic
81	C₂×D₄	C₂³	1	1	yes	characteristic
82	C₂²×C₄	—	2	1	no	—
83	C₂²×C₄	—	2	1	no	—
84	C₂²×C₄	C₂³	1	1	yes	characteristic
85	C₂²×C₄	C₂³	1	1	yes	characteristic
86	C₂²×C₄	C₂³	1	1	yes	characteristic
87	SD₁₆	—	4	1	no	—
88	SD₁₆	—	4	1	no	—
89	D₈	—	4	1	no	—
90	D₈	—	4	1	no	—
91	M₁₆	—	4	1	no	—
92	C₂×C₈	—	4	1	no	—
93	C₂×C₈	C₂³	1	1	yes	characteristic
94	C₂×C₈	C₂³	1	1	yes	characteristic
95	C₂×C₈	C₂³	1	1	yes	characteristic
96	C₂×C₈	C₂³	1	1	yes	characteristic
97	C₄⋊C₄	—	2	1	no	—
98	C₄⋊C₄	—	2	1	no	—
99	C₄⋊C₄	—	2	1	no	—
100	C₄⋊C₄	C₂³	1	1	yes	characteristic
101	C₄⋊C₄	C₂³	1	1	yes	characteristic
102	C₄⋊C₄	C₂³	1	1	yes	characteristic
103	C₄⋊C₄	C₂³	1	1	yes	characteristic
104	C₂²⋊C₄	—	2	1	no	—
105	C₂²⋊C₄	—	2	1	no	—
106	C₂²⋊C₄	—	2	1	no	—
107	C₂²⋊C₄	—	2	1	no	—
108	C₂²⋊C₄	—	2	1	no	—
109	C₂²⋊C₄	—	2	1	no	—
110	C₂²⋊C₄	—	2	1	no	—
111	C₂²⋊C₄	—	2	1	no	—
112	C₂²⋊C₄	—	2	1	no	—
113	C₄²	—	2	1	no	—
114	C₂³	—	2	1	no	—
115	C₂³	—	2	1	no	—
116	C₂³	—	2	1	no	—
117	C₂³	—	4	2	no	—
118	C₂³	—	8	1	no	—
119	C₂³	—	8	1	no	—
120	C₂³	C₂×D₄	1	1	yes	characteristic
121	C₂³	C₂×D₄	1	1	yes	characteristic
122	C₂³	C₂×D₄	1	1	yes	characteristic
123	Q₈	—	2	1	no	—
124	Q₈	—	2	1	no	—
125	Q₈	—	4	1	no	—
126	D₄	—	2	1	no	—
127	D₄	—	2	1	no	—
128	D₄	—	2	1	no	—
129	D₄	—	2	1	no	—
130	D₄	—	2	1	no	—
131	D₄	—	2	1	no	—
132	D₄	—	4	1	no	—
133	D₄	—	4	1	no	—
134	D₄	—	4	1	no	—
135	D₄	—	4	2	no	—
136	D₄	—	4	2	no	—
137	D₄	—	8	1	no	—
138	D₄	—	8	1	no	—
139	D₄	—	8	1	no	—
140	D₄	—	8	1	no	—
141	D₄	—	8	1	no	—
142	D₄	—	8	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	8	1	no	—
156	C₂×C₄	C₂×D₄	1	1	yes	characteristic
157	C₂×C₄	C₂×D₄	1	1	yes	characteristic
158	C₂×C₄	C₂×D₄	1	1	yes	characteristic
159	C₂×C₄	C₂⁴	1	1	yes	characteristic
160	C₈	—	2	1	no	—
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	—	2	1	no	—
167	C₂²	—	2	1	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	—	2	1	no	—
172	C₂²	—	2	1	no	—
173	C₂²	—	4	1	no	—
174	C₂²	—	4	1	no	—
175	C₂²	—	4	1	no	—
176	C₂²	—	4	1	no	—
177	C₂²	—	4	1	no	—
178	C₂²	—	4	1	no	—
179	C₂²	—	4	1	no	—
180	C₂²	—	4	1	no	—
181	C₂²	—	4	1	no	—
182	C₂²	—	8	4	no	—
183	C₂²	C₂²×D₄	1	1	yes	characteristic
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	4	1	no	—
188	C₄	—	4	1	no	—
189	C₄	—	4	1	no	—
190	C₄	—	4	1	no	—
191	C₄	—	4	1	no	—
192	C₄	H(5,𝔽₂)	1	1	yes	characteristic
193	C₄	H(5,𝔽₂)	1	1	yes	characteristic
194	C₂	—	4	1	no	—
195	C₂	—	4	1	no	—
196	C₂	—	4	1	no	—
197	C₂	—	8	1	no	—
198	C₂	—	8	1	no	—
199	C₂	—	8	1	no	—
200	C₂	#534	1	1	yes	characteristic
201	C₂	#576	1	1	yes	characteristic
202	C₂	#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #537	C₂	semidirect
2. #535	C₂	semidirect
3. #485	C₂	semidirect
4. #482	C₂	semidirect
5. #481	C₂	semidirect
6. #480	C₂	semidirect
7. #469	C₂	semidirect
8. #468	C₂	semidirect
9. #466	C₂	semidirect
10. #465	C₂	semidirect
11. #450	C₂	semidirect
12. #449	C₂	semidirect
13. #449	C₂	semidirect
14. #447	C₂	semidirect
15. #408	C₂	semidirect
17. C₂×#41	C₂²	semidirect
23. C₂×M₁₆	C₂²	semidirect
24. C₂²×C₈	C₂²	semidirect
29. C₂²⋊Q₈	C₂²	semidirect
38. #121	C₂²	semidirect
42. C₂×C₄⋊C₄	C₂²	semidirect
43. #107	C₂²	semidirect
44. #107	C₂²	semidirect
45. #106	C₂²	semidirect
46. #106	C₂²	semidirect
47. #103	C₂²	semidirect
48. #103	C₂²	semidirect
53. #102	C₂²	semidirect
54. #102	C₂²	semidirect
57. C₂²⋊C₈	C₂²	semidirect
58. C₂²⋊C₈	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#537	2
#535	2
#485	2
#482	2
#481	2
#480	2
#469	2
#468	2
#466	2
#465	2
#450	2
#449	2
#447	2
#408	2