Finite Group Databasebeta

Subgroups — #3179

Order 128.

187 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 187 conjugacy classes of subgroups — too many to draw. Only its 88 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	#542	C₂	1	1	yes	maximal, characteristic, complemented
2	#540	C₂	1	1	yes	maximal, characteristic, complemented
3	#501	C₂	1	1	yes	maximal, characteristic, complemented
4	#496	C₂	1	1	yes	maximal, characteristic, complemented
5	#488	C₂	1	1	yes	maximal, characteristic, complemented
6	#486	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	#442	C₂	1	1	yes	maximal, characteristic, complemented
12	#440	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×D₈	C₂	1	1	yes	maximal, characteristic, complemented
15	#405	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×SD₁₆	C₂²	1	1	yes	characteristic
17	C₂×SD₁₆	C₂²	1	1	yes	characteristic
18	C₂×D₈	C₂²	1	1	yes	characteristic
19	C₂×D₈	C₂²	1	1	yes	characteristic
20	C₂×M₁₆	C₂²	1	1	yes	characteristic
21	C₂²×C₈	C₂²	1	1	yes	characteristic
22	C₄⋊Q₈	C₂²	1	1	yes	characteristic
23	#127	C₂²	1	1	yes	characteristic
24	#126	—	2	1	no	—
25	#125	C₂²	1	1	yes	characteristic
26	#124	—	2	1	no	—
27	#124	C₂²	1	1	yes	characteristic
28	#123	—	2	1	no	—
29	#123	—	2	1	no	—
30	#123	—	2	1	no	—
31	C₂²⋊Q₈	—	2	1	no	—
32	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
33	#121	—	2	1	no	—
34	#121	—	2	1	no	—
35	#121	C₂²	1	1	yes	characteristic
36	#121	C₂²	1	1	yes	characteristic
37	#121	C₂²	1	1	yes	characteristic
38	C₄×Q₈	C₂²	1	1	yes	characteristic
39	C₄×D₄	C₂²	1	1	yes	characteristic
40	C₄×D₄	C₂²	1	1	yes	characteristic
41	C₄×D₄	C₂²	1	1	yes	characteristic
42	#117	C₂²	1	1	yes	characteristic
43	#107	C₂²	1	1	yes	characteristic
44	#107	C₂²	1	1	yes	characteristic
45	#106	C₂²	1	1	yes	characteristic
46	#106	C₂²	1	1	yes	characteristic
47	#103	C₂²	1	1	yes	characteristic
48	#103	C₂²	1	1	yes	characteristic
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
54	#102	C₂²	1	1	yes	characteristic
55	#97	C₂²	1	1	yes	characteristic
56	#97	C₂²	1	1	yes	characteristic
57	C₄×C₈	C₂²	1	1	yes	characteristic
58	C₄×C₈	C₂²	1	1	yes	characteristic
59	C₂×Q₈	—	2	1	no	—
60	C₂×Q₈	C₂³	1	1	yes	characteristic
61	C₂×D₄	—	2	1	no	—
62	C₂×D₄	—	2	1	no	—
63	C₂×D₄	—	2	1	no	—
64	C₂×D₄	—	2	1	no	—
65	C₂×D₄	—	2	1	no	—
66	C₂×D₄	C₂³	1	1	yes	characteristic
67	C₂×D₄	C₂³	1	1	yes	characteristic
68	C₂×D₄	C₂³	1	1	yes	characteristic
69	C₂²×C₄	—	2	1	no	—
70	C₂²×C₄	—	2	1	no	—
71	C₂²×C₄	—	2	1	no	—
72	C₂²×C₄	C₂³	1	1	yes	characteristic
73	SD₁₆	—	2	2	no	—
74	SD₁₆	—	2	2	no	—
75	D₈	—	2	2	no	—
76	D₈	—	2	2	no	—
77	M₁₆	—	2	2	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	C₂³	1	1	yes	characteristic
80	C₂×C₈	C₂³	1	1	yes	characteristic
81	C₂×C₈	C₂³	1	1	yes	characteristic
82	C₂×C₈	C₂³	1	1	yes	characteristic
83	C₄⋊C₄	—	2	1	no	—
84	C₄⋊C₄	—	2	1	no	—
85	C₄⋊C₄	—	2	1	no	—
86	C₄⋊C₄	—	2	1	no	—
87	C₄⋊C₄	—	2	1	no	—
88	C₄⋊C₄	D₄	1	1	yes	characteristic
89	C₄⋊C₄	D₄	1	1	yes	characteristic
90	C₄⋊C₄	C₂³	1	1	yes	characteristic
91	C₄⋊C₄	C₂³	1	1	yes	characteristic
92	C₄⋊C₄	C₂³	1	1	yes	characteristic
93	C₄⋊C₄	C₂³	1	1	yes	characteristic
94	C₂²⋊C₄	—	2	1	no	—
95	C₂²⋊C₄	—	2	1	no	—
96	C₂²⋊C₄	—	2	1	no	—
97	C₂²⋊C₄	—	2	1	no	—
98	C₂²⋊C₄	—	2	1	no	—
99	C₂²⋊C₄	—	2	1	no	—
100	C₂²⋊C₄	—	2	1	no	—
101	C₂²⋊C₄	—	2	1	no	—
102	C₂²⋊C₄	—	2	1	no	—
103	C₂²⋊C₄	D₄	1	1	yes	characteristic
104	C₂²⋊C₄	D₄	1	1	yes	characteristic
105	C₄²	—	2	1	no	—
106	C₄²	C₂³	1	1	yes	characteristic
107	C₄²	C₂³	1	1	yes	characteristic
108	C₂³	—	2	1	no	—
109	C₂³	—	2	1	no	—
110	C₂³	—	2	1	no	—
111	C₂³	C₂×D₄	1	1	yes	characteristic
112	Q₈	—	2	1	no	—
113	Q₈	—	2	1	no	—
114	Q₈	—	8	1	no	—
115	D₄	—	2	1	no	—
116	D₄	—	2	1	no	—
117	D₄	—	2	1	no	—
118	D₄	—	2	1	no	—
119	D₄	—	2	1	no	—
120	D₄	—	2	1	no	—
121	D₄	—	8	1	no	—
122	D₄	—	8	1	no	—
123	D₄	—	8	1	no	—
124	D₄	—	8	1	no	—
125	D₄	—	8	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	8	1	no	—
134	C₂×C₄	—	8	1	no	—
135	C₂×C₄	—	8	1	no	—
136	C₂×C₄	C₂×D₄	1	1	yes	characteristic
137	C₂×C₄	C₂×D₄	1	1	yes	characteristic
138	C₂×C₄	C₂×D₄	1	1	yes	characteristic
139	C₂×C₄	C₂×D₄	1	1	yes	characteristic
140	C₂×C₄	C₂×D₄	1	1	yes	characteristic
141	C₂×C₄	C₂⁴	1	1	yes	characteristic
142	C₈	—	2	1	no	—
143	C₈	—	2	1	no	—
144	C₈	#41	1	2	yes	—
145	C₈	#41	1	2	yes	—
146	C₂²	—	2	1	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	4	1	no	—
150	C₂²	—	4	1	no	—
151	C₂²	—	4	1	no	—
152	C₂²	—	4	1	no	—
153	C₂²	—	4	1	no	—
154	C₂²	—	4	1	no	—
155	C₂²	—	4	1	no	—
156	C₂²	—	4	1	no	—
157	C₂²	—	4	1	no	—
158	C₂²	C₂²×D₄	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	4	1	no	—
165	C₄	—	4	1	no	—
166	C₄	—	4	1	no	—
167	C₄	—	4	1	no	—
168	C₄	—	4	1	no	—
169	C₄	C₂×#41	1	1	yes	characteristic
170	C₄	C₂×#41	1	1	yes	characteristic
171	C₂	—	4	1	no	—
172	C₂	—	8	1	no	—
173	C₂	—	8	1	no	—
174	C₂	—	8	1	no	—
175	C₂	#532	1	1	yes	characteristic
176	C₂	#576	1	1	yes	characteristic
177	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 ◁ #117 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #542	C₂	semidirect
2. #540	C₂	semidirect
3. #501	C₂	semidirect
4. #496	C₂	semidirect
5. #488	C₂	semidirect
6. #486	C₂	semidirect
7. #469	C₂	semidirect
8. #468	C₂	semidirect
9. #466	C₂	semidirect
10. #465	C₂	semidirect
11. #442	C₂	semidirect
12. #440	C₂	semidirect
13. C₄×SD₁₆	C₂	semidirect
14. C₄×D₈	C₂	semidirect
15. #405	C₂	semidirect

Maximal subgroups

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

subgroup	index
#542	2
#540	2
#501	2
#496	2
#488	2
#486	2
#469	2
#468	2
#466	2
#465	2
#442	2
#440	2
C₄×SD₁₆	2
C₄×D₈	2
#405	2