Finite Group Databasebeta

Subgroups — #3121

Order 128.

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

normal subgroups only

This group has 185 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
2	#519	C₂	1	1	yes	maximal, characteristic, complemented
3	#485	C₂	1	1	yes	maximal, characteristic, complemented
4	#482	C₂	1	1	yes	maximal, characteristic
5	#477	C₂	1	1	yes	maximal, characteristic, complemented
6	#475	C₂	1	1	yes	maximal, characteristic, complemented
7	#464	C₂	1	1	yes	maximal, characteristic, complemented
8	#464	C₂	1	1	yes	maximal, characteristic, complemented
9	#451	C₂	1	1	yes	maximal, characteristic, complemented
10	#448	C₂	1	1	yes	maximal, characteristic
11	#441	C₂	1	1	yes	maximal, characteristic, complemented
12	#440	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
15	#433	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Q₈	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆	C₂²	1	1	yes	characteristic
19	C₂×SD₁₆	C₂²	1	1	yes	characteristic
20	C₂×SD₁₆	C₂²	1	1	yes	characteristic
21	C₄⋊Q₈	C₂²	1	1	yes	characteristic
22	#126	—	2	1	no	—
23	#124	—	2	1	no	—
24	#124	C₂²	1	1	yes	characteristic
25	#123	—	2	1	no	—
26	C₂²⋊Q₈	—	2	1	no	—
27	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
28	#121	C₂²	1	1	yes	characteristic
29	C₄×Q₈	—	2	1	no	—
30	C₄×Q₈	—	2	1	no	—
31	C₄×Q₈	C₂²	1	1	yes	characteristic
32	C₄×Q₈	C₂²	1	1	yes	characteristic
33	C₄×Q₈	C₂²	1	1	yes	characteristic
34	C₄×Q₈	C₂²	1	1	yes	characteristic
35	C₄×Q₈	C₂²	1	1	yes	characteristic
36	C₄×D₄	C₂²	1	1	yes	characteristic
37	#117	—	2	1	no	—
38	#117	—	2	1	no	—
39	#117	C₂²	1	1	yes	characteristic
40	#117	C₂²	1	1	yes	characteristic
41	#107	C₂²	1	1	yes	characteristic
42	#107	C₂²	1	1	yes	characteristic
43	#106	C₂²	1	1	yes	characteristic
44	#106	C₂²	1	1	yes	characteristic
45	C₄⋊C₈	C₂²	1	1	yes	characteristic
46	C₄⋊C₈	C₂²	1	1	yes	characteristic
47	#103	C₂²	1	1	yes	characteristic
48	#103	C₂²	1	1	yes	characteristic
49	#103	C₂²	1	1	yes	characteristic
50	#103	C₂²	1	1	yes	characteristic
51	#103	C₂²	1	1	yes	characteristic
52	#103	C₂²	1	1	yes	characteristic
53	#102	C₂²	1	1	yes	characteristic
54	#102	C₂²	1	1	yes	characteristic
55	C₂²⋊C₈	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈	C₂²	1	1	yes	characteristic
57	#97	C₂²	1	1	yes	characteristic
58	C₄×C₈	C₂²	1	1	yes	characteristic
59	C₂×Q₈	—	2	2	no	—
60	C₂×Q₈	—	2	2	no	—
61	C₂×Q₈	—	2	1	no	—
62	C₂×Q₈	—	2	1	no	—
63	C₂×Q₈	C₂³	1	1	yes	characteristic
64	C₂×Q₈	C₂³	1	1	yes	characteristic
65	C₂×Q₈	C₂³	1	1	yes	characteristic
66	C₂×D₄	—	2	1	no	—
67	C₂×D₄	C₂³	1	1	yes	characteristic
68	C₂²×C₄	—	2	1	no	—
69	C₂²×C₄	—	2	1	no	—
70	C₂²×C₄	C₂³	1	1	yes	characteristic
71	Q₁₆	—	2	2	no	—
72	Q₁₆	—	2	2	no	—
73	SD₁₆	—	2	2	no	—
74	SD₁₆	—	2	2	no	—
75	C₂×C₈	C₂³	1	1	yes	characteristic
76	C₂×C₈	C₂³	1	1	yes	characteristic
77	C₂×C₈	C₂³	1	1	yes	characteristic
78	C₂×C₈	C₂³	1	1	yes	characteristic
79	C₄⋊C₄	—	2	1	no	—
80	C₄⋊C₄	—	2	1	no	—
81	C₄⋊C₄	—	2	1	no	—
82	C₄⋊C₄	—	2	1	no	—
83	C₄⋊C₄	—	2	1	no	—
84	C₄⋊C₄	—	2	1	no	—
85	C₄⋊C₄	—	2	1	no	—
86	C₄⋊C₄	D₄	1	1	yes	characteristic
87	C₄⋊C₄	D₄	1	1	yes	characteristic
88	C₄⋊C₄	C₂³	1	1	yes	characteristic
89	C₄⋊C₄	C₂³	1	1	yes	characteristic
90	C₄⋊C₄	C₂³	1	1	yes	characteristic
91	C₄⋊C₄	C₂³	1	1	yes	characteristic
92	C₂²⋊C₄	—	2	1	no	—
93	C₂²⋊C₄	—	2	1	no	—
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	D₄	1	1	yes	characteristic
99	C₂²⋊C₄	D₄	1	1	yes	characteristic
100	C₄²	—	2	1	no	—
101	C₄²	—	2	1	no	—
102	C₄²	—	2	1	no	—
103	C₄²	—	2	1	no	—
104	C₄²	—	2	1	no	—
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	C₂×D₄	1	1	yes	characteristic
110	Q₈	—	2	4	no	—
111	Q₈	—	2	1	no	—
112	Q₈	—	2	1	no	—
113	Q₈	—	2	1	no	—
114	Q₈	—	2	1	no	—
115	Q₈	—	8	1	no	—
116	Q₈	#41	1	2	yes	—
117	Q₈	#41	1	2	yes	—
118	D₄	—	2	1	no	—
119	D₄	—	2	1	no	—
120	D₄	—	8	1	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	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	—	8	1	no	—
132	C₂×C₄	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄	C₂×D₄	1	1	yes	characteristic
137	C₂×C₄	C₂⁴	1	1	yes	characteristic
138	C₈	—	2	1	no	—
139	C₈	—	2	1	no	—
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	1	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	4	1	no	—
146	C₂²	—	4	1	no	—
147	C₂²	—	4	1	no	—
148	C₂²	C₂²×D₄	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	4	1	no	—
157	C₄	—	4	1	no	—
158	C₄	—	4	1	no	—
159	C₄	—	4	1	no	—
160	C₄	—	4	1	no	—
161	C₄	C₂×#41	1	1	yes	characteristic
162	C₄	C₂×#41	1	1	yes	characteristic
163	C₂	—	4	1	no	—
164	C₂	—	8	1	no	—
165	C₂	#525	1	1	yes	characteristic
166	C₂	#577	1	1	yes	characteristic
167	C₂	#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	#121	C₂²
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
2. #519	C₂	semidirect
3. #485	C₂	semidirect
5. #477	C₂	semidirect
6. #475	C₂	semidirect
7. #464	C₂	semidirect
8. #464	C₂	semidirect
9. #451	C₂	semidirect
11. #441	C₂	semidirect
12. #440	C₂	semidirect
13. C₄×Q₁₆	C₂	semidirect
14. C₄×SD₁₆	C₂	semidirect
15. #433	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
#519	2
#485	2
#482	2
#477	2
#475	2
#464	2
#451	2
#448	2
#441	2
#440	2
C₄×Q₁₆	2
C₄×SD₁₆	2
#433	2