Finite Group Databasebeta

Subgroups — #3241

Order 128.

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

normal subgroups only

This group has 166 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	#544	C₂	1	1	yes	maximal, characteristic, complemented
2	#540	C₂	1	1	yes	maximal, characteristic
3	#501	C₂	1	1	yes	maximal, characteristic, complemented
4	#499	C₂	1	1	yes	maximal, characteristic, complemented
5	#490	C₂	1	1	yes	maximal, characteristic, complemented
6	#486	C₂	1	1	yes	maximal, characteristic, complemented
7	#482	C₂	1	1	yes	maximal, characteristic, complemented
8	#482	C₂	1	1	yes	maximal, characteristic, complemented
9	#481	C₂	1	1	yes	maximal, characteristic, complemented
10	#480	C₂	1	1	yes	maximal, characteristic, complemented
11	#478	C₂	1	1	yes	maximal, characteristic, complemented
12	#478	C₂	1	1	yes	maximal, characteristic, complemented
13	#476	C₂	1	1	yes	maximal, characteristic, complemented
14	#474	C₂	1	1	yes	maximal, characteristic, complemented
15	#433	C₂	1	1	yes	maximal, characteristic, complemented
16	C₄⋊Q₈	—	2	1	no	—
17	C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	#127	C₂²	1	1	yes	characteristic
20	#125	—	2	1	no	—
21	#125	C₂²	1	1	yes	characteristic
22	#125	C₂²	1	1	yes	characteristic
23	#125	C₂²	1	1	yes	characteristic
24	#123	—	2	1	no	—
25	#123	—	2	1	no	—
26	C₂²⋊Q₈	—	2	1	no	—
27	C₂²⋊Q₈	—	2	1	no	—
28	#121	—	2	1	no	—
29	#121	—	2	1	no	—
30	#121	C₂²	1	1	yes	characteristic
31	#121	C₂²	1	1	yes	characteristic
32	C₄×D₄	C₂²	1	1	yes	characteristic
33	C₄×D₄	C₂²	1	1	yes	characteristic
34	#117	C₂²	1	1	yes	characteristic
35	#117	C₂²	1	1	yes	characteristic
36	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
37	#107	C₂²	1	1	yes	characteristic
38	#107	C₂²	1	1	yes	characteristic
39	#107	C₂²	1	1	yes	characteristic
40	#107	C₂²	1	1	yes	characteristic
41	#106	C₂²	1	1	yes	characteristic
42	#106	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	#102	C₂²	1	1	yes	characteristic
48	#102	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	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	1	no	—
60	C₂×Q₈	—	2	1	no	—
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	C₂³	1	1	yes	characteristic
66	C₂×D₄	C₂³	1	1	yes	characteristic
67	C₂²×C₄	—	2	1	no	—
68	C₂²×C₄	—	2	1	no	—
69	C₂²×C₄	—	2	1	no	—
70	C₂²×C₄	C₂³	1	1	yes	characteristic
71	C₂×C₈	C₂³	1	1	yes	characteristic
72	C₂×C₈	C₂³	1	1	yes	characteristic
73	C₂×C₈	C₂³	1	1	yes	characteristic
74	C₂×C₈	C₂³	1	1	yes	characteristic
75	C₄⋊C₄	—	2	1	no	—
76	C₄⋊C₄	—	2	1	no	—
77	C₄⋊C₄	—	2	1	no	—
78	C₄⋊C₄	—	2	1	no	—
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	D₄	1	1	yes	characteristic
85	C₄⋊C₄	D₄	1	1	yes	characteristic
86	C₄⋊C₄	C₂³	1	1	yes	characteristic
87	C₄⋊C₄	C₂³	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	D₄	1	1	yes	characteristic
98	C₂²⋊C₄	D₄	1	1	yes	characteristic
99	C₄²	—	2	1	no	—
100	C₄²	C₂³	1	1	yes	characteristic
101	C₄²	C₂³	1	1	yes	characteristic
102	C₂³	—	2	1	no	—
103	C₂³	—	2	1	no	—
104	C₂³	C₂×D₄	1	1	yes	characteristic
105	Q₈	—	8	1	no	—
106	Q₈	—	8	1	no	—
107	D₄	—	2	1	no	—
108	D₄	—	2	1	no	—
109	D₄	—	2	1	no	—
110	D₄	—	2	1	no	—
111	D₄	—	8	1	no	—
112	D₄	—	8	1	no	—
113	D₄	—	8	1	no	—
114	D₄	—	8	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	8	1	no	—
124	C₂×C₄	—	8	1	no	—
125	C₂×C₄	—	8	1	no	—
126	C₂×C₄	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄	C₂⁴	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	2	1	no	—
134	C₈	—	2	1	no	—
135	C₈	—	2	1	no	—
136	C₂²	—	2	1	no	—
137	C₂²	—	2	1	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	4	1	no	—
140	C₂²	—	4	1	no	—
141	C₂²	—	4	1	no	—
142	C₂²	—	4	1	no	—
143	C₂²	—	4	1	no	—
144	C₂²	—	4	1	no	—
145	C₂²	C₂²×D₄	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	4	1	no	—
152	C₄	—	4	1	no	—
153	C₄	—	4	1	no	—
154	C₄	—	4	1	no	—
155	C₄	—	4	1	no	—
156	C₄	—	4	1	no	—
157	C₄	2₋¹⁺⁴	1	1	yes	characteristic
158	C₄	2₋¹⁺⁴	1	1	yes	characteristic
159	C₂	—	4	1	no	—
160	C₂	—	8	1	no	—
161	C₂	—	8	1	no	—
162	C₂	#536	1	1	yes	characteristic
163	C₂	#576	1	1	yes	characteristic
164	C₂	#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	#540	C₂
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. #544	C₂	semidirect
3. #501	C₂	semidirect
4. #499	C₂	semidirect
5. #490	C₂	semidirect
6. #486	C₂	semidirect
7. #482	C₂	semidirect
8. #482	C₂	semidirect
9. #481	C₂	semidirect
10. #480	C₂	semidirect
11. #478	C₂	semidirect
12. #478	C₂	semidirect
13. #476	C₂	semidirect
14. #474	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
#544	2
#540	2
#501	2
#499	2
#490	2
#486	2
#482	2
#481	2
#480	2
#478	2
#476	2
#474	2
#433	2