Finite Group Databasebeta

Subgroups — #3344

Order 128.

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

normal subgroups only

This group has 172 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	#554	C₂	1	1	yes	maximal, characteristic
2	#552	C₂	1	1	yes	maximal, characteristic
3	#501	C₂	1	1	yes	maximal, characteristic, complemented
4	#485	C₂	1	1	yes	maximal, characteristic, complemented
5	#484	C₂	1	1	yes	maximal, characteristic, complemented
6	#483	C₂	1	1	yes	maximal, characteristic, complemented
7	#483	C₂	1	1	yes	maximal, characteristic, complemented
8	#479	C₂	1	1	yes	maximal, characteristic, complemented
9	#475	C₂	1	1	yes	maximal, characteristic, complemented
10	#470	C₂	1	1	yes	maximal, characteristic, complemented
11	#467	C₂	1	1	yes	maximal, characteristic, complemented
12	#441	C₂	1	1	yes	maximal, characteristic, complemented
13	#435	C₂	1	1	yes	maximal, characteristic
14	#428	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#106	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₁₆	C₂²	1	1	yes	characteristic
17	C₂×M₁₆	C₂²	1	1	yes	characteristic
18	C₂²×C₈	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	#126	—	2	1	no	—
21	#125	—	2	1	no	—
22	#125	—	2	1	no	—
23	#125	C₂²	1	1	yes	characteristic
24	#123	—	2	1	no	—
25	C₂²⋊Q₈	—	2	1	no	—
26	C₂²⋊Q₈	—	2	1	no	—
27	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
28	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
29	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
30	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
31	C₄×Q₈	C₂²	1	1	yes	characteristic
32	C₄×Q₈	C₂²	1	1	yes	characteristic
33	C₄×D₄	—	2	1	no	—
34	C₄×D₄	C₂²	1	1	yes	characteristic
35	#117	—	2	1	no	—
36	#117	C₂²	1	1	yes	characteristic
37	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
38	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
40	#107	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	#106	C₂²	1	1	yes	characteristic
46	#106	C₂²	1	1	yes	characteristic
47	#106	C₂²	1	1	yes	characteristic
48	#106	C₂²	1	1	yes	characteristic
49	C₄⋊C₈	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	#103	C₂²	1	1	yes	characteristic
54	#103	C₂²	1	1	yes	characteristic
55	#103	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈	C₂²	1	1	yes	characteristic
57	C₂²⋊C₈	C₂²	1	1	yes	characteristic
58	#97	C₂²	1	1	yes	characteristic
59	C₂×Q₈	—	2	1	no	—
60	C₂×Q₈	C₂³	1	1	yes	characteristic
61	C₂×Q₈	C₂³	1	1	yes	characteristic
62	C₂×D₄	D₄	1	1	yes	characteristic
63	C₂²×C₄	—	2	1	no	—
64	C₂²×C₄	—	2	1	no	—
65	C₂²×C₄	—	2	1	no	—
66	C₂²×C₄	C₂³	1	1	yes	characteristic
67	C₂²×C₄	C₂³	1	1	yes	characteristic
68	Q₁₆	—	2	2	no	—
69	M₁₆	—	2	2	no	—
70	C₂×C₈	—	2	2	no	—
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	—	2	1	no	—
85	C₄⋊C₄	—	2	1	no	—
86	C₄⋊C₄	—	2	1	no	—
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	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	D₄	1	1	yes	characteristic
100	C₂²⋊C₄	D₄	1	1	yes	characteristic
101	C₄²	—	2	1	no	—
102	C₄²	—	2	1	no	—
103	C₄²	—	2	1	no	—
104	C₄²	C₂³	1	1	yes	characteristic
105	C₂³	C₂×D₄	1	1	yes	characteristic
106	C₂³	C₂×D₄	1	1	yes	characteristic
107	Q₈	—	2	1	no	—
108	Q₈	—	2	1	no	—
109	Q₈	—	2	1	no	—
110	Q₈	—	2	1	no	—
111	Q₈	—	8	1	no	—
112	D₄	—	2	2	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	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	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	8	1	no	—
127	C₂×C₄	—	8	1	no	—
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	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄	C₂⁴	1	1	yes	characteristic
133	C₈	—	2	1	no	—
134	C₈	—	2	1	no	—
135	C₈	—	2	1	no	—
136	C₈	#41	1	2	yes	—
137	C₂²	—	2	1	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	#137	1	2	yes	—
143	C₂²	C₂²×D₄	1	1	yes	characteristic
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	4	1	no	—
149	C₄	—	4	1	no	—
150	C₄	—	4	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	C₂×#41	1	1	yes	characteristic
157	C₄	2₋¹⁺⁴	1	1	yes	characteristic
158	C₂	—	2	2	no	—
159	C₂	—	4	1	no	—
160	C₂	#547	1	1	yes	characteristic
161	C₂	C₂×#137	1	1	yes	characteristic
162	C₂	#577	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. #501	C₂	semidirect
4. #485	C₂	semidirect
5. #484	C₂	semidirect
6. #483	C₂	semidirect
7. #483	C₂	semidirect
8. #479	C₂	semidirect
9. #475	C₂	semidirect
10. #470	C₂	semidirect
11. #467	C₂	semidirect
12. #441	C₂	semidirect
14. #428	C₂	semidirect
15. C₂×#106	C₂	semidirect

Maximal subgroups

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

subgroup	index
#554	2
#552	2
#501	2
#485	2
#484	2
#483	2
#479	2
#475	2
#470	2
#467	2
#441	2
#435	2
#428	2
C₂×#106	2