Finite Group Databasebeta

Subgroups — #3312

Order 128.

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

normal subgroups only

This group has 188 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	#552	C₂	1	1	yes	maximal, characteristic
2	D₄×Q₈	C₂	1	1	yes	maximal, characteristic
3	#491	C₂	1	1	yes	maximal, characteristic, complemented
4	#485	C₂	1	1	yes	maximal, characteristic, complemented
5	#483	C₂	1	1	yes	maximal, characteristic, complemented
6	#479	C₂	1	1	yes	maximal, characteristic, complemented
7	#470	C₂	1	1	yes	maximal, characteristic, complemented
8	#467	C₂	1	1	yes	maximal, characteristic, complemented
9	#462	C₂	1	1	yes	maximal, characteristic, complemented
10	#451	C₂	1	1	yes	maximal, characteristic, complemented
11	#451	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	#419	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#103	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Q₈	C₂²	1	1	yes	characteristic
17	C₂²×Q₈	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆	C₂²	1	1	yes	characteristic
19	C₂×Q₁₆	C₂²	1	1	yes	characteristic
20	C₂×Q₁₆	C₂²	1	1	yes	characteristic
21	C₂×M₁₆	C₂²	1	1	yes	characteristic
22	C₂²×C₈	C₂²	1	1	yes	characteristic
23	C₄⋊Q₈	—	2	1	no	—
24	C₄⋊Q₈	C₂²	1	1	yes	characteristic
25	#126	—	2	1	no	—
26	#125	—	2	1	no	—
27	#125	C₂²	1	1	yes	characteristic
28	#123	—	2	1	no	—
29	C₂²⋊Q₈	—	2	1	no	—
30	C₂²⋊Q₈	—	2	1	no	—
31	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
32	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
33	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
34	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
35	C₄×Q₈	C₂²	1	1	yes	characteristic
36	C₄×Q₈	C₂²	1	1	yes	characteristic
37	C₄×D₄	—	2	1	no	—
38	C₄×D₄	C₂²	1	1	yes	characteristic
39	#117	—	2	1	no	—
40	#117	C₂²	1	1	yes	characteristic
41	C₂×C₄⋊C₄	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	#103	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	#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	—	2	1	no	—
61	C₂×Q₈	—	2	2	no	—
62	C₂×Q₈	—	2	2	no	—
63	C₂×Q₈	—	4	1	no	—
64	C₂×Q₈	—	4	1	no	—
65	C₂×Q₈	C₂³	1	1	yes	characteristic
66	C₂×Q₈	C₂³	1	1	yes	characteristic
67	C₂×Q₈	C₂³	1	1	yes	characteristic
68	C₂×Q₈	C₂³	1	1	yes	characteristic
69	C₂×D₄	D₄	1	1	yes	characteristic
70	C₂²×C₄	—	2	1	no	—
71	C₂²×C₄	—	2	1	no	—
72	C₂²×C₄	—	2	1	no	—
73	C₂²×C₄	C₂³	1	1	yes	characteristic
74	C₂²×C₄	C₂³	1	1	yes	characteristic
75	Q₁₆	—	2	2	no	—
76	Q₁₆	—	4	1	no	—
77	Q₁₆	—	4	1	no	—
78	M₁₆	—	4	1	no	—
79	C₂×C₈	—	4	1	no	—
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	C₂×C₈	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
89	C₄⋊C₄	—	2	1	no	—
90	C₄⋊C₄	—	2	1	no	—
91	C₄⋊C₄	—	2	1	no	—
92	C₄⋊C₄	—	2	1	no	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	D₄	1	1	yes	characteristic
96	C₄⋊C₄	C₂³	1	1	yes	characteristic
97	C₄⋊C₄	C₂³	1	1	yes	characteristic
98	C₄⋊C₄	C₂³	1	1	yes	characteristic
99	C₄⋊C₄	C₂³	1	1	yes	characteristic
100	C₂²⋊C₄	—	2	1	no	—
101	C₂²⋊C₄	—	2	1	no	—
102	C₂²⋊C₄	—	2	1	no	—
103	C₂²⋊C₄	—	2	1	no	—
104	C₂²⋊C₄	—	2	1	no	—
105	C₂²⋊C₄	D₄	1	1	yes	characteristic
106	C₂²⋊C₄	D₄	1	1	yes	characteristic
107	C₄²	—	2	1	no	—
108	C₄²	—	2	1	no	—
109	C₄²	—	2	1	no	—
110	C₄²	C₂³	1	1	yes	characteristic
111	C₂³	C₂×D₄	1	1	yes	characteristic
112	C₂³	C₂×D₄	1	1	yes	characteristic
113	Q₈	—	2	1	no	—
114	Q₈	—	2	1	no	—
115	Q₈	—	2	1	no	—
116	Q₈	—	2	1	no	—
117	Q₈	—	2	1	no	—
118	Q₈	—	2	1	no	—
119	Q₈	—	2	1	no	—
120	Q₈	—	4	2	no	—
121	Q₈	—	4	2	no	—
122	Q₈	#41	1	2	yes	—
123	D₄	—	2	2	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	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	C₂×D₄	1	1	yes	characteristic
140	C₂×C₄	C₂×D₄	1	1	yes	characteristic
141	C₂×C₄	C₂×D₄	1	1	yes	characteristic
142	C₂×C₄	C₂×D₄	1	1	yes	characteristic
143	C₂×C₄	C₂⁴	1	1	yes	characteristic
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	2	1	no	—
147	C₈	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	—	2	1	no	—
153	C₂²	#137	1	2	yes	—
154	C₂²	C₂²×D₄	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	2	no	—
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	4	1	no	—
161	C₄	—	4	1	no	—
162	C₄	—	4	1	no	—
163	C₄	—	4	1	no	—
164	C₄	—	4	1	no	—
165	C₄	—	4	1	no	—
166	C₄	—	4	1	no	—
167	C₄	C₂×#41	1	1	yes	characteristic
168	C₄	H(5,𝔽₂)	1	1	yes	characteristic
169	C₂	—	2	2	no	—
170	C₂	—	4	1	no	—
171	C₂	#546	1	1	yes	characteristic
172	C₂	C₂×#137	1	1	yes	characteristic
173	C₂	#578	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. #491	C₂	semidirect
4. #485	C₂	semidirect
5. #483	C₂	semidirect
6. #479	C₂	semidirect
7. #470	C₂	semidirect
8. #467	C₂	semidirect
9. #462	C₂	semidirect
10. #451	C₂	semidirect
11. #451	C₂	semidirect
12. #441	C₂	semidirect
14. #419	C₂	semidirect
15. C₂×#103	C₂	semidirect

Maximal subgroups

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

subgroup	index
#552	2
D₄×Q₈	2
#491	2
#485	2
#483	2
#479	2
#470	2
#467	2
#462	2
#451	2
#441	2
#435	2
#419	2
C₂×#103	2