Finite Group Databasebeta

Subgroups — #3313

Order 128.

190 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 86 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	#556	C₂	1	1	yes	maximal, characteristic, complemented
2	#548	C₂	1	1	yes	maximal, characteristic
3	#489	C₂	1	1	yes	maximal, characteristic, complemented
4	#485	C₂	1	1	yes	maximal, characteristic, complemented
5	#482	C₂	1	1	yes	maximal, characteristic, complemented
6	#477	C₂	1	1	yes	maximal, characteristic, complemented
7	#468	C₂	1	1	yes	maximal, characteristic, complemented
8	#467	C₂	1	1	yes	maximal, characteristic, complemented
9	#464	C₂	1	1	yes	maximal, characteristic, complemented
10	#452	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, complemented
14	#419	C₂	1	1	yes	maximal, characteristic, complemented
15	#416	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#41	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×SD₁₆	C₂²	1	1	yes	characteristic
21	C₂×M₁₆	C₂²	1	1	yes	characteristic
22	C₂²×C₈	C₂²	1	1	yes	characteristic
23	C₄⋊Q₈	C₂²	1	1	yes	characteristic
24	#126	—	2	1	no	—
25	#126	—	2	1	no	—
26	#124	—	2	1	no	—
27	#124	—	2	1	no	—
28	#124	C₂²	1	1	yes	characteristic
29	C₂²⋊Q₈	—	2	1	no	—
30	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
31	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
32	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
33	#121	—	2	1	no	—
34	#121	C₂²	1	1	yes	characteristic
35	C₄×Q₈	—	2	1	no	—
36	C₄×Q₈	C₂²	1	1	yes	characteristic
37	C₄×Q₈	C₂²	1	1	yes	characteristic
38	C₄×D₄	—	2	1	no	—
39	C₄×D₄	C₂²	1	1	yes	characteristic
40	#117	C₂²	1	1	yes	characteristic
41	#117	C₂²	1	1	yes	characteristic
42	#107	C₂²	1	1	yes	characteristic
43	#107	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	#102	C₂²	1	1	yes	characteristic
54	#102	C₂²	1	1	yes	characteristic
55	#102	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	#41	—	2	2	no	—
60	#41	—	4	1	no	—
61	C₂×Q₈	—	2	1	no	—
62	C₂×Q₈	—	2	1	no	—
63	C₂×Q₈	—	2	2	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×D₄	—	2	1	no	—
69	C₂×D₄	—	2	1	no	—
70	C₂×D₄	D₄	1	1	yes	characteristic
71	C₂×D₄	C₂³	1	1	yes	characteristic
72	C₂²×C₄	—	2	1	no	—
73	C₂²×C₄	—	2	1	no	—
74	C₂²×C₄	C₂³	1	1	yes	characteristic
75	C₂²×C₄	C₂³	1	1	yes	characteristic
76	Q₁₆	—	2	2	no	—
77	Q₁₆	—	4	1	no	—
78	SD₁₆	—	4	1	no	—
79	M₁₆	—	4	1	no	—
80	C₂×C₈	—	4	1	no	—
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	C₂×C₈	C₂³	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
92	C₄⋊C₄	C₂³	1	1	yes	characteristic
93	C₄⋊C₄	C₂³	1	1	yes	characteristic
94	C₄⋊C₄	C₂³	1	1	yes	characteristic
95	C₄⋊C₄	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
104	C₂²⋊C₄	D₄	1	1	yes	characteristic
105	C₂²⋊C₄	D₄	1	1	yes	characteristic
106	C₄²	—	2	1	no	—
107	C₄²	—	2	1	no	—
108	C₄²	—	2	1	no	—
109	C₄²	—	2	1	no	—
110	C₄²	C₂³	1	1	yes	characteristic
111	C₂³	—	2	1	no	—
112	C₂³	C₂×D₄	1	1	yes	characteristic
113	C₂³	C₂×D₄	1	1	yes	characteristic
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	—	4	2	no	—
120	Q₈	—	4	2	no	—
121	Q₈	#41	1	2	yes	—
122	D₄	—	2	1	no	—
123	D₄	—	2	1	no	—
124	D₄	—	4	1	no	—
125	D₄	—	4	2	no	—
126	D₄	—	4	2	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	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	4	2	no	—
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	C₂×D₄	1	1	yes	characteristic
144	C₂×C₄	C₂⁴	1	1	yes	characteristic
145	C₈	—	2	1	no	—
146	C₈	—	2	1	no	—
147	C₈	—	2	1	no	—
148	C₈	—	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	—	2	1	no	—
154	C₂²	—	2	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	2	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	—	4	1	no	—
170	C₄	C₂×#41	1	1	yes	characteristic
171	C₄	H(5,𝔽₂)	1	1	yes	characteristic
172	C₂	—	4	1	no	—
173	C₂	—	4	1	no	—
174	C₂	—	8	1	no	—
175	C₂	#546	1	1	yes	characteristic
176	C₂	#575	1	1	yes	characteristic
177	C₂	#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	#548	C₂
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
1. #556	C₂	semidirect
3. #489	C₂	semidirect
4. #485	C₂	semidirect
5. #482	C₂	semidirect
6. #477	C₂	semidirect
7. #468	C₂	semidirect
8. #467	C₂	semidirect
9. #464	C₂	semidirect
10. #452	C₂	semidirect
11. #451	C₂	semidirect
12. #441	C₂	semidirect
13. #435	C₂	semidirect
14. #419	C₂	semidirect
15. #416	C₂	semidirect

Maximal subgroups

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

subgroup	index
#556	2
#548	2
#489	2
#485	2
#482	2
#477	2
#468	2
#467	2
#464	2
#452	2
#451	2
#441	2
#435	2
#419	2
#416	2