Finite Group Databasebeta

Subgroups — #3201

Order 128.

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

normal subgroups only

This group has 195 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	#537	C₂	1	1	yes	maximal, characteristic
2	#536	C₂	1	1	yes	maximal, characteristic, complemented
3	#485	C₂	1	1	yes	maximal, characteristic, complemented
4	#484	C₂	1	1	yes	maximal, characteristic, complemented
5	#483	C₂	1	1	yes	maximal, characteristic, complemented
6	#482	C₂	1	1	yes	maximal, characteristic, complemented
7	#470	C₂	1	1	yes	maximal, characteristic, complemented
8	#468	C₂	1	1	yes	maximal, characteristic, complemented
9	#467	C₂	1	1	yes	maximal, characteristic, complemented
10	#465	C₂	1	1	yes	maximal, characteristic, complemented
11	#452	C₂	1	1	yes	maximal, characteristic, complemented
12	#452	C₂	1	1	yes	maximal, characteristic, complemented
13	#451	C₂	1	1	yes	maximal, characteristic, complemented
14	#448	C₂	1	1	yes	maximal, characteristic, complemented
15	#408	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#41	C₂²	1	1	yes	characteristic
17	C₂×#41	C₂²	1	1	yes	characteristic
18	C₂²×Q₈	C₂²	1	1	yes	characteristic
19	C₂×Q₁₆	C₂²	1	1	yes	characteristic
20	C₂×Q₁₆	C₂²	1	1	yes	characteristic
21	C₂×SD₁₆	C₂²	1	1	yes	characteristic
22	C₂×SD₁₆	C₂²	1	1	yes	characteristic
23	C₂×M₁₆	C₂²	1	1	yes	characteristic
24	C₂²×C₈	C₂²	1	1	yes	characteristic
25	C₄⋊Q₈	—	2	1	no	—
26	#124	—	2	1	no	—
27	#123	—	2	1	no	—
28	#123	—	2	1	no	—
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	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
34	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
35	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
36	#121	—	2	1	no	—
37	#121	—	2	1	no	—
38	#121	—	2	1	no	—
39	#121	C₂²	1	1	yes	characteristic
40	#121	C₂²	1	1	yes	characteristic
41	#117	C₂²	1	1	yes	characteristic
42	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
43	#107	C₂²	1	1	yes	characteristic
44	#107	C₂²	1	1	yes	characteristic
45	#106	C₂²	1	1	yes	characteristic
46	#106	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	C₂²⋊C₈	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈	C₂²	1	1	yes	characteristic
59	#41	—	4	1	no	—
60	#41	—	4	1	no	—
61	#41	—	4	1	no	—
62	#41	—	4	1	no	—
63	C₂×Q₈	—	2	1	no	—
64	C₂×Q₈	—	4	1	no	—
65	C₂×Q₈	—	4	1	no	—
66	C₂×Q₈	D₄	1	1	yes	characteristic
67	C₂×Q₈	C₂³	1	1	yes	characteristic
68	C₂×Q₈	C₂³	1	1	yes	characteristic
69	C₂×Q₈	C₂³	1	1	yes	characteristic
70	C₂×D₄	—	2	1	no	—
71	C₂×D₄	—	2	1	no	—
72	C₂×D₄	—	2	1	no	—
73	C₂×D₄	D₄	1	1	yes	characteristic
74	C₂×D₄	D₄	1	1	yes	characteristic
75	C₂×D₄	D₄	1	1	yes	characteristic
76	C₂×D₄	C₂³	1	1	yes	characteristic
77	C₂²×C₄	—	2	1	no	—
78	C₂²×C₄	—	2	1	no	—
79	C₂²×C₄	—	2	1	no	—
80	C₂²×C₄	C₂³	1	1	yes	characteristic
81	C₂²×C₄	C₂³	1	1	yes	characteristic
82	C₂²×C₄	C₂³	1	1	yes	characteristic
83	Q₁₆	—	4	1	no	—
84	Q₁₆	—	4	1	no	—
85	SD₁₆	—	4	1	no	—
86	SD₁₆	—	4	1	no	—
87	M₁₆	—	4	1	no	—
88	C₂×C₈	—	4	1	no	—
89	C₂×C₈	C₂³	1	1	yes	characteristic
90	C₂×C₈	C₂³	1	1	yes	characteristic
91	C₂×C₈	C₂³	1	1	yes	characteristic
92	C₂×C₈	C₂³	1	1	yes	characteristic
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	2	1	no	—
98	C₄⋊C₄	—	2	1	no	—
99	C₄⋊C₄	—	2	1	no	—
100	C₄⋊C₄	C₂³	1	1	yes	characteristic
101	C₄⋊C₄	C₂³	1	1	yes	characteristic
102	C₄⋊C₄	C₂³	1	1	yes	characteristic
103	C₄⋊C₄	C₂³	1	1	yes	characteristic
104	C₂²⋊C₄	—	2	1	no	—
105	C₂²⋊C₄	—	2	1	no	—
106	C₂²⋊C₄	—	2	1	no	—
107	C₂²⋊C₄	—	2	1	no	—
108	C₂²⋊C₄	—	2	1	no	—
109	C₂²⋊C₄	—	2	1	no	—
110	C₂²⋊C₄	—	2	1	no	—
111	C₂²⋊C₄	—	2	1	no	—
112	C₂²⋊C₄	—	2	1	no	—
113	C₄²	—	2	1	no	—
114	C₂³	—	2	1	no	—
115	C₂³	C₂×D₄	1	1	yes	characteristic
116	C₂³	C₂×D₄	1	1	yes	characteristic
117	C₂³	C₂×D₄	1	1	yes	characteristic
118	Q₈	—	2	1	no	—
119	Q₈	—	2	1	no	—
120	Q₈	—	2	1	no	—
121	Q₈	—	2	1	no	—
122	Q₈	—	2	1	no	—
123	Q₈	—	2	1	no	—
124	Q₈	—	4	1	no	—
125	Q₈	—	4	2	no	—
126	D₄	—	2	1	no	—
127	D₄	—	2	1	no	—
128	D₄	—	4	1	no	—
129	D₄	—	4	1	no	—
130	D₄	—	4	1	no	—
131	D₄	—	4	2	no	—
132	D₄	—	8	1	no	—
133	D₄	—	8	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	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	8	1	no	—
150	C₂×C₄	C₂×D₄	1	1	yes	characteristic
151	C₂×C₄	C₂×D₄	1	1	yes	characteristic
152	C₂×C₄	C₂×D₄	1	1	yes	characteristic
153	C₂×C₄	C₂⁴	1	1	yes	characteristic
154	C₈	—	2	1	no	—
155	C₈	—	2	1	no	—
156	C₈	—	2	1	no	—
157	C₈	—	2	1	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	—	2	1	no	—
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	—	2	1	no	—
167	C₂²	—	4	1	no	—
168	C₂²	—	4	1	no	—
169	C₂²	—	4	1	no	—
170	C₂²	C₂²×D₄	1	1	yes	characteristic
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	4	1	no	—
175	C₄	—	4	1	no	—
176	C₄	—	4	1	no	—
177	C₄	—	4	1	no	—
178	C₄	—	4	1	no	—
179	C₄	—	4	1	no	—
180	C₄	—	4	1	no	—
181	C₄	H(5,𝔽₂)	1	1	yes	characteristic
182	C₄	H(5,𝔽₂)	1	1	yes	characteristic
183	C₂	—	4	1	no	—
184	C₂	—	4	1	no	—
185	C₂	—	4	1	no	—
186	C₂	—	8	1	no	—
187	C₂	#534	1	1	yes	characteristic
188	C₂	#577	1	1	yes	characteristic
189	C₂	#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	#537	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #536	C₂	semidirect
3. #485	C₂	semidirect
4. #484	C₂	semidirect
5. #483	C₂	semidirect
6. #482	C₂	semidirect
7. #470	C₂	semidirect
8. #468	C₂	semidirect
9. #467	C₂	semidirect
10. #465	C₂	semidirect
11. #452	C₂	semidirect
12. #452	C₂	semidirect
13. #451	C₂	semidirect
14. #448	C₂	semidirect
15. #408	C₂	semidirect

Maximal subgroups

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

subgroup	index
#537	2
#536	2
#485	2
#484	2
#483	2
#482	2
#470	2
#468	2
#467	2
#465	2
#452	2
#451	2
#448	2
#408	2