Finite Group Databasebeta

Subgroups — #3269

Order 128.

239 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 239 conjugacy classes of subgroups — too many to draw. Only its 94 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	C₂×#137	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#135	C₂	1	1	yes	maximal, characteristic, complemented
3	#548	C₂	1	1	yes	maximal, characteristic, complemented
4	#547	C₂	1	1	yes	maximal, characteristic, complemented
5	#497	C₂	1	1	yes	maximal, characteristic, complemented
6	#468	C₂	1	1	yes	maximal, characteristic, complemented
7	#467	C₂	1	1	yes	maximal, characteristic, complemented
8	#463	C₂	1	1	yes	maximal, characteristic, complemented
9	#462	C₂	1	1	yes	maximal, characteristic, complemented
10	#452	C₂	1	1	yes	maximal, characteristic, complemented
11	#452	C₂	1	1	yes	maximal, characteristic, complemented
12	#451	C₂	1	1	yes	maximal, characteristic, complemented
13	#449	C₂	1	1	yes	maximal, characteristic, complemented
14	#440	C₂	1	1	yes	maximal, characteristic, complemented
15	#435	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#41	C₂²	1	1	yes	characteristic
17	C₂×#41	C₂²	1	1	yes	characteristic
18	C₂×#41	C₂²	1	1	yes	characteristic
19	C₂²×Q₈	C₂²	1	1	yes	characteristic
20	#137	—	2	4	no	—
21	#135	—	2	4	no	—
22	C₂×Q₁₆	C₂²	1	1	yes	characteristic
23	C₂×Q₁₆	C₂²	1	1	yes	characteristic
24	C₂×Q₁₆	C₂²	1	1	yes	characteristic
25	C₂×Q₁₆	C₂²	1	1	yes	characteristic
26	C₂×SD₁₆	C₂²	1	1	yes	characteristic
27	C₂×SD₁₆	C₂²	1	1	yes	characteristic
28	C₂×SD₁₆	C₂²	1	1	yes	characteristic
29	C₂×SD₁₆	C₂²	1	1	yes	characteristic
30	C₂×D₈	C₂²	1	1	yes	characteristic
31	C₂×M₁₆	C₂²	1	1	yes	characteristic
32	C₂²×C₈	C₂²	1	1	yes	characteristic
33	C₄⋊Q₈	C₂²	1	1	yes	characteristic
34	#124	—	2	1	no	—
35	#124	C₂²	1	1	yes	characteristic
36	#123	—	2	1	no	—
37	#123	—	2	1	no	—
38	C₂²⋊Q₈	—	2	1	no	—
39	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
40	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
41	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
42	#121	—	2	1	no	—
43	#121	—	2	1	no	—
44	#121	C₂²	1	1	yes	characteristic
45	C₄×Q₈	C₂²	1	1	yes	characteristic
46	C₄×D₄	—	2	1	no	—
47	C₄×D₄	C₂²	1	1	yes	characteristic
48	C₄×D₄	C₂²	1	1	yes	characteristic
49	C₂×C₄⋊C₄	—	2	1	no	—
50	#107	C₂²	1	1	yes	characteristic
51	C₄⋊C₈	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	#102	C₂²	1	1	yes	characteristic
57	#102	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈	C₂²	1	1	yes	characteristic
59	C₂²⋊C₈	C₂²	1	1	yes	characteristic
60	#97	C₂²	1	1	yes	characteristic
61	#41	—	2	2	no	—
62	#41	—	2	2	no	—
63	#41	—	2	2	no	—
64	#41	—	4	1	no	—
65	#41	—	4	1	no	—
66	#41	—	4	1	no	—
67	C₂×Q₈	—	2	1	no	—
68	C₂×Q₈	—	2	2	no	—
69	C₂×Q₈	—	4	1	no	—
70	C₂×Q₈	C₂³	1	1	yes	characteristic
71	C₂×Q₈	C₂³	1	1	yes	characteristic
72	C₂×Q₈	C₂³	1	1	yes	characteristic
73	C₂×Q₈	C₂³	1	1	yes	characteristic
74	C₂×D₄	—	2	1	no	—
75	C₂×D₄	—	2	1	no	—
76	C₂×D₄	—	2	1	no	—
77	C₂×D₄	—	2	1	no	—
78	C₂×D₄	D₄	1	1	yes	characteristic
79	C₂×D₄	C₂³	1	1	yes	characteristic
80	C₂×D₄	C₂³	1	1	yes	characteristic
81	C₂²×C₄	—	2	1	no	—
82	C₂²×C₄	—	2	1	no	—
83	C₂²×C₄	—	2	1	no	—
84	C₂²×C₄	—	2	1	no	—
85	C₂²×C₄	—	2	1	no	—
86	C₂²×C₄	C₂³	1	1	yes	characteristic
87	C₂²×C₄	C₂³	1	1	yes	characteristic
88	Q₁₆	—	2	2	no	—
89	Q₁₆	—	2	2	no	—
90	Q₁₆	—	2	2	no	—
91	Q₁₆	—	4	1	no	—
92	SD₁₆	—	2	2	no	—
93	SD₁₆	—	2	2	no	—
94	SD₁₆	—	4	1	no	—
95	SD₁₆	D₄	1	4	yes	complemented
96	D₈	—	2	2	no	—
97	M₁₆	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	C₂³	1	1	yes	characteristic
100	C₂×C₈	C₂³	1	1	yes	characteristic
101	C₂×C₈	C₂³	1	1	yes	characteristic
102	C₂×C₈	C₂³	1	1	yes	characteristic
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	D₄	1	1	yes	characteristic
110	C₄⋊C₄	C₂³	1	1	yes	characteristic
111	C₄⋊C₄	C₂³	1	1	yes	characteristic
112	C₂²⋊C₄	—	2	1	no	—
113	C₂²⋊C₄	—	2	1	no	—
114	C₂²⋊C₄	—	2	1	no	—
115	C₂²⋊C₄	—	2	1	no	—
116	C₂²⋊C₄	—	2	1	no	—
117	C₂²⋊C₄	—	2	1	no	—
118	C₂²⋊C₄	—	2	1	no	—
119	C₂²⋊C₄	D₄	1	1	yes	characteristic
120	C₂²⋊C₄	D₄	1	1	yes	characteristic
121	C₄²	—	2	1	no	—
122	C₄²	C₂³	1	1	yes	characteristic
123	C₂³	—	2	1	no	—
124	C₂³	—	2	1	no	—
125	C₂³	C₂×D₄	1	1	yes	characteristic
126	C₂³	C₂×D₄	1	1	yes	characteristic
127	Q₈	—	2	1	no	—
128	Q₈	—	2	1	no	—
129	Q₈	—	2	1	no	—
130	Q₈	—	2	1	no	—
131	Q₈	—	2	1	no	—
132	Q₈	—	2	1	no	—
133	Q₈	—	2	1	no	—
134	Q₈	—	4	2	no	—
135	Q₈	C₂×D₄	1	2	yes	—
136	D₄	—	2	1	no	—
137	D₄	—	2	1	no	—
138	D₄	—	2	1	no	—
139	D₄	—	4	1	no	—
140	D₄	—	4	2	no	—
141	D₄	—	4	2	no	—
142	D₄	—	4	2	no	—
143	D₄	—	4	2	no	—
144	D₄	C₂×D₄	1	2	yes	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	C₂×D₄	1	1	yes	characteristic
161	C₂×C₄	C₂×D₄	1	1	yes	characteristic
162	C₂×C₄	C₂×D₄	1	1	yes	characteristic
163	C₂×C₄	C₂×D₄	1	1	yes	characteristic
164	C₂×C₄	C₂⁴	1	1	yes	characteristic
165	C₈	—	2	1	no	—
166	C₈	—	2	1	no	—
167	C₈	—	2	1	no	—
168	C₈	C₂×D₄	1	2	yes	—
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	2	1	no	—
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	—	4	1	no	—
177	C₂²	—	4	1	no	—
178	C₂²	—	4	1	no	—
179	C₂²	—	4	1	no	—
180	C₂²	—	4	1	no	—
181	C₂²	C₂²×D₄	1	1	yes	characteristic
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	2	2	no	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	4	1	no	—
188	C₄	—	4	1	no	—
189	C₄	—	4	1	no	—
190	C₄	—	4	1	no	—
191	C₄	—	4	1	no	—
192	C₄	C₂²×D₄	1	1	yes	characteristic
193	C₄	H(5,𝔽₂)	1	1	yes	characteristic
194	C₂	—	4	1	no	—
195	C₂	—	4	1	no	—
196	C₂	—	4	2	no	—
197	C₂	—	8	1	no	—
198	C₂	D₄²	1	1	yes	characteristic
199	C₂	#575	1	1	yes	characteristic
200	C₂	#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
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. C₂×#137	C₂	semidirect
2. C₂×#135	C₂	semidirect
3. #548	C₂	semidirect
4. #547	C₂	semidirect
5. #497	C₂	semidirect
6. #468	C₂	semidirect
7. #467	C₂	semidirect
8. #463	C₂	semidirect
9. #462	C₂	semidirect
10. #452	C₂	semidirect
11. #452	C₂	semidirect
12. #451	C₂	semidirect
13. #449	C₂	semidirect
14. #440	C₂	semidirect
15. #435	C₂	semidirect
95. SD₁₆	D₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#137	2
C₂×#135	2
#548	2
#547	2
#497	2
#468	2
#467	2
#463	2
#462	2
#452	2
#451	2
#449	2
#440	2
#435	2