Finite Group Databasebeta

Subgroups — #3224

Order 128.

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

normal subgroups only

This group has 204 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	#542	C₂	1	1	yes	maximal, characteristic, complemented
2	#535	C₂	1	1	yes	maximal, characteristic, complemented
3	#496	C₂	1	1	yes	maximal, characteristic, complemented
4	#495	C₂	1	1	yes	maximal, characteristic, complemented
5	#489	C₂	1	1	yes	maximal, characteristic, complemented
6	#486	C₂	1	1	yes	maximal, characteristic, complemented
7	#464	C₂	1	1	yes	maximal, characteristic, complemented
8	#463	C₂	1	1	yes	maximal, characteristic, complemented
9	#460	C₂	1	1	yes	maximal, characteristic, complemented
10	#459	C₂	1	1	yes	maximal, characteristic, complemented
11	#452	C₂	1	1	yes	maximal, characteristic, complemented
12	#450	C₂	1	1	yes	maximal, characteristic, complemented
13	#449	C₂	1	1	yes	maximal, characteristic, complemented
14	#447	C₂	1	1	yes	maximal, characteristic, complemented
15	#433	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#41	C₂²	1	1	yes	characteristic
17	C₂²×D₄	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆	C₂²	1	1	yes	characteristic, complemented
19	C₂×SD₁₆	C₂²	1	1	yes	characteristic
20	C₂×SD₁₆	C₂²	1	1	yes	characteristic
21	C₂×SD₁₆	C₂²	1	1	yes	characteristic, complemented
22	C₂×SD₁₆	C₂²	1	1	yes	characteristic, complemented
23	C₂×D₈	C₂²	1	1	yes	characteristic
24	C₂×D₈	C₂²	1	1	yes	characteristic
25	C₂×D₈	C₂²	1	1	yes	characteristic, complemented
26	C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
27	#127	C₂²	1	1	yes	characteristic
28	#127	C₂²	1	1	yes	characteristic
29	#126	—	2	1	no	—
30	#124	—	2	1	no	—
31	#124	C₂²	1	1	yes	characteristic
32	#124	C₂²	1	1	yes	characteristic
33	#124	C₂²	1	1	yes	characteristic, complemented
34	#123	—	2	1	no	—
35	C₂²⋊Q₈	—	2	1	no	—
36	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
37	#121	—	2	1	no	—
38	#121	—	2	1	no	—
39	#121	C₂²	1	1	yes	characteristic
40	#120	—	2	1	no	—
41	#120	—	2	1	no	—
42	C₄×Q₈	C₂²	1	1	yes	characteristic, complemented
43	C₄×D₄	C₂²	1	1	yes	characteristic, complemented
44	#117	C₂²	1	1	yes	characteristic
45	C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
46	C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
47	#103	C₂²	1	1	yes	characteristic, complemented
48	#103	C₂²	1	1	yes	characteristic, complemented
49	#102	C₂²	1	1	yes	characteristic
50	#102	C₂²	1	1	yes	characteristic
51	#102	C₂²	1	1	yes	characteristic
52	#102	C₂²	1	1	yes	characteristic
53	#102	C₂²	1	1	yes	characteristic, complemented
54	#102	C₂²	1	1	yes	characteristic, complemented
55	C₂²⋊C₈	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈	C₂²	1	1	yes	characteristic
57	#97	C₂²	1	1	yes	characteristic, complemented
58	C₄×C₈	C₂²	1	1	yes	characteristic, complemented
59	C₂⁴	—	2	1	no	—
60	#41	—	4	1	no	—
61	#41	—	4	1	no	—
62	C₂×Q₈	—	2	1	no	—
63	C₂×Q₈	C₂³	1	1	yes	characteristic
64	C₂×Q₈	C₂³	1	1	yes	characteristic
65	C₂×D₄	—	2	1	no	—
66	C₂×D₄	—	2	1	no	—
67	C₂×D₄	—	2	1	no	—
68	C₂×D₄	—	2	1	no	—
69	C₂×D₄	—	2	1	no	—
70	C₂×D₄	—	2	1	no	—
71	C₂×D₄	—	2	1	no	—
72	C₂×D₄	—	4	1	no	—
73	C₂×D₄	—	4	1	no	—
74	C₂×D₄	C₂³	1	1	yes	characteristic
75	C₂×D₄	C₂³	1	1	yes	characteristic
76	C₂×D₄	C₂³	1	1	yes	characteristic
77	C₂×D₄	C₂³	1	1	yes	characteristic
78	C₂²×C₄	—	2	1	no	—
79	C₂²×C₄	—	2	1	no	—
80	C₂²×C₄	C₂³	1	1	yes	characteristic
81	Q₁₆	—	4	1	no	—
82	SD₁₆	—	4	1	no	—
83	SD₁₆	—	4	1	no	—
84	SD₁₆	—	4	1	no	—
85	SD₁₆	—	4	1	no	—
86	D₈	—	4	1	no	—
87	D₈	—	4	1	no	—
88	D₈	—	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	D₄	1	1	yes	characteristic
97	C₄⋊C₄	D₄	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	—	2	1	no	—
106	C₂²⋊C₄	—	2	1	no	—
107	C₂²⋊C₄	—	2	1	no	—
108	C₂²⋊C₄	—	2	1	no	—
109	C₂²⋊C₄	D₄	1	1	yes	characteristic
110	C₂²⋊C₄	D₄	1	1	yes	characteristic
111	C₄²	—	2	1	no	—
112	C₄²	C₂³	1	1	yes	characteristic
113	C₄²	C₂³	1	1	yes	characteristic
114	C₂³	—	2	1	no	—
115	C₂³	—	2	1	no	—
116	C₂³	—	2	1	no	—
117	C₂³	—	2	1	no	—
118	C₂³	—	4	2	no	—
119	C₂³	—	8	1	no	—
120	C₂³	—	8	1	no	—
121	C₂³	C₂×D₄	1	1	yes	characteristic
122	Q₈	—	2	1	no	—
123	Q₈	—	2	1	no	—
124	Q₈	—	2	1	no	—
125	Q₈	—	2	1	no	—
126	Q₈	—	8	1	no	—
127	D₄	—	2	1	no	—
128	D₄	—	2	1	no	—
129	D₄	—	2	1	no	—
130	D₄	—	2	1	no	—
131	D₄	—	2	1	no	—
132	D₄	—	2	1	no	—
133	D₄	—	2	1	no	—
134	D₄	—	2	1	no	—
135	D₄	—	4	2	no	—
136	D₄	—	4	2	no	—
137	D₄	—	8	1	no	—
138	D₄	—	8	1	no	—
139	D₄	—	8	1	no	—
140	D₄	—	8	1	no	—
141	D₄	—	8	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	—	2	1	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	C₂×D₄	1	1	yes	characteristic
154	C₂×C₄	C₂×D₄	1	1	yes	characteristic
155	C₂×C₄	C₂⁴	1	1	yes	characteristic
156	C₈	—	2	1	no	—
157	C₈	—	2	1	no	—
158	C₈	—	2	1	no	—
159	C₈	—	2	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	—	2	1	no	—
162	C₂²	—	2	1	no	—
163	C₂²	—	4	1	no	—
164	C₂²	—	4	1	no	—
165	C₂²	—	4	1	no	—
166	C₂²	—	4	1	no	—
167	C₂²	—	4	1	no	—
168	C₂²	—	4	1	no	—
169	C₂²	—	4	1	no	—
170	C₂²	—	4	1	no	—
171	C₂²	—	4	1	no	—
172	C₂²	—	4	1	no	—
173	C₂²	—	4	1	no	—
174	C₂²	—	4	1	no	—
175	C₂²	—	8	4	no	—
176	C₂²	C₂²×D₄	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	4	1	no	—
183	C₄	—	4	1	no	—
184	C₄	—	4	1	no	—
185	C₄	—	4	1	no	—
186	C₄	H(5,𝔽₂)	1	1	yes	characteristic
187	C₄	H(5,𝔽₂)	1	1	yes	characteristic
188	C₂	—	4	1	no	—
189	C₂	—	8	1	no	—
190	C₂	—	8	1	no	—
191	C₂	—	8	1	no	—
192	C₂	—	8	1	no	—
193	C₂	#535	1	1	yes	characteristic
194	C₂	#576	1	1	yes	characteristic
195	C₂	#577	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 ◁ #117 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #542	C₂	semidirect
2. #535	C₂	semidirect
3. #496	C₂	semidirect
4. #495	C₂	semidirect
5. #489	C₂	semidirect
6. #486	C₂	semidirect
7. #464	C₂	semidirect
8. #463	C₂	semidirect
9. #460	C₂	semidirect
10. #459	C₂	semidirect
11. #452	C₂	semidirect
12. #450	C₂	semidirect
13. #449	C₂	semidirect
14. #447	C₂	semidirect
15. #433	C₂	semidirect
18. C₂×Q₁₆	C₂²	semidirect
21. C₂×SD₁₆	C₂²	semidirect
22. C₂×SD₁₆	C₂²	semidirect
25. C₂×D₈	C₂²	semidirect
26. C₄⋊Q₈	C₂²	semidirect
33. #124	C₂²	semidirect
42. C₄×Q₈	C₂²	semidirect
43. C₄×D₄	C₂²	semidirect
45. C₄⋊C₈	C₂²	semidirect
46. C₄⋊C₈	C₂²	semidirect
47. #103	C₂²	semidirect
48. #103	C₂²	semidirect
53. #102	C₂²	semidirect
54. #102	C₂²	semidirect
57. #97	C₂²	semidirect
58. C₄×C₈	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#542	2
#535	2
#496	2
#495	2
#489	2
#486	2
#464	2
#463	2
#460	2
#459	2
#452	2
#450	2
#449	2
#447	2
#433	2