Finite Group Databasebeta

Subgroups — #3347

Order 128.

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

normal subgroups only

This group has 180 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	#555	C₂	1	1	yes	maximal, characteristic, complemented
2	#553	C₂	1	1	yes	maximal, characteristic, complemented
3	#501	C₂	1	1	yes	maximal, characteristic, complemented
4	#482	C₂	1	1	yes	maximal, characteristic, complemented
5	#482	C₂	1	1	yes	maximal, characteristic, complemented
6	#482	C₂	1	1	yes	maximal, characteristic, complemented
7	#480	C₂	1	1	yes	maximal, characteristic, complemented
8	#478	C₂	1	1	yes	maximal, characteristic, complemented
9	#474	C₂	1	1	yes	maximal, characteristic, complemented
10	#469	C₂	1	1	yes	maximal, characteristic, complemented
11	#466	C₂	1	1	yes	maximal, characteristic, complemented
12	#442	C₂	1	1	yes	maximal, characteristic, complemented
13	#435	C₂	1	1	yes	maximal, characteristic, complemented
14	#428	C₂	1	1	yes	maximal, characteristic, complemented
15	#427	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₈	C₂²	1	1	yes	characteristic
17	C₂×M₁₆	C₂²	1	1	yes	characteristic
18	C₂²×C₈	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	#126	—	2	1	no	—
21	#125	C₂²	1	1	yes	characteristic
22	#124	—	2	1	no	—
23	#124	—	2	1	no	—
24	#123	—	2	1	no	—
25	#121	—	2	1	no	—
26	#121	—	2	1	no	—
27	#121	C₂²	1	1	yes	characteristic
28	#121	C₂²	1	1	yes	characteristic
29	#121	C₂²	1	1	yes	characteristic
30	#121	C₂²	1	1	yes	characteristic
31	C₄×D₄	—	2	1	no	—
32	C₄×D₄	C₂²	1	1	yes	characteristic
33	C₄×D₄	C₂²	1	1	yes	characteristic
34	C₄×D₄	C₂²	1	1	yes	characteristic
35	#117	—	2	1	no	—
36	#117	C₂²	1	1	yes	characteristic
37	#117	C₂²	1	1	yes	characteristic
38	#117	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
40	#107	C₂²	1	1	yes	characteristic
41	#107	C₂²	1	1	yes	characteristic
42	#107	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	#106	C₂²	1	1	yes	characteristic
48	#106	C₂²	1	1	yes	characteristic
49	C₄⋊C₈	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
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	C₂×Q₈	—	2	1	no	—
60	C₂×D₄	—	2	1	no	—
61	C₂×D₄	—	2	1	no	—
62	C₂×D₄	—	2	1	no	—
63	C₂×D₄	—	2	1	no	—
64	C₂×D₄	D₄	1	1	yes	characteristic
65	C₂×D₄	C₂³	1	1	yes	characteristic
66	C₂×D₄	C₂³	1	1	yes	characteristic
67	C₂²×C₄	—	2	1	no	—
68	C₂²×C₄	—	2	1	no	—
69	C₂²×C₄	—	2	1	no	—
70	C₂²×C₄	C₂³	1	1	yes	characteristic
71	C₂²×C₄	C₂³	1	1	yes	characteristic
72	D₈	—	2	2	no	—
73	M₁₆	—	2	2	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	C₂³	1	1	yes	characteristic
76	C₂×C₈	C₂³	1	1	yes	characteristic
77	C₂×C₈	C₂³	1	1	yes	characteristic
78	C₂×C₈	C₂³	1	1	yes	characteristic
79	C₄⋊C₄	—	2	1	no	—
80	C₄⋊C₄	—	2	1	no	—
81	C₄⋊C₄	—	2	1	no	—
82	C₄⋊C₄	—	2	1	no	—
83	C₄⋊C₄	D₄	1	1	yes	characteristic
84	C₄⋊C₄	C₂³	1	1	yes	characteristic
85	C₄⋊C₄	C₂³	1	1	yes	characteristic
86	C₄⋊C₄	C₂³	1	1	yes	characteristic
87	C₄⋊C₄	C₂³	1	1	yes	characteristic
88	C₄⋊C₄	C₂³	1	1	yes	characteristic
89	C₄⋊C₄	C₂³	1	1	yes	characteristic
90	C₂²⋊C₄	—	2	1	no	—
91	C₂²⋊C₄	—	2	1	no	—
92	C₂²⋊C₄	—	2	1	no	—
93	C₂²⋊C₄	—	2	1	no	—
94	C₂²⋊C₄	—	2	1	no	—
95	C₂²⋊C₄	—	2	1	no	—
96	C₂²⋊C₄	—	2	1	no	—
97	C₂²⋊C₄	—	2	1	no	—
98	C₂²⋊C₄	—	2	1	no	—
99	C₂²⋊C₄	D₄	1	1	yes	characteristic
100	C₂²⋊C₄	D₄	1	1	yes	characteristic
101	C₄²	—	2	1	no	—
102	C₄²	—	2	1	no	—
103	C₄²	—	2	1	no	—
104	C₄²	C₂³	1	1	yes	characteristic
105	C₂³	—	2	1	no	—
106	C₂³	—	2	1	no	—
107	C₂³	C₂×D₄	1	1	yes	characteristic
108	C₂³	C₂×D₄	1	1	yes	characteristic
109	Q₈	—	8	1	no	—
110	D₄	—	2	1	no	—
111	D₄	—	2	1	no	—
112	D₄	—	2	1	no	—
113	D₄	—	2	1	no	—
114	D₄	—	4	1	no	—
115	D₄	—	4	2	no	—
116	D₄	—	8	1	no	—
117	D₄	—	8	1	no	—
118	D₄	—	8	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	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	—	4	2	no	—
130	C₂×C₄	—	8	1	no	—
131	C₂×C₄	—	8	1	no	—
132	C₂×C₄	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄	C₂⁴	1	1	yes	characteristic
137	C₈	—	2	1	no	—
138	C₈	—	2	1	no	—
139	C₈	—	2	1	no	—
140	C₈	#41	1	2	yes	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	1	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	—	4	1	no	—
148	C₂²	—	4	1	no	—
149	C₂²	—	4	1	no	—
150	C₂²	—	4	1	no	—
151	C₂²	—	4	1	no	—
152	C₂²	—	4	1	no	—
153	C₂²	C₂²×D₄	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	4	1	no	—
159	C₄	—	4	1	no	—
160	C₄	—	4	1	no	—
161	C₄	—	4	1	no	—
162	C₄	—	4	1	no	—
163	C₄	—	4	1	no	—
164	C₄	C₂×#41	1	1	yes	characteristic
165	C₄	2₋¹⁺⁴	1	1	yes	characteristic
166	C₂	—	4	1	no	—
167	C₂	—	4	1	no	—
168	C₂	—	8	1	no	—
169	C₂	—	8	1	no	—
170	C₂	#547	1	1	yes	characteristic
171	C₂	#575	1	1	yes	characteristic
172	C₂	#576	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. #555	C₂	semidirect
2. #553	C₂	semidirect
3. #501	C₂	semidirect
4. #482	C₂	semidirect
5. #482	C₂	semidirect
6. #482	C₂	semidirect
7. #480	C₂	semidirect
8. #478	C₂	semidirect
9. #474	C₂	semidirect
10. #469	C₂	semidirect
11. #466	C₂	semidirect
12. #442	C₂	semidirect
13. #435	C₂	semidirect
14. #428	C₂	semidirect
15. #427	C₂	semidirect

Maximal subgroups

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

subgroup	index
#555	2
#553	2
#501	2
#482	2
#480	2
#478	2
#474	2
#469	2
#466	2
#442	2
#435	2
#428	2
#427	2