Finite Group Databasebeta

Subgroups — #3178

Order 128.

175 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 175 conjugacy classes of subgroups — too many to draw. Only its 88 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
2	#541	C₂	1	1	yes	maximal, characteristic, complemented
3	#501	C₂	1	1	yes	maximal, characteristic, complemented
4	#497	C₂	1	1	yes	maximal, characteristic, complemented
5	#488	C₂	1	1	yes	maximal, characteristic, complemented
6	#487	C₂	1	1	yes	maximal, characteristic, complemented
7	#470	C₂	1	1	yes	maximal, characteristic, complemented
8	#468	C₂	1	1	yes	maximal, characteristic
9	#467	C₂	1	1	yes	maximal, characteristic, complemented
10	#465	C₂	1	1	yes	maximal, characteristic
11	#441	C₂	1	1	yes	maximal, characteristic, complemented
12	#440	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
15	#405	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₁₆	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆	C₂²	1	1	yes	characteristic
18	C₂×SD₁₆	C₂²	1	1	yes	characteristic
19	C₂×SD₁₆	C₂²	1	1	yes	characteristic
20	C₂×M₁₆	C₂²	1	1	yes	characteristic
21	C₂²×C₈	C₂²	1	1	yes	characteristic
22	C₄⋊Q₈	C₂²	1	1	yes	characteristic
23	C₄⋊Q₈	C₂²	1	1	yes	characteristic
24	#126	—	2	1	no	—
25	#126	—	2	1	no	—
26	#126	—	2	1	no	—
27	#125	—	2	1	no	—
28	#125	—	2	1	no	—
29	#125	C₂²	1	1	yes	characteristic
30	#124	—	2	1	no	—
31	#124	C₂²	1	1	yes	characteristic
32	#123	—	2	1	no	—
33	C₂²⋊Q₈	—	2	1	no	—
34	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
35	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
36	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
37	#121	C₂²	1	1	yes	characteristic
38	C₄×Q₈	C₂²	1	1	yes	characteristic
39	C₄×Q₈	C₂²	1	1	yes	characteristic
40	C₄×Q₈	C₂²	1	1	yes	characteristic
41	C₄×D₄	C₂²	1	1	yes	characteristic
42	#117	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	#97	C₂²	1	1	yes	characteristic
56	#97	C₂²	1	1	yes	characteristic
57	C₄×C₈	C₂²	1	1	yes	characteristic
58	C₄×C₈	C₂²	1	1	yes	characteristic
59	C₂×Q₈	—	2	1	no	—
60	C₂×Q₈	C₂³	1	1	yes	characteristic
61	C₂×Q₈	C₂³	1	1	yes	characteristic
62	C₂×Q₈	C₂³	1	1	yes	characteristic
63	C₂×D₄	—	2	1	no	—
64	C₂×D₄	C₂³	1	1	yes	characteristic
65	C₂²×C₄	—	2	1	no	—
66	C₂²×C₄	C₂³	1	1	yes	characteristic
67	Q₁₆	—	2	2	no	—
68	Q₁₆	—	2	2	no	—
69	SD₁₆	—	2	2	no	—
70	SD₁₆	—	2	2	no	—
71	M₁₆	—	2	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	C₂³	1	1	yes	characteristic
74	C₂×C₈	C₂³	1	1	yes	characteristic
75	C₂×C₈	C₂³	1	1	yes	characteristic
76	C₂×C₈	C₂³	1	1	yes	characteristic
77	C₄⋊C₄	—	2	1	no	—
78	C₄⋊C₄	—	2	1	no	—
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	—	2	1	no	—
84	C₄⋊C₄	—	2	1	no	—
85	C₄⋊C₄	—	2	1	no	—
86	C₄⋊C₄	—	2	1	no	—
87	C₄⋊C₄	—	2	1	no	—
88	C₄⋊C₄	D₄	1	1	yes	characteristic
89	C₄⋊C₄	D₄	1	1	yes	characteristic
90	C₄⋊C₄	C₂³	1	1	yes	characteristic
91	C₄⋊C₄	C₂³	1	1	yes	characteristic
92	C₄⋊C₄	C₂³	1	1	yes	characteristic
93	C₄⋊C₄	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
100	C₂²⋊C₄	—	2	1	no	—
101	C₂²⋊C₄	D₄	1	1	yes	characteristic
102	C₂²⋊C₄	D₄	1	1	yes	characteristic
103	C₄²	—	2	1	no	—
104	C₄²	—	2	1	no	—
105	C₄²	—	2	1	no	—
106	C₄²	C₂³	1	1	yes	characteristic
107	C₄²	C₂³	1	1	yes	characteristic
108	C₂³	—	2	1	no	—
109	C₂³	C₂×D₄	1	1	yes	characteristic
110	Q₈	—	2	1	no	—
111	Q₈	—	2	1	no	—
112	Q₈	—	2	1	no	—
113	Q₈	—	2	1	no	—
114	Q₈	—	2	1	no	—
115	Q₈	—	2	1	no	—
116	Q₈	—	8	1	no	—
117	D₄	—	2	1	no	—
118	D₄	—	2	1	no	—
119	D₄	—	8	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	—	8	1	no	—
130	C₂×C₄	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄	C₂×D₄	1	1	yes	characteristic
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	C₂⁴	1	1	yes	characteristic
136	C₈	—	2	1	no	—
137	C₈	—	2	1	no	—
138	C₈	#41	1	2	yes	—
139	C₈	#41	1	2	yes	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	4	1	no	—
144	C₂²	—	4	1	no	—
145	C₂²	—	4	1	no	—
146	C₂²	C₂²×D₄	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	4	1	no	—
153	C₄	—	4	1	no	—
154	C₄	—	4	1	no	—
155	C₄	—	4	1	no	—
156	C₄	—	4	1	no	—
157	C₄	—	4	1	no	—
158	C₄	—	4	1	no	—
159	C₄	C₂×#41	1	1	yes	characteristic
160	C₄	C₂×#41	1	1	yes	characteristic
161	C₂	—	4	1	no	—
162	C₂	—	8	1	no	—
163	C₂	#532	1	1	yes	characteristic
164	C₂	#577	1	1	yes	characteristic
165	C₂	#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	#121	C₂²
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
2. #541	C₂	semidirect
3. #501	C₂	semidirect
4. #497	C₂	semidirect
5. #488	C₂	semidirect
6. #487	C₂	semidirect
7. #470	C₂	semidirect
9. #467	C₂	semidirect
11. #441	C₂	semidirect
12. #440	C₂	semidirect
13. C₄×Q₁₆	C₂	semidirect
14. C₄×SD₁₆	C₂	semidirect
15. #405	C₂	semidirect

Maximal subgroups

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

subgroup	index
#542	2
#541	2
#501	2
#497	2
#488	2
#487	2
#470	2
#468	2
#467	2
#465	2
#441	2
#440	2
C₄×Q₁₆	2
C₄×SD₁₆	2
#405	2