Finite Group Databasebeta

Subgroups — #2802

Order 128.

180 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 96 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	#400	C₂	1	1	yes	maximal, characteristic
2	#400	C₂	1	1	yes	maximal, characteristic
3	#398	C₂	1	1	yes	maximal, characteristic
4	#398	C₂	1	1	yes	maximal, characteristic
5	#398	C₂	1	1	yes	maximal, characteristic
6	#398	C₂	1	1	yes	maximal, characteristic
7	#389	C₂	1	1	yes	maximal, characteristic
8	#389	C₂	1	1	yes	maximal, characteristic
9	#389	C₂	1	1	yes	maximal, characteristic
10	#389	C₂	1	1	yes	maximal, characteristic
11	#389	C₂	1	1	yes	maximal, characteristic
12	#387	C₂	1	1	yes	maximal, characteristic
13	#387	C₂	1	1	yes	maximal, characteristic
14	#387	C₂	1	1	yes	maximal, characteristic
15	#387	C₂	1	1	yes	maximal, characteristic
16	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
32	C₂×C₄²	C₂²	1	1	yes	characteristic
33	C₂×C₄²	C₂²	1	1	yes	characteristic
34	C₂×C₄²	C₂²	1	1	yes	characteristic
35	#95	C₂²	1	1	yes	characteristic
36	#95	C₂²	1	1	yes	characteristic
37	#95	C₂²	1	1	yes	characteristic
38	#95	C₂²	1	1	yes	characteristic
39	#95	C₂²	1	1	yes	characteristic
40	#95	C₂²	1	1	yes	characteristic
41	#95	C₂²	1	1	yes	characteristic
42	#95	C₂²	1	1	yes	characteristic
43	#95	C₂²	1	1	yes	characteristic
44	#95	C₂²	1	1	yes	characteristic
45	#95	C₂²	1	1	yes	characteristic
46	#95	C₂²	1	1	yes	characteristic
47	#95	C₂²	1	1	yes	characteristic
48	#95	C₂²	1	1	yes	characteristic
49	#95	C₂²	1	1	yes	characteristic
50	#95	C₂²	1	1	yes	characteristic
51	C₂²×C₄	C₂³	1	1	yes	characteristic
52	C₂²×C₄	C₂³	1	1	yes	characteristic
53	C₂²×C₄	C₂³	1	1	yes	characteristic
54	C₂²×C₄	C₂³	1	1	yes	characteristic
55	C₂²×C₄	C₂³	1	1	yes	characteristic
56	C₂²×C₄	C₂³	1	1	yes	characteristic
57	C₂²×C₄	C₂³	1	1	yes	characteristic
58	C₂²×C₄	C₂³	1	1	yes	characteristic
59	C₂²×C₄	C₂³	1	1	yes	characteristic
60	C₂²×C₄	C₂³	1	1	yes	characteristic
61	C₂²×C₄	C₂³	1	1	yes	characteristic
62	C₂²×C₄	C₂³	1	1	yes	characteristic
63	C₂²×C₄	C₂³	1	1	yes	characteristic
64	C₂²×C₄	C₂³	1	1	yes	characteristic
65	C₂²×C₄	C₂³	1	1	yes	characteristic
66	C₄⋊C₄	—	2	2	no	—
67	C₄⋊C₄	—	2	2	no	—
68	C₄⋊C₄	—	2	2	no	—
69	C₄⋊C₄	—	2	2	no	—
70	C₄⋊C₄	—	2	2	no	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	4	1	no	—
73	C₄⋊C₄	—	4	1	no	—
74	C₄⋊C₄	—	4	1	no	—
75	C₄⋊C₄	—	4	1	no	—
76	C₄⋊C₄	—	4	1	no	—
77	C₄⋊C₄	—	4	1	no	—
78	C₄⋊C₄	—	4	1	no	—
79	C₄⋊C₄	—	4	1	no	—
80	C₄⋊C₄	—	4	1	no	—
81	C₄⋊C₄	Q₈	1	4	yes	—
82	C₄²	—	4	1	no	—
83	C₄²	—	4	1	no	—
84	C₄²	—	4	1	no	—
85	C₂³	C₂⁴	1	1	yes	characteristic
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	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	C₂×Q₈	1	2	yes	—
127	C₂×C₄	C₂×Q₈	1	2	yes	—
128	C₂×C₄	C₂×Q₈	1	2	yes	—
129	C₂×C₄	#41	1	2	yes	—
130	C₂×C₄	#41	1	2	yes	—
131	C₂²	C₂²×Q₈	1	1	yes	characteristic
132	C₂²	C₂×#41	1	1	yes	characteristic
133	C₂²	C₂×#41	1	1	yes	characteristic
134	C₂²	H(5,𝔽₂)	1	1	yes	characteristic
135	C₂²	H(5,𝔽₂)	1	1	yes	characteristic
136	C₂²	2₋¹⁺⁴	1	1	yes	characteristic
137	C₂²	2₋¹⁺⁴	1	1	yes	characteristic
138	C₄	—	2	2	no	—
139	C₄	—	2	2	no	—
140	C₄	—	2	2	no	—
141	C₄	—	2	2	no	—
142	C₄	—	2	2	no	—
143	C₄	—	4	1	no	—
144	C₄	—	4	1	no	—
145	C₄	—	4	1	no	—
146	C₄	—	4	1	no	—
147	C₄	—	4	1	no	—
148	C₄	—	4	1	no	—
149	C₄	—	4	1	no	—
150	C₄	—	4	1	no	—
151	C₄	—	4	1	no	—
152	C₄	—	4	1	no	—
153	C₂	#539	1	1	yes	characteristic
154	C₂	#541	1	1	yes	characteristic
155	C₂	#544	1	1	yes	characteristic
156	C₂	#553	1	1	yes	characteristic
157	C₂	#554	1	1	yes	characteristic
158	C₂	#557	1	1	yes	characteristic
159	C₂	#562	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂³	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

Maximal subgroups

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

subgroup	index
#400	2
#398	2
#389	2
#387	2