Finite Group Databasebeta

Subgroups — #3225

Order 128.

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

normal subgroups only

This group has 188 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	#536	C₂	1	1	yes	maximal, characteristic, complemented
3	#497	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	#487	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	#462	C₂	1	1	yes	maximal, characteristic, complemented
10	#461	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
14	#448	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×Q₈	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆	C₂²	1	1	yes	characteristic
19	C₂×Q₁₆	C₂²	1	1	yes	characteristic
20	C₂×Q₁₆	C₂²	1	1	yes	characteristic
21	C₂×SD₁₆	C₂²	1	1	yes	characteristic
22	C₂×SD₁₆	C₂²	1	1	yes	characteristic
23	C₂×SD₁₆	C₂²	1	1	yes	characteristic
24	C₂×SD₁₆	C₂²	1	1	yes	characteristic
25	C₂×D₈	C₂²	1	1	yes	characteristic
26	C₄⋊Q₈	C₂²	1	1	yes	characteristic
27	C₄⋊Q₈	C₂²	1	1	yes	characteristic
28	C₄⋊Q₈	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
34	#123	—	2	1	no	—
35	#123	—	2	1	no	—
36	#123	—	2	1	no	—
37	C₂²⋊Q₈	—	2	1	no	—
38	C₂²⋊Q₈	—	2	1	no	—
39	C₂²⋊Q₈	—	2	1	no	—
40	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
41	#121	C₂²	1	1	yes	characteristic
42	C₄×Q₈	C₂²	1	1	yes	characteristic
43	C₄×D₄	C₂²	1	1	yes	characteristic
44	#117	C₂²	1	1	yes	characteristic
45	C₄⋊C₈	C₂²	1	1	yes	characteristic
46	C₄⋊C₈	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	C₂²⋊C₈	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈	C₂²	1	1	yes	characteristic
57	#97	C₂²	1	1	yes	characteristic
58	C₄×C₈	C₂²	1	1	yes	characteristic
59	#41	—	4	1	no	—
60	#41	—	4	1	no	—
61	C₂×Q₈	—	2	1	no	—
62	C₂×Q₈	—	2	1	no	—
63	C₂×Q₈	—	4	1	no	—
64	C₂×Q₈	—	4	1	no	—
65	C₂×Q₈	C₂³	1	1	yes	characteristic
66	C₂×Q₈	C₂³	1	1	yes	characteristic
67	C₂×Q₈	C₂³	1	1	yes	characteristic
68	C₂×Q₈	C₂³	1	1	yes	characteristic
69	C₂×D₄	—	2	1	no	—
70	C₂×D₄	—	2	1	no	—
71	C₂×D₄	C₂³	1	1	yes	characteristic
72	C₂×D₄	C₂³	1	1	yes	characteristic
73	C₂²×C₄	—	2	1	no	—
74	C₂²×C₄	—	2	1	no	—
75	C₂²×C₄	—	2	1	no	—
76	C₂²×C₄	C₂³	1	1	yes	characteristic
77	Q₁₆	—	4	1	no	—
78	Q₁₆	—	4	1	no	—
79	Q₁₆	—	4	1	no	—
80	SD₁₆	—	4	1	no	—
81	SD₁₆	—	4	1	no	—
82	SD₁₆	—	4	1	no	—
83	SD₁₆	—	4	1	no	—
84	D₈	—	4	1	no	—
85	C₂×C₈	C₂³	1	1	yes	characteristic
86	C₂×C₈	C₂³	1	1	yes	characteristic
87	C₂×C₈	C₂³	1	1	yes	characteristic
88	C₂×C₈	C₂³	1	1	yes	characteristic
89	C₄⋊C₄	—	2	1	no	—
90	C₄⋊C₄	—	2	1	no	—
91	C₄⋊C₄	—	2	1	no	—
92	C₄⋊C₄	—	2	1	no	—
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	C₂×D₄	1	1	yes	characteristic
117	Q₈	—	2	1	no	—
118	Q₈	—	2	1	no	—
119	Q₈	—	2	1	no	—
120	Q₈	—	2	1	no	—
121	Q₈	—	2	1	no	—
122	Q₈	—	2	1	no	—
123	Q₈	—	2	1	no	—
124	Q₈	—	2	1	no	—
125	Q₈	—	4	2	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	—	4	2	no	—
132	D₄	—	8	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	8	1	no	—
144	C₂×C₄	C₂×D₄	1	1	yes	characteristic
145	C₂×C₄	C₂×D₄	1	1	yes	characteristic
146	C₂×C₄	C₂×D₄	1	1	yes	characteristic
147	C₂×C₄	C₂×D₄	1	1	yes	characteristic
148	C₂×C₄	C₂×D₄	1	1	yes	characteristic
149	C₂×C₄	C₂⁴	1	1	yes	characteristic
150	C₈	—	2	1	no	—
151	C₈	—	2	1	no	—
152	C₈	—	2	1	no	—
153	C₈	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	—	2	1	no	—
157	C₂²	—	4	1	no	—
158	C₂²	—	4	1	no	—
159	C₂²	—	4	1	no	—
160	C₂²	—	4	1	no	—
161	C₂²	—	4	1	no	—
162	C₂²	—	4	1	no	—
163	C₂²	C₂²×D₄	1	1	yes	characteristic
164	C₄	—	2	1	no	—
165	C₄	—	2	1	no	—
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	4	1	no	—
170	C₄	—	4	1	no	—
171	C₄	—	4	1	no	—
172	C₄	—	4	1	no	—
173	C₄	—	4	1	no	—
174	C₄	—	4	1	no	—
175	C₄	H(5,𝔽₂)	1	1	yes	characteristic
176	C₄	H(5,𝔽₂)	1	1	yes	characteristic
177	C₂	—	4	1	no	—
178	C₂	—	8	1	no	—
179	C₂	—	8	1	no	—
180	C₂	#535	1	1	yes	characteristic
181	C₂	#577	1	1	yes	characteristic
182	C₂	#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	#449	C₂
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. #536	C₂	semidirect
3. #497	C₂	semidirect
4. #495	C₂	semidirect
5. #489	C₂	semidirect
6. #487	C₂	semidirect
7. #464	C₂	semidirect
8. #463	C₂	semidirect
9. #462	C₂	semidirect
10. #461	C₂	semidirect
11. #452	C₂	semidirect
12. #451	C₂	semidirect
14. #448	C₂	semidirect
15. #433	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
#536	2
#497	2
#495	2
#489	2
#487	2
#464	2
#463	2
#462	2
#461	2
#452	2
#451	2
#449	2
#448	2
#433	2