Finite Group Databasebeta

Subgroups — #3002

Order 128.

382 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 110 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	C₂²×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#135	C₂	1	2	yes	maximal, complemented
3	#525	C₂	1	1	yes	maximal, characteristic, complemented
4	#452	C₂	1	2	yes	maximal, complemented
5	#451	C₂	1	1	yes	maximal, characteristic, complemented
6	#450	C₂	1	1	yes	maximal, characteristic, complemented
7	#449	C₂	1	2	yes	maximal, complemented
8	#448	C₂	1	1	yes	maximal, characteristic, complemented
9	#447	C₂	1	1	yes	maximal, characteristic, complemented
10	#416	C₂	1	2	yes	maximal, complemented
11	C₂×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41	—	2	8	no	—
13	C₂×#41	—	2	1	no	—
14	C₂×#41	—	2	1	no	—
15	C₂×#41	C₂²	1	2	yes	—
16	C₂×#41	C₂²	1	2	yes	—
17	C₂×#41	C₂²	1	1	yes	characteristic, complemented
18	C₂²×Q₈	C₂²	1	1	yes	characteristic
19	C₂²×D₄	—	2	1	no	—
20	C₂²×D₄	C₂²	1	1	yes	characteristic
21	C₂³×C₄	—	2	1	no	—
22	C₂³×C₄	C₂²	1	1	yes	characteristic
23	#135	—	2	8	no	—
24	C₂×Q₁₆	C₂²	1	2	yes	complemented
25	C₂×SD₁₆	C₂²	1	2	yes	complemented
26	C₂×SD₁₆	C₂²	1	2	yes	complemented
27	C₂×D₈	C₂²	1	2	yes	complemented
28	C₂²×C₈	C₂²	1	2	yes	complemented
29	#123	—	2	1	no	—
30	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
31	C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
32	#121	C₂²	1	1	yes	characteristic
33	#121	C₂²	1	1	yes	characteristic, complemented
34	#120	—	2	1	no	—
35	C₄×D₄	—	2	1	no	—
36	C₄×D₄	—	2	1	no	—
37	#117	C₂²	1	1	yes	characteristic, complemented
38	#103	C₂²	1	2	yes	complemented
39	#103	C₂²	1	2	yes	complemented
40	#102	C₂²	1	2	yes	complemented
41	#102	C₂²	1	2	yes	complemented
42	C₂²⋊C₈	C₂²	1	1	yes	characteristic
43	C₂²⋊C₈	C₂²	1	1	yes	characteristic
44	C₂²⋊C₈	C₂²	1	2	yes	complemented
45	C₂⁴	—	2	1	no	—
46	C₂⁴	D₄	1	1	yes	characteristic
47	#41	—	2	8	no	—
48	#41	—	2	8	no	—
49	#41	—	2	4	no	—
50	#41	—	2	4	no	—
51	#41	—	2	4	no	—
52	#41	—	2	4	no	—
53	#41	D₄	1	8	yes	complemented
54	C₂×Q₈	—	2	4	no	—
55	C₂×Q₈	—	2	1	no	—
56	C₂×Q₈	C₂³	1	1	yes	characteristic
57	C₂×Q₈	C₂³	1	2	yes	—
58	C₂×D₄	—	2	8	no	—
59	C₂×D₄	—	2	2	no	—
60	C₂×D₄	—	2	2	no	—
61	C₂×D₄	—	2	4	no	—
62	C₂×D₄	—	2	1	no	—
63	C₂×D₄	—	2	1	no	—
64	C₂×D₄	—	2	1	no	—
65	C₂×D₄	—	2	1	no	—
66	C₂×D₄	C₂³	1	1	yes	characteristic
67	C₂×D₄	C₂³	1	2	yes	—
68	C₂²×C₄	—	2	8	no	—
69	C₂²×C₄	—	2	1	no	—
70	C₂²×C₄	—	2	2	no	—
71	C₂²×C₄	—	2	4	no	—
72	C₂²×C₄	—	2	2	no	—
73	C₂²×C₄	—	2	1	no	—
74	C₂²×C₄	D₄	1	1	yes	characteristic
75	C₂²×C₄	D₄	1	1	yes	characteristic
76	C₂²×C₄	D₄	1	1	yes	characteristic
77	C₂²×C₄	C₂³	1	1	yes	characteristic
78	C₂²×C₄	C₂³	1	1	yes	characteristic
79	C₂²×C₄	C₂³	1	1	yes	characteristic
80	Q₁₆	—	2	4	no	—
81	SD₁₆	—	2	4	no	—
82	SD₁₆	—	2	4	no	—
83	D₈	—	2	4	no	—
84	C₂×C₈	—	2	4	no	—
85	C₂×C₈	C₂³	1	2	yes	—
86	C₂×C₈	C₂³	1	2	yes	—
87	C₄⋊C₄	—	2	1	no	—
88	C₄⋊C₄	—	2	1	no	—
89	C₄⋊C₄	C₂³	1	1	yes	characteristic
90	C₄⋊C₄	C₂³	1	1	yes	characteristic
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	C₄²	—	2	1	no	—
97	C₂³	—	2	4	no	—
98	C₂³	—	2	2	no	—
99	C₂³	—	2	2	no	—
100	C₂³	—	2	2	no	—
101	C₂³	—	2	1	no	—
102	C₂³	—	4	1	no	—
103	C₂³	—	4	2	no	—
104	C₂³	—	4	2	no	—
105	C₂³	C₂×D₄	1	1	yes	characteristic
106	C₂³	C₂×D₄	1	1	yes	characteristic
107	C₂³	C₂×D₄	1	1	yes	characteristic
108	Q₈	—	2	4	no	—
109	Q₈	—	2	2	no	—
110	Q₈	—	2	2	no	—
111	Q₈	C₂×D₄	1	4	yes	—
112	D₄	—	2	8	no	—
113	D₄	—	2	4	no	—
114	D₄	—	2	8	no	—
115	D₄	—	2	2	no	—
116	D₄	—	2	2	no	—
117	D₄	—	4	2	no	—
118	D₄	—	4	2	no	—
119	D₄	—	4	2	no	—
120	D₄	C₂×D₄	1	4	yes	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	8	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	8	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄	C₂×D₄	1	4	yes	—
137	C₂×C₄	C₂×D₄	1	1	yes	characteristic
138	C₂×C₄	C₂⁴	1	1	yes	characteristic
139	C₈	—	2	2	no	—
140	C₈	—	2	2	no	—
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	4	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	4	4	no	—
150	C₂²	—	4	8	no	—
151	C₂²	—	4	2	no	—
152	C₂²	—	4	2	no	—
153	C₂²	—	4	1	no	—
154	C₂²	—	4	1	no	—
155	C₂²	—	4	1	no	—
156	C₂²	#120	1	2	yes	—
157	C₂²	#135	1	2	yes	complemented
158	C₂²	C₂²×D₄	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	2	4	no	—
161	C₄	—	2	1	no	—
162	C₄	—	4	1	no	—
163	C₄	—	4	1	no	—
164	C₄	—	4	1	no	—
165	C₄	#120	1	2	yes	—
166	C₄	C₂²×D₄	1	2	yes	—
167	C₂	—	2	2	no	—
168	C₂	—	2	2	no	—
169	C₂	—	4	1	no	—
170	C₂	—	4	4	no	—
171	C₂	—	8	1	no	—
172	C₂	C₂×#120	1	1	yes	characteristic
173	C₂	C₂×#135	1	1	yes	characteristic
174	C₂	#575	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
socle	C₂²	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂²×#41	C₂	semidirect
2. C₂×#135	C₂	semidirect
3. #525	C₂	semidirect
4. #452	C₂	semidirect
5. #451	C₂	semidirect
6. #450	C₂	semidirect
7. #449	C₂	semidirect
8. #448	C₂	semidirect
9. #447	C₂	semidirect
10. #416	C₂	semidirect
11. C₂×C₂²⋊C₈	C₂	semidirect
17. C₂×#41	C₂²	semidirect
24. C₂×Q₁₆	C₂²	semidirect
25. C₂×SD₁₆	C₂²	semidirect
26. C₂×SD₁₆	C₂²	semidirect
27. C₂×D₈	C₂²	semidirect
28. C₂²×C₈	C₂²	semidirect
31. C₂²⋊Q₈	C₂²	semidirect
33. #121	C₂²	semidirect
37. #117	C₂²	semidirect
38. #103	C₂²	semidirect
39. #103	C₂²	semidirect
40. #102	C₂²	semidirect
41. #102	C₂²	semidirect
44. C₂²⋊C₈	C₂²	semidirect
53. #41	D₄	semidirect
157. C₂²	#135	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#41	2
C₂×#135	2
#525	2
#452	2
#451	2
#450	2
#449	2
#448	2
#447	2
#416	2
C₂×C₂²⋊C₈	2