Finite Group Databasebeta

Subgroups — #3118

Order 128.

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

normal subgroups only

This group has 203 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, complemented
2	#518	C₂	1	1	yes	maximal, characteristic, complemented
3	#485	C₂	1	1	yes	maximal, characteristic, complemented
4	#482	C₂	1	1	yes	maximal, characteristic, complemented
5	#476	C₂	1	1	yes	maximal, characteristic, complemented
6	#474	C₂	1	1	yes	maximal, characteristic, complemented
7	#463	C₂	1	1	yes	maximal, characteristic, complemented
8	#463	C₂	1	1	yes	maximal, characteristic, complemented
9	#450	C₂	1	1	yes	maximal, characteristic, complemented
10	#447	C₂	1	1	yes	maximal, characteristic
11	#442	C₂	1	1	yes	maximal, characteristic, complemented
12	#440	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×D₈	C₂	1	1	yes	maximal, characteristic, complemented
15	#433	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×D₄	C₂²	1	1	yes	characteristic
17	C₂×SD₁₆	C₂²	1	1	yes	characteristic
18	C₂×SD₁₆	C₂²	1	1	yes	characteristic
19	C₂×D₈	C₂²	1	1	yes	characteristic
20	C₂×D₈	C₂²	1	1	yes	characteristic
21	C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
22	#126	—	2	1	no	—
23	#124	—	2	1	no	—
24	#124	C₂²	1	1	yes	characteristic, complemented
25	#123	—	2	1	no	—
26	C₂²⋊Q₈	—	2	1	no	—
27	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
28	#121	C₂²	1	1	yes	characteristic
29	C₄×Q₈	C₂²	1	1	yes	characteristic, complemented
30	C₄×D₄	—	2	1	no	—
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	C₄×D₄	C₂²	1	1	yes	characteristic
36	C₄×D₄	C₂²	1	1	yes	characteristic, complemented
37	#117	C₂²	1	1	yes	characteristic
38	#117	C₂²	1	1	yes	characteristic
39	C₂×C₂²⋊C₄	—	2	1	no	—
40	C₂×C₂²⋊C₄	—	2	1	no	—
41	#107	C₂²	1	1	yes	characteristic, complemented
42	#107	C₂²	1	1	yes	characteristic, complemented
43	#106	C₂²	1	1	yes	characteristic, complemented
44	#106	C₂²	1	1	yes	characteristic, complemented
45	C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
46	C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
47	#103	C₂²	1	1	yes	characteristic, complemented
48	#103	C₂²	1	1	yes	characteristic, complemented
49	#102	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, complemented
54	#102	C₂²	1	1	yes	characteristic, complemented
55	C₂²⋊C₈	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈	C₂²	1	1	yes	characteristic
57	#97	C₂²	1	1	yes	characteristic, complemented
58	C₄×C₈	C₂²	1	1	yes	characteristic, complemented
59	C₂⁴	—	2	1	no	—
60	C₂×Q₈	—	2	1	no	—
61	C₂×Q₈	C₂³	1	1	yes	characteristic
62	C₂×D₄	—	2	2	no	—
63	C₂×D₄	—	2	2	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	1	yes	characteristic
68	C₂×D₄	C₂³	1	1	yes	characteristic
69	C₂²×C₄	—	2	1	no	—
70	C₂²×C₄	—	2	1	no	—
71	C₂²×C₄	—	2	1	no	—
72	C₂²×C₄	—	2	1	no	—
73	C₂²×C₄	—	2	1	no	—
74	C₂²×C₄	C₂³	1	1	yes	characteristic
75	SD₁₆	—	2	2	no	—
76	SD₁₆	—	2	2	no	—
77	D₈	—	2	2	no	—
78	D₈	—	2	2	no	—
79	C₂×C₈	C₂³	1	1	yes	characteristic
80	C₂×C₈	C₂³	1	1	yes	characteristic
81	C₂×C₈	C₂³	1	1	yes	characteristic
82	C₂×C₈	C₂³	1	1	yes	characteristic
83	C₄⋊C₄	—	2	1	no	—
84	C₄⋊C₄	—	2	1	no	—
85	C₄⋊C₄	—	2	1	no	—
86	C₄⋊C₄	D₄	1	1	yes	characteristic
87	C₄⋊C₄	D₄	1	1	yes	characteristic
88	C₄⋊C₄	C₂³	1	1	yes	characteristic
89	C₄⋊C₄	C₂³	1	1	yes	characteristic
90	C₄⋊C₄	C₂³	1	1	yes	characteristic
91	C₄⋊C₄	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
100	C₂²⋊C₄	—	2	1	no	—
101	C₂²⋊C₄	—	2	1	no	—
102	C₂²⋊C₄	D₄	1	1	yes	characteristic
103	C₂²⋊C₄	D₄	1	1	yes	characteristic
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	4	no	—
109	C₂³	—	2	1	no	—
110	C₂³	—	2	1	no	—
111	C₂³	—	2	1	no	—
112	C₂³	—	8	1	no	—
113	C₂³	—	8	1	no	—
114	C₂³	C₂×D₄	1	1	yes	characteristic
115	Q₈	—	2	1	no	—
116	Q₈	—	2	1	no	—
117	Q₈	—	8	1	no	—
118	D₄	—	2	4	no	—
119	D₄	—	2	1	no	—
120	D₄	—	2	1	no	—
121	D₄	—	2	1	no	—
122	D₄	—	2	1	no	—
123	D₄	—	8	1	no	—
124	D₄	#41	1	2	yes	—
125	D₄	#41	1	2	yes	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	8	1	no	—
134	C₂×C₄	—	8	1	no	—
135	C₂×C₄	—	8	1	no	—
136	C₂×C₄	—	8	1	no	—
137	C₂×C₄	—	8	1	no	—
138	C₂×C₄	C₂×D₄	1	1	yes	characteristic
139	C₂×C₄	C₂×D₄	1	1	yes	characteristic
140	C₂×C₄	C₂×D₄	1	1	yes	characteristic
141	C₂×C₄	C₂×D₄	1	1	yes	characteristic
142	C₂×C₄	C₂×D₄	1	1	yes	characteristic
143	C₂×C₄	C₂⁴	1	1	yes	characteristic
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	2	1	no	—
147	C₈	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	2	no	—
150	C₂²	—	2	2	no	—
151	C₂²	—	2	1	no	—
152	C₂²	—	2	1	no	—
153	C₂²	—	4	1	no	—
154	C₂²	—	4	1	no	—
155	C₂²	—	4	1	no	—
156	C₂²	—	4	1	no	—
157	C₂²	—	4	1	no	—
158	C₂²	—	4	1	no	—
159	C₂²	—	4	1	no	—
160	C₂²	—	8	4	no	—
161	C₂²	C₂²×D₄	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	2	1	no	—
166	C₄	—	2	1	no	—
167	C₄	—	4	1	no	—
168	C₄	—	4	1	no	—
169	C₄	—	4	1	no	—
170	C₄	—	4	1	no	—
171	C₄	—	4	1	no	—
172	C₄	C₂×#41	1	1	yes	characteristic
173	C₄	C₂×#41	1	1	yes	characteristic
174	C₂	—	4	1	no	—
175	C₂	—	4	2	no	—
176	C₂	—	4	2	no	—
177	C₂	—	8	1	no	—
178	C₂	#525	1	1	yes	characteristic
179	C₂	#576	1	1	yes	characteristic
180	C₂	#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	#447	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. #518	C₂	semidirect
3. #485	C₂	semidirect
4. #482	C₂	semidirect
5. #476	C₂	semidirect
6. #474	C₂	semidirect
7. #463	C₂	semidirect
8. #463	C₂	semidirect
9. #450	C₂	semidirect
11. #442	C₂	semidirect
12. #440	C₂	semidirect
13. C₄×SD₁₆	C₂	semidirect
14. C₄×D₈	C₂	semidirect
15. #433	C₂	semidirect
21. C₄⋊Q₈	C₂²	semidirect
24. #124	C₂²	semidirect
29. C₄×Q₈	C₂²	semidirect
36. C₄×D₄	C₂²	semidirect
41. #107	C₂²	semidirect
42. #107	C₂²	semidirect
43. #106	C₂²	semidirect
44. #106	C₂²	semidirect
45. C₄⋊C₈	C₂²	semidirect
46. C₄⋊C₈	C₂²	semidirect
47. #103	C₂²	semidirect
48. #103	C₂²	semidirect
53. #102	C₂²	semidirect
54. #102	C₂²	semidirect
57. #97	C₂²	semidirect
58. C₄×C₈	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#542	2
#518	2
#485	2
#482	2
#476	2
#474	2
#463	2
#450	2
#447	2
#442	2
#440	2
C₄×SD₁₆	2
C₄×D₈	2
#433	2