Subgroups — C₂×ΓL(1,𝔽₈)×C₂²⋊Q₈ #13060684

Order 1344.

644 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₄×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
3	ΓL(1,𝔽₈)×C₂²⋊Q₈	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊C₄×ΓL(1,𝔽₈)	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊C₄×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂²⋊C₄×ΓL(1,𝔽₈)	C₂	1	2	yes	maximal
7	C₂×C₇×C₂²⋊Q₈	C₃	1	1	yes	maximal, characteristic, complemented, Hall
8	C₂⁴×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
9	C₂×Q₈×ΓL(1,𝔽₈)	—	2	4	no	—
10	C₂×Q₈×ΓL(1,𝔽₈)	C₂²	1	4	yes	complemented
11	C₂²×C₄×ΓL(1,𝔽₈)	—	2	4	no	—
12	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
13	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	4	yes	—
14	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
15	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
16	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	4	yes	complemented
18	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	8	yes	complemented
19	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	8	yes	—
20	C₂²×C₇×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
21	C₂³×C₄×C₇	C₂×C₃	1	1	yes	characteristic
22	C₇×C₂²⋊Q₈	C₂×C₃	1	8	yes	complemented
23	C₂×C₇×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂×C₇×C₄⋊C₄	C₂×C₃	1	2	yes	complemented
25	C₂×C₇×C₂²⋊C₄	C₂×C₃	1	2	yes	—
26	C₂×C₃×C₂²⋊Q₈	—	7	1	no	maximal, Hall
27	C₂³×ΓL(1,𝔽₈)	—	2	2	no	—
28	C₂³×ΓL(1,𝔽₈)	—	2	2	no	—
29	C₂³×ΓL(1,𝔽₈)	Q₈	1	4	yes	complemented
30	C₂³×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
31	C₂³×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
32	Q₈×ΓL(1,𝔽₈)	—	2	8	no	—
33	C₂×C₄×ΓL(1,𝔽₈)	—	2	2	no	—
34	C₂×C₄×ΓL(1,𝔽₈)	—	2	4	no	—
35	C₂×C₄×ΓL(1,𝔽₈)	—	2	4	no	—
36	C₂×C₄×ΓL(1,𝔽₈)	—	2	8	no	—
37	C₂×C₄×ΓL(1,𝔽₈)	D₄	1	4	yes	—
38	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	4	yes	—
39	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	4	yes	—
40	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
41	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
42	C₂⁴×C₇	C₂²×C₃	1	1	yes	characteristic
43	C₂×C₇×Q₈	—	2	4	no	—
44	C₂×C₇×Q₈	C₂²×C₃	1	4	yes	complemented
45	C₂²×C₄×C₇	—	2	4	no	—
46	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	—
47	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	—
48	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
49	C₂²×C₄×C₇	C₂²×C₃	1	4	yes	—
50	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
51	C₇×C₄⋊C₄	C₂²×C₃	1	4	yes	complemented
52	C₇×C₄⋊C₄	C₂²×C₃	1	8	yes	complemented
53	C₇×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
54	C₂²×C₃×Q₈	—	7	1	no	—
55	C₂³×C₃×C₄	—	7	1	no	—
56	C₃×C₂²⋊Q₈	—	7	8	no	—
57	C₂×C₃×C₄⋊C₄	—	7	1	no	—
58	C₂×C₃×C₄⋊C₄	—	7	2	no	—
59	C₂×C₃×C₂²⋊C₄	—	7	2	no	—
60	C₂²×ΓL(1,𝔽₈)	—	2	8	no	—
61	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
62	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
63	C₂²×ΓL(1,𝔽₈)	C₂×D₄	1	2	yes	—
64	C₂²×ΓL(1,𝔽₈)	C₂×Q₈	1	2	yes	—
65	C₂²×ΓL(1,𝔽₈)	C₂×Q₈	1	4	yes	complemented
66	C₂²×ΓL(1,𝔽₈)	#41	1	2	yes	—
67	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
68	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
69	C₄×ΓL(1,𝔽₈)	—	2	4	no	—
70	C₄×ΓL(1,𝔽₈)	—	2	4	no	—
71	C₄×ΓL(1,𝔽₈)	C₂×D₄	1	4	yes	—
72	C₂×C₂²⋊Q₈	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
73	C₂³×C₇	—	2	2	no	—
74	C₂³×C₇	—	2	2	no	—
75	C₂³×C₇	C₃×Q₈	1	4	yes	complemented
76	C₂³×C₇	C₂³×C₃	1	2	yes	—
77	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
78	C₇×Q₈	—	2	8	no	—
79	C₂×C₄×C₇	—	2	2	no	—
80	C₂×C₄×C₇	—	2	4	no	—
81	C₂×C₄×C₇	—	2	4	no	—
82	C₂×C₄×C₇	—	2	8	no	—
83	C₂×C₄×C₇	C₃×D₄	1	4	yes	—
84	C₂×C₄×C₇	C₂³×C₃	1	4	yes	—
85	C₂×C₄×C₇	C₂³×C₃	1	4	yes	—
86	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
87	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
88	C₂⁴×C₃	—	7	1	no	—
89	C₂×C₃×Q₈	—	7	4	no	—
90	C₂×C₃×Q₈	—	14	4	no	—
91	C₂²×C₃×C₄	—	7	1	no	—
92	C₂²×C₃×C₄	—	7	2	no	—
93	C₂²×C₃×C₄	—	7	2	no	—
94	C₂²×C₃×C₄	—	7	1	no	—
95	C₂²×C₃×C₄	—	7	4	no	—
96	C₂²×C₃×C₄	—	14	4	no	—
97	C₃×C₄⋊C₄	—	7	4	no	—
98	C₃×C₄⋊C₄	—	7	8	no	—
99	C₃×C₂²⋊C₄	—	7	8	no	—
100	C₂×ΓL(1,𝔽₈)	—	2	4	no	—
101	C₂×ΓL(1,𝔽₈)	C₂²⋊Q₈	1	4	yes	direct product factor
102	C₂×ΓL(1,𝔽₈)	C₂²×D₄	1	1	yes	characteristic
103	C₂×ΓL(1,𝔽₈)	C₂²×Q₈	1	1	yes	characteristic
104	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
105	C₂²×Q₈	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
106	C₂³×C₄	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
107	C₂²⋊Q₈	C₂×ΓL(1,𝔽₈)	1	8	yes	direct product factor
108	C₂×C₄⋊C₄	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
109	C₂×C₄⋊C₄	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
110	C₂×C₂²⋊C₄	C₂×ΓL(1,𝔽₈)	1	2	yes	—
111	C₂²×C₇	—	2	8	no	—
112	C₂²×C₇	—	2	2	no	—
113	C₂²×C₇	—	2	2	no	—
114	C₂²×C₇	C₂×C₃×D₄	1	2	yes	—
115	C₂²×C₇	C₂×C₃×Q₈	1	2	yes	—
116	C₂²×C₇	C₂×C₃×Q₈	1	4	yes	complemented
117	C₂²×C₇	C₃×#41	1	2	yes	—
118	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
119	C₄×C₇	—	2	2	no	—
120	C₄×C₇	—	2	4	no	—
121	C₄×C₇	—	2	4	no	—
122	C₄×C₇	C₂×C₃×D₄	1	4	yes	—
123	C₂³×C₃	—	7	4	no	—
124	C₂³×C₃	—	7	2	no	—
125	C₂³×C₃	—	7	1	no	—
126	C₂³×C₃	—	14	2	no	—
127	C₂³×C₃	—	14	2	no	—
128	C₃×Q₈	—	14	8	no	—
129	C₂×C₃×C₄	—	7	4	no	—
130	C₂×C₃×C₄	—	7	2	no	—
131	C₂×C₃×C₄	—	7	4	no	—
132	C₂×C₃×C₄	—	7	4	no	—
133	C₂×C₃×C₄	—	7	2	no	—
134	C₂×C₃×C₄	—	14	8	no	—
135	C₂×C₃×C₄	—	14	4	no	—
136	C₂×C₃×C₄	—	14	2	no	—
137	C₂×C₃×C₄	—	14	4	no	—
138	ΓL(1,𝔽₈)	C₂×C₂²⋊Q₈	1	1	yes	characteristic, direct product factor, Hall
139	C₂⁴	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
140	C₂×Q₈	—	2	4	no	—
141	C₂×Q₈	C₂²×ΓL(1,𝔽₈)	1	4	yes	complemented
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	C₂²×ΓL(1,𝔽₈)	1	4	yes	—
144	C₂²×C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
145	C₂²×C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
146	C₂²×C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
147	C₂²×C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
148	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	4	yes	complemented
149	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	8	yes	complemented
150	C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	1	8	yes	—
151	C₂×C₇	—	2	4	no	—
152	C₂×C₇	C₃×C₂²⋊Q₈	1	4	yes	complemented
153	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
154	C₂×C₇	C₂²×C₃×Q₈	1	1	yes	characteristic
155	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
156	C₂²×C₃	—	7	2	no	—
157	C₂²×C₃	—	7	2	no	—
158	C₂²×C₃	—	7	4	no	—
159	C₂²×C₃	—	7	2	no	—
160	C₂²×C₃	—	7	1	no	—
161	C₂²×C₃	—	14	2	no	—
162	C₂²×C₃	—	14	2	no	—
163	C₂²×C₃	—	14	8	no	—
164	C₃×C₄	—	7	4	no	—
165	C₃×C₄	—	14	2	no	—
166	C₃×C₄	—	14	4	no	—
167	C₃×C₄	—	14	4	no	—
168	C₂³	—	2	2	no	—
169	C₂³	—	2	2	no	—
170	C₂³	Q₈×ΓL(1,𝔽₈)	1	4	yes	complemented
171	C₂³	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
172	C₂³	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
173	Q₈	—	2	8	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	D₄×ΓL(1,𝔽₈)	1	4	yes	—
179	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	4	yes	—
180	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	4	yes	—
181	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
182	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
183	C₇	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
184	C₂×C₃	—	7	1	no	—
185	C₂×C₃	—	7	1	no	—
186	C₂×C₃	—	7	1	no	—
187	C₂×C₃	—	7	4	no	—
188	C₂×C₃	—	14	4	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	2	8	no	—
192	C₂²	C₂×D₄×ΓL(1,𝔽₈)	1	2	yes	—
193	C₂²	C₂×Q₈×ΓL(1,𝔽₈)	1	2	yes	—
194	C₂²	C₂×Q₈×ΓL(1,𝔽₈)	1	4	yes	complemented
195	C₂²	#41×ΓL(1,𝔽₈)	1	2	yes	—
196	C₂²	C₂⁴×ΓL(1,𝔽₈)	1	1	yes	characteristic
197	C₄	—	2	2	no	—
198	C₄	—	2	4	no	—
199	C₄	—	2	4	no	—
200	C₄	C₂×D₄×ΓL(1,𝔽₈)	1	4	yes	—
201	C₃	—	7	1	no	Sylow
202	C₂	—	2	4	no	—
203	C₂	ΓL(1,𝔽₈)×C₂²⋊Q₈	1	4	yes	direct product factor
204	C₂	C₂²×D₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
205	C₂	C₂²×Q₈×ΓL(1,𝔽₈)	1	1	yes	characteristic
206	C₂	C₂×#41×ΓL(1,𝔽₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×ΓL(1,𝔽₈)
commutator, cosocle	C₂²×C₇	C₂⁴×C₃
Frattini	C₂²	C₂⁴×ΓL(1,𝔽₈)
Fitting	C₂×C₇×C₂²⋊Q₈	C₃
socle	C₂³×C₇	C₂³×C₃
prime radical	C₂⁴×ΓL(1,𝔽₈)	C₂²
hypercenter	C₂×C₂²⋊Q₈	ΓL(1,𝔽₈)
hypocenter	C₇	C₂×C₃×C₂²⋊Q₈
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×Q₈×ΓL(1,𝔽₈)	C₂	semidirect
3. ΓL(1,𝔽₈)×C₂²⋊Q₈	C₂	direct (N × H)
4. C₂×C₄⋊C₄×ΓL(1,𝔽₈)	C₂	semidirect
5. C₂×C₄⋊C₄×ΓL(1,𝔽₈)	C₂	semidirect
7. C₂×C₇×C₂²⋊Q₈	C₃	semidirect
10. C₂×Q₈×ΓL(1,𝔽₈)	C₂²	semidirect
17. C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	semidirect
18. C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	semidirect
20. C₂²×C₇×Q₈	C₂×C₃	semidirect
22. C₇×C₂²⋊Q₈	C₂×C₃	semidirect
23. C₂×C₇×C₄⋊C₄	C₂×C₃	semidirect
24. C₂×C₇×C₄⋊C₄	C₂×C₃	semidirect
29. C₂³×ΓL(1,𝔽₈)	Q₈	semidirect
44. C₂×C₇×Q₈	C₂²×C₃	semidirect
51. C₇×C₄⋊C₄	C₂²×C₃	semidirect
52. C₇×C₄⋊C₄	C₂²×C₃	semidirect
65. C₂²×ΓL(1,𝔽₈)	C₂×Q₈	semidirect
72. C₂×C₂²⋊Q₈	ΓL(1,𝔽₈)	direct (N × H)
75. C₂³×C₇	C₃×Q₈	semidirect
101. C₂×ΓL(1,𝔽₈)	C₂²⋊Q₈	direct (N × H)
105. C₂²×Q₈	C₂×ΓL(1,𝔽₈)	semidirect
107. C₂²⋊Q₈	C₂×ΓL(1,𝔽₈)	direct (N × H)
108. C₂×C₄⋊C₄	C₂×ΓL(1,𝔽₈)	semidirect
109. C₂×C₄⋊C₄	C₂×ΓL(1,𝔽₈)	semidirect
116. C₂²×C₇	C₂×C₃×Q₈	semidirect
138. ΓL(1,𝔽₈)	C₂×C₂²⋊Q₈	direct (N × H)
141. C₂×Q₈	C₂²×ΓL(1,𝔽₈)	semidirect
148. C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	semidirect
149. C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	semidirect
152. C₂×C₇	C₃×C₂²⋊Q₈	semidirect
170. C₂³	Q₈×ΓL(1,𝔽₈)	semidirect
183. C₇	C₂×C₃×C₂²⋊Q₈	semidirect
194. C₂²	C₂×Q₈×ΓL(1,𝔽₈)	semidirect
203. C₂	ΓL(1,𝔽₈)×C₂²⋊Q₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Q₈×ΓL(1,𝔽₈)	2
C₂³×C₄×ΓL(1,𝔽₈)	2
ΓL(1,𝔽₈)×C₂²⋊Q₈	2
C₂×C₄⋊C₄×ΓL(1,𝔽₈)	2
C₂×C₂²⋊C₄×ΓL(1,𝔽₈)	2
C₂×C₇×C₂²⋊Q₈	3
C₂×C₃×C₂²⋊Q₈	7

Subgroups — C2×ΓL(1,𝔽8)×C22⋊Q8 #13060684

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×ΓL(1,𝔽₈)×C₂²⋊Q₈ #13060684