Finite Group Databasebeta

← C2×ΓL(1,𝔽8)×#104 #13054951

Subgroups — C2×ΓL(1,𝔽8)×#104 #13054951

Order 1344.

340 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×ΓL(1,𝔽8)C211yesmaximal, characteristic
2C2×M16×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
3C2×C42×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
4ΓL(1,𝔽8)×#104C214yesmaximal, direct product factor
5C2×C7×#104C311yesmaximal, characteristic, complemented, Hall
6#41×ΓL(1,𝔽8)—21no—
7#41×ΓL(1,𝔽8)—21no—
8#41×ΓL(1,𝔽8)C412yescomplemented
9#41×ΓL(1,𝔽8)C2212yes—
10C2×Q8×ΓL(1,𝔽8)C411yescharacteristic, complemented
11C2×D4×ΓL(1,𝔽8)—21no—
12C2×D4×ΓL(1,𝔽8)C411yescharacteristic, complemented
13C22×C4×ΓL(1,𝔽8)—21no—
14C22×C4×ΓL(1,𝔽8)—21no—
15C22×C4×ΓL(1,𝔽8)C2211yescharacteristic
16M16×ΓL(1,𝔽8)—21no—
17M16×ΓL(1,𝔽8)C2212yescomplemented
18C2×C8×ΓL(1,𝔽8)—21no—
19C42×ΓL(1,𝔽8)—21no—
20C42×ΓL(1,𝔽8)C2212yescomplemented
21C2×C7×#41C2×C311yescharacteristic
22C2×C7×M16C2×C311yescharacteristic, complemented
23C2×C42×C7C2×C311yescharacteristic, complemented
24C7×#104C2×C314yescomplemented
25C2×C3×#104—71nomaximal, Hall
26C23×ΓL(1,𝔽8)—21no—
27C23×ΓL(1,𝔽8)D411yescharacteristic
28Q8×ΓL(1,𝔽8)—21no—
29Q8×ΓL(1,𝔽8)C2×C412yescomplemented
30D4×ΓL(1,𝔽8)—22no—
31D4×ΓL(1,𝔽8)—22no—
32D4×ΓL(1,𝔽8)—21no—
33D4×ΓL(1,𝔽8)C2×C412yescomplemented
34C2×C4×ΓL(1,𝔽8)—24no—
35C2×C4×ΓL(1,𝔽8)—21no—
36C2×C4×ΓL(1,𝔽8)—22no—
37C2×C4×ΓL(1,𝔽8)—22no—
38C2×C4×ΓL(1,𝔽8)—22no—
39C2×C4×ΓL(1,𝔽8)C2×C411yescharacteristic
40C2×C4×ΓL(1,𝔽8)C2×C411yescharacteristic
41C2×C4×ΓL(1,𝔽8)D411yescharacteristic
42C2×C4×ΓL(1,𝔽8)D411yescharacteristic
43C2×C4×ΓL(1,𝔽8)D411yescharacteristic
44C2×C4×ΓL(1,𝔽8)C2311yescharacteristic
45C8×ΓL(1,𝔽8)—22no—
46C7×#41—21no—
47C7×#41—21no—
48C7×#41C3×C412yescomplemented
49C7×#41C22×C312yes—
50C2×C7×Q8C3×C411yescharacteristic, complemented
51C2×C7×D4—21no—
52C2×C7×D4C3×C411yescharacteristic, complemented
53C22×C4×C7—21no—
54C22×C4×C7—21no—
55C22×C4×C7C22×C311yescharacteristic
56C7×M16—21no—
57C7×M16C22×C312yescomplemented
58C2×C7×C8—21no—
59C42×C7—21no—
60C42×C7C22×C312yescomplemented
61C2×C3×#41—71no—
62C2×C3×M16—71no—
63C2×C3×C42—71no—
64C3×#104—74no—
65C22×ΓL(1,𝔽8)—22no—
66C22×ΓL(1,𝔽8)—22no—
67C22×ΓL(1,𝔽8)—42no—
68C22×ΓL(1,𝔽8)C22⋊C411yescharacteristic
69C22×ΓL(1,𝔽8)C22⋊C411yescharacteristic
70C22×ΓL(1,𝔽8)C2×D411yescharacteristic
71C4×ΓL(1,𝔽8)—24no—
72C4×ΓL(1,𝔽8)—22no—
73C4×ΓL(1,𝔽8)C22⋊C411yescharacteristic
74C4×ΓL(1,𝔽8)C22⋊C411yescharacteristic
75C4×ΓL(1,𝔽8)C22×C411yescharacteristic
76C4×ΓL(1,𝔽8)C2×D411yescharacteristic
77C2×#104ΓL(1,𝔽8)11yescharacteristic, direct product factor, Sylow
78C23×C7—21no—
79C23×C7C3×D411yescharacteristic
80C7×Q8—21no—
81C7×Q8C2×C3×C412yescomplemented
82C7×D4—22no—
83C7×D4—22no—
84C7×D4—21no—
85C7×D4C2×C3×C412yescomplemented
86C2×C4×C7—24no—
87C2×C4×C7—21no—
88C2×C4×C7—22no—
89C2×C4×C7—22no—
90C2×C4×C7—22no—
91C2×C4×C7C2×C3×C411yescharacteristic
92C2×C4×C7C2×C3×C411yescharacteristic
93C2×C4×C7C3×D411yescharacteristic
94C2×C4×C7C3×D411yescharacteristic
95C2×C4×C7C3×D411yescharacteristic
96C2×C4×C7C23×C311yescharacteristic
97C7×C8—22no—
98C3×#41—72no—
99C3×#41—72no—
100C3×#41—141no—
101C3×#41—141no—
102C2×C3×Q8—71no—
103C2×C3×D4—71no—
104C2×C3×D4—141no—
105C22×C3×C4—71no—
106C22×C3×C4—141no—
107C22×C3×C4—141no—
108C3×M16—72no—
109C3×M16—141no—
110C2×C3×C8—141no—
111C3×C42—72no—
112C3×C42—141no—
113C2×ΓL(1,𝔽8)—21no—
114C2×ΓL(1,𝔽8)—21no—
115C2×ΓL(1,𝔽8)—42no—
116C2×ΓL(1,𝔽8)#10412yesdirect product factor
117C2×ΓL(1,𝔽8)C2×C22⋊C411yescharacteristic
118C2×#41C2×ΓL(1,𝔽8)11yescharacteristic
119C2×M16C2×ΓL(1,𝔽8)11yescharacteristic, complemented
120C2×C42C2×ΓL(1,𝔽8)11yescharacteristic, complemented
121#104C2×ΓL(1,𝔽8)14yesdirect product factor
122C22×C7—22no—
123C22×C7—22no—
124C22×C7—42no—
125C22×C7C3×C22⋊C411yescharacteristic
126C22×C7C3×C22⋊C411yescharacteristic
127C22×C7C2×C3×D411yescharacteristic
128C4×C7—24no—
129C4×C7—22no—
130C4×C7C3×C22⋊C411yescharacteristic
131C4×C7C3×C22⋊C411yescharacteristic
132C4×C7C22×C3×C411yescharacteristic
133C4×C7C2×C3×D411yescharacteristic
134C23×C3—71no—
135C23×C3—141no—
136C3×Q8—72no—
137C3×Q8—141no—
138C3×D4—72no—
139C3×D4—141no—
140C3×D4—142no—
141C3×D4—142no—
142C2×C3×C4—71no—
143C2×C3×C4—71no—
144C2×C3×C4—71no—
145C2×C3×C4—71no—
146C2×C3×C4—71no—
147C2×C3×C4—71no—
148C2×C3×C4—142no—
149C2×C3×C4—142no—
150C2×C3×C4—142no—
151C2×C3×C4—141no—
152C2×C3×C4—144no—
153C3×C8—142no—
154ΓL(1,𝔽8)C2×#10411yescharacteristic, direct product factor, Hall
155#41—21no—
156#41—21no—
157#41C4×ΓL(1,𝔽8)12yescomplemented
158#41C22×ΓL(1,𝔽8)12yes—
159C2×Q8C4×ΓL(1,𝔽8)11yescharacteristic, complemented
160C2×D4—21no—
161C2×D4C4×ΓL(1,𝔽8)11yescharacteristic, complemented
162C22×C4—21no—
163C22×C4—21no—
164C22×C4C22×ΓL(1,𝔽8)11yescharacteristic
165M16—21no—
166M16C22×ΓL(1,𝔽8)12yescomplemented
167C2×C8—21no—
168C42—21no—
169C42C22×ΓL(1,𝔽8)12yescomplemented
170C2×C7—21no—
171C2×C7—21no—
172C2×C7—42no—
173C2×C7C3×#10412yescomplemented
174C2×C7C2×C3×C22⋊C411yescharacteristic
175C22×C3—71no—
176C22×C3—71no—
177C22×C3—71no—
178C22×C3—142no—
179C22×C3—142no—
180C22×C3—282no—
181C3×C4—71no—
182C3×C4—71no—
183C3×C4—71no—
184C3×C4—71no—
185C3×C4—142no—
186C3×C4—144no—
187C23—21no—
188C23D4×ΓL(1,𝔽8)11yescharacteristic
189Q8—21no—
190Q8C2×C4×ΓL(1,𝔽8)12yescomplemented
191D4—22no—
192D4—22no—
193D4—21no—
194D4C2×C4×ΓL(1,𝔽8)12yescomplemented
195C2×C4—24no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4C2×C4×ΓL(1,𝔽8)11yescharacteristic
201C2×C4C2×C4×ΓL(1,𝔽8)11yescharacteristic
202C2×C4D4×ΓL(1,𝔽8)11yescharacteristic
203C2×C4D4×ΓL(1,𝔽8)11yescharacteristic
204C2×C4D4×ΓL(1,𝔽8)11yescharacteristic
205C2×C4C23×ΓL(1,𝔽8)11yescharacteristic
206C8—22no—
207C7C2×C3×#10411yescharacteristic, complemented, Sylow
208C2×C3—72no—
209C2×C3—71no—
210C2×C3—141no—
211C2×C3—141no—
212C2×C3—282no—
213C22—22no—
214C22—22no—
215C22—42no—
216C22C22⋊C4×ΓL(1,𝔽8)11yescharacteristic
217C22C22⋊C4×ΓL(1,𝔽8)11yescharacteristic
218C22C2×D4×ΓL(1,𝔽8)11yescharacteristic
219C4—24no—
220C4—22no—
221C4C22⋊C4×ΓL(1,𝔽8)11yescharacteristic
222C4C22⋊C4×ΓL(1,𝔽8)11yescharacteristic
223C4C22×C4×ΓL(1,𝔽8)11yescharacteristic
224C4C2×D4×ΓL(1,𝔽8)11yescharacteristic
225C3—71noSylow
226C2—21no—
227C2—21no—
228C2—42no—
229C2ΓL(1,𝔽8)×#10412yesdirect product factor
230C2C2×C22⋊C4×ΓL(1,𝔽8)11yescharacteristic
subgroupG/H
centerC2×C4D4×ΓL(1,𝔽8)
commutatorC4×C7C22×C3×C4
FrattiniC2×C4C23×ΓL(1,𝔽8)
FittingC2×C7×#104C3
socleC22×C7C3×C22⋊C4
prime radicalC2×#41×ΓL(1,𝔽8)C2
cosocleC2×C4×C7C23×C3
hypercenterC2×#104ΓL(1,𝔽8)
hypocenterC7C2×C3×#104
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C2×M16×ΓL(1,𝔽8)C2semidirect
3. C2×C42×ΓL(1,𝔽8)C2semidirect
4. ΓL(1,𝔽8)×#104C2direct (N × H)
5. C2×C7×#104C3semidirect
8. #41×ΓL(1,𝔽8)C4semidirect
10. C2×Q8×ΓL(1,𝔽8)C4semidirect
12. C2×D4×ΓL(1,𝔽8)C4semidirect
17. M16×ΓL(1,𝔽8)C22semidirect
20. C42×ΓL(1,𝔽8)C22semidirect
22. C2×C7×M16C2×C3semidirect
23. C2×C42×C7C2×C3semidirect
24. C7×#104C2×C3semidirect
29. Q8×ΓL(1,𝔽8)C2×C4semidirect
33. D4×ΓL(1,𝔽8)C2×C4semidirect
48. C7×#41C3×C4semidirect
50. C2×C7×Q8C3×C4semidirect
52. C2×C7×D4C3×C4semidirect
57. C7×M16C22×C3semidirect
60. C42×C7C22×C3semidirect
77. C2×#104ΓL(1,𝔽8)direct (N × H)
81. C7×Q8C2×C3×C4semidirect
85. C7×D4C2×C3×C4semidirect
116. C2×ΓL(1,𝔽8)#104direct (N × H)
119. C2×M16C2×ΓL(1,𝔽8)semidirect
120. C2×C42C2×ΓL(1,𝔽8)semidirect
121. #104C2×ΓL(1,𝔽8)direct (N × H)
154. ΓL(1,𝔽8)C2×#104direct (N × H)
157. #41C4×ΓL(1,𝔽8)semidirect
159. C2×Q8C4×ΓL(1,𝔽8)semidirect
161. C2×D4C4×ΓL(1,𝔽8)semidirect
166. M16C22×ΓL(1,𝔽8)semidirect
169. C42C22×ΓL(1,𝔽8)semidirect
173. C2×C7C3×#104semidirect
190. Q8C2×C4×ΓL(1,𝔽8)semidirect
194. D4C2×C4×ΓL(1,𝔽8)semidirect
207. C7C2×C3×#104semidirect
229. C2ΓL(1,𝔽8)×#104direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#41×ΓL(1,𝔽8)2
C2×M16×ΓL(1,𝔽8)2
C2×C42×ΓL(1,𝔽8)2
ΓL(1,𝔽8)×#1042
C2×C7×#1043
C2×C3×#1047