Finite Group Databasebeta

← #3265

Subgroups — #3265

Order 128.

273 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 273 conjugacy classes of subgroups — too many to draw. Only its 96 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×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
2C22×SD16C211yesmaximal, characteristic, complemented
3#548C211yesmaximal, characteristic, complemented
4D42C211yesmaximal, characteristic, complemented
5#496C211yesmaximal, characteristic, complemented
6#468C211yesmaximal, characteristic, complemented
7#466C211yesmaximal, characteristic, complemented
8#463C211yesmaximal, characteristic, complemented
9#460C211yesmaximal, characteristic, complemented
10#450C211yesmaximal, characteristic, complemented
11#449C211yesmaximal, characteristic, complemented
12#448C211yesmaximal, characteristic, complemented
13#447C211yesmaximal, characteristic, complemented
14#440C211yesmaximal, characteristic, complemented
15#435C211yesmaximal, characteristic, complemented
16C2×#41C2211yescharacteristic, complemented
17C22×Q8C2211yescharacteristic, complemented
18C22×D4—21no—
19C22×D4C2211yescharacteristic
20C22×D4C2211yescharacteristic
21AGL(1,ℤ/8)—24no—
22C2×SD16—24no—
23C2×SD16C2211yescharacteristic
24C2×SD16C2211yescharacteristic
25C2×SD16C2211yescharacteristic, complemented
26C2×SD16C2211yescharacteristic, complemented
27C2×SD16C2211yescharacteristic, complemented
28C2×SD16C2211yescharacteristic, complemented
29C2×D8C2211yescharacteristic
30C2×D8C2211yescharacteristic, complemented
31C2×D8C2211yescharacteristic, complemented
32C2×M16C2211yescharacteristic, complemented
33C22×C8C2211yescharacteristic, complemented
34#127C2211yescharacteristic
35#124—21no—
36#124C2211yescharacteristic, complemented
37C22⋊Q8—21no—
38C22⋊Q8C2211yescharacteristic, complemented
39#121—21no—
40#121—21no—
41#121C2211yescharacteristic
42#121C2211yescharacteristic
43#121C2211yescharacteristic, complemented
44#120—21no—
45#120—21no—
46C4×Q8C2211yescharacteristic, complemented
47C4×D4—21no—
48C4×D4C2211yescharacteristic
49C4×D4C2211yescharacteristic
50#107C2211yescharacteristic, complemented
51C4⋊C8C2211yescharacteristic, complemented
52#103C2211yescharacteristic, complemented
53#103C2211yescharacteristic, complemented
54#102C2211yescharacteristic
55#102C2211yescharacteristic, complemented
56#102C2211yescharacteristic, complemented
57#102C2211yescharacteristic, complemented
58C22⋊C8C2211yescharacteristic, complemented
59C22⋊C8C2211yescharacteristic, complemented
60#97C2211yescharacteristic, complemented
61C24—21no—
62C24—21no—
63#41—22no—
64#41—41no—
65C2×Q8—21no—
66C2×Q8—22no—
67C2×Q8—41no—
68C2×Q8C2311yescharacteristic
69C2×Q8C2311yescharacteristic
70C2×D4—22no—
71C2×D4—22no—
72C2×D4—21no—
73C2×D4—21no—
74C2×D4—21no—
75C2×D4—21no—
76C2×D4—21no—
77C2×D4—21no—
78C2×D4—21no—
79C2×D4—21no—
80C2×D4—41no—
81C2×D4—41no—
82C2×D4—44no—
83C2×D4D411yescharacteristic
84C2×D4C2311yescharacteristic
85C2×D4C2311yescharacteristic
86C2×D4C2311yescharacteristic
87C2×D4C2311yescharacteristic
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4C2311yescharacteristic
92C22×C4C2311yescharacteristic
93SD16—22no—
94SD16—22no—
95SD16—22no—
96SD16—22no—
97SD16—41no—
98SD16D414yescomplemented
99D8—22no—
100D8—22no—
101D8—41no—
102M16—22no—
103C2×C8—22no—
104C2×C8C2311yescharacteristic
105C2×C8C2311yescharacteristic
106C2×C8C2311yescharacteristic
107C2×C8C2311yescharacteristic
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4D411yescharacteristic
111C4⋊C4C2311yescharacteristic
112C4⋊C4C2311yescharacteristic
113C22⋊C4—21no—
114C22⋊C4—21no—
115C22⋊C4—21no—
116C22⋊C4—21no—
117C22⋊C4—21no—
118C22⋊C4—21no—
119C22⋊C4—21no—
120C22⋊C4D411yescharacteristic
121C22⋊C4D411yescharacteristic
122C42—21no—
123C42C2311yescharacteristic
124C23—21no—
125C23—21no—
126C23—21no—
127C23—21no—
128C23—42no—
129C23—42no—
130C23—42no—
131C23—42no—
132C23—81no—
133C23—81no—
134C23C2×D411yescharacteristic
135C23C2×D411yescharacteristic
136Q8—21no—
137Q8—21no—
138Q8—21no—
139Q8—42no—
140Q8C2×D412yescomplemented
141D4—22no—
142D4—21no—
143D4—21no—
144D4—21no—
145D4—21no—
146D4—21no—
147D4—21no—
148D4—21no—
149D4—42no—
150D4—42no—
151D4—42no—
152D4—42no—
153D4—42no—
154D4—42no—
155D4—42no—
156D4—81no—
157D4C2×D412yes—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4C2×D411yescharacteristic
170C2×C4C2×D411yescharacteristic
171C2×C4C2×D411yescharacteristic
172C2×C4C2×D411yescharacteristic
173C2×C4C2411yescharacteristic
174C8—21no—
175C8—21no—
176C8—21no—
177C8C2×D412yescomplemented
178C22—21no—
179C22—21no—
180C22—21no—
181C22—22no—
182C22—21no—
183C22—21no—
184C22—41no—
185C22—41no—
186C22—41no—
187C22—41no—
188C22—41no—
189C22—41no—
190C22—41no—
191C22—41no—
192C22—41no—
193C22—41no—
194C22—41no—
195C22—84no—
196C22—84no—
197C22AGL(1,ℤ/8)12yescomplemented
198C22C22×D411yescharacteristic
199C4—21no—
200C4—21no—
201C4—22no—
202C4—21no—
203C4—21no—
204C4—41no—
205C4—41no—
206C4—41no—
207C4C22×D411yescharacteristic
208C4H(5,𝔽2)11yescharacteristic
209C2—22no—
210C2—41no—
211C2—42no—
212C2—81no—
213C2—81no—
214C2—81no—
215C2D4211yescharacteristic
216C2C2×AGL(1,ℤ/8)11yescharacteristic
217C2#57711yescharacteristic
subgroupG/H
center, socleC22C22×D4
commutator, Frattini, cosocleC2×C4C24
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×D4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×AGL(1,ℤ/8)C2semidirect
2. C22×SD16C2semidirect
3. #548C2semidirect
4. D42C2semidirect
5. #496C2semidirect
6. #468C2semidirect
7. #466C2semidirect
8. #463C2semidirect
9. #460C2semidirect
10. #450C2semidirect
11. #449C2semidirect
12. #448C2semidirect
13. #447C2semidirect
14. #440C2semidirect
15. #435C2semidirect
16. C2×#41C22semidirect
17. C22×Q8C22semidirect
25. C2×SD16C22semidirect
26. C2×SD16C22semidirect
27. C2×SD16C22semidirect
28. C2×SD16C22semidirect
30. C2×D8C22semidirect
31. C2×D8C22semidirect
32. C2×M16C22semidirect
33. C22×C8C22semidirect
36. #124C22semidirect
38. C22⋊Q8C22semidirect
43. #121C22semidirect
46. C4×Q8C22semidirect
50. #107C22semidirect
51. C4⋊C8C22semidirect
52. #103C22semidirect
53. #103C22semidirect
55. #102C22semidirect
56. #102C22semidirect
57. #102C22semidirect
58. C22⋊C8C22semidirect
59. C22⋊C8C22semidirect
60. #97C22semidirect
98. SD16D4semidirect
140. Q8C2×D4semidirect
177. C8C2×D4semidirect
197. C22AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
C2×AGL(1,ℤ/8)2
C22×SD162
#5482
D422
#4962
#4682
#4662
#4632
#4602
#4502
#4492
#4482
#4472
#4402
#4352