Finite Group Databasebeta

← #41003644

Subgroups — #41003644

Order 26944.

268 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 99 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
1#27212412C211yesmaximal, characteristic, complemented
2#27212411C211yesmaximal, characteristic, complemented
3#27212394C211yesmaximal, characteristic, complemented
4#27212393C212yesmaximal, complemented
5#27212390C212yesmaximal, complemented
6#27212387C212yesmaximal, complemented
7#27212386C211yesmaximal, characteristic, complemented
8C2×#20269756C211yesmaximal, characteristic, complemented
9D8×D421C212yesmaximal, complemented
10#27212374C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C421C211yesmaximal, characteristic, complemented
12#20269772C2212yescomplemented
13#20269771—21no—
14#20269771C2211yescharacteristic
15#20269771C2211yescharacteristic, complemented
16#20269770—21no—
17#20269770C2212yescomplemented
18C2×D1684—21no—
19C2×D1684C2211yescharacteristic, complemented
20C2×C22⋊D421—22no—
21D4×D421—21no—
22D4×D421—22no—
23D4×D421C2212yescomplemented
24D4×D421C2212yescomplemented
25#20269761C2211yescharacteristic, complemented
26#20269760C2212yescomplemented
27#20269759C2211yescharacteristic, complemented
28#20269758C2211yescharacteristic, complemented
29D3368C2212yescomplemented
30#20269756C2211yescharacteristic, complemented
31#20269756C2212yescomplemented
32#20269756C2212yescomplemented
33#20269754C2212yescomplemented
34C2×#16809890C2211yescharacteristic, complemented
35C8×D421C2212yescomplemented
36#41×C421C2211yescharacteristic, complemented
37C2×D4×C421C2211yescharacteristic, complemented
38SD16×C421C2212yescomplemented
39D8×C421C2212yescomplemented
40M16×C421C2211yescharacteristic, complemented
41C22×D421—21no—
42C22×D421—22no—
43C22×D421—22no—
44Dic842D411yescharacteristic
45Dic842C2311yescharacteristic
46D1684—22no—
47D1684—21no—
48D1684D411yescharacteristic
49D1684C2312yes—
50D1684C2311yescharacteristic
51C22⋊D421—21no—
52C22⋊D421—22no—
53C22⋊D421—22no—
54C22⋊D421—22no—
55C22⋊D421D412yes—
56C2×Dic421—22no—
57C4×D421—21no—
58C4×D421—22no—
59C4×D421C2312yes—
60#16809890C2311yescharacteristic
61#16809890C2311yescharacteristic
62C23×C421—21no—
63Q8×C421C2311yescharacteristic
64D4×C421—21no—
65D4×C421—21no—
66D4×C421C2311yescharacteristic
67D4×C421C2312yes—
68C2×C4×C421—21no—
69C2×C4×C421C2311yescharacteristic
70C8×C421C2312yes—
71C22×C421—21no—
72C22×C421—22no—
73C22×C421—81no—
74C22×C421C2×D411yescharacteristic
75C2×D421—21no—
76C2×D421—22no—
77C2×D421—82no—
78C2×D421—82no—
79C2×D421—81no—
80C2×D421C2×D412yes—
81C4×C421—21no—
82C4×C421C2×D411yescharacteristic
83C4×C421C2411yescharacteristic
84Dic421—21no—
85Dic421C2×D412yes—
86C2×C421—21no—
87C2×C421—41no—
88C2×C421—42no—
89C2×C421C22×D411yescharacteristic
90D421—22no—
91D421—41no—
92D421—42no—
93C421#57611yescharacteristic, complemented, Sylow
94#576—4211nomaximal, Sylow
95H(5,𝔽2)—4211no—
96H(5,𝔽2)—4211no—
97AGL(1,ℤ/8)—4212no—
98AGL(1,ℤ/8)—4212no—
99AGL(1,ℤ/8)—4211no—
100AGL(1,ℤ/8)D42111yescharacteristic, complemented
101#135—4211no—
102#135—4212no—
103C2×D8—4211no—
104C2×D8—4212no—
105#131—4211no—
106#41—8421no—
107#41—8421no—
108#41—4212no—
109#41—4212no—
110#41—4211no—
111#41—4211no—
112#41C2×D42111yescharacteristic, complemented
113C2×D4—8421no—
114C2×D4—8422no—
115C2×D4—8421no—
116C2×D4—8422no—
117C2×D4—4212no—
118C2×D4—4212no—
119C2×D4—4211no—
120C2×D4C2×D42111yescharacteristic, complemented
121Q16—4211no—
122SD16—4211no—
123SD16—4211no—
124SD16—4212no—
125SD16C2×D42112yescomplemented
126D8—4212no—
127D8—4211no—
128D8—4212no—
129D8—4212no—
130D8C2×D42112yescomplemented
131M16—4212no—
132M16C2×D42111yescharacteristic, complemented
133C2×C8—4211no—
134C2×C8—4212no—
135C23—21no—
136C23—8422no—
137C23—8422no—
138C23—8421no—
139Q8—4211no—
140Q8—4211no—
141Q8C22×D42111yescharacteristic
142D4—21no—
143D4—21no—
144D4—8422no—
145D4—8422no—
146D4—8421no—
147D4—8422no—
148D4—8422no—
149D4—8421no—
150D4—4212no—
151D4—4211no—
152D4—4212no—
153D4—4211no—
154D4C22×D42112yes—
155D4C22×D42111yescharacteristic
156C2×C4—21no—
157C2×C4—8422no—
158C2×C4—8422no—
159C2×C4—8421no—
160C2×C4—4212no—
161C2×C4C22×D42111yescharacteristic
162C8—4211no—
163C8—4211no—
164C8C22×D42112yes—
165C22—21no—
166C22—22no—
167C22—81no—
168C22—33681no—
169C22—33682no—
170C22—33682no—
171C22—8422no—
172C22—8421no—
173C22—4212no—
174C22D4×D42111yescharacteristic
175C4—21no—
176C4—8421no—
177C4—4212no—
178C4D4×D42111yescharacteristic
179C4C23×D42111yescharacteristic
180C2—21no—
181C2—42no—
182C2—41no—
183C2—8422no—
184C2—16842no—
185C2—16841no—
186C2C2×D4×D42111yescharacteristic
subgroupG/H
centerC2C2×D4×D421
commutator, cosocleC4×C421C24
FrattiniC4C23×D421
FittingAGL(1,ℤ/8)×C421C2
socleC2×C421C22×D4
hypercenterAGL(1,ℤ/8)D421
hypocenterC421#576
LCSG ▷ C4×C421 ▷ C2×C421 ▷ C421
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #27212412C2semidirect
2. #27212411C2semidirect
3. #27212394C2semidirect
4. #27212393C2semidirect
5. #27212390C2semidirect
6. #27212387C2semidirect
7. #27212386C2semidirect
8. C2×#20269756C2semidirect
9. D8×D421C2semidirect
10. #27212374C2semidirect
11. AGL(1,ℤ/8)×C421C2semidirect
12. #20269772C22semidirect
15. #20269771C22semidirect
17. #20269770C22semidirect
19. C2×D1684C22semidirect
23. D4×D421C22semidirect
24. D4×D421C22semidirect
25. #20269761C22semidirect
26. #20269760C22semidirect
27. #20269759C22semidirect
28. #20269758C22semidirect
29. D3368C22semidirect
30. #20269756C22semidirect
31. #20269756C22semidirect
32. #20269756C22semidirect
33. #20269754C22semidirect
34. C2×#16809890C22semidirect
35. C8×D421C22semidirect
36. #41×C421C22semidirect
37. C2×D4×C421C22semidirect
38. SD16×C421C22semidirect
39. D8×C421C22semidirect
40. M16×C421C22semidirect
93. C421#576semidirect
100. AGL(1,ℤ/8)D421semidirect
112. #41C2×D421semidirect
120. C2×D4C2×D421semidirect
125. SD16C2×D421semidirect
130. D8C2×D421semidirect
132. M16C2×D421semidirect

Maximal subgroups

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

subgroupindex
#272124122
#272124112
#272123942
#272123932
#272123902
#272123872
#272123862
C2×#202697562
D8×D4212
#272123742
AGL(1,ℤ/8)×C4212
#576421