Finite Group Databasebeta

← #27213942

Subgroups — #27213942

Order 13504.

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

normal subgroups only

This group has 262 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
1C2×#16809961C211yesmaximal, characteristic, complemented
2#41×D211C211yesmaximal, characteristic, complemented
3#20270010C211yesmaximal, characteristic, complemented
4#20270007C211yesmaximal, characteristic, complemented
5#20270006C212yesmaximal, complemented
6#20270003C211yesmaximal, characteristic, complemented
7#20270001C212yesmaximal, complemented
8#20269998C212yesmaximal, complemented
9#20269996C212yesmaximal, complemented
10M16×D211C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C211C211yesmaximal, characteristic, complemented
12#16809963C2211yescharacteristic, complemented
13#16809962—21no—
14#16809962C2211yescharacteristic
15#16809961—21no—
16#16809961—21no—
17#16809961—21no—
18#16809961C2212yes—
19#16809961C2211yescharacteristic, complemented
20#16809961C2212yescomplemented
21C2×Dic422C2211yescharacteristic, complemented
22Q8×D211C2211yescharacteristic, complemented
23C2×C22⋊D211—21no—
24D4×D211—21no—
25D4×D211C2211yescharacteristic, complemented
26C22×Dic211—21no—
27C2×C4×D211—21no—
28C2×C4×D211C2211yescharacteristic
29Dic844C2212yescomplemented
30#16809952C2212yescomplemented
31#16809951C2212yescomplemented
32#16809950C2212yescomplemented
33#16809950C2212yescomplemented
34#16809947C2212yescomplemented
35#16809946C2211yescharacteristic, complemented
36#16809945C2212yescomplemented
37C8×D211C2212yescomplemented
38#41×C211C2211yescharacteristic
39C2×D4×C211C2211yescharacteristic, complemented
40SD16×C211C2212yescomplemented
41D8×C211C2212yescomplemented
42M16×C211C2211yescharacteristic, complemented
43C22×D211—21no—
44C22×D211D411yescharacteristic
45Dic422—21no—
46Dic422—21no—
47Dic422C2311yescharacteristic
48Dic422C2312yes—
49D844—21no—
50D844C2311yescharacteristic
51C22⋊D211—21no—
52C22⋊D211—21no—
53C22⋊D211—22no—
54C22⋊D211—22no—
55C22⋊D211—21no—
56C2×Dic211—21no—
57C2×Dic211—21no—
58C2×Dic211—22no—
59C2×Dic211—22no—
60C2×Dic211D411yescharacteristic
61C4×D211—21no—
62C4×D211—21no—
63C4×D211—21no—
64C4×D211D411yescharacteristic
65C4×D211D411yescharacteristic
66C4×D211C2311yescharacteristic
67C4×D211C2311yescharacteristic
68C211⋊C8C2312yes—
69C23×C211—21no—
70Q8×C211C2311yescharacteristic
71D4×C211—21no—
72D4×C211—21no—
73D4×C211C2311yescharacteristic
74D4×C211C2312yes—
75C2×C4×C211—21no—
76C2×C4×C211C2311yescharacteristic
77C8×C211C2312yes—
78C22×C211—21no—
79C22×C211—22no—
80C22×C211—42no—
81C22×C211C2×D411yescharacteristic
82C2×D211—21no—
83C2×D211—41no—
84C2×D211—42no—
85C2×D211C2×D411yescharacteristic
86C2×D211C2×D411yescharacteristic
87C4×C211—21no—
88C4×C211C2×D411yescharacteristic
89C4×C211C2411yescharacteristic
90Dic211—21no—
91Dic211—22no—
92Dic211C2×D411yescharacteristic
93Dic211C2×D411yescharacteristic
94C2×C211—21no—
95C2×C211—41no—
96C2×C211—42no—
97C2×C211C22×D411yescharacteristic
98D211—21no—
99D211—21no—
100D211—41no—
101C211#57511yescharacteristic, complemented, Sylow
102#575—2111nomaximal, Sylow
103C2×#41—2111no—
104C2×#41—2111no—
105#137—2111no—
106#137—2112no—
107#137—2111no—
108AGL(1,ℤ/8)—2111no—
109AGL(1,ℤ/8)—2112no—
110AGL(1,ℤ/8)D21111yescharacteristic, complemented
111#135—2112no—
112#135—2112no—
113C2×M16—2111no—
114#41—4221no—
115#41—4221no—
116#41—4221no—
117#41—4221no—
118#41—2111no—
119#41—2112no—
120#41—2111no—
121#41—2112no—
122#41—2111no—
123#41C2×D21111yescharacteristic
124C2×Q8—2111no—
125C2×Q8—2111no—
126C2×D4—4221no—
127C2×D4—4221no—
128C2×D4—2111no—
129C2×D4C2×D21111yescharacteristic, complemented
130C22×C4—4221no—
131C22×C4—4221no—
132C22×C4—2111no—
133Q16—2112no—
134Q16—2112no—
135SD16—2112no—
136SD16—2112no—
137SD16—2112no—
138SD16C2×D21112yescomplemented
139D8—2112no—
140D8C2×D21112yescomplemented
141M16—2111no—
142M16—2112no—
143M16C2×D21111yescharacteristic, complemented
144C2×C8—2112no—
145C23—21no—
146C23—4221no—
147C23—2111no—
148Q8—4221no—
149Q8—4221no—
150Q8—2111no—
151Q8—2112no—
152Q8C22×D21111yescharacteristic
153D4—21no—
154D4—21no—
155D4—4221no—
156D4—4221no—
157D4—4222no—
158D4—4222no—
159D4—4221no—
160D4—4221no—
161D4—2111no—
162D4C22×D21112yes—
163D4C22×D21111yescharacteristic
164C2×C4—21no—
165C2×C4—4221no—
166C2×C4—4221no—
167C2×C4—4221no—
168C2×C4—4221no—
169C2×C4—4222no—
170C2×C4—4222no—
171C2×C4—4221no—
172C2×C4—2111no—
173C2×C4—2111no—
174C2×C4—2111no—
175C2×C4—2111no—
176C2×C4—2111no—
177C2×C4C22×D21111yescharacteristic
178C8—2112no—
179C8C22×D21112yes—
180C22—21no—
181C22—22no—
182C22—42no—
183C22—8441no—
184C22—8442no—
185C22—4221no—
186C22—2111no—
187C22—2111no—
188C22D4×D21111yescharacteristic
189C4—21no—
190C4—4221no—
191C4—4222no—
192C4—2111no—
193C4—2111no—
194C4D4×D21111yescharacteristic
195C4C23×D21111yescharacteristic
196C2—21no—
197C2—42no—
198C2—41no—
199C2—8441no—
200C2—4221no—
201C2—4221no—
202C2C2×D4×D21111yescharacteristic
subgroupG/H
centerC2C2×D4×D211
commutator, cosocleC4×C211C24
FrattiniC4C23×D211
FittingAGL(1,ℤ/8)×C211C2
socleC2×C211C22×D4
hypercenterAGL(1,ℤ/8)D211
hypocenterC211#575
LCSG ▷ C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#16809961C2semidirect
2. #41×D211C2semidirect
3. #20270010C2semidirect
4. #20270007C2semidirect
5. #20270006C2semidirect
6. #20270003C2semidirect
7. #20270001C2semidirect
8. #20269998C2semidirect
9. #20269996C2semidirect
10. M16×D211C2semidirect
11. AGL(1,ℤ/8)×C211C2semidirect
12. #16809963C22semidirect
19. #16809961C22semidirect
20. #16809961C22semidirect
21. C2×Dic422C22semidirect
22. Q8×D211C22semidirect
25. D4×D211C22semidirect
29. Dic844C22semidirect
30. #16809952C22semidirect
31. #16809951C22semidirect
32. #16809950C22semidirect
33. #16809950C22semidirect
34. #16809947C22semidirect
35. #16809946C22semidirect
36. #16809945C22semidirect
37. C8×D211C22semidirect
39. C2×D4×C211C22semidirect
40. SD16×C211C22semidirect
41. D8×C211C22semidirect
42. M16×C211C22semidirect
101. C211#575semidirect
110. AGL(1,ℤ/8)D211semidirect
129. C2×D4C2×D211semidirect
138. SD16C2×D211semidirect
140. D8C2×D211semidirect
143. M16C2×D211semidirect

Maximal subgroups

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

subgroupindex
C2×#168099612
#41×D2112
#202700102
#202700072
#202700062
#202700032
#202700012
#202699982
#202699962
M16×D2112
AGL(1,ℤ/8)×C2112
#575211