Finite Group Databasebeta

← #37527239

Subgroups — #37527239

Order 22976.

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×#19153340C211yesmaximal, characteristic, complemented
2#41×D359C211yesmaximal, characteristic, complemented
3#26040048C211yesmaximal, characteristic, complemented
4#26040045C211yesmaximal, characteristic, complemented
5#26040044C212yesmaximal, complemented
6#26040041C211yesmaximal, characteristic, complemented
7#26040039C212yesmaximal, complemented
8#26040036C212yesmaximal, complemented
9#26040034C212yesmaximal, complemented
10M16×D359C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C359C211yesmaximal, characteristic, complemented
12#19153342C2211yescharacteristic, complemented
13#19153341—21no—
14#19153341C2211yescharacteristic
15#19153340—21no—
16#19153340—21no—
17#19153340—21no—
18#19153340C2212yes—
19#19153340C2211yescharacteristic, complemented
20#19153340C2212yescomplemented
21C2×Dic718C2211yescharacteristic, complemented
22Q8×D359C2211yescharacteristic, complemented
23C2×C22⋊D359—21no—
24D4×D359—21no—
25D4×D359C2211yescharacteristic, complemented
26C22×Dic359—21no—
27C2×C4×D359—21no—
28C2×C4×D359C2211yescharacteristic
29Dic1436C2212yescomplemented
30#19153331C2212yescomplemented
31#19153330C2212yescomplemented
32#19153329C2212yescomplemented
33#19153329C2212yescomplemented
34#19153326C2212yescomplemented
35#19153325C2211yescharacteristic, complemented
36#19153324C2212yescomplemented
37C8×D359C2212yescomplemented
38#41×C359C2211yescharacteristic
39C2×D4×C359C2211yescharacteristic, complemented
40SD16×C359C2212yescomplemented
41D8×C359C2212yescomplemented
42M16×C359C2211yescharacteristic, complemented
43C22×D359—21no—
44C22×D359D411yescharacteristic
45Dic718—21no—
46Dic718—21no—
47Dic718C2311yescharacteristic
48Dic718C2312yes—
49D1436—21no—
50D1436C2311yescharacteristic
51C22⋊D359—21no—
52C22⋊D359—21no—
53C22⋊D359—22no—
54C22⋊D359—22no—
55C22⋊D359—21no—
56C2×Dic359—21no—
57C2×Dic359—21no—
58C2×Dic359—22no—
59C2×Dic359—22no—
60C2×Dic359D411yescharacteristic
61C4×D359—21no—
62C4×D359—21no—
63C4×D359—21no—
64C4×D359D411yescharacteristic
65C4×D359D411yescharacteristic
66C4×D359C2311yescharacteristic
67C4×D359C2311yescharacteristic
68C359⋊C8C2312yes—
69C23×C359—21no—
70Q8×C359C2311yescharacteristic
71D4×C359—21no—
72D4×C359—21no—
73D4×C359C2311yescharacteristic
74D4×C359C2312yes—
75C2×C4×C359—21no—
76C2×C4×C359C2311yescharacteristic
77C8×C359C2312yes—
78C22×C359—21no—
79C22×C359—22no—
80C22×C359—42no—
81C22×C359C2×D411yescharacteristic
82C2×D359—21no—
83C2×D359—41no—
84C2×D359—42no—
85C2×D359C2×D411yescharacteristic
86C2×D359C2×D411yescharacteristic
87C4×C359—21no—
88C4×C359C2×D411yescharacteristic
89C4×C359C2411yescharacteristic
90Dic359—21no—
91Dic359—22no—
92Dic359C2×D411yescharacteristic
93Dic359C2×D411yescharacteristic
94C2×C359—21no—
95C2×C359—41no—
96C2×C359—42no—
97C2×C359C22×D411yescharacteristic
98D359—21no—
99D359—21no—
100D359—41no—
101C359#57511yescharacteristic, complemented, Sylow
102#575—3591nomaximal, Sylow
103C2×#41—3591no—
104C2×#41—3591no—
105#137—3591no—
106#137—3592no—
107#137—3591no—
108AGL(1,ℤ/8)—3591no—
109AGL(1,ℤ/8)—3592no—
110AGL(1,ℤ/8)D35911yescharacteristic, complemented
111#135—3592no—
112#135—3592no—
113C2×M16—3591no—
114#41—3591no—
115#41—3592no—
116#41—3591no—
117#41—3592no—
118#41—3591no—
119#41—7181no—
120#41—7181no—
121#41—7181no—
122#41—7181no—
123#41C2×D35911yescharacteristic
124C2×Q8—3591no—
125C2×Q8—3591no—
126C2×D4—3591no—
127C2×D4—7181no—
128C2×D4—7181no—
129C2×D4C2×D35911yescharacteristic, complemented
130C22×C4—3591no—
131C22×C4—7181no—
132C22×C4—7181no—
133Q16—3592no—
134Q16—3592no—
135SD16—3592no—
136SD16—3592no—
137SD16—3592no—
138SD16C2×D35912yescomplemented
139D8—3592no—
140D8C2×D35912yescomplemented
141M16—3591no—
142M16—3592no—
143M16C2×D35911yescharacteristic, complemented
144C2×C8—3592no—
145C23—21no—
146C23—3591no—
147C23—7181no—
148Q8—3591no—
149Q8—3592no—
150Q8—7181no—
151Q8—7181no—
152Q8C22×D35911yescharacteristic
153D4—21no—
154D4—21no—
155D4—3591no—
156D4—7181no—
157D4—7181no—
158D4—7181no—
159D4—7182no—
160D4—7182no—
161D4—7181no—
162D4C22×D35912yes—
163D4C22×D35911yescharacteristic
164C2×C4—21no—
165C2×C4—3591no—
166C2×C4—3591no—
167C2×C4—3591no—
168C2×C4—3591no—
169C2×C4—3591no—
170C2×C4—7181no—
171C2×C4—7181no—
172C2×C4—7181no—
173C2×C4—7182no—
174C2×C4—7182no—
175C2×C4—7181no—
176C2×C4—7181no—
177C2×C4C22×D35911yescharacteristic
178C8—3592no—
179C8C22×D35912yes—
180C22—21no—
181C22—22no—
182C22—42no—
183C22—3591no—
184C22—3591no—
185C22—14361no—
186C22—14362no—
187C22—7181no—
188C22D4×D35911yescharacteristic
189C4—21no—
190C4—3591no—
191C4—3591no—
192C4—7181no—
193C4—7182no—
194C4D4×D35911yescharacteristic
195C4C23×D35911yescharacteristic
196C2—21no—
197C2—42no—
198C2—41no—
199C2—14361no—
200C2—7181no—
201C2—7181no—
202C2C2×D4×D35911yescharacteristic
subgroupG/H
centerC2C2×D4×D359
commutator, cosocleC4×C359C24
FrattiniC4C23×D359
FittingAGL(1,ℤ/8)×C359C2
socleC2×C359C22×D4
hypercenterAGL(1,ℤ/8)D359
hypocenterC359#575
LCSG ▷ C4×C359 ▷ C2×C359 ▷ C359
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#19153340C2semidirect
2. #41×D359C2semidirect
3. #26040048C2semidirect
4. #26040045C2semidirect
5. #26040044C2semidirect
6. #26040041C2semidirect
7. #26040039C2semidirect
8. #26040036C2semidirect
9. #26040034C2semidirect
10. M16×D359C2semidirect
11. AGL(1,ℤ/8)×C359C2semidirect
12. #19153342C22semidirect
19. #19153340C22semidirect
20. #19153340C22semidirect
21. C2×Dic718C22semidirect
22. Q8×D359C22semidirect
25. D4×D359C22semidirect
29. Dic1436C22semidirect
30. #19153331C22semidirect
31. #19153330C22semidirect
32. #19153329C22semidirect
33. #19153329C22semidirect
34. #19153326C22semidirect
35. #19153325C22semidirect
36. #19153324C22semidirect
37. C8×D359C22semidirect
39. C2×D4×C359C22semidirect
40. SD16×C359C22semidirect
41. D8×C359C22semidirect
42. M16×C359C22semidirect
101. C359#575semidirect
110. AGL(1,ℤ/8)D359semidirect
129. C2×D4C2×D359semidirect
138. SD16C2×D359semidirect
140. D8C2×D359semidirect
143. M16C2×D359semidirect

Maximal subgroups

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

subgroupindex
C2×#191533402
#41×D3592
#260400482
#260400452
#260400442
#260400412
#260400392
#260400362
#260400342
M16×D3592
AGL(1,ℤ/8)×C3592
#575359