Finite Group Databasebeta

← #37554982

Subgroups — #37554982

Order 23488.

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×#19154576C211yesmaximal, characteristic, complemented
2#41×D367C211yesmaximal, characteristic, complemented
3#26044294C211yesmaximal, characteristic, complemented
4#26044291C211yesmaximal, characteristic, complemented
5#26044290C212yesmaximal, complemented
6#26044287C211yesmaximal, characteristic, complemented
7#26044285C212yesmaximal, complemented
8#26044282C212yesmaximal, complemented
9#26044280C212yesmaximal, complemented
10M16×D367C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C367C211yesmaximal, characteristic, complemented
12#19154578C2211yescharacteristic, complemented
13#19154577—21no—
14#19154577C2211yescharacteristic
15#19154576—21no—
16#19154576—21no—
17#19154576—21no—
18#19154576C2212yes—
19#19154576C2211yescharacteristic, complemented
20#19154576C2212yescomplemented
21C2×Dic734C2211yescharacteristic, complemented
22Q8×D367C2211yescharacteristic, complemented
23C2×C22⋊D367—21no—
24D4×D367—21no—
25D4×D367C2211yescharacteristic, complemented
26C22×Dic367—21no—
27C2×C4×D367—21no—
28C2×C4×D367C2211yescharacteristic
29Dic1468C2212yescomplemented
30#19154567C2212yescomplemented
31#19154566C2212yescomplemented
32#19154565C2212yescomplemented
33#19154565C2212yescomplemented
34#19154562C2212yescomplemented
35#19154561C2211yescharacteristic, complemented
36#19154560C2212yescomplemented
37C8×D367C2212yescomplemented
38#41×C367C2211yescharacteristic
39C2×D4×C367C2211yescharacteristic, complemented
40SD16×C367C2212yescomplemented
41D8×C367C2212yescomplemented
42M16×C367C2211yescharacteristic, complemented
43C22×D367—21no—
44C22×D367D411yescharacteristic
45Dic734—21no—
46Dic734—21no—
47Dic734C2311yescharacteristic
48Dic734C2312yes—
49D1468—21no—
50D1468C2311yescharacteristic
51C22⋊D367—21no—
52C22⋊D367—21no—
53C22⋊D367—22no—
54C22⋊D367—22no—
55C22⋊D367—21no—
56C2×Dic367—21no—
57C2×Dic367—21no—
58C2×Dic367—22no—
59C2×Dic367—22no—
60C2×Dic367D411yescharacteristic
61C4×D367—21no—
62C4×D367—21no—
63C4×D367—21no—
64C4×D367D411yescharacteristic
65C4×D367D411yescharacteristic
66C4×D367C2311yescharacteristic
67C4×D367C2311yescharacteristic
68C367⋊C8C2312yes—
69C23×C367—21no—
70Q8×C367C2311yescharacteristic
71D4×C367—21no—
72D4×C367—21no—
73D4×C367C2311yescharacteristic
74D4×C367C2312yes—
75C2×C4×C367—21no—
76C2×C4×C367C2311yescharacteristic
77C8×C367C2312yes—
78C22×C367—21no—
79C22×C367—22no—
80C22×C367—42no—
81C22×C367C2×D411yescharacteristic
82C2×D367—21no—
83C2×D367—41no—
84C2×D367—42no—
85C2×D367C2×D411yescharacteristic
86C2×D367C2×D411yescharacteristic
87C4×C367—21no—
88C4×C367C2×D411yescharacteristic
89C4×C367C2411yescharacteristic
90Dic367—21no—
91Dic367—22no—
92Dic367C2×D411yescharacteristic
93Dic367C2×D411yescharacteristic
94C2×C367—21no—
95C2×C367—41no—
96C2×C367—42no—
97C2×C367C22×D411yescharacteristic
98D367—21no—
99D367—21no—
100D367—41no—
101C367#57511yescharacteristic, complemented, Sylow
102#575—3671nomaximal, Sylow
103C2×#41—3671no—
104C2×#41—3671no—
105#137—3671no—
106#137—3672no—
107#137—3671no—
108AGL(1,ℤ/8)—3671no—
109AGL(1,ℤ/8)—3672no—
110AGL(1,ℤ/8)D36711yescharacteristic, complemented
111#135—3672no—
112#135—3672no—
113C2×M16—3671no—
114#41—3671no—
115#41—3672no—
116#41—3671no—
117#41—3672no—
118#41—3671no—
119#41—7341no—
120#41—7341no—
121#41—7341no—
122#41—7341no—
123#41C2×D36711yescharacteristic
124C2×Q8—3671no—
125C2×Q8—3671no—
126C2×D4—3671no—
127C2×D4—7341no—
128C2×D4—7341no—
129C2×D4C2×D36711yescharacteristic, complemented
130C22×C4—3671no—
131C22×C4—7341no—
132C22×C4—7341no—
133Q16—3672no—
134Q16—3672no—
135SD16—3672no—
136SD16—3672no—
137SD16—3672no—
138SD16C2×D36712yescomplemented
139D8—3672no—
140D8C2×D36712yescomplemented
141M16—3671no—
142M16—3672no—
143M16C2×D36711yescharacteristic, complemented
144C2×C8—3672no—
145C23—21no—
146C23—3671no—
147C23—7341no—
148Q8—3671no—
149Q8—3672no—
150Q8—7341no—
151Q8—7341no—
152Q8C22×D36711yescharacteristic
153D4—21no—
154D4—21no—
155D4—3671no—
156D4—7341no—
157D4—7341no—
158D4—7341no—
159D4—7342no—
160D4—7342no—
161D4—7341no—
162D4C22×D36712yes—
163D4C22×D36711yescharacteristic
164C2×C4—21no—
165C2×C4—3671no—
166C2×C4—3671no—
167C2×C4—3671no—
168C2×C4—3671no—
169C2×C4—3671no—
170C2×C4—7341no—
171C2×C4—7341no—
172C2×C4—7341no—
173C2×C4—7342no—
174C2×C4—7342no—
175C2×C4—7341no—
176C2×C4—7341no—
177C2×C4C22×D36711yescharacteristic
178C8—3672no—
179C8C22×D36712yes—
180C22—21no—
181C22—22no—
182C22—42no—
183C22—3671no—
184C22—3671no—
185C22—14681no—
186C22—14682no—
187C22—7341no—
188C22D4×D36711yescharacteristic
189C4—21no—
190C4—3671no—
191C4—3671no—
192C4—7341no—
193C4—7342no—
194C4D4×D36711yescharacteristic
195C4C23×D36711yescharacteristic
196C2—21no—
197C2—42no—
198C2—41no—
199C2—14681no—
200C2—7341no—
201C2—7341no—
202C2C2×D4×D36711yescharacteristic
subgroupG/H
centerC2C2×D4×D367
commutator, cosocleC4×C367C24
FrattiniC4C23×D367
FittingAGL(1,ℤ/8)×C367C2
socleC2×C367C22×D4
hypercenterAGL(1,ℤ/8)D367
hypocenterC367#575
LCSG ▷ C4×C367 ▷ C2×C367 ▷ C367
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#19154576C2semidirect
2. #41×D367C2semidirect
3. #26044294C2semidirect
4. #26044291C2semidirect
5. #26044290C2semidirect
6. #26044287C2semidirect
7. #26044285C2semidirect
8. #26044282C2semidirect
9. #26044280C2semidirect
10. M16×D367C2semidirect
11. AGL(1,ℤ/8)×C367C2semidirect
12. #19154578C22semidirect
19. #19154576C22semidirect
20. #19154576C22semidirect
21. C2×Dic734C22semidirect
22. Q8×D367C22semidirect
25. D4×D367C22semidirect
29. Dic1468C22semidirect
30. #19154567C22semidirect
31. #19154566C22semidirect
32. #19154565C22semidirect
33. #19154565C22semidirect
34. #19154562C22semidirect
35. #19154561C22semidirect
36. #19154560C22semidirect
37. C8×D367C22semidirect
39. C2×D4×C367C22semidirect
40. SD16×C367C22semidirect
41. D8×C367C22semidirect
42. M16×C367C22semidirect
101. C367#575semidirect
110. AGL(1,ℤ/8)D367semidirect
129. C2×D4C2×D367semidirect
138. SD16C2×D367semidirect
140. D8C2×D367semidirect
143. M16C2×D367semidirect

Maximal subgroups

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

subgroupindex
C2×#191545762
#41×D3672
#260442942
#260442912
#260442902
#260442872
#260442852
#260442822
#260442802
M16×D3672
AGL(1,ℤ/8)×C3672
#575367