Finite Group Databasebeta

← #38786910

Subgroups — #38786910

Order 25664.

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×#20264365C211yesmaximal, characteristic, complemented
2#41×D401C211yesmaximal, characteristic, complemented
3#27165770C211yesmaximal, characteristic, complemented
4#27165767C211yesmaximal, characteristic, complemented
5#27165766C212yesmaximal, complemented
6#27165763C211yesmaximal, characteristic, complemented
7#27165761C212yesmaximal, complemented
8#27165758C212yesmaximal, complemented
9#27165756C212yesmaximal, complemented
10M16×D401C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C401C211yesmaximal, characteristic, complemented
12#20264367C2211yescharacteristic, complemented
13#20264366—21no—
14#20264366C2211yescharacteristic
15#20264365—21no—
16#20264365—21no—
17#20264365—21no—
18#20264365C2212yes—
19#20264365C2211yescharacteristic, complemented
20#20264365C2212yescomplemented
21C2×Dic802C2211yescharacteristic, complemented
22Q8×D401C2211yescharacteristic, complemented
23C2×C22⋊D401—21no—
24D4×D401—21no—
25D4×D401C2211yescharacteristic, complemented
26C22×Dic401—21no—
27C2×C4×D401—21no—
28C2×C4×D401C2211yescharacteristic
29Dic1604C2212yescomplemented
30#20264356C2212yescomplemented
31#20264355C2212yescomplemented
32#20264354C2212yescomplemented
33#20264354C2212yescomplemented
34#20264351C2212yescomplemented
35#20264350C2211yescharacteristic, complemented
36#20264349C2212yescomplemented
37C8×D401C2212yescomplemented
38#41×C401C2211yescharacteristic
39C2×D4×C401C2211yescharacteristic, complemented
40SD16×C401C2212yescomplemented
41D8×C401C2212yescomplemented
42M16×C401C2211yescharacteristic, complemented
43C22×D401—21no—
44C22×D401D411yescharacteristic
45Dic802—21no—
46Dic802—21no—
47Dic802C2311yescharacteristic
48Dic802C2312yes—
49D1604—21no—
50D1604C2311yescharacteristic
51C22⋊D401—21no—
52C22⋊D401—21no—
53C22⋊D401—22no—
54C22⋊D401—22no—
55C22⋊D401—21no—
56C2×Dic401—21no—
57C2×Dic401—21no—
58C2×Dic401—22no—
59C2×Dic401—22no—
60C2×Dic401D411yescharacteristic
61C4×D401—21no—
62C4×D401—21no—
63C4×D401—21no—
64C4×D401D411yescharacteristic
65C4×D401D411yescharacteristic
66C4×D401C2311yescharacteristic
67C4×D401C2311yescharacteristic
68#15691999C2312yes—
69C23×C401—21no—
70Q8×C401C2311yescharacteristic
71D4×C401—21no—
72D4×C401—21no—
73D4×C401C2311yescharacteristic
74D4×C401C2312yes—
75C2×C4×C401—21no—
76C2×C4×C401C2311yescharacteristic
77C8×C401C2312yes—
78C22×C401—21no—
79C22×C401—22no—
80C22×C401—42no—
81C22×C401C2×D411yescharacteristic
82C2×D401—21no—
83C2×D401—41no—
84C2×D401—42no—
85C2×D401C2×D411yescharacteristic
86C2×D401C2×D411yescharacteristic
87C4×C401—21no—
88C4×C401C2×D411yescharacteristic
89C4×C401C2411yescharacteristic
90Dic401—21no—
91Dic401—22no—
92Dic401C2×D411yescharacteristic
93Dic401C2×D411yescharacteristic
94C2×C401—21no—
95C2×C401—41no—
96C2×C401—42no—
97C2×C401C22×D411yescharacteristic
98D401—21no—
99D401—21no—
100D401—41no—
101C401#57511yescharacteristic, complemented, Sylow
102#575—4011nomaximal, Sylow
103C2×#41—4011no—
104C2×#41—4011no—
105#137—4011no—
106#137—4012no—
107#137—4011no—
108AGL(1,ℤ/8)—4011no—
109AGL(1,ℤ/8)—4012no—
110AGL(1,ℤ/8)D40111yescharacteristic, complemented
111#135—4012no—
112#135—4012no—
113C2×M16—4011no—
114#41—8021no—
115#41—8021no—
116#41—8021no—
117#41—8021no—
118#41—4011no—
119#41—4012no—
120#41—4011no—
121#41—4012no—
122#41—4011no—
123#41C2×D40111yescharacteristic
124C2×Q8—4011no—
125C2×Q8—4011no—
126C2×D4—8021no—
127C2×D4—8021no—
128C2×D4—4011no—
129C2×D4C2×D40111yescharacteristic, complemented
130C22×C4—8021no—
131C22×C4—8021no—
132C22×C4—4011no—
133Q16—4012no—
134Q16—4012no—
135SD16—4012no—
136SD16—4012no—
137SD16—4012no—
138SD16C2×D40112yescomplemented
139D8—4012no—
140D8C2×D40112yescomplemented
141M16—4011no—
142M16—4012no—
143M16C2×D40111yescharacteristic, complemented
144C2×C8—4012no—
145C23—21no—
146C23—8021no—
147C23—4011no—
148Q8—8021no—
149Q8—8021no—
150Q8—4011no—
151Q8—4012no—
152Q8C22×D40111yescharacteristic
153D4—21no—
154D4—21no—
155D4—8021no—
156D4—8021no—
157D4—8022no—
158D4—8022no—
159D4—8021no—
160D4—8021no—
161D4—4011no—
162D4C22×D40112yes—
163D4C22×D40111yescharacteristic
164C2×C4—21no—
165C2×C4—8021no—
166C2×C4—8021no—
167C2×C4—8021no—
168C2×C4—8021no—
169C2×C4—8022no—
170C2×C4—8022no—
171C2×C4—8021no—
172C2×C4—4011no—
173C2×C4—4011no—
174C2×C4—4011no—
175C2×C4—4011no—
176C2×C4—4011no—
177C2×C4C22×D40111yescharacteristic
178C8—4012no—
179C8C22×D40112yes—
180C22—21no—
181C22—22no—
182C22—42no—
183C22—8021no—
184C22—16041no—
185C22—16042no—
186C22—4011no—
187C22—4011no—
188C22D4×D40111yescharacteristic
189C4—21no—
190C4—8021no—
191C4—8022no—
192C4—4011no—
193C4—4011no—
194C4D4×D40111yescharacteristic
195C4C23×D40111yescharacteristic
196C2—21no—
197C2—42no—
198C2—41no—
199C2—8021no—
200C2—8021no—
201C2—16041no—
202C2C2×D4×D40111yescharacteristic
subgroupG/H
centerC2C2×D4×D401
commutator, cosocleC4×C401C24
FrattiniC4C23×D401
FittingAGL(1,ℤ/8)×C401C2
socleC2×C401C22×D4
hypercenterAGL(1,ℤ/8)D401
hypocenterC401#575
LCSG ▷ C4×C401 ▷ C2×C401 ▷ C401
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#20264365C2semidirect
2. #41×D401C2semidirect
3. #27165770C2semidirect
4. #27165767C2semidirect
5. #27165766C2semidirect
6. #27165763C2semidirect
7. #27165761C2semidirect
8. #27165758C2semidirect
9. #27165756C2semidirect
10. M16×D401C2semidirect
11. AGL(1,ℤ/8)×C401C2semidirect
12. #20264367C22semidirect
19. #20264365C22semidirect
20. #20264365C22semidirect
21. C2×Dic802C22semidirect
22. Q8×D401C22semidirect
25. D4×D401C22semidirect
29. Dic1604C22semidirect
30. #20264356C22semidirect
31. #20264355C22semidirect
32. #20264354C22semidirect
33. #20264354C22semidirect
34. #20264351C22semidirect
35. #20264350C22semidirect
36. #20264349C22semidirect
37. C8×D401C22semidirect
39. C2×D4×C401C22semidirect
40. SD16×C401C22semidirect
41. D8×C401C22semidirect
42. M16×C401C22semidirect
101. C401#575semidirect
110. AGL(1,ℤ/8)D401semidirect
129. C2×D4C2×D401semidirect
138. SD16C2×D401semidirect
140. D8C2×D401semidirect
143. M16C2×D401semidirect

Maximal subgroups

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

subgroupindex
C2×#202643652
#41×D4012
#271657702
#271657672
#271657662
#271657632
#271657612
#271657582
#271657562
M16×D4012
AGL(1,ℤ/8)×C4012
#575401