Finite Group Databasebeta

← #28384376

Subgroups — #28384376

Order 14272.

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×#16813169C211yesmaximal, characteristic, complemented
2#41×D223C211yesmaximal, characteristic, complemented
3#20297337C211yesmaximal, characteristic, complemented
4#20297334C211yesmaximal, characteristic, complemented
5#20297333C212yesmaximal, complemented
6#20297330C211yesmaximal, characteristic, complemented
7#20297328C212yesmaximal, complemented
8#20297325C212yesmaximal, complemented
9#20297323C212yesmaximal, complemented
10M16×D223C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C223C211yesmaximal, characteristic, complemented
12#16813171C2211yescharacteristic, complemented
13#16813170—21no—
14#16813170C2211yescharacteristic
15#16813169—21no—
16#16813169—21no—
17#16813169—21no—
18#16813169C2212yes—
19#16813169C2211yescharacteristic, complemented
20#16813169C2212yescomplemented
21C2×Dic446C2211yescharacteristic, complemented
22Q8×D223C2211yescharacteristic, complemented
23C2×C22⋊D223—21no—
24D4×D223—21no—
25D4×D223C2211yescharacteristic, complemented
26C22×Dic223—21no—
27C2×C4×D223—21no—
28C2×C4×D223C2211yescharacteristic
29Dic892C2212yescomplemented
30#16813160C2212yescomplemented
31#16813159C2212yescomplemented
32#16813158C2212yescomplemented
33#16813158C2212yescomplemented
34#16813155C2212yescomplemented
35#16813154C2211yescharacteristic, complemented
36#16813153C2212yescomplemented
37C8×D223C2212yescomplemented
38#41×C223C2211yescharacteristic
39C2×D4×C223C2211yescharacteristic, complemented
40SD16×C223C2212yescomplemented
41D8×C223C2212yescomplemented
42M16×C223C2211yescharacteristic, complemented
43C22×D223—21no—
44C22×D223D411yescharacteristic
45Dic446—21no—
46Dic446—21no—
47Dic446C2311yescharacteristic
48Dic446C2312yes—
49D892—21no—
50D892C2311yescharacteristic
51C22⋊D223—21no—
52C22⋊D223—21no—
53C22⋊D223—22no—
54C22⋊D223—22no—
55C22⋊D223—21no—
56C2×Dic223—21no—
57C2×Dic223—21no—
58C2×Dic223—22no—
59C2×Dic223—22no—
60C2×Dic223D411yescharacteristic
61C4×D223—21no—
62C4×D223—21no—
63C4×D223—21no—
64C4×D223D411yescharacteristic
65C4×D223D411yescharacteristic
66C4×D223C2311yescharacteristic
67C4×D223C2311yescharacteristic
68C223⋊C8C2312yes—
69C23×C223—21no—
70Q8×C223C2311yescharacteristic
71D4×C223—21no—
72D4×C223—21no—
73D4×C223C2311yescharacteristic
74D4×C223C2312yes—
75C2×C4×C223—21no—
76C2×C4×C223C2311yescharacteristic
77C8×C223C2312yes—
78C22×C223—21no—
79C22×C223—22no—
80C22×C223—42no—
81C22×C223C2×D411yescharacteristic
82C2×D223—21no—
83C2×D223—41no—
84C2×D223—42no—
85C2×D223C2×D411yescharacteristic
86C2×D223C2×D411yescharacteristic
87C4×C223—21no—
88C4×C223C2×D411yescharacteristic
89C4×C223C2411yescharacteristic
90Dic223—21no—
91Dic223—22no—
92Dic223C2×D411yescharacteristic
93Dic223C2×D411yescharacteristic
94C2×C223—21no—
95C2×C223—41no—
96C2×C223—42no—
97C2×C223C22×D411yescharacteristic
98D223—21no—
99D223—21no—
100D223—41no—
101C223#57511yescharacteristic, complemented, Sylow
102#575—2231nomaximal, Sylow
103C2×#41—2231no—
104C2×#41—2231no—
105#137—2231no—
106#137—2232no—
107#137—2231no—
108AGL(1,ℤ/8)—2231no—
109AGL(1,ℤ/8)—2232no—
110AGL(1,ℤ/8)D22311yescharacteristic, complemented
111#135—2232no—
112#135—2232no—
113C2×M16—2231no—
114#41—4461no—
115#41—4461no—
116#41—4461no—
117#41—4461no—
118#41—2231no—
119#41—2232no—
120#41—2232no—
121#41—2231no—
122#41—2231no—
123#41C2×D22311yescharacteristic
124C2×Q8—2231no—
125C2×Q8—2231no—
126C2×D4—4461no—
127C2×D4—4461no—
128C2×D4—2231no—
129C2×D4C2×D22311yescharacteristic, complemented
130C22×C4—4461no—
131C22×C4—4461no—
132C22×C4—2231no—
133Q16—2232no—
134Q16—2232no—
135SD16—2232no—
136SD16—2232no—
137SD16—2232no—
138SD16C2×D22312yescomplemented
139D8—2232no—
140D8C2×D22312yescomplemented
141M16—2231no—
142M16—2232no—
143M16C2×D22311yescharacteristic, complemented
144C2×C8—2232no—
145C23—21no—
146C23—4461no—
147C23—2231no—
148Q8—4461no—
149Q8—4461no—
150Q8—2231no—
151Q8—2232no—
152Q8C22×D22311yescharacteristic
153D4—21no—
154D4—21no—
155D4—4461no—
156D4—4461no—
157D4—4462no—
158D4—4462no—
159D4—4461no—
160D4—4461no—
161D4—2231no—
162D4C22×D22312yes—
163D4C22×D22311yescharacteristic
164C2×C4—21no—
165C2×C4—4461no—
166C2×C4—4461no—
167C2×C4—4461no—
168C2×C4—4461no—
169C2×C4—4462no—
170C2×C4—4462no—
171C2×C4—4461no—
172C2×C4—2231no—
173C2×C4—2231no—
174C2×C4—2231no—
175C2×C4—2231no—
176C2×C4—2231no—
177C2×C4C22×D22311yescharacteristic
178C8—2232no—
179C8C22×D22312yes—
180C22—21no—
181C22—22no—
182C22—42no—
183C22—8921no—
184C22—8922no—
185C22—4461no—
186C22—2231no—
187C22—2231no—
188C22D4×D22311yescharacteristic
189C4—21no—
190C4—4461no—
191C4—4462no—
192C4—2231no—
193C4—2231no—
194C4D4×D22311yescharacteristic
195C4C23×D22311yescharacteristic
196C2—21no—
197C2—42no—
198C2—41no—
199C2—8921no—
200C2—4461no—
201C2—4461no—
202C2C2×D4×D22311yescharacteristic
subgroupG/H
centerC2C2×D4×D223
commutator, cosocleC4×C223C24
FrattiniC4C23×D223
FittingAGL(1,ℤ/8)×C223C2
socleC2×C223C22×D4
hypercenterAGL(1,ℤ/8)D223
hypocenterC223#575
LCSG ▷ C4×C223 ▷ C2×C223 ▷ C223
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#16813169C2semidirect
2. #41×D223C2semidirect
3. #20297337C2semidirect
4. #20297334C2semidirect
5. #20297333C2semidirect
6. #20297330C2semidirect
7. #20297328C2semidirect
8. #20297325C2semidirect
9. #20297323C2semidirect
10. M16×D223C2semidirect
11. AGL(1,ℤ/8)×C223C2semidirect
12. #16813171C22semidirect
19. #16813169C22semidirect
20. #16813169C22semidirect
21. C2×Dic446C22semidirect
22. Q8×D223C22semidirect
25. D4×D223C22semidirect
29. Dic892C22semidirect
30. #16813160C22semidirect
31. #16813159C22semidirect
32. #16813158C22semidirect
33. #16813158C22semidirect
34. #16813155C22semidirect
35. #16813154C22semidirect
36. #16813153C22semidirect
37. C8×D223C22semidirect
39. C2×D4×C223C22semidirect
40. SD16×C223C22semidirect
41. D8×C223C22semidirect
42. M16×C223C22semidirect
101. C223#575semidirect
110. AGL(1,ℤ/8)D223semidirect
129. C2×D4C2×D223semidirect
138. SD16C2×D223semidirect
140. D8C2×D223semidirect
143. M16C2×D223semidirect

Maximal subgroups

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

subgroupindex
C2×#168131692
#41×D2232
#202973372
#202973342
#202973332
#202973302
#202973282
#202973252
#202973232
M16×D2232
AGL(1,ℤ/8)×C2232
#575223