Finite Group Databasebeta

← #28384379

Subgroups — #28384379

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×#16813171C211yesmaximal, characteristic, complemented
2#41×D223C211yesmaximal, characteristic, complemented
3#20297336C212yesmaximal, complemented
4#20297335C211yesmaximal, characteristic, complemented
5#20297332C211yesmaximal, characteristic, complemented
6#20297331C212yesmaximal, complemented
7#20297329C211yesmaximal, characteristic, complemented
8#20297327C212yesmaximal, complemented
9#20297325C212yesmaximal, complemented
10M16×D223C211yesmaximal, characteristic, complemented
11#137×C223C211yesmaximal, characteristic, complemented
12#16813171—21no—
13#16813171—21no—
14#16813171C2212yes—
15#16813171C2212yescomplemented
16#16813171C2211yescharacteristic, complemented
17#16813170—21no—
18#16813170C2211yescharacteristic
19#16813169—21no—
20#16813169C2211yescharacteristic, complemented
21Q8×D223C2211yescharacteristic, complemented
22C2×D892—21no—
23C2×D892C2211yescharacteristic, complemented
24D4×D223—21no—
25D4×D223C2211yescharacteristic, complemented
26C2×C4×D223—21no—
27C2×C4×D223—21no—
28C2×C4×D223C2211yescharacteristic
29#16813160C2212yescomplemented
30#16813159C2212yescomplemented
31#16813157C2212yescomplemented
32#16813157C2212yescomplemented
33D1784C2212yescomplemented
34#16813155C2212yescomplemented
35#16813154C2211yescharacteristic, complemented
36#16813153C2212yescomplemented
37C8×D223C2212yescomplemented
38#41×C223C2211yescharacteristic
39C2×Q8×C223C2211yescharacteristic, complemented
40Q16×C223C2212yescomplemented
41SD16×C223C2212yescomplemented
42M16×C223C2211yescharacteristic, complemented
43C22×D223—21no—
44C22×D223—21no—
45C22×D223D411yescharacteristic
46Dic446—21no—
47Dic446C2311yescharacteristic
48D892—21no—
49D892—22no—
50D892—22no—
51D892—21no—
52D892C2312yes—
53D892C2311yescharacteristic
54C22⋊D223—21no—
55C22⋊D223—21no—
56C22⋊D223—21no—
57C2×Dic223—21no—
58C2×Dic223D411yescharacteristic
59C4×D223—21no—
60C4×D223—21no—
61C4×D223—21no—
62C4×D223—22no—
63C4×D223—22no—
64C4×D223D411yescharacteristic
65C4×D223D411yescharacteristic
66C4×D223C2311yescharacteristic
67C4×D223C2311yescharacteristic
68C223⋊C8C2312yes—
69Q8×C223—21no—
70Q8×C223C2312yes—
71Q8×C223C2311yescharacteristic
72D4×C223—21no—
73D4×C223C2311yescharacteristic
74C2×C4×C223—21no—
75C2×C4×C223—21no—
76C2×C4×C223C2311yescharacteristic
77C8×C223C2312yes—
78C22×C223—21no—
79C22×C223C2×D411yescharacteristic
80C2×D223—21no—
81C2×D223—22no—
82C2×D223—41no—
83C2×D223—42no—
84C2×D223—42no—
85C2×D223C2×D411yescharacteristic
86C2×D223C2×D411yescharacteristic
87C4×C223—21no—
88C4×C223—22no—
89C4×C223C2×D411yescharacteristic
90C4×C223C2411yescharacteristic
91Dic223—21no—
92Dic223C2×D411yescharacteristic
93Dic223C2×D411yescharacteristic
94C2×C223—21no—
95C2×C223—41no—
96C2×C223C22×D411yescharacteristic
97D223—21no—
98D223—21no—
99D223—41no—
100D223—42no—
101C223#57511yescharacteristic, complemented, Sylow
102#575—2231nomaximal, Sylow
103C2×#41—2231no—
104C2×#41—2231no—
105#137—2231no—
106#137—2232no—
107#137D22311yescharacteristic, complemented
108AGL(1,ℤ/8)—2231no—
109AGL(1,ℤ/8)—2232no—
110AGL(1,ℤ/8)—2231no—
111#135—2232no—
112#135—2232no—
113C2×M16—2231no—
114#41—4461no—
115#41—4461no—
116#41—4461no—
117#41—4461no—
118#41—2232no—
119#41—2231no—
120#41—2232no—
121#41—2231no—
122#41—2231no—
123#41C2×D22311yescharacteristic
124C2×Q8—2231no—
125C2×Q8C2×D22311yescharacteristic, complemented
126C2×D4—4461no—
127C2×D4—4461no—
128C2×D4—2231no—
129C2×D4—2231no—
130C22×C4—4461no—
131C22×C4—4461no—
132C22×C4—2231no—
133Q16—2232no—
134Q16C2×D22312yescomplemented
135SD16—2232no—
136SD16—2232no—
137SD16—2232no—
138SD16C2×D22312yescomplemented
139D8—2232no—
140D8—2232no—
141M16—2231no—
142M16—2232no—
143M16C2×D22311yescharacteristic, complemented
144C2×C8—2232no—
145C23—4461no—
146C23—4461no—
147C23—2231no—
148Q8—21no—
149Q8—4461no—
150Q8—2231no—
151Q8C22×D22311yescharacteristic
152Q8C22×D22312yes—
153D4—21no—
154D4—4461no—
155D4—4462no—
156D4—4462no—
157D4—4461no—
158D4—4461no—
159D4—4461no—
160D4—4461no—
161D4—2232no—
162D4—2231no—
163D4C22×D22311yescharacteristic
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—4461no—
167C2×C4—4461no—
168C2×C4—4462no—
169C2×C4—4462no—
170C2×C4—4461no—
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—8922no—
182C22—8921no—
183C22—8922no—
184C22—4462no—
185C22—4461no—
186C22—2231no—
187C22—2231no—
188C22D4×D22311yescharacteristic
189C4—21no—
190C4—22no—
191C4—4461no—
192C4—2231no—
193C4—2231no—
194C4D4×D22311yescharacteristic
195C4C23×D22311yescharacteristic
196C2—21no—
197C2—41no—
198C2—8922no—
199C2—8921no—
200C2—4461no—
201C2—4461no—
202C2C2×D4×D22311yescharacteristic
subgroupG/H
centerC2C2×D4×D223
commutator, cosocleC4×C223C24
FrattiniC4C23×D223
Fitting#137×C223C2
socleC2×C223C22×D4
hypercenter#137D223
hypocenterC223#575
LCSG ▷ C4×C223 ▷ C2×C223 ▷ C223
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. C2×#16813171C2semidirect
2. #41×D223C2semidirect
3. #20297336C2semidirect
4. #20297335C2semidirect
5. #20297332C2semidirect
6. #20297331C2semidirect
7. #20297329C2semidirect
8. #20297327C2semidirect
9. #20297325C2semidirect
10. M16×D223C2semidirect
11. #137×C223C2semidirect
15. #16813171C22semidirect
16. #16813171C22semidirect
20. #16813169C22semidirect
21. Q8×D223C22semidirect
23. C2×D892C22semidirect
25. D4×D223C22semidirect
29. #16813160C22semidirect
30. #16813159C22semidirect
31. #16813157C22semidirect
32. #16813157C22semidirect
33. D1784C22semidirect
34. #16813155C22semidirect
35. #16813154C22semidirect
36. #16813153C22semidirect
37. C8×D223C22semidirect
39. C2×Q8×C223C22semidirect
40. Q16×C223C22semidirect
41. SD16×C223C22semidirect
42. M16×C223C22semidirect
101. C223#575semidirect
107. #137D223semidirect
125. C2×Q8C2×D223semidirect
134. Q16C2×D223semidirect
138. SD16C2×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×#168131712
#41×D2232
#202973362
#202973352
#202973322
#202973312
#202973292
#202973272
#202973252
M16×D2232
#137×C2232
#575223