Finite Group Databasebeta

← #37566053

Subgroups — #37566053

Order 24256.

258 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 258 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
1#27131379C211yesmaximal, characteristic, complemented
2#27131377C211yesmaximal, characteristic, complemented
3#27131363C212yesmaximal, complemented
4#27131359C211yesmaximal, characteristic, complemented
5#27131358C212yesmaximal, complemented
6#27131354C212yesmaximal, complemented
7#27131351C211yesmaximal, characteristic, complemented
8C2×#20258593C211yesmaximal, characteristic, complemented
9SD16×D379C212yesmaximal, complemented
10#27131339C211yesmaximal, characteristic, complemented
11#137×C379C211yesmaximal, characteristic, complemented
12#20258607—21no—
13#20258607C2212yes—
14#20258607C2212yescomplemented
15#20258606—21no—
16#20258606—22no—
17#20258606C2211yescharacteristic
18#20258606C2211yescharacteristic, complemented
19Q8×D379—21no—
20Q8×D379C2212yescomplemented
21C2×D1516—21no—
22C2×D1516C2211yescharacteristic
23D4×D379—22no—
24D4×D379C2212yes—
25#20258596C2211yescharacteristic, complemented
26#20258595C2212yescomplemented
27#20258594C2211yescharacteristic, complemented
28#20258593C2212yescomplemented
29#20258593C2212yescomplemented
30#20258593C2211yescharacteristic, complemented
31D3032C2212yescomplemented
32#20258591C2211yescharacteristic, complemented
33#20258589C2212yescomplemented
34C2×C379⋊C8C2211yescharacteristic, complemented
35C8×D379C2212yescomplemented
36#41×C379C2211yescharacteristic
37C2×Q8×C379C2211yescharacteristic, complemented
38Q16×C379C2212yescomplemented
39SD16×C379C2212yescomplemented
40M16×C379C2211yescharacteristic, complemented
41C22×D379—21no—
42C22×D379—22no—
43Dic758—22no—
44Dic758D411yescharacteristic
45Dic758C2311yescharacteristic
46D1516—22no—
47D1516—22no—
48D1516—21no—
49D1516D411yescharacteristic
50D1516C2311yescharacteristic
51D1516C2312yes—
52C22⋊D379—21no—
53C22⋊D379—22no—
54C22⋊D379D412yes—
55C4×D379—22no—
56C4×D379—21no—
57C4×D379—22no—
58C4×D379—22no—
59C4×D379C2312yes—
60C379⋊C8C2311yescharacteristic
61C379⋊C8C2311yescharacteristic
62Q8×C379—21no—
63Q8×C379C2312yes—
64Q8×C379C2311yescharacteristic
65D4×C379—21no—
66D4×C379C2311yescharacteristic
67C2×C4×C379—21no—
68C2×C4×C379—21no—
69C2×C4×C379C2311yescharacteristic
70C8×C379C2312yes—
71C22×C379—21no—
72C22×C379C2×D411yescharacteristic
73C2×D379—21no—
74C2×D379—22no—
75C2×D379—82no—
76C2×D379—81no—
77C2×D379C2×D412yes—
78C4×C379—21no—
79C4×C379—22no—
80C4×C379C2×D411yescharacteristic
81C4×C379C2411yescharacteristic
82Dic379—21no—
83Dic379C2×D412yes—
84C2×C379—21no—
85C2×C379—41no—
86C2×C379C22×D411yescharacteristic
87D379—22no—
88D379—41no—
89D379—42no—
90C379#57711yescharacteristic, complemented, Sylow
91#577—3791nomaximal, Sylow
922−1+4—3791no—
93H(5,𝔽2)—3791no—
94#137—3792no—
95#137D37911yescharacteristic, complemented
96AGL(1,ℤ/8)—3792no—
97AGL(1,ℤ/8)—3791no—
98#135—3791no—
99#135—3792no—
100C2×SD16—3791no—
101C2×SD16—3792no—
102#131—3791no—
103#41—3791no—
104#41—3792no—
105#41—3792no—
106#41—3791no—
107#41—7582no—
108#41—7581no—
109#41—7581no—
110#41C2×D37911yescharacteristic
111C2×Q8—3792no—
112C2×Q8—7581no—
113C2×Q8C2×D37911yescharacteristic, complemented
114C2×D4—3792no—
115C2×D4—3791no—
116C2×D4—7581no—
117C2×D4—7582no—
118Q16—3791no—
119Q16C2×D37912yescomplemented
120SD16—3792no—
121SD16—3792no—
122SD16—3791no—
123SD16—3791no—
124SD16—3792no—
125SD16C2×D37912yescomplemented
126D8—3791no—
127D8—3792no—
128M16—3792no—
129M16C2×D37911yescharacteristic, complemented
130C2×C8—3791no—
131C2×C8—3792no—
132C23—7582no—
133C23—7581no—
134Q8—21no—
135Q8—3791no—
136Q8—3791no—
137Q8—7582no—
138Q8C22×D37912yes—
139Q8C22×D37911yescharacteristic
140D4—21no—
141D4—3792no—
142D4—3791no—
143D4—3791no—
144D4—3792no—
145D4—7582no—
146D4—7582no—
147D4—7582no—
148D4—7581no—
149D4—7581no—
150D4C22×D37911yescharacteristic
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—3792no—
154C2×C4—7582no—
155C2×C4—7582no—
156C2×C4—7582no—
157C2×C4—7581no—
158C2×C4C22×D37911yescharacteristic
159C8—3791no—
160C8—3791no—
161C8C22×D37912yes—
162C22—21no—
163C22—3792no—
164C22—30321no—
165C22—30322no—
166C22—7581no—
167C22—7582no—
168C22D4×D37911yescharacteristic
169C4—21no—
170C4—22no—
171C4—3792no—
172C4—7581no—
173C4D4×D37911yescharacteristic
174C4C23×D37911yescharacteristic
175C2—21no—
176C2—41no—
177C2—15162no—
178C2—15161no—
179C2—7582no—
180C2C2×D4×D37911yescharacteristic
subgroupG/H
centerC2C2×D4×D379
commutator, cosocleC4×C379C24
FrattiniC4C23×D379
Fitting#137×C379C2
socleC2×C379C22×D4
hypercenter#137D379
hypocenterC379#577
LCSG ▷ C4×C379 ▷ C2×C379 ▷ C379
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. #27131379C2semidirect
2. #27131377C2semidirect
3. #27131363C2semidirect
4. #27131359C2semidirect
5. #27131358C2semidirect
6. #27131354C2semidirect
7. #27131351C2semidirect
8. C2×#20258593C2semidirect
9. SD16×D379C2semidirect
10. #27131339C2semidirect
11. #137×C379C2semidirect
14. #20258607C22semidirect
18. #20258606C22semidirect
20. Q8×D379C22semidirect
25. #20258596C22semidirect
26. #20258595C22semidirect
27. #20258594C22semidirect
28. #20258593C22semidirect
29. #20258593C22semidirect
30. #20258593C22semidirect
31. D3032C22semidirect
32. #20258591C22semidirect
33. #20258589C22semidirect
34. C2×C379⋊C8C22semidirect
35. C8×D379C22semidirect
37. C2×Q8×C379C22semidirect
38. Q16×C379C22semidirect
39. SD16×C379C22semidirect
40. M16×C379C22semidirect
90. C379#577semidirect
95. #137D379semidirect
113. C2×Q8C2×D379semidirect
119. Q16C2×D379semidirect
125. SD16C2×D379semidirect
129. M16C2×D379semidirect

Maximal subgroups

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

subgroupindex
#271313792
#271313772
#271313632
#271313592
#271313582
#271313542
#271313512
C2×#202585932
SD16×D3792
#271313392
#137×C3792
#577379