Finite Group Databasebeta

← #38767983

Subgroups — #38767983

Order 25472.

214 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 55 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#27164728C211yesmaximal, characteristic
2#27164397C211yesmaximal, characteristic, complemented
3#27164305C211yesmaximal, characteristic
4#27163766C211yesmaximal, characteristic, complemented
5Q8⋊(C199⋊C8)C211yesmaximal, characteristic, complemented
6#27163745C211yesmaximal, characteristic
7#477×C199C211yesmaximal, characteristic, complemented
8C2×#15691876—22no—
9#20263955C2211yescharacteristic
10#20263947—21no—
11#20263933—22no—
12#20263931—21no—
13C4×D796—21no—
14C4×D796C2211yescharacteristic
15C4⋊C4×D199—21no—
16#20263886—21no—
17C4⋊(C199⋊C8)C2211yescharacteristic
18#20263875—21no—
19C4×C199⋊C8C2211yescharacteristic
20C4⋊Q8×C199C2211yescharacteristic
21C4×Q8×C199C2211yescharacteristic
22#107×C199—21no—
23C4⋊C8×C199C2211yescharacteristic
24#103×C199—21no—
25C2×Dic398D412yes—
26C2×D796D411yescharacteristic
27C2×C4×D199—22no—
28C2×C4×D199—21no—
29#15691876—44no—
30C2×C199⋊C8—21no—
31C2×C199⋊C8—21no—
32C4⋊Dic199—21no—
33C4⋊Dic199—21no—
34C4⋊Dic199D411yescharacteristic
35Dic199⋊C4—22no—
36Dic199⋊C4—22no—
37(C2×D199)⋊C4—22no—
38(C2×D199)⋊C4—21no—
39C2×Q8×C199—21no—
40C2×Q8×C199D411yescharacteristic
41C2×C8×C199—21no—
42C4⋊C4×C199—21no—
43C4⋊C4×C199—21no—
44C4⋊C4×C199—21no—
45C4⋊C4×C199D411yescharacteristic
46C42×C199—21no—
47C42×C199C2311yescharacteristic
48C22×D199—21no—
49Dic398—42no—
50D796—21no—
51D796Q1612yescomplemented
52C2×Dic199—21no—
53C2×Dic199—22no—
54C4×D199—41no—
55C4×D199—41no—
56C4×D199—82no—
57C199⋊C8—22no—
58C199⋊C8—41no—
59Q8×C199—21no—
60Q8×C199—21no—
61Q8×C199—42no—
62C2×C4×C199—21no—
63C2×C4×C199—21no—
64C2×C4×C199—41no—
65C2×C4×C199C2×D411yescharacteristic
66C2×C4×C199C2×D411yescharacteristic
67C2×C4×C199C2×D411yescharacteristic
68C8×C199—41no—
69C22×C199#12011yescharacteristic
70C2×D199—22no—
71C2×D199—41no—
72C2×D199—41no—
73C4×C199—21no—
74C4×C199—41no—
75C4×C199—41no—
76C4×C199—81no—
77C4×C199C2×Q1611yescharacteristic
78C4×C199#13511yescharacteristic
79C4×C199#13711yescharacteristic
80C4×C199#13711yescharacteristic
81Dic199—41no—
82Dic199—42no—
83C2×C199#45111yescharacteristic
84C2×C199#45211yescharacteristic
85C2×C199#45311yescharacteristic
86D199—42no—
87C199#163211yescharacteristic, complemented, Sylow
88#1632—1991nomaximal, Sylow
89#554—1991no—
90#494—1991no—
91#477D19911yescharacteristic, complemented
92#476—1991no—
93#333—1991no—
94Q8⋊C8—1991no—
95D4⋊C8—1991no—
96C2×Q16—3982no—
97C4⋊Q8—1991no—
98C4⋊Q8C2×D19911yescharacteristic
99#125—3981no—
100C22⋊Q8—3982no—
101C22⋊Q8—3981no—
102C4×Q8C2×D19911yescharacteristic
103C4×D4—3981no—
104C4×D4—1991no—
105C2×C4⋊C4—3981no—
106#107—21no—
107#106—3981no—
108C4⋊C8—1991no—
109C4⋊C8C2×D19911yescharacteristic
110#103—21no—
111#102—3981no—
112C4×C8—1991no—
113C2×Q8—21no—
114C2×Q8—1992no—
115C2×Q8C22⋊D19911yescharacteristic
116C2×D4—1991no—
117C22×C4—3982no—
118C22×C4—3981no—
119Q16—7964no—
120C2×C8—21no—
121C2×C8—3981no—
122C2×C8—3981no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—3982no—
127C4⋊C4—3982no—
128C4⋊C4—3981no—
129C4⋊C4—3981no—
130C4⋊C4—1991no—
131C4⋊C4C22⋊D19911yescharacteristic
132C22⋊C4—3982no—
133C22⋊C4—3981no—
134C42—21no—
135C42C22×D19911yescharacteristic
136C23—3981no—
137Q8—21no—
138Q8—21no—
139Q8—42no—
140Q8—7962no—
141D4—3981no—
142D4—1992no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—41no—
146C2×C4—7961no—
147C2×C4—7961no—
148C2×C4—15922no—
149C2×C4—3981no—
150C2×C4—3982no—
151C2×C4D4×D19911yescharacteristic
152C2×C4D4×D19911yescharacteristic
153C2×C4C2×C22⋊D19911yescharacteristic
154C8—41no—
155C8—7961no—
156C8—3982no—
157C22—7961no—
158C22—7961no—
159C22—3982no—
160C22#2026392311yescharacteristic
161C4—21no—
162C4—41no—
163C4—41no—
164C4—81no—
165C4—7961no—
166C4—7962no—
167C4Q16×D19911yescharacteristic
168C4#2026397511yescharacteristic
169C4#2026398611yescharacteristic
170C4#2026398711yescharacteristic
171C2—7962no—
172C2#2716417211yescharacteristic
173C2#2716417811yescharacteristic
174C2#2716418411yescharacteristic
subgroupG/H
centerC22#20263923
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#477×C199C2
socleC22×C199#120
prime radicalC2×D796D4
hypercenter#477D199
hypocenterC199#1632
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
2. #27164397C2semidirect
4. #27163766C2semidirect
5. Q8⋊(C199⋊C8)C2semidirect
7. #477×C199C2semidirect
51. D796Q16semidirect
87. C199#1632semidirect
91. #477D199semidirect

Maximal subgroups

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

subgroupindex
#271647282
#271643972
#271643052
#271637662
Q8⋊(C199⋊C8)2
#271637452
#477×C1992
#1632199