Finite Group Databasebeta

← #38768035

Subgroups — #38768035

Order 25472.

212 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 53 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#27164753C211yesmaximal, characteristic
2#27164366C211yesmaximal, characteristic
3#27164225C211yesmaximal, characteristic, complemented
4#27163761C211yesmaximal, characteristic
5Q8⋊(C199⋊C8)C211yesmaximal, characteristic, complemented
6#27163748C211yesmaximal, characteristic, complemented
7#475×C199C211yesmaximal, characteristic, complemented
8C2×#15691876—21no—
9#20263952C2211yescharacteristic
10#20263944—21no—
11#20263943—21no—
12#20263940—22no—
13C4×Dic398C2211yescharacteristic
14C4×D796—21no—
15#20263917—21no—
16C4⋊(C199⋊C8)C2211yescharacteristic
17#20263879—21no—
18#20263875—22no—
19C4×C199⋊C8C2211yescharacteristic
20C4⋊Q8×C199C2211yescharacteristic
21C4×Q8×C199C2211yescharacteristic
22#106×C199—21no—
23C4⋊C8×C199C2211yescharacteristic
24#103×C199—21no—
25C2×Dic398D411yescharacteristic
26C2×D796—21no—
27C2×D796—21no—
28C2×D796D412yes—
29C2×C4×D199—22no—
30#15691876—42no—
31C2×C199⋊C8—21no—
32C2×C199⋊C8—21no—
33C4⋊Dic199D411yescharacteristic
34Dic199⋊C4—21no—
35(C2×D199)⋊C4—22no—
36(C2×D199)⋊C4—22no—
37(C2×D199)⋊C4—22no—
38C4×Dic199—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—22no—
49Dic398—21no—
50Dic398—21no—
51D796—42no—
52D796—41no—
53D796—41no—
54D796—41no—
55D796—41no—
56C2×Dic199—21no—
57C2×Dic199—21no—
58C4×D199—82no—
59C199⋊C8—41no—
60C199⋊C8—41no—
61Q8×C199—21no—
62Q8×C199—21no—
63Q8×C199—41no—
64Q8×C199—41no—
65C2×C4×C199—21no—
66C2×C4×C199—21no—
67C2×C4×C199—41no—
68C2×C4×C199C2×D411yescharacteristic
69C2×C4×C199C2×D411yescharacteristic
70C2×C4×C199C2×D411yescharacteristic
71C8×C199—41no—
72C22×C199#12011yescharacteristic
73C2×D199—42no—
74C2×D199—42no—
75C2×D199—42no—
76C4×C199—21no—
77C4×C199—41no—
78C4×C199—41no—
79C4×C199—81no—
80C4×C199#13511yescharacteristic
81C4×C199#13511yescharacteristic
82C4×C199AGL(1,ℤ/8)11yescharacteristic
83C4×C199#13711yescharacteristic
84Dic199—41no—
85Dic199—41no—
86C2×C199#44911yescharacteristic
87C2×C199#45211yescharacteristic
88C2×C199#45511yescharacteristic
89D199—82no—
90C199#164011yescharacteristic, complemented, Sylow
91#1640—1991nomaximal, Sylow
92#559—1991no—
93#486—1991no—
94#475D19911yescharacteristic, complemented
95#462—1991no—
96#331—1991no—
97Q8⋊C8—1991no—
98Q8⋊C8—1991no—
99C2×Q16—3981no—
100C4⋊Q8C2×D19911yescharacteristic
101#127—1991no—
102#124—3981no—
103#124—3981no—
104#123—3982no—
105C4×Q8—1991no—
106C4×Q8C2×D19911yescharacteristic
107C4×D4—3981no—
108C4×D4—3981no—
109#106—21no—
110C4⋊C8—1991no—
111C4⋊C8C2×D19911yescharacteristic
112#103—21no—
113#103—3981no—
114#102—3982no—
115C4×C8—1991no—
116C2×Q8—21no—
117C2×Q8—1991no—
118C2×Q8C22⋊D19911yescharacteristic
119C2×D4—3981no—
120C2×D4—3981no—
121C2×D4—1992no—
122C22×C4—3982no—
123Q16—7962no—
124C2×C8—21no—
125C2×C8—3981no—
126C2×C8—3981no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—3981no—
131C4⋊C4—1991no—
132C4⋊C4C22⋊D19911yescharacteristic
133C22⋊C4—3982no—
134C22⋊C4—3982no—
135C22⋊C4—3982no—
136C42—21no—
137C42—3981no—
138C42C22×D19911yescharacteristic
139C23—3982no—
140Q8—21no—
141Q8—21no—
142Q8—41no—
143Q8—41no—
144Q8—3981no—
145Q8—3981no—
146D4—7962no—
147D4—7961no—
148D4—7961no—
149D4—7961no—
150D4—7961no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—15922no—
155C2×C4—3981no—
156C2×C4—3981no—
157C2×C4D4×D19911yescharacteristic
158C2×C4D4×D19911yescharacteristic
159C2×C4C2×C22⋊D19911yescharacteristic
160C8—41no—
161C8—7961no—
162C8—7961no—
163C22—7962no—
164C22—7962no—
165C22—7962no—
166C22#2026392311yescharacteristic
167C4—21no—
168C4—41no—
169C4—41no—
170C4—81no—
171C4—7961no—
172C4—7961no—
173C4#2026397511yescharacteristic
174C4#2026397611yescharacteristic
175C4#2026398211yescharacteristic
176C4#2026398611yescharacteristic
177C2—15922no—
178C2#2716416211yescharacteristic
179C2#2716417811yescharacteristic
180C2#2716419411yescharacteristic
subgroupG/H
centerC22#20263923
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#475×C199C2
socleC22×C199#120
prime radical#20263952C22
hypercenter#475D199
hypocenterC199#1640
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
3. #27164225C2semidirect
5. Q8⋊(C199⋊C8)C2semidirect
6. #27163748C2semidirect
7. #475×C199C2semidirect
90. C199#1640semidirect
94. #475D199semidirect

Maximal subgroups

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

subgroupindex
#271647532
#271643662
#271642252
#271637612
Q8⋊(C199⋊C8)2
#271637482
#475×C1992
#1640199