Finite Group Databasebeta

← #38768052

Subgroups — #38768052

Order 25472.

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

normal subgroups only

This group has 250 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#27164689C211yesmaximal, characteristic
2#27164365C211yesmaximal, characteristic, complemented
3#27164304C211yesmaximal, characteristic, complemented
4#27164223C211yesmaximal, characteristic, complemented
5#27163761C211yesmaximal, characteristic, complemented
6#27163750C211yesmaximal, characteristic, complemented
7D4⋊C8×C199C211yesmaximal, characteristic, complemented
8C2×#15691886—21no—
9C2×#15691874—21no—
10#20263955C2211yescharacteristic
11#20263952C2211yescharacteristic
12#20263951—21no—
13#20263929—21no—
14#20263928—22no—
15C4×Dic398C2211yescharacteristic
16#20263917—21no—
17#20263887—21no—
18C4⋊(C199⋊C8)C2211yescharacteristic
19#20263880—21no—
20#20263878—21no—
21#20263877—22no—
22C4×D4×C199C2211yescharacteristic
23C4⋊C8×C199C2211yescharacteristic
24C4×C8×C199C2211yescharacteristic
25#15691886—42no—
26#15691886—42no—
27C2×Dic398—21no—
28C2×Dic398D411yescharacteristic
29C2×D796—21no—
30C2×D796—21no—
31C2×D796D412yes—
32C2×C22⋊D199—22no—
33C2×C22⋊D199—22no—
34C2×C4×D199—22no—
35#15691874—42no—
36C2×C199⋊C8—21no—
37C4⋊Dic199—21no—
38C4⋊Dic199—21no—
39C4⋊Dic199D411yescharacteristic
40Dic199⋊C4—21no—
41(C2×D199)⋊C4—22no—
42C4×Dic199—21no—
43C2×D4×C199D411yescharacteristic
44C22×C4×C199—21no—
45C2×C8×C199—21no—
46C2×C8×C199—21no—
47C4⋊C4×C199D411yescharacteristic
48C22⋊C4×C199—21no—
49C42×C199C2311yescharacteristic
50C22×D199—22no—
51Dic398—21no—
52Dic398—41no—
53Dic398—41no—
54Dic398SD1612yescomplemented
55D796—42no—
56D796—41no—
57D796—41no—
58D796—41no—
59D796—41no—
60C22⋊D199—44no—
61C22⋊D199—82no—
62C2×Dic199—21no—
63C2×Dic199—21no—
64C2×Dic199—41no—
65C4×D199—44no—
66C199⋊C8—41no—
67C23×C199—21no—
68D4×C199—21no—
69D4×C199—21no—
70C2×C4×C199—21no—
71C2×C4×C199—41no—
72C2×C4×C199—41no—
73C2×C4×C199C2×D411yescharacteristic
74C2×C4×C199C2×D411yescharacteristic
75C2×C4×C199C2×D411yescharacteristic
76C8×C199—41no—
77C8×C199—41no—
78C22×C199—41no—
79C22×C199—41no—
80C22×C199—41no—
81C22×C199#12011yescharacteristic
82C2×D199—42no—
83C2×D199—42no—
84C2×D199—42no—
85C4×C199—21no—
86C4×C199—41no—
87C4×C199C2×SD1611yescharacteristic
88C4×C199#13511yescharacteristic
89C4×C199AGL(1,ℤ/8)11yescharacteristic
90C4×C199AGL(1,ℤ/8)11yescharacteristic
91Dic199—22no—
92Dic199—41no—
93Dic199—81no—
94C2×C199—81no—
95C2×C199#44911yescharacteristic
96C2×C199#45011yescharacteristic
97C2×C199#45511yescharacteristic
98D199—82no—
99C199#164211yescharacteristic, complemented, Sylow
100#1642—1991nomaximal, Sylow
101#550—1991no—
102#486—1991no—
103#475—1991no—
104#461—1991no—
105#331—1991no—
106Q8⋊C8—1991no—
107D4⋊C8D19911yescharacteristic, complemented
108C2×#41—3981no—
109C2×SD16—3981no—
110C4⋊Q8—1991no—
111#127—3981no—
112#127—1991no—
113#121—3982no—
114#121—3981no—
115C4×Q8—1991no—
116C4×D4—3981no—
117C4×D4C2×D19911yescharacteristic
118#106—3981no—
119C4⋊C8—1991no—
120C4⋊C8C2×D19911yescharacteristic
121#103—3981no—
122#103—3981no—
123#102—3982no—
124C4×C8C2×D19911yescharacteristic
125#41—7962no—
126#41—7962no—
127C2×Q8—3981no—
128C2×Q8—1991no—
129C2×D4—3982no—
130C2×D4—3982no—
131C2×D4—3981no—
132C2×D4—3981no—
133C2×D4—1992no—
134C2×D4D79611yescharacteristic
135C22×C4—21no—
136C22×C4—3982no—
137SD16—7962no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—3981no—
141C4⋊C4—3981no—
142C4⋊C4—3981no—
143C4⋊C4—3981no—
144C4⋊C4—1991no—
145C4⋊C4D79611yescharacteristic
146C22⋊C4—21no—
147C22⋊C4—3982no—
148C42—3981no—
149C42C22×D19911yescharacteristic
150C23—21no—
151C23—3982no—
152Q8—7961no—
153Q8—7961no—
154Q8—3981no—
155Q8—1992no—
156D4—21no—
157D4—21no—
158D4—7962no—
159D4—7964no—
160D4—7961no—
161D4—7961no—
162D4—7961no—
163D4—7961no—
164D4—15922no—
165C2×C4—21no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—7964no—
169C2×C4—7961no—
170C2×C4—3981no—
171C2×C4—3981no—
172C2×C4D4×D19911yescharacteristic
173C2×C4D4×D19911yescharacteristic
174C2×C4C2×D79611yescharacteristic
175C8—41no—
176C8—41no—
177C8—7961no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—7962no—
182C22—7962no—
183C22—7962no—
184C22#2026392511yescharacteristic
185C4—21no—
186C4—41no—
187C4—7961no—
188C4—15921no—
189C4—3982no—
190C4SD16×D19911yescharacteristic
191C4#2026397711yescharacteristic
192C4#2026397911yescharacteristic
193C4#2026398311yescharacteristic
194C2—81no—
195C2—15922no—
196C2#2716416511yescharacteristic
197C2#2716417011yescharacteristic
198C2#2716419711yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
FittingD4⋊C8×C199C2
socleC22×C199#120
prime radical#27164689C2
hypercenterD4⋊C8D199
hypocenterC199#1642
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #27164365C2semidirect
3. #27164304C2semidirect
4. #27164223C2semidirect
5. #27163761C2semidirect
6. #27163750C2semidirect
7. D4⋊C8×C199C2semidirect
54. Dic398SD16semidirect
99. C199#1642semidirect
107. D4⋊C8D199semidirect

Maximal subgroups

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

subgroupindex
#271646892
#271643652
#271643042
#271642232
#271637612
#271637502
D4⋊C8×C1992
#1642199