Finite Group Databasebeta

← #38767978

Subgroups — #38767978

Order 25472.

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

normal subgroups only

This group has 256 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#27164688C211yesmaximal, characteristic
2#27164390C211yesmaximal, characteristic, complemented
3#27164314C211yesmaximal, characteristic, complemented
4#27164309C211yesmaximal, characteristic, complemented
5#27163748C211yesmaximal, characteristic, complemented
6#27163745C211yesmaximal, characteristic, complemented
7#331×C199C211yesmaximal, characteristic, complemented
8C2×#15691885—21no—
9C2×#15691874—22no—
10#20263955C2211yescharacteristic
11#20263951—21no—
12#20263927—22no—
13#20263926—21no—
14C4×Dic398C2211yescharacteristic
15C4×D796C2211yescharacteristic
16D4×Dic199—21no—
17#20263889—21no—
18#20263887—21no—
19#20263880—21no—
20#20263877—21no—
21C4×C199⋊C8C2211yescharacteristic
22#127×C199C2211yescharacteristic
23C4⋊C8×C199C2211yescharacteristic
24C4⋊C8×C199C2211yescharacteristic
25#15691885—42no—
26#15691885—42no—
27C2×Dic398—21no—
28C2×Dic398D411yescharacteristic
29C2×D796D411yescharacteristic
30C2×C22⋊D199—22no—
31C2×C22⋊D199—22no—
32C22×Dic199—22no—
33C2×C4×D199—21no—
34#15691874—44no—
35C2×C199⋊C8—21no—
36C4⋊Dic199—21no—
37C4⋊Dic199—21no—
38C4⋊Dic199D411yescharacteristic
39C4⋊Dic199D411yescharacteristic
40Dic199⋊C4—21no—
41C22⋊Dic199—22no—
42(C2×D199)⋊C4—21no—
43C4×Dic199—21no—
44C2×D4×C199—21no—
45C2×D4×C199—21no—
46C2×D4×C199D412yes—
47C2×C8×C199—21no—
48C2×C8×C199—21no—
49C42×C199C2311yescharacteristic
50C22×D199—21no—
51Dic398—21no—
52Dic398—42no—
53Dic398SD1612yescomplemented
54D796—21no—
55D796—21no—
56C22⋊D199—44no—
57C22⋊D199—82no—
58C2×Dic199—21no—
59C2×Dic199—21no—
60C2×Dic199—21no—
61C2×Dic199—44no—
62C2×Dic199—41no—
63C4×D199—41no—
64C4×D199—41no—
65C199⋊C8—22no—
66C23×C199—22no—
67D4×C199—42no—
68D4×C199—41no—
69D4×C199—41no—
70D4×C199—41no—
71D4×C199—41no—
72C2×C4×C199C2×D411yescharacteristic
73C2×C4×C199C2×D411yescharacteristic
74C2×C4×C199C2×D411yescharacteristic
75C8×C199—41no—
76C8×C199—41no—
77C22×C199—42no—
78C22×C199—42no—
79C22×C199—42no—
80C22×C199#12011yescharacteristic
81C2×D199—41no—
82C2×D199—41no—
83C2×D199—41no—
84C4×C199—21no—
85C4×C199C2×SD1611yescharacteristic
86C4×C199#13511yescharacteristic
87C4×C199AGL(1,ℤ/8)11yescharacteristic
88C4×C199#13711yescharacteristic
89Dic199—22no—
90Dic199—41no—
91Dic199—41no—
92Dic199—81no—
93C2×C199—82no—
94C2×C199#45011yescharacteristic
95C2×C199#45211yescharacteristic
96C2×C199#45311yescharacteristic
97D199—81no—
98C199#163111yescharacteristic, complemented, Sylow
99#1631—1991nomaximal, Sylow
100#550—1991no—
101#492—1991no—
102#477—1991no—
103#476—1991no—
104#331D19911yescharacteristic, complemented
105Q8⋊C8—1991no—
106D4⋊C8—1991no—
107C2×#41—3981no—
108C2×SD16—3982no—
109C4⋊Q8—1991no—
110#127—3981no—
111#127C2×D19911yescharacteristic
112#121—3982no—
113#121—3981no—
114C4×Q8—1991no—
115C4×D4—3981no—
116C4×D4—1991no—
117#107—3981no—
118#106—3981no—
119C4⋊C8C2×D19911yescharacteristic
120C4⋊C8C2×D19911yescharacteristic
121#103—3981no—
122#102—3981no—
123C4×C8—1991no—
124#41—7962no—
125#41—7962no—
126C2×Q8—3981no—
127C2×Q8—1991no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—3982no—
131C2×D4—3982no—
132C2×D4—1991no—
133C2×D4D79612yes—
134C22×C4—3982no—
135C22×C4—3981no—
136SD16—7964no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—3981no—
140C4⋊C4—3981no—
141C4⋊C4—3981no—
142C4⋊C4—3981no—
143C4⋊C4—1991no—
144C4⋊C4—1991no—
145C22⋊C4—3982no—
146C22⋊C4—3981no—
147C42—3981no—
148C42C22×D19911yescharacteristic
149C23—22no—
150C23—3981no—
151Q8—7962no—
152Q8—3981no—
153Q8—1992no—
154D4—42no—
155D4—41no—
156D4—41no—
157D4—41no—
158D4—41no—
159D4—7964no—
160D4—15922no—
161D4—3981no—
162D4—3981no—
163C2×C4—7964no—
164C2×C4—7961no—
165C2×C4—7961no—
166C2×C4—7961no—
167C2×C4—3981no—
168C2×C4—3981no—
169C2×C4—3981no—
170C2×C4D4×D19911yescharacteristic
171C2×C4D4×D19911yescharacteristic
172C2×C4C2×D79611yescharacteristic
173C8—41no—
174C8—41no—
175C8—3982no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—7961no—
180C22—7961no—
181C22—7961no—
182C22#2026392511yescharacteristic
183C4—21no—
184C4—7961no—
185C4—7961no—
186C4—15921no—
187C4—3982no—
188C4SD16×D19911yescharacteristic
189C4#2026397411yescharacteristic
190C4#2026397911yescharacteristic
191C4#2026398411yescharacteristic
192C2—82no—
193C2—15921no—
194C2#2716417011yescharacteristic
195C2#2716418111yescharacteristic
196C2#2716418811yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#331×C199C2
socleC22×C199#120
prime radical#27164688C2
hypercenter#331D199
hypocenterC199#1631
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #27164390C2semidirect
3. #27164314C2semidirect
4. #27164309C2semidirect
5. #27163748C2semidirect
6. #27163745C2semidirect
7. #331×C199C2semidirect
53. Dic398SD16semidirect
98. C199#1631semidirect
104. #331D199semidirect

Maximal subgroups

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

subgroupindex
#271646882
#271643902
#271643142
#271643092
#271637482
#271637452
#331×C1992
#1631199