Finite Group Databasebeta

← #38768112

Subgroups — #38768112

Order 25472.

234 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 59 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
1Q8×D796C211yesmaximal, characteristic
2#27164414C211yesmaximal, characteristic, complemented
3#27164220C211yesmaximal, characteristic
4#27163765C211yesmaximal, characteristic, complemented
5Q8⋊(C199⋊C8)C211yesmaximal, characteristic, complemented
6#27163745C211yesmaximal, characteristic
7#475×C199C211yesmaximal, characteristic, complemented
8C2×Q8×D199—21no—
9C2×#15691873—21no—
10#20263955—21no—
11#20263955C2211yescharacteristic
12#20263935—22no—
13#20263935—21no—
14C4×D796—21no—
15C4×D796C2211yescharacteristic
16#20263886—22no—
17C4⋊(C199⋊C8)C2211yescharacteristic
18#20263879—21no—
19C4×C199⋊C8C2211yescharacteristic
20C4⋊Q8×C199C2211yescharacteristic
21C4×Q8×C199C2211yescharacteristic
22#106×C199—21no—
23C4⋊C8×C199C2211yescharacteristic
24#103×C199—21no—
25C2×Dic398—22no—
26C2×Dic398—21no—
27Q8×D199—42no—
28Q8×D199—42no—
29C2×D796D411yescharacteristic
30C2×C4×D199—22no—
31C2×C4×D199—21no—
32#15691873—42no—
33C2×C199⋊C8—21no—
34C2×C199⋊C8—21no—
35C4⋊Dic199—22no—
36C4⋊Dic199—21no—
37C4⋊Dic199D411yescharacteristic
38C4⋊Dic199D412yes—
39(C2×D199)⋊C4—22no—
40(C2×D199)⋊C4—21no—
41C2×Q8×C199—21no—
42C2×Q8×C199D411yescharacteristic
43C2×C8×C199—21no—
44C4⋊C4×C199—21no—
45C4⋊C4×C199—21no—
46C4⋊C4×C199—21no—
47C4⋊C4×C199D411yescharacteristic
48C42×C199—21no—
49C42×C199C2311yescharacteristic
50C22×D199—21no—
51Dic398—44no—
52Dic398—41no—
53Dic398—41no—
54D796—21no—
55D796SD1612yes—
56C2×Dic199—22no—
57C2×Dic199—21no—
58C4×D199—44no—
59C4×D199—41no—
60C4×D199—41no—
61C199⋊C8—22no—
62C199⋊C8—41no—
63Q8×C199—21no—
64Q8×C199—41no—
65Q8×C199—41no—
66Q8×C199SD1612yescomplemented
67C2×C4×C199—21no—
68C2×C4×C199—21no—
69C2×C4×C199—41no—
70C2×C4×C199C2×D411yescharacteristic
71C2×C4×C199C2×D411yescharacteristic
72C2×C4×C199C2×D411yescharacteristic
73C8×C199—41no—
74C22×C199#12011yescharacteristic
75C2×D199—22no—
76C2×D199—41no—
77C2×D199—41no—
78C4×C199—22no—
79C4×C199—21no—
80C4×C199—41no—
81C4×C199—81no—
82C4×C199C2×SD1611yescharacteristic
83C4×C199C2×SD1611yescharacteristic
84C4×C199AGL(1,ℤ/8)11yescharacteristic
85C4×C199#13711yescharacteristic
86Dic199—42no—
87Dic199—41no—
88C2×C199#44811yescharacteristic
89C2×C199#45011yescharacteristic
90C2×C199#45611yescharacteristic
91D199—42no—
92C199#165111yescharacteristic, complemented, Sylow
93#1651—1991nomaximal, Sylow
94D4×Q8—1991no—
95#498—1991no—
96#475D19911yescharacteristic, complemented
97#461—1991no—
98#332—1991no—
99Q8⋊C8—1991no—
100D4⋊C8—1991no—
101C22×Q8—3981no—
102C2×SD16—3981no—
103C4⋊Q8—3981no—
104C4⋊Q8—1991no—
105C4⋊Q8C2×D19911yescharacteristic
106C22⋊Q8—3982no—
107C22⋊Q8—3981no—
108C4×Q8C2×D19911yescharacteristic
109C4×D4—3981no—
110C4×D4—1991no—
111#106—21no—
112#106—3982no—
113C4⋊C8—1991no—
114C4⋊C8C2×D19911yescharacteristic
115#103—21no—
116#103—3981no—
117C4×C8—1991no—
118C2×Q8—21no—
119C2×Q8—7962no—
120C2×Q8—7962no—
121C2×Q8—3982no—
122C2×Q8—3981no—
123C2×Q8C22⋊D19911yescharacteristic
124C2×D4—1991no—
125C22×C4—3982no—
126C22×C4—3981no—
127SD16—7962no—
128C2×C8—21no—
129C2×C8—3981no—
130C2×C8—3981no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—3982no—
135C4⋊C4—3981no—
136C4⋊C4—1991no—
137C4⋊C4—1992no—
138C4⋊C4C22⋊D19911yescharacteristic
139C22⋊C4—3982no—
140C22⋊C4—3981no—
141C42—21no—
142C42C22×D19911yescharacteristic
143C23—3981no—
144Q8—21no—
145Q8—41no—
146Q8—41no—
147Q8—7964no—
148Q8—7961no—
149Q8—7961no—
150Q8#1569187312yescomplemented
151D4—3981no—
152D4—1992no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—41no—
156C2×C4—7964no—
157C2×C4—7961no—
158C2×C4—7961no—
159C2×C4—3982no—
160C2×C4—3981no—
161C2×C4D4×D19911yescharacteristic
162C2×C4D4×D19911yescharacteristic
163C2×C4C2×C22⋊D19911yescharacteristic
164C8—41no—
165C8—7961no—
166C8—3982no—
167C22—7961no—
168C22—7961no—
169C22—3982no—
170C22#2026392311yescharacteristic
171C4—22no—
172C4—21no—
173C4—41no—
174C4—81no—
175C4—7962no—
176C4—7961no—
177C4SD16×D19911yescharacteristic
178C4C2×#1569187311yescharacteristic
179C4#2026398011yescharacteristic
180C4#2026398411yescharacteristic
181C2—7962no—
182C2#2716415511yescharacteristic
183C2#2716416811yescharacteristic
184C2#2716419911yescharacteristic
subgroupG/H
centerC22#20263923
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#475×C199C2
socleC22×C199#120
prime radicalC2×D796D4
hypercenter#475D199
hypocenterC199#1651
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #27164414C2semidirect
4. #27163765C2semidirect
5. Q8⋊(C199⋊C8)C2semidirect
7. #475×C199C2semidirect
66. Q8×C199SD16semidirect
92. C199#1651semidirect
96. #475D199semidirect
150. Q8#15691873semidirect

Maximal subgroups

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

subgroupindex
Q8×D7962
#271644142
#271642202
#271637652
Q8⋊(C199⋊C8)2
#271637452
#475×C1992
#1651199