Finite Group Databasebeta

← #38768000

Subgroups — #38768000

Order 25472.

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

normal subgroups only

This group has 234 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#27164673C211yesmaximal, characteristic
2#27164368C211yesmaximal, characteristic, complemented
3#27164305C211yesmaximal, characteristic, complemented
4#27163763C211yesmaximal, characteristic, complemented
5D4⋊(C199⋊C8)C211yesmaximal, characteristic, complemented
6#27163745C211yesmaximal, characteristic, complemented
7#476×C199C211yesmaximal, characteristic, complemented
8C2×#15691886—21no—
9#20263955C2211yescharacteristic
10#20263937—22no—
11#20263934—21no—
12#20263926—21no—
13C4×D796C2211yescharacteristic
14C4⋊C4×D199—21no—
15#20263886—21no—
16C4⋊(C199⋊C8)C2211yescharacteristic
17#20263879—22no—
18#20263875—21no—
19C4×C199⋊C8C2211yescharacteristic
20C4⋊Q8×C199C2211yescharacteristic
21C4×D4×C199C2211yescharacteristic
22#106×C199—21no—
23C4⋊C8×C199C2211yescharacteristic
24#102×C199—21no—
25#15691886—42no—
26#15691886—42no—
27C2×Dic398—21no—
28C2×D796D411yescharacteristic
29C2×C22⋊D199—22no—
30C2×C4×D199—22no—
31C2×C4×D199—21no—
32C2×C199⋊C8—21no—
33C2×C199⋊C8—21no—
34C4⋊Dic199—21no—
35C4⋊Dic199D412yes—
36C4⋊Dic199D411yescharacteristic
37Dic199⋊C4—22no—
38C22⋊Dic199—22no—
39(C2×D199)⋊C4—21no—
40C2×Q8×C199—21no—
41C2×D4×C199D411yescharacteristic
42C22×C4×C199—21no—
43C2×C8×C199—21no—
44C4⋊C4×C199—21no—
45C4⋊C4×C199—21no—
46C4⋊C4×C199D411yescharacteristic
47C22⋊C4×C199—21no—
48C42×C199C2311yescharacteristic
49C22×D199—21no—
50Dic398—41no—
51Dic398—41no—
52D796—21no—
53D796SD1612yes—
54C22⋊D199—44no—
55C2×Dic199—21no—
56C2×Dic199—22no—
57C4×D199—44no—
58C4×D199—41no—
59C4×D199—41no—
60C199⋊C8—41no—
61C199⋊C8—41no—
62C23×C199—21no—
63Q8×C199—42no—
64D4×C199—21no—
65D4×C199—21no—
66C2×C4×C199—21no—
67C2×C4×C199—41no—
68C2×C4×C199—41no—
69C2×C4×C199—41no—
70C2×C4×C199C2×D411yescharacteristic
71C2×C4×C199C2×D411yescharacteristic
72C2×C4×C199C2×D411yescharacteristic
73C8×C199—41no—
74C22×C199—41no—
75C22×C199—41no—
76C22×C199—41no—
77C22×C199#12011yescharacteristic
78C2×D199—22no—
79C2×D199—41no—
80C2×D199—41no—
81C4×C199—21no—
82C4×C199—41no—
83C4×C199—81no—
84C4×C199C2×SD1611yescharacteristic
85C4×C199#13511yescharacteristic
86C4×C199AGL(1,ℤ/8)11yescharacteristic
87C4×C199#13711yescharacteristic
88Dic199—42no—
89Dic199—41no—
90C2×C199—81no—
91C2×C199#45011yescharacteristic
92C2×C199#45211yescharacteristic
93C2×C199#45411yescharacteristic
94D199—42no—
95C199#163511yescharacteristic, complemented, Sylow
96#1635—1991nomaximal, Sylow
97#547—1991no—
98#487—1991no—
99#476—1991no—
100#476D19911yescharacteristic, complemented
101#332—1991no—
102D4⋊C8—1991no—
103D4⋊C8—1991no—
104C2×#41—3981no—
105C4⋊Q8—1991no—
106C4⋊Q8C2×D19911yescharacteristic
107#123—3982no—
108C22⋊Q8—3981no—
109#121—3981no—
110C4×D4—1991no—
111C4×D4C2×D19911yescharacteristic
112C2×C4⋊C4—3981no—
113#106—21no—
114#106—3981no—
115C4⋊C8—1991no—
116C4⋊C8C2×D19911yescharacteristic
117#103—3982no—
118#102—21no—
119#102—3981no—
120C4×C8—1991no—
121#41—7962no—
122#41—7962no—
123C2×Q8—21no—
124C2×Q8—3981no—
125C2×D4—3982no—
126C2×D4—1991no—
127C2×D4C22⋊D19911yescharacteristic
128C22×C4—21no—
129C22×C4—3982no—
130C22×C4—3981no—
131C2×C8—21no—
132C2×C8—3981no—
133C2×C8—3981no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—3982no—
137C4⋊C4—3981no—
138C4⋊C4—1992no—
139C4⋊C4—1991no—
140C4⋊C4C22⋊D19911yescharacteristic
141C22⋊C4—21no—
142C22⋊C4—3982no—
143C22⋊C4—3981no—
144C42C22×D19911yescharacteristic
145C23—21no—
146C23—3981no—
147Q8—42no—
148Q8—7961no—
149Q8—7961no—
150D4—21no—
151D4—21no—
152D4—7964no—
153D4—3981no—
154D4—1992no—
155C2×C4—21no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—7964no—
160C2×C4—7961no—
161C2×C4—7961no—
162C2×C4—3981no—
163C2×C4—3982no—
164C2×C4D4×D19911yescharacteristic
165C2×C4D4×D19911yescharacteristic
166C2×C4C2×C22⋊D19911yescharacteristic
167C8—41no—
168C8—7961no—
169C8—7961no—
170C22—41no—
171C22—41no—
172C22—41no—
173C22—7961no—
174C22—7961no—
175C22—3982no—
176C22#2026392311yescharacteristic
177C4—21no—
178C4—41no—
179C4—81no—
180C4—7962no—
181C4—7961no—
182C4SD16×D19911yescharacteristic
183C4#2026397511yescharacteristic
184C4#2026398211yescharacteristic
185C4#2026398611yescharacteristic
186C2—81no—
187C2—7962no—
188C2#2716416711yescharacteristic
189C2#2716417811yescharacteristic
190C2#2716418911yescharacteristic
subgroupG/H
centerC22#20263923
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#476×C199C2
socleC22×C199#120
prime radical#27164673C2
hypercenter#476D199
hypocenterC199#1635
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
2. #27164368C2semidirect
3. #27164305C2semidirect
4. #27163763C2semidirect
5. D4⋊(C199⋊C8)C2semidirect
6. #27163745C2semidirect
7. #476×C199C2semidirect
95. C199#1635semidirect
100. #476D199semidirect

Maximal subgroups

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

subgroupindex
#271646732
#271643682
#271643052
#271637632
D4⋊(C199⋊C8)2
#271637452
#476×C1992
#1635199