Finite Group Databasebeta

← #38767936

Subgroups — #38767936

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#27164671C211yesmaximal, characteristic, complemented
2#27164210C211yesmaximal, characteristic, complemented
3#27164210C211yesmaximal, characteristic, complemented
4#27163764C211yesmaximal, characteristic, complemented
5#27163747C211yesmaximal, characteristic, complemented
6#27163747C211yesmaximal, characteristic, complemented
7#486×C199C211yesmaximal, characteristic, complemented
8C2×#15691887—21no—
9C2×#15691871—21no—
10C2×#15691871—21no—
11#20263940—22no—
12#20263931—21no—
13#20263930—21no—
14C4×D796C2211yescharacteristic
15C4×D796C2211yescharacteristic
16C4⋊C4×D199—21no—
17C4⋊(C199⋊C8)C2211yescharacteristic
18C4⋊(C199⋊C8)C2211yescharacteristic
19#20263876—21no—
20#20263876—21no—
21C4⋊Q8×C199C2211yescharacteristic
22#127×C199C2211yescharacteristic
23#102×C199—22no—
24C4×C8×C199C2211yescharacteristic
25#15691887—42no—
26#15691887—42no—
27C2×D796—22no—
28C2×D796D411yescharacteristic
29C2×D796D411yescharacteristic
30C2×C4×D199—22no—
31C2×C4×D199—21no—
32C2×C4×D199—21no—
33#15691871—42no—
34#15691871—42no—
35C2×C199⋊C8—21no—
36C2×C199⋊C8—21no—
37C4⋊Dic199D411yescharacteristic
38C4⋊Dic199D411yescharacteristic
39Dic199⋊C4—22no—
40(C2×D199)⋊C4—22no—
41(C2×D199)⋊C4—21no—
42(C2×D199)⋊C4—21no—
43C2×Q8×C199—21no—
44C2×D4×C199—21no—
45C2×D4×C199—21no—
46C2×D4×C199—21no—
47C2×C8×C199—21no—
48C4⋊C4×C199—21no—
49C4⋊C4×C199D412yes—
50C42×C199C2311yescharacteristic
51C22×D199—21no—
52C22×D199—21no—
53D796—21no—
54D796—21no—
55D796—21no—
56D796—44no—
57D796D812yescomplemented
58C2×Dic199—21no—
59C2×Dic199—21no—
60C4×D199—44no—
61C4×D199—41no—
62C4×D199—41no—
63C4×D199—41no—
64C4×D199—41no—
65C199⋊C8—41no—
66C199⋊C8—41no—
67C23×C199—41no—
68Q8×C199—41no—
69Q8×C199—41no—
70D4×C199—42no—
71D4×C199—41no—
72D4×C199—41no—
73D4×C199—41no—
74D4×C199—41no—
75C2×C4×C199—22no—
76C2×C4×C199C2×D411yescharacteristic
77C2×C4×C199C2×D411yescharacteristic
78C2×C4×C199C2×D411yescharacteristic
79C8×C199—41no—
80C22×C199—81no—
81C22×C199—81no—
82C22×C199—81no—
83C22×C199#12011yescharacteristic
84C2×D199—22no—
85C2×D199—41no—
86C2×D199—41no—
87C2×D199—41no—
88C2×D199—41no—
89C2×D199—41no—
90C4×C199—21no—
91C4×C199—42no—
92C4×C199C2×D811yescharacteristic
93C4×C199#13511yescharacteristic
94C4×C199AGL(1,ℤ/8)11yescharacteristic
95C4×C199AGL(1,ℤ/8)11yescharacteristic
96Dic199—41no—
97Dic199—41no—
98C2×C199—161no—
99C2×C199#44711yescharacteristic
100C2×C199#44911yescharacteristic
101C2×C199#45411yescharacteristic
102D199—42no—
103D199—81no—
104C199#162511yescharacteristic, complemented, Sylow
105#1625—1991nomaximal, Sylow
106#547—1991no—
107#486D19911yescharacteristic, complemented
108#459—1991no—
109#459—1991no—
110#332—1991no—
111D4⋊C8—1991no—
112D4⋊C8—1991no—
113C2×#41—3981no—
114C2×D8—3981no—
115C2×D8—3981no—
116C4⋊Q8C2×D19911yescharacteristic
117#127C2×D19911yescharacteristic
118#123—3982no—
119C22⋊Q8—3981no—
120#121—3981no—
121C4×D4—1991no—
122C4×D4—1991no—
123C2×C4⋊C4—3981no—
124C4⋊C8—1991no—
125C4⋊C8—1991no—
126#102—22no—
127#102—3981no—
128#102—3981no—
129C4×C8C2×D19911yescharacteristic
130#41—7962no—
131#41—7962no—
132C2×Q8—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—3982no—
137C2×D4—1991no—
138C2×D4—1991no—
139C22×C4—3982no—
140C22×C4—3981no—
141C22×C4—3981no—
142D8—7962no—
143D8—7962no—
144C2×C8—21no—
145C2×C8—3981no—
146C2×C8—3981no—
147C4⋊C4—21no—
148C4⋊C4—3982no—
149C4⋊C4—1991no—
150C4⋊C4—1991no—
151C4⋊C4C22⋊D19912yes—
152C22⋊C4—3982no—
153C22⋊C4—3981no—
154C22⋊C4—3981no—
155C42C22×D19911yescharacteristic
156C23—41no—
157C23—3981no—
158C23—3981no—
159Q8—41no—
160Q8—41no—
161D4—42no—
162D4—41no—
163D4—41no—
164D4—41no—
165D4—41no—
166D4—7964no—
167D4—3981no—
168D4—3981no—
169D4—3981no—
170D4—1992no—
171C2×C4—22no—
172C2×C4—7964no—
173C2×C4—7961no—
174C2×C4—7961no—
175C2×C4—7961no—
176C2×C4—7961no—
177C2×C4—3981no—
178C2×C4—3981no—
179C2×C4D4×D19911yescharacteristic
180C2×C4D4×D19911yescharacteristic
181C2×C4C2×C22⋊D19911yescharacteristic
182C8—41no—
183C8—7961no—
184C8—7961no—
185C22—81no—
186C22—81no—
187C22—81no—
188C22—7961no—
189C22—7961no—
190C22—7961no—
191C22—7961no—
192C22—7961no—
193C22—3982no—
194C22#2026392311yescharacteristic
195C4—21no—
196C4—42no—
197C4—7961no—
198C4—7961no—
199C4D8×D19911yescharacteristic
200C4#2026397611yescharacteristic
201C4#2026397911yescharacteristic
202C4#2026398211yescharacteristic
203C2—161no—
204C2—7962no—
205C2—15921no—
206C2#2716415011yescharacteristic
207C2#2716416211yescharacteristic
208C2#2716419011yescharacteristic
subgroupG/H
centerC22#20263923
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#486×C199C2
socleC22×C199#120
hypercenter#486D199
hypocenterC199#1625
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #27164671C2semidirect
2. #27164210C2semidirect
3. #27164210C2semidirect
4. #27163764C2semidirect
5. #27163747C2semidirect
6. #27163747C2semidirect
7. #486×C199C2semidirect
57. D796D8semidirect
104. C199#1625semidirect
107. #486D199semidirect

Maximal subgroups

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

subgroupindex
#271646712
#271642102
#271637642
#271637472
#486×C1992
#1625199