Finite Group Databasebeta

← #38768044

Subgroups — #38768044

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#27164688C211yesmaximal, characteristic
2#27164364C211yesmaximal, characteristic, complemented
3#27164299C211yesmaximal, characteristic, complemented
4#27164229C211yesmaximal, characteristic, complemented
5#27163748C211yesmaximal, characteristic, complemented
6#27163745C211yesmaximal, characteristic, complemented
7#331×C199C211yesmaximal, characteristic, complemented
8C2×#15691885—21no—
9C2×Dic796—21no—
10#20263955C2211yescharacteristic
11#20263951—21no—
12#20263927—22no—
13#20263926—21no—
14C4×Dic398C2211yescharacteristic
15C4×D796C2211yescharacteristic
16D4×Dic199—21no—
17#20263889—21no—
18#20263880—21no—
19#20263877—21no—
20#20263876—22no—
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—
34Dic796—42no—
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—41no—
53Dic398—41no—
54Dic398D812yes—
55D796—21no—
56D796—21no—
57C22⋊D199—44no—
58C22⋊D199—82no—
59C2×Dic199—21no—
60C2×Dic199—21no—
61C2×Dic199—21no—
62C2×Dic199—44no—
63C2×Dic199—41no—
64C4×D199—41no—
65C4×D199—41no—
66C199⋊C8—41no—
67C23×C199—22no—
68D4×C199—42no—
69D4×C199—41no—
70D4×C199—41no—
71D4×C199—41no—
72D4×C199—41no—
73C2×C4×C199C2×D411yescharacteristic
74C2×C4×C199C2×D411yescharacteristic
75C2×C4×C199C2×D411yescharacteristic
76C8×C199—41no—
77C8×C199—41no—
78C22×C199—42no—
79C22×C199—42no—
80C22×C199—42no—
81C22×C199#12011yescharacteristic
82C2×D199—41no—
83C2×D199—41no—
84C2×D199—41no—
85C4×C199—21no—
86C4×C199C2×D811yescharacteristic
87C4×C199#13511yescharacteristic
88C4×C199AGL(1,ℤ/8)11yescharacteristic
89C4×C199#13711yescharacteristic
90Dic199—22no—
91Dic199—41no—
92Dic199—41no—
93Dic199—81no—
94C2×C199—82no—
95C2×C199#44711yescharacteristic
96C2×C199#45211yescharacteristic
97C2×C199#45511yescharacteristic
98D199—81no—
99C199#164111yescharacteristic, complemented, Sylow
100#1641—1991nomaximal, Sylow
101#550—1991no—
102#486—1991no—
103#474—1991no—
104#462—1991no—
105#331D19911yescharacteristic, complemented
106Q8⋊C8—1991no—
107D4⋊C8—1991no—
108C2×#41—3981no—
109C2×Q16—3981no—
110C4⋊Q8—1991no—
111#127—3981no—
112#127C2×D19911yescharacteristic
113#121—3982no—
114#121—3981no—
115C4×Q8—1991no—
116C4×D4—3981no—
117C4×D4—1991no—
118#107—3981no—
119C4⋊C8C2×D19911yescharacteristic
120C4⋊C8C2×D19911yescharacteristic
121#103—3981no—
122#102—3981no—
123#102—3982no—
124C4×C8—1991no—
125#41—7962no—
126#41—7962no—
127C2×Q8—3981no—
128C2×Q8—1991no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—3982no—
132C2×D4—3982no—
133C2×D4—1991no—
134C2×D4D79612yes—
135C22×C4—3982no—
136C22×C4—3981no—
137Q16—7962no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—3981no—
141C4⋊C4—3981no—
142C4⋊C4—3981no—
143C4⋊C4—3981no—
144C4⋊C4—1991no—
145C4⋊C4—1991no—
146C22⋊C4—3982no—
147C22⋊C4—3981no—
148C42—3981no—
149C42C22×D19911yescharacteristic
150C23—22no—
151C23—3981no—
152Q8—7961no—
153Q8—7961no—
154Q8—3981no—
155Q8—1992no—
156D4—42no—
157D4—41no—
158D4—41no—
159D4—41no—
160D4—41no—
161D4—7964no—
162D4—15922no—
163D4—3981no—
164D4—3981no—
165C2×C4—7964no—
166C2×C4—7961no—
167C2×C4—7961no—
168C2×C4—7961no—
169C2×C4—3981no—
170C2×C4—3981no—
171C2×C4—3981no—
172C2×C4D4×D19911yescharacteristic
173C2×C4D4×D19911yescharacteristic
174C2×C4C2×D79611yescharacteristic
175C8—41no—
176C8—41no—
177C8—7961no—
178C22—42no—
179C22—42no—
180C22—42no—
181C22—7961no—
182C22—7961no—
183C22—7961no—
184C22#2026392511yescharacteristic
185C4—21no—
186C4—7961no—
187C4—7961no—
188C4—15921no—
189C4—3982no—
190C4D8×D19911yescharacteristic
191C4#2026397411yescharacteristic
192C4#2026398211yescharacteristic
193C4#2026398411yescharacteristic
194C2—82no—
195C2—15921no—
196C2#2716415311yescharacteristic
197C2#2716418111yescharacteristic
198C2#2716419811yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#331×C199C2
socleC22×C199#120
prime radical#27164688C2
hypercenter#331D199
hypocenterC199#1641
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. #27164364C2semidirect
3. #27164299C2semidirect
4. #27164229C2semidirect
5. #27163748C2semidirect
6. #27163745C2semidirect
7. #331×C199C2semidirect
99. C199#1641semidirect
105. #331D199semidirect

Maximal subgroups

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

subgroupindex
#271646882
#271643642
#271642992
#271642292
#271637482
#271637452
#331×C1992
#1641199