Finite Group Databasebeta

← #38767954

Subgroups — #38767954

Order 25472.

250 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 57 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#27164683C211yesmaximal, characteristic, complemented
2#27164366C211yesmaximal, characteristic, complemented
3#27164219C211yesmaximal, characteristic, complemented
4#27164214C211yesmaximal, characteristic
5#27163748C211yesmaximal, characteristic, complemented
6#27163745C211yesmaximal, characteristic, complemented
7#332×C199C211yesmaximal, characteristic, complemented
8C2×Q8×D199—21no—
9C2×#15691875—21no—
10C2×D1592—21no—
11#20263954—21no—
12#20263952C2211yescharacteristic
13#20263933—22no—
14#20263931—21no—
15C4×Dic398C2211yescharacteristic
16C4×D796C2211yescharacteristic
17#20263917—21no—
18#20263877—21no—
19#20263877—21no—
20#20263875—22no—
21C4×C199⋊C8C2211yescharacteristic
22C4⋊Q8×C199C2211yescharacteristic
23C4⋊C8×C199C2211yescharacteristic
24C4⋊C8×C199C2211yescharacteristic
25C2×Dic398—22no—
26C2×Dic398D411yescharacteristic
27Q8×D199—42no—
28Q8×D199—42no—
29C2×D796—21no—
30C2×D796—21no—
31C2×D796—21no—
32C2×D796D411yescharacteristic
33C2×C4×D199—22no—
34C2×C4×D199—21no—
35#15691875—42no—
36D1592—42no—
37C2×C199⋊C8—21no—
38C4⋊Dic199D411yescharacteristic
39C4⋊Dic199D411yescharacteristic
40Dic199⋊C4—22no—
41Dic199⋊C4—21no—
42(C2×D199)⋊C4—22no—
43(C2×D199)⋊C4—21no—
44C4×Dic199—21no—
45C2×Q8×C199—21no—
46C2×C8×C199—21no—
47C2×C8×C199—21no—
48C4⋊C4×C199—21no—
49C4⋊C4×C199D412yes—
50C42×C199C2311yescharacteristic
51C22×D199—21no—
52C22×D199—41no—
53Dic398—21no—
54Dic398—44no—
55Dic398D812yescomplemented
56D796—21no—
57D796—42no—
58D796—41no—
59D796—41no—
60D796—41no—
61D796—41no—
62D796SD1612yescomplemented
63C2×Dic199—21no—
64C2×Dic199—21no—
65C2×Dic199—21no—
66C4×D199—44no—
67C4×D199—41no—
68C4×D199—41no—
69C199⋊C8—41no—
70Q8×C199—41no—
71Q8×C199—41no—
72C2×C4×C199—22no—
73C2×C4×C199C2×D411yescharacteristic
74C2×C4×C199C2×D411yescharacteristic
75C2×C4×C199C2×D411yescharacteristic
76C8×C199—41no—
77C8×C199—41no—
78C22×C199#12011yescharacteristic
79C2×D199—22no—
80C2×D199—41no—
81C2×D199—41no—
82C2×D199—81no—
83C2×D199—81no—
84C2×D199—81no—
85C4×C199—21no—
86C4×C199—42no—
87C4×C199C2×D811yescharacteristic
88C4×C199C2×SD1611yescharacteristic
89C4×C199AGL(1,ℤ/8)11yescharacteristic
90C4×C199#13711yescharacteristic
91Dic199—22no—
92Dic199—41no—
93Dic199—41no—
94C2×C199#44711yescharacteristic
95C2×C199#44811yescharacteristic
96C2×C199#45411yescharacteristic
97D199—42no—
98D199—161no—
99C199#162711yescharacteristic, complemented, Sylow
100#1627—1991nomaximal, Sylow
101D4×Q8—1991no—
102#486—1991no—
103#460—1991no—
104#459—1991no—
105#332D19911yescharacteristic, complemented
106Q8⋊C8—1991no—
107D4⋊C8—1991no—
108C22×Q8—3981no—
109C2×SD16—3981no—
110C2×D8—3981no—
111C4⋊Q8—3981no—
112C4⋊Q8C2×D19911yescharacteristic
113#127—1991no—
114C22⋊Q8—3981no—
115C22⋊Q8—3982no—
116C4×Q8—1991no—
117C4×D4—3981no—
118C4×D4—1991no—
119C4⋊C8C2×D19911yescharacteristic
120C4⋊C8C2×D19911yescharacteristic
121#102—3981no—
122#102—3981no—
123#102—3982no—
124C4×C8—1991no—
125C2×Q8—21no—
126C2×Q8—7962no—
127C2×Q8—7962no—
128C2×Q8—3982no—
129C2×Q8—1991no—
130C2×D4—3981no—
131C2×D4—3981no—
132C2×D4—3981no—
133C2×D4—1991no—
134C22×C4—3982no—
135C22×C4—3981no—
136SD16—7962no—
137D8—7962no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—3981no—
141C4⋊C4—21no—
142C4⋊C4—3982no—
143C4⋊C4—3981no—
144C4⋊C4—1991no—
145C4⋊C4—1991no—
146C4⋊C4D79612yes—
147C22⋊C4—3982no—
148C22⋊C4—3981no—
149C42—3981no—
150C42C22×D19911yescharacteristic
151C23—7961no—
152C23—3981no—
153Q8—41no—
154Q8—41no—
155Q8—7964no—
156Q8—3981no—
157Q8—1992no—
158D4—7962no—
159D4—7961no—
160D4—7961no—
161D4—7961no—
162D4—7961no—
163D4—3981no—
164D4—1992no—
165C2×C4—22no—
166C2×C4—7964no—
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—7961no—
179C22—7961no—
180C22—15921no—
181C22—15921no—
182C22—15921no—
183C22—3982no—
184C22#2026392511yescharacteristic
185C4—21no—
186C4—42no—
187C4—7961no—
188C4—7961no—
189C4—3982no—
190C4D8×D19911yescharacteristic
191C4SD16×D19911yescharacteristic
192C4#2026398211yescharacteristic
193C4#2026398711yescharacteristic
194C2—7962no—
195C2—31841no—
196C2#2716415311yescharacteristic
197C2#2716415811yescharacteristic
198C2#2716419311yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#332×C199C2
socleC22×C199#120
prime radical#27164214C2
hypercenter#332D199
hypocenterC199#1627
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
1. #27164683C2semidirect
2. #27164366C2semidirect
3. #27164219C2semidirect
5. #27163748C2semidirect
6. #27163745C2semidirect
7. #332×C199C2semidirect
55. Dic398D8semidirect
62. D796SD16semidirect
99. C199#1627semidirect
105. #332D199semidirect

Maximal subgroups

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

subgroupindex
#271646832
#271643662
#271642192
#271642142
#271637482
#271637452
#332×C1992
#1627199