Finite Group Databasebeta

← #38768085

Subgroups — #38768085

Order 25472.

240 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 61 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#27164421C211yesmaximal, characteristic, complemented
2#27164298C211yesmaximal, characteristic, complemented
3#27164223C211yesmaximal, characteristic, complemented
4#27163764C211yesmaximal, characteristic, complemented
5D4⋊(C199⋊C8)C211yesmaximal, characteristic, complemented
6D4⋊(C199⋊C8)C211yesmaximal, characteristic, complemented
7#547×C199C211yesmaximal, characteristic
8C2×#15691874—21no—
9#20263955C2211yescharacteristic
10#20263888—22no—
11#20263888—21no—
12C4⋊(C199⋊C8)C2211yescharacteristic
13C4⋊(C199⋊C8)C2211yescharacteristic
14#20263878—21no—
15#20263876—21no—
16C4×C199⋊C8C2211yescharacteristic
17C2×#41×C199—21no—
18C4⋊Q8×C199C2211yescharacteristic
19#123×C199—22no—
20C22⋊Q8×C199—21no—
21#121×C199—21no—
22C4×D4×C199C2211yescharacteristic
23C4×D4×C199C2211yescharacteristic
24C2×C4⋊C4×C199—21no—
25C2×Dic398—21no—
26#15691874—42no—
27C2×C199⋊C8—21no—
28C2×C199⋊C8—21no—
29C2×C199⋊C8—21no—
30C4⋊Dic199—21no—
31C4⋊Dic199—21no—
32#41×C199—42no—
33#41×C199—42no—
34C2×Q8×C199—21no—
35C2×D4×C199—22no—
36C2×D4×C199D411yescharacteristic
37C2×D4×C199D411yescharacteristic
38C22×C4×C199—22no—
39C22×C4×C199—21no—
40C22×C4×C199—21no—
41C4⋊C4×C199—22no—
42C4⋊C4×C199—21no—
43C4⋊C4×C199D411yescharacteristic
44C4⋊C4×C199D411yescharacteristic
45C4⋊C4×C199D412yes—
46C22⋊C4×C199—22no—
47C22⋊C4×C199—21no—
48C22⋊C4×C199—21no—
49C42×C199C2311yescharacteristic
50Dic398—41no—
51Dic398—41no—
52C2×Dic199—41no—
53C199⋊C8—22no—
54C199⋊C8—41no—
55C199⋊C8—41no—
56C23×C199—21no—
57C23×C199—21no—
58Q8×C199—41no—
59Q8×C199—41no—
60D4×C199—21no—
61D4×C199—21no—
62D4×C199—21no—
63D4×C199—44no—
64D4×C199D812yes—
65C2×C4×C199—21no—
66C2×C4×C199—21no—
67C2×C4×C199—22no—
68C2×C4×C199—44no—
69C2×C4×C199—41no—
70C2×C4×C199—41no—
71C2×C4×C199—41no—
72C2×C4×C199—41no—
73C2×C4×C199C2×D411yescharacteristic
74C2×C4×C199C2×D411yescharacteristic
75C2×C4×C199C2×D411yescharacteristic
76C22×C199—22no—
77C22×C199—41no—
78C22×C199—41no—
79C22×C199—41no—
80C22×C199—41no—
81C22×C199—41no—
82C22×C199#12011yescharacteristic
83C4×C199—21no—
84C4×C199—41no—
85C4×C199—42no—
86C4×C199—41no—
87C4×C199C2×D811yescharacteristic
88C4×C199#13511yescharacteristic
89C4×C199AGL(1,ℤ/8)11yescharacteristic
90C4×C199#13711yescharacteristic
91Dic199—81no—
92C2×C199—42no—
93C2×C199—81no—
94C2×C199#44711yescharacteristic
95C2×C199#45211yescharacteristic
96C2×C199#45611yescharacteristic
97C199#164711yescharacteristic, complemented, Sylow
98#1647—1991nomaximal, Sylow
99#547D19911yescharacteristic
100#500—1991no—
101#474—1991no—
102#461—1991no—
103#332—1991no—
104D4⋊C8—1991no—
105D4⋊C8—1991no—
106C2×#41—21no—
107C2×SD16—3981no—
108C4⋊Q8—1991no—
109C4⋊Q8C2×D19911yescharacteristic
110#123—22no—
111C22⋊Q8—21no—
112#121—21no—
113C4×D4C2×D19911yescharacteristic
114C4×D4C2×D19911yescharacteristic
115C2×C4⋊C4—21no—
116#107—3981no—
117#107—3982no—
118C4⋊C8—1991no—
119C4⋊C8—1991no—
120#103—3981no—
121#102—3981no—
122C4×C8—1991no—
123#41—42no—
124#41—42no—
125C2×Q8—21no—
126C2×Q8—3981no—
127C2×D4—22no—
128C2×D4C22⋊D19911yescharacteristic
129C2×D4C22⋊D19911yescharacteristic
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—21no—
133SD16—7962no—
134C2×C8—3981no—
135C2×C8—3981no—
136C2×C8—3981no—
137C4⋊C4—22no—
138C4⋊C4—21no—
139C4⋊C4—3981no—
140C4⋊C4—3981no—
141C4⋊C4C22⋊D19911yescharacteristic
142C4⋊C4C22⋊D19911yescharacteristic
143C4⋊C4C22⋊D19912yes—
144C22⋊C4—22no—
145C22⋊C4—21no—
146C22⋊C4—21no—
147C42C22×D19911yescharacteristic
148C23—21no—
149C23—21no—
150Q8—41no—
151Q8—41no—
152Q8—7961no—
153Q8—7961no—
154D4—21no—
155D4—21no—
156D4—21no—
157D4—44no—
158D4#1569187112yes—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—44no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—7961no—
168C2×C4C2×C22⋊D19911yescharacteristic
169C2×C4C2×C22⋊D19911yescharacteristic
170C2×C4C2×C22⋊D19911yescharacteristic
171C8—7961no—
172C8—7961no—
173C8—3982no—
174C22—22no—
175C22—41no—
176C22—41no—
177C22—41no—
178C22—41no—
179C22—41no—
180C22#2026392411yescharacteristic
181C4—21no—
182C4—41no—
183C4—41no—
184C4—42no—
185C4—15921no—
186C4C2×#1569187111yescharacteristic
187C4#2026397511yescharacteristic
188C4#2026398111yescharacteristic
189C4#2026398511yescharacteristic
190C2—42no—
191C2—81no—
192C2#2716415211yescharacteristic
193C2#2716418011yescharacteristic
194C2#2716420111yescharacteristic
subgroupG/H
centerC22#20263924
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting, prime radical#547×C199C2
socleC22×C199#120
hypercenter#547D199
hypocenterC199#1647
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #27164421C2semidirect
2. #27164298C2semidirect
3. #27164223C2semidirect
4. #27163764C2semidirect
5. D4⋊(C199⋊C8)C2semidirect
6. D4⋊(C199⋊C8)C2semidirect
97. C199#1647semidirect

Maximal subgroups

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

subgroupindex
#271644212
#271642982
#271642232
#271637642
D4⋊(C199⋊C8)2
#547×C1992
#1647199