Finite Group Databasebeta

← #2888⋊C9 #11919537

Subgroups — #2888⋊C9 #11919537

Order 1152.

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

normal subgroups only

This group has 336 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
1C2×#63700C211yesmaximal, characteristic
2#568⋊C9C212yesmaximal, complemented
3C3×#2888C311yesmaximal, characteristic
4#63700—22no—
5C2×#41⋊C9C412yes—
6C2×#41⋊C9C2211yescharacteristic
7#131⋊C9—22no—
8C9×#101—41nomaximal
9C2×C3×H(5,𝔽2)C2×C311yescharacteristic
10C3×#568C2×C312yes—
11C3×#413—31no—
12C3×#413—31no—
13C2×C3×#100—32no—
14C2×Q8×C9—41no—
15#41⋊C9—22no—
16#41⋊C9D412yes—
17C2×Q8⋊C9C2×C411yescharacteristic
18C9×M16—42no—
19#2888C911yescharacteristic, complemented, Sylow
20C3×H(5,𝔽2)—22no—
21C3×H(5,𝔽2)—62no—
22C2×C3×#41—31no—
23C2×C3×#41C3×C412yes—
24C2×C3×#41C22×C311yescharacteristic
25C22×C3×D4—32no—
26C22×C3×D4—31no—
27C3×#131—22no—
28C3×#131—62no—
29C2×C3×M16—32no—
30C2×C3×M16—32no—
31C3×#101—31no—
32C3×#101A411yescharacteristic
33C3×#100—32no—
34C3×#100—32no—
35Q8×C9—82no—
36C2×C4×C9—42no—
37C2×C4×C9—41no—
38Q8⋊C9C22⋊C411yescharacteristic
39C8×C9—82no—
40C2×H(5,𝔽2)C2×C911yescharacteristic
41#568C2×C912yes—
42#413—31no—
43#413—31no—
44C2×#100—32no—
45C24×C3—32no—
46C3×#41—22no—
47C3×#41—32no—
48C3×#41—62no—
49C3×#41—62no—
50C3×#41—62no—
51C3×#41C3×D412yes—
52C2×C3×Q8C2×C3×C411yescharacteristic
53C2×C3×Q8C2×A411yescharacteristic
54C2×C3×D4—31no—
55C2×C3×D4—32no—
56C2×C3×D4—32no—
57C2×C3×D4—31no—
58C2×C3×D4—62no—
59C2×C3×D4—62no—
60C2×C3×D4—64no—
61C2×C3×D4—64no—
62C22×C3×C4—31no—
63C22×C3×C4—32no—
64C3×M16—32no—
65C3×M16—62no—
66C3×M16—62no—
67C3×M16C2×A412yes—
68C2×C3×C8—62no—
69C2×C3×C8—62no—
70C22×C9—41no—
71C4×C9—42no—
72C4×C9—82no—
73H(5,𝔽2)—22no—
74H(5,𝔽2)—62no—
75C2×#41—31no—
76C2×#41C4×C912yes—
77C2×#41C22×C911yescharacteristic
78C22×D4—31no—
79C22×D4—32no—
80#131—22no—
81#131—62no—
82C2×M16—32no—
83C2×M16—32no—
84#101—31no—
85#101C22⋊C911yescharacteristic
86#100—32no—
87#100—32no—
88C23×C3—31no—
89C23×C3—32no—
90C23×C3—64no—
91C23×C3—242no—
92C3×Q8—22no—
93C3×Q8—31no—
94C3×Q8C3×C22⋊C411yescharacteristic
95C3×D4—32no—
96C3×D4—32no—
97C3×D4—64no—
98C3×D4—62no—
99C3×D4—62no—
100C3×D4—62no—
101C2×C3×C4—31no—
102C2×C3×C4—32no—
103C2×C3×C4—32no—
104C2×C3×C4—62no—
105C2×C3×C4—62no—
106C2×C3×C4C4×A412yes—
107C2×C3×C4C22×A411yescharacteristic
108C3×C8—22no—
109C3×C8—62no—
110C2×C9—41no—
111C2×C9—81no—
112C24—32no—
113#41—22no—
114#41—32no—
115#41—62no—
116#41—62no—
117#41—62no—
118#41D4×C912yes—
119C2×Q8C2×C4×C911yescharacteristic
120C2×Q8C2×C22⋊C911yescharacteristic
121C2×D4—31no—
122C2×D4—32no—
123C2×D4—32no—
124C2×D4—31no—
125C2×D4—62no—
126C2×D4—62no—
127C2×D4—64no—
128C2×D4—64no—
129C22×C4—31no—
130C22×C4—32no—
131M16—32no—
132M16—62no—
133M16—62no—
134M16C2×C22⋊C912yes—
135C2×C8—62no—
136C2×C8—62no—
137C22×C3—32no—
138C22×C3—62no—
139C22×C3—121no—
140C22×C3—122no—
141C22×C3—244no—
142C22×C3C2×C4×A411yescharacteristic
143C3×C4—22no—
144C3×C4—31no—
145C3×C4—31no—
146C3×C4D4×A412yes—
147C9—41noSylow
148C23—31no—
149C23—32no—
150C23—64no—
151C23—242no—
152Q8—22no—
153Q8—31no—
154Q8C9×C22⋊C411yescharacteristic
155D4—32no—
156D4—32no—
157D4—64no—
158D4—62no—
159D4—62no—
160D4—62no—
161C2×C4—31no—
162C2×C4—32no—
163C2×C4—32no—
164C2×C4—62no—
165C2×C4—62no—
166C2×C4C4×C22⋊C912yes—
167C2×C4C22×C22⋊C911yescharacteristic
168C8—22no—
169C8—62no—
170C2×C3—21no—
171C2×C3—62no—
172C2×C3—122no—
173C2×C3A4×C22⋊C411yescharacteristic
174C22—32no—
175C22—62no—
176C22—121no—
177C22—122no—
178C22—244no—
179C22C2×C4×C22⋊C911yescharacteristic
180C4—22no—
181C4—31no—
182C4—31no—
183C4D4×C22⋊C912yes—
184C3#2888⋊C311yescharacteristic
185C2—21no—
186C2—62no—
187C2—122no—
188C2C22⋊C4×C22⋊C911yescharacteristic
subgroupG/H
center, socleC2×C3A4×C22⋊C4
commutatorC2×Q8C2×C4×C9
FrattiniC2×C3×C4C22×A4
FittingC3×#2888C3
prime radicalC2×C3×H(5,𝔽2)C2×C3
cosocleC2×C3×#41C22×C3
hypercenterC3×#101A4
hypocenterQ8C9×C22⋊C4
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#101

Complements

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

NH ≅ G/Ntype
2. #568⋊C9C2semidirect
19. #2888C9semidirect

Maximal subgroups

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

subgroupindex
C2×#637002
#568⋊C92
C3×#28883
C9×#1014