Finite Group Databasebeta

← #10588222⋊C3 #13135631

Subgroups — #10588222⋊C3 #13135631

Order 1728.

312 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 56 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
13−1+2×#127C211yesmaximal, characteristic
2#63393⋊C3C212yesmaximal, complemented
3C3×#4545—31nomaximal
4#10588222C311yesmaximal, characteristic, complemented
5C2×D4×3−1+2—22no—
6C2×D4×3−1+2C411yescharacteristic, complemented
7C2×D4×3−1+2C2211yescharacteristic
8C42×3−1+2C411yescharacteristic, complemented
9#3676⋊C3—22no—
10C32×#127D311yescharacteristic
11C3×#834—32no—
12C9×#127—21no—
13C9×#127C2×C311yescharacteristic
14#63393C2×C312yescomplemented
15C23×3−1+2—21no—
16C23×3−1+2D412yes—
17D4×3−1+2—41no—
18D4×3−1+2—44no—
19D4×3−1+2—41no—
20C2×C4×3−1+2—21no—
21C2×C4×3−1+2C2×C411yescharacteristic
22C2×3−1+2⋊C4—42no—
23C3×#353—91no—
24#4545—31no—
25C2×C32×D4—22no—
26C2×C32×D4Dic311yescharacteristic
27C2×C32×D4C2×D311yescharacteristic
28C32×C42Dic311yescharacteristic
29C3×#217—62no—
30C2×D4×C9—24no—
31C2×D4×C9—21no—
32C2×D4×C9—21no—
33C2×D4×C9—22no—
34C2×D4×C9C3×C411yescharacteristic, complemented
35C2×D4×C9C22×C311yescharacteristic
36C42×C9—21no—
37C42×C9C3×C411yescharacteristic, complemented
38#3676—22no—
39C22×3−1+2—22no—
40C22×3−1+2—42no—
41C22×3−1+2—42no—
42C22×3−1+2—41no—
43C22×3−1+2C22⋊C411yescharacteristic
44C4×3−1+2—22no—
45C4×3−1+2—21no—
463−1+2⋊C4—82no—
47C3×#127—31no—
48C3×#127C3×D311yescharacteristic
49C3×#99—92no—
50#834—32no—
51C23×C32—21no—
52C23×C32#6712yes—
53C32×D4—41no—
54C32×D4—41no—
55C32×D4—44no—
56C2×C32×C4—21no—
57C2×C32×C4C2×Dic311yescharacteristic
58C2×C3×Dic3—122no—
59C23×C9—22no—
60C23×C9—21no—
61C23×C9—22no—
62C23×C9C3×D412yes—
63D4×C9—41no—
64D4×C9—41no—
65D4×C9—42no—
66D4×C9—42no—
67D4×C9—44no—
68D4×C9—48no—
69C2×C4×C9—22no—
70C2×C4×C9—21no—
71C2×C4×C9—21no—
72C2×C4×C9C2×C3×C411yescharacteristic
73C2×Dic9—42no—
74#353—91noSylow
75C2×3−1+2—21no—
76C2×3−1+2—42no—
77C2×3−1+2—81no—
78C2×3−1+2#9911yescharacteristic
79C2×C3×D4—22no—
80C2×C3×D4—31no—
81C2×C3×D4—31no—
82C2×C3×D4—62no—
83C2×C3×D4C3×Dic311yescharacteristic
84C2×C3×D4C2×C3×D311yescharacteristic
85C3×C22⋊C4—182no—
86C3×C42—31no—
87C3×C42C3×Dic311yescharacteristic
88#217—62no—
89C22×C32—22no—
90C22×C32—42no—
91C22×C32—42no—
92C22×C32—41no—
93C22×C32#21711yescharacteristic
94C32×C4—22no—
95C32×C4—21no—
96C3×Dic3—242no—
97C22×C9—22no—
98C22×C9—21no—
99C22×C9—42no—
100C22×C9—42no—
101C22×C9—44no—
102C22×C9—44no—
103C22×C9—42no—
104C22×C9—42no—
105C22×C9—41no—
106C22×C9C3×C22⋊C411yescharacteristic
107C4×C9—24no—
108C4×C9—22no—
109C4×C9—22no—
110C4×C9—21no—
111Dic9—82no—
112#127AGL(1,ℤ/9)11yescharacteristic
113#99—92no—
1143−1+2#35311yescharacteristic, complemented, Sylow
115C23×C3—21no—
116C23×C3—32no—
117C23×C3—61no—
118C23×C3C3×#6712yes—
119C3×D4—41no—
120C3×D4—41no—
121C3×D4—44no—
122C3×D4—121no—
123C3×D4—121no—
124C3×D4—124no—
125C2×C3×C4—21no—
126C2×C3×C4—31no—
127C2×C3×C4—61no—
128C2×C3×C4—362no—
129C2×C3×C4C2×C3×Dic311yescharacteristic
130C2×Dic3—122no—
131C2×C32—21no—
132C2×C32—42no—
133C2×C32—81no—
134C2×C32#83411yescharacteristic
135C2×C9—22no—
136C2×C9—21no—
137C2×C9—21no—
138C2×C9—42no—
139C2×C9—82no—
140C2×C9—82no—
141C2×C9—81no—
142C2×C9C3×#9911yescharacteristic
143C2×D4—22no—
144C2×D43−1+2⋊C411yescharacteristic, complemented
145C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
146C22⋊C4—182no—
147C423−1+2⋊C411yescharacteristic, complemented
148C22×C3—22no—
149C22×C3—31no—
150C22×C3—42no—
151C22×C3—42no—
152C22×C3—41no—
153C22×C3—62no—
154C22×C3—121no—
155C22×C3—122no—
156C22×C3—122no—
157C22×C3C3×#21711yescharacteristic
158C3×C4—22no—
159C3×C4—21no—
160C3×C4—61no—
161C3×C4—62no—
162C3×C4—722no—
163Dic3—242no—
164C32#454511yescharacteristic
165C9—21no—
166C9C3×#35311yescharacteristic, complemented
167C23—21no—
168C23#611⋊C312yes—
169D4—41no—
170D4—41no—
171D4—44no—
172C2×C4—21no—
173C2×C4—362no—
174C2×C4C2×3−1+2⋊C411yescharacteristic
175C2×C3—21no—
176C2×C3—31no—
177C2×C3—42no—
178C2×C3—61no—
179C2×C3—81no—
180C2×C3—122no—
181C2×C3—241no—
182C2×C3C3×#83411yescharacteristic
183C22—22no—
184C22—42no—
185C22—42no—
186C22—41no—
187C22#3676⋊C311yescharacteristic
188C4—22no—
189C4—21no—
190C4—722no—
191C3—31no—
192C3C3×#454511yescharacteristic
193C2—21no—
194C2—42no—
195C2—81no—
196C2#63393⋊C311yescharacteristic
subgroupG/H
centerC2#63393⋊C3
commutatorC2×C4×C9C2×C3×C4
FrattiniC2×C3×D4C2×C3×D3
Fitting3−1+2×#127C2
socleC2×C3C3×#834
prime radicalC32×#127D3
cosocleC2×D4×C9C22×C3
hypercenter#127AGL(1,ℤ/9)
hypocenterC9C3×#353
LCSG ▷ C2×C4×C9 ▷ C22×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #127

Complements

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

NH ≅ G/Ntype
2. #63393⋊C3C2semidirect
4. #10588222C3semidirect
6. C2×D4×3−1+2C4semidirect
8. C42×3−1+2C4semidirect
14. #63393C2×C3semidirect
34. C2×D4×C9C3×C4semidirect
37. C42×C9C3×C4semidirect
114. 3−1+2#353semidirect
144. C2×D43−1+2⋊C4semidirect
147. C423−1+2⋊C4semidirect
166. C9C3×#353semidirect

Maximal subgroups

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

subgroupindex
3−1+2×#1272
#63393⋊C32
C3×#45453
#105882223