Finite Group Databasebeta

← #13137265

Subgroups — #13137265

Order 1728.

412 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C3×#63437C211yesmaximal, characteristic
2C9×#869C211yesmaximal, characteristic, complemented
3C2×#88427C212yesmaximal, complemented
4C2×#88425C212yesmaximal
5#11720516C211yesmaximal, characteristic, complemented
6#10590376—31nomaximal
7#10588771—31nomaximal
8C2×C3×D36C2211yescharacteristic
9C2×C9×Dic6C2212yes—
10C2×C3×Dic18C2211yescharacteristic
11C3×C42×C9C2211yescharacteristic
12C2×C3⋊(C9⋊C8)C2212yes—
13C3×(C2×D9)⋊C4—22no—
14C9×C4⋊Dic3—22no—
15#88427—24no—
16#88425—24no—
17C3×#875—31no—
18C3×#869D311yescharacteristic
19C2×#3718—32no—
20C2×#3715—32no—
21#63630—31no—
22C9×C4⋊Q8—31no—
23C2×#3675—32no—
24C2×#3674—32no—
25#63437D311yescharacteristic
26#63362—31no—
27#63362—61no—
28C22×C3×D9—21no—
29C2×C3×C4×C9D412yes—
30C2×C3×C4×C9C2311yescharacteristic
31C2×C9×Dic3—22no—
32C2×C3×Dic9—21no—
33C3×D36—22no—
34C9×Dic6—22no—
35C9×Dic6—22no—
36C3×Dic18—22no—
37C3⋊(C9⋊C8)D414yes—
38#5086—91no—
39#5060—91no—
40C2×C3×D12—31no—
41C2×C3×Dic6—31no—
42C2×C3×Dic6C2×D312yes—
43C32×C42C2×D311yescharacteristic
44C2×C3⋊(C3⋊C8)—32no—
45C3×(C2×D3)⋊C4—62no—
46C3×C4⋊Dic3—22no—
47#3718—64no—
48#3715—64no—
49C2×Q8×C9—32no—
50C2×D36C2×D311yescharacteristic
51C2×Dic18C2×D311yescharacteristic
52C9×C4⋊C4—62no—
53C42×C9—21no—
54C42×C9C2×D311yescharacteristic
55#3675—64no—
56#3674—64no—
57(C2×D9)⋊C4—22no—
58C2×C9⋊C8—32no—
59C2×C9⋊C8—62no—
60C22×C3×C9C2×D411yescharacteristic
61C2×C3×D9—41no—
62C2×C3×D9—42no—
63C3×C4×C9—22no—
64C3×C4×C9C2×D412yes—
65C9×Dic3—42no—
66C3×Dic9—41no—
67C3×C4⋊Q8—31no—
68C3×#124—91no—
69C2×#216—92no—
70C2×#216—92no—
71C2×#215—92no—
72C2×#214—92no—
73#875—31no—
74#869D911yescharacteristic, complemented
75#803—91no—
76#803—91no—
77#803—181no—
78C22×C3×D3—61no—
79C2×C32×C4#6712yes—
80C2×C32×C4C22×D311yescharacteristic
81C2×C3×Dic3—22no—
82C2×C3×Dic3—61no—
83C3×D12—62no—
84C3×Dic6—22no—
85C3×Dic6—22no—
86C3×Dic6—62no—
87C22×D9—21no—
88C3⋊(C3⋊C8)—34no—
89Q8×C9—62no—
90Q8×C9—62no—
91C2×C4×C9—22no—
92C2×C4×C9—21no—
93C2×C4×C9—62no—
94C2×C4×C9#6712yes—
95C2×C4×C9C22×D311yescharacteristic
96C2×Dic9—21no—
97D36—22no—
98Dic18—22no—
99C9⋊C8—34no—
100C9⋊C8—64no—
101#497—271noSylow
102C2×C3×C9#12711yescharacteristic
103C2×C3×C9#13712yes—
104C3×D9—81no—
105C2×C3×Q8—32no—
106C2×C3×Q8—91no—
107C2×C3×D4—91no—
108C2×D12—31no—
109C2×Dic6—31no—
110C2×Dic6C2×D912yes—
111C3×C4⋊C4—62no—
112C3×C22⋊C4—182no—
113C3×C42—21no—
114C3×C42C2×D911yescharacteristic
115C3×C42D3211yescharacteristic
116#216—184no—
117#216—184no—
118#215—184no—
119#214—184no—
120(C2×D3)⋊C4—62no—
121C4⋊Dic3—22no—
122C2×C3⋊C8—92no—
123C2×C3⋊C8—92no—
124C2×C3⋊C8—182no—
125C22×C32C2×#6711yescharacteristic
126C2×C3×D3—121no—
127C2×C3×D3—122no—
128C32×C4—22no—
129C32×C4D3×D412yes—
130C3×Dic3—42no—
131C3×Dic3—121no—
132C22×C9—21no—
133C22×C9C2×#6711yescharacteristic
134C2×D9—41no—
135C2×D9—42no—
136C4×C9—22no—
137C4×C9—22no—
138C4×C9—42no—
139C4×C9—122no—
140C4×C9D3×D412yes—
141Dic9—41no—
142C2×Q16—272no—
143C2×SD16—272no—
144C4⋊Q8—31no—
145#124—91no—
146#97—271no—
147C3×C9#49711yescharacteristic, complemented, Sylow
148C23×C3—181no—
149C22×D3—61no—
150C3×Q8—62no—
151C3×Q8—62no—
152C3×Q8—182no—
153C3×D4—182no—
154C2×C3×C4—22no—
155C2×C3×C4—21no—
156C2×C3×C4—62no—
157C2×C3×C4—181no—
158C2×C3×C4#61112yes—
159C2×C3×C4C22×D911yescharacteristic
160C2×C3×C4#62512yes—
161C2×C3×C4C2×D3211yescharacteristic
162C2×Dic3—22no—
163C2×Dic3—61no—
164D12—62no—
165Dic6—22no—
166Dic6—22no—
167Dic6—62no—
168C3⋊C8—94no—
169C3⋊C8—94no—
170C3⋊C8—184no—
171C2×C32#94011yescharacteristic
172C2×C32#94212yes—
173C3×D3—241no—
174C2×C9—22no—
175C2×C9—21no—
176C2×C9#94011yescharacteristic
177C2×C9#95112yes—
178D9—81no—
179C2×Q8—32no—
180C2×Q8—91no—
181C2×D4—91no—
182Q16—544no—
183SD16—544no—
184C2×C8—272no—
185C4⋊C4—62no—
186C22⋊C4—182no—
187C42D3×D911yescharacteristic
188C22×C3—21no—
189C22×C3—361no—
190C22×C3—362no—
191C22×C3C2×#61111yescharacteristic
192C22×C3C2×#62511yescharacteristic
193C2×D3—122no—
194C2×D3—121no—
195C3×C4—22no—
196C3×C4—22no—
197C3×C4—22no—
198C3×C4—42no—
199C3×C4—122no—
200C3×C4—361no—
201C3×C4D4×D912yes—
202C3×C4#380212yes—
203Dic3—42no—
204Dic3—121no—
205C32#508611yescharacteristic
206C9—21no—
207C9#506011yescharacteristic, complemented
208C23—181no—
209Q8—62no—
210Q8—62no—
211Q8—182no—
212D4—182no—
213C2×C4—62no—
214C2×C4—181no—
215C2×C4#619412yes—
216C2×C4C2×D3×D911yescharacteristic
217C8—274no—
218C2×C3—21no—
219C2×C3—22no—
220C2×C3—721no—
221C2×C3#6350211yescharacteristic
222C2×C3#6350512yes—
223C2×C3#6382712yes—
224C2×C3#6391411yescharacteristic
225D3—241no—
226C22—361no—
227C22—362no—
228C22C2×#619411yescharacteristic
229C4—22no—
230C4—122no—
231C4—361no—
232C4#8864612yes—
233C3—21no—
234C3#1058877111yescharacteristic, complemented
235C3#1059037611yescharacteristic
236C2—721no—
237C2#1172089312yes—
238C2#1172104311yescharacteristic
subgroupG/H
centerC22C2×#6194
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C42×C9C22
socleC22×C32C2×#67
prime radicalC2×C3×D36C22
hypercenterC42D3×D9
hypocenterC3×C9#497
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C9×#869C2semidirect
3. C2×#88427C2semidirect
5. #11720516C2semidirect
74. #869D9semidirect
147. C3×C9#497semidirect
207. C9#5060semidirect
234. C3#10588771semidirect

Maximal subgroups

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

subgroupindex
C3×#634372
C9×#8692
C2×#884272
C2×#884252
#117205162
#105903763
#105887713