Finite Group Databasebeta

← #13157226

Subgroups — #13157226

Order 1728.

476 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 45 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×#64147C211yesmaximal, characteristic
2#11723106C211yesmaximal, characteristic, complemented
3#11723103C211yesmaximal, characteristic, complemented
4#10590209—31nomaximal
5C22×C3⋊#157—21no—
6C2×C33×D4C411yescharacteristic, complemented
7C2×C3×#638—21no—
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C3⋊#622—21no—
10C3×#3757—21no—
11C3×C4⋊(C3⋊Dic3)C411yescharacteristic, complemented
12#64147D311yescharacteristic
13C3×#938—62no—
14#63781—31no—
15#63778—31no—
16C23×C33—21no—
17C22×C3×C3⋊D3D411yescharacteristic
18C2×C3⋊#157—22no—
19C2×C3⋊#157—41no—
20C2×C3⋊#157—41no—
21C33×D4—41no—
22C2×C33×C4C2×C411yescharacteristic
23C3×#638—41no—
24C3×#638—41no—
25C2×C3×C3⋊Dic3—21no—
26C2×C3×C3⋊Dic3D411yescharacteristic
27C3×C4×C3⋊D3—41no—
28C3⋊#622—41no—
29#4537—91nomaximal
30C22×#157—61no—
31C2×C32×D4—22no—
32C2×C32×D4—42no—
33C2×C32×D4Dic311yescharacteristic, complemented
34C2×#638—21no—
35C2×C4×C3⋊D3C2×D311yescharacteristic
36C2×C3×#67—64no—
37C2×C3×C4×D3—62no—
38C2×#622—61no—
39#3757—21no—
40C4⋊(C3⋊Dic3)Dic311yescharacteristic, complemented
41C3×#217—64no—
42C3×C4⋊Dic3—62no—
43C22×C33—41no—
44C22×C33—41no—
45C22×C33—41no—
46C22×C33C22⋊C411yescharacteristic
47C2×C3×C3⋊D3—21no—
48C2×C3×C3⋊D3—21no—
49C2×C3×C3⋊D3D811yescharacteristic
50C2×C3×C3⋊D3SD1611yescharacteristic
51C3⋊#157—42no—
52C33×C4—21no—
53C3×C3⋊Dic3—21no—
54C3×C3⋊Dic3—41no—
55C3×#121—91no—
56#938—62no—
57C3⋊#95—91no—
58#830—91no—
59C23×C32—21no—
60C23×C32—24no—
61C23×C32—44no—
62C22×C3⋊D3#6711yescharacteristic
63C22×C3×D3—62no—
64C2×#157—62no—
65C2×#157—121no—
66C2×#157—121no—
67C32×D4—41no—
68C32×D4—44no—
69C32×D4—48no—
70C2×C32×C4—22no—
71C2×C32×C4—42no—
72C2×C32×C4C2×Dic311yescharacteristic
73#638—41no—
74#638—41no—
75C2×C3⋊Dic3—21no—
76C2×C3⋊Dic3#6711yescharacteristic
77C4×C3⋊D3—41no—
78C3×#67—124no—
79C3×#67—124no—
80C2×C3×Dic3—62no—
81C2×C3×Dic3—64no—
82C3×C4×D3—124no—
83#622—121no—
84#327—271noSylow
85C2×C33—81no—
86C2×C33#9911yescharacteristic
87C2×C33#10211yescharacteristic
88C2×C33#10411yescharacteristic
89C3×C3⋊D3—21no—
90C3×C3⋊D3—21no—
91C2×C3×D4—22no—
92C2×C3×D4—42no—
93C2×C3×D4—181no—
94C2×C3×D4#15711yescharacteristic, complemented
95C22×C3×C4—91no—
96C2×#67—64no—
97C22×Dic3—181no—
98C2×C4×D3—62no—
99C3×C4⋊C4—91no—
100C3×C22⋊C4—181no—
101#217—64no—
102C4⋊Dic3—62no—
103C2×C3⋊C8—181no—
104C22×C32—22no—
105C22×C32—42no—
106C22×C32—44no—
107C22×C32—48no—
108C22×C32—41no—
109C22×C32—41no—
110C22×C32—41no—
111C22×C32—44no—
112C22×C32—44no—
113C22×C32—84no—
114C22×C32—84no—
115C22×C32#21711yescharacteristic
116C2×C3⋊D3—21no—
117C2×C3⋊D3—21no—
118C2×C3⋊D3#21311yescharacteristic
119C2×C3⋊D3#21411yescharacteristic
120C2×C3×D3—62no—
121C2×C3×D3—62no—
122C2×C3×D3—122no—
123C2×C3×D3—122no—
124#157—122no—
125C32×C4—21no—
126C32×C4—24no—
127C32×C4—44no—
128C3⋊Dic3—21no—
129C3⋊Dic3—41no—
130C3×Dic3—122no—
131C3×Dic3—124no—
132#121—91no—
133C22⋊C8—271no—
134#95—271no—
135C33#32711yescharacteristic, complemented, Sylow
136C23×C3—24no—
137C23×C3—21no—
138C23×C3—44no—
139C23×C3—91no—
140C22×D3—62no—
141C3×D4—41no—
142C3×D4—44no—
143C3×D4—48no—
144C3×D4—361no—
145C3×D4—361no—
146C2×C3×C4—22no—
147C2×C3×C4—42no—
148C2×C3×C4—91no—
149C2×C3×C4—181no—
150C2×C3×C4—361no—
151C2×C3×C4C2×#15711yescharacteristic
152#67—124no—
153#67—124no—
154C2×Dic3—62no—
155C2×Dic3—64no—
156C2×Dic3—182no—
157C2×Dic3—361no—
158C2×Dic3—361no—
159C4×D3—124no—
160C3⋊C8—361no—
161C2×C32—22no—
162C2×C32—22no—
163C2×C32—22no—
164C2×C32—42no—
165C2×C32—42no—
166C2×C32—42no—
167C2×C32—84no—
168C2×C32—88no—
169C2×C32—81no—
170C2×C32#83211yescharacteristic
171C2×C32#83411yescharacteristic
172C2×C32#83711yescharacteristic
173C3⋊D3—21no—
174C3⋊D3—21no—
175C3×D3—122no—
176C3×D3—122no—
177C2×D4—181no—
178C2×D4C3⋊#15711yescharacteristic, complemented
179C22×C4—91no—
180C22×C4—541no—
181C2×C8—541no—
182C4⋊C4—91no—
183C22⋊C4—181no—
184C22×C3—22no—
185C22×C3—44no—
186C22×C3—44no—
187C22×C3—44no—
188C22×C3—48no—
189C22×C3—41no—
190C22×C3—41no—
191C22×C3—41no—
192C22×C3—42no—
193C22×C3—84no—
194C22×C3—84no—
195C22×C3—91no—
196C22×C3—91no—
197C22×C3—181no—
198C22×C3—181no—
199C22×C3C22⋊#15711yescharacteristic
200C2×D3—62no—
201C2×D3—62no—
202C2×D3—122no—
203C2×D3—122no—
204C3×C4—24no—
205C3×C4—21no—
206C3×C4—44no—
207C3×C4—181no—
208C3×C4—361no—
209Dic3—122no—
210Dic3—124no—
211Dic3—362no—
212C32—22no—
213C32—42no—
214C32#453711yescharacteristic, complemented
215C23—21no—
216C23—91no—
217D4—41no—
218D4—361no—
219D4—361no—
220C2×C4—91no—
221C2×C4—181no—
222C2×C4—361no—
223C2×C4—542no—
224C2×C4—1081no—
225C2×C4—1081no—
226C2×C4C2×C3⋊#15711yescharacteristic
227C8—1081no—
228C2×C3—22no—
229C2×C3—22no—
230C2×C3—22no—
231C2×C3—42no—
232C2×C3—42no—
233C2×C3—42no—
234C2×C3—84no—
235C2×C3—88no—
236C2×C3—81no—
237C2×C3—181no—
238C2×C3—181no—
239C2×C3#6378211yescharacteristic
240C2×C3#6378311yescharacteristic
241C2×C3#6378611yescharacteristic
242D3—122no—
243D3—122no—
244C22—41no—
245C22—41no—
246C22—41no—
247C22—91no—
248C22—91no—
249C22—181no—
250C22—181no—
251C22#8899411yescharacteristic
252C4—21no—
253C4—181no—
254C4—361no—
255C4—1082no—
256C3—22no—
257C3—42no—
258C3#1059020911yescharacteristic, complemented
259C2—81no—
260C2—181no—
261C2—181no—
262C2#1172310711yescharacteristic
263C2#1172310811yescharacteristic
264C2#1172311111yescharacteristic
subgroupG/H
centerC22#88994
commutatorC2×C33×C4C2×C4
FrattiniC2×C4C2×C3⋊#157
FittingC2×C33×D4C4
socleC22×C33C22⋊C4
prime radicalC3×#64147C2
cosocleC2×C3×C4×C3⋊D3C22
hypercenterC2×D4C3⋊#157
hypocenterC33#327
LCSG ▷ C2×C33×C4 ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11723106C2semidirect
3. #11723103C2semidirect
6. C2×C33×D4C4semidirect
11. C3×C4⋊(C3⋊Dic3)C4semidirect
33. C2×C32×D4Dic3semidirect
40. C4⋊(C3⋊Dic3)Dic3semidirect
94. C2×C3×D4#157semidirect
135. C33#327semidirect
178. C2×D4C3⋊#157semidirect
214. C32#4537semidirect
258. C3#10590209semidirect

Maximal subgroups

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

subgroupindex
C3×#641472
#117231062
#117231032
#105902093
#45379