Finite Group Databasebeta

← #13173371

Subgroups — #13173371

Order 1728.

436 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 105 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#11724514C211yesmaximal, characteristic
2C2×#88988C211yesmaximal, characteristic
3C3×M16×C3⋊D3C211yesmaximal, characteristic
4#11723092C212yesmaximal, complemented
5C8×C3⋊#157C212yesmaximal, complemented
6#10593775—31nomaximal
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8#88994C412yes—
9C2×C3⋊#622C412yescomplemented
10C4⋊(C3⋊#157)C412yes—
11C4×C3⋊#157C2212yes—
12#88988C412yescomplemented
13#88988C2211yescharacteristic
14#88988C2211yescharacteristic
15C33×M16C411yescharacteristic, complemented
16C3×#3748C411yescharacteristic, complemented
17C3×#3743C412yescomplemented
18C3×C8×C3⋊D3C2212yes—
19#64286—31no—
20C2×#3787—31no—
21M16×C3⋊D3D311yescharacteristic
22C3×D3×M16—62no—
23#63767—32no—
24C8×#157—32no—
25C22×C3×C3⋊D3C2×C411yescharacteristic
26C2×C3⋊#157C2×C414yes—
27C2×C33×C4C2×C411yescharacteristic
28C2×C3×C3⋊Dic3C2×C411yescharacteristic
29C3×C4×C3⋊D3C2×C411yescharacteristic
30C3×C4×C3⋊D3C2×C411yescharacteristic
31C3×C4×C3⋊D3C2×C411yescharacteristic
32C3×C4×C3⋊D3C2311yescharacteristic
33C3⋊#622C2×C412yes—
34C3⋊#622C2×C412yes—
35C33×C8C2×C412yes—
36C3×C3⋊(C3⋊C8)C2×C412yes—
37#5122—91nomaximal
38C2×C4×C3⋊D3C2×D311yescharacteristic
39C2×C3×C4×D3—62no—
40C22⋊#157—32no—
41C2×#622—32no—
42C4⋊#157—32no—
43C4×#157—32no—
44#3787—32no—
45#3787—31no—
46#3787—31no—
47C32×M16—22no—
48C32×M16—42no—
49C32×M16Dic311yescharacteristic, complemented
50#3748Dic311yescharacteristic, complemented
51#3743Dic312yescomplemented
52C8×C3⋊D3C2×D312yes—
53C3×#208—62no—
54C3×ΓL(1,𝔽25)—64no—
55C3×D3×C8—64no—
56C22×C33C4211yescharacteristic
57C2×C3×C3⋊D3—22no—
58C2×C3×C3⋊D3C4211yescharacteristic
59C2×C3×C3⋊D3C22×C411yescharacteristic
60C3⋊#157—24no—
61C33×C4C4211yescharacteristic
62C33×C4C22×C411yescharacteristic
63C3×C3⋊Dic3C4211yescharacteristic
64C3×C3⋊Dic3C22×C411yescharacteristic
65C2×C3×M16—91no—
66#926—91no—
67C22×C3⋊C8—91no—
68D3×M16—62no—
69#815—92no—
70C8×Dic3—92no—
71C22×C3⋊D3C4×D311yescharacteristic
72C22×C3×D3—62no—
73C2×#157—34no—
74C2×C32×C4—22no—
75C2×C32×C4—42no—
76C2×C32×C4C2×Dic311yescharacteristic
77C2×C3⋊Dic3C4×D311yescharacteristic
78C4×C3⋊D3C4×D311yescharacteristic
79C4×C3⋊D3C4×D311yescharacteristic
80C4×C3⋊D3C2×Dic311yescharacteristic
81C4×C3⋊D3C22×D311yescharacteristic
82C2×C3×Dic3—62no—
83C3×C4×D3—62no—
84C3×C4×D3—62no—
85C3×C4×D3—62no—
86C3×C4×D3—62no—
87#622—32no—
88#622—32no—
89C32×C8—24no—
90C32×C8—44no—
91C32×C8C2×Dic312yes—
92C3⋊(C3⋊C8)C2×Dic312yes—
93C3×C3⋊C8—64no—
94#405—271noSylow
95C2×C33—21no—
96C2×C33C2×C4211yescharacteristic
97C3×C3⋊D3—21no—
98C3×C3⋊D3#13112yes—
99C22×C3×C4—91no—
100C2×C4×D3—62no—
101C3×M16—22no—
102C3×M16—42no—
103C3×M16—91no—
104C3×M16—92no—
105C3×M16#15711yescharacteristic, complemented
106C2×C3×C8—92no—
107#217—92no—
108C4⋊Dic3—92no—
109C4×Dic3—92no—
110#208—62no—
111C2×C3⋊C8—91no—
112C2×C3⋊C8—91no—
113C2×C3⋊C8—92no—
114C2×C3⋊C8—92no—
115ΓL(1,𝔽25)—64no—
116D3×C8—64no—
117C22×C32—22no—
118C22×C32—42no—
119C22×C32C4×Dic311yescharacteristic
120C2×C3⋊D3—22no—
121C2×C3⋊D3C4×Dic311yescharacteristic
122C2×C3⋊D3C2×C4×D311yescharacteristic
123C2×C3×D3—62no—
124C2×C3×D3—62no—
125C2×C3×D3—124no—
126#157—64no—
127C32×C4—22no—
128C32×C4—22no—
129C32×C4—42no—
130C32×C4—42no—
131C32×C4C4×Dic311yescharacteristic
132C32×C4C22×Dic311yescharacteristic
133C3⋊Dic3C4×Dic311yescharacteristic
134C3⋊Dic3C2×C4×D311yescharacteristic
135C3×Dic3—62no—
136C3×Dic3—62no—
137C2×M16—91no—
138C22×C8—271no—
139#117—271no—
140#97—272no—
141C4×C8—272no—
142C33#40511yescharacteristic, complemented, Sylow
143C23×C3—91no—
144C22×D3—62no—
145C2×C3×C4—22no—
146C2×C3×C4—42no—
147C2×C3×C4—91no—
148C2×C3×C4—91no—
149C2×C3×C4—91no—
150C2×C3×C4—91no—
151C2×C3×C4—91no—
152C2×C3×C4C2×#15711yescharacteristic
153C2×Dic3—62no—
154C2×Dic3—94no—
155C4×D3—62no—
156C4×D3—62no—
157C4×D3—62no—
158C4×D3—62no—
159C3×C8—24no—
160C3×C8—44no—
161C3×C8—92no—
162C3×C8C2×#15712yes—
163C3⋊C8—64no—
164C3⋊C8—92no—
165C3⋊C8—92no—
166C2×C32—22no—
167C2×C32—21no—
168C2×C32—42no—
169C2×C32—42no—
170C2×C32—82no—
171C2×C32C2×C4×Dic311yescharacteristic
172C3⋊D3—21no—
173C3⋊D3#90712yes—
174C3×D3—64no—
175C3×D3—122no—
176C22×C4—91no—
177M16—91no—
178M16—92no—
179M16C3⋊#15711yescharacteristic, complemented
180C2×C8—92no—
181C2×C8—271no—
182C2×C8—271no—
183C2×C8—272no—
184C2×C8—272no—
185C4⋊C4—272no—
186C22⋊C4—272no—
187C42—272no—
188C22×C3—22no—
189C22×C3—42no—
190C22×C3—91no—
191C22×C3—91no—
192C22×C3—182no—
193C22×C3C4×#15711yescharacteristic
194C2×D3—62no—
195C2×D3—62no—
196C2×D3—124no—
197C3×C4—22no—
198C3×C4—22no—
199C3×C4—42no—
200C3×C4—42no—
201C3×C4—91no—
202C3×C4—91no—
203C3×C4C4×#15711yescharacteristic
204C3×C4C22×#15711yescharacteristic
205Dic3—62no—
206Dic3—62no—
207Dic3—184no—
208C32—22no—
209C32—42no—
210C32#512211yescharacteristic, complemented
211C23—91no—
212C2×C4—91no—
213C2×C4—91no—
214C2×C4—91no—
215C2×C4—91no—
216C2×C4—91no—
217C2×C4—274no—
218C2×C4C2×C3⋊#15711yescharacteristic
219C8—92no—
220C8—272no—
221C8—272no—
222C8C2×C3⋊#15712yes—
223C2×C3—21no—
224C2×C3—22no—
225C2×C3—42no—
226C2×C3—42no—
227C2×C3—82no—
228C2×C3—92no—
229C2×C3—181no—
230C2×C3C2×C4×#15711yescharacteristic
231D3—64no—
232D3—122no—
233C22—91no—
234C22—91no—
235C22—182no—
236C22C4×C3⋊#15711yescharacteristic
237C4—91no—
238C4—91no—
239C4—544no—
240C4C4×C3⋊#15711yescharacteristic
241C4C22×C3⋊#15711yescharacteristic
242C3—22no—
243C3—42no—
244C3#1059377511yescharacteristic, complemented
245C2—21no—
246C2—92no—
247C2—181no—
248C2C2×C4×C3⋊#15711yescharacteristic
subgroupG/H
center, FrattiniC4C22×C3⋊#157
commutator, socleC2×C33C2×C42
FittingC33×M16C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC3×C4×C3⋊D3C23
hypercenterM16C3⋊#157
hypocenterC33#405
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ M16

Complements

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

NH ≅ G/Ntype
4. #11723092C2semidirect
5. C8×C3⋊#157C2semidirect
9. C2×C3⋊#622C4semidirect
12. #88988C4semidirect
15. C33×M16C4semidirect
16. C3×#3748C4semidirect
17. C3×#3743C4semidirect
49. C32×M16Dic3semidirect
50. #3748Dic3semidirect
51. #3743Dic3semidirect
105. C3×M16#157semidirect
142. C33#405semidirect
179. M16C3⋊#157semidirect
210. C32#5122semidirect
244. C3#10593775semidirect

Maximal subgroups

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

subgroupindex
#117245142
C2×#889882
C3×M16×C3⋊D32
#117230922
C8×C3⋊#1572
#105937753
#51229