Finite Group Databasebeta

← #13166290

Subgroups — #13166290

Order 1728.

436 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 102 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×C3×#3787C211yesmaximal, characteristic
3C3⋊D3×#208C211yesmaximal, characteristic
4#11722705C212yesmaximal, complemented
5#11722703C212yesmaximal, complemented
6#10593775—31nomaximal
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8#88994C412yes—
9C4⋊(C3⋊#157)C412yes—
10C4×C3⋊#157C2212yes—
11C2×C3×#622C412yescomplemented
12C3×#3787C412yescomplemented
13C3×#3787C2211yescharacteristic
14C3×#3787C2211yescharacteristic
15#88855C411yescharacteristic, complemented
16C32×#208C411yescharacteristic, complemented
17#88787C412yescomplemented
18C3⋊D3×C3⋊C8C2212yes—
19#64286—31no—
20C2×#3787D311yescharacteristic
21M16×C3⋊D3—31no—
22D3×#208—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—
35C3⋊(C3⋊(C3⋊C8))C2×C412yes—
36C32×C3⋊C8C2×C412yes—
37#5101—91nomaximal
38C2×C4×C3⋊D3C2×D311yescharacteristic
39C2×C3×C4×D3—62no—
40C22⋊#157—32no—
41C2×#622Dic312yescomplemented
42C4⋊#157—32no—
43C4×#157—32no—
44#3787Dic312yescomplemented
45#3787C2×D311yescharacteristic
46#3787C2×D311yescharacteristic
47C32×M16—31no—
48#3748—31no—
49#3748—62no—
50#3748—122no—
51#3743—32no—
52C8×C3⋊D3—32no—
53C3×#208—22no—
54#3711—64no—
55D3×C3⋊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
65C22×C3×C8—91no—
66#926—91no—
67C2×#208—91no—
68D3×M16—182no—
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×C4C4×D311yescharacteristic
77C2×C3⋊Dic3C2×Dic311yescharacteristic
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#622C2×Dic312yes—
88#622C2×Dic312yes—
89C32×C8—32no—
90C3⋊(C3⋊C8)—32no—
91C3⋊(C3⋊C8)—64no—
92C3⋊(C3⋊C8)—124no—
93C3×C3⋊C8—24no—
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—62no—
102C2×C3×C8—91no—
103C2×C3×C8—91no—
104C2×C3×C8—92no—
105C2×C3×C8—92no—
106#217—92no—
107C4⋊Dic3—92no—
108C4×Dic3—92no—
109#208—91no—
110#208—92no—
111#208—182no—
112#208—362no—
113#208#15711yescharacteristic, complemented
114C2×C3⋊C8—92no—
115ΓL(1,𝔽25)—184no—
116D3×C8—184no—
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×C4C2×C4×D311yescharacteristic
133C3⋊Dic3C4×Dic311yescharacteristic
134C3⋊Dic3C22×Dic311yescharacteristic
135C3×Dic3—62no—
136C3×Dic3—62no—
137C2×M16—271no—
138C22×C8—91no—
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—64no—
160C3×C8—92no—
161C3×C8—92no—
162C3⋊C8—92no—
163C3⋊C8—184no—
164C3⋊C8—364no—
165C3⋊C8C2×#15712yes—
166C2×C32—22no—
167C2×C32—21no—
168C2×C32—42no—
169C2×C32—42no—
170C2×C32—82no—
171C2×C32C2×C4×Dic311yescharacteristic
172C3⋊D3—21no—
173C3⋊D3#90112yes—
174C3×D3—64no—
175C3×D3—122no—
176C22×C4—91no—
177M16—31no—
178M16—272no—
179M16—271no—
180C2×C8—91no—
181C2×C8—91no—
182C2×C8—92no—
183C2×C8—92no—
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#510111yescharacteristic, complemented
211C23—91no—
212C2×C4—91no—
213C2×C4—91no—
214C2×C4—91no—
215C2×C4—91no—
216C2×C4—91no—
217C2×C4—274no—
218C2×C4D3×#15711yescharacteristic
219C8—32no—
220C8—92no—
221C8—92no—
222C8—272no—
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—
236C22Dic3×#15711yescharacteristic
237C4—91no—
238C4—91no—
239C4—544no—
240C4Dic3×#15711yescharacteristic
241C4C2×D3×#15711yescharacteristic
242C3—22no—
243C3—42no—
244C3#1059377511yescharacteristic, complemented
245C2—21no—
246C2—92no—
247C2—181no—
248C2C2×Dic3×#15711yescharacteristic
subgroupG/H
center, FrattiniC4C2×D3×#157
commutator, socleC2×C33C2×C42
FittingC2×C33×C4C2×C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC3×C4×C3⋊D3C23
hypercenterC2×C4D3×#157
hypocenterC33#405
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722705C2semidirect
5. #11722703C2semidirect
11. C2×C3×#622C4semidirect
12. C3×#3787C4semidirect
15. #88855C4semidirect
16. C32×#208C4semidirect
17. #88787C4semidirect
41. C2×#622Dic3semidirect
44. #3787Dic3semidirect
113. #208#157semidirect
142. C33#405semidirect
210. C32#5101semidirect
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×C3×#37872
C3⋊D3×#2082
#117227052
#117227032
#105937753
#51019