Finite Group Databasebeta

← #13137887

Subgroups — #13137887

Order 1728.

518 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 71 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#11721755C211yesmaximal, characteristic, complemented
2#11721144C211yesmaximal, characteristic
3C2×#88438C212yesmaximal, complemented
4#63636⋊C3C212yesmaximal, complemented
5#11720304C211yesmaximal, characteristic, complemented
6#10590549—32nomaximal
7C2×#6229C2211yescharacteristic
8C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
9C2×#6198—21no—
10C2×#6198—21no—
11C4⋊C4×H(3,𝔽3)C2211yescharacteristic
12#88540—21no—
13C4×#1084—21no—
14C4×#1084C2211yescharacteristic
15C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
16#88438—24no—
17C3×#887—32no—
18C3×#887—61no—
19#63911—32no—
20C2×#3717—32no—
21C2×#3715—32no—
22#63636D312yescomplemented
23C3×#809—32no—
24Dic3⋊(C3⋊C8)—32no—
25C22×(C3⋊D3)⋊C3—22no—
26C2×C4×H(3,𝔽3)—21no—
27C2×C4×H(3,𝔽3)C2311yescharacteristic
28C2×#1084—21no—
29C2×#1084D411yescharacteristic
30C2×#1084D411yescharacteristic
31#6229—21no—
32#6229D412yescomplemented
33C4⋊((C3⋊D3)⋊C3)—22no—
34C4⋊((C3⋊D3)⋊C3)—22no—
35#6198—42no—
36#6198—42no—
37(C3⋊(C3⋊C8))⋊C3—22no—
38#5027—91no—
39#4762—92no—
40C2×C4⋊(C3⋊D3)C2×D312yes—
41C2×C3×D12—32no—
42C2×C3×Dic6—32no—
43C2×C3×Dic6—61no—
44C2×#626—62no—
45C2×#626—62no—
46C32×C4⋊C4—21no—
47C32×C4⋊C4C2×D312yes—
48C2×C3⋊(C3⋊C8)C2×D312yes—
49C3×Dic3⋊C4—62no—
50C3×Dic3⋊C4—62no—
51C3×C4×Dic3—32no—
52C3×C4×Dic3—62no—
53C3×C4×Dic3—61no—
54C3×C4×Dic3—62no—
55C2×C3×C3⋊C8—32no—
56#3717—64no—
57#3715—64no—
58C22×H(3,𝔽3)C2×D411yescharacteristic
59C2×(C3⋊D3)⋊C3—42no—
60C2×(C3⋊D3)⋊C3—42no—
61C2×(C3⋊D3)⋊C3—42no—
62C4×H(3,𝔽3)—41no—
63C4×H(3,𝔽3)C2×D411yescharacteristic
64C4×H(3,𝔽3)#4111yescharacteristic
65#1084—22no—
66#1084—21no—
67#1084SD1612yescomplemented
68C3×C4×Q8—91no—
69C2×#215—92no—
70#940—92no—
71C2×#214—92no—
72C2×#204—92no—
73#887—32no—
74#887—61no—
75#874—91no—
76C3×#102—92no—
77Dic3⋊C8—92no—
78#809—32no—
79#809—92no—
80C4⋊(C3⋊C8)—91no—
81C22×C3⋊D3—22no—
82C22×C3×D3—62no—
83C2×C32×C4—22no—
84C2×C32×C4—22no—
85C2×C32×C4—21no—
86C2×C32×C4C22×D312yes—
87C4⋊(C3⋊D3)—22no—
88C4⋊(C3⋊D3)—22no—
89C2×C3×Dic3—32no—
90C2×C3×Dic3—32no—
91C2×C3×Dic3—61no—
92C2×C3×Dic3—61no—
93C2×C3×Dic3—62no—
94C2×C3×Dic3—62no—
95C3×D12—62no—
96C3×D12—62no—
97C3×Dic6—34no—
98C3×Dic6—62no—
99C3×Dic6—62no—
100C3×Dic6—62no—
101#626—124no—
102#626—124no—
103C3⋊(C3⋊C8)—22no—
104C3×C3⋊C8—62no—
105#460—271noSylow
106C2×H(3,𝔽3)#12111yescharacteristic
107C2×H(3,𝔽3)C2×SD1611yescharacteristic
108C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
109(C3⋊D3)⋊C3—82no—
110C2×C3×Q8—91no—
111C2×C3×D4—92no—
112C2×#67—182no—
113C2×#67—182no—
114C2×D12—32no—
115C2×D12—92no—
116C2×D12—181no—
117C2×D12—181no—
118C2×Dic6—32no—
119C2×Dic6—61no—
120C2×C3×C8—92no—
121C3×C4⋊C4—32no—
122C3×C4⋊C4—61no—
123C3×C4⋊C4—181no—
124C3×C4⋊C4D3211yescharacteristic
125C3×C42—91no—
126C3×C42—181no—
127#215—184no—
128#214—184no—
129Dic3⋊C4—62no—
130Dic3⋊C4—62no—
131C4×Dic3—32no—
132C4×Dic3—61no—
133C4×Dic3—62no—
134C4×Dic3—62no—
135C2×C3⋊C8—32no—
136C2×C3⋊C8—92no—
137#204—184no—
138C22×C32—21no—
139C22×C32D3×D412yes—
140C2×C3⋊D3—42no—
141C2×C3⋊D3—42no—
142C2×C3⋊D3—42no—
143C2×C3×D3—122no—
144C2×C3×D3—122no—
145C2×C3×D3—122no—
146C32×C4—21no—
147C32×C4—21no—
148C32×C4—42no—
149C32×C4—42no—
150C32×C4#23512yes—
151C32×C4D3×D412yes—
152C3×Dic3—34no—
153C3×Dic3—62no—
154C3×Dic3—64no—
155C3×Dic3—64no—
156C3×Dic3—62no—
157C3×Dic3—121no—
158C2×SD16—272no—
159#127—271no—
160C4×Q8—91no—
161C4⋊C8—271no—
162#102—92no—
163H(3,𝔽3)#46011yescharacteristic, complemented, Sylow
164C23×C3—182no—
165C22×D3—62no—
166C22×D3—182no—
167C3×Q8—92no—
168C3×Q8—181no—
169C3×D4—182no—
170C3×D4—182no—
171C2×C3×C4—21no—
172C2×C3×C4—32no—
173C2×C3×C4—62no—
174C2×C3×C4—62no—
175C2×C3×C4—61no—
176C2×C3×C4—91no—
177C2×C3×C4—91no—
178C2×C3×C4—181no—
179C2×C3×C4C2×D3211yescharacteristic
180#67—364no—
181#67—364no—
182C2×Dic3—32no—
183C2×Dic3—32no—
184C2×Dic3—61no—
185C2×Dic3—61no—
186C2×Dic3—62no—
187C2×Dic3—62no—
188D12—62no—
189D12—62no—
190D12—182no—
191D12—182no—
192D12—362no—
193D12—362no—
194Dic6—34no—
195Dic6—62no—
196Dic6—62no—
197Dic6—62no—
198C3×C8—182no—
199C3⋊C8—62no—
200C3⋊C8—182no—
201C2×C32—21no—
202C2×C32—21no—
203C2×C32—21no—
204C2×C32#88412yes—
205C2×C32#91112yes—
206C2×C32D3×SD1612yes—
207C3⋊D3—82no—
208C3×D3—242no—
209C2×Q8—91no—
210C2×D4—92no—
211C2×D4—541no—
212C2×D4—541no—
213SD16—544no—
214C2×C8—92no—
215C4⋊C4—181no—
216C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
217C42—91no—
218C42—181no—
219C22×C3—32no—
220C22×C3—61no—
221C22×C3—362no—
222C22×C3—362no—
223C22×C3—362no—
224C22×C3#381111yescharacteristic
225C2×D3—122no—
226C2×D3—122no—
227C2×D3—122no—
228C2×D3—362no—
229C2×D3—362no—
230C2×D3—362no—
231C3×C4—32no—
232C3×C4—32no—
233C3×C4—41no—
234C3×C4—61no—
235C3×C4—61no—
236C3×C4—92no—
237C3×C4—122no—
238C3×C4—122no—
239C3×C4—182no—
240C3×C4—181no—
241C3×C4#379811yescharacteristic
242C3×C4#380211yescharacteristic
243Dic3—34no—
244Dic3—62no—
245Dic3—64no—
246Dic3—62no—
247Dic3—64no—
248Dic3—121no—
249C32—21no—
250C32#476212yescomplemented
251C23—182no—
252Q8—92no—
253Q8—181no—
254D4—182no—
255D4—182no—
256D4—1082no—
257D4—1082no—
258C2×C4—21no—
259C2×C4—91no—
260C2×C4—91no—
261C2×C4—181no—
262C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
263C8—182no—
264C2×C3—32no—
265C2×C3—32no—
266C2×C3—32no—
267C2×C3—61no—
268C2×C3—61no—
269C2×C3—61no—
270C2×C3—722no—
271C2×C3#6391211yescharacteristic
272C2×C3#6392711yescharacteristic
273C2×C3#6394011yescharacteristic
274D3—242no—
275D3—722no—
276C22—362no—
277C22—362no—
278C22—362no—
279C22#8867711yescharacteristic
280C4—41no—
281C4—92no—
282C4—182no—
283C4—181no—
284C4#8865011yescharacteristic
285C4#8865511yescharacteristic
286C3—32no—
287C3—61no—
288C3#1059055311yescharacteristic
289C2—722no—
290C2#1172114211yescharacteristic
291C2#1172119111yescharacteristic
292C2#1172122811yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3#3811
prime radical#11721144C2
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#460
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11721755C2semidirect
3. C2×#88438C2semidirect
4. #63636⋊C3C2semidirect
5. #11720304C2semidirect
22. #63636D3semidirect
32. #6229D4semidirect
67. #1084SD16semidirect
163. H(3,𝔽3)#460semidirect
250. C32#4762semidirect

Maximal subgroups

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

subgroupindex
#117217552
#117211442
C2×#884382
#63636⋊C32
#117203042
#105905493