Finite Group Databasebeta

← #13161493

Subgroups — #13161493

Order 1728.

554 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 111 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×#64079C211yesmaximal, characteristic
2C3×#64078C211yesmaximal, characteristic
3C3×#63630C211yesmaximal, characteristic
4#11722076C214yesmaximal
5#10590295—31nomaximal
6#10590291—31nomaximal
7#10590289—31nomaximal
8C2×C3×#634C2212yes—
9C33×C42C2211yescharacteristic
10C3×C4⋊(C3⋊Dic3)—22no—
11C3×C4⋊(C3⋊Dic3)C2212yes—
12C3×#3750—22no—
13C2×C3×C3⋊(C3⋊C8)C2212yes—
14#64079D311yescharacteristic
15#64078D311yescharacteristic
16C3×#870—31no—
17C3×#870—31no—
18C3×#870—61no—
19C3×#869—31no—
20C3×#869—31no—
21C3×#869—61no—
22#63630D311yescharacteristic
23C3×#803—31no—
24C3×#803—31no—
25C3×#803—61no—
26#63567—34no—
27#63566—34no—
28#63565—34no—
29C2×C33×C4D412yes—
30C2×C33×C4C2311yescharacteristic
31C2×C3×C3⋊Dic3—22no—
32C2×C3×C3⋊Dic3—22no—
33C3×#634—24no—
34C3×C3⋊(C3⋊C8)—22no—
35#5082—91no—
36#5069—91no—
37#4703—91no—
38C2×#634C2×D312yes—
39C2×C3×Dic6—32no—
40C2×C3×Dic6—32no—
41C2×C3×Dic6—62no—
42C32×C42—21no—
43C32×C42—21no—
44C32×C42—21no—
45C32×C42—41no—
46C32×C42C2×D311yescharacteristic
47C32×C42C2×D311yescharacteristic
48C32×C42C2×D311yescharacteristic
49C4⋊(C3⋊Dic3)—22no—
50C4⋊(C3⋊Dic3)C2×D312yes—
51#3750—22no—
52C2×C3⋊(C3⋊C8)C2×D312yes—
53C3×C4⋊Dic3—32no—
54C3×C4⋊Dic3—32no—
55C3×C4⋊Dic3—62no—
56C3×C4⋊Dic3—64no—
57C3×C4⋊Dic3—62no—
58C3×C4⋊Dic3—62no—
59C3×Dic3⋊C4—64no—
60C3×Dic3⋊C4—62no—
61C3×Dic3⋊C4—62no—
62C2×C3×C3⋊C8—32no—
63C2×C3×C3⋊C8—32no—
64C2×C3×C3⋊C8—62no—
65C22×C33C2×D411yescharacteristic
66C33×C4—22no—
67C33×C4#4112yes—
68C3×C3⋊Dic3—42no—
69C3×C3⋊Dic3—42no—
70C3×C4⋊Q8—91no—
71C3×#125—91no—
72#870—31no—
73#870—31no—
74#870—61no—
75#869—31no—
76#869—31no—
77#869—61no—
78C3×#97—91no—
79#835—94no—
80#816—94no—
81#810—94no—
82#803—31no—
83#803—31no—
84#803—61no—
85C2×C32×C4—22no—
86C2×C32×C4—22no—
87C2×C32×C4—22no—
88C2×C32×C4—21no—
89C2×C32×C4—21no—
90C2×C32×C4—21no—
91C2×C32×C4—42no—
92C2×C32×C4—41no—
93C2×C32×C4D1212yes—
94C2×C32×C4#6712yes—
95C2×C32×C4#6712yes—
96C2×C32×C4C22×D311yescharacteristic
97C2×C32×C4C22×D311yescharacteristic
98C2×C32×C4C22×D311yescharacteristic
99C2×C3⋊Dic3—22no—
100C2×C3⋊Dic3—22no—
101#634—24no—
102C2×C3×Dic3—64no—
103C2×C3×Dic3—64no—
104C2×C3×Dic3—62no—
105C2×C3×Dic3—62no—
106C2×C3×Dic3—62no—
107C2×C3×Dic3—62no—
108C3×Dic6—64no—
109C3×Dic6—64no—
110C3×Dic6—124no—
111C3⋊(C3⋊C8)—22no—
112C3×C3⋊C8—64no—
113C3×C3⋊C8—62no—
114C3×C3⋊C8—62no—
115#491—271noSylow
116C2×C33#12411yescharacteristic
117C2×C33#13712yes—
118C2×C3×Q8—92no—
119C2×Dic6—32no—
120C2×Dic6—32no—
121C2×Dic6—62no—
122C2×C3×C8—92no—
123C3×C4⋊C4—92no—
124C3×C4⋊C4—182no—
125C3×C4⋊C4—182no—
126C3×C42—21no—
127C3×C42—21no—
128C3×C42—21no—
129C3×C42—41no—
130C3×C42D3211yescharacteristic
131C3×C42D3211yescharacteristic
132C3×C42D3211yescharacteristic
133C4⋊Dic3—32no—
134C4⋊Dic3—32no—
135C4⋊Dic3—62no—
136C4⋊Dic3—64no—
137C4⋊Dic3—62no—
138C4⋊Dic3—62no—
139Dic3⋊C4—64no—
140Dic3⋊C4—62no—
141Dic3⋊C4—62no—
142C2×C3⋊C8—32no—
143C2×C3⋊C8—32no—
144C2×C3⋊C8—62no—
145C22×C32—21no—
146C22×C32—21no—
147C22×C32—21no—
148C22×C32—41no—
149C22×C32C2×D1211yescharacteristic
150C22×C32C2×#6711yescharacteristic
151C22×C32C2×#6711yescharacteristic
152C32×C4—22no—
153C32×C4—24no—
154C32×C4—22no—
155C32×C4—22no—
156C32×C4—22no—
157C32×C4—22no—
158C32×C4—22no—
159C32×C4—42no—
160C32×C4—42no—
161C32×C4—42no—
162C32×C4—44no—
163C32×C4#23512yes—
164C32×C4#23712yes—
165C32×C4#23912yes—
166C3⋊Dic3—42no—
167C3⋊Dic3—42no—
168C3×Dic3—124no—
169C3×Dic3—124no—
170C3×Dic3—122no—
171C3×Dic3—122no—
172C3×Dic3—122no—
173C3×Dic3—122no—
174C4⋊Q8—91no—
175#125—91no—
176#103—274no—
177#97—91no—
178C33#49111yescharacteristic, complemented, Sylow
179C3×Q8—184no—
180C2×C3×C4—22no—
181C2×C3×C4—22no—
182C2×C3×C4—22no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—42no—
187C2×C3×C4—41no—
188C2×C3×C4—182no—
189C2×C3×C4—182no—
190C2×C3×C4#62512yes—
191C2×C3×C4#62612yes—
192C2×C3×C4#62612yes—
193C2×C3×C4C2×D3211yescharacteristic
194C2×C3×C4C2×D3211yescharacteristic
195C2×C3×C4C2×D3211yescharacteristic
196C2×Dic3—64no—
197C2×Dic3—64no—
198C2×Dic3—62no—
199C2×Dic3—62no—
200C2×Dic3—62no—
201C2×Dic3—62no—
202Dic6—64no—
203Dic6—64no—
204Dic6—124no—
205C3×C8—182no—
206C3⋊C8—64no—
207C3⋊C8—62no—
208C3⋊C8—62no—
209C2×C32—22no—
210C2×C32—22no—
211C2×C32—22no—
212C2×C32—21no—
213C2×C32—21no—
214C2×C32—21no—
215C2×C32—41no—
216C2×C32—42no—
217C2×C32#87511yescharacteristic
218C2×C32#90912yes—
219C2×C32#93611yescharacteristic
220C2×C32#94212yes—
221C2×C32#94711yescharacteristic
222C2×C32#95112yes—
223C2×Q8—92no—
224C2×C8—92no—
225C4⋊C4—92no—
226C4⋊C4—182no—
227C4⋊C4—182no—
228C42#111311yescharacteristic
229C22×C3—21no—
230C22×C3—21no—
231C22×C3—21no—
232C22×C3—41no—
233C22×C3C2×#62511yescharacteristic
234C22×C3C2×#62611yescharacteristic
235C22×C3C2×#62611yescharacteristic
236C3×C4—24no—
237C3×C4—22no—
238C3×C4—22no—
239C3×C4—22no—
240C3×C4—22no—
241C3×C4—22no—
242C3×C4—22no—
243C3×C4—42no—
244C3×C4—42no—
245C3×C4—44no—
246C3×C4—42no—
247C3×C4—362no—
248C3×C4—362no—
249C3×C4#379512yes—
250C3×C4#379612yes—
251C3×C4#379912yes—
252Dic3—124no—
253Dic3—124no—
254Dic3—122no—
255Dic3—122no—
256Dic3—122no—
257Dic3—122no—
258C32—21no—
259C32—21no—
260C32—21no—
261C32—41no—
262C32#470311yescharacteristic, complemented
263C32#506911yescharacteristic, complemented
264C32#508211yescharacteristic, complemented
265Q8—184no—
266C2×C4—182no—
267C2×C4—182no—
268C2×C4#629512yes—
269C2×C4C2×#111311yescharacteristic
270C8—182no—
271C2×C3—22no—
272C2×C3—22no—
273C2×C3—22no—
274C2×C3—21no—
275C2×C3—21no—
276C2×C3—21no—
277C2×C3—42no—
278C2×C3—41no—
279C2×C3#6382712yes—
280C2×C3#6383112yes—
281C2×C3#6383312yes—
282C2×C3#6386011yescharacteristic
283C2×C3#6386311yescharacteristic
284C2×C3#6386411yescharacteristic
285C22C2×#629511yescharacteristic
286C4—22no—
287C4—362no—
288C4—362no—
289C4#8904112yes—
290C3—21no—
291C3—21no—
292C3—21no—
293C3—41no—
294C3#1059028911yescharacteristic, complemented
295C3#1059029111yescharacteristic, complemented
296C3#1059029511yescharacteristic, complemented
297C2#1172366312yes—
298C2#1172367711yescharacteristic
subgroupG/H
centerC22C2×#6295
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC33×C42C22
socle, prime radicalC22×C33C2×D4
hypercenterC42#1113
hypocenterC33#491
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
178. C33#491semidirect
262. C32#4703semidirect
263. C32#5069semidirect
264. C32#5082semidirect
294. C3#10590289semidirect
295. C3#10590291semidirect
296. C3#10590295semidirect

Maximal subgroups

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

subgroupindex
C3×#640792
C3×#640782
C3×#636302
#117220762
#105902953
#105902913
#105902893