Finite Group Databasebeta

← C3×#10590575 #13157568

Subgroups — C3×#10590575 #13157568

Order 1728.

710 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4⋊C4×C3⋊D3C211yesmaximal, characteristic
2C3×#63900C211yesmaximal, characteristic
3C2×C3×#3710C211yesmaximal, characteristic
4C32×#808C212yesmaximal, complemented
5C3×#63573C212yesmaximal, complemented
6C3×D3×#106—32nomaximal
7#10590575C311yesmaximal, characteristic, direct product factor
8C2×C3×C4×C3⋊D3—21no—
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C3×#624—21no—
11C33×C4⋊C4C2211yescharacteristic
12C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
13C3×#3750—21no—
14C32×C4⋊Dic3C2212yes—
15C2×C32×C3⋊C8C2212yes—
16C3×#3724—21no—
17C3×#3710C414yescomplemented
18C4⋊C4×C3⋊D3C2×C311yescharacteristic
19C2×C3×D3×C8—32no—
20C3×D3×C4⋊C4—32no—
21C3×D3×C4⋊C4—32no—
22C3×D3×C4⋊C4—61no—
23#63900C2×C311yescharacteristic
24C2×#3710C2×C311yescharacteristic
25C32×#106—32no—
26C3×#817—32no—
27C3×#808—22no—
28C3×#808—32no—
29C3×#808D312yescomplemented
30C3×#808C2×C312yescomplemented
31#63573C2×C312yescomplemented
32C22×C3×C3⋊D3D411yescharacteristic
33C2×C33×C4—21no—
34C2×C33×C4C2311yescharacteristic
35C2×C3×C3⋊Dic3—21no—
36C2×C3×C3⋊Dic3D411yescharacteristic
37C3×C4×C3⋊D3—42no—
38C3×C4×C3⋊D3C2×C412yes—
39C3×C4×C3⋊D3Q812yes—
40C2×C32×Dic3—22no—
41C3×#624—42no—
42C32×C3⋊C8C2×C414yescomplemented
43C2×C3×#106—91no—
44D3×#106—32no—
45C2×C4×C3⋊D3—21no—
46C2×C4×C3⋊D3C22×C311yescharacteristic
47C2×C3×C4×D3—32no—
48C2×C3×C4×D3—61no—
49C2×C3×C4×D3—62no—
50C2×C3×C4×D3—62no—
51C2×C3×C4×D3—62no—
52C2×#624—21no—
53C2×C32×C8—32no—
54C32×C4⋊C4—21no—
55C32×C4⋊C4—22no—
56C32×C4⋊C4—32no—
57C32×C4⋊C4—41no—
58C32×C4⋊C4C2×D312yes—
59C32×C4⋊C4C22×C311yescharacteristic
60C4⋊(C3⋊Dic3)C22×C311yescharacteristic
61#3750—21no—
62C3×C4⋊Dic3—22no—
63C3×C4⋊Dic3—32no—
64C3×C4⋊Dic3—61no—
65C3×C4⋊Dic3C2×D312yes—
66C3×C4⋊Dic3C22×C312yes—
67C3×Dic3⋊C4—62no—
68C3×Dic3⋊C4—62no—
69C3×Dic3⋊C4—62no—
70C2×C3×C3⋊C8—22no—
71C2×C3×C3⋊C8C2×D312yes—
72C2×C3×C3⋊C8C22×C312yes—
73C3×D3×C8—38no—
74#3724—21no—
75#3710C3×C414yescomplemented
76C22×C33C2×D411yescharacteristic
77C2×C3×C3⋊D3C4⋊C412yes—
78C2×C3×C3⋊D3SD1612yes—
79C2×C3×C3⋊D3SD1612yes—
80C33×C4—41no—
81C33×C4C22×C411yescharacteristic
82C33×C4C2×Q811yescharacteristic
83C3×C3⋊Dic3—41no—
84C3×C3⋊Dic3C4⋊C412yes—
85C32×Dic3—42no—
86C22×C3×C8—91no—
87C2×C3×C4⋊C4—91no—
88C2×C3×C4⋊C4—91no—
89C2×D3×C8—32no—
90D3×C4⋊C4—32no—
91D3×C4⋊C4—32no—
92D3×C4⋊C4—61no—
93C3×#106—32no—
94C3×#106—32no—
95C3×#106—62no—
96C3×#106—92no—
97#817—32no—
98#808—32no—
99#808C3×D312yescomplemented
100C22×C3⋊D3C3×D411yescharacteristic
101C22×C3×D3—32no—
102C22×C3×D3—61no—
103C2×C32×C4—21no—
104C2×C32×C4—22no—
105C2×C32×C4—22no—
106C2×C32×C4—21no—
107C2×C32×C4—22no—
108C2×C32×C4—41no—
109C2×C32×C4—42no—
110C2×C32×C4—42no—
111C2×C32×C4—62no—
112C2×C32×C4C22×D312yes—
113C2×C32×C4C23×C311yescharacteristic
114C2×C3⋊Dic3—21no—
115C2×C3⋊Dic3C3×D411yescharacteristic
116C4×C3⋊D3—42no—
117C4×C3⋊D3C2×C3×C412yes—
118C4×C3⋊D3C3×Q812yes—
119C2×C3×Dic3—22no—
120C2×C3×Dic3—22no—
121C2×C3×Dic3—32no—
122C2×C3×Dic3—42no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—62no—
125C2×C3×Dic3—62no—
126C3×C4×D3—34no—
127C3×C4×D3—34no—
128C3×C4×D3—62no—
129C3×C4×D3—62no—
130C3×C4×D3—124no—
131C3×C4×D3—124no—
132C3×C4×D3—124no—
133#624—42no—
134C32×C8—34no—
135C3×C3⋊C8—24no—
136C3×C3⋊C8C4×D314yescomplemented
137C3×C3⋊C8C2×C3×C414yescomplemented
138C2×#106—91noSylow
139C2×C33C2×C4⋊C411yescharacteristic
140C2×C33C2×SD1611yescharacteristic
141C2×C33C2×SD1611yescharacteristic
142C3×C3⋊D3#10614yescomplemented
143C22×C3×C4—91no—
144C22×C3×C4—181no—
145C22×C3×C4—181no—
146C2×C4×D3—32no—
147C2×C4×D3—61no—
148C2×C4×D3—62no—
149C2×C4×D3—62no—
150C2×C4×D3—62no—
151C2×C3×C8—32no—
152C2×C3×C8—32no—
153C2×C3×C8—62no—
154C2×C3×C8—94no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4—22no—
157C3×C4⋊C4—32no—
158C3×C4⋊C4—32no—
159C3×C4⋊C4—41no—
160C3×C4⋊C4—62no—
161C3×C4⋊C4—91no—
162C3×C4⋊C4—181no—
163C3×C4⋊C4—181no—
164C3×C4⋊C4D3211yescharacteristic
165C3×C4⋊C4C2×C3×D312yes—
166C4⋊Dic3—32no—
167C4⋊Dic3—61no—
168C4⋊Dic3C2×C3×D312yes—
169Dic3⋊C4—62no—
170Dic3⋊C4—62no—
171Dic3⋊C4—62no—
172C2×C3⋊C8C2×C3×D312yes—
173D3×C8—38no—
174C22×C32—21no—
175C22×C32—22no—
176C22×C32—41no—
177C22×C32D3×D412yes—
178C22×C32C2×C3×D411yescharacteristic
179C2×C3⋊D3C3×C4⋊C412yes—
180C2×C3⋊D3C3×SD1612yes—
181C2×C3⋊D3C3×SD1612yes—
182C2×C3×D3—34no—
183C2×C3×D3—34no—
184C2×C3×D3—34no—
185C2×C3×D3—62no—
186C2×C3×D3—62no—
187C2×C3×D3—62no—
188C32×C4—21no—
189C32×C4—21no—
190C32×C4—22no—
191C32×C4—22no—
192C32×C4—41no—
193C32×C4—41no—
194C32×C4—41no—
195C32×C4—42no—
196C32×C4—42no—
197C32×C4—82no—
198C32×C4—82no—
199C32×C4—122no—
200C32×C4C2×C4×D312yes—
201C32×C4D3×Q812yes—
202C32×C4C22×C3×C411yescharacteristic
203C32×C4C2×C3×Q811yescharacteristic
204C3⋊Dic3—41no—
205C3⋊Dic3C3×C4⋊C412yes—
206C3×Dic3—34no—
207C3×Dic3—42no—
208C3×Dic3—42no—
209C3×Dic3—62no—
210C3×Dic3—82no—
211C3×Dic3—122no—
212C3×Dic3—122no—
213C22×C8—91no—
214C2×C4⋊C4—91no—
215C2×C4⋊C4—91no—
216#106—32no—
217#106—92no—
218C33C2×#10611yescharacteristic, complemented, Sylow
219C23×C3—91no—
220C22×D3—32no—
221C22×D3—61no—
222C2×C3×C4—22no—
223C2×C3×C4—21no—
224C2×C3×C4—22no—
225C2×C3×C4—22no—
226C2×C3×C4—21no—
227C2×C3×C4—42no—
228C2×C3×C4—42no—
229C2×C3×C4—41no—
230C2×C3×C4—62no—
231C2×C3×C4—62no—
232C2×C3×C4—91no—
233C2×C3×C4—92no—
234C2×C3×C4—92no—
235C2×C3×C4—122no—
236C2×C3×C4—181no—
237C2×C3×C4—362no—
238C2×C3×C4—362no—
239C2×C3×C4C2×D3211yescharacteristic
240C2×C3×C4C22×C3×D312yes—
241C2×Dic3—22no—
242C2×Dic3—32no—
243C2×Dic3—61no—
244C2×Dic3—62no—
245C2×Dic3—62no—
246C4×D3—34no—
247C4×D3—34no—
248C4×D3—62no—
249C4×D3—62no—
250C4×D3—124no—
251C4×D3—124no—
252C4×D3—124no—
253C3×C8—34no—
254C3×C8—34no—
255C3×C8—64no—
256C3⋊C8C3×C4×D314yescomplemented
257C2×C32—22no—
258C2×C32—22no—
259C2×C32—22no—
260C2×C32—21no—
261C2×C32—21no—
262C2×C32—21no—
263C2×C32—41no—
264C2×C32—41no—
265C2×C32—41no—
266C2×C32D3×C4⋊C412yes—
267C2×C32D3×SD1612yes—
268C2×C32D3×SD1612yes—
269C2×C32C2×C3×C4⋊C411yescharacteristic
270C2×C32C2×C3×SD1611yescharacteristic
271C2×C32C2×C3×SD1611yescharacteristic
272C3⋊D3C3×#10614yescomplemented
273C3×D3—38no—
274C3×D3—64no—
275C22×C4—91no—
276C22×C4—181no—
277C22×C4—181no—
278C2×C8—32no—
279C2×C8—94no—
280C4⋊C4—32no—
281C4⋊C4—91no—
282C4⋊C4—181no—
283C4⋊C4—181no—
284C4⋊C4C3×D3211yescharacteristic
285C22×C3—21no—
286C22×C3—22no—
287C22×C3—41no—
288C22×C3—92no—
289C22×C3—92no—
290C22×C3—92no—
291C22×C3#381111yescharacteristic
292C22×C3C3×D3×D412yes—
293C2×D3—34no—
294C2×D3—34no—
295C2×D3—34no—
296C2×D3—62no—
297C2×D3—62no—
298C2×D3—62no—
299C3×C4—21no—
300C3×C4—22no—
301C3×C4—22no—
302C3×C4—21no—
303C3×C4—42no—
304C3×C4—42no—
305C3×C4—41no—
306C3×C4—41no—
307C3×C4—41no—
308C3×C4—82no—
309C3×C4—82no—
310C3×C4—92no—
311C3×C4—122no—
312C3×C4—122no—
313C3×C4—242no—
314C3×C4—361no—
315C3×C4#379711yescharacteristic
316C3×C4C4×D3211yescharacteristic
317C3×C4C2×C3×C4×D312yes—
318C3×C4C3×D3×Q812yes—
319Dic3—34no—
320Dic3—42no—
321Dic3—62no—
322Dic3—122no—
323Dic3—122no—
324C32—21no—
325C32—22no—
326C32—41no—
327C32D3×#10612yescomplemented
328C32C2×C3×#10611yescharacteristic, complemented
329C23—91no—
330C2×C4—21no—
331C2×C4—62no—
332C2×C4—91no—
333C2×C4—92no—
334C2×C4—92no—
335C2×C4—181no—
336C2×C4—362no—
337C2×C4—362no—
338C2×C4C2×C3×D3211yescharacteristic
339C8—34no—
340C2×C3—22no—
341C2×C3—21no—
342C2×C3—21no—
343C2×C3—22no—
344C2×C3—21no—
345C2×C3—22no—
346C2×C3—41no—
347C2×C3—41no—
348C2×C3—41no—
349C2×C3—94no—
350C2×C3#6388311yescharacteristic
351C2×C3#6393011yescharacteristic
352C2×C3#6394011yescharacteristic
353C2×C3C3×D3×C4⋊C412yes—
354C2×C3C3×D3×SD1612yes—
355C2×C3C3×D3×SD1612yes—
356D3—38no—
357D3—64no—
358C22—92no—
359C22—92no—
360C22—92no—
361C22C3×#381111yescharacteristic
362C4—41no—
363C4—92no—
364C4—122no—
365C4—361no—
366C4C3×#379711yescharacteristic
367C4C3×C4×D3211yescharacteristic
368C3—21no—
369C3—22no—
370C3—41no—
371C3#1059057511yescharacteristic, direct product factor
372C3C3×D3×#10612yescomplemented
373C2—94no—
374C2C3×#6388311yescharacteristic
375C2C3×#6393011yescharacteristic
376C2C3×#6394011yescharacteristic
subgroupG/H
centerC22×C3#3811
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C32×C4C23×C3
hypercenterC3×C4⋊C4D32
hypocenterC32C2×C3×#106
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. C32×#808C2semidirect
5. C3×#63573C2semidirect
7. #10590575C3direct (N × H)
17. C3×#3710C4semidirect
29. C3×#808D3semidirect
30. C3×#808C2×C3semidirect
31. #63573C2×C3semidirect
42. C32×C3⋊C8C2×C4semidirect
75. #3710C3×C4semidirect
99. #808C3×D3semidirect
136. C3×C3⋊C8C4×D3semidirect
137. C3×C3⋊C8C2×C3×C4semidirect
142. C3×C3⋊D3#106semidirect
218. C33C2×#106semidirect
256. C3⋊C8C3×C4×D3semidirect
272. C3⋊D3C3×#106semidirect
327. C32D3×#106semidirect
328. C32C2×C3×#106semidirect
371. C3#10590575direct (N × H)
372. C3C3×D3×#106semidirect

Maximal subgroups

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

subgroupindex
C3×C4⋊C4×C3⋊D32
C3×#639002
C2×C3×#37102
C32×#8082
C3×#635732
C3×D3×#1063
#105905753