Finite Group Databasebeta

← C3×#10591115 #13158146

Subgroups — C3×#10591115 #13158146

Order 1728.

802 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3748C211yesmaximal, characteristic
2C2×C32×C4×Dic3C211yesmaximal, characteristic
3C2×C32×#208C211yesmaximal, characteristic
4C3×Dic3⋊(C3⋊C8)C214yesmaximal, complemented
5C3×#5118—31nomaximal
6C3×#4924—31nomaximal
7#10591115C311yesmaximal, characteristic, direct product factor
8C22×C33×C4C2211yescharacteristic
9C22×C32×Dic3C412yes—
10C3×#3748—22no—
11C2×C3×C3⋊(C3⋊C8)C2212yes—
12C32×C4×Dic3C411yescharacteristic
13C32×C4×Dic3C411yescharacteristic
14C32×C4×Dic3C2212yes—
15C32×#208—22no—
16C2×C32×C3⋊C8C2212yes—
17C2×C32×M16—31no—
18C2×C32×C42—31no—
19C2×#3748C2×C311yescharacteristic
20C2×C3×C4×Dic3—21no—
21C2×C3×C4×Dic3D311yescharacteristic
22C2×C3×C4×Dic3C2×C311yescharacteristic
23C2×C3×#208—21no—
24C2×C3×#208—31no—
25C2×C3×#208—31no—
26C2×C3×#208—61no—
27C2×C3×#208D311yescharacteristic
28C2×C3×#208C2×C311yescharacteristic
29C3×Dic3⋊C8—34no—
30C3×C4⋊(C3⋊C8)—34no—
31Dic3⋊(C3⋊C8)C2×C314yescomplemented
32C23×C33C2×C411yescharacteristic
33C2×C33×C4C2×C411yescharacteristic
34C2×C33×C4D412yes—
35C2×C33×C4Q812yes—
36C2×C33×C4C2311yescharacteristic
37C2×C32×Dic3—24no—
38C2×C32×Dic3C2×C412yes—
39C2×C32×Dic3C2×C412yes—
40C3×C3⋊(C3⋊C8)—22no—
41C32×C3⋊C8—22no—
42C3×#423—91no—
43#5118—31no—
44#4924—31no—
45C22×C32×C4—21no—
46C22×C32×C4—21no—
47C22×C32×C4—21no—
48C22×C32×C4—32no—
49C22×C32×C4—41no—
50C22×C32×C4C2×D311yescharacteristic
51C22×C32×C4C2×D311yescharacteristic
52C22×C32×C4C22×C311yescharacteristic
53C22×C3×Dic3—22no—
54C22×C3×Dic3Dic312yes—
55C22×C3×Dic3C3×C412yes—
56C32×M16—62no—
57C2×C32×C8—32no—
58C32×C42—31no—
59C32×C42—31no—
60C32×C42—32no—
61#3748—22no—
62C2×C3⋊(C3⋊C8)C22×C312yes—
63C3×C4×Dic3—21no—
64C3×C4×Dic3—21no—
65C3×C4×Dic3—22no—
66C3×C4×Dic3Dic311yescharacteristic
67C3×C4×Dic3Dic311yescharacteristic
68C3×C4×Dic3C3×C411yescharacteristic
69C3×C4×Dic3C3×C411yescharacteristic
70C3×C4×Dic3C2×D312yes—
71C3×C4×Dic3C22×C312yes—
72C3×#208—24no—
73C3×#208—22no—
74C3×#208—22no—
75C3×#208—64no—
76C3×#208—62no—
77C3×#208—62no—
78C2×C3×C3⋊C8—22no—
79C2×C3×C3⋊C8—32no—
80C2×C3×C3⋊C8—32no—
81C2×C3×C3⋊C8—62no—
82C2×C3×C3⋊C8C2×D312yes—
83C2×C3×C3⋊C8C22×C312yes—
84C22×C33—22no—
85C22×C33C4⋊C412yes—
86C22×C33C22×C411yescharacteristic
87C33×C4C4⋊C412yes—
88C33×C4C2×D411yescharacteristic
89C33×C4C2×Q811yescharacteristic
90C32×Dic3—22no—
91C32×Dic3M1614yescomplemented
92C2×C3×M16—31no—
93C2×C3×M16—31no—
94C2×C3×M16—61no—
95C2×C3×M16—91no—
96C2×C3×C42—31no—
97C2×C3×C42—31no—
98C2×C3×C42—61no—
99C2×C4×Dic3C3×D311yescharacteristic
100C2×#208—31no—
101C2×#208—31no—
102C2×#208—61no—
103C2×#208C3×D311yescharacteristic
104C3×C4⋊C8—94no—
105Dic3⋊C8—34no—
106C4⋊(C3⋊C8)—34no—
107C23×C32—21no—
108C23×C32—21no—
109C23×C32—21no—
110C23×C32—41no—
111C23×C32C4×D311yescharacteristic
112C23×C32C2×Dic311yescharacteristic
113C23×C32C2×C3×C411yescharacteristic
114C2×C32×C4—21no—
115C2×C32×C4—22no—
116C2×C32×C4—22no—
117C2×C32×C4—22no—
118C2×C32×C4—22no—
119C2×C32×C4—21no—
120C2×C32×C4—22no—
121C2×C32×C4—21no—
122C2×C32×C4—22no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4—32no—
127C2×C32×C4—32no—
128C2×C32×C4—41no—
129C2×C32×C4—42no—
130C2×C32×C4—42no—
131C2×C32×C4—41no—
132C2×C32×C4—64no—
133C2×C32×C4Dic612yes—
134C2×C32×C4Dic612yes—
135C2×C32×C4C4×D311yescharacteristic
136C2×C32×C4D1212yes—
137C2×C32×C4C2×Dic311yescharacteristic
138C2×C32×C4#6712yes—
139C2×C32×C4C2×C3×C411yescharacteristic
140C2×C32×C4C3×D412yes—
141C2×C32×C4C3×Q812yes—
142C2×C32×C4C22×D311yescharacteristic
143C2×C32×C4C22×D311yescharacteristic
144C2×C32×C4C23×C311yescharacteristic
145C2×C3×Dic3—28no—
146C2×C3×Dic3—24no—
147C2×C3×Dic3—24no—
148C2×C3×Dic3—22no—
149C2×C3×Dic3—22no—
150C2×C3×Dic3C2×Dic312yes—
151C2×C3×Dic3C2×Dic312yes—
152C2×C3×Dic3C2×C3×C412yes—
153C2×C3×Dic3C2×C3×C412yes—
154C32×C8—62no—
155C3⋊(C3⋊C8)—22no—
156C3×C3⋊C8—24no—
157C3×C3⋊C8—22no—
158C3×C3⋊C8—22no—
159C3×C3⋊C8—64no—
160C3×C3⋊C8—62no—
161C3×C3⋊C8—62no—
162#423—91noSylow
163C2×C33—22no—
164C2×C33C2×C4⋊C411yescharacteristic
165C2×C33C2×M1612yes—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—32no—
170C22×C3×C4—32no—
171C22×C3×C4—41no—
172C22×C3×C4—62no—
173C22×C3×C4D3211yescharacteristic
174C22×C3×C4C2×C3×D311yescharacteristic
175C22×C3×C4C2×C3×D311yescharacteristic
176C22×Dic3C3×Dic312yes—
177C3×M16—64no—
178C3×M16—62no—
179C3×M16—62no—
180C3×M16—182no—
181C2×C3×C8—32no—
182C2×C3×C8—32no—
183C2×C3×C8—62no—
184C2×C3×C8—92no—
185C3×C42—31no—
186C3×C42—31no—
187C3×C42—31no—
188C3×C42—31no—
189C3×C42—32no—
190C3×C42—32no—
191C3×C42—61no—
192C3×C42—61no—
193C3×C42—62no—
194C4×Dic3C3×Dic311yescharacteristic
195C4×Dic3C3×Dic311yescharacteristic
196C4×Dic3C2×C3×D312yes—
197#208—22no—
198#208—64no—
199#208—62no—
200#208—62no—
201C2×C3⋊C8—32no—
202C2×C3⋊C8—32no—
203C2×C3⋊C8—62no—
204C2×C3⋊C8C2×C3×D312yes—
205C22×C32—22no—
206C22×C32—24no—
207C22×C32—22no—
208C22×C32—22no—
209C22×C32—22no—
210C22×C32—22no—
211C22×C32—22no—
212C22×C32—21no—
213C22×C32—21no—
214C22×C32—21no—
215C22×C32—42no—
216C22×C32—41no—
217C22×C32—42no—
218C22×C32—42no—
219C22×C32—44no—
220C22×C32Dic3⋊C412yes—
221C22×C32C4⋊Dic312yes—
222C22×C32C3×C4⋊C412yes—
223C22×C32C2×C4×D311yescharacteristic
224C22×C32C22×Dic311yescharacteristic
225C22×C32C22×C3×C411yescharacteristic
226C32×C4—22no—
227C32×C4—22no—
228C32×C4—22no—
229C32×C4—21no—
230C32×C4—21no—
231C32×C4—21no—
232C32×C4—21no—
233C32×C4—21no—
234C32×C4—21no—
235C32×C4—34no—
236C32×C4—42no—
237C32×C4—41no—
238C32×C4—41no—
239C32×C4—62no—
240C32×C4Dic3⋊C412yes—
241C32×C4C4⋊Dic312yes—
242C32×C4C3×C4⋊C412yes—
243C32×C4C2×Dic611yescharacteristic
244C32×C4C2×Dic611yescharacteristic
245C32×C4C2×D1211yescharacteristic
246C32×C4C2×#6711yescharacteristic
247C32×C4C2×C3×D411yescharacteristic
248C32×C4C2×C3×Q811yescharacteristic
249C3×Dic3—24no—
250C3×Dic3—22no—
251C3×Dic3—22no—
252C3×Dic3—24no—
253C3×Dic3#20814yescomplemented
254C3×Dic3C3×M1614yescomplemented
255C2×M16—31no—
256C2×M16—91no—
257C2×C42—31no—
258C4⋊C8—94no—
259C33#42311yescharacteristic, complemented, Sylow
260C23×C3—21no—
261C23×C3—21no—
262C23×C3—21no—
263C23×C3—41no—
264C23×C3D3×Dic311yescharacteristic
265C23×C3C3×C4×D311yescharacteristic
266C23×C3C2×C3×Dic311yescharacteristic
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—32no—
280C2×C3×C4—32no—
281C2×C3×C4—32no—
282C2×C3×C4—32no—
283C2×C3×C4—41no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—41no—
287C2×C3×C4—62no—
288C2×C3×C4—62no—
289C2×C3×C4—64no—
290C2×C3×C4—64no—
291C2×C3×C4—68no—
292C2×C3×C4D3×Dic311yescharacteristic
293C2×C3×C4#62612yes—
294C2×C3×C4#62712yes—
295C2×C3×C4C3×Dic612yes—
296C2×C3×C4C3×Dic612yes—
297C2×C3×C4C3×C4×D311yescharacteristic
298C2×C3×C4C3×D1212yes—
299C2×C3×C4C2×C3×Dic311yescharacteristic
300C2×C3×C4C3×#6712yes—
301C2×C3×C4C2×D3211yescharacteristic
302C2×C3×C4C22×C3×D311yescharacteristic
303C2×C3×C4C22×C3×D311yescharacteristic
304C2×Dic3—24no—
305C2×Dic3C2×C3×Dic312yes—
306C2×Dic3C2×C3×Dic312yes—
307C3×C8—64no—
308C3×C8—62no—
309C3×C8—62no—
310C3×C8—182no—
311C3⋊C8—22no—
312C3⋊C8—64no—
313C3⋊C8—62no—
314C3⋊C8—62no—
315C2×C32—22no—
316C2×C32—24no—
317C2×C32—22no—
318C2×C32—22no—
319C2×C32—22no—
320C2×C32—22no—
321C2×C32—21no—
322C2×C32—21no—
323C2×C32—22no—
324C2×C32—21no—
325C2×C32—41no—
326C2×C32—42no—
327C2×C32—42no—
328C2×C32—42no—
329C2×C32—44no—
330C2×C32D3×M1612yes—
331C2×C32C2×#20812yes—
332C2×C32C2×Dic3⋊C411yescharacteristic
333C2×C32C2×C4⋊Dic311yescharacteristic
334C2×C32C2×C3×C4⋊C411yescharacteristic
335C2×C32C2×C3×M1612yes—
336C22×C4—32no—
337C22×C4C3×D3211yescharacteristic
338M16—62no—
339M16—182no—
340C2×C8—32no—
341C2×C8—92no—
342C42—31no—
343C42—31no—
344C42—32no—
345C22×C3—24no—
346C22×C3—22no—
347C22×C3—22no—
348C22×C3—22no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—22no—
353C22×C3—22no—
354C22×C3—22no—
355C22×C3—42no—
356C22×C3—42no—
357C22×C3—44no—
358C22×C3—41no—
359C22×C3—42no—
360C22×C3Dic3⋊Dic312yes—
361C22×C3C3×Dic3⋊C412yes—
362C22×C3C3×C4⋊Dic312yes—
363C22×C3C2×D3×Dic311yescharacteristic
364C22×C3C2×C3×C4×D311yescharacteristic
365C22×C3C22×C3×Dic311yescharacteristic
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—21no—
371C3×C4—21no—
372C3×C4—22no—
373C3×C4—22no—
374C3×C4—22no—
375C3×C4—34no—
376C3×C4—34no—
377C3×C4—41no—
378C3×C4—41no—
379C3×C4—42no—
380C3×C4—64no—
381C3×C4—64no—
382C3×C4—62no—
383C3×C4—62no—
384C3×C4Dic3⋊Dic312yes—
385C3×C4C3×Dic3⋊C412yes—
386C3×C4C3×C4⋊Dic312yes—
387C3×C4C2×#62611yescharacteristic
388C3×C4C2×#62711yescharacteristic
389C3×C4C2×C3×Dic611yescharacteristic
390C3×C4C2×C3×Dic611yescharacteristic
391C3×C4C2×C3×D1211yescharacteristic
392C3×C4C2×C3×#6711yescharacteristic
393Dic3—22no—
394Dic3C3×#20814yescomplemented
395C32—21no—
396C32—21no—
397C32—21no—
398C32—41no—
399C32#492411yescharacteristic, complemented
400C32#511811yescharacteristic, complemented
401C32C3×#42311yescharacteristic, complemented
402C23C3×D3×Dic311yescharacteristic
403C2×C4—32no—
404C2×C4—32no—
405C2×C4—64no—
406C2×C4C3×D3×Dic311yescharacteristic
407C2×C4C3×#62612yes—
408C2×C4C3×#62712yes—
409C2×C4C2×C3×D3211yescharacteristic
410C8—62no—
411C8—182no—
412C2×C3—24no—
413C2×C3—22no—
414C2×C3—22no—
415C2×C3—22no—
416C2×C3—21no—
417C2×C3—21no—
418C2×C3—21no—
419C2×C3—22no—
420C2×C3—22no—
421C2×C3—22no—
422C2×C3—42no—
423C2×C3—42no—
424C2×C3—44no—
425C2×C3—41no—
426C2×C3—42no—
427C2×C3D3×#20812yes—
428C2×C3C2×Dic3⋊Dic311yescharacteristic
429C2×C3C3×D3×M1612yes—
430C2×C3C2×C3×#20812yes—
431C2×C3C2×C3×Dic3⋊C411yescharacteristic
432C2×C3C2×C3×C4⋊Dic311yescharacteristic
433C22—22no—
434C22C3×Dic3⋊Dic312yes—
435C22C2×C3×D3×Dic311yescharacteristic
436C4—34no—
437C4—62no—
438C4C3×Dic3⋊Dic312yes—
439C4C2×C3×#62611yescharacteristic
440C4C2×C3×#62711yescharacteristic
441C3—21no—
442C3—21no—
443C3—21no—
444C3—41no—
445C3#1059111511yescharacteristic, direct product factor
446C3C3×#492411yescharacteristic, complemented
447C3C3×#511811yescharacteristic, complemented
448C2—22no—
449C2C3×D3×#20812yes—
450C2C2×C3×Dic3⋊Dic311yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D32
commutatorC22×C32C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C22×C4
prime radicalC23×C33C2×C4
cosocleC2×C32×C4C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#423
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
4. C3×Dic3⋊(C3⋊C8)C2semidirect
7. #10591115C3direct (N × H)
31. Dic3⋊(C3⋊C8)C2×C3semidirect
91. C32×Dic3M16semidirect
253. C3×Dic3#208semidirect
254. C3×Dic3C3×M16semidirect
259. C33#423semidirect
394. Dic3C3×#208semidirect
399. C32#4924semidirect
400. C32#5118semidirect
401. C32C3×#423semidirect
445. C3#10591115direct (N × H)
446. C3C3×#4924semidirect
447. C3C3×#5118semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37482
C2×C32×C4×Dic32
C2×C32×#2082
C3×Dic3⋊(C3⋊C8)2
C3×#51183
C3×#49243
#105911153