Finite Group Databasebeta

← C3×#10590413 #13157523

Subgroups — C3×#10590413 #13157523

Order 1728.

498 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#890C211yesmaximal, characteristic
2C3×#63869C211yesmaximal, characteristic
3C3×#63635C211yesmaximal, characteristic
4C32×#807C211yesmaximal, characteristic
5C3×#63575C211yesmaximal, characteristic
6C3×#63574C211yesmaximal, characteristic
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic
8C3×#5065—31nomaximal
9C3×#4878—31nomaximal
10#10590413C311yesmaximal, characteristic, direct product factor
11C33×C4⋊C4C2211yescharacteristic
12C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
13C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
14C32×C4⋊Dic3—21no—
15C32×C4⋊Dic3C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×Dic3⋊C4—21no—
18C32×C4×Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×#3724—21no—
21C3×Dic3⋊Dic3—21no—
22C32×#125—31no—
23C3×#890—21no—
24C3×#890—31no—
25C3×#890D311yescharacteristic
26C3×#890C2×C311yescharacteristic
27C3×#870—31no—
28#63869C2×C311yescharacteristic
29C32×#107—31no—
30#63635C2×C311yescharacteristic
31C3×#818—31no—
32C3×C8×Dic3—31no—
33C3×#808—31no—
34C3×#808—31no—
35C3×#808—31no—
36C3×#808—31no—
37C3×#808—61no—
38C3×#807—21no—
39C3×#807—31no—
40C3×#807D311yescharacteristic
41C3×#807C2×C311yescharacteristic
42C3×C4×C3⋊C8—31no—
43#63575C2×C311yescharacteristic
44#63574C2×C311yescharacteristic
45Dic3×C3⋊C8C2×C311yescharacteristic
46C2×C33×C4—21no—
47C2×C33×C4C2311yescharacteristic
48C2×C3×C3⋊Dic3—21no—
49C2×C32×Dic3—21no—
50C2×C32×Dic3—21no—
51C2×C32×Dic3D411yescharacteristic
52C2×C32×Dic3D411yescharacteristic
53C3×C3⋊(C3⋊C8)Q812yes—
54C32×C3⋊C8Q812yes—
55C3×#499—91no—
56#5065—31no—
57#4878—31no—
58C2×C32×C8—31no—
59C32×C4⋊C4—21no—
60C32×C4⋊C4—21no—
61C32×C4⋊C4—21no—
62C32×C4⋊C4—31no—
63C32×C4⋊C4—31no—
64C32×C4⋊C4—41no—
65C32×C4⋊C4—61no—
66C32×C4⋊C4—61no—
67C32×C4⋊C4C2×D311yescharacteristic
68C32×C4⋊C4C2×D311yescharacteristic
69C32×C4⋊C4C22×C311yescharacteristic
70C32×C42—31no—
71C4⋊(C3⋊Dic3)C22×C311yescharacteristic
72C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
73C3×C4⋊Dic3—22no—
74C3×C4⋊Dic3—21no—
75C3×C4⋊Dic3—21no—
76C3×C4⋊Dic3—21no—
77C3×C4⋊Dic3—21no—
78C3×C4⋊Dic3—31no—
79C3×C4⋊Dic3—31no—
80C3×C4⋊Dic3—61no—
81C3×C4⋊Dic3—61no—
82C3×C4⋊Dic3C2×D311yescharacteristic
83C3×C4⋊Dic3C2×D311yescharacteristic
84C3×C4⋊Dic3C22×C311yescharacteristic
85C3×C4⋊Dic3C22×C311yescharacteristic
86C3×Dic3⋊C4—22no—
87C3×Dic3⋊C4—21no—
88C3×Dic3⋊C4—21no—
89C3×Dic3⋊C4—61no—
90C3×Dic3⋊C4—61no—
91C3×Dic3⋊C4—61no—
92C3×C4×Dic3—21no—
93C3×C4×Dic3C2×D311yescharacteristic
94C3×C4×Dic3C22×C311yescharacteristic
95C2×C3×C3⋊C8—21no—
96C2×C3×C3⋊C8—31no—
97C2×C3×C3⋊C8—31no—
98C2×C3×C3⋊C8—61no—
99C2×C3×C3⋊C8C2×D311yescharacteristic
100C2×C3×C3⋊C8C22×C311yescharacteristic
101#3724—21no—
102Dic3⋊Dic3—21no—
103C22×C33C2×D411yescharacteristic
104C33×C4—41no—
105C33×C4C2×Q811yescharacteristic
106C33×C4C2×Q811yescharacteristic
107C3×C3⋊Dic3—41no—
108C32×Dic3—21no—
109C32×Dic3—21no—
110C32×Dic3—41no—
111C32×Dic3—41no—
112C3×#125—31no—
113C3×#125—31no—
114C3×#125—61no—
115C3×#125—91no—
116#890—31no—
117#890C3×D311yescharacteristic
118#870—31no—
119C3×#107—31no—
120C3×#107—31no—
121C3×#107—61no—
122C3×#107—91no—
123C3×#106—91no—
124C3×#106—91no—
125C3×C4×C8—91no—
126#818—31no—
127C8×Dic3—31no—
128#808—31no—
129#808—31no—
130#808—31no—
131#808—31no—
132#808—61no—
133#807—31no—
134#807C3×D311yescharacteristic
135C4×C3⋊C8—31no—
136C2×C32×C4—21no—
137C2×C32×C4—22no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—31no—
144C2×C32×C4—31no—
145C2×C32×C4—41no—
146C2×C32×C4—41no—
147C2×C32×C4—41no—
148C2×C32×C4—42no—
149C2×C32×C4—61no—
150C2×C32×C4—61no—
151C2×C32×C4C22×D311yescharacteristic
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C23×C311yescharacteristic
154C2×C3⋊Dic3—21no—
155C2×C3×Dic3—22no—
156C2×C3×Dic3—21no—
157C2×C3×Dic3—21no—
158C2×C3×Dic3—21no—
159C2×C3×Dic3—21no—
160C2×C3×Dic3—21no—
161C2×C3×Dic3—21no—
162C2×C3×Dic3—41no—
163C2×C3×Dic3—61no—
164C2×C3×Dic3—61no—
165C2×C3×Dic3—121no—
166C2×C3×Dic3#6711yescharacteristic
167C2×C3×Dic3#6711yescharacteristic
168C2×C3×Dic3C3×D411yescharacteristic
169C2×C3×Dic3C3×D411yescharacteristic
170C32×C8—32no—
171C3⋊(C3⋊C8)C3×Q812yes—
172C3×C3⋊C8—22no—
173C3×C3⋊C8—32no—
174C3×C3⋊C8—32no—
175C3×C3⋊C8—62no—
176C3×C3⋊C8Dic612yes—
177C3×C3⋊C8C3×Q812yes—
178#499—91noSylow
179C2×C33C4⋊Q811yescharacteristic
180C2×C33#13511yescharacteristic
181C2×C33#13511yescharacteristic
182C2×C3×C8—31no—
183C2×C3×C8—31no—
184C2×C3×C8—61no—
185C2×C3×C8—91no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—31no—
190C3×C4⋊C4—31no—
191C3×C4⋊C4—31no—
192C3×C4⋊C4—31no—
193C3×C4⋊C4—41no—
194C3×C4⋊C4—61no—
195C3×C4⋊C4—61no—
196C3×C4⋊C4—61no—
197C3×C4⋊C4—61no—
198C3×C4⋊C4—61no—
199C3×C4⋊C4—61no—
200C3×C4⋊C4—62no—
201C3×C4⋊C4—62no—
202C3×C4⋊C4—91no—
203C3×C4⋊C4—181no—
204C3×C4⋊C4—181no—
205C3×C4⋊C4D3211yescharacteristic
206C3×C4⋊C4C2×C3×D311yescharacteristic
207C3×C4⋊C4C2×C3×D311yescharacteristic
208C3×C42—31no—
209C3×C42—31no—
210C3×C42—61no—
211C4⋊Dic3—21no—
212C4⋊Dic3—31no—
213C4⋊Dic3—31no—
214C4⋊Dic3—61no—
215C4⋊Dic3—61no—
216C4⋊Dic3C2×C3×D311yescharacteristic
217C4⋊Dic3C2×C3×D311yescharacteristic
218Dic3⋊C4—21no—
219Dic3⋊C4—61no—
220Dic3⋊C4—61no—
221Dic3⋊C4—61no—
222C4×Dic3C2×C3×D311yescharacteristic
223C2×C3⋊C8—31no—
224C2×C3⋊C8—31no—
225C2×C3⋊C8—61no—
226C2×C3⋊C8C2×C3×D311yescharacteristic
227C22×C32—21no—
228C22×C32—21no—
229C22×C32—21no—
230C22×C32—41no—
231C22×C32D3×D411yescharacteristic
232C22×C32C2×#6711yescharacteristic
233C22×C32C2×C3×D411yescharacteristic
234C32×C4—21no—
235C32×C4—21no—
236C32×C4—21no—
237C32×C4—21no—
238C32×C4—21no—
239C32×C4—21no—
240C32×C4—41no—
241C32×C4—41no—
242C32×C4—41no—
243C32×C4—42no—
244C32×C4—41no—
245C32×C4—41no—
246C32×C4—61no—
247C32×C4—61no—
248C32×C4—81no—
249C32×C4—81no—
250C32×C4—82no—
251C32×C4—121no—
252C32×C4—121no—
253C32×C4C2×Dic611yescharacteristic
254C32×C4D3×Q811yescharacteristic
255C32×C4D3×Q811yescharacteristic
256C32×C4D3×Q811yescharacteristic
257C32×C4C2×C3×Q811yescharacteristic
258C32×C4C2×C3×Q811yescharacteristic
259C3⋊Dic3—41no—
260C3×Dic3—21no—
261C3×Dic3—21no—
262C3×Dic3—21no—
263C3×Dic3—21no—
264C3×Dic3—41no—
265C3×Dic3—41no—
266C3×Dic3—42no—
267C3×Dic3—41no—
268C3×Dic3—41no—
269C3×Dic3—41no—
270C3×Dic3—41no—
271C3×Dic3—81no—
272C3×Dic3—121no—
273C3×Dic3—121no—
274C3×Dic3—241no—
275#125—31no—
276#125—91no—
277#107—31no—
278#107—91no—
279#106—91no—
280#106—91no—
281C4×C8—91no—
282C33#49911yescharacteristic, complemented, Sylow
283C2×C3×C4—22no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—31no—
291C2×C3×C4—31no—
292C2×C3×C4—31no—
293C2×C3×C4—31no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4—42no—
297C2×C3×C4—41no—
298C2×C3×C4—61no—
299C2×C3×C4—61no—
300C2×C3×C4—62no—
301C2×C3×C4—61no—
302C2×C3×C4—61no—
303C2×C3×C4—61no—
304C2×C3×C4—61no—
305C2×C3×C4—121no—
306C2×C3×C4—181no—
307C2×C3×C4C2×D3211yescharacteristic
308C2×C3×C4C22×C3×D311yescharacteristic
309C2×C3×C4C22×C3×D311yescharacteristic
310C2×Dic3—21no—
311C2×Dic3—21no—
312C2×Dic3—61no—
313C2×Dic3—61no—
314C2×Dic3—121no—
315C2×Dic3C3×#6711yescharacteristic
316C2×Dic3C3×#6711yescharacteristic
317C3×C8—32no—
318C3×C8—32no—
319C3×C8—62no—
320C3×C8—92no—
321C3⋊C8—32no—
322C3⋊C8—32no—
323C3⋊C8—62no—
324C3⋊C8C3×Dic612yes—
325C2×C32—21no—
326C2×C32—21no—
327C2×C32—21no—
328C2×C32—21no—
329C2×C32—21no—
330C2×C32—21no—
331C2×C32—21no—
332C2×C32—21no—
333C2×C32—21no—
334C2×C32—41no—
335C2×C32—41no—
336C2×C32—41no—
337C2×C32#88811yescharacteristic
338C2×C32#91211yescharacteristic
339C2×C32#91911yescharacteristic
340C2×C32#94411yescharacteristic
341C2×C32#95011yescharacteristic
342C2×C32#95011yescharacteristic
343C2×C32C3×C4⋊Q811yescharacteristic
344C2×C32C3×#13511yescharacteristic
345C2×C32C3×#13511yescharacteristic
346C2×C8—31no—
347C2×C8—91no—
348C4⋊C4—31no—
349C4⋊C4—31no—
350C4⋊C4—61no—
351C4⋊C4—61no—
352C4⋊C4—91no—
353C4⋊C4—181no—
354C4⋊C4—181no—
355C4⋊C4C3×D3211yescharacteristic
356C42—31no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—41no—
361C22×C3D3×#6711yescharacteristic
362C22×C3C3×D3×D411yescharacteristic
363C22×C3C2×C3×#6711yescharacteristic
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—42no—
371C3×C4—41no—
372C3×C4—41no—
373C3×C4—41no—
374C3×C4—41no—
375C3×C4—41no—
376C3×C4—61no—
377C3×C4—61no—
378C3×C4—61no—
379C3×C4—61no—
380C3×C4—81no—
381C3×C4—81no—
382C3×C4—82no—
383C3×C4—121no—
384C3×C4—121no—
385C3×C4—122no—
386C3×C4—121no—
387C3×C4—121no—
388C3×C4—121no—
389C3×C4—121no—
390C3×C4—241no—
391C3×C4—361no—
392C3×C4D3×Dic611yescharacteristic
393C3×C4#379711yescharacteristic
394C3×C4C2×C3×Dic611yescharacteristic
395C3×C4C3×D3×Q811yescharacteristic
396C3×C4C3×D3×Q811yescharacteristic
397C3×C4C3×D3×Q811yescharacteristic
398Dic3—21no—
399Dic3—21no—
400Dic3—41no—
401Dic3—41no—
402Dic3—121no—
403Dic3—121no—
404Dic3—241no—
405C32—21no—
406C32—21no—
407C32—21no—
408C32—41no—
409C32#487811yescharacteristic, complemented
410C32#506511yescharacteristic, complemented
411C32C3×#49911yescharacteristic, complemented
412C2×C4—21no—
413C2×C4—31no—
414C2×C4—31no—
415C2×C4—61no—
416C2×C4—61no—
417C2×C4—181no—
418C2×C4C2×C3×D3211yescharacteristic
419C8—32no—
420C8—92no—
421C2×C3—21no—
422C2×C3—21no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—21no—
429C2×C3—21no—
430C2×C3—41no—
431C2×C3—41no—
432C2×C3—41no—
433C2×C3#6383911yescharacteristic
434C2×C3#6393311yescharacteristic
435C2×C3#6394911yescharacteristic
436C2×C3C3×#88811yescharacteristic
437C2×C3C3×#91211yescharacteristic
438C2×C3C3×#91911yescharacteristic
439C2×C3C3×#94411yescharacteristic
440C2×C3C3×#95011yescharacteristic
441C2×C3C3×#95011yescharacteristic
442C22C3×D3×#6711yescharacteristic
443C4—41no—
444C4—61no—
445C4—61no—
446C4—121no—
447C4—121no—
448C4—361no—
449C4C3×D3×Dic611yescharacteristic
450C4C3×#379711yescharacteristic
451C3—21no—
452C3—21no—
453C3—21no—
454C3—41no—
455C3#1059041311yescharacteristic, direct product factor
456C3C3×#487811yescharacteristic, complemented
457C3C3×#506511yescharacteristic, complemented
458C2C3×#6383911yescharacteristic
459C2C3×#6393311yescharacteristic
460C2C3×#6394911yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socle, prime radicalC22×C33C2×D4
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#499
LCSG ▷ C2×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
10. #10590413C3direct (N × H)
282. C33#499semidirect
409. C32#4878semidirect
410. C32#5065semidirect
411. C32C3×#499semidirect
455. C3#10590413direct (N × H)
456. C3C3×#4878semidirect
457. C3C3×#5065semidirect

Maximal subgroups

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

subgroupindex
C32×#8902
C3×#638692
C3×#636352
C32×#8072
C3×#635752
C3×#635742
C3×Dic3×C3⋊C82
C3×#50653
C3×#48783
#105904133