Finite Group Databasebeta

← C3×#10590485 #13157606

Subgroups — C3×#10590485 #13157606

Order 1728.

606 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64096C211yesmaximal, characteristic
2C3×#63843C211yesmaximal, characteristic
3C2×C3×#3719C211yesmaximal, characteristic
4C32×#810C211yesmaximal, characteristic
5C32×#808C211yesmaximal, characteristic
6C3×#63567C211yesmaximal, characteristic
7C3×#63555C211yesmaximal, characteristic
8C3×#4853—31nomaximal
9C3×#4789—31nomaximal
10#10590485C311yesmaximal, characteristic, direct product factor
11C2×C3×#634C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C33×C4⋊C4C2211yescharacteristic
14C3×#3750—21no—
15C3×C4×C3⋊Dic3—21no—
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×Dic3⋊Dic3—21no—
21C3×Dic32—21no—
22C3×#3719C414yescomplemented
23#64096C2×C311yescharacteristic
24C3×Q8×Dic3—31no—
25C2×C3×#216—31no—
26C2×C3×Dic12—31no—
27C3×#887—31no—
28C3×#887—31no—
29C3×#887—31no—
30C3×#887—61no—
31#63843C2×C311yescharacteristic
32C2×#3719C2×C311yescharacteristic
33C32×#106—31no—
34C32×#103—31no—
35C3×#816—31no—
36C3×#815—31no—
37C3×#815—31no—
38C3×#810—21no—
39C3×#810—31no—
40C3×#810D311yescharacteristic
41C3×#810C2×C311yescharacteristic
42C3×#808—21no—
43C3×#808D311yescharacteristic
44C3×#808C2×C311yescharacteristic
45#63567C2×C311yescharacteristic
46#63555C2×C311yescharacteristic
47C2×C33×C4—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3D411yescharacteristic
51C2×C3×C3⋊Dic3D411yescharacteristic
52C3×#634—21no—
53C3×#634C2×C412yes—
54C2×C32×Dic3—21no—
55C2×C32×Dic3—21no—
56C32×Dic6—21no—
57C32×Dic6C2×C412yes—
58C32×C3⋊C8—21no—
59C32×C3⋊C8C2×C412yes—
60C3×#441—91no—
61#4853—31no—
62#4789—31no—
63C2×C32×Q8—31no—
64C2×#634C22×C311yescharacteristic
65C2×C3×Dic6—21no—
66C2×C3×Dic6—31no—
67C2×C3×Dic6—31no—
68C2×C3×Dic6—61no—
69C2×C3×Dic6C2×D311yescharacteristic
70C2×C3×Dic6C22×C311yescharacteristic
71C2×C32×C8—31no—
72C2×C32×C8—31no—
73C32×C4⋊C4—21no—
74C32×C4⋊C4—21no—
75C32×C4⋊C4—21no—
76C32×C4⋊C4—31no—
77C32×C4⋊C4—41no—
78C32×C4⋊C4C2×D311yescharacteristic
79C32×C4⋊C4C2×D311yescharacteristic
80C32×C4⋊C4C22×C311yescharacteristic
81#3750—21no—
82C4×C3⋊Dic3—21no—
83C4×C3⋊Dic3C22×C311yescharacteristic
84C3×#216—34no—
85C3×C4⋊Dic3—21no—
86C3×C4⋊Dic3—61no—
87C3×C4⋊Dic3C2×D311yescharacteristic
88C3×C4⋊Dic3C22×C311yescharacteristic
89C3×Dic3⋊C4—61no—
90C3×Dic3⋊C4—61no—
91C3×Dic3⋊C4—61no—
92C3×Dic3⋊C4—62no—
93C3×C4×Dic3—31no—
94C3×C4×Dic3—31no—
95C3×C4×Dic3—61no—
96C3×C4×Dic3—61no—
97C3×C4×Dic3—61no—
98C3×C4×Dic3—61no—
99C3×C4×Dic3—61no—
100C3×C4×Dic3—62no—
101C2×C3×C3⋊C8—21no—
102C2×C3×C3⋊C8—21no—
103C2×C3×C3⋊C8C2×D311yescharacteristic
104C2×C3×C3⋊C8C2×D311yescharacteristic
105C2×C3×C3⋊C8C22×C311yescharacteristic
106C2×C3×C3⋊C8C22×C311yescharacteristic
107C3×Dic12—34no—
108Dic3⋊Dic3—21no—
109Dic32—21no—
110#3719C3×C414yescomplemented
111C22×C33C2×D411yescharacteristic
112C33×C4—41no—
113C33×C4C22×C411yescharacteristic
114C33×C4#4111yescharacteristic
115C3×C3⋊Dic3—22no—
116C3×C3⋊Dic3—21no—
117C3×C3⋊Dic3—21no—
118C32×Dic3—22no—
119C32×Dic3—41no—
120C2×C3×Q16—91no—
121C3×C4×Q8—91no—
122C3×C4×Q8—91no—
123Q8×Dic3—31no—
124C2×#216—31no—
125C2×Dic12—31no—
126#887—31no—
127#887—31no—
128#887—31no—
129#887—61no—
130C3×#106—31no—
131C3×#106—31no—
132C3×#106—61no—
133C3×#103—31no—
134C3×#103—31no—
135C3×#103—61no—
136C3×#103—91no—
137C3×#97—91no—
138#816—31no—
139#815—31no—
140#815—31no—
141#810—31no—
142#810C3×D311yescharacteristic
143#808C3×D311yescharacteristic
144C32×Q8—32no—
145C32×Q8—61no—
146C2×C32×C4—21no—
147C2×C32×C4—22no—
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4—21no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—41no—
154C2×C32×C4—41no—
155C2×C32×C4—41no—
156C2×C32×C4—42no—
157C2×C32×C4—61no—
158C2×C32×C4—61no—
159C2×C32×C4C22×D311yescharacteristic
160C2×C32×C4C22×D311yescharacteristic
161C2×C32×C4C23×C311yescharacteristic
162C2×C3⋊Dic3—21no—
163C2×C3⋊Dic3C3×D411yescharacteristic
164C2×C3⋊Dic3C3×D411yescharacteristic
165#634—21no—
166#634C2×C3×C412yes—
167C2×C3×Dic3—21no—
168C2×C3×Dic3—21no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—21no—
171C2×C3×Dic3—31no—
172C2×C3×Dic3—31no—
173C2×C3×Dic3—31no—
174C2×C3×Dic3—31no—
175C2×C3×Dic3—41no—
176C2×C3×Dic3—41no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—62no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—61no—
182C3×Dic6—21no—
183C3×Dic6—21no—
184C3×Dic6—22no—
185C3×Dic6—32no—
186C3×Dic6—32no—
187C3×Dic6—41no—
188C3×Dic6—62no—
189C3×Dic6—62no—
190C3×Dic6—61no—
191C3×Dic6—61no—
192C3×Dic6C4×D312yes—
193C3×Dic6C2×C3×C412yes—
194C32×C8—32no—
195C32×C8—61no—
196C3×C3⋊C8—21no—
197C3×C3⋊C8—21no—
198C3×C3⋊C8—22no—
199C3×C3⋊C8—41no—
200C3×C3⋊C8C4×D312yes—
201C3×C3⋊C8C2×C3×C412yes—
202#441—91noSylow
203C2×C33C4×D411yescharacteristic
204C2×C33#13711yescharacteristic
205C2×C33#13711yescharacteristic
206C2×C3×Q8—31no—
207C2×C3×Q8—31no—
208C2×C3×Q8—61no—
209C2×C3×Q8—91no—
210C2×Dic6—31no—
211C2×Dic6—31no—
212C2×Dic6—61no—
213C2×Dic6C2×C3×D311yescharacteristic
214C3×Q16—94no—
215C2×C3×C8—31no—
216C2×C3×C8—31no—
217C2×C3×C8—31no—
218C2×C3×C8—31no—
219C2×C3×C8—61no—
220C2×C3×C8—61no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—31no—
225C3×C4⋊C4—31no—
226C3×C4⋊C4—41no—
227C3×C4⋊C4—61no—
228C3×C4⋊C4—181no—
229C3×C4⋊C4—181no—
230C3×C4⋊C4D3211yescharacteristic
231C3×C4⋊C4C2×C3×D311yescharacteristic
232C3×C4⋊C4C2×C3×D311yescharacteristic
233C3×C42—91no—
234C3×C42—181no—
235C3×C42—181no—
236#216—34no—
237C4⋊Dic3—61no—
238C4⋊Dic3C2×C3×D311yescharacteristic
239Dic3⋊C4—61no—
240Dic3⋊C4—61no—
241Dic3⋊C4—61no—
242Dic3⋊C4—62no—
243C4×Dic3—31no—
244C4×Dic3—31no—
245C4×Dic3—61no—
246C4×Dic3—61no—
247C4×Dic3—61no—
248C4×Dic3—61no—
249C4×Dic3—61no—
250C4×Dic3—62no—
251C2×C3⋊C8C2×C3×D311yescharacteristic
252C2×C3⋊C8C2×C3×D311yescharacteristic
253Dic12—34no—
254C22×C32—21no—
255C22×C32—21no—
256C22×C32—21no—
257C22×C32—41no—
258C22×C32D3×D411yescharacteristic
259C22×C32D3×D411yescharacteristic
260C22×C32C2×C3×D411yescharacteristic
261C32×C4—21no—
262C32×C4—21no—
263C32×C4—21no—
264C32×C4—21no—
265C32×C4—21no—
266C32×C4—21no—
267C32×C4—41no—
268C32×C4—41no—
269C32×C4—41no—
270C32×C4—42no—
271C32×C4—41no—
272C32×C4—41no—
273C32×C4—62no—
274C32×C4—81no—
275C32×C4—81no—
276C32×C4—82no—
277C32×C4—121no—
278C32×C4C2×C4×D311yescharacteristic
279C32×C4C2×C4×D311yescharacteristic
280C32×C4#23711yescharacteristic
281C32×C4#23911yescharacteristic
282C32×C4C22×C3×C411yescharacteristic
283C32×C4C3×#4111yescharacteristic
284C3⋊Dic3—22no—
285C3⋊Dic3—21no—
286C3⋊Dic3—21no—
287C3×Dic3—22no—
288C3×Dic3—22no—
289C3×Dic3—42no—
290C3×Dic3—41no—
291C3×Dic3—41no—
292C3×Dic3—64no—
293C3×Dic3—62no—
294C3×Dic3—62no—
295C3×Dic3—62no—
296C3×Dic3—61no—
297C3×Dic3—61no—
298C3×Dic3—61no—
299C3×Dic3—61no—
300C3×Dic3—81no—
301C3×Dic3—121no—
302C2×Q16—91no—
303C4×Q8—91no—
304C4×Q8—91no—
305#106—31no—
306#103—31no—
307#103—91no—
308#97—91no—
309C33#44111yescharacteristic, complemented, Sylow
310C3×Q8—32no—
311C3×Q8—32no—
312C3×Q8—62no—
313C3×Q8—61no—
314C3×Q8—61no—
315C3×Q8—92no—
316C3×Q8—121no—
317C3×Q8—181no—
318C2×C3×C4—22no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—42no—
328C2×C3×C4—41no—
329C2×C3×C4—61no—
330C2×C3×C4—61no—
331C2×C3×C4—61no—
332C2×C3×C4—61no—
333C2×C3×C4—91no—
334C2×C3×C4—91no—
335C2×C3×C4—121no—
336C2×C3×C4—121no—
337C2×C3×C4—181no—
338C2×C3×C4C2×D3211yescharacteristic
339C2×C3×C4C22×C3×D311yescharacteristic
340C2×C3×C4C22×C3×D311yescharacteristic
341C2×Dic3—21no—
342C2×Dic3—21no—
343C2×Dic3—31no—
344C2×Dic3—31no—
345C2×Dic3—31no—
346C2×Dic3—31no—
347C2×Dic3—61no—
348C2×Dic3—61no—
349C2×Dic3—62no—
350C2×Dic3—61no—
351C2×Dic3—61no—
352Dic6—21no—
353Dic6—32no—
354Dic6—32no—
355Dic6—62no—
356Dic6—62no—
357Dic6—61no—
358Dic6—61no—
359Dic6C3×C4×D312yes—
360C3×C8—32no—
361C3×C8—32no—
362C3×C8—62no—
363C3×C8—61no—
364C3×C8—61no—
365C3×C8—121no—
366C3⋊C8—21no—
367C3⋊C8C3×C4×D312yes—
368C2×C32—21no—
369C2×C32—21no—
370C2×C32—21no—
371C2×C32—21no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32—21no—
375C2×C32—21no—
376C2×C32—21no—
377C2×C32—41no—
378C2×C32—41no—
379C2×C32—41no—
380C2×C32#67⋊C411yescharacteristic
381C2×C32#89311yescharacteristic
382C2×C32#91511yescharacteristic
383C2×C32#91511yescharacteristic
384C2×C32#91511yescharacteristic
385C2×C32#91811yescharacteristic
386C2×C32C3×C4×D411yescharacteristic
387C2×C32C3×#13711yescharacteristic
388C2×C32C3×#13711yescharacteristic
389C2×Q8—31no—
390C2×Q8—91no—
391Q16—94no—
392C2×C8—31no—
393C2×C8—31no—
394C4⋊C4—31no—
395C4⋊C4—181no—
396C4⋊C4—181no—
397C4⋊C4C3×D3211yescharacteristic
398C42—91no—
399C42—181no—
400C42—181no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—41no—
405C22×C3#381111yescharacteristic
406C22×C3C3×D3×D411yescharacteristic
407C22×C3C3×D3×D411yescharacteristic
408C3×C4—21no—
409C3×C4—21no—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—42no—
415C3×C4—41no—
416C3×C4—41no—
417C3×C4—41no—
418C3×C4—41no—
419C3×C4—41no—
420C3×C4—62no—
421C3×C4—62no—
422C3×C4—81no—
423C3×C4—81no—
424C3×C4—82no—
425C3×C4—122no—
426C3×C4—121no—
427C3×C4—121no—
428C3×C4—182no—
429C3×C4—181no—
430C3×C4—181no—
431C3×C4—241no—
432C3×C4#379611yescharacteristic
433C3×C4C4×D3211yescharacteristic
434C3×C4C2×C3×C4×D311yescharacteristic
435C3×C4C2×C3×C4×D311yescharacteristic
436C3×C4C3×#23711yescharacteristic
437C3×C4C3×#23911yescharacteristic
438Dic3—22no—
439Dic3—41no—
440Dic3—64no—
441Dic3—62no—
442Dic3—62no—
443Dic3—62no—
444Dic3—61no—
445Dic3—61no—
446Dic3—61no—
447Dic3—61no—
448Dic3—121no—
449C32—21no—
450C32—21no—
451C32—21no—
452C32—41no—
453C32#478911yescharacteristic, complemented
454C32#485311yescharacteristic, complemented
455C32C3×#44111yescharacteristic, complemented
456Q8—32no—
457Q8—61no—
458Q8—92no—
459Q8—181no—
460C2×C4—21no—
461C2×C4—61no—
462C2×C4—61no—
463C2×C4—91no—
464C2×C4—91no—
465C2×C4—181no—
466C2×C4C2×C3×D3211yescharacteristic
467C8—32no—
468C8—61no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—41no—
479C2×C3—41no—
480C2×C3—41no—
481C2×C3#6388111yescharacteristic
482C2×C3#6393111yescharacteristic
483C2×C3#6395111yescharacteristic
484C2×C3C3×#67⋊C411yescharacteristic
485C2×C3C3×#89311yescharacteristic
486C2×C3C3×#91511yescharacteristic
487C2×C3C3×#91511yescharacteristic
488C2×C3C3×#91511yescharacteristic
489C2×C3C3×#91811yescharacteristic
490C22C3×#381111yescharacteristic
491C4—41no—
492C4—62no—
493C4—121no—
494C4—181no—
495C4—182no—
496C4—181no—
497C4C3×#379611yescharacteristic
498C4C3×C4×D3211yescharacteristic
499C3—21no—
500C3—21no—
501C3—21no—
502C3—41no—
503C3#1059048511yescharacteristic, direct product factor
504C3C3×#478911yescharacteristic, complemented
505C3C3×#485311yescharacteristic, complemented
506C2C3×#6388111yescharacteristic
507C2C3×#6393111yescharacteristic
508C2C3×#6395111yescharacteristic
subgroupG/H
centerC22×C3#3811
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socle, prime radicalC22×C33C2×D4
cosocleC2×C32×C4C23×C3
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#441
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
10. #10590485C3direct (N × H)
22. C3×#3719C4semidirect
110. #3719C3×C4semidirect
309. C33#441semidirect
453. C32#4789semidirect
454. C32#4853semidirect
455. C32C3×#441semidirect
503. C3#10590485direct (N × H)
504. C3C3×#4789semidirect
505. C3C3×#4853semidirect

Maximal subgroups

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

subgroupindex
C3×#640962
C3×#638432
C2×C3×#37192
C32×#8102
C32×#8082
C3×#635672
C3×#635552
C3×#48533
C3×#47893
#105904853