Finite Group Databasebeta

← C3×#10590524 #13157583

Subgroups — C3×#10590524 #13157583

Order 1728.

618 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64106C211yesmaximal, characteristic
2C3×#63871C211yesmaximal, characteristic
3C2×C3×#3719C211yesmaximal, characteristic, complemented
4C2×C3×#3712C211yesmaximal, characteristic
5C32×#810C211yesmaximal, characteristic, complemented
6C32×#808C211yesmaximal, characteristic, complemented
7C3×#63567C211yesmaximal, characteristic, complemented
8C3×#4867—31nomaximal
9C3×#4816—31nomaximal
10#10590524C311yesmaximal, characteristic, direct product factor
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C33×C4⋊C4C2211yescharacteristic
15C3×#3752—21no—
16C3×C4⋊(C3⋊Dic3)—21no—
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×Dic3⋊Dic3—21no—
21C3×#3722—21no—
22C3×#3719—22no—
23C3×#3712—22no—
24#64106C2×C311yescharacteristic
25C3×#946—31no—
26C2×C3×#216—31no—
27C2×C3×Dic12—31no—
28C2×C3×ΓL(1,𝔽25)—31no—
29C2×C3×ΓL(1,𝔽25)—31no—
30C3×#897—31no—
31C3×#897—31no—
32C3×#897—31no—
33C3×#897—61no—
34#63871C2×C311yescharacteristic
35C2×#3719C2×C311yescharacteristic, complemented
36C2×#3712C2×C311yescharacteristic
37C32×#106—31no—
38C32×#103—31no—
39C3×#816—31no—
40C3×#810—21no—
41C3×#810—31no—
42C3×#810D311yescharacteristic, complemented
43C3×#810C2×C311yescharacteristic, complemented
44C3×#808—21no—
45C3×#808D311yescharacteristic, complemented
46C3×#808C2×C311yescharacteristic, complemented
47#63567C2×C311yescharacteristic, complemented
48C22×C3×C3⋊D3D411yescharacteristic
49C2×C33×C4—21no—
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3D411yescharacteristic
53C3×C4×C3⋊D3—21no—
54C3×C4×C3⋊D3—21no—
55C3×#634—21no—
56C3×#634—21no—
57C2×C32×Dic3—21no—
58C2×C32×Dic3—21no—
59C32×Dic6—21no—
60C32×Dic6—21no—
61C32×C3⋊C8—21no—
62C32×C3⋊C8D412yes—
63C3×#470—91no—
64#4867—31no—
65#4816—31no—
66C2×C32×Q8—31no—
67C2×C4×C3⋊D3C22×C311yescharacteristic
68C2×#634C22×C311yescharacteristic
69C2×C3×C4×D3—31no—
70C2×C3×C4×D3—31no—
71C2×C3×C4×D3—61no—
72C2×C3×Dic6—21no—
73C2×C3×Dic6—31no—
74C2×C3×Dic6—31no—
75C2×C3×Dic6—61no—
76C2×C3×Dic6C2×D311yescharacteristic
77C2×C3×Dic6C22×C311yescharacteristic
78C2×C32×C8—31no—
79C2×C32×C8—31no—
80C32×C4⋊C4—21no—
81C32×C4⋊C4—21no—
82C32×C4⋊C4—21no—
83C32×C4⋊C4—31no—
84C32×C4⋊C4—41no—
85C32×C4⋊C4C2×D311yescharacteristic
86C32×C4⋊C4C2×D311yescharacteristic
87C32×C4⋊C4C22×C311yescharacteristic
88#3752—21no—
89C4⋊(C3⋊Dic3)—21no—
90C3×#216—62no—
91C3×(C2×D3)⋊C4—61no—
92C3×(C2×D3)⋊C4—61no—
93C3×(C2×D3)⋊C4—61no—
94C3×(C2×D3)⋊C4—61no—
95C3×(C2×D3)⋊C4—62no—
96C3×C4⋊Dic3—21no—
97C3×C4⋊Dic3—61no—
98C3×C4⋊Dic3—61no—
99C3×C4⋊Dic3—61no—
100C3×C4⋊Dic3—62no—
101C3×C4⋊Dic3C2×D311yescharacteristic
102C3×C4⋊Dic3C22×C311yescharacteristic
103C3×Dic3⋊C4—61no—
104C2×C3×C3⋊C8—21no—
105C2×C3×C3⋊C8—21no—
106C2×C3×C3⋊C8C2×D311yescharacteristic
107C2×C3×C3⋊C8C2×D311yescharacteristic
108C2×C3×C3⋊C8C22×C311yescharacteristic
109C2×C3×C3⋊C8C22×C311yescharacteristic
110C3×Dic12—62no—
111C3×ΓL(1,𝔽25)—62no—
112C3×ΓL(1,𝔽25)—62no—
113Dic3⋊Dic3—21no—
114#3722—21no—
115#3719—22no—
116#3712—22no—
117C22×C33C2×D411yescharacteristic
118C2×C3×C3⋊D3—21no—
119C2×C3×C3⋊D3—21no—
120C2×C3×C3⋊D3—21no—
121C33×C4—41no—
122C33×C4C2×D411yescharacteristic
123C33×C4#4111yescharacteristic
124C3×C3⋊Dic3—21no—
125C3×C3⋊Dic3—41no—
126C32×Dic3—41no—
127C32×Dic3—41no—
128C2×C3×Q16—91no—
129C2×C3×M16—91no—
130C3×C22⋊Q8—91no—
131C3×C22⋊Q8—91no—
132#946—31no—
133C2×#216—31no—
134C2×Dic12—31no—
135C2×ΓL(1,𝔽25)—31no—
136C2×ΓL(1,𝔽25)—31no—
137#897—31no—
138#897—31no—
139#897—31no—
140#897—61no—
141C3×#106—31no—
142C3×#106—31no—
143C3×#106—61no—
144C3×#103—31no—
145C3×#103—31no—
146C3×#103—61no—
147C3×#103—91no—
148#816—31no—
149#810—31no—
150#810C3×D311yescharacteristic, complemented
151#808C3×D311yescharacteristic, complemented
152C22×C3⋊D3C3×D411yescharacteristic
153C22×C3×D3—31no—
154C22×C3×D3—31no—
155C22×C3×D3—61no—
156C32×Q8—61no—
157C32×Q8—61no—
158C2×C32×C4—21no—
159C2×C32×C4—22no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—41no—
166C2×C32×C4—41no—
167C2×C32×C4—41no—
168C2×C32×C4—42no—
169C2×C32×C4—61no—
170C2×C32×C4—61no—
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C23×C311yescharacteristic
174C2×C3⋊Dic3—21no—
175C2×C3⋊Dic3C3×D411yescharacteristic
176C4×C3⋊D3—21no—
177C4×C3⋊D3—21no—
178#634—21no—
179#634—21no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—31no—
185C2×C3×Dic3—31no—
186C2×C3×Dic3—41no—
187C2×C3×Dic3—41no—
188C2×C3×Dic3—61no—
189C2×C3×Dic3—62no—
190C2×C3×Dic3—61no—
191C2×C3×Dic3—61no—
192C3×C4×D3—61no—
193C3×C4×D3—61no—
194C3×C4×D3—61no—
195C3×C4×D3—61no—
196C3×C4×D3—121no—
197C3×C4×D3—121no—
198C3×Dic6—22no—
199C3×Dic6—21no—
200C3×Dic6—21no—
201C3×Dic6—21no—
202C3×Dic6—21no—
203C3×Dic6—41no—
204C3×Dic6—61no—
205C3×Dic6—61no—
206C3×Dic6—61no—
207C3×Dic6—61no—
208C3×Dic6—121no—
209C3×Dic6—121no—
210C32×C8—32no—
211C32×C8—61no—
212C3×C3⋊C8—21no—
213C3×C3⋊C8—21no—
214C3×C3⋊C8—22no—
215C3×C3⋊C8—41no—
216C3×C3⋊C8D1212yes—
217C3×C3⋊C8C3×D412yes—
218#470—91noSylow
219C2×C33#12111yescharacteristic
220C2×C33#13711yescharacteristic
221C2×C33#13711yescharacteristic
222C3×C3⋊D3—41no—
223C2×C3×Q8—31no—
224C2×C3×Q8—31no—
225C2×C3×Q8—61no—
226C2×C3×Q8—91no—
227C22×C3×C4—91no—
228C2×C4×D3—31no—
229C2×C4×D3—31no—
230C2×C4×D3—61no—
231C2×Dic6—31no—
232C2×Dic6—31no—
233C2×Dic6—61no—
234C2×Dic6C2×C3×D311yescharacteristic
235C3×Q16—182no—
236C3×M16—182no—
237C2×C3×C8—31no—
238C2×C3×C8—31no—
239C2×C3×C8—31no—
240C2×C3×C8—31no—
241C2×C3×C8—61no—
242C2×C3×C8—61no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—31no—
247C3×C4⋊C4—31no—
248C3×C4⋊C4—41no—
249C3×C4⋊C4—61no—
250C3×C4⋊C4—181no—
251C3×C4⋊C4—181no—
252C3×C4⋊C4D3211yescharacteristic
253C3×C4⋊C4C2×C3×D311yescharacteristic
254C3×C4⋊C4C2×C3×D311yescharacteristic
255C3×C22⋊C4—181no—
256C3×C22⋊C4—181no—
257#216—62no—
258(C2×D3)⋊C4—61no—
259(C2×D3)⋊C4—61no—
260(C2×D3)⋊C4—61no—
261(C2×D3)⋊C4—61no—
262(C2×D3)⋊C4—62no—
263C4⋊Dic3—61no—
264C4⋊Dic3—61no—
265C4⋊Dic3—61no—
266C4⋊Dic3—62no—
267C4⋊Dic3C2×C3×D311yescharacteristic
268Dic3⋊C4—61no—
269C2×C3⋊C8C2×C3×D311yescharacteristic
270C2×C3⋊C8C2×C3×D311yescharacteristic
271Dic12—62no—
272ΓL(1,𝔽25)—62no—
273ΓL(1,𝔽25)—62no—
274C22×C32—21no—
275C22×C32—21no—
276C22×C32—21no—
277C22×C32—41no—
278C22×C32D3×D411yescharacteristic
279C22×C32D3×D411yescharacteristic
280C22×C32C2×C3×D411yescharacteristic
281C2×C3⋊D3—21no—
282C2×C3⋊D3—21no—
283C2×C3⋊D3—21no—
284C2×C3×D3—62no—
285C2×C3×D3—61no—
286C2×C3×D3—61no—
287C2×C3×D3—61no—
288C2×C3×D3—61no—
289C2×C3×D3—61no—
290C2×C3×D3—61no—
291C2×C3×D3—121no—
292C2×C3×D3—121no—
293C32×C4—21no—
294C32×C4—21no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4—21no—
298C32×C4—21no—
299C32×C4—41no—
300C32×C4—41no—
301C32×C4—41no—
302C32×C4—42no—
303C32×C4—41no—
304C32×C4—41no—
305C32×C4—81no—
306C32×C4—81no—
307C32×C4—82no—
308C32×C4—121no—
309C32×C4—121no—
310C32×C4C2×D1211yescharacteristic
311C32×C4#23511yescharacteristic
312C32×C4D3×D411yescharacteristic
313C32×C4#23911yescharacteristic
314C32×C4C2×C3×D411yescharacteristic
315C32×C4C3×#4111yescharacteristic
316C3⋊Dic3—21no—
317C3⋊Dic3—41no—
318C3×Dic3—42no—
319C3×Dic3—41no—
320C3×Dic3—41no—
321C3×Dic3—41no—
322C3×Dic3—41no—
323C3×Dic3—61no—
324C3×Dic3—61no—
325C3×Dic3—81no—
326C3×Dic3—121no—
327C3×Dic3—122no—
328C3×Dic3—121no—
329C3×Dic3—121no—
330C2×Q16—91no—
331C2×M16—91no—
332C22⋊Q8—91no—
333C22⋊Q8—91no—
334#106—31no—
335#103—31no—
336#103—91no—
337C33#47011yescharacteristic, complemented, Sylow
338C23×C3—91no—
339C22×D3—31no—
340C22×D3—31no—
341C22×D3—61no—
342C3×Q8—62no—
343C3×Q8—61no—
344C3×Q8—61no—
345C3×Q8—61no—
346C3×Q8—61no—
347C3×Q8—121no—
348C3×Q8—181no—
349C3×Q8—181no—
350C2×C3×C4—22no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—42no—
360C2×C3×C4—41no—
361C2×C3×C4—61no—
362C2×C3×C4—61no—
363C2×C3×C4—61no—
364C2×C3×C4—61no—
365C2×C3×C4—91no—
366C2×C3×C4—121no—
367C2×C3×C4—121no—
368C2×C3×C4—181no—
369C2×C3×C4—181no—
370C2×C3×C4—181no—
371C2×C3×C4C2×D3211yescharacteristic
372C2×C3×C4C22×C3×D311yescharacteristic
373C2×C3×C4C22×C3×D311yescharacteristic
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—31no—
377C2×Dic3—31no—
378C2×Dic3—61no—
379C2×Dic3—62no—
380C2×Dic3—61no—
381C2×Dic3—61no—
382C4×D3—61no—
383C4×D3—61no—
384C4×D3—61no—
385C4×D3—61no—
386C4×D3—121no—
387C4×D3—121no—
388Dic6—21no—
389Dic6—21no—
390Dic6—61no—
391Dic6—61no—
392Dic6—61no—
393Dic6—61no—
394Dic6—121no—
395Dic6—121no—
396C3×C8—32no—
397C3×C8—32no—
398C3×C8—62no—
399C3×C8—61no—
400C3×C8—61no—
401C3×C8—121no—
402C3⋊C8—21no—
403C3⋊C8C3×D1212yes—
404C2×C32—21no—
405C2×C32—21no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—21no—
409C2×C32—21no—
410C2×C32—21no—
411C2×C32—21no—
412C2×C32—21no—
413C2×C32—41no—
414C2×C32—41no—
415C2×C32—41no—
416C2×C32#88411yescharacteristic
417C2×C32#89511yescharacteristic
418C2×C32#91511yescharacteristic
419C2×C32#91511yescharacteristic
420C2×C32#91511yescharacteristic
421C2×C32#91811yescharacteristic
422C2×C32C3×#12111yescharacteristic
423C2×C32C3×#13711yescharacteristic
424C2×C32C3×#13711yescharacteristic
425C3⋊D3—41no—
426C3×D3—122no—
427C3×D3—121no—
428C3×D3—121no—
429C2×Q8—31no—
430C2×Q8—91no—
431C22×C4—91no—
432Q16—182no—
433M16—182no—
434C2×C8—31no—
435C2×C8—31no—
436C4⋊C4—31no—
437C4⋊C4—181no—
438C4⋊C4—181no—
439C4⋊C4C3×D3211yescharacteristic
440C22⋊C4—181no—
441C22⋊C4—181no—
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—41no—
446C22×C3—181no—
447C22×C3—181no—
448C22×C3—181no—
449C22×C3#381111yescharacteristic
450C22×C3C3×D3×D411yescharacteristic
451C22×C3C3×D3×D411yescharacteristic
452C2×D3—62no—
453C2×D3—61no—
454C2×D3—61no—
455C2×D3—61no—
456C2×D3—61no—
457C2×D3—61no—
458C2×D3—61no—
459C2×D3—121no—
460C2×D3—121no—
461C3×C4—21no—
462C3×C4—21no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—42no—
468C3×C4—41no—
469C3×C4—41no—
470C3×C4—41no—
471C3×C4—41no—
472C3×C4—41no—
473C3×C4—81no—
474C3×C4—81no—
475C3×C4—82no—
476C3×C4—122no—
477C3×C4—121no—
478C3×C4—121no—
479C3×C4—121no—
480C3×C4—121no—
481C3×C4—181no—
482C3×C4—241no—
483C3×C4—361no—
484C3×C4#379911yescharacteristic
485C3×C4D3×D1211yescharacteristic
486C3×C4C2×C3×D1211yescharacteristic
487C3×C4C3×#23511yescharacteristic
488C3×C4C3×D3×D411yescharacteristic
489C3×C4C3×#23911yescharacteristic
490Dic3—41no—
491Dic3—41no—
492Dic3—61no—
493Dic3—61no—
494Dic3—121no—
495Dic3—122no—
496Dic3—121no—
497Dic3—121no—
498C32—21no—
499C32—21no—
500C32—21no—
501C32—41no—
502C32#481611yescharacteristic, complemented
503C32#486711yescharacteristic, complemented
504C32C3×#47011yescharacteristic, complemented
505C23—91no—
506Q8—61no—
507Q8—61no—
508Q8—181no—
509Q8—181no—
510C2×C4—21no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—91no—
514C2×C4—181no—
515C2×C4—181no—
516C2×C4—181no—
517C2×C4C2×C3×D3211yescharacteristic
518C8—32no—
519C8—61no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—21no—
525C2×C3—21no—
526C2×C3—21no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—41no—
530C2×C3—41no—
531C2×C3—41no—
532C2×C3—361no—
533C2×C3#6391011yescharacteristic
534C2×C3#6393111yescharacteristic
535C2×C3#6395111yescharacteristic
536C2×C3C3×#88411yescharacteristic
537C2×C3C3×#89511yescharacteristic
538C2×C3C3×#91511yescharacteristic
539C2×C3C3×#91511yescharacteristic
540C2×C3C3×#91511yescharacteristic
541C2×C3C3×#91811yescharacteristic
542D3—122no—
543D3—121no—
544D3—121no—
545C22—181no—
546C22—181no—
547C22—181no—
548C22C3×#381111yescharacteristic
549C4—41no—
550C4—121no—
551C4—121no—
552C4—181no—
553C4—361no—
554C4C3×#379911yescharacteristic
555C4C3×D3×D1211yescharacteristic
556C3—21no—
557C3—21no—
558C3—21no—
559C3—41no—
560C3#1059052411yescharacteristic, direct product factor
561C3C3×#481611yescharacteristic, complemented
562C3C3×#486711yescharacteristic, complemented
563C2—361no—
564C2C3×#6391011yescharacteristic
565C2C3×#6393111yescharacteristic
566C2C3×#6395111yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#470
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
3. C2×C3×#3719C2semidirect
5. C32×#810C2semidirect
6. C32×#808C2semidirect
7. C3×#63567C2semidirect
10. #10590524C3direct (N × H)
35. C2×#3719C2×C3semidirect
42. C3×#810D3semidirect
43. C3×#810C2×C3semidirect
45. C3×#808D3semidirect
46. C3×#808C2×C3semidirect
47. #63567C2×C3semidirect
150. #810C3×D3semidirect
151. #808C3×D3semidirect
337. C33#470semidirect
502. C32#4816semidirect
503. C32#4867semidirect
504. C32C3×#470semidirect
560. C3#10590524direct (N × H)
561. C3C3×#4816semidirect
562. C3C3×#4867semidirect

Maximal subgroups

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

subgroupindex
C3×#641062
C3×#638712
C2×C3×#37192
C2×C3×#37122
C32×#8102
C32×#8082
C3×#635672
C3×#48673
C3×#48163
#105905243