Finite Group Databasebeta

← C3×#10590520 #13157496

Subgroups — C3×#10590520 #13157496

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
1C32×#897C211yesmaximal, characteristic
2C3×#63856C211yesmaximal, characteristic
3C2×C3×#3718C211yesmaximal, characteristic, complemented
4C2×C3×#3711C211yesmaximal, characteristic
5C3×#63635C211yesmaximal, characteristic, complemented
6C32×#810C211yesmaximal, characteristic, complemented
7C3×#63566C211yesmaximal, characteristic, complemented
8C3×#5052—31nomaximal
9C3×#4816—31nomaximal
10#10590520C311yesmaximal, characteristic, direct product factor
11C2×C32×C4×D3C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C33×C4⋊C4C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×(C2×D3)⋊C4—21no—
18C32×C4⋊Dic3—21no—
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×#3724—21no—
21C3×(C2×D3)⋊Dic3—21no—
22C3×#3718—22no—
23C3×#3711—22no—
24C32×C22⋊Q8—31no—
25C3×#946—31no—
26C2×C3×#216—31no—
27C2×C3×#216—31no—
28C3×#924—31no—
29C2×C3×#208—31no—
30C2×C3×ΓL(1,𝔽25)—31no—
31C3×#897—21no—
32C3×#897D311yescharacteristic
33C3×#897C2×C311yescharacteristic
34#63856C2×C311yescharacteristic
35C2×#3718C2×C311yescharacteristic, complemented
36C2×#3711C2×C311yescharacteristic
37C32×#103—31no—
38#63635C2×C311yescharacteristic, complemented
39C3×#835—31no—
40C3×#816—31no—
41C3×#810—21no—
42C3×#810D311yescharacteristic, complemented
43C3×#810C2×C311yescharacteristic, complemented
44C3×#808—31no—
45C3×#808—31no—
46C3×#808—61no—
47#63566C2×C311yescharacteristic, complemented
48C22×C32×D3D411yescharacteristic
49C2×C33×C4—21no—
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3—21no—
54C2×C32×Dic3D411yescharacteristic
55C32×C4×D3—21no—
56C32×C4×D3—21no—
57C32×Dic6—21no—
58C32×Dic6—21no—
59C32×Dic6—21no—
60C32×Dic6—21no—
61C3×C3⋊(C3⋊C8)D412yes—
62C32×C3⋊C8—21no—
63C3×#470—91no—
64#5052—31no—
65#4816—31no—
66C2×C32×Q8—31no—
67C2×C32×Q8—31no—
68C22×C32×C4—31no—
69C2×C3×C4×D3—21no—
70C2×C3×C4×D3C2×D311yescharacteristic
71C2×C3×C4×D3C22×C311yescharacteristic
72C2×C3×Dic6—21no—
73C2×C3×Dic6—21no—
74C2×C3×Dic6C2×D311yescharacteristic
75C2×C3×Dic6C2×D311yescharacteristic
76C2×C3×Dic6C22×C311yescharacteristic
77C2×C3×Dic6C22×C311yescharacteristic
78C2×C32×C8—31no—
79C32×C4⋊C4—21no—
80C32×C4⋊C4—21no—
81C32×C4⋊C4—21no—
82C32×C4⋊C4—41no—
83C32×C4⋊C4—61no—
84C32×C4⋊C4C2×D311yescharacteristic
85C32×C4⋊C4C2×D311yescharacteristic
86C32×C4⋊C4C22×C311yescharacteristic
87C32×C22⋊C4—61no—
88C4⋊(C3⋊Dic3)C22×C311yescharacteristic
89C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
90C3×#217—61no—
91C3×#216—62no—
92C3×#216—62no—
93C3×(C2×D3)⋊C4—22no—
94C3×(C2×D3)⋊C4—21no—
95C3×(C2×D3)⋊C4—21no—
96C3×(C2×D3)⋊C4—61no—
97C3×C4⋊Dic3—22no—
98C3×C4⋊Dic3—21no—
99C3×C4⋊Dic3—21no—
100C3×C4⋊Dic3—31no—
101C3×C4⋊Dic3—31no—
102C3×C4⋊Dic3—61no—
103C3×Dic3⋊C4—61no—
104C3×Dic3⋊C4—61no—
105C3×#208—62no—
106C2×C3×C3⋊C8—21no—
107C2×C3×C3⋊C8—31no—
108C2×C3×C3⋊C8—31no—
109C2×C3×C3⋊C8—61no—
110C2×C3×C3⋊C8C2×D311yescharacteristic
111C2×C3×C3⋊C8C22×C311yescharacteristic
112C3×ΓL(1,𝔽25)—62no—
113#3724—21no—
114(C2×D3)⋊Dic3—21no—
115#3718—22no—
116#3711—22no—
117C22×C33C2×D411yescharacteristic
118C2×C32×D3—21no—
119C2×C32×D3—21no—
120C2×C32×D3—21no—
121C33×C4—41no—
122C33×C4C2×D411yescharacteristic
123C33×C4#4111yescharacteristic
124C3×C3⋊Dic3—41no—
125C32×Dic3—21no—
126C32×Dic3—41no—
127C32×Dic3—41no—
128C2×C3×Q16—91no—
129C2×C3×M16—91no—
130C3×C22⋊Q8—31no—
131C3×C22⋊Q8—31no—
132C3×C22⋊Q8—61no—
133C3×C22⋊Q8—91no—
134#946—31no—
135C2×#216—31no—
136C2×#216—31no—
137#924—31no—
138C2×#208—31no—
139C2×ΓL(1,𝔽25)—31no—
140#897C3×D311yescharacteristic
141C3×#106—91no—
142C3×#103—31no—
143C3×#103—31no—
144C3×#103—61no—
145C3×#103—91no—
146#835—31no—
147#816—31no—
148#810C3×D311yescharacteristic, complemented
149#808—31no—
150#808—31no—
151#808—61no—
152C23×C32—31no—
153C22×C3×D3—21no—
154C22×C3×D3#6711yescharacteristic
155C22×C3×D3C3×D411yescharacteristic
156C32×Q8—61no—
157C32×Q8—61no—
158C32×Q8—61no—
159C32×Q8—61no—
160C2×C32×C4—21no—
161C2×C32×C4—22no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—31no—
168C2×C32×C4—41no—
169C2×C32×C4—41no—
170C2×C32×C4—41no—
171C2×C32×C4—42no—
172C2×C32×C4—61no—
173C2×C32×C4—61no—
174C2×C32×C4—61no—
175C2×C32×C4—61no—
176C2×C32×C4C22×D311yescharacteristic
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C23×C311yescharacteristic
179C2×C3⋊Dic3—21no—
180C2×C3×Dic3—22no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—21no—
185C2×C3×Dic3—21no—
186C2×C3×Dic3—41no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3—61no—
189C2×C3×Dic3—121no—
190C2×C3×Dic3#6711yescharacteristic
191C2×C3×Dic3C3×D411yescharacteristic
192C3×C4×D3—21no—
193C3×C4×D3—21no—
194C3×C4×D3—21no—
195C3×C4×D3—21no—
196C3×C4×D3—41no—
197C3×C4×D3—41no—
198C3×Dic6—22no—
199C3×Dic6—21no—
200C3×Dic6—21no—
201C3×Dic6—21no—
202C3×Dic6—21no—
203C3×Dic6—21no—
204C3×Dic6—21no—
205C3×Dic6—21no—
206C3×Dic6—21no—
207C3×Dic6—41no—
208C3×Dic6—41no—
209C3×Dic6—41no—
210C32×C8—61no—
211C3⋊(C3⋊C8)C3×D412yes—
212C3×C3⋊C8—21no—
213C3×C3⋊C8—21no—
214C3×C3⋊C8—32no—
215C3×C3⋊C8—32no—
216C3×C3⋊C8—41no—
217C3×C3⋊C8—62no—
218#470—91noSylow
219C2×C33#12111yescharacteristic
220C2×C33#13711yescharacteristic
221C2×C33#13711yescharacteristic
222C32×D3—41no—
223C2×C3×Q8—31no—
224C2×C3×Q8—31no—
225C2×C3×Q8—31no—
226C2×C3×Q8—31no—
227C2×C3×Q8—61no—
228C2×C3×Q8—61no—
229C22×C3×C4—31no—
230C22×C3×C4—31no—
231C22×C3×C4—61no—
232C2×C4×D3C2×C3×D311yescharacteristic
233C2×Dic6C2×C3×D311yescharacteristic
234C2×Dic6C2×C3×D311yescharacteristic
235C3×Q16—182no—
236C3×M16—182no—
237C2×C3×C8—31no—
238C2×C3×C8—31no—
239C2×C3×C8—61no—
240C2×C3×C8—91no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—41no—
245C3×C4⋊C4—61no—
246C3×C4⋊C4—61no—
247C3×C4⋊C4—62no—
248C3×C4⋊C4—91no—
249C3×C4⋊C4—181no—
250C3×C4⋊C4D3211yescharacteristic
251C3×C4⋊C4C2×C3×D311yescharacteristic
252C3×C4⋊C4C2×C3×D311yescharacteristic
253C3×C22⋊C4—61no—
254C3×C22⋊C4—61no—
255C3×C22⋊C4—62no—
256C3×C22⋊C4—181no—
257#217—61no—
258#216—62no—
259#216—62no—
260(C2×D3)⋊C4—21no—
261(C2×D3)⋊C4—61no—
262C4⋊Dic3—21no—
263C4⋊Dic3—31no—
264C4⋊Dic3—31no—
265C4⋊Dic3—61no—
266Dic3⋊C4—61no—
267Dic3⋊C4—61no—
268#208—62no—
269C2×C3⋊C8—31no—
270C2×C3⋊C8—31no—
271C2×C3⋊C8—61no—
272C2×C3⋊C8C2×C3×D311yescharacteristic
273ΓL(1,𝔽25)—62no—
274C22×C32—21no—
275C22×C32—21no—
276C22×C32—21no—
277C22×C32—41no—
278C22×C32—61no—
279C22×C32—61no—
280C22×C32—61no—
281C22×C32D3×D411yescharacteristic
282C22×C32C2×#6711yescharacteristic
283C22×C32C2×C3×D411yescharacteristic
284C2×C3×D3—22no—
285C2×C3×D3—21no—
286C2×C3×D3—21no—
287C2×C3×D3—21no—
288C2×C3×D3—21no—
289C2×C3×D3—21no—
290C2×C3×D3—21no—
291C2×C3×D3—41no—
292C2×C3×D3—41no—
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—61no—
306C32×C4—81no—
307C32×C4—81no—
308C32×C4—82no—
309C32×C4—121no—
310C32×C4—121no—
311C32×C4#23511yescharacteristic
312C32×C4D3×D411yescharacteristic
313C32×C4D3×D411yescharacteristic
314C32×C4#23711yescharacteristic
315C32×C4C2×C3×D411yescharacteristic
316C32×C4C3×#4111yescharacteristic
317C3⋊Dic3—41no—
318C3×Dic3—21no—
319C3×Dic3—21no—
320C3×Dic3—41no—
321C3×Dic3—42no—
322C3×Dic3—42no—
323C3×Dic3—41no—
324C3×Dic3—41no—
325C3×Dic3—41no—
326C3×Dic3—41no—
327C3×Dic3—121no—
328C3×Dic3—121no—
329C3×Dic3—241no—
330C2×Q16—91no—
331C2×M16—91no—
332C22⋊Q8—31no—
333C22⋊Q8—91no—
334#106—91no—
335#103—31no—
336#103—91no—
337C33#47011yescharacteristic, complemented, Sylow
338C23×C3—31no—
339C23×C3—31no—
340C23×C3—61no—
341C22×D3C3×#6711yescharacteristic
342C3×Q8—62no—
343C3×Q8—61no—
344C3×Q8—61no—
345C3×Q8—61no—
346C3×Q8—61no—
347C3×Q8—61no—
348C3×Q8—61no—
349C3×Q8—61no—
350C3×Q8—61no—
351C3×Q8—121no—
352C3×Q8—121no—
353C3×Q8—121no—
354C2×C3×C4—22no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—31no—
362C2×C3×C4—31no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—42no—
366C2×C3×C4—41no—
367C2×C3×C4—61no—
368C2×C3×C4—61no—
369C2×C3×C4—61no—
370C2×C3×C4—61no—
371C2×C3×C4—61no—
372C2×C3×C4—62no—
373C2×C3×C4—61no—
374C2×C3×C4—61no—
375C2×C3×C4—61no—
376C2×C3×C4—61no—
377C2×C3×C4—121no—
378C2×C3×C4—121no—
379C2×C3×C4—121no—
380C2×C3×C4—181no—
381C2×C3×C4C2×D3211yescharacteristic
382C2×C3×C4C22×C3×D311yescharacteristic
383C2×C3×C4C22×C3×D311yescharacteristic
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—61no—
387C2×Dic3—61no—
388C2×Dic3—121no—
389C2×Dic3C3×#6711yescharacteristic
390C4×D3—21no—
391C4×D3—21no—
392Dic6—21no—
393Dic6—21no—
394Dic6—21no—
395Dic6—21no—
396C3×C8—61no—
397C3×C8—61no—
398C3×C8—92no—
399C3×C8—121no—
400C3⋊C8—21no—
401C3⋊C8—32no—
402C3⋊C8—32no—
403C3⋊C8—62no—
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—121no—
417C2×C32#88411yescharacteristic
418C2×C32#91511yescharacteristic
419C2×C32#91811yescharacteristic
420C2×C32#93911yescharacteristic
421C2×C32#94211yescharacteristic
422C2×C32#95111yescharacteristic
423C2×C32C3×#12111yescharacteristic
424C2×C32C3×#13711yescharacteristic
425C2×C32C3×#13711yescharacteristic
426C3×D3—42no—
427C3×D3—41no—
428C3×D3—41no—
429C2×Q8—31no—
430C2×Q8—31no—
431C22×C4—31no—
432Q16—182no—
433M16—182no—
434C2×C8—31no—
435C2×C8—91no—
436C4⋊C4—61no—
437C4⋊C4—91no—
438C4⋊C4—181no—
439C4⋊C4C3×D3211yescharacteristic
440C22⋊C4—61no—
441C22⋊C4—181no—
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—41no—
446C22×C3—62no—
447C22×C3—61no—
448C22×C3—61no—
449C22×C3—61no—
450C22×C3—61no—
451C22×C3—61no—
452C22×C3—61no—
453C22×C3—121no—
454C22×C3—121no—
455C22×C3D3×#6711yescharacteristic
456C22×C3C3×D3×D411yescharacteristic
457C22×C3C2×C3×#6711yescharacteristic
458C2×D3—21no—
459C2×D3—21no—
460C2×D3—21no—
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—61no—
474C3×C4—61no—
475C3×C4—81no—
476C3×C4—81no—
477C3×C4—82no—
478C3×C4—121no—
479C3×C4—122no—
480C3×C4—122no—
481C3×C4—121no—
482C3×C4—121no—
483C3×C4—121no—
484C3×C4—121no—
485C3×C4—361no—
486C3×C4#379511yescharacteristic
487C3×C4#380211yescharacteristic
488C3×C4C3×#23511yescharacteristic
489C3×C4C3×D3×D411yescharacteristic
490C3×C4C3×D3×D411yescharacteristic
491C3×C4C3×#23711yescharacteristic
492Dic3—21no—
493Dic3—41no—
494Dic3—41no—
495Dic3—121no—
496Dic3—121no—
497Dic3—241no—
498C32—21no—
499C32—21no—
500C32—21no—
501C32—41no—
502C32#481611yescharacteristic, complemented
503C32#505211yescharacteristic, complemented
504C32C3×#47011yescharacteristic, complemented
505C23—31no—
506Q8—61no—
507Q8—61no—
508Q8—61no—
509Q8—61no—
510C2×C4—21no—
511C2×C4—31no—
512C2×C4—61no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—61no—
516C2×C4—181no—
517C2×C4C2×C3×D3211yescharacteristic
518C8—61no—
519C8—92no—
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—122no—
533C2×C3—121no—
534C2×C3—121no—
535C2×C3#6388511yescharacteristic
536C2×C3#6392911yescharacteristic
537C2×C3#6394811yescharacteristic
538C2×C3C3×#88411yescharacteristic
539C2×C3C3×#91511yescharacteristic
540C2×C3C3×#91811yescharacteristic
541C2×C3C3×#93911yescharacteristic
542C2×C3C3×#94211yescharacteristic
543C2×C3C3×#95111yescharacteristic
544D3—41no—
545C22—61no—
546C22—61no—
547C22—61no—
548C22C3×D3×#6711yescharacteristic
549C4—41no—
550C4—61no—
551C4—121no—
552C4—121no—
553C4—361no—
554C4C3×#379511yescharacteristic
555C4C3×#380211yescharacteristic
556C3—21no—
557C3—21no—
558C3—21no—
559C3—41no—
560C3#1059052011yescharacteristic, direct product factor
561C3C3×#481611yescharacteristic, complemented
562C3C3×#505211yescharacteristic, complemented
563C2—121no—
564C2C3×#6388511yescharacteristic
565C2C3×#6392911yescharacteristic
566C2C3×#6394811yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC22×C32×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×#3718C2semidirect
5. C3×#63635C2semidirect
6. C32×#810C2semidirect
7. C3×#63566C2semidirect
10. #10590520C3direct (N × H)
35. C2×#3718C2×C3semidirect
38. #63635C2×C3semidirect
42. C3×#810D3semidirect
43. C3×#810C2×C3semidirect
47. #63566C2×C3semidirect
148. #810C3×D3semidirect
337. C33#470semidirect
502. C32#4816semidirect
503. C32#5052semidirect
504. C32C3×#470semidirect
560. C3#10590520direct (N × H)
561. C3C3×#4816semidirect
562. C3C3×#5052semidirect

Maximal subgroups

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

subgroupindex
C32×#8972
C3×#638562
C2×C3×#37182
C2×C3×#37112
C3×#636352
C32×#8102
C3×#635662
C3×#50523
C3×#48163
#105905203