Finite Group Databasebeta

← C3×#10590506 #13157580

Subgroups — C3×#10590506 #13157580

Order 1728.

666 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64106C211yesmaximal, characteristic, complemented
2C3×#63916C211yesmaximal, characteristic
3C2×C3×#3716C211yesmaximal, characteristic, complemented
4C2×C3×#3712C211yesmaximal, characteristic, complemented
5C32×#809C211yesmaximal, characteristic, complemented
6C32×#807C211yesmaximal, characteristic, complemented
7C3×#63567C211yesmaximal, characteristic, complemented
8C3×#4886—31nomaximal
9C3×#4780—31nomaximal
10#10590506C311yesmaximal, characteristic, direct product factor
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C3×#626—21no—
15C33×C4⋊C4C2211yescharacteristic
16C3×#3752—21no—
17C3×C4⋊(C3⋊Dic3)—21no—
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×(C2×D3)⋊Dic3—21no—
22C3×#3716—22no—
23C3×#3712—22no—
24#64106C2×C311yescharacteristic, complemented
25C3×#938—31no—
26C2×C3×#214—31no—
27C2×C3×#204—31no—
28C2×C3×ΓL(1,𝔽25)—31no—
29C2×C3×ΓL(1,𝔽25)—31no—
30C3×#897—31no—
31C3×#897—31no—
32C3×#897—61no—
33C3×#895—31no—
34#63916C2×C311yescharacteristic
35C2×#3716C2×C311yescharacteristic, complemented
36C2×#3712C2×C311yescharacteristic, complemented
37C32×#107—31no—
38C32×#102—31no—
39C3×#816—31no—
40C3×#810—31no—
41C3×#809—21no—
42C3×#809D311yescharacteristic, complemented
43C3×#809C2×C311yescharacteristic, complemented
44C3×#807—21no—
45C3×#807D311yescharacteristic, complemented
46C3×#807C2×C311yescharacteristic, complemented
47#63567C2×C311yescharacteristic, complemented
48C22×C3×C3⋊D3D411yescharacteristic
49C22×C32×D3—21no—
50C2×C33×C4—21no—
51C2×C33×C4C2311yescharacteristic
52C2×C3×C3⋊Dic3—21no—
53C2×C3×C3⋊Dic3D411yescharacteristic
54C3×C4×C3⋊D3—21no—
55C3×C4×C3⋊D3—21no—
56C3×#634—21no—
57C3×#634—21no—
58C2×C32×Dic3—21no—
59C32×D12—21no—
60C32×D12—21no—
61C3×#626—42no—
62C32×C3⋊C8—21no—
63C32×C3⋊C8D412yescomplemented
64C3×#468—91no—
65#4886—31no—
66#4780—31no—
67C2×C32×D4—31no—
68C2×C4×C3⋊D3C22×C311yescharacteristic
69C2×#634C22×C311yescharacteristic
70C2×C3×#67—61no—
71C2×C3×D12—21no—
72C2×C3×D12—61no—
73C2×C3×D12C2×D311yescharacteristic
74C2×C3×D12C22×C311yescharacteristic
75C2×C3×C4×D3—31no—
76C2×C3×C4×D3—31no—
77C2×C3×C4×D3—61no—
78C2×C3×Dic6—31no—
79C2×C3×Dic6—31no—
80C2×C3×Dic6—61no—
81C2×#626—21no—
82C2×C32×C8—31no—
83C2×C32×C8—31no—
84C32×C4⋊C4—21no—
85C32×C4⋊C4—21no—
86C32×C4⋊C4—21no—
87C32×C4⋊C4—31no—
88C32×C4⋊C4—41no—
89C32×C4⋊C4C2×D311yescharacteristic
90C32×C4⋊C4C2×D311yescharacteristic
91C32×C4⋊C4C22×C311yescharacteristic
92#3752—21no—
93C4⋊(C3⋊Dic3)—21no—
94C3×#217—61no—
95C3×#214—62no—
96C3×(C2×D3)⋊C4—61no—
97C3×(C2×D3)⋊C4—61no—
98C3×(C2×D3)⋊C4—61no—
99C3×(C2×D3)⋊C4—62no—
100C3×C4⋊Dic3—21no—
101C3×C4⋊Dic3—61no—
102C3×C4⋊Dic3—61no—
103C3×C4⋊Dic3—62no—
104C3×C4⋊Dic3C2×D311yescharacteristic
105C3×C4⋊Dic3C22×C311yescharacteristic
106C2×C3×C3⋊C8—21no—
107C2×C3×C3⋊C8—21no—
108C2×C3×C3⋊C8C2×D311yescharacteristic
109C2×C3×C3⋊C8C2×D311yescharacteristic
110C2×C3×C3⋊C8C22×C311yescharacteristic
111C2×C3×C3⋊C8C22×C311yescharacteristic
112C3×#204—62no—
113C3×ΓL(1,𝔽25)—62no—
114C3×ΓL(1,𝔽25)—62no—
115(C2×D3)⋊Dic3—21no—
116#3716—22no—
117#3712—22no—
118C22×C33C2×D411yescharacteristic
119C2×C3×C3⋊D3—21no—
120C2×C3×C3⋊D3—21no—
121C2×C3×C3⋊D3—21no—
122C2×C32×D3—41no—
123C2×C32×D3—41no—
124C2×C32×D3—41no—
125C33×C4—41no—
126C33×C4C2×D411yescharacteristic
127C33×C4#4111yescharacteristic
128C3×C3⋊Dic3—21no—
129C3×C3⋊Dic3—41no—
130C32×Dic3—41no—
131C2×C3×SD16—91no—
132C2×C3×M16—91no—
133C3×C22⋊Q8—91no—
134C3×#121—91no—
135#938—31no—
136C2×#214—31no—
137C2×#204—31no—
138C2×ΓL(1,𝔽25)—31no—
139C2×ΓL(1,𝔽25)—31no—
140#897—31no—
141#897—31no—
142#897—61no—
143#895—31no—
144C3×#107—31no—
145C3×#107—31no—
146C3×#107—61no—
147C3×#103—91no—
148C3×#102—31no—
149C3×#102—31no—
150C3×#102—61no—
151#816—31no—
152#810—31no—
153#809C3×D311yescharacteristic, complemented
154#807C3×D311yescharacteristic, complemented
155C23×C32—61no—
156C22×C3⋊D3C3×D411yescharacteristic
157C22×C3×D3—21no—
158C22×C3×D3—21no—
159C22×C3×D3—31no—
160C22×C3×D3—31no—
161C22×C3×D3—41no—
162C22×C3×D3—61no—
163C32×D4—61no—
164C32×D4—61no—
165C2×C32×C4—21no—
166C2×C32×C4—22no—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C32×C4—21no—
171C2×C32×C4—21no—
172C2×C32×C4—41no—
173C2×C32×C4—41no—
174C2×C32×C4—41no—
175C2×C32×C4—42no—
176C2×C32×C4—61no—
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C22×D311yescharacteristic
179C2×C32×C4C23×C311yescharacteristic
180C2×C3⋊Dic3—21no—
181C2×C3⋊Dic3C3×D411yescharacteristic
182C4×C3⋊D3—21no—
183C4×C3⋊D3—21no—
184#634—21no—
185#634—21no—
186C3×#67—122no—
187C2×C3×Dic3—21no—
188C2×C3×Dic3—21no—
189C2×C3×Dic3—31no—
190C2×C3×Dic3—31no—
191C2×C3×Dic3—41no—
192C2×C3×Dic3—61no—
193C2×C3×Dic3—62no—
194C2×C3×Dic3—61no—
195C2×C3×Dic3—61no—
196C3×D12—22no—
197C3×D12—21no—
198C3×D12—21no—
199C3×D12—21no—
200C3×D12—21no—
201C3×D12—41no—
202C3×D12—122no—
203C3×C4×D3—61no—
204C3×C4×D3—61no—
205C3×C4×D3—61no—
206C3×C4×D3—61no—
207C3×C4×D3—121no—
208C3×C4×D3—121no—
209C3×Dic6—61no—
210C3×Dic6—61no—
211C3×Dic6—61no—
212C3×Dic6—61no—
213C3×Dic6—121no—
214C3×Dic6—121no—
215#626—42no—
216C32×C8—32no—
217C32×C8—61no—
218C3×C3⋊C8—21no—
219C3×C3⋊C8—21no—
220C3×C3⋊C8—22no—
221C3×C3⋊C8—41no—
222C3×C3⋊C8D1212yescomplemented
223C3×C3⋊C8C3×D412yescomplemented
224#468—91noSylow
225C2×C33#12111yescharacteristic
226C2×C33AGL(1,ℤ/8)11yescharacteristic
227C2×C33#13711yescharacteristic
228C3×C3⋊D3—41no—
229C32×D3—81no—
230C2×C3×Q8—91no—
231C2×C3×D4—31no—
232C2×C3×D4—31no—
233C2×C3×D4—61no—
234C2×C3×D4—181no—
235C22×C3×C4—91no—
236C2×#67—61no—
237C2×D12—61no—
238C2×D12C2×C3×D311yescharacteristic
239C2×C4×D3—31no—
240C2×C4×D3—31no—
241C2×C4×D3—61no—
242C2×Dic6—31no—
243C2×Dic6—31no—
244C2×Dic6—61no—
245C3×SD16—182no—
246C3×M16—182no—
247C2×C3×C8—31no—
248C2×C3×C8—31no—
249C2×C3×C8—31no—
250C2×C3×C8—31no—
251C2×C3×C8—61no—
252C2×C3×C8—61no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—31no—
257C3×C4⋊C4—31no—
258C3×C4⋊C4—41no—
259C3×C4⋊C4—61no—
260C3×C4⋊C4—181no—
261C3×C4⋊C4D3211yescharacteristic
262C3×C4⋊C4C2×C3×D311yescharacteristic
263C3×C4⋊C4C2×C3×D311yescharacteristic
264C3×C22⋊C4—181no—
265C3×C22⋊C4—181no—
266#217—61no—
267#214—62no—
268(C2×D3)⋊C4—61no—
269(C2×D3)⋊C4—61no—
270(C2×D3)⋊C4—61no—
271(C2×D3)⋊C4—62no—
272C4⋊Dic3—61no—
273C4⋊Dic3—61no—
274C4⋊Dic3—62no—
275C4⋊Dic3C2×C3×D311yescharacteristic
276C2×C3⋊C8C2×C3×D311yescharacteristic
277C2×C3⋊C8C2×C3×D311yescharacteristic
278#204—62no—
279ΓL(1,𝔽25)—62no—
280ΓL(1,𝔽25)—62no—
281C22×C32—21no—
282C22×C32—21no—
283C22×C32—21no—
284C22×C32—41no—
285C22×C32—121no—
286C22×C32—121no—
287C22×C32—121no—
288C22×C32D3×D411yescharacteristic
289C22×C32D3×D411yescharacteristic
290C22×C32C2×C3×D411yescharacteristic
291C2×C3⋊D3—21no—
292C2×C3⋊D3—21no—
293C2×C3⋊D3—21no—
294C2×C3×D3—42no—
295C2×C3×D3—41no—
296C2×C3×D3—41no—
297C2×C3×D3—41no—
298C2×C3×D3—41no—
299C2×C3×D3—41no—
300C2×C3×D3—41no—
301C2×C3×D3—62no—
302C2×C3×D3—61no—
303C2×C3×D3—61no—
304C2×C3×D3—61no—
305C2×C3×D3—61no—
306C2×C3×D3—61no—
307C2×C3×D3—61no—
308C2×C3×D3—81no—
309C2×C3×D3—81no—
310C2×C3×D3—121no—
311C2×C3×D3—121no—
312C32×C4—21no—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—21no—
316C32×C4—21no—
317C32×C4—21no—
318C32×C4—41no—
319C32×C4—41no—
320C32×C4—41no—
321C32×C4—42no—
322C32×C4—41no—
323C32×C4—41no—
324C32×C4—81no—
325C32×C4—81no—
326C32×C4—82no—
327C32×C4—121no—
328C32×C4C2×D1211yescharacteristic
329C32×C4#23511yescharacteristic
330C32×C4D3×D411yescharacteristic
331C32×C4#23911yescharacteristic
332C32×C4C2×C3×D411yescharacteristic
333C32×C4C3×#4111yescharacteristic
334C3⋊Dic3—21no—
335C3⋊Dic3—41no—
336C3×Dic3—41no—
337C3×Dic3—41no—
338C3×Dic3—61no—
339C3×Dic3—61no—
340C3×Dic3—81no—
341C3×Dic3—121no—
342C3×Dic3—122no—
343C3×Dic3—121no—
344C3×Dic3—121no—
345C2×SD16—91no—
346C2×M16—91no—
347C22⋊Q8—91no—
348#121—91no—
349#107—31no—
350#103—91no—
351#102—31no—
352C33#46811yescharacteristic, complemented, Sylow
353C23×C3—61no—
354C23×C3—61no—
355C23×C3—91no—
356C23×C3—121no—
357C22×D3—21no—
358C22×D3—31no—
359C22×D3—31no—
360C22×D3—61no—
361C3×Q8—181no—
362C3×Q8—181no—
363C3×D4—62no—
364C3×D4—61no—
365C3×D4—61no—
366C3×D4—61no—
367C3×D4—61no—
368C3×D4—121no—
369C3×D4—362no—
370C2×C3×C4—22no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—42no—
380C2×C3×C4—41no—
381C2×C3×C4—61no—
382C2×C3×C4—61no—
383C2×C3×C4—91no—
384C2×C3×C4—121no—
385C2×C3×C4—181no—
386C2×C3×C4—181no—
387C2×C3×C4—181no—
388C2×C3×C4C2×D3211yescharacteristic
389C2×C3×C4C22×C3×D311yescharacteristic
390C2×C3×C4C22×C3×D311yescharacteristic
391#67—122no—
392C2×Dic3—21no—
393C2×Dic3—31no—
394C2×Dic3—31no—
395C2×Dic3—61no—
396C2×Dic3—62no—
397C2×Dic3—61no—
398C2×Dic3—61no—
399D12—21no—
400D12—21no—
401D12—122no—
402C4×D3—61no—
403C4×D3—61no—
404C4×D3—61no—
405C4×D3—61no—
406C4×D3—121no—
407C4×D3—121no—
408Dic6—61no—
409Dic6—61no—
410Dic6—61no—
411Dic6—61no—
412Dic6—121no—
413Dic6—121no—
414C3×C8—32no—
415C3×C8—32no—
416C3×C8—62no—
417C3×C8—61no—
418C3×C8—61no—
419C3×C8—121no—
420C3⋊C8—21no—
421C3⋊C8C3×D1212yescomplemented
422C2×C32—21no—
423C2×C32—21no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—21no—
430C2×C32—21no—
431C2×C32—41no—
432C2×C32—41no—
433C2×C32—41no—
434C2×C32—241no—
435C2×C32#88411yescharacteristic
436C2×C32#89511yescharacteristic
437C2×C32#91111yescharacteristic
438C2×C32#91111yescharacteristic
439C2×C32#91511yescharacteristic
440C2×C32#91811yescharacteristic
441C2×C32C3×#12111yescharacteristic
442C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
443C2×C32C3×#13711yescharacteristic
444C3⋊D3—41no—
445C3×D3—82no—
446C3×D3—81no—
447C3×D3—81no—
448C3×D3—122no—
449C3×D3—121no—
450C3×D3—121no—
451C2×Q8—91no—
452C2×D4—31no—
453C2×D4—181no—
454C22×C4—91no—
455SD16—182no—
456M16—182no—
457C2×C8—31no—
458C2×C8—31no—
459C4⋊C4—31no—
460C4⋊C4—181no—
461C4⋊C4C3×D3211yescharacteristic
462C22⋊C4—181no—
463C22⋊C4—181no—
464C22×C3—21no—
465C22×C3—21no—
466C22×C3—21no—
467C22×C3—41no—
468C22×C3—122no—
469C22×C3—121no—
470C22×C3—121no—
471C22×C3—121no—
472C22×C3—121no—
473C22×C3—121no—
474C22×C3—121no—
475C22×C3—181no—
476C22×C3—181no—
477C22×C3—181no—
478C22×C3—241no—
479C22×C3—241no—
480C22×C3#381111yescharacteristic
481C22×C3C3×D3×D411yescharacteristic
482C22×C3C3×D3×D411yescharacteristic
483C2×D3—41no—
484C2×D3—41no—
485C2×D3—41no—
486C2×D3—62no—
487C2×D3—61no—
488C2×D3—61no—
489C2×D3—61no—
490C2×D3—61no—
491C2×D3—61no—
492C2×D3—61no—
493C2×D3—121no—
494C2×D3—121no—
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—21no—
498C3×C4—21no—
499C3×C4—21no—
500C3×C4—21no—
501C3×C4—42no—
502C3×C4—41no—
503C3×C4—41no—
504C3×C4—41no—
505C3×C4—41no—
506C3×C4—41no—
507C3×C4—81no—
508C3×C4—81no—
509C3×C4—82no—
510C3×C4—121no—
511C3×C4—121no—
512C3×C4—181no—
513C3×C4—241no—
514C3×C4—361no—
515C3×C4#379911yescharacteristic
516C3×C4D3×D1211yescharacteristic
517C3×C4C2×C3×D1211yescharacteristic
518C3×C4C3×#23511yescharacteristic
519C3×C4C3×D3×D411yescharacteristic
520C3×C4C3×#23911yescharacteristic
521Dic3—41no—
522Dic3—61no—
523Dic3—61no—
524Dic3—121no—
525Dic3—122no—
526Dic3—121no—
527Dic3—121no—
528C32—21no—
529C32—21no—
530C32—21no—
531C32—41no—
532C32#478011yescharacteristic, complemented
533C32#488611yescharacteristic, complemented
534C32C3×#46811yescharacteristic, complemented
535C23—61no—
536C23—91no—
537Q8—181no—
538Q8—181no—
539D4—61no—
540D4—61no—
541D4—362no—
542C2×C4—21no—
543C2×C4—61no—
544C2×C4—91no—
545C2×C4—181no—
546C2×C4—181no—
547C2×C4—181no—
548C2×C4C2×C3×D3211yescharacteristic
549C8—32no—
550C8—61no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—21no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3—41no—
561C2×C3—41no—
562C2×C3—41no—
563C2×C3—242no—
564C2×C3—241no—
565C2×C3—241no—
566C2×C3—361no—
567C2×C3#6391011yescharacteristic
568C2×C3#6392711yescharacteristic
569C2×C3#6395111yescharacteristic
570C2×C3C3×#88411yescharacteristic
571C2×C3C3×#89511yescharacteristic
572C2×C3C3×#91111yescharacteristic
573C2×C3C3×#91111yescharacteristic
574C2×C3C3×#91511yescharacteristic
575C2×C3C3×#91811yescharacteristic
576D3—81no—
577D3—122no—
578D3—121no—
579D3—121no—
580C22—121no—
581C22—121no—
582C22—121no—
583C22—181no—
584C22—181no—
585C22—181no—
586C22C3×#381111yescharacteristic
587C4—41no—
588C4—121no—
589C4—181no—
590C4—361no—
591C4C3×#379911yescharacteristic
592C4C3×D3×D1211yescharacteristic
593C3—21no—
594C3—21no—
595C3—21no—
596C3—41no—
597C3#1059050611yescharacteristic, direct product factor
598C3C3×#478011yescharacteristic, complemented
599C3C3×#488611yescharacteristic, complemented
600C2—241no—
601C2—361no—
602C2C3×#6391011yescharacteristic
603C2C3×#6392711yescharacteristic
604C2C3×#6395111yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC3×#63916C2
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#468
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
1. C3×#64106C2semidirect
3. C2×C3×#3716C2semidirect
4. C2×C3×#3712C2semidirect
5. C32×#809C2semidirect
6. C32×#807C2semidirect
7. C3×#63567C2semidirect
10. #10590506C3direct (N × H)
24. #64106C2×C3semidirect
35. C2×#3716C2×C3semidirect
36. C2×#3712C2×C3semidirect
42. C3×#809D3semidirect
43. C3×#809C2×C3semidirect
45. C3×#807D3semidirect
46. C3×#807C2×C3semidirect
47. #63567C2×C3semidirect
63. C32×C3⋊C8D4semidirect
153. #809C3×D3semidirect
154. #807C3×D3semidirect
222. C3×C3⋊C8D12semidirect
223. C3×C3⋊C8C3×D4semidirect
352. C33#468semidirect
421. C3⋊C8C3×D12semidirect
532. C32#4780semidirect
533. C32#4886semidirect
534. C32C3×#468semidirect
597. C3#10590506direct (N × H)
598. C3C3×#4780semidirect
599. C3C3×#4886semidirect

Maximal subgroups

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

subgroupindex
C3×#641062
C3×#639162
C2×C3×#37162
C2×C3×#37122
C32×#8092
C32×#8072
C3×#635672
C3×#48863
C3×#47803
#105905063