Finite Group Databasebeta

← C3×#10591618 #13158621

Subgroups — C3×#10591618 #13158621

Order 1728.

810 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3755C211yesmaximal, characteristic, complemented
2C32×#944C211yesmaximal, characteristic, complemented
3C2×C32×#215C211yesmaximal, characteristic, complemented
4C3×#63911C211yesmaximal, characteristic
5C2×C3×#3717C211yesmaximal, characteristic, complemented
6C2×C3×#3715C211yesmaximal, characteristic, complemented
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic, complemented
8C3×#5175—31nomaximal
9C3×#5085—31nomaximal
10#10591618C311yesmaximal, characteristic, direct product factor
11C2×C33×Q8C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×C3×#626—22no—
16C3×#3755—22no—
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×#215—22no—
19C32×Dic3⋊C4—22no—
20C32×C4×Dic3C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C3×#3717—22no—
23C3×#3715—22no—
24C2×C32×SD16—31no—
25C32×C4⋊Q8—31no—
26C2×#3755C2×C311yescharacteristic, complemented
27C3×#944—21no—
28C3×#944D311yescharacteristic, complemented
29C3×#944C2×C311yescharacteristic, complemented
30C2×C3×#215—21no—
31C2×C3×#215—31no—
32C2×C3×#215—31no—
33C2×C3×#215—31no—
34C2×C3×#215—31no—
35C2×C3×#215—61no—
36C2×C3×#215D311yescharacteristic, complemented
37C2×C3×#215C2×C311yescharacteristic, complemented
38C3×#940—31no—
39C2×C3×#214—31no—
40C2×C3×#204—31no—
41C3×#874—31no—
42#63911C2×C311yescharacteristic
43C2×#3717C2×C311yescharacteristic, complemented
44C2×#3715C2×C311yescharacteristic, complemented
45C3×C8×Dic3—31no—
46C3×C4×C3⋊C8—31no—
47Dic3×C3⋊C8C2×C311yescharacteristic, complemented
48C22×C3×C3⋊D3—21no—
49C22×C32×D3—21no—
50C33×Q8—21no—
51C33×Q8—21no—
52C2×C33×C4—21no—
53C2×C33×C4C2311yescharacteristic
54C3×C4⋊(C3⋊D3)—21no—
55C3×C4⋊(C3⋊D3)—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3D412yes—
58C32×D12—21no—
59C32×D12—21no—
60C32×Dic6—21no—
61C32×Dic6—21no—
62C3×#626—44no—
63C3×C3⋊(C3⋊C8)D412yescomplemented
64C32×C3⋊C8D412yescomplemented
65C3×#492—91no—
66#5175—31no—
67#5085—31no—
68C2×C32×Q8—21no—
69C2×C32×Q8—21no—
70C2×C32×Q8—21no—
71C2×C32×Q8—31no—
72C2×C32×Q8—41no—
73C2×C32×Q8C2×D311yescharacteristic
74C2×C32×Q8C2×D311yescharacteristic
75C2×C32×Q8C22×C311yescharacteristic
76C2×C32×D4—31no—
77C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
78C2×C3×#67—62no—
79C2×C3×D12—21no—
80C2×C3×D12—31no—
81C2×C3×D12—31no—
82C2×C3×D12—61no—
83C2×C3×D12—62no—
84C2×C3×D12C2×D311yescharacteristic
85C2×C3×D12C22×C311yescharacteristic
86C2×C3×Dic6—21no—
87C2×C3×Dic6C2×D311yescharacteristic
88C2×C3×Dic6C22×C311yescharacteristic
89C2×#626—22no—
90C32×SD16—62no—
91C2×C32×C8—31no—
92C32×C4⋊C4—62no—
93C32×C42—31no—
94#3755—22no—
95C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
96C3×#215—24no—
97C3×#215—22no—
98C3×#215—22no—
99C3×#215—62no—
100C3×#215—62no—
101C3×#215—62no—
102C3×#215—64no—
103C3×#215—62no—
104C3×#214—62no—
105C3×Dic3⋊C4—24no—
106C3×Dic3⋊C4—22no—
107C3×Dic3⋊C4—22no—
108C3×C4×Dic3—21no—
109C3×C4×Dic3C2×D311yescharacteristic
110C3×C4×Dic3C22×C311yescharacteristic
111C2×C3×C3⋊C8—21no—
112C2×C3×C3⋊C8—31no—
113C2×C3×C3⋊C8—31no—
114C2×C3×C3⋊C8—61no—
115C2×C3×C3⋊C8C2×D311yescharacteristic
116C2×C3×C3⋊C8C22×C311yescharacteristic
117C3×#204—62no—
118#3717—22no—
119#3715—22no—
120C22×C33C2×D411yescharacteristic
121C2×C3×C3⋊D3—41no—
122C2×C3×C3⋊D3—42no—
123C2×C32×D3—41no—
124C2×C32×D3—42no—
125C33×C4—41no—
126C33×C4C2×D411yescharacteristic
127C33×C4C2×D411yescharacteristic
128C32×Dic3—41no—
129C32×Dic3SD1614yescomplemented
130C2×C3×SD16—31no—
131C2×C3×SD16—31no—
132C2×C3×SD16—61no—
133C2×C3×SD16—91no—
134C2×C3×SD16—91no—
135C2×C3×SD16—91no—
136C3×C4⋊Q8—31no—
137C3×C4⋊Q8—31no—
138C3×C4⋊Q8—61no—
139C3×#127—91no—
140#944C3×D311yescharacteristic, complemented
141C2×#215—31no—
142C2×#215—31no—
143C2×#215—31no—
144C2×#215—31no—
145C2×#215—61no—
146C2×#215C3×D311yescharacteristic, complemented
147#940—31no—
148C2×#214—31no—
149C2×#204—31no—
150#874—31no—
151C3×C4×C8—91no—
152C8×Dic3—31no—
153C4×C3⋊C8—31no—
154C23×C32—61no—
155C22×C3⋊D3—21no—
156C22×C3×D3—21no—
157C22×C3×D3—21no—
158C22×C3×D3—41no—
159C22×C3×D3—61no—
160C22×C3×D3—61no—
161C22×C3×D3—121no—
162C32×Q8—21no—
163C32×Q8—21no—
164C32×Q8—22no—
165C32×Q8—22no—
166C32×Q8—21no—
167C32×Q8—21no—
168C32×Q8—21no—
169C32×Q8—21no—
170C32×Q8—41no—
171C32×Q8—41no—
172C32×Q8—41no—
173C32×Q8—41no—
174C32×Q8—42no—
175C32×Q8—42no—
176C32×Q8—61no—
177C32×Q8—61no—
178C32×D4—61no—
179C32×D4—61no—
180C2×C32×C4—21no—
181C2×C32×C4—22no—
182C2×C32×C4—21no—
183C2×C32×C4—21no—
184C2×C32×C4—21no—
185C2×C32×C4—21no—
186C2×C32×C4—21no—
187C2×C32×C4—32no—
188C2×C32×C4—41no—
189C2×C32×C4—41no—
190C2×C32×C4—41no—
191C2×C32×C4—42no—
192C2×C32×C4—61no—
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C22×D311yescharacteristic
195C2×C32×C4C23×C311yescharacteristic
196C4⋊(C3⋊D3)—21no—
197C4⋊(C3⋊D3)—21no—
198C3×#67—124no—
199C2×C3×Dic3—22no—
200C2×C3×Dic3—21no—
201C2×C3×Dic3—21no—
202C2×C3×Dic3—22no—
203C2×C3×Dic3#6712yes—
204C2×C3×Dic3C3×D412yes—
205C3×D12—22no—
206C3×D12—21no—
207C3×D12—21no—
208C3×D12—21no—
209C3×D12—21no—
210C3×D12—41no—
211C3×D12—62no—
212C3×D12—61no—
213C3×D12—61no—
214C3×D12—61no—
215C3×D12—61no—
216C3×D12—121no—
217C3×D12—124no—
218C3×Dic6—21no—
219C3×Dic6—21no—
220C3×Dic6—21no—
221C3×Dic6—21no—
222C3×Dic6—41no—
223C3×Dic6—41no—
224#626—44no—
225C32×C8—32no—
226C3⋊(C3⋊C8)C3×D412yescomplemented
227C3×C3⋊C8—22no—
228C3×C3⋊C8—32no—
229C3×C3⋊C8—32no—
230C3×C3⋊C8—62no—
231C3×C3⋊C8#6712yescomplemented
232C3×C3⋊C8C3×D412yescomplemented
233#492—91noSylow
234C2×C33#12711yescharacteristic
235C2×C33C2×SD1612yes—
236C3×C3⋊D3—81no—
237C32×D3—81no—
238C2×C3×Q8—21no—
239C2×C3×Q8—21no—
240C2×C3×Q8—21no—
241C2×C3×Q8—31no—
242C2×C3×Q8—31no—
243C2×C3×Q8—41no—
244C2×C3×Q8—61no—
245C2×C3×Q8D3211yescharacteristic
246C2×C3×Q8C2×C3×D311yescharacteristic
247C2×C3×Q8C2×C3×D311yescharacteristic
248C2×C3×D4—31no—
249C2×C3×D4—31no—
250C2×C3×D4—61no—
251C2×C3×D4—91no—
252C2×C3×D4—182no—
253C2×#67—62no—
254C2×D12—31no—
255C2×D12—31no—
256C2×D12—61no—
257C2×D12—62no—
258C2×D12C2×C3×D311yescharacteristic
259C2×Dic6C2×C3×D311yescharacteristic
260C3×SD16—62no—
261C3×SD16—62no—
262C3×SD16—64no—
263C3×SD16—182no—
264C3×SD16—182no—
265C3×SD16—182no—
266C2×C3×C8—31no—
267C2×C3×C8—31no—
268C2×C3×C8—61no—
269C2×C3×C8—91no—
270C3×C4⋊C4—62no—
271C3×C4⋊C4—62no—
272C3×C4⋊C4—64no—
273C3×C42—31no—
274C3×C42—31no—
275C3×C42—61no—
276#215—22no—
277#215—62no—
278#215—62no—
279#215—62no—
280#215—64no—
281#215—62no—
282#214—62no—
283Dic3⋊C4—22no—
284C4×Dic3C2×C3×D311yescharacteristic
285C2×C3⋊C8—31no—
286C2×C3⋊C8—31no—
287C2×C3⋊C8—61no—
288C2×C3⋊C8C2×C3×D311yescharacteristic
289#204—62no—
290C22×C32—21no—
291C22×C32—21no—
292C22×C32—21no—
293C22×C32—41no—
294C22×C32—121no—
295C22×C32—122no—
296C22×C32D3×D411yescharacteristic
297C22×C32C2×#6711yescharacteristic
298C22×C32C2×C3×D411yescharacteristic
299C2×C3⋊D3—41no—
300C2×C3⋊D3—42no—
301C2×C3×D3—42no—
302C2×C3×D3—42no—
303C2×C3×D3—41no—
304C2×C3×D3—41no—
305C2×C3×D3—42no—
306C2×C3×D3—82no—
307C2×C3×D3—122no—
308C2×C3×D3—122no—
309C2×C3×D3—121no—
310C2×C3×D3—121no—
311C2×C3×D3—122no—
312C2×C3×D3—242no—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—21no—
316C32×C4—21no—
317C32×C4—21no—
318C32×C4—21no—
319C32×C4—34no—
320C32×C4—41no—
321C32×C4—41no—
322C32×C4—42no—
323C32×C4—42no—
324C32×C4—42no—
325C32×C4—44no—
326C32×C4—41no—
327C32×C4—41no—
328C32×C4—41no—
329C32×C4—121no—
330C32×C4D3×D411yescharacteristic
331C32×C4D3×D411yescharacteristic
332C32×C4D3×D411yescharacteristic
333C32×C4C2×#6711yescharacteristic
334C32×C4C2×C3×D411yescharacteristic
335C32×C4C2×C3×D411yescharacteristic
336C3×Dic3—24no—
337C3×Dic3—42no—
338C3×Dic3—41no—
339C3×Dic3—41no—
340C3×Dic3#21514yescomplemented
341C3×Dic3C3×SD1614yescomplemented
342C2×SD16—31no—
343C2×SD16—91no—
344C2×SD16—91no—
345C2×SD16—91no—
346C4⋊Q8—31no—
347#127—91no—
348C4×C8—91no—
349C33#49211yescharacteristic, complemented, Sylow
350C23×C3—61no—
351C23×C3—61no—
352C23×C3—121no—
353C23×C3—181no—
354C22×D3—21no—
355C22×D3—61no—
356C22×D3—61no—
357C22×D3—121no—
358C3×Q8—22no—
359C3×Q8—22no—
360C3×Q8—21no—
361C3×Q8—21no—
362C3×Q8—21no—
363C3×Q8—21no—
364C3×Q8—21no—
365C3×Q8—21no—
366C3×Q8—41no—
367C3×Q8—41no—
368C3×Q8—41no—
369C3×Q8—41no—
370C3×Q8—42no—
371C3×Q8—42no—
372C3×Q8—61no—
373C3×Q8—61no—
374C3×Q8—61no—
375C3×Q8—61no—
376C3×Q8—121no—
377C3×Q8—121no—
378C3×D4—62no—
379C3×D4—61no—
380C3×D4—61no—
381C3×D4—61no—
382C3×D4—61no—
383C3×D4—121no—
384C3×D4—181no—
385C3×D4—181no—
386C3×D4—364no—
387C2×C3×C4—22no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—32no—
395C2×C3×C4—32no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—42no—
399C2×C3×C4—41no—
400C2×C3×C4—62no—
401C2×C3×C4—62no—
402C2×C3×C4—61no—
403C2×C3×C4—61no—
404C2×C3×C4C2×D3211yescharacteristic
405C2×C3×C4C22×C3×D311yescharacteristic
406C2×C3×C4C22×C3×D311yescharacteristic
407#67—124no—
408C2×Dic3—21no—
409C2×Dic3C3×#6712yes—
410D12—21no—
411D12—21no—
412D12—62no—
413D12—61no—
414D12—61no—
415D12—61no—
416D12—61no—
417D12—121no—
418D12—124no—
419Dic6—21no—
420Dic6—21no—
421C3×C8—32no—
422C3×C8—32no—
423C3×C8—62no—
424C3×C8—92no—
425C3⋊C8—32no—
426C3⋊C8—32no—
427C3⋊C8—62no—
428C3⋊C8C3×#6712yescomplemented
429C2×C32—22no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—22no—
433C2×C32—22no—
434C2×C32—21no—
435C2×C32—41no—
436C2×C32—42no—
437C2×C32—241no—
438C2×C32D3×SD1612yes—
439C2×C32#94011yescharacteristic
440C2×C32#94011yescharacteristic
441C2×C32C2×#21512yes—
442C2×C32C3×#12711yescharacteristic
443C2×C32C2×C3×SD1612yes—
444C3⋊D3—81no—
445C3×D3—82no—
446C3×D3—81no—
447C3×D3—81no—
448C3×D3—242no—
449C3×D3—241no—
450C3×D3—241no—
451C2×Q8—31no—
452C2×Q8C3×D3211yescharacteristic
453C2×D4—31no—
454C2×D4—91no—
455C2×D4—182no—
456SD16—62no—
457SD16—182no—
458SD16—182no—
459SD16—182no—
460C2×C8—31no—
461C2×C8—91no—
462C4⋊C4—62no—
463C42—31no—
464C22×C3—21no—
465C22×C3—21no—
466C22×C3—21no—
467C22×C3—41no—
468C22×C3—122no—
469C22×C3—121no—
470C22×C3—122no—
471C22×C3—122no—
472C22×C3—121no—
473C22×C3—242no—
474C22×C3—362no—
475C22×C3—361no—
476C22×C3D3×#6711yescharacteristic
477C22×C3C3×D3×D411yescharacteristic
478C22×C3C2×C3×#6711yescharacteristic
479C2×D3—42no—
480C2×D3—41no—
481C2×D3—122no—
482C2×D3—122no—
483C2×D3—121no—
484C2×D3—121no—
485C2×D3—122no—
486C2×D3—242no—
487C3×C4—21no—
488C3×C4—21no—
489C3×C4—21no—
490C3×C4—21no—
491C3×C4—21no—
492C3×C4—21no—
493C3×C4—34no—
494C3×C4—34no—
495C3×C4—42no—
496C3×C4—42no—
497C3×C4—42no—
498C3×C4—44no—
499C3×C4—41no—
500C3×C4—41no—
501C3×C4—41no—
502C3×C4—41no—
503C3×C4—41no—
504C3×C4—64no—
505C3×C4—122no—
506C3×C4—121no—
507C3×C4—121no—
508C3×C4D3×#6711yescharacteristic
509C3×C4#381111yescharacteristic
510C3×C4C3×D3×D411yescharacteristic
511C3×C4C3×D3×D411yescharacteristic
512C3×C4C3×D3×D411yescharacteristic
513C3×C4C2×C3×#6711yescharacteristic
514Dic3—41no—
515Dic3C3×#21514yescomplemented
516C32—21no—
517C32—21no—
518C32—21no—
519C32—41no—
520C32#508511yescharacteristic, complemented
521C32#517511yescharacteristic, complemented
522C32C3×#49211yescharacteristic, complemented
523C23—61no—
524C23—181no—
525Q8—21no—
526Q8—21no—
527Q8—61no—
528Q8—61no—
529D4—61no—
530D4—61no—
531D4—181no—
532D4—181no—
533D4—364no—
534C2×C4—21no—
535C2×C4—32no—
536C2×C4—61no—
537C2×C4C2×C3×D3211yescharacteristic
538C8—32no—
539C8—92no—
540C2×C3—21no—
541C2×C3—21no—
542C2×C3—21no—
543C2×C3—22no—
544C2×C3—22no—
545C2×C3—22no—
546C2×C3—41no—
547C2×C3—42no—
548C2×C3—242no—
549C2×C3—241no—
550C2×C3—241no—
551C2×C3—721no—
552C2×C3D3×#21512yes—
553C2×C3#6397911yescharacteristic
554C2×C3C3×D3×SD1612yes—
555C2×C3C3×#94011yescharacteristic
556C2×C3C3×#94011yescharacteristic
557C2×C3C2×C3×#21512yes—
558D3—81no—
559D3—242no—
560D3—241no—
561D3—241no—
562C22—121no—
563C22—122no—
564C22—361no—
565C22—362no—
566C22C3×D3×#6711yescharacteristic
567C4—34no—
568C4—41no—
569C4—121no—
570C4C3×D3×#6711yescharacteristic
571C4C3×#381111yescharacteristic
572C3—21no—
573C3—21no—
574C3—21no—
575C3—41no—
576C3#1059161811yescharacteristic, direct product factor
577C3C3×#508511yescharacteristic, complemented
578C3C3×#517511yescharacteristic, complemented
579C2—241no—
580C2—721no—
581C2C3×D3×#21512yes—
582C2C3×#6397911yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC3×#63911C2
hypercenterC2×C3×Q8D32
hypocenterC32C3×#492
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×C3×#3755C2semidirect
2. C32×#944C2semidirect
3. C2×C32×#215C2semidirect
5. C2×C3×#3717C2semidirect
6. C2×C3×#3715C2semidirect
7. C3×Dic3×C3⋊C8C2semidirect
10. #10591618C3direct (N × H)
26. C2×#3755C2×C3semidirect
28. C3×#944D3semidirect
29. C3×#944C2×C3semidirect
36. C2×C3×#215D3semidirect
37. C2×C3×#215C2×C3semidirect
43. C2×#3717C2×C3semidirect
44. C2×#3715C2×C3semidirect
47. Dic3×C3⋊C8C2×C3semidirect
63. C3×C3⋊(C3⋊C8)D4semidirect
64. C32×C3⋊C8D4semidirect
129. C32×Dic3SD16semidirect
140. #944C3×D3semidirect
146. C2×#215C3×D3semidirect
226. C3⋊(C3⋊C8)C3×D4semidirect
231. C3×C3⋊C8#67semidirect
232. C3×C3⋊C8C3×D4semidirect
340. C3×Dic3#215semidirect
341. C3×Dic3C3×SD16semidirect
349. C33#492semidirect
428. C3⋊C8C3×#67semidirect
515. Dic3C3×#215semidirect
520. C32#5085semidirect
521. C32#5175semidirect
522. C32C3×#492semidirect
576. C3#10591618direct (N × H)
577. C3C3×#5085semidirect
578. C3C3×#5175semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37552
C32×#9442
C2×C32×#2152
C3×#639112
C2×C3×#37172
C2×C3×#37152
C3×Dic3×C3⋊C82
C3×#51753
C3×#50853
#105916183