Finite Group Databasebeta

← C2×#11723720 #13178785

Subgroups — C2×#11723720 #13178785

Order 1728.

2106 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#638C211yesmaximal, characteristic
2C23×C3×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C2×C3×#3757C211yesmaximal, characteristic, complemented
4C2×#88815C212yesmaximal, complemented
5#11723720C218yesmaximal, direct product factor
6C2×#88814C212yesmaximal, complemented
7C2×#63966—31nomaximal
8C2×#63962—31nomaximal
9C2×#63961—31nomaximal
10C24×C33C2211yescharacteristic
11C23×C3×C3⋊D3C2211yescharacteristic
12C2×C3×#638—24no—
13C2×C3×#638C2214yes—
14C22×C3×C3⋊Dic3—24no—
15C22×C3×C3⋊Dic3C2212yes—
16C22×C3×C3⋊Dic3C2211yescharacteristic
17C22×C3×C3⋊Dic3C2212yes—
18C22×C3×C3⋊Dic3C2214yescomplemented
19C3×#3757C2214yescomplemented
20#88815C2218yescomplemented
21#88814C2218yescomplemented
22C22×#638D311yescharacteristic
23C23×C3⋊Dic3D311yescharacteristic, complemented
24C22×C3×#67—31no—
25C22×C3×#67—31no—
26C22×C3×#67—61no—
27C23×C3×Dic3—31no—
28C23×C3×Dic3—31no—
29C23×C3×Dic3—61no—
30C2×#3757D311yescharacteristic, complemented
31C2×C3×#217—31no—
32C2×C3×#217—31no—
33C2×C3×#217—61no—
34C2×#3724—32no—
35#63966—38no—
36C2×Dic3⋊Dic3—32no—
37C2×Dic3⋊Dic3—32no—
38C2×#3722—32no—
39#63962—38no—
40#63961—38no—
41C2×(C2×D3)⋊Dic3—32no—
42C2×(C2×D3)⋊Dic3—32no—
43C23×C33—24no—
44C23×C33D414yes—
45C23×C33C2312yes—
46C23×C33C2311yescharacteristic
47C22×C3×C3⋊D3—22no—
48C22×C3×C3⋊D3—24no—
49C22×C3×C3⋊D3C2312yes—
50C3×#638—28no—
51C2×C3×C3⋊Dic3—24no—
52C2×C3×C3⋊Dic3—28no—
53C2×C3×C3⋊Dic3—22no—
54C2×C3×C3⋊Dic3—24no—
55C2×C3×C3⋊Dic3C2314yes—
56C2×C3×C3⋊Dic3C2314yes—
57C2×C3×C3⋊Dic3C2312yes—
58C2×#935—91no—
59C2×#929—91no—
60C2×#886—91no—
61C24×C32—21no—
62C24×C32—21no—
63C24×C32—21no—
64C24×C32—41no—
65C24×C32C2×D311yescharacteristic
66C24×C32C2×D311yescharacteristic
67C24×C32C2×D311yescharacteristic
68C23×C3⋊D3C2×D311yescharacteristic
69C23×C3×D3—31no—
70C23×C3×D3—31no—
71C23×C3×D3—61no—
72C2×#638—24no—
73C2×#638C2×D314yes—
74C22×C3⋊Dic3—24no—
75C22×C3⋊Dic3C2×D312yes—
76C22×C3⋊Dic3C2×D311yescharacteristic
77C22×C3⋊Dic3C2×D312yes—
78C22×C3⋊Dic3C2×D314yescomplemented
79C2×C3×#67—34no—
80C2×C3×#67—34no—
81C2×C3×#67—64no—
82C2×C3×#67—68no—
83C2×C3×#67—64no—
84C2×C3×#67—64no—
85C22×C3×Dic3—32no—
86C22×C3×Dic3—32no—
87C22×C3×Dic3—32no—
88C22×C3×Dic3—34no—
89C22×C3×Dic3—32no—
90C22×C3×Dic3—34no—
91C22×C3×Dic3—31no—
92C22×C3×Dic3—31no—
93C22×C3×Dic3—62no—
94C22×C3×Dic3—64no—
95C22×C3×Dic3—61no—
96C22×C3×Dic3—68no—
97C22×C3×Dic3—64no—
98C22×C3×Dic3—64no—
99C22×C3×Dic3—62no—
100#3757C2×D314yescomplemented
101C3×#217—34no—
102C3×#217—34no—
103C3×#217—64no—
104#3724—38no—
105Dic3⋊Dic3—38no—
106Dic3⋊Dic3—38no—
107#3722—38no—
108(C2×D3)⋊Dic3—38no—
109(C2×D3)⋊Dic3—38no—
110C22×C33—28no—
111C22×C33—24no—
112C22×C33C2×D414yes—
113C22×C33C2×D412yes—
114C22×C33#4114yes—
115C22×C33C2411yescharacteristic
116C2×C3×C3⋊D3—22no—
117C2×C3×C3⋊D3—24no—
118C2×C3×C3⋊D3—44no—
119C3×C3⋊Dic3—24no—
120C3×C3⋊Dic3—22no—
121C3×C3⋊Dic3—24no—
122C22×C3×D4—91no—
123C23×C3×C4—91no—
124C22×#67—31no—
125C22×#67—31no—
126C22×#67—61no—
127C23×Dic3—31no—
128C23×Dic3—31no—
129C23×Dic3—61no—
130C2×C3×C22⋊C4—91no—
131C2×#217—31no—
132C2×#217—31no—
133C2×#217—61no—
134C2×#217—92no—
135#935—98no—
136#929—98no—
137C2×(C2×D3)⋊C4—92no—
138C2×(C2×D3)⋊C4—92no—
139C2×C4⋊Dic3—92no—
140C2×Dic3⋊C4—92no—
141C2×Dic3⋊C4—92no—
142#886—98no—
143C23×C32—24no—
144C23×C32—28no—
145C23×C32—24no—
146C23×C32—24no—
147C23×C32—24no—
148C23×C32—24no—
149C23×C32—24no—
150C23×C32—21no—
151C23×C32—22no—
152C23×C32—21no—
153C23×C32—21no—
154C23×C32—22no—
155C23×C32—22no—
156C23×C32—44no—
157C23×C32—42no—
158C23×C32—41no—
159C23×C32—44no—
160C23×C32—44no—
161C23×C32—48no—
162C23×C32D1214yes—
163C23×C32#6714yes—
164C23×C32#6714yes—
165C23×C32C22×D312yes—
166C23×C32C22×D312yes—
167C23×C32C22×D312yes—
168C23×C32C22×D311yescharacteristic
169C23×C32C22×D311yescharacteristic
170C23×C32C22×D311yescharacteristic
171C22×C3⋊D3—22no—
172C22×C3⋊D3—24no—
173C22×C3⋊D3C22×D312yes—
174C22×C3×D3—32no—
175C22×C3×D3—32no—
176C22×C3×D3—62no—
177C22×C3×D3—64no—
178C22×C3×D3—62no—
179C22×C3×D3—64no—
180C22×C3×D3—64no—
181C22×C3×D3—62no—
182C22×C3×D3—124no—
183#638—28no—
184C2×C3⋊Dic3—24no—
185C2×C3⋊Dic3—28no—
186C2×C3⋊Dic3—22no—
187C2×C3⋊Dic3—24no—
188C2×C3⋊Dic3C22×D314yes—
189C2×C3⋊Dic3C22×D314yes—
190C2×C3⋊Dic3C22×D312yes—
191C3×#67—616no—
192C3×#67—68no—
193C3×#67—68no—
194C2×C3×Dic3—34no—
195C2×C3×Dic3—34no—
196C2×C3×Dic3—34no—
197C2×C3×Dic3—34no—
198C2×C3×Dic3—32no—
199C2×C3×Dic3—32no—
200C2×C3×Dic3—64no—
201C2×C3×Dic3—62no—
202C2×C3×Dic3—64no—
203C2×C3×Dic3—68no—
204C2×C3×Dic3—616no—
205C2×C3×Dic3—64no—
206C2×C3×Dic3—68no—
207C2×C3×Dic3—64no—
208C2×C3×Dic3—62no—
209C2×C3×Dic3—64no—
210C2×C3×Dic3—64no—
211C2×C3×Dic3—68no—
212C2×C3×Dic3—64no—
213C2×C3×Dic3—62no—
214C2×C3×Dic3—124no—
215C2×#123—271noSylow
216C2×C33—24no—
217C2×C33#12314yescomplemented
218C2×C33C22×D411yescharacteristic
219C2×C33C2×#4112yes—
220C3×C3⋊D3—42no—
221C24×C3—21no—
222C24×C3—21no—
223C24×C3—21no—
224C24×C3—41no—
225C24×C3—91no—
226C24×C3D3211yescharacteristic
227C24×C3D3211yescharacteristic
228C24×C3D3211yescharacteristic
229C23×D3—31no—
230C23×D3—31no—
231C23×D3—61no—
232C2×C3×D4—94no—
233C2×C3×D4—184no—
234C22×C3×C4—91no—
235C22×C3×C4—92no—
236C22×C3×C4—92no—
237C22×C3×C4—94no—
238C22×C3×C4—184no—
239C2×#67—34no—
240C2×#67—34no—
241C2×#67—64no—
242C2×#67—68no—
243C2×#67—64no—
244C2×#67—64no—
245C22×Dic3—34no—
246C22×Dic3—31no—
247C22×Dic3—32no—
248C22×Dic3—32no—
249C22×Dic3—32no—
250C22×Dic3—31no—
251C22×Dic3—32no—
252C22×Dic3—34no—
253C22×Dic3—62no—
254C22×Dic3—64no—
255C22×Dic3—61no—
256C22×Dic3—62no—
257C22×Dic3—68no—
258C22×Dic3—64no—
259C22×Dic3—64no—
260C3×C22⋊C4—94no—
261#217—34no—
262#217—34no—
263#217—64no—
264#217—98no—
265(C2×D3)⋊C4—98no—
266(C2×D3)⋊C4—98no—
267C4⋊Dic3—98no—
268Dic3⋊C4—98no—
269Dic3⋊C4—98no—
270C22×C32—28no—
271C22×C32—24no—
272C22×C32—28no—
273C22×C32—216no—
274C22×C32—28no—
275C22×C32—28no—
276C22×C32—24no—
277C22×C32—24no—
278C22×C32—21no—
279C22×C32—21no—
280C22×C32—21no—
281C22×C32—24no—
282C22×C32—24no—
283C22×C32—24no—
284C22×C32—22no—
285C22×C32—24no—
286C22×C32—24no—
287C22×C32—22no—
288C22×C32—22no—
289C22×C32—24no—
290C22×C32—42no—
291C22×C32—44no—
292C22×C32—44no—
293C22×C32—41no—
294C22×C32—44no—
295C22×C32—44no—
296C22×C32—48no—
297C22×C32—48no—
298C22×C32—416no—
299C22×C32—48no—
300C22×C32C2×D1214yes—
301C22×C32C2×D1212yes—
302C22×C32#23514yes—
303C22×C32#23714yes—
304C22×C32#23714yes—
305C22×C32C2×#6714yes—
306C22×C32C2×#6714yes—
307C22×C32C2×#6712yes—
308C22×C32C2×#6712yes—
309C22×C32C23×D311yescharacteristic
310C22×C32C23×D311yescharacteristic
311C22×C32C23×D311yescharacteristic
312C2×C3⋊D3—22no—
313C2×C3⋊D3—24no—
314C2×C3⋊D3—44no—
315C2×C3×D3—64no—
316C2×C3×D3—62no—
317C2×C3×D3—64no—
318C2×C3×D3—64no—
319C2×C3×D3—62no—
320C2×C3×D3—124no—
321C2×C3×D3—128no—
322C2×C3×D3—124no—
323C2×C3×D3—124no—
324C3⋊Dic3—24no—
325C3⋊Dic3—22no—
326C3⋊Dic3—24no—
327C3×Dic3—68no—
328C3×Dic3—64no—
329C3×Dic3—64no—
330C3×Dic3—62no—
331C3×Dic3—64no—
332C3×Dic3—64no—
333C3×Dic3—64no—
334C3×Dic3—62no—
335C3×Dic3—124no—
336C22×D4—91no—
337C23×C4—91no—
338#123—278no—
339C2×C4⋊C4—272no—
340C2×C22⋊C4—91no—
341C2×C22⋊C4—272no—
342C33C2×#12311yescharacteristic, complemented, Sylow
343C23×C3—28no—
344C23×C3—24no—
345C23×C3—24no—
346C23×C3—24no—
347C23×C3—24no—
348C23×C3—24no—
349C23×C3—24no—
350C23×C3—22no—
351C23×C3—22no—
352C23×C3—22no—
353C23×C3—21no—
354C23×C3—21no—
355C23×C3—21no—
356C23×C3—44no—
357C23×C3—44no—
358C23×C3—48no—
359C23×C3—44no—
360C23×C3—42no—
361C23×C3—41no—
362C23×C3—92no—
363C23×C3—184no—
364C23×C3—182no—
365C23×C3#62514yes—
366C23×C3#62614yes—
367C23×C3#62614yes—
368C23×C3C2×D3212yes—
369C23×C3C2×D3212yes—
370C23×C3C2×D3212yes—
371C23×C3C2×D3211yescharacteristic
372C23×C3C2×D3211yescharacteristic
373C23×C3C2×D3211yescharacteristic
374C22×D3—32no—
375C22×D3—32no—
376C22×D3—62no—
377C22×D3—64no—
378C22×D3—64no—
379C22×D3—64no—
380C22×D3—62no—
381C22×D3—62no—
382C22×D3—124no—
383C3×D4—188no—
384C2×C3×C4—92no—
385C2×C3×C4—94no—
386C2×C3×C4—94no—
387C2×C3×C4—184no—
388C2×C3×C4—188no—
389C2×C3×C4—184no—
390C2×C3×C4—182no—
391#67—616no—
392#67—68no—
393#67—68no—
394C2×Dic3—32no—
395C2×Dic3—34no—
396C2×Dic3—34no—
397C2×Dic3—34no—
398C2×Dic3—32no—
399C2×Dic3—34no—
400C2×Dic3—64no—
401C2×Dic3—62no—
402C2×Dic3—64no—
403C2×Dic3—68no—
404C2×Dic3—616no—
405C2×Dic3—64no—
406C2×Dic3—64no—
407C2×Dic3—64no—
408C2×Dic3—68no—
409C2×Dic3—64no—
410C2×Dic3—62no—
411C2×Dic3—68no—
412C2×Dic3—64no—
413C2×Dic3—62no—
414C2×Dic3—124no—
415C2×C32—24no—
416C2×C32—28no—
417C2×C32—24no—
418C2×C32—24no—
419C2×C32—22no—
420C2×C32—22no—
421C2×C32—22no—
422C2×C32—21no—
423C2×C32—21no—
424C2×C32—21no—
425C2×C32—24no—
426C2×C32—24no—
427C2×C32—24no—
428C2×C32—44no—
429C2×C32—41no—
430C2×C32—42no—
431C2×C32—44no—
432C2×C32—44no—
433C2×C32—48no—
434C2×C32#88614yescomplemented
435C2×C32#92914yescomplemented
436C2×C32#93514yescomplemented
437C2×C32C22×D1211yescharacteristic
438C2×C32C2×#23512yes—
439C2×C32C2×#23712yes—
440C2×C32C2×#23712yes—
441C2×C32C22×#6711yescharacteristic
442C2×C32C22×#6711yescharacteristic
443C3⋊D3—42no—
444C3×D3—124no—
445C3×D3—122no—
446C3×D3—122no—
447C24—91no—
448C24#111311yescharacteristic
449C2×D4—94no—
450C2×D4—184no—
451C22×C4—94no—
452C22×C4—92no—
453C22×C4—92no—
454C22×C4—91no—
455C22×C4—184no—
456C4⋊C4—278no—
457C22⋊C4—94no—
458C22⋊C4—278no—
459C22×C3—28no—
460C22×C3—216no—
461C22×C3—24no—
462C22×C3—24no—
463C22×C3—24no—
464C22×C3—28no—
465C22×C3—28no—
466C22×C3—28no—
467C22×C3—21no—
468C22×C3—21no—
469C22×C3—21no—
470C22×C3—24no—
471C22×C3—24no—
472C22×C3—24no—
473C22×C3—24no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—22no—
477C22×C3—24no—
478C22×C3—24no—
479C22×C3—44no—
480C22×C3—44no—
481C22×C3—48no—
482C22×C3—48no—
483C22×C3—48no—
484C22×C3—416no—
485C22×C3—41no—
486C22×C3—44no—
487C22×C3—44no—
488C22×C3—42no—
489C22×C3—184no—
490C22×C3—182no—
491C22×C3—364no—
492C22×C3#380414yes—
493C22×C3#380414yes—
494C22×C3#380514yes—
495C22×C3C2×#62514yes—
496C22×C3C2×#62512yes—
497C22×C3C2×#62614yes—
498C22×C3C2×#62614yes—
499C22×C3C2×#62612yes—
500C22×C3C2×#62612yes—
501C22×C3C22×D3211yescharacteristic
502C22×C3C22×D3211yescharacteristic
503C22×C3C22×D3211yescharacteristic
504C2×D3—64no—
505C2×D3—64no—
506C2×D3—64no—
507C2×D3—62no—
508C2×D3—62no—
509C2×D3—124no—
510C2×D3—128no—
511C2×D3—124no—
512C2×D3—124no—
513C3×C4—184no—
514C3×C4—184no—
515C3×C4—182no—
516Dic3—68no—
517Dic3—64no—
518Dic3—64no—
519Dic3—64no—
520Dic3—64no—
521Dic3—62no—
522Dic3—64no—
523Dic3—62no—
524Dic3—124no—
525C32—21no—
526C32—21no—
527C32—21no—
528C32—41no—
529C32C2×#88611yescharacteristic, complemented
530C32C2×#92911yescharacteristic, complemented
531C32C2×#93511yescharacteristic, complemented
532C23—24no—
533C23—92no—
534C23—184no—
535C23—182no—
536C23#629514yes—
537C23C2×#111312yes—
538C23C2×#111311yescharacteristic
539D4—188no—
540C2×C4—94no—
541C2×C4—94no—
542C2×C4—92no—
543C2×C4—184no—
544C2×C4—188no—
545C2×C4—184no—
546C2×C4—182no—
547C2×C3—28no—
548C2×C3—24no—
549C2×C3—24no—
550C2×C3—24no—
551C2×C3—22no—
552C2×C3—22no—
553C2×C3—22no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—24no—
558C2×C3—24no—
559C2×C3—24no—
560C2×C3—44no—
561C2×C3—44no—
562C2×C3—48no—
563C2×C3—42no—
564C2×C3—41no—
565C2×C3—44no—
566C2×C3—362no—
567C2×C3#6396114yescomplemented
568C2×C3#6396214yescomplemented
569C2×C3#6396614yescomplemented
570C2×C3C2×#380412yes—
571C2×C3C2×#380412yes—
572C2×C3C2×#380512yes—
573C2×C3C22×#62511yescharacteristic
574C2×C3C22×#62611yescharacteristic
575C2×C3C22×#62611yescharacteristic
576D3—124no—
577D3—122no—
578D3—122no—
579C22—24no—
580C22—28no—
581C22—184no—
582C22—182no—
583C22—364no—
584C22#8904614yes—
585C22C2×#629514yes—
586C22C2×#629512yes—
587C22C22×#111311yescharacteristic
588C4—184no—
589C4—184no—
590C4—182no—
591C3—21no—
592C3—21no—
593C3—21no—
594C3—41no—
595C3C2×#6396111yescharacteristic, complemented
596C3C2×#6396211yescharacteristic, complemented
597C3C2×#6396611yescharacteristic, complemented
598C2—24no—
599C2—362no—
600C2#1172372014yesdirect product factor
601C2C2×#8904612yes—
602C2C22×#629511yescharacteristic
subgroupG/H
centerC23C2×#1113
commutator, cosocleC22×C33C24
FrattiniC22C22×#1113
FittingC24×C33C22
socleC23×C33C23
prime radicalC22×C3×#638C2
hypercenterC24#1113
hypocenterC33C2×#123
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C23×C3×C3⋊Dic3C2semidirect
3. C2×C3×#3757C2semidirect
4. C2×#88815C2semidirect
5. #11723720C2direct (N × H)
6. C2×#88814C2semidirect
18. C22×C3×C3⋊Dic3C22semidirect
19. C3×#3757C22semidirect
20. #88815C22semidirect
21. #88814C22semidirect
23. C23×C3⋊Dic3D3semidirect
30. C2×#3757D3semidirect
78. C22×C3⋊Dic3C2×D3semidirect
100. #3757C2×D3semidirect
217. C2×C33#123semidirect
342. C33C2×#123semidirect
434. C2×C32#886semidirect
435. C2×C32#929semidirect
436. C2×C32#935semidirect
529. C32C2×#886semidirect
530. C32C2×#929semidirect
531. C32C2×#935semidirect
567. C2×C3#63961semidirect
568. C2×C3#63962semidirect
569. C2×C3#63966semidirect
595. C3C2×#63961semidirect
596. C3C2×#63962semidirect
597. C3C2×#63966semidirect
600. C2#11723720direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6382
C23×C3×C3⋊Dic32
C2×C3×#37572
C2×#888152
#117237202
C2×#888142
C2×#639663
C2×#639623
C2×#639613