Finite Group Databasebeta

← C2×#11722948 #13172303

Subgroups — C2×#11722948 #13172303

Order 1728.

3490 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×D32C211yesmaximal, characteristic, complemented
2C22×#6292C211yesmaximal, characteristic, complemented
3C22×#6291C212yesmaximal, complemented
4C2×#88814C211yesmaximal, characteristic, complemented
5C2×C3×(C2×D3)⋊Dic3C212yesmaximal, complemented
6#11722948C218yesmaximal, direct product factor
7C2×#63977—32nomaximal
8C2×#63923—31nomaximal
9C23×C3⋊(C3⋊D3)C2211yescharacteristic, complemented
10C23×C3×C3⋊D3C2211yescharacteristic
11C23×C32×D3C2212yes—
12C22×C3×D32—24no—
13C22×C3×D32—28no—
14C22×C3×D32C2214yescomplemented
15C22×C3×C3⋊Dic3C2212yescomplemented
16C22×C32×Dic3C2211yescharacteristic, complemented
17C2×#6292—24no—
18C2×#6292C2214yescomplemented
19C2×#6291—28no—
20C2×#6291C2218yescomplemented
21#88814C2214yescomplemented
22C3×(C2×D3)⋊Dic3C2218yescomplemented
23C24×C3×D3—32no—
24C23×D32D311yescharacteristic, complemented
25C22×#638—32no—
26C22×C4⋊(C3⋊D3)—31no—
27C22×#626—32no—
28C22×#626—32no—
29C22×#626—32no—
30C22×#626—62no—
31C22×#626—61no—
32C2×C3×#217—32no—
33C2×C3×(C2×D3)⋊C4—32no—
34#63977—316no—
35C2×#3722—31no—
36C2×(C2×D3)⋊Dic3—32no—
37C2×(C2×D3)⋊Dic3D312yescomplemented
38#63923—38no—
39C23×C33C2311yescharacteristic
40C22×C3⋊(C3⋊D3)—24no—
41C22×C3⋊(C3⋊D3)—22no—
42C22×C3⋊(C3⋊D3)C2312yescomplemented
43C22×C3×C3⋊D3—24no—
44C22×C3×C3⋊D3D414yescomplemented
45C22×C3×C3⋊D3C2312yes—
46C22×C32×D3—24no—
47C22×C32×D3—24no—
48C22×C32×D3D418yescomplemented
49C22×C32×D3C2314yes—
50C2×C3×D32—232no—
51C2×C3×D32—28no—
52C2×C3×D32—216no—
53C2×C3×C3⋊Dic3—24no—
54C2×C3×C3⋊Dic3C2314yescomplemented
55C2×C32×Dic3—22no—
56C2×C32×Dic3C2312yescomplemented
57#6292—28no—
58#6291—216no—
59C2×#953—91no—
60C2×#882—92no—
61C24×C32—32no—
62C23×C3⋊D3—31no—
63C23×C3⋊D3—32no—
64C23×C3⋊D3—62no—
65C23×C3⋊D3—61no—
66C23×C3⋊D3—121no—
67C23×C3⋊D3C2×D311yescharacteristic
68C23×C3×D3—22no—
69C23×C3×D3—32no—
70C23×C3×D3—38no—
71C23×C3×D3—68no—
72C23×C3×D3—68no—
73C23×C3×D3—68no—
74C23×C3×D3—61no—
75C23×C3×D3C2×D312yes—
76C23×C3×D3C2×D312yes—
77C22×D32—24no—
78C22×D32—28no—
79C22×D32C2×D314yescomplemented
80C22×C32×C4—31no—
81C2×#638—38no—
82C2×#638—68no—
83C22×C3⋊Dic3C2×D312yescomplemented
84C2×C4⋊(C3⋊D3)—34no—
85C2×C4⋊(C3⋊D3)—64no—
86C22×C3×Dic3—21no—
87C22×C3×Dic3—32no—
88C22×C3×Dic3—32no—
89C22×C3×Dic3—62no—
90C22×C3×Dic3C2×D312yescomplemented
91C2×#626—38no—
92C2×#626—38no—
93C2×#626—38no—
94C2×#626—68no—
95C2×#626—64no—
96C2×#626—68no—
97C2×#626—68no—
98C2×#626—616no—
99C2×#626—68no—
100C2×#626—68no—
101C3×#217—38no—
102C3×(C2×D3)⋊C4—38no—
103#3722—34no—
104(C2×D3)⋊Dic3—38no—
105(C2×D3)⋊Dic3C2×D318yescomplemented
106C22×C33C2×D414yes—
107C22×C33C2×D412yes—
108C22×C33C2411yescharacteristic
109C2×C3⋊(C3⋊D3)—24no—
110C2×C3⋊(C3⋊D3)—22no—
111C2×C3⋊(C3⋊D3)—44no—
112C2×C3×C3⋊D3—28no—
113C2×C3×C3⋊D3—24no—
114C2×C3×C3⋊D3C2×D414yescomplemented
115C2×C32×D3—216no—
116C2×C32×D3—24no—
117C2×C32×D3—24no—
118C2×C32×D3C2×D418yescomplemented
119C3×D32—232no—
120C3×C3⋊Dic3—24no—
121C32×Dic3—22no—
122C25×C3—91no—
123C24×D3—32no—
124C22×#67—92no—
125C22×#67—92no—
126C22×#67—91no—
127C22×#67—182no—
128C22×D12—92no—
129C22×D12—92no—
130C22×D12—181no—
131C2×C3×C22⋊C4—92no—
132#953—98no—
133C2×#217—32no—
134C2×#217—91no—
135C2×(C2×D3)⋊C4—32no—
136C2×(C2×D3)⋊C4—92no—
137#882—916no—
138C23×C32—22no—
139C23×C32—21no—
140C23×C32—38no—
141C23×C32—34no—
142C23×C32—41no—
143C23×C32—64no—
144C23×C32—64no—
145C23×C32C22×D312yes—
146C23×C32C22×D311yescharacteristic
147C22×C3⋊D3—24no—
148C22×C3⋊D3—32no—
149C22×C3⋊D3—34no—
150C22×C3⋊D3—64no—
151C22×C3⋊D3—62no—
152C22×C3⋊D3—64no—
153C22×C3⋊D3—62no—
154C22×C3⋊D3—64no—
155C22×C3⋊D3—68no—
156C22×C3⋊D3—64no—
157C22×C3⋊D3—64no—
158C22×C3⋊D3—64no—
159C22×C3⋊D3—122no—
160C22×C3⋊D3—124no—
161C22×C3⋊D3—124no—
162C22×C3⋊D3—124no—
163C22×C3⋊D3—124no—
164C22×C3⋊D3—128no—
165C22×C3⋊D3#6714yescomplemented
166C22×C3⋊D3C22×D312yes—
167C22×C3×D3—24no—
168C22×C3×D3—24no—
169C22×C3×D3—24no—
170C22×C3×D3—24no—
171C22×C3×D3—28no—
172C22×C3×D3—24no—
173C22×C3×D3—38no—
174C22×C3×D3—34no—
175C22×C3×D3—44no—
176C22×C3×D3—44no—
177C22×C3×D3—64no—
178C22×C3×D3—62no—
179C22×C3×D3—616no—
180C22×C3×D3—616no—
181C22×C3×D3—664no—
182C22×C3×D3—64no—
183C22×C3×D3—64no—
184C22×C3×D3—616no—
185C22×C3×D3—124no—
186C22×C3×D3D1218yescomplemented
187C22×C3×D3#6718yescomplemented
188C22×C3×D3C22×D314yes—
189C22×C3×D3C22×D314yes—
190C2×D32—232no—
191C2×D32—28no—
192C2×D32—216no—
193C2×C32×C4—32no—
194C2×C32×C4—62no—
195#638—616no—
196C2×C3⋊Dic3—24no—
197C2×C3⋊Dic3C22×D314yescomplemented
198C4⋊(C3⋊D3)—68no—
199C2×C3×Dic3—24no—
200C2×C3×Dic3—24no—
201C2×C3×Dic3—22no—
202C2×C3×Dic3—34no—
203C2×C3×Dic3—34no—
204C2×C3×Dic3—64no—
205C2×C3×Dic3—68no—
206C2×C3×Dic3—64no—
207C2×C3×Dic3—64no—
208C2×C3×Dic3C22×D314yescomplemented
209#626—616no—
210#626—616no—
211#626—632no—
212#626—616no—
213#626—616no—
214C2×#120—271noSylow
215C2×C33#12014yescomplemented
216C2×C33C22×D412yes—
217C2×C33C22×D411yescharacteristic
218C3⋊(C3⋊D3)—42no—
219C3×C3⋊D3—24no—
220C32×D3—28no—
221C24×C3—32no—
222C24×C3—32no—
223C24×C3—62no—
224C24×C3—94no—
225C24×C3—91no—
226C24×C3—188no—
227C24×C3—184no—
228C23×D3—32no—
229C23×D3—38no—
230C23×D3—61no—
231C23×D3—68no—
232C23×D3—68no—
233C23×D3—68no—
234C23×D3—92no—
235C23×D3—91no—
236C23×D3—181no—
237C23×D3—182no—
238C23×D3—361no—
239C23×D3D3212yes—
240C22×C3×C4—32no—
241C22×C3×C4—61no—
242C22×C3×C4—92no—
243C2×#67—98no—
244C2×#67—98no—
245C2×#67—94no—
246C2×#67—188no—
247C2×#67—184no—
248C2×#67—188no—
249C2×#67—188no—
250C2×#67—1816no—
251C22×Dic3—32no—
252C22×Dic3—32no—
253C22×Dic3—62no—
254C22×Dic3D3211yescharacteristic, complemented
255C2×D12—98no—
256C2×D12—98no—
257C2×D12—184no—
258C2×D12—188no—
259C2×D12—188no—
260C2×D12—188no—
261C3×C22⋊C4—98no—
262#217—38no—
263#217—94no—
264(C2×D3)⋊C4—38no—
265(C2×D3)⋊C4—98no—
266C22×C32—21no—
267C22×C32—22no—
268C22×C32—24no—
269C22×C32—24no—
270C22×C32—24no—
271C22×C32—24no—
272C22×C32—22no—
273C22×C32—38no—
274C22×C32—44no—
275C22×C32—42no—
276C22×C32—41no—
277C22×C32—616no—
278C22×C32—64no—
279C22×C32—64no—
280C22×C32C2×D1214yes—
281C22×C32D3×D414yes—
282C22×C32D3×D414yes—
283C22×C32C2×#6714yes—
284C22×C32C2×#6712yes—
285C22×C32C23×D312yes—
286C22×C32C23×D311yescharacteristic
287C2×C3⋊D3—28no—
288C2×C3⋊D3—24no—
289C2×C3⋊D3—64no—
290C2×C3⋊D3—62no—
291C2×C3⋊D3—64no—
292C2×C3⋊D3—68no—
293C2×C3⋊D3—64no—
294C2×C3⋊D3—64no—
295C2×C3⋊D3—64no—
296C2×C3⋊D3—124no—
297C2×C3⋊D3—124no—
298C2×C3⋊D3—124no—
299C2×C3⋊D3—124no—
300C2×C3⋊D3—128no—
301C2×C3⋊D3—128no—
302C2×C3⋊D3—1216no—
303C2×C3⋊D3—1216no—
304C2×C3⋊D3—128no—
305C2×C3⋊D3—124no—
306C2×C3⋊D3C2×#6714yescomplemented
307C2×C3×D3—216no—
308C2×C3×D3—216no—
309C2×C3×D3—24no—
310C2×C3×D3—24no—
311C2×C3×D3—24no—
312C2×C3×D3—24no—
313C2×C3×D3—28no—
314C2×C3×D3—38no—
315C2×C3×D3—44no—
316C2×C3×D3—44no—
317C2×C3×D3—416no—
318C2×C3×D3—64no—
319C2×C3×D3—664no—
320C2×C3×D3—616no—
321C2×C3×D3—64no—
322C2×C3×D3—64no—
323C2×C3×D3—124no—
324C2×C3×D3—128no—
325C2×C3×D3C2×D1218yescomplemented
326C2×C3×D3C2×#6718yescomplemented
327D32—232no—
328C32×C4—62no—
329C3⋊Dic3—24no—
330C3×Dic3—24no—
331C3×Dic3—24no—
332C3×Dic3—68no—
333C3×Dic3—64no—
334C3×Dic3—64no—
335C25—91no—
336C22×D4—271no—
337C22×D4—272no—
338#120—278no—
339C2×C22⋊C4—92no—
340C2×C22⋊C4—271no—
341C33C2×#12011yescharacteristic, complemented, Sylow
342C23×C3—22no—
343C23×C3—21no—
344C23×C3—34no—
345C23×C3—34no—
346C23×C3—38no—
347C23×C3—38no—
348C23×C3—41no—
349C23×C3—64no—
350C23×C3—68no—
351C23×C3—64no—
352C23×C3—64no—
353C23×C3—64no—
354C23×C3—64no—
355C23×C3—92no—
356C23×C3—94no—
357C23×C3—124no—
358C23×C3—124no—
359C23×C3—1816no—
360C23×C3—188no—
361C23×C3—1832no—
362C23×C3—184no—
363C23×C3C2×D3211yescharacteristic
364C23×C3C2×D3212yes—
365C22×D3—24no—
366C22×D3—24no—
367C22×D3—34no—
368C22×D3—38no—
369C22×D3—64no—
370C22×D3—62no—
371C22×D3—616no—
372C22×D3—616no—
373C22×D3—664no—
374C22×D3—64no—
375C22×D3—64no—
376C22×D3—616no—
377C22×D3—94no—
378C22×D3—92no—
379C22×D3—124no—
380C22×D3—182no—
381C22×D3—184no—
382C22×D3—184no—
383C22×D3—188no—
384C22×D3—184no—
385C22×D3—184no—
386C22×D3—184no—
387C22×D3—184no—
388C22×D3—182no—
389C22×D3—362no—
390C22×D3—364no—
391C22×D3—364no—
392C22×D3—364no—
393C22×D3—368no—
394C22×D3—364no—
395C22×D3#62618yescomplemented
396C22×D3C2×D3214yes—
397C2×C3×C4—34no—
398C2×C3×C4—62no—
399C2×C3×C4—64no—
400C2×C3×C4—64no—
401C2×C3×C4—94no—
402C2×C3×C4—184no—
403#67—188no—
404#67—1816no—
405#67—1816no—
406#67—1832no—
407C2×Dic3—22no—
408C2×Dic3—34no—
409C2×Dic3—34no—
410C2×Dic3—64no—
411C2×Dic3—68no—
412C2×Dic3—64no—
413C2×Dic3—64no—
414C2×Dic3C2×D3212yescomplemented
415D12—1816no—
416D12—1816no—
417D12—1816no—
418C2×C32—22no—
419C2×C32—22no—
420C2×C32—22no—
421C2×C32—22no—
422C2×C32—21no—
423C2×C32—24no—
424C2×C32—28no—
425C2×C32—44no—
426C2×C32—42no—
427C2×C32—41no—
428C2×C32—68no—
429C2×C32#88218yescomplemented
430C2×C32#95314yescomplemented
431C2×C32C22×D1212yes—
432C2×C32C2×D3×D412yes—
433C2×C32C2×D3×D412yes—
434C2×C32C22×#6712yes—
435C2×C32C22×#6711yescharacteristic
436C3⋊D3—24no—
437C3⋊D3—128no—
438C3⋊D3—124no—
439C3⋊D3—128no—
440C3⋊D3—124no—
441C3⋊D3—122no—
442C3×D3—28no—
443C3×D3—28no—
444C3×D3—48no—
445C3×D3—68no—
446C3×D3—124no—
447C24—32no—
448C24—94no—
449C24—91no—
450C24—188no—
451C24—184no—
452C24—271no—
453C2×D4—274no—
454C2×D4—278no—
455C2×D4—544no—
456C2×D4—548no—
457C22×C4—31no—
458C22×C4—92no—
459C22⋊C4—98no—
460C22⋊C4—274no—
461C22×C3—22no—
462C22×C3—21no—
463C22×C3—24no—
464C22×C3—24no—
465C22×C3—24no—
466C22×C3—22no—
467C22×C3—24no—
468C22×C3—38no—
469C22×C3—38no—
470C22×C3—41no—
471C22×C3—44no—
472C22×C3—42no—
473C22×C3—68no—
474C22×C3—64no—
475C22×C3—64no—
476C22×C3—64no—
477C22×C3—64no—
478C22×C3—616no—
479C22×C3—616no—
480C22×C3—94no—
481C22×C3—124no—
482C22×C3—124no—
483C22×C3—1216no—
484C22×C3—1832no—
485C22×C3—184no—
486C22×C3—188no—
487C22×C3D3×D1214yes—
488C22×C3C2×#62614yes—
489C22×C3D3×#6714yes—
490C22×C3D3×#6714yes—
491C22×C3#381112yes—
492C22×C3C22×D3211yescharacteristic
493C22×C3C22×D3212yes—
494C2×D3—24no—
495C2×D3—24no—
496C2×D3—216no—
497C2×D3—38no—
498C2×D3—64no—
499C2×D3—664no—
500C2×D3—64no—
501C2×D3—64no—
502C2×D3—616no—
503C2×D3—124no—
504C2×D3—128no—
505C2×D3—184no—
506C2×D3—188no—
507C2×D3—184no—
508C2×D3—184no—
509C2×D3—184no—
510C2×D3—184no—
511C2×D3—182no—
512C2×D3—364no—
513C2×D3—364no—
514C2×D3—364no—
515C2×D3—368no—
516C2×D3—364no—
517C2×D3—368no—
518C2×D3—3616no—
519C2×D3—3616no—
520C2×D3—364no—
521C2×D3—368no—
522C2×D3C2×#62618yescomplemented
523C3×C4—64no—
524C3×C4—64no—
525C3×C4—184no—
526Dic3—22no—
527Dic3—68no—
528Dic3—64no—
529Dic3—64no—
530C32—22no—
531C32—21no—
532C32—41no—
533C32C2×#88212yescomplemented
534C32C2×#95311yescharacteristic, complemented
535C23—34no—
536C23—38no—
537C23—64no—
538C23—64no—
539C23—92no—
540C23—94no—
541C23—1816no—
542C23—188no—
543C23—1832no—
544C23—184no—
545C23—272no—
546C23—544no—
547C23—542no—
548C23D3311yescharacteristic
549D4—548no—
550D4—5416no—
551C2×C4—32no—
552C2×C4—62no—
553C2×C4—94no—
554C2×C4—184no—
555C2×C3—22no—
556C2×C3—22no—
557C2×C3—22no—
558C2×C3—21no—
559C2×C3—22no—
560C2×C3—24no—
561C2×C3—28no—
562C2×C3—42no—
563C2×C3—41no—
564C2×C3—44no—
565C2×C3—68no—
566C2×C3—68no—
567C2×C3—128no—
568C2×C3—184no—
569C2×C3#6392314yescomplemented
570C2×C3#6397718yescomplemented
571C2×C3C2×D3×D1212yes—
572C2×C3C22×#62612yes—
573C2×C3C2×D3×#6712yes—
574C2×C3C2×D3×#6712yes—
575C2×C3C2×#381111yescharacteristic
576D3—28no—
577D3—68no—
578D3—124no—
579D3—364no—
580D3—368no—
581D3—368no—
582D3—364no—
583D3—362no—
584C22—38no—
585C22—64no—
586C22—64no—
587C22—616no—
588C22—94no—
589C22—1832no—
590C22—184no—
591C22—188no—
592C22—544no—
593C22—542no—
594C22—1084no—
595C22D3×#62614yes—
596C22#8895212yes—
597C22C2×D3311yescharacteristic
598C4—62no—
599C4—184no—
600C3—21no—
601C3—22no—
602C3—41no—
603C3C2×#6392311yescharacteristic, complemented
604C3C2×#6397712yescomplemented
605C2—68no—
606C2—184no—
607C2—1082no—
608C2#1172294814yesdirect product factor
609C2C2×D3×#62612yes—
610C2C2×#8895211yescharacteristic
subgroupG/H
center, hypercenterC23D33
commutator, cosocleC22×C33C24
FrattiniC22C2×D33
Fitting, socleC23×C33C23
hypocenterC33C2×#120
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C23×C3×D32C2semidirect
2. C22×#6292C2semidirect
3. C22×#6291C2semidirect
4. C2×#88814C2semidirect
5. C2×C3×(C2×D3)⋊Dic3C2semidirect
6. #11722948C2direct (N × H)
9. C23×C3⋊(C3⋊D3)C22semidirect
14. C22×C3×D32C22semidirect
15. C22×C3×C3⋊Dic3C22semidirect
16. C22×C32×Dic3C22semidirect
18. C2×#6292C22semidirect
20. C2×#6291C22semidirect
21. #88814C22semidirect
22. C3×(C2×D3)⋊Dic3C22semidirect
24. C23×D32D3semidirect
37. C2×(C2×D3)⋊Dic3D3semidirect
42. C22×C3⋊(C3⋊D3)C23semidirect
44. C22×C3×C3⋊D3D4semidirect
48. C22×C32×D3D4semidirect
54. C2×C3×C3⋊Dic3C23semidirect
56. C2×C32×Dic3C23semidirect
79. C22×D32C2×D3semidirect
83. C22×C3⋊Dic3C2×D3semidirect
90. C22×C3×Dic3C2×D3semidirect
105. (C2×D3)⋊Dic3C2×D3semidirect
114. C2×C3×C3⋊D3C2×D4semidirect
118. C2×C32×D3C2×D4semidirect
165. C22×C3⋊D3#67semidirect
186. C22×C3×D3D12semidirect
187. C22×C3×D3#67semidirect
197. C2×C3⋊Dic3C22×D3semidirect
208. C2×C3×Dic3C22×D3semidirect
215. C2×C33#120semidirect
254. C22×Dic3D32semidirect
306. C2×C3⋊D3C2×#67semidirect
325. C2×C3×D3C2×D12semidirect
326. C2×C3×D3C2×#67semidirect
341. C33C2×#120semidirect
395. C22×D3#626semidirect
414. C2×Dic3C2×D32semidirect
429. C2×C32#882semidirect
430. C2×C32#953semidirect
522. C2×D3C2×#626semidirect
533. C32C2×#882semidirect
534. C32C2×#953semidirect
569. C2×C3#63923semidirect
570. C2×C3#63977semidirect
603. C3C2×#63923semidirect
604. C3C2×#63977semidirect
608. C2#11722948direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×D322
C22×#62922
C22×#62912
C2×#888142
C2×C3×(C2×D3)⋊Dic32
#117229482
C2×#639773
C2×#639233