Finite Group Databasebeta

← C2×#11721312 #13152091

Subgroups — C2×#11721312 #13152091

Order 1728.

1727 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6235C211yesmaximal, characteristic, complemented
2C22×#638⋊C3C211yesmaximal, characteristic, complemented
3C22×#6200C211yesmaximal, characteristic, complemented
4C22×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic, complemented
5C2×#3757⋊C3C211yesmaximal, characteristic, complemented
6#11721312C218yesmaximal, direct product factor
7C2×#88451C211yesmaximal, characteristic, complemented
8C2×#88449C211yesmaximal, characteristic, complemented
9C2×#63970—31nomaximal
10C2×#63958—31nomaximal
11C24×H(3,𝔽3)C2211yescharacteristic
12C23×#272C2211yescharacteristic
13C23×(C3⋊D3)⋊C3C2211yescharacteristic
14C2×#6235—24no—
15C2×#6235C2214yescomplemented
16C22×#1084C2211yescharacteristic
17C2×#638⋊C3—24no—
18C2×#638⋊C3C2214yescomplemented
19C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
21C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
22C2×#6200—24no—
23C2×#6200C2214yescomplemented
24C2×((C3⋊D3)⋊C3)⋊C4—24no—
25C2×((C3⋊D3)⋊C3)⋊C4C2214yescomplemented
26#3757⋊C3C2214yescomplemented
27#88451C2214yescomplemented
28#88449C2214yescomplemented
29C22×#638D311yescharacteristic, complemented
30C22×C3×#67—31no—
31C22×C3×#67—31no—
32C22×C3×#67—31no—
33C22×C3×#67—61no—
34C22×#626—31no—
35C22×#625—31no—
36C22×#624—31no—
37C22×D3×Dic3—31no—
38C2×#3757D311yescharacteristic, complemented
39C2×C3×#217—31no—
40#63970—38no—
41C2×Dic3⋊Dic3—31no—
42C2×Dic3⋊Dic3—31no—
43C2×(C2×D3)⋊Dic3—31no—
44C2×(C2×D3)⋊Dic3—31no—
45#63958—38no—
46C23×H(3,𝔽3)—22no—
47C23×H(3,𝔽3)—22no—
48C23×H(3,𝔽3)—22no—
49C23×H(3,𝔽3)C2312yes—
50C23×H(3,𝔽3)C2311yescharacteristic
51C22×#272—22no—
52C22×#272—22no—
53C22×#272—22no—
54C22×#272C2312yes—
55C22×(C3⋊D3)⋊C3—22no—
56C22×(C3⋊D3)⋊C3—22no—
57C22×(C3⋊D3)⋊C3D414yescomplemented
58C22×(C3⋊D3)⋊C3C2312yes—
59#6235—28no—
60C2×#1084D414yescomplemented
61C2×#1084C2312yes—
62#638⋊C3—28no—
63C2×(C3⋊Dic3)⋊C3—22no—
64C2×(C3⋊Dic3)⋊C3—22no—
65C2×(C3⋊Dic3)⋊C3—22no—
66C2×(C3⋊Dic3)⋊C3C2312yes—
67C2×(C3⋊Dic3)⋊C3C2312yes—
68C2×(C3⋊Dic3)⋊C3C2312yes—
69#6200—28no—
70((C3⋊D3)⋊C3)⋊C4—28no—
71C2×#939—91no—
72C2×#938—91no—
73C2×#884—91no—
74C24×C32—21no—
75C24×C32C2×D311yescharacteristic
76C24×C32C2×D311yescharacteristic
77C23×C3⋊D3C2×D311yescharacteristic
78C23×C3×D3—31no—
79C23×C3×D3—31no—
80C23×C3×D3—31no—
81C23×C3×D3—61no—
82C2×#638—24no—
83C2×#638C2×D314yescomplemented
84C22×C3⋊Dic3C2×D311yescharacteristic
85C22×C3⋊Dic3C2×D311yescharacteristic
86C22×C3⋊Dic3C2×D311yescharacteristic
87C2×C3×#67—34no—
88C2×C3×#67—34no—
89C2×C3×#67—34no—
90C2×C3×#67—64no—
91C2×C3×#67—64no—
92C2×C3×#67—64no—
93C2×C3×#67—64no—
94C2×C3×#67—68no—
95C22×C3×Dic3—31no—
96C22×C3×Dic3—31no—
97C22×C3×Dic3—31no—
98C22×C3×Dic3—31no—
99C22×C3×Dic3—31no—
100C22×C3×Dic3—61no—
101C2×#626—34no—
102C2×#626—64no—
103C2×#625—34no—
104C2×#625—64no—
105C2×#624—34no—
106C2×#624—64no—
107C2×D3×Dic3—34no—
108C2×D3×Dic3—64no—
109#3757C2×D314yescomplemented
110C3×#217—34no—
111Dic3⋊Dic3—34no—
112Dic3⋊Dic3—34no—
113(C2×D3)⋊Dic3—34no—
114(C2×D3)⋊Dic3—34no—
115C22×H(3,𝔽3)—22no—
116C22×H(3,𝔽3)—22no—
117C22×H(3,𝔽3)—22no—
118C22×H(3,𝔽3)—44no—
119C22×H(3,𝔽3)C2×D412yes—
120C22×H(3,𝔽3)C2×D412yes—
121C22×H(3,𝔽3)#4112yes—
122C22×H(3,𝔽3)C2411yescharacteristic
123C2×#272—22no—
124C2×#272—22no—
125C2×#272—22no—
126C2×#272—44no—
127C2×(C3⋊D3)⋊C3—28no—
128C2×(C3⋊D3)⋊C3—22no—
129C2×(C3⋊D3)⋊C3—22no—
130C2×(C3⋊D3)⋊C3C2×D414yescomplemented
131#1084C2×D414yescomplemented
132(C3⋊Dic3)⋊C3—22no—
133(C3⋊Dic3)⋊C3—22no—
134(C3⋊Dic3)⋊C3—22no—
135C22×C3×D4—91no—
136C22×C3×D4—91no—
137C22×#67—31no—
138C22×#67—31no—
139C22×#67—31no—
140C22×#67—61no—
141C22×#67—91no—
142C22×#67—91no—
143C22×#67—91no—
144C23×Dic3—91no—
145C22×D12—91no—
146C22×C4×D3—91no—
147C22×C4×D3—91no—
148C2×C3×C22⋊C4—91no—
149C2×#217—31no—
150C2×#217—91no—
151C2×#217—91no—
152#939—98no—
153#938—98no—
154C2×(C2×D3)⋊C4—91no—
155C2×(C2×D3)⋊C4—91no—
156C2×C4⋊Dic3—91no—
157C2×Dic3⋊C4—91no—
158C2×Dic3⋊C4—91no—
159#884—98no—
160C23×C32—24no—
161C23×C32—22no—
162C23×C32—22no—
163C23×C32—22no—
164C23×C32—22no—
165C23×C32—22no—
166C23×C32—22no—
167C23×C32—21no—
168C23×C32—22no—
169C23×C32—42no—
170C23×C32—42no—
171C23×C32C22×D312yes—
172C23×C32C22×D312yes—
173C23×C32C22×D311yescharacteristic
174C23×C32C22×D311yescharacteristic
175C22×C3⋊D3—22no—
176C22×C3⋊D3—22no—
177C22×C3⋊D3#6714yescomplemented
178C22×C3⋊D3C22×D312yes—
179C22×C3×D3—34no—
180C22×C3×D3—32no—
181C22×C3×D3—32no—
182C22×C3×D3—32no—
183C22×C3×D3—62no—
184C22×C3×D3—62no—
185C22×C3×D3—62no—
186C22×C3×D3—64no—
187C22×C3×D3—62no—
188C22×C3×D3—62no—
189C22×C3×D3—62no—
190C22×C3×D3—62no—
191C22×C3×D3—62no—
192C22×C3×D3—62no—
193C22×C3×D3—122no—
194C22×C3×D3—122no—
195#638—28no—
196C2×C3⋊Dic3—22no—
197C2×C3⋊Dic3—22no—
198C2×C3⋊Dic3—22no—
199C2×C3⋊Dic3C22×D312yes—
200C2×C3⋊Dic3C22×D312yes—
201C2×C3⋊Dic3C22×D312yes—
202C3×#67—68no—
203C3×#67—68no—
204C3×#67—68no—
205C3×#67—616no—
206C2×C3×Dic3—34no—
207C2×C3×Dic3—34no—
208C2×C3×Dic3—32no—
209C2×C3×Dic3—32no—
210C2×C3×Dic3—32no—
211C2×C3×Dic3—32no—
212C2×C3×Dic3—32no—
213C2×C3×Dic3—64no—
214C2×C3×Dic3—62no—
215C2×C3×Dic3—62no—
216C2×C3×Dic3—62no—
217C2×C3×Dic3—62no—
218#626—68no—
219#625—68no—
220#624—68no—
221D3×Dic3—68no—
222C2×#121—271noSylow
223C2×H(3,𝔽3)—42no—
224C2×H(3,𝔽3)#12114yescomplemented
225C2×H(3,𝔽3)C22×D411yescharacteristic
226C2×H(3,𝔽3)C22×D411yescharacteristic
227C2×H(3,𝔽3)C2×#4111yescharacteristic
228#272—42no—
229(C3⋊D3)⋊C3—24no—
230C24×C3—31no—
231C24×C3—31no—
232C24×C3—61no—
233C24×C3—91no—
234C24×C3—91no—
235C24×C3D3211yescharacteristic
236C23×D3—31no—
237C23×D3—31no—
238C23×D3—31no—
239C23×D3—61no—
240C23×D3—91no—
241C2×C3×D4—94no—
242C2×C3×D4—94no—
243C2×C3×D4—184no—
244C2×C3×D4—184no—
245C22×C3×C4—91no—
246C22×C3×C4—91no—
247C22×C3×C4—91no—
248C22×C3×C4—91no—
249C2×#67—34no—
250C2×#67—34no—
251C2×#67—34no—
252C2×#67—64no—
253C2×#67—64no—
254C2×#67—64no—
255C2×#67—64no—
256C2×#67—68no—
257C2×#67—94no—
258C2×#67—94no—
259C2×#67—94no—
260C2×#67—184no—
261C2×#67—184no—
262C2×#67—184no—
263C22×Dic3—31no—
264C22×Dic3—31no—
265C22×Dic3—31no—
266C22×Dic3—31no—
267C22×Dic3—31no—
268C22×Dic3—61no—
269C22×Dic3—91no—
270C22×Dic3—91no—
271C22×Dic3—91no—
272C22×Dic3—94no—
273C22×Dic3—184no—
274C2×D12—94no—
275C2×D12—184no—
276C2×C4×D3—94no—
277C2×C4×D3—94no—
278C2×C4×D3—184no—
279C2×C4×D3—184no—
280C3×C22⋊C4—94no—
281#217—34no—
282#217—94no—
283#217—94no—
284(C2×D3)⋊C4—94no—
285(C2×D3)⋊C4—94no—
286C4⋊Dic3—94no—
287Dic3⋊C4—94no—
288Dic3⋊C4—94no—
289C22×C32—24no—
290C22×C32—22no—
291C22×C32—22no—
292C22×C32—22no—
293C22×C32—22no—
294C22×C32—22no—
295C22×C32—22no—
296C22×C32—21no—
297C22×C32—22no—
298C22×C32—22no—
299C22×C32—22no—
300C22×C32—42no—
301C22×C32—42no—
302C22×C32—48no—
303C22×C32—44no—
304C22×C32—44no—
305C22×C32#23512yes—
306C22×C32D3×D412yes—
307C22×C32D3×D412yes—
308C22×C32D3×D412yes—
309C22×C32#23712yes—
310C22×C32C2×#6712yes—
311C22×C32C23×D311yescharacteristic
312C22×C32C23×D311yescharacteristic
313C2×C3⋊D3—28no—
314C2×C3⋊D3—22no—
315C2×C3⋊D3—22no—
316C2×C3⋊D3C2×#6714yescomplemented
317C2×C3×D3—34no—
318C2×C3×D3—68no—
319C2×C3×D3—62no—
320C2×C3×D3—62no—
321C2×C3×D3—64no—
322C2×C3×D3—62no—
323C2×C3×D3—62no—
324C2×C3×D3—62no—
325C2×C3×D3—62no—
326C2×C3×D3—62no—
327C2×C3×D3—62no—
328C2×C3×D3—122no—
329C2×C3×D3—122no—
330C2×C3×D3—128no—
331C2×C3×D3—124no—
332C2×C3×D3—124no—
333C3⋊Dic3—22no—
334C3⋊Dic3—22no—
335C3⋊Dic3—22no—
336C3×Dic3—34no—
337C3×Dic3—34no—
338C3×Dic3—64no—
339C3×Dic3—62no—
340C3×Dic3—62no—
341C3×Dic3—62no—
342C22×D4—91no—
343C22×D4—91no—
344C22×D4—271no—
345C23×C4—271no—
346#121—278no—
347C2×C4⋊C4—271no—
348C2×C22⋊C4—91no—
349C2×C22⋊C4—271no—
350H(3,𝔽3)C2×#12111yescharacteristic, complemented, Sylow
351C23×C3—22no—
352C23×C3—22no—
353C23×C3—22no—
354C23×C3—32no—
355C23×C3—32no—
356C23×C3—31no—
357C23×C3—31no—
358C23×C3—62no—
359C23×C3—62no—
360C23×C3—62no—
361C23×C3—62no—
362C23×C3—62no—
363C23×C3—62no—
364C23×C3—64no—
365C23×C3—62no—
366C23×C3—61no—
367C23×C3—92no—
368C23×C3—94no—
369C23×C3—92no—
370C23×C3—122no—
371C23×C3—122no—
372C23×C3—182no—
373C23×C3—182no—
374C23×C3—182no—
375C23×C3—182no—
376C23×C3—182no—
377C23×C3C2×D3212yes—
378C23×C3C2×D3211yescharacteristic
379C22×D3—32no—
380C22×D3—34no—
381C22×D3—32no—
382C22×D3—32no—
383C22×D3—62no—
384C22×D3—62no—
385C22×D3—62no—
386C22×D3—62no—
387C22×D3—62no—
388C22×D3—62no—
389C22×D3—62no—
390C22×D3—62no—
391C22×D3—62no—
392C22×D3—64no—
393C22×D3—94no—
394C22×D3—92no—
395C22×D3—122no—
396C22×D3—122no—
397C22×D3—182no—
398C22×D3—182no—
399C3×D4—188no—
400C3×D4—188no—
401C2×C3×C4—92no—
402C2×C3×C4—94no—
403C2×C3×C4—92no—
404C2×C3×C4—92no—
405C2×C3×C4—92no—
406C2×C3×C4—182no—
407C2×C3×C4—182no—
408C2×C3×C4—182no—
409#67—68no—
410#67—68no—
411#67—68no—
412#67—616no—
413#67—188no—
414#67—188no—
415#67—188no—
416C2×Dic3—32no—
417C2×Dic3—32no—
418C2×Dic3—34no—
419C2×Dic3—34no—
420C2×Dic3—32no—
421C2×Dic3—32no—
422C2×Dic3—32no—
423C2×Dic3—62no—
424C2×Dic3—64no—
425C2×Dic3—62no—
426C2×Dic3—62no—
427C2×Dic3—62no—
428C2×Dic3—92no—
429C2×Dic3—92no—
430C2×Dic3—92no—
431C2×Dic3—182no—
432C2×Dic3—182no—
433C2×Dic3—182no—
434C2×Dic3—188no—
435D12—188no—
436C4×D3—188no—
437C4×D3—188no—
438C2×C32—21no—
439C2×C32—21no—
440C2×C32—21no—
441C2×C32—24no—
442C2×C32—44no—
443C2×C32—42no—
444C2×C32—42no—
445C2×C32#88414yescomplemented
446C2×C32#93914yescomplemented
447C2×C32C2×#23511yescharacteristic
448C2×C32C2×D3×D411yescharacteristic
449C2×C32C2×D3×D411yescharacteristic
450C2×C32C2×D3×D411yescharacteristic
451C2×C32C2×#23711yescharacteristic
452C2×C32C22×#6711yescharacteristic
453C3⋊D3—24no—
454C3×D3—64no—
455C3×D3—124no—
456C3×D3—122no—
457C3×D3—122no—
458C24—91no—
459C24—91no—
460C24((C3⋊D3)⋊C3)⋊C211yescharacteristic
461C2×D4—94no—
462C2×D4—94no—
463C2×D4—184no—
464C2×D4—184no—
465C2×D4—274no—
466C2×D4—544no—
467C22×C4—91no—
468C22×C4—91no—
469C22×C4—91no—
470C22×C4—91no—
471C22×C4—274no—
472C22×C4—544no—
473C4⋊C4—274no—
474C22⋊C4—94no—
475C22⋊C4—274no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3—31no—
480C22×C3—31no—
481C22×C3—32no—
482C22×C3—32no—
483C22×C3—32no—
484C22×C3—32no—
485C22×C3—32no—
486C22×C3—32no—
487C22×C3—44no—
488C22×C3—62no—
489C22×C3—62no—
490C22×C3—62no—
491C22×C3—62no—
492C22×C3—62no—
493C22×C3—62no—
494C22×C3—64no—
495C22×C3—61no—
496C22×C3—62no—
497C22×C3—62no—
498C22×C3—62no—
499C22×C3—94no—
500C22×C3—122no—
501C22×C3—122no—
502C22×C3—124no—
503C22×C3—124no—
504C22×C3—128no—
505C22×C3—182no—
506C22×C3—182no—
507C22×C3—188no—
508C22×C3—182no—
509C22×C3—182no—
510C22×C3—182no—
511C22×C3—364no—
512C22×C3#380412yes—
513C22×C3D3×#6712yes—
514C22×C3#381112yes—
515C22×C3C22×D3211yescharacteristic
516C2×D3—34no—
517C2×D3—62no—
518C2×D3—62no—
519C2×D3—68no—
520C2×D3—62no—
521C2×D3—62no—
522C2×D3—62no—
523C2×D3—62no—
524C2×D3—62no—
525C2×D3—62no—
526C2×D3—64no—
527C2×D3—94no—
528C2×D3—122no—
529C2×D3—122no—
530C2×D3—124no—
531C2×D3—124no—
532C2×D3—128no—
533C2×D3—182no—
534C2×D3—182no—
535C2×D3—188no—
536C3×C4—94no—
537C3×C4—182no—
538C3×C4—182no—
539C3×C4—182no—
540Dic3—34no—
541Dic3—34no—
542Dic3—64no—
543Dic3—62no—
544Dic3—62no—
545Dic3—62no—
546Dic3—182no—
547Dic3—182no—
548Dic3—182no—
549C32—21no—
550C32C2×#88411yescharacteristic, complemented
551C32C2×#93911yescharacteristic, complemented
552C23—22no—
553C23—22no—
554C23—22no—
555C23—92no—
556C23—94no—
557C23—92no—
558C23—182no—
559C23—182no—
560C23—182no—
561C23—182no—
562C23—182no—
563C23C2×((C3⋊D3)⋊C3)⋊C212yes—
564C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
565D4—188no—
566D4—188no—
567D4—548no—
568C2×C4—92no—
569C2×C4—94no—
570C2×C4—92no—
571C2×C4—92no—
572C2×C4—92no—
573C2×C4—182no—
574C2×C4—182no—
575C2×C4—182no—
576C2×C4—548no—
577C2×C3—31no—
578C2×C3—31no—
579C2×C3—31no—
580C2×C3—31no—
581C2×C3—31no—
582C2×C3—31no—
583C2×C3—34no—
584C2×C3—34no—
585C2×C3—42no—
586C2×C3—61no—
587C2×C3—61no—
588C2×C3—61no—
589C2×C3—64no—
590C2×C3—122no—
591C2×C3—122no—
592C2×C3—124no—
593C2×C3—184no—
594C2×C3—362no—
595C2×C3#6397114yes—
596C2×C3C2×#380411yescharacteristic
597C2×C3C2×D3×#6711yescharacteristic
598C2×C3C2×#381111yescharacteristic
599D3—64no—
600D3—122no—
601D3—122no—
602D3—124no—
603D3—184no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—44no—
608C22—94no—
609C22—182no—
610C22—182no—
611C22—188no—
612C22—182no—
613C22—182no—
614C22—182no—
615C22—364no—
616C22#8867112yes—
617C22#8867612yes—
618C22#8867712yes—
619C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
620C4—94no—
621C4—182no—
622C4—182no—
623C4—182no—
624C3—31no—
625C3—31no—
626C3—61no—
627C3C2×#6397111yescharacteristic
628C2—42no—
629C2—184no—
630C2—362no—
631C2#1172131214yesdirect product factor
632C2C2×#8867111yescharacteristic
633C2C2×#8867611yescharacteristic
634C2C2×#8867711yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
FrattiniC22×C3C22×D32
FittingC24×H(3,𝔽3)C22
socleC23×C3C2×D32
hypercenterC24((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#121
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C22×#6235C2semidirect
2. C22×#638⋊C3C2semidirect
3. C22×#6200C2semidirect
4. C22×((C3⋊D3)⋊C3)⋊C4C2semidirect
5. C2×#3757⋊C3C2semidirect
6. #11721312C2direct (N × H)
7. C2×#88451C2semidirect
8. C2×#88449C2semidirect
15. C2×#6235C22semidirect
18. C2×#638⋊C3C22semidirect
23. C2×#6200C22semidirect
25. C2×((C3⋊D3)⋊C3)⋊C4C22semidirect
26. #3757⋊C3C22semidirect
27. #88451C22semidirect
28. #88449C22semidirect
29. C22×#638D3semidirect
38. C2×#3757D3semidirect
57. C22×(C3⋊D3)⋊C3D4semidirect
60. C2×#1084D4semidirect
83. C2×#638C2×D3semidirect
109. #3757C2×D3semidirect
130. C2×(C3⋊D3)⋊C3C2×D4semidirect
131. #1084C2×D4semidirect
177. C22×C3⋊D3#67semidirect
224. C2×H(3,𝔽3)#121semidirect
316. C2×C3⋊D3C2×#67semidirect
350. H(3,𝔽3)C2×#121semidirect
445. C2×C32#884semidirect
446. C2×C32#939semidirect
550. C32C2×#884semidirect
551. C32C2×#939semidirect
631. C2#11721312direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62352
C22×#638⋊C32
C22×#62002
C22×((C3⋊D3)⋊C3)⋊C42
C2×#3757⋊C32
#117213122
C2×#884512
C2×#884492
C2×#639703
C2×#639583