Finite Group Databasebeta

← C2×#11721293 #13152082

Subgroups — C2×#11721293 #13152082

Order 1728.

1727 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6235C211yesmaximal, characteristic
2C23×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
3C22×#6199C211yesmaximal, characteristic, complemented
4C2×#3757⋊C3C211yesmaximal, characteristic, complemented
5C2×#88451C211yesmaximal, characteristic, complemented
6C2×#88449C211yesmaximal, characteristic, complemented
7C2×#88447C211yesmaximal, characteristic, complemented
8#11721293C218yesmaximal, direct product factor
9C2×#67⋊Dic3—31nomaximal
10C2×Dic3×#67—31nomaximal
11C24×H(3,𝔽3)C2211yescharacteristic
12C23×#272C2211yescharacteristic
13C2×#6235C418yescomplemented
14C2×#6235C2214yes—
15C22×#1084C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3—24no—
17C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
21C22×(C3⋊Dic3)⋊C3C2214yescomplemented
22C2×#6199—24no—
23C2×#6199C2214yescomplemented
24#3757⋊C3C2214yescomplemented
25#88451C2214yescomplemented
26#88449C2214yescomplemented
27#88447C2214yescomplemented
28C23×C3⋊Dic3D311yescharacteristic, complemented
29C22×C3×#67—31no—
30C22×C3×#67—31no—
31C22×C3×#67—61no—
32C23×C3×Dic3—31no—
33C22×D3×Dic3—31no—
34C22×D3×Dic3—31no—
35C2×#3757D311yescharacteristic, complemented
36C2×C3×#217—31no—
37#67⋊Dic3—38no—
38Dic3×#67—38no—
39C2×Dic3⋊Dic3—31no—
40C2×Dic3⋊Dic3—31no—
41C2×(C2×D3)⋊Dic3—31no—
42C2×(C2×D3)⋊Dic3—31no—
43C2×Dic32—31no—
44C2×Dic32—31no—
45C23×H(3,𝔽3)—22no—
46C23×H(3,𝔽3)—22no—
47C23×H(3,𝔽3)C2×C414yescomplemented
48C23×H(3,𝔽3)C2312yes—
49C23×H(3,𝔽3)C2311yescharacteristic
50C22×#272—22no—
51C22×#272—22no—
52C22×#272C2×C414yescomplemented
53C22×#272C2312yes—
54#6235C2×C4116yescomplemented
55C2×#1084C2×C414yescomplemented
56C2×#1084C2312yes—
57C2×(C3⋊Dic3)⋊C3—22no—
58C2×(C3⋊Dic3)⋊C3—22no—
59C2×(C3⋊Dic3)⋊C3—22no—
60C2×(C3⋊Dic3)⋊C3—28no—
61C2×(C3⋊Dic3)⋊C3D414yescomplemented
62C2×(C3⋊Dic3)⋊C3C2312yes—
63C2×(C3⋊Dic3)⋊C3C2312yes—
64C2×(C3⋊Dic3)⋊C3C2312yes—
65C2×(C3⋊Dic3)⋊C3C2312yes—
66#6199—28no—
67C2×D4×Dic3—91no—
68C2×C4×#67—91no—
69C2×#67⋊C4—91no—
70C24×C32—21no—
71C24×C32C2×D311yescharacteristic
72C24×C32C2×D311yescharacteristic
73C23×C3×D3—31no—
74C23×C3×D3—31no—
75C23×C3×D3—61no—
76C22×C3⋊Dic3—24no—
77C22×C3⋊Dic3C2×D311yescharacteristic
78C22×C3⋊Dic3C2×D311yescharacteristic
79C22×C3⋊Dic3C2×D311yescharacteristic
80C22×C3⋊Dic3C2×D311yescharacteristic
81C22×C3⋊Dic3C2×D314yescomplemented
82C2×C3×#67—38no—
83C2×C3×#67—38no—
84C2×C3×#67—34no—
85C2×C3×#67—34no—
86C2×C3×#67—68no—
87C2×C3×#67—64no—
88C22×C3×Dic3—31no—
89C22×C3×Dic3—31no—
90C22×C3×Dic3—31no—
91C22×C3×Dic3—31no—
92C22×C3×Dic3—31no—
93C22×C3×Dic3—31no—
94C22×C3×Dic3—34no—
95C22×C3×Dic3—64no—
96C22×C3×Dic3—61no—
97C2×D3×Dic3—34no—
98C2×D3×Dic3—34no—
99C2×D3×Dic3—64no—
100C2×D3×Dic3—64no—
101#3757C2×D314yescomplemented
102C3×#217—34no—
103Dic3⋊Dic3—34no—
104Dic3⋊Dic3—34no—
105(C2×D3)⋊Dic3—34no—
106(C2×D3)⋊Dic3—34no—
107Dic32—34no—
108Dic32—34no—
109C22×H(3,𝔽3)—28no—
110C22×H(3,𝔽3)—22no—
111C22×H(3,𝔽3)—22no—
112C22×H(3,𝔽3)C22×C412yes—
113C22×H(3,𝔽3)C22×C414yescomplemented
114C22×H(3,𝔽3)C2×D412yes—
115C22×H(3,𝔽3)#4112yes—
116C22×H(3,𝔽3)C2411yescharacteristic
117C2×#272—28no—
118C2×#272—22no—
119C2×#272—22no—
120C2×#272C22×C414yescomplemented
121#1084C22×C414yescomplemented
122(C3⋊Dic3)⋊C3—22no—
123(C3⋊Dic3)⋊C3—22no—
124(C3⋊Dic3)⋊C3—22no—
125(C3⋊Dic3)⋊C3C2×D414yescomplemented
126C22×C3×D4—91no—
127C23×C3×C4—91no—
128C22×#67—31no—
129C22×#67—31no—
130C22×#67—61no—
131C23×Dic3—31no—
132C23×Dic3—91no—
133C23×Dic3—91no—
134C22×C4×D3—91no—
135C22×C4×D3—91no—
136C2×C3×C22⋊C4—91no—
137C2×#217—31no—
138C2×#217—91no—
139C2×#217—91no—
140D4×Dic3—98no—
141C4×#67—98no—
142C2×(C2×D3)⋊C4—91no—
143C2×(C2×D3)⋊C4—91no—
144C2×C4⋊Dic3—91no—
145C2×Dic3⋊C4—91no—
146C2×Dic3⋊C4—91no—
147C2×C4×Dic3—91no—
148C2×C4×Dic3—91no—
149C2×C4×Dic3—91no—
150#67⋊C4—98no—
151C23×C32—22no—
152C23×C32—22no—
153C23×C32—22no—
154C23×C32—22no—
155C23×C32—24no—
156C23×C32—21no—
157C23×C32—22no—
158C23×C32—42no—
159C23×C32—42no—
160C23×C32C4×D314yescomplemented
161C23×C32C4×D314yescomplemented
162C23×C32C22×D312yes—
163C23×C32C22×D312yes—
164C23×C32C22×D311yescharacteristic
165C23×C32C22×D311yescharacteristic
166C22×C3×D3—34no—
167C22×C3×D3—34no—
168C22×C3×D3—32no—
169C22×C3×D3—32no—
170C22×C3×D3—64no—
171C22×C3×D3—62no—
172C22×C3×D3—62no—
173C22×C3×D3—62no—
174C22×C3×D3—62no—
175C22×C3×D3—62no—
176C22×C3×D3—122no—
177C22×C3×D3—122no—
178C2×C3⋊Dic3—22no—
179C2×C3⋊Dic3—22no—
180C2×C3⋊Dic3—22no—
181C2×C3⋊Dic3—28no—
182C2×C3⋊Dic3#6714yescomplemented
183C2×C3⋊Dic3C22×D312yes—
184C2×C3⋊Dic3C22×D312yes—
185C2×C3⋊Dic3C22×D312yes—
186C2×C3⋊Dic3C22×D312yes—
187C3×#67—316no—
188C3×#67—316no—
189C3×#67—616no—
190C2×C3×Dic3—34no—
191C2×C3×Dic3—34no—
192C2×C3×Dic3—34no—
193C2×C3×Dic3—32no—
194C2×C3×Dic3—32no—
195C2×C3×Dic3—32no—
196C2×C3×Dic3—32no—
197C2×C3×Dic3—32no—
198C2×C3×Dic3—32no—
199C2×C3×Dic3—64no—
200C2×C3×Dic3—62no—
201C2×C3×Dic3—68no—
202C2×C3×Dic3—62no—
203C2×C3×Dic3—62no—
204C2×C3×Dic3—62no—
205D3×Dic3—68no—
206D3×Dic3—68no—
207C2×C4×D4—271noSylow
208C2×H(3,𝔽3)—24no—
209C2×H(3,𝔽3)C4×D414yescomplemented
210C2×H(3,𝔽3)C23×C411yescharacteristic
211C2×H(3,𝔽3)C22×D411yescharacteristic
212C2×H(3,𝔽3)C2×#4111yescharacteristic
213#272—24no—
214C24×C3—31no—
215C24×C3—31no—
216C24×C3—61no—
217C24×C3—91no—
218C24×C3D3211yescharacteristic
219C23×D3—31no—
220C23×D3—31no—
221C23×D3—61no—
222C2×C3×D4—98no—
223C2×C3×D4—94no—
224C22×C3×C4—91no—
225C22×C3×C4—91no—
226C22×C3×C4—91no—
227C22×C3×C4—91no—
228C22×C3×C4—91no—
229C22×C3×C4—94no—
230C22×C3×C4—184no—
231C2×#67—34no—
232C2×#67—34no—
233C2×#67—38no—
234C2×#67—38no—
235C2×#67—68no—
236C2×#67—64no—
237C22×Dic3—31no—
238C22×Dic3—31no—
239C22×Dic3—31no—
240C22×Dic3—31no—
241C22×Dic3—31no—
242C22×Dic3—31no—
243C22×Dic3—34no—
244C22×Dic3—61no—
245C22×Dic3—64no—
246C22×Dic3—91no—
247C22×Dic3—94no—
248C22×Dic3—91no—
249C22×Dic3—91no—
250C22×Dic3—91no—
251C22×Dic3—94no—
252C22×Dic3—184no—
253C22×Dic3—184no—
254C2×C4×D3—94no—
255C2×C4×D3—94no—
256C2×C4×D3—184no—
257C2×C4×D3—184no—
258C3×C22⋊C4—94no—
259#217—34no—
260#217—94no—
261#217—94no—
262(C2×D3)⋊C4—94no—
263(C2×D3)⋊C4—94no—
264C4⋊Dic3—94no—
265Dic3⋊C4—94no—
266Dic3⋊C4—94no—
267C4×Dic3—94no—
268C4×Dic3—94no—
269C4×Dic3—94no—
270C22×C32—28no—
271C22×C32—28no—
272C22×C32—22no—
273C22×C32—22no—
274C22×C32—22no—
275C22×C32—22no—
276C22×C32—21no—
277C22×C32—22no—
278C22×C32—22no—
279C22×C32—24no—
280C22×C32—22no—
281C22×C32—42no—
282C22×C32—42no—
283C22×C32—48no—
284C22×C32C2×C4×D312yes—
285C22×C32C2×C4×D312yes—
286C22×C32C2×C4×D314yescomplemented
287C22×C32C2×C4×D314yescomplemented
288C22×C32#23512yes—
289C22×C32D3×D412yes—
290C22×C32#23712yes—
291C22×C32C2×#6712yes—
292C22×C32C23×D311yescharacteristic
293C22×C32C23×D311yescharacteristic
294C2×C3×D3—34no—
295C2×C3×D3—34no—
296C2×C3×D3—64no—
297C2×C3×D3—68no—
298C2×C3×D3—68no—
299C2×C3×D3—62no—
300C2×C3×D3—62no—
301C2×C3×D3—62no—
302C2×C3×D3—62no—
303C2×C3×D3—122no—
304C2×C3×D3—122no—
305C2×C3×D3—128no—
306C3⋊Dic3—22no—
307C3⋊Dic3—22no—
308C3⋊Dic3—22no—
309C3⋊Dic3C2×#6714yescomplemented
310C3×Dic3—34no—
311C3×Dic3—34no—
312C3×Dic3—34no—
313C3×Dic3—64no—
314C3×Dic3—62no—
315C3×Dic3—62no—
316C3×Dic3—62no—
317C22×D4—91no—
318C23×C4—91no—
319C23×C4—271no—
320C4×D4—278no—
321C2×C4⋊C4—271no—
322C2×C22⋊C4—91no—
323C2×C22⋊C4—271no—
324C2×C42—271no—
325H(3,𝔽3)C2×C4×D411yescharacteristic, complemented, Sylow
326C23×C3—22no—
327C23×C3—22no—
328C23×C3—34no—
329C23×C3—34no—
330C23×C3—32no—
331C23×C3—32no—
332C23×C3—31no—
333C23×C3—31no—
334C23×C3—62no—
335C23×C3—62no—
336C23×C3—62no—
337C23×C3—62no—
338C23×C3—64no—
339C23×C3—62no—
340C23×C3—61no—
341C23×C3—92no—
342C23×C3—94no—
343C23×C3—122no—
344C23×C3—122no—
345C23×C3—182no—
346C23×C3—182no—
347C23×C3#62414yes—
348C23×C3C2×D3212yes—
349C23×C3C2×D3211yescharacteristic
350C22×D3—32no—
351C22×D3—32no—
352C22×D3—34no—
353C22×D3—34no—
354C22×D3—62no—
355C22×D3—64no—
356C22×D3—62no—
357C22×D3—62no—
358C22×D3—62no—
359C22×D3—62no—
360C22×D3—122no—
361C22×D3—122no—
362C3×D4—916no—
363C2×C3×C4—92no—
364C2×C3×C4—94no—
365C2×C3×C4—92no—
366C2×C3×C4—94no—
367C2×C3×C4—92no—
368C2×C3×C4—92no—
369C2×C3×C4—92no—
370C2×C3×C4—188no—
371C2×C3×C4—182no—
372C2×C3×C4—182no—
373C2×C3×C4—182no—
374#67—316no—
375#67—316no—
376#67—616no—
377C2×Dic3—32no—
378C2×Dic3—32no—
379C2×Dic3—34no—
380C2×Dic3—34no—
381C2×Dic3—32no—
382C2×Dic3—34no—
383C2×Dic3—32no—
384C2×Dic3—32no—
385C2×Dic3—32no—
386C2×Dic3—62no—
387C2×Dic3—64no—
388C2×Dic3—68no—
389C2×Dic3—62no—
390C2×Dic3—62no—
391C2×Dic3—62no—
392C2×Dic3—94no—
393C2×Dic3—92no—
394C2×Dic3—92no—
395C2×Dic3—92no—
396C2×Dic3—92no—
397C2×Dic3—188no—
398C2×Dic3—182no—
399C2×Dic3—182no—
400C2×Dic3—182no—
401C2×Dic3—188no—
402C4×D3—188no—
403C4×D3—188no—
404C2×C32—24no—
405C2×C32—24no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—21no—
409C2×C32—24no—
410C2×C32—44no—
411C2×C32#67⋊C414yescomplemented
412C2×C32C4×#6714yescomplemented
413C2×C32C22×C4×D311yescharacteristic
414C2×C32C22×C4×D311yescharacteristic
415C2×C32C2×#23511yescharacteristic
416C2×C32C2×D3×D411yescharacteristic
417C2×C32C2×#23711yescharacteristic
418C2×C32C22×#6711yescharacteristic
419C3×D3—64no—
420C3×D3—64no—
421C3×D3—124no—
422C24—91no—
423C24((C3⋊D3)⋊C3)⋊C211yescharacteristic
424C2×D4—98no—
425C2×D4—94no—
426C22×C4—91no—
427C22×C4—91no—
428C22×C4—94no—
429C22×C4—91no—
430C22×C4—91no—
431C22×C4—91no—
432C22×C4—184no—
433C22×C4—274no—
434C22×C4—544no—
435C4⋊C4—274no—
436C22⋊C4—94no—
437C22⋊C4—274no—
438C42—274no—
439C22×C3—22no—
440C22×C3—22no—
441C22×C3—28no—
442C22×C3—31no—
443C22×C3—31no—
444C22×C3—32no—
445C22×C3—32no—
446C22×C3—32no—
447C22×C3—32no—
448C22×C3—34no—
449C22×C3—34no—
450C22×C3—32no—
451C22×C3—32no—
452C22×C3—62no—
453C22×C3—62no—
454C22×C3—62no—
455C22×C3—62no—
456C22×C3—68no—
457C22×C3—68no—
458C22×C3—61no—
459C22×C3—62no—
460C22×C3—62no—
461C22×C3—64no—
462C22×C3—62no—
463C22×C3—94no—
464C22×C3—122no—
465C22×C3—122no—
466C22×C3—128no—
467C22×C3—182no—
468C22×C3—182no—
469C22×C3—188no—
470C22×C3#380412yes—
471C22×C3C2×#62412yes—
472C22×C3C2×#62414yes—
473C22×C3D3×#6712yes—
474C22×C3C22×D3211yescharacteristic
475C2×D3—34no—
476C2×D3—34no—
477C2×D3—64no—
478C2×D3—62no—
479C2×D3—62no—
480C2×D3—62no—
481C2×D3—62no—
482C2×D3—68no—
483C2×D3—68no—
484C2×D3—122no—
485C2×D3—122no—
486C2×D3—128no—
487C3×C4—94no—
488C3×C4—94no—
489C3×C4—182no—
490C3×C4—182no—
491C3×C4—182no—
492Dic3—34no—
493Dic3—34no—
494Dic3—34no—
495Dic3—64no—
496Dic3—62no—
497Dic3—62no—
498Dic3—62no—
499Dic3—94no—
500Dic3—182no—
501Dic3—182no—
502Dic3—182no—
503C32—21no—
504C32C2×#67⋊C411yescharacteristic, complemented
505C32C2×C4×#6711yescharacteristic, complemented
506C23—22no—
507C23—22no—
508C23—92no—
509C23—94no—
510C23—182no—
511C23—182no—
512C23#619914yescomplemented
513C23C2×((C3⋊D3)⋊C3)⋊C212yes—
514C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
515D4—916no—
516C2×C4—92no—
517C2×C4—92no—
518C2×C4—94no—
519C2×C4—94no—
520C2×C4—92no—
521C2×C4—92no—
522C2×C4—92no—
523C2×C4—188no—
524C2×C4—182no—
525C2×C4—182no—
526C2×C4—182no—
527C2×C4—548no—
528C2×C3—24no—
529C2×C3—31no—
530C2×C3—31no—
531C2×C3—31no—
532C2×C3—31no—
533C2×C3—31no—
534C2×C3—31no—
535C2×C3—34no—
536C2×C3—34no—
537C2×C3—64no—
538C2×C3—64no—
539C2×C3—61no—
540C2×C3—61no—
541C2×C3—61no—
542C2×C3—64no—
543C2×C3—124no—
544C2×C3—184no—
545C2×C3#6395214yes—
546C2×C3C2×#380411yescharacteristic
547C2×C3C22×#62411yescharacteristic
548C2×C3C2×D3×#6711yescharacteristic
549D3—64no—
550D3—64no—
551D3—124no—
552C22—22no—
553C22—22no—
554C22—28no—
555C22—94no—
556C22—182no—
557C22—182no—
558C22—188no—
559C22#8867112yes—
560C22C2×#619912yes—
561C22C2×#619914yescomplemented
562C22#8867612yes—
563C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
564C4—94no—
565C4—94no—
566C4—182no—
567C4—182no—
568C4—182no—
569C3—31no—
570C3—31no—
571C3—61no—
572C3C2×#6395211yescharacteristic
573C2—24no—
574C2—184no—
575C2#1172129314yesdirect product factor
576C2C2×#8867111yescharacteristic
577C2C22×#619911yescharacteristic
578C2C2×#8867611yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
FrattiniC22×C3C22×D32
FittingC24×H(3,𝔽3)C22
socleC23×C3C2×D32
prime radicalC22×#6235C2
cosocleC22×H(3,𝔽3)C24
hypercenterC24((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×C4×D4
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C23×(C3⋊Dic3)⋊C3C2semidirect
3. C22×#6199C2semidirect
4. C2×#3757⋊C3C2semidirect
5. C2×#88451C2semidirect
6. C2×#88449C2semidirect
7. C2×#88447C2semidirect
8. #11721293C2direct (N × H)
13. C2×#6235C4semidirect
21. C22×(C3⋊Dic3)⋊C3C22semidirect
23. C2×#6199C22semidirect
24. #3757⋊C3C22semidirect
25. #88451C22semidirect
26. #88449C22semidirect
27. #88447C22semidirect
28. C23×C3⋊Dic3D3semidirect
35. C2×#3757D3semidirect
47. C23×H(3,𝔽3)C2×C4semidirect
52. C22×#272C2×C4semidirect
54. #6235C2×C4semidirect
55. C2×#1084C2×C4semidirect
61. C2×(C3⋊Dic3)⋊C3D4semidirect
81. C22×C3⋊Dic3C2×D3semidirect
101. #3757C2×D3semidirect
113. C22×H(3,𝔽3)C22×C4semidirect
120. C2×#272C22×C4semidirect
121. #1084C22×C4semidirect
125. (C3⋊Dic3)⋊C3C2×D4semidirect
160. C23×C32C4×D3semidirect
161. C23×C32C4×D3semidirect
182. C2×C3⋊Dic3#67semidirect
209. C2×H(3,𝔽3)C4×D4semidirect
286. C22×C32C2×C4×D3semidirect
287. C22×C32C2×C4×D3semidirect
309. C3⋊Dic3C2×#67semidirect
325. H(3,𝔽3)C2×C4×D4semidirect
411. C2×C32#67⋊C4semidirect
412. C2×C32C4×#67semidirect
504. C32C2×#67⋊C4semidirect
505. C32C2×C4×#67semidirect
512. C23#6199semidirect
561. C22C2×#6199semidirect
575. C2#11721293direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62352
C23×(C3⋊Dic3)⋊C32
C22×#61992
C2×#3757⋊C32
C2×#884512
C2×#884492
C2×#884472
#117212932
C2×#67⋊Dic33
C2×Dic3×#673