Finite Group Databasebeta

← C2×#64269⋊C3 #13164448

Subgroups — C2×#64269⋊C3 #13164448

Order 1728.

1208 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 1208 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C22×(C3⋊S4)⋊C3C211yesmaximal, characteristic, complemented
3#64269⋊C3C214yesmaximal, direct product factor
4C2×(C3⋊SL(2,ℤ/4))⋊C3C211yesmaximal, characteristic, complemented
5C2×#64269C311yesmaximal, characteristic, complemented
6C2×C3×#953—31nomaximal
7C22×C22⋊H(3,𝔽3)—22no—
8C22×C22⋊H(3,𝔽3)C2211yescharacteristic
9C22×C22⋊H(3,𝔽3)C2212yescomplemented
10C2×(C3⋊S4)⋊C3—22no—
11C2×(C3⋊S4)⋊C3C2212yescomplemented
12C2×#638⋊C3—41nomaximal
13(C3⋊SL(2,ℤ/4))⋊C3C2212yescomplemented
14C25×C32D311yescharacteristic, complemented
15C23×C3×A4—21no—
16C23×C3×A4C2×C311yescharacteristic, complemented
17C22×C3⋊S4C2×C311yescharacteristic, complemented
18C22×C3×#67—31no—
19C22×C3×#67—31no—
20C22×C3×#67—31no—
21#64269C2×C314yescomplemented
22C2×C3⋊SL(2,ℤ/4)C2×C311yescharacteristic, complemented
23C3×#953—34no—
24C3×#953—34no—
25C2×C3×#217—31no—
26C2×C3×#217—31no—
27C2×C3×#217—31no—
28C23×H(3,𝔽3)—41no—
29C22×(C3⋊D3)⋊C3—41no—
30C2×C22⋊H(3,𝔽3)—22no—
31C2×C22⋊H(3,𝔽3)D412yescomplemented
32C2×C22⋊H(3,𝔽3)C2311yescharacteristic
33(C3⋊S4)⋊C3—22no—
34#638⋊C3—44no—
35C2×(C3⋊Dic3)⋊C3—41no—
36C2×GL(2,ℤ/4)—91no—
37C2×C3×#120—91no—
38C2×#953—31no—
39C24×C32—22no—
40C24×C32—31no—
41C24×C32—31no—
42C24×C32—32no—
43C24×C32—62no—
44C24×C32C2×D311yescharacteristic
45C24×C32C2×D312yescomplemented
46C23×C3×D3—31no—
47C22×C3×A4—24no—
48C22×C3×A4—21no—
49C22×C3×A4—22no—
50C22×C3×A4—22no—
51C22×C3×A4C22×C311yescharacteristic
52C22×C3×A4C22×C312yescomplemented
53C2×C3⋊S4—22no—
54C2×C3⋊S4C22×C312yescomplemented
55C2×#638—41no—
56C2×C3×#67—32no—
57C2×C3×#67—34no—
58C2×C3×#67—34no—
59C2×C3×#67—32no—
60C2×C3×#67—62no—
61C2×C3×#67—64no—
62C2×C3×#67—62no—
63C2×C3×#67—61no—
64C2×C3×#67—62no—
65C2×C3×#67—61no—
66C22×C3×Dic3—31no—
67C22×C3×Dic3—31no—
68C22×C3×Dic3—31no—
69C3⋊SL(2,ℤ/4)C22×C312yescomplemented
70C3×#217—32no—
71C3×#217—34no—
72C3×#217—34no—
73C3×#217—32no—
74C22×H(3,𝔽3)—41no—
75C22×H(3,𝔽3)—42no—
76C22×H(3,𝔽3)—82no—
77C2×(C3⋊D3)⋊C3—42no—
78C2×(C3⋊D3)⋊C3—82no—
79C22⋊H(3,𝔽3)C2×D411yescharacteristic, complemented
80(C3⋊Dic3)⋊C3—42no—
81C25×C3—31no—
82C25×C3C3×D311yescharacteristic
83C23×A4—31no—
84C23×A4—61no—
85C22×S4—91no—
86C22×C3×D4—91no—
87C22×C3×D4—91no—
88C22×C3×D4—91no—
89C22×#67—31no—
90C22×#67—31no—
91C22×#67—31no—
92GL(2,ℤ/4)—94no—
93C2×SL(2,ℤ/4)—91no—
94C3×#120—94no—
95C3×#120—94no—
96C2×C3×C22⋊C4—91no—
97C2×C3×C22⋊C4—91no—
98C2×C3×C22⋊C4—91no—
99#953—34no—
100#953—34no—
101C2×#217—31no—
102C2×#217—31no—
103C2×#217—31no—
104C23×C32—22no—
105C23×C32—31no—
106C23×C32—31no—
107C23×C32—32no—
108C23×C32—32no—
109C23×C32—34no—
110C23×C32—31no—
111C23×C32—32no—
112C23×C32—32no—
113C23×C32—41no—
114C23×C32—62no—
115C23×C32—62no—
116C23×C32—62no—
117C23×C32—61no—
118C23×C32—64no—
119C23×C32—62no—
120C23×C32—64no—
121C23×C32—81no—
122C23×C32#6712yescomplemented
123C23×C32S411yescharacteristic, complemented
124C23×C32C22×D311yescharacteristic
125C22×C3⋊D3—41no—
126C22×C3×D3—32no—
127C22×C3×D3—62no—
128C22×C3×D3—62no—
129C22×C3×D3—61no—
130C22×C3×D3—61no—
131C2×C3×A4—24no—
132C2×C3×A4—22no—
133C2×C3×A4—22no—
134C2×C3×A4—21no—
135C2×C3×A4C3×D412yescomplemented
136C2×C3×A4C23×C311yescharacteristic
137C3⋊S4—22no—
138#638—44no—
139C2×C3⋊Dic3—41no—
140C3×#67—64no—
141C3×#67—62no—
142C3×#67—68no—
143C3×#67—64no—
144C3×#67—62no—
145C3×#67—64no—
146C2×C3×Dic3—32no—
147C2×C3×Dic3—32no—
148C2×C3×Dic3—32no—
149C2×C3×Dic3—62no—
150C2×C3×Dic3—62no—
151C2×C3×Dic3—61no—
152C2×C3×Dic3—61no—
153C2×#120—91noSylow
154C2×H(3,𝔽3)—42no—
155C2×H(3,𝔽3)—41no—
156C2×H(3,𝔽3)—82no—
157(C3⋊D3)⋊C3—82no—
158C24×C3—22no—
159C24×C3—31no—
160C24×C3—31no—
161C24×C3—31no—
162C24×C3—31no—
163C24×C3—32no—
164C24×C3—32no—
165C24×C3—31no—
166C24×C3—32no—
167C24×C3—62no—
168C24×C3—61no—
169C24×C3—61no—
170C24×C3—62no—
171C24×C3—62no—
172C24×C3—64no—
173C24×C3—62no—
174C24×C3—91no—
175C24×C3C2×C3×D311yescharacteristic
176C24×C3C2×C3×D312yes—
177C23×D3—31no—
178C22×A4—31no—
179C22×A4—32no—
180C22×A4—62no—
181C22×A4—64no—
182C22×A4—61no—
183C22×A4—62no—
184C2×S4—92no—
185C2×S4—182no—
186C2×C3×D4—92no—
187C2×C3×D4—94no—
188C2×C3×D4—94no—
189C2×C3×D4—92no—
190C2×C3×D4—182no—
191C2×C3×D4—184no—
192C2×C3×D4—182no—
193C2×C3×D4—181no—
194C2×C3×D4—182no—
195C2×C3×D4—181no—
196C22×C3×C4—91no—
197C22×C3×C4—91no—
198C22×C3×C4—91no—
199C2×#67—32no—
200C2×#67—34no—
201C2×#67—34no—
202C2×#67—32no—
203C2×#67—62no—
204C2×#67—64no—
205C2×#67—62no—
206C2×#67—61no—
207C2×#67—62no—
208C2×#67—61no—
209C2×#67—361no—
210C22×Dic3—31no—
211C22×Dic3—31no—
212C22×Dic3—31no—
213SL(2,ℤ/4)—92no—
214C3×C22⋊C4—92no—
215C3×C22⋊C4—94no—
216C3×C22⋊C4—94no—
217C3×C22⋊C4—92no—
218#217—34no—
219#217—32no—
220#217—34no—
221#217—32no—
222C22×C32—22no—
223C22×C32—32no—
224C22×C32—31no—
225C22×C32—32no—
226C22×C32—32no—
227C22×C32—31no—
228C22×C32—31no—
229C22×C32—34no—
230C22×C32—32no—
231C22×C32—41no—
232C22×C32—42no—
233C22×C32—62no—
234C22×C32—62no—
235C22×C32—64no—
236C22×C32—61no—
237C22×C32—62no—
238C22×C32—64no—
239C22×C32—62no—
240C22×C32—82no—
241C22×C32—84no—
242C22×C32—81no—
243C22×C32—82no—
244C22×C32C2×#6711yescharacteristic, complemented
245C22×C32C2×S411yescharacteristic
246C22×C32C2×S412yescomplemented
247C2×C3⋊D3—42no—
248C2×C3⋊D3—82no—
249C2×C3×D3—62no—
250C2×C3×D3—62no—
251C2×C3×D3—62no—
252C2×C3×D3—122no—
253C2×C3×D3—122no—
254C3×A4—21no—
255C3×A4C2×C3×D411yescharacteristic, complemented
256C3⋊Dic3—42no—
257C3×Dic3—62no—
258C3×Dic3—62no—
259C3×Dic3—62no—
260C25(C3⋊D3)⋊C311yescharacteristic, complemented
261C22×D4—91no—
262C22×D4—91no—
263C22×D4—91no—
264#120—94no—
265#120—94no—
266C2×C22⋊C4—91no—
267C2×C22⋊C4—91no—
268C2×C22⋊C4—91no—
269H(3,𝔽3)—41noSylow
270C23×C3—22no—
271C23×C3—31no—
272C23×C3—31no—
273C23×C3—31no—
274C23×C3—32no—
275C23×C3—32no—
276C23×C3—31no—
277C23×C3—32no—
278C23×C3—32no—
279C23×C3—34no—
280C23×C3—32no—
281C23×C3—34no—
282C23×C3—31no—
283C23×C3—32no—
284C23×C3—32no—
285C23×C3—31no—
286C23×C3—32no—
287C23×C3—32no—
288C23×C3—31no—
289C23×C3—31no—
290C23×C3—62no—
291C23×C3—61no—
292C23×C3—61no—
293C23×C3—62no—
294C23×C3—62no—
295C23×C3—61no—
296C23×C3—62no—
297C23×C3—62no—
298C23×C3—64no—
299C23×C3—62no—
300C23×C3—62no—
301C23×C3—62no—
302C23×C3—61no—
303C23×C3—64no—
304C23×C3—62no—
305C23×C3—62no—
306C23×C3—62no—
307C23×C3—64no—
308C23×C3—62no—
309C23×C3—61no—
310C23×C3—62no—
311C23×C3—62no—
312C23×C3—64no—
313C23×C3—64no—
314C23×C3—64no—
315C23×C3—64no—
316C23×C3—64no—
317C23×C3—68no—
318C23×C3—64no—
319C23×C3—68no—
320C23×C3—92no—
321C23×C3—121no—
322C23×C3—181no—
323C23×C3—182no—
324C23×C3—182no—
325C23×C3—181no—
326C23×C3—241no—
327C23×C3C3×#6712yes—
328C23×C3C3×S411yescharacteristic
329C23×C3C22×C3×D311yescharacteristic
330C22×D3—32no—
331C22×D3—62no—
332C22×D3—62no—
333C22×D3—61no—
334C22×D3—61no—
335C22×D3—361no—
336C2×A4—32no—
337C2×A4—31no—
338C2×A4—62no—
339C2×A4—64no—
340C2×A4—62no—
341C2×A4—61no—
342S4—182no—
343C3×D4—184no—
344C3×D4—182no—
345C3×D4—188no—
346C3×D4—184no—
347C3×D4—184no—
348C3×D4—182no—
349C2×C3×C4—92no—
350C2×C3×C4—92no—
351C2×C3×C4—92no—
352C2×C3×C4—182no—
353C2×C3×C4—182no—
354C2×C3×C4—181no—
355C2×C3×C4—181no—
356#67—64no—
357#67—62no—
358#67—68no—
359#67—64no—
360#67—64no—
361#67—62no—
362#67—364no—
363C2×Dic3—32no—
364C2×Dic3—32no—
365C2×Dic3—32no—
366C2×Dic3—62no—
367C2×Dic3—62no—
368C2×Dic3—61no—
369C2×Dic3—61no—
370C2×Dic3—361no—
371C2×C32—22no—
372C2×C32—31no—
373C2×C32—31no—
374C2×C32—32no—
375C2×C32—42no—
376C2×C32—41no—
377C2×C32—62no—
378C2×C32—82no—
379C2×C32—84no—
380C2×C32—82no—
381C2×C32—81no—
382C2×C32GL(2,ℤ/4)12yescomplemented
383C2×C32C22×S411yescharacteristic
384C3⋊D3—82no—
385C3×D3—122no—
386C24—22no—
387C24—31no—
388C24—31no—
389C24—32no—
390C24—62no—
391C24—91no—
392C24C2×(C3⋊D3)⋊C311yescharacteristic
393C24C2×(C3⋊D3)⋊C312yescomplemented
394C2×D4—92no—
395C2×D4—94no—
396C2×D4—94no—
397C2×D4—92no—
398C2×D4—182no—
399C2×D4—184no—
400C2×D4—182no—
401C2×D4—181no—
402C2×D4—182no—
403C2×D4—181no—
404C22×C4—91no—
405C22×C4—91no—
406C22×C4—91no—
407C22⋊C4—92no—
408C22⋊C4—94no—
409C22⋊C4—94no—
410C22⋊C4—92no—
411C22×C3—22no—
412C22×C3—32no—
413C22×C3—31no—
414C22×C3—31no—
415C22×C3—31no—
416C22×C3—32no—
417C22×C3—32no—
418C22×C3—32no—
419C22×C3—32no—
420C22×C3—31no—
421C22×C3—31no—
422C22×C3—34no—
423C22×C3—32no—
424C22×C3—31no—
425C22×C3—34no—
426C22×C3—32no—
427C22×C3—32no—
428C22×C3—31no—
429C22×C3—32no—
430C22×C3—31no—
431C22×C3—62no—
432C22×C3—62no—
433C22×C3—61no—
434C22×C3—61no—
435C22×C3—61no—
436C22×C3—62no—
437C22×C3—62no—
438C22×C3—62no—
439C22×C3—64no—
440C22×C3—62no—
441C22×C3—62no—
442C22×C3—64no—
443C22×C3—61no—
444C22×C3—62no—
445C22×C3—62no—
446C22×C3—62no—
447C22×C3—64no—
448C22×C3—61no—
449C22×C3—62no—
450C22×C3—62no—
451C22×C3—64no—
452C22×C3—64no—
453C22×C3—64no—
454C22×C3—62no—
455C22×C3—64no—
456C22×C3—64no—
457C22×C3—68no—
458C22×C3—64no—
459C22×C3—68no—
460C22×C3—62no—
461C22×C3—121no—
462C22×C3—122no—
463C22×C3—182no—
464C22×C3—182no—
465C22×C3—182no—
466C22×C3—242no—
467C22×C3—244no—
468C22×C3—241no—
469C22×C3—242no—
470C22×C3—362no—
471C22×C3—362no—
472C22×C3C2×C3×#6711yescharacteristic
473C22×C3C2×C3×S411yescharacteristic
474C22×C3C2×C3×S412yes—
475C2×D3—62no—
476C2×D3—62no—
477C2×D3—62no—
478C2×D3—122no—
479C2×D3—122no—
480C2×D3—362no—
481C2×D3—722no—
482A4—31no—
483A4—61no—
484C3×C4—182no—
485C3×C4—182no—
486C3×C4—182no—
487Dic3—62no—
488Dic3—62no—
489Dic3—62no—
490Dic3—362no—
491C32—41no—
492C32—81no—
493C32C2×GL(2,ℤ/4)11yescharacteristic, complemented
494C23—22no—
495C23—31no—
496C23—31no—
497C23—32no—
498C23—32no—
499C23—34no—
500C23—31no—
501C23—32no—
502C23—32no—
503C23—62no—
504C23—62no—
505C23—62no—
506C23—61no—
507C23—64no—
508C23—62no—
509C23—64no—
510C23—92no—
511C23—181no—
512C23—182no—
513C23—182no—
514C23—181no—
515C23#638⋊C312yescomplemented
516C23(C3⋊S4)⋊C311yescharacteristic, complemented
517C23C22×(C3⋊D3)⋊C311yescharacteristic
518D4—184no—
519D4—182no—
520D4—188no—
521D4—184no—
522D4—184no—
523D4—182no—
524C2×C4—92no—
525C2×C4—92no—
526C2×C4—92no—
527C2×C4—182no—
528C2×C4—182no—
529C2×C4—181no—
530C2×C4—181no—
531C2×C3—22no—
532C2×C3—31no—
533C2×C3—31no—
534C2×C3—32no—
535C2×C3—31no—
536C2×C3—31no—
537C2×C3—32no—
538C2×C3—31no—
539C2×C3—32no—
540C2×C3—62no—
541C2×C3—61no—
542C2×C3—61no—
543C2×C3—62no—
544C2×C3—64no—
545C2×C3—62no—
546C2×C3—62no—
547C2×C3—121no—
548C2×C3—122no—
549C2×C3—242no—
550C2×C3—244no—
551C2×C3—241no—
552C2×C3—242no—
553C2×C3—362no—
554C2×C3C3×GL(2,ℤ/4)12yes—
555C2×C3C22×C3×S411yescharacteristic
556D3—122no—
557D3—722no—
558C22—22no—
559C22—32no—
560C22—31no—
561C22—31no—
562C22—31no—
563C22—32no—
564C22—32no—
565C22—34no—
566C22—32no—
567C22—62no—
568C22—62no—
569C22—64no—
570C22—61no—
571C22—62no—
572C22—64no—
573C22—62no—
574C22—182no—
575C22—182no—
576C22—182no—
577C22—362no—
578C22—362no—
579C22C2×#638⋊C311yescharacteristic, complemented
580C22C2×(C3⋊S4)⋊C311yescharacteristic
581C22C2×(C3⋊S4)⋊C312yescomplemented
582C4—182no—
583C4—182no—
584C4—182no—
585C3—31no—
586C3—121no—
587C3—241no—
588C3C2×C3×GL(2,ℤ/4)11yescharacteristic
589C2—22no—
590C2—31no—
591C2—31no—
592C2—32no—
593C2—62no—
594C2—362no—
595C2#64269⋊C312yesdirect product factor
596C2C22×(C3⋊S4)⋊C311yescharacteristic
subgroupG/H
centerC22C2×(C3⋊S4)⋊C3
commutator, cosocleC2×C3×A4C23×C3
FrattiniC2×C3C22×C3×S4
FittingC25×C32D3
socleC24×C3C2×C3×D3
hypercenterC23(C3⋊S4)⋊C3
hypocenterC3×A4C2×C3×D4
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊H(3,𝔽3)C2semidirect
2. C22×(C3⋊S4)⋊C3C2semidirect
3. #64269⋊C3C2direct (N × H)
4. C2×(C3⋊SL(2,ℤ/4))⋊C3C2semidirect
5. C2×#64269C3semidirect
9. C22×C22⋊H(3,𝔽3)C22semidirect
11. C2×(C3⋊S4)⋊C3C22semidirect
13. (C3⋊SL(2,ℤ/4))⋊C3C22semidirect
14. C25×C32D3semidirect
16. C23×C3×A4C2×C3semidirect
17. C22×C3⋊S4C2×C3semidirect
21. #64269C2×C3semidirect
22. C2×C3⋊SL(2,ℤ/4)C2×C3semidirect
31. C2×C22⋊H(3,𝔽3)D4semidirect
45. C24×C32C2×D3semidirect
52. C22×C3×A4C22×C3semidirect
54. C2×C3⋊S4C22×C3semidirect
69. C3⋊SL(2,ℤ/4)C22×C3semidirect
79. C22⋊H(3,𝔽3)C2×D4semidirect
122. C23×C32#67semidirect
123. C23×C32S4semidirect
135. C2×C3×A4C3×D4semidirect
244. C22×C32C2×#67semidirect
246. C22×C32C2×S4semidirect
255. C3×A4C2×C3×D4semidirect
260. C25(C3⋊D3)⋊C3semidirect
382. C2×C32GL(2,ℤ/4)semidirect
393. C24C2×(C3⋊D3)⋊C3semidirect
493. C32C2×GL(2,ℤ/4)semidirect
515. C23#638⋊C3semidirect
516. C23(C3⋊S4)⋊C3semidirect
579. C22C2×#638⋊C3semidirect
581. C22C2×(C3⋊S4)⋊C3semidirect
595. C2#64269⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊H(3,𝔽3)2
C22×(C3⋊S4)⋊C32
#64269⋊C32
C2×(C3⋊SL(2,ℤ/4))⋊C32
C2×#642693
C2×C3×#9533
C2×#638⋊C34