Finite Group Databasebeta

← C3×#85494 #11922780

Subgroups — C3×#85494 #11922780

Order 1152.

1056 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 92 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
1C2×C3×D4×A4C211yesmaximal, characteristic, complemented
2C2×C3×GL(2,ℤ/4)C212yesmaximal, complemented
3C2×C3×C4×S4C211yesmaximal, characteristic, complemented
4C3×#5432C212yesmaximal, complemented
5C3×C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
6#85494C311yesmaximal, characteristic, direct product factor
7C3×#2408—31nomaximal
8C23×C3×A4C2212yes—
9C22×C3×S4C2211yescharacteristic
10C3×D4×A4—22no—
11C2×C3×C4×A4C2211yescharacteristic
12C3×GL(2,ℤ/4)—24no—
13C2×C3×SL(2,ℤ/4)C2212yes—
14C2×C3×SL(2,ℤ/4)C2211yescharacteristic
15C3×C4×S4—22no—
16C3×#938—41nomaximal
17C23×C3×D4D311yescharacteristic, complemented
18C2×D4×A4—21no—
19C2×D4×A4C2×C311yescharacteristic, complemented
20C2×GL(2,ℤ/4)C2×C312yescomplemented
21C2×C4×S4C2×C311yescharacteristic, complemented
22C2×C3×#121—31no—
23C2×C3×#121—31no—
24C2×C3×#120—32no—
25C2×C3×C4×D4—31no—
26#5432C2×C312yescomplemented
27C4⋊SL(2,ℤ/4)C2×C311yescharacteristic, complemented
28C3×#394—32no—
29C3×#392—32no—
30C3×#390—31no—
31C3×#390—31no—
32C3×#380—31no—
33C3×#379—32no—
34C22×C3×A4—22no—
35C22×C3×A4—22no—
36C22×C3×A4—22no—
37C22×C3×A4C2311yescharacteristic
38C2×C3×S4—21no—
39C2×C3×S4—21no—
40C2×C3×S4D412yescomplemented
41C2×C32×D4—41no—
42C2×C3×#67—42no—
43C2×C3×C4×D3—41no—
44C3×C4×A4D412yescomplemented
45C3×SL(2,ℤ/4)—22no—
46C3×SL(2,ℤ/4)—21no—
47C3×#217—42no—
48C3×C4⋊Dic3—41no—
49#2408—31noSylow
50C25×C3C2×D312yes—
51C23×A4—22no—
52C23×A4C22×C312yes—
53C22×S4C22×C311yescharacteristic
54C22×C3×D4—22no—
55C22×C3×D4—31no—
56C22×C3×D4—31no—
57C22×C3×D4—31no—
58C22×C3×D4—32no—
59C22×C3×D4—32no—
60C22×C3×D4—32no—
61C22×C3×D4—62no—
62C23×C3×C4—31no—
63C23×C3×C4C2×D311yescharacteristic
64D4×A4—24no—
65D4×A4—22no—
66C2×C4×A4—21no—
67C2×C4×A4C22×C311yescharacteristic
68GL(2,ℤ/4)—24no—
69C2×SL(2,ℤ/4)C22×C311yescharacteristic
70C2×SL(2,ℤ/4)C22×C312yes—
71C4×S4—22no—
72C3×#121—64no—
73C3×#121—61no—
74C3×#121—62no—
75C3×#121—61no—
76C3×#120—64no—
77C3×#120—64no—
78C3×C4×D4—62no—
79C3×C4×D4—62no—
80C2×C3×C4⋊C4—31no—
81C2×C3×C4⋊C4—31no—
82C2×C3×C4⋊C4—31no—
83C2×C3×C22⋊C4—32no—
84C2×C3×C22⋊C4—32no—
85C2×C3×C22⋊C4—32no—
86C2×C3×C22⋊C4—32no—
87C2×C3×C22⋊C4—31no—
88C2×C3×C22⋊C4—31no—
89C2×C3×C22⋊C4—32no—
90C2×C3×C22⋊C4—32no—
91C2×C3×C22⋊C4—32no—
92C2×C3×C42—31no—
93#938—41no—
94C3×#95—32no—
95C23×C32—42no—
96C22×C3×D3—41no—
97C2×C3×A4—42no—
98C2×C3×A4C2×D411yescharacteristic
99C2×C3×A4C2×D411yescharacteristic
100C2×C3×A4#4111yescharacteristic
101C3×S4—22no—
102C32×D4—82no—
103C2×C32×C4—41no—
104C3×#67—84no—
105C2×C3×Dic3—42no—
106C2×C3×Dic3—41no—
107C3×C4×D3—82no—
108C23×D4C3×D311yescharacteristic, complemented
109C2×#121—31no—
110C2×#121—31no—
111C2×#120—32no—
112C2×C4×D4—31no—
113#394—32no—
114#392—32no—
115#390—31no—
116#390—31no—
117#380—31no—
118#379—32no—
119C24×C3—22no—
120C24×C3—22no—
121C24×C3—22no—
122C24×C3—32no—
123C24×C3—31no—
124C24×C3—32no—
125C24×C3—62no—
126C24×C3—62no—
127C24×C3—62no—
128C24×C3C22×D311yescharacteristic
129C22×A4—24no—
130C22×A4—22no—
131C22×A4—22no—
132C22×A4—22no—
133C22×A4—21no—
134C22×A4—42no—
135C22×A4—42no—
136C22×A4C23×C311yescharacteristic
137C2×S4—21no—
138C2×S4—21no—
139C2×S4C3×D412yescomplemented
140C2×C3×D4—32no—
141C2×C3×D4—34no—
142C2×C3×D4—32no—
143C2×C3×D4—32no—
144C2×C3×D4—31no—
145C2×C3×D4—41no—
146C2×C3×D4—62no—
147C2×C3×D4—62no—
148C2×C3×D4—61no—
149C2×C3×D4—62no—
150C2×C3×D4—64no—
151C2×C3×D4—61no—
152C2×C3×D4—61no—
153C2×C3×D4—64no—
154C2×C3×D4—61no—
155C2×C3×D4—61no—
156C2×C3×D4—64no—
157C2×C3×D4—64no—
158C2×C3×D4—64no—
159C2×C3×D4—62no—
160C2×C3×D4—62no—
161C2×C3×D4—81no—
162C2×C3×D4—122no—
163C2×C3×D4—122no—
164C2×C3×D4S411yescharacteristic, complemented
165C22×C3×C4—31no—
166C22×C3×C4—31no—
167C22×C3×C4—32no—
168C22×C3×C4—31no—
169C22×C3×C4—32no—
170C22×C3×C4—32no—
171C22×C3×C4—31no—
172C22×C3×C4—31no—
173C22×C3×C4—62no—
174C22×C3×C4—61no—
175C22×C3×C4—62no—
176C22×C3×C4—61no—
177C22×C3×C4#6712yescomplemented
178C2×#67—42no—
179C2×C4×D3—41no—
180C4×A4—22no—
181C4×A4C3×D412yescomplemented
182SL(2,ℤ/4)—21no—
183SL(2,ℤ/4)—22no—
184C3×C4⋊C4—62no—
185C3×C4⋊C4—61no—
186C3×C4⋊C4—62no—
187C3×C4⋊C4—61no—
188C3×C22⋊C4—62no—
189C3×C22⋊C4—64no—
190C3×C22⋊C4—64no—
191C3×C22⋊C4—61no—
192C3×C22⋊C4—62no—
193C3×C22⋊C4—64no—
194C3×C22⋊C4—62no—
195C3×C22⋊C4—62no—
196C3×C22⋊C4—61no—
197C3×C22⋊C4—62no—
198C3×C22⋊C4—122no—
199C3×C22⋊C4—122no—
200C3×C42—62no—
201#217—42no—
202C4⋊Dic3—41no—
203C22×C32—41no—
204C22×C32—82no—
205C22×C32—82no—
206C22×C32—82no—
207C2×C3×D3—42no—
208C2×C3×D3—81no—
209C2×C3×D3—81no—
210C3×A4#12111yescharacteristic, complemented
211C32×C4—42no—
212C3×Dic3—81no—
213C3×Dic3—82no—
214C25C2×C3×D312yes—
215C22×D4—22no—
216C22×D4—31no—
217C22×D4—31no—
218C22×D4—31no—
219C22×D4—32no—
220C22×D4—32no—
221C22×D4—32no—
222C22×D4—62no—
223C23×C4—31no—
224C23×C4C2×C3×D311yescharacteristic
225#121—64no—
226#121—61no—
227#121—62no—
228#121—61no—
229#120—64no—
230#120—64no—
231C4×D4—62no—
232C4×D4—62no—
233C2×C4⋊C4—31no—
234C2×C4⋊C4—31no—
235C2×C4⋊C4—31no—
236C2×C22⋊C4—31no—
237C2×C22⋊C4—32no—
238C2×C22⋊C4—32no—
239C2×C22⋊C4—32no—
240C2×C22⋊C4—32no—
241C2×C22⋊C4—31no—
242C2×C22⋊C4—32no—
243C2×C22⋊C4—32no—
244C2×C22⋊C4—32no—
245C2×C42—31no—
246#95—32no—
247C23×C3—31no—
248C23×C3—31no—
249C23×C3—31no—
250C23×C3—31no—
251C23×C3—34no—
252C23×C3—32no—
253C23×C3—32no—
254C23×C3—42no—
255C23×C3—42no—
256C23×C3—62no—
257C23×C3—62no—
258C23×C3—62no—
259C23×C3—62no—
260C23×C3—62no—
261C23×C3—62no—
262C23×C3—62no—
263C23×C3—62no—
264C23×C3—62no—
265C23×C3—62no—
266C23×C3—61no—
267C23×C3—61no—
268C23×C3—61no—
269C23×C3—61no—
270C23×C3—61no—
271C23×C3—62no—
272C23×C3—61no—
273C23×C3—82no—
274C23×C3—122no—
275C23×C3—122no—
276C23×C3—122no—
277C23×C3—124no—
278C23×C3D3×D411yescharacteristic
279C23×C3#23711yescharacteristic
280C23×C3C2×#6711yescharacteristic
281C23×C3C2×S412yes—
282C22×D3—41no—
283C2×A4—21no—
284C2×A4—21no—
285C2×A4—21no—
286C2×A4—44no—
287C2×A4—42no—
288C2×A4C2×C3×D411yescharacteristic
289C2×A4C2×C3×D411yescharacteristic
290C2×A4C3×#4111yescharacteristic
291S4—22no—
292C3×D4—22no—
293C3×D4—64no—
294C3×D4—64no—
295C3×D4—62no—
296C3×D4—62no—
297C3×D4—62no—
298C3×D4—62no—
299C3×D4—62no—
300C3×D4—82no—
301C3×D4—84no—
302C3×D4—124no—
303C3×D4—128no—
304C3×D4—124no—
305C2×C3×C4—31no—
306C2×C3×C4—32no—
307C2×C3×C4—32no—
308C2×C3×C4—32no—
309C2×C3×C4—32no—
310C2×C3×C4—41no—
311C2×C3×C4—61no—
312C2×C3×C4—61no—
313C2×C3×C4—61no—
314C2×C3×C4—61no—
315C2×C3×C4—61no—
316C2×C3×C4—62no—
317C2×C3×C4—62no—
318C2×C3×C4—61no—
319C2×C3×C4—62no—
320C2×C3×C4—61no—
321C2×C3×C4—61no—
322C2×C3×C4—62no—
323C2×C3×C4—62no—
324C2×C3×C4—61no—
325C2×C3×C4—62no—
326C2×C3×C4—81no—
327C2×C3×C4—122no—
328C2×C3×C4—122no—
329C2×C3×C4C2×S411yescharacteristic
330#67—84no—
331C2×Dic3—42no—
332C2×Dic3—41no—
333C4×D3—82no—
334C2×C32—41no—
335C2×C32—41no—
336C2×C32—41no—
337C2×C32—162no—
338C3×D3—82no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—32no—
343C24—31no—
344C24—32no—
345C24—62no—
346C24—62no—
347C24—62no—
348C24C22×C3×D311yescharacteristic
349C2×D4—32no—
350C2×D4—34no—
351C2×D4—32no—
352C2×D4—32no—
353C2×D4—31no—
354C2×D4—62no—
355C2×D4—62no—
356C2×D4—61no—
357C2×D4—62no—
358C2×D4—64no—
359C2×D4—61no—
360C2×D4—64no—
361C2×D4—61no—
362C2×D4—61no—
363C2×D4—61no—
364C2×D4—64no—
365C2×D4—64no—
366C2×D4—64no—
367C2×D4—62no—
368C2×D4—62no—
369C2×D4—122no—
370C2×D4—122no—
371C2×D4C3×S411yescharacteristic, complemented
372C22×C4—31no—
373C22×C4—31no—
374C22×C4—32no—
375C22×C4—31no—
376C22×C4—32no—
377C22×C4—32no—
378C22×C4—31no—
379C22×C4—31no—
380C22×C4—62no—
381C22×C4—61no—
382C22×C4—62no—
383C22×C4—61no—
384C22×C4C3×#6712yescomplemented
385C4⋊C4—62no—
386C4⋊C4—61no—
387C4⋊C4—62no—
388C4⋊C4—61no—
389C22⋊C4—62no—
390C22⋊C4—64no—
391C22⋊C4—64no—
392C22⋊C4—61no—
393C22⋊C4—62no—
394C22⋊C4—64no—
395C22⋊C4—62no—
396C22⋊C4—62no—
397C22⋊C4—62no—
398C22⋊C4—61no—
399C22⋊C4—122no—
400C22⋊C4—122no—
401C42—62no—
402C22×C3—22no—
403C22×C3—22no—
404C22×C3—22no—
405C22×C3—31no—
406C22×C3—31no—
407C22×C3—31no—
408C22×C3—31no—
409C22×C3—31no—
410C22×C3—31no—
411C22×C3—31no—
412C22×C3—31no—
413C22×C3—31no—
414C22×C3—41no—
415C22×C3—61no—
416C22×C3—62no—
417C22×C3—62no—
418C22×C3—64no—
419C22×C3—64no—
420C22×C3—62no—
421C22×C3—62no—
422C22×C3—62no—
423C22×C3—62no—
424C22×C3—82no—
425C22×C3—82no—
426C22×C3—82no—
427C22×C3—84no—
428C22×C3—81no—
429C22×C3—122no—
430C22×C3—122no—
431C22×C3—122no—
432C22×C3—122no—
433C22×C3—122no—
434C22×C3—122no—
435C22×C3—121no—
436C22×C3—121no—
437C22×C3—121no—
438C22×C3—121no—
439C22×C3—162no—
440C22×C3—162no—
441C22×C3#93811yescharacteristic, complemented
442C22×C3C22×S411yescharacteristic
443C2×D3—42no—
444C2×D3—81no—
445C2×D3—81no—
446A4—21no—
447A4C3×#12111yescharacteristic, complemented
448C3×C4—32no—
449C3×C4—42no—
450C3×C4—62no—
451C3×C4—82no—
452C3×C4—121no—
453C3×C4—122no—
454C3×C4—122no—
455C3×C4—121no—
456C3×C4GL(2,ℤ/4)12yescomplemented
457Dic3—81no—
458Dic3—82no—
459C32—41noSylow
460C23—31no—
461C23—31no—
462C23—31no—
463C23—31no—
464C23—34no—
465C23—32no—
466C23—32no—
467C23—42no—
468C23—62no—
469C23—62no—
470C23—62no—
471C23—62no—
472C23—62no—
473C23—62no—
474C23—62no—
475C23—62no—
476C23—62no—
477C23—62no—
478C23—61no—
479C23—61no—
480C23—61no—
481C23—61no—
482C23—61no—
483C23—62no—
484C23—61no—
485C23—122no—
486C23—122no—
487C23—122no—
488C23—124no—
489C23C3×D3×D411yescharacteristic
490C23C3×#23711yescharacteristic
491C23C2×C3×#6711yescharacteristic
492C23C2×C3×S412yes—
493D4—22no—
494D4—64no—
495D4—64no—
496D4—62no—
497D4—62no—
498D4—62no—
499D4—62no—
500D4—62no—
501D4—124no—
502D4—128no—
503D4—124no—
504C2×C4—31no—
505C2×C4—32no—
506C2×C4—32no—
507C2×C4—32no—
508C2×C4—32no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—61no—
514C2×C4—62no—
515C2×C4—62no—
516C2×C4—61no—
517C2×C4—62no—
518C2×C4—61no—
519C2×C4—61no—
520C2×C4—62no—
521C2×C4—62no—
522C2×C4—61no—
523C2×C4—62no—
524C2×C4—122no—
525C2×C4—122no—
526C2×C4C2×C3×S411yescharacteristic
527C2×C3—31no—
528C2×C3—31no—
529C2×C3—31no—
530C2×C3—31no—
531C2×C3—42no—
532C2×C3—41no—
533C2×C3—41no—
534C2×C3—41no—
535C2×C3—81no—
536C2×C3—81no—
537C2×C3—81no—
538C2×C3—122no—
539C2×C3—122no—
540C2×C3—162no—
541C2×C3—164no—
542C2×C3D4×S411yescharacteristic
543C2×C3#591411yescharacteristic
544C2×C3C2×GL(2,ℤ/4)11yescharacteristic
545D3—82no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—31no—
550C22—31no—
551C22—31no—
552C22—31no—
553C22—31no—
554C22—31no—
555C22—31no—
556C22—31no—
557C22—31no—
558C22—61no—
559C22—62no—
560C22—62no—
561C22—64no—
562C22—64no—
563C22—62no—
564C22—62no—
565C22—62no—
566C22—62no—
567C22—122no—
568C22—122no—
569C22—122no—
570C22—122no—
571C22—122no—
572C22—122no—
573C22—121no—
574C22—121no—
575C22—121no—
576C22—121no—
577C22C3×#93811yescharacteristic, complemented
578C22C22×C3×S411yescharacteristic
579C4—32no—
580C4—62no—
581C4—122no—
582C4—121no—
583C4—122no—
584C4—121no—
585C4C3×GL(2,ℤ/4)12yescomplemented
586C3—41no—
587C3—81no—
588C3#8549411yescharacteristic, direct product factor
589C2—31no—
590C2—31no—
591C2—31no—
592C2—31no—
593C2—42no—
594C2—122no—
595C2—122no—
596C2C3×D4×S411yescharacteristic
597C2C3×#591411yescharacteristic
598C2C2×C3×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC22×C3C22×S4
commutator, cosocleC22×A4C23×C3
FrattiniC22C22×C3×S4
FittingC23×C3×D4D3
socleC24×C3C22×D3
hypercenterC2×C3×D4S4
hypocenterA4C3×#121
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×D4×A4C2semidirect
2. C2×C3×GL(2,ℤ/4)C2semidirect
3. C2×C3×C4×S4C2semidirect
4. C3×#5432C2semidirect
5. C3×C4⋊SL(2,ℤ/4)C2semidirect
6. #85494C3direct (N × H)
17. C23×C3×D4D3semidirect
19. C2×D4×A4C2×C3semidirect
20. C2×GL(2,ℤ/4)C2×C3semidirect
21. C2×C4×S4C2×C3semidirect
26. #5432C2×C3semidirect
27. C4⋊SL(2,ℤ/4)C2×C3semidirect
40. C2×C3×S4D4semidirect
44. C3×C4×A4D4semidirect
108. C23×D4C3×D3semidirect
139. C2×S4C3×D4semidirect
164. C2×C3×D4S4semidirect
177. C22×C3×C4#67semidirect
181. C4×A4C3×D4semidirect
210. C3×A4#121semidirect
371. C2×D4C3×S4semidirect
384. C22×C4C3×#67semidirect
441. C22×C3#938semidirect
447. A4C3×#121semidirect
456. C3×C4GL(2,ℤ/4)semidirect
577. C22C3×#938semidirect
585. C4C3×GL(2,ℤ/4)semidirect
588. C3#85494direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×A42
C2×C3×GL(2,ℤ/4)2
C2×C3×C4×S42
C3×#54322
C3×C4⋊SL(2,ℤ/4)2
#854943
C3×#24083
C3×#9384