Finite Group Databasebeta

← C3×#85298 #11922583

Subgroups — C3×#85298 #11922583

Order 1152.

788 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 788 conjugacy classes of subgroups — too many to draw. Only its 80 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×GL(2,ℤ/4)C211yesmaximal, characteristic
2C2×C3×C4×S4C211yesmaximal, characteristic, complemented
3C3×A4×C22⋊C4C211yesmaximal, characteristic, complemented
4C3×#5432C211yesmaximal, characteristic, complemented
5C3×(C2×S4)⋊C4C211yesmaximal, characteristic, complemented
6C3×C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
7C3×SL(2,ℤ/4)⋊C4C211yesmaximal, characteristic, complemented
8#85298C311yesmaximal, characteristic, direct product factor
9C3×#2418—31nomaximal
10C23×C3×A4C2211yescharacteristic
11C22×C3×S4C2211yescharacteristic
12C2×C3×C4×A4C2211yescharacteristic
13C2×C3×C4×A4C2211yescharacteristic
14C3×GL(2,ℤ/4)—22no—
15C2×C3×SL(2,ℤ/4)C2211yescharacteristic
16C2×C3×SL(2,ℤ/4)C2211yescharacteristic
17C2×C3×SL(2,ℤ/4)C2211yescharacteristic
18C3×C4×S4—22no—
19C3×#883—41nomaximal
20C2×GL(2,ℤ/4)C2×C311yescharacteristic
21C2×C4×S4C2×C311yescharacteristic, complemented
22C2×C3×#123—31no—
23C2×C3×#120—31no—
24C2×C3×C4×D4—31no—
25C22×C3×C22⋊C4D311yescharacteristic, complemented
26A4×C22⋊C4—21no—
27A4×C22⋊C4C2×C311yescharacteristic, complemented
28#5432C2×C311yescharacteristic, complemented
29(C2×S4)⋊C4C2×C311yescharacteristic, complemented
30C4⋊SL(2,ℤ/4)C2×C311yescharacteristic, complemented
31SL(2,ℤ/4)⋊C4C2×C311yescharacteristic, complemented
32C3×#399—31no—
33C3×#397—31no—
34C3×#394—31no—
35C3×#394—31no—
36C3×#388—31no—
37C3×#388—31no—
38C3×#386—31no—
39C3×#386—31no—
40C3×#385—31no—
41C3×#380—31no—
42C3×#379—31no—
43C22×C3×A4—21no—
44C22×C3×A4—21no—
45C22×C3×A4—21no—
46C22×C3×A4C2311yescharacteristic
47C2×C3×S4—21no—
48C2×C3×S4—21no—
49C2×C3×S4D412yes—
50C2×C3×#67—41no—
51C2×C3×C4×D3—41no—
52C3×C4×A4—21no—
53C3×C4×A4—21no—
54C3×SL(2,ℤ/4)—21no—
55C3×SL(2,ℤ/4)—21no—
56C3×SL(2,ℤ/4)—21no—
57C32×C22⋊C4—41no—
58C3×#217—41no—
59C3×(C2×D3)⋊C4—41no—
60C3×C4⋊Dic3—41no—
61C3×Dic3⋊C4—41no—
62#2418—31noSylow
63C25×C3C2×D311yescharacteristic
64C23×A4—21no—
65C23×A4C22×C311yescharacteristic
66C22×S4C22×C311yescharacteristic
67C22×C3×D4—31no—
68C22×C3×D4—31no—
69C22×C3×D4—31no—
70C23×C3×C4—31no—
71C23×C3×C4C2×D311yescharacteristic
72C23×C3×C4C2×D311yescharacteristic
73C2×C4×A4—21no—
74C2×C4×A4—21no—
75C2×C4×A4C22×C311yescharacteristic
76C2×C4×A4C22×C311yescharacteristic
77GL(2,ℤ/4)—22no—
78C2×SL(2,ℤ/4)C22×C311yescharacteristic
79C2×SL(2,ℤ/4)C22×C311yescharacteristic
80C2×SL(2,ℤ/4)C22×C311yescharacteristic
81C4×S4—22no—
82C3×#123—61no—
83C3×#123—61no—
84C3×#123—61no—
85C3×#123—61no—
86C3×#120—62no—
87C3×#120—62no—
88C3×C4×D4—62no—
89C3×C4×D4—62no—
90C2×C3×C4⋊C4—31no—
91C2×C3×C4⋊C4—31no—
92C2×C3×C4⋊C4—31no—
93C2×C3×C4⋊C4—31no—
94C2×C3×C4⋊C4—31no—
95C2×C3×C22⋊C4—31no—
96C2×C3×C22⋊C4—31no—
97C2×C3×C22⋊C4—31no—
98C2×C3×C22⋊C4—31no—
99C2×C3×C22⋊C4—31no—
100C2×C3×C22⋊C4—31no—
101C2×C3×C22⋊C4—31no—
102C2×C3×C22⋊C4—31no—
103C2×C3×C22⋊C4—31no—
104C2×C3×C22⋊C4—31no—
105C2×C3×C22⋊C4—31no—
106C2×C3×C22⋊C4—31no—
107C2×C3×C22⋊C4—31no—
108C2×C3×C22⋊C4—31no—
109C2×C3×C22⋊C4—31no—
110C2×C3×C22⋊C4—31no—
111C2×C3×C42—31no—
112#883—41no—
113C3×#95—31no—
114C3×#95—31no—
115C3×#95—31no—
116C3×#95—31no—
117C3×#95—31no—
118C3×#95—31no—
119C23×C32—41no—
120C22×C3×D3—41no—
121C2×C3×A4—41no—
122C2×C3×A4C2×D411yescharacteristic
123C2×C3×A4#4111yescharacteristic
124C2×C3×A4#4111yescharacteristic
125C3×S4—22no—
126C2×C32×C4—41no—
127C2×C32×C4—41no—
128C3×#67—82no—
129C2×C3×Dic3—41no—
130C2×C3×Dic3—41no—
131C2×C3×Dic3—41no—
132C3×C4×D3—82no—
133C2×#123—31no—
134C2×#120—31no—
135C2×C4×D4—31no—
136C22×C22⋊C4C3×D311yescharacteristic, complemented
137#399—31no—
138#397—31no—
139#394—31no—
140#394—31no—
141#388—31no—
142#388—31no—
143#386—31no—
144#386—31no—
145#385—31no—
146#380—31no—
147#379—31no—
148C24×C3—21no—
149C24×C3—21no—
150C24×C3—21no—
151C24×C3—31no—
152C24×C3—31no—
153C24×C3—31no—
154C24×C3—61no—
155C24×C3—61no—
156C24×C3—61no—
157C24×C3C22×D311yescharacteristic
158C22×A4—22no—
159C22×A4—21no—
160C22×A4—21no—
161C22×A4—21no—
162C22×A4—21no—
163C22×A4—41no—
164C22×A4—41no—
165C22×A4C23×C311yescharacteristic
166C2×S4—21no—
167C2×S4—21no—
168C2×S4C3×D412yes—
169C2×C3×D4—32no—
170C2×C3×D4—32no—
171C2×C3×D4—61no—
172C2×C3×D4—61no—
173C2×C3×D4—61no—
174C2×C3×D4—61no—
175C2×C3×D4—62no—
176C2×C3×D4—62no—
177C2×C3×D4—61no—
178C2×C3×D4—62no—
179C2×C3×D4—61no—
180C2×C3×D4—121no—
181C2×C3×D4—121no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—31no—
185C22×C3×C4—31no—
186C22×C3×C4—31no—
187C22×C3×C4—31no—
188C22×C3×C4—31no—
189C22×C3×C4—31no—
190C22×C3×C4—31no—
191C22×C3×C4—31no—
192C22×C3×C4—31no—
193C22×C3×C4—31no—
194C22×C3×C4—31no—
195C22×C3×C4—61no—
196C22×C3×C4—61no—
197C22×C3×C4—61no—
198C22×C3×C4—62no—
199C22×C3×C4—61no—
200C22×C3×C4—121no—
201C2×#67—41no—
202C2×C4×D3—41no—
203C4×A4—21no—
204C4×A4—21no—
205C4×A4—41no—
206C4×A4—41no—
207SL(2,ℤ/4)—21no—
208SL(2,ℤ/4)—21no—
209SL(2,ℤ/4)—21no—
210C3×C4⋊C4—62no—
211C3×C4⋊C4—61no—
212C3×C4⋊C4—61no—
213C3×C4⋊C4—61no—
214C3×C4⋊C4—61no—
215C3×C4⋊C4—121no—
216C3×C4⋊C4—121no—
217C3×C22⋊C4—31no—
218C3×C22⋊C4—31no—
219C3×C22⋊C4—31no—
220C3×C22⋊C4—41no—
221C3×C22⋊C4—61no—
222C3×C22⋊C4—62no—
223C3×C22⋊C4—62no—
224C3×C22⋊C4—61no—
225C3×C22⋊C4—61no—
226C3×C22⋊C4—62no—
227C3×C22⋊C4—62no—
228C3×C22⋊C4—61no—
229C3×C22⋊C4—61no—
230C3×C22⋊C4—61no—
231C3×C22⋊C4—61no—
232C3×C22⋊C4—61no—
233C3×C22⋊C4—61no—
234C3×C22⋊C4—61no—
235C3×C22⋊C4—61no—
236C3×C22⋊C4—81no—
237C3×C22⋊C4—121no—
238C3×C22⋊C4—121no—
239C3×C22⋊C4—121no—
240C3×C22⋊C4S411yescharacteristic, complemented
241C3×C42—62no—
242#217—41no—
243(C2×D3)⋊C4—41no—
244C4⋊Dic3—41no—
245Dic3⋊C4—41no—
246C22×C32—41no—
247C22×C32—81no—
248C22×C32—81no—
249C22×C32—81no—
250C2×C3×D3—42no—
251C2×C3×D3—81no—
252C2×C3×D3—81no—
253C3×A4#12311yescharacteristic, complemented
254C32×C4—81no—
255C32×C4—81no—
256C3×Dic3—81no—
257C3×Dic3—81no—
258C3×Dic3—81no—
259C25C2×C3×D311yescharacteristic
260C22×D4—31no—
261C22×D4—31no—
262C22×D4—31no—
263C23×C4—31no—
264C23×C4C2×C3×D311yescharacteristic
265C23×C4C2×C3×D311yescharacteristic
266#123—61no—
267#123—61no—
268#123—61no—
269#123—61no—
270#120—62no—
271#120—62no—
272C4×D4—62no—
273C4×D4—62no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—31no—
279C2×C22⋊C4—31no—
280C2×C22⋊C4—31no—
281C2×C22⋊C4—31no—
282C2×C22⋊C4—31no—
283C2×C22⋊C4—31no—
284C2×C22⋊C4—31no—
285C2×C22⋊C4—31no—
286C2×C22⋊C4—31no—
287C2×C22⋊C4—31no—
288C2×C22⋊C4—31no—
289C2×C22⋊C4—31no—
290C2×C22⋊C4—31no—
291C2×C22⋊C4—31no—
292C2×C22⋊C4—31no—
293C2×C22⋊C4—31no—
294C2×C22⋊C4—31no—
295C2×C42—31no—
296#95—31no—
297#95—31no—
298#95—31no—
299#95—31no—
300#95—31no—
301#95—31no—
302C23×C3—31no—
303C23×C3—31no—
304C23×C3—31no—
305C23×C3—31no—
306C23×C3—32no—
307C23×C3—32no—
308C23×C3—31no—
309C23×C3—41no—
310C23×C3—41no—
311C23×C3—61no—
312C23×C3—61no—
313C23×C3—61no—
314C23×C3—61no—
315C23×C3—61no—
316C23×C3—61no—
317C23×C3—61no—
318C23×C3—61no—
319C23×C3—61no—
320C23×C3—61no—
321C23×C3—61no—
322C23×C3—61no—
323C23×C3—61no—
324C23×C3—61no—
325C23×C3—61no—
326C23×C3—61no—
327C23×C3—61no—
328C23×C3—81no—
329C23×C3—121no—
330C23×C3—121no—
331C23×C3—121no—
332C23×C3—122no—
333C23×C3#23511yescharacteristic
334C23×C3D3×D411yescharacteristic
335C23×C3#23711yescharacteristic
336C23×C3C2×S411yescharacteristic
337C22×D3—41no—
338C2×A4—21no—
339C2×A4—21no—
340C2×A4—21no—
341C2×A4—42no—
342C2×A4—41no—
343C2×A4C2×C3×D411yescharacteristic
344C2×A4C3×#4111yescharacteristic
345C2×A4C3×#4111yescharacteristic
346S4—22no—
347C3×D4—62no—
348C3×D4—62no—
349C3×D4—62no—
350C3×D4—62no—
351C3×D4—122no—
352C3×D4—124no—
353C3×D4—122no—
354C2×C3×C4—32no—
355C2×C3×C4—31no—
356C2×C3×C4—31no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—61no—
360C2×C3×C4—61no—
361C2×C3×C4—61no—
362C2×C3×C4—61no—
363C2×C3×C4—61no—
364C2×C3×C4—61no—
365C2×C3×C4—61no—
366C2×C3×C4—61no—
367C2×C3×C4—61no—
368C2×C3×C4—61no—
369C2×C3×C4—61no—
370C2×C3×C4—61no—
371C2×C3×C4—61no—
372C2×C3×C4—61no—
373C2×C3×C4—61no—
374C2×C3×C4—61no—
375C2×C3×C4—61no—
376C2×C3×C4—61no—
377C2×C3×C4—61no—
378C2×C3×C4—61no—
379C2×C3×C4—61no—
380C2×C3×C4—61no—
381C2×C3×C4—61no—
382C2×C3×C4—61no—
383C2×C3×C4—61no—
384C2×C3×C4—61no—
385C2×C3×C4—61no—
386C2×C3×C4—61no—
387C2×C3×C4—81no—
388C2×C3×C4—81no—
389C2×C3×C4—122no—
390C2×C3×C4—122no—
391C2×C3×C4C2×S411yescharacteristic
392C2×C3×C4C2×S411yescharacteristic
393#67—82no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C4×D3—82no—
398C2×C32—41no—
399C2×C32—41no—
400C2×C32—41no—
401C2×C32—161no—
402C3×D3—82no—
403C24—21no—
404C24—21no—
405C24—21no—
406C24—31no—
407C24—31no—
408C24—31no—
409C24—61no—
410C24—61no—
411C24—61no—
412C24C22×C3×D311yescharacteristic
413C2×D4—32no—
414C2×D4—32no—
415C2×D4—61no—
416C2×D4—61no—
417C2×D4—61no—
418C2×D4—61no—
419C2×D4—62no—
420C2×D4—62no—
421C2×D4—61no—
422C2×D4—62no—
423C2×D4—61no—
424C2×D4—121no—
425C2×D4—121no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—31no—
429C22×C4—31no—
430C22×C4—31no—
431C22×C4—31no—
432C22×C4—31no—
433C22×C4—31no—
434C22×C4—31no—
435C22×C4—31no—
436C22×C4—31no—
437C22×C4—31no—
438C22×C4—31no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—62no—
443C22×C4—61no—
444C22×C4—121no—
445C4⋊C4—62no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—121no—
451C4⋊C4—121no—
452C22⋊C4—31no—
453C22⋊C4—31no—
454C22⋊C4—31no—
455C22⋊C4—61no—
456C22⋊C4—62no—
457C22⋊C4—62no—
458C22⋊C4—61no—
459C22⋊C4—62no—
460C22⋊C4—61no—
461C22⋊C4—61no—
462C22⋊C4—61no—
463C22⋊C4—61no—
464C22⋊C4—62no—
465C22⋊C4—61no—
466C22⋊C4—61no—
467C22⋊C4—61no—
468C22⋊C4—61no—
469C22⋊C4—61no—
470C22⋊C4—121no—
471C22⋊C4—121no—
472C22⋊C4—121no—
473C22⋊C4C3×S411yescharacteristic, complemented
474C42—62no—
475C22×C3—21no—
476C22×C3—21no—
477C22×C3—21no—
478C22×C3—31no—
479C22×C3—31no—
480C22×C3—31no—
481C22×C3—31no—
482C22×C3—31no—
483C22×C3—31no—
484C22×C3—31no—
485C22×C3—31no—
486C22×C3—31no—
487C22×C3—41no—
488C22×C3—61no—
489C22×C3—62no—
490C22×C3—62no—
491C22×C3—62no—
492C22×C3—62no—
493C22×C3—61no—
494C22×C3—62no—
495C22×C3—61no—
496C22×C3—61no—
497C22×C3—81no—
498C22×C3—81no—
499C22×C3—81no—
500C22×C3—82no—
501C22×C3—81no—
502C22×C3—121no—
503C22×C3—121no—
504C22×C3—121no—
505C22×C3—121no—
506C22×C3—121no—
507C22×C3—121no—
508C22×C3—121no—
509C22×C3—121no—
510C22×C3—121no—
511C22×C3—121no—
512C22×C3—161no—
513C22×C3—161no—
514C22×C3#88311yescharacteristic, complemented
515C22×C3C22×S411yescharacteristic
516C2×D3—42no—
517C2×D3—81no—
518C2×D3—81no—
519A4—21no—
520A4C3×#12311yescharacteristic, complemented
521C3×C4—21no—
522C3×C4—21no—
523C3×C4—62no—
524C3×C4—61no—
525C3×C4—61no—
526C3×C4—81no—
527C3×C4—81no—
528C3×C4—121no—
529C3×C4—121no—
530C3×C4—121no—
531C3×C4—121no—
532C3×C4—121no—
533C3×C4—121no—
534C3×C4—161no—
535C3×C4—161no—
536Dic3—81no—
537Dic3—81no—
538Dic3—81no—
539C32—41noSylow
540C23—31no—
541C23—31no—
542C23—31no—
543C23—31no—
544C23—32no—
545C23—32no—
546C23—31no—
547C23—41no—
548C23—61no—
549C23—61no—
550C23—61no—
551C23—61no—
552C23—61no—
553C23—61no—
554C23—61no—
555C23—61no—
556C23—61no—
557C23—61no—
558C23—61no—
559C23—61no—
560C23—61no—
561C23—61no—
562C23—61no—
563C23—61no—
564C23—61no—
565C23—121no—
566C23—121no—
567C23—121no—
568C23—122no—
569C23C3×#23511yescharacteristic
570C23C3×D3×D411yescharacteristic
571C23C3×#23711yescharacteristic
572C23C2×C3×S411yescharacteristic
573D4—62no—
574D4—62no—
575D4—62no—
576D4—62no—
577D4—122no—
578D4—124no—
579D4—122no—
580C2×C4—32no—
581C2×C4—31no—
582C2×C4—31no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—61no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—61no—
604C2×C4—61no—
605C2×C4—61no—
606C2×C4—61no—
607C2×C4—61no—
608C2×C4—61no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—122no—
612C2×C4—122no—
613C2×C4C2×C3×S411yescharacteristic
614C2×C4C2×C3×S411yescharacteristic
615C2×C3—31no—
616C2×C3—31no—
617C2×C3—31no—
618C2×C3—31no—
619C2×C3—41no—
620C2×C3—41no—
621C2×C3—41no—
622C2×C3—41no—
623C2×C3—81no—
624C2×C3—81no—
625C2×C3—81no—
626C2×C3—121no—
627C2×C3—122no—
628C2×C3—161no—
629C2×C3—162no—
630C2×C3#591211yescharacteristic
631C2×C3D4×S411yescharacteristic
632C2×C3#591411yescharacteristic
633D3—82no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—31no—
638C22—31no—
639C22—31no—
640C22—31no—
641C22—31no—
642C22—31no—
643C22—31no—
644C22—31no—
645C22—31no—
646C22—61no—
647C22—62no—
648C22—62no—
649C22—62no—
650C22—62no—
651C22—61no—
652C22—62no—
653C22—61no—
654C22—61no—
655C22—121no—
656C22—121no—
657C22—121no—
658C22—121no—
659C22—121no—
660C22—121no—
661C22—121no—
662C22—121no—
663C22—121no—
664C22—121no—
665C22C3×#88311yescharacteristic, complemented
666C22C22×C3×S411yescharacteristic
667C4—21no—
668C4—21no—
669C4—62no—
670C4—61no—
671C4—61no—
672C4—121no—
673C4—121no—
674C4—121no—
675C4—121no—
676C4—121no—
677C4—121no—
678C3—41no—
679C3—81no—
680C3#8529811yescharacteristic, direct product factor
681C2—31no—
682C2—31no—
683C2—31no—
684C2—31no—
685C2—41no—
686C2—121no—
687C2—122no—
688C2C3×#591211yescharacteristic
689C2C3×D4×S411yescharacteristic
690C2C3×#591411yescharacteristic
subgroupG/H
centerC22×C3C22×S4
commutator, cosocleC22×A4C23×C3
FrattiniC22C22×C3×S4
FittingC22×C3×C22⋊C4D3
socleC24×C3C22×D3
prime radicalC2×C3×GL(2,ℤ/4)C2
hypercenterC3×C22⋊C4S4
hypocenterA4C3×#123
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22×C3 ◁ C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4×S4C2semidirect
3. C3×A4×C22⋊C4C2semidirect
4. C3×#5432C2semidirect
5. C3×(C2×S4)⋊C4C2semidirect
6. C3×C4⋊SL(2,ℤ/4)C2semidirect
7. C3×SL(2,ℤ/4)⋊C4C2semidirect
8. #85298C3direct (N × H)
21. C2×C4×S4C2×C3semidirect
25. C22×C3×C22⋊C4D3semidirect
27. A4×C22⋊C4C2×C3semidirect
28. #5432C2×C3semidirect
29. (C2×S4)⋊C4C2×C3semidirect
30. C4⋊SL(2,ℤ/4)C2×C3semidirect
31. SL(2,ℤ/4)⋊C4C2×C3semidirect
136. C22×C22⋊C4C3×D3semidirect
240. C3×C22⋊C4S4semidirect
253. C3×A4#123semidirect
473. C22⋊C4C3×S4semidirect
514. C22×C3#883semidirect
520. A4C3×#123semidirect
665. C22C3×#883semidirect
680. C3#85298direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×GL(2,ℤ/4)2
C2×C3×C4×S42
C3×A4×C22⋊C42
C3×#54322
C3×(C2×S4)⋊C42
C3×C4⋊SL(2,ℤ/4)2
C3×SL(2,ℤ/4)⋊C42
#852983
C3×#24183
C3×#8834