Finite Group Databasebeta

← C3×#85311 #11922587

Subgroups — C3×#85311 #11922587

Order 1152.

700 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 84 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×C4×S4C211yesmaximal, characteristic
2C2×C3×A4⋊Q8C211yesmaximal, characteristic, complemented
3C3×A4×C4⋊C4C211yesmaximal, characteristic, complemented
4C3×(C2×S4)⋊C4C211yesmaximal, characteristic
5C3×(C2×S4)⋊C4C211yesmaximal, characteristic
6C3×SL(2,ℤ/4)⋊C4C211yesmaximal, characteristic, complemented
7C3×SL(2,ℤ/4)⋊C4C211yesmaximal, characteristic, complemented
8#85311C311yesmaximal, characteristic, direct product factor
9C3×#2449—31nomaximal
10C22×C3×S4C2211yescharacteristic
11C2×C3×C4×A4C2211yescharacteristic
12C2×C3×C4×A4C2211yescharacteristic
13C2×C3×C4×A4C2211yescharacteristic
14C2×C3×SL(2,ℤ/4)C2211yescharacteristic
15C2×C3×SL(2,ℤ/4)C2211yescharacteristic
16C2×C3×SL(2,ℤ/4)C2211yescharacteristic
17C3×C4×S4—22no—
18C3×A4⋊Q8—22no—
19C3×#896—41nomaximal
20C2×C4×S4C2×C311yescharacteristic
21C2×A4⋊Q8C2×C311yescharacteristic, complemented
22C2×C3×C22⋊Q8—31no—
23C2×C3×C22⋊Q8—31no—
24C2×C3×C4×D4—31no—
25C22×C3×C4⋊C4D311yescharacteristic, complemented
26A4×C4⋊C4—21no—
27A4×C4⋊C4C2×C311yescharacteristic, complemented
28(C2×S4)⋊C4C2×C311yescharacteristic
29(C2×S4)⋊C4C2×C311yescharacteristic
30SL(2,ℤ/4)⋊C4C2×C311yescharacteristic, complemented
31SL(2,ℤ/4)⋊C4C2×C311yescharacteristic, complemented
32C3×#399—31no—
33C3×#398—31no—
34C3×#396—31no—
35C3×#395—31no—
36C3×#389—31no—
37C3×#387—31no—
38C3×#386—31no—
39C3×#386—31no—
40C3×#385—31no—
41C3×#385—31no—
42C3×#385—31no—
43C22×C3×A4C2311yescharacteristic
44C2×C3×S4—21no—
45C2×C3×S4—21no—
46C2×C3×S4Q812yescomplemented
47C2×C3×C4×D3—41no—
48C2×C3×Dic6—41no—
49C3×C4×A4—21no—
50C3×C4×A4—21no—
51C3×C4×A4—21no—
52C3×SL(2,ℤ/4)—21no—
53C3×SL(2,ℤ/4)—21no—
54C3×SL(2,ℤ/4)D412yes—
55C32×C4⋊C4—41no—
56C3×(C2×D3)⋊C4—41no—
57C3×(C2×D3)⋊C4—41no—
58C3×Dic3⋊C4—41no—
59C3×Dic3⋊C4—41no—
60#2449—31noSylow
61C22×S4C22×C311yescharacteristic
62C22×C3×Q8—31no—
63C22×C3×D4—31no—
64C23×C3×C4—31no—
65C23×C3×C4C2×D311yescharacteristic
66C23×C3×C4C2×D311yescharacteristic
67C23×C3×C4C2×D311yescharacteristic
68C2×C4×A4—21no—
69C2×C4×A4—21no—
70C2×C4×A4—21no—
71C2×C4×A4C22×C311yescharacteristic
72C2×C4×A4C22×C311yescharacteristic
73C2×C4×A4C22×C311yescharacteristic
74C2×SL(2,ℤ/4)C22×C311yescharacteristic
75C2×SL(2,ℤ/4)C22×C311yescharacteristic
76C2×SL(2,ℤ/4)C22×C311yescharacteristic
77C4×S4—22no—
78A4⋊Q8—22no—
79C3×C22⋊Q8—62no—
80C3×C22⋊Q8—62no—
81C3×C22⋊Q8—61no—
82C3×C22⋊Q8—61no—
83C3×C22⋊Q8—61no—
84C3×C22⋊Q8—61no—
85C3×C4×D4—62no—
86C3×C4×D4—62no—
87C2×C3×C4⋊C4—31no—
88C2×C3×C4⋊C4—31no—
89C2×C3×C4⋊C4—31no—
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×C4⋊C4—31no—
96C2×C3×C4⋊C4—31no—
97C2×C3×C4⋊C4—31no—
98C2×C3×C4⋊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×C42—31no—
108#896—41no—
109C3×#95—31no—
110C3×#95—31no—
111C3×#95—31no—
112C3×#95—31no—
113C3×#95—31no—
114C3×#95—31no—
115C3×#95—31no—
116C3×#95—31no—
117C22×C3×D3—41no—
118C2×C3×A4C2×D411yescharacteristic
119C2×C3×A4C2×Q811yescharacteristic
120C2×C3×A4#4111yescharacteristic
121C3×S4—22no—
122C2×C32×C4—41no—
123C2×C32×C4—41no—
124C2×C32×C4—41no—
125C2×C3×Dic3—41no—
126C2×C3×Dic3—41no—
127C2×C3×Dic3—41no—
128C3×C4×D3—82no—
129C3×Dic6—82no—
130C2×C22⋊Q8—31no—
131C2×C22⋊Q8—31no—
132C2×C4×D4—31no—
133C22×C4⋊C4C3×D311yescharacteristic, complemented
134#399—31no—
135#398—31no—
136#396—31no—
137#395—31no—
138#389—31no—
139#387—31no—
140#386—31no—
141#386—31no—
142#385—31no—
143#385—31no—
144#385—31no—
145C24×C3—31no—
146C24×C3C22×D311yescharacteristic
147C22×A4—21no—
148C22×A4C23×C311yescharacteristic
149C2×S4—21no—
150C2×S4—21no—
151C2×S4C3×Q812yescomplemented
152C2×C3×Q8—62no—
153C2×C3×Q8—61no—
154C2×C3×Q8—61no—
155C2×C3×Q8—121no—
156C2×C3×D4—32no—
157C2×C3×D4—32no—
158C2×C3×D4—61no—
159C2×C3×D4—61no—
160C2×C3×D4—61no—
161C2×C3×D4—61no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—31no—
166C22×C3×C4—31no—
167C22×C3×C4—31no—
168C22×C3×C4—31no—
169C22×C3×C4—31no—
170C22×C3×C4—31no—
171C22×C3×C4—31no—
172C22×C3×C4—31no—
173C22×C3×C4—31no—
174C22×C3×C4—31no—
175C22×C3×C4—31no—
176C22×C3×C4—31no—
177C22×C3×C4—31no—
178C22×C3×C4—61no—
179C22×C3×C4—62no—
180C22×C3×C4—61no—
181C22×C3×C4—61no—
182C22×C3×C4—61no—
183C22×C3×C4—61no—
184C22×C3×C4—121no—
185C2×C4×D3—41no—
186C2×Dic6—41no—
187C4×A4—22no—
188C4×A4—21no—
189C4×A4—21no—
190C4×A4—21no—
191C4×A4—41no—
192C4×A4—41no—
193SL(2,ℤ/4)—21no—
194SL(2,ℤ/4)—21no—
195SL(2,ℤ/4)C3×D412yes—
196C3×C4⋊C4—31no—
197C3×C4⋊C4—31no—
198C3×C4⋊C4—31no—
199C3×C4⋊C4—41no—
200C3×C4⋊C4—61no—
201C3×C4⋊C4—62no—
202C3×C4⋊C4—62no—
203C3×C4⋊C4—62no—
204C3×C4⋊C4—62no—
205C3×C4⋊C4—62no—
206C3×C4⋊C4—61no—
207C3×C4⋊C4—61no—
208C3×C4⋊C4—61no—
209C3×C4⋊C4—61no—
210C3×C4⋊C4—81no—
211C3×C4⋊C4—121no—
212C3×C4⋊C4S411yescharacteristic, complemented
213C3×C22⋊C4—32no—
214C3×C22⋊C4—32no—
215C3×C22⋊C4—62no—
216C3×C22⋊C4—62no—
217C3×C22⋊C4—61no—
218C3×C22⋊C4—61no—
219C3×C22⋊C4—61no—
220C3×C22⋊C4—61no—
221C3×C22⋊C4—61no—
222C3×C22⋊C4—61no—
223C3×C22⋊C4—61no—
224C3×C22⋊C4—61no—
225C3×C22⋊C4—121no—
226C3×C42—62no—
227(C2×D3)⋊C4—41no—
228(C2×D3)⋊C4—41no—
229Dic3⋊C4—41no—
230Dic3⋊C4—41no—
231C22×C32—41no—
232C2×C3×D3—42no—
233C2×C3×D3—81no—
234C2×C3×D3—81no—
235C3×A4C22⋊Q811yescharacteristic, complemented
236C32×C4—81no—
237C32×C4—81no—
238C32×C4—81no—
239C3×Dic3—42no—
240C3×Dic3—81no—
241C3×Dic3—81no—
242C22×Q8—31no—
243C22×D4—31no—
244C23×C4—31no—
245C23×C4C2×C3×D311yescharacteristic
246C23×C4C2×C3×D311yescharacteristic
247C23×C4C2×C3×D311yescharacteristic
248C22⋊Q8—62no—
249C22⋊Q8—62no—
250C22⋊Q8—61no—
251C22⋊Q8—61no—
252C22⋊Q8—61no—
253C22⋊Q8—61no—
254C4×D4—62no—
255C4×D4—62no—
256C2×C4⋊C4—31no—
257C2×C4⋊C4—31no—
258C2×C4⋊C4—31no—
259C2×C4⋊C4—31no—
260C2×C4⋊C4—31no—
261C2×C4⋊C4—31no—
262C2×C4⋊C4—31no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—31no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4—31no—
268C2×C22⋊C4—31no—
269C2×C22⋊C4—31no—
270C2×C22⋊C4—31no—
271C2×C22⋊C4—31no—
272C2×C22⋊C4—31no—
273C2×C22⋊C4—31no—
274C2×C22⋊C4—31no—
275C2×C22⋊C4—31no—
276C2×C42—31no—
277#95—31no—
278#95—31no—
279#95—31no—
280#95—31no—
281#95—31no—
282#95—31no—
283#95—31no—
284#95—31no—
285C23×C3—31no—
286C23×C3—31no—
287C23×C3—31no—
288C23×C3—31no—
289C23×C3—32no—
290C23×C3—61no—
291C23×C3—61no—
292C23×C3—61no—
293C23×C3—61no—
294C23×C3—61no—
295C23×C3—61no—
296C23×C3#23511yescharacteristic
297C23×C3D3×D411yescharacteristic
298C23×C3D3×Q811yescharacteristic
299C22×D3—41no—
300C2×A4—21no—
301C2×A4—21no—
302C2×A4—21no—
303C2×A4C2×C3×D411yescharacteristic
304C2×A4C2×C3×Q811yescharacteristic
305C2×A4C3×#4111yescharacteristic
306S4—22no—
307C3×Q8—122no—
308C3×Q8—122no—
309C3×D4—62no—
310C3×D4—62no—
311C3×D4—62no—
312C3×D4—62no—
313C2×C3×C4—32no—
314C2×C3×C4—32no—
315C2×C3×C4—32no—
316C2×C3×C4—32no—
317C2×C3×C4—31no—
318C2×C3×C4—31no—
319C2×C3×C4—31no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—61no—
324C2×C3×C4—61no—
325C2×C3×C4—61no—
326C2×C3×C4—61no—
327C2×C3×C4—61no—
328C2×C3×C4—61no—
329C2×C3×C4—61no—
330C2×C3×C4—61no—
331C2×C3×C4—61no—
332C2×C3×C4—61no—
333C2×C3×C4—61no—
334C2×C3×C4—61no—
335C2×C3×C4—61no—
336C2×C3×C4—61no—
337C2×C3×C4—61no—
338C2×C3×C4—61no—
339C2×C3×C4—61no—
340C2×C3×C4—61no—
341C2×C3×C4—61no—
342C2×C3×C4—61no—
343C2×C3×C4—61no—
344C2×C3×C4—61no—
345C2×C3×C4—61no—
346C2×C3×C4—61no—
347C2×C3×C4—61no—
348C2×C3×C4—61no—
349C2×C3×C4—61no—
350C2×C3×C4—61no—
351C2×C3×C4—61no—
352C2×C3×C4—81no—
353C2×C3×C4—81no—
354C2×C3×C4—81no—
355C2×C3×C4—122no—
356C2×C3×C4—122no—
357C2×C3×C4C2×S411yescharacteristic
358C2×C3×C4C2×S411yescharacteristic
359C2×C3×C4C2×S411yescharacteristic
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C4×D3—82no—
364Dic6—82no—
365C2×C32—41no—
366C2×C32—41no—
367C2×C32—41no—
368C3×D3—82no—
369C24—31no—
370C24C22×C3×D311yescharacteristic
371C2×Q8—62no—
372C2×Q8—61no—
373C2×Q8—61no—
374C2×Q8—121no—
375C2×D4—32no—
376C2×D4—32no—
377C2×D4—61no—
378C2×D4—61no—
379C2×D4—61no—
380C2×D4—61no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—31no—
385C22×C4—31no—
386C22×C4—31no—
387C22×C4—31no—
388C22×C4—31no—
389C22×C4—31no—
390C22×C4—31no—
391C22×C4—31no—
392C22×C4—31no—
393C22×C4—31no—
394C22×C4—31no—
395C22×C4—31no—
396C22×C4—31no—
397C22×C4—61no—
398C22×C4—62no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—121no—
404C4⋊C4—31no—
405C4⋊C4—31no—
406C4⋊C4—31no—
407C4⋊C4—61no—
408C4⋊C4—62no—
409C4⋊C4—62no—
410C4⋊C4—62no—
411C4⋊C4—62no—
412C4⋊C4—61no—
413C4⋊C4—61no—
414C4⋊C4—62no—
415C4⋊C4—61no—
416C4⋊C4—61no—
417C4⋊C4—121no—
418C4⋊C4C3×S411yescharacteristic, complemented
419C22⋊C4—32no—
420C22⋊C4—32no—
421C22⋊C4—62no—
422C22⋊C4—61no—
423C22⋊C4—61no—
424C22⋊C4—62no—
425C22⋊C4—61no—
426C22⋊C4—61no—
427C22⋊C4—61no—
428C22⋊C4—61no—
429C22⋊C4—61no—
430C22⋊C4—61no—
431C22⋊C4—121no—
432C42—62no—
433C22×C3—31no—
434C22×C3—31no—
435C22×C3—31no—
436C22×C3—31no—
437C22×C3—31no—
438C22×C3—31no—
439C22×C3—31no—
440C22×C3—31no—
441C22×C3—31no—
442C22×C3—41no—
443C22×C3—61no—
444C22×C3—62no—
445C22×C3—62no—
446C22×C3—62no—
447C22×C3—81no—
448C22×C3—121no—
449C22×C3—121no—
450C22×C3—121no—
451C22×C3—121no—
452C22×C3#89611yescharacteristic, complemented
453C22×C3C22×S411yescharacteristic
454C2×D3—42no—
455C2×D3—81no—
456C2×D3—81no—
457A4—21no—
458A4C3×C22⋊Q811yescharacteristic, complemented
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—62no—
463C3×C4—62no—
464C3×C4—61no—
465C3×C4—61no—
466C3×C4—61no—
467C3×C4—62no—
468C3×C4—81no—
469C3×C4—81no—
470C3×C4—81no—
471C3×C4—82no—
472C3×C4—121no—
473C3×C4—121no—
474C3×C4—121no—
475C3×C4—121no—
476C3×C4—161no—
477C3×C4—161no—
478Dic3—42no—
479Dic3—81no—
480Dic3—81no—
481C32—41noSylow
482C23—31no—
483C23—31no—
484C23—31no—
485C23—31no—
486C23—32no—
487C23—61no—
488C23—61no—
489C23—61no—
490C23—61no—
491C23—61no—
492C23—61no—
493C23C3×#23511yescharacteristic
494C23C3×D3×D411yescharacteristic
495C23C3×D3×Q811yescharacteristic
496Q8—122no—
497Q8—122no—
498D4—62no—
499D4—62no—
500D4—62no—
501D4—62no—
502C2×C4—32no—
503C2×C4—32no—
504C2×C4—32no—
505C2×C4—32no—
506C2×C4—31no—
507C2×C4—31no—
508C2×C4—31no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—61no—
520C2×C4—61no—
521C2×C4—61no—
522C2×C4—61no—
523C2×C4—61no—
524C2×C4—61no—
525C2×C4—61no—
526C2×C4—61no—
527C2×C4—61no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—122no—
539C2×C4—122no—
540C2×C4C2×C3×S411yescharacteristic
541C2×C4C2×C3×S411yescharacteristic
542C2×C4C2×C3×S411yescharacteristic
543C2×C3—31no—
544C2×C3—31no—
545C2×C3—31no—
546C2×C3—31no—
547C2×C3—41no—
548C2×C3—41no—
549C2×C3—41no—
550C2×C3—81no—
551C2×C3—81no—
552C2×C3—81no—
553C2×C3—122no—
554C2×C3#591211yescharacteristic
555C2×C3D4×S411yescharacteristic
556C2×C3Q8×S411yescharacteristic
557D3—82no—
558C22—31no—
559C22—31no—
560C22—31no—
561C22—31no—
562C22—31no—
563C22—31no—
564C22—31no—
565C22—31no—
566C22—31no—
567C22—61no—
568C22—62no—
569C22—62no—
570C22—62no—
571C22—121no—
572C22—121no—
573C22—121no—
574C22—121no—
575C22C3×#89611yescharacteristic, complemented
576C22C22×C3×S411yescharacteristic
577C4—21no—
578C4—21no—
579C4—21no—
580C4—62no—
581C4—62no—
582C4—61no—
583C4—61no—
584C4—61no—
585C4—62no—
586C4—121no—
587C4—121no—
588C4—121no—
589C4—121no—
590C3—41no—
591C3—81no—
592C3#8531111yescharacteristic, direct product factor
593C2—31no—
594C2—31no—
595C2—31no—
596C2—31no—
597C2—122no—
598C2C3×#591211yescharacteristic
599C2C3×D4×S411yescharacteristic
600C2C3×Q8×S411yescharacteristic
subgroupG/H
centerC22×C3C22×S4
commutator, cosocleC22×A4C23×C3
FrattiniC22C22×C3×S4
FittingC22×C3×C4⋊C4D3
socleC24×C3C22×D3
prime radicalC22×C3×S4C22
hypercenterC3×C4⋊C4S4
hypocenterA4C3×C22⋊Q8
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22×C3 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×A4⋊Q8C2semidirect
3. C3×A4×C4⋊C4C2semidirect
6. C3×SL(2,ℤ/4)⋊C4C2semidirect
7. C3×SL(2,ℤ/4)⋊C4C2semidirect
8. #85311C3direct (N × H)
21. C2×A4⋊Q8C2×C3semidirect
25. C22×C3×C4⋊C4D3semidirect
27. A4×C4⋊C4C2×C3semidirect
30. SL(2,ℤ/4)⋊C4C2×C3semidirect
31. SL(2,ℤ/4)⋊C4C2×C3semidirect
46. C2×C3×S4Q8semidirect
133. C22×C4⋊C4C3×D3semidirect
151. C2×S4C3×Q8semidirect
212. C3×C4⋊C4S4semidirect
235. C3×A4C22⋊Q8semidirect
418. C4⋊C4C3×S4semidirect
452. C22×C3#896semidirect
458. A4C3×C22⋊Q8semidirect
575. C22C3×#896semidirect
592. C3#85311direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×S42
C2×C3×A4⋊Q82
C3×A4×C4⋊C42
C3×(C2×S4)⋊C42
C3×SL(2,ℤ/4)⋊C42
#853113
C3×#24493
C3×#8964