Finite Group Databasebeta

← C3×#71847 #11796232

Subgroups — C3×#71847 #11796232

Order 1152.

733 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#534C211yesmaximal, characteristic
2C2×C3×#834C211yesmaximal, characteristic, complemented
3C2×C3×#833C211yesmaximal, characteristic, complemented
4C3×#4539C212yesmaximal, complemented
5C3×#4536C212yesmaximal, complemented
6C3×#2121—31nomaximal
7#71847C311yesmaximal, characteristic, direct product factor
8C22×C32×D4C411yescharacteristic, complemented
9C22×C32×D4C2211yescharacteristic
10C32×#123—21no—
11C32×#123C411yescharacteristic, complemented
12C32×#123C2211yescharacteristic
13C32×#121—21no—
14C32×#121—21no—
15C32×#120—21no—
16C32×#120C411yescharacteristic, complemented
17C32×#120C2211yescharacteristic
18C2×C32×C22⋊C4C411yescharacteristic, complemented
19C2×C3×#217—21no—
20C2×C3×#208—21no—
21C3×#834—21no—
22C3×#834C2212yes—
23C3×#833—21no—
24C3×#833C2212yes—
25C3×#534—21no—
26C3×#534D311yescharacteristic
27C3×#534C2×C311yescharacteristic
28C2×C3×#100—31no—
29C2×C3×#99—31no—
30C2×#834C2×C311yescharacteristic, complemented
31C2×#833C2×C311yescharacteristic, complemented
32C3×#352—32no—
33C3×AΣL(1,𝔽16)—32no—
34#4539C2×C312yescomplemented
35#4536C2×C312yescomplemented
36C24×C32C2×C411yescharacteristic
37C24×C32C2×C411yescharacteristic, complemented
38C24×C32D411yescharacteristic
39C2×C32×D4—21no—
40C2×C32×D4—21no—
41C2×C32×D4—21no—
42C2×C32×D4—21no—
43C2×C32×D4—41no—
44C2×C32×D4—41no—
45C2×C32×D4—41no—
46C2×C32×D4—41no—
47C2×C32×D4C2×C411yescharacteristic
48C2×C32×D4D411yescharacteristic
49C2×C32×D4D411yescharacteristic
50C2×C32×D4C2311yescharacteristic
51C22×C32×C4—21no—
52C22×C32×C4C2×C411yescharacteristic, complemented
53C22×C32×C4D411yescharacteristic
54C22×C3×Dic3—41no—
55C32×C4⋊C4—21no—
56C32×C4⋊C4—21no—
57C32×C22⋊C4—21no—
58C32×C22⋊C4—21no—
59C32×C22⋊C4—21no—
60C32×C22⋊C4—21no—
61C32×C22⋊C4—21no—
62C32×C22⋊C4C2×C411yescharacteristic, complemented
63C32×C22⋊C4C2×C411yescharacteristic, complemented
64C3×#217—22no—
65C3×#217—41no—
66C3×#208—22no—
67C3×#208—41no—
68C2×C3×C3⋊C8—41no—
69#2121—31noSylow
70C22×C3×D4—21no—
71C22×C3×D4—21no—
72C22×C3×D4Dic311yescharacteristic, complemented
73C22×C3×D4C3×C411yescharacteristic, complemented
74C22×C3×D4C2×D311yescharacteristic
75C22×C3×D4C22×C311yescharacteristic
76C2×C3×M16—61no—
77C3×#123—22no—
78C3×#123—21no—
79C3×#123—21no—
80C3×#123—21no—
81C3×#123—21no—
82C3×#123Dic311yescharacteristic, complemented
83C3×#123C3×C411yescharacteristic, complemented
84C3×#123C2×D311yescharacteristic
85C3×#123C22×C311yescharacteristic
86C3×#121—22no—
87C3×#121—22no—
88C3×#121—21no—
89C3×#121—21no—
90C3×#121—21no—
91C3×#121—21no—
92C3×#120—22no—
93C3×#120—21no—
94C3×#120—21no—
95C3×#120—21no—
96C3×#120—21no—
97C3×#120Dic311yescharacteristic, complemented
98C3×#120C3×C411yescharacteristic, complemented
99C3×#120C2×D311yescharacteristic
100C3×#120C22×C311yescharacteristic
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—61no—
103C2×C3×C22⋊C4Dic311yescharacteristic, complemented
104C2×C3×C22⋊C4C3×C411yescharacteristic, complemented
105C2×#217—21no—
106C2×#208—21no—
107C3×#100—32no—
108C3×#100—61no—
109C3×#99—32no—
110C3×#99—61no—
111#834—21no—
112#834C22×C312yes—
113#833—21no—
114#833C22×C312yes—
115C23×C32—21no—
116C23×C32—21no—
117C23×C32—21no—
118C23×C32—21no—
119C23×C32—21no—
120C23×C32—41no—
121C23×C32—81no—
122C23×C32—81no—
123C23×C32—81no—
124C23×C32C22⋊C411yescharacteristic
125C23×C32C22⋊C411yescharacteristic
126C23×C32C22⋊C411yescharacteristic
127C23×C32C22×C411yescharacteristic
128C23×C32C2×D411yescharacteristic
129C32×D4—41no—
130C32×D4—41no—
131C32×D4—41no—
132C32×D4—41no—
133C32×D4—42no—
134C32×D4—41no—
135C32×D4—41no—
136C32×D4—81no—
137C32×D4—81no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—41no—
144C2×C32×C4—81no—
145C2×C32×C4C22⋊C411yescharacteristic
146C2×C32×C4C2×D411yescharacteristic
147C2×C3×Dic3—42no—
148C2×C3×Dic3—82no—
149C3×C3⋊C8—42no—
150#534C3×D311yescharacteristic
151C2×#100—31no—
152C2×#99—31no—
153#352—32no—
154AΣL(1,𝔽16)—32no—
155C24×C3—21no—
156C24×C3—21no—
157C24×C3—21no—
158C24×C3C2×Dic311yescharacteristic
159C24×C3C2×Dic311yescharacteristic, complemented
160C24×C3#6711yescharacteristic
161C24×C3C2×C3×C411yescharacteristic
162C24×C3C2×C3×C411yescharacteristic, complemented
163C24×C3C3×D411yescharacteristic
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C2×C3×D4—21no—
167C2×C3×D4—21no—
168C2×C3×D4—21no—
169C2×C3×D4—21no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—21no—
173C2×C3×D4—21no—
174C2×C3×D4—21no—
175C2×C3×D4—21no—
176C2×C3×D4—21no—
177C2×C3×D4—21no—
178C2×C3×D4—21no—
179C2×C3×D4—21no—
180C2×C3×D4—41no—
181C2×C3×D4—41no—
182C2×C3×D4—41no—
183C2×C3×D4—41no—
184C2×C3×D4—42no—
185C2×C3×D4—42no—
186C2×C3×D4—42no—
187C2×C3×D4—42no—
188C2×C3×D4—41no—
189C2×C3×D4—41no—
190C2×C3×D4—41no—
191C2×C3×D4—41no—
192C2×C3×D4C2×Dic311yescharacteristic
193C2×C3×D4#6711yescharacteristic
194C2×C3×D4#6711yescharacteristic
195C2×C3×D4C2×C3×C411yescharacteristic
196C2×C3×D4C3×D411yescharacteristic
197C2×C3×D4C3×D411yescharacteristic
198C2×C3×D4C22×D311yescharacteristic
199C2×C3×D4C23×C311yescharacteristic
200C22×C3×C4—22no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—121no—
206C22×C3×C4C2×Dic311yescharacteristic, complemented
207C22×C3×C4#6711yescharacteristic
208C22×C3×C4C2×C3×C411yescharacteristic, complemented
209C22×C3×C4C3×D411yescharacteristic
210C22×Dic3—41no—
211C3×M16—62no—
212C3×M16—121no—
213C2×C3×C8—121no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C22⋊C4—22no—
221C3×C22⋊C4—22no—
222C3×C22⋊C4—22no—
223C3×C22⋊C4—22no—
224C3×C22⋊C4—22no—
225C3×C22⋊C4—21no—
226C3×C22⋊C4—21no—
227C3×C22⋊C4—21no—
228C3×C22⋊C4—21no—
229C3×C22⋊C4—21no—
230C3×C22⋊C4—21no—
231C3×C22⋊C4—21no—
232C3×C22⋊C4—21no—
233C3×C22⋊C4—21no—
234C3×C22⋊C4—21no—
235C3×C22⋊C4—21no—
236C3×C22⋊C4—21no—
237C3×C22⋊C4—62no—
238C3×C22⋊C4—121no—
239C3×C22⋊C4C2×Dic311yescharacteristic, complemented
240C3×C22⋊C4C2×Dic311yescharacteristic, complemented
241C3×C22⋊C4C2×C3×C411yescharacteristic, complemented
242C3×C22⋊C4C2×C3×C411yescharacteristic, complemented
243#217—22no—
244#217—41no—
245#208—22no—
246#208—41no—
247C2×C3⋊C8—41no—
248C22×C32—21no—
249C22×C32—21no—
250C22×C32—21no—
251C22×C32—21no—
252C22×C32—41no—
253C22×C32—41no—
254C22×C32—41no—
255C22×C32—41no—
256C22×C32—41no—
257C22×C32—41no—
258C22×C32—41no—
259C22×C32—41no—
260C22×C32—42no—
261C22×C32—81no—
262C22×C32—81no—
263C22×C32—81no—
264C22×C32—82no—
265C22×C32#9911yescharacteristic
266C22×C32#9911yescharacteristic
267C22×C32C2×C22⋊C411yescharacteristic
268C32×C4—21no—
269C32×C4—21no—
270C32×C4—41no—
271C32×C4—41no—
272C32×C4—41no—
273C3×Dic3—82no—
274C22×D4C3×Dic311yescharacteristic, complemented
275C22×D4C2×C3×D311yescharacteristic
276C2×M16—61no—
277#123—21no—
278#123C3×Dic311yescharacteristic, complemented
279#123C2×C3×D311yescharacteristic
280#121—21no—
281#121—21no—
282#120—21no—
283#120C3×Dic311yescharacteristic, complemented
284#120C2×C3×D311yescharacteristic
285C2×C22⋊C4—61no—
286C2×C22⋊C4C3×Dic311yescharacteristic, complemented
287#100—32no—
288#100—61no—
289#99—32no—
290#99—61no—
291C23×C3—22no—
292C23×C3—22no—
293C23×C3—21no—
294C23×C3—21no—
295C23×C3—21no—
296C23×C3—21no—
297C23×C3—21no—
298C23×C3—21no—
299C23×C3—21no—
300C23×C3—21no—
301C23×C3—22no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—21no—
306C23×C3—21no—
307C23×C3—21no—
308C23×C3—21no—
309C23×C3—41no—
310C23×C3—41no—
311C23×C3—42no—
312C23×C3—41no—
313C23×C3—41no—
314C23×C3—81no—
315C23×C3—81no—
316C23×C3—81no—
317C23×C3—81no—
318C23×C3—81no—
319C23×C3—81no—
320C23×C3—82no—
321C23×C3—82no—
322C23×C3—82no—
323C23×C3#21711yescharacteristic
324C23×C3#21711yescharacteristic
325C23×C3#21711yescharacteristic
326C23×C3C3×C22⋊C411yescharacteristic
327C23×C3C3×C22⋊C411yescharacteristic
328C23×C3C3×C22⋊C411yescharacteristic
329C23×C3C22×Dic311yescharacteristic
330C23×C3C2×#6711yescharacteristic
331C23×C3C22×C3×C411yescharacteristic
332C23×C3C2×C3×D411yescharacteristic
333C3×D4—41no—
334C3×D4—41no—
335C3×D4—41no—
336C3×D4—41no—
337C3×D4—41no—
338C3×D4—41no—
339C3×D4—41no—
340C3×D4—41no—
341C3×D4—42no—
342C3×D4—42no—
343C3×D4—42no—
344C3×D4—42no—
345C3×D4—44no—
346C3×D4—42no—
347C3×D4—42no—
348C3×D4—42no—
349C3×D4—42no—
350C3×D4—41no—
351C3×D4—41no—
352C3×D4—41no—
353C3×D4—41no—
354C3×D4—81no—
355C3×D4—81no—
356C3×D4—81no—
357C3×D4—81no—
358C3×D4—82no—
359C3×D4—82no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—42no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—81no—
381C2×C3×C4—81no—
382C2×C3×C4—82no—
383C2×C3×C4—122no—
384C2×C3×C4—242no—
385C2×C3×C4#21711yescharacteristic
386C2×C3×C4C3×C22⋊C411yescharacteristic
387C2×C3×C4C2×#6711yescharacteristic
388C2×C3×C4C2×C3×D411yescharacteristic
389C2×Dic3—42no—
390C2×Dic3—82no—
391C3×C8—122no—
392C3⋊C8—42no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32—21no—
396C2×C32—41no—
397C2×C32—41no—
398C2×C32—41no—
399C2×C32—41no—
400C2×C32—81no—
401C2×C32C2×#9911yescharacteristic
402C24C2×C3×Dic311yescharacteristic
403C24C2×C3×Dic311yescharacteristic, complemented
404C24C3×#6711yescharacteristic
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—41no—
410C2×D4—41no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4C2×C3×Dic311yescharacteristic
414C2×D4C3×#6711yescharacteristic
415C2×D4C3×#6711yescharacteristic
416C2×D4C22×C3×D311yescharacteristic
417C22×C4—21no—
418C22×C4—121no—
419C22×C4C2×C3×Dic311yescharacteristic, complemented
420C22×C4C3×#6711yescharacteristic
421M16—62no—
422M16—121no—
423C2×C8—121no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C22⋊C4—21no—
427C22⋊C4—21no—
428C22⋊C4—21no—
429C22⋊C4—21no—
430C22⋊C4—21no—
431C22⋊C4—62no—
432C22⋊C4—121no—
433C22⋊C4C2×C3×Dic311yescharacteristic, complemented
434C22⋊C4C2×C3×Dic311yescharacteristic, complemented
435C22×C3—22no—
436C22×C3—22no—
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—21no—
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—41no—
449C22×C3—41no—
450C22×C3—41no—
451C22×C3—41no—
452C22×C3—41no—
453C22×C3—41no—
454C22×C3—41no—
455C22×C3—41no—
456C22×C3—41no—
457C22×C3—41no—
458C22×C3—41no—
459C22×C3—41no—
460C22×C3—42no—
461C22×C3—42no—
462C22×C3—42no—
463C22×C3—42no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3—42no—
468C22×C3—42no—
469C22×C3—42no—
470C22×C3—41no—
471C22×C3—41no—
472C22×C3—41no—
473C22×C3—41no—
474C22×C3—41no—
475C22×C3—41no—
476C22×C3—82no—
477C22×C3—81no—
478C22×C3—81no—
479C22×C3—81no—
480C22×C3—81no—
481C22×C3—81no—
482C22×C3—81no—
483C22×C3—82no—
484C22×C3—82no—
485C22×C3—82no—
486C22×C3—82no—
487C22×C3—82no—
488C22×C3—84no—
489C22×C3#83411yescharacteristic
490C22×C3#83411yescharacteristic
491C22×C3C3×#9911yescharacteristic
492C22×C3C3×#9911yescharacteristic
493C22×C3C2×#21711yescharacteristic
494C22×C3C2×C3×C22⋊C411yescharacteristic
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—21no—
498C3×C4—21no—
499C3×C4—41no—
500C3×C4—41no—
501C3×C4—42no—
502C3×C4—42no—
503C3×C4—42no—
504C3×C4—41no—
505C3×C4—41no—
506C3×C4—41no—
507C3×C4—41no—
508C3×C4—41no—
509C3×C4—41no—
510C3×C4—242no—
511Dic3—82no—
512C32#212111yescharacteristic, complemented, Sylow
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—41no—
519C23—81no—
520C23—81no—
521C23—81no—
522C23C3×#21711yescharacteristic
523C23C3×#21711yescharacteristic
524C23C3×#21711yescharacteristic
525C23C22×C3×Dic311yescharacteristic
526C23C2×C3×#6711yescharacteristic
527D4—41no—
528D4—41no—
529D4—41no—
530D4—41no—
531D4—42no—
532D4—41no—
533D4—41no—
534D4—81no—
535D4—81no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—41no—
542C2×C4—81no—
543C2×C4—122no—
544C2×C4—242no—
545C2×C4C3×#21711yescharacteristic
546C2×C4C2×C3×#6711yescharacteristic
547C8—122no—
548C2×C3—22no—
549C2×C3—21no—
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—21no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3—41no—
557C2×C3—41no—
558C2×C3—42no—
559C2×C3—42no—
560C2×C3—41no—
561C2×C3—41no—
562C2×C3—41no—
563C2×C3—41no—
564C2×C3—41no—
565C2×C3—41no—
566C2×C3—41no—
567C2×C3—41no—
568C2×C3—81no—
569C2×C3—81no—
570C2×C3—82no—
571C2×C3—81no—
572C2×C3—81no—
573C2×C3C2×#83411yescharacteristic
574C2×C3C2×C3×#9911yescharacteristic
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—42no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—82no—
592C22C3×#83411yescharacteristic
593C22C3×#83411yescharacteristic
594C22C2×C3×#21711yescharacteristic
595C4—21no—
596C4—21no—
597C4—41no—
598C4—41no—
599C4—41no—
600C4—242no—
601C3—21no—
602C3#7184711yescharacteristic, direct product factor
603C3C3×#212111yescharacteristic, complemented
604C2—21no—
605C2—21no—
606C2—21no—
607C2—41no—
608C2—41no—
609C2—41no—
610C2—41no—
611C2—81no—
612C2C2×C3×#83411yescharacteristic
subgroupG/H
centerC2×C3C2×#834
commutatorC23×C3C22×C3×C4
FrattiniC2×D4C22×C3×D3
Fitting, prime radicalC32×#534C2
socleC2×C32C2×#99
cosocleC2×C3×D4C23×C3
hypercenterC3×#534D3
hypocenterC3C3×#2121
LCSG ▷ C23×C3 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C22×C3×D4 ◁ C3×#534

Complements

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

NH ≅ G/Ntype
2. C2×C3×#834C2semidirect
3. C2×C3×#833C2semidirect
4. C3×#4539C2semidirect
5. C3×#4536C2semidirect
7. #71847C3direct (N × H)
8. C22×C32×D4C4semidirect
11. C32×#123C4semidirect
16. C32×#120C4semidirect
18. C2×C32×C22⋊C4C4semidirect
30. C2×#834C2×C3semidirect
31. C2×#833C2×C3semidirect
34. #4539C2×C3semidirect
35. #4536C2×C3semidirect
37. C24×C32C2×C4semidirect
52. C22×C32×C4C2×C4semidirect
62. C32×C22⋊C4C2×C4semidirect
63. C32×C22⋊C4C2×C4semidirect
72. C22×C3×D4Dic3semidirect
73. C22×C3×D4C3×C4semidirect
82. C3×#123Dic3semidirect
83. C3×#123C3×C4semidirect
97. C3×#120Dic3semidirect
98. C3×#120C3×C4semidirect
103. C2×C3×C22⋊C4Dic3semidirect
104. C2×C3×C22⋊C4C3×C4semidirect
159. C24×C3C2×Dic3semidirect
162. C24×C3C2×C3×C4semidirect
206. C22×C3×C4C2×Dic3semidirect
208. C22×C3×C4C2×C3×C4semidirect
239. C3×C22⋊C4C2×Dic3semidirect
240. C3×C22⋊C4C2×Dic3semidirect
241. C3×C22⋊C4C2×C3×C4semidirect
242. C3×C22⋊C4C2×C3×C4semidirect
274. C22×D4C3×Dic3semidirect
278. #123C3×Dic3semidirect
283. #120C3×Dic3semidirect
286. C2×C22⋊C4C3×Dic3semidirect
403. C24C2×C3×Dic3semidirect
419. C22×C4C2×C3×Dic3semidirect
433. C22⋊C4C2×C3×Dic3semidirect
434. C22⋊C4C2×C3×Dic3semidirect
512. C32#2121semidirect
602. C3#71847direct (N × H)
603. C3C3×#2121semidirect

Maximal subgroups

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

subgroupindex
C32×#5342
C2×C3×#8342
C2×C3×#8332
C3×#45392
C3×#45362
C3×#21213
#718473