Finite Group Databasebeta

← C3×#80034 #11878692

Subgroups — C3×#80034 #11878692

Order 1152.

935 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#944C211yesmaximal, characteristic, complemented
2C3×#5596C211yesmaximal, characteristic
3C32×#448C211yesmaximal, characteristic, complemented
4C3×#5227C211yesmaximal, characteristic, complemented
5C3×#5175C212yesmaximal, complemented
6C3×#5162C212yesmaximal, complemented
7C3×#5040C211yesmaximal, characteristic, complemented
8C3×#4818C212yesmaximal, complemented
9C3×#4792C212yesmaximal, complemented
10C3×#4738C211yesmaximal, characteristic, complemented
11C3×#4718C211yesmaximal, characteristic, complemented
12C3×#3206—31nomaximal
13#80034C311yesmaximal, characteristic, direct product factor
14C22×C32×Q8C2211yescharacteristic
15C2×C3×#237—21no—
16C22×C3×Dic6C2211yescharacteristic
17C2×C32×SD16C2212yes—
18C32×#121C2211yescharacteristic
19C3×#944—21no—
20C3×#944—21no—
21C3×#944C2212yes—
22C3×#944C2212yes—
23C2×C3×#215C2212yes—
24C3×#940C2211yescharacteristic
25C3×#939—21no—
26C3×D4×Dic3C2211yescharacteristic
27C2×C3×#214C2212yes—
28C3×#930C2211yescharacteristic
29C2×C3×Dic3⋊C4—21no—
30C2×C3×Dic3⋊C4—21no—
31C2×C3×C4×Dic3C2211yescharacteristic
32C2×C3×#204C2212yes—
33C3×#893C2211yescharacteristic
34C3×#888C2211yescharacteristic
35C3×#885—21no—
36C3×#67⋊C4—21no—
37C32×#103C2212yes—
38C32×C22⋊C8C2211yescharacteristic
39C3×#835C2212yes—
40C3×#830C2211yescharacteristic
41C3×#816C2212yes—
42C3×Dic3⋊C8C2212yes—
43C3×C8×Dic3C2212yes—
44C3×#810C2212yes—
45C3×#532—31no—
46C2×C3×C4⋊Q8—31no—
47C2×#944C2×C311yescharacteristic, complemented
48#5596C2×C311yescharacteristic
49C3×#492—32no—
50C3×#487—32no—
51C3×#461—32no—
52C3×#461—32no—
53C3×#448—21no—
54C3×#448—31no—
55C3×#448—31no—
56C3×#448—31no—
57C3×#448D311yescharacteristic, complemented
58C3×#448C2×C311yescharacteristic, complemented
59C3×#431—31no—
60#5227C2×C311yescharacteristic, complemented
61#5175C2×C312yescomplemented
62#5162C2×C312yescomplemented
63#5040C2×C311yescharacteristic, complemented
64#4818C2×C312yescomplemented
65#4792C2×C312yescomplemented
66#4738C2×C311yescharacteristic, complemented
67#4718C2×C311yescharacteristic, complemented
68C2×C32×Q8—21no—
69C2×C32×Q8—42no—
70C2×C32×Q8C2312yes—
71C2×C32×D4—21no—
72C2×C32×D4C2311yescharacteristic
73C22×C32×C4—21no—
74C22×C32×C4C2311yescharacteristic
75C2×C3×#67—21no—
76C2×C3×#67—21no—
77C2×C3×#67—21no—
78C22×C3×Dic3—21no—
79C22×C3×Dic3—21no—
80C22×C3×Dic3D411yescharacteristic
81C22×C3×Dic3D411yescharacteristic
82C3×#237—44no—
83C2×C3×D12C2311yescharacteristic
84C2×C3×C4×D3—21no—
85C2×C3×Dic6—21no—
86C2×C3×Dic6—21no—
87C2×C3×Dic6—42no—
88C2×C3×Dic6C2312yes—
89C32×SD16—42no—
90C2×C32×C8C2312yes—
91C32×C4⋊C4C2311yescharacteristic
92C32×C22⋊C4—21no—
93C3×#217—21no—
94C3×#215—42no—
95C3×#214—42no—
96C3×(C2×D3)⋊C4—21no—
97C3×(C2×D3)⋊C4—21no—
98C3×C4⋊Dic3C2311yescharacteristic
99C3×Dic3⋊C4—21no—
100C3×Dic3⋊C4—22no—
101C3×Dic3⋊C4—22no—
102C3×Dic3⋊C4—22no—
103C3×Dic3⋊C4—22no—
104C3×C4×Dic3D411yescharacteristic
105C3×C4×Dic3D411yescharacteristic
106C3×C4×Dic3C2311yescharacteristic
107C3×C4×Dic3C2311yescharacteristic
108C2×C3×C3⋊C8C2312yes—
109C3×#204—42no—
110#3206—31noSylow
111C2×C3×#41—61no—
112C22×C3×Q8—21no—
113C22×C3×Q8—31no—
114C22×C3×Q8C2×D311yescharacteristic
115C22×C3×Q8C22×C311yescharacteristic
116C2×#237—21no—
117C22×Dic6C22×C311yescharacteristic
118C2×C3×SD16—22no—
119C2×C3×SD16—32no—
120C2×C3×SD16—32no—
121C2×C3×SD16—32no—
122C2×C3×SD16C2×D312yes—
123C2×C3×SD16C22×C312yes—
124C3×C4⋊Q8—31no—
125C3×C4⋊Q8—32no—
126C3×C4⋊Q8—32no—
127C3×C4⋊Q8—61no—
128C3×C4⋊Q8—61no—
129C3×#127—31no—
130C3×#124—61no—
131C3×#121—21no—
132C3×#121—31no—
133C3×#121—61no—
134C3×#121C2×D311yescharacteristic
135C3×#121C22×C311yescharacteristic
136C3×C4×D4—31no—
137C3×C4×D4—31no—
138C3×C4×D4—61no—
139C2×C3×C4⋊C4—61no—
140C2×C3×C4⋊C4—61no—
141C2×C3×C42—31no—
142#944—21no—
143#944—21no—
144#944C22×C312yes—
145#944C22×C312yes—
146C2×#215C22×C312yes—
147#940C22×C311yescharacteristic
148#939—21no—
149D4×Dic3C22×C311yescharacteristic
150C2×#214C22×C312yes—
151#930C22×C311yescharacteristic
152C2×Dic3⋊C4—21no—
153C2×Dic3⋊C4—21no—
154C2×C4×Dic3C22×C311yescharacteristic
155C2×#204C22×C312yes—
156#893C22×C311yescharacteristic
157#888C22×C311yescharacteristic
158#885—21no—
159#67⋊C4—21no—
160C3×C4⋊C8—32no—
161C3×#103—22no—
162C3×#103—32no—
163C3×#103—32no—
164C3×#103—32no—
165C3×#103C2×D312yes—
166C3×#103C22×C312yes—
167C3×C22⋊C8—21no—
168C3×C22⋊C8—31no—
169C3×C22⋊C8C2×D311yescharacteristic
170C3×C22⋊C8C22×C311yescharacteristic
171C3×C4×C8—32no—
172#835C22×C312yes—
173#830C22×C311yescharacteristic
174#816C22×C312yes—
175Dic3⋊C8C22×C312yes—
176C8×Dic3C22×C312yes—
177#810C22×C312yes—
178C23×C32—21no—
179C23×C32C2×D411yescharacteristic
180C22×C3×D3—21no—
181C32×Q8—22no—
182C32×Q8—22no—
183C32×Q8—42no—
184C32×D4—22no—
185C32×D4—42no—
186C2×C32×C4—22no—
187C2×C32×C4—22no—
188C2×C32×C4—21no—
189C2×C32×C4—42no—
190C2×C32×C4C2×D411yescharacteristic
191C2×C32×C4C2411yescharacteristic
192C3×#67—42no—
193C3×#67—42no—
194C3×#67—42no—
195C2×C3×Dic3—22no—
196C2×C3×Dic3—22no—
197C2×C3×Dic3—21no—
198C2×C3×Dic3—42no—
199C2×C3×Dic3—42no—
200C2×C3×Dic3SD1614yescomplemented
201C2×C3×Dic3C2×D411yescharacteristic
202C2×C3×Dic3C2×D411yescharacteristic
203C2×C3×Dic3C2×D411yescharacteristic
204C2×C3×Dic3C2×D411yescharacteristic
205C3×D12—22no—
206C3×C4×D3—42no—
207C3×Dic6—22no—
208C3×Dic6—22no—
209C3×Dic6—42no—
210C3×Dic6—42no—
211C32×C8—22no—
212C3×C3⋊C8—22no—
213#532—31no—
214C2×C4⋊Q8—31no—
215#492—32no—
216#487—32no—
217#461—32no—
218#461—32no—
219#448—31no—
220#448—31no—
221#448—31no—
222#448C3×D311yescharacteristic, complemented
223#431—31no—
224C3×#41—124no—
225C2×C3×Q8—21no—
226C2×C3×Q8—21no—
227C2×C3×Q8—22no—
228C2×C3×Q8—32no—
229C2×C3×Q8—41no—
230C2×C3×Q8—42no—
231C2×C3×Q8—42no—
232C2×C3×Q8—44no—
233C2×C3×Q8—61no—
234C2×C3×Q8—61no—
235C2×C3×Q8—122no—
236C2×C3×Q8C22×D312yes—
237C2×C3×Q8C23×C312yes—
238C2×C3×D4—21no—
239C2×C3×D4—21no—
240C2×C3×D4—21no—
241C2×C3×D4—31no—
242C2×C3×D4—41no—
243C2×C3×D4—61no—
244C2×C3×D4—61no—
245C2×C3×D4—61no—
246C2×C3×D4C22×D311yescharacteristic
247C2×C3×D4C23×C311yescharacteristic
248C22×C3×C4—21no—
249C22×C3×C4—21no—
250C22×C3×C4—21no—
251C22×C3×C4—31no—
252C22×C3×C4—31no—
253C22×C3×C4—41no—
254C22×C3×C4—61no—
255C22×C3×C4—61no—
256C22×C3×C4—61no—
257C22×C3×C4C22×D311yescharacteristic
258C22×C3×C4C23×C311yescharacteristic
259C2×#67—21no—
260C2×#67—21no—
261C2×#67—21no—
262C22×Dic3—21no—
263C22×Dic3—21no—
264C22×Dic3C3×D411yescharacteristic
265C22×Dic3C3×D411yescharacteristic
266#237—44no—
267C2×D12C23×C311yescharacteristic
268C2×C4×D3—21no—
269C2×Dic6—21no—
270C2×Dic6—21no—
271C2×Dic6—42no—
272C2×Dic6C23×C312yes—
273C3×SD16—42no—
274C3×SD16—42no—
275C3×SD16—44no—
276C3×SD16—122no—
277C3×SD16—122no—
278C3×SD16—122no—
279C2×C3×C8—22no—
280C2×C3×C8—32no—
281C2×C3×C8C22×D312yes—
282C2×C3×C8C23×C312yes—
283C3×C4⋊C4—21no—
284C3×C4⋊C4—31no—
285C3×C4⋊C4—62no—
286C3×C4⋊C4—62no—
287C3×C4⋊C4—62no—
288C3×C4⋊C4—62no—
289C3×C4⋊C4—61no—
290C3×C4⋊C4C22×D311yescharacteristic
291C3×C4⋊C4C23×C311yescharacteristic
292C3×C22⋊C4—21no—
293C3×C22⋊C4—21no—
294C3×C22⋊C4—41no—
295C3×C22⋊C4—61no—
296C3×C22⋊C4—61no—
297C3×C22⋊C4—61no—
298C3×C42—31no—
299C3×C42—31no—
300C3×C42—31no—
301C3×C42—31no—
302#217—21no—
303#215—42no—
304#214—42no—
305(C2×D3)⋊C4—21no—
306(C2×D3)⋊C4—21no—
307C4⋊Dic3C23×C311yescharacteristic
308Dic3⋊C4—21no—
309Dic3⋊C4—22no—
310Dic3⋊C4—22no—
311Dic3⋊C4—22no—
312Dic3⋊C4—22no—
313C4×Dic3C3×D411yescharacteristic
314C4×Dic3C3×D411yescharacteristic
315C4×Dic3C23×C311yescharacteristic
316C4×Dic3C23×C311yescharacteristic
317C2×C3⋊C8C23×C312yes—
318#204—42no—
319C22×C32—21no—
320C22×C32—21no—
321C22×C32—41no—
322C22×C32—41no—
323C22×C32—41no—
324C22×C32C2×SD1612yes—
325C22×C32C22×D411yescharacteristic
326C2×C3×D3—41no—
327C2×C3×D3—41no—
328C2×C3×D3—41no—
329C32×C4—21no—
330C32×C4—42no—
331C32×C4—41no—
332C32×C4H(5,𝔽2)12yes—
333C3×Dic3—21no—
334C3×Dic3—42no—
335C3×Dic3—41no—
336C3×Dic3C2×SD1612yes—
337C3×Dic3C2×SD1612yes—
338C3×Dic3#13712yes—
339C2×#41—61no—
340C22×Q8—31no—
341C22×Q8C2×C3×D311yescharacteristic
342C2×SD16—32no—
343C2×SD16—32no—
344C2×SD16—32no—
345C2×SD16C2×C3×D312yes—
346C4⋊Q8—31no—
347C4⋊Q8—32no—
348C4⋊Q8—32no—
349C4⋊Q8—61no—
350C4⋊Q8—61no—
351#127—31no—
352#124—61no—
353#121—31no—
354#121—61no—
355#121C2×C3×D311yescharacteristic
356C4×D4—31no—
357C4×D4—31no—
358C4×D4—61no—
359C2×C4⋊C4—61no—
360C2×C4⋊C4—61no—
361C2×C42—31no—
362C4⋊C8—32no—
363#103—32no—
364#103—32no—
365#103—32no—
366#103C2×C3×D312yes—
367C22⋊C8—31no—
368C22⋊C8C2×C3×D311yescharacteristic
369C4×C8—32no—
370C23×C3—21no—
371C23×C3—21no—
372C23×C3—21no—
373C23×C3—41no—
374C23×C3—61no—
375C23×C3D3×D411yescharacteristic
376C23×C3C2×C3×D411yescharacteristic
377C22×D3—21no—
378C3×Q8—24no—
379C3×Q8—22no—
380C3×Q8—22no—
381C3×Q8—22no—
382C3×Q8—22no—
383C3×Q8—42no—
384C3×Q8—42no—
385C3×Q8—42no—
386C3×Q8—44no—
387C3×Q8—62no—
388C3×Q8—62no—
389C3×Q8—122no—
390C3×Q8—122no—
391C3×D4—22no—
392C3×D4—22no—
393C3×D4—42no—
394C3×D4—42no—
395C3×D4—42no—
396C3×D4—62no—
397C3×D4—82no—
398C3×D4—122no—
399C3×D4—122no—
400C3×D4—122no—
401C2×C3×C4—22no—
402C2×C3×C4—22no—
403C2×C3×C4—21no—
404C2×C3×C4—22no—
405C2×C3×C4—22no—
406C2×C3×C4—21no—
407C2×C3×C4—21no—
408C2×C3×C4—21no—
409C2×C3×C4—34no—
410C2×C3×C4—31no—
411C2×C3×C4—31no—
412C2×C3×C4—31no—
413C2×C3×C4—31no—
414C2×C3×C4—42no—
415C2×C3×C4—42no—
416C2×C3×C4—41no—
417C2×C3×C4—42no—
418C2×C3×C4—42no—
419C2×C3×C4—44no—
420C2×C3×C4—62no—
421C2×C3×C4—61no—
422C2×C3×C4—62no—
423C2×C3×C4—122no—
424C2×C3×C4—122no—
425C2×C3×C4—122no—
426C2×C3×C4D3×D411yescharacteristic
427C2×C3×C4C2×C3×D411yescharacteristic
428C2×C3×C4C23×D311yescharacteristic
429C2×C3×C4C24×C311yescharacteristic
430#67—42no—
431#67—42no—
432#67—42no—
433C2×Dic3—22no—
434C2×Dic3—22no—
435C2×Dic3—21no—
436C2×Dic3—42no—
437C2×Dic3—42no—
438C2×Dic3C3×SD1614yescomplemented
439C2×Dic3C2×C3×D411yescharacteristic
440C2×Dic3C2×C3×D411yescharacteristic
441C2×Dic3C2×C3×D411yescharacteristic
442C2×Dic3C2×C3×D411yescharacteristic
443D12—22no—
444C4×D3—42no—
445Dic6—22no—
446Dic6—22no—
447Dic6—42no—
448Dic6—42no—
449C3×C8—22no—
450C3×C8—22no—
451C3×C8—42no—
452C3×C8—62no—
453C3⋊C8—22no—
454C2×C32—22no—
455C2×C32—81no—
456C2×C32#53511yescharacteristic
457C2×C32C22×SD1611yescharacteristic
458C2×C32C2×#13711yescharacteristic
459C3×D3—81no—
460#41—124no—
461C2×Q8—21no—
462C2×Q8—32no—
463C2×Q8—42no—
464C2×Q8—61no—
465C2×Q8—61no—
466C2×Q8—122no—
467C2×Q8C22×C3×D312yes—
468C2×D4—21no—
469C2×D4—31no—
470C2×D4—61no—
471C2×D4—61no—
472C2×D4—61no—
473C2×D4C22×C3×D311yescharacteristic
474C22×C4—21no—
475C22×C4—31no—
476C22×C4—31no—
477C22×C4—61no—
478C22×C4—61no—
479C22×C4—61no—
480C22×C4C22×C3×D311yescharacteristic
481SD16—42no—
482SD16—122no—
483SD16—122no—
484SD16—122no—
485C2×C8—32no—
486C2×C8C22×C3×D312yes—
487C4⋊C4—31no—
488C4⋊C4—62no—
489C4⋊C4—62no—
490C4⋊C4—62no—
491C4⋊C4—62no—
492C4⋊C4—61no—
493C4⋊C4C22×C3×D311yescharacteristic
494C22⋊C4—21no—
495C22⋊C4—61no—
496C22⋊C4—61no—
497C22⋊C4—61no—
498C42—31no—
499C42—31no—
500C42—31no—
501C42—31no—
502C22×C3—21no—
503C22×C3—21no—
504C22×C3—21no—
505C22×C3—21no—
506C22×C3—22no—
507C22×C3—21no—
508C22×C3—41no—
509C22×C3—41no—
510C22×C3—41no—
511C22×C3—41no—
512C22×C3—41no—
513C22×C3—41no—
514C22×C3—41no—
515C22×C3—41no—
516C22×C3—81no—
517C22×C3—81no—
518C22×C3—81no—
519C22×C3—121no—
520C22×C3—121no—
521C22×C3—121no—
522C22×C3D3×SD1612yes—
523C22×C3C2×C3×SD1612yes—
524C22×C3C2×D3×D411yescharacteristic
525C22×C3C22×C3×D411yescharacteristic
526C2×D3—41no—
527C2×D3—41no—
528C2×D3—41no—
529C3×C4—21no—
530C3×C4—21no—
531C3×C4—22no—
532C3×C4—32no—
533C3×C4—32no—
534C3×C4—32no—
535C3×C4—41no—
536C3×C4—44no—
537C3×C4—41no—
538C3×C4—41no—
539C3×C4—42no—
540C3×C4—42no—
541C3×C4—61no—
542C3×C4—81no—
543C3×C4—121no—
544C3×C4—122no—
545C3×C4#100412yes—
546C3×C4C3×H(5,𝔽2)12yes—
547Dic3—21no—
548Dic3—41no—
549Dic3—42no—
550Dic3C2×C3×SD1612yes—
551Dic3C2×C3×SD1612yes—
552Dic3C3×#13712yes—
553C32#320611yescharacteristic, complemented, Sylow
554C23—21no—
555C23—61no—
556C23C3×D3×D411yescharacteristic
557Q8—22no—
558Q8—22no—
559Q8—42no—
560Q8—62no—
561Q8—62no—
562Q8—122no—
563Q8—122no—
564D4—22no—
565D4—42no—
566D4—62no—
567D4—122no—
568D4—122no—
569D4—122no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—34no—
574C2×C4—31no—
575C2×C4—31no—
576C2×C4—31no—
577C2×C4—31no—
578C2×C4—42no—
579C2×C4—62no—
580C2×C4—61no—
581C2×C4—62no—
582C2×C4—122no—
583C2×C4—122no—
584C2×C4—122no—
585C2×C4C3×D3×D411yescharacteristic
586C2×C4C23×C3×D311yescharacteristic
587C8—22no—
588C8—62no—
589C2×C3—22no—
590C2×C3—22no—
591C2×C3—21no—
592C2×C3—21no—
593C2×C3—21no—
594C2×C3—42no—
595C2×C3—81no—
596C2×C3—81no—
597C2×C3—161no—
598C2×C3—241no—
599C2×C3#559211yescharacteristic
600C2×C3C2×D3×SD1611yescharacteristic
601C2×C3D3×#13711yescharacteristic
602C2×C3C3×#53511yescharacteristic
603C2×C3C22×C3×SD1611yescharacteristic
604C2×C3C2×C3×#13711yescharacteristic
605D3—81no—
606C22—21no—
607C22—21no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—121no—
612C22—121no—
613C22—121no—
614C22C3×D3×SD1612yes—
615C22C2×C3×D3×D411yescharacteristic
616C4—21no—
617C4—32no—
618C4—32no—
619C4—32no—
620C4—41no—
621C4—42no—
622C4—61no—
623C4—121no—
624C4—122no—
625C4C3×#100412yes—
626C3—21no—
627C3#8003411yescharacteristic, direct product factor
628C3C3×#320611yescharacteristic, complemented
629C2—22no—
630C2—81no—
631C2—241no—
632C2C3×#559211yescharacteristic
633C2C2×C3×D3×SD1611yescharacteristic
634C2C3×D3×#13711yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#448C2
socleC22×C32C22×D4
prime radicalC3×#5596C2
hypercenterC3×#448D3
hypocenterC3C3×#3206
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#448

Complements

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

NH ≅ G/Ntype
1. C2×C3×#944C2semidirect
3. C32×#448C2semidirect
4. C3×#5227C2semidirect
5. C3×#5175C2semidirect
6. C3×#5162C2semidirect
7. C3×#5040C2semidirect
8. C3×#4818C2semidirect
9. C3×#4792C2semidirect
10. C3×#4738C2semidirect
11. C3×#4718C2semidirect
13. #80034C3direct (N × H)
47. C2×#944C2×C3semidirect
57. C3×#448D3semidirect
58. C3×#448C2×C3semidirect
60. #5227C2×C3semidirect
61. #5175C2×C3semidirect
62. #5162C2×C3semidirect
63. #5040C2×C3semidirect
64. #4818C2×C3semidirect
65. #4792C2×C3semidirect
66. #4738C2×C3semidirect
67. #4718C2×C3semidirect
200. C2×C3×Dic3SD16semidirect
222. #448C3×D3semidirect
438. C2×Dic3C3×SD16semidirect
553. C32#3206semidirect
627. C3#80034direct (N × H)
628. C3C3×#3206semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9442
C3×#55962
C32×#4482
C3×#52272
C3×#51752
C3×#51622
C3×#50402
C3×#48182
C3×#47922
C3×#47382
C3×#47182
C3×#32063
#800343