Finite Group Databasebeta

← C3×#70414 #11784960

Subgroups — C3×#70414 #11784960

Order 1152.

1253 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#1009C211yesmaximal, characteristic
2C32×#421C211yesmaximal, characteristic, complemented
3C3×#5236C211yesmaximal, characteristic, complemented
4C3×#5132C211yesmaximal, characteristic, complemented
5C3×#5131C211yesmaximal, characteristic, complemented
6C3×#5006C211yesmaximal, characteristic, complemented
7C3×#5001C211yesmaximal, characteristic, complemented
8C3×#1795—31nomaximal
9#70414C311yesmaximal, characteristic, direct product factor
10C2×C32×#41C2211yescharacteristic
11C3×#1009—22no—
12C3×#1009—22no—
13C3×#1009—22no—
14C3×#1009—22no—
15C2×C3×#239C412yescomplemented
16C2×C3×D3×D4—22no—
17C2×C3×D3×D4C412yescomplemented
18C2×C3×#235—21no—
19C2×C3×#235C2211yescharacteristic
20C22×C3×D12—21no—
21C22×C3×D12C2211yescharacteristic
22C2×C32×M16C2211yescharacteristic, complemented
23C32×#117C2211yescharacteristic, complemented
24C3×#926C2211yescharacteristic, complemented
25C2×C3×#208C2211yescharacteristic, complemented
26C32×#104—22no—
27C3×#837—22no—
28C3×#824—21no—
29C3×#823—21no—
30C3×#821—22no—
31C3×#809—22no—
32C3×#806—22no—
33C2×C3×H(5,𝔽2)—31no—
34C2×#1009C2×C311yescharacteristic
35C3×#421—21no—
36C3×#421—31no—
37C3×#421D311yescharacteristic, complemented
38C3×#421C2×C311yescharacteristic, complemented
39C3×#418—31no—
40C3×#418—31no—
41C3×#413—31no—
42C3×#410—31no—
43#5236C2×C311yescharacteristic, complemented
44#5132C2×C311yescharacteristic, complemented
45#5131C2×C311yescharacteristic, complemented
46#5006C2×C311yescharacteristic, complemented
47#5001C2×C311yescharacteristic, complemented
48C23×C3×D3—21no—
49C23×C3×D3—22no—
50C32×#41—21no—
51C32×#41—21no—
52C32×#41D412yes—
53C32×#41D412yes—
54C2×C32×Q8C2×C411yescharacteristic
55C2×C32×D4—21no—
56C2×C32×D4C2×C411yescharacteristic
57C22×C32×C4—21no—
58C22×C32×C4C2311yescharacteristic
59C2×C3×#67—22no—
60C2×C3×#67D412yes—
61C3×#239—24no—
62C3×#239—24no—
63C3×D3×D4—28no—
64C3×D3×D4—28no—
65C3×D3×D4—24no—
66C3×D3×D4—24no—
67C3×#235—24no—
68C3×#235—24no—
69C3×#235—21no—
70C3×#235—21no—
71C3×#235—21no—
72C3×#235—21no—
73C2×C3×D12—24no—
74C2×C3×D12—24no—
75C2×C3×D12—22no—
76C2×C3×D12—21no—
77C2×C3×D12—21no—
78C2×C3×D12—21no—
79C2×C3×D12C2×C412yes—
80C2×C3×D12D411yescharacteristic
81C2×C3×D12D412yes—
82C2×C3×D12D412yes—
83C2×C3×C4×D3—22no—
84C2×C3×C4×D3C2×C412yes—
85C2×C3×Dic6D411yescharacteristic
86C32×M16—21no—
87C32×M16—21no—
88C2×C32×C8—21no—
89C32×C4⋊C4—21no—
90C32×C22⋊C4—21no—
91C32×C42—21no—
92C3×#217—21no—
93C3×C4⋊Dic3—21no—
94C3×C4×Dic3—21no—
95C3×#208—21no—
96C3×#208—21no—
97C2×C3×C3⋊C8—21no—
98#1795—31noSylow
99C3×H(5,𝔽2)—62no—
100C3×H(5,𝔽2)—62no—
101C3×H(5,𝔽2)—62no—
102C3×H(5,𝔽2)—62no—
103C2×C3×#41—21no—
104C2×C3×#41—31no—
105C2×C3×#41—32no—
106C2×C3×#41—61no—
107C2×C3×#41C2×D311yescharacteristic
108C2×C3×#41C22×C311yescharacteristic
109C22×C3×D4—32no—
110C22×C3×D4—31no—
111C22×C3×D4—61no—
112C22×C3×D4—62no—
113#1009—22no—
114#1009—22no—
115#1009—22no—
116#1009—22no—
117C2×#239C3×C412yescomplemented
118C2×D3×D4—22no—
119C2×D3×D4C3×C412yescomplemented
120C2×#235—21no—
121C2×#235C22×C311yescharacteristic
122C22×D12—21no—
123C22×D12C22×C311yescharacteristic
124C2×C3×M16—21no—
125C2×C3×M16—31no—
126C2×C3×M16C2×D311yescharacteristic, complemented
127C2×C3×M16C22×C311yescharacteristic, complemented
128C3×#117—21no—
129C3×#117—31no—
130C3×#117C2×D311yescharacteristic, complemented
131C3×#117C22×C311yescharacteristic, complemented
132#926C22×C311yescharacteristic, complemented
133C2×#208C22×C311yescharacteristic, complemented
134C3×#104—24no—
135C3×#104—22no—
136C3×#104—22no—
137C3×#104—62no—
138C3×#102—62no—
139C3×#102—62no—
140C3×#101—61no—
141C3×#100—61no—
142C3×#99—62no—
143#837—22no—
144#824—21no—
145#823—21no—
146#821—22no—
147#809—22no—
148#806—22no—
149C23×C32—21no—
150C23×C32C2×D411yescharacteristic
151C22×C3×D3—22no—
152C22×C3×D3—28no—
153C22×C3×D3—22no—
154C22×C3×D3—22no—
155C22×C3×D3—82no—
156C22×C3×D3—84no—
157C22×C3×D3C22⋊C412yes—
158C32×Q8—21no—
159C32×Q8C22⋊C412yes—
160C32×D4—22no—
161C32×D4—22no—
162C32×D4—21no—
163C32×D4C22⋊C412yes—
164C2×C32×C4—21no—
165C2×C32×C4—22no—
166C2×C32×C4—22no—
167C2×C32×C4—41no—
168C2×C32×C4C22×C411yescharacteristic
169C2×C32×C4C2×D411yescharacteristic
170C2×C32×C4C2×D411yescharacteristic
171C2×C32×C4C2×D411yescharacteristic
172C2×C32×C4C2×D411yescharacteristic
173C2×C32×C4C2×D411yescharacteristic
174C3×#67—28no—
175C3×#67—22no—
176C3×#67—22no—
177C2×C3×Dic3—41no—
178C2×C3×Dic3C22⋊C412yes—
179C3×D12—28no—
180C3×D12—22no—
181C3×D12—21no—
182C3×D12—22no—
183C3×D12—21no—
184C3×D12—22no—
185C3×D12C22⋊C414yes—
186C3×C4×D3—28no—
187C3×C4×D3—22no—
188C3×C4×D3—22no—
189C3×Dic6—21no—
190C3×Dic6—21no—
191C32×C8—41no—
192C3×C3⋊C8—41no—
193C2×H(5,𝔽2)—31no—
194#421—31no—
195#421C3×D311yescharacteristic, complemented
196#418—31no—
197#418—31no—
198#413—31no—
199#410—31no—
200C24×C3—62no—
201C24×C3—61no—
202C23×D3—21no—
203C23×D3—22no—
204C3×#41—21no—
205C3×#41—21no—
206C3×#41—21no—
207C3×#41—21no—
208C3×#41—22no—
209C3×#41—22no—
210C3×#41—41no—
211C3×#41—41no—
212C3×#41—64no—
213C3×#41—64no—
214C3×#41—64no—
215C3×#41—64no—
216C3×#41—61no—
217C3×#41—61no—
218C3×#41—61no—
219C3×#41—61no—
220C3×#41D1212yes—
221C3×#41#6712yes—
222C3×#41C3×D412yes—
223C3×#41C3×D412yes—
224C2×C3×Q8—21no—
225C2×C3×Q8—31no—
226C2×C3×Q8C4×D311yescharacteristic
227C2×C3×Q8C2×C3×C411yescharacteristic
228C2×C3×D4—21no—
229C2×C3×D4—21no—
230C2×C3×D4—21no—
231C2×C3×D4—32no—
232C2×C3×D4—32no—
233C2×C3×D4—32no—
234C2×C3×D4—32no—
235C2×C3×D4—31no—
236C2×C3×D4—41no—
237C2×C3×D4—62no—
238C2×C3×D4—64no—
239C2×C3×D4—64no—
240C2×C3×D4—62no—
241C2×C3×D4—64no—
242C2×C3×D4—68no—
243C2×C3×D4—68no—
244C2×C3×D4—64no—
245C2×C3×D4—61no—
246C2×C3×D4—61no—
247C2×C3×D4—61no—
248C2×C3×D4C4×D311yescharacteristic
249C2×C3×D4C2×C3×C411yescharacteristic
250C22×C3×C4—21no—
251C22×C3×C4—21no—
252C22×C3×C4—21no—
253C22×C3×C4—32no—
254C22×C3×C4—41no—
255C22×C3×C4—62no—
256C22×C3×C4C22×D311yescharacteristic
257C22×C3×C4C23×C311yescharacteristic
258C2×#67—22no—
259C2×#67C3×D412yes—
260#239—24no—
261#239—24no—
262D3×D4—28no—
263D3×D4—28no—
264D3×D4—24no—
265D3×D4—24no—
266#235—24no—
267#235—24no—
268#235—21no—
269#235—21no—
270#235—21no—
271#235—21no—
272C2×D12—24no—
273C2×D12—24no—
274C2×D12—22no—
275C2×D12—21no—
276C2×D12—21no—
277C2×D12—21no—
278C2×D12C2×C3×C412yes—
279C2×D12C3×D411yescharacteristic
280C2×D12C3×D412yes—
281C2×D12C3×D412yes—
282C2×C4×D3—22no—
283C2×C4×D3C2×C3×C412yes—
284C2×Dic6C3×D411yescharacteristic
285C3×M16—22no—
286C3×M16—21no—
287C3×M16—21no—
288C3×M16—21no—
289C3×M16—21no—
290C3×M16—41no—
291C3×M16—61no—
292C3×M16—61no—
293C2×C3×C8—21no—
294C2×C3×C8—21no—
295C2×C3×C8—41no—
296C2×C3×C8—61no—
297C3×C4⋊C4—21no—
298C3×C4⋊C4—21no—
299C3×C4⋊C4—41no—
300C3×C4⋊C4—61no—
301C3×C22⋊C4—21no—
302C3×C22⋊C4—21no—
303C3×C22⋊C4—41no—
304C3×C22⋊C4—61no—
305C3×C42—22no—
306C3×C42—21no—
307C3×C42—21no—
308C3×C42—61no—
309#217—21no—
310C4⋊Dic3—21no—
311C4×Dic3—21no—
312#208—21no—
313#208—21no—
314C2×C3⋊C8—21no—
315C22×C32—22no—
316C22×C32—41no—
317C22×C32—42no—
318C22×C32C2×C22⋊C411yescharacteristic
319C22×C32#12011yescharacteristic
320C22×C32#12011yescharacteristic
321C2×C3×D3—22no—
322C2×C3×D3—24no—
323C2×C3×D3—44no—
324C2×C3×D3—42no—
325C2×C3×D3—82no—
326C2×C3×D3—82no—
327C2×C3×D3—88no—
328C32×C4—22no—
329C32×C4—81no—
330C32×C4C2×C22⋊C411yescharacteristic
331C32×C4C2×C22⋊C411yescharacteristic
332C32×C4#12011yescharacteristic
333C32×C4#12011yescharacteristic
334C3×Dic3—22no—
335C3×Dic3—81no—
336H(5,𝔽2)—62no—
337H(5,𝔽2)—62no—
338H(5,𝔽2)—62no—
339H(5,𝔽2)—62no—
340C2×#41—31no—
341C2×#41—32no—
342C2×#41—61no—
343C2×#41C2×C3×D311yescharacteristic
344C22×D4—32no—
345C22×D4—31no—
346C22×D4—61no—
347C22×D4—62no—
348C2×M16—31no—
349C2×M16C2×C3×D311yescharacteristic, complemented
350#117—31no—
351#117C2×C3×D311yescharacteristic, complemented
352#104—22no—
353#104—62no—
354#102—62no—
355#102—62no—
356#101—61no—
357#100—61no—
358#99—62no—
359C23×C3—21no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—32no—
363C23×C3—41no—
364C23×C3—68no—
365C23×C3—62no—
366C23×C3—62no—
367C23×C3—62no—
368C23×C3—242no—
369C23×C3—244no—
370C23×C3D3×D411yescharacteristic
371C23×C3C2×C3×D411yescharacteristic
372C22×D3—22no—
373C22×D3—28no—
374C22×D3—22no—
375C22×D3—22no—
376C22×D3—82no—
377C22×D3—84no—
378C22×D3C3×C22⋊C412yes—
379C3×Q8—21no—
380C3×Q8—21no—
381C3×Q8—22no—
382C3×Q8—41no—
383C3×Q8—61no—
384C3×Q8—61no—
385C3×Q8(C2×D3)⋊C412yes—
386C3×Q8C3×C22⋊C412yes—
387C3×D4—22no—
388C3×D4—22no—
389C3×D4—22no—
390C3×D4—22no—
391C3×D4—21no—
392C3×D4—21no—
393C3×D4—22no—
394C3×D4—34no—
395C3×D4—41no—
396C3×D4—42no—
397C3×D4—42no—
398C3×D4—68no—
399C3×D4—68no—
400C3×D4—62no—
401C3×D4—62no—
402C3×D4—62no—
403C3×D4—61no—
404C3×D4—62no—
405C3×D4—62no—
406C3×D4—61no—
407C3×D4(C2×D3)⋊C412yes—
408C3×D4C3×C22⋊C412yes—
409C2×C3×C4—21no—
410C2×C3×C4—21no—
411C2×C3×C4—22no—
412C2×C3×C4—22no—
413C2×C3×C4—22no—
414C2×C3×C4—22no—
415C2×C3×C4—21no—
416C2×C3×C4—21no—
417C2×C3×C4—21no—
418C2×C3×C4—21no—
419C2×C3×C4—21no—
420C2×C3×C4—21no—
421C2×C3×C4—32no—
422C2×C3×C4—42no—
423C2×C3×C4—42no—
424C2×C3×C4—41no—
425C2×C3×C4—42no—
426C2×C3×C4—41no—
427C2×C3×C4—41no—
428C2×C3×C4—68no—
429C2×C3×C4—62no—
430C2×C3×C4—62no—
431C2×C3×C4—121no—
432C2×C3×C4C2×C4×D311yescharacteristic
433C2×C3×C4C2×D1211yescharacteristic
434C2×C3×C4D3×D411yescharacteristic
435C2×C3×C4D3×D411yescharacteristic
436C2×C3×C4D3×D411yescharacteristic
437C2×C3×C4C2×#6711yescharacteristic
438C2×C3×C4C22×C3×C411yescharacteristic
439C2×C3×C4C2×C3×D411yescharacteristic
440C2×C3×C4C2×C3×D411yescharacteristic
441C2×C3×C4C2×C3×D411yescharacteristic
442C2×C3×C4C2×C3×D411yescharacteristic
443C2×C3×C4C2×C3×D411yescharacteristic
444#67—28no—
445#67—22no—
446#67—22no—
447C2×Dic3—41no—
448C2×Dic3C3×C22⋊C412yes—
449D12—28no—
450D12—22no—
451D12—21no—
452D12—22no—
453D12—21no—
454D12—22no—
455D12C3×C22⋊C414yes—
456C4×D3—28no—
457C4×D3—22no—
458C4×D3—22no—
459Dic6—21no—
460Dic6—21no—
461C3×C8—42no—
462C3×C8—41no—
463C3×C8—41no—
464C3×C8—121no—
465C3⋊C8—41no—
466C2×C32—21no—
467C2×C32—21no—
468C2×C32—21no—
469C2×C32—42no—
470C2×C32#37911yescharacteristic
471C3×D3—44no—
472C3×D3—42no—
473C24—62no—
474C24—61no—
475#41—21no—
476#41—21no—
477#41—64no—
478#41—64no—
479#41—64no—
480#41—64no—
481#41—61no—
482#41—61no—
483#41—61no—
484#41—61no—
485#41C3×D1212yes—
486#41C3×#6712yes—
487C2×Q8—31no—
488C2×Q8C3×C4×D311yescharacteristic
489C2×D4—21no—
490C2×D4—32no—
491C2×D4—32no—
492C2×D4—32no—
493C2×D4—32no—
494C2×D4—31no—
495C2×D4—62no—
496C2×D4—64no—
497C2×D4—64no—
498C2×D4—62no—
499C2×D4—64no—
500C2×D4—68no—
501C2×D4—68no—
502C2×D4—64no—
503C2×D4—61no—
504C2×D4—61no—
505C2×D4—61no—
506C2×D4C3×C4×D311yescharacteristic
507C22×C4—21no—
508C22×C4—32no—
509C22×C4—62no—
510C22×C4C22×C3×D311yescharacteristic
511M16—21no—
512M16—21no—
513M16—61no—
514M16—61no—
515C2×C8—21no—
516C2×C8—61no—
517C4⋊C4—21no—
518C4⋊C4—61no—
519C22⋊C4—21no—
520C22⋊C4—61no—
521C42—21no—
522C42—61no—
523C22×C3—22no—
524C22×C3—22no—
525C22×C3—21no—
526C22×C3—21no—
527C22×C3—21no—
528C22×C3—42no—
529C22×C3—41no—
530C22×C3—41no—
531C22×C3—42no—
532C22×C3—42no—
533C22×C3—64no—
534C22×C3—62no—
535C22×C3—81no—
536C22×C3—82no—
537C22×C3—122no—
538C22×C3—124no—
539C22×C3—242no—
540C22×C3—242no—
541C22×C3—248no—
542C22×C3D3×C22⋊C411yescharacteristic
543C22×C3#88211yescharacteristic
544C22×C3#93711yescharacteristic
545C22×C3C2×C3×C22⋊C411yescharacteristic
546C22×C3C3×#12011yescharacteristic
547C22×C3C3×#12011yescharacteristic
548C2×D3—22no—
549C2×D3—24no—
550C2×D3—44no—
551C2×D3—42no—
552C2×D3—82no—
553C2×D3—82no—
554C2×D3—88no—
555C3×C4—22no—
556C3×C4—22no—
557C3×C4—21no—
558C3×C4—21no—
559C3×C4—21no—
560C3×C4—21no—
561C3×C4—42no—
562C3×C4—62no—
563C3×C4—81no—
564C3×C4—81no—
565C3×C4—82no—
566C3×C4—241no—
567C3×C4D3×C22⋊C411yescharacteristic
568C3×C4#88211yescharacteristic
569C3×C4C2×(C2×D3)⋊C411yescharacteristic
570C3×C4#93711yescharacteristic
571C3×C4C2×C3×C22⋊C411yescharacteristic
572C3×C4C2×C3×C22⋊C411yescharacteristic
573C3×C4C3×#12011yescharacteristic
574C3×C4C3×#12011yescharacteristic
575Dic3—22no—
576Dic3—81no—
577C32#179511yescharacteristic, complemented, Sylow
578C23—21no—
579C23—32no—
580C23—68no—
581C23—62no—
582C23—62no—
583C23—62no—
584C23—242no—
585C23—244no—
586C23C3×D3×D411yescharacteristic
587Q8—21no—
588Q8—61no—
589Q8—61no—
590Q8C3×(C2×D3)⋊C412yes—
591D4—22no—
592D4—22no—
593D4—21no—
594D4—34no—
595D4—68no—
596D4—68no—
597D4—62no—
598D4—62no—
599D4—62no—
600D4—61no—
601D4—62no—
602D4—62no—
603D4—61no—
604D4C3×(C2×D3)⋊C412yes—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—32no—
609C2×C4—41no—
610C2×C4—68no—
611C2×C4—62no—
612C2×C4—62no—
613C2×C4—121no—
614C2×C4C2×C3×C4×D311yescharacteristic
615C2×C4C2×C3×D1211yescharacteristic
616C2×C4C3×D3×D411yescharacteristic
617C2×C4C3×D3×D411yescharacteristic
618C2×C4C3×D3×D411yescharacteristic
619C2×C4C2×C3×#6711yescharacteristic
620C8—41no—
621C8—121no—
622C2×C3—21no—
623C2×C3—21no—
624C2×C3—21no—
625C2×C3—21no—
626C2×C3—21no—
627C2×C3—21no—
628C2×C3—21no—
629C2×C3—41no—
630C2×C3—41no—
631C2×C3—41no—
632C2×C3—42no—
633C2×C3—42no—
634C2×C3—82no—
635C2×C3—124no—
636C2×C3—122no—
637C2×C3#495511yescharacteristic
638C2×C3C3×#37911yescharacteristic
639D3—44no—
640D3—42no—
641C22—22no—
642C22—41no—
643C22—42no—
644C22—64no—
645C22—62no—
646C22—122no—
647C22—124no—
648C22—242no—
649C22—242no—
650C22—248no—
651C22C3×D3×C22⋊C411yescharacteristic
652C22C3×#88211yescharacteristic
653C22C3×#93711yescharacteristic
654C4—22no—
655C4—62no—
656C4—81no—
657C4—241no—
658C4C3×D3×C22⋊C411yescharacteristic
659C4C3×#88211yescharacteristic
660C4C2×C3×(C2×D3)⋊C411yescharacteristic
661C4C3×#93711yescharacteristic
662C3—21no—
663C3#7041411yescharacteristic, direct product factor
664C3C3×#179511yescharacteristic, complemented
665C2—21no—
666C2—21no—
667C2—21no—
668C2—42no—
669C2—124no—
670C2—122no—
671C2C3×#495511yescharacteristic
subgroupG/H
centerC2×C3#4955
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#421C2
socleC2×C32#379
prime radicalC2×C3×#1009C2
cosocleC22×C3×C4C23×C3
hypercenterC3×#421D3
hypocenterC3C3×#1795
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C3×#421

Complements

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

NH ≅ G/Ntype
2. C32×#421C2semidirect
3. C3×#5236C2semidirect
4. C3×#5132C2semidirect
5. C3×#5131C2semidirect
6. C3×#5006C2semidirect
7. C3×#5001C2semidirect
9. #70414C3direct (N × H)
15. C2×C3×#239C4semidirect
17. C2×C3×D3×D4C4semidirect
22. C2×C32×M16C22semidirect
23. C32×#117C22semidirect
24. C3×#926C22semidirect
25. C2×C3×#208C22semidirect
37. C3×#421D3semidirect
38. C3×#421C2×C3semidirect
43. #5236C2×C3semidirect
44. #5132C2×C3semidirect
45. #5131C2×C3semidirect
46. #5006C2×C3semidirect
47. #5001C2×C3semidirect
117. C2×#239C3×C4semidirect
119. C2×D3×D4C3×C4semidirect
126. C2×C3×M16C2×D3semidirect
127. C2×C3×M16C22×C3semidirect
130. C3×#117C2×D3semidirect
131. C3×#117C22×C3semidirect
132. #926C22×C3semidirect
133. C2×#208C22×C3semidirect
195. #421C3×D3semidirect
349. C2×M16C2×C3×D3semidirect
351. #117C2×C3×D3semidirect
577. C32#1795semidirect
663. C3#70414direct (N × H)
664. C3C3×#1795semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#10092
C32×#4212
C3×#52362
C3×#51322
C3×#51312
C3×#50062
C3×#50012
C3×#17953
#704143