Finite Group Databasebeta

← C3×#83816 #11868164

Subgroups — C3×#83816 #11868164

Order 1152.

1021 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#568C211yesmaximal, characteristic, complemented
2C2×C3×#951C211yesmaximal, characteristic, complemented
3C3×#5822C211yesmaximal, characteristic
4C2×C3×#909C211yesmaximal, characteristic, complemented
5C3×#5143C214yesmaximal, complemented
6C3×#5141C214yesmaximal, complemented
7C2×C3×#824C211yesmaximal, characteristic, complemented
8C3×#5132C211yesmaximal, characteristic, complemented
9C3×#5125C211yesmaximal, characteristic, complemented
10C3×#3069—31nomaximal
11#83816C311yesmaximal, characteristic, direct product factor
12C2×C32×#41C2211yescharacteristic
13C2×C3×#235C2211yescharacteristic
14C22×C3×Dic6C2211yescharacteristic
15C32×#131—22no—
16C32×#131C2214yes—
17C2×C32×M16—22no—
18C2×C32×M16C2211yescharacteristic
19C2×C32×M16C2211yescharacteristic
20C3×#951—22no—
21C3×#951C2212yes—
22C3×#951C2214yes—
23C3×#950—22no—
24C3×#949C2212yes—
25C2×C3×#216—21no—
26C3×#942—21no—
27C2×C3×#214—21no—
28C3×#932—21no—
29C2×C3×#208C2211yescharacteristic
30C2×C3×#208C2211yescharacteristic
31C3×#909—22no—
32C3×#909—22no—
33C2×C3×Dic12C2212yescomplemented
34C2×C3×#204C2212yescomplemented
35C3×#824C2212yescomplemented
36C3×#824C2214yescomplemented
37C3×#823C2212yescomplemented
38C3×#819C2214yescomplemented
39C3×#575—31no—
40C2×C3×#137—31no—
41C2×C3×#137—31no—
42C3×#568—21no—
43C3×#568D311yescharacteristic, complemented
44C3×#568C2×C311yescharacteristic, complemented
45C2×#951C2×C311yescharacteristic, complemented
46#5822C2×C311yescharacteristic
47C2×#909C2×C311yescharacteristic, complemented
48C3×#473—34no—
49C3×#471—34no—
50C3×#430—31no—
51C3×#413—31no—
52C2×C3×#101—31no—
53#5143C2×C314yescomplemented
54#5141C2×C314yescomplemented
55C2×#824C2×C311yescharacteristic, complemented
56#5132C2×C311yescharacteristic, complemented
57#5125C2×C311yescharacteristic, complemented
58C32×#41—21no—
59C32×#41—21no—
60C32×#41D412yes—
61C32×#41C2312yes—
62C2×C32×Q8D411yescharacteristic
63C2×C32×D4—21no—
64C2×C32×D4D411yescharacteristic
65C22×C32×C4—21no—
66C22×C32×C4C2311yescharacteristic
67C2×C3×#67—21no—
68C22×C3×Dic3—21no—
69C3×#235—21no—
70C3×#235—21no—
71C3×#235—21no—
72C3×#235—21no—
73C2×C3×D12C2311yescharacteristic
74C2×C3×C4×D3—21no—
75C2×C3×Dic6—21no—
76C2×C3×Dic6—21no—
77C2×C3×Dic6—21no—
78C2×C3×Dic6—21no—
79C2×C3×Dic6—21no—
80C2×C3×Dic6C2312yes—
81C2×C3×Dic6C2311yescharacteristic
82C32×M16—21no—
83C32×M16—24no—
84C32×M16D412yes—
85C32×M16D412yes—
86C32×M16C2312yes—
87C2×C32×C8—21no—
88C2×C32×C8—24no—
89C2×C32×C8C2312yes—
90C3×#216—22no—
91C3×#216—24no—
92C3×#215—22no—
93C3×#214—22no—
94C3×#214—24no—
95C3×#213—22no—
96C3×#208—21no—
97C3×#208—21no—
98C3×#208C2312yes—
99C3×#208C2312yes—
100C2×C3×C3⋊C8—21no—
101C2×C3×C3⋊C8—21no—
102C3×Dic12—24no—
103C3×#204—24no—
104#3069—31noSylow
105C2×C3×#41—21no—
106C2×C3×#41—31no—
107C2×C3×#41C2×D311yescharacteristic
108C2×C3×#41C22×C311yescharacteristic
109C22×C3×Q8—31no—
110C2×#235C22×C311yescharacteristic
111C22×Dic6C22×C311yescharacteristic
112C3×#137—34no—
113C3×#137—32no—
114C3×#137—62no—
115C3×#137—61no—
116C3×#137—62no—
117C3×#137—62no—
118C3×AGL(1,ℤ/8)—32no—
119C3×AGL(1,ℤ/8)—61no—
120C3×#135—62no—
121C2×C3×Q16—32no—
122C2×C3×Q16—61no—
123C2×C3×SD16—32no—
124C2×C3×SD16—61no—
125C3×#131—24no—
126C3×#131—22no—
127C3×#131—22no—
128C3×#131—24no—
129C3×#131C2×D314yes—
130C3×#131C22×C314yes—
131C2×C3×M16—24no—
132C2×C3×M16—22no—
133C2×C3×M16—22no—
134C2×C3×M16—21no—
135C2×C3×M16—21no—
136C2×C3×M16—31no—
137C2×C3×M16—31no—
138C2×C3×M16C2×D311yescharacteristic
139C2×C3×M16C2×D311yescharacteristic
140C2×C3×M16C22×C311yescharacteristic
141C2×C3×M16C22×C311yescharacteristic
142#951—22no—
143#951C22×C314yes—
144#951C22×C312yes—
145#950—22no—
146#949C22×C312yes—
147C2×#216—21no—
148#942—21no—
149C2×#214—21no—
150#932—21no—
151C2×#208C22×C311yescharacteristic
152C2×#208C22×C311yescharacteristic
153#909—22no—
154#909—22no—
155C2×Dic12C22×C312yescomplemented
156C2×#204C22×C312yescomplemented
157C3×#108—34no—
158C3×#101—34no—
159C3×#101—32no—
160C3×#100—32no—
161#824C22×C312yescomplemented
162#824C22×C314yescomplemented
163#823C22×C312yescomplemented
164#819C22×C314yescomplemented
165C23×C32—21no—
166C23×C32#4111yescharacteristic
167C22×C3×D3—21no—
168C32×Q8—21no—
169C32×Q8C2×D412yes—
170C32×D4—22no—
171C32×D4—22no—
172C32×D4—21no—
173C32×D4C2×D412yes—
174C2×C32×C4—21no—
175C2×C32×C4—22no—
176C2×C32×C4—22no—
177C2×C32×C4C2×D411yescharacteristic
178C2×C32×C4C2×D411yescharacteristic
179C2×C32×C4C2×D411yescharacteristic
180C2×C32×C4C2×D411yescharacteristic
181C2×C32×C4#4111yescharacteristic
182C2×C32×C4C2411yescharacteristic
183C3×#67—41no—
184C3×#67—41no—
185C2×C3×Dic3—22no—
186C2×C3×Dic3—21no—
187C2×C3×Dic3—41no—
188C2×C3×Dic3—41no—
189C3×D12—21no—
190C3×D12—21no—
191C3×C4×D3—41no—
192C3×C4×D3—41no—
193C3×Dic6—22no—
194C3×Dic6—21no—
195C3×Dic6—22no—
196C3×Dic6—21no—
197C3×Dic6—41no—
198C3×Dic6—41no—
199C32×C8—22no—
200C32×C8C2×D414yes—
201C3×C3⋊C8—22no—
202C3×C3⋊C8—22no—
203#575—31no—
204C2×#137—31no—
205C2×#137—31no—
206#568C3×D311yescharacteristic, complemented
207#473—34no—
208#471—34no—
209#430—31no—
210#413—31no—
211C2×#101—31no—
212C3×#41—22no—
213C3×#41—22no—
214C3×#41—21no—
215C3×#41—21no—
216C3×#41—21no—
217C3×#41—21no—
218C3×#41—22no—
219C3×#41—22no—
220C3×#41—61no—
221C3×#41—61no—
222C3×#41—61no—
223C3×#41—61no—
224C3×#41D1212yes—
225C3×#41C3×D412yes—
226C3×#41C22×D312yes—
227C3×#41C23×C312yes—
228C2×C3×Q8—21no—
229C2×C3×Q8—32no—
230C2×C3×Q8—31no—
231C2×C3×Q8—61no—
232C2×C3×Q8—61no—
233C2×C3×Q8—61no—
234C2×C3×Q8—61no—
235C2×C3×Q8—61no—
236C2×C3×Q8D1211yescharacteristic
237C2×C3×Q8C3×D411yescharacteristic
238C2×C3×D4—22no—
239C2×C3×D4—21no—
240C2×C3×D4—21no—
241C2×C3×D4—21no—
242C2×C3×D4—31no—
243C2×C3×D4—61no—
244C2×C3×D4D1211yescharacteristic
245C2×C3×D4C3×D411yescharacteristic
246C22×C3×C4—22no—
247C22×C3×C4—21no—
248C22×C3×C4—21no—
249C22×C3×C4—21no—
250C22×C3×C4—61no—
251C22×C3×C4—61no—
252C22×C3×C4C22×D311yescharacteristic
253C22×C3×C4C23×C311yescharacteristic
254C2×#67—21no—
255C22×Dic3—21no—
256#235—21no—
257#235—21no—
258#235—21no—
259#235—21no—
260C2×D12C23×C311yescharacteristic
261C2×C4×D3—21no—
262C2×Dic6—21no—
263C2×Dic6—21no—
264C2×Dic6—21no—
265C2×Dic6—21no—
266C2×Dic6—21no—
267C2×Dic6C23×C312yes—
268C2×Dic6C23×C311yescharacteristic
269C3×Q16—64no—
270C3×Q16—62no—
271C3×Q16—64no—
272C3×SD16—62no—
273C3×SD16—64no—
274C3×SD16—62no—
275C3×SD16—64no—
276C3×D8—62no—
277C3×M16—22no—
278C3×M16—28no—
279C3×M16—21no—
280C3×M16—21no—
281C3×M16—24no—
282C3×M16—24no—
283C3×M16—22no—
284C3×M16—22no—
285C3×M16—22no—
286C3×M16—32no—
287C3×M16—32no—
288C3×M16—61no—
289C3×M16—61no—
290C3×M16#6712yes—
291C3×M16#6712yes—
292C3×M16C3×D412yes—
293C3×M16C3×D412yes—
294C3×M16C22×D312yes—
295C3×M16C23×C312yes—
296C2×C3×C8—22no—
297C2×C3×C8—28no—
298C2×C3×C8—21no—
299C2×C3×C8—21no—
300C2×C3×C8—24no—
301C2×C3×C8—24no—
302C2×C3×C8—22no—
303C2×C3×C8—61no—
304C2×C3×C8—61no—
305C2×C3×C8C22×D312yes—
306C2×C3×C8C23×C312yes—
307#216—22no—
308#216—24no—
309#215—22no—
310#214—22no—
311#214—24no—
312#213—22no—
313#208—21no—
314#208—21no—
315#208C23×C312yes—
316#208C23×C312yes—
317C2×C3⋊C8—21no—
318C2×C3⋊C8—21no—
319Dic12—24no—
320#204—24no—
321C22×C32—22no—
322C22×C32—41no—
323C22×C32—81no—
324C22×C32#12111yescharacteristic
325C22×C32#12111yescharacteristic
326C22×C32C2×#4111yescharacteristic
327C2×C3×D3—41no—
328C2×C3×D3—81no—
329C32×C4—22no—
330C32×C4#12111yescharacteristic
331C32×C4#12111yescharacteristic
332C32×C4C22×D411yescharacteristic
333C32×C4C22×D411yescharacteristic
334C3×Dic3—42no—
335C3×Dic3—41no—
336C2×#41—31no—
337C2×#41C2×C3×D311yescharacteristic
338C22×Q8—31no—
339#137—34no—
340#137—32no—
341#137—62no—
342#137—61no—
343#137—62no—
344#137—62no—
345AGL(1,ℤ/8)—32no—
346AGL(1,ℤ/8)—61no—
347#135—62no—
348C2×Q16—32no—
349C2×Q16—61no—
350C2×SD16—32no—
351C2×SD16—61no—
352#131—22no—
353#131C2×C3×D314yes—
354C2×M16—22no—
355C2×M16—31no—
356C2×M16—31no—
357C2×M16C2×C3×D311yescharacteristic
358C2×M16C2×C3×D311yescharacteristic
359#108—34no—
360#101—34no—
361#101—32no—
362#100—32no—
363C23×C3—22no—
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—21no—
367C23×C3—61no—
368C23×C3#23511yescharacteristic
369C23×C3C3×#4111yescharacteristic
370C22×D3—21no—
371C3×Q8—22no—
372C3×Q8—21no—
373C3×Q8—21no—
374C3×Q8—22no—
375C3×Q8—62no—
376C3×Q8—61no—
377C3×Q8—62no—
378C3×Q8—61no—
379C3×Q8—121no—
380C3×Q8—121no—
381C3×Q8C2×D1212yes—
382C3×Q8C2×C3×D412yes—
383C3×D4—24no—
384C3×D4—24no—
385C3×D4—22no—
386C3×D4—22no—
387C3×D4—22no—
388C3×D4—22no—
389C3×D4—22no—
390C3×D4—21no—
391C3×D4—21no—
392C3×D4—22no—
393C3×D4—61no—
394C3×D4—61no—
395C3×D4—121no—
396C3×D4—121no—
397C3×D4C2×D1212yes—
398C3×D4C2×C3×D412yes—
399C2×C3×C4—24no—
400C2×C3×C4—22no—
401C2×C3×C4—24no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—22no—
405C2×C3×C4—22no—
406C2×C3×C4—22no—
407C2×C3×C4—22no—
408C2×C3×C4—21no—
409C2×C3×C4—21no—
410C2×C3×C4—21no—
411C2×C3×C4—21no—
412C2×C3×C4—21no—
413C2×C3×C4—21no—
414C2×C3×C4—61no—
415C2×C3×C4—62no—
416C2×C3×C4—121no—
417C2×C3×C4—121no—
418C2×C3×C4—121no—
419C2×C3×C4—121no—
420C2×C3×C4C2×D1211yescharacteristic
421C2×C3×C4C2×D1211yescharacteristic
422C2×C3×C4#23511yescharacteristic
423C2×C3×C4C2×#6711yescharacteristic
424C2×C3×C4C2×#6711yescharacteristic
425C2×C3×C4C2×C3×D411yescharacteristic
426C2×C3×C4C2×C3×D411yescharacteristic
427C2×C3×C4C2×C3×D411yescharacteristic
428C2×C3×C4C2×C3×D411yescharacteristic
429C2×C3×C4C3×#4111yescharacteristic
430C2×C3×C4C23×D311yescharacteristic
431C2×C3×C4C24×C311yescharacteristic
432#67—41no—
433#67—41no—
434C2×Dic3—22no—
435C2×Dic3—21no—
436C2×Dic3—41no—
437C2×Dic3—41no—
438D12—21no—
439D12—21no—
440C4×D3—41no—
441C4×D3—41no—
442Dic6—22no—
443Dic6—21no—
444Dic6—22no—
445Dic6—21no—
446Dic6—41no—
447Dic6—41no—
448C3×C8—24no—
449C3×C8—22no—
450C3×C8—22no—
451C3×C8—24no—
452C3×C8—62no—
453C3×C8—62no—
454C3×C8C2×#6714yes—
455C3×C8C2×C3×D414yes—
456C3⋊C8—22no—
457C3⋊C8—22no—
458C2×C32—21no—
459C2×C32—21no—
460C2×C32—21no—
461C2×C32—42no—
462C2×C32C2×#12111yescharacteristic
463C3×D3—81no—
464#41—21no—
465#41—21no—
466#41—61no—
467#41—61no—
468#41—61no—
469#41—61no—
470#41C3×D1212yes—
471#41C22×C3×D312yes—
472C2×Q8—32no—
473C2×Q8—31no—
474C2×Q8—61no—
475C2×Q8—61no—
476C2×Q8—61no—
477C2×Q8—61no—
478C2×Q8—61no—
479C2×Q8C3×D1211yescharacteristic
480C2×D4—21no—
481C2×D4—31no—
482C2×D4—61no—
483C2×D4C3×D1211yescharacteristic
484C22×C4—21no—
485C22×C4—61no—
486C22×C4—61no—
487C22×C4C22×C3×D311yescharacteristic
488Q16—64no—
489Q16—62no—
490Q16—64no—
491SD16—62no—
492SD16—64no—
493SD16—62no—
494SD16—64no—
495D8—62no—
496M16—21no—
497M16—24no—
498M16—32no—
499M16—32no—
500M16—61no—
501M16—61no—
502M16C3×#6712yes—
503M16C3×#6712yes—
504M16C22×C3×D312yes—
505C2×C8—21no—
506C2×C8—24no—
507C2×C8—61no—
508C2×C8—61no—
509C2×C8C22×C3×D312yes—
510C22×C3—24no—
511C22×C3—22no—
512C22×C3—22no—
513C22×C3—21no—
514C22×C3—21no—
515C22×C3—21no—
516C22×C3—41no—
517C22×C3—41no—
518C22×C3—81no—
519C22×C3—81no—
520C22×C3—81no—
521C22×C3—82no—
522C22×C3—121no—
523C22×C3—241no—
524C22×C3#93011yescharacteristic
525C22×C3#93011yescharacteristic
526C22×C3C3×#12111yescharacteristic
527C22×C3C3×#12111yescharacteristic
528C22×C3C2×#23511yescharacteristic
529C22×C3C2×C3×#4111yescharacteristic
530C2×D3—41no—
531C2×D3—81no—
532C3×C4—24no—
533C3×C4—22no—
534C3×C4—22no—
535C3×C4—21no—
536C3×C4—21no—
537C3×C4—21no—
538C3×C4—21no—
539C3×C4—121no—
540C3×C4—122no—
541C3×C4#93011yescharacteristic
542C3×C4#93011yescharacteristic
543C3×C4C3×#12111yescharacteristic
544C3×C4C3×#12111yescharacteristic
545C3×C4C22×D1211yescharacteristic
546C3×C4C22×#6711yescharacteristic
547C3×C4C22×C3×D411yescharacteristic
548C3×C4C22×C3×D411yescharacteristic
549Dic3—42no—
550Dic3—41no—
551C32#306911yescharacteristic, complemented, Sylow
552C23—21no—
553C23—61no—
554C23C3×#23511yescharacteristic
555Q8—21no—
556Q8—62no—
557Q8—61no—
558Q8—62no—
559Q8—61no—
560Q8—121no—
561Q8—121no—
562Q8C2×C3×D1212yes—
563D4—22no—
564D4—22no—
565D4—21no—
566D4—61no—
567D4—61no—
568D4—121no—
569D4—121no—
570D4C2×C3×D1212yes—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—61no—
575C2×C4—62no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4C2×C3×D1211yescharacteristic
581C2×C4C2×C3×D1211yescharacteristic
582C2×C4C3×#23511yescharacteristic
583C2×C4C2×C3×#6711yescharacteristic
584C2×C4C2×C3×#6711yescharacteristic
585C2×C4C23×C3×D311yescharacteristic
586C8—22no—
587C8—62no—
588C8—62no—
589C8C2×C3×#6714yes—
590C2×C3—21no—
591C2×C3—21no—
592C2×C3—21no—
593C2×C3—21no—
594C2×C3—21no—
595C2×C3—21no—
596C2×C3—21no—
597C2×C3—41no—
598C2×C3—41no—
599C2×C3—41no—
600C2×C3—44no—
601C2×C3—42no—
602C2×C3—42no—
603C2×C3—241no—
604C2×C3C2×#93011yescharacteristic
605C2×C3C2×C3×#12111yescharacteristic
606D3—81no—
607C22—22no—
608C22—41no—
609C22—81no—
610C22—121no—
611C22—241no—
612C22C3×#93011yescharacteristic
613C22C3×#93011yescharacteristic
614C22C2×C3×#23511yescharacteristic
615C4—22no—
616C4—121no—
617C4—122no—
618C4C3×#93011yescharacteristic
619C4C3×#93011yescharacteristic
620C4C22×C3×D1211yescharacteristic
621C4C22×C3×#6711yescharacteristic
622C3—21no—
623C3#8381611yescharacteristic, direct product factor
624C3C3×#306911yescharacteristic, complemented
625C2—21no—
626C2—21no—
627C2—21no—
628C2—42no—
629C2—241no—
630C2C2×C3×#93011yescharacteristic
subgroupG/H
centerC2×C3C2×#930
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#568C2
socleC2×C32C2×#121
prime radicalC3×#5822C2
hypercenterC3×#568D3
hypocenterC3C3×#3069
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C3×#568

Complements

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

NH ≅ G/Ntype
1. C32×#568C2semidirect
2. C2×C3×#951C2semidirect
4. C2×C3×#909C2semidirect
5. C3×#5143C2semidirect
6. C3×#5141C2semidirect
7. C2×C3×#824C2semidirect
8. C3×#5132C2semidirect
9. C3×#5125C2semidirect
11. #83816C3direct (N × H)
33. C2×C3×Dic12C22semidirect
34. C2×C3×#204C22semidirect
35. C3×#824C22semidirect
36. C3×#824C22semidirect
37. C3×#823C22semidirect
38. C3×#819C22semidirect
43. C3×#568D3semidirect
44. C3×#568C2×C3semidirect
45. C2×#951C2×C3semidirect
47. C2×#909C2×C3semidirect
53. #5143C2×C3semidirect
54. #5141C2×C3semidirect
55. C2×#824C2×C3semidirect
56. #5132C2×C3semidirect
57. #5125C2×C3semidirect
155. C2×Dic12C22×C3semidirect
156. C2×#204C22×C3semidirect
161. #824C22×C3semidirect
162. #824C22×C3semidirect
163. #823C22×C3semidirect
164. #819C22×C3semidirect
206. #568C3×D3semidirect
551. C32#3069semidirect
623. C3#83816direct (N × H)
624. C3C3×#3069semidirect

Maximal subgroups

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

subgroupindex
C32×#5682
C2×C3×#9512
C3×#58222
C2×C3×#9092
C3×#51432
C3×#51412
C2×C3×#8242
C3×#51322
C3×#51252
C3×#30693
#838163