Finite Group Databasebeta

← C3×#80366 #11882228

Subgroups — C3×#80366 #11882228

Order 1152.

848 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5716C211yesmaximal, characteristic
2C3×#5578C211yesmaximal, characteristic, complemented
3C32×#460C211yesmaximal, characteristic, complemented
4C3×#5171C212yesmaximal, complemented
5C3×#5162C212yesmaximal, complemented
6C3×#5075C211yesmaximal, characteristic, complemented
7C3×#5020C211yesmaximal, characteristic, complemented
8C3×#4851C211yesmaximal, characteristic, complemented
9C3×D3×C4⋊C8C211yesmaximal, characteristic, complemented
10C3×#4778C212yesmaximal, complemented
11C3×#4761C212yesmaximal, complemented
12C3×#3246—31nomaximal
13#80366C311yesmaximal, characteristic, direct product factor
14C2×C3×#237—21no—
15C2×C32×SD16C2212yes—
16C32×#127C2211yescharacteristic
17C32×C4×Q8C2211yescharacteristic
18C3×#946C2211yescharacteristic
19C3×#944C2211yescharacteristic
20C3×#938—21no—
21C3×#938C2212yes—
22C3×#936—21no—
23C3×D4×Dic3—21no—
24C3×D4×Dic3C2212yes—
25C2×C3×#214C2212yes—
26C2×C3×D3×C8C2212yes—
27C3×#897C2211yescharacteristic
28C3×#896—21no—
29C3×#889—21no—
30C3×#888C2211yescharacteristic
31C3×#872—21no—
32C3×C42×D3C2211yescharacteristic
33C3×#869C2211yescharacteristic
34C3×C4×Dic6—21no—
35C3×C4×Dic6C2211yescharacteristic
36C32×C4⋊C8C2211yescharacteristic
37C32×#102C2212yes—
38C3×#835C2212yes—
39C3×#832C2212yes—
40C3×#817C2212yes—
41C3×#816C2212yes—
42C3×Dic3⋊C8C2212yes—
43C3×#808C2212yes—
44C3×C4⋊(C3⋊C8)C2211yescharacteristic
45C3×#533—31no—
46C3×#532—31no—
47#5716C2×C311yescharacteristic
48#5578C2×C311yescharacteristic, complemented
49C3×#476—32no—
50C3×#475—31no—
51C3×#475—31no—
52C3×#465—32no—
53C3×#465—32no—
54C3×#461—32no—
55C3×#460—21no—
56C3×#460—31no—
57C3×#460D311yescharacteristic, complemented
58C3×#460C2×C311yescharacteristic, complemented
59C2×C3×C4⋊C8—31no—
60#5171C2×C312yescomplemented
61#5162C2×C312yescomplemented
62#5075C2×C311yescharacteristic, complemented
63#5020C2×C311yescharacteristic, complemented
64#4851C2×C311yescharacteristic, complemented
65D3×C4⋊C8C2×C311yescharacteristic, complemented
66#4778C2×C312yescomplemented
67#4761C2×C312yescomplemented
68C2×C32×Q8C2311yescharacteristic
69C2×C32×D4—21no—
70C2×C32×D4—21no—
71C2×C32×D4C2312yes—
72C2×C3×#67—22no—
73C22×C3×Dic3—22no—
74C3×#237—44no—
75C2×C3×C4×D3D411yescharacteristic
76C2×C3×C4×D3D411yescharacteristic
77C2×C3×C4×D3C2311yescharacteristic
78C2×C3×Dic6—21no—
79C2×C3×Dic6—21no—
80C2×C3×Dic6C2311yescharacteristic
81C32×SD16—42no—
82C2×C32×C8C2312yes—
83C32×C4⋊C4—21no—
84C32×C4⋊C4C2311yescharacteristic
85C32×C42—21no—
86C32×C42C2311yescharacteristic
87C3×#217—22no—
88C3×#217—22no—
89C3×#214—42no—
90C3×(C2×D3)⋊C4—21no—
91C3×(C2×D3)⋊C4—21no—
92C3×C4⋊Dic3—21no—
93C3×C4⋊Dic3—21no—
94C3×C4⋊Dic3C2311yescharacteristic
95C3×C4⋊Dic3C2312yes—
96C3×Dic3⋊C4—21no—
97C3×Dic3⋊C4—21no—
98C3×Dic3⋊C4—21no—
99C3×Dic3⋊C4—21no—
100C3×Dic3⋊C4—21no—
101C3×C4×Dic3—21no—
102C3×C4×Dic3D411yescharacteristic
103C3×C4×Dic3D411yescharacteristic
104C3×C4×Dic3C2311yescharacteristic
105C2×C3×C3⋊C8C2312yes—
106C3×D3×C8—42no—
107#3246—31noSylow
108C2×C3×#41—61no—
109C2×#237—21no—
110C2×C3×SD16—22no—
111C2×C3×SD16—32no—
112C2×C3×SD16C2×D312yes—
113C2×C3×SD16C22×C312yes—
114C22×C3×C8—32no—
115C3×C4⋊Q8—31no—
116C3×C4⋊Q8—31no—
117C3×C4⋊Q8—31no—
118C3×#127—21no—
119C3×#127C2×D311yescharacteristic
120C3×#127C22×C311yescharacteristic
121C3×#125—61no—
122C3×#124—61no—
123C3×C22⋊Q8—31no—
124C3×C22⋊Q8—31no—
125C3×C22⋊Q8—61no—
126C3×#121—32no—
127C3×#121—61no—
128C3×C4×Q8—21no—
129C3×C4×Q8—31no—
130C3×C4×Q8—61no—
131C3×C4×Q8C2×D311yescharacteristic
132C3×C4×Q8C22×C311yescharacteristic
133C3×C4×D4—32no—
134C3×C4×D4—61no—
135C3×#117—61no—
136C2×C3×C42—31no—
137#946C22×C311yescharacteristic
138#944C22×C311yescharacteristic
139#938—21no—
140#938C22×C312yes—
141#936—21no—
142D4×Dic3—21no—
143D4×Dic3C22×C312yes—
144C2×#214C22×C312yes—
145C2×D3×C8C22×C312yes—
146#897C22×C311yescharacteristic
147#896—21no—
148#889—21no—
149#888C22×C311yescharacteristic
150#872—21no—
151C42×D3C22×C311yescharacteristic
152#869C22×C311yescharacteristic
153C4×Dic6—21no—
154C4×Dic6C22×C311yescharacteristic
155C3×#106—32no—
156C3×#106—32no—
157C3×C4⋊C8—21no—
158C3×C4⋊C8—32no—
159C3×C4⋊C8—31no—
160C3×C4⋊C8C2×D311yescharacteristic
161C3×C4⋊C8C22×C311yescharacteristic
162C3×#103—32no—
163C3×#103—32no—
164C3×#102—22no—
165C3×#102—32no—
166C3×#102C2×D312yes—
167C3×#102C22×C312yes—
168#835C22×C312yes—
169#832C22×C312yes—
170#817C22×C312yes—
171#816C22×C312yes—
172Dic3⋊C8C22×C312yes—
173#808C22×C312yes—
174C4⋊(C3⋊C8)C22×C311yescharacteristic
175C23×C32—22no—
176C22×C3×D3C2×D411yescharacteristic
177C32×Q8—21no—
178C32×Q8—21no—
179C32×D4—22no—
180C32×D4—22no—
181C32×D4—42no—
182C32×D4—42no—
183C2×C32×C4—21no—
184C2×C32×C4—21no—
185C2×C32×C4C2×D411yescharacteristic
186C2×C32×C4C2×D411yescharacteristic
187C2×C32×C4C2411yescharacteristic
188C3×#67—44no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—22no—
192C2×C3×Dic3—44no—
193C2×C3×Dic3C2×D411yescharacteristic
194C2×C3×Dic3C2×D411yescharacteristic
195C2×C3×Dic3C2×D411yescharacteristic
196C3×C4×D3—21no—
197C3×C4×D3—21no—
198C3×C4×D3—41no—
199C3×C4×D3SD1614yescomplemented
200C3×Dic6—21no—
201C3×Dic6—21no—
202C3×Dic6—42no—
203C3×Dic6—42no—
204C32×C8—22no—
205C3×C3⋊C8—22no—
206#533—31no—
207#532—31no—
208#476—32no—
209#475—31no—
210#475—31no—
211#465—32no—
212#465—32no—
213#461—32no—
214#460—31no—
215#460C3×D311yescharacteristic, complemented
216C2×C4⋊C8—31no—
217C3×#41—124no—
218C2×C3×Q8—21no—
219C2×C3×Q8—31no—
220C2×C3×Q8—61no—
221C2×C3×Q8—61no—
222C2×C3×Q8C22×D311yescharacteristic
223C2×C3×Q8C23×C311yescharacteristic
224C2×C3×D4—21no—
225C2×C3×D4—21no—
226C2×C3×D4—21no—
227C2×C3×D4—21no—
228C2×C3×D4—22no—
229C2×C3×D4—41no—
230C2×C3×D4—41no—
231C2×C3×D4—62no—
232C2×C3×D4C22×D312yes—
233C2×C3×D4C23×C312yes—
234C22×C3×C4—31no—
235C22×C3×C4—31no—
236C22×C3×C4—31no—
237C22×C3×C4—62no—
238C2×#67—22no—
239C22×Dic3—22no—
240#237—44no—
241C2×C4×D3C3×D411yescharacteristic
242C2×C4×D3C3×D411yescharacteristic
243C2×C4×D3C23×C311yescharacteristic
244C2×Dic6—21no—
245C2×Dic6—21no—
246C2×Dic6C23×C311yescharacteristic
247C3×SD16—42no—
248C3×SD16—42no—
249C3×SD16—44no—
250C3×SD16—122no—
251C2×C3×C8—22no—
252C2×C3×C8—32no—
253C2×C3×C8—122no—
254C2×C3×C8C22×D312yes—
255C2×C3×C8C23×C312yes—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—21no—
259C3×C4⋊C4—31no—
260C3×C4⋊C4—32no—
261C3×C4⋊C4—41no—
262C3×C4⋊C4—61no—
263C3×C4⋊C4—61no—
264C3×C4⋊C4—61no—
265C3×C4⋊C4—61no—
266C3×C4⋊C4—61no—
267C3×C4⋊C4—61no—
268C3×C4⋊C4—61no—
269C3×C4⋊C4C22×D311yescharacteristic
270C3×C4⋊C4C23×C311yescharacteristic
271C3×C22⋊C4—61no—
272C3×C22⋊C4—61no—
273C3×C22⋊C4—62no—
274C3×C22⋊C4—62no—
275C3×C42—21no—
276C3×C42—21no—
277C3×C42—21no—
278C3×C42—31no—
279C3×C42—31no—
280C3×C42—31no—
281C3×C42—41no—
282C3×C42—61no—
283C3×C42C22×D311yescharacteristic
284C3×C42C23×C311yescharacteristic
285#217—22no—
286#217—22no—
287#214—42no—
288(C2×D3)⋊C4—21no—
289(C2×D3)⋊C4—21no—
290C4⋊Dic3—21no—
291C4⋊Dic3—21no—
292C4⋊Dic3C23×C312yes—
293C4⋊Dic3C23×C311yescharacteristic
294Dic3⋊C4—21no—
295Dic3⋊C4—21no—
296Dic3⋊C4—21no—
297Dic3⋊C4—21no—
298Dic3⋊C4—21no—
299C4×Dic3—21no—
300C4×Dic3C3×D411yescharacteristic
301C4×Dic3C3×D411yescharacteristic
302C4×Dic3C23×C311yescharacteristic
303C2×C3⋊C8C23×C312yes—
304D3×C8—42no—
305C22×C32—42no—
306C22×C32—42no—
307C22×C32—42no—
308C22×C32C22×D411yescharacteristic
309C2×C3×D3—21no—
310C2×C3×D3—21no—
311C2×C3×D3C2×SD1612yes—
312C32×C4—21no—
313C32×C4—41no—
314C32×C4—41no—
315C32×C4C2×SD1612yes—
316C32×C4H(5,𝔽2)11yescharacteristic
317C32×C42−1+411yescharacteristic
318C3×Dic3—21no—
319C3×Dic3—21no—
320C3×Dic3—41no—
321C3×Dic3—42no—
322C3×Dic3—41no—
323C3×Dic3C2×SD1612yes—
324C2×#41—61no—
325C2×SD16—32no—
326C2×SD16C2×C3×D312yes—
327C22×C8—32no—
328C4⋊Q8—31no—
329C4⋊Q8—31no—
330C4⋊Q8—31no—
331#127C2×C3×D311yescharacteristic
332#125—61no—
333#124—61no—
334C22⋊Q8—31no—
335C22⋊Q8—31no—
336C22⋊Q8—61no—
337#121—32no—
338#121—61no—
339C4×Q8—31no—
340C4×Q8—61no—
341C4×Q8C2×C3×D311yescharacteristic
342C4×D4—32no—
343C4×D4—61no—
344#117—61no—
345C2×C42—31no—
346#106—32no—
347#106—32no—
348C4⋊C8—32no—
349C4⋊C8—31no—
350C4⋊C8C2×C3×D311yescharacteristic
351#103—32no—
352#103—32no—
353#102—32no—
354#102C2×C3×D312yes—
355C23×C3—22no—
356C23×C3—22no—
357C23×C3—31no—
358C23×C3—42no—
359C22×D3C2×C3×D411yescharacteristic
360C3×Q8—22no—
361C3×Q8—21no—
362C3×Q8—21no—
363C3×Q8—21no—
364C3×Q8—21no—
365C3×Q8—41no—
366C3×Q8—61no—
367C3×Q8—61no—
368C3×Q8—122no—
369C3×Q8—122no—
370C3×D4—22no—
371C3×D4—22no—
372C3×D4—22no—
373C3×D4—22no—
374C3×D4—42no—
375C3×D4—42no—
376C3×D4—42no—
377C3×D4—42no—
378C3×D4—42no—
379C3×D4—42no—
380C3×D4—82no—
381C3×D4—82no—
382C3×D4—124no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—31no—
391C2×C3×C4—34no—
392C2×C3×C4—31no—
393C2×C3×C4—31no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—61no—
397C2×C3×C4—61no—
398C2×C3×C4—61no—
399C2×C3×C4—61no—
400C2×C3×C4—62no—
401C2×C3×C4—121no—
402C2×C3×C4—124no—
403C2×C3×C4D3×D411yescharacteristic
404C2×C3×C4D3×D411yescharacteristic
405C2×C3×C4C2×C3×D411yescharacteristic
406C2×C3×C4C2×C3×D411yescharacteristic
407C2×C3×C4C23×D311yescharacteristic
408C2×C3×C4C24×C311yescharacteristic
409#67—44no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—22no—
413C2×Dic3—44no—
414C2×Dic3C2×C3×D411yescharacteristic
415C2×Dic3C2×C3×D411yescharacteristic
416C2×Dic3C2×C3×D411yescharacteristic
417C4×D3—21no—
418C4×D3—21no—
419C4×D3—41no—
420C4×D3C3×SD1614yescomplemented
421Dic6—21no—
422Dic6—21no—
423Dic6—42no—
424Dic6—42no—
425C3×C8—22no—
426C3×C8—22no—
427C3×C8—42no—
428C3×C8—62no—
429C3⋊C8—22no—
430C2×C32—82no—
431C2×C32#53711yescharacteristic
432C2×C32C22×SD1611yescharacteristic
433C2×C32#57511yescharacteristic
434C3×D3—22no—
435#41—124no—
436C2×Q8—31no—
437C2×Q8—61no—
438C2×Q8—61no—
439C2×Q8C22×C3×D311yescharacteristic
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—62no—
443C2×D4C22×C3×D312yes—
444C22×C4—31no—
445C22×C4—31no—
446C22×C4—31no—
447C22×C4—62no—
448SD16—42no—
449SD16—122no—
450C2×C8—32no—
451C2×C8—122no—
452C2×C8C22×C3×D312yes—
453C4⋊C4—21no—
454C4⋊C4—31no—
455C4⋊C4—32no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4C22×C3×D311yescharacteristic
464C22⋊C4—61no—
465C22⋊C4—61no—
466C22⋊C4—62no—
467C22⋊C4—62no—
468C42—21no—
469C42—31no—
470C42—31no—
471C42—31no—
472C42—61no—
473C42C22×C3×D311yescharacteristic
474C22×C3—21no—
475C22×C3—32no—
476C22×C3—42no—
477C22×C3—42no—
478C22×C3—42no—
479C22×C3—42no—
480C22×C3—42no—
481C22×C3—42no—
482C22×C3—61no—
483C22×C3—61no—
484C22×C3—82no—
485C22×C3—82no—
486C22×C3—82no—
487C22×C3C2×D3×D411yescharacteristic
488C22×C3C22×C3×D411yescharacteristic
489C2×D3—21no—
490C2×D3—21no—
491C2×D3C2×C3×SD1612yes—
492C3×C4—21no—
493C3×C4—21no—
494C3×C4—22no—
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—32no—
498C3×C4—41no—
499C3×C4—42no—
500C3×C4—41no—
501C3×C4—41no—
502C3×C4—41no—
503C3×C4—41no—
504C3×C4—61no—
505C3×C4—61no—
506C3×C4—81no—
507C3×C4—122no—
508C3×C4—121no—
509C3×C4—121no—
510C3×C4D3×SD1612yes—
511C3×C4C2×C3×SD1612yes—
512C3×C4#100411yescharacteristic
513C3×C4#101011yescharacteristic
514C3×C4C3×H(5,𝔽2)11yescharacteristic
515C3×C4C3×2−1+411yescharacteristic
516Dic3—21no—
517Dic3—21no—
518Dic3—42no—
519Dic3—41no—
520Dic3—41no—
521Dic3C2×C3×SD1612yes—
522C32#324611yescharacteristic, complemented, Sylow
523C23—22no—
524C23—31no—
525Q8—21no—
526Q8—21no—
527Q8—61no—
528Q8—61no—
529Q8—122no—
530Q8—122no—
531D4—22no—
532D4—22no—
533D4—42no—
534D4—42no—
535D4—124no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—31no—
539C2×C4—34no—
540C2×C4—31no—
541C2×C4—31no—
542C2×C4—61no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—61no—
546C2×C4—62no—
547C2×C4—121no—
548C2×C4—124no—
549C2×C4C3×D3×D411yescharacteristic
550C2×C4C3×D3×D411yescharacteristic
551C2×C4C23×C3×D311yescharacteristic
552C8—22no—
553C8—62no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—62no—
558C2×C3—82no—
559C2×C3—82no—
560C2×C3—162no—
561C2×C3#560811yescharacteristic
562C2×C3C2×D3×SD1611yescharacteristic
563C2×C3#577311yescharacteristic
564C2×C3C3×#53711yescharacteristic
565C2×C3C22×C3×SD1611yescharacteristic
566C2×C3C3×#57511yescharacteristic
567D3—22no—
568C22—32no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—61no—
573C22—61no—
574C22C2×C3×D3×D411yescharacteristic
575C4—21no—
576C4—32no—
577C4—41no—
578C4—41no—
579C4—61no—
580C4—61no—
581C4—122no—
582C4—121no—
583C4—121no—
584C4C3×D3×SD1612yes—
585C4C3×#100411yescharacteristic
586C4C3×#101011yescharacteristic
587C3—21no—
588C3#8036611yescharacteristic, direct product factor
589C3C3×#324611yescharacteristic, complemented
590C2—62no—
591C2—82no—
592C2C3×#560811yescharacteristic
593C2C2×C3×D3×SD1611yescharacteristic
594C2C3×#577311yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#460C2
socleC22×C32C22×D4
prime radicalC3×#5716C2
hypercenterC3×#460D3
hypocenterC3C3×#3246
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#460

Complements

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

NH ≅ G/Ntype
2. C3×#5578C2semidirect
3. C32×#460C2semidirect
4. C3×#5171C2semidirect
5. C3×#5162C2semidirect
6. C3×#5075C2semidirect
7. C3×#5020C2semidirect
8. C3×#4851C2semidirect
9. C3×D3×C4⋊C8C2semidirect
10. C3×#4778C2semidirect
11. C3×#4761C2semidirect
13. #80366C3direct (N × H)
48. #5578C2×C3semidirect
57. C3×#460D3semidirect
58. C3×#460C2×C3semidirect
60. #5171C2×C3semidirect
61. #5162C2×C3semidirect
62. #5075C2×C3semidirect
63. #5020C2×C3semidirect
64. #4851C2×C3semidirect
65. D3×C4⋊C8C2×C3semidirect
66. #4778C2×C3semidirect
67. #4761C2×C3semidirect
199. C3×C4×D3SD16semidirect
215. #460C3×D3semidirect
420. C4×D3C3×SD16semidirect
522. C32#3246semidirect
588. C3#80366direct (N × H)
589. C3C3×#3246semidirect

Maximal subgroups

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

subgroupindex
C3×#57162
C3×#55782
C32×#4602
C3×#51712
C3×#51622
C3×#50752
C3×#50202
C3×#48512
C3×D3×C4⋊C82
C3×#47782
C3×#47612
C3×#32463
#803663