Finite Group Databasebeta

← C3×#70094 #11796235

Subgroups — C3×#70094 #11796235

Order 1152.

653 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5804C211yesmaximal, characteristic
2C2×C32×#100C211yesmaximal, characteristic, complemented
3C2×C3×#834C211yesmaximal, characteristic, complemented
4C3×#4476C212yesmaximal, complemented
5C3×#4475C212yesmaximal, complemented
6C3×#2121—31nomaximal
7#70094C311yesmaximal, characteristic, direct product factor
8C22×C32×D4C2211yescharacteristic
9C22×C3×#67C411yescharacteristic, complemented
10C2×C32×M16—21no—
11C3×#953—21no—
12C2×C3×#217—21no—
13C2×C3×#217C411yescharacteristic, complemented
14C3×#939—21no—
15C3×#939—21no—
16C3×#937C411yescharacteristic, complemented
17C3×#937C2211yescharacteristic
18C3×#935C411yescharacteristic, complemented
19C3×#935C2211yescharacteristic
20C3×#929—21no—
21C32×#100—21no—
22C32×#100C2212yes—
23C3×#834—21no—
24C3×#834C2212yes—
25C3×#534—31no—
26#5804C2×C311yescharacteristic
27C2×C3×#100—21no—
28C2×C3×#100D311yescharacteristic, complemented
29C2×C3×#100C2×C311yescharacteristic, complemented
30C2×C3×#99—31no—
31C2×#834C2×C311yescharacteristic, complemented
32C3×#352—32no—
33C3×AΣL(1,𝔽16)—32no—
34#4476C2×C312yescomplemented
35#4475C2×C312yescomplemented
36C24×C32C2×C411yescharacteristic
37C24×C32D411yescharacteristic
38C23×C3×D3C2×C411yescharacteristic, complemented
39C2×C32×D4—41no—
40C2×C32×D4—41no—
41C2×C32×D4C2×C411yescharacteristic
42C2×C32×D4D411yescharacteristic
43C2×C32×D4D411yescharacteristic
44C2×C32×D4C2311yescharacteristic
45C22×C32×C4D411yescharacteristic
46C2×C3×#67—21no—
47C2×C3×#67—21no—
48C2×C3×#67—21no—
49C2×C3×#67—21no—
50C2×C3×#67—41no—
51C2×C3×#67—41no—
52C22×C3×Dic3—21no—
53C22×C3×Dic3—41no—
54C22×C3×Dic3C2×C411yescharacteristic, complemented
55C32×M16—22no—
56C32×M16—41no—
57C2×C32×C8—41no—
58C3×#217—21no—
59C3×#217—21no—
60C3×#217—21no—
61C3×#217—22no—
62C3×#217—41no—
63C3×#217C2×C411yescharacteristic, complemented
64C3×#217C2×C411yescharacteristic, complemented
65C3×(C2×D3)⋊C4—21no—
66C3×(C2×D3)⋊C4—21no—
67C3×Dic3⋊C4—21no—
68C3×Dic3⋊C4—21no—
69#2121—31noSylow
70C22×C3×D4—21no—
71C22×C3×D4—31no—
72C22×C3×D4C2×D311yescharacteristic
73C22×C3×D4C22×C311yescharacteristic
74C22×#67C3×C411yescharacteristic, complemented
75C2×C3×M16—22no—
76C2×C3×M16—21no—
77C2×C3×M16—21no—
78C3×#123—31no—
79C3×#123—31no—
80C3×#123—61no—
81C3×#121—61no—
82C3×#121—61no—
83C3×#120—31no—
84C3×#120—31no—
85C3×#120—61no—
86C2×C3×C22⋊C4—31no—
87C2×C3×C22⋊C4—61no—
88#953—21no—
89C2×#217—21no—
90C2×#217C3×C411yescharacteristic, complemented
91#939—21no—
92#939—21no—
93#937C3×C411yescharacteristic, complemented
94#937C22×C311yescharacteristic
95#935C3×C411yescharacteristic, complemented
96#935C22×C311yescharacteristic
97#929—21no—
98C3×#100—22no—
99C3×#100—21no—
100C3×#100—21no—
101C3×#100—22no—
102C3×#100C2×D312yes—
103C3×#100C22×C312yes—
104C3×#99—32no—
105C3×#99—61no—
106#834—21no—
107#834C22×C312yes—
108C23×C32—21no—
109C23×C32—21no—
110C23×C32—21no—
111C23×C32—21no—
112C23×C32—81no—
113C23×C32—81no—
114C23×C32C22⋊C411yescharacteristic
115C23×C32C22⋊C411yescharacteristic
116C23×C32C22⋊C411yescharacteristic
117C23×C32C22×C411yescharacteristic
118C23×C32C2×D411yescharacteristic
119C22×C3×D3—21no—
120C22×C3×D3—41no—
121C22×C3×D3—81no—
122C32×D4—41no—
123C32×D4—41no—
124C32×D4—41no—
125C32×D4—41no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4C22⋊C411yescharacteristic
129C2×C32×C4C2×D411yescharacteristic
130C3×#67—42no—
131C3×#67—41no—
132C3×#67—41no—
133C3×#67—81no—
134C3×#67—81no—
135C2×C3×Dic3—21no—
136C2×C3×Dic3—21no—
137C2×C3×Dic3—21no—
138C2×C3×Dic3—41no—
139C2×C3×Dic3—42no—
140C2×C3×Dic3—81no—
141C2×C3×Dic3—82no—
142C32×C8—42no—
143#534—31no—
144C2×#100C3×D311yescharacteristic, complemented
145C2×#99—31no—
146#352—32no—
147AΣL(1,𝔽16)—32no—
148C24×C3—21no—
149C24×C3—21no—
150C24×C3—31no—
151C24×C3C4×D311yescharacteristic
152C24×C3D1211yescharacteristic
153C24×C3C2×C3×C411yescharacteristic
154C24×C3C3×D411yescharacteristic
155C23×D3C2×C3×C411yescharacteristic, complemented
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4—41no—
161C2×C3×D4—41no—
162C2×C3×D4—41no—
163C2×C3×D4—41no—
164C2×C3×D4—42no—
165C2×C3×D4—42no—
166C2×C3×D4—61no—
167C2×C3×D4—61no—
168C2×C3×D4—61no—
169C2×C3×D4—61no—
170C2×C3×D4—121no—
171C2×C3×D4—121no—
172C2×C3×D4C4×D311yescharacteristic
173C2×C3×D4D1211yescharacteristic
174C2×C3×D4#6711yescharacteristic
175C2×C3×D4C2×C3×C411yescharacteristic
176C2×C3×D4C3×D411yescharacteristic
177C2×C3×D4C3×D411yescharacteristic
178C2×C3×D4C22×D311yescharacteristic
179C2×C3×D4C23×C311yescharacteristic
180C22×C3×C4—21no—
181C22×C3×C4—31no—
182C22×C3×C4—61no—
183C22×C3×C4—121no—
184C22×C3×C4#6711yescharacteristic
185C22×C3×C4C3×D411yescharacteristic
186C2×#67—21no—
187C2×#67—21no—
188C2×#67—21no—
189C2×#67—21no—
190C2×#67—41no—
191C2×#67—41no—
192C22×Dic3—21no—
193C22×Dic3—41no—
194C22×Dic3C2×C3×C411yescharacteristic, complemented
195C3×M16—24no—
196C3×M16—22no—
197C3×M16—22no—
198C3×M16—42no—
199C3×M16—41no—
200C3×M16—41no—
201C2×C3×C8—42no—
202C2×C3×C8—41no—
203C2×C3×C8—41no—
204C3×C4⋊C4—61no—
205C3×C4⋊C4—61no—
206C3×C22⋊C4—31no—
207C3×C22⋊C4—31no—
208C3×C22⋊C4—61no—
209C3×C22⋊C4—61no—
210C3×C22⋊C4—61no—
211C3×C22⋊C4—61no—
212C3×C22⋊C4—61no—
213C3×C22⋊C4—62no—
214C3×C22⋊C4—121no—
215#217—21no—
216#217—21no—
217#217—21no—
218#217—22no—
219#217—41no—
220#217C2×C3×C411yescharacteristic, complemented
221#217C2×C3×C411yescharacteristic, complemented
222(C2×D3)⋊C4—21no—
223(C2×D3)⋊C4—21no—
224Dic3⋊C4—21no—
225Dic3⋊C4—21no—
226C22×C32—21no—
227C22×C32—21no—
228C22×C32—21no—
229C22×C32—21no—
230C22×C32—41no—
231C22×C32—41no—
232C22×C32—41no—
233C22×C32—41no—
234C22×C32—41no—
235C22×C32—42no—
236C22×C32—81no—
237C22×C32—81no—
238C22×C32—81no—
239C22×C32#9911yescharacteristic
240C22×C32#9911yescharacteristic
241C22×C32C2×C22⋊C411yescharacteristic
242C2×C3×D3—41no—
243C2×C3×D3—41no—
244C2×C3×D3—41no—
245C2×C3×D3—82no—
246C32×C4—21no—
247C32×C4—21no—
248C3×Dic3—41no—
249C3×Dic3—41no—
250C3×Dic3—41no—
251C3×Dic3—82no—
252C22×D4—31no—
253C22×D4C2×C3×D311yescharacteristic
254C2×M16—21no—
255#123—31no—
256#123—31no—
257#123—61no—
258#121—61no—
259#121—61no—
260#120—31no—
261#120—31no—
262#120—61no—
263C2×C22⋊C4—31no—
264C2×C22⋊C4—61no—
265#100—21no—
266#100C2×C3×D312yes—
267#99—32no—
268#99—61no—
269C23×C3—21no—
270C23×C3—21no—
271C23×C3—21no—
272C23×C3—21no—
273C23×C3—21no—
274C23×C3—21no—
275C23×C3—21no—
276C23×C3—21no—
277C23×C3—21no—
278C23×C3—21no—
279C23×C3—21no—
280C23×C3—21no—
281C23×C3—21no—
282C23×C3—41no—
283C23×C3—41no—
284C23×C3—41no—
285C23×C3—41no—
286C23×C3—61no—
287C23×C3—81no—
288C23×C3—81no—
289C23×C3—81no—
290C23×C3—81no—
291C23×C3—82no—
292C23×C3—82no—
293C23×C3—121no—
294C23×C3—241no—
295C23×C3(C2×D3)⋊C411yescharacteristic
296C23×C3(C2×D3)⋊C411yescharacteristic
297C23×C3(C2×D3)⋊C411yescharacteristic
298C23×C3C3×C22⋊C411yescharacteristic
299C23×C3C3×C22⋊C411yescharacteristic
300C23×C3C3×C22⋊C411yescharacteristic
301C23×C3C2×C4×D311yescharacteristic
302C23×C3C2×D1211yescharacteristic
303C23×C3C22×C3×C411yescharacteristic
304C23×C3C2×C3×D411yescharacteristic
305C22×D3—21no—
306C22×D3—41no—
307C22×D3—81no—
308C3×D4—41no—
309C3×D4—41no—
310C3×D4—41no—
311C3×D4—41no—
312C3×D4—41no—
313C3×D4—41no—
314C3×D4—41no—
315C3×D4—41no—
316C3×D4—42no—
317C3×D4—42no—
318C3×D4—42no—
319C3×D4—42no—
320C3×D4—122no—
321C3×D4—121no—
322C3×D4—121no—
323C3×D4—241no—
324C3×D4—241no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—61no—
334C2×C3×C4—61no—
335C2×C3×C4—61no—
336C2×C3×C4—121no—
337C2×C3×C4—122no—
338C2×C3×C4—241no—
339C2×C3×C4—242no—
340C2×C3×C4(C2×D3)⋊C411yescharacteristic
341C2×C3×C4C3×C22⋊C411yescharacteristic
342C2×C3×C4C2×#6711yescharacteristic
343C2×C3×C4C2×C3×D411yescharacteristic
344#67—42no—
345#67—41no—
346#67—41no—
347#67—81no—
348#67—81no—
349C2×Dic3—21no—
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—41no—
353C2×Dic3—42no—
354C2×Dic3—81no—
355C2×Dic3—82no—
356C3×C8—44no—
357C3×C8—42no—
358C3×C8—42no—
359C2×C32—21no—
360C2×C32—21no—
361C2×C32—21no—
362C2×C32—41no—
363C2×C32—41no—
364C2×C32—41no—
365C2×C32—41no—
366C2×C32C2×#9911yescharacteristic
367C3×D3—81no—
368C24—31no—
369C24C3×C4×D311yescharacteristic
370C24C3×D1211yescharacteristic
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—61no—
374C2×D4—61no—
375C2×D4—61no—
376C2×D4—61no—
377C2×D4—121no—
378C2×D4—121no—
379C2×D4C3×C4×D311yescharacteristic
380C2×D4C3×D1211yescharacteristic
381C2×D4C3×#6711yescharacteristic
382C2×D4C22×C3×D311yescharacteristic
383C22×C4—31no—
384C22×C4—61no—
385C22×C4—121no—
386C22×C4C3×#6711yescharacteristic
387M16—22no—
388M16—41no—
389C2×C8—41no—
390C4⋊C4—61no—
391C4⋊C4—61no—
392C22⋊C4—31no—
393C22⋊C4—31no—
394C22⋊C4—61no—
395C22⋊C4—61no—
396C22⋊C4—61no—
397C22⋊C4—61no—
398C22⋊C4—61no—
399C22⋊C4—62no—
400C22⋊C4—121no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—21no—
412C22×C3—41no—
413C22×C3—41no—
414C22×C3—41no—
415C22×C3—41no—
416C22×C3—41no—
417C22×C3—41no—
418C22×C3—41no—
419C22×C3—41no—
420C22×C3—41no—
421C22×C3—41no—
422C22×C3—41no—
423C22×C3—41no—
424C22×C3—41no—
425C22×C3—41no—
426C22×C3—42no—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—81no—
430C22×C3—81no—
431C22×C3—81no—
432C22×C3—81no—
433C22×C3—82no—
434C22×C3—81no—
435C22×C3—81no—
436C22×C3—81no—
437C22×C3—81no—
438C22×C3—81no—
439C22×C3—81no—
440C22×C3—82no—
441C22×C3—82no—
442C22×C3—82no—
443C22×C3—121no—
444C22×C3—121no—
445C22×C3—121no—
446C22×C3—242no—
447C22×C3#80611yescharacteristic
448C22×C3#80611yescharacteristic
449C22×C3C3×#9911yescharacteristic
450C22×C3C3×#9911yescharacteristic
451C22×C3C2×(C2×D3)⋊C411yescharacteristic
452C22×C3C2×C3×C22⋊C411yescharacteristic
453C2×D3—41no—
454C2×D3—41no—
455C2×D3—41no—
456C2×D3—82no—
457C3×C4—22no—
458C3×C4—22no—
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—21no—
463C3×C4—121no—
464C3×C4—121no—
465C3×C4—121no—
466C3×C4—242no—
467Dic3—41no—
468Dic3—41no—
469Dic3—41no—
470Dic3—82no—
471C32#212111yescharacteristic, complemented, Sylow
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—61no—
477C23—81no—
478C23—81no—
479C23—121no—
480C23—241no—
481C23C3×(C2×D3)⋊C411yescharacteristic
482C23C3×(C2×D3)⋊C411yescharacteristic
483C23C3×(C2×D3)⋊C411yescharacteristic
484C23C2×C3×C4×D311yescharacteristic
485C23C2×C3×D1211yescharacteristic
486D4—41no—
487D4—41no—
488D4—41no—
489D4—41no—
490D4—122no—
491D4—121no—
492D4—121no—
493D4—241no—
494D4—241no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—61no—
498C2×C4—61no—
499C2×C4—61no—
500C2×C4—121no—
501C2×C4—122no—
502C2×C4—241no—
503C2×C4—242no—
504C2×C4C3×(C2×D3)⋊C411yescharacteristic
505C2×C4C2×C3×#6711yescharacteristic
506C8—42no—
507C2×C3—22no—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—21no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—41no—
516C2×C3—41no—
517C2×C3—41no—
518C2×C3—41no—
519C2×C3—41no—
520C2×C3—41no—
521C2×C3—41no—
522C2×C3—41no—
523C2×C3—41no—
524C2×C3—41no—
525C2×C3—81no—
526C2×C3—81no—
527C2×C3—81no—
528C2×C3—81no—
529C2×C3—241no—
530C2×C3C2×#80611yescharacteristic
531C2×C3C2×C3×#9911yescharacteristic
532D3—81no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—42no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22—121no—
547C22—121no—
548C22—121no—
549C22—242no—
550C22C3×#80611yescharacteristic
551C22C3×#80611yescharacteristic
552C22C2×C3×(C2×D3)⋊C411yescharacteristic
553C4—21no—
554C4—21no—
555C4—121no—
556C4—121no—
557C4—121no—
558C4—242no—
559C3—21no—
560C3#7009411yescharacteristic, direct product factor
561C3C3×#212111yescharacteristic, complemented
562C2—21no—
563C2—21no—
564C2—21no—
565C2—41no—
566C2—41no—
567C2—41no—
568C2—41no—
569C2—241no—
570C2C2×C3×#80611yescharacteristic
subgroupG/H
centerC2×C3C2×#806
commutatorC23×C3C22×C3×C4
FrattiniC2×D4C22×C3×D3
FittingC2×C32×#100C2
socleC2×C32C2×#99
prime radicalC3×#5804C2
cosocleC2×C3×D4C23×C3
hypercenterC2×C3×#100D3
hypocenterC3C3×#2121
LCSG ▷ C23×C3 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C22×C3×D4 ◁ C2×C3×#100

Complements

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

NH ≅ G/Ntype
2. C2×C32×#100C2semidirect
3. C2×C3×#834C2semidirect
4. C3×#4476C2semidirect
5. C3×#4475C2semidirect
7. #70094C3direct (N × H)
9. C22×C3×#67C4semidirect
13. C2×C3×#217C4semidirect
16. C3×#937C4semidirect
18. C3×#935C4semidirect
28. C2×C3×#100D3semidirect
29. C2×C3×#100C2×C3semidirect
31. C2×#834C2×C3semidirect
34. #4476C2×C3semidirect
35. #4475C2×C3semidirect
38. C23×C3×D3C2×C4semidirect
54. C22×C3×Dic3C2×C4semidirect
63. C3×#217C2×C4semidirect
64. C3×#217C2×C4semidirect
74. C22×#67C3×C4semidirect
90. C2×#217C3×C4semidirect
93. #937C3×C4semidirect
95. #935C3×C4semidirect
144. C2×#100C3×D3semidirect
155. C23×D3C2×C3×C4semidirect
194. C22×Dic3C2×C3×C4semidirect
220. #217C2×C3×C4semidirect
221. #217C2×C3×C4semidirect
471. C32#2121semidirect
560. C3#70094direct (N × H)
561. C3C3×#2121semidirect

Maximal subgroups

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

subgroupindex
C3×#58042
C2×C32×#1002
C2×C3×#8342
C3×#44762
C3×#44752
C3×#21213
#700943