Finite Group Databasebeta

← C3×#68304 #11775286

Subgroups — C3×#68304 #11775286

Order 1152.

697 conjugacy classes of subgroups.

The subgroup lattice has 603 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×D3×#121C211yesmaximal, characteristic
2C3×#4726C211yesmaximal, characteristic, complemented
3C3×#4664C211yesmaximal, characteristic, complemented
4C32×#327C211yesmaximal, characteristic, complemented
5C3×#4537C211yesmaximal, characteristic, complemented
6C3×#4470C211yesmaximal, characteristic, complemented
7C3×#4451C211yesmaximal, characteristic, complemented
8C3×#1510—31nomaximal
9#68304C311yesmaximal, characteristic, direct product factor
10C2×C3×D3×D4—21no—
11C2×C3×D3×D4C411yescharacteristic, complemented
12C22×C3×C4×D3C2211yescharacteristic
13C32×#121C2211yescharacteristic
14C3×#939—21no—
15C3×#938C411yescharacteristic, complemented
16C3×#930C2211yescharacteristic
17C2×C3×(C2×D3)⋊C4—21no—
18C2×C3×ΓL(1,𝔽25)—21no—
19C3×#895C411yescharacteristic, complemented
20C3×D3×C4⋊C4C411yescharacteristic, complemented
21C3×#884—21no—
22C3×D3×C22⋊C4—21no—
23C32×C22⋊C8C2211yescharacteristic, complemented
24C32×#95C2211yescharacteristic, complemented
25C3×C3⋊#95—21no—
26C3×C3⋊#95C2211yescharacteristic, complemented
27C3×#830C2211yescharacteristic, complemented
28C3×(C2×D3)⋊C8—21no—
29C2×C3×#121—31no—
30D3×#121C2×C311yescharacteristic
31C3×#407—31no—
32C3×#381—31no—
33#4726C2×C311yescharacteristic, complemented
34#4664C2×C311yescharacteristic, complemented
35C3×#327—21no—
36C3×#327—31no—
37C3×#327—31no—
38C3×#327—31no—
39C3×#327D311yescharacteristic, complemented
40C3×#327C2×C311yescharacteristic, complemented
41#4537C2×C311yescharacteristic, complemented
42#4470C2×C311yescharacteristic, complemented
43#4451C2×C311yescharacteristic, complemented
44C23×C3×D3—21no—
45C23×C3×D3D411yescharacteristic
46C2×C32×D4—21no—
47C2×C32×D4C2×C411yescharacteristic
48C22×C32×C4—21no—
49C22×C32×C4C2311yescharacteristic
50C2×C3×#67—21no—
51C2×C3×#67—21no—
52C2×C3×#67—21no—
53C22×C3×Dic3—21no—
54C22×C3×Dic3D411yescharacteristic
55C3×D3×D4—42no—
56C3×D3×D4—42no—
57C3×D3×D4—42no—
58C2×C3×D12—21no—
59C2×C3×D12C2×C411yescharacteristic
60C2×C3×C4×D3—21no—
61C2×C3×C4×D3—42no—
62C2×C3×C4×D3C2×C411yescharacteristic
63C2×C3×C4×D3C2×C411yescharacteristic
64C2×C3×C4×D3D411yescharacteristic
65C2×C3×C4×D3D411yescharacteristic
66C2×C32×C8—21no—
67C32×C4⋊C4C2×C411yescharacteristic
68C32×C22⋊C4—21no—
69C3×#217—21no—
70C3×(C2×D3)⋊C4—21no—
71C3×(C2×D3)⋊C4—21no—
72C3×(C2×D3)⋊C4—42no—
73C3×C4⋊Dic3C2×C411yescharacteristic
74C3×Dic3⋊C4—21no—
75C2×C3×C3⋊C8—21no—
76C3×ΓL(1,𝔽25)—42no—
77#1510—31noSylow
78C22×C3×D4—31no—
79C22×C3×D4—61no—
80C23×C3×C4—31no—
81C2×D3×D4—21no—
82C2×D3×D4C3×C411yescharacteristic, complemented
83C22×C4×D3C22×C311yescharacteristic
84C2×C3×M16—61no—
85C3×#121—21no—
86C3×#121—31no—
87C3×#121—31no—
88C3×#121—31no—
89C3×#121—61no—
90C3×#121—61no—
91C3×#121C2×D311yescharacteristic
92C3×#121C22×C311yescharacteristic
93C2×C3×C4⋊C4—31no—
94C2×C3×C22⋊C4—61no—
95C2×C3×C22⋊C4—61no—
96#939—21no—
97#938C3×C411yescharacteristic, complemented
98#930C22×C311yescharacteristic
99C2×(C2×D3)⋊C4—21no—
100C2×ΓL(1,𝔽25)—21no—
101#895C3×C411yescharacteristic, complemented
102D3×C4⋊C4C3×C411yescharacteristic, complemented
103#884—21no—
104D3×C22⋊C4—21no—
105C3×C22⋊C8—21no—
106C3×C22⋊C8—31no—
107C3×C22⋊C8—61no—
108C3×C22⋊C8C2×D311yescharacteristic, complemented
109C3×C22⋊C8C22×C311yescharacteristic, complemented
110C3×#95—21no—
111C3×#95—31no—
112C3×#95—61no—
113C3×#95C2×D311yescharacteristic, complemented
114C3×#95C22×C311yescharacteristic, complemented
115C3⋊#95—21no—
116C3⋊#95C22×C311yescharacteristic, complemented
117#830C22×C311yescharacteristic, complemented
118(C2×D3)⋊C8—21no—
119C23×C32—21no—
120C23×C32C2×D411yescharacteristic
121C22×C3×D3—21no—
122C22×C3×D3—21no—
123C22×C3×D3—22no—
124C22×C3×D3—21no—
125C22×C3×D3—41no—
126C22×C3×D3—42no—
127C22×C3×D3—42no—
128C22×C3×D3—42no—
129C22×C3×D3C22⋊C411yescharacteristic
130C22×C3×D3C22⋊C411yescharacteristic
131C32×D4—41no—
132C32×D4—41no—
133C32×D4—41no—
134C2×C32×C4—21no—
135C2×C32×C4—41no—
136C2×C32×C4—41no—
137C2×C32×C4—41no—
138C2×C32×C4—41no—
139C2×C32×C4C22×C411yescharacteristic
140C2×C32×C4C2×D411yescharacteristic
141C3×#67—42no—
142C3×#67—42no—
143C3×#67—41no—
144C3×#67—41no—
145C2×C3×Dic3—21no—
146C2×C3×Dic3—41no—
147C2×C3×Dic3—41no—
148C2×C3×Dic3—41no—
149C2×C3×Dic3—41no—
150C2×C3×Dic3C22⋊C411yescharacteristic
151C2×C3×Dic3C22⋊C411yescharacteristic
152C3×D12—41no—
153C3×D12—42no—
154C3×C4×D3—22no—
155C3×C4×D3—22no—
156C3×C4×D3—41no—
157C3×C4×D3—41no—
158C3×C4×D3—42no—
159C32×C8—41no—
160C3×C3⋊C8—41no—
161C2×#121—31no—
162#407—31no—
163#381—31no—
164#327—31no—
165#327—31no—
166#327—31no—
167#327C3×D311yescharacteristic, complemented
168C24×C3—31no—
169C24×C3—61no—
170C23×D3—21no—
171C23×D3C3×D411yescharacteristic
172C2×C3×D4—21no—
173C2×C3×D4—21no—
174C2×C3×D4—21no—
175C2×C3×D4—31no—
176C2×C3×D4—41no—
177C2×C3×D4—61no—
178C2×C3×D4—61no—
179C2×C3×D4—61no—
180C2×C3×D4—61no—
181C2×C3×D4—122no—
182C2×C3×D4—122no—
183C2×C3×D4—122no—
184C2×C3×D4C4×D311yescharacteristic
185C2×C3×D4C2×C3×C411yescharacteristic
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—31no—
190C22×C3×C4—31no—
191C22×C3×C4—31no—
192C22×C3×C4—31no—
193C22×C3×C4—31no—
194C22×C3×C4—41no—
195C22×C3×C4—61no—
196C22×C3×C4—61no—
197C22×C3×C4—122no—
198C22×C3×C4C22×D311yescharacteristic
199C22×C3×C4C23×C311yescharacteristic
200C2×#67—21no—
201C2×#67—21no—
202C2×#67—21no—
203C22×Dic3—21no—
204C22×Dic3C3×D411yescharacteristic
205D3×D4—42no—
206D3×D4—42no—
207D3×D4—42no—
208C2×D12—21no—
209C2×D12C2×C3×C411yescharacteristic
210C2×C4×D3—21no—
211C2×C4×D3—42no—
212C2×C4×D3C2×C3×C411yescharacteristic
213C2×C4×D3C2×C3×C411yescharacteristic
214C2×C4×D3C3×D411yescharacteristic
215C2×C4×D3C3×D411yescharacteristic
216C3×M16—122no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—41no—
220C2×C3×C8—61no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—31no—
223C3×C4⋊C4—61no—
224C3×C4⋊C4C4×D311yescharacteristic
225C3×C4⋊C4C2×C3×C411yescharacteristic
226C3×C22⋊C4—21no—
227C3×C22⋊C4—21no—
228C3×C22⋊C4—41no—
229C3×C22⋊C4—61no—
230C3×C22⋊C4—61no—
231C3×C22⋊C4—61no—
232C3×C22⋊C4—122no—
233#217—21no—
234(C2×D3)⋊C4—21no—
235(C2×D3)⋊C4—21no—
236(C2×D3)⋊C4—42no—
237C4⋊Dic3C2×C3×C411yescharacteristic
238Dic3⋊C4—21no—
239C2×C3⋊C8—21no—
240ΓL(1,𝔽25)—42no—
241C22×C32—21no—
242C22×C32—21no—
243C22×C32—41no—
244C22×C32—41no—
245C22×C32—41no—
246C22×C32C2×C22⋊C411yescharacteristic
247C22×C32AGL(1,ℤ/8)11yescharacteristic
248C22×C32AGL(1,ℤ/8)11yescharacteristic
249C2×C3×D3—21no—
250C2×C3×D3—21no—
251C2×C3×D3—21no—
252C2×C3×D3—21no—
253C2×C3×D3—21no—
254C2×C3×D3—42no—
255C2×C3×D3—42no—
256C2×C3×D3—41no—
257C2×C3×D3—41no—
258C2×C3×D3—41no—
259C2×C3×D3—84no—
260C2×C3×D3#9912yes—
261C32×C4—21no—
262C32×C4—21no—
263C32×C4—41no—
264C32×C4—81no—
265C3×Dic3—21no—
266C3×Dic3—21no—
267C3×Dic3—41no—
268C3×Dic3—81no—
269C22×D4—31no—
270C22×D4—61no—
271C23×C4—31no—
272C2×M16—61no—
273#121—31no—
274#121—31no—
275#121—31no—
276#121—61no—
277#121—61no—
278#121C2×C3×D311yescharacteristic
279C2×C4⋊C4—31no—
280C2×C22⋊C4—61no—
281C2×C22⋊C4—61no—
282C22⋊C8—31no—
283C22⋊C8—61no—
284C22⋊C8C2×C3×D311yescharacteristic, complemented
285#95—31no—
286#95—61no—
287#95C2×C3×D311yescharacteristic, complemented
288C23×C3—21no—
289C23×C3—21no—
290C23×C3—21no—
291C23×C3—31no—
292C23×C3—31no—
293C23×C3—41no—
294C23×C3—62no—
295C23×C3—61no—
296C23×C3—61no—
297C23×C3—61no—
298C23×C3—121no—
299C23×C3—122no—
300C23×C3—122no—
301C23×C3—122no—
302C23×C3D3×D411yescharacteristic
303C23×C3C2×C3×D411yescharacteristic
304C22×D3—21no—
305C22×D3—21no—
306C22×D3—22no—
307C22×D3—21no—
308C22×D3—41no—
309C22×D3—42no—
310C22×D3—42no—
311C22×D3—42no—
312C22×D3C3×C22⋊C411yescharacteristic
313C22×D3C3×C22⋊C411yescharacteristic
314C3×D4—41no—
315C3×D4—41no—
316C3×D4—41no—
317C3×D4—41no—
318C3×D4—41no—
319C3×D4—41no—
320C3×D4—81no—
321C3×D4—81no—
322C3×D4—81no—
323C3×D4—121no—
324C3×D4—122no—
325C3×D4—122no—
326C3×D4—122no—
327C3×D4—121no—
328C3×D4—121no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—31no—
334C2×C3×C4—31no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—42no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—62no—
346C2×C3×C4—62no—
347C2×C3×C4—61no—
348C2×C3×C4—81no—
349C2×C3×C4—81no—
350C2×C3×C4—81no—
351C2×C3×C4—121no—
352C2×C3×C4—121no—
353C2×C3×C4—122no—
354C2×C3×C4—121no—
355C2×C3×C4—121no—
356C2×C3×C4—121no—
357C2×C3×C4—121no—
358C2×C3×C4C2×C4×D311yescharacteristic
359C2×C3×C4D3×D411yescharacteristic
360C2×C3×C4C22×C3×C411yescharacteristic
361C2×C3×C4C2×C3×D411yescharacteristic
362#67—42no—
363#67—42no—
364#67—41no—
365#67—41no—
366C2×Dic3—21no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3C3×C22⋊C411yescharacteristic
372C2×Dic3C3×C22⋊C411yescharacteristic
373D12—41no—
374D12—42no—
375C4×D3—22no—
376C4×D3—22no—
377C4×D3—41no—
378C4×D3—41no—
379C4×D3—42no—
380C3×C8—41no—
381C3×C8—41no—
382C3×C8—81no—
383C3×C8—121no—
384C3⋊C8—41no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—81no—
388C2×C32C2×#9911yescharacteristic
389C2×C32#41811yescharacteristic
390C2×C32#42111yescharacteristic
391C3×D3—22no—
392C3×D3—41no—
393C3×D3—81no—
394C24—31no—
395C24—61no—
396C2×D4—21no—
397C2×D4—31no—
398C2×D4—61no—
399C2×D4—61no—
400C2×D4—61no—
401C2×D4—61no—
402C2×D4—122no—
403C2×D4—122no—
404C2×D4—122no—
405C2×D4C3×C4×D311yescharacteristic
406C22×C4—21no—
407C22×C4—31no—
408C22×C4—31no—
409C22×C4—31no—
410C22×C4—31no—
411C22×C4—31no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—122no—
415C22×C4C22×C3×D311yescharacteristic
416M16—122no—
417C2×C8—21no—
418C2×C8—61no—
419C4⋊C4—31no—
420C4⋊C4—61no—
421C4⋊C4C3×C4×D311yescharacteristic
422C22⋊C4—21no—
423C22⋊C4—61no—
424C22⋊C4—61no—
425C22⋊C4—61no—
426C22⋊C4—122no—
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—21no—
434C22×C3—32no—
435C22×C3—41no—
436C22×C3—41no—
437C22×C3—41no—
438C22×C3—41no—
439C22×C3—41no—
440C22×C3—41no—
441C22×C3—41no—
442C22×C3—41no—
443C22×C3—61no—
444C22×C3—61no—
445C22×C3—61no—
446C22×C3—61no—
447C22×C3—61no—
448C22×C3—81no—
449C22×C3—81no—
450C22×C3—81no—
451C22×C3—122no—
452C22×C3—122no—
453C22×C3—121no—
454C22×C3—121no—
455C22×C3—121no—
456C22×C3—244no—
457C22×C3D3×C22⋊C411yescharacteristic
458C22×C3#91111yescharacteristic
459C22×C3#91411yescharacteristic
460C22×C3C2×C3×C22⋊C411yescharacteristic
461C22×C3C3×AGL(1,ℤ/8)11yescharacteristic
462C22×C3C3×AGL(1,ℤ/8)11yescharacteristic
463C2×D3—21no—
464C2×D3—21no—
465C2×D3—21no—
466C2×D3—21no—
467C2×D3—21no—
468C2×D3—42no—
469C2×D3—42no—
470C2×D3—41no—
471C2×D3—41no—
472C2×D3—41no—
473C2×D3—84no—
474C2×D3C3×#9912yes—
475C3×C4—21no—
476C3×C4—21no—
477C3×C4—21no—
478C3×C4—21no—
479C3×C4—41no—
480C3×C4—41no—
481C3×C4—41no—
482C3×C4—41no—
483C3×C4—61no—
484C3×C4—61no—
485C3×C4—81no—
486C3×C4—81no—
487C3×C4—81no—
488C3×C4—82no—
489C3×C4—121no—
490C3×C4—241no—
491Dic3—21no—
492Dic3—21no—
493Dic3—41no—
494Dic3—81no—
495C32#151011yescharacteristic, complemented, Sylow
496C23—21no—
497C23—31no—
498C23—31no—
499C23—62no—
500C23—61no—
501C23—61no—
502C23—61no—
503C23—121no—
504C23—122no—
505C23—122no—
506C23—122no—
507C23C3×D3×D411yescharacteristic
508D4—41no—
509D4—41no—
510D4—41no—
511D4—121no—
512D4—122no—
513D4—122no—
514D4—122no—
515D4—121no—
516D4—121no—
517C2×C4—21no—
518C2×C4—31no—
519C2×C4—31no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—62no—
525C2×C4—62no—
526C2×C4—61no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4—122no—
530C2×C4—121no—
531C2×C4—121no—
532C2×C4—121no—
533C2×C4—121no—
534C2×C4C2×C3×C4×D311yescharacteristic
535C2×C4C3×D3×D411yescharacteristic
536C8—41no—
537C8—121no—
538C2×C3—21no—
539C2×C3—21no—
540C2×C3—21no—
541C2×C3—21no—
542C2×C3—21no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—41no—
546C2×C3—41no—
547C2×C3—62no—
548C2×C3—81no—
549C2×C3—81no—
550C2×C3—121no—
551C2×C3—161no—
552C2×C3—241no—
553C2×C3D3×#9911yescharacteristic
554C2×C3#477011yescharacteristic
555C2×C3#482111yescharacteristic
556C2×C3C2×C3×#9911yescharacteristic
557C2×C3C3×#41811yescharacteristic
558C2×C3C3×#42111yescharacteristic
559D3—22no—
560D3—41no—
561D3—81no—
562C22—21no—
563C22—21no—
564C22—32no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—61no—
569C22—61no—
570C22—61no—
571C22—61no—
572C22—61no—
573C22—122no—
574C22—122no—
575C22—121no—
576C22—121no—
577C22—121no—
578C22—244no—
579C22C3×D3×C22⋊C411yescharacteristic
580C22C3×#91111yescharacteristic
581C22C3×#91411yescharacteristic
582C4—21no—
583C4—21no—
584C4—41no—
585C4—61no—
586C4—61no—
587C4—81no—
588C4—121no—
589C4—241no—
590C3—21no—
591C3#6830411yescharacteristic, direct product factor
592C3C3×#151011yescharacteristic, complemented
593C2—21no—
594C2—21no—
595C2—62no—
596C2—81no—
597C2—121no—
598C2—241no—
599C2C3×D3×#9911yescharacteristic
600C2C3×#477011yescharacteristic
601C2C3×#482111yescharacteristic
subgroupG/H
centerC22×C3D3×C22⋊C4
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#327C2
socleC22×C32C2×C22⋊C4
prime radicalC3×D3×#121C2
cosocleC22×C3×C4C23×C3
hypercenterC3×#327D3
hypocenterC3C3×#1510
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#327

Complements

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

NH ≅ G/Ntype
2. C3×#4726C2semidirect
3. C3×#4664C2semidirect
4. C32×#327C2semidirect
5. C3×#4537C2semidirect
6. C3×#4470C2semidirect
7. C3×#4451C2semidirect
9. #68304C3direct (N × H)
11. C2×C3×D3×D4C4semidirect
15. C3×#938C4semidirect
19. C3×#895C4semidirect
20. C3×D3×C4⋊C4C4semidirect
23. C32×C22⋊C8C22semidirect
24. C32×#95C22semidirect
26. C3×C3⋊#95C22semidirect
27. C3×#830C22semidirect
33. #4726C2×C3semidirect
34. #4664C2×C3semidirect
39. C3×#327D3semidirect
40. C3×#327C2×C3semidirect
41. #4537C2×C3semidirect
42. #4470C2×C3semidirect
43. #4451C2×C3semidirect
82. C2×D3×D4C3×C4semidirect
97. #938C3×C4semidirect
101. #895C3×C4semidirect
102. D3×C4⋊C4C3×C4semidirect
108. C3×C22⋊C8C2×D3semidirect
109. C3×C22⋊C8C22×C3semidirect
113. C3×#95C2×D3semidirect
114. C3×#95C22×C3semidirect
116. C3⋊#95C22×C3semidirect
117. #830C22×C3semidirect
167. #327C3×D3semidirect
284. C22⋊C8C2×C3×D3semidirect
287. #95C2×C3×D3semidirect
495. C32#1510semidirect
591. C3#68304direct (N × H)
592. C3C3×#1510semidirect

Maximal subgroups

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

subgroupindex
C3×D3×#1212
C3×#47262
C3×#46642
C32×#3272
C3×#45372
C3×#44702
C3×#44512
C3×#15103
#683043