Finite Group Databasebeta

← C3×#25833 #10644440

Subgroups — C3×#25833 #10644440

Order 768.

934 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3504C211yesmaximal, characteristic
2C3×#3165C211yesmaximal, characteristic, complemented
3C3×#3079C211yesmaximal, characteristic
4C3×#2660C211yesmaximal, characteristic, complemented
5C3×#2061C212yesmaximal, complemented
6C3×#2060C212yesmaximal, complemented
7C3×#1998C211yesmaximal, characteristic, complemented
8C3×#1995C211yesmaximal, characteristic, complemented
9C3×#1860C212yesmaximal
10C3×#1859C212yesmaximal
11C3×#1759C211yesmaximal, characteristic
12#25833C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#557—21no—
14C3×#557—21no—
15C3×#554—21no—
16C3×#554—21no—
17C3×#554—22no—
18C3×#554—22no—
19C3×#544—21no—
20C3×#544—21no—
21C3×#544C2211yescharacteristic
22C3×#544C2211yescharacteristic
23C3×#533—21no—
24C3×#533C2211yescharacteristic
25C2×C3×C4⋊Q8C2211yescharacteristic
26C2×C3×#125—21no—
27C2×C3×#125C2211yescharacteristic
28C3×#520C2211yescharacteristic
29C3×#520C2211yescharacteristic
30C3×C4×#41C2211yescharacteristic
31C3×#501—21no—
32C3×#501—21no—
33C3×#500—21no—
34C3×#498—21no—
35C3×#477—21no—
36C3×#476—21no—
37C3×#475—21no—
38C3×#474—21no—
39C3×#428C2212yes—
40C3×#428C2212yes—
41C3×#428C2212yes—
42C3×#423C2211yescharacteristic
43C2×C3×#104C2212yes—
44C3×#417C2212yes—
45C3×C4×M16C2211yescharacteristic
46C3×#398—22no—
47C3×#395—22no—
48C3×#384C2212yes—
49C3×#382C2212yes—
50C3×#376C2211yescharacteristic
51C3×#343C2212yes—
52C3×#343C2212yes—
53C3×#339C2212yes—
54C3×#339C2212yes—
55C3×#339C2212yes—
56#3504C2×C311yescharacteristic
57#3165C2×C311yescharacteristic, complemented
58#3079C2×C311yescharacteristic
59#2660C2×C311yescharacteristic, complemented
60#2061C2×C312yescomplemented
61#2060C2×C312yescomplemented
62#1998C2×C311yescharacteristic, complemented
63#1995C2×C311yescharacteristic, complemented
64#1860C2×C312yes—
65#1859C2×C312yes—
66#1759C2×C311yescharacteristic
67C2×C3×#41C2311yescharacteristic
68C22×C3×Q8—21no—
69C2×C3×M16C2312yes—
70C2×C3×M16C2312yes—
71C3×C4⋊Q8—22no—
72C3×C4⋊Q8—22no—
73C3×C4⋊Q8—21no—
74C3×C4⋊Q8—21no—
75C3×C4⋊Q8—21no—
76C3×C4⋊Q8—21no—
77C3×C4⋊Q8—21no—
78C3×C4⋊Q8—21no—
79C3×C4⋊Q8—21no—
80C3×C4⋊Q8—21no—
81C3×C4⋊Q8D411yescharacteristic
82C3×C4⋊Q8D411yescharacteristic
83C3×#125—22no—
84C3×#125—22no—
85C3×#125—22no—
86C3×#125—22no—
87C3×#125—21no—
88C3×#125—21no—
89C3×#125—21no—
90C3×#125—21no—
91C3×#125—21no—
92C3×#125—21no—
93C3×#125—21no—
94C3×#125—21no—
95C3×#125D411yescharacteristic
96C3×#125D411yescharacteristic
97C3×C22⋊Q8—22no—
98C3×C22⋊Q8—22no—
99C3×C22⋊Q8—22no—
100C3×C22⋊Q8—22no—
101C3×C22⋊Q8—22no—
102C3×C22⋊Q8—22no—
103C3×C22⋊Q8—21no—
104C3×C22⋊Q8—21no—
105C3×C22⋊Q8—22no—
106C3×C22⋊Q8—22no—
107C3×C4×Q8—22no—
108C3×C4×Q8—22no—
109C3×C4×Q8—21no—
110C3×C4×Q8—21no—
111C3×C4×Q8—21no—
112C3×C4×Q8—21no—
113C3×C4×Q8D411yescharacteristic
114C3×C4×Q8D411yescharacteristic
115C3×C4×D4—21no—
116C3×C4×D4—21no—
117C3×C4×D4—21no—
118C3×C4×D4—21no—
119C3×C4×D4—22no—
120C3×C4×D4—22no—
121C3×C4×D4D411yescharacteristic
122C3×C4×D4D411yescharacteristic
123C3×#117—21no—
124C3×#117—21no—
125C3×#117—21no—
126C3×#117C2311yescharacteristic
127C3×#117C2311yescharacteristic
128C3×#117C2311yescharacteristic
129C2×C3×C4⋊C4—22no—
130C2×C3×C4⋊C4—22no—
131C2×C3×C4⋊C4—21no—
132C2×C3×C4⋊C4—21no—
133C2×C3×C4⋊C4—21no—
134C2×C3×C4⋊C4—24no—
135C2×C3×C4⋊C4D412yes—
136C2×C3×C4⋊C4D412yes—
137C2×C3×C4⋊C4C2312yes—
138C2×C3×C4⋊C4C2311yescharacteristic
139C2×C3×C4⋊C4C2311yescharacteristic
140C2×C3×C42—21no—
141C2×C3×C42C2312yes—
142C2×C3×C42C2311yescharacteristic
143C3×#107—22no—
144C3×#107—22no—
145C3×#107—22no—
146C3×#106—22no—
147C3×#106—22no—
148C3×#106—22no—
149C3×C4⋊C8—21no—
150C3×C4⋊C8—21no—
151C3×#104Q818yescomplemented
152C3×#103—22no—
153C3×#102—22no—
154C3×#97—21no—
155C3×C4×C8—21no—
156C3×#95—21no—
157C3×#95—21no—
158C3×#95—24no—
159#557—21no—
160#557—21no—
161#554—21no—
162#554—21no—
163#554—22no—
164#554—22no—
165#544—21no—
166#544—21no—
167#544C22×C311yescharacteristic
168#544C22×C311yescharacteristic
169#533—21no—
170#533C22×C311yescharacteristic
171C2×C4⋊Q8C22×C311yescharacteristic
172C2×#125—21no—
173C2×#125C22×C311yescharacteristic
174#520C22×C311yescharacteristic
175#520C22×C311yescharacteristic
176C4×#41C22×C311yescharacteristic
177#501—21no—
178#501—21no—
179#500—21no—
180#498—21no—
181#477—21no—
182#476—21no—
183#475—21no—
184#474—21no—
185#428C22×C312yes—
186#428C22×C312yes—
187#428C22×C312yes—
188#423C22×C311yescharacteristic
189C2×#104C22×C312yes—
190#417C22×C312yes—
191C4×M16C22×C311yescharacteristic
192#398—22no—
193#395—22no—
194#384C22×C312yes—
195#382C22×C312yes—
196#376C22×C311yescharacteristic
197#343C22×C312yes—
198#343C22×C312yes—
199#339C22×C312yes—
200#339C22×C312yes—
201#339C22×C312yes—
202C3×#41—22no—
203C3×#41C2×Q814yes—
204C2×C3×Q8—22no—
205C2×C3×Q8—22no—
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—41no—
209C2×C3×Q8—41no—
210C2×C3×Q8—82no—
211C2×C3×Q8C2×D411yescharacteristic
212C2×C3×D4—21no—
213C2×C3×D4C2×D411yescharacteristic
214C22×C3×C4—21no—
215C22×C3×C4—22no—
216C22×C3×C4—22no—
217C22×C3×C4—22no—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—22no—
221C22×C3×C4C2×D412yes—
222C22×C3×C4C2×D412yes—
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4C2411yescharacteristic
226C3×M16—22no—
227C3×M16—24no—
228C3×M16C2×Q814yes—
229C2×C3×C8—22no—
230C2×C3×C8—22no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4—22no—
236C3×C4⋊C4—22no—
237C3×C4⋊C4—22no—
238C3×C4⋊C4—22no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—22no—
242C3×C4⋊C4—22no—
243C3×C4⋊C4—22no—
244C3×C4⋊C4—22no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—22no—
256C3×C4⋊C4—22no—
257C3×C4⋊C4—41no—
258C3×C4⋊C4—41no—
259C3×C4⋊C4—84no—
260C3×C4⋊C4C2×D411yescharacteristic
261C3×C4⋊C4C2×D411yescharacteristic
262C3×C4⋊C4C2×D411yescharacteristic
263C3×C4⋊C4C2×D411yescharacteristic
264C3×C4⋊C4C2×D411yescharacteristic
265C3×C4⋊C4C2×D411yescharacteristic
266C3×C22⋊C4—22no—
267C3×C22⋊C4—22no—
268C3×C22⋊C4—21no—
269C3×C22⋊C4—21no—
270C3×C22⋊C4—21no—
271C3×C22⋊C4—21no—
272C3×C22⋊C4—21no—
273C3×C42—22no—
274C3×C42—22no—
275C3×C42—21no—
276C3×C42—21no—
277C3×C42—21no—
278C3×C42—21no—
279C3×C42—21no—
280C3×C42—22no—
281C3×C42C2×D411yescharacteristic
282C3×C42C2×D411yescharacteristic
283C3×C42C2×D411yescharacteristic
284C3×C42C2×D411yescharacteristic
285C3×C42C2×Q814yes—
286C2×#41C23×C311yescharacteristic
287C22×Q8—21no—
288C2×M16C23×C312yes—
289C2×M16C23×C312yes—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—21no—
293C4⋊Q8—21no—
294C4⋊Q8—21no—
295C4⋊Q8—21no—
296C4⋊Q8—21no—
297C4⋊Q8—21no—
298C4⋊Q8—21no—
299C4⋊Q8—21no—
300C4⋊Q8C3×D411yescharacteristic
301C4⋊Q8C3×D411yescharacteristic
302#125—22no—
303#125—22no—
304#125—22no—
305#125—22no—
306#125—21no—
307#125—21no—
308#125—21no—
309#125—21no—
310#125—21no—
311#125—21no—
312#125—21no—
313#125—21no—
314#125C3×D411yescharacteristic
315#125C3×D411yescharacteristic
316C22⋊Q8—22no—
317C22⋊Q8—22no—
318C22⋊Q8—22no—
319C22⋊Q8—22no—
320C22⋊Q8—22no—
321C22⋊Q8—22no—
322C22⋊Q8—21no—
323C22⋊Q8—21no—
324C22⋊Q8—22no—
325C22⋊Q8—22no—
326C4×Q8—22no—
327C4×Q8—22no—
328C4×Q8—21no—
329C4×Q8—21no—
330C4×Q8—21no—
331C4×Q8—21no—
332C4×Q8C3×D411yescharacteristic
333C4×Q8C3×D411yescharacteristic
334C4×D4—21no—
335C4×D4—21no—
336C4×D4—21no—
337C4×D4—21no—
338C4×D4—22no—
339C4×D4—22no—
340C4×D4C3×D411yescharacteristic
341C4×D4C3×D411yescharacteristic
342#117—21no—
343#117—21no—
344#117—21no—
345#117C23×C311yescharacteristic
346#117C23×C311yescharacteristic
347#117C23×C311yescharacteristic
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4C3×D412yes—
355C2×C4⋊C4C3×D412yes—
356C2×C4⋊C4C23×C312yes—
357C2×C4⋊C4C23×C311yescharacteristic
358C2×C4⋊C4C23×C311yescharacteristic
359C2×C42—21no—
360C2×C42C23×C312yes—
361C2×C42C23×C311yescharacteristic
362#107—22no—
363#107—22no—
364#107—22no—
365#106—22no—
366#106—22no—
367#106—22no—
368C4⋊C8—21no—
369C4⋊C8—21no—
370#104C3×Q818yescomplemented
371#103—22no—
372#102—22no—
373#97—21no—
374C4×C8—21no—
375#95—21no—
376#95—21no—
377#95—24no—
378C23×C3—21no—
379C23×C3C22×D411yescharacteristic
380C3×Q8—21no—
381C3×Q8—81no—
382C3×Q8—81no—
383C3×Q8—82no—
384C3×Q8—82no—
385C3×Q8—82no—
386C3×Q8—82no—
387C3×Q8C4⋊Q812yes—
388C3×D4—24no—
389C3×D4—21no—
390C3×D4C4⋊Q812yes—
391C2×C3×C4—21no—
392C2×C3×C4—24no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—22no—
398C2×C3×C4—22no—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
401C2×C3×C4—24no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—42no—
405C2×C3×C4—42no—
406C2×C3×C4—42no—
407C2×C3×C4—42no—
408C2×C3×C4—44no—
409C2×C3×C4—42no—
410C2×C3×C4—81no—
411C2×C3×C4—81no—
412C2×C3×C4—82no—
413C2×C3×C4—82no—
414C2×C3×C4#12011yescharacteristic
415C2×C3×C4#12011yescharacteristic
416C2×C3×C4#12011yescharacteristic
417C2×C3×C4#12011yescharacteristic
418C2×C3×C4C22×D411yescharacteristic
419C2×C3×C4C22×D411yescharacteristic
420C2×C3×C4C22×Q812yes—
421C2×C3×C42−1+412yes—
422C3×C8—24no—
423C3×C8—42no—
424#41—22no—
425#41C2×C3×Q814yes—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—41no—
431C2×Q8—41no—
432C2×Q8—82no—
433C2×Q8C2×C3×D411yescharacteristic
434C2×D4—21no—
435C2×D4C2×C3×D411yescharacteristic
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4C2×C3×D412yes—
444C22×C4C2×C3×D412yes—
445C22×C4C2×C3×D411yescharacteristic
446C22×C4C2×C3×D411yescharacteristic
447C22×C4C24×C311yescharacteristic
448M16—22no—
449M16—24no—
450M16C2×C3×Q814yes—
451C2×C8—22no—
452C2×C8—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—84no—
482C4⋊C4C2×C3×D411yescharacteristic
483C4⋊C4C2×C3×D411yescharacteristic
484C4⋊C4C2×C3×D411yescharacteristic
485C4⋊C4C2×C3×D411yescharacteristic
486C4⋊C4C2×C3×D411yescharacteristic
487C4⋊C4C2×C3×D411yescharacteristic
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—21no—
491C22⋊C4—21no—
492C22⋊C4—21no—
493C22⋊C4—21no—
494C22⋊C4—21no—
495C42—22no—
496C42—22no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—22no—
503C42C2×C3×D411yescharacteristic
504C42C2×C3×D411yescharacteristic
505C42C2×C3×D411yescharacteristic
506C42C2×C3×D411yescharacteristic
507C42C2×C3×Q814yes—
508C22×C3—21no—
509C22×C3—21no—
510C22×C3—22no—
511C22×C3—41no—
512C22×C3—41no—
513C22×C3C2×#12011yescharacteristic
514C22×C3D4×Q812yes—
515C3×C4—22no—
516C3×C4—21no—
517C3×C4—21no—
518C3×C4—21no—
519C3×C4—21no—
520C3×C4—44no—
521C3×C4—42no—
522C3×C4—41no—
523C3×C4—41no—
524C3×C4—41no—
525C3×C4—42no—
526C3×C4—41no—
527C3×C4—41no—
528C3×C4—41no—
529C3×C4—82no—
530C3×C4C2×C4⋊Q811yescharacteristic
531C3×C4#53611yescharacteristic
532C3×C4D4×Q812yes—
533C23—21no—
534C23C22×C3×D411yescharacteristic
535Q8—21no—
536Q8—81no—
537Q8—81no—
538Q8—82no—
539Q8—82no—
540Q8—82no—
541Q8—82no—
542Q8C3×C4⋊Q812yes—
543D4—24no—
544D4—21no—
545D4C3×C4⋊Q812yes—
546C2×C4—21no—
547C2×C4—24no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—24no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4C3×#12011yescharacteristic
570C2×C4C3×#12011yescharacteristic
571C2×C4C3×#12011yescharacteristic
572C2×C4C3×#12011yescharacteristic
573C2×C4C22×C3×D411yescharacteristic
574C2×C4C22×C3×D411yescharacteristic
575C2×C4C22×C3×Q812yes—
576C2×C4C3×2−1+412yes—
577C8—24no—
578C8—42no—
579C2×C3—22no—
580C2×C3—42no—
581C2×C3#243411yescharacteristic
582C2×C3#302111yescharacteristic
583C2×C3#302211yescharacteristic
584C22—21no—
585C22—21no—
586C22—22no—
587C22—41no—
588C22—41no—
589C22C2×C3×#12011yescharacteristic
590C22C3×D4×Q812yes—
591C4—22no—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—21no—
596C4—44no—
597C4—42no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—42no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—82no—
606C4C2×C3×C4⋊Q811yescharacteristic
607C4C3×#53611yescharacteristic
608C4C3×D4×Q812yes—
609C3#2583311yescharacteristic, direct product factor, Sylow
610C2—22no—
611C2—42no—
612C2C3×#243411yescharacteristic
613C2C3×#302111yescharacteristic
614C2C3×#302211yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3165C2semidirect
4. C3×#2660C2semidirect
5. C3×#2061C2semidirect
6. C3×#2060C2semidirect
7. C3×#1998C2semidirect
8. C3×#1995C2semidirect
12. #25833C3direct (N × H)
57. #3165C2×C3semidirect
59. #2660C2×C3semidirect
60. #2061C2×C3semidirect
61. #2060C2×C3semidirect
62. #1998C2×C3semidirect
63. #1995C2×C3semidirect
151. C3×#104Q8semidirect
370. #104C3×Q8semidirect
609. C3#25833direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35042
C3×#31652
C3×#30792
C3×#26602
C3×#20612
C3×#20602
C3×#19982
C3×#19952
C3×#18602
C3×#18592
C3×#17592
#258333