Finite Group Databasebeta

← C3×#25692 #10644299

Subgroups — C3×#25692 #10644299

Order 768.

976 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3510C211yesmaximal, characteristic
2C3×#3165C211yesmaximal, characteristic, complemented
3C3×#3079C211yesmaximal, characteristic, complemented
4C3×#2640C211yesmaximal, characteristic, complemented
5C3×#2038C212yesmaximal, complemented
6C3×#2037C212yesmaximal
7C3×#1995C211yesmaximal, characteristic, complemented
8C3×#1967C211yesmaximal, characteristic, complemented
9C3×#1857C212yesmaximal, complemented
10C3×#1856C212yesmaximal, complemented
11C3×#1759C211yesmaximal, characteristic, complemented
12#25692C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#559—21no—
14C3×#559—21no—
15C3×#556—21no—
16C3×#556—21no—
17C3×#551—22no—
18C3×#551—22no—
19C3×#542—21no—
20C3×#542—21no—
21C3×#536C2211yescharacteristic
22C3×#535C2211yescharacteristic
23C3×#533C2211yescharacteristic
24C2×C3×C4⋊Q8C2211yescharacteristic
25C3×#529—21no—
26C3×#529—21no—
27C2×C3×#124C2211yescharacteristic
28C3×#519C2211yescharacteristic
29C3×#518C2211yescharacteristic
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—
45C2×C3×#101C2212yes—
46C2×C3×#100C2212yes—
47C3×C4×M16C2211yescharacteristic
48C3×#397—22no—
49C3×#393—22no—
50C3×#391C2212yes—
51C3×#390C2212yes—
52C3×#376C2211yescharacteristic
53C3×#339C2212yes—
54C3×#339C2212yes—
55C3×#339C2212yes—
56#3510C2×C311yescharacteristic
57#3165C2×C311yescharacteristic, complemented
58#3079C2×C311yescharacteristic, complemented
59#2640C2×C311yescharacteristic, complemented
60#2038C2×C312yescomplemented
61#2037C2×C312yes—
62#1995C2×C311yescharacteristic, complemented
63#1967C2×C311yescharacteristic, complemented
64#1857C2×C312yescomplemented
65#1856C2×C312yescomplemented
66#1759C2×C311yescharacteristic, complemented
67C2×C3×#41C2311yescharacteristic
68C22×C3×Q8—21no—
69C22×C3×Q8C2311yescharacteristic
70C22×C3×D4C2311yescharacteristic
71C2×C3×M16C2312yes—
72C2×C3×M16C2312yes—
73C3×C4⋊Q8—21no—
74C3×C4⋊Q8—21no—
75C3×C4⋊Q8—21no—
76C3×C4⋊Q8—21no—
77C3×C4⋊Q8—21no—
78C3×C4⋊Q8D411yescharacteristic
79C3×C4⋊Q8D411yescharacteristic
80C3×#127—21no—
81C3×#126—22no—
82C3×#126—22no—
83C3×#126—22no—
84C3×#126—22no—
85C3×#125D411yescharacteristic
86C3×#125D411yescharacteristic
87C3×#124—22no—
88C3×#124—22no—
89C3×#124—21no—
90C3×#124—21no—
91C3×#124—21no—
92C3×#124—21no—
93C3×#124—21no—
94C3×#124—21no—
95C3×#123—22no—
96C3×#123—22no—
97C3×#123—22no—
98C3×#123—22no—
99C3×#123—22no—
100C3×C22⋊Q8—22no—
101C3×C22⋊Q8—22no—
102C3×C22⋊Q8—21no—
103C3×C22⋊Q8—21no—
104C3×C22⋊Q8—21no—
105C3×C22⋊Q8—21no—
106C3×#121—21no—
107C3×#121—21no—
108C3×#120—22no—
109C3×C4×Q8—21no—
110C3×C4×Q8—21no—
111C3×C4×Q8—21no—
112C3×C4×Q8—21no—
113C3×C4×Q8—21no—
114C3×C4×Q8—21no—
115C3×C4×Q8D411yescharacteristic
116C3×C4×Q8D411yescharacteristic
117C3×C4×D4—22no—
118C3×C4×D4—22no—
119C3×C4×D4—21no—
120C3×C4×D4—21no—
121C3×C4×D4—21no—
122C3×C4×D4—21no—
123C3×C4×D4—21no—
124C3×C4×D4—21no—
125C3×C4×D4D411yescharacteristic
126C3×C4×D4D411yescharacteristic
127C3×#117—22no—
128C3×#117—22no—
129C3×#117—22no—
130C3×#117—21no—
131C3×#117C2311yescharacteristic
132C3×#117C2311yescharacteristic
133C3×#117C2311yescharacteristic
134C2×C3×C4⋊C4—21no—
135C2×C3×C4⋊C4C2312yes—
136C2×C3×C22⋊C4—22no—
137C2×C3×C22⋊C4—24no—
138C2×C3×C22⋊C4D412yes—
139C2×C3×C22⋊C4D412yes—
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×#104—24no—
152C3×#103—22no—
153C3×#102—22no—
154C3×#101—24no—
155C3×#100—24no—
156C3×#97—21no—
157C3×C4×C8—21no—
158C3×#95—21no—
159C3×#95—21no—
160C3×#95—24no—
161#559—21no—
162#559—21no—
163#556—21no—
164#556—21no—
165#551—22no—
166#551—22no—
167#542—21no—
168#542—21no—
169#536C22×C311yescharacteristic
170#535C22×C311yescharacteristic
171#533C22×C311yescharacteristic
172C2×C4⋊Q8C22×C311yescharacteristic
173#529—21no—
174#529—21no—
175C2×#124C22×C311yescharacteristic
176#519C22×C311yescharacteristic
177#518C22×C311yescharacteristic
178C4×#41C22×C311yescharacteristic
179#501—21no—
180#501—21no—
181#500—21no—
182#498—21no—
183#477—21no—
184#476—21no—
185#475—21no—
186#474—21no—
187#428C22×C312yes—
188#428C22×C312yes—
189#428C22×C312yes—
190#423C22×C311yescharacteristic
191C2×#104C22×C312yes—
192#417C22×C312yes—
193C2×#101C22×C312yes—
194C2×#100C22×C312yes—
195C4×M16C22×C311yescharacteristic
196#397—22no—
197#393—22no—
198#391C22×C312yes—
199#390C22×C312yes—
200#376C22×C311yescharacteristic
201#339C22×C312yes—
202#339C22×C312yes—
203#339C22×C312yes—
204C24×C3C2×D412yes—
205C3×#41—22no—
206C3×#41—22no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8—22no—
212C2×C3×Q8—22no—
213C2×C3×Q8—41no—
214C2×C3×Q8—41no—
215C2×C3×Q8—82no—
216C2×C3×Q8C2×D411yescharacteristic
217C2×C3×D4—21no—
218C2×C3×D4—22no—
219C2×C3×D4—22no—
220C2×C3×D4—21no—
221C2×C3×D4—21no—
222C2×C3×D4—22no—
223C2×C3×D4—22no—
224C2×C3×D4C2×D411yescharacteristic
225C22×C3×C4—22no—
226C22×C3×C4—22no—
227C22×C3×C4—21no—
228C22×C3×C4—21no—
229C22×C3×C4—21no—
230C22×C3×C4—22no—
231C22×C3×C4—22no—
232C22×C3×C4C2×D412yes—
233C22×C3×C4C2×D411yescharacteristic
234C22×C3×C4C2×D411yescharacteristic
235C22×C3×C4C2411yescharacteristic
236C3×M16—22no—
237C3×M16—22no—
238C3×M16—22no—
239C3×M16#4114yes—
240C2×C3×C8—22no—
241C2×C3×C8—22no—
242C3×C4⋊C4—22no—
243C3×C4⋊C4—22no—
244C3×C4⋊C4—22no—
245C3×C4⋊C4—22no—
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—22no—
255C3×C4⋊C4—22no—
256C3×C4⋊C4—41no—
257C3×C4⋊C4—41no—
258C3×C4⋊C4C2×D411yescharacteristic
259C3×C4⋊C4C2×D411yescharacteristic
260C3×C4⋊C4C2×D411yescharacteristic
261C3×C4⋊C4C2×D411yescharacteristic
262C3×C4⋊C4C2×D411yescharacteristic
263C3×C4⋊C4C2×D411yescharacteristic
264C3×C22⋊C4—22no—
265C3×C22⋊C4—22no—
266C3×C22⋊C4—22no—
267C3×C22⋊C4—22no—
268C3×C22⋊C4—22no—
269C3×C22⋊C4—22no—
270C3×C22⋊C4—21no—
271C3×C22⋊C4—21no—
272C3×C22⋊C4—21no—
273C3×C22⋊C4—22no—
274C3×C22⋊C4—22no—
275C3×C22⋊C4—22no—
276C3×C22⋊C4—22no—
277C3×C22⋊C4—21no—
278C3×C22⋊C4—21no—
279C3×C22⋊C4—84no—
280C3×C42—22no—
281C3×C42—22no—
282C3×C42—21no—
283C3×C42—21no—
284C3×C42—21no—
285C3×C42—21no—
286C3×C42—21no—
287C3×C42—22no—
288C3×C42—22no—
289C3×C42C2×D411yescharacteristic
290C3×C42C2×D411yescharacteristic
291C3×C42C2×D411yescharacteristic
292C3×C42C2×D411yescharacteristic
293C2×#41C23×C311yescharacteristic
294C22×Q8—21no—
295C22×Q8C23×C311yescharacteristic
296C22×D4C23×C311yescharacteristic
297C2×M16C23×C312yes—
298C2×M16C23×C312yes—
299C4⋊Q8—21no—
300C4⋊Q8—21no—
301C4⋊Q8—21no—
302C4⋊Q8—21no—
303C4⋊Q8—21no—
304C4⋊Q8C3×D411yescharacteristic
305C4⋊Q8C3×D411yescharacteristic
306#127—21no—
307#126—22no—
308#126—22no—
309#126—22no—
310#126—22no—
311#125C3×D411yescharacteristic
312#125C3×D411yescharacteristic
313#124—22no—
314#124—22no—
315#124—21no—
316#124—21no—
317#124—21no—
318#124—21no—
319#124—21no—
320#124—21no—
321#123—22no—
322#123—22no—
323#123—22no—
324#123—22no—
325#123—22no—
326C22⋊Q8—22no—
327C22⋊Q8—22no—
328C22⋊Q8—21no—
329C22⋊Q8—21no—
330C22⋊Q8—21no—
331C22⋊Q8—21no—
332#121—21no—
333#121—21no—
334#120—22no—
335C4×Q8—21no—
336C4×Q8—21no—
337C4×Q8—21no—
338C4×Q8—21no—
339C4×Q8—21no—
340C4×Q8—21no—
341C4×Q8C3×D411yescharacteristic
342C4×Q8C3×D411yescharacteristic
343C4×D4—22no—
344C4×D4—22no—
345C4×D4—21no—
346C4×D4—21no—
347C4×D4—21no—
348C4×D4—21no—
349C4×D4—21no—
350C4×D4—21no—
351C4×D4C3×D411yescharacteristic
352C4×D4C3×D411yescharacteristic
353#117—22no—
354#117—22no—
355#117—22no—
356#117—21no—
357#117C23×C311yescharacteristic
358#117C23×C311yescharacteristic
359#117C23×C311yescharacteristic
360C2×C4⋊C4—21no—
361C2×C4⋊C4C23×C312yes—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4C3×D412yes—
365C2×C22⋊C4C3×D412yes—
366C2×C42—21no—
367C2×C42C23×C312yes—
368C2×C42C23×C311yescharacteristic
369#107—22no—
370#107—22no—
371#107—22no—
372#106—22no—
373#106—22no—
374#106—22no—
375C4⋊C8—21no—
376C4⋊C8—21no—
377#104—24no—
378#103—22no—
379#102—22no—
380#101—24no—
381#100—24no—
382#97—21no—
383C4×C8—21no—
384#95—21no—
385#95—21no—
386#95—24no—
387C23×C3—22no—
388C23×C3—21no—
389C23×C3—24no—
390C23×C3—42no—
391C23×C3—42no—
392C23×C3C22×D411yescharacteristic
393C3×Q8—21no—
394C3×Q8—21no—
395C3×Q8—41no—
396C3×Q8—41no—
397C3×Q8—42no—
398C3×Q8—81no—
399C3×Q8—81no—
400C3×Q8—82no—
401C3×Q8—82no—
402C3×D4—21no—
403C3×D4—21no—
404C3×D4—42no—
405C3×D4—41no—
406C3×D4—41no—
407C3×D4—42no—
408C3×D4—82no—
409C3×D4—82no—
410C2×C3×C4—21no—
411C2×C3×C4—21no—
412C2×C3×C4—21no—
413C2×C3×C4—21no—
414C2×C3×C4—22no—
415C2×C3×C4—21no—
416C2×C3×C4—21no—
417C2×C3×C4—21no—
418C2×C3×C4—24no—
419C2×C3×C4—41no—
420C2×C3×C4—41no—
421C2×C3×C4—42no—
422C2×C3×C4—42no—
423C2×C3×C4—44no—
424C2×C3×C4—42no—
425C2×C3×C4—42no—
426C2×C3×C4—42no—
427C2×C3×C4—81no—
428C2×C3×C4—81no—
429C2×C3×C4—82no—
430C2×C3×C4—82no—
431C2×C3×C4#12011yescharacteristic
432C2×C3×C4#12011yescharacteristic
433C2×C3×C4#12011yescharacteristic
434C2×C3×C4#12011yescharacteristic
435C2×C3×C4C22×D411yescharacteristic
436C2×C3×C4C22×D411yescharacteristic
437C2×C3×C4C2×#4112yes—
438C2×C3×C42−1+412yes—
439C3×C8—24no—
440C3×C8—42no—
441C24C2×C3×D412yes—
442#41—22no—
443#41—22no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—41no—
451C2×Q8—41no—
452C2×Q8—82no—
453C2×Q8C2×C3×D411yescharacteristic
454C2×D4—21no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4C2×C3×D411yescharacteristic
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4C2×C3×D412yes—
470C22×C4C2×C3×D411yescharacteristic
471C22×C4C2×C3×D411yescharacteristic
472C22×C4C24×C311yescharacteristic
473M16—22no—
474M16—22no—
475M16—22no—
476M16C3×#4114yes—
477C2×C8—22no—
478C2×C8—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4C2×C3×D411yescharacteristic
496C4⋊C4C2×C3×D411yescharacteristic
497C4⋊C4C2×C3×D411yescharacteristic
498C4⋊C4C2×C3×D411yescharacteristic
499C4⋊C4C2×C3×D411yescharacteristic
500C4⋊C4C2×C3×D411yescharacteristic
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—21no—
515C22⋊C4—21no—
516C22⋊C4—84no—
517C42—22no—
518C42—22no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—22no—
525C42—22no—
526C42C2×C3×D411yescharacteristic
527C42C2×C3×D411yescharacteristic
528C42C2×C3×D411yescharacteristic
529C42C2×C3×D411yescharacteristic
530C22×C3—21no—
531C22×C3—21no—
532C22×C3—41no—
533C22×C3—42no—
534C22×C3—41no—
535C22×C3—42no—
536C22×C3—41no—
537C22×C3—42no—
538C22×C3—84no—
539C22×C3C2×#12011yescharacteristic
540C22×C3#54712yes—
541C3×C4—21no—
542C3×C4—21no—
543C3×C4—21no—
544C3×C4—21no—
545C3×C4—42no—
546C3×C4—41no—
547C3×C4—41no—
548C3×C4—41no—
549C3×C4—41no—
550C3×C4—41no—
551C3×C4—41no—
552C3×C4—41no—
553C3×C4—82no—
554C3×C4—82no—
555C3×C4#53211yescharacteristic
556C3×C4#53611yescharacteristic
557C3×C4#54812yes—
558C23—22no—
559C23—21no—
560C23—24no—
561C23—42no—
562C23—42no—
563C23C22×C3×D411yescharacteristic
564Q8—21no—
565Q8—21no—
566Q8—41no—
567Q8—41no—
568Q8—42no—
569Q8—81no—
570Q8—81no—
571Q8—82no—
572Q8—82no—
573D4—21no—
574D4—21no—
575D4—42no—
576D4—41no—
577D4—41no—
578D4—42no—
579D4—82no—
580D4—82no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—24no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—81no—
599C2×C4—81no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4C3×#12011yescharacteristic
603C2×C4C3×#12011yescharacteristic
604C2×C4C3×#12011yescharacteristic
605C2×C4C3×#12011yescharacteristic
606C2×C4C22×C3×D411yescharacteristic
607C2×C4C22×C3×D411yescharacteristic
608C2×C4C2×C3×#4112yes—
609C2×C4C3×2−1+412yes—
610C8—24no—
611C8—42no—
612C2×C3—22no—
613C2×C3—81no—
614C2×C3—82no—
615C2×C3#243211yescharacteristic
616C2×C3#302111yescharacteristic
617C2×C3#302211yescharacteristic
618C22—21no—
619C22—21no—
620C22—41no—
621C22—42no—
622C22—41no—
623C22—42no—
624C22—41no—
625C22—42no—
626C22—84no—
627C22C2×C3×#12011yescharacteristic
628C22C3×#54712yes—
629C4—21no—
630C4—21no—
631C4—21no—
632C4—21no—
633C4—42no—
634C4—41no—
635C4—41no—
636C4—41no—
637C4—41no—
638C4—41no—
639C4—41no—
640C4—41no—
641C4—82no—
642C4—82no—
643C4C3×#53211yescharacteristic
644C4C3×#53611yescharacteristic
645C4C3×#54812yes—
646C3#2569211yescharacteristic, direct product factor, Sylow
647C2—22no—
648C2—82no—
649C2—81no—
650C2C3×#243211yescharacteristic
651C2C3×#302111yescharacteristic
652C2C3×#302211yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#535C22
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
3. C3×#3079C2semidirect
4. C3×#2640C2semidirect
5. C3×#2038C2semidirect
7. C3×#1995C2semidirect
8. C3×#1967C2semidirect
9. C3×#1857C2semidirect
10. C3×#1856C2semidirect
11. C3×#1759C2semidirect
12. #25692C3direct (N × H)
57. #3165C2×C3semidirect
58. #3079C2×C3semidirect
59. #2640C2×C3semidirect
60. #2038C2×C3semidirect
62. #1995C2×C3semidirect
63. #1967C2×C3semidirect
64. #1857C2×C3semidirect
65. #1856C2×C3semidirect
66. #1759C2×C3semidirect
646. C3#25692direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35102
C3×#31652
C3×#30792
C3×#26402
C3×#20382
C3×#20372
C3×#19952
C3×#19672
C3×#18572
C3×#18562
C3×#17592
#256923