Finite Group Databasebeta

← C3×#25604 #10644211

Subgroups — C3×#25604 #10644211

Order 768.

1178 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#529C211yesmaximal, characteristic
2C2×C3×#497C211yesmaximal, characteristic, complemented
3C2×C3×#491C211yesmaximal, characteristic, complemented
4C3×#2857C211yesmaximal, characteristic, complemented
5C3×#2001C212yesmaximal, complemented
6C3×#2000C212yesmaximal
7C3×#1992C211yesmaximal, characteristic, complemented
8C3×#1959C211yesmaximal, characteristic, complemented
9C3×#1878C212yesmaximal, complemented
10C3×#1877C212yesmaximal, complemented
11C3×#1749C211yesmaximal, characteristic, complemented
12#25604C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2212yes—
14C22×C3×SD16C2212yes—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C3×#529—24no—
17C3×#529—24no—
18C2×C3×#126—21no—
19C2×C3×#125C2211yescharacteristic
20C2×C3×#124C2211yescharacteristic
21C2×C3×#123—21no—
22C2×C3×C22⋊Q8C2211yescharacteristic
23C2×C3×C22⋊Q8C2212yes—
24C2×C3×#121C2211yescharacteristic
25C2×C3×C4×Q8C2211yescharacteristic
26C2×C3×C4×D4—21no—
27C2×C3×C4×D4C2211yescharacteristic
28C2×C3×#117—21no—
29C22×C3×C42C2211yescharacteristic
30C3×#497—24no—
31C3×#491—24no—
32C2×C3×#103C2212yes—
33C2×C3×#103C2212yes—
34C2×C3×#103C2212yes—
35C2×C3×#102C2212yes—
36C2×C3×C22⋊C8C2212yes—
37C2×C3×#97C2211yescharacteristic
38C2×C3×#97C2211yescharacteristic
39C3×#391C2212yes—
40C3×#389—21no—
41C3×#389—21no—
42C3×#384C2211yescharacteristic
43C3×#380—21no—
44C3×#380—21no—
45C3×#340C2212yes—
46C3×#340C2212yes—
47C3×#336C2212yes—
48C2×#529C2×C311yescharacteristic
49C2×#497C2×C311yescharacteristic, complemented
50C2×#491C2×C311yescharacteristic, complemented
51#2857C2×C311yescharacteristic, complemented
52#2001C2×C312yescomplemented
53#2000C2×C312yes—
54#1992C2×C311yescharacteristic, complemented
55#1959C2×C311yescharacteristic, complemented
56#1878C2×C312yescomplemented
57#1877C2×C312yescomplemented
58#1749C2×C311yescharacteristic, complemented
59C22×C3×Q8C2312yes—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4—21no—
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4—21no—
64C23×C3×C4D411yescharacteristic
65C23×C3×C4D411yescharacteristic
66C23×C3×C4C2311yescharacteristic
67C2×C3×Q16—24no—
68C2×C3×Q16—24no—
69C2×C3×Q16—42no—
70C2×C3×SD16—24no—
71C2×C3×SD16—24no—
72C2×C3×SD16—42no—
73C22×C3×C8C2312yes—
74C22×C3×C8C2312yes—
75C3×C4⋊Q8—22no—
76C3×C4⋊Q8—22no—
77C3×#126—28no—
78C3×#125—22no—
79C3×#125D414yes—
80C3×#124—22no—
81C3×#124D414yes—
82C3×#123—28no—
83C3×C22⋊Q8—22no—
84C3×C22⋊Q8—22no—
85C3×C22⋊Q8—42no—
86C3×C22⋊Q8—42no—
87C3×#121—22no—
88C3×#121—22no—
89C3×C4×Q8—22no—
90C3×C4×Q8—22no—
91C3×C4×D4—28no—
92C3×C4×D4—22no—
93C3×C4×D4—22no—
94C3×#117—28no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4—22no—
104C2×C3×C4⋊C4C2312yes—
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—21no—
109C2×C3×C22⋊C4—21no—
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4—22no—
113C2×C3×C42—21no—
114C2×C3×C42—22no—
115C2×C3×C42—22no—
116C2×C3×C42D411yescharacteristic
117C2×C3×C42D411yescharacteristic
118C2×C3×C42C2311yescharacteristic
119C2×C3×C42C2311yescharacteristic
120C3×#103—24no—
121C3×#103—24no—
122C3×#103—42no—
123C3×#102—42no—
124C3×C22⋊C8—24no—
125C3×#97—22no—
126C3×#97—22no—
127C3×#95—22no—
128C3×#95—22no—
129C22×Q16C22×C312yes—
130C22×SD16C22×C312yes—
131C2×C4⋊Q8C22×C311yescharacteristic
132#529—24no—
133#529—24no—
134C2×#126—21no—
135C2×#125C22×C311yescharacteristic
136C2×#124C22×C311yescharacteristic
137C2×#123—21no—
138C2×C22⋊Q8C22×C311yescharacteristic
139C2×C22⋊Q8C22×C312yes—
140C2×#121C22×C311yescharacteristic
141C2×C4×Q8C22×C311yescharacteristic
142C2×C4×D4—21no—
143C2×C4×D4C22×C311yescharacteristic
144C2×#117—21no—
145C22×C42C22×C311yescharacteristic
146#497—24no—
147#491—24no—
148C2×#103C22×C312yes—
149C2×#103C22×C312yes—
150C2×#103C22×C312yes—
151C2×#102C22×C312yes—
152C2×C22⋊C8C22×C312yes—
153C2×#97C22×C311yescharacteristic
154C2×#97C22×C311yescharacteristic
155#391C22×C312yes—
156#389—21no—
157#389—21no—
158#384C22×C311yescharacteristic
159#380—21no—
160#380—21no—
161#340C22×C312yes—
162#340C22×C312yes—
163#336C22×C312yes—
164C24×C3—21no—
165C24×C3C2×D411yescharacteristic
166C2×C3×Q8—21no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—22no—
174C2×C3×Q8—22no—
175C2×C3×Q8C2×D412yes—
176C2×C3×D4—24no—
177C2×C3×D4—21no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—42no—
181C2×C3×D4—42no—
182C2×C3×D4C2×D412yes—
183C22×C3×C4—24no—
184C22×C3×C4—22no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—22no—
194C22×C3×C4—22no—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—41no—
198C22×C3×C4—41no—
199C22×C3×C4—42no—
200C22×C3×C4—42no—
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C3×Q16—48no—
208C3×SD16—48no—
209C2×C3×C8—22no—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C2×C3×C8—22no—
213C2×C3×C8—22no—
214C2×C3×C8#4114yes—
215C3×C4⋊C4—24no—
216C3×C4⋊C4—24no—
217C3×C4⋊C4—24no—
218C3×C4⋊C4—24no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—42no—
224C3×C4⋊C4—42no—
225C3×C4⋊C4—44no—
226C3×C4⋊C4—42no—
227C3×C4⋊C4—42no—
228C3×C4⋊C4C2×D412yes—
229C3×C4⋊C4C2×D412yes—
230C3×C22⋊C4—24no—
231C3×C22⋊C4—24no—
232C3×C22⋊C4—24no—
233C3×C22⋊C4—24no—
234C3×C22⋊C4—24no—
235C3×C22⋊C4—44no—
236C3×C42—24no—
237C3×C42—22no—
238C3×C42—22no—
239C3×C42—21no—
240C3×C42—21no—
241C3×C42C2×D412yes—
242C3×C42C2×D412yes—
243C22×Q8C23×C312yes—
244C22×Q8C23×C311yescharacteristic
245C22×D4—21no—
246C22×D4C23×C311yescharacteristic
247C23×C4—21no—
248C23×C4C3×D411yescharacteristic
249C23×C4C3×D411yescharacteristic
250C23×C4C23×C311yescharacteristic
251C2×Q16—24no—
252C2×Q16—24no—
253C2×Q16—42no—
254C2×SD16—24no—
255C2×SD16—24no—
256C2×SD16—42no—
257C22×C8C23×C312yes—
258C22×C8C23×C312yes—
259C4⋊Q8—22no—
260C4⋊Q8—22no—
261#126—28no—
262#125—22no—
263#125C3×D414yes—
264#124—22no—
265#124C3×D414yes—
266#123—28no—
267C22⋊Q8—22no—
268C22⋊Q8—22no—
269C22⋊Q8—42no—
270C22⋊Q8—42no—
271#121—22no—
272#121—22no—
273C4×Q8—22no—
274C4×Q8—22no—
275C4×D4—28no—
276C4×D4—22no—
277C4×D4—22no—
278#117—28no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4C23×C312yes—
289C2×C4⋊C4C23×C311yescharacteristic
290C2×C4⋊C4C23×C311yescharacteristic
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—22no—
297C2×C42—21no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42C3×D411yescharacteristic
301C2×C42C3×D411yescharacteristic
302C2×C42C23×C311yescharacteristic
303C2×C42C23×C311yescharacteristic
304#103—24no—
305#103—24no—
306#103—42no—
307#102—42no—
308C22⋊C8—24no—
309#97—22no—
310#97—22no—
311#95—22no—
312#95—22no—
313C23×C3—21no—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—22no—
320C23×C3—41no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3—41no—
324C23×C3—41no—
325C23×C3—41no—
326C23×C3#13712yes—
327C23×C3C22×D411yescharacteristic
328C3×Q8—24no—
329C3×Q8—42no—
330C3×Q8—42no—
331C3×Q8—42no—
332C3×Q8—42no—
333C3×Q8—42no—
334C3×D4—24no—
335C3×D4—48no—
336C3×D4—42no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—44no—
353C2×C3×C4—44no—
354C2×C3×C4—48no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4#12012yes—
370C2×C3×C4#12012yes—
371C2×C3×C4#13712yes—
372C2×C3×C4C22×D411yescharacteristic
373C2×C3×C4C22×D411yescharacteristic
374C2×C3×C4C2×#4112yes—
375C2×C3×C4H(5,𝔽2)12yes—
376C3×C8—24no—
377C3×C8—42no—
378C24—21no—
379C24C2×C3×D411yescharacteristic
380C2×Q8—21no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8C2×C3×D412yes—
390C2×D4—24no—
391C2×D4—21no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4C2×C3×D412yes—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4C2×C3×D411yescharacteristic
416C22×C4C2×C3×D411yescharacteristic
417C22×C4C2×C3×D411yescharacteristic
418C22×C4C2×C3×D411yescharacteristic
419C22×C4C2×C3×D411yescharacteristic
420C22×C4C24×C311yescharacteristic
421Q16—48no—
422SD16—48no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8C3×#4114yes—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—44no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4C2×C3×D412yes—
443C4⋊C4C2×C3×D412yes—
444C22⋊C4—24no—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—44no—
450C42—24no—
451C42—22no—
452C42—22no—
453C42—21no—
454C42—21no—
455C42C2×C3×D412yes—
456C42C2×C3×D412yes—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—22no—
460C22×C3—41no—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3—41no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3—81no—
468C22×C3—81no—
469C22×C3—81no—
470C22×C3—81no—
471C22×C3C2×#12011yescharacteristic
472C22×C3#53211yescharacteristic
473C22×C3#53511yescharacteristic
474C22×C3C2×#13711yescharacteristic
475C22×C3C2×#13711yescharacteristic
476C22×C3#57511yescharacteristic
477C22×C3#57511yescharacteristic
478C3×C4—22no—
479C3×C4—22no—
480C3×C4—22no—
481C3×C4—41no—
482C3×C4—42no—
483C3×C4—41no—
484C3×C4—42no—
485C3×C4—42no—
486C3×C4—82no—
487C3×C4—82no—
488C3×C4#54612yes—
489C3×C4#55012yes—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—22no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23C3×#13712yes—
504C23C22×C3×D411yescharacteristic
505Q8—24no—
506Q8—42no—
507Q8—42no—
508Q8—42no—
509Q8—42no—
510Q8—42no—
511D4—24no—
512D4—48no—
513D4—42no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—48no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4C3×#12012yes—
547C2×C4C3×#12012yes—
548C2×C4C3×#13712yes—
549C2×C4C22×C3×D411yescharacteristic
550C2×C4C22×C3×D411yescharacteristic
551C2×C4C2×C3×#4112yes—
552C2×C4C3×H(5,𝔽2)12yes—
553C8—24no—
554C8—42no—
555C2×C3—42no—
556C2×C3—82no—
557C2×C3#243111yescharacteristic
558C2×C3#300711yescharacteristic
559C2×C3#300711yescharacteristic
560C2×C3#317211yescharacteristic
561C2×C3#318111yescharacteristic
562C2×C3#321811yescharacteristic
563C2×C3#321911yescharacteristic
564C22—22no—
565C22—22no—
566C22—22no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22C2×C3×#12011yescharacteristic
579C22C3×#53211yescharacteristic
580C22C3×#53511yescharacteristic
581C22C2×C3×#13711yescharacteristic
582C22C2×C3×#13711yescharacteristic
583C22C3×#57511yescharacteristic
584C22C3×#57511yescharacteristic
585C4—22no—
586C4—22no—
587C4—22no—
588C4—41no—
589C4—42no—
590C4—41no—
591C4—42no—
592C4—42no—
593C4—82no—
594C4—82no—
595C4C3×#54612yes—
596C4C3×#55012yes—
597C3#2560411yescharacteristic, direct product factor, Sylow
598C2—42no—
599C2—82no—
600C2C3×#243111yescharacteristic
601C2C3×#300711yescharacteristic
602C2C3×#300711yescharacteristic
603C2C3×#317211yescharacteristic
604C2C3×#318111yescharacteristic
605C2C3×#321811yescharacteristic
606C2C3×#321911yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#497C2semidirect
3. C2×C3×#491C2semidirect
4. C3×#2857C2semidirect
5. C3×#2001C2semidirect
7. C3×#1992C2semidirect
8. C3×#1959C2semidirect
9. C3×#1878C2semidirect
10. C3×#1877C2semidirect
11. C3×#1749C2semidirect
12. #25604C3direct (N × H)
49. C2×#497C2×C3semidirect
50. C2×#491C2×C3semidirect
51. #2857C2×C3semidirect
52. #2001C2×C3semidirect
54. #1992C2×C3semidirect
55. #1959C2×C3semidirect
56. #1878C2×C3semidirect
57. #1877C2×C3semidirect
58. #1749C2×C3semidirect
597. C3#25604direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5292
C2×C3×#4972
C2×C3×#4912
C3×#28572
C3×#20012
C3×#20002
C3×#19922
C3×#19592
C3×#18782
C3×#18772
C3×#17492
#256043