Finite Group Databasebeta

← C3×#25582 #10644189

Subgroups — C3×#25582 #10644189

Order 768.

1190 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#553C211yesmaximal, characteristic
2C2×C3×#491C211yesmaximal, characteristic, complemented
3C2×C3×#461C211yesmaximal, characteristic, complemented
4C3×#2542C211yesmaximal, characteristic, complemented
5C3×#2000C212yesmaximal, complemented
6C3×#2000C212yesmaximal, complemented
7C3×#1989C211yesmaximal, characteristic, complemented
8C3×#1955C211yesmaximal, characteristic, complemented
9C3×#1873C212yesmaximal, complemented
10C3×#1872C212yesmaximal, complemented
11C3×#1762C211yesmaximal, characteristic, complemented
12#25582C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2212yes—
14C3×#553—24no—
15C3×#553—24no—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#126—21no—
18C2×C3×#125C2211yescharacteristic
19C2×C3×#123—21no—
20C2×C3×C22⋊Q8C2212yes—
21C2×C3×C22⋊Q8C2212yes—
22C2×C3×#121—21no—
23C2×C3×#121C2211yescharacteristic
24C2×C3×#121C2211yescharacteristic
25C2×C3×C4×D4—21no—
26C2×C3×C4×D4C2211yescharacteristic
27C2×C3×C4×D4C2211yescharacteristic
28C2×C3×#117C2211yescharacteristic
29C22×C3×C4⋊C4C2211yescharacteristic
30C3×#491—24no—
31C3×#461—24no—
32C2×C3×C4⋊C8C2211yescharacteristic
33C2×C3×#103C2212yes—
34C2×C3×#103C2212yes—
35C2×C3×#103C2212yes—
36C2×C3×#102C2212yes—
37C2×C3×C22⋊C8C2212yes—
38C2×C3×C22⋊C8C2212yes—
39C2×C3×#97C2211yescharacteristic
40C3×#399—21no—
41C3×#399—21no—
42C3×#390—21no—
43C3×#384—21no—
44C3×C4×C4⋊C4C2211yescharacteristic
45C3×#340C2212yes—
46C3×#340C2212yes—
47C3×#340C2212yes—
48C2×#553C2×C311yescharacteristic
49C2×#491C2×C311yescharacteristic, complemented
50C2×#461C2×C311yescharacteristic, complemented
51#2542C2×C311yescharacteristic, complemented
52#2000C2×C312yescomplemented
53#2000C2×C312yescomplemented
54#1989C2×C311yescharacteristic, complemented
55#1955C2×C311yescharacteristic, complemented
56#1873C2×C312yescomplemented
57#1872C2×C312yescomplemented
58#1762C2×C311yescharacteristic, complemented
59C22×C3×Q8C2312yes—
60C22×C3×D4—21no—
61C22×C3×D4—21no—
62C22×C3×D4D411yescharacteristic
63C22×C3×D4C2311yescharacteristic
64C23×C3×C4—21no—
65C23×C3×C4—21no—
66C23×C3×C4C2311yescharacteristic
67C23×C3×C4C2311yescharacteristic
68C2×C3×SD16—24no—
69C2×C3×SD16—42no—
70C2×C3×SD16—42no—
71C22×C3×C8C2312yes—
72C22×C3×C8C2312yes—
73C3×C4⋊Q8—22no—
74C3×C4⋊Q8—22no—
75C3×#126—28no—
76C3×#125—22no—
77C3×#125D414yes—
78C3×#123—28no—
79C3×C22⋊Q8—24no—
80C3×C22⋊Q8—24no—
81C3×C22⋊Q8—42no—
82C3×C22⋊Q8—42no—
83C3×#121—28no—
84C3×#121—22no—
85C3×#121—22no—
86C3×#121—22no—
87C3×#121—22no—
88C3×C4×D4—28no—
89C3×C4×D4—22no—
90C3×C4×D4—22no—
91C3×C4×D4—22no—
92C3×C4×D4D414yes—
93C3×#117—22no—
94C3×#117—22no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—22no—
99C2×C3×C4⋊C4—22no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—22no—
102C2×C3×C4⋊C4—22no—
103C2×C3×C4⋊C4—21no—
104C2×C3×C4⋊C4D411yescharacteristic
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C4⋊C4C2312yes—
109C2×C3×C22⋊C4—21no—
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4—21no—
113C2×C3×C22⋊C4—22no—
114C2×C3×C22⋊C4—22no—
115C2×C3×C22⋊C4D411yescharacteristic
116C2×C3×C22⋊C4D411yescharacteristic
117C2×C3×C42—21no—
118C2×C3×C42—21no—
119C2×C3×C42C2311yescharacteristic
120C3×C4⋊C8—22no—
121C3×#103—24no—
122C3×#103—24no—
123C3×#103—24no—
124C3×#102—24no—
125C3×C22⋊C8—42no—
126C3×C22⋊C8—42no—
127C3×#97—22no—
128C3×#95—21no—
129C22×SD16C22×C312yes—
130#553—24no—
131#553—24no—
132C2×C4⋊Q8C22×C311yescharacteristic
133C2×#126—21no—
134C2×#125C22×C311yescharacteristic
135C2×#123—21no—
136C2×C22⋊Q8C22×C312yes—
137C2×C22⋊Q8C22×C312yes—
138C2×#121—21no—
139C2×#121C22×C311yescharacteristic
140C2×#121C22×C311yescharacteristic
141C2×C4×D4—21no—
142C2×C4×D4C22×C311yescharacteristic
143C2×C4×D4C22×C311yescharacteristic
144C2×#117C22×C311yescharacteristic
145C22×C4⋊C4C22×C311yescharacteristic
146#491—24no—
147#461—24no—
148C2×C4⋊C8C22×C311yescharacteristic
149C2×#103C22×C312yes—
150C2×#103C22×C312yes—
151C2×#103C22×C312yes—
152C2×#102C22×C312yes—
153C2×C22⋊C8C22×C312yes—
154C2×C22⋊C8C22×C312yes—
155C2×#97C22×C311yescharacteristic
156#399—21no—
157#399—21no—
158#390—21no—
159#384—21no—
160C4×C4⋊C4C22×C311yescharacteristic
161#340C22×C312yes—
162#340C22×C312yes—
163#340C22×C312yes—
164C24×C3—21no—
165C24×C3C2×D411yescharacteristic
166C24×C3C2×D411yescharacteristic
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×D4—24no—
174C2×C3×D4—22no—
175C2×C3×D4—24no—
176C2×C3×D4—22no—
177C2×C3×D4—21no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—41no—
181C2×C3×D4—42no—
182C2×C3×D4—42no—
183C2×C3×D4—82no—
184C2×C3×D4C2×D412yes—
185C22×C3×C4—22no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—24no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—22no—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—22no—
198C22×C3×C4—24no—
199C22×C3×C4—22no—
200C22×C3×C4—82no—
201C22×C3×C4—82no—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C3×SD16—48no—
208C2×C3×C8—22no—
209C2×C3×C8—22no—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C2×C3×C8—22no—
213C2×C3×C8—22no—
214C3×C4⋊C4—24no—
215C3×C4⋊C4—24no—
216C3×C4⋊C4—24no—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—44no—
223C3×C4⋊C4—44no—
224C3×C4⋊C4—42no—
225C3×C4⋊C4—42no—
226C3×C4⋊C4C2×D412yes—
227C3×C4⋊C4C2×D412yes—
228C3×C4⋊C4C2×D412yes—
229C3×C4⋊C4#4114yes—
230C3×C22⋊C4—24no—
231C3×C22⋊C4—24no—
232C3×C22⋊C4—24no—
233C3×C22⋊C4—24no—
234C3×C22⋊C4—22no—
235C3×C22⋊C4—22no—
236C3×C22⋊C4—44no—
237C3×C22⋊C4—44no—
238C3×C42—24no—
239C3×C42—42no—
240C3×C42C2×D412yes—
241C3×C42C2×D412yes—
242C22×Q8C23×C312yes—
243C22×D4—21no—
244C22×D4—21no—
245C22×D4C3×D411yescharacteristic
246C22×D4C23×C311yescharacteristic
247C23×C4—21no—
248C23×C4—21no—
249C23×C4C23×C311yescharacteristic
250C23×C4C23×C311yescharacteristic
251C2×SD16—24no—
252C2×SD16—42no—
253C2×SD16—42no—
254C22×C8C23×C312yes—
255C22×C8C23×C312yes—
256C4⋊Q8—22no—
257C4⋊Q8—22no—
258#126—28no—
259#125—22no—
260#125C3×D414yes—
261#123—28no—
262C22⋊Q8—24no—
263C22⋊Q8—24no—
264C22⋊Q8—42no—
265C22⋊Q8—42no—
266#121—28no—
267#121—22no—
268#121—22no—
269#121—22no—
270#121—22no—
271C4×D4—28no—
272C4×D4—22no—
273C4×D4—22no—
274C4×D4—22no—
275C4×D4C3×D414yes—
276#117—22no—
277#117—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4C3×D411yescharacteristic
288C2×C4⋊C4C23×C311yescharacteristic
289C2×C4⋊C4C23×C311yescharacteristic
290C2×C4⋊C4C23×C311yescharacteristic
291C2×C4⋊C4C23×C312yes—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4C3×D411yescharacteristic
299C2×C22⋊C4C3×D411yescharacteristic
300C2×C42—21no—
301C2×C42—21no—
302C2×C42C23×C311yescharacteristic
303C4⋊C8—22no—
304#103—24no—
305#103—24no—
306#103—24no—
307#102—24no—
308C22⋊C8—42no—
309C22⋊C8—42no—
310#97—22no—
311#95—21no—
312C23×C3—21no—
313C23×C3—22no—
314C23×C3—21no—
315C23×C3—22no—
316C23×C3—21no—
317C23×C3—22no—
318C23×C3—22no—
319C23×C3—21no—
320C23×C3—21no—
321C23×C3—22no—
322C23×C3—42no—
323C23×C3—41no—
324C23×C3—41no—
325C23×C3—42no—
326C23×C3#13712yes—
327C23×C3C22×D411yescharacteristic
328C3×Q8—42no—
329C3×Q8—42no—
330C3×Q8—42no—
331C3×Q8—42no—
332C3×D4—22no—
333C3×D4—22no—
334C3×D4—44no—
335C3×D4—48no—
336C3×D4—42no—
337C3×D4—84no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—22no—
341C2×C3×C4—21no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—21no—
345C2×C3×C4—22no—
346C2×C3×C4—24no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—41no—
350C2×C3×C4—42no—
351C2×C3×C4—41no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—84no—
365C2×C3×C4—84no—
366C2×C3×C4#12012yes—
367C2×C3×C4#12012yes—
368C2×C3×C4C22×D411yescharacteristic
369C2×C3×C4C22×D411yescharacteristic
370C2×C3×C4C2×#4112yes—
371C2×C3×C4H(5,𝔽2)12yes—
372C3×C8—42no—
373C3×C8—42no—
374C24—21no—
375C24C2×C3×D411yescharacteristic
376C24C2×C3×D411yescharacteristic
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×D4—24no—
384C2×D4—22no—
385C2×D4—24no—
386C2×D4—22no—
387C2×D4—21no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—41no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4—82no—
394C2×D4C2×C3×D412yes—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—24no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—82no—
411C22×C4—82no—
412C22×C4C2×C3×D411yescharacteristic
413C22×C4C2×C3×D411yescharacteristic
414C22×C4C2×C3×D411yescharacteristic
415C22×C4C2×C3×D411yescharacteristic
416C22×C4C24×C311yescharacteristic
417SD16—48no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—22no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4C2×C3×D412yes—
437C4⋊C4C2×C3×D412yes—
438C4⋊C4C2×C3×D412yes—
439C4⋊C4C3×#4114yes—
440C22⋊C4—24no—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C42—24no—
449C42—42no—
450C42C2×C3×D412yes—
451C42C2×C3×D412yes—
452C22×C3—22no—
453C22×C3—24no—
454C22×C3—41no—
455C22×C3—41no—
456C22×C3—42no—
457C22×C3—41no—
458C22×C3—42no—
459C22×C3—41no—
460C22×C3—42no—
461C22×C3—42no—
462C22×C3—41no—
463C22×C3—42no—
464C22×C3—42no—
465C22×C3—82no—
466C22×C3—82no—
467C22×C3C2×#12011yescharacteristic
468C22×C3#54612yes—
469C22×C3C2×#13711yescharacteristic
470C22×C3#57511yescharacteristic
471C22×C3#57712yes—
472C3×C4—22no—
473C3×C4—22no—
474C3×C4—41no—
475C3×C4—41no—
476C3×C4—42no—
477C3×C4—42no—
478C3×C4—42no—
479C3×C4—44no—
480C3×C4—82no—
481C3×C4#53211yescharacteristic
482C3×C4#53511yescharacteristic
483C3×C4#55012yes—
484C23—21no—
485C23—22no—
486C23—21no—
487C23—22no—
488C23—21no—
489C23—21no—
490C23—22no—
491C23—21no—
492C23—22no—
493C23—22no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—42no—
498C23C3×#13712yes—
499C23C22×C3×D411yescharacteristic
500Q8—42no—
501Q8—42no—
502Q8—42no—
503Q8—42no—
504D4—22no—
505D4—22no—
506D4—44no—
507D4—48no—
508D4—42no—
509D4—84no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—84no—
537C2×C4—84no—
538C2×C4C3×#12012yes—
539C2×C4C3×#12012yes—
540C2×C4C22×C3×D411yescharacteristic
541C2×C4C22×C3×D411yescharacteristic
542C2×C4C2×C3×#4112yes—
543C2×C4C3×H(5,𝔽2)12yes—
544C8—42no—
545C8—42no—
546C2×C3—42no—
547C2×C3—81no—
548C2×C3—82no—
549C2×C3#243111yescharacteristic
550C2×C3#300711yescharacteristic
551C2×C3#300911yescharacteristic
552C2×C3#330712yes—
553C2×C3#332012yes—
554C22—22no—
555C22—24no—
556C22—41no—
557C22—41no—
558C22—42no—
559C22—41no—
560C22—42no—
561C22—41no—
562C22—42no—
563C22—42no—
564C22—41no—
565C22—42no—
566C22—42no—
567C22—82no—
568C22—82no—
569C22C2×C3×#12011yescharacteristic
570C22C3×#54612yes—
571C22C2×C3×#13711yescharacteristic
572C22C3×#57511yescharacteristic
573C22C3×#57712yes—
574C4—22no—
575C4—22no—
576C4—41no—
577C4—41no—
578C4—42no—
579C4—42no—
580C4—42no—
581C4—44no—
582C4—82no—
583C4C3×#53211yescharacteristic
584C4C3×#53511yescharacteristic
585C4C3×#55012yes—
586C3#2558211yescharacteristic, direct product factor, Sylow
587C2—42no—
588C2—81no—
589C2—82no—
590C2C3×#243111yescharacteristic
591C2C3×#300711yescharacteristic
592C2C3×#300911yescharacteristic
593C2C3×#330712yes—
594C2C3×#332012yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#553C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#491C2semidirect
3. C2×C3×#461C2semidirect
4. C3×#2542C2semidirect
5. C3×#2000C2semidirect
6. C3×#2000C2semidirect
7. C3×#1989C2semidirect
8. C3×#1955C2semidirect
9. C3×#1873C2semidirect
10. C3×#1872C2semidirect
11. C3×#1762C2semidirect
12. #25582C3direct (N × H)
49. C2×#491C2×C3semidirect
50. C2×#461C2×C3semidirect
51. #2542C2×C3semidirect
52. #2000C2×C3semidirect
53. #2000C2×C3semidirect
54. #1989C2×C3semidirect
55. #1955C2×C3semidirect
56. #1873C2×C3semidirect
57. #1872C2×C3semidirect
58. #1762C2×C3semidirect
586. C3#25582direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5532
C2×C3×#4912
C2×C3×#4612
C3×#25422
C3×#20002
C3×#19892
C3×#19552
C3×#18732
C3×#18722
C3×#17622
#255823