Finite Group Databasebeta

← C3×#29607 #10648214

Subgroups — C3×#29607 #10648214

Order 768.

1390 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#537C211yesmaximal, characteristic, complemented
2C2×C3×#480C211yesmaximal, characteristic, complemented
3C2×C3×#449C211yesmaximal, characteristic, complemented
4C3×#2615C211yesmaximal, characteristic, complemented
5C3×#2036C212yesmaximal, complemented
6C3×#2033C212yesmaximal, complemented
7C3×#2000C212yesmaximal, complemented
8C3×#1999C212yesmaximal, complemented
9C3×#1824C211yesmaximal, characteristic, complemented
10C3×#1788C211yesmaximal, characteristic, complemented
11C3×#1780C211yesmaximal, characteristic, complemented
12#29607C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C22×C3×#41C2211yescharacteristic
15C22×C3×SD16C2211yescharacteristic
16C22×C3×D8C2211yescharacteristic
17C3×#537—24no—
18C3×#537—24no—
19C2×C3×C22⋊Q8—21no—
20C2×C3×C22⋊Q8C2212yes—
21C2×C3×#121—21no—
22C2×C3×#121—22no—
23C2×C3×#121C2212yes—
24C2×C3×#121C2211yescharacteristic
25C2×C3×#121C2212yes—
26C22×C3×C4⋊C4C2211yescharacteristic
27C3×#480—24no—
28C3×#449—24no—
29C2×C3×#107C2212yes—
30C2×C3×#103C2211yescharacteristic
31C2×C3×#103C2212yes—
32C2×C3×#102C2212yes—
33C2×C3×#102C2211yescharacteristic
34C2×C3×#102C2212yes—
35C2×C3×C22⋊C8C2211yescharacteristic
36C2×C3×C22⋊C8C2211yescharacteristic
37C2×C3×C22⋊C8C2212yes—
38C2×C3×C22⋊C8C2212yes—
39C3×#393—21no—
40C3×#392—22no—
41C3×#392—21no—
42C3×#380C2212yes—
43C3×#380C2211yescharacteristic
44C3×#340C2212yes—
45C3×#340C2212yes—
46C2×#537C2×C311yescharacteristic, complemented
47C2×#480C2×C311yescharacteristic, complemented
48C2×#449C2×C311yescharacteristic, complemented
49#2615C2×C311yescharacteristic, complemented
50#2036C2×C312yescomplemented
51#2033C2×C312yescomplemented
52#2000C2×C312yescomplemented
53#1999C2×C312yescomplemented
54#1824C2×C311yescharacteristic, complemented
55#1788C2×C311yescharacteristic, complemented
56#1780C2×C311yescharacteristic, complemented
57C2×C3×#41—22no—
58C2×C3×#41—22no—
59C2×C3×#41—22no—
60C2×C3×#41—42no—
61C2×C3×#41—42no—
62C2×C3×#41—42no—
63C2×C3×#41—42no—
64C2×C3×#41D414yescomplemented
65C22×C3×Q8D411yescharacteristic
66C22×C3×Q8C2311yescharacteristic
67C22×C3×D4—21no—
68C22×C3×D4—22no—
69C22×C3×D4—22no—
70C22×C3×D4—21no—
71C22×C3×D4D412yes—
72C22×C3×D4D411yescharacteristic
73C22×C3×D4C2311yescharacteristic
74C22×C3×D4C2311yescharacteristic
75C23×C3×C4—21no—
76C23×C3×C4—21no—
77C23×C3×C4C2312yes—
78C23×C3×C4C2311yescharacteristic
79C2×C3×SD16—22no—
80C2×C3×SD16—42no—
81C2×C3×D8—22no—
82C2×C3×D8—42no—
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2312yes—
86C3×C22⋊Q8—28no—
87C3×C22⋊Q8—24no—
88C3×C22⋊Q8—24no—
89C3×#121—216no—
90C3×#121—28no—
91C3×#121—24no—
92C3×#121—24no—
93C3×#121—22no—
94C3×#121—22no—
95C3×#121—42no—
96C3×#121—42no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—22no—
99C2×C3×C4⋊C4—22no—
100C2×C3×C4⋊C4D414yescomplemented
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C4⋊C4C2312yes—
103C2×C3×C4⋊C4C2312yes—
104C2×C3×C22⋊C4—22no—
105C2×C3×C22⋊C4—22no—
106C2×C3×C22⋊C4—22no—
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—22no—
109C2×C3×C22⋊C4—21no—
110C3×#107—24no—
111C3×#103—22no—
112C3×#103—42no—
113C3×#102—22no—
114C3×#102—24no—
115C3×#102—42no—
116C3×C22⋊C8—22no—
117C3×C22⋊C8—22no—
118C3×C22⋊C8—42no—
119C3×C22⋊C8—42no—
120C3×#95—22no—
121C3×#95—21no—
122C22×#41C22×C311yescharacteristic
123C22×#41C22×C311yescharacteristic
124C22×SD16C22×C311yescharacteristic
125C22×D8C22×C311yescharacteristic
126#537—24no—
127#537—24no—
128C2×C22⋊Q8—21no—
129C2×C22⋊Q8C22×C312yes—
130C2×#121—21no—
131C2×#121—22no—
132C2×#121C22×C311yescharacteristic
133C2×#121C22×C312yes—
134C2×#121C22×C312yes—
135C22×C4⋊C4C22×C311yescharacteristic
136#480—24no—
137#449—24no—
138C2×#107C22×C312yes—
139C2×#103C22×C311yescharacteristic
140C2×#103C22×C312yes—
141C2×#102C22×C312yes—
142C2×#102C22×C311yescharacteristic
143C2×#102C22×C312yes—
144C2×C22⋊C8C22×C311yescharacteristic
145C2×C22⋊C8C22×C311yescharacteristic
146C2×C22⋊C8C22×C312yes—
147C2×C22⋊C8C22×C312yes—
148#393—21no—
149#392—22no—
150#392—21no—
151#380C22×C311yescharacteristic
152#380C22×C312yes—
153#340C22×C312yes—
154#340C22×C312yes—
155C24×C3—21no—
156C24×C3—21no—
157C24×C3C2×D412yes—
158C24×C3C2×D411yescharacteristic
159C3×#41—44no—
160C3×#41—44no—
161C3×#41—44no—
162C3×#41—44no—
163C3×#41—48no—
164C3×#41—48no—
165C2×C3×Q8—22no—
166C2×C3×Q8—21no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8—42no—
170C2×C3×Q8C2×D412yes—
171C2×C3×D4—24no—
172C2×C3×D4—24no—
173C2×C3×D4—28no—
174C2×C3×D4—22no—
175C2×C3×D4—21no—
176C2×C3×D4—21no—
177C2×C3×D4—21no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—42no—
183C2×C3×D4—42no—
184C2×C3×D4—42no—
185C2×C3×D4—44no—
186C2×C3×D4—44no—
187C2×C3×D4—42no—
188C2×C3×D4—82no—
189C2×C3×D4—82no—
190C2×C3×D4—82no—
191C2×C3×D4—82no—
192C2×C3×D4—82no—
193C2×C3×D4C2×D412yes—
194C22×C3×C4—22no—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—21no—
198C22×C3×C4—24no—
199C22×C3×C4—24no—
200C22×C3×C4—22no—
201C22×C3×C4—22no—
202C22×C3×C4—22no—
203C22×C3×C4—22no—
204C22×C3×C4—22no—
205C22×C3×C4—22no—
206C22×C3×C4—21no—
207C22×C3×C4—22no—
208C22×C3×C4—44no—
209C22×C3×C4—82no—
210C22×C3×C4C2×D412yes—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D412yes—
213C22×C3×C4C2×D412yes—
214C22×C3×C4C2411yescharacteristic
215C3×SD16—44no—
216C3×D8—44no—
217C2×C3×C8—22no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—22no—
221C2×C3×C8—22no—
222C2×C3×C8—22no—
223C2×C3×C8—22no—
224C3×C4⋊C4—28no—
225C3×C4⋊C4—24no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4C2×D414yes—
232C3×C22⋊C4—28no—
233C3×C22⋊C4—28no—
234C3×C22⋊C4—44no—
235C3×C22⋊C4—42no—
236C3×C22⋊C4—82no—
237C3×C22⋊C4—81no—
238C2×#41—22no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—42no—
242C2×#41—42no—
243C2×#41—42no—
244C2×#41—42no—
245C2×#41C3×D414yescomplemented
246C22×Q8C3×D411yescharacteristic
247C22×Q8C23×C311yescharacteristic
248C22×D4—21no—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—21no—
252C22×D4C3×D412yes—
253C22×D4C3×D411yescharacteristic
254C22×D4C23×C311yescharacteristic
255C22×D4C23×C311yescharacteristic
256C23×C4—21no—
257C23×C4—21no—
258C23×C4C23×C312yes—
259C23×C4C23×C311yescharacteristic
260C2×SD16—22no—
261C2×SD16—42no—
262C2×D8—22no—
263C2×D8—42no—
264C22×C8C23×C311yescharacteristic
265C22×C8C23×C311yescharacteristic
266C22×C8C23×C312yes—
267C22⋊Q8—28no—
268C22⋊Q8—24no—
269C22⋊Q8—24no—
270#121—216no—
271#121—28no—
272#121—24no—
273#121—24no—
274#121—22no—
275#121—22no—
276#121—42no—
277#121—42no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4C3×D414yescomplemented
282C2×C4⋊C4C23×C312yes—
283C2×C4⋊C4C23×C311yescharacteristic
284C2×C4⋊C4C23×C312yes—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—21no—
291#107—24no—
292#103—22no—
293#103—42no—
294#102—22no—
295#102—24no—
296#102—42no—
297C22⋊C8—22no—
298C22⋊C8—22no—
299C22⋊C8—42no—
300C22⋊C8—42no—
301#95—22no—
302#95—21no—
303C23×C3—22no—
304C23×C3—22no—
305C23×C3—21no—
306C23×C3—21no—
307C23×C3—22no—
308C23×C3—22no—
309C23×C3—22no—
310C23×C3—22no—
311C23×C3—22no—
312C23×C3—22no—
313C23×C3—22no—
314C23×C3—22no—
315C23×C3—42no—
316C23×C3—42no—
317C23×C3—41no—
318C23×C3—41no—
319C23×C3—41no—
320C23×C3—42no—
321C23×C3—42no—
322C23×C3—41no—
323C23×C3—41no—
324C23×C3#12012yes—
325C23×C3C22×D411yescharacteristic
326C3×Q8—22no—
327C3×Q8—22no—
328C3×Q8—44no—
329C3×Q8—42no—
330C3×D4—22no—
331C3×D4—22no—
332C3×D4—44no—
333C3×D4—48no—
334C3×D4—48no—
335C3×D4—41no—
336C3×D4—41no—
337C3×D4—42no—
338C3×D4—42no—
339C3×D4—88no—
340C3×D4—84no—
341C3×D4—88no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—24no—
346C2×C3×C4—24no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—48no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—41no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—84no—
365C2×C3×C4#12012yes—
366C2×C3×C4C22×D411yescharacteristic
367C2×C3×C4C22×D411yescharacteristic
368C2×C3×C4H(5,𝔽2)12yes—
369C2×C3×C4H(5,𝔽2)12yes—
370C3×C8—41no—
371C3×C8—41no—
372C3×C8—42no—
373C24—21no—
374C24—21no—
375C24C2×C3×D412yes—
376C24C2×C3×D411yescharacteristic
377#41—44no—
378#41—44no—
379#41—44no—
380#41—44no—
381#41—48no—
382#41—48no—
383C2×Q8—22no—
384C2×Q8—21no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—42no—
388C2×Q8C2×C3×D412yes—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—28no—
392C2×D4—22no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—42no—
406C2×D4—82no—
407C2×D4—82no—
408C2×D4—82no—
409C2×D4—82no—
410C2×D4—82no—
411C2×D4C2×C3×D412yes—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—44no—
427C22×C4—82no—
428C22×C4C2×C3×D412yes—
429C22×C4C2×C3×D411yescharacteristic
430C22×C4C2×C3×D412yes—
431C22×C4C2×C3×D412yes—
432C22×C4C24×C311yescharacteristic
433SD16—44no—
434D8—44no—
435C2×C8—22no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C4⋊C4—28no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4C2×C3×D414yes—
450C22⋊C4—28no—
451C22⋊C4—28no—
452C22⋊C4—44no—
453C22⋊C4—42no—
454C22⋊C4—82no—
455C22⋊C4—81no—
456C22×C3—22no—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—42no—
460C22×C3—42no—
461C22×C3—42no—
462C22×C3—42no—
463C22×C3—42no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3—42no—
468C22×C3—42no—
469C22×C3—42no—
470C22×C3—42no—
471C22×C3—82no—
472C22×C3—82no—
473C22×C3—81no—
474C22×C3—82no—
475C22×C3—82no—
476C22×C3—81no—
477C22×C3—81no—
478C22×C3C2×#12011yescharacteristic
479C22×C3#53412yes—
480C22×C3#57612yes—
481C22×C3#57711yescharacteristic
482C22×C3#57811yescharacteristic
483C3×C4—22no—
484C3×C4—42no—
485C3×C4—42no—
486C3×C4—44no—
487C3×C4—82no—
488C3×C4—81no—
489C3×C4#53411yescharacteristic
490C3×C4#53411yescharacteristic
491C3×C4#53512yes—
492C23—22no—
493C23—21no—
494C23—21no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—42no—
505C23—41no—
506C23—41no—
507C23—42no—
508C23—42no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—42no—
513C23C3×#12012yes—
514C23C22×C3×D411yescharacteristic
515Q8—22no—
516Q8—22no—
517Q8—44no—
518Q8—42no—
519D4—22no—
520D4—22no—
521D4—44no—
522D4—48no—
523D4—48no—
524D4—41no—
525D4—41no—
526D4—42no—
527D4—42no—
528D4—88no—
529D4—84no—
530D4—88no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—48no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—84no—
554C2×C4C3×#12012yes—
555C2×C4C22×C3×D411yescharacteristic
556C2×C4C22×C3×D411yescharacteristic
557C2×C4C3×H(5,𝔽2)12yes—
558C2×C4C3×H(5,𝔽2)12yes—
559C8—41no—
560C8—41no—
561C8—42no—
562C2×C3—42no—
563C2×C3—82no—
564C2×C3—82no—
565C2×C3—161no—
566C2×C3#261311yescharacteristic
567C2×C3#300811yescharacteristic
568C2×C3#301311yescharacteristic
569C2×C3#319812yes—
570C2×C3#320312yes—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22C2×C3×#12011yescharacteristic
594C22C3×#53412yes—
595C22C3×#57612yes—
596C22C3×#57711yescharacteristic
597C22C3×#57811yescharacteristic
598C4—22no—
599C4—44no—
600C4—42no—
601C4—42no—
602C4—82no—
603C4—81no—
604C4C3×#53411yescharacteristic
605C4C3×#53411yescharacteristic
606C4C3×#53512yes—
607C3#2960711yescharacteristic, direct product factor, Sylow
608C2—42no—
609C2—82no—
610C2—82no—
611C2—161no—
612C2C3×#261311yescharacteristic
613C2C3×#300811yescharacteristic
614C2C3×#301311yescharacteristic
615C2C3×#319812yes—
616C2C3×#320312yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×#41 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#537C2semidirect
2. C2×C3×#480C2semidirect
3. C2×C3×#449C2semidirect
4. C3×#2615C2semidirect
5. C3×#2036C2semidirect
6. C3×#2033C2semidirect
7. C3×#2000C2semidirect
8. C3×#1999C2semidirect
9. C3×#1824C2semidirect
10. C3×#1788C2semidirect
11. C3×#1780C2semidirect
12. #29607C3direct (N × H)
46. C2×#537C2×C3semidirect
47. C2×#480C2×C3semidirect
48. C2×#449C2×C3semidirect
49. #2615C2×C3semidirect
50. #2036C2×C3semidirect
51. #2033C2×C3semidirect
52. #2000C2×C3semidirect
53. #1999C2×C3semidirect
54. #1824C2×C3semidirect
55. #1788C2×C3semidirect
56. #1780C2×C3semidirect
64. C2×C3×#41D4semidirect
100. C2×C3×C4⋊C4D4semidirect
245. C2×#41C3×D4semidirect
281. C2×C4⋊C4C3×D4semidirect
607. C3#29607direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5372
C2×C3×#4802
C2×C3×#4492
C3×#26152
C3×#20362
C3×#20332
C3×#20002
C3×#19992
C3×#18242
C3×#17882
C3×#17802
#296073