Finite Group Databasebeta

← C3×#18690 #10637297

Subgroups — C3×#18690 #10637297

Order 768.

878 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3544C211yesmaximal, characteristic
2C3×#3239C211yesmaximal, characteristic, complemented
3C3×#3218C211yesmaximal, characteristic, complemented
4C2×C3×#487C211yesmaximal, characteristic, complemented
5C3×#1646C212yesmaximal, complemented
6C3×#1645C212yesmaximal, complemented
7C3×#1644C212yesmaximal, complemented
8C3×#1643C212yesmaximal, complemented
9C3×#1558C211yesmaximal, characteristic, complemented
10C3×#1494C211yesmaximal, characteristic, complemented
11C3×#1494C211yesmaximal, characteristic, complemented
12#18690C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#563—22no—
14C3×Q82C2211yescharacteristic
15C3×#555C2211yescharacteristic
16C3×#554C2211yescharacteristic
17C3×#554C2211yescharacteristic
18C3×#552—21no—
19C3×#552—21no—
20C3×#547—21no—
21C3×#547—21no—
22C3×#544—21no—
23C3×#544—21no—
24C3×#542—21no—
25C3×#542—21no—
26C3×#536—22no—
27C2×C3×C4⋊Q8C2211yescharacteristic
28C2×C3×C4⋊Q8C2211yescharacteristic
29C3×#529C2211yescharacteristic
30C3×#529C2211yescharacteristic
31C3×#501—21no—
32C3×#497—21no—
33C3×#491—21no—
34C3×#489—21no—
35C3×#487—22no—
36C3×#487C2212yes—
37C3×#487C2212yes—
38C3×#484—21no—
39C3×#481—21no—
40C3×#475C2212yes—
41C3×#474C2212yes—
42C3×#462C2212yes—
43C3×#461C2212yes—
44C3×#451—21no—
45C3×#448—21no—
46C3×#432C2211yescharacteristic
47C3×#432C2211yescharacteristic
48C2×C3×#103—22no—
49C2×C3×C4×C8C2211yescharacteristic
50C3×#333C2212yes—
51C3×#333C2212yes—
52C3×Q8⋊C8C2212yes—
53C3×Q8⋊C8C2212yes—
54C3×D4⋊C8C2212yes—
55C3×D4⋊C8C2212yes—
56#3544C2×C311yescharacteristic
57#3239C2×C311yescharacteristic, complemented
58#3218C2×C311yescharacteristic, complemented
59C2×#487C2×C311yescharacteristic, complemented
60#1646C2×C312yescomplemented
61#1645C2×C312yescomplemented
62#1644C2×C312yescomplemented
63#1643C2×C312yescomplemented
64#1558C2×C311yescharacteristic, complemented
65#1494C2×C311yescharacteristic, complemented
66#1494C2×C311yescharacteristic, complemented
67C2×C3×#41—22no—
68C22×C3×Q8—21no—
69C22×C3×Q8D412yes—
70C2×C3×Q16—22no—
71C2×C3×SD16—22no—
72C22×C3×C8—21no—
73C3×C4⋊Q8—22no—
74C3×C4⋊Q8—22no—
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⋊Q8—21no—
82C3×C4⋊Q8D412yes—
83C3×C4⋊Q8C2312yes—
84C3×C4⋊Q8C2311yescharacteristic
85C3×C4⋊Q8C2311yescharacteristic
86C3×#126—22no—
87C3×#126—22no—
88C3×#126—21no—
89C3×#126—21no—
90C3×#125—22no—
91C3×#125—22no—
92C3×#125—21no—
93C3×#125—21no—
94C3×#125D411yescharacteristic
95C3×#125D411yescharacteristic
96C3×#124—22no—
97C3×#124—22no—
98C3×#124D411yescharacteristic
99C3×#124D411yescharacteristic
100C3×#123—22no—
101C3×#123—22no—
102C3×#123—22no—
103C3×#123—22no—
104C3×#123—21no—
105C3×#123—21no—
106C3×C22⋊Q8—22no—
107C3×C22⋊Q8—22no—
108C3×C22⋊Q8—22no—
109C3×C22⋊Q8—22no—
110C3×C22⋊Q8—22no—
111C3×C22⋊Q8—22no—
112C3×C22⋊Q8—21no—
113C3×C22⋊Q8—21no—
114C3×C22⋊Q8D411yescharacteristic
115C3×C22⋊Q8D411yescharacteristic
116C3×#121—21no—
117C3×#121—21no—
118C3×#121D411yescharacteristic
119C3×#121D411yescharacteristic
120C3×C4×Q8—21no—
121C3×C4×Q8—21no—
122C3×C4×Q8C2311yescharacteristic
123C3×C4×Q8C2311yescharacteristic
124C3×C4×D4—21no—
125C3×C4×D4—21no—
126C3×C4×D4—21no—
127C3×C4×D4—21no—
128C3×C4×D4C2311yescharacteristic
129C3×C4×D4C2311yescharacteristic
130C3×#117—22no—
131C3×#117—21no—
132C3×#117—21no—
133C3×#117—21no—
134C3×#117—21no—
135C2×C3×C4⋊C4—21no—
136C2×C3×C4⋊C4—21no—
137C2×C3×C4⋊C4—21no—
138C2×C3×C4⋊C4—21no—
139C2×C3×C4⋊C4—21no—
140C2×C3×C4⋊C4—21no—
141C2×C3×C42C2311yescharacteristic
142C3×#107—22no—
143C3×#106—22no—
144C3×C4⋊C8C2312yes—
145C3×C4⋊C8C2312yes—
146C3×#103—24no—
147C3×#103—24no—
148C3×#103—22no—
149C3×#103—22no—
150C3×#103—22no—
151C3×#102—22no—
152C3×C22⋊C8—21no—
153C3×C22⋊C8—21no—
154C3×#97—21no—
155C3×#97—21no—
156C3×C4×C8—21no—
157C3×C4×C8C2312yes—
158#563—22no—
159Q82C22×C311yescharacteristic
160#555C22×C311yescharacteristic
161#554C22×C311yescharacteristic
162#554C22×C311yescharacteristic
163#552—21no—
164#552—21no—
165#547—21no—
166#547—21no—
167#544—21no—
168#544—21no—
169#542—21no—
170#542—21no—
171#536—22no—
172C2×C4⋊Q8C22×C311yescharacteristic
173C2×C4⋊Q8C22×C311yescharacteristic
174#529C22×C311yescharacteristic
175#529C22×C311yescharacteristic
176#501—21no—
177#497—21no—
178#491—21no—
179#489—21no—
180#487—22no—
181#487C22×C312yes—
182#487C22×C312yes—
183#484—21no—
184#481—21no—
185#475C22×C312yes—
186#474C22×C312yes—
187#462C22×C312yes—
188#461C22×C312yes—
189#451—21no—
190#448—21no—
191#432C22×C311yescharacteristic
192#432C22×C311yescharacteristic
193C2×#103—22no—
194C2×C4×C8C22×C311yescharacteristic
195#333C22×C312yes—
196#333C22×C312yes—
197Q8⋊C8C22×C312yes—
198Q8⋊C8C22×C312yes—
199D4⋊C8C22×C312yes—
200D4⋊C8C22×C312yes—
201C3×#41—84no—
202C2×C3×Q8—22no—
203C2×C3×Q8—22no—
204C2×C3×Q8—22no—
205C2×C3×Q8—44no—
206C2×C3×Q8—41no—
207C2×C3×Q8—82no—
208C2×C3×Q8C2×D411yescharacteristic
209C2×C3×Q8C2×D411yescharacteristic
210C2×C3×Q8C2×D412yes—
211C2×C3×Q8C2×D412yes—
212C2×C3×D4—21no—
213C2×C3×D4—21no—
214C2×C3×D4—22no—
215C2×C3×D4C2×D411yescharacteristic
216C2×C3×D4C2×D411yescharacteristic
217C22×C3×C4—22no—
218C22×C3×C4—22no—
219C22×C3×C4—22no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—41no—
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4C2×D411yescharacteristic
226C3×Q16—82no—
227C3×SD16—82no—
228C2×C3×C8—22no—
229C2×C3×C8—22no—
230C2×C3×C8—22no—
231C2×C3×C8—42no—
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—22no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
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—22no—
253C3×C4⋊C4—22no—
254C3×C4⋊C4—41no—
255C3×C4⋊C4—41no—
256C3×C4⋊C4C2×D411yescharacteristic
257C3×C4⋊C4C2×D411yescharacteristic
258C3×C4⋊C4C2×D411yescharacteristic
259C3×C4⋊C4C2×D411yescharacteristic
260C3×C22⋊C4—22no—
261C3×C22⋊C4—22no—
262C3×C22⋊C4—22no—
263C3×C22⋊C4—22no—
264C3×C22⋊C4—21no—
265C3×C22⋊C4—21no—
266C3×C22⋊C4—21no—
267C3×C22⋊C4—21no—
268C3×C22⋊C4—21no—
269C3×C22⋊C4—21no—
270C3×C22⋊C4—21no—
271C3×C22⋊C4—21no—
272C3×C22⋊C4—21no—
273C3×C22⋊C4—21no—
274C3×C42—22no—
275C3×C42—22no—
276C3×C42—21no—
277C3×C42—21no—
278C3×C42C2×D411yescharacteristic
279C3×C42C2×D411yescharacteristic
280C3×C42C2×D411yescharacteristic
281C3×C42C2411yescharacteristic
282C2×#41—22no—
283C22×Q8—21no—
284C22×Q8C3×D412yes—
285C2×Q16—22no—
286C2×SD16—22no—
287C22×C8—21no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—21no—
291C4⋊Q8—21no—
292C4⋊Q8—21no—
293C4⋊Q8—21no—
294C4⋊Q8—21no—
295C4⋊Q8—21no—
296C4⋊Q8—21no—
297C4⋊Q8C3×D412yes—
298C4⋊Q8C23×C312yes—
299C4⋊Q8C23×C311yescharacteristic
300C4⋊Q8C23×C311yescharacteristic
301#126—22no—
302#126—22no—
303#126—21no—
304#126—21no—
305#125—22no—
306#125—22no—
307#125—21no—
308#125—21no—
309#125C3×D411yescharacteristic
310#125C3×D411yescharacteristic
311#124—22no—
312#124—22no—
313#124C3×D411yescharacteristic
314#124C3×D411yescharacteristic
315#123—22no—
316#123—22no—
317#123—22no—
318#123—22no—
319#123—21no—
320#123—21no—
321C22⋊Q8—22no—
322C22⋊Q8—22no—
323C22⋊Q8—22no—
324C22⋊Q8—22no—
325C22⋊Q8—22no—
326C22⋊Q8—22no—
327C22⋊Q8—21no—
328C22⋊Q8—21no—
329C22⋊Q8C3×D411yescharacteristic
330C22⋊Q8C3×D411yescharacteristic
331#121—21no—
332#121—21no—
333#121C3×D411yescharacteristic
334#121C3×D411yescharacteristic
335C4×Q8—21no—
336C4×Q8—21no—
337C4×Q8C23×C311yescharacteristic
338C4×Q8C23×C311yescharacteristic
339C4×D4—21no—
340C4×D4—21no—
341C4×D4—21no—
342C4×D4—21no—
343C4×D4C23×C311yescharacteristic
344C4×D4C23×C311yescharacteristic
345#117—22no—
346#117—21no—
347#117—21no—
348#117—21no—
349#117—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C42C23×C311yescharacteristic
357#107—22no—
358#106—22no—
359C4⋊C8C23×C312yes—
360C4⋊C8C23×C312yes—
361#103—24no—
362#103—24no—
363#103—22no—
364#103—22no—
365#103—22no—
366#102—22no—
367C22⋊C8—21no—
368C22⋊C8—21no—
369#97—21no—
370#97—21no—
371C4×C8—21no—
372C4×C8C23×C312yes—
373C23×C3—21no—
374C23×C3—21no—
375C23×C3#12011yescharacteristic
376C3×Q8—22no—
377C3×Q8—22no—
378C3×Q8—42no—
379C3×Q8—42no—
380C3×Q8—42no—
381C3×Q8—42no—
382C3×Q8—82no—
383C3×Q8—82no—
384C3×Q8—82no—
385C3×D4—22no—
386C3×D4—22no—
387C3×D4—42no—
388C3×D4—42no—
389C3×D4—82no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—22no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—22no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—22no—
401C2×C3×C4—44no—
402C2×C3×C4—42no—
403C2×C3×C4—42no—
404C2×C3×C4—42no—
405C2×C3×C4—82no—
406C2×C3×C4—82no—
407C2×C3×C4—82no—
408C2×C3×C4#12011yescharacteristic
409C2×C3×C4#12011yescharacteristic
410C2×C3×C4#12011yescharacteristic
411C2×C3×C4C22×D411yescharacteristic
412C2×C3×C4C22×D411yescharacteristic
413C2×C3×C4C22×D411yescharacteristic
414C3×C8—42no—
415C3×C8—42no—
416C3×C8—42no—
417#41—84no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—44no—
422C2×Q8—41no—
423C2×Q8—82no—
424C2×Q8C2×C3×D411yescharacteristic
425C2×Q8C2×C3×D411yescharacteristic
426C2×Q8C2×C3×D412yes—
427C2×Q8C2×C3×D412yes—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—22no—
431C2×D4C2×C3×D411yescharacteristic
432C2×D4C2×C3×D411yescharacteristic
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—41no—
439C22×C4C2×C3×D411yescharacteristic
440C22×C4C2×C3×D411yescharacteristic
441C22×C4C2×C3×D411yescharacteristic
442Q16—82no—
443SD16—82no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—42no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4C2×C3×D411yescharacteristic
473C4⋊C4C2×C3×D411yescharacteristic
474C4⋊C4C2×C3×D411yescharacteristic
475C4⋊C4C2×C3×D411yescharacteristic
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—21no—
481C22⋊C4—21no—
482C22⋊C4—21no—
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—21no—
489C22⋊C4—21no—
490C42—22no—
491C42—22no—
492C42—21no—
493C42—21no—
494C42C2×C3×D411yescharacteristic
495C42C2×C3×D411yescharacteristic
496C42C2×C3×D411yescharacteristic
497C42C24×C311yescharacteristic
498C22×C3—21no—
499C22×C3—21no—
500C22×C3—41no—
501C22×C3—41no—
502C22×C3—41no—
503C22×C3—41no—
504C22×C3—41no—
505C22×C3—41no—
506C22×C3#45512yes—
507C22×C3C2×#12011yescharacteristic
508C3×C4—21no—
509C3×C4—21no—
510C3×C4—21no—
511C3×C4—21no—
512C3×C4—41no—
513C3×C4—41no—
514C3×C4—41no—
515C3×C4—41no—
516C3×C4—42no—
517C3×C4—42no—
518C3×C4—41no—
519C3×C4—41no—
520C3×C4—82no—
521C3×C4#57712yes—
522C3×C4#57812yes—
523C23—21no—
524C23—21no—
525C23C3×#12011yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—42no—
529Q8—42no—
530Q8—42no—
531Q8—42no—
532Q8—82no—
533Q8—82no—
534Q8—82no—
535D4—22no—
536D4—22no—
537D4—42no—
538D4—42no—
539D4—82no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4C3×#12011yescharacteristic
559C2×C4C3×#12011yescharacteristic
560C2×C4C3×#12011yescharacteristic
561C2×C4C22×C3×D411yescharacteristic
562C2×C4C22×C3×D411yescharacteristic
563C2×C4C22×C3×D411yescharacteristic
564C8—42no—
565C8—42no—
566C8—42no—
567C2×C3—22no—
568C2×C3—81no—
569C2×C3—81no—
570C2×C3#301011yescharacteristic
571C2×C3#301111yescharacteristic
572C2×C3C2×#45511yescharacteristic
573C22—21no—
574C22—21no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22C3×#45512yes—
582C22C2×C3×#12011yescharacteristic
583C4—21no—
584C4—21no—
585C4—21no—
586C4—21no—
587C4—41no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—42no—
592C4—42no—
593C4—41no—
594C4—41no—
595C4—82no—
596C4C3×#57712yes—
597C4C3×#57812yes—
598C3#1869011yescharacteristic, direct product factor, Sylow
599C2—22no—
600C2—81no—
601C2—81no—
602C2C3×#301011yescharacteristic
603C2C3×#301111yescharacteristic
604C2C2×C3×#45511yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC42C24×C3
prime radicalC3×#3544C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3239C2semidirect
3. C3×#3218C2semidirect
4. C2×C3×#487C2semidirect
5. C3×#1646C2semidirect
6. C3×#1645C2semidirect
7. C3×#1644C2semidirect
8. C3×#1643C2semidirect
9. C3×#1558C2semidirect
10. C3×#1494C2semidirect
11. C3×#1494C2semidirect
12. #18690C3direct (N × H)
57. #3239C2×C3semidirect
58. #3218C2×C3semidirect
59. C2×#487C2×C3semidirect
60. #1646C2×C3semidirect
61. #1645C2×C3semidirect
62. #1644C2×C3semidirect
63. #1643C2×C3semidirect
64. #1558C2×C3semidirect
65. #1494C2×C3semidirect
66. #1494C2×C3semidirect
598. C3#18690direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35442
C3×#32392
C3×#32182
C2×C3×#4872
C3×#16462
C3×#16452
C3×#16442
C3×#16432
C3×#15582
C3×#14942
#186903