Finite Group Databasebeta

← C3×#19092 #10637699

Subgroups — C3×#19092 #10637699

Order 768.

816 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3563C211yesmaximal, characteristic, complemented
2C3×#3235C211yesmaximal, characteristic, complemented
3C3×#3234C211yesmaximal, characteristic, complemented
4C3×#3219C211yesmaximal, characteristic
5C3×#1663C212yesmaximal, complemented
6C3×#1660C212yesmaximal, complemented
7C3×#1658C212yesmaximal, complemented
8C3×#1657C212yesmaximal, complemented
9C3×#1537C211yesmaximal, characteristic, complemented
10C3×#1535C211yesmaximal, characteristic, complemented
11C3×#1534C211yesmaximal, characteristic, complemented
12#19092C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#564C2212yes—
14C3×#563C2212yes—
15C3×#557—21no—
16C3×#554—21no—
17C3×#552—21no—
18C3×#552—22no—
19C3×#552—21no—
20C3×#544—21no—
21C3×#544—21no—
22C3×#541—21no—
23C3×#541—22no—
24C3×#539—21no—
25C3×#533C2211yescharacteristic
26C3×#532C2211yescharacteristic
27C3×#529C2211yescharacteristic
28C2×C3×#125C2211yescharacteristic
29C3×#501C2212yes—
30C3×#501C2212yes—
31C3×#497C2211yescharacteristic
32C3×#496C2211yescharacteristic
33C3×#491C2211yescharacteristic
34C3×#490C2211yescharacteristic
35C3×#490C2212yes—
36C3×#490C2212yes—
37C3×#482—22no—
38C3×#481—21no—
39C3×#480—21no—
40C3×#478—22no—
41C3×#476—21no—
42C3×#474—21no—
43C3×#464—21no—
44C3×#463—21no—
45C3×#452—21no—
46C3×#449—21no—
47C3×#432C2211yescharacteristic
48C3×#432C2211yescharacteristic
49C3×#432C2211yescharacteristic
50C3×#330C2212yes—
51C3×#330C2212yes—
52C3×#330C2212yes—
53C3×#330C2212yes—
54C3×#330C2212yes—
55C3×#329C2212yes—
56#3563C2×C311yescharacteristic, complemented
57#3235C2×C311yescharacteristic, complemented
58#3234C2×C311yescharacteristic, complemented
59#3219C2×C311yescharacteristic
60#1663C2×C312yescomplemented
61#1660C2×C312yescomplemented
62#1658C2×C312yescomplemented
63#1657C2×C312yescomplemented
64#1537C2×C311yescharacteristic, complemented
65#1535C2×C311yescharacteristic, complemented
66#1534C2×C311yescharacteristic, complemented
67C2×C3×#41—21no—
68C2×C3×Q16—21no—
69C2×C3×SD16—21no—
70C2×C3×SD16—21no—
71C2×C3×D8—21no—
72C3×C4⋊Q8—22no—
73C3×C4⋊Q8—21no—
74C3×C4⋊Q8—22no—
75C3×C4⋊Q8D411yescharacteristic
76C3×C4⋊Q8D411yescharacteristic
77C3×C4⋊Q8C2311yescharacteristic
78C3×#127C2311yescharacteristic
79C3×#126—22no—
80C3×#126—22no—
81C3×#126—22no—
82C3×#126—22no—
83C3×#126—21no—
84C3×#125—22no—
85C3×#125—22no—
86C3×#125—21no—
87C3×#125—22no—
88C3×#125—22no—
89C3×#125—22no—
90C3×#125—22no—
91C3×#125—22no—
92C3×#125—22no—
93C3×#125—21no—
94C3×#125—21no—
95C3×#125D412yes—
96C3×#125C2312yes—
97C3×#125C2312yes—
98C3×#125C2311yescharacteristic
99C3×#124—21no—
100C3×#124C2311yescharacteristic
101C3×#123—22no—
102C3×#123—22no—
103C3×#123—21no—
104C3×C22⋊Q8—22no—
105C3×C22⋊Q8—21no—
106C3×C22⋊Q8—21no—
107C3×C22⋊Q8—22no—
108C3×C22⋊Q8—21no—
109C3×C22⋊Q8—21no—
110C3×C22⋊Q8—22no—
111C3×C22⋊Q8—22no—
112C3×C22⋊Q8D411yescharacteristic
113C3×#121—21no—
114C3×#121—21no—
115C3×#121D411yescharacteristic
116C3×C4×Q8—21no—
117C3×C4×Q8—22no—
118C3×C4×Q8—21no—
119C3×C4×Q8—21no—
120C3×C4×Q8—21no—
121C3×C4×Q8D411yescharacteristic
122C3×C4×D4—22no—
123C3×C4×D4—21no—
124C3×C4×D4—21no—
125C3×C4×D4—21no—
126C3×C4×D4D411yescharacteristic
127C3×#117—22no—
128C3×#117—21no—
129C3×#117—22no—
130C3×#117—21no—
131C3×#117D412yes—
132C2×C3×C4⋊C4—21no—
133C2×C3×C4⋊C4—21no—
134C2×C3×C4⋊C4—21no—
135C2×C3×C4⋊C4D411yescharacteristic
136C2×C3×C4⋊C4D411yescharacteristic
137C2×C3×C42C2311yescharacteristic
138C3×#107—22no—
139C3×#107—22no—
140C3×#106—22no—
141C3×#106—22no—
142C3×C4⋊C8—21no—
143C3×C4⋊C8—21no—
144C3×C4⋊C8—21no—
145C3×#103—21no—
146C3×#103—21no—
147C3×#102—21no—
148C3×#102—21no—
149C3×#102—22no—
150C3×#102—22no—
151C3×#102—22no—
152C3×#102—22no—
153C3×C22⋊C8—21no—
154C3×C22⋊C8—21no—
155C3×C22⋊C8—21no—
156C3×#97C2312yes—
157C3×#97C2312yes—
158C3×#97C2311yescharacteristic
159C3×#97C2311yescharacteristic
160#564C22×C312yes—
161#563C22×C312yes—
162#557—21no—
163#554—21no—
164#552—21no—
165#552—22no—
166#552—21no—
167#544—21no—
168#544—21no—
169#541—21no—
170#541—22no—
171#539—21no—
172#533C22×C311yescharacteristic
173#532C22×C311yescharacteristic
174#529C22×C311yescharacteristic
175C2×#125C22×C311yescharacteristic
176#501C22×C312yes—
177#501C22×C312yes—
178#497C22×C311yescharacteristic
179#496C22×C311yescharacteristic
180#491C22×C311yescharacteristic
181#490C22×C312yes—
182#490C22×C312yes—
183#490C22×C311yescharacteristic
184#482—22no—
185#481—21no—
186#480—21no—
187#478—22no—
188#476—21no—
189#474—21no—
190#464—21no—
191#463—21no—
192#452—21no—
193#449—21no—
194#432C22×C311yescharacteristic
195#432C22×C311yescharacteristic
196#432C22×C311yescharacteristic
197#330C22×C312yes—
198#330C22×C312yes—
199#330C22×C312yes—
200#330C22×C312yes—
201#330C22×C312yes—
202#329C22×C312yes—
203C3×#41—81no—
204C3×#41—81no—
205C2×C3×Q8—22no—
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8C2×D411yescharacteristic
210C2×C3×D4—21no—
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4—21no—
214C2×C3×D4—41no—
215C2×C3×D4C2×D411yescharacteristic
216C22×C3×C4—21no—
217C22×C3×C4—22no—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—41no—
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2×D411yescharacteristic
224C3×Q16—81no—
225C3×SD16—81no—
226C3×SD16—81no—
227C3×D8—81no—
228C2×C3×C8—22no—
229C2×C3×C8—22no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
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—22no—
243C3×C4⋊C4—22no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—22no—
249C3×C4⋊C4—22no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—22no—
253C3×C4⋊C4—22no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4C2×D411yescharacteristic
257C3×C4⋊C4C2×D411yescharacteristic
258C3×C4⋊C4C2×D412yes—
259C3×C4⋊C4C2×D412yes—
260C3×C4⋊C4C2×D412yes—
261C3×C4⋊C4C2×D412yes—
262C3×C22⋊C4—22no—
263C3×C22⋊C4—22no—
264C3×C22⋊C4—22no—
265C3×C22⋊C4—21no—
266C3×C22⋊C4—22no—
267C3×C22⋊C4—21no—
268C3×C22⋊C4—21no—
269C3×C22⋊C4—21no—
270C3×C22⋊C4—21no—
271C3×C22⋊C4—41no—
272C3×C22⋊C4—41no—
273C3×C42—22no—
274C3×C42—22no—
275C3×C42—22no—
276C3×C42—22no—
277C3×C42—21no—
278C3×C42C2×D411yescharacteristic
279C3×C42C2×D411yescharacteristic
280C3×C42C2×D411yescharacteristic
281C3×C42C2411yescharacteristic
282C2×#41—21no—
283C2×Q16—21no—
284C2×SD16—21no—
285C2×SD16—21no—
286C2×D8—21no—
287C4⋊Q8—22no—
288C4⋊Q8—21no—
289C4⋊Q8—22no—
290C4⋊Q8C3×D411yescharacteristic
291C4⋊Q8C3×D411yescharacteristic
292C4⋊Q8C23×C311yescharacteristic
293#127C23×C311yescharacteristic
294#126—22no—
295#126—22no—
296#126—22no—
297#126—22no—
298#126—21no—
299#125—22no—
300#125—22no—
301#125—21no—
302#125—22no—
303#125—22no—
304#125—22no—
305#125—22no—
306#125—22no—
307#125—22no—
308#125—21no—
309#125—21no—
310#125C3×D412yes—
311#125C23×C312yes—
312#125C23×C311yescharacteristic
313#125C23×C312yes—
314#124—21no—
315#124C23×C311yescharacteristic
316#123—22no—
317#123—22no—
318#123—21no—
319C22⋊Q8—22no—
320C22⋊Q8—21no—
321C22⋊Q8—21no—
322C22⋊Q8—22no—
323C22⋊Q8—21no—
324C22⋊Q8—21no—
325C22⋊Q8—22no—
326C22⋊Q8—22no—
327C22⋊Q8C3×D411yescharacteristic
328#121—21no—
329#121—21no—
330#121C3×D411yescharacteristic
331C4×Q8—21no—
332C4×Q8—22no—
333C4×Q8—21no—
334C4×Q8—21no—
335C4×Q8—21no—
336C4×Q8C3×D411yescharacteristic
337C4×D4—22no—
338C4×D4—21no—
339C4×D4—21no—
340C4×D4—21no—
341C4×D4C3×D411yescharacteristic
342#117—22no—
343#117—21no—
344#117—22no—
345#117—21no—
346#117C3×D412yes—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4C3×D411yescharacteristic
351C2×C4⋊C4C3×D411yescharacteristic
352C2×C42C23×C311yescharacteristic
353#107—22no—
354#107—22no—
355#106—22no—
356#106—22no—
357C4⋊C8—21no—
358C4⋊C8—21no—
359C4⋊C8—21no—
360#103—21no—
361#103—21no—
362#102—21no—
363#102—21no—
364#102—22no—
365#102—22no—
366#102—22no—
367#102—22no—
368C22⋊C8—21no—
369C22⋊C8—21no—
370C22⋊C8—21no—
371#97C23×C312yes—
372#97C23×C312yes—
373#97C23×C311yescharacteristic
374#97C23×C311yescharacteristic
375C23×C3—21no—
376C23×C3—41no—
377C23×C3#12011yescharacteristic
378C3×Q8—41no—
379C3×Q8—42no—
380C3×Q8—41no—
381C3×Q8—41no—
382C3×Q8—81no—
383C3×Q8—82no—
384C3×D4—41no—
385C3×D4—41no—
386C3×D4—41no—
387C3×D4—42no—
388C3×D4—42no—
389C3×D4—42no—
390C3×D4—82no—
391C2×C3×C4—22no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—22no—
396C2×C3×C4—22no—
397C2×C3×C4—22no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—42no—
402C2×C3×C4—42no—
403C2×C3×C4—41no—
404C2×C3×C4—82no—
405C2×C3×C4—81no—
406C2×C3×C4—82no—
407C2×C3×C4#12011yescharacteristic
408C2×C3×C4#12011yescharacteristic
409C2×C3×C4#12011yescharacteristic
410C2×C3×C4C22×D411yescharacteristic
411C2×C3×C4C22×D411yescharacteristic
412C2×C3×C4C22×D411yescharacteristic
413C3×C8—42no—
414C3×C8—42no—
415C3×C8—41no—
416C3×C8—41no—
417#41—81no—
418#41—81no—
419C2×Q8—22no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8C2×C3×D411yescharacteristic
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—41no—
429C2×D4C2×C3×D411yescharacteristic
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—41no—
435C22×C4C2×C3×D411yescharacteristic
436C22×C4C2×C3×D411yescharacteristic
437C22×C4C2×C3×D411yescharacteristic
438Q16—81no—
439SD16—81no—
440SD16—81no—
441D8—81no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—21no—
445C2×C8—21no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4C2×C3×D411yescharacteristic
471C4⋊C4C2×C3×D411yescharacteristic
472C4⋊C4C2×C3×D412yes—
473C4⋊C4C2×C3×D412yes—
474C4⋊C4C2×C3×D412yes—
475C4⋊C4C2×C3×D412yes—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—21no—
480C22⋊C4—22no—
481C22⋊C4—21no—
482C22⋊C4—21no—
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—21no—
492C42C2×C3×D411yescharacteristic
493C42C2×C3×D411yescharacteristic
494C42C2×C3×D411yescharacteristic
495C42C24×C311yescharacteristic
496C22×C3—21no—
497C22×C3—21no—
498C22×C3—21no—
499C22×C3—41no—
500C22×C3—41no—
501C22×C3—41no—
502C22×C3—81no—
503C22×C3—81no—
504C22×C3—81no—
505C22×C3C2×#12011yescharacteristic
506C3×C4—21no—
507C3×C4—21no—
508C3×C4—21no—
509C3×C4—21no—
510C3×C4—21no—
511C3×C4—21no—
512C3×C4—41no—
513C3×C4—42no—
514C3×C4—41no—
515C3×C4—42no—
516C3×C4—42no—
517C3×C4—42no—
518C3×C4—41no—
519C3×C4—81no—
520C23—21no—
521C23—41no—
522C23C3×#12011yescharacteristic
523Q8—41no—
524Q8—42no—
525Q8—41no—
526Q8—41no—
527Q8—81no—
528Q8—82no—
529D4—41no—
530D4—41no—
531D4—41no—
532D4—42no—
533D4—42no—
534D4—42no—
535D4—82no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—82no—
550C2×C4—81no—
551C2×C4—82no—
552C2×C4C3×#12011yescharacteristic
553C2×C4C3×#12011yescharacteristic
554C2×C4C3×#12011yescharacteristic
555C2×C4C22×C3×D411yescharacteristic
556C2×C4C22×C3×D411yescharacteristic
557C2×C4C22×C3×D411yescharacteristic
558C8—41no—
559C8—41no—
560C8—42no—
561C8—42no—
562C2×C3—41no—
563C2×C3—81no—
564C2×C3—161no—
565C2×C3#302011yescharacteristic
566C2×C3#302011yescharacteristic
567C2×C3#302111yescharacteristic
568C22—21no—
569C22—21no—
570C22—21no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22C2×C3×#12011yescharacteristic
578C4—21no—
579C4—21no—
580C4—21no—
581C4—21no—
582C4—21no—
583C4—21no—
584C4—41no—
585C4—42no—
586C4—41no—
587C4—42no—
588C4—42no—
589C4—42no—
590C4—41no—
591C4—81no—
592C3#1909211yescharacteristic, direct product factor, Sylow
593C2—41no—
594C2—81no—
595C2—161no—
596C2C3×#302011yescharacteristic
597C2C3×#302011yescharacteristic
598C2C3×#302111yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC42C24×C3
prime radicalC3×#3219C2
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
1. C3×#3563C2semidirect
2. C3×#3235C2semidirect
3. C3×#3234C2semidirect
5. C3×#1663C2semidirect
6. C3×#1660C2semidirect
7. C3×#1658C2semidirect
8. C3×#1657C2semidirect
9. C3×#1537C2semidirect
10. C3×#1535C2semidirect
11. C3×#1534C2semidirect
12. #19092C3direct (N × H)
56. #3563C2×C3semidirect
57. #3235C2×C3semidirect
58. #3234C2×C3semidirect
60. #1663C2×C3semidirect
61. #1660C2×C3semidirect
62. #1658C2×C3semidirect
63. #1657C2×C3semidirect
64. #1537C2×C3semidirect
65. #1535C2×C3semidirect
66. #1534C2×C3semidirect
592. C3#19092direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35632
C3×#32352
C3×#32342
C3×#32192
C3×#16632
C3×#16602
C3×#16582
C3×#16572
C3×#15372
C3×#15352
C3×#15342
#190923