Finite Group Databasebeta

← C3×#19089 #10637696

Subgroups — C3×#19089 #10637696

Order 768.

850 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3559C211yesmaximal, characteristic
2C3×#3240C211yesmaximal, characteristic, complemented
3C3×#3236C211yesmaximal, characteristic, complemented
4C3×#3216C211yesmaximal, characteristic, complemented
5C3×#1664C212yesmaximal, complemented
6C3×#1662C212yesmaximal, complemented
7C3×#1661C212yesmaximal, complemented
8C3×#1660C212yesmaximal, complemented
9C3×#1536C211yesmaximal, characteristic, complemented
10C3×#1535C211yesmaximal, characteristic, complemented
11C3×#1534C211yesmaximal, characteristic, complemented
12#19089C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#563C2212yes—
14C3×#562C2212yes—
15C3×#554—21no—
16C3×#553—21no—
17C3×#553—21no—
18C3×#547—22no—
19C3×#547—21no—
20C3×#544—21no—
21C3×#540—21no—
22C3×#539—22no—
23C3×#539—21no—
24C3×#537—21no—
25C3×#533C2211yescharacteristic
26C3×#532C2211yescharacteristic
27C3×#529C2211yescharacteristic
28C2×C3×#125C2211yescharacteristic
29C3×#501C2212yes—
30C3×#501C2211yescharacteristic
31C3×#501C2211yescharacteristic
32C3×#497C2212yes—
33C3×#491C2212yes—
34C3×#489C2212yes—
35C3×#489C2211yescharacteristic
36C3×#489C2211yescharacteristic
37C3×#485—21no—
38C3×#484—21no—
39C3×#483—21no—
40C3×#482—21no—
41C3×#479—22no—
42C3×#479—21no—
43C3×#478—21no—
44C3×#462—21no—
45C3×#460—21no—
46C3×#452—22no—
47C3×#432C2211yescharacteristic
48C3×#432C2211yescharacteristic
49C3×#432C2211yescharacteristic
50C3×#330C2212yes—
51C3×#330C2212yes—
52C3×#330C2212yes—
53C3×#329C2212yes—
54C3×#329C2212yes—
55C3×#329C2212yes—
56#3559C2×C311yescharacteristic
57#3240C2×C311yescharacteristic, complemented
58#3236C2×C311yescharacteristic, complemented
59#3216C2×C311yescharacteristic, complemented
60#1664C2×C312yescomplemented
61#1662C2×C312yescomplemented
62#1661C2×C312yescomplemented
63#1660C2×C312yescomplemented
64#1536C2×C311yescharacteristic, complemented
65#1535C2×C311yescharacteristic, complemented
66#1534C2×C311yescharacteristic, complemented
67C2×C3×#41—22no—
68C2×C3×#41D412yes—
69C2×C3×Q16—22no—
70C2×C3×SD16—22no—
71C3×C4⋊Q8—21no—
72C3×C4⋊Q8—22no—
73C3×C4⋊Q8D411yescharacteristic
74C3×C4⋊Q8C2311yescharacteristic
75C3×C4⋊Q8C2311yescharacteristic
76C3×#127D411yescharacteristic
77C3×#126—22no—
78C3×#126—22no—
79C3×#126—21no—
80C3×#125—22no—
81C3×#125—21no—
82C3×#125—21no—
83C3×#125—21no—
84C3×#125—21no—
85C3×#125D412yes—
86C3×#125C2312yes—
87C3×#125C2311yescharacteristic
88C3×#124C2311yescharacteristic
89C3×#124C2312yes—
90C3×#123—22no—
91C3×#123—22no—
92C3×#123—22no—
93C3×#123—22no—
94C3×#123—21no—
95C3×#123—22no—
96C3×#123—22no—
97C3×C22⋊Q8—22no—
98C3×C22⋊Q8—22no—
99C3×C22⋊Q8—22no—
100C3×C22⋊Q8—22no—
101C3×C22⋊Q8—22no—
102C3×C22⋊Q8—22no—
103C3×C22⋊Q8—21no—
104C3×C22⋊Q8—21no—
105C3×C22⋊Q8D411yescharacteristic
106C3×#121—22no—
107C3×#121—22no—
108C3×#121—21no—
109C3×#121—21no—
110C3×#121—22no—
111C3×#121—22no—
112C3×#121—21no—
113C3×#121—21no—
114C3×#121D411yescharacteristic
115C3×C4×Q8—21no—
116C3×C4×Q8—21no—
117C3×C4×Q8D411yescharacteristic
118C3×C4×D4—22no—
119C3×C4×D4—22no—
120C3×C4×D4—21no—
121C3×C4×D4—21no—
122C3×C4×D4—21no—
123C3×C4×D4—21no—
124C3×C4×D4—21no—
125C3×C4×D4D411yescharacteristic
126C3×#117—21no—
127C3×#117—21no—
128C3×#117—21no—
129C2×C3×C4⋊C4—22no—
130C2×C3×C4⋊C4—21no—
131C2×C3×C4⋊C4—21no—
132C2×C3×C4⋊C4—21no—
133C2×C3×C4⋊C4D411yescharacteristic
134C2×C3×C4⋊C4D411yescharacteristic
135C2×C3×C42C2311yescharacteristic
136C3×#107—21no—
137C3×#107—21no—
138C3×#107—22no—
139C3×#106—21no—
140C3×#106—21no—
141C3×#106—22no—
142C3×C4⋊C8—21no—
143C3×C4⋊C8—21no—
144C3×C4⋊C8—21no—
145C3×#103—22no—
146C3×#103—21no—
147C3×#103—21no—
148C3×#103—22no—
149C3×#103—22no—
150C3×#102—22no—
151C3×#102—21no—
152C3×#102—21no—
153C3×C22⋊C8—21no—
154C3×C22⋊C8—21no—
155C3×C22⋊C8—21no—
156C3×#97C2312yes—
157C3×#97C2311yescharacteristic
158C3×#97C2311yescharacteristic
159C3×#97C2312yes—
160#563C22×C312yes—
161#562C22×C312yes—
162#554—21no—
163#553—21no—
164#553—21no—
165#547—22no—
166#547—21no—
167#544—21no—
168#540—21no—
169#539—22no—
170#539—21no—
171#537—21no—
172#533C22×C311yescharacteristic
173#532C22×C311yescharacteristic
174#529C22×C311yescharacteristic
175C2×#125C22×C311yescharacteristic
176#501C22×C311yescharacteristic
177#501C22×C312yes—
178#501C22×C311yescharacteristic
179#497C22×C312yes—
180#491C22×C312yes—
181#489C22×C311yescharacteristic
182#489C22×C311yescharacteristic
183#489C22×C312yes—
184#485—21no—
185#484—21no—
186#483—21no—
187#482—21no—
188#479—22no—
189#479—21no—
190#478—21no—
191#462—21no—
192#460—21no—
193#452—22no—
194#432C22×C311yescharacteristic
195#432C22×C311yescharacteristic
196#432C22×C311yescharacteristic
197#330C22×C312yes—
198#330C22×C312yes—
199#330C22×C312yes—
200#329C22×C312yes—
201#329C22×C312yes—
202#329C22×C312yes—
203C3×#41—84no—
204C3×#41—82no—
205C2×C3×Q8—22no—
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8C2×D411yescharacteristic
209C2×C3×Q8C2×D412yes—
210C2×C3×D4—22no—
211C2×C3×D4—21no—
212C2×C3×D4—22no—
213C2×C3×D4—22no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4C2×D411yescharacteristic
217C2×C3×D4C2×D412yes—
218C22×C3×C4—21no—
219C22×C3×C4—22no—
220C22×C3×C4—22no—
221C22×C3×C4—22no—
222C22×C3×C4—21no—
223C22×C3×C4—21no—
224C22×C3×C4—22no—
225C22×C3×C4C2×D411yescharacteristic
226C22×C3×C4C2×D411yescharacteristic
227C22×C3×C4C2×D411yescharacteristic
228C3×Q16—82no—
229C3×SD16—82no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—22no—
233C2×C3×C8—22no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4—22no—
236C3×C4⋊C4—22no—
237C3×C4⋊C4—22no—
238C3×C4⋊C4—22no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—22no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—22no—
247C3×C4⋊C4—22no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—41no—
255C3×C4⋊C4—41no—
256C3×C4⋊C4C2×D412yes—
257C3×C4⋊C4C2×D412yes—
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—22no—
265C3×C22⋊C4—21no—
266C3×C22⋊C4—21no—
267C3×C22⋊C4—21no—
268C3×C22⋊C4—21no—
269C3×C22⋊C4—22no—
270C3×C22⋊C4—22no—
271C3×C22⋊C4—21no—
272C3×C22⋊C4—41no—
273C3×C42—22no—
274C3×C42—21no—
275C3×C42—41no—
276C3×C42C2×D411yescharacteristic
277C3×C42C2×D411yescharacteristic
278C3×C42C2×D411yescharacteristic
279C3×C42C2411yescharacteristic
280C2×#41—22no—
281C2×#41C3×D412yes—
282C2×Q16—22no—
283C2×SD16—22no—
284C4⋊Q8—21no—
285C4⋊Q8—22no—
286C4⋊Q8C3×D411yescharacteristic
287C4⋊Q8C23×C311yescharacteristic
288C4⋊Q8C23×C311yescharacteristic
289#127C3×D411yescharacteristic
290#126—22no—
291#126—22no—
292#126—21no—
293#125—22no—
294#125—21no—
295#125—21no—
296#125—21no—
297#125—21no—
298#125C3×D412yes—
299#125C23×C312yes—
300#125C23×C311yescharacteristic
301#124C23×C311yescharacteristic
302#124C23×C312yes—
303#123—22no—
304#123—22no—
305#123—22no—
306#123—22no—
307#123—21no—
308#123—22no—
309#123—22no—
310C22⋊Q8—22no—
311C22⋊Q8—22no—
312C22⋊Q8—22no—
313C22⋊Q8—22no—
314C22⋊Q8—22no—
315C22⋊Q8—22no—
316C22⋊Q8—21no—
317C22⋊Q8—21no—
318C22⋊Q8C3×D411yescharacteristic
319#121—22no—
320#121—22no—
321#121—21no—
322#121—21no—
323#121—22no—
324#121—22no—
325#121—21no—
326#121—21no—
327#121C3×D411yescharacteristic
328C4×Q8—21no—
329C4×Q8—21no—
330C4×Q8C3×D411yescharacteristic
331C4×D4—22no—
332C4×D4—22no—
333C4×D4—21no—
334C4×D4—21no—
335C4×D4—21no—
336C4×D4—21no—
337C4×D4—21no—
338C4×D4C3×D411yescharacteristic
339#117—21no—
340#117—21no—
341#117—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4C3×D411yescharacteristic
347C2×C4⋊C4C3×D411yescharacteristic
348C2×C42C23×C311yescharacteristic
349#107—21no—
350#107—21no—
351#107—22no—
352#106—21no—
353#106—21no—
354#106—22no—
355C4⋊C8—21no—
356C4⋊C8—21no—
357C4⋊C8—21no—
358#103—22no—
359#103—21no—
360#103—21no—
361#103—22no—
362#103—22no—
363#102—22no—
364#102—21no—
365#102—21no—
366C22⋊C8—21no—
367C22⋊C8—21no—
368C22⋊C8—21no—
369#97C23×C312yes—
370#97C23×C311yescharacteristic
371#97C23×C311yescharacteristic
372#97C23×C312yes—
373C23×C3—22no—
374C23×C3—21no—
375C23×C3#12011yescharacteristic
376C3×Q8—42no—
377C3×Q8—41no—
378C3×Q8—41no—
379C3×Q8—41no—
380C3×Q8—42no—
381C3×Q8—82no—
382C3×D4—42no—
383C3×D4—41no—
384C3×D4—42no—
385C3×D4—44no—
386C3×D4—42no—
387C3×D4—81no—
388C3×D4—82no—
389C3×D4—82no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—21no—
395C2×C3×C4—22no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—42no—
400C2×C3×C4—42no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—82no—
404C2×C3×C4—81no—
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—41no—
416C3×C8—41no—
417C3×C8—42no—
418#41—84no—
419#41—82no—
420C2×Q8—22no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8C2×C3×D411yescharacteristic
424C2×Q8C2×C3×D412yes—
425C2×D4—22no—
426C2×D4—21no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4C2×C3×D411yescharacteristic
432C2×D4C2×C3×D412yes—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4C2×C3×D411yescharacteristic
441C22×C4C2×C3×D411yescharacteristic
442C22×C4C2×C3×D411yescharacteristic
443Q16—82no—
444SD16—82no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—22no—
448C2×C8—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4C2×C3×D412yes—
472C4⋊C4C2×C3×D412yes—
473C4⋊C4C2×C3×D411yescharacteristic
474C4⋊C4C2×C3×D411yescharacteristic
475C22⋊C4—22no—
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—22no—
485C22⋊C4—22no—
486C22⋊C4—21no—
487C22⋊C4—41no—
488C42—22no—
489C42—21no—
490C42—41no—
491C42C2×C3×D411yescharacteristic
492C42C2×C3×D411yescharacteristic
493C42C2×C3×D411yescharacteristic
494C42C24×C311yescharacteristic
495C22×C3—21no—
496C22×C3—21no—
497C22×C3—21no—
498C22×C3—42no—
499C22×C3—41no—
500C22×C3—41no—
501C22×C3—42no—
502C22×C3—42no—
503C22×C3—41no—
504C22×C3C2×#12011yescharacteristic
505C3×C4—21no—
506C3×C4—21no—
507C3×C4—21no—
508C3×C4—21no—
509C3×C4—21no—
510C3×C4—21no—
511C3×C4—42no—
512C3×C4—42no—
513C3×C4—41no—
514C3×C4—41no—
515C3×C4—42no—
516C3×C4—41no—
517C3×C4—81no—
518C3×C4—81no—
519C23—22no—
520C23—21no—
521C23C3×#12011yescharacteristic
522Q8—42no—
523Q8—41no—
524Q8—41no—
525Q8—41no—
526Q8—42no—
527Q8—82no—
528D4—42no—
529D4—41no—
530D4—42no—
531D4—44no—
532D4—42no—
533D4—81no—
534D4—82no—
535D4—82no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—82no—
550C2×C4—81no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4C3×#12011yescharacteristic
555C2×C4C3×#12011yescharacteristic
556C2×C4C3×#12011yescharacteristic
557C2×C4C22×C3×D411yescharacteristic
558C2×C4C22×C3×D411yescharacteristic
559C2×C4C22×C3×D411yescharacteristic
560C8—42no—
561C8—42no—
562C8—41no—
563C8—41no—
564C2×C3—41no—
565C2×C3—82no—
566C2×C3—81no—
567C2×C3#301911yescharacteristic
568C2×C3#302011yescharacteristic
569C2×C3#302211yescharacteristic
570C22—21no—
571C22—21no—
572C22—21no—
573C22—42no—
574C22—41no—
575C22—41no—
576C22—42no—
577C22—42no—
578C22—41no—
579C22C2×C3×#12011yescharacteristic
580C4—21no—
581C4—21no—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—42no—
587C4—42no—
588C4—41no—
589C4—41no—
590C4—42no—
591C4—41no—
592C4—81no—
593C4—81no—
594C3#1908911yescharacteristic, direct product factor, Sylow
595C2—41no—
596C2—82no—
597C2—81no—
598C2C3×#301911yescharacteristic
599C2C3×#302011yescharacteristic
600C2C3×#302211yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC42C24×C3
prime radicalC3×#3559C2
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×#3240C2semidirect
3. C3×#3236C2semidirect
4. C3×#3216C2semidirect
5. C3×#1664C2semidirect
6. C3×#1662C2semidirect
7. C3×#1661C2semidirect
8. C3×#1660C2semidirect
9. C3×#1536C2semidirect
10. C3×#1535C2semidirect
11. C3×#1534C2semidirect
12. #19089C3direct (N × H)
57. #3240C2×C3semidirect
58. #3236C2×C3semidirect
59. #3216C2×C3semidirect
60. #1664C2×C3semidirect
61. #1662C2×C3semidirect
62. #1661C2×C3semidirect
63. #1660C2×C3semidirect
64. #1536C2×C3semidirect
65. #1535C2×C3semidirect
66. #1534C2×C3semidirect
594. C3#19089direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35592
C3×#32402
C3×#32362
C3×#32162
C3×#16642
C3×#16622
C3×#16612
C3×#16602
C3×#15362
C3×#15352
C3×#15342
#190893