Finite Group Databasebeta

← C3×#16707 #10635314

Subgroups — C3×#16707 #10635314

Order 768.

1306 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#1445C212yesmaximal, complemented
2C3×#1442C212yesmaximal, complemented
3C3×#1442C212yesmaximal, complemented
4C3×#1442C212yesmaximal, complemented
5C3×#1439C211yesmaximal, characteristic
6C3×#1439C212yesmaximal
7C3×#1439C211yesmaximal, characteristic
8C3×C22⋊#95C211yesmaximal, characteristic
9C3×C22⋊#95C212yesmaximal
10#16707C311yesmaximal, characteristic, direct product factor, Sylow
11C24×C3×C4C2211yescharacteristic
12C22×C3×C4⋊C4C2212yes—
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C22⋊C4C2212yes—
16C22×C3×C22⋊C4C2212yes—
17C22×C3×C22⋊C4C2211yescharacteristic
18C22×C3×C22⋊C4C2211yescharacteristic
19C2×C3×#95C2212yes—
20C2×C3×#95C2212yes—
21C2×C3×#95C2212yes—
22C2×C3×#95C2212yes—
23C2×C3×#95C2212yes—
24C2×C3×#95C2212yes—
25C2×C3×#95C2212yes—
26C2×C3×#95C2212yes—
27C2×C3×#95C2212yes—
28C2×C3×#95C2211yescharacteristic
29C2×C3×#95C2212yes—
30C2×C3×#95C2212yes—
31C2×C3×#95C2211yescharacteristic
32#1445C2×C312yescomplemented
33#1442C2×C312yescomplemented
34#1442C2×C312yescomplemented
35#1442C2×C312yescomplemented
36#1439C2×C311yescharacteristic
37#1439C2×C312yes—
38#1439C2×C311yescharacteristic
39C22⋊#95C2×C312yes—
40C22⋊#95C2×C311yescharacteristic
41C25×C3C2311yescharacteristic
42C23×C3×C4—22no—
43C23×C3×C4—24no—
44C23×C3×C4—41no—
45C23×C3×C4—42no—
46C23×C3×C4—41no—
47C23×C3×C4C2312yes—
48C23×C3×C4C2312yes—
49C23×C3×C4C2312yes—
50C23×C3×C4C2312yes—
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2312yes—
54C23×C3×C4C2312yes—
55C2×C3×C4⋊C4—24no—
56C2×C3×C4⋊C4—42no—
57C2×C3×C4⋊C4—42no—
58C2×C3×C4⋊C4—42no—
59C2×C3×C4⋊C4—42no—
60C2×C3×C4⋊C4—41no—
61C2×C3×C4⋊C4—41no—
62C2×C3×C22⋊C4—24no—
63C2×C3×C22⋊C4—22no—
64C2×C3×C22⋊C4—24no—
65C2×C3×C22⋊C4—24no—
66C2×C3×C22⋊C4—41no—
67C2×C3×C22⋊C4—41no—
68C2×C3×C22⋊C4—42no—
69C2×C3×C22⋊C4—42no—
70C2×C3×C22⋊C4—41no—
71C2×C3×C22⋊C4—41no—
72C2×C3×C22⋊C4—42no—
73C2×C3×C22⋊C4Q814yes—
74C3×#95—24no—
75C3×#95—24no—
76C3×#95—24no—
77C3×#95—41no—
78C3×#95—42no—
79C3×#95—41no—
80C3×#95—42no—
81C3×#95—42no—
82C3×#95—42no—
83C3×#95—42no—
84C3×#95—42no—
85C3×#95—42no—
86C3×#95—42no—
87C24×C4C22×C311yescharacteristic
88C22×C4⋊C4C22×C312yes—
89C22×C4⋊C4C22×C311yescharacteristic
90C22×C4⋊C4C22×C311yescharacteristic
91C22×C22⋊C4C22×C312yes—
92C22×C22⋊C4C22×C312yes—
93C22×C22⋊C4C22×C311yescharacteristic
94C22×C22⋊C4C22×C311yescharacteristic
95C2×#95C22×C312yes—
96C2×#95C22×C312yes—
97C2×#95C22×C312yes—
98C2×#95C22×C312yes—
99C2×#95C22×C312yes—
100C2×#95C22×C312yes—
101C2×#95C22×C312yes—
102C2×#95C22×C312yes—
103C2×#95C22×C312yes—
104C2×#95C22×C311yescharacteristic
105C2×#95C22×C312yes—
106C2×#95C22×C312yes—
107C2×#95C22×C311yescharacteristic
108C24×C3—22no—
109C24×C3—22no—
110C24×C3—21no—
111C24×C3—22no—
112C24×C3—21no—
113C24×C3—22no—
114C24×C3—21no—
115C24×C3—21no—
116C24×C3C2×Q812yes—
117C24×C3#4112yes—
118C24×C3#4112yes—
119C24×C3C2411yescharacteristic
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—44no—
169C22×C3×C4—48no—
170C22×C3×C4—48no—
171C22×C3×C4—44no—
172C22×C3×C4—44no—
173C22×C3×C4C2×Q812yes—
174C22×C3×C4C2×Q812yes—
175C22×C3×C4#4114yes—
176C22×C3×C4#4114yes—
177C3×C4⋊C4—48no—
178C3×C4⋊C4—44no—
179C3×C4⋊C4—44no—
180C3×C22⋊C4—44no—
181C3×C22⋊C4—48no—
182C3×C22⋊C4—48no—
183C3×C22⋊C4—44no—
184C25C23×C311yescharacteristic
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—41no—
188C23×C4—42no—
189C23×C4—41no—
190C23×C4C23×C312yes—
191C23×C4C23×C312yes—
192C23×C4C23×C312yes—
193C23×C4C23×C312yes—
194C23×C4C23×C311yescharacteristic
195C23×C4C23×C311yescharacteristic
196C23×C4C23×C312yes—
197C23×C4C23×C312yes—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4C3×Q814yes—
217#95—24no—
218#95—24no—
219#95—24no—
220#95—41no—
221#95—42no—
222#95—41no—
223#95—42no—
224#95—42no—
225#95—42no—
226#95—42no—
227#95—42no—
228#95—42no—
229#95—42no—
230C23×C3—24no—
231C23×C3—22no—
232C23×C3—24no—
233C23×C3—22no—
234C23×C3—22no—
235C23×C3—22no—
236C23×C3—22no—
237C23×C3—22no—
238C23×C3—24no—
239C23×C3—24no—
240C23×C3—24no—
241C23×C3—24no—
242C23×C3—42no—
243C23×C3—42no—
244C23×C3—42no—
245C23×C3—42no—
246C23×C3—41no—
247C23×C3—42no—
248C23×C3—41no—
249C23×C3—41no—
250C23×C3—41no—
251C23×C3—42no—
252C23×C3#12514yes—
253C23×C3C22×Q811yescharacteristic
254C23×C3C2×#4111yescharacteristic
255C23×C3C2×#4111yescharacteristic
256C23×C3C2×#4112yes—
257C23×C3C2×#4112yes—
258C23×C3H(5,𝔽2)11yescharacteristic
259C23×C3H(5,𝔽2)11yescharacteristic
260C23×C3H(5,𝔽2)12yes—
261C23×C3H(5,𝔽2)11yescharacteristic
262C23×C3H(5,𝔽2)12yes—
263C23×C32−1+411yescharacteristic
264C2×C3×C4—24no—
265C2×C3×C4—24no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—24no—
269C2×C3×C4—24no—
270C2×C3×C4—24no—
271C2×C3×C4—24no—
272C2×C3×C4—24no—
273C2×C3×C4—24no—
274C2×C3×C4—416no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—41no—
283C2×C3×C4—42no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C24—22no—
318C24—22no—
319C24—21no—
320C24—22no—
321C24—21no—
322C24—22no—
323C24—21no—
324C24—21no—
325C24C2×C3×Q812yes—
326C24C3×#4112yes—
327C24C3×#4112yes—
328C24C24×C311yescharacteristic
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—44no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4C2×C3×Q812yes—
383C22×C4C2×C3×Q812yes—
384C22×C4C3×#4114yes—
385C22×C4C3×#4114yes—
386C4⋊C4—48no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C22⋊C4—44no—
390C22⋊C4—48no—
391C22⋊C4—44no—
392C22⋊C4—48no—
393C22×C3—24no—
394C22×C3—28no—
395C22×C3—42no—
396C22×C3—44no—
397C22×C3—42no—
398C22×C3—44no—
399C22×C3—42no—
400C22×C3—42no—
401C22×C3—44no—
402C22×C3—42no—
403C22×C3—42no—
404C22×C3C2×#12511yescharacteristic
405C22×C3#52912yes—
406C22×C3#53812yes—
407C22×C3#53811yescharacteristic
408C22×C3#53811yescharacteristic
409C22×C3#53812yes—
410C22×C3#53812yes—
411C22×C3#53911yescharacteristic
412C22×C3#53911yescharacteristic
413C22×C3#54012yes—
414C22×C3#54012yes—
415C22×C3#54311yescharacteristic
416C22×C3#54411yescharacteristic
417C22×C3#55112yes—
418C22×C3#55111yescharacteristic
419C22×C3#55112yes—
420C22×C3#55212yes—
421C22×C3#55412yes—
422C22×C3#55412yes—
423C22×C3#55911yescharacteristic
424C22×C3#56011yescharacteristic
425C22×C3H(3,𝔽4)11yescharacteristic
426C22×C3#56212yes—
427C3×C4—44no—
428C3×C4—44no—
429C3×C4—42no—
430C3×C4—42no—
431C3×C4—82no—
432C3×C4—82no—
433C3×C4—82no—
434C3×C4—82no—
435C23—24no—
436C23—22no—
437C23—24no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—41no—
456C23—42no—
457C23C3×#12514yes—
458C23C22×C3×Q811yescharacteristic
459C23C2×C3×#4112yes—
460C23C2×C3×#4111yescharacteristic
461C23C2×C3×#4111yescharacteristic
462C23C2×C3×#4112yes—
463C23C3×H(5,𝔽2)11yescharacteristic
464C23C3×H(5,𝔽2)12yes—
465C23C3×H(5,𝔽2)12yes—
466C23C3×H(5,𝔽2)11yescharacteristic
467C23C3×H(5,𝔽2)11yescharacteristic
468C23C3×2−1+411yescharacteristic
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—416no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C3—44no—
523C2×C3#250512yes—
524C2×C3#251711yescharacteristic
525C2×C3#264611yescharacteristic
526C2×C3#264812yes—
527C2×C3#273611yescharacteristic
528C2×C3#274312yes—
529C2×C3#274811yescharacteristic
530C2×C3#275211yescharacteristic
531C2×C3#276812yes—
532C2×C3#280411yescharacteristic
533C2×C3#282411yescharacteristic
534C22—24no—
535C22—28no—
536C22—44no—
537C22—42no—
538C22—44no—
539C22—42no—
540C22—42no—
541C22—44no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22C2×C3×#12511yescharacteristic
546C22C3×#52912yes—
547C22C3×#53811yescharacteristic
548C22C3×#53811yescharacteristic
549C22C3×#53812yes—
550C22C3×#53812yes—
551C22C3×#53812yes—
552C22C3×#53911yescharacteristic
553C22C3×#53911yescharacteristic
554C22C3×#54012yes—
555C22C3×#54012yes—
556C22C3×#54311yescharacteristic
557C22C3×#54411yescharacteristic
558C22C3×#55112yes—
559C22C3×#55111yescharacteristic
560C22C3×#55112yes—
561C22C3×#55212yes—
562C22C3×#55412yes—
563C22C3×#55412yes—
564C22C3×#55911yescharacteristic
565C22C3×#56011yescharacteristic
566C22C3×H(3,𝔽4)11yescharacteristic
567C22C3×#56212yes—
568C4—44no—
569C4—44no—
570C4—42no—
571C4—42no—
572C4—82no—
573C4—82no—
574C4—82no—
575C4—82no—
576C3#1670711yescharacteristic, direct product factor, Sylow
577C2—44no—
578C2C3×#250512yes—
579C2C3×#251711yescharacteristic
580C2C3×#264611yescharacteristic
581C2C3×#264812yes—
582C2C3×#273611yescharacteristic
583C2C3×#274312yes—
584C2C3×#274811yescharacteristic
585C2C3×#275211yescharacteristic
586C2C3×#276812yes—
587C2C3×#280411yescharacteristic
588C2C3×#282411yescharacteristic
subgroupG/H
center, socleC24×C3C24
commutator, Frattini, cosocleC24C24×C3
prime radicalC25×C3C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#1445C2semidirect
2. C3×#1442C2semidirect
3. C3×#1442C2semidirect
4. C3×#1442C2semidirect
10. #16707C3direct (N × H)
32. #1445C2×C3semidirect
33. #1442C2×C3semidirect
34. #1442C2×C3semidirect
35. #1442C2×C3semidirect
576. C3#16707direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#14452
C3×#14422
C3×#14392
C3×C22⋊#952
#167073