Finite Group Databasebeta

← C3×#11069 #10629676

Subgroups — C3×#11069 #10629676

Order 768.

2538 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2315C211yesmaximal, characteristic, complemented
2C2×C3×#379C212yesmaximal
3C2×C3×#379C211yesmaximal, characteristic
4C2×C3×#342C212yesmaximal, complemented
5C2×C3×#342C211yesmaximal, characteristic, complemented
6#11069C311yesmaximal, characteristic, direct product factor, Sylow
7C26×C3C2211yescharacteristic
8C22×C3×C22⋊C4—24no—
9C22×C3×C22⋊C4—22no—
10C22×C3×C22⋊C4C2212yes—
11C22×C3×C22⋊C4C2211yescharacteristic
12C22×C3×C22⋊C4C2212yes—
13C22×C3×C22⋊C4C2211yescharacteristic
14C3×#385—24no—
15C3×#380—22no—
16C3×#379—24no—
17C3×#379—22no—
18C3×#379—24no—
19C3×#379C414yescomplemented
20C3×#342—22no—
21C3×#342—24no—
22#2315C2×C311yescharacteristic, complemented
23C2×#379C2×C312yes—
24C2×#379C2×C311yescharacteristic
25C2×#342C2×C312yescomplemented
26C2×#342C2×C311yescharacteristic, complemented
27C25×C3—21no—
28C25×C3—22no—
29C25×C3—21no—
30C25×C3—22no—
31C25×C3—22no—
32C25×C3—21no—
33C25×C3—21no—
34C25×C3—22no—
35C25×C3—48no—
36C25×C3C2×C412yes—
37C25×C3D412yes—
38C25×C3Q812yes—
39C25×C3C2311yescharacteristic
40C23×C3×C4—21no—
41C23×C3×C4—22no—
42C23×C3×C4D414yes—
43C23×C3×C4D412yes—
44C2×C3×C4⋊C4—24no—
45C2×C3×C22⋊C4—22no—
46C2×C3×C22⋊C4—22no—
47C2×C3×C22⋊C4—22no—
48C2×C3×C22⋊C4—22no—
49C2×C3×C22⋊C4—22no—
50C2×C3×C22⋊C4—22no—
51C2×C3×C22⋊C4—22no—
52C2×C3×C22⋊C4—22no—
53C2×C3×C22⋊C4—216no—
54C2×C3×C22⋊C4—22no—
55C2×C3×C22⋊C4—21no—
56C2×C3×C22⋊C4—22no—
57C2×C3×C22⋊C4—22no—
58C2×C3×C22⋊C4—22no—
59C2×C3×C22⋊C4—22no—
60C2×C3×C22⋊C4—22no—
61C2×C3×C22⋊C4—22no—
62C2×C3×C22⋊C4—28no—
63C2×C3×C22⋊C4—48no—
64C2×C3×C22⋊C4—416no—
65C2×C3×C22⋊C4C2×C412yes—
66C2×C3×C22⋊C4C2×C412yes—
67C2×C3×C22⋊C4D414yes—
68C2×C3×C22⋊C4D412yes—
69C3×#95—24no—
70C26C22×C311yescharacteristic
71C22×C22⋊C4—24no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4C22×C312yes—
74C22×C22⋊C4C22×C311yescharacteristic
75C22×C22⋊C4C22×C312yes—
76C22×C22⋊C4C22×C311yescharacteristic
77#385—24no—
78#380—22no—
79#379—24no—
80#379—24no—
81#379—22no—
82#379C3×C414yescomplemented
83#342—22no—
84#342—24no—
85C24×C3—28no—
86C24×C3—216no—
87C24×C3—22no—
88C24×C3—24no—
89C24×C3—24no—
90C24×C3—22no—
91C24×C3—22no—
92C24×C3—24no—
93C24×C3—22no—
94C24×C3—24no—
95C24×C3—22no—
96C24×C3—22no—
97C24×C3—24no—
98C24×C3—24no—
99C24×C3—48no—
100C24×C3—48no—
101C24×C3—48no—
102C24×C3—44no—
103C24×C3—44no—
104C24×C3—48no—
105C24×C3—48no—
106C24×C3—416no—
107C24×C3—416no—
108C24×C3—48no—
109C24×C3—48no—
110C24×C3—416no—
111C24×C3—41no—
112C24×C3—42no—
113C24×C3—42no—
114C24×C3—42no—
115C24×C3—41no—
116C24×C3—42no—
117C24×C3—42no—
118C24×C3—41no—
119C24×C3—42no—
120C24×C3—41no—
121C24×C3C4⋊C414yescomplemented
122C24×C3C22×C411yescharacteristic
123C24×C3C2×D411yescharacteristic
124C24×C3C2×D412yes—
125C24×C3C2×D411yescharacteristic
126C24×C3C2×D411yescharacteristic
127C24×C3C2×D412yes—
128C24×C3C2×Q811yescharacteristic
129C24×C3#4111yescharacteristic
130C24×C3#4111yescharacteristic
131C24×C3#4112yes—
132C24×C3#4112yes—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—24no—
137C22×C3×C4—24no—
138C22×C3×C4—24no—
139C22×C3×C4—24no—
140C22×C3×C4—44no—
141C22×C3×C4—48no—
142C22×C3×C4—44no—
143C22×C3×C4—48no—
144C22×C3×C4—42no—
145C22×C3×C4—41no—
146C22×C3×C4—42no—
147C22×C3×C4—41no—
148C22×C3×C4C22⋊C414yescomplemented
149C3×C4⋊C4—84no—
150C3×C22⋊C4—24no—
151C3×C22⋊C4—24no—
152C3×C22⋊C4—24no—
153C3×C22⋊C4—24no—
154C3×C22⋊C4—24no—
155C3×C22⋊C4—416no—
156C3×C22⋊C4—432no—
157C3×C22⋊C4—42no—
158C3×C22⋊C4—42no—
159C3×C22⋊C4—42no—
160C3×C22⋊C4—42no—
161C3×C22⋊C4—42no—
162C3×C22⋊C4—42no—
163C3×C22⋊C4—42no—
164C25—21no—
165C25—22no—
166C25—21no—
167C25—22no—
168C25—22no—
169C25—21no—
170C25—21no—
171C25—22no—
172C25—48no—
173C25C2×C3×C412yes—
174C25C3×D412yes—
175C25C3×Q812yes—
176C25C23×C311yescharacteristic
177C23×C4—21no—
178C23×C4—22no—
179C23×C4C3×D414yes—
180C23×C4C3×D412yes—
181C2×C4⋊C4—24no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—216no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—48no—
201C2×C22⋊C4—416no—
202C2×C22⋊C4C2×C3×C412yes—
203C2×C22⋊C4C2×C3×C412yes—
204C2×C22⋊C4C3×D414yes—
205C2×C22⋊C4C3×D412yes—
206#95—24no—
207C23×C3—24no—
208C23×C3—28no—
209C23×C3—28no—
210C23×C3—216no—
211C23×C3—216no—
212C23×C3—28no—
213C23×C3—28no—
214C23×C3—216no—
215C23×C3—24no—
216C23×C3—28no—
217C23×C3—48no—
218C23×C3—42no—
219C23×C3—42no—
220C23×C3—42no—
221C23×C3—44no—
222C23×C3—42no—
223C23×C3—44no—
224C23×C3—42no—
225C23×C3—42no—
226C23×C3—44no—
227C23×C3—44no—
228C23×C3—416no—
229C23×C3—48no—
230C23×C3—48no—
231C23×C3—48no—
232C23×C3—416no—
233C23×C3—48no—
234C23×C3—44no—
235C23×C3—48no—
236C23×C3—416no—
237C23×C3—44no—
238C23×C3—48no—
239C23×C3—48no—
240C23×C3—416no—
241C23×C3—48no—
242C23×C3—48no—
243C23×C3—416no—
244C23×C3—416no—
245C23×C3—416no—
246C23×C3—416no—
247C23×C3—416no—
248C23×C3—42no—
249C23×C3—44no—
250C23×C3—42no—
251C23×C3—42no—
252C23×C3—42no—
253C23×C3—44no—
254C23×C3—44no—
255C23×C3—42no—
256C23×C3—42no—
257C23×C3—48no—
258C23×C3C2×C22⋊C411yescharacteristic
259C23×C3C2×C4⋊C411yescharacteristic
260C23×C3#11711yescharacteristic
261C23×C3C4×D412yes—
262C23×C3C4×D412yes—
263C23×C3#12012yes—
264C23×C3#12011yescharacteristic
265C23×C3#12111yescharacteristic
266C23×C3#12112yes—
267C23×C3#12112yes—
268C23×C3#12111yescharacteristic
269C23×C3#12112yes—
270C23×C3C22⋊Q811yescharacteristic
271C23×C3C22⋊Q812yes—
272C23×C3C22⋊Q811yescharacteristic
273C23×C3C22⋊Q812yes—
274C23×C3#12311yescharacteristic
275C23×C3#12312yes—
276C23×C3#12312yes—
277C23×C3#12412yes—
278C23×C3#12411yescharacteristic
279C23×C3#12612yes—
280C23×C3#12711yescharacteristic
281C2×C3×C4—24no—
282C2×C3×C4—44no—
283C2×C3×C4—44no—
284C2×C3×C4—44no—
285C2×C3×C4—48no—
286C2×C3×C4—48no—
287C2×C3×C4—42no—
288C2×C3×C4—44no—
289C2×C3×C4—44no—
290C2×C3×C4—48no—
291C2×C3×C4—82no—
292C2×C3×C4—84no—
293C2×C3×C4—82no—
294C2×C3×C4—82no—
295C2×C3×C4—84no—
296C2×C3×C4—84no—
297C24—216no—
298C24—28no—
299C24—22no—
300C24—24no—
301C24—24no—
302C24—22no—
303C24—22no—
304C24—24no—
305C24—22no—
306C24—24no—
307C24—22no—
308C24—22no—
309C24—24no—
310C24—24no—
311C24—48no—
312C24—48no—
313C24—48no—
314C24—44no—
315C24—48no—
316C24—44no—
317C24—416no—
318C24—48no—
319C24—48no—
320C24—48no—
321C24—416no—
322C24—416no—
323C24—41no—
324C24—42no—
325C24—42no—
326C24—42no—
327C24—41no—
328C24—42no—
329C24—42no—
330C24—41no—
331C24—42no—
332C24—41no—
333C24C3×C4⋊C414yescomplemented
334C24C22×C3×C411yescharacteristic
335C24C2×C3×D411yescharacteristic
336C24C2×C3×D412yes—
337C24C2×C3×D411yescharacteristic
338C24C2×C3×D411yescharacteristic
339C24C2×C3×D412yes—
340C24C2×C3×Q811yescharacteristic
341C24C3×#4111yescharacteristic
342C24C3×#4111yescharacteristic
343C24C3×#4112yes—
344C24C3×#4112yes—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—44no—
353C22×C4—48no—
354C22×C4—48no—
355C22×C4—44no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4C3×C22⋊C414yescomplemented
361C4⋊C4—84no—
362C22⋊C4—24no—
363C22⋊C4—24no—
364C22⋊C4—24no—
365C22⋊C4—24no—
366C22⋊C4—24no—
367C22⋊C4—416no—
368C22⋊C4—432no—
369C22⋊C4—42no—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22×C3—22no—
377C22×C3—24no—
378C22×C3—22no—
379C22×C3—24no—
380C22×C3—24no—
381C22×C3—24no—
382C22×C3—22no—
383C22×C3—22no—
384C22×C3—22no—
385C22×C3—24no—
386C22×C3—22no—
387C22×C3—24no—
388C22×C3—28no—
389C22×C3—216no—
390C22×C3—41no—
391C22×C3—41no—
392C22×C3—42no—
393C22×C3—42no—
394C22×C3—41no—
395C22×C3—41no—
396C22×C3—42no—
397C22×C3—42no—
398C22×C3—42no—
399C22×C3—42no—
400C22×C3—48no—
401C22×C3—416no—
402C22×C3—44no—
403C22×C3—48no—
404C22×C3—416no—
405C22×C3—416no—
406C22×C3—48no—
407C22×C3—48no—
408C22×C3—48no—
409C22×C3—44no—
410C22×C3—48no—
411C22×C3—48no—
412C22×C3#38011yescharacteristic
413C22×C3#38512yes—
414C22×C3#38611yescharacteristic
415C22×C3#38812yes—
416C22×C3#39011yescharacteristic
417C22×C3#39211yescharacteristic
418C22×C3#39311yescharacteristic
419C22×C3#39412yes—
420C22×C3#39612yes—
421C22×C3#39711yescharacteristic
422C22×C3#39911yescharacteristic
423C22×C3B(4,𝔽2)14yes—
424C3×C4—44no—
425C3×C4—82no—
426C3×C4—84no—
427C3×C4—84no—
428C23—28no—
429C23—24no—
430C23—216no—
431C23—28no—
432C23—28no—
433C23—28no—
434C23—216no—
435C23—216no—
436C23—24no—
437C23—28no—
438C23—48no—
439C23—42no—
440C23—42no—
441C23—44no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—44no—
447C23—44no—
448C23—48no—
449C23—416no—
450C23—48no—
451C23—44no—
452C23—48no—
453C23—416no—
454C23—44no—
455C23—48no—
456C23—44no—
457C23—416no—
458C23—48no—
459C23—48no—
460C23—48no—
461C23—416no—
462C23—48no—
463C23—48no—
464C23—416no—
465C23—416no—
466C23—416no—
467C23—416no—
468C23—416no—
469C23—42no—
470C23—44no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—44no—
475C23—44no—
476C23—42no—
477C23—42no—
478C23—48no—
479C23C2×C3×C22⋊C411yescharacteristic
480C23C2×C3×C4⋊C411yescharacteristic
481C23C3×#11711yescharacteristic
482C23C3×C4×D412yes—
483C23C3×C4×D412yes—
484C23C3×#12012yes—
485C23C3×#12011yescharacteristic
486C23C3×#12111yescharacteristic
487C23C3×#12112yes—
488C23C3×#12112yes—
489C23C3×#12111yescharacteristic
490C23C3×#12112yes—
491C23C3×C22⋊Q811yescharacteristic
492C23C3×C22⋊Q812yes—
493C23C3×C22⋊Q811yescharacteristic
494C23C3×C22⋊Q812yes—
495C23C3×#12311yescharacteristic
496C23C3×#12312yes—
497C23C3×#12312yes—
498C23C3×#12412yes—
499C23C3×#12411yescharacteristic
500C23C3×#12612yes—
501C23C3×#12711yescharacteristic
502C2×C4—24no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—48no—
506C2×C4—44no—
507C2×C4—48no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—48no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—84no—
516C2×C4—84no—
517C2×C4—84no—
518C2×C3—21no—
519C2×C3—22no—
520C2×C3—21no—
521C2×C3—22no—
522C2×C3—22no—
523C2×C3—22no—
524C2×C3—21no—
525C2×C3—21no—
526C2×C3—48no—
527C2×C3#143911yescharacteristic
528C2×C3#188912yes—
529C2×C3#202112yes—
530C2×C3#202912yes—
531C22—24no—
532C22—24no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—24no—
537C22—22no—
538C22—24no—
539C22—22no—
540C22—24no—
541C22—22no—
542C22—24no—
543C22—28no—
544C22—216no—
545C22—41no—
546C22—41no—
547C22—42no—
548C22—42no—
549C22—41no—
550C22—41no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—48no—
556C22—416no—
557C22—44no—
558C22—48no—
559C22—416no—
560C22—416no—
561C22—48no—
562C22—48no—
563C22—48no—
564C22—48no—
565C22—48no—
566C22—44no—
567C22C3×#38011yescharacteristic
568C22C3×#38512yes—
569C22C3×#38611yescharacteristic
570C22C3×#38812yes—
571C22C3×#39011yescharacteristic
572C22C3×#39211yescharacteristic
573C22C3×#39311yescharacteristic
574C22C3×#39412yes—
575C22C3×#39612yes—
576C22C3×#39711yescharacteristic
577C22C3×#39911yescharacteristic
578C22C3×B(4,𝔽2)14yes—
579C4—44no—
580C4—82no—
581C4—84no—
582C4—84no—
583C3#1106911yescharacteristic, direct product factor, Sylow
584C2—21no—
585C2—22no—
586C2—21no—
587C2—22no—
588C2—22no—
589C2—22no—
590C2—21no—
591C2—21no—
592C2—48no—
593C2C3×#143911yescharacteristic
594C2C3×#188912yes—
595C2C3×#202112yes—
596C2C3×#202912yes—
subgroupG/H
center, socleC23×C3#120
commutatorC24C22×C3×C4
Frattini, cosocleC25C23×C3
prime radicalC26×C3C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C25×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#2315C2semidirect
4. C2×C3×#342C2semidirect
5. C2×C3×#342C2semidirect
6. #11069C3direct (N × H)
19. C3×#379C4semidirect
22. #2315C2×C3semidirect
25. C2×#342C2×C3semidirect
26. C2×#342C2×C3semidirect
82. #379C3×C4semidirect
121. C24×C3C4⋊C4semidirect
148. C22×C3×C4C22⋊C4semidirect
333. C24C3×C4⋊C4semidirect
360. C22×C4C3×C22⋊C4semidirect
583. C3#11069direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#23152
C2×C3×#3792
C2×C3×#3422
#110693