Finite Group Databasebeta

← C3×#33020 #10651627

Subgroups — C3×#33020 #10651627

Order 768.

1824 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
1C2×C3×#518C211yesmaximal, characteristic, complemented
2C3×#2405C212yesmaximal, complemented
3C3×#2403C211yesmaximal, characteristic
4C3×#1889C214yesmaximal, complemented
5C3×#1867C214yesmaximal, complemented
6C3×#1819C212yesmaximal, complemented
7C3×#1781C211yesmaximal, characteristic, complemented
8#33020C311yesmaximal, characteristic, direct product factor, Sylow
9C23×C3×D4C2211yescharacteristic
10C23×C3×D4C2211yescharacteristic
11C2×C3×#123—22no—
12C2×C3×#120—21no—
13C2×C3×#120C2212yes—
14C3×#518—22no—
15C3×#518—24no—
16C3×#518C2214yes—
17C2×C3×C4×D4—22no—
18C2×C3×C4×D4—22no—
19C2×C3×#117—21no—
20C22×C3×C22⋊C4—21no—
21C22×C3×C22⋊C4C2212yes—
22C2×C3×#99C2214yes—
23C3×#394—24no—
24C3×#392—22no—
25C3×#386—21no—
26C3×#386—22no—
27C3×#386C2212yes—
28C3×#385C2214yescomplemented
29C3×#379C2211yescharacteristic
30C3×#379C2212yes—
31C3×#379C2214yescomplemented
32C3×#342C2218yescomplemented
33C2×#518C2×C311yescharacteristic, complemented
34#2405C2×C312yescomplemented
35#2403C2×C311yescharacteristic
36#1889C2×C314yescomplemented
37#1867C2×C314yescomplemented
38#1819C2×C312yescomplemented
39#1781C2×C311yescharacteristic, complemented
40C25×C3D411yescharacteristic
41C25×C3C2311yescharacteristic
42C25×C3C2311yescharacteristic
43C22×C3×D4—22no—
44C22×C3×D4—22no—
45C22×C3×D4—28no—
46C22×C3×D4—21no—
47C22×C3×D4—22no—
48C22×C3×D4—21no—
49C22×C3×D4—22no—
50C22×C3×D4—22no—
51C22×C3×D4—44no—
52C22×C3×D4D412yes—
53C22×C3×D4C2312yes—
54C23×C3×C4—21no—
55C23×C3×C4—21no—
56C23×C3×C4D411yescharacteristic
57C23×C3×C4D412yescomplemented
58C23×C3×C4D412yescomplemented
59C23×C3×C4C2311yescharacteristic
60C3×#123—48no—
61C3×#120—28no—
62C3×#120—44no—
63C3×C4×D4—28no—
64C3×C4×D4—28no—
65C3×C4×D4—28no—
66C3×C4×D4—28no—
67C3×#117—24no—
68C3×#117—24no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—24no—
72C2×C3×C22⋊C4—22no—
73C2×C3×C22⋊C4—24no—
74C2×C3×C22⋊C4—22no—
75C2×C3×C22⋊C4—24no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—21no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—24no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4—24no—
87C2×C3×C22⋊C4D414yescomplemented
88C2×C3×C22⋊C4C2314yes—
89C2×C3×C22⋊C4C2312yes—
90C2×C3×C22⋊C4C2314yes—
91C2×C3×C42—22no—
92C3×#99—28no—
93C3×#95—24no—
94C3×#95—22no—
95C23×D4C22×C311yescharacteristic
96C23×D4C22×C311yescharacteristic
97C2×#123—22no—
98C2×#120—21no—
99C2×#120C22×C312yes—
100#518—22no—
101#518—24no—
102#518C22×C314yes—
103C2×C4×D4—22no—
104C2×C4×D4—22no—
105C2×#117—21no—
106C22×C22⋊C4—21no—
107C22×C22⋊C4C22×C312yes—
108C2×#99C22×C314yes—
109#394—24no—
110#392—22no—
111#386—21no—
112#386—22no—
113#386C22×C312yes—
114#385C22×C314yescomplemented
115#379C22×C311yescharacteristic
116#379C22×C312yes—
117#379C22×C314yescomplemented
118#342C22×C318yescomplemented
119C24×C3—21no—
120C24×C3—21no—
121C24×C3—24no—
122C24×C3—24no—
123C24×C3—22no—
124C24×C3—21no—
125C24×C3—24no—
126C24×C3—42no—
127C24×C3—42no—
128C24×C3—42no—
129C24×C3—81no—
130C24×C3—81no—
131C24×C3—81no—
132C24×C3C2×D411yescharacteristic
133C24×C3C2×D412yes—
134C24×C3C2×D411yescharacteristic
135C24×C3C2×D412yes—
136C24×C3C2×D412yes—
137C24×C3C2411yescharacteristic
138C2×C3×D4—216no—
139C2×C3×D4—216no—
140C2×C3×D4—24no—
141C2×C3×D4—24no—
142C2×C3×D4—24no—
143C2×C3×D4—24no—
144C2×C3×D4—24no—
145C2×C3×D4—28no—
146C2×C3×D4—24no—
147C2×C3×D4—42no—
148C2×C3×D4—44no—
149C2×C3×D4—44no—
150C2×C3×D4—42no—
151C2×C3×D4—44no—
152C2×C3×D4—42no—
153C2×C3×D4—41no—
154C2×C3×D4—42no—
155C2×C3×D4—41no—
156C2×C3×D4—44no—
157C2×C3×D4—44no—
158C2×C3×D4—82no—
159C2×C3×D4#4118yes—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—24no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—21no—
166C22×C3×C4—22no—
167C22×C3×C4—24no—
168C22×C3×C4—24no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—24no—
172C22×C3×C4—22no—
173C22×C3×C4—42no—
174C22×C3×C4—44no—
175C22×C3×C4—82no—
176C22×C3×C4—82no—
177C22×C3×C4—82no—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D414yes—
180C22×C3×C4C2×D414yes—
181C3×C4⋊C4—24no—
182C3×C4⋊C4—24no—
183C3×C4⋊C4—48no—
184C3×C22⋊C4—28no—
185C3×C22⋊C4—24no—
186C3×C22⋊C4—24no—
187C3×C22⋊C4—24no—
188C3×C22⋊C4—24no—
189C3×C22⋊C4—28no—
190C3×C22⋊C4—24no—
191C3×C22⋊C4—44no—
192C3×C22⋊C4—42no—
193C3×C22⋊C4—48no—
194C3×C22⋊C4—44no—
195C3×C22⋊C4—44no—
196C3×C22⋊C4—84no—
197C3×C22⋊C4—82no—
198C3×C22⋊C4—81no—
199C3×C42—28no—
200C25C3×D411yescharacteristic
201C25C23×C311yescharacteristic
202C25C23×C311yescharacteristic
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—28no—
206C22×D4—21no—
207C22×D4—22no—
208C22×D4—21no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—44no—
212C22×D4C3×D412yes—
213C22×D4C23×C312yes—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4C3×D411yescharacteristic
217C23×C4C3×D412yescomplemented
218C23×C4C3×D412yescomplemented
219C23×C4C23×C311yescharacteristic
220#123—48no—
221#120—28no—
222#120—44no—
223C4×D4—28no—
224C4×D4—28no—
225C4×D4—28no—
226C4×D4—28no—
227#117—24no—
228#117—24no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—24no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4C3×D414yescomplemented
248C2×C22⋊C4C23×C312yes—
249C2×C22⋊C4C23×C314yes—
250C2×C22⋊C4C23×C314yes—
251C2×C42—22no—
252#99—28no—
253#95—24no—
254#95—22no—
255C23×C3—24no—
256C23×C3—24no—
257C23×C3—22no—
258C23×C3—22no—
259C23×C3—24no—
260C23×C3—24no—
261C23×C3—22no—
262C23×C3—22no—
263C23×C3—22no—
264C23×C3—24no—
265C23×C3—24no—
266C23×C3—22no—
267C23×C3—41no—
268C23×C3—42no—
269C23×C3—41no—
270C23×C3—41no—
271C23×C3—42no—
272C23×C3—41no—
273C23×C3—42no—
274C23×C3—41no—
275C23×C3—42no—
276C23×C3—42no—
277C23×C3—42no—
278C23×C3—44no—
279C23×C3—42no—
280C23×C3—41no—
281C23×C3—42no—
282C23×C3—42no—
283C23×C3—41no—
284C23×C3—42no—
285C23×C3—44no—
286C23×C3—81no—
287C23×C3—81no—
288C23×C3—88no—
289C23×C3—88no—
290C23×C3—88no—
291C23×C3#12011yescharacteristic
292C23×C3#12012yes—
293C23×C3#12011yescharacteristic
294C23×C3#12314yescomplemented
295C23×C3C22×D411yescharacteristic
296C23×C3C22×D412yes—
297C23×C3C2×#4114yes—
298C3×D4—216no—
299C3×D4—216no—
300C3×D4—48no—
301C3×D4—44no—
302C3×D4—44no—
303C3×D4—88no—
304C2×C3×C4—24no—
305C2×C3×C4—24no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—24no—
312C2×C3×C4—42no—
313C2×C3×C4—41no—
314C2×C3×C4—44no—
315C2×C3×C4—41no—
316C2×C3×C4—44no—
317C2×C3×C4—44no—
318C2×C3×C4—44no—
319C2×C3×C4—44no—
320C2×C3×C4—42no—
321C2×C3×C4—44no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—44no—
325C2×C3×C4—82no—
326C2×C3×C4—82no—
327C2×C3×C4—84no—
328C2×C3×C4—84no—
329C2×C3×C4#12314yescomplemented
330C24—21no—
331C24—21no—
332C24—24no—
333C24—24no—
334C24—22no—
335C24—21no—
336C24—24no—
337C24—42no—
338C24—42no—
339C24—42no—
340C24—81no—
341C24—81no—
342C24—81no—
343C24C2×C3×D411yescharacteristic
344C24C2×C3×D412yes—
345C24C2×C3×D411yescharacteristic
346C24C2×C3×D412yes—
347C24C2×C3×D412yes—
348C24C24×C311yescharacteristic
349C2×D4—216no—
350C2×D4—216no—
351C2×D4—24no—
352C2×D4—24no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—28no—
357C2×D4—24no—
358C2×D4—42no—
359C2×D4—44no—
360C2×D4—44no—
361C2×D4—42no—
362C2×D4—44no—
363C2×D4—42no—
364C2×D4—41no—
365C2×D4—42no—
366C2×D4—41no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—82no—
370C2×D4C3×#4118yes—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—42no—
385C22×C4—44no—
386C22×C4—82no—
387C22×C4—82no—
388C22×C4—82no—
389C22×C4C2×C3×D412yes—
390C22×C4C2×C3×D414yes—
391C22×C4C2×C3×D414yes—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—48no—
395C22⋊C4—28no—
396C22⋊C4—24no—
397C22⋊C4—24no—
398C22⋊C4—24no—
399C22⋊C4—24no—
400C22⋊C4—28no—
401C22⋊C4—24no—
402C22⋊C4—44no—
403C22⋊C4—42no—
404C22⋊C4—48no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—84no—
408C22⋊C4—82no—
409C22⋊C4—81no—
410C42—28no—
411C22×C3—21no—
412C22×C3—22no—
413C22×C3—21no—
414C22×C3—21no—
415C22×C3—22no—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—22no—
419C22×C3—24no—
420C22×C3—22no—
421C22×C3—21no—
422C22×C3—22no—
423C22×C3—21no—
424C22×C3—44no—
425C22×C3—48no—
426C22×C3—44no—
427C22×C3—44no—
428C22×C3—48no—
429C22×C3—44no—
430C22×C3—48no—
431C22×C3—42no—
432C22×C3—42no—
433C22×C3—42no—
434C22×C3—82no—
435C22×C3—82no—
436C22×C3—84no—
437C22×C3—82no—
438C22×C3—84no—
439C22×C3—84no—
440C22×C3—81no—
441C22×C3—81no—
442C22×C3B(4,𝔽2)12yes—
443C22×C3C2×#12011yescharacteristic
444C22×C3C2×#12312yes—
445C22×C3#52514yes—
446C3×C4—24no—
447C3×C4—42no—
448C3×C4—44no—
449C3×C4—44no—
450C3×C4—84no—
451C3×C4—82no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—24no—
459C23—22no—
460C23—22no—
461C23—24no—
462C23—24no—
463C23—22no—
464C23—41no—
465C23—42no—
466C23—41no—
467C23—41no—
468C23—42no—
469C23—41no—
470C23—42no—
471C23—41no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—44no—
476C23—42no—
477C23—41no—
478C23—42no—
479C23—42no—
480C23—41no—
481C23—42no—
482C23—44no—
483C23—81no—
484C23—81no—
485C23—88no—
486C23—88no—
487C23—88no—
488C23C3×#12011yescharacteristic
489C23C3×#12012yes—
490C23C3×#12011yescharacteristic
491C23C3×#12314yescomplemented
492C23C22×C3×D411yescharacteristic
493C23C22×C3×D412yes—
494C23C2×C3×#4114yes—
495D4—216no—
496D4—216no—
497D4—48no—
498D4—44no—
499D4—44no—
500D4—88no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—41no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—84no—
525C2×C4—84no—
526C2×C4C3×#12314yescomplemented
527C2×C3—22no—
528C2×C3—24no—
529C2×C3—22no—
530C2×C3—42no—
531C2×C3—41no—
532C2×C3—44no—
533C2×C3—82no—
534C2×C3#284911yescharacteristic
535C2×C3C2×B(4,𝔽2)11yescharacteristic
536C2×C3#302611yescharacteristic
537C22—22no—
538C22—21no—
539C22—21no—
540C22—22no—
541C22—21no—
542C22—21no—
543C22—22no—
544C22—21no—
545C22—22no—
546C22—21no—
547C22—21no—
548C22—22no—
549C22—24no—
550C22—44no—
551C22—44no—
552C22—48no—
553C22—44no—
554C22—48no—
555C22—44no—
556C22—48no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—82no—
561C22—82no—
562C22—84no—
563C22—82no—
564C22—84no—
565C22—84no—
566C22—81no—
567C22—81no—
568C22C3×B(4,𝔽2)12yes—
569C22C2×C3×#12011yescharacteristic
570C22C2×C3×#12312yes—
571C22C3×#52514yes—
572C4—24no—
573C4—42no—
574C4—44no—
575C4—44no—
576C4—82no—
577C4—84no—
578C3#3302011yescharacteristic, direct product factor, Sylow
579C2—22no—
580C2—24no—
581C2—22no—
582C2—41no—
583C2—42no—
584C2—44no—
585C2—82no—
586C2C3×#284911yescharacteristic
587C2C2×C3×B(4,𝔽2)11yescharacteristic
588C2C3×#302611yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC24C24×C3
prime radicalC3×#2403C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C25×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#518C2semidirect
2. C3×#2405C2semidirect
4. C3×#1889C2semidirect
5. C3×#1867C2semidirect
6. C3×#1819C2semidirect
7. C3×#1781C2semidirect
8. #33020C3direct (N × H)
28. C3×#385C22semidirect
31. C3×#379C22semidirect
32. C3×#342C22semidirect
33. C2×#518C2×C3semidirect
34. #2405C2×C3semidirect
36. #1889C2×C3semidirect
37. #1867C2×C3semidirect
38. #1819C2×C3semidirect
39. #1781C2×C3semidirect
57. C23×C3×C4D4semidirect
58. C23×C3×C4D4semidirect
87. C2×C3×C22⋊C4D4semidirect
114. #385C22×C3semidirect
117. #379C22×C3semidirect
118. #342C22×C3semidirect
217. C23×C4C3×D4semidirect
218. C23×C4C3×D4semidirect
247. C2×C22⋊C4C3×D4semidirect
294. C23×C3#123semidirect
329. C2×C3×C4#123semidirect
491. C23C3×#123semidirect
526. C2×C4C3×#123semidirect
578. C3#33020direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5182
C3×#24052
C3×#24032
C3×#18892
C3×#18672
C3×#18192
C3×#17812
#330203