Finite Group Databasebeta

← C3×#23935 #10642542

Subgroups — C3×#23935 #10642542

Order 768.

1276 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#533C211yesmaximal, characteristic, complemented
2C2×C3×#452C212yesmaximal
3C3×#2297C211yesmaximal, characteristic
4C3×#2024C211yesmaximal, characteristic, complemented
5C3×#2023C211yesmaximal, characteristic, complemented
6C3×#2016C211yesmaximal, characteristic, complemented
7C3×#2013C211yesmaximal, characteristic, complemented
8C3×#1880C212yesmaximal, complemented
9C3×#1878C212yesmaximal, complemented
10C3×#1873C212yesmaximal, complemented
11C3×#1817C211yesmaximal, characteristic, complemented
12#23935C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C22×C3×Q16C2212yes—
15C22×C3×SD16C2212yes—
16C3×#533—28no—
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×#125—21no—
20C2×C3×C22⋊Q8—21no—
21C2×C3×C22⋊Q8C2212yes—
22C2×C3×C4×Q8—21no—
23C2×C3×C4×Q8C2212yes—
24C2×C3×#117—21no—
25C2×C3×#117C2211yescharacteristic
26C22×C3×C42C2211yescharacteristic
27C3×#452—28no—
28C2×C3×#103C2212yes—
29C2×C3×#103C2212yes—
30C2×C3×#103C2212yes—
31C2×C3×#102C2212yes—
32C2×C3×C22⋊C8C2212yes—
33C3×#391C2211yescharacteristic
34C3×#391C2211yescharacteristic
35C3×#390C2211yescharacteristic
36C3×#390C2211yescharacteristic
37C3×#386—21no—
38C3×#386—21no—
39C3×C4×C22⋊C4—21no—
40C3×C4×C22⋊C4—21no—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#336C2211yescharacteristic
44C3×#336C2211yescharacteristic
45C3×#336C2212yes—
46C2×#533C2×C311yescharacteristic, complemented
47C2×#452C2×C312yes—
48#2297C2×C311yescharacteristic
49#2024C2×C311yescharacteristic, complemented
50#2023C2×C311yescharacteristic, complemented
51#2016C2×C311yescharacteristic, complemented
52#2013C2×C311yescharacteristic, complemented
53#1880C2×C312yescomplemented
54#1878C2×C312yescomplemented
55#1873C2×C312yescomplemented
56#1817C2×C311yescharacteristic, complemented
57C2×C3×#41—22no—
58C2×C3×#41—22no—
59C2×C3×#41—24no—
60C2×C3×#41—42no—
61C22×C3×Q8—21no—
62C22×C3×Q8C2312yes—
63C22×C3×Q8C2311yescharacteristic
64C22×C3×D4—21no—
65C22×C3×D4C2311yescharacteristic
66C23×C3×C4—21no—
67C23×C3×C4D411yescharacteristic
68C23×C3×C4D411yescharacteristic
69C23×C3×C4C2311yescharacteristic
70C2×C3×Q16—24no—
71C2×C3×Q16—24no—
72C2×C3×Q16—42no—
73C2×C3×SD16—24no—
74C2×C3×SD16—24no—
75C2×C3×SD16—42no—
76C22×C3×C8C2312yes—
77C22×C3×C8C2312yes—
78C3×C4⋊Q8—24no—
79C3×C4⋊Q8—24no—
80C3×#125—28no—
81C3×C22⋊Q8—28no—
82C3×C22⋊Q8—24no—
83C3×C22⋊Q8D418yes—
84C3×C4×Q8—28no—
85C3×C4×Q8—24no—
86C3×C4×Q8—24no—
87C3×#117—28no—
88C3×#117—42no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—22no—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2312yes—
97C2×C3×C22⋊C4—22no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C42—24no—
104C2×C3×C42—22no—
105C2×C3×C42—21no—
106C2×C3×C42D411yescharacteristic
107C2×C3×C42D411yescharacteristic
108C2×C3×C42C2311yescharacteristic
109C2×C3×C42C2311yescharacteristic
110C3×#103—24no—
111C3×#103—42no—
112C3×#103—42no—
113C3×#102—24no—
114C3×C22⋊C8—24no—
115C3×#95—21no—
116C3×#95—21no—
117C3×#95—21no—
118C3×#95—21no—
119C22×#41C22×C311yescharacteristic
120C22×Q16C22×C312yes—
121C22×SD16C22×C312yes—
122#533—28no—
123C2×C4⋊Q8C22×C311yescharacteristic
124C2×C4⋊Q8C22×C311yescharacteristic
125C2×#125—21no—
126C2×C22⋊Q8—21no—
127C2×C22⋊Q8C22×C312yes—
128C2×C4×Q8—21no—
129C2×C4×Q8C22×C312yes—
130C2×#117—21no—
131C2×#117C22×C311yescharacteristic
132C22×C42C22×C311yescharacteristic
133#452—28no—
134C2×#103C22×C312yes—
135C2×#103C22×C312yes—
136C2×#103C22×C312yes—
137C2×#102C22×C312yes—
138C2×C22⋊C8C22×C312yes—
139#391C22×C311yescharacteristic
140#391C22×C311yescharacteristic
141#390C22×C311yescharacteristic
142#390C22×C311yescharacteristic
143#386—21no—
144#386—21no—
145C4×C22⋊C4—21no—
146C4×C22⋊C4—21no—
147#340C22×C312yes—
148#340C22×C312yes—
149#336C22×C312yes—
150#336C22×C311yescharacteristic
151#336C22×C311yescharacteristic
152C24×C3—21no—
153C24×C3C2×D411yescharacteristic
154C3×#41—48no—
155C3×#41—48no—
156C2×C3×Q8—24no—
157C2×C3×Q8—22no—
158C2×C3×Q8—21no—
159C2×C3×Q8—22no—
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—42no—
166C2×C3×Q8—42no—
167C2×C3×Q8C2×D414yes—
168C2×C3×Q8#4112yes—
169C2×C3×D4—24no—
170C2×C3×D4—22no—
171C2×C3×D4—21no—
172C2×C3×D4—22no—
173C2×C3×D4—42no—
174C2×C3×D4—42no—
175C2×C3×D4#4112yes—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—22no—
182C22×C3×C4—24no—
183C22×C3×C4—22no—
184C22×C3×C4—21no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—41no—
191C22×C3×C4—42no—
192C22×C3×C4—42no—
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D414yes—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2411yescharacteristic
200C3×Q16—48no—
201C3×SD16—48no—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C3×C4⋊C4—24no—
209C3×C4⋊C4—24no—
210C3×C4⋊C4—28no—
211C3×C4⋊C4—24no—
212C3×C4⋊C4—28no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4C2×D414yes—
217C3×C22⋊C4—28no—
218C3×C22⋊C4—42no—
219C3×C22⋊C4—42no—
220C3×C22⋊C4—42no—
221C3×C22⋊C4—42no—
222C3×C22⋊C4—42no—
223C3×C42—28no—
224C3×C42—22no—
225C3×C42—22no—
226C3×C42—22no—
227C3×C42—22no—
228C3×C42—42no—
229C2×#41—22no—
230C2×#41—22no—
231C2×#41—24no—
232C2×#41—42no—
233C22×Q8—21no—
234C22×Q8C23×C312yes—
235C22×Q8C23×C311yescharacteristic
236C22×D4—21no—
237C22×D4C23×C311yescharacteristic
238C23×C4—21no—
239C23×C4C3×D411yescharacteristic
240C23×C4C3×D411yescharacteristic
241C23×C4C23×C311yescharacteristic
242C2×Q16—24no—
243C2×Q16—24no—
244C2×Q16—42no—
245C2×SD16—24no—
246C2×SD16—24no—
247C2×SD16—42no—
248C22×C8C23×C312yes—
249C22×C8C23×C312yes—
250C4⋊Q8—24no—
251C4⋊Q8—24no—
252#125—28no—
253C22⋊Q8—28no—
254C22⋊Q8—24no—
255C22⋊Q8C3×D418yes—
256C4×Q8—28no—
257C4×Q8—24no—
258C4×Q8—24no—
259#117—28no—
260#117—42no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4C23×C311yescharacteristic
267C2×C4⋊C4C23×C311yescharacteristic
268C2×C4⋊C4C23×C312yes—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C42—22no—
276C2×C42—24no—
277C2×C42—21no—
278C2×C42C3×D411yescharacteristic
279C2×C42C3×D411yescharacteristic
280C2×C42C23×C311yescharacteristic
281C2×C42C23×C311yescharacteristic
282#103—24no—
283#103—42no—
284#103—42no—
285#102—24no—
286C22⋊C8—24no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—21no—
291C23×C3—21no—
292C23×C3—21no—
293C23×C3—21no—
294C23×C3—21no—
295C23×C3—21no—
296C23×C3—21no—
297C23×C3—22no—
298C23×C3—41no—
299C23×C3—41no—
300C23×C3—41no—
301C23×C3—41no—
302C23×C3—41no—
303C23×C3—41no—
304C23×C3#12012yes—
305C23×C3C22×D411yescharacteristic
306C3×Q8—24no—
307C3×Q8—24no—
308C3×Q8—24no—
309C3×Q8—48no—
310C3×Q8—42no—
311C3×Q8—42no—
312C3×Q8—42no—
313C3×D4—24no—
314C3×D4—48no—
315C3×D4—42no—
316C2×C3×C4—28no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—21no—
328C2×C3×C4—24no—
329C2×C3×C4—24no—
330C2×C3×C4—22no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—44no—
334C2×C3×C4—48no—
335C2×C3×C4—41no—
336C2×C3×C4—42no—
337C2×C3×C4—41no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4#12012yes—
348C2×C3×C4#13512yes—
349C2×C3×C4#13712yes—
350C2×C3×C4#13712yes—
351C2×C3×C4C22×D412yes—
352C2×C3×C4C2×#4111yescharacteristic
353C2×C3×C4C2×#4111yescharacteristic
354C2×C3×C4H(5,𝔽2)12yes—
355C3×C8—42no—
356C3×C8—42no—
357C24—21no—
358C24C2×C3×D411yescharacteristic
359#41—48no—
360#41—48no—
361C2×Q8—24no—
362C2×Q8—22no—
363C2×Q8—21no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—42no—
371C2×Q8—42no—
372C2×Q8C2×C3×D414yes—
373C2×Q8C3×#4112yes—
374C2×D4—24no—
375C2×D4—22no—
376C2×D4—21no—
377C2×D4—22no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4C3×#4112yes—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—41no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4C2×C3×D411yescharacteristic
399C22×C4C2×C3×D411yescharacteristic
400C22×C4C2×C3×D414yes—
401C22×C4C2×C3×D411yescharacteristic
402C22×C4C2×C3×D411yescharacteristic
403C22×C4C2×C3×D411yescharacteristic
404C22×C4C24×C311yescharacteristic
405Q16—48no—
406SD16—48no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—28no—
416C4⋊C4—24no—
417C4⋊C4—28no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4C2×C3×D414yes—
422C22⋊C4—28no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C22⋊C4—42no—
426C22⋊C4—42no—
427C22⋊C4—42no—
428C42—28no—
429C42—22no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—42no—
434C22×C3—22no—
435C22×C3—22no—
436C22×C3—22no—
437C22×C3—41no—
438C22×C3—41no—
439C22×C3—41no—
440C22×C3—41no—
441C22×C3—42no—
442C22×C3—42no—
443C22×C3—42no—
444C22×C3—81no—
445C22×C3—81no—
446C22×C3—81no—
447C22×C3—81no—
448C22×C3C2×#12011yescharacteristic
449C22×C3#52511yescharacteristic
450C22×C3#53511yescharacteristic
451C22×C3C2×#13511yescharacteristic
452C22×C3C2×#13711yescharacteristic
453C22×C3C2×#13711yescharacteristic
454C22×C3#57511yescharacteristic
455C3×C4—22no—
456C3×C4—22no—
457C3×C4—22no—
458C3×C4—22no—
459C3×C4—42no—
460C3×C4—44no—
461C3×C4—41no—
462C3×C4—44no—
463C3×C4—81no—
464C3×C4—81no—
465C3×C4#54612yes—
466C3×C4#54612yes—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—22no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23C3×#12012yes—
481C23C22×C3×D411yescharacteristic
482Q8—24no—
483Q8—24no—
484Q8—24no—
485Q8—48no—
486Q8—42no—
487Q8—42no—
488Q8—42no—
489D4—24no—
490D4—48no—
491D4—42no—
492C2×C4—28no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—44no—
510C2×C4—48no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4C3×#12012yes—
524C2×C4C3×#13512yes—
525C2×C4C3×#13712yes—
526C2×C4C3×#13712yes—
527C2×C4C22×C3×D412yes—
528C2×C4C2×C3×#4111yescharacteristic
529C2×C4C2×C3×#4111yescharacteristic
530C2×C4C3×H(5,𝔽2)12yes—
531C8—42no—
532C8—42no—
533C2×C3—42no—
534C2×C3—82no—
535C2×C3#240411yescharacteristic
536C2×C3#300211yescharacteristic
537C2×C3#300411yescharacteristic
538C2×C3#310711yescharacteristic
539C2×C3#311611yescharacteristic
540C2×C3#321011yescharacteristic
541C2×C3#321611yescharacteristic
542C22—22no—
543C22—22no—
544C22—22no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22C2×C3×#12011yescharacteristic
557C22C3×#52511yescharacteristic
558C22C3×#53511yescharacteristic
559C22C2×C3×#13511yescharacteristic
560C22C2×C3×#13711yescharacteristic
561C22C2×C3×#13711yescharacteristic
562C22C3×#57511yescharacteristic
563C4—22no—
564C4—22no—
565C4—22no—
566C4—22no—
567C4—42no—
568C4—44no—
569C4—41no—
570C4—44no—
571C4—81no—
572C4—81no—
573C4C3×#54612yes—
574C4C3×#54612yes—
575C3#2393511yescharacteristic, direct product factor, Sylow
576C2—42no—
577C2—82no—
578C2C3×#240411yescharacteristic
579C2C3×#300211yescharacteristic
580C2C3×#300411yescharacteristic
581C2C3×#310711yescharacteristic
582C2C3×#311611yescharacteristic
583C2C3×#321011yescharacteristic
584C2C3×#321611yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#533C2semidirect
4. C3×#2024C2semidirect
5. C3×#2023C2semidirect
6. C3×#2016C2semidirect
7. C3×#2013C2semidirect
8. C3×#1880C2semidirect
9. C3×#1878C2semidirect
10. C3×#1873C2semidirect
11. C3×#1817C2semidirect
12. #23935C3direct (N × H)
46. C2×#533C2×C3semidirect
49. #2024C2×C3semidirect
50. #2023C2×C3semidirect
51. #2016C2×C3semidirect
52. #2013C2×C3semidirect
53. #1880C2×C3semidirect
54. #1878C2×C3semidirect
55. #1873C2×C3semidirect
56. #1817C2×C3semidirect
575. C3#23935direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5332
C2×C3×#4522
C3×#22972
C3×#20242
C3×#20232
C3×#20162
C3×#20132
C3×#18802
C3×#18782
C3×#18732
C3×#18172
#239353