Finite Group Databasebeta

← C3×#23589 #10642196

Subgroups — C3×#23589 #10642196

Order 768.

1138 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3460C211yesmaximal, characteristic
2C3×C4×#137C212yesmaximal, complemented
3C3×C4×#124C211yesmaximal, characteristic, complemented
4C3×#2061C211yesmaximal, characteristic, complemented
5C3×#2040C211yesmaximal, characteristic, complemented
6C3×#2037C212yesmaximal, complemented
7C3×#2028C211yesmaximal, characteristic, complemented
8C3×#2007C211yesmaximal, characteristic, complemented
9C3×#2004C212yesmaximal, complemented
10C3×C4×#104C212yesmaximal, complemented
11C3×C4×#100C211yesmaximal, characteristic, complemented
12#23589C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#137C2212yes—
14C3×#544C2211yescharacteristic
15C3×#542—22no—
16C3×#540—22no—
17C3×#538—22no—
18C3×#537C2211yescharacteristic
19C3×#535C2212yes—
20C3×#532—22no—
21C3×#529—22no—
22C2×C3×#124C2211yescharacteristic
23C3×#525—22no—
24C3×C4×#41C2212yes—
25C2×C3×C4×Q8C2211yescharacteristic
26C2×C3×C4×D4C2211yescharacteristic
27C3×#441—22no—
28C3×#440—22no—
29C3×C4×Q16—22no—
30C3×C4×SD16—22no—
31C3×#428C2212yes—
32C2×C3×#104C2212yes—
33C2×C3×#104C2212yes—
34C3×#419C2212yes—
35C3×#417C2212yes—
36C2×C3×#100C2212yes—
37C3×C4×M16C2212yes—
38C3×#391C2211yescharacteristic
39C3×#390C2211yescharacteristic
40C3×#388—22no—
41C3×#382C2211yescharacteristic
42C3×C4×C22⋊C4—22no—
43C3×C43C2211yescharacteristic
44C3×#343C2211yescharacteristic
45C3×#343C2211yescharacteristic
46C3×#339C2212yes—
47C3×#339C2212yes—
48#3460C2×C311yescharacteristic
49C4×#137C2×C312yescomplemented
50C4×#124C2×C311yescharacteristic, complemented
51#2061C2×C311yescharacteristic, complemented
52#2040C2×C311yescharacteristic, complemented
53#2037C2×C312yescomplemented
54#2028C2×C311yescharacteristic, complemented
55#2007C2×C311yescharacteristic, complemented
56#2004C2×C312yescomplemented
57C4×#104C2×C312yescomplemented
58C4×#100C2×C311yescharacteristic, complemented
59C2×C3×#41—22no—
60C2×C3×#41C2312yes—
61C22×C3×Q8C2311yescharacteristic
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4D412yes—
64C3×#137—24no—
65C3×#137—24no—
66C2×C3×Q16—22no—
67C2×C3×SD16—22no—
68C2×C3×M16C2312yes—
69C2×C3×M16C2312yes—
70C3×C4⋊Q8—22no—
71C3×#127—22no—
72C3×#126—24no—
73C3×#125—22no—
74C3×#124—24no—
75C3×#124—22no—
76C3×#124—22no—
77C3×#124—22no—
78C3×#123—24no—
79C3×#123—24no—
80C3×C22⋊Q8—22no—
81C3×C22⋊Q8—22no—
82C3×C22⋊Q8—22no—
83C3×#121—22no—
84C3×#121—24no—
85C3×#121—22no—
86C3×#121—22no—
87C3×#121—22no—
88C3×#121—22no—
89C3×#120—24no—
90C3×C4×Q8—22no—
91C3×C4×Q8—22no—
92C3×C4×Q8D412yes—
93C3×C4×Q8D412yes—
94C3×C4×D4—24no—
95C3×C4×D4—22no—
96C3×C4×D4—22no—
97C3×C4×D4—22no—
98C3×C4×D4—22no—
99C3×C4×D4D412yes—
100C3×C4×D4D412yes—
101C3×#117—22no—
102C3×#117—22no—
103C3×#117C2312yes—
104C2×C3×C4⋊C4—21no—
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C22⋊C4—22no—
108C2×C3×C22⋊C4—22no—
109C2×C3×C22⋊C4D412yes—
110C2×C3×C42—22no—
111C2×C3×C42—21no—
112C2×C3×C42—21no—
113C2×C3×C42—21no—
114C2×C3×C42C2311yescharacteristic
115C2×C3×C42C2311yescharacteristic
116C2×C3×C42C2311yescharacteristic
117C3×#107—22no—
118C3×#106—22no—
119C3×#104—24no—
120C3×#104—24no—
121C3×#103—22no—
122C3×#103—22no—
123C3×#103—22no—
124C3×#102—22no—
125C3×#100—24no—
126C3×#97—22no—
127C3×C4×C8—22no—
128C3×#95—22no—
129C3×#95—22no—
130C2×#137C22×C312yes—
131#544C22×C311yescharacteristic
132#542—22no—
133#540—22no—
134#538—22no—
135#537C22×C311yescharacteristic
136#535C22×C312yes—
137#532—22no—
138#529—22no—
139C2×#124C22×C311yescharacteristic
140#525—22no—
141C4×#41C22×C312yes—
142C2×C4×Q8C22×C311yescharacteristic
143C2×C4×D4C22×C311yescharacteristic
144#441—22no—
145#440—22no—
146C4×Q16—22no—
147C4×SD16—22no—
148#428C22×C312yes—
149C2×#104C22×C312yes—
150C2×#104C22×C312yes—
151#419C22×C312yes—
152#417C22×C312yes—
153C2×#100C22×C312yes—
154C4×M16C22×C312yes—
155#391C22×C311yescharacteristic
156#390C22×C311yescharacteristic
157#388—22no—
158#382C22×C311yescharacteristic
159C4×C22⋊C4—22no—
160C43C22×C311yescharacteristic
161#343C22×C311yescharacteristic
162#343C22×C311yescharacteristic
163#339C22×C312yes—
164#339C22×C312yes—
165C24×C3C2×D412yes—
166C3×#41—22no—
167C3×#41—22no—
168C3×#41—22no—
169C3×#41—22no—
170C3×#41—84no—
171C2×C3×Q8—22no—
172C2×C3×Q8—21no—
173C2×C3×Q8—22no—
174C2×C3×Q8—22no—
175C2×C3×Q8C2×D412yes—
176C2×C3×Q8#4112yescomplemented
177C2×C3×D4—22no—
178C2×C3×D4—24no—
179C2×C3×D4—24no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4C2×D412yes—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—21no—
195C22×C3×C4—44no—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D412yes—
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2411yescharacteristic
200C3×Q16—28no—
201C3×SD16—28no—
202C3×M16—22no—
203C3×M16—22no—
204C3×M16—22no—
205C3×M16#4114yescomplemented
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—24no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—42no—
216C3×C4⋊C4C2×D412yes—
217C3×C4⋊C4C2×D412yes—
218C3×C22⋊C4—28no—
219C3×C22⋊C4—24no—
220C3×C22⋊C4—24no—
221C3×C22⋊C4—24no—
222C3×C22⋊C4—22no—
223C3×C22⋊C4—24no—
224C3×C22⋊C4—22no—
225C3×C22⋊C4—22no—
226C3×C22⋊C4—44no—
227C3×C42—22no—
228C3×C42—22no—
229C3×C42—22no—
230C3×C42—21no—
231C3×C42—22no—
232C3×C42—42no—
233C3×C42—42no—
234C3×C42—41no—
235C3×C42—41no—
236C3×C42C2×D412yes—
237C3×C42C2×D412yes—
238C3×C42#4112yescomplemented
239C2×#41—22no—
240C2×#41C23×C312yes—
241C22×Q8C23×C311yescharacteristic
242C22×D4C23×C311yescharacteristic
243C23×C4C3×D412yes—
244#137—24no—
245#137—24no—
246C2×Q16—22no—
247C2×SD16—22no—
248C2×M16C23×C312yes—
249C2×M16C23×C312yes—
250C4⋊Q8—22no—
251#127—22no—
252#126—24no—
253#125—22no—
254#124—24no—
255#124—22no—
256#124—22no—
257#124—22no—
258#123—24no—
259#123—24no—
260C22⋊Q8—22no—
261C22⋊Q8—22no—
262C22⋊Q8—22no—
263#121—22no—
264#121—24no—
265#121—22no—
266#121—22no—
267#121—22no—
268#121—22no—
269#120—24no—
270C4×Q8—22no—
271C4×Q8—22no—
272C4×Q8C3×D412yes—
273C4×Q8C3×D412yes—
274C4×D4—24no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4C3×D412yes—
280C4×D4C3×D412yes—
281#117—22no—
282#117—22no—
283#117C23×C312yes—
284C2×C4⋊C4—21no—
285C2×C4⋊C4C23×C311yescharacteristic
286C2×C4⋊C4C23×C311yescharacteristic
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4C3×D412yes—
290C2×C42—22no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42C23×C311yescharacteristic
295C2×C42C23×C311yescharacteristic
296C2×C42C23×C311yescharacteristic
297#107—22no—
298#106—22no—
299#104—24no—
300#104—24no—
301#103—22no—
302#103—22no—
303#103—22no—
304#102—22no—
305#100—24no—
306#97—22no—
307C4×C8—22no—
308#95—22no—
309#95—22no—
310C23×C3—22no—
311C23×C3—22no—
312C23×C3—22no—
313C23×C3—22no—
314C23×C3—42no—
315C23×C3—42no—
316C23×C3C22×D411yescharacteristic
317C3×Q8—24no—
318C3×Q8—22no—
319C3×Q8—22no—
320C3×Q8—42no—
321C3×Q8—82no—
322C3×D4—22no—
323C3×D4—22no—
324C3×D4—42no—
325C3×D4—42no—
326C3×D4—42no—
327C3×D4—42no—
328C3×D4—84no—
329C3×D4—82no—
330C3×D4—84no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—44no—
349C2×C3×C4—44no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—41no—
353C2×C3×C4—82no—
354C2×C3×C4—82no—
355C2×C3×C4#12012yes—
356C2×C3×C4#12011yescharacteristic
357C2×C3×C4#12011yescharacteristic
358C2×C3×C4C22×D412yes—
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4C2×#4112yes—
361C2×C3×C4C2×#4111yescharacteristic
362C3×C8—24no—
363C3×C8—42no—
364C24C2×C3×D412yes—
365#41—22no—
366#41—22no—
367#41—22no—
368#41—22no—
369#41—84no—
370C2×Q8—22no—
371C2×Q8—21no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8C2×C3×D412yes—
375C2×Q8C3×#4112yescomplemented
376C2×D4—22no—
377C2×D4—24no—
378C2×D4—24no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4C2×C3×D412yes—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—44no—
395C22×C4C2×C3×D411yescharacteristic
396C22×C4C2×C3×D412yes—
397C22×C4C2×C3×D411yescharacteristic
398C22×C4C24×C311yescharacteristic
399Q16—28no—
400SD16—28no—
401M16—22no—
402M16—22no—
403M16—22no—
404M16C3×#4114yescomplemented
405C2×C8—22no—
406C2×C8—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—42no—
415C4⋊C4C2×C3×D412yes—
416C4⋊C4C2×C3×D412yes—
417C22⋊C4—28no—
418C22⋊C4—24no—
419C22⋊C4—24no—
420C22⋊C4—24no—
421C22⋊C4—22no—
422C22⋊C4—24no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—44no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—21no—
430C42—22no—
431C42—42no—
432C42—42no—
433C42—41no—
434C42—41no—
435C42C2×C3×D412yes—
436C42C2×C3×D412yes—
437C42C3×#4112yescomplemented
438C22×C3—21no—
439C22×C3—21no—
440C22×C3—44no—
441C22×C3—42no—
442C22×C3—42no—
443C22×C3—42no—
444C22×C3—42no—
445C22×C3—42no—
446C22×C3—42no—
447C22×C3—82no—
448C22×C3C2×#12011yescharacteristic
449C22×C3#52511yescharacteristic
450C22×C3#53211yescharacteristic
451C3×C4—21no—
452C3×C4—22no—
453C3×C4—42no—
454C3×C4—42no—
455C3×C4—42no—
456C3×C4—42no—
457C3×C4—42no—
458C3×C4—42no—
459C3×C4—42no—
460C3×C4—81no—
461C3×C4#45512yes—
462C3×C4#52512yes—
463C3×C4#53212yes—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—42no—
469C23—42no—
470C23C22×C3×D411yescharacteristic
471Q8—24no—
472Q8—22no—
473Q8—22no—
474Q8—42no—
475Q8—82no—
476D4—22no—
477D4—22no—
478D4—42no—
479D4—42no—
480D4—42no—
481D4—42no—
482D4—82no—
483D4—84no—
484D4—84no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4C3×#12012yes—
510C2×C4C3×#12011yescharacteristic
511C2×C4C3×#12011yescharacteristic
512C2×C4C22×C3×D412yes—
513C2×C4C2×C3×#4111yescharacteristic
514C2×C4C2×C3×#4112yes—
515C2×C4C2×C3×#4111yescharacteristic
516C8—24no—
517C8—42no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—82no—
521C2×C3—82no—
522C2×C3#238811yescharacteristic
523C2×C3C2×#45511yescharacteristic
524C2×C3#301811yescharacteristic
525C22—21no—
526C22—21no—
527C22—44no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—82no—
535C22C2×C3×#12011yescharacteristic
536C22C3×#52511yescharacteristic
537C22C3×#53211yescharacteristic
538C4—21no—
539C4—22no—
540C4—42no—
541C4—42no—
542C4—42no—
543C4—42no—
544C4—42no—
545C4—42no—
546C4—42no—
547C4—81no—
548C4C3×#45512yes—
549C4C3×#52512yes—
550C4C3×#53212yes—
551C3#2358911yescharacteristic, direct product factor, Sylow
552C2—21no—
553C2—21no—
554C2—82no—
555C2—82no—
556C2C3×#238811yescharacteristic
557C2C2×C3×#45511yescharacteristic
558C2C3×#301811yescharacteristic
subgroupG/H
centerC2×C3×C4#120
commutator, Frattini, cosocleC22×C4C24×C3
socleC22×C3C2×#120
prime radicalC3×#3460C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×C4×#137C2semidirect
3. C3×C4×#124C2semidirect
4. C3×#2061C2semidirect
5. C3×#2040C2semidirect
6. C3×#2037C2semidirect
7. C3×#2028C2semidirect
8. C3×#2007C2semidirect
9. C3×#2004C2semidirect
10. C3×C4×#104C2semidirect
11. C3×C4×#100C2semidirect
12. #23589C3direct (N × H)
49. C4×#137C2×C3semidirect
50. C4×#124C2×C3semidirect
51. #2061C2×C3semidirect
52. #2040C2×C3semidirect
53. #2037C2×C3semidirect
54. #2028C2×C3semidirect
55. #2007C2×C3semidirect
56. #2004C2×C3semidirect
57. C4×#104C2×C3semidirect
58. C4×#100C2×C3semidirect
176. C2×C3×Q8#41semidirect
205. C3×M16#41semidirect
238. C3×C42#41semidirect
375. C2×Q8C3×#41semidirect
404. M16C3×#41semidirect
437. C42C3×#41semidirect
551. C3#23589direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34602
C3×C4×#1372
C3×C4×#1242
C3×#20612
C3×#20402
C3×#20372
C3×#20282
C3×#20072
C3×#20042
C3×C4×#1042
C3×C4×#1002
#235893