Finite Group Databasebeta

← C3×#78837 #11864497

Subgroups — C3×#78837 #11864497

Order 1152.

1479 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#1010C211yesmaximal, characteristic
2C2×C3×#951C211yesmaximal, characteristic, complemented
3C22×C3×Dic12C211yesmaximal, characteristic, complemented
4C3×#5526C211yesmaximal, characteristic, complemented
5C32×#416C211yesmaximal, characteristic, complemented
6C3×#5110C211yesmaximal, characteristic, complemented
7C3×#5004C211yesmaximal, characteristic, complemented
8C3×#4809C214yesmaximal, complemented
9C3×#4783C214yesmaximal, complemented
10C3×#3011—31nomaximal
11#78837C311yesmaximal, characteristic, direct product factor
12C2×C32×#41C2211yescharacteristic
13C3×#1010—24no—
14C3×#1010—24no—
15C2×C3×D3×Q8C2212yes—
16C2×C3×#237—22no—
17C2×C3×#237C2212yes—
18C2×C3×#235—21no—
19C2×C3×#235C2211yescharacteristic
20C22×C3×Dic6—21no—
21C22×C3×Dic6C2211yescharacteristic
22C22×C3×Dic6C2211yescharacteristic
23C22×C32×C8C2211yescharacteristic
24C32×#117C2211yescharacteristic
25C3×#951—24no—
26C2×C3×#216C2212yes—
27C2×C3×#214C2212yes—
28C2×C3×#208C2211yescharacteristic
29C2×C3×Dic12—24no—
30C2×C3×Dic12C2214yes—
31C3×#897C2214yes—
32C3×#886—22no—
33C3×#875—21no—
34C3×#869—21no—
35C32×#103C2212yes—
36C32×#102C2212yes—
37C3×(C2×D3)⋊C8C2214yes—
38C3×#810C2214yes—
39C2×C3×2−1+4—31no—
40C2×#1010C2×C311yescharacteristic
41C2×C3×#137—31no—
42C22×C3×Q16—31no—
43C3×#536—31no—
44C2×#951C2×C311yescharacteristic, complemented
45C22×Dic12C2×C311yescharacteristic, complemented
46#5526C2×C311yescharacteristic, complemented
47C3×#452—34no—
48C3×#451—34no—
49C3×#419—31no—
50C3×#416—21no—
51C3×#416D311yescharacteristic, complemented
52C3×#416C2×C311yescharacteristic, complemented
53C3×#408—31no—
54#5110C2×C311yescharacteristic, complemented
55#5004C2×C311yescharacteristic, complemented
56#4809C2×C314yescomplemented
57#4783C2×C314yescomplemented
58C32×#41—22no—
59C32×#41D414yes—
60C2×C32×Q8C2311yescharacteristic
61C2×C32×D4—21no—
62C2×C32×D4C2311yescharacteristic
63C22×C32×C4—21no—
64C22×C32×C4C2311yescharacteristic
65C2×C3×#67—22no—
66C2×C3×#67D412yes—
67C22×C3×Dic3—21no—
68C22×C3×Dic3—22no—
69C22×C3×Dic3—21no—
70C3×D3×Q8—24no—
71C3×D3×Q8—24no—
72C3×#237—216no—
73C3×#237—24no—
74C3×#237—24no—
75C3×#235—28no—
76C3×#235—22no—
77C3×#235—22no—
78C2×C3×D12D411yescharacteristic
79C2×C3×C4×D3—22no—
80C2×C3×C4×D3C2312yes—
81C2×C3×Dic6—28no—
82C2×C3×Dic6—21no—
83C2×C3×Dic6—22no—
84C2×C3×Dic6—22no—
85C2×C3×Dic6—22no—
86C2×C3×Dic6—22no—
87C2×C3×Dic6—21no—
88C2×C3×Dic6D411yescharacteristic
89C2×C3×Dic6D414yes—
90C2×C3×Dic6C2312yes—
91C2×C3×Dic6C2312yes—
92C2×C32×C8—22no—
93C2×C32×C8C2312yes—
94C32×C4⋊C4C2312yes—
95C32×C22⋊C4—21no—
96C32×C42—21no—
97C3×#216—24no—
98C3×#214—24no—
99C3×(C2×D3)⋊C4—24no—
100C3×C4⋊Dic3—24no—
101C3×#208—22no—
102C2×C3×C3⋊C8C2312yes—
103C3×Dic12—28no—
104#3011—31noSylow
105C3×2−1+4—64no—
106C3×2−1+4—64no—
107C2×C3×#41—21no—
108C2×C3×#41—31no—
109C2×C3×#41—32no—
110C2×C3×#41—61no—
111C2×C3×#41—62no—
112C2×C3×#41C2×D311yescharacteristic
113C2×C3×#41C22×C311yescharacteristic
114C22×C3×Q8—32no—
115C22×C3×Q8—31no—
116C22×C3×Q8—31no—
117C22×C3×Q8—61no—
118#1010—24no—
119#1010—24no—
120C2×D3×Q8C22×C312yes—
121C2×#237—22no—
122C2×#237C22×C312yes—
123C2×#235—21no—
124C2×#235C22×C311yescharacteristic
125C22×Dic6—21no—
126C22×Dic6C22×C311yescharacteristic
127C22×Dic6C22×C311yescharacteristic
128C3×#137—64no—
129C2×C3×Q16—32no—
130C2×C3×Q16—34no—
131C2×C3×Q16—64no—
132C2×C3×SD16—32no—
133C2×C3×M16—31no—
134C22×C3×C8—21no—
135C22×C3×C8C2×D311yescharacteristic
136C22×C3×C8C22×C311yescharacteristic
137C3×C4⋊Q8—61no—
138C3×#124—61no—
139C3×#123—62no—
140C3×C22⋊Q8—34no—
141C3×#117—21no—
142C3×#117C2×D311yescharacteristic
143C3×#117C22×C311yescharacteristic
144#951—24no—
145C2×#216C22×C312yes—
146C2×#214C22×C312yes—
147C2×#208C22×C311yescharacteristic
148C2×Dic12—24no—
149C2×Dic12C22×C314yes—
150#897C22×C314yes—
151#886—22no—
152#875—21no—
153#869—21no—
154C3×#103—22no—
155C3×#103—34no—
156C3×#103C2×D312yes—
157C3×#103C22×C312yes—
158C3×#102—22no—
159C3×#102C2×D312yes—
160C3×#102C22×C312yes—
161C3×C22⋊C8—34no—
162(C2×D3)⋊C8C22×C314yes—
163#810C22×C314yes—
164C23×C32—21no—
165C23×C32C2×D411yescharacteristic
166C22×C3×D3C2×D412yes—
167C32×Q8—21no—
168C32×Q8C2×D412yes—
169C32×D4—24no—
170C32×D4—21no—
171C32×D4C2×D412yes—
172C2×C32×C4—21no—
173C2×C32×C4—24no—
174C2×C32×C4—22no—
175C2×C32×C4C2×D412yes—
176C2×C32×C4C2×D411yescharacteristic
177C2×C32×C4C2×D412yes—
178C2×C32×C4C2411yescharacteristic
179C3×#67—28no—
180C3×#67—24no—
181C2×C3×Dic3—28no—
182C2×C3×Dic3—24no—
183C2×C3×Dic3—22no—
184C2×C3×Dic3—22no—
185C2×C3×Dic3—42no—
186C2×C3×Dic3C2×D412yes—
187C3×D12—22no—
188C3×C4×D3—28no—
189C3×C4×D3—22no—
190C3×C4×D3—22no—
191C3×Dic6—28no—
192C3×Dic6—22no—
193C3×Dic6—24no—
194C3×Dic6—22no—
195C3×Dic6—22no—
196C3×Dic6—22no—
197C3×Dic6—42no—
198C3×Dic6C2×D414yes—
199C32×C8—22no—
200C3×C3⋊C8—22no—
201C2×2−1+4—31no—
202C2×#137—31no—
203C22×Q16—31no—
204#536—31no—
205#452—34no—
206#451—34no—
207#419—31no—
208#416C3×D311yescharacteristic, complemented
209#408—31no—
210C3×#41—22no—
211C3×#41—22no—
212C3×#41—24no—
213C3×#41—42no—
214C3×#41—68no—
215C3×#41—64no—
216C3×#41—616no—
217C3×#41—64no—
218C3×#41—62no—
219C3×#41—62no—
220C3×#41D1214yes—
221C3×#41C3×D414yes—
222C2×C3×Q8—21no—
223C2×C3×Q8—31no—
224C2×C3×Q8—34no—
225C2×C3×Q8—32no—
226C2×C3×Q8—32no—
227C2×C3×Q8—62no—
228C2×C3×Q8—68no—
229C2×C3×Q8—64no—
230C2×C3×Q8—64no—
231C2×C3×Q8—62no—
232C2×C3×Q8—61no—
233C2×C3×Q8—61no—
234C2×C3×Q8—62no—
235C2×C3×Q8—62no—
236C2×C3×Q8C22×D311yescharacteristic
237C2×C3×Q8C23×C311yescharacteristic
238C2×C3×D4—21no—
239C2×C3×D4—21no—
240C2×C3×D4—21no—
241C2×C3×D4—32no—
242C2×C3×D4—31no—
243C2×C3×D4—41no—
244C2×C3×D4—62no—
245C2×C3×D4C22×D311yescharacteristic
246C2×C3×D4C23×C311yescharacteristic
247C22×C3×C4—21no—
248C22×C3×C4—21no—
249C22×C3×C4—21no—
250C22×C3×C4—32no—
251C22×C3×C4—41no—
252C22×C3×C4—62no—
253C22×C3×C4—62no—
254C22×C3×C4—61no—
255C22×C3×C4—61no—
256C22×C3×C4C22×D311yescharacteristic
257C22×C3×C4C23×C311yescharacteristic
258C2×#67—22no—
259C2×#67C3×D412yes—
260C22×Dic3—21no—
261C22×Dic3—22no—
262C22×Dic3—21no—
263D3×Q8—24no—
264D3×Q8—24no—
265#237—216no—
266#237—24no—
267#237—24no—
268#235—28no—
269#235—22no—
270#235—22no—
271C2×D12C3×D411yescharacteristic
272C2×C4×D3—22no—
273C2×C4×D3C23×C312yes—
274C2×Dic6—28no—
275C2×Dic6—21no—
276C2×Dic6—22no—
277C2×Dic6—22no—
278C2×Dic6—22no—
279C2×Dic6—22no—
280C2×Dic6—21no—
281C2×Dic6C3×D411yescharacteristic
282C2×Dic6C3×D414yes—
283C2×Dic6C23×C312yes—
284C2×Dic6C23×C312yes—
285C3×Q16—68no—
286C3×Q16—64no—
287C3×SD16—64no—
288C3×M16—62no—
289C2×C3×C8—22no—
290C2×C3×C8—22no—
291C2×C3×C8—22no—
292C2×C3×C8—32no—
293C2×C3×C8—42no—
294C2×C3×C8C22×D312yes—
295C2×C3×C8C23×C312yes—
296C3×C4⋊C4—22no—
297C3×C4⋊C4—64no—
298C3×C4⋊C4C22×D312yes—
299C3×C4⋊C4C23×C312yes—
300C3×C22⋊C4—21no—
301C3×C22⋊C4—21no—
302C3×C22⋊C4—41no—
303C3×C22⋊C4—64no—
304C3×C42—21no—
305C3×C42—21no—
306C3×C42—41no—
307#216—24no—
308#214—24no—
309(C2×D3)⋊C4—24no—
310C4⋊Dic3—24no—
311#208—22no—
312C2×C3⋊C8C23×C312yes—
313Dic12—28no—
314C22×C32—22no—
315C22×C32—22no—
316C22×C32—42no—
317C22×C32#12012yes—
318C22×C32C22×D411yescharacteristic
319C2×C3×D3—22no—
320C2×C3×D3—24no—
321C32×C4—22no—
322C32×C4—42no—
323C32×C4#12012yes—
324C32×C4C22×D411yescharacteristic
325C32×C4C22×D411yescharacteristic
326C3×Dic3—22no—
327C3×Dic3—24no—
328C3×Dic3—42no—
3292−1+4—64no—
3302−1+4—64no—
331C2×#41—31no—
332C2×#41—32no—
333C2×#41—61no—
334C2×#41—62no—
335C2×#41C2×C3×D311yescharacteristic
336C22×Q8—32no—
337C22×Q8—31no—
338C22×Q8—31no—
339C22×Q8—61no—
340#137—64no—
341C2×Q16—32no—
342C2×Q16—34no—
343C2×Q16—64no—
344C2×SD16—32no—
345C2×M16—31no—
346C22×C8C2×C3×D311yescharacteristic
347C4⋊Q8—61no—
348#124—61no—
349#123—62no—
350C22⋊Q8—34no—
351#117C2×C3×D311yescharacteristic
352#103—34no—
353#103C2×C3×D312yes—
354#102C2×C3×D312yes—
355C22⋊C8—34no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—32no—
360C23×C3—41no—
361C23×C3D3×D411yescharacteristic
362C23×C3C2×C3×D411yescharacteristic
363C22×D3C2×C3×D412yes—
364C3×Q8—21no—
365C3×Q8—21no—
366C3×Q8—22no—
367C3×Q8—34no—
368C3×Q8—41no—
369C3×Q8—68no—
370C3×Q8—62no—
371C3×Q8—64no—
372C3×Q8—62no—
373C3×Q8—62no—
374C3×Q8—62no—
375C3×Q8—122no—
376C3×Q8C2×D1212yes—
377C3×Q8C2×C3×D412yes—
378C3×D4—24no—
379C3×D4—24no—
380C3×D4—21no—
381C3×D4—21no—
382C3×D4—22no—
383C3×D4—41no—
384C3×D4—44no—
385C3×D4—68no—
386C3×D4—64no—
387C3×D4—62no—
388C3×D4C2×D1212yes—
389C3×D4C2×C3×D412yes—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—24no—
393C2×C3×C4—24no—
394C2×C3×C4—22no—
395C2×C3×C4—22no—
396C2×C3×C4—21no—
397C2×C3×C4—22no—
398C2×C3×C4—21no—
399C2×C3×C4—22no—
400C2×C3×C4—32no—
401C2×C3×C4—44no—
402C2×C3×C4—41no—
403C2×C3×C4—42no—
404C2×C3×C4—68no—
405C2×C3×C4—68no—
406C2×C3×C4—64no—
407C2×C3×C4—62no—
408C2×C3×C4—62no—
409C2×C3×C4—62no—
410C2×C3×C4—62no—
411C2×C3×C4—122no—
412C2×C3×C4C2×D1212yes—
413C2×C3×C4D3×D411yescharacteristic
414C2×C3×C4D3×D412yes—
415C2×C3×C4C2×C3×D412yes—
416C2×C3×C4C2×C3×D411yescharacteristic
417C2×C3×C4C2×C3×D412yes—
418C2×C3×C4C23×D311yescharacteristic
419C2×C3×C4C24×C311yescharacteristic
420#67—28no—
421#67—24no—
422C2×Dic3—28no—
423C2×Dic3—24no—
424C2×Dic3—22no—
425C2×Dic3—22no—
426C2×Dic3—42no—
427C2×Dic3C2×C3×D412yes—
428D12—22no—
429C4×D3—28no—
430C4×D3—22no—
431C4×D3—22no—
432Dic6—28no—
433Dic6—22no—
434Dic6—24no—
435Dic6—22no—
436Dic6—22no—
437Dic6—22no—
438Dic6—42no—
439Dic6C2×C3×D414yes—
440C3×C8—22no—
441C3×C8—22no—
442C3×C8—42no—
443C3×C8—62no—
444C3⋊C8—22no—
445C2×C32—22no—
446C2×C32—42no—
447C2×C32C2×#12011yescharacteristic
448C2×C32#57812yes—
449C3×D3—42no—
450#41—22no—
451#41—68no—
452#41—64no—
453#41—616no—
454#41—64no—
455#41—62no—
456#41—62no—
457#41C3×D1214yes—
458C2×Q8—31no—
459C2×Q8—32no—
460C2×Q8—32no—
461C2×Q8—34no—
462C2×Q8—62no—
463C2×Q8—68no—
464C2×Q8—64no—
465C2×Q8—64no—
466C2×Q8—62no—
467C2×Q8—61no—
468C2×Q8—61no—
469C2×Q8—62no—
470C2×Q8—62no—
471C2×Q8C22×C3×D311yescharacteristic
472C2×D4—21no—
473C2×D4—32no—
474C2×D4—31no—
475C2×D4—62no—
476C2×D4C22×C3×D311yescharacteristic
477C22×C4—21no—
478C22×C4—32no—
479C22×C4—62no—
480C22×C4—62no—
481C22×C4—61no—
482C22×C4—61no—
483C22×C4C22×C3×D311yescharacteristic
484Q16—68no—
485Q16—64no—
486SD16—64no—
487M16—62no—
488C2×C8—22no—
489C2×C8—32no—
490C2×C8C22×C3×D312yes—
491C4⋊C4—64no—
492C4⋊C4C22×C3×D312yes—
493C22⋊C4—21no—
494C22⋊C4—64no—
495C42—21no—
496C22×C3—22no—
497C22×C3—22no—
498C22×C3—22no—
499C22×C3—22no—
500C22×C3—22no—
501C22×C3—21no—
502C22×C3—42no—
503C22×C3—42no—
504C22×C3—42no—
505C22×C3—42no—
506C22×C3—62no—
507C22×C3—64no—
508C22×C3—82no—
509C22×C3#88212yes—
510C22×C3C3×#12012yes—
511C22×C3C2×D3×D411yescharacteristic
512C22×C3C22×C3×D411yescharacteristic
513C2×D3—24no—
514C2×D3—22no—
515C3×C4—22no—
516C3×C4—22no—
517C3×C4—22no—
518C3×C4—21no—
519C3×C4—21no—
520C3×C4—42no—
521C3×C4—42no—
522C3×C4—42no—
523C3×C4—62no—
524C3×C4—64no—
525C3×C4—82no—
526C3×C4—122no—
527C3×C4#88212yes—
528C3×C4C3×#12012yes—
529C3×C4C22×D1211yescharacteristic
530C3×C4C2×D3×D411yescharacteristic
531C3×C4C22×C3×D411yescharacteristic
532C3×C4C22×C3×D411yescharacteristic
533Dic3—22no—
534Dic3—24no—
535Dic3—42no—
536C32#301111yescharacteristic, complemented, Sylow
537C23—21no—
538C23—32no—
539C23C3×D3×D411yescharacteristic
540Q8—21no—
541Q8—34no—
542Q8—68no—
543Q8—62no—
544Q8—64no—
545Q8—62no—
546Q8—62no—
547Q8—62no—
548Q8—122no—
549Q8C2×C3×D1212yes—
550D4—24no—
551D4—21no—
552D4—68no—
553D4—64no—
554D4—62no—
555D4C2×C3×D1212yes—
556C2×C4—21no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—32no—
560C2×C4—68no—
561C2×C4—68no—
562C2×C4—64no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—62no—
566C2×C4—62no—
567C2×C4—122no—
568C2×C4C2×C3×D1212yes—
569C2×C4C3×D3×D411yescharacteristic
570C2×C4C3×D3×D412yes—
571C2×C4C23×C3×D311yescharacteristic
572C8—22no—
573C8—62no—
574C2×C3—22no—
575C2×C3—22no—
576C2×C3—21no—
577C2×C3—22no—
578C2×C3—42no—
579C2×C3—42no—
580C2×C3—42no—
581C2×C3—82no—
582C2×C3—122no—
583C2×C3C2×#88211yescharacteristic
584C2×C3#577112yes—
585C2×C3C2×C3×#12011yescharacteristic
586C2×C3C3×#57812yes—
587D3—42no—
588C22—22no—
589C22—22no—
590C22—42no—
591C22—62no—
592C22—64no—
593C22C3×#88212yes—
594C22C2×C3×D3×D411yescharacteristic
595C4—22no—
596C4—42no—
597C4—62no—
598C4—64no—
599C4—122no—
600C4C3×#88212yes—
601C4C22×C3×D1211yescharacteristic
602C4C2×C3×D3×D411yescharacteristic
603C3—21no—
604C3#7883711yescharacteristic, direct product factor
605C3C3×#301111yescharacteristic, complemented
606C2—22no—
607C2—42no—
608C2—122no—
609C2C2×C3×#88211yescharacteristic
610C2C3×#577112yes—
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#416C2
socleC22×C32C22×D4
prime radicalC2×C3×#1010C2
hypercenterC3×#416D3
hypocenterC3C3×#3011
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#416

Complements

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

NH ≅ G/Ntype
2. C2×C3×#951C2semidirect
3. C22×C3×Dic12C2semidirect
4. C3×#5526C2semidirect
5. C32×#416C2semidirect
6. C3×#5110C2semidirect
7. C3×#5004C2semidirect
8. C3×#4809C2semidirect
9. C3×#4783C2semidirect
11. #78837C3direct (N × H)
44. C2×#951C2×C3semidirect
45. C22×Dic12C2×C3semidirect
46. #5526C2×C3semidirect
51. C3×#416D3semidirect
52. C3×#416C2×C3semidirect
54. #5110C2×C3semidirect
55. #5004C2×C3semidirect
56. #4809C2×C3semidirect
57. #4783C2×C3semidirect
208. #416C3×D3semidirect
536. C32#3011semidirect
604. C3#78837direct (N × H)
605. C3C3×#3011semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#10102
C2×C3×#9512
C22×C3×Dic122
C3×#55262
C32×#4162
C3×#51102
C3×#50042
C3×#48092
C3×#47832
C3×#30113
#788373