Finite Group Databasebeta

← C3×#84117 #11864488

Subgroups — C3×#84117 #11864488

Order 1152.

1797 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×2−1+4C211yesmaximal, characteristic
2C3×#5825C211yesmaximal, characteristic, complemented
3C2×C3×#951C211yesmaximal, characteristic, complemented
4C22×C3×#216C211yesmaximal, characteristic, complemented
5C3×#5239C214yesmaximal, complemented
6C3×#5234C211yesmaximal, characteristic, complemented
7C3×#5233C211yesmaximal, characteristic, complemented
8C3×#5228C214yesmaximal, complemented
9C3×#5225C211yesmaximal, characteristic, complemented
10C3×#3011—31nomaximal
11#84117C311yesmaximal, characteristic, direct product factor
12C32×2−1+4—24no—
13C32×2−1+4—24no—
14C2×C32×#41—22no—
15C2×C32×#41—21no—
16C2×C32×#41C2212yes—
17C2×C32×#41C2211yescharacteristic
18C2×C32×#41C2211yescharacteristic
19C22×C32×Q8—21no—
20C22×C32×Q8C2212yes—
21C22×C32×Q8C2211yescharacteristic
22C22×C3×Dic6C2211yescharacteristic
23C3×#951—24no—
24C3×#944—21no—
25C2×C3×#216—24no—
26C2×C3×#216C2214yes—
27C2×C3×#216C2212yes—
28C3×#936—21no—
29C3×#935—22no—
30C2×C3×#214C2212yes—
31C3×#926C2211yescharacteristic
32C3×#924C2214yes—
33C2×C3×#208C2211yescharacteristic
34C22×C3×C3⋊C8C2211yescharacteristic
35C3×#835C2214yes—
36C3×#835C2212yes—
37C3×#832C2212yes—
38C3×#830C2214yes—
39C2×C3×2−1+4—21no—
40C2×C3×2−1+4D311yescharacteristic
41C2×C3×2−1+4C2×C311yescharacteristic
42C2×C3×#137—31no—
43C22×C3×Q16—31no—
44C3×#536—31no—
45#5825C2×C311yescharacteristic, complemented
46C2×#951C2×C311yescharacteristic, complemented
47C22×#216C2×C311yescharacteristic, complemented
48C3×#452—34no—
49C3×#451—34no—
50C3×#419—31no—
51C3×#416—31no—
52C3×#408—31no—
53#5239C2×C314yescomplemented
54#5234C2×C311yescharacteristic, complemented
55#5233C2×C311yescharacteristic, complemented
56#5228C2×C314yescomplemented
57#5225C2×C311yescharacteristic, complemented
58C32×#41—216no—
59C32×#41—28no—
60C32×#41—24no—
61C32×#41—24no—
62C32×#41—22no—
63C32×#41—22no—
64C32×#41—22no—
65C32×#41D414yes—
66C2×C32×Q8—28no—
67C2×C32×Q8—24no—
68C2×C32×Q8—24no—
69C2×C32×Q8—22no—
70C2×C32×Q8—21no—
71C2×C32×Q8—22no—
72C2×C32×Q8D414yes—
73C2×C32×Q8D411yescharacteristic
74C2×C32×Q8C2311yescharacteristic
75C2×C32×Q8C2312yes—
76C2×C32×D4—22no—
77C2×C32×D4—21no—
78C2×C32×D4D412yes—
79C2×C32×D4D411yescharacteristic
80C2×C32×D4C2311yescharacteristic
81C22×C32×C4—22no—
82C22×C32×C4—21no—
83C22×C32×C4—22no—
84C22×C32×C4—21no—
85C22×C32×C4C2312yes—
86C22×C32×C4C2311yescharacteristic
87C22×C3×Dic3—21no—
88C2×C3×Dic6—22no—
89C2×C3×Dic6—22no—
90C2×C3×Dic6—21no—
91C2×C3×Dic6C2312yes—
92C3×#217—24no—
93C3×#217—21no—
94C3×#216—28no—
95C3×#216—24no—
96C3×#214—24no—
97C3×C4⋊Dic3C2312yes—
98C3×Dic3⋊C4—24no—
99C3×C4×Dic3—21no—
100C3×#208—22no—
101C2×C3×C3⋊C8—22no—
102C2×C3×C3⋊C8C2312yes—
103C2×C3×C3⋊C8C2312yes—
104#3011—31noSylow
105C3×2−1+4—28no—
106C3×2−1+4—28no—
107C3×2−1+4—24no—
108C3×2−1+4—24no—
109C3×2−1+4—24no—
110C3×2−1+4—24no—
111C2×C3×#41—24no—
112C2×C3×#41—22no—
113C2×C3×#41—22no—
114C2×C3×#41—21no—
115C2×C3×#41—22no—
116C2×C3×#41—21no—
117C2×C3×#41—22no—
118C2×C3×#41—21no—
119C2×C3×#41—21no—
120C2×C3×#41C2×D312yes—
121C2×C3×#41C2×D311yescharacteristic
122C2×C3×#41C2×D311yescharacteristic
123C2×C3×#41C22×C312yes—
124C2×C3×#41C22×C311yescharacteristic
125C2×C3×#41C22×C311yescharacteristic
126C22×C3×Q8—22no—
127C22×C3×Q8—21no—
128C22×C3×Q8—21no—
129C22×C3×Q8—22no—
130C22×C3×Q8—21no—
131C22×C3×Q8—31no—
132C22×C3×Q8C2×D312yes—
133C22×C3×Q8C2×D311yescharacteristic
134C22×C3×Q8C22×C312yes—
135C22×C3×Q8C22×C311yescharacteristic
136C22×Dic6C22×C311yescharacteristic
137C3×#137—64no—
138C2×C3×Q16—34no—
139C2×C3×Q16—32no—
140C2×C3×Q16—64no—
141C2×C3×SD16—32no—
142C2×C3×M16—31no—
143C22×C3×C8—31no—
144C3×C4⋊Q8—61no—
145C3×#124—61no—
146C3×#123—62no—
147C3×C22⋊Q8—34no—
148C3×#117—31no—
149#951—24no—
150#944—21no—
151C2×#216—24no—
152C2×#216C22×C312yes—
153C2×#216C22×C314yes—
154#936—21no—
155#935—22no—
156C2×#214C22×C312yes—
157#926C22×C311yescharacteristic
158#924C22×C314yes—
159C2×#208C22×C311yescharacteristic
160C22×C3⋊C8C22×C311yescharacteristic
161C3×#103—32no—
162C3×#103—34no—
163C3×#102—32no—
164C3×C22⋊C8—34no—
165#835C22×C314yes—
166#835C22×C312yes—
167#832C22×C312yes—
168#830C22×C314yes—
169C23×C32—21no—
170C23×C32C2×D412yes—
171C23×C32C2×D411yescharacteristic
172C32×Q8—28no—
173C32×Q8—24no—
174C32×Q8—22no—
175C32×Q8—21no—
176C32×Q8—22no—
177C32×Q8C2×D412yes—
178C32×Q8C2×D414yes—
179C32×D4—28no—
180C32×D4—24no—
181C32×D4—24no—
182C32×D4—22no—
183C32×D4—21no—
184C32×D4C2×D412yes—
185C2×C32×C4—28no—
186C2×C32×C4—28no—
187C2×C32×C4—24no—
188C2×C32×C4—22no—
189C2×C32×C4—21no—
190C2×C32×C4—24no—
191C2×C32×C4—22no—
192C2×C32×C4—22no—
193C2×C32×C4C2×D412yes—
194C2×C32×C4C2×D411yescharacteristic
195C2×C32×C4C2×D412yes—
196C2×C32×C4C2×D412yes—
197C2×C32×C4C2411yescharacteristic
198C2×C3×Dic3—22no—
199C2×C3×Dic3—22no—
200C2×C3×Dic3—42no—
201C3×Dic6—22no—
202C3×Dic6—22no—
203C3×Dic6—42no—
204C3×C3⋊C8—22no—
205C3×C3⋊C8—22no—
206C2×2−1+4C3×D311yescharacteristic
207C2×#137—31no—
208C22×Q16—31no—
209#536—31no—
210#452—34no—
211#451—34no—
212#419—31no—
213#416—31no—
214#408—31no—
215C3×#41—232no—
216C3×#41—216no—
217C3×#41—216no—
218C3×#41—216no—
219C3×#41—28no—
220C3×#41—28no—
221C3×#41—28no—
222C3×#41—28no—
223C3×#41—24no—
224C3×#41—24no—
225C3×#41—24no—
226C3×#41—24no—
227C3×#41—24no—
228C3×#41—24no—
229C3×#41—24no—
230C3×#41—22no—
231C3×#41—22no—
232C3×#41—22no—
233C3×#41—22no—
234C3×#41—22no—
235C3×#41—22no—
236C3×#41—24no—
237C3×#41#6714yes—
238C3×#41C3×D414yes—
239C2×C3×Q8—24no—
240C2×C3×Q8—22no—
241C2×C3×Q8—216no—
242C2×C3×Q8—28no—
243C2×C3×Q8—28no—
244C2×C3×Q8—28no—
245C2×C3×Q8—28no—
246C2×C3×Q8—24no—
247C2×C3×Q8—24no—
248C2×C3×Q8—24no—
249C2×C3×Q8—24no—
250C2×C3×Q8—24no—
251C2×C3×Q8—22no—
252C2×C3×Q8—21no—
253C2×C3×Q8—22no—
254C2×C3×Q8—22no—
255C2×C3×Q8—21no—
256C2×C3×Q8—22no—
257C2×C3×Q8—21no—
258C2×C3×Q8—24no—
259C2×C3×Q8—21no—
260C2×C3×Q8—22no—
261C2×C3×Q8—32no—
262C2×C3×Q8—61no—
263C2×C3×Q8—62no—
264C2×C3×Q8—62no—
265C2×C3×Q8#6714yes—
266C2×C3×Q8#6711yescharacteristic
267C2×C3×Q8C3×D414yes—
268C2×C3×Q8C3×D411yescharacteristic
269C2×C3×Q8C22×D311yescharacteristic
270C2×C3×Q8C22×D312yes—
271C2×C3×Q8C23×C311yescharacteristic
272C2×C3×Q8C23×C312yes—
273C2×C3×D4—24no—
274C2×C3×D4—22no—
275C2×C3×D4—22no—
276C2×C3×D4—21no—
277C2×C3×D4—22no—
278C2×C3×D4—21no—
279C2×C3×D4—22no—
280C2×C3×D4—21no—
281C2×C3×D4—21no—
282C2×C3×D4#6712yes—
283C2×C3×D4#6711yescharacteristic
284C2×C3×D4C3×D412yes—
285C2×C3×D4C3×D411yescharacteristic
286C2×C3×D4C22×D311yescharacteristic
287C2×C3×D4C23×C311yescharacteristic
288C22×C3×C4—24no—
289C22×C3×C4—22no—
290C22×C3×C4—24no—
291C22×C3×C4—22no—
292C22×C3×C4—22no—
293C22×C3×C4—21no—
294C22×C3×C4—22no—
295C22×C3×C4—21no—
296C22×C3×C4—22no—
297C22×C3×C4—21no—
298C22×C3×C4—22no—
299C22×C3×C4—21no—
300C22×C3×C4—22no—
301C22×C3×C4—21no—
302C22×C3×C4—61no—
303C22×C3×C4C22×D312yes—
304C22×C3×C4C22×D311yescharacteristic
305C22×C3×C4C23×C312yes—
306C22×C3×C4C23×C311yescharacteristic
307C22×Dic3—21no—
308C2×Dic6—22no—
309C2×Dic6—22no—
310C2×Dic6—21no—
311C2×Dic6C23×C312yes—
312C3×Q16—68no—
313C3×Q16—64no—
314C3×SD16—64no—
315C3×M16—62no—
316C2×C3×C8—32no—
317C2×C3×C8—32no—
318C2×C3×C8—62no—
319C3×C4⋊C4—32no—
320C3×C4⋊C4—64no—
321C3×C22⋊C4—61no—
322C3×C22⋊C4—64no—
323C3×C42—61no—
324#217—24no—
325#217—21no—
326#216—28no—
327#216—24no—
328#214—24no—
329C4⋊Dic3C23×C312yes—
330Dic3⋊C4—24no—
331C4×Dic3—21no—
332#208—22no—
333C2×C3⋊C8—22no—
334C2×C3⋊C8C23×C312yes—
335C2×C3⋊C8C23×C312yes—
336C22×C32—22no—
337C22×C32—22no—
338C22×C32—22no—
339C22×C32—24no—
340C22×C32—42no—
341C22×C32#12012yes—
342C22×C32C22×D411yescharacteristic
343C32×C4—24no—
344C32×C4—22no—
345C32×C4—22no—
346C32×C4#12012yes—
347C32×C4C22×D411yescharacteristic
348C32×C4C22×D411yescharacteristic
349C3×Dic3—42no—
350C3×Dic3—42no—
3512−1+4—24no—
3522−1+4—24no—
353C2×#41—22no—
354C2×#41—21no—
355C2×#41C2×C3×D312yes—
356C2×#41C2×C3×D311yescharacteristic
357C2×#41C2×C3×D311yescharacteristic
358C22×Q8—21no—
359C22×Q8—31no—
360C22×Q8C2×C3×D312yes—
361C22×Q8C2×C3×D311yescharacteristic
362#137—64no—
363C2×Q16—34no—
364C2×Q16—32no—
365C2×Q16—64no—
366C2×SD16—32no—
367C2×M16—31no—
368C22×C8—31no—
369C4⋊Q8—61no—
370#124—61no—
371#123—62no—
372C22⋊Q8—34no—
373#117—31no—
374#103—32no—
375#103—34no—
376#102—32no—
377C22⋊C8—34no—
378C23×C3—22no—
379C23×C3—21no—
380C23×C3—21no—
381C23×C3—22no—
382C23×C3—21no—
383C23×C3C2×#6712yes—
384C23×C3C2×#6711yescharacteristic
385C23×C3C2×C3×D412yes—
386C23×C3C2×C3×D411yescharacteristic
387C3×Q8—216no—
388C3×Q8—28no—
389C3×Q8—28no—
390C3×Q8—28no—
391C3×Q8—24no—
392C3×Q8—22no—
393C3×Q8—24no—
394C3×Q8—24no—
395C3×Q8—22no—
396C3×Q8—24no—
397C3×Q8—22no—
398C3×Q8—21no—
399C3×Q8—22no—
400C3×Q8—21no—
401C3×Q8—22no—
402C3×Q8—22no—
403C3×Q8—24no—
404C3×Q8—62no—
405C3×Q8—62no—
406C3×Q8—122no—
407C3×Q8C2×#6712yes—
408C3×Q8C2×#6714yes—
409C3×Q8C2×C3×D412yes—
410C3×Q8C2×C3×D414yes—
411C3×D4—216no—
412C3×D4—28no—
413C3×D4—28no—
414C3×D4—28no—
415C3×D4—28no—
416C3×D4—24no—
417C3×D4—22no—
418C3×D4—24no—
419C3×D4—24no—
420C3×D4—24no—
421C3×D4—24no—
422C3×D4—22no—
423C3×D4—22no—
424C3×D4—21no—
425C3×D4—21no—
426C3×D4—22no—
427C3×D4C2×#6712yes—
428C3×D4C2×C3×D412yes—
429C2×C3×C4—24no—
430C2×C3×C4—22no—
431C2×C3×C4—216no—
432C2×C3×C4—216no—
433C2×C3×C4—28no—
434C2×C3×C4—28no—
435C2×C3×C4—28no—
436C2×C3×C4—28no—
437C2×C3×C4—28no—
438C2×C3×C4—28no—
439C2×C3×C4—24no—
440C2×C3×C4—24no—
441C2×C3×C4—24no—
442C2×C3×C4—24no—
443C2×C3×C4—22no—
444C2×C3×C4—21no—
445C2×C3×C4—22no—
446C2×C3×C4—21no—
447C2×C3×C4—24no—
448C2×C3×C4—24no—
449C2×C3×C4—22no—
450C2×C3×C4—22no—
451C2×C3×C4—22no—
452C2×C3×C4—22no—
453C2×C3×C4—21no—
454C2×C3×C4—22no—
455C2×C3×C4—21no—
456C2×C3×C4—22no—
457C2×C3×C4—22no—
458C2×C3×C4—62no—
459C2×C3×C4—62no—
460C2×C3×C4—122no—
461C2×C3×C4C2×#6712yes—
462C2×C3×C4C2×#6711yescharacteristic
463C2×C3×C4C2×#6712yes—
464C2×C3×C4C2×#6712yes—
465C2×C3×C4C2×C3×D412yes—
466C2×C3×C4C2×C3×D411yescharacteristic
467C2×C3×C4C2×C3×D412yes—
468C2×C3×C4C2×C3×D412yes—
469C2×C3×C4C23×D311yescharacteristic
470C2×C3×C4C24×C311yescharacteristic
471C2×Dic3—22no—
472C2×Dic3—22no—
473C2×Dic3—42no—
474Dic6—22no—
475Dic6—22no—
476Dic6—42no—
477C3×C8—62no—
478C3×C8—62no—
479C3⋊C8—22no—
480C3⋊C8—22no—
481C2×C32—22no—
482C2×C32—42no—
483C2×C32—42no—
484C2×C32C2×#12011yescharacteristic
485C2×C32#57812yes—
486#41—216no—
487#41—28no—
488#41—24no—
489#41—24no—
490#41—22no—
491#41—22no—
492#41—22no—
493#41C3×#6714yes—
494C2×Q8—28no—
495C2×Q8—24no—
496C2×Q8—24no—
497C2×Q8—22no—
498C2×Q8—21no—
499C2×Q8—22no—
500C2×Q8—32no—
501C2×Q8—61no—
502C2×Q8—62no—
503C2×Q8—62no—
504C2×Q8C3×#6714yes—
505C2×Q8C3×#6711yescharacteristic
506C2×Q8C22×C3×D311yescharacteristic
507C2×Q8C22×C3×D312yes—
508C2×D4—22no—
509C2×D4—21no—
510C2×D4C3×#6712yes—
511C2×D4C3×#6711yescharacteristic
512C2×D4C22×C3×D311yescharacteristic
513C22×C4—22no—
514C22×C4—21no—
515C22×C4—22no—
516C22×C4—21no—
517C22×C4—61no—
518C22×C4C22×C3×D312yes—
519C22×C4C22×C3×D311yescharacteristic
520Q16—68no—
521Q16—64no—
522SD16—64no—
523M16—62no—
524C2×C8—32no—
525C2×C8—32no—
526C2×C8—62no—
527C4⋊C4—32no—
528C4⋊C4—64no—
529C22⋊C4—61no—
530C22⋊C4—64no—
531C42—61no—
532C22×C3—22no—
533C22×C3—22no—
534C22×C3—24no—
535C22×C3—24no—
536C22×C3—24no—
537C22×C3—24no—
538C22×C3—22no—
539C22×C3—22no—
540C22×C3—22no—
541C22×C3—22no—
542C22×C3—22no—
543C22×C3—21no—
544C22×C3—42no—
545C22×C3—44no—
546C22×C3—44no—
547C22×C3—42no—
548C22×C3—42no—
549C22×C3#95312yes—
550C22×C3C3×#12012yes—
551C22×C3C22×#6711yescharacteristic
552C22×C3C22×C3×D411yescharacteristic
553C3×C4—28no—
554C3×C4—24no—
555C3×C4—24no—
556C3×C4—24no—
557C3×C4—22no—
558C3×C4—22no—
559C3×C4—24no—
560C3×C4—22no—
561C3×C4—22no—
562C3×C4—22no—
563C3×C4—21no—
564C3×C4—21no—
565C3×C4—122no—
566C3×C4—122no—
567C3×C4#95312yes—
568C3×C4C3×#12012yes—
569C3×C4C22×#6711yescharacteristic
570C3×C4C22×#6711yescharacteristic
571C3×C4C22×C3×D411yescharacteristic
572C3×C4C22×C3×D411yescharacteristic
573Dic3—42no—
574Dic3—42no—
575C32#301111yescharacteristic, complemented, Sylow
576C23—21no—
577C23C2×C3×#6712yes—
578C23C2×C3×#6711yescharacteristic
579Q8—28no—
580Q8—24no—
581Q8—22no—
582Q8—21no—
583Q8—22no—
584Q8—62no—
585Q8—62no—
586Q8—122no—
587Q8C2×C3×#6712yes—
588Q8C2×C3×#6714yes—
589D4—28no—
590D4—24no—
591D4—24no—
592D4—22no—
593D4—21no—
594D4C2×C3×#6712yes—
595C2×C4—28no—
596C2×C4—28no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—62no—
604C2×C4—62no—
605C2×C4—122no—
606C2×C4C2×C3×#6712yes—
607C2×C4C2×C3×#6711yescharacteristic
608C2×C4C2×C3×#6712yes—
609C2×C4C2×C3×#6712yes—
610C2×C4C23×C3×D311yescharacteristic
611C8—62no—
612C8—62no—
613C2×C3—22no—
614C2×C3—22no—
615C2×C3—22no—
616C2×C3—21no—
617C2×C3—42no—
618C2×C3—44no—
619C2×C3—44no—
620C2×C3—42no—
621C2×C3—42no—
622C2×C3—42no—
623C2×C3—42no—
624C2×C3#583812yes—
625C2×C3C2×#95311yescharacteristic
626C2×C3C2×C3×#12011yescharacteristic
627C2×C3C3×#57812yes—
628C22—22no—
629C22—24no—
630C22—22no—
631C22—22no—
632C22—42no—
633C22C3×#95312yes—
634C22C22×C3×#6711yescharacteristic
635C4—24no—
636C4—22no—
637C4—22no—
638C4—122no—
639C4—122no—
640C4C3×#95312yes—
641C4C22×C3×#6711yescharacteristic
642C4C22×C3×#6711yescharacteristic
643C3—21no—
644C3#8411711yescharacteristic, direct product factor
645C3C3×#301111yescharacteristic, complemented
646C2—22no—
647C2—42no—
648C2—42no—
649C2C3×#583812yes—
650C2C2×C3×#95311yescharacteristic
subgroupG/H
centerC22×C3C22×#67
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
Fitting, prime radicalC2×C32×2−1+4C2
socleC22×C32C22×D4
hypercenterC2×C3×2−1+4D3
hypocenterC3C3×#3011
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C2×C3×#41 ◁ C2×C3×2−1+4

Complements

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

NH ≅ G/Ntype
2. C3×#5825C2semidirect
3. C2×C3×#951C2semidirect
4. C22×C3×#216C2semidirect
5. C3×#5239C2semidirect
6. C3×#5234C2semidirect
7. C3×#5233C2semidirect
8. C3×#5228C2semidirect
9. C3×#5225C2semidirect
11. #84117C3direct (N × H)
45. #5825C2×C3semidirect
46. C2×#951C2×C3semidirect
47. C22×#216C2×C3semidirect
53. #5239C2×C3semidirect
54. #5234C2×C3semidirect
55. #5233C2×C3semidirect
56. #5228C2×C3semidirect
57. #5225C2×C3semidirect
575. C32#3011semidirect
644. C3#84117direct (N × H)
645. C3C3×#3011semidirect

Maximal subgroups

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

subgroupindex
C2×C32×2−1+42
C3×#58252
C2×C3×#9512
C22×C3×#2162
C3×#52392
C3×#52342
C3×#52332
C3×#52282
C3×#52252
C3×#30113
#841173