Finite Group Databasebeta

← C3×#86243 #11906001

Subgroups — C3×#86243 #11906001

Order 1152.

3107 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#1010C212yesmaximal, complemented
2C32×#532C211yesmaximal, characteristic, complemented
3C3×#5825C212yesmaximal, complemented
4C3×D3×C4⋊Q8C212yesmaximal, complemented
5C3×#5716C212yesmaximal, complemented
6C3×#5675C214yesmaximal, complemented
7C3×#5611C218yesmaximal, complemented
8C3×#5551C214yesmaximal, complemented
9C3×D4×Dic6C214yesmaximal, complemented
10C3×C4×#235C211yesmaximal, characteristic, complemented
11C2×C3×#869C211yesmaximal, characteristic, complemented
12C3×#3499—31nomaximal
13#86243C311yesmaximal, characteristic, direct product factor
14C2×C32×#41C2212yes—
15C3×#1010—216no—
16C2×C3×D3×Q8C2214yes—
17C2×C3×#237C2218yes—
18C2×C3×#237C2214yes—
19C2×C3×#235C2214yes—
20C2×C3×#235C2211yescharacteristic
21C22×C3×Dic6C2214yes—
22C22×C3×Dic6C2212yes—
23C32×C4⋊Q8C2211yescharacteristic
24C32×#127C2211yescharacteristic
25C32×#124C2212yes—
26C32×#121C2214yes—
27C32×C4×D4C2214yes—
28C2×C32×C42C2211yescharacteristic
29C3×#944C2214yes—
30C3×#938C2218yes—
31C3×#936C2214yes—
32C3×#935C2218yes—
33C3×D4×Dic3C2218yes—
34C3×C4×#67C2214yes—
35C3×#926C2211yescharacteristic
36C2×C3×C4⋊Dic3C2214yes—
37C3×#924C2218yes—
38C3×D3×C4⋊C4C2218yes—
39C3×#888C2218yes—
40C3×#883C22116yes—
41C3×#878C22116yes—
42C3×C4×D12C2212yes—
43C3×#872C2212yes—
44C3×C42×D3C2212yes—
45C3×#869C2214yes—
46C3×#869C2212yes—
47C3×#869C2212yes—
48C3×C4×Dic6C2212yes—
49C2×C3×2−1+4—32no—
50C2×#1010C2×C312yescomplemented
51C3×D4×Q8—34no—
52C3×#547—38no—
53C3×#547—34no—
54C3×#536—32no—
55C3×#536—34no—
56C3×#532—21no—
57C3×#532—32no—
58C3×#532D311yescharacteristic, complemented
59C3×#532C2×C311yescharacteristic, complemented
60C2×C3×C4⋊Q8—32no—
61C2×C3×C4⋊Q8—31no—
62C3×C4×#41—31no—
63#5825C2×C312yescomplemented
64D3×C4⋊Q8C2×C312yescomplemented
65#5716C2×C312yescomplemented
66#5675C2×C314yescomplemented
67#5611C2×C318yescomplemented
68#5551C2×C314yescomplemented
69D4×Dic6C2×C314yescomplemented
70C4×#235C2×C311yescharacteristic, complemented
71C2×#869C2×C311yescharacteristic, complemented
72C32×#41—28no—
73C2×C32×Q8C2312yes—
74C2×C32×D4C2314yes—
75C2×C32×D4C2314yes—
76C2×C32×D4C2312yes—
77C22×C32×C4C2314yes—
78C22×C32×C4C2312yes—
79C22×C32×C4C2311yescharacteristic
80C2×C3×#67C2318yes—
81C2×C3×#67C2312yes—
82C22×C3×Dic3C2318yes—
83C22×C3×Dic3C2314yes—
84C3×D3×Q8—216no—
85C3×#237—232no—
86C3×#237—216no—
87C3×#235—216no—
88C3×#235D418yescomplemented
89C2×C3×D12C2311yescharacteristic
90C2×C3×C4×D3C2318yes—
91C2×C3×C4×D3C2314yes—
92C2×C3×C4×D3C2312yes—
93C2×C3×Dic6—28no—
94C2×C3×Dic6—216no—
95C2×C3×Dic6C2318yes—
96C2×C3×Dic6C2314yes—
97C2×C3×Dic6C2314yes—
98C2×C3×Dic6C2311yescharacteristic
99C32×C4⋊C4C2314yes—
100C32×C22⋊C4C2318yes—
101C32×C42C2312yes—
102C32×C42C2311yescharacteristic
103C32×C42C2311yescharacteristic
104C3×#217C23116yes—
105C3×#217C2312yes—
106C3×(C2×D3)⋊C4C2314yes—
107C3×C4⋊Dic3C2318yes—
108C3×C4⋊Dic3C2318yes—
109C3×C4⋊Dic3C2312yes—
110C3×Dic3⋊C4C2314yes—
111C3×Dic3⋊C4C23116yes—
112C3×C4×Dic3C2314yes—
113C3×C4×Dic3C2312yes—
114#3499—31noSylow
115C3×2−1+4—616no—
116C2×C3×#41—22no—
117C2×C3×#41—34no—
118C2×C3×#41—38no—
119C2×C3×#41—31no—
120C2×C3×#41—34no—
121C2×C3×#41C2×D312yes—
122C2×C3×#41C22×C312yes—
123C22×C3×Q8—34no—
124C22×C3×Q8—32no—
125C22×C3×Q8—34no—
126#1010—216no—
127C2×D3×Q8C22×C314yes—
128C2×#237C22×C314yes—
129C2×#237C22×C318yes—
130C2×#235C22×C314yes—
131C2×#235C22×C311yescharacteristic
132C22×Dic6C22×C312yes—
133C22×Dic6C22×C314yes—
134C3×C4⋊Q8—21no—
135C3×C4⋊Q8—34no—
136C3×C4⋊Q8—38no—
137C3×C4⋊Q8—32no—
138C3×C4⋊Q8—32no—
139C3×C4⋊Q8—34no—
140C3×C4⋊Q8C2×D311yescharacteristic
141C3×C4⋊Q8C22×C311yescharacteristic
142C3×#127—21no—
143C3×#127C2×D311yescharacteristic
144C3×#127C22×C311yescharacteristic
145C3×#124—22no—
146C3×#124—34no—
147C3×#124C2×D312yes—
148C3×#124C22×C312yes—
149C3×#123—38no—
150C3×#123—316no—
151C3×C22⋊Q8—38no—
152C3×C22⋊Q8—316no—
153C3×#121—24no—
154C3×#121—38no—
155C3×#121C2×D314yes—
156C3×#121C22×C314yes—
157C3×C4×Q8—32no—
158C3×C4×D4—24no—
159C3×C4×D4—38no—
160C3×C4×D4—34no—
161C3×C4×D4—32no—
162C3×C4×D4C2×D314yes—
163C3×C4×D4C22×C314yes—
164C3×#117—31no—
165C3×#117—32no—
166C2×C3×C4⋊C4—34no—
167C2×C3×C4⋊C4—38no—
168C2×C3×C42—21no—
169C2×C3×C42—32no—
170C2×C3×C42C2×D311yescharacteristic
171C2×C3×C42C22×C311yescharacteristic
172#944C22×C314yes—
173#938C22×C318yes—
174#936C22×C314yes—
175#935C22×C318yes—
176D4×Dic3C22×C318yes—
177C4×#67C22×C314yes—
178#926C22×C311yescharacteristic
179C2×C4⋊Dic3C22×C314yes—
180#924C22×C318yes—
181D3×C4⋊C4C22×C318yes—
182#888C22×C318yes—
183#883C22×C3116yes—
184#878C22×C3116yes—
185C4×D12C22×C312yes—
186#872C22×C312yes—
187C42×D3C22×C312yes—
188#869C22×C312yes—
189#869C22×C312yes—
190#869C22×C314yes—
191C4×Dic6C22×C312yes—
192C23×C32C2414yes—
193C23×C32C2411yescharacteristic
194C22×C3×D3C2412yes—
195C32×Q8—24no—
196C32×D4—28no—
197C32×D4—24no—
198C32×D4—28no—
199C2×C32×C4—28no—
200C2×C32×C4—24no—
201C2×C32×C4C2×D414yes—
202C2×C32×C4C2414yes—
203C2×C32×C4C2412yes—
204C2×C32×C4C2411yescharacteristic
205C2×C32×C4C2411yescharacteristic
206C2×C32×C4C2412yes—
207C3×#67—216no—
208C3×#67C2×D418yes—
209C2×C3×Dic3—28no—
210C2×C3×Dic3—216no—
211C2×C3×Dic3C2414yes—
212C2×C3×Dic3C2418yes—
213C2×C3×Dic3C2412yes—
214C3×D12C2×D414yes—
215C3×C4×D3—216no—
216C3×C4×D3—28no—
217C3×C4×D3C2×D418yes—
218C3×Dic6—28no—
219C3×Dic6—28no—
220C3×Dic6—216no—
221C3×Dic6C2×D414yes—
222C2×2−1+4—32no—
223D4×Q8—34no—
224#547—38no—
225#547—34no—
226#536—32no—
227#536—34no—
228#532—32no—
229#532C3×D311yescharacteristic, complemented
230C2×C4⋊Q8—32no—
231C2×C4⋊Q8—31no—
232C4×#41—31no—
233C3×#41—28no—
234C3×#41—28no—
235C3×#41—38no—
236C3×#41—48no—
237C3×#41—616no—
238C3×#41—616no—
239C3×#41—632no—
240C2×C3×Q8—22no—
241C2×C3×Q8—31no—
242C2×C3×Q8—38no—
243C2×C3×Q8—34no—
244C2×C3×Q8—34no—
245C2×C3×Q8—616no—
246C2×C3×Q8—616no—
247C2×C3×Q8—68no—
248C2×C3×Q8C22×D312yes—
249C2×C3×Q8C23×C312yes—
250C2×C3×D4—24no—
251C2×C3×D4—24no—
252C2×C3×D4—22no—
253C2×C3×D4—32no—
254C2×C3×D4—38no—
255C2×C3×D4—31no—
256C2×C3×D4C22×D314yes—
257C2×C3×D4C22×D314yes—
258C2×C3×D4C22×D312yes—
259C2×C3×D4C23×C314yes—
260C2×C3×D4C23×C314yes—
261C2×C3×D4C23×C312yes—
262C22×C3×C4—21no—
263C22×C3×C4—22no—
264C22×C3×C4—24no—
265C22×C3×C4—32no—
266C22×C3×C4—34no—
267C22×C3×C4—38no—
268C22×C3×C4—34no—
269C22×C3×C4—38no—
270C22×C3×C4C22×D312yes—
271C22×C3×C4C22×D314yes—
272C22×C3×C4C22×D311yescharacteristic
273C22×C3×C4C23×C314yes—
274C22×C3×C4C23×C312yes—
275C22×C3×C4C23×C311yescharacteristic
276C2×#67C23×C318yes—
277C2×#67C23×C312yes—
278C22×Dic3C23×C318yes—
279C22×Dic3C23×C314yes—
280D3×Q8—216no—
281#237—232no—
282#237—216no—
283#235—216no—
284#235C3×D418yescomplemented
285C2×D12C23×C311yescharacteristic
286C2×C4×D3C23×C314yes—
287C2×C4×D3C23×C318yes—
288C2×C4×D3C23×C312yes—
289C2×Dic6—28no—
290C2×Dic6—216no—
291C2×Dic6C23×C314yes—
292C2×Dic6C23×C314yes—
293C2×Dic6C23×C318yes—
294C2×Dic6C23×C311yescharacteristic
295C3×C4⋊C4—24no—
296C3×C4⋊C4—316no—
297C3×C4⋊C4—34no—
298C3×C4⋊C4—38no—
299C3×C4⋊C4—38no—
300C3×C4⋊C4—32no—
301C3×C4⋊C4C22×D314yes—
302C3×C4⋊C4C23×C314yes—
303C3×C22⋊C4—28no—
304C3×C22⋊C4—32no—
305C3×C22⋊C4—316no—
306C3×C22⋊C4—34no—
307C3×C22⋊C4C22×D318yes—
308C3×C22⋊C4C23×C318yes—
309C3×C42—22no—
310C3×C42—21no—
311C3×C42—21no—
312C3×C42—32no—
313C3×C42—34no—
314C3×C42C22×D311yescharacteristic
315C3×C42C22×D311yescharacteristic
316C3×C42C22×D312yes—
317C3×C42C23×C312yes—
318C3×C42C23×C311yescharacteristic
319C3×C42C23×C311yescharacteristic
320#217C23×C3116yes—
321#217C23×C312yes—
322(C2×D3)⋊C4C23×C314yes—
323C4⋊Dic3C23×C312yes—
324C4⋊Dic3C23×C318yes—
325C4⋊Dic3C23×C318yes—
326Dic3⋊C4C23×C314yes—
327Dic3⋊C4C23×C3116yes—
328C4×Dic3C23×C314yes—
329C4×Dic3C23×C312yes—
330C22×C32—22no—
331C22×C32—24no—
332C22×C32—28no—
333C22×C32C22×D412yes—
334C22×C32C2511yescharacteristic
335C2×C3×D3—24no—
336C2×C3×D3C22×D414yes—
337C32×C4—24no—
338C32×C4—22no—
339C32×C4C22×D412yes—
340C32×C4C22×D412yes—
341C32×C42−1+414yescomplemented
342C3×Dic3—28no—
343C3×Dic3—24no—
344C3×Dic3C22×D414yes—
3452−1+4—616no—
346C2×#41—34no—
347C2×#41—38no—
348C2×#41—31no—
349C2×#41—34no—
350C2×#41C2×C3×D312yes—
351C22×Q8—34no—
352C22×Q8—32no—
353C22×Q8—34no—
354C4⋊Q8—34no—
355C4⋊Q8—38no—
356C4⋊Q8—32no—
357C4⋊Q8—32no—
358C4⋊Q8—34no—
359C4⋊Q8C2×C3×D311yescharacteristic
360#127C2×C3×D311yescharacteristic
361#124—34no—
362#124C2×C3×D312yes—
363#123—38no—
364#123—316no—
365C22⋊Q8—38no—
366C22⋊Q8—316no—
367#121—38no—
368#121C2×C3×D314yes—
369C4×Q8—32no—
370C4×D4—38no—
371C4×D4—34no—
372C4×D4—32no—
373C4×D4C2×C3×D314yes—
374#117—31no—
375#117—32no—
376C2×C4⋊C4—34no—
377C2×C4⋊C4—38no—
378C2×C42—32no—
379C2×C42C2×C3×D311yescharacteristic
380C23×C3—21no—
381C23×C3—24no—
382C23×C3—32no—
383C23×C3C23×D314yes—
384C23×C3C23×D311yescharacteristic
385C23×C3C24×C314yes—
386C23×C3C24×C311yescharacteristic
387C22×D3C24×C312yes—
388C3×Q8—24no—
389C3×Q8—24no—
390C3×Q8—34no—
391C3×Q8—44no—
392C3×Q8—616no—
393C3×Q8—68no—
394C3×Q8—68no—
395C3×D4—28no—
396C3×D4—28no—
397C3×D4—24no—
398C3×D4—28no—
399C3×D4—24no—
400C3×D4—28no—
401C3×D4—38no—
402C3×D4—34no—
403C3×D4—48no—
404C3×D4—44no—
405C3×D4—48no—
406C3×D4—616no—
407C2×C3×C4—28no—
408C2×C3×C4—24no—
409C2×C3×C4—28no—
410C2×C3×C4—24no—
411C2×C3×C4—22no—
412C2×C3×C4—21no—
413C2×C3×C4—21no—
414C2×C3×C4—22no—
415C2×C3×C4—24no—
416C2×C3×C4—24no—
417C2×C3×C4—32no—
418C2×C3×C4—38no—
419C2×C3×C4—34no—
420C2×C3×C4—38no—
421C2×C3×C4—44no—
422C2×C3×C4—48no—
423C2×C3×C4—616no—
424C2×C3×C4—616no—
425C2×C3×C4—68no—
426C2×C3×C4—68no—
427C2×C3×C4D3×D414yes—
428C2×C3×C4C2×C3×D414yes—
429C2×C3×C4C23×D312yes—
430C2×C3×C4C23×D314yes—
431C2×C3×C4C23×D311yescharacteristic
432C2×C3×C4C23×D311yescharacteristic
433C2×C3×C4C23×D312yes—
434C2×C3×C4C24×C314yes—
435C2×C3×C4C24×C312yes—
436C2×C3×C4C24×C311yescharacteristic
437C2×C3×C4C24×C311yescharacteristic
438C2×C3×C4C24×C312yes—
439#67—216no—
440#67C2×C3×D418yes—
441C2×Dic3—28no—
442C2×Dic3—216no—
443C2×Dic3C24×C314yes—
444C2×Dic3C24×C318yes—
445C2×Dic3C24×C312yes—
446D12C2×C3×D414yes—
447C4×D3—216no—
448C4×D3—28no—
449C4×D3C2×C3×D418yes—
450Dic6—28no—
451Dic6—28no—
452Dic6—216no—
453Dic6C2×C3×D414yes—
454C2×C32—22no—
455C2×C32—44no—
456C2×C32C23×D411yescharacteristic
457C2×C32C2×2−1+412yes—
458C3×D3—24no—
459#41—28no—
460#41—38no—
461#41—616no—
462#41—616no—
463#41—632no—
464C2×Q8—31no—
465C2×Q8—38no—
466C2×Q8—34no—
467C2×Q8—34no—
468C2×Q8—616no—
469C2×Q8—616no—
470C2×Q8—68no—
471C2×Q8C22×C3×D312yes—
472C2×D4—32no—
473C2×D4—38no—
474C2×D4—31no—
475C2×D4C22×C3×D314yes—
476C2×D4C22×C3×D314yes—
477C2×D4C22×C3×D312yes—
478C22×C4—32no—
479C22×C4—34no—
480C22×C4—38no—
481C22×C4—34no—
482C22×C4—38no—
483C22×C4C22×C3×D312yes—
484C22×C4C22×C3×D314yes—
485C22×C4C22×C3×D311yescharacteristic
486C4⋊C4—38no—
487C4⋊C4—316no—
488C4⋊C4—38no—
489C4⋊C4—34no—
490C4⋊C4—32no—
491C4⋊C4C22×C3×D314yes—
492C22⋊C4—32no—
493C22⋊C4—316no—
494C22⋊C4—34no—
495C22⋊C4C22×C3×D318yes—
496C42—32no—
497C42—34no—
498C42C22×C3×D311yescharacteristic
499C42C22×C3×D311yescharacteristic
500C42C22×C3×D312yes—
501C22×C3—22no—
502C22×C3—22no—
503C22×C3—24no—
504C22×C3—28no—
505C22×C3—24no—
506C22×C3—28no—
507C22×C3—22no—
508C22×C3—21no—
509C22×C3—34no—
510C22×C3—42no—
511C22×C3—44no—
512C22×C3—48no—
513C22×C3—64no—
514C22×C3C2×D3×D412yes—
515C22×C3C22×C3×D412yes—
516C22×C3C24×D311yescharacteristic
517C22×C3C25×C311yescharacteristic
518C2×D3—24no—
519C2×D3C22×C3×D414yes—
520C3×C4—24no—
521C3×C4—22no—
522C3×C4—24no—
523C3×C4—22no—
524C3×C4—22no—
525C3×C4—22no—
526C3×C4—24no—
527C3×C4—34no—
528C3×C4—44no—
529C3×C4—42no—
530C3×C4—68no—
531C3×C4—64no—
532C3×C4C2×D3×D412yes—
533C3×C4C2×D3×D412yes—
534C3×C4#101014yescomplemented
535C3×C4C22×C3×D412yes—
536C3×C4C22×C3×D412yes—
537C3×C4C3×2−1+414yescomplemented
538Dic3—28no—
539Dic3—24no—
540Dic3C22×C3×D414yes—
541C32#349911yescharacteristic, complemented, Sylow
542C23—32no—
543C23C23×C3×D314yes—
544C23C23×C3×D311yescharacteristic
545Q8—24no—
546Q8—34no—
547Q8—616no—
548Q8—68no—
549Q8—68no—
550D4—28no—
551D4—24no—
552D4—28no—
553D4—38no—
554D4—34no—
555D4—616no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—32no—
559C2×C4—38no—
560C2×C4—34no—
561C2×C4—38no—
562C2×C4—616no—
563C2×C4—616no—
564C2×C4—68no—
565C2×C4—68no—
566C2×C4C3×D3×D414yes—
567C2×C4C23×C3×D312yes—
568C2×C4C23×C3×D314yes—
569C2×C4C23×C3×D311yescharacteristic
570C2×C4C23×C3×D311yescharacteristic
571C2×C4C23×C3×D312yes—
572C2×C3—22no—
573C2×C3—22no—
574C2×C3—21no—
575C2×C3—22no—
576C2×C3—42no—
577C2×C3—44no—
578C2×C3—44no—
579C2×C3—64no—
580C2×C3—84no—
581C2×C3C22×D3×D411yescharacteristic
582C2×C3C2×#101012yes—
583C2×C3C23×C3×D411yescharacteristic
584C2×C3C2×C3×2−1+412yes—
585D3—24no—
586C22—22no—
587C22—24no—
588C22—28no—
589C22—34no—
590C22—64no—
591C22C2×C3×D3×D412yes—
592C22C24×C3×D311yescharacteristic
593C4—24no—
594C4—22no—
595C4—34no—
596C4—68no—
597C4—64no—
598C4C2×C3×D3×D412yes—
599C4C2×C3×D3×D412yes—
600C4C3×#101014yescomplemented
601C3—21no—
602C3#8624311yescharacteristic, direct product factor
603C3C3×#349911yescharacteristic, complemented
604C2—22no—
605C2—44no—
606C2—64no—
607C2C22×C3×D3×D411yescharacteristic
608C2C2×C3×#101012yes—
subgroupG/H
centerC22×C3C24×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC32×#532C2
socleC22×C32C25
hypercenterC3×#532D3
hypocenterC3C3×#3499
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×#532

Complements

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

NH ≅ G/Ntype
1. C2×C3×#1010C2semidirect
2. C32×#532C2semidirect
3. C3×#5825C2semidirect
4. C3×D3×C4⋊Q8C2semidirect
5. C3×#5716C2semidirect
6. C3×#5675C2semidirect
7. C3×#5611C2semidirect
8. C3×#5551C2semidirect
9. C3×D4×Dic6C2semidirect
10. C3×C4×#235C2semidirect
11. C2×C3×#869C2semidirect
13. #86243C3direct (N × H)
50. C2×#1010C2×C3semidirect
58. C3×#532D3semidirect
59. C3×#532C2×C3semidirect
63. #5825C2×C3semidirect
64. D3×C4⋊Q8C2×C3semidirect
65. #5716C2×C3semidirect
66. #5675C2×C3semidirect
67. #5611C2×C3semidirect
68. #5551C2×C3semidirect
69. D4×Dic6C2×C3semidirect
70. C4×#235C2×C3semidirect
71. C2×#869C2×C3semidirect
88. C3×#235D4semidirect
229. #532C3×D3semidirect
284. #235C3×D4semidirect
341. C32×C42−1+4semidirect
534. C3×C4#1010semidirect
537. C3×C4C3×2−1+4semidirect
541. C32#3499semidirect
600. C4C3×#1010semidirect
602. C3#86243direct (N × H)
603. C3C3×#3499semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#10102
C32×#5322
C3×#58252
C3×D3×C4⋊Q82
C3×#57162
C3×#56752
C3×#56112
C3×#55512
C3×D4×Dic62
C3×C4×#2352
C2×C3×#8692
C3×#34993
#862433