Finite Group Databasebeta

← C3×#86214 #11906010

Subgroups — C3×#86214 #11906010

Order 1152.

3097 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#1007C212yesmaximal, complemented
2C2×C32×C4⋊Q8C211yesmaximal, characteristic, complemented
3C3×#5816C212yesmaximal, complemented
4C3×#5727C214yesmaximal, complemented
5C3×#5726C214yesmaximal, complemented
6C3×#5725C212yesmaximal, complemented
7C3×#5724C214yesmaximal, complemented
8C3×D3×C4⋊Q8C212yesmaximal, complemented
9C3×#5507C218yesmaximal, complemented
10C3×#5477C211yesmaximal, characteristic, complemented
11C3×C4×#235C211yesmaximal, characteristic, complemented
12C3×#3499—31nomaximal
13#86214C311yesmaximal, characteristic, direct product factor
14C22×C32×Q8C2212yes—
15C3×#1007—216no—
16C2×C3×#239C2214yes—
17C2×C3×#239C2214yes—
18C2×C3×D3×Q8C2214yes—
19C2×C3×D3×Q8C2214yes—
20C2×C3×#235C2218yes—
21C2×C3×#235C2212yes—
22C2×C3×#235C2211yescharacteristic
23C32×C4⋊Q8C2212yes—
24C32×C4⋊Q8C2212yes—
25C32×C4⋊Q8C2214yes—
26C2×C32×C4⋊C4C2214yes—
27C2×C32×C42C2211yescharacteristic
28C3×#947C2214yes—
29C3×#946C2218yes—
30C3×#944C2214yes—
31C3×#930C2214yes—
32C3×#929C2218yes—
33C3×C4×#67C2214yes—
34C3×#926C2211yescharacteristic
35C3×#896C22116yes—
36C3×#895C2218yes—
37C3×#894C22116yes—
38C3×#893C2218yes—
39C3×D3×C4⋊C4C2218yes—
40C3×#888C2218yes—
41C3×#875C2212yes—
42C3×#874C2211yescharacteristic
43C3×C4×D12C2212yes—
44C3×C4×D12C2214yes—
45C3×#872C2212yes—
46C3×C42×D3C2212yes—
47C3×#869C2211yescharacteristic
48C3×C4×Dic6C2212yes—
49C2×C3×2−1+4—32no—
50C2×#1007C2×C312yescomplemented
51C3×D4×Q8—34no—
52C3×#547—34no—
53C3×#547—38no—
54C3×#536—32no—
55C3×#536—34no—
56C3×#532—32no—
57C3×#532—31no—
58C2×C3×C4⋊Q8—21no—
59C2×C3×C4⋊Q8—32no—
60C2×C3×C4⋊Q8D311yescharacteristic, complemented
61C2×C3×C4⋊Q8C2×C311yescharacteristic, complemented
62C3×C4×#41—31no—
63#5816C2×C312yescomplemented
64#5727C2×C314yescomplemented
65#5726C2×C314yescomplemented
66#5725C2×C312yescomplemented
67#5724C2×C314yescomplemented
68D3×C4⋊Q8C2×C312yescomplemented
69#5507C2×C318yescomplemented
70#5477C2×C311yescharacteristic, complemented
71C4×#235C2×C311yescharacteristic, complemented
72C2×C32×Q8—28no—
73C2×C32×Q8C2314yes—
74C2×C32×Q8C2314yes—
75C22×C32×C4C2312yes—
76C22×C32×C4C2314yes—
77C22×C32×C4C2311yescharacteristic
78C2×C3×#67C2314yes—
79C2×C3×#67C2312yes—
80C3×#239—216no—
81C3×#239—216no—
82C3×D3×Q8—216no—
83C3×D3×Q8—216no—
84C3×#235—28no—
85C3×#235—232no—
86C3×#235D418yescomplemented
87C2×C3×D12C2314yes—
88C2×C3×D12C2318yes—
89C2×C3×D12C2312yes—
90C2×C3×D12C2311yescharacteristic
91C2×C3×C4×D3C2318yes—
92C2×C3×C4×D3C2314yes—
93C2×C3×C4×D3C2318yes—
94C2×C3×C4×D3C2314yes—
95C2×C3×C4×D3C2312yes—
96C2×C3×Dic6C2318yes—
97C2×C3×Dic6C2312yes—
98C2×C3×Dic6C2311yescharacteristic
99C32×C4⋊C4C2318yes—
100C32×C4⋊C4C2318yes—
101C32×C42C2312yes—
102C32×C42C2311yescharacteristic
103C32×C42C2311yescharacteristic
104C3×#217C2312yes—
105C3×(C2×D3)⋊C4C23116yes—
106C3×(C2×D3)⋊C4C2314yes—
107C3×(C2×D3)⋊C4C2318yes—
108C3×C4⋊Dic3C2314yes—
109C3×C4⋊Dic3C2312yes—
110C3×Dic3⋊C4C23116yes—
111C3×Dic3⋊C4C2314yes—
112C3×C4×Dic3C2314yes—
113C3×C4×Dic3C2312yes—
114#3499—31noSylow
115C3×2−1+4—616no—
116C2×C3×#41—34no—
117C2×C3×#41—34no—
118C2×C3×#41—31no—
119C2×C3×#41—32no—
120C2×C3×#41—38no—
121C22×C3×Q8—22no—
122C22×C3×Q8—34no—
123C22×C3×Q8—34no—
124C22×C3×Q8C2×D312yes—
125C22×C3×Q8C22×C312yes—
126#1007—216no—
127C2×#239C22×C314yes—
128C2×#239C22×C314yes—
129C2×D3×Q8C22×C314yes—
130C2×D3×Q8C22×C314yes—
131C2×#235C22×C312yes—
132C2×#235C22×C318yes—
133C2×#235C22×C311yescharacteristic
134C3×C4⋊Q8—22no—
135C3×C4⋊Q8—22no—
136C3×C4⋊Q8—24no—
137C3×C4⋊Q8—34no—
138C3×C4⋊Q8—38no—
139C3×C4⋊Q8—31no—
140C3×C4⋊Q8C2×D312yes—
141C3×C4⋊Q8C2×D312yes—
142C3×C4⋊Q8C2×D314yes—
143C3×C4⋊Q8C22×C312yes—
144C3×C4⋊Q8C22×C312yes—
145C3×C4⋊Q8C22×C314yes—
146C3×#127—31no—
147C3×#124—34no—
148C3×#124—32no—
149C3×#123—38no—
150C3×#123—316no—
151C3×C22⋊Q8—38no—
152C3×C22⋊Q8—316no—
153C3×#121—34no—
154C3×#121—38no—
155C3×C4×Q8—32no—
156C3×C4×D4—34no—
157C3×C4×D4—38no—
158C3×C4×D4—32no—
159C3×C4×D4—34no—
160C3×#117—31no—
161C3×#117—32no—
162C2×C3×C4⋊C4—24no—
163C2×C3×C4⋊C4—38no—
164C2×C3×C4⋊C4C2×D314yes—
165C2×C3×C4⋊C4C22×C314yes—
166C2×C3×C42—21no—
167C2×C3×C42—32no—
168C2×C3×C42C2×D311yescharacteristic
169C2×C3×C42C22×C311yescharacteristic
170#947C22×C314yes—
171#946C22×C318yes—
172#944C22×C314yes—
173#930C22×C314yes—
174#929C22×C318yes—
175C4×#67C22×C314yes—
176#926C22×C311yescharacteristic
177#896C22×C3116yes—
178#895C22×C318yes—
179#894C22×C3116yes—
180#893C22×C318yes—
181D3×C4⋊C4C22×C318yes—
182#888C22×C318yes—
183#875C22×C312yes—
184#874C22×C311yescharacteristic
185C4×D12C22×C312yes—
186C4×D12C22×C314yes—
187#872C22×C312yes—
188C42×D3C22×C312yes—
189#869C22×C311yescharacteristic
190C4×Dic6C22×C312yes—
191C23×C32C2411yescharacteristic
192C22×C3×D3C2414yes—
193C22×C3×D3C2412yes—
194C32×Q8—28no—
195C32×Q8—28no—
196C2×C32×C4—24no—
197C2×C32×C4—28no—
198C2×C32×C4C2×D414yes—
199C2×C32×C4C2412yes—
200C2×C32×C4C2418yes—
201C2×C32×C4C2411yescharacteristic
202C2×C32×C4C2411yescharacteristic
203C2×C32×C4C2412yes—
204C3×#67—28no—
205C3×#67C2×D418yes—
206C2×C3×Dic3C2414yes—
207C2×C3×Dic3C2414yes—
208C2×C3×Dic3C2412yes—
209C3×D12—216no—
210C3×D12—24no—
211C3×D12—28no—
212C3×D12C2×D414yes—
213C3×C4×D3—216no—
214C3×C4×D3—28no—
215C3×C4×D3—28no—
216C3×C4×D3—216no—
217C3×C4×D3C2×D418yes—
218C3×Dic6—24no—
219C3×Dic6—216no—
220C3×Dic6C2×D414yes—
221C2×2−1+4—32no—
222D4×Q8—34no—
223#547—34no—
224#547—38no—
225#536—32no—
226#536—34no—
227#532—32no—
228#532—31no—
229C2×C4⋊Q8—32no—
230C2×C4⋊Q8C3×D311yescharacteristic, complemented
231C4×#41—31no—
232C3×#41—38no—
233C3×#41—632no—
234C3×#41—616no—
235C3×#41—616no—
236C3×#41—68no—
237C2×C3×Q8—28no—
238C2×C3×Q8—28no—
239C2×C3×Q8—24no—
240C2×C3×Q8—24no—
241C2×C3×Q8—31no—
242C2×C3×Q8—38no—
243C2×C3×Q8—32no—
244C2×C3×Q8—48no—
245C2×C3×Q8—616no—
246C2×C3×Q8—616no—
247C2×C3×Q8C22×D314yes—
248C2×C3×Q8C22×D314yes—
249C2×C3×Q8C23×C314yes—
250C2×C3×Q8C23×C314yes—
251C2×C3×D4—32no—
252C2×C3×D4—34no—
253C2×C3×D4—31no—
254C2×C3×D4—34no—
255C2×C3×D4—38no—
256C2×C3×D4—32no—
257C22×C3×C4—22no—
258C22×C3×C4—21no—
259C22×C3×C4—24no—
260C22×C3×C4—32no—
261C22×C3×C4—34no—
262C22×C3×C4—38no—
263C22×C3×C4—38no—
264C22×C3×C4—34no—
265C22×C3×C4C22×D312yes—
266C22×C3×C4C22×D314yes—
267C22×C3×C4C22×D311yescharacteristic
268C22×C3×C4C23×C312yes—
269C22×C3×C4C23×C314yes—
270C22×C3×C4C23×C311yescharacteristic
271C2×#67C23×C314yes—
272C2×#67C23×C312yes—
273#239—216no—
274#239—216no—
275D3×Q8—216no—
276D3×Q8—216no—
277#235—28no—
278#235—232no—
279#235C3×D418yescomplemented
280C2×D12C23×C314yes—
281C2×D12C23×C312yes—
282C2×D12C23×C318yes—
283C2×D12C23×C311yescharacteristic
284C2×C4×D3C23×C314yes—
285C2×C4×D3C23×C314yes—
286C2×C4×D3C23×C318yes—
287C2×C4×D3C23×C318yes—
288C2×C4×D3C23×C312yes—
289C2×Dic6C23×C312yes—
290C2×Dic6C23×C318yes—
291C2×Dic6C23×C311yescharacteristic
292C3×C4⋊C4—28no—
293C3×C4⋊C4—28no—
294C3×C4⋊C4—316no—
295C3×C4⋊C4—34no—
296C3×C4⋊C4—34no—
297C3×C4⋊C4—32no—
298C3×C4⋊C4C22×D318yes—
299C3×C4⋊C4C22×D318yes—
300C3×C4⋊C4C23×C318yes—
301C3×C4⋊C4C23×C318yes—
302C3×C22⋊C4—32no—
303C3×C22⋊C4—34no—
304C3×C22⋊C4—316no—
305C3×C22⋊C4—38no—
306C3×C42—21no—
307C3×C42—21no—
308C3×C42—22no—
309C3×C42—32no—
310C3×C42—34no—
311C3×C42C22×D311yescharacteristic
312C3×C42C22×D311yescharacteristic
313C3×C42C22×D312yes—
314C3×C42C23×C312yes—
315C3×C42C23×C311yescharacteristic
316C3×C42C23×C311yescharacteristic
317#217C23×C312yes—
318(C2×D3)⋊C4C23×C314yes—
319(C2×D3)⋊C4C23×C318yes—
320(C2×D3)⋊C4C23×C3116yes—
321C4⋊Dic3C23×C312yes—
322C4⋊Dic3C23×C314yes—
323Dic3⋊C4C23×C314yes—
324Dic3⋊C4C23×C3116yes—
325C4×Dic3C23×C314yes—
326C4×Dic3C23×C312yes—
327C22×C32—22no—
328C22×C32C22×D412yes—
329C22×C32C2511yescharacteristic
330C2×C3×D3—24no—
331C2×C3×D3—24no—
332C2×C3×D3—28no—
333C2×C3×D3C22×D414yes—
334C32×C4—22no—
335C32×C4—28no—
336C32×C4C22×D412yes—
337C32×C4C22×D412yes—
338C32×C42−1+414yescomplemented
339C3×Dic3—24no—
340C3×Dic3—24no—
341C3×Dic3C22×D414yes—
3422−1+4—616no—
343C2×#41—34no—
344C2×#41—34no—
345C2×#41—31no—
346C2×#41—32no—
347C2×#41—38no—
348C22×Q8—34no—
349C22×Q8—34no—
350C22×Q8C2×C3×D312yes—
351C4⋊Q8—34no—
352C4⋊Q8—38no—
353C4⋊Q8—31no—
354C4⋊Q8C2×C3×D312yes—
355C4⋊Q8C2×C3×D312yes—
356C4⋊Q8C2×C3×D314yes—
357#127—31no—
358#124—34no—
359#124—32no—
360#123—38no—
361#123—316no—
362C22⋊Q8—38no—
363C22⋊Q8—316no—
364#121—34no—
365#121—38no—
366C4×Q8—32no—
367C4×D4—34no—
368C4×D4—38no—
369C4×D4—32no—
370C4×D4—34no—
371#117—31no—
372#117—32no—
373C2×C4⋊C4—38no—
374C2×C4⋊C4C2×C3×D314yes—
375C2×C42—32no—
376C2×C42C2×C3×D311yescharacteristic
377C23×C3—21no—
378C23×C3—32no—
379C23×C3—34no—
380C23×C3C23×D311yescharacteristic
381C23×C3C24×C311yescharacteristic
382C22×D3C24×C314yes—
383C22×D3C24×C312yes—
384C3×Q8—28no—
385C3×Q8—28no—
386C3×Q8—28no—
387C3×Q8—28no—
388C3×Q8—34no—
389C3×Q8—48no—
390C3×Q8—48no—
391C3×Q8—616no—
392C3×Q8—64no—
393C3×D4—38no—
394C3×D4—34no—
395C3×D4—616no—
396C3×D4—68no—
397C3×D4—64no—
398C3×D4—68no—
399C2×C3×C4—28no—
400C2×C3×C4—28no—
401C2×C3×C4—28no—
402C2×C3×C4—24no—
403C2×C3×C4—24no—
404C2×C3×C4—22no—
405C2×C3×C4—22no—
406C2×C3×C4—21no—
407C2×C3×C4—21no—
408C2×C3×C4—28no—
409C2×C3×C4—24no—
410C2×C3×C4—32no—
411C2×C3×C4—38no—
412C2×C3×C4—34no—
413C2×C3×C4—34no—
414C2×C3×C4—48no—
415C2×C3×C4—616no—
416C2×C3×C4—616no—
417C2×C3×C4—68no—
418C2×C3×C4—68no—
419C2×C3×C4D3×D414yes—
420C2×C3×C4C2×C3×D414yes—
421C2×C3×C4C23×D312yes—
422C2×C3×C4C23×D318yes—
423C2×C3×C4C23×D311yescharacteristic
424C2×C3×C4C23×D311yescharacteristic
425C2×C3×C4C23×D312yes—
426C2×C3×C4C24×C312yes—
427C2×C3×C4C24×C318yes—
428C2×C3×C4C24×C311yescharacteristic
429C2×C3×C4C24×C311yescharacteristic
430C2×C3×C4C24×C312yes—
431#67—28no—
432#67C2×C3×D418yes—
433C2×Dic3C24×C314yes—
434C2×Dic3C24×C314yes—
435C2×Dic3C24×C312yes—
436D12—216no—
437D12—24no—
438D12—28no—
439D12C2×C3×D414yes—
440C4×D3—216no—
441C4×D3—28no—
442C4×D3—28no—
443C4×D3—216no—
444C4×D3C2×C3×D418yes—
445Dic6—24no—
446Dic6—216no—
447Dic6C2×C3×D414yes—
448C2×C32—22no—
449C2×C32C23×D411yescharacteristic
450C2×C32C2×2−1+412yes—
451C3×D3—24no—
452C3×D3—44no—
453#41—38no—
454#41—632no—
455#41—616no—
456#41—616no—
457#41—68no—
458C2×Q8—28no—
459C2×Q8—31no—
460C2×Q8—38no—
461C2×Q8—32no—
462C2×Q8—616no—
463C2×Q8—616no—
464C2×Q8C22×C3×D314yes—
465C2×Q8C22×C3×D314yes—
466C2×D4—32no—
467C2×D4—34no—
468C2×D4—31no—
469C2×D4—34no—
470C2×D4—38no—
471C2×D4—32no—
472C22×C4—32no—
473C22×C4—34no—
474C22×C4—38no—
475C22×C4—38no—
476C22×C4—34no—
477C22×C4C22×C3×D312yes—
478C22×C4C22×C3×D314yes—
479C22×C4C22×C3×D311yescharacteristic
480C4⋊C4—34no—
481C4⋊C4—316no—
482C4⋊C4—34no—
483C4⋊C4—32no—
484C4⋊C4C22×C3×D318yes—
485C4⋊C4C22×C3×D318yes—
486C22⋊C4—32no—
487C22⋊C4—34no—
488C22⋊C4—316no—
489C22⋊C4—38no—
490C42—32no—
491C42—34no—
492C42C22×C3×D311yescharacteristic
493C42C22×C3×D311yescharacteristic
494C42C22×C3×D312yes—
495C22×C3—24no—
496C22×C3—22no—
497C22×C3—22no—
498C22×C3—22no—
499C22×C3—21no—
500C22×C3—34no—
501C22×C3—64no—
502C22×C3—64no—
503C22×C3—68no—
504C22×C3C2×D3×D412yes—
505C22×C3C22×C3×D412yes—
506C22×C3C24×D311yescharacteristic
507C22×C3C25×C311yescharacteristic
508C2×D3—24no—
509C2×D3—24no—
510C2×D3—28no—
511C2×D3C22×C3×D414yes—
512C3×C4—24no—
513C3×C4—28no—
514C3×C4—22no—
515C3×C4—28no—
516C3×C4—22no—
517C3×C4—22no—
518C3×C4—22no—
519C3×C4—24no—
520C3×C4—34no—
521C3×C4—48no—
522C3×C4—64no—
523C3×C4—64no—
524C3×C4C2×D3×D412yes—
525C3×C4C2×D3×D412yes—
526C3×C4#100714yescomplemented
527C3×C4C22×C3×D412yes—
528C3×C4C22×C3×D412yes—
529C3×C4C3×2−1+414yescomplemented
530Dic3—24no—
531Dic3—24no—
532Dic3C22×C3×D414yes—
533C32#349911yescharacteristic, complemented, Sylow
534C23—32no—
535C23—34no—
536C23C23×C3×D311yescharacteristic
537Q8—28no—
538Q8—28no—
539Q8—34no—
540Q8—616no—
541Q8—64no—
542D4—38no—
543D4—34no—
544D4—616no—
545D4—68no—
546D4—64no—
547D4—68no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—32no—
551C2×C4—38no—
552C2×C4—34no—
553C2×C4—34no—
554C2×C4—616no—
555C2×C4—616no—
556C2×C4—68no—
557C2×C4—68no—
558C2×C4C3×D3×D414yes—
559C2×C4C23×C3×D312yes—
560C2×C4C23×C3×D318yes—
561C2×C4C23×C3×D311yescharacteristic
562C2×C4C23×C3×D311yescharacteristic
563C2×C4C23×C3×D312yes—
564C2×C3—24no—
565C2×C3—22no—
566C2×C3—22no—
567C2×C3—21no—
568C2×C3—22no—
569C2×C3—64no—
570C2×C3—124no—
571C2×C3C22×D3×D411yescharacteristic
572C2×C3C2×#100712yes—
573C2×C3C23×C3×D411yescharacteristic
574C2×C3C2×C3×2−1+412yes—
575D3—24no—
576D3—44no—
577C22—22no—
578C22—34no—
579C22—64no—
580C22—64no—
581C22—68no—
582C22C2×C3×D3×D412yes—
583C22C24×C3×D311yescharacteristic
584C4—28no—
585C4—22no—
586C4—34no—
587C4—64no—
588C4—64no—
589C4C2×C3×D3×D412yes—
590C4C2×C3×D3×D412yes—
591C4C3×#100714yescomplemented
592C3—21no—
593C3#8621411yescharacteristic, direct product factor
594C3C3×#349911yescharacteristic, complemented
595C2—22no—
596C2—64no—
597C2—124no—
598C2C22×C3×D3×D411yescharacteristic
599C2C2×C3×#100712yes—
subgroupG/H
centerC22×C3C24×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC2×C32×C4⋊Q8C2
socleC22×C32C25
hypercenterC2×C3×C4⋊Q8D3
hypocenterC3C3×#3499
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C2×C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#1007C2semidirect
2. C2×C32×C4⋊Q8C2semidirect
3. C3×#5816C2semidirect
4. C3×#5727C2semidirect
5. C3×#5726C2semidirect
6. C3×#5725C2semidirect
7. C3×#5724C2semidirect
8. C3×D3×C4⋊Q8C2semidirect
9. C3×#5507C2semidirect
10. C3×#5477C2semidirect
11. C3×C4×#235C2semidirect
13. #86214C3direct (N × H)
50. C2×#1007C2×C3semidirect
60. C2×C3×C4⋊Q8D3semidirect
61. C2×C3×C4⋊Q8C2×C3semidirect
63. #5816C2×C3semidirect
64. #5727C2×C3semidirect
65. #5726C2×C3semidirect
66. #5725C2×C3semidirect
67. #5724C2×C3semidirect
68. D3×C4⋊Q8C2×C3semidirect
69. #5507C2×C3semidirect
70. #5477C2×C3semidirect
71. C4×#235C2×C3semidirect
86. C3×#235D4semidirect
230. C2×C4⋊Q8C3×D3semidirect
279. #235C3×D4semidirect
338. C32×C42−1+4semidirect
526. C3×C4#1007semidirect
529. C3×C4C3×2−1+4semidirect
533. C32#3499semidirect
591. C4C3×#1007semidirect
593. C3#86214direct (N × H)
594. C3C3×#3499semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#10072
C2×C32×C4⋊Q82
C3×#58162
C3×#57272
C3×#57262
C3×#57252
C3×#57242
C3×D3×C4⋊Q82
C3×#55072
C3×#54772
C3×C4×#2352
C3×#34993
#862143