Finite Group Databasebeta

← C3×#85858 #11904389

Subgroups — C3×#85858 #11904389

Order 1152.

3289 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#1004C211yesmaximal, characteristic, complemented
2C22×C3×#237C211yesmaximal, characteristic, complemented
3C32×#518C211yesmaximal, characteristic, complemented
4C3×#5555C218yesmaximal, complemented
5C3×#5554C218yesmaximal, complemented
6C3×#5526C212yesmaximal, complemented
7C3×#5524C212yesmaximal, complemented
8C3×#5493C214yesmaximal, complemented
9C2×C3×#886C214yesmaximal, complemented
10C3×#3485—31nomaximal
11#85858C311yesmaximal, characteristic, direct product factor
12C22×C32×D4C2211yescharacteristic
13C22×C3×#67C2214yes—
14C22×C3×#67C2212yescomplemented
15C23×C3×Dic3C2212yescomplemented
16C3×#1004—28no—
17C2×C3×#237—28no—
18C2×C3×#237C2218yescomplemented
19C2×C3×#237C2214yescomplemented
20C2×C3×D3×D4C2214yescomplemented
21C2×C3×#235C2212yes—
22C22×C3×C4×D3C2211yescharacteristic, complemented
23C22×C3×Dic6C2211yescharacteristic, complemented
24C32×C4×D4C2218yescomplemented
25C32×#117C2212yescomplemented
26C2×C32×C22⋊C4C2214yescomplemented
27C3×#930C22116yescomplemented
28C2×C3×(C2×D3)⋊C4C2218yescomplemented
29C2×C3×C4⋊Dic3C2218yescomplemented
30C3×#897C2214yescomplemented
31C3×#897C2218yescomplemented
32C3×#895C2214yescomplemented
33C3×#886C2214yescomplemented
34C3×#886C2218yescomplemented
35C3×#886C22116yescomplemented
36C3×#886C2218yescomplemented
37C3×#882C2214yescomplemented
38C3×#882C2218yescomplemented
39C3×#875C2214yescomplemented
40C3×C4×D12C2218yescomplemented
41C3×#869C2214yescomplemented
42C2×C3×H(5,𝔽2)—31no—
43C22×C3×#41—31no—
44C2×#1004C2×C311yescharacteristic, complemented
45C22×#237C2×C311yescharacteristic, complemented
46C3×#547—38no—
47C3×#546—38no—
48C3×#536—32no—
49C3×#534—34no—
50C3×#525—32no—
51C2×C3×#123—34no—
52C3×#518—21no—
53C3×#518D311yescharacteristic, complemented
54C3×#518C2×C311yescharacteristic, complemented
55#5555C2×C318yescomplemented
56#5554C2×C318yescomplemented
57#5526C2×C312yescomplemented
58#5524C2×C312yescomplemented
59#5493C2×C314yescomplemented
60C2×#886C2×C314yescomplemented
61C24×C32C2312yes—
62C23×C3×D3C2311yescharacteristic
63C23×C3×D3C2314yes—
64C2×C32×D4D418yescomplemented
65C2×C32×D4C2314yes—
66C22×C32×C4C2318yes—
67C22×C32×C4C2311yescharacteristic
68C2×C3×#67—28no—
69C2×C3×#67—216no—
70C2×C3×#67C2314yes—
71C2×C3×#67C2318yes—
72C2×C3×#67C2318yes—
73C22×C3×Dic3—28no—
74C22×C3×Dic3C2312yes—
75C22×C3×Dic3C2311yescharacteristic
76C22×C3×Dic3C2314yes—
77C22×C3×Dic3C2318yes—
78C3×#237—232no—
79C3×#237—216no—
80C3×D3×D4—216no—
81C3×#235—28no—
82C2×C3×D12C2312yes—
83C2×C3×D12C2318yes—
84C2×C3×C4×D3—24no—
85C2×C3×C4×D3C2314yes—
86C2×C3×C4×D3C2314yes—
87C2×C3×Dic6—24no—
88C2×C3×Dic6C2314yes—
89C2×C3×Dic6C2312yes—
90C32×C4⋊C4C2314yes—
91C32×C22⋊C4C2314yes—
92C32×C22⋊C4C2318yes—
93C32×C42C2314yes—
94C3×(C2×D3)⋊C4C23116yes—
95C3×(C2×D3)⋊C4C2318yes—
96C3×(C2×D3)⋊C4C2318yes—
97C3×C4⋊Dic3C2318yes—
98C3×C4⋊Dic3C23116yes—
99#3485—31noSylow
100C3×H(5,𝔽2)—68no—
101C2×C3×#41—38no—
102C2×C3×#41—34no—
103C2×C3×#41—32no—
104C2×C3×#41—68no—
105C22×C3×Q8—31no—
106C22×C3×D4—21no—
107C22×C3×D4—32no—
108C22×C3×D4—34no—
109C22×C3×D4—34no—
110C22×C3×D4C2×D311yescharacteristic
111C22×C3×D4C22×C311yescharacteristic
112C23×C3×C4—32no—
113C23×C3×C4—31no—
114C22×#67C22×C314yes—
115C22×#67C22×C312yescomplemented
116C23×Dic3C22×C312yescomplemented
117#1004—28no—
118C2×#237—28no—
119C2×#237C22×C318yescomplemented
120C2×#237C22×C314yescomplemented
121C2×D3×D4C22×C314yescomplemented
122C2×#235C22×C312yes—
123C22×C4×D3C22×C311yescharacteristic, complemented
124C22×Dic6C22×C311yescharacteristic, complemented
125C3×C4⋊Q8—34no—
126C3×#124—34no—
127C3×#123—34no—
128C3×#123—38no—
129C3×#123—316no—
130C3×#123—38no—
131C3×C22⋊Q8—34no—
132C3×C22⋊Q8—38no—
133C3×#121—316no—
134C3×#121—34no—
135C3×#120—34no—
136C3×#120—38no—
137C3×C4×D4—28no—
138C3×C4×D4—38no—
139C3×C4×D4C2×D318yescomplemented
140C3×C4×D4C22×C318yescomplemented
141C3×#117—22no—
142C3×#117C2×D312yescomplemented
143C3×#117C22×C312yescomplemented
144C2×C3×C4⋊C4—38no—
145C2×C3×C22⋊C4—24no—
146C2×C3×C22⋊C4—38no—
147C2×C3×C22⋊C4C2×D314yescomplemented
148C2×C3×C22⋊C4C22×C314yescomplemented
149#930C22×C3116yescomplemented
150C2×(C2×D3)⋊C4C22×C318yescomplemented
151C2×C4⋊Dic3C22×C318yescomplemented
152#897C22×C314yescomplemented
153#897C22×C318yescomplemented
154#895C22×C314yescomplemented
155#886C22×C314yescomplemented
156#886C22×C318yescomplemented
157#886C22×C3116yescomplemented
158#886C22×C318yescomplemented
159#882C22×C314yescomplemented
160#882C22×C318yescomplemented
161#875C22×C314yescomplemented
162C4×D12C22×C318yescomplemented
163#869C22×C314yescomplemented
164C23×C32—44no—
165C23×C32C2×D418yes—
166C23×C32C2411yescharacteristic
167C23×C32C2414yes—
168C22×C3×D3—22no—
169C22×C3×D3—28no—
170C22×C3×D3—42no—
171C22×C3×D3—48no—
172C22×C3×D3C2414yes—
173C22×C3×D3C2412yes—
174C32×D4C2×D4116yes—
175C2×C32×C4—48no—
176C2×C32×C4C2×D414yes—
177C2×C32×C4C2418yes—
178C2×C32×C4C2412yes—
179C3×#67—28no—
180C3×#67—216no—
181C3×#67—216no—
182C2×C3×Dic3—24no—
183C2×C3×Dic3—22no—
184C2×C3×Dic3—216no—
185C2×C3×Dic3—28no—
186C2×C3×Dic3C2414yes—
187C2×C3×Dic3C2414yes—
188C2×C3×Dic3C2412yes—
189C3×D12—24no—
190C3×D12—48no—
191C3×C4×D3—28no—
192C3×C4×D3—28no—
193C3×Dic6—28no—
194C3×Dic6—24no—
195C2×H(5,𝔽2)—31no—
196C22×#41—31no—
197#547—38no—
198#546—38no—
199#536—32no—
200#534—34no—
201#525—32no—
202C2×#123—34no—
203#518C3×D311yescharacteristic, complemented
204C24×C3—22no—
205C24×C3—34no—
206C24×C3—31no—
207C24×C3C22×D312yes—
208C24×C3C23×C312yes—
209C23×D3C23×C311yescharacteristic
210C23×D3C23×C314yes—
211C3×#41—632no—
212C3×#41—616no—
213C3×#41—68no—
214C2×C3×Q8—34no—
215C2×C3×Q8—32no—
216C2×C3×Q8—64no—
217C2×C3×D4—28no—
218C2×C3×D4—24no—
219C2×C3×D4—38no—
220C2×C3×D4—34no—
221C2×C3×D4—38no—
222C2×C3×D4—32no—
223C2×C3×D4—38no—
224C2×C3×D4—68no—
225C2×C3×D4—616no—
226C2×C3×D4—616no—
227C2×C3×D4D1218yescomplemented
228C2×C3×D4C3×D418yescomplemented
229C2×C3×D4C22×D314yes—
230C2×C3×D4C23×C314yes—
231C22×C3×C4—21no—
232C22×C3×C4—28no—
233C22×C3×C4—38no—
234C22×C3×C4—34no—
235C22×C3×C4—31no—
236C22×C3×C4—32no—
237C22×C3×C4—34no—
238C22×C3×C4—34no—
239C22×C3×C4—68no—
240C22×C3×C4—64no—
241C22×C3×C4C22×D318yes—
242C22×C3×C4C22×D311yescharacteristic
243C22×C3×C4C23×C318yes—
244C22×C3×C4C23×C311yescharacteristic
245C2×#67—28no—
246C2×#67—216no—
247C2×#67C23×C314yes—
248C2×#67C23×C318yes—
249C2×#67C23×C318yes—
250C22×Dic3—28no—
251C22×Dic3C23×C312yes—
252C22×Dic3C23×C311yescharacteristic
253C22×Dic3C23×C314yes—
254C22×Dic3C23×C318yes—
255#237—232no—
256#237—216no—
257D3×D4—216no—
258#235—28no—
259C2×D12C23×C312yes—
260C2×D12C23×C318yes—
261C2×C4×D3—24no—
262C2×C4×D3C23×C314yes—
263C2×C4×D3C23×C314yes—
264C2×Dic6—24no—
265C2×Dic6C23×C312yes—
266C2×Dic6C23×C314yes—
267C3×C4⋊C4—24no—
268C3×C4⋊C4—38no—
269C3×C4⋊C4—316no—
270C3×C4⋊C4C22×D314yes—
271C3×C4⋊C4C23×C314yes—
272C3×C22⋊C4—24no—
273C3×C22⋊C4—28no—
274C3×C22⋊C4—38no—
275C3×C22⋊C4—316no—
276C3×C22⋊C4—38no—
277C3×C22⋊C4C22×D314yes—
278C3×C22⋊C4C22×D318yes—
279C3×C22⋊C4C23×C314yes—
280C3×C22⋊C4C23×C318yes—
281C3×C42—24no—
282C3×C42C22×D314yes—
283C3×C42C23×C314yes—
284(C2×D3)⋊C4C23×C318yes—
285(C2×D3)⋊C4C23×C318yes—
286(C2×D3)⋊C4C23×C3116yes—
287C4⋊Dic3C23×C3116yes—
288C4⋊Dic3C23×C318yes—
289C22×C32—28no—
290C22×C32—22no—
291C22×C32—48no—
292C22×C32C22×D412yes—
293C22×C32C22×D418yes—
294C22×C32C2511yescharacteristic
295C2×C3×D3—22no—
296C2×C3×D3—24no—
297C2×C3×D3—24no—
298C2×C3×D3—28no—
299C2×C3×D3—44no—
300C2×C3×D3—88no—
301C32×C4—28no—
302C32×C4C22×D414yes—
303C3×Dic3—24no—
304C3×Dic3—24no—
305C3×Dic3—22no—
306H(5,𝔽2)—68no—
307C2×#41—38no—
308C2×#41—34no—
309C2×#41—32no—
310C2×#41—68no—
311C22×Q8—31no—
312C22×D4—34no—
313C22×D4—32no—
314C22×D4—34no—
315C22×D4C2×C3×D311yescharacteristic
316C23×C4—32no—
317C23×C4—31no—
318C4⋊Q8—34no—
319#124—34no—
320#123—34no—
321#123—38no—
322#123—316no—
323#123—38no—
324C22⋊Q8—34no—
325C22⋊Q8—38no—
326#121—316no—
327#121—34no—
328#120—34no—
329#120—38no—
330C4×D4—38no—
331C4×D4C2×C3×D318yescomplemented
332#117C2×C3×D312yescomplemented
333C2×C4⋊C4—38no—
334C2×C22⋊C4—38no—
335C2×C22⋊C4C2×C3×D314yescomplemented
336C23×C3—21no—
337C23×C3—24no—
338C23×C3—28no—
339C23×C3—34no—
340C23×C3—32no—
341C23×C3—44no—
342C23×C3—44no—
343C23×C3—62no—
344C23×C3—68no—
345C23×C3—84no—
346C23×C3—122no—
347C23×C3—128no—
348C23×C3C2×D1218yes—
349C23×C3C2×C3×D418yes—
350C23×C3C23×D311yescharacteristic
351C23×C3C23×D314yes—
352C23×C3C24×C311yescharacteristic
353C23×C3C24×C314yes—
354C22×D3—22no—
355C22×D3—28no—
356C22×D3—42no—
357C22×D3—48no—
358C22×D3C24×C314yes—
359C22×D3C24×C312yes—
360C3×Q8—68no—
361C3×Q8—64no—
362C3×D4—216no—
363C3×D4—68no—
364C3×D4—616no—
365C3×D4—616no—
366C3×D4—64no—
367C3×D4—128no—
368C3×D4C2×D12116yes—
369C3×D4C2×C3×D4116yes—
370C2×C3×C4—22no—
371C2×C3×C4—28no—
372C2×C3×C4—24no—
373C2×C3×C4—32no—
374C2×C3×C4—34no—
375C2×C3×C4—34no—
376C2×C3×C4—48no—
377C2×C3×C4—48no—
378C2×C3×C4—62no—
379C2×C3×C4—64no—
380C2×C3×C4—616no—
381C2×C3×C4—68no—
382C2×C3×C4—68no—
383C2×C3×C4—68no—
384C2×C3×C4—88no—
385C2×C3×C4C2×D1214yes—
386C2×C3×C4C2×C3×D414yes—
387C2×C3×C4C23×D318yes—
388C2×C3×C4C23×D312yes—
389C2×C3×C4C24×C318yes—
390C2×C3×C4C24×C312yes—
391#67—28no—
392#67—216no—
393#67—216no—
394C2×Dic3—24no—
395C2×Dic3—22no—
396C2×Dic3—216no—
397C2×Dic3—28no—
398C2×Dic3C24×C314yes—
399C2×Dic3C24×C314yes—
400C2×Dic3C24×C312yes—
401D12—24no—
402D12—48no—
403C4×D3—28no—
404C4×D3—28no—
405Dic6—28no—
406Dic6—24no—
407C2×C32—22no—
408C2×C32—28no—
409C2×C32C23×D411yescharacteristic
410C2×C32#58512yes—
411C3×D3—42no—
412C3×D3—44no—
413C24—34no—
414C24—31no—
415C24C22×C3×D312yes—
416#41—632no—
417#41—616no—
418#41—68no—
419C2×Q8—34no—
420C2×Q8—32no—
421C2×Q8—64no—
422C2×D4—38no—
423C2×D4—34no—
424C2×D4—38no—
425C2×D4—32no—
426C2×D4—38no—
427C2×D4—68no—
428C2×D4—616no—
429C2×D4—616no—
430C2×D4C3×D1218yescomplemented
431C2×D4C22×C3×D314yes—
432C22×C4—38no—
433C22×C4—34no—
434C22×C4—31no—
435C22×C4—32no—
436C22×C4—34no—
437C22×C4—34no—
438C22×C4—68no—
439C22×C4—64no—
440C22×C4C22×C3×D318yes—
441C22×C4C22×C3×D311yescharacteristic
442C4⋊C4—38no—
443C4⋊C4—316no—
444C4⋊C4C22×C3×D314yes—
445C22⋊C4—38no—
446C22⋊C4—316no—
447C22⋊C4—38no—
448C22⋊C4C22×C3×D314yes—
449C22⋊C4C22×C3×D318yes—
450C42C22×C3×D314yes—
451C22×C3—28no—
452C22×C3—22no—
453C22×C3—28no—
454C22×C3—22no—
455C22×C3—22no—
456C22×C3—28no—
457C22×C3—21no—
458C22×C3—48no—
459C22×C3—42no—
460C22×C3—48no—
461C22×C3—48no—
462C22×C3—62no—
463C22×C3—64no—
464C22×C3—64no—
465C22×C3—68no—
466C22×C3—88no—
467C22×C3—124no—
468C22×C3—248no—
469C22×C3C22×D1212yes—
470C22×C3C22×D1218yes—
471C22×C3C22×C3×D412yes—
472C22×C3C22×C3×D418yes—
473C22×C3C24×D311yescharacteristic
474C22×C3C25×C311yescharacteristic
475C2×D3—22no—
476C2×D3—24no—
477C2×D3—24no—
478C2×D3—28no—
479C2×D3—44no—
480C2×D3—88no—
481C3×C4—28no—
482C3×C4—28no—
483C3×C4—24no—
484C3×C4—48no—
485C3×C4—62no—
486C3×C4—64no—
487C3×C4—64no—
488C3×C4C22×D1214yes—
489C3×C4C22×C3×D414yes—
490Dic3—24no—
491Dic3—24no—
492Dic3—22no—
493C32#348511yescharacteristic, complemented, Sylow
494C23—34no—
495C23—32no—
496C23—44no—
497C23—62no—
498C23—68no—
499C23—122no—
500C23—128no—
501C23C2×C3×D1218yes—
502C23C23×C3×D311yescharacteristic
503C23C23×C3×D314yes—
504Q8—68no—
505Q8—64no—
506D4—68no—
507D4—616no—
508D4—616no—
509D4—64no—
510D4—128no—
511D4C2×C3×D12116yes—
512C2×C4—32no—
513C2×C4—34no—
514C2×C4—34no—
515C2×C4—48no—
516C2×C4—62no—
517C2×C4—64no—
518C2×C4—616no—
519C2×C4—68no—
520C2×C4—68no—
521C2×C4—68no—
522C2×C4C2×C3×D1214yes—
523C2×C4C23×C3×D318yes—
524C2×C4C23×C3×D312yes—
525C2×C3—22no—
526C2×C3—28no—
527C2×C3—22no—
528C2×C3—28no—
529C2×C3—21no—
530C2×C3—22no—
531C2×C3—42no—
532C2×C3—48no—
533C2×C3—122no—
534C2×C3—124no—
535C2×C3C23×D1211yescharacteristic
536C2×C3#596612yes—
537C2×C3C23×C3×D411yescharacteristic
538C2×C3C3×#58512yes—
539D3—42no—
540D3—44no—
541C22—28no—
542C22—22no—
543C22—48no—
544C22—62no—
545C22—64no—
546C22—64no—
547C22—68no—
548C22—124no—
549C22—248no—
550C22C22×C3×D1212yes—
551C22C22×C3×D1218yes—
552C22C24×C3×D311yescharacteristic
553C4—28no—
554C4—62no—
555C4—64no—
556C4—64no—
557C4C22×C3×D1214yes—
558C3—21no—
559C3#8585811yescharacteristic, direct product factor
560C3C3×#348511yescharacteristic, complemented
561C2—22no—
562C2—28no—
563C2—122no—
564C2—124no—
565C2C23×C3×D1211yescharacteristic
566C2C3×#596612yes—
subgroupG/H
centerC22×C3C24×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC32×#518C2
socleC22×C32C25
hypercenterC3×#518D3
hypocenterC3C3×#3485
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×#518

Complements

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

NH ≅ G/Ntype
1. C2×C3×#1004C2semidirect
2. C22×C3×#237C2semidirect
3. C32×#518C2semidirect
4. C3×#5555C2semidirect
5. C3×#5554C2semidirect
6. C3×#5526C2semidirect
7. C3×#5524C2semidirect
8. C3×#5493C2semidirect
9. C2×C3×#886C2semidirect
11. #85858C3direct (N × H)
14. C22×C3×#67C22semidirect
15. C23×C3×Dic3C22semidirect
18. C2×C3×#237C22semidirect
19. C2×C3×#237C22semidirect
20. C2×C3×D3×D4C22semidirect
22. C22×C3×C4×D3C22semidirect
23. C22×C3×Dic6C22semidirect
24. C32×C4×D4C22semidirect
25. C32×#117C22semidirect
26. C2×C32×C22⋊C4C22semidirect
27. C3×#930C22semidirect
28. C2×C3×(C2×D3)⋊C4C22semidirect
29. C2×C3×C4⋊Dic3C22semidirect
30. C3×#897C22semidirect
31. C3×#897C22semidirect
32. C3×#895C22semidirect
33. C3×#886C22semidirect
34. C3×#886C22semidirect
35. C3×#886C22semidirect
36. C3×#886C22semidirect
37. C3×#882C22semidirect
38. C3×#882C22semidirect
39. C3×#875C22semidirect
40. C3×C4×D12C22semidirect
41. C3×#869C22semidirect
44. C2×#1004C2×C3semidirect
45. C22×#237C2×C3semidirect
53. C3×#518D3semidirect
54. C3×#518C2×C3semidirect
55. #5555C2×C3semidirect
56. #5554C2×C3semidirect
57. #5526C2×C3semidirect
58. #5524C2×C3semidirect
59. #5493C2×C3semidirect
60. C2×#886C2×C3semidirect
64. C2×C32×D4D4semidirect
115. C22×#67C22×C3semidirect
116. C23×Dic3C22×C3semidirect
119. C2×#237C22×C3semidirect
120. C2×#237C22×C3semidirect
121. C2×D3×D4C22×C3semidirect
123. C22×C4×D3C22×C3semidirect
124. C22×Dic6C22×C3semidirect
139. C3×C4×D4C2×D3semidirect
140. C3×C4×D4C22×C3semidirect
142. C3×#117C2×D3semidirect
143. C3×#117C22×C3semidirect
147. C2×C3×C22⋊C4C2×D3semidirect
148. C2×C3×C22⋊C4C22×C3semidirect
149. #930C22×C3semidirect
150. C2×(C2×D3)⋊C4C22×C3semidirect
151. C2×C4⋊Dic3C22×C3semidirect
152. #897C22×C3semidirect
153. #897C22×C3semidirect
154. #895C22×C3semidirect
155. #886C22×C3semidirect
156. #886C22×C3semidirect
157. #886C22×C3semidirect
158. #886C22×C3semidirect
159. #882C22×C3semidirect
160. #882C22×C3semidirect
161. #875C22×C3semidirect
162. C4×D12C22×C3semidirect
163. #869C22×C3semidirect
203. #518C3×D3semidirect
227. C2×C3×D4D12semidirect
228. C2×C3×D4C3×D4semidirect
331. C4×D4C2×C3×D3semidirect
332. #117C2×C3×D3semidirect
335. C2×C22⋊C4C2×C3×D3semidirect
430. C2×D4C3×D12semidirect
493. C32#3485semidirect
559. C3#85858direct (N × H)
560. C3C3×#3485semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#10042
C22×C3×#2372
C32×#5182
C3×#55552
C3×#55542
C3×#55262
C3×#55242
C3×#54932
C2×C3×#8862
C3×#34853
#858583