Finite Group Databasebeta

← C3×#83679 #11898418

Subgroups — C3×#83679 #11898418

Order 1152.

850 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#556C211yesmaximal, characteristic, complemented
2C3×#5585C211yesmaximal, characteristic, complemented
3C3×#5090C211yesmaximal, characteristic, complemented
4C3×#5085C211yesmaximal, characteristic, complemented
5C3×#5059C212yesmaximal, complemented
6C3×#5057C212yesmaximal, complemented
7C3×#5028C212yesmaximal, complemented
8C3×C4×#215C212yesmaximal, complemented
9C3×Q8×C3⋊C8C211yesmaximal, characteristic, complemented
10C3×C4×#214C211yesmaximal, characteristic, complemented
11C3×#5015C211yesmaximal, characteristic
12C3×#3394—31nomaximal
13#83679C311yesmaximal, characteristic, direct product factor
14C32×C4⋊Q8C2211yescharacteristic
15C32×#126—22no—
16C32×#124C2212yes—
17C32×C22⋊Q8—21no—
18C32×C4×Q8C2212yes—
19C32×C4×Q8C2211yescharacteristic
20C32×C4×D4C2211yescharacteristic
21C32×#117—21no—
22C3×#947—21no—
23C2×C3×#216C2212yes—
24C2×C3×#215C2212yes—
25C2×C3×#215C2212yes—
26C2×C3×#214C2211yescharacteristic
27C2×C3×#213C2212yes—
28C3×#894—22no—
29C3×#893—21no—
30C3×#875C2212yes—
31C3×#874C2211yescharacteristic
32C3×C4×D12C2212yes—
33C3×C4×Dic6C2211yescharacteristic
34C3×#835C2212yes—
35C3×#832C2211yescharacteristic
36C3×#810C2211yescharacteristic
37C3×#809C2212yes—
38C3×#808C2211yescharacteristic
39C3×C4⋊(C3⋊C8)C2211yescharacteristic
40C3×C4⋊(C3⋊C8)C2212yes—
41C3×C4×C3⋊C8C2211yescharacteristic
42C3×C4×C3⋊C8C2212yes—
43C3×#559—31no—
44C3×#556—21no—
45C3×#556D311yescharacteristic, complemented
46C3×#556C2×C311yescharacteristic, complemented
47#5585C2×C311yescharacteristic, complemented
48C3×#495—32no—
49C3×#492—31no—
50C3×#464—32no—
51C3×#463—32no—
52C3×#462—31no—
53C3×#459—31no—
54C3×C8×Q8—31no—
55C3×C4×SD16—32no—
56C3×C4×SD16—31no—
57#5090C2×C311yescharacteristic, complemented
58#5085C2×C311yescharacteristic, complemented
59#5059C2×C312yescomplemented
60#5057C2×C312yescomplemented
61#5028C2×C312yescomplemented
62C4×#215C2×C312yescomplemented
63Q8×C3⋊C8C2×C311yescharacteristic, complemented
64C4×#214C2×C311yescharacteristic, complemented
65#5015C2×C311yescharacteristic
66C2×C32×Q8D411yescharacteristic
67C2×C32×Q8C2312yes—
68C2×C32×D4C2311yescharacteristic
69C22×C32×C4—21no—
70C2×C3×D12—21no—
71C2×C3×D12—21no—
72C2×C3×D12C2312yes—
73C2×C3×C4×D3—22no—
74C2×C3×Dic6C2311yescharacteristic
75C32×C4⋊C4—21no—
76C32×C4⋊C4—22no—
77C32×C4⋊C4—21no—
78C32×C4⋊C4D411yescharacteristic
79C32×C4⋊C4D412yes—
80C32×C4⋊C4C2311yescharacteristic
81C32×C22⋊C4—22no—
82C32×C22⋊C4—21no—
83C32×C22⋊C4—22no—
84C32×C42—22no—
85C32×C42C2311yescharacteristic
86C32×C42C2312yes—
87C3×#216—42no—
88C3×#215—24no—
89C3×#215—42no—
90C3×#214—22no—
91C3×#213—42no—
92C3×(C2×D3)⋊C4—22no—
93C3×(C2×D3)⋊C4—22no—
94C3×(C2×D3)⋊C4—22no—
95C3×C4⋊Dic3C2311yescharacteristic
96C3×Dic3⋊C4—21no—
97C3×C4×Dic3—21no—
98C2×C3×C3⋊C8C2312yes—
99C2×C3×C3⋊C8C2311yescharacteristic
100C2×C3×C3⋊C8C2311yescharacteristic
101#3394—31noSylow
102C2×C3×Q16—32no—
103C2×C3×SD16—32no—
104C2×C3×SD16—31no—
105C2×C3×SD16—32no—
106C2×C3×D8—32no—
107C3×C4⋊Q8—21no—
108C3×C4⋊Q8C2×D311yescharacteristic
109C3×C4⋊Q8C22×C311yescharacteristic
110C3×#127—31no—
111C3×#126—24no—
112C3×#126—22no—
113C3×#126—22no—
114C3×#124—22no—
115C3×#124—32no—
116C3×#124—61no—
117C3×#124C2×D312yes—
118C3×#124C22×C312yes—
119C3×#123—62no—
120C3×C22⋊Q8—22no—
121C3×C22⋊Q8—21no—
122C3×C22⋊Q8—21no—
123C3×C4×Q8—22no—
124C3×C4×Q8—21no—
125C3×C4×Q8—31no—
126C3×C4×Q8C2×D312yes—
127C3×C4×Q8C2×D311yescharacteristic
128C3×C4×Q8C22×C312yes—
129C3×C4×Q8C22×C311yescharacteristic
130C3×C4×D4—21no—
131C3×C4×D4—32no—
132C3×C4×D4—61no—
133C3×C4×D4C2×D311yescharacteristic
134C3×C4×D4C22×C311yescharacteristic
135C3×#117—22no—
136C3×#117—21no—
137C3×#117—21no—
138#947—21no—
139C2×#216C22×C312yes—
140C2×#215C22×C312yes—
141C2×#215C22×C312yes—
142C2×#214C22×C311yescharacteristic
143C2×#213C22×C312yes—
144#894—22no—
145#893—21no—
146#875C22×C312yes—
147#874C22×C311yescharacteristic
148C4×D12C22×C312yes—
149C4×Dic6C22×C311yescharacteristic
150C3×#106—31no—
151C3×C4⋊C8—31no—
152C3×C4⋊C8—32no—
153C3×#103—32no—
154C3×#103—31no—
155C3×#102—31no—
156C3×#102—32no—
157C3×C4×C8—31no—
158C3×C4×C8—32no—
159#835C22×C312yes—
160#832C22×C311yescharacteristic
161#810C22×C311yescharacteristic
162#809C22×C312yes—
163#808C22×C311yescharacteristic
164C4⋊(C3⋊C8)C22×C311yescharacteristic
165C4⋊(C3⋊C8)C22×C312yes—
166C4×C3⋊C8C22×C311yescharacteristic
167C4×C3⋊C8C22×C312yes—
168C23×C32—21no—
169C22×C3×D3—22no—
170C32×Q8—22no—
171C32×Q8—22no—
172C32×Q8—22no—
173C32×D4—21no—
174C32×D4—21no—
175C2×C32×C4—21no—
176C2×C32×C4—22no—
177C2×C32×C4—42no—
178C2×C32×C4C2×D411yescharacteristic
179C2×C32×C4C2×D412yes—
180C2×C32×C4C2×D412yes—
181C2×C32×C4C2×D411yescharacteristic
182C2×C32×C4C2411yescharacteristic
183C2×C3×Dic3—21no—
184C2×C3×Dic3—21no—
185C3×D12—22no—
186C3×D12—22no—
187C3×D12—42no—
188C3×D12—81no—
189C3×C4×D3—82no—
190C3×Dic6—21no—
191C3×Dic6—21no—
192C3×C3⋊C8—22no—
193C3×C3⋊C8—21no—
194C3×C3⋊C8#4112yes—
195#559—31no—
196#556C3×D311yescharacteristic, complemented
197#495—32no—
198#492—31no—
199#464—32no—
200#463—32no—
201#462—31no—
202#459—31no—
203C8×Q8—31no—
204C4×SD16—32no—
205C4×SD16—31no—
206C2×C3×Q8—21no—
207C2×C3×Q8—22no—
208C2×C3×Q8—31no—
209C2×C3×Q8#6711yescharacteristic
210C2×C3×Q8C3×D411yescharacteristic
211C2×C3×Q8C22×D312yes—
212C2×C3×Q8C23×C312yes—
213C2×C3×D4—21no—
214C2×C3×D4—32no—
215C2×C3×D4—61no—
216C2×C3×D4—61no—
217C2×C3×D4C22×D311yescharacteristic
218C2×C3×D4C23×C311yescharacteristic
219C22×C3×C4—22no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—62no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12C23×C312yes—
226C2×C4×D3—22no—
227C2×Dic6C23×C311yescharacteristic
228C3×Q16—122no—
229C3×SD16—62no—
230C3×SD16—64no—
231C3×SD16—122no—
232C3×D8—122no—
233C2×C3×C8—31no—
234C2×C3×C8—32no—
235C2×C3×C8—31no—
236C3×C4⋊C4—22no—
237C3×C4⋊C4—24no—
238C3×C4⋊C4—22no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—22no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—22no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—22no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—31no—
249C3×C4⋊C4—61no—
250C3×C4⋊C4#6711yescharacteristic
251C3×C4⋊C4#6712yes—
252C3×C4⋊C4C3×D411yescharacteristic
253C3×C4⋊C4C3×D412yes—
254C3×C4⋊C4C22×D311yescharacteristic
255C3×C4⋊C4C23×C311yescharacteristic
256C3×C22⋊C4—24no—
257C3×C22⋊C4—22no—
258C3×C22⋊C4—24no—
259C3×C22⋊C4—22no—
260C3×C22⋊C4—21no—
261C3×C22⋊C4—22no—
262C3×C22⋊C4—22no—
263C3×C22⋊C4—21no—
264C3×C22⋊C4—22no—
265C3×C22⋊C4—62no—
266C3×C22⋊C4—62no—
267C3×C22⋊C4—62no—
268C3×C42—24no—
269C3×C42—22no—
270C3×C42—22no—
271C3×C42—21no—
272C3×C42—22no—
273C3×C42—61no—
274C3×C42C22×D312yes—
275C3×C42C22×D311yescharacteristic
276C3×C42C23×C311yescharacteristic
277C3×C42C23×C312yes—
278#216—42no—
279#215—24no—
280#215—42no—
281#214—22no—
282#213—42no—
283(C2×D3)⋊C4—22no—
284(C2×D3)⋊C4—22no—
285(C2×D3)⋊C4—22no—
286C4⋊Dic3C23×C311yescharacteristic
287Dic3⋊C4—21no—
288C4×Dic3—21no—
289C2×C3⋊C8C23×C312yes—
290C2×C3⋊C8C23×C311yescharacteristic
291C2×C3⋊C8C23×C311yescharacteristic
292C22×C32—41no—
293C22×C32—41no—
294C22×C32—41no—
295C22×C32C22×D411yescharacteristic
296C2×C3×D3—42no—
297C2×C3×D3—42no—
298C2×C3×D3—42no—
299C32×C4—22no—
300C32×C4—22no—
301C32×C4—21no—
302C32×C4—42no—
303C32×C4—41no—
304C32×C4#13512yes—
305C32×C4C2×#4111yescharacteristic
306C32×C4H(5,𝔽2)11yescharacteristic
307C3×Dic3—41no—
308C3×Dic3—41no—
309C2×Q16—32no—
310C2×SD16—32no—
311C2×SD16—31no—
312C2×SD16—32no—
313C2×D8—32no—
314C4⋊Q8C2×C3×D311yescharacteristic
315#127—31no—
316#126—22no—
317#124—32no—
318#124—61no—
319#124C2×C3×D312yes—
320#123—62no—
321C22⋊Q8—21no—
322C4×Q8—31no—
323C4×Q8C2×C3×D312yes—
324C4×Q8C2×C3×D311yescharacteristic
325C4×D4—32no—
326C4×D4—61no—
327C4×D4C2×C3×D311yescharacteristic
328#117—21no—
329#106—31no—
330C4⋊C8—31no—
331C4⋊C8—32no—
332#103—32no—
333#103—31no—
334#102—31no—
335#102—32no—
336C4×C8—31no—
337C4×C8—32no—
338C23×C3—22no—
339C23×C3—21no—
340C23×C3—21no—
341C23×C3—62no—
342C22×D3—22no—
343C3×Q8—22no—
344C3×Q8—22no—
345C3×Q8—22no—
346C3×Q8—22no—
347C3×Q8—22no—
348C3×Q8—22no—
349C3×Q8—42no—
350C3×Q8—42no—
351C3×Q8—42no—
352C3×Q8—61no—
353C3×Q8—61no—
354C3×D4—21no—
355C3×D4—21no—
356C3×D4—21no—
357C3×D4—21no—
358C3×D4—41no—
359C3×D4—41no—
360C3×D4—62no—
361C3×D4—62no—
362C3×D4—122no—
363C3×D4—241no—
364C2×C3×C4—22no—
365C2×C3×C4—24no—
366C2×C3×C4—21no—
367C2×C3×C4—22no—
368C2×C3×C4—21no—
369C2×C3×C4—22no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—22no—
373C2×C3×C4—21no—
374C2×C3×C4—22no—
375C2×C3×C4—44no—
376C2×C3×C4—42no—
377C2×C3×C4—42no—
378C2×C3×C4—61no—
379C2×C3×C4—61no—
380C2×C3×C4—242no—
381C2×C3×C4C2×#6712yes—
382C2×C3×C4C2×#6712yes—
383C2×C3×C4C2×#6711yescharacteristic
384C2×C3×C4C2×#6711yescharacteristic
385C2×C3×C4C2×C3×D411yescharacteristic
386C2×C3×C4C2×C3×D412yes—
387C2×C3×C4C2×C3×D412yes—
388C2×C3×C4C2×C3×D411yescharacteristic
389C2×C3×C4C23×D311yescharacteristic
390C2×C3×C4C24×C311yescharacteristic
391C2×Dic3—21no—
392C2×Dic3—21no—
393D12—22no—
394D12—22no—
395D12—42no—
396D12—81no—
397C4×D3—82no—
398Dic6—21no—
399Dic6—21no—
400C3×C8—32no—
401C3×C8—62no—
402C3×C8—61no—
403C3⋊C8—22no—
404C3⋊C8—21no—
405C3⋊C8C3×#4112yes—
406C2×C32—81no—
407C2×C32#55011yescharacteristic
408C2×C32C2×#13511yescharacteristic
409C2×C32#57711yescharacteristic
410C3×D3—82no—
411C2×Q8—31no—
412C2×Q8C3×#6711yescharacteristic
413C2×Q8C22×C3×D312yes—
414C2×D4—32no—
415C2×D4—61no—
416C2×D4—61no—
417C2×D4C22×C3×D311yescharacteristic
418C22×C4—21no—
419C22×C4—62no—
420Q16—122no—
421SD16—62no—
422SD16—64no—
423SD16—122no—
424D8—122no—
425C2×C8—31no—
426C2×C8—32no—
427C2×C8—31no—
428C4⋊C4—21no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—31no—
432C4⋊C4—61no—
433C4⋊C4C3×#6711yescharacteristic
434C4⋊C4C3×#6712yes—
435C4⋊C4C22×C3×D311yescharacteristic
436C22⋊C4—22no—
437C22⋊C4—21no—
438C22⋊C4—22no—
439C22⋊C4—62no—
440C22⋊C4—62no—
441C22⋊C4—62no—
442C42—22no—
443C42—61no—
444C42C22×C3×D312yes—
445C42C22×C3×D311yescharacteristic
446C22×C3—21no—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—42no—
450C22×C3—41no—
451C22×C3—41no—
452C22×C3—41no—
453C22×C3—41no—
454C22×C3—41no—
455C22×C3—41no—
456C22×C3—122no—
457C22×C3—122no—
458C22×C3—122no—
459C22×C3C22×#6711yescharacteristic
460C22×C3C22×C3×D411yescharacteristic
461C2×D3—42no—
462C2×D3—42no—
463C2×D3—42no—
464C3×C4—22no—
465C3×C4—22no—
466C3×C4—21no—
467C3×C4—22no—
468C3×C4—22no—
469C3×C4—21no—
470C3×C4—22no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—42no—
474C3×C4—42no—
475C3×C4—41no—
476C3×C4—42no—
477C3×C4—44no—
478C3×C4—41no—
479C3×C4—41no—
480C3×C4—42no—
481C3×C4—42no—
482C3×C4—121no—
483C3×C4—121no—
484C3×C4#95012yes—
485C3×C4C3×#13512yes—
486C3×C4D3×#4111yescharacteristic
487C3×C4#100911yescharacteristic
488C3×C4C2×C3×#4111yescharacteristic
489C3×C4C3×H(5,𝔽2)11yescharacteristic
490Dic3—41no—
491Dic3—41no—
492C32#339411yescharacteristic, complemented, Sylow
493C23—21no—
494C23—62no—
495Q8—22no—
496Q8—22no—
497Q8—22no—
498Q8—61no—
499Q8—61no—
500D4—21no—
501D4—21no—
502D4—62no—
503D4—62no—
504D4—122no—
505D4—241no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—42no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—242no—
512C2×C4C2×C3×#6712yes—
513C2×C4C2×C3×#6712yes—
514C2×C4C2×C3×#6711yescharacteristic
515C2×C4C2×C3×#6711yescharacteristic
516C2×C4C23×C3×D311yescharacteristic
517C8—32no—
518C8—62no—
519C8—61no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—82no—
524C2×C3—81no—
525C2×C3—81no—
526C2×C3—242no—
527C2×C3C2×#95011yescharacteristic
528C2×C3#583411yescharacteristic
529C2×C3#583711yescharacteristic
530C2×C3C3×#55011yescharacteristic
531C2×C3C2×C3×#13511yescharacteristic
532C2×C3C3×#57711yescharacteristic
533D3—82no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—122no—
538C22—122no—
539C22—122no—
540C22C22×C3×#6711yescharacteristic
541C4—22no—
542C4—22no—
543C4—21no—
544C4—41no—
545C4—42no—
546C4—121no—
547C4—121no—
548C4C3×#95012yes—
549C4C3×D3×#4111yescharacteristic
550C4C3×#100911yescharacteristic
551C3—21no—
552C3#8367911yescharacteristic, direct product factor
553C3C3×#339411yescharacteristic, complemented
554C2—81no—
555C2—242no—
556C2C2×C3×#95011yescharacteristic
557C2C3×#583411yescharacteristic
558C2C3×#583711yescharacteristic
subgroupG/H
centerC22×C3C22×#67
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#556C2
socleC22×C32C22×D4
prime radicalC3×#5015C2
hypercenterC3×#556D3
hypocenterC3C3×#3394
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C4×Q8 ◁ C3×#556

Complements

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

NH ≅ G/Ntype
1. C32×#556C2semidirect
2. C3×#5585C2semidirect
3. C3×#5090C2semidirect
4. C3×#5085C2semidirect
5. C3×#5059C2semidirect
6. C3×#5057C2semidirect
7. C3×#5028C2semidirect
8. C3×C4×#215C2semidirect
9. C3×Q8×C3⋊C8C2semidirect
10. C3×C4×#214C2semidirect
13. #83679C3direct (N × H)
45. C3×#556D3semidirect
46. C3×#556C2×C3semidirect
47. #5585C2×C3semidirect
57. #5090C2×C3semidirect
58. #5085C2×C3semidirect
59. #5059C2×C3semidirect
60. #5057C2×C3semidirect
61. #5028C2×C3semidirect
62. C4×#215C2×C3semidirect
63. Q8×C3⋊C8C2×C3semidirect
64. C4×#214C2×C3semidirect
196. #556C3×D3semidirect
492. C32#3394semidirect
552. C3#83679direct (N × H)
553. C3C3×#3394semidirect

Maximal subgroups

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

subgroupindex
C32×#5562
C3×#55852
C3×#50902
C3×#50852
C3×#50592
C3×#50572
C3×#50282
C3×C4×#2152
C3×Q8×C3⋊C82
C3×C4×#2142
C3×#50152
C3×#33943
#836793