Finite Group Databasebeta

← C3×#83717 #11895610

Subgroups — C3×#83717 #11895610

Order 1152.

814 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#559C211yesmaximal, characteristic
2C3×#5567C211yesmaximal, characteristic, complemented
3C3×#5075C211yesmaximal, characteristic
4C3×#5070C211yesmaximal, characteristic
5C3×#5056C212yesmaximal, complemented
6C3×#5054C212yesmaximal, complemented
7C3×C4×#216C211yesmaximal, characteristic, complemented
8C3×Q8×C3⋊C8C211yesmaximal, characteristic, complemented
9C3×#5020C211yesmaximal, characteristic, complemented
10C3×C4×#214C212yesmaximal, complemented
11C3×#5009C212yesmaximal, complemented
12C3×#3365—31nomaximal
13#83717C311yesmaximal, characteristic, direct product factor
14C32×#127C2211yescharacteristic
15C32×#124—21no—
16C32×#124C2212yes—
17C32×#123—22no—
18C32×C4×Q8C2211yescharacteristic
19C32×C4×Q8C2211yescharacteristic
20C32×C4×D4—21no—
21C32×C4×D4C2212yes—
22C3×#944—21no—
23C2×C3×#216C2211yescharacteristic
24C2×C3×#214C2212yes—
25C3×#889—22no—
26C3×#887—21no—
27C3×#870C2212yes—
28C3×#869C2211yescharacteristic
29C3×C4×Dic6C2212yes—
30C3×C4×Dic6C2211yescharacteristic
31C3×#835C2212yes—
32C3×#835C2211yescharacteristic
33C3×#832C2212yes—
34C3×#832C2212yes—
35C3×#832C2212yes—
36C3×#810C2211yescharacteristic
37C3×#808C2212yes—
38C3×#807C2211yescharacteristic
39C3×C4⋊(C3⋊C8)C2212yes—
40C3×C4⋊(C3⋊C8)C2211yescharacteristic
41C3×C4×C3⋊C8C2212yes—
42C3×C4×C3⋊C8C2211yescharacteristic
43C3×#559—21no—
44C3×#559D311yescharacteristic
45C3×#559C2×C311yescharacteristic
46C3×#557—31no—
47#5567C2×C311yescharacteristic, complemented
48C3×#488—32no—
49C3×#486—31no—
50C3×#478—32no—
51C3×#475—31no—
52C3×#464—32no—
53C3×#460—31no—
54C3×C8×Q8—31no—
55C3×C4×Q16—31no—
56C3×C4×SD16—32no—
57#5075C2×C311yescharacteristic
58#5070C2×C311yescharacteristic
59#5056C2×C312yescomplemented
60#5054C2×C312yescomplemented
61C4×#216C2×C311yescharacteristic, complemented
62Q8×C3⋊C8C2×C311yescharacteristic, complemented
63#5020C2×C311yescharacteristic, complemented
64C4×#214C2×C312yescomplemented
65#5009C2×C312yescomplemented
66C2×C32×Q8D411yescharacteristic
67C2×C32×Q8C2311yescharacteristic
68C2×C32×D4—21no—
69C2×C32×D4—21no—
70C2×C32×D4C2312yes—
71C22×C32×C4—22no—
72C2×C3×Dic6—21no—
73C2×C3×Dic6C2311yescharacteristic
74C32×C4⋊C4—21no—
75C32×C4⋊C4D411yescharacteristic
76C32×C4⋊C4D412yes—
77C32×C4⋊C4C2311yescharacteristic
78C32×C22⋊C4—22no—
79C32×C22⋊C4—22no—
80C32×C22⋊C4—22no—
81C32×C42—21no—
82C32×C42C2312yes—
83C32×C42C2311yescharacteristic
84C3×#216—22no—
85C3×#214—24no—
86C3×C4⋊Dic3—21no—
87C3×C4⋊Dic3C2311yescharacteristic
88C3×C4⋊Dic3C2312yes—
89C3×Dic3⋊C4—22no—
90C3×Dic3⋊C4—21no—
91C3×Dic3⋊C4—22no—
92C3×Dic3⋊C4—22no—
93C3×C4×Dic3—22no—
94C3×C4×Dic3—21no—
95C2×C3×C3⋊C8C2312yes—
96C2×C3×C3⋊C8C2311yescharacteristic
97C2×C3×C3⋊C8C2311yescharacteristic
98#3365—31noSylow
99C2×C3×Q16—31no—
100C2×C3×SD16—32no—
101C3×C4⋊Q8—31no—
102C3×C4⋊Q8—61no—
103C3×#127—21no—
104C3×#127C2×D311yescharacteristic
105C3×#127C22×C311yescharacteristic
106C3×#125—32no—
107C3×#125—62no—
108C3×#124—22no—
109C3×#124—21no—
110C3×#124—21no—
111C3×#124—22no—
112C3×#124C2×D312yes—
113C3×#124C22×C312yes—
114C3×#123—24no—
115C3×#123—22no—
116C3×#123—22no—
117C3×C4×Q8—21no—
118C3×C4×Q8—21no—
119C3×C4×Q8—32no—
120C3×C4×Q8—31no—
121C3×C4×Q8—61no—
122C3×C4×Q8C2×D311yescharacteristic
123C3×C4×Q8C2×D311yescharacteristic
124C3×C4×Q8C22×C311yescharacteristic
125C3×C4×Q8C22×C311yescharacteristic
126C3×C4×D4—22no—
127C3×C4×D4—21no—
128C3×C4×D4—21no—
129C3×C4×D4—22no—
130C3×C4×D4C2×D312yes—
131C3×C4×D4C22×C312yes—
132#944—21no—
133C2×#216C22×C311yescharacteristic
134C2×#214C22×C312yes—
135#889—22no—
136#887—21no—
137#870C22×C312yes—
138#869C22×C311yescharacteristic
139C4×Dic6C22×C312yes—
140C4×Dic6C22×C311yescharacteristic
141C3×#107—31no—
142C3×#106—32no—
143C3×C4⋊C8—31no—
144C3×C4⋊C8—32no—
145C3×#103—31no—
146C3×#103—32no—
147C3×#103—31no—
148C3×#102—32no—
149C3×#102—32no—
150C3×#102—32no—
151C3×C4×C8—32no—
152C3×C4×C8—31no—
153#835C22×C312yes—
154#835C22×C311yescharacteristic
155#832C22×C312yes—
156#832C22×C312yes—
157#832C22×C312yes—
158#810C22×C311yescharacteristic
159#808C22×C312yes—
160#807C22×C311yescharacteristic
161C4⋊(C3⋊C8)C22×C312yes—
162C4⋊(C3⋊C8)C22×C311yescharacteristic
163C4×C3⋊C8C22×C312yes—
164C4×C3⋊C8C22×C311yescharacteristic
165C23×C32—22no—
166C32×Q8—22no—
167C32×Q8—21no—
168C32×Q8—21no—
169C32×D4—22no—
170C32×D4—22no—
171C32×D4—42no—
172C32×D4—81no—
173C2×C32×C4—21no—
174C2×C32×C4—21no—
175C2×C32×C4—82no—
176C2×C32×C4C2×D412yes—
177C2×C32×C4C2×D412yes—
178C2×C32×C4C2×D411yescharacteristic
179C2×C32×C4C2×D411yescharacteristic
180C2×C32×C4C2411yescharacteristic
181C2×C3×Dic3—21no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—22no—
184C3×Dic6—21no—
185C3×Dic6—42no—
186C3×Dic6#4112yes—
187C3×C3⋊C8—22no—
188C3×C3⋊C8—21no—
189C3×C3⋊C8—21no—
190#559C3×D311yescharacteristic
191#557—31no—
192#488—32no—
193#486—31no—
194#478—32no—
195#475—31no—
196#464—32no—
197#460—31no—
198C8×Q8—31no—
199C4×Q16—31no—
200C4×SD16—32no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—31no—
204C2×C3×Q8—61no—
205C2×C3×Q8#6711yescharacteristic
206C2×C3×Q8C3×D411yescharacteristic
207C2×C3×Q8C22×D311yescharacteristic
208C2×C3×Q8C23×C311yescharacteristic
209C2×C3×D4—22no—
210C2×C3×D4—22no—
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4—21no—
214C2×C3×D4—21no—
215C2×C3×D4—22no—
216C2×C3×D4C22×D312yes—
217C2×C3×D4C23×C312yes—
218C22×C3×C4—24no—
219C22×C3×C4—22no—
220C22×C3×C4—22no—
221C2×Dic6—21no—
222C2×Dic6C23×C311yescharacteristic
223C3×Q16—62no—
224C3×SD16—64no—
225C2×C3×C8—31no—
226C2×C3×C8—31no—
227C2×C3×C8—32no—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—32no—
235C3×C4⋊C4—31no—
236C3×C4⋊C4—61no—
237C3×C4⋊C4—62no—
238C3×C4⋊C4—62no—
239C3×C4⋊C4—62no—
240C3×C4⋊C4—61no—
241C3×C4⋊C4#6711yescharacteristic
242C3×C4⋊C4#6712yes—
243C3×C4⋊C4C3×D411yescharacteristic
244C3×C4⋊C4C3×D412yes—
245C3×C4⋊C4C22×D311yescharacteristic
246C3×C4⋊C4C23×C311yescharacteristic
247C3×C22⋊C4—24no—
248C3×C22⋊C4—24no—
249C3×C22⋊C4—24no—
250C3×C22⋊C4—22no—
251C3×C22⋊C4—22no—
252C3×C22⋊C4—22no—
253C3×C22⋊C4—22no—
254C3×C22⋊C4—22no—
255C3×C22⋊C4—22no—
256C3×C42—22no—
257C3×C42—21no—
258C3×C42—21no—
259C3×C42—22no—
260C3×C42—21no—
261C3×C42—62no—
262C3×C42—61no—
263C3×C42C22×D312yes—
264C3×C42C22×D311yescharacteristic
265C3×C42C23×C311yescharacteristic
266C3×C42C23×C312yes—
267#216—22no—
268#214—24no—
269C4⋊Dic3—21no—
270C4⋊Dic3C23×C311yescharacteristic
271C4⋊Dic3C23×C312yes—
272Dic3⋊C4—21no—
273Dic3⋊C4—22no—
274Dic3⋊C4—22no—
275Dic3⋊C4—22no—
276C4×Dic3—22no—
277C4×Dic3—21no—
278C2×C3⋊C8C23×C311yescharacteristic
279C2×C3⋊C8C23×C311yescharacteristic
280C2×C3⋊C8C23×C312yes—
281C22×C32—42no—
282C22×C32—42no—
283C22×C32—42no—
284C22×C32C22×D411yescharacteristic
285C32×C4—22no—
286C32×C4—22no—
287C32×C4—21no—
288C32×C4—41no—
289C32×C4—41no—
290C32×C4#13512yes—
291C32×C4C2×#4111yescharacteristic
292C32×C42−1+411yescharacteristic
293C3×Dic3—22no—
294C3×Dic3—42no—
295C3×Dic3—41no—
296C2×Q16—31no—
297C2×SD16—32no—
298C4⋊Q8—31no—
299C4⋊Q8—61no—
300#127C2×C3×D311yescharacteristic
301#125—32no—
302#125—62no—
303#124—21no—
304#124C2×C3×D312yes—
305#123—22no—
306C4×Q8—32no—
307C4×Q8—31no—
308C4×Q8—61no—
309C4×Q8C2×C3×D311yescharacteristic
310C4×Q8C2×C3×D311yescharacteristic
311C4×D4—21no—
312C4×D4C2×C3×D312yes—
313#107—31no—
314#106—32no—
315C4⋊C8—31no—
316C4⋊C8—32no—
317#103—31no—
318#103—32no—
319#103—31no—
320#102—32no—
321#102—32no—
322#102—32no—
323C4×C8—32no—
324C4×C8—31no—
325C23×C3—24no—
326C23×C3—22no—
327C23×C3—22no—
328C3×Q8—22no—
329C3×Q8—22no—
330C3×Q8—21no—
331C3×Q8—21no—
332C3×Q8—21no—
333C3×Q8—21no—
334C3×Q8—32no—
335C3×Q8—42no—
336C3×Q8—41no—
337C3×Q8—41no—
338C3×Q8—61no—
339C3×Q8—122no—
340C3×D4—22no—
341C3×D4—22no—
342C3×D4—22no—
343C3×D4—22no—
344C3×D4—42no—
345C3×D4—42no—
346C3×D4—44no—
347C3×D4—42no—
348C3×D4—42no—
349C3×D4—81no—
350C3×D4—81no—
351C3×D4—82no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—21no—
363C2×C3×C4—61no—
364C2×C3×C4—61no—
365C2×C3×C4—62no—
366C2×C3×C4—82no—
367C2×C3×C4—82no—
368C2×C3×C4—84no—
369C2×C3×C4C2×#6712yes—
370C2×C3×C4C2×#6712yes—
371C2×C3×C4C2×#6711yescharacteristic
372C2×C3×C4C2×#6711yescharacteristic
373C2×C3×C4C2×C3×D412yes—
374C2×C3×C4C2×C3×D412yes—
375C2×C3×C4C2×C3×D411yescharacteristic
376C2×C3×C4C2×C3×D411yescharacteristic
377C2×C3×C4C23×D311yescharacteristic
378C2×C3×C4C24×C311yescharacteristic
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—22no—
382Dic6—21no—
383Dic6—42no—
384Dic6C3×#4112yes—
385C3×C8—62no—
386C3×C8—61no—
387C3×C8—61no—
388C3⋊C8—22no—
389C3⋊C8—21no—
390C3⋊C8—21no—
391C2×C32—82no—
392C2×C32#54811yescharacteristic
393C2×C32C2×#13511yescharacteristic
394C2×C32#57711yescharacteristic
395C2×Q8—31no—
396C2×Q8—61no—
397C2×Q8C3×#6711yescharacteristic
398C2×Q8C22×C3×D311yescharacteristic
399C2×D4—21no—
400C2×D4—21no—
401C2×D4C22×C3×D312yes—
402C22×C4—22no—
403Q16—62no—
404SD16—64no—
405C2×C8—31no—
406C2×C8—31no—
407C2×C8—32no—
408C4⋊C4—21no—
409C4⋊C4—32no—
410C4⋊C4—31no—
411C4⋊C4—61no—
412C4⋊C4—62no—
413C4⋊C4—62no—
414C4⋊C4—62no—
415C4⋊C4—61no—
416C4⋊C4C3×#6711yescharacteristic
417C4⋊C4C3×#6712yes—
418C4⋊C4C22×C3×D311yescharacteristic
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C42—21no—
423C42—62no—
424C42—61no—
425C42C22×C3×D312yes—
426C42C22×C3×D311yescharacteristic
427C22×C3—21no—
428C22×C3—44no—
429C22×C3—44no—
430C22×C3—44no—
431C22×C3—42no—
432C22×C3—42no—
433C22×C3—42no—
434C22×C3—42no—
435C22×C3—42no—
436C22×C3—42no—
437C22×C3C22×#6711yescharacteristic
438C22×C3C22×C3×D411yescharacteristic
439C3×C4—22no—
440C3×C4—22no—
441C3×C4—21no—
442C3×C4—22no—
443C3×C4—22no—
444C3×C4—21no—
445C3×C4—22no—
446C3×C4—21no—
447C3×C4—21no—
448C3×C4—42no—
449C3×C4—41no—
450C3×C4—42no—
451C3×C4—42no—
452C3×C4—42no—
453C3×C4—41no—
454C3×C4—41no—
455C3×C4—41no—
456C3×C4—41no—
457C3×C4—62no—
458C3×C4—122no—
459C3×C4—121no—
460C3×C4#95012yes—
461C3×C4C3×#13512yes—
462C3×C4D3×#4111yescharacteristic
463C3×C4#101011yescharacteristic
464C3×C4C2×C3×#4111yescharacteristic
465C3×C4C3×2−1+411yescharacteristic
466Dic3—22no—
467Dic3—42no—
468Dic3—41no—
469C32#336511yescharacteristic, complemented, Sylow
470C23—22no—
471Q8—22no—
472Q8—21no—
473Q8—21no—
474Q8—32no—
475Q8—61no—
476Q8—122no—
477D4—22no—
478D4—22no—
479D4—42no—
480D4—81no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—61no—
484C2×C4—61no—
485C2×C4—62no—
486C2×C4—82no—
487C2×C4C2×C3×#6712yes—
488C2×C4C2×C3×#6712yes—
489C2×C4C2×C3×#6711yescharacteristic
490C2×C4C2×C3×#6711yescharacteristic
491C2×C4C23×C3×D311yescharacteristic
492C8—62no—
493C8—61no—
494C8—61no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—84no—
499C2×C3—82no—
500C2×C3—82no—
501C2×C3C2×#95011yescharacteristic
502C2×C3#583111yescharacteristic
503C2×C3#583611yescharacteristic
504C2×C3C3×#54811yescharacteristic
505C2×C3C2×C3×#13511yescharacteristic
506C2×C3C3×#57711yescharacteristic
507C22—42no—
508C22—42no—
509C22—42no—
510C22C22×C3×#6711yescharacteristic
511C4—22no—
512C4—22no—
513C4—21no—
514C4—41no—
515C4—41no—
516C4—62no—
517C4—122no—
518C4—121no—
519C4C3×#95012yes—
520C4C3×D3×#4111yescharacteristic
521C4C3×#101011yescharacteristic
522C3—21no—
523C3#8371711yescharacteristic, direct product factor
524C3C3×#336511yescharacteristic, complemented
525C2—82no—
526C2C2×C3×#95011yescharacteristic
527C2C3×#583111yescharacteristic
528C2C3×#583611yescharacteristic
subgroupG/H
centerC22×C3C22×#67
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#559C2
socleC22×C32C22×D4
prime radicalC32×#127C22
hypercenterC3×#559D3
hypocenterC3C3×#3365
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C4×Q8 ◁ C3×#559

Complements

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

NH ≅ G/Ntype
2. C3×#5567C2semidirect
5. C3×#5056C2semidirect
6. C3×#5054C2semidirect
7. C3×C4×#216C2semidirect
8. C3×Q8×C3⋊C8C2semidirect
9. C3×#5020C2semidirect
10. C3×C4×#214C2semidirect
11. C3×#5009C2semidirect
13. #83717C3direct (N × H)
47. #5567C2×C3semidirect
59. #5056C2×C3semidirect
60. #5054C2×C3semidirect
61. C4×#216C2×C3semidirect
62. Q8×C3⋊C8C2×C3semidirect
63. #5020C2×C3semidirect
64. C4×#214C2×C3semidirect
65. #5009C2×C3semidirect
469. C32#3365semidirect
523. C3#83717direct (N × H)
524. C3C3×#3365semidirect

Maximal subgroups

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

subgroupindex
C32×#5592
C3×#55672
C3×#50752
C3×#50702
C3×#50562
C3×#50542
C3×C4×#2162
C3×Q8×C3⋊C82
C3×#50202
C3×C4×#2142
C3×#50092
C3×#33653
#837173