Finite Group Databasebeta

← C3×#10591543 #13158646

Subgroups — C3×#10591543 #13158646

Order 1728.

596 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64151C211yesmaximal, characteristic
2C2×C32×#216C211yesmaximal, characteristic, complemented
3C3×#63833C211yesmaximal, characteristic
4C3×#63817C211yesmaximal, characteristic
5C32×#836C211yesmaximal, characteristic, complemented
6C3×#63577C211yesmaximal, characteristic, complemented
7C3×#63560C211yesmaximal, characteristic, complemented
8C3×#5192—31nomaximal
9C3×#4755—31nomaximal
10#10591543C311yesmaximal, characteristic, direct product factor
11C2×C33×Q8C2211yescharacteristic
12C3×#3828C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C3×#3756—21no—
15C3×#3755—21no—
16C3×#3748C2211yescharacteristic
17C32×#216—21no—
18C32×#216—21no—
19C32×#208C2211yescharacteristic
20C32×#208C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C3×#3719—21no—
23C3×#3717—21no—
24C3×#3712—21no—
25C3×#3710—21no—
26C2×C32×Q16—31no—
27#64151C2×C311yescharacteristic
28C2×C3×#216—21no—
29C2×C3×#216D311yescharacteristic, complemented
30C2×C3×#216C2×C311yescharacteristic, complemented
31C3×#942—31no—
32C3×#942—31no—
33C3×#942—31no—
34C3×#942—61no—
35C3×#909—31no—
36C3×#907—31no—
37C3×#901—31no—
38#63833C2×C311yescharacteristic
39#63817C2×C311yescharacteristic
40C32×#101—31no—
41C3×#836—21no—
42C3×#836—31no—
43C3×#836D311yescharacteristic, complemented
44C3×#836C2×C311yescharacteristic, complemented
45C3×#824—31no—
46C3×#822—31no—
47C3×#819—31no—
48#63577C2×C311yescharacteristic, complemented
49#63560C2×C311yescharacteristic, complemented
50C33×Q8—21no—
51C33×Q8—21no—
52C2×C33×C4—21no—
53C2×C33×C4C2311yescharacteristic
54C3×#638—21no—
55C3×C4⋊(C3⋊D3)D411yescharacteristic
56C3×C4×C3⋊D3—21no—
57C3×#634D411yescharacteristic
58C2×C32×Dic3—21no—
59C32×Dic6—21no—
60C32×Dic6—21no—
61C3×C3⋊(C3⋊C8)—21no—
62C32×C3⋊C8—21no—
63C32×C3⋊C8—21no—
64C32×C3⋊C8D411yescharacteristic
65C32×C3⋊C8D411yescharacteristic
66C3×#473—91no—
67#5192—31no—
68#4755—31no—
69C2×C32×Q8—21no—
70C2×C32×Q8—21no—
71C2×C32×Q8—21no—
72C2×C32×Q8—31no—
73C2×C32×Q8—41no—
74C2×C32×Q8C2×D311yescharacteristic
75C2×C32×Q8C2×D311yescharacteristic
76C2×C32×Q8C22×C311yescharacteristic
77#3828C22×C311yescharacteristic
78C3×#235—31no—
79C3×#235—31no—
80C3×#235—61no—
81C2×C3×Dic6—21no—
82C2×C3×Dic6C2×D311yescharacteristic
83C2×C3×Dic6C22×C311yescharacteristic
84C32×Q16—61no—
85C32×Q16—61no—
86C32×M16—31no—
87C32×M16—31no—
88C2×C32×C8—31no—
89#3756—21no—
90#3755—21no—
91#3748C22×C311yescharacteristic
92C3×#216—22no—
93C3×#216—22no—
94C3×#216—21no—
95C3×#216—21no—
96C3×#216—21no—
97C3×#216—21no—
98C3×#216—61no—
99C3×#216—61no—
100C3×#216—61no—
101C3×#216—62no—
102C3×#215—61no—
103C3×#215—61no—
104C3×#215—61no—
105C3×#215—62no—
106C3×#208—21no—
107C3×#208—21no—
108C3×#208—31no—
109C3×#208—31no—
110C3×#208—61no—
111C3×#208C2×D311yescharacteristic
112C3×#208C2×D311yescharacteristic
113C3×#208C22×C311yescharacteristic
114C3×#208C22×C311yescharacteristic
115C2×C3×C3⋊C8—21no—
116C2×C3×C3⋊C8C2×D311yescharacteristic
117C2×C3×C3⋊C8C22×C311yescharacteristic
118C3×Dic12—61no—
119C3×#204—61no—
120C3×ΓL(1,𝔽25)—61no—
121C3×ΓL(1,𝔽25)—61no—
122C3×D3×C8—61no—
123C3×D3×C8—61no—
124#3719—21no—
125#3717—21no—
126#3712—21no—
127#3710—21no—
128C22×C33#4111yescharacteristic
129C2×C3×C3⋊D3—21no—
130C33×C4—41no—
131C33×C4C2×D411yescharacteristic
132C33×C4C2×D411yescharacteristic
133C3×C3⋊Dic3—21no—
134C32×Dic3—41no—
135C3×#137—91no—
136C3×#137—91no—
137C2×C3×Q16—31no—
138C2×C3×Q16—31no—
139C2×C3×Q16—61no—
140C3×#131—91no—
141C2×#216C3×D311yescharacteristic, complemented
142#942—31no—
143#942—31no—
144#942—31no—
145#942—61no—
146#909—31no—
147#907—31no—
148#901—31no—
149C3×#108—91no—
150C3×#101—31no—
151C3×#101—31no—
152C3×#101—61no—
153C3×#101—91no—
154#836—31no—
155#836C3×D311yescharacteristic, complemented
156#824—31no—
157#822—31no—
158#819—31no—
159C32×Q8—21no—
160C32×Q8—21no—
161C32×Q8—22no—
162C32×Q8—22no—
163C32×Q8—21no—
164C32×Q8—21no—
165C32×Q8—21no—
166C32×Q8—21no—
167C32×Q8—41no—
168C32×Q8—41no—
169C32×Q8—41no—
170C32×Q8—41no—
171C32×Q8—42no—
172C32×Q8—42no—
173C32×Q8—61no—
174C32×Q8—61no—
175C2×C32×C4—21no—
176C2×C32×C4—22no—
177C2×C32×C4—21no—
178C2×C32×C4—21no—
179C2×C32×C4—21no—
180C2×C32×C4—21no—
181C2×C32×C4—21no—
182C2×C32×C4—41no—
183C2×C32×C4—41no—
184C2×C32×C4—41no—
185C2×C32×C4—42no—
186C2×C32×C4—61no—
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D311yescharacteristic
189C2×C32×C4C23×C311yescharacteristic
190#638—21no—
191C4⋊(C3⋊D3)C3×D411yescharacteristic
192C4×C3⋊D3—21no—
193#634C3×D411yescharacteristic
194C3×#67—61no—
195C3×#67—61no—
196C3×#67—121no—
197C2×C3×Dic3—21no—
198C2×C3×Dic3—21no—
199C2×C3×Dic3—41no—
200C3×D12—31no—
201C3×D12—31no—
202C3×D12—61no—
203C3×C4×D3—61no—
204C3×C4×D3—61no—
205C3×C4×D3—121no—
206C3×Dic6—22no—
207C3×Dic6—21no—
208C3×Dic6—21no—
209C3×Dic6—21no—
210C3×Dic6—21no—
211C3×Dic6—31no—
212C3×Dic6—31no—
213C3×Dic6—41no—
214C3×Dic6—61no—
215C32×C8—31no—
216C32×C8—31no—
217C32×C8—61no—
218C32×C8—61no—
219C3⋊(C3⋊C8)—21no—
220C3×C3⋊C8—21no—
221C3×C3⋊C8—21no—
222C3×C3⋊C8—21no—
223C3×C3⋊C8—21no—
224C3×C3⋊C8—21no—
225C3×C3⋊C8—21no—
226C3×C3⋊C8—41no—
227C3×C3⋊C8—41no—
228C3×C3⋊C8—62no—
229C3×C3⋊C8—61no—
230C3×C3⋊C8—61no—
231C3×C3⋊C8#6711yescharacteristic
232C3×C3⋊C8#6711yescharacteristic
233C3×C3⋊C8C3×D411yescharacteristic
234C3×C3⋊C8C3×D411yescharacteristic
235#473—91noSylow
236C2×C33—21no—
237C2×C33#12111yescharacteristic
238C3×C3⋊D3—41no—
239C3×#41—91no—
240C2×C3×Q8—21no—
241C2×C3×Q8—21no—
242C2×C3×Q8—21no—
243C2×C3×Q8—31no—
244C2×C3×Q8—31no—
245C2×C3×Q8—41no—
246C2×C3×Q8—61no—
247C2×C3×Q8D3211yescharacteristic
248C2×C3×Q8C2×C3×D311yescharacteristic
249C2×C3×Q8C2×C3×D311yescharacteristic
250#235—31no—
251#235—31no—
252#235—61no—
253C2×Dic6C2×C3×D311yescharacteristic
254C3×Q16—61no—
255C3×Q16—61no—
256C3×Q16—61no—
257C3×Q16—61no—
258C3×Q16—62no—
259C3×Q16—62no—
260C3×Q16—181no—
261C3×Q16—181no—
262C3×SD16—181no—
263C3×SD16—181no—
264C3×M16—31no—
265C3×M16—31no—
266C3×M16—31no—
267C3×M16—31no—
268C3×M16—61no—
269C3×M16—61no—
270C3×M16—91no—
271C3×M16—181no—
272C2×C3×C8—31no—
273C2×C3×C8—31no—
274C2×C3×C8—61no—
275C2×C3×C8—181no—
276#216—21no—
277#216—21no—
278#216—61no—
279#216—61no—
280#216—61no—
281#216—62no—
282#215—61no—
283#215—61no—
284#215—61no—
285#215—62no—
286#208—31no—
287#208—31no—
288#208—61no—
289#208C2×C3×D311yescharacteristic
290#208C2×C3×D311yescharacteristic
291C2×C3⋊C8C2×C3×D311yescharacteristic
292Dic12—61no—
293#204—61no—
294ΓL(1,𝔽25)—61no—
295ΓL(1,𝔽25)—61no—
296D3×C8—61no—
297D3×C8—61no—
298C22×C32—21no—
299C22×C32—21no—
300C22×C32—21no—
301C22×C32—41no—
302C22×C32#23511yescharacteristic
303C22×C32#23711yescharacteristic
304C22×C32C3×#4111yescharacteristic
305C2×C3⋊D3—21no—
306C2×C3×D3—61no—
307C2×C3×D3—61no—
308C2×C3×D3—121no—
309C32×C4—21no—
310C32×C4—21no—
311C32×C4—21no—
312C32×C4—21no—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—41no—
316C32×C4—41no—
317C32×C4—42no—
318C32×C4—42no—
319C32×C4—42no—
320C32×C4—44no—
321C32×C4—41no—
322C32×C4—41no—
323C32×C4—41no—
324C32×C4—121no—
325C32×C4D3×D411yescharacteristic
326C32×C4D3×D411yescharacteristic
327C32×C4D3×D411yescharacteristic
328C32×C4C2×#6711yescharacteristic
329C32×C4C2×C3×D411yescharacteristic
330C32×C4C2×C3×D411yescharacteristic
331C3⋊Dic3—21no—
332C3×Dic3—42no—
333C3×Dic3—41no—
334C3×Dic3—41no—
335C3×Dic3—61no—
336C3×Dic3—61no—
337C3×Dic3—121no—
338#137—91no—
339#137—91no—
340C2×Q16—31no—
341#131—91no—
342#108—91no—
343#101—31no—
344#101—91no—
345C33#47311yescharacteristic, complemented, Sylow
346C3×Q8—22no—
347C3×Q8—22no—
348C3×Q8—21no—
349C3×Q8—21no—
350C3×Q8—21no—
351C3×Q8—21no—
352C3×Q8—21no—
353C3×Q8—21no—
354C3×Q8—41no—
355C3×Q8—41no—
356C3×Q8—41no—
357C3×Q8—41no—
358C3×Q8—42no—
359C3×Q8—42no—
360C3×Q8—62no—
361C3×Q8—61no—
362C3×Q8—61no—
363C3×Q8—61no—
364C3×Q8—61no—
365C3×Q8—91no—
366C3×Q8—121no—
367C3×D4—91no—
368C3×D4—181no—
369C2×C3×C4—22no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—42no—
379C2×C3×C4—41no—
380C2×C3×C4—61no—
381C2×C3×C4—61no—
382C2×C3×C4—121no—
383C2×C3×C4—181no—
384C2×C3×C4C2×D3211yescharacteristic
385C2×C3×C4C22×C3×D311yescharacteristic
386C2×C3×C4C22×C3×D311yescharacteristic
387#67—61no—
388#67—61no—
389#67—121no—
390C2×Dic3—21no—
391D12—31no—
392D12—31no—
393D12—61no—
394C4×D3—61no—
395C4×D3—61no—
396C4×D3—121no—
397Dic6—21no—
398Dic6—21no—
399Dic6—31no—
400Dic6—31no—
401Dic6—61no—
402C3×C8—31no—
403C3×C8—31no—
404C3×C8—31no—
405C3×C8—31no—
406C3×C8—61no—
407C3×C8—61no—
408C3×C8—61no—
409C3×C8—61no—
410C3×C8—61no—
411C3×C8—61no—
412C3×C8—121no—
413C3×C8—121no—
414C3×C8—181no—
415C3⋊C8—21no—
416C3⋊C8—21no—
417C3⋊C8—62no—
418C3⋊C8—61no—
419C3⋊C8—61no—
420C3⋊C8C3×#6711yescharacteristic
421C3⋊C8C3×#6711yescharacteristic
422C2×C32—21no—
423C2×C32—22no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—41no—
430C2×C32—41no—
431C2×C32—41no—
432C2×C32—42no—
433C2×C32#88411yescharacteristic
434C2×C32#93811yescharacteristic
435C2×C32C3×#12111yescharacteristic
436C3⋊D3—41no—
437C3×D3—121no—
438C3×D3—121no—
439C3×D3—241no—
440#41—91no—
441C2×Q8—31no—
442C2×Q8C3×D3211yescharacteristic
443Q16—61no—
444Q16—61no—
445Q16—181no—
446Q16—181no—
447SD16—181no—
448SD16—181no—
449M16—31no—
450M16—31no—
451M16—91no—
452M16—181no—
453C2×C8—31no—
454C2×C8—181no—
455C22×C3—21no—
456C22×C3—21no—
457C22×C3—21no—
458C22×C3—41no—
459C22×C3—181no—
460C22×C3#380411yescharacteristic
461C22×C3C3×#23511yescharacteristic
462C22×C3C3×#23711yescharacteristic
463C2×D3—61no—
464C2×D3—61no—
465C2×D3—121no—
466C3×C4—21no—
467C3×C4—21no—
468C3×C4—21no—
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—42no—
473C3×C4—42no—
474C3×C4—42no—
475C3×C4—44no—
476C3×C4—41no—
477C3×C4—41no—
478C3×C4—41no—
479C3×C4—41no—
480C3×C4—41no—
481C3×C4—122no—
482C3×C4—121no—
483C3×C4—121no—
484C3×C4—181no—
485C3×C4D3×#6711yescharacteristic
486C3×C4#381111yescharacteristic
487C3×C4C3×D3×D411yescharacteristic
488C3×C4C3×D3×D411yescharacteristic
489C3×C4C3×D3×D411yescharacteristic
490C3×C4C2×C3×#6711yescharacteristic
491Dic3—41no—
492Dic3—61no—
493Dic3—61no—
494Dic3—121no—
495C32—21no—
496C32—21no—
497C32—21no—
498C32—41no—
499C32#475511yescharacteristic, complemented
500C32#519211yescharacteristic, complemented
501C32C3×#47311yescharacteristic, complemented
502Q8—21no—
503Q8—21no—
504Q8—61no—
505Q8—61no—
506Q8—91no—
507D4—91no—
508D4—181no—
509C2×C4—21no—
510C2×C4—61no—
511C2×C4—181no—
512C2×C4C2×C3×D3211yescharacteristic
513C8—31no—
514C8—31no—
515C8—61no—
516C8—61no—
517C8—181no—
518C2×C3—22no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—21no—
525C2×C3—41no—
526C2×C3—41no—
527C2×C3—42no—
528C2×C3—41no—
529C2×C3—361no—
530C2×C3#6395811yescharacteristic
531C2×C3C3×#88411yescharacteristic
532C2×C3C3×#93811yescharacteristic
533D3—121no—
534D3—121no—
535D3—241no—
536C22—181no—
537C22C3×#380411yescharacteristic
538C4—41no—
539C4—121no—
540C4—181no—
541C4C3×D3×#6711yescharacteristic
542C4C3×#381111yescharacteristic
543C3—21no—
544C3—21no—
545C3—21no—
546C3—41no—
547C3#1059154311yescharacteristic, direct product factor
548C3C3×#475511yescharacteristic, complemented
549C3C3×#519211yescharacteristic, complemented
550C2—21no—
551C2—361no—
552C2C3×#6395811yescharacteristic
subgroupG/H
centerC2×C3#63958
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×Q8C22
socleC2×C33#121
prime radicalC3×#3828C22
hypercenterC2×C3×Q8D32
hypocenterC32C3×#473
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
2. C2×C32×#216C2semidirect
5. C32×#836C2semidirect
6. C3×#63577C2semidirect
7. C3×#63560C2semidirect
10. #10591543C3direct (N × H)
29. C2×C3×#216D3semidirect
30. C2×C3×#216C2×C3semidirect
43. C3×#836D3semidirect
44. C3×#836C2×C3semidirect
48. #63577C2×C3semidirect
49. #63560C2×C3semidirect
141. C2×#216C3×D3semidirect
155. #836C3×D3semidirect
345. C33#473semidirect
499. C32#4755semidirect
500. C32#5192semidirect
501. C32C3×#473semidirect
547. C3#10591543direct (N × H)
548. C3C3×#4755semidirect
549. C3C3×#5192semidirect

Maximal subgroups

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

subgroupindex
C3×#641512
C2×C32×#2162
C3×#638332
C3×#638172
C32×#8362
C3×#635772
C3×#635602
C3×#51923
C3×#47553
#105915433