Finite Group Databasebeta

← C3×#10590342 #13157407

Subgroups — C3×#10590342 #13157407

Order 1728.

574 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×#63829C211yesmaximal, characteristic
2C3×#63817C211yesmaximal, characteristic, complemented
3C3×#63814C211yesmaximal, characteristic, complemented
4C3×#63632C211yesmaximal, characteristic, complemented
5C32×#805C211yesmaximal, characteristic, complemented
6C32×#803C211yesmaximal, characteristic, complemented
7C3×#63577C211yesmaximal, characteristic, complemented
8C3×#4823—31nomaximal
9C3×#4717—31nomaximal
10#10590342C311yesmaximal, characteristic, direct product factor
11C3×#3828C2211yescharacteristic
12C32×#235C2211yescharacteristic
13C33×C42C2211yescharacteristic
14C3×#3748C2211yescharacteristic
15C32×#208C2211yescharacteristic
16C2×C32×C3⋊C8C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18C3×#3719C411yescharacteristic, complemented
19C3×#3717C411yescharacteristic, complemented
20C3×#3716C411yescharacteristic, complemented
21C3×#3714C411yescharacteristic, complemented
22C3×#3712—21no—
23C3×#3711—21no—
24C3×#3710—21no—
25C3×D3×C3⋊C8—21no—
26C3×#950—31no—
27C3×#948—31no—
28C3×#907—31no—
29C3×#904—31no—
30C3×#901—31no—
31C3×#901—31no—
32#63829C2×C311yescharacteristic
33#63817C2×C311yescharacteristic, complemented
34#63814C2×C311yescharacteristic, complemented
35C32×#104—31no—
36C32×#97—31no—
37#63632C2×C311yescharacteristic, complemented
38C3×#822—31no—
39C3×#819—31no—
40C3×#805—21no—
41C3×#805—31no—
42C3×#805—31no—
43C3×#805—61no—
44C3×#805D311yescharacteristic, complemented
45C3×#805C2×C311yescharacteristic, complemented
46C3×#803—21no—
47C3×#803D311yescharacteristic, complemented
48C3×#803C2×C311yescharacteristic, complemented
49#63577C2×C311yescharacteristic, complemented
50C2×C33×C4—21no—
51C2×C33×C4C2311yescharacteristic
52C3×#638—21no—
53C3×C4⋊(C3⋊D3)C2×C411yescharacteristic
54C3×C4×C3⋊D3—21no—
55C3×#634C2×C411yescharacteristic
56C32×#67—21no—
57C32×D12C2×C411yescharacteristic
58C32×C4×D3—21no—
59C32×Dic6C2×C411yescharacteristic
60C3×C3⋊(C3⋊C8)—21no—
61C32×C3⋊C8—21no—
62C32×C3⋊C8C2×C411yescharacteristic
63C32×C3⋊C8C2×C411yescharacteristic
64C32×C3⋊C8D411yescharacteristic
65C32×C3⋊C8D411yescharacteristic
66C3×#444—91no—
67#4823—31no—
68#4717—31no—
69C32×#41—31no—
70#3828C22×C311yescharacteristic
71C3×#235—21no—
72C3×#235—31no—
73C3×#235—31no—
74C3×#235—61no—
75C3×#235C2×D311yescharacteristic
76C3×#235C22×C311yescharacteristic
77C32×M16—31no—
78C2×C32×C8—31no—
79C2×C32×C8—31no—
80C32×C42—21no—
81C32×C42—21no—
82C32×C42—21no—
83C32×C42—41no—
84C32×C42C2×D311yescharacteristic
85C32×C42C2×D311yescharacteristic
86C32×C42C22×C311yescharacteristic
87#3748C22×C311yescharacteristic
88C3×#216—31no—
89C3×#215—31no—
90C3×#214—31no—
91C3×#213—31no—
92C3×#208—21no—
93C3×#208—31no—
94C3×#208—31no—
95C3×#208—61no—
96C3×#208—61no—
97C3×#208C2×D311yescharacteristic
98C3×#208C22×C311yescharacteristic
99C2×C3×C3⋊C8—21no—
100C2×C3×C3⋊C8—21no—
101C2×C3×C3⋊C8—61no—
102C2×C3×C3⋊C8C2×D311yescharacteristic
103C2×C3×C3⋊C8C2×D311yescharacteristic
104C2×C3×C3⋊C8C22×C311yescharacteristic
105C2×C3×C3⋊C8C22×C311yescharacteristic
106C3×Dic12—31no—
107C3×D24—31no—
108C3×#204—31no—
109C3×#204—31no—
110C3×ΓL(1,𝔽25)—61no—
111C3×ΓL(1,𝔽25)—61no—
112C3×ΓL(1,𝔽25)—61no—
113C3×D3×C8—61no—
114C3×D3×C8—61no—
115C3×D3×C8—61no—
116#3719C3×C411yescharacteristic, complemented
117#3717C3×C411yescharacteristic, complemented
118#3716C3×C411yescharacteristic, complemented
119#3714C3×C411yescharacteristic, complemented
120#3712—21no—
121#3711—21no—
122#3710—21no—
123D3×C3⋊C8—21no—
124C22×C33#4111yescharacteristic
125C2×C3×C3⋊D3—21no—
126C2×C32×D3—21no—
127C33×C4—41no—
128C33×C4C22×C411yescharacteristic
129C33×C4C2×D411yescharacteristic
130C3×C3⋊Dic3—21no—
131C32×Dic3—21no—
132C3×#135—91no—
133C3×#131—91no—
134C3×#131—91no—
135#950—31no—
136#948—31no—
137#907—31no—
138#904—31no—
139#901—31no—
140#901—31no—
141C3×#108—91no—
142C3×#104—31no—
143C3×#104—31no—
144C3×#104—61no—
145C3×#104—91no—
146C3×#97—31no—
147C3×#97—31no—
148C3×#97—61no—
149#822—31no—
150#819—31no—
151#805—31no—
152#805—31no—
153#805—61no—
154#805C3×D311yescharacteristic, complemented
155#803C3×D311yescharacteristic, complemented
156C32×Q8—31no—
157C32×D4—31no—
158C32×D4—61no—
159C2×C32×C4—21no—
160C2×C32×C4—22no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—41no—
167C2×C32×C4—41no—
168C2×C32×C4—41no—
169C2×C32×C4—42no—
170C2×C32×C4—61no—
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C23×C311yescharacteristic
174#638—21no—
175C4⋊(C3⋊D3)C2×C3×C411yescharacteristic
176C4×C3⋊D3—21no—
177#634C2×C3×C411yescharacteristic
178C3×#67—21no—
179C3×#67—21no—
180C3×#67—41no—
181C3×#67—61no—
182C3×#67—61no—
183C3×#67—121no—
184C3×D12—21no—
185C3×D12—31no—
186C3×D12—31no—
187C3×D12—61no—
188C3×D12C4×D311yescharacteristic
189C3×D12C2×C3×C411yescharacteristic
190C3×C4×D3—21no—
191C3×C4×D3—21no—
192C3×C4×D3—41no—
193C3×C4×D3—61no—
194C3×C4×D3—61no—
195C3×C4×D3—121no—
196C3×Dic6—21no—
197C3×Dic6—31no—
198C3×Dic6—31no—
199C3×Dic6—61no—
200C3×Dic6C4×D311yescharacteristic
201C3×Dic6C2×C3×C411yescharacteristic
202C32×C8—31no—
203C32×C8—31no—
204C32×C8—31no—
205C32×C8—31no—
206C32×C8—61no—
207C3⋊(C3⋊C8)—21no—
208C3×C3⋊C8—21no—
209C3×C3⋊C8—21no—
210C3×C3⋊C8—21no—
211C3×C3⋊C8—21no—
212C3×C3⋊C8—21no—
213C3×C3⋊C8—21no—
214C3×C3⋊C8—41no—
215C3×C3⋊C8—61no—
216C3×C3⋊C8—61no—
217C3×C3⋊C8—121no—
218C3×C3⋊C8C4×D311yescharacteristic
219C3×C3⋊C8C4×D311yescharacteristic
220C3×C3⋊C8D1211yescharacteristic
221C3×C3⋊C8D1211yescharacteristic
222C3×C3⋊C8C2×C3×C411yescharacteristic
223C3×C3⋊C8C2×C3×C411yescharacteristic
224C3×C3⋊C8C3×D411yescharacteristic
225C3×C3⋊C8C3×D411yescharacteristic
226#444—91noSylow
227C2×C33—21no—
228C2×C33C4×D411yescharacteristic
229C3×C3⋊D3—41no—
230C32×D3—41no—
231C3×#41—31no—
232C3×#41—31no—
233C3×#41—61no—
234C3×#41—91no—
235#235—31no—
236#235—31no—
237#235—61no—
238#235C2×C3×D311yescharacteristic
239C3×Q16—91no—
240C3×SD16—91no—
241C3×SD16—91no—
242C3×D8—91no—
243C3×M16—31no—
244C3×M16—31no—
245C3×M16—61no—
246C3×M16—91no—
247C3×M16—181no—
248C3×M16—181no—
249C2×C3×C8—31no—
250C2×C3×C8—31no—
251C2×C3×C8—31no—
252C2×C3×C8—31no—
253C2×C3×C8—61no—
254C2×C3×C8—61no—
255C2×C3×C8—181no—
256C2×C3×C8—181no—
257C3×C42—21no—
258C3×C42—21no—
259C3×C42—21no—
260C3×C42—41no—
261C3×C42D3211yescharacteristic
262C3×C42C2×C3×D311yescharacteristic
263C3×C42C2×C3×D311yescharacteristic
264#216—31no—
265#215—31no—
266#214—31no—
267#213—31no—
268#208—31no—
269#208—31no—
270#208—61no—
271#208—61no—
272#208C2×C3×D311yescharacteristic
273C2×C3⋊C8—61no—
274C2×C3⋊C8C2×C3×D311yescharacteristic
275C2×C3⋊C8C2×C3×D311yescharacteristic
276Dic12—31no—
277D24—31no—
278#204—31no—
279#204—31no—
280ΓL(1,𝔽25)—61no—
281ΓL(1,𝔽25)—61no—
282ΓL(1,𝔽25)—61no—
283D3×C8—61no—
284D3×C8—61no—
285D3×C8—61no—
286C22×C32—21no—
287C22×C32—21no—
288C22×C32—21no—
289C22×C32—41no—
290C22×C32—61no—
291C22×C32#23511yescharacteristic
292C22×C32#23711yescharacteristic
293C22×C32C3×#4111yescharacteristic
294C2×C3⋊D3—21no—
295C2×C3×D3—21no—
296C2×C3×D3—21no—
297C2×C3×D3—41no—
298C2×C3×D3—61no—
299C2×C3×D3—61no—
300C2×C3×D3—121no—
301C32×C4—21no—
302C32×C4—21no—
303C32×C4—21no—
304C32×C4—21no—
305C32×C4—21no—
306C32×C4—21no—
307C32×C4—41no—
308C32×C4—41no—
309C32×C4—41no—
310C32×C4—42no—
311C32×C4—41no—
312C32×C4—41no—
313C32×C4—42no—
314C32×C4—42no—
315C32×C4—44no—
316C32×C4—61no—
317C32×C4C2×C4×D311yescharacteristic
318C32×C4C2×C4×D311yescharacteristic
319C32×C4C2×D1211yescharacteristic
320C32×C4D3×D411yescharacteristic
321C32×C4C22×C3×C411yescharacteristic
322C32×C4C2×C3×D411yescharacteristic
323C3⋊Dic3—21no—
324C3×Dic3—21no—
325C3×Dic3—21no—
326C3×Dic3—41no—
327C3×Dic3—61no—
328C3×Dic3—61no—
329C3×Dic3—121no—
330#135—91no—
331#131—91no—
332#131—91no—
333#108—91no—
334#104—31no—
335#104—91no—
336#97—31no—
337C33#44411yescharacteristic, complemented, Sylow
338C3×Q8—31no—
339C3×Q8—31no—
340C3×Q8—61no—
341C3×Q8—91no—
342C3×D4—31no—
343C3×D4—31no—
344C3×D4—61no—
345C3×D4—61no—
346C3×D4—61no—
347C3×D4—91no—
348C3×D4—121no—
349C3×D4—181no—
350C2×C3×C4—22no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—42no—
360C2×C3×C4—41no—
361C2×C3×C4—61no—
362C2×C3×C4—61no—
363C2×C3×C4—121no—
364C2×C3×C4—181no—
365C2×C3×C4C2×D3211yescharacteristic
366C2×C3×C4C22×C3×D311yescharacteristic
367C2×C3×C4C22×C3×D311yescharacteristic
368#67—21no—
369#67—61no—
370#67—61no—
371#67—121no—
372D12—31no—
373D12—31no—
374D12—61no—
375D12C3×C4×D311yescharacteristic
376C4×D3—21no—
377C4×D3—61no—
378C4×D3—61no—
379C4×D3—121no—
380Dic6—31no—
381Dic6—31no—
382Dic6—61no—
383Dic6C3×C4×D311yescharacteristic
384C3×C8—31no—
385C3×C8—31no—
386C3×C8—31no—
387C3×C8—31no—
388C3×C8—31no—
389C3×C8—31no—
390C3×C8—31no—
391C3×C8—31no—
392C3×C8—61no—
393C3×C8—61no—
394C3×C8—61no—
395C3×C8—61no—
396C3×C8—61no—
397C3×C8—61no—
398C3×C8—121no—
399C3×C8—181no—
400C3⋊C8—21no—
401C3⋊C8—61no—
402C3⋊C8—61no—
403C3⋊C8—121no—
404C3⋊C8C3×C4×D311yescharacteristic
405C3⋊C8C3×C4×D311yescharacteristic
406C3⋊C8C3×D1211yescharacteristic
407C3⋊C8C3×D1211yescharacteristic
408C2×C32—21no—
409C2×C32—22no—
410C2×C32—21no—
411C2×C32—21no—
412C2×C32—21no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32—41no—
416C2×C32—41no—
417C2×C32—41no—
418C2×C32—42no—
419C2×C32—121no—
420C2×C32C4×D1211yescharacteristic
421C2×C32#67⋊C411yescharacteristic
422C2×C32C3×C4×D411yescharacteristic
423C3⋊D3—41no—
424C3×D3—41no—
425C3×D3—41no—
426C3×D3—81no—
427C3×D3—121no—
428C3×D3—121no—
429C3×D3—241no—
430#41—31no—
431#41—91no—
432Q16—91no—
433SD16—91no—
434SD16—91no—
435D8—91no—
436M16—31no—
437M16—91no—
438M16—181no—
439M16—181no—
440C2×C8—31no—
441C2×C8—31no—
442C2×C8—181no—
443C2×C8—181no—
444C42C3×D3211yescharacteristic
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—41no—
449C22×C3—61no—
450C22×C3—61no—
451C22×C3—121no—
452C22×C3—181no—
453C22×C3#380411yescharacteristic
454C22×C3C3×#23511yescharacteristic
455C22×C3C3×#23711yescharacteristic
456C2×D3—21no—
457C2×D3—61no—
458C2×D3—61no—
459C2×D3—121no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—21no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—42no—
467C3×C4—41no—
468C3×C4—41no—
469C3×C4—41no—
470C3×C4—42no—
471C3×C4—42no—
472C3×C4—44no—
473C3×C4—41no—
474C3×C4—41no—
475C3×C4—61no—
476C3×C4—61no—
477C3×C4—121no—
478C3×C4—181no—
479C3×C4C4×D3211yescharacteristic
480C3×C4D3×D1211yescharacteristic
481C3×C4C2×C3×C4×D311yescharacteristic
482C3×C4C2×C3×C4×D311yescharacteristic
483C3×C4C2×C3×D1211yescharacteristic
484C3×C4C3×D3×D411yescharacteristic
485Dic3—21no—
486Dic3—61no—
487Dic3—61no—
488Dic3—121no—
489C32—21no—
490C32—21no—
491C32—21no—
492C32—41no—
493C32#471711yescharacteristic, complemented
494C32#482311yescharacteristic, complemented
495C32C3×#44411yescharacteristic, complemented
496Q8—31no—
497Q8—91no—
498D4—31no—
499D4—61no—
500D4—91no—
501D4—181no—
502C2×C4—21no—
503C2×C4—61no—
504C2×C4—181no—
505C2×C4C2×C3×D3211yescharacteristic
506C8—31no—
507C8—31no—
508C8—31no—
509C8—31no—
510C8—61no—
511C8—181no—
512C2×C3—22no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—41no—
520C2×C3—41no—
521C2×C3—42no—
522C2×C3—41no—
523C2×C3—121no—
524C2×C3—121no—
525C2×C3—241no—
526C2×C3—361no—
527C2×C3#6388011yescharacteristic
528C2×C3C3×C4×D1211yescharacteristic
529C2×C3C3×#67⋊C411yescharacteristic
530D3—41no—
531D3—121no—
532D3—121no—
533D3—241no—
534C22—61no—
535C22—181no—
536C22C3×#380411yescharacteristic
537C4—41no—
538C4—61no—
539C4—181no—
540C4C3×C4×D3211yescharacteristic
541C4C3×D3×D1211yescharacteristic
542C3—21no—
543C3—21no—
544C3—21no—
545C3—41no—
546C3#1059034211yescharacteristic, direct product factor
547C3C3×#471711yescharacteristic, complemented
548C3C3×#482311yescharacteristic, complemented
549C2—21no—
550C2—121no—
551C2—361no—
552C2C3×#6388011yescharacteristic
subgroupG/H
centerC3×C4D3×D12
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC2×C33C4×D4
prime radicalC3×#63829C2
cosocleC2×C32×C4C23×C3
hypercenterC3×C42D32
hypocenterC32C3×#444
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
2. C3×#63817C2semidirect
3. C3×#63814C2semidirect
4. C3×#63632C2semidirect
5. C32×#805C2semidirect
6. C32×#803C2semidirect
7. C3×#63577C2semidirect
10. #10590342C3direct (N × H)
18. C3×#3719C4semidirect
19. C3×#3717C4semidirect
20. C3×#3716C4semidirect
21. C3×#3714C4semidirect
33. #63817C2×C3semidirect
34. #63814C2×C3semidirect
37. #63632C2×C3semidirect
44. C3×#805D3semidirect
45. C3×#805C2×C3semidirect
47. C3×#803D3semidirect
48. C3×#803C2×C3semidirect
49. #63577C2×C3semidirect
116. #3719C3×C4semidirect
117. #3717C3×C4semidirect
118. #3716C3×C4semidirect
119. #3714C3×C4semidirect
154. #805C3×D3semidirect
155. #803C3×D3semidirect
337. C33#444semidirect
493. C32#4717semidirect
494. C32#4823semidirect
495. C32C3×#444semidirect
546. C3#10590342direct (N × H)
547. C3C3×#4717semidirect
548. C3C3×#4823semidirect

Maximal subgroups

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

subgroupindex
C3×#638292
C3×#638172
C3×#638142
C3×#636322
C32×#8052
C32×#8032
C3×#635772
C3×#48233
C3×#47173
#105903423