Finite Group Databasebeta

← C3×#10591580 #13158656

Subgroups — C3×#10591580 #13158656

Order 1728.

640 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64153C211yesmaximal, characteristic
2C2×C32×#216C211yesmaximal, characteristic
3C3×#63843C211yesmaximal, characteristic
4C2×C3×#3719C211yesmaximal, characteristic
5C32×#835C211yesmaximal, characteristic
6C3×#63572C211yesmaximal, characteristic
7C3×#63567C211yesmaximal, characteristic
8C3×#5182—31nomaximal
9C3×#4795—31nomaximal
10#10591580C311yesmaximal, characteristic, direct product factor
11C2×C33×Q8C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C3×#3750—21no—
15C3×#3750—21no—
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C32×#216—22no—
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×Dic3⋊Dic3—21no—
22C3×Dic32—21no—
23C3×#3719—22no—
24C2×C32×Q16—31no—
25#64153C2×C311yescharacteristic
26C3×Q8×Dic3—31no—
27C3×#944—31no—
28C3×#944—31no—
29C3×#944—61no—
30C2×C3×#216—21no—
31C2×C3×#216—31no—
32C2×C3×#216D311yescharacteristic
33C2×C3×#216C2×C311yescharacteristic
34C2×C3×Dic12—31no—
35C3×#887—31no—
36#63843C2×C311yescharacteristic
37C2×#3719C2×C311yescharacteristic
38C32×#103—31no—
39C3×#835—21no—
40C3×#835D311yescharacteristic
41C3×#835C2×C311yescharacteristic
42C3×#816—31no—
43C3×Dic3⋊C8—31no—
44C3×Dic3⋊C8—31no—
45C3×#810—31no—
46#63572C2×C311yescharacteristic
47#63567C2×C311yescharacteristic
48C33×Q8—21no—
49C33×Q8—21no—
50C2×C33×C4—21no—
51C2×C33×C4C2311yescharacteristic
52C2×C3×C3⋊Dic3—21no—
53C2×C3×C3⋊Dic3D411yescharacteristic
54C2×C3×C3⋊Dic3D411yescharacteristic
55C3×#634—21no—
56C3×#634—21no—
57C2×C32×Dic3—21no—
58C2×C32×Dic3—21no—
59C32×Dic6—21no—
60C32×Dic6D412yes—
61C32×C3⋊C8—21no—
62C32×C3⋊C8—21no—
63C3×#462—91no—
64#5182—31no—
65#4795—31no—
66C2×C32×Q8—21no—
67C2×C32×Q8—21no—
68C2×C32×Q8—21no—
69C2×C32×Q8—31no—
70C2×C32×Q8—41no—
71C2×C32×Q8C2×D311yescharacteristic
72C2×C32×Q8C2×D311yescharacteristic
73C2×C32×Q8C22×C311yescharacteristic
74C2×#634C22×C311yescharacteristic
75C2×C3×Dic6—21no—
76C2×C3×Dic6—31no—
77C2×C3×Dic6—31no—
78C2×C3×Dic6—61no—
79C2×C3×Dic6C2×D311yescharacteristic
80C2×C3×Dic6C22×C311yescharacteristic
81C32×Q16—62no—
82C2×C32×C8—31no—
83C2×C32×C8—31no—
84C32×C4⋊C4—31no—
85#3750—21no—
86#3750—21no—
87C4×C3⋊Dic3C22×C311yescharacteristic
88C3×#216—24no—
89C3×#216—22no—
90C3×#216—22no—
91C3×#216—62no—
92C3×C4⋊Dic3—21no—
93C3×C4⋊Dic3—61no—
94C3×C4⋊Dic3C2×D311yescharacteristic
95C3×C4⋊Dic3C22×C311yescharacteristic
96C3×Dic3⋊C4—61no—
97C3×Dic3⋊C4—61no—
98C3×Dic3⋊C4—61no—
99C3×Dic3⋊C4—61no—
100C3×Dic3⋊C4—61no—
101C3×Dic3⋊C4—62no—
102C3×Dic3⋊C4—62no—
103C3×C4×Dic3—31no—
104C3×C4×Dic3—31no—
105C3×C4×Dic3—61no—
106C3×C4×Dic3—61no—
107C3×C4×Dic3—61no—
108C2×C3×C3⋊C8—21no—
109C2×C3×C3⋊C8—21no—
110C2×C3×C3⋊C8C2×D311yescharacteristic
111C2×C3×C3⋊C8C2×D311yescharacteristic
112C2×C3×C3⋊C8C22×C311yescharacteristic
113C2×C3×C3⋊C8C22×C311yescharacteristic
114C3×Dic12—62no—
115Dic3⋊Dic3—21no—
116Dic32—21no—
117#3719—22no—
118C22×C33C2×D411yescharacteristic
119C33×C4—41no—
120C33×C4C2×D411yescharacteristic
121C33×C4#4111yescharacteristic
122C3×C3⋊Dic3—21no—
123C3×C3⋊Dic3—41no—
124C3×C3⋊Dic3Q1612yescomplemented
125C32×Dic3—22no—
126C32×Dic3—41no—
127C2×C3×Q16—31no—
128C2×C3×Q16—31no—
129C2×C3×Q16—61no—
130C2×C3×Q16—91no—
131C3×C4⋊Q8—91no—
132C3×C4×Q8—91no—
133Q8×Dic3—31no—
134#944—31no—
135#944—31no—
136#944—61no—
137C2×#216—31no—
138C2×#216C3×D311yescharacteristic
139C2×Dic12—31no—
140#887—31no—
141C3×C4⋊C8—91no—
142C3×#103—31no—
143C3×#103—31no—
144C3×#103—61no—
145C3×#103—91no—
146#835C3×D311yescharacteristic
147#816—31no—
148Dic3⋊C8—31no—
149Dic3⋊C8—31no—
150#810—31no—
151C32×Q8—21no—
152C32×Q8—21no—
153C32×Q8—22no—
154C32×Q8—22no—
155C32×Q8—21no—
156C32×Q8—21no—
157C32×Q8—21no—
158C32×Q8—21no—
159C32×Q8—32no—
160C32×Q8—41no—
161C32×Q8—41no—
162C32×Q8—41no—
163C32×Q8—41no—
164C32×Q8—42no—
165C32×Q8—42no—
166C32×Q8—61no—
167C2×C32×C4—21no—
168C2×C32×C4—22no—
169C2×C32×C4—21no—
170C2×C32×C4—21no—
171C2×C32×C4—21no—
172C2×C32×C4—21no—
173C2×C32×C4—21no—
174C2×C32×C4—41no—
175C2×C32×C4—41no—
176C2×C32×C4—41no—
177C2×C32×C4—42no—
178C2×C32×C4—61no—
179C2×C32×C4—61no—
180C2×C32×C4C22×D311yescharacteristic
181C2×C32×C4C22×D311yescharacteristic
182C2×C32×C4C23×C311yescharacteristic
183C2×C3⋊Dic3—21no—
184C2×C3⋊Dic3C3×D411yescharacteristic
185C2×C3⋊Dic3C3×D411yescharacteristic
186#634—21no—
187#634—21no—
188C2×C3×Dic3—21no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—21no—
192C2×C3×Dic3—31no—
193C2×C3×Dic3—31no—
194C2×C3×Dic3—31no—
195C2×C3×Dic3—31no—
196C2×C3×Dic3—41no—
197C2×C3×Dic3—41no—
198C2×C3×Dic3—61no—
199C2×C3×Dic3—61no—
200C2×C3×Dic3—62no—
201C2×C3×Dic3—61no—
202C2×C3×Dic3—61no—
203C3×Dic6—21no—
204C3×Dic6—21no—
205C3×Dic6—22no—
206C3×Dic6—41no—
207C3×Dic6—61no—
208C3×Dic6—61no—
209C3×Dic6—61no—
210C3×Dic6—61no—
211C3×Dic6—121no—
212C3×Dic6—121no—
213C3×Dic6#6712yes—
214C3×Dic6C3×D412yes—
215C32×C8—61no—
216C32×C8—61no—
217C3×C3⋊C8—22no—
218C3×C3⋊C8—21no—
219C3×C3⋊C8—21no—
220C3×C3⋊C8—21no—
221C3×C3⋊C8—21no—
222C3×C3⋊C8—41no—
223#462—91noSylow
224C2×C33#12111yescharacteristic
225C2×C33C2×Q1611yescharacteristic
226C2×C33#13711yescharacteristic
227C2×C3×Q8—21no—
228C2×C3×Q8—21no—
229C2×C3×Q8—21no—
230C2×C3×Q8—31no—
231C2×C3×Q8—31no—
232C2×C3×Q8—41no—
233C2×C3×Q8—61no—
234C2×C3×Q8—91no—
235C2×C3×Q8D3211yescharacteristic
236C2×C3×Q8C2×C3×D311yescharacteristic
237C2×C3×Q8C2×C3×D311yescharacteristic
238C2×Dic6—31no—
239C2×Dic6—31no—
240C2×Dic6—61no—
241C2×Dic6C2×C3×D311yescharacteristic
242C3×Q16—62no—
243C3×Q16—62no—
244C3×Q16—64no—
245C3×Q16—182no—
246C2×C3×C8—31no—
247C2×C3×C8—31no—
248C2×C3×C8—31no—
249C2×C3×C8—31no—
250C2×C3×C8—61no—
251C2×C3×C8—61no—
252C3×C4⋊C4—31no—
253C3×C4⋊C4—31no—
254C3×C4⋊C4—61no—
255C3×C4⋊C4—181no—
256C3×C4⋊C4—181no—
257C3×C4⋊C4—181no—
258C3×C42—91no—
259C3×C42—181no—
260#216—22no—
261#216—62no—
262C4⋊Dic3—61no—
263C4⋊Dic3C2×C3×D311yescharacteristic
264Dic3⋊C4—61no—
265Dic3⋊C4—61no—
266Dic3⋊C4—61no—
267Dic3⋊C4—61no—
268Dic3⋊C4—61no—
269Dic3⋊C4—62no—
270Dic3⋊C4—62no—
271C4×Dic3—31no—
272C4×Dic3—31no—
273C4×Dic3—61no—
274C4×Dic3—61no—
275C4×Dic3—61no—
276C2×C3⋊C8C2×C3×D311yescharacteristic
277C2×C3⋊C8C2×C3×D311yescharacteristic
278Dic12—62no—
279C22×C32—21no—
280C22×C32—21no—
281C22×C32—21no—
282C22×C32—41no—
283C22×C32D3×D411yescharacteristic
284C22×C32D3×D411yescharacteristic
285C22×C32C2×C3×D411yescharacteristic
286C32×C4—21no—
287C32×C4—21no—
288C32×C4—21no—
289C32×C4—21no—
290C32×C4—21no—
291C32×C4—21no—
292C32×C4—41no—
293C32×C4—41no—
294C32×C4—42no—
295C32×C4—42no—
296C32×C4—42no—
297C32×C4—44no—
298C32×C4—41no—
299C32×C4—41no—
300C32×C4—41no—
301C32×C4—62no—
302C32×C4—121no—
303C32×C4#23511yescharacteristic
304C32×C4D3×D411yescharacteristic
305C32×C4#23711yescharacteristic
306C32×C4C2×#6711yescharacteristic
307C32×C4C2×C3×D411yescharacteristic
308C32×C4C3×#4111yescharacteristic
309C3⋊Dic3—21no—
310C3⋊Dic3—41no—
311C3⋊Dic3C3×Q1612yescomplemented
312C3×Dic3—22no—
313C3×Dic3—22no—
314C3×Dic3—32no—
315C3×Dic3—32no—
316C3×Dic3—42no—
317C3×Dic3—41no—
318C3×Dic3—41no—
319C3×Dic3—62no—
320C3×Dic3—62no—
321C3×Dic3—61no—
322C3×Dic3—61no—
323C3×Dic3—81no—
324C3×Dic3—122no—
325C3×Dic3—121no—
326C3×Dic3—121no—
327C2×Q16—31no—
328C2×Q16—91no—
329C4⋊Q8—91no—
330C4×Q8—91no—
331C4⋊C8—91no—
332#103—31no—
333#103—91no—
334C33#46211yescharacteristic, complemented, Sylow
335C3×Q8—22no—
336C3×Q8—22no—
337C3×Q8—21no—
338C3×Q8—21no—
339C3×Q8—21no—
340C3×Q8—21no—
341C3×Q8—21no—
342C3×Q8—21no—
343C3×Q8—32no—
344C3×Q8—32no—
345C3×Q8—41no—
346C3×Q8—41no—
347C3×Q8—41no—
348C3×Q8—41no—
349C3×Q8—42no—
350C3×Q8—42no—
351C3×Q8—62no—
352C3×Q8—61no—
353C3×Q8—61no—
354C3×Q8—121no—
355C3×Q8—181no—
356C3×Q8—181no—
357C2×C3×C4—22no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—42no—
367C2×C3×C4—41no—
368C2×C3×C4—61no—
369C2×C3×C4—61no—
370C2×C3×C4—61no—
371C2×C3×C4—61no—
372C2×C3×C4—91no—
373C2×C3×C4—91no—
374C2×C3×C4—121no—
375C2×C3×C4—121no—
376C2×C3×C4—181no—
377C2×C3×C4C2×D3211yescharacteristic
378C2×C3×C4C22×C3×D311yescharacteristic
379C2×C3×C4C22×C3×D311yescharacteristic
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—31no—
383C2×Dic3—31no—
384C2×Dic3—31no—
385C2×Dic3—31no—
386C2×Dic3—61no—
387C2×Dic3—61no—
388C2×Dic3—62no—
389C2×Dic3—61no—
390C2×Dic3—61no—
391Dic6—21no—
392Dic6—61no—
393Dic6—61no—
394Dic6—61no—
395Dic6—61no—
396Dic6—121no—
397Dic6—121no—
398Dic6C3×#6712yes—
399C3×C8—62no—
400C3×C8—61no—
401C3×C8—61no—
402C3×C8—61no—
403C3×C8—61no—
404C3×C8—121no—
405C3⋊C8—21no—
406C3⋊C8—21no—
407C2×C32—21no—
408C2×C32—21no—
409C2×C32—21no—
410C2×C32—21no—
411C2×C32—21no—
412C2×C32—21no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32—21no—
416C2×C32—41no—
417C2×C32—41no—
418C2×C32—41no—
419C2×C32#88411yescharacteristic
420C2×C32#91511yescharacteristic
421C2×C32D3×Q1611yescharacteristic
422C2×C32D3×Q1611yescharacteristic
423C2×C32#91811yescharacteristic
424C2×C32#93911yescharacteristic
425C2×C32C3×#12111yescharacteristic
426C2×C32C2×C3×Q1611yescharacteristic
427C2×C32C3×#13711yescharacteristic
428C2×Q8—31no—
429C2×Q8—91no—
430C2×Q8C3×D3211yescharacteristic
431Q16—62no—
432Q16—182no—
433C2×C8—31no—
434C2×C8—31no—
435C4⋊C4—31no—
436C4⋊C4—181no—
437C4⋊C4—181no—
438C4⋊C4—181no—
439C42—91no—
440C42—181no—
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—41no—
445C22×C3#381111yescharacteristic
446C22×C3C3×D3×D411yescharacteristic
447C22×C3C3×D3×D411yescharacteristic
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—21no—
453C3×C4—21no—
454C3×C4—42no—
455C3×C4—42no—
456C3×C4—42no—
457C3×C4—44no—
458C3×C4—41no—
459C3×C4—41no—
460C3×C4—41no—
461C3×C4—41no—
462C3×C4—41no—
463C3×C4—62no—
464C3×C4—62no—
465C3×C4—92no—
466C3×C4—122no—
467C3×C4—121no—
468C3×C4—121no—
469C3×C4—181no—
470C3×C4—241no—
471C3×C4—361no—
472C3×C4#380411yescharacteristic
473C3×C4D3×#6711yescharacteristic
474C3×C4C3×#23511yescharacteristic
475C3×C4C3×D3×D411yescharacteristic
476C3×C4C3×#23711yescharacteristic
477C3×C4C2×C3×#6711yescharacteristic
478Dic3—22no—
479Dic3—32no—
480Dic3—32no—
481Dic3—41no—
482Dic3—62no—
483Dic3—62no—
484Dic3—61no—
485Dic3—61no—
486Dic3—122no—
487Dic3—121no—
488Dic3—121no—
489C32—21no—
490C32—21no—
491C32—21no—
492C32—41no—
493C32#479511yescharacteristic, complemented
494C32#518211yescharacteristic, complemented
495C32C3×#46211yescharacteristic, complemented
496Q8—21no—
497Q8—21no—
498Q8—32no—
499Q8—61no—
500Q8—181no—
501Q8—181no—
502C2×C4—21no—
503C2×C4—61no—
504C2×C4—61no—
505C2×C4—91no—
506C2×C4—91no—
507C2×C4—181no—
508C2×C4C2×C3×D3211yescharacteristic
509C8—61no—
510C8—61no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—41no—
521C2×C3—41no—
522C2×C3—41no—
523C2×C3#6394411yescharacteristic
524C2×C3#6395111yescharacteristic
525C2×C3#6397111yescharacteristic
526C2×C3C3×#88411yescharacteristic
527C2×C3C3×#91511yescharacteristic
528C2×C3C3×D3×Q1611yescharacteristic
529C2×C3C3×D3×Q1611yescharacteristic
530C2×C3C3×#91811yescharacteristic
531C2×C3C3×#93911yescharacteristic
532C22C3×#381111yescharacteristic
533C4—41no—
534C4—62no—
535C4—92no—
536C4—121no—
537C4—181no—
538C4—361no—
539C4C3×#380411yescharacteristic
540C4C3×D3×#6711yescharacteristic
541C3—21no—
542C3—21no—
543C3—21no—
544C3—41no—
545C3#1059158011yescharacteristic, direct product factor
546C3C3×#479511yescharacteristic, complemented
547C3C3×#518211yescharacteristic, complemented
548C2C3×#6394411yescharacteristic
549C2C3×#6395111yescharacteristic
550C2C3×#6397111yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×Q8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×C3×Q8D32
hypocenterC32C3×#462
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
10. #10591580C3direct (N × H)
124. C3×C3⋊Dic3Q16semidirect
311. C3⋊Dic3C3×Q16semidirect
334. C33#462semidirect
493. C32#4795semidirect
494. C32#5182semidirect
495. C32C3×#462semidirect
545. C3#10591580direct (N × H)
546. C3C3×#4795semidirect
547. C3C3×#5182semidirect

Maximal subgroups

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

subgroupindex
C3×#641532
C2×C32×#2162
C3×#638432
C2×C3×#37192
C32×#8352
C3×#635722
C3×#635672
C3×#51823
C3×#47953
#105915803