Finite Group Databasebeta

← C3×#10590899 #13157866

Subgroups — C3×#10590899 #13157866

Order 1728.

686 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3744C211yesmaximal, characteristic
2C2×C32×Dic12C211yesmaximal, characteristic, complemented
3C3×#63866C211yesmaximal, characteristic, complemented
4C3×#63864C211yesmaximal, characteristic
5C2×C3×#3719C211yesmaximal, characteristic, complemented
6C2×C3×#3717C211yesmaximal, characteristic
7C32×#815C211yesmaximal, characteristic, complemented
8C3×#5190—31nomaximal
9C3×#4715—31nomaximal
10#10590899C311yesmaximal, characteristic, direct product factor
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×C33×C8C2211yescharacteristic
16C3×#3744—22no—
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19C32×Dic12—22no—
20C3×Dic3⋊Dic3—22no—
21C3×#3722—22no—
22C3×#3719—22no—
23C3×#3717—22no—
24C2×C32×Q16—31no—
25C2×#3744C2×C311yescharacteristic
26C3×#947—31no—
27C3×#944—31no—
28C2×C3×#216—31no—
29C2×C3×#215—31no—
30C2×C3×Dic12—21no—
31C2×C3×Dic12—31no—
32C2×C3×Dic12D311yescharacteristic, complemented
33C2×C3×Dic12C2×C311yescharacteristic, complemented
34C2×C3×#204—31no—
35C2×C3×#204—31no—
36C2×C3×#204—31no—
37C2×C3×#204—61no—
38C3×#875—31no—
39C3×#869—31no—
40#63866C2×C311yescharacteristic, complemented
41#63864C2×C311yescharacteristic
42C2×#3719C2×C311yescharacteristic, complemented
43C2×#3717C2×C311yescharacteristic
44C32×#97—31no—
45C3×#815—21no—
46C3×#815D311yescharacteristic, complemented
47C3×#815C2×C311yescharacteristic, complemented
48C22×C3×C3⋊D3—21no—
49C2×C33×C4C2311yescharacteristic
50C2×C3×C3⋊Dic3—21no—
51C3×C4⋊(C3⋊D3)—21no—
52C3×C4⋊(C3⋊D3)—21no—
53C3×#634—21no—
54C3×#634—21no—
55C2×C32×Dic3—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3D412yes—
58C32×Dic6—21no—
59C32×Dic6—21no—
60C32×Dic6—21no—
61C32×Dic6—21no—
62C33×C8D412yes—
63C32×C3⋊C8D412yes—
64C3×#497—91no—
65#5190—31no—
66#4715—31no—
67C2×C32×Q8—31no—
68C2×C32×Q8—31no—
69C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
70C2×#634C22×C311yescharacteristic
71C2×C3×D12—31no—
72C2×C3×D12—31no—
73C2×C3×D12—61no—
74C2×C3×Dic6—21no—
75C2×C3×Dic6—21no—
76C2×C3×Dic6—31no—
77C2×C3×Dic6—31no—
78C2×C3×Dic6—61no—
79C2×C3×Dic6C2×D311yescharacteristic
80C2×C3×Dic6C2×D311yescharacteristic
81C2×C3×Dic6C22×C311yescharacteristic
82C2×C3×Dic6C22×C311yescharacteristic
83C32×Q16—62no—
84C2×C32×C8—21no—
85C2×C32×C8—21no—
86C2×C32×C8—21no—
87C2×C32×C8—31no—
88C2×C32×C8—41no—
89C2×C32×C8C2×D311yescharacteristic
90C2×C32×C8C2×D311yescharacteristic
91C2×C32×C8C22×C311yescharacteristic
92C32×C42—31no—
93#3744—22no—
94C3×#216—62no—
95C3×#215—62no—
96C3×(C2×D3)⋊C4—62no—
97C3×(C2×D3)⋊C4—62no—
98C3×C4⋊Dic3—62no—
99C3×Dic3⋊C4—62no—
100C3×C4×Dic3—21no—
101C3×C4×Dic3C2×D311yescharacteristic
102C3×C4×Dic3C22×C311yescharacteristic
103C2×C3×C3⋊C8—21no—
104C2×C3×C3⋊C8C2×D311yescharacteristic
105C2×C3×C3⋊C8C22×C311yescharacteristic
106C3×Dic12—24no—
107C3×Dic12—22no—
108C3×Dic12—22no—
109C3×Dic12—62no—
110C3×#204—64no—
111C3×#204—62no—
112C3×#204—62no—
113C3×#204—62no—
114Dic3⋊Dic3—22no—
115#3722—22no—
116#3719—22no—
117#3717—22no—
118C22×C33C2×D411yescharacteristic
119C2×C3×C3⋊D3—41no—
120C2×C3×C3⋊D3—42no—
121C33×C4C2×D411yescharacteristic
122C33×C4C2×D411yescharacteristic
123C3×C3⋊Dic3—41no—
124C32×Dic3—22no—
125C32×Dic3—41no—
126C32×Dic3—41no—
127C2×C3×Q16—31no—
128C2×C3×Q16—31no—
129C2×C3×Q16—61no—
130C2×C3×Q16—91no—
131C2×C3×SD16—91no—
132C2×C3×SD16—91no—
133C3×C4⋊Q8—91no—
134C3×#124—91no—
135#947—31no—
136#944—31no—
137C2×#216—31no—
138C2×#215—31no—
139C2×Dic12—31no—
140C2×Dic12C3×D311yescharacteristic, complemented
141C2×#204—31no—
142C2×#204—31no—
143C2×#204—31no—
144C2×#204—61no—
145#875—31no—
146#869—31no—
147C3×#97—31no—
148C3×#97—31no—
149C3×#97—61no—
150#815C3×D311yescharacteristic, complemented
151C22×C3⋊D3—21no—
152C22×C3×D3—61no—
153C22×C3×D3—61no—
154C22×C3×D3—121no—
155C32×Q8—61no—
156C32×Q8—61no—
157C32×Q8—61no—
158C32×Q8—61no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—32no—
163C2×C32×C4—41no—
164C2×C32×C4—61no—
165C2×C32×C4—61no—
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C23×C311yescharacteristic
169C2×C3⋊Dic3—21no—
170C4⋊(C3⋊D3)—21no—
171C4⋊(C3⋊D3)—21no—
172#634—21no—
173#634—21no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—22no—
179C2×C3×Dic3—41no—
180C2×C3×Dic3—41no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—61no—
183C2×C3×Dic3—121no—
184C2×C3×Dic3D1212yes—
185C2×C3×Dic3C3×D412yes—
186C3×D12—62no—
187C3×D12—61no—
188C3×D12—61no—
189C3×D12—61no—
190C3×D12—61no—
191C3×D12—121no—
192C3×Dic6—22no—
193C3×Dic6—22no—
194C3×Dic6—21no—
195C3×Dic6—21no—
196C3×Dic6—21no—
197C3×Dic6—21no—
198C3×Dic6—21no—
199C3×Dic6—21no—
200C3×Dic6—21no—
201C3×Dic6—21no—
202C3×Dic6—41no—
203C3×Dic6—41no—
204C3×Dic6—62no—
205C3×Dic6—61no—
206C3×Dic6—61no—
207C3×Dic6—61no—
208C3×Dic6—61no—
209C3×Dic6—121no—
210C32×C8—22no—
211C32×C8—22no—
212C32×C8—22no—
213C32×C8—32no—
214C32×C8—42no—
215C32×C8D1212yes—
216C32×C8#6712yes—
217C32×C8C3×D412yes—
218C3×C3⋊C8—22no—
219C3×C3⋊C8D1212yes—
220C3×C3⋊C8C3×D412yes—
221#497—91noSylow
222C2×C33#12711yescharacteristic
223C2×C33#13712yes—
224C3×C3⋊D3—81no—
225C2×C3×Q8—31no—
226C2×C3×Q8—31no—
227C2×C3×Q8—31no—
228C2×C3×Q8—31no—
229C2×C3×Q8—61no—
230C2×C3×Q8—61no—
231C2×C3×Q8—91no—
232C2×C3×D4—91no—
233C2×D12—31no—
234C2×D12—31no—
235C2×D12—61no—
236C2×Dic6—31no—
237C2×Dic6—31no—
238C2×Dic6—61no—
239C2×Dic6C2×C3×D311yescharacteristic
240C2×Dic6C2×C3×D311yescharacteristic
241C3×Q16—64no—
242C3×Q16—62no—
243C3×Q16—62no—
244C3×Q16—182no—
245C3×SD16—182no—
246C3×SD16—182no—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8—31no—
251C2×C3×C8—31no—
252C2×C3×C8—41no—
253C2×C3×C8—61no—
254C2×C3×C8D3211yescharacteristic
255C2×C3×C8C2×C3×D311yescharacteristic
256C2×C3×C8C2×C3×D311yescharacteristic
257C3×C4⋊C4—182no—
258C3×C22⋊C4—182no—
259C3×C42—31no—
260C3×C42—31no—
261C3×C42—61no—
262#216—62no—
263#215—62no—
264(C2×D3)⋊C4—62no—
265(C2×D3)⋊C4—62no—
266C4⋊Dic3—62no—
267Dic3⋊C4—62no—
268C4×Dic3C2×C3×D311yescharacteristic
269C2×C3⋊C8C2×C3×D311yescharacteristic
270Dic12—22no—
271Dic12—62no—
272#204—64no—
273#204—62no—
274#204—62no—
275#204—62no—
276C22×C32—21no—
277C22×C32—21no—
278C22×C32—21no—
279C22×C32—41no—
280C22×C32C2×D1211yescharacteristic
281C22×C32D3×D411yescharacteristic
282C22×C32C2×C3×D411yescharacteristic
283C2×C3⋊D3—41no—
284C2×C3⋊D3—42no—
285C2×C3×D3—122no—
286C2×C3×D3—121no—
287C2×C3×D3—122no—
288C2×C3×D3—122no—
289C2×C3×D3—121no—
290C2×C3×D3—242no—
291C32×C4—21no—
292C32×C4—21no—
293C32×C4—21no—
294C32×C4—21no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4—41no—
298C32×C4—41no—
299C32×C4—62no—
300C32×C4—121no—
301C32×C4—121no—
302C32×C4C2×D1211yescharacteristic
303C32×C4C2×D1211yescharacteristic
304C32×C4D3×D411yescharacteristic
305C32×C4C2×#6711yescharacteristic
306C32×C4C2×C3×D411yescharacteristic
307C32×C4C2×C3×D411yescharacteristic
308C3⋊Dic3—41no—
309C3×Dic3—22no—
310C3×Dic3—22no—
311C3×Dic3—42no—
312C3×Dic3—42no—
313C3×Dic3—42no—
314C3×Dic3—41no—
315C3×Dic3—41no—
316C3×Dic3—41no—
317C3×Dic3—41no—
318C3×Dic3—122no—
319C3×Dic3—121no—
320C3×Dic3—121no—
321C2×Q16—31no—
322C2×Q16—91no—
323C2×SD16—91no—
324C2×SD16—91no—
325C4⋊Q8—91no—
326#124—91no—
327#97—31no—
328C33#49711yescharacteristic, complemented, Sylow
329C23×C3—181no—
330C22×D3—61no—
331C22×D3—61no—
332C22×D3—121no—
333C3×Q8—62no—
334C3×Q8—62no—
335C3×Q8—61no—
336C3×Q8—61no—
337C3×Q8—61no—
338C3×Q8—61no—
339C3×Q8—61no—
340C3×Q8—61no—
341C3×Q8—61no—
342C3×Q8—61no—
343C3×Q8—121no—
344C3×Q8—121no—
345C3×Q8—181no—
346C3×Q8—181no—
347C3×D4—181no—
348C3×D4—181no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—32no—
353C2×C3×C4—32no—
354C2×C3×C4—41no—
355C2×C3×C4—62no—
356C2×C3×C4—61no—
357C2×C3×C4—61no—
358C2×C3×C4—61no—
359C2×C3×C4—61no—
360C2×C3×C4—121no—
361C2×C3×C4—121no—
362C2×C3×C4—181no—
363C2×C3×C4C2×D3211yescharacteristic
364C2×C3×C4C22×C3×D311yescharacteristic
365C2×C3×C4C22×C3×D311yescharacteristic
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—61no—
369C2×Dic3—61no—
370C2×Dic3—121no—
371C2×Dic3C3×D1212yes—
372D12—62no—
373D12—61no—
374D12—61no—
375D12—61no—
376D12—61no—
377D12—121no—
378Dic6—21no—
379Dic6—21no—
380Dic6—21no—
381Dic6—21no—
382Dic6—62no—
383Dic6—61no—
384Dic6—61no—
385Dic6—61no—
386Dic6—61no—
387Dic6—121no—
388C3×C8—22no—
389C3×C8—22no—
390C3×C8—22no—
391C3×C8—32no—
392C3×C8—32no—
393C3×C8—42no—
394C3×C8—62no—
395C3×C8#62612yes—
396C3×C8C3×D1212yes—
397C3×C8C3×#6712yes—
398C3⋊C8C3×D1212yes—
399C2×C32—22no—
400C2×C32—22no—
401C2×C32—22no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32—21no—
405C2×C32—41no—
406C2×C32—42no—
407C2×C32#87411yescharacteristic
408C2×C32#90912yes—
409C2×C32#91812yes—
410C2×C32#94011yescharacteristic
411C2×C32C3×#12711yescharacteristic
412C2×C32C3×#13712yes—
413C3⋊D3—81no—
414C3×D3—242no—
415C3×D3—241no—
416C3×D3—241no—
417C2×Q8—31no—
418C2×Q8—31no—
419C2×Q8—91no—
420C2×D4—91no—
421Q16—62no—
422Q16—182no—
423SD16—182no—
424SD16—182no—
425C2×C8—31no—
426C2×C8C3×D3211yescharacteristic
427C4⋊C4—182no—
428C22⋊C4—182no—
429C42—31no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—41no—
434C22×C3—362no—
435C22×C3—361no—
436C22×C3D3×D1211yescharacteristic
437C22×C3C2×C3×D1211yescharacteristic
438C22×C3C3×D3×D411yescharacteristic
439C2×D3—122no—
440C2×D3—122no—
441C2×D3—122no—
442C2×D3—121no—
443C2×D3—121no—
444C2×D3—242no—
445C3×C4—21no—
446C3×C4—21no—
447C3×C4—21no—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—41no—
452C3×C4—41no—
453C3×C4—62no—
454C3×C4—62no—
455C3×C4—122no—
456C3×C4—122no—
457C3×C4—122no—
458C3×C4—121no—
459C3×C4—121no—
460C3×C4—121no—
461C3×C4—121no—
462C3×C4—361no—
463C3×C4D3×D1211yescharacteristic
464C3×C4C2×#62611yescharacteristic
465C3×C4C2×C3×D1211yescharacteristic
466C3×C4C2×C3×D1211yescharacteristic
467C3×C4C3×D3×D411yescharacteristic
468C3×C4C2×C3×#6711yescharacteristic
469Dic3—22no—
470Dic3—41no—
471Dic3—41no—
472Dic3—122no—
473Dic3—121no—
474Dic3—121no—
475C32—21no—
476C32—21no—
477C32—21no—
478C32—41no—
479C32#471511yescharacteristic, complemented
480C32#519011yescharacteristic, complemented
481C32C3×#49711yescharacteristic, complemented
482C23—181no—
483Q8—61no—
484Q8—61no—
485Q8—61no—
486Q8—61no—
487Q8—181no—
488Q8—181no—
489D4—181no—
490D4—181no—
491C2×C4—32no—
492C2×C4—61no—
493C2×C4—61no—
494C2×C4—181no—
495C2×C4C2×C3×D3211yescharacteristic
496C8—32no—
497C8C3×#62612yes—
498C2×C3—22no—
499C2×C3—22no—
500C2×C3—22no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—42no—
505C2×C3—41no—
506C2×C3—721no—
507C2×C3#6380112yes—
508C2×C3#6391111yescharacteristic
509C2×C3C3×#87411yescharacteristic
510C2×C3C3×#90912yes—
511C2×C3C3×#91812yes—
512C2×C3C3×#94011yescharacteristic
513D3—242no—
514D3—241no—
515D3—241no—
516C22—362no—
517C22—361no—
518C22C3×D3×D1211yescharacteristic
519C4—62no—
520C4—121no—
521C4—121no—
522C4—361no—
523C4C3×D3×D1211yescharacteristic
524C4C2×C3×#62611yescharacteristic
525C3—21no—
526C3—21no—
527C3—21no—
528C3—41no—
529C3#1059089911yescharacteristic, direct product factor
530C3C3×#471511yescharacteristic, complemented
531C3C3×#519011yescharacteristic, complemented
532C2—721no—
533C2C3×#6380112yes—
534C2C3×#6391111yescharacteristic
subgroupG/H
centerC22×C3D3×D12
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC2×C3×C8D32
hypocenterC32C3×#497
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
2. C2×C32×Dic12C2semidirect
3. C3×#63866C2semidirect
5. C2×C3×#3719C2semidirect
7. C32×#815C2semidirect
10. #10590899C3direct (N × H)
32. C2×C3×Dic12D3semidirect
33. C2×C3×Dic12C2×C3semidirect
40. #63866C2×C3semidirect
42. C2×#3719C2×C3semidirect
46. C3×#815D3semidirect
47. C3×#815C2×C3semidirect
140. C2×Dic12C3×D3semidirect
150. #815C3×D3semidirect
328. C33#497semidirect
479. C32#4715semidirect
480. C32#5190semidirect
481. C32C3×#497semidirect
529. C3#10590899direct (N × H)
530. C3C3×#4715semidirect
531. C3C3×#5190semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37442
C2×C32×Dic122
C3×#638662
C3×#638642
C2×C3×#37192
C2×C3×#37172
C32×#8152
C3×#51903
C3×#47153
#105908993