Finite Group Databasebeta

← C3×#10591358 #13158309

Subgroups — C3×#10591358 #13158309

Order 1728.

914 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3750C211yesmaximal, characteristic
2C2×C32×C4⋊Dic3C211yesmaximal, characteristic
3C2×C3×Dic3⋊Dic3C212yesmaximal
4C32×C3⋊#95C211yesmaximal, characteristic
5C3×#63579C212yesmaximal
6C3×#4983—31nomaximal
7C3×#4661—31nomaximal
8#10591358C311yesmaximal, characteristic, direct product factor
9C22×C33×C4C2211yescharacteristic
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C3×C3⋊Dic3C2211yescharacteristic
12C22×C32×Dic3C2212yes—
13C22×C32×Dic3C2212yes—
14C3×#3750—22no—
15C32×C4⋊Dic3—22no—
16C3×Dic3⋊Dic3—24no—
17C2×C32×C4⋊C4—31no—
18C2×#3750C2×C311yescharacteristic
19C2×C3×C4⋊Dic3—21no—
20C2×C3×C4⋊Dic3—32no—
21C2×C3×C4⋊Dic3D311yescharacteristic
22C2×C3×C4⋊Dic3C2×C311yescharacteristic
23C2×C3×Dic3⋊C4—31no—
24C2×C3×Dic3⋊C4—31no—
25C2×C3×Dic3⋊C4—32no—
26C2×C3×Dic3⋊C4—61no—
27C2×Dic3⋊Dic3C2×C312yes—
28C32×#95—31no—
29C3×C3⋊#95—21no—
30C3×C3⋊#95—32no—
31C3×C3⋊#95—32no—
32C3×C3⋊#95D311yescharacteristic
33C3×C3⋊#95C2×C311yescharacteristic
34#63579C2×C312yes—
35C23×C33C2311yescharacteristic
36C2×C33×C4—22no—
37C2×C33×C4D412yes—
38C2×C3×C3⋊Dic3—21no—
39C2×C3×C3⋊Dic3—21no—
40C2×C3×C3⋊Dic3—21no—
41C2×C3×C3⋊Dic3—22no—
42C2×C3×C3⋊Dic3D412yes—
43C2×C32×Dic3—22no—
44C2×C32×Dic3—22no—
45C2×C32×Dic3—22no—
46C2×C32×Dic3—22no—
47C2×C32×Dic3—22no—
48C2×C32×Dic3Q814yes—
49C3×#398—91no—
50#4983—31no—
51#4661—31no—
52C22×C32×C4—21no—
53C22×C32×C4—21no—
54C22×C32×C4—21no—
55C22×C32×C4—32no—
56C22×C32×C4—32no—
57C22×C32×C4—41no—
58C22×C32×C4C2×D311yescharacteristic
59C22×C32×C4C2×D311yescharacteristic
60C22×C32×C4C22×C311yescharacteristic
61C22×C3⋊Dic3C22×C311yescharacteristic
62C22×C3⋊Dic3C22×C311yescharacteristic
63C22×C3×Dic3—22no—
64C22×C3×Dic3—22no—
65C22×C3×Dic3—31no—
66C22×C3×Dic3—31no—
67C22×C3×Dic3—31no—
68C22×C3×Dic3—31no—
69C22×C3×Dic3—61no—
70C22×C3×Dic3—61no—
71C22×C3×Dic3C2×D312yes—
72C22×C3×Dic3C2×D312yes—
73C22×C3×Dic3C22×C312yes—
74C22×C3×Dic3C22×C312yes—
75C32×C4⋊C4—62no—
76#3750—22no—
77C3×C4⋊Dic3—24no—
78C3×C4⋊Dic3—22no—
79C3×C4⋊Dic3—22no—
80C3×C4⋊Dic3—64no—
81C3×Dic3⋊C4—64no—
82C3×Dic3⋊C4—64no—
83C3×Dic3⋊C4—62no—
84C3×Dic3⋊C4—62no—
85Dic3⋊Dic3—24no—
86C22×C33C2×D411yescharacteristic
87C22×C33C2×D411yescharacteristic
88C22×C33C2×Q812yes—
89C22×C33#4111yescharacteristic
90C22×C33#4112yes—
91C33×C4—22no—
92C3×C3⋊Dic3—22no—
93C3×C3⋊Dic3—41no—
94C32×Dic3—24no—
95C32×Dic3—42no—
96C2×C3×C4⋊C4—31no—
97C2×C3×C4⋊C4—31no—
98C2×C3×C4⋊C4—61no—
99C2×C3×C4⋊C4—92no—
100C2×C3×C4⋊C4—91no—
101C2×C4⋊Dic3—32no—
102C2×C4⋊Dic3C3×D311yescharacteristic
103C2×Dic3⋊C4—31no—
104C2×Dic3⋊C4—31no—
105C2×Dic3⋊C4—32no—
106C2×Dic3⋊C4—61no—
107C3×#95—31no—
108C3×#95—31no—
109C3×#95—61no—
110C3×#95—92no—
111C3⋊#95—32no—
112C3⋊#95—32no—
113C3⋊#95C3×D311yescharacteristic
114C23×C32—21no—
115C23×C32—21no—
116C23×C32—21no—
117C23×C32—41no—
118C23×C32C22×D311yescharacteristic
119C23×C32C22×D311yescharacteristic
120C23×C32C23×C311yescharacteristic
121C2×C32×C4—22no—
122C2×C32×C4—24no—
123C2×C32×C4—22no—
124C2×C32×C4—22no—
125C2×C32×C4—22no—
126C2×C32×C4—22no—
127C2×C32×C4—22no—
128C2×C32×C4—34no—
129C2×C32×C4—42no—
130C2×C32×C4—42no—
131C2×C32×C4—42no—
132C2×C32×C4—44no—
133C2×C32×C4—62no—
134C2×C32×C4—62no—
135C2×C32×C4—62no—
136C2×C32×C4—62no—
137C2×C32×C4—62no—
138C2×C32×C4D1212yes—
139C2×C32×C4#6712yes—
140C2×C32×C4C3×D412yes—
141C2×C3⋊Dic3—21no—
142C2×C3⋊Dic3—21no—
143C2×C3⋊Dic3—21no—
144C2×C3⋊Dic3—22no—
145C2×C3⋊Dic3C3×D412yes—
146C2×C3×Dic3—22no—
147C2×C3×Dic3—22no—
148C2×C3×Dic3—22no—
149C2×C3×Dic3—22no—
150C2×C3×Dic3—24no—
151C2×C3×Dic3—22no—
152C2×C3×Dic3—22no—
153C2×C3×Dic3—22no—
154C2×C3×Dic3—22no—
155C2×C3×Dic3—22no—
156C2×C3×Dic3—22no—
157C2×C3×Dic3—24no—
158C2×C3×Dic3—32no—
159C2×C3×Dic3—32no—
160C2×C3×Dic3—42no—
161C2×C3×Dic3—42no—
162C2×C3×Dic3—42no—
163C2×C3×Dic3—42no—
164C2×C3×Dic3—62no—
165C2×C3×Dic3—62no—
166C2×C3×Dic3—62no—
167C2×C3×Dic3—61no—
168C2×C3×Dic3—61no—
169C2×C3×Dic3—61no—
170C2×C3×Dic3—61no—
171C2×C3×Dic3—62no—
172C2×C3×Dic3—61no—
173C2×C3×Dic3—62no—
174C2×C3×Dic3—62no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—122no—
177C2×C3×Dic3Dic614yes—
178C2×C3×Dic3C3×Q814yes—
179#398—91noSylow
180C2×C33#12111yescharacteristic
181C2×C33C22⋊Q812yes—
182C2×C33#12311yescharacteristic
183C2×C33#12512yes—
184C2×C33C4⋊Q811yescharacteristic
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—32no—
189C22×C3×C4—32no—
190C22×C3×C4—32no—
191C22×C3×C4—32no—
192C22×C3×C4—41no—
193C22×C3×C4—62no—
194C22×C3×C4—62no—
195C22×C3×C4—91no—
196C22×C3×C4—91no—
197C22×C3×C4D3211yescharacteristic
198C22×C3×C4C2×C3×D311yescharacteristic
199C22×C3×C4C2×C3×D311yescharacteristic
200C22×Dic3—31no—
201C22×Dic3—31no—
202C22×Dic3—31no—
203C22×Dic3—31no—
204C22×Dic3—61no—
205C22×Dic3—61no—
206C22×Dic3C2×C3×D312yes—
207C22×Dic3C2×C3×D312yes—
208C3×C4⋊C4—64no—
209C3×C4⋊C4—62no—
210C3×C4⋊C4—62no—
211C3×C4⋊C4—184no—
212C3×C4⋊C4—182no—
213C4⋊Dic3—22no—
214C4⋊Dic3—64no—
215Dic3⋊C4—64no—
216Dic3⋊C4—64no—
217Dic3⋊C4—62no—
218Dic3⋊C4—62no—
219C22×C32—22no—
220C22×C32—21no—
221C22×C32—21no—
222C22×C32—22no—
223C22×C32—21no—
224C22×C32—22no—
225C22×C32—21no—
226C22×C32—22no—
227C22×C32—21no—
228C22×C32—21no—
229C22×C32—21no—
230C22×C32—22no—
231C22×C32—22no—
232C22×C32—21no—
233C22×C32—21no—
234C22×C32—41no—
235C22×C32—41no—
236C22×C32—42no—
237C22×C32—42no—
238C22×C32—41no—
239C22×C32C2×Dic612yes—
240C22×C32C2×D1211yescharacteristic
241C22×C32#23512yes—
242C22×C32D3×D411yescharacteristic
243C22×C32D3×D411yescharacteristic
244C22×C32#23711yescharacteristic
245C22×C32#23712yes—
246C22×C32D3×Q812yes—
247C22×C32#23911yescharacteristic
248C22×C32C2×#6711yescharacteristic
249C22×C32C2×C3×D411yescharacteristic
250C22×C32C2×C3×D411yescharacteristic
251C22×C32C2×C3×Q812yes—
252C22×C32C3×#4112yes—
253C22×C32C3×#4111yescharacteristic
254C32×C4—22no—
255C32×C4—24no—
256C32×C4—22no—
257C32×C4—22no—
258C32×C4—42no—
259C32×C4—42no—
260C32×C4—44no—
261C32×C4—64no—
262C32×C4—122no—
263C3⋊Dic3—22no—
264C3⋊Dic3—41no—
265C3×Dic3—24no—
266C3×Dic3—24no—
267C3×Dic3—44no—
268C3×Dic3—44no—
269C3×Dic3—42no—
270C3×Dic3—42no—
271C3×Dic3—64no—
272C3×Dic3—62no—
273C3×Dic3—62no—
274C3×Dic3—122no—
275C3×Dic3—121no—
276C3×Dic3—121no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—92no—
279C2×C4⋊C4—91no—
280#95—31no—
281#95—92no—
282C33#39811yescharacteristic, complemented, Sylow
283C23×C3—21no—
284C23×C3—21no—
285C23×C3—21no—
286C23×C3—41no—
287C23×C3C2×D3211yescharacteristic
288C23×C3C22×C3×D311yescharacteristic
289C23×C3C22×C3×D311yescharacteristic
290C2×C3×C4—24no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—34no—
298C2×C3×C4—34no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—44no—
302C2×C3×C4—42no—
303C2×C3×C4—64no—
304C2×C3×C4—62no—
305C2×C3×C4—62no—
306C2×C3×C4—62no—
307C2×C3×C4—62no—
308C2×C3×C4—64no—
309C2×C3×C4—62no—
310C2×C3×C4—62no—
311C2×C3×C4—62no—
312C2×C3×C4—62no—
313C2×C3×C4—62no—
314C2×C3×C4—62no—
315C2×C3×C4—92no—
316C2×C3×C4—122no—
317C2×C3×C4—122no—
318C2×C3×C4—122no—
319C2×C3×C4—122no—
320C2×C3×C4—181no—
321C2×C3×C4—181no—
322C2×C3×C4—182no—
323C2×C3×C4—181no—
324C2×C3×C4#62612yes—
325C2×C3×C4C3×D1212yes—
326C2×C3×C4C3×#6712yes—
327C2×Dic3—22no—
328C2×Dic3—22no—
329C2×Dic3—22no—
330C2×Dic3—22no—
331C2×Dic3—22no—
332C2×Dic3—32no—
333C2×Dic3—32no—
334C2×Dic3—62no—
335C2×Dic3—62no—
336C2×Dic3—62no—
337C2×Dic3—61no—
338C2×Dic3—61no—
339C2×Dic3—61no—
340C2×Dic3—61no—
341C2×Dic3—62no—
342C2×Dic3—61no—
343C2×Dic3—62no—
344C2×Dic3—62no—
345C2×Dic3—61no—
346C2×Dic3—122no—
347C2×Dic3C3×Dic614yes—
348C2×C32—22no—
349C2×C32—21no—
350C2×C32—22no—
351C2×C32—22no—
352C2×C32—21no—
353C2×C32—22no—
354C2×C32—21no—
355C2×C32—21no—
356C2×C32—21no—
357C2×C32—22no—
358C2×C32—21no—
359C2×C32—22no—
360C2×C32—21no—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32—41no—
364C2×C32—42no—
365C2×C32—41no—
366C2×C32—42no—
367C2×C32—41no—
368C2×C32#86911yescharacteristic
369C2×C32#87812yes—
370C2×C32#88611yescharacteristic
371C2×C32#88912yes—
372C2×C32#89012yes—
373C2×C32#89511yescharacteristic
374C2×C32#89612yes—
375C2×C32#92911yescharacteristic
376C2×C32#93811yescharacteristic
377C2×C32#94411yescharacteristic
378C2×C32C3×#12111yescharacteristic
379C2×C32C3×C22⋊Q812yes—
380C2×C32C3×#12311yescharacteristic
381C2×C32C3×#12512yes—
382C2×C32C3×C4⋊Q811yescharacteristic
383C22×C4—32no—
384C22×C4—32no—
385C22×C4—91no—
386C22×C4—91no—
387C22×C4C3×D3211yescharacteristic
388C4⋊C4—62no—
389C4⋊C4—184no—
390C4⋊C4—182no—
391C22×C3—22no—
392C22×C3—22no—
393C22×C3—22no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—22no—
401C22×C3—22no—
402C22×C3—22no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—42no—
407C22×C3—41no—
408C22×C3—41no—
409C22×C3—42no—
410C22×C3—41no—
411C22×C3D3×Dic612yes—
412C22×C3#379611yescharacteristic
413C22×C3#380211yescharacteristic
414C22×C3#380412yes—
415C22×C3C2×#62611yescharacteristic
416C22×C3C2×C3×Dic612yes—
417C22×C3C2×C3×D1211yescharacteristic
418C22×C3C3×#23512yes—
419C22×C3C3×D3×D411yescharacteristic
420C22×C3C3×D3×D411yescharacteristic
421C22×C3C3×#23711yescharacteristic
422C22×C3C3×#23712yes—
423C22×C3C3×D3×Q812yes—
424C22×C3C3×#23911yescharacteristic
425C22×C3C2×C3×#6711yescharacteristic
426C3×C4—24no—
427C3×C4—22no—
428C3×C4—22no—
429C3×C4—22no—
430C3×C4—42no—
431C3×C4—42no—
432C3×C4—44no—
433C3×C4—64no—
434C3×C4—64no—
435C3×C4—124no—
436C3×C4—124no—
437C3×C4—122no—
438C3×C4—122no—
439C3×C4—182no—
440C3×C4—361no—
441Dic3—24no—
442Dic3—42no—
443Dic3—64no—
444Dic3—62no—
445Dic3—62no—
446Dic3—122no—
447Dic3—121no—
448Dic3—121no—
449C32—21no—
450C32—21no—
451C32—21no—
452C32—41no—
453C32#466111yescharacteristic, complemented
454C32#498311yescharacteristic, complemented
455C32C3×#39811yescharacteristic, complemented
456C23C2×C3×D3211yescharacteristic
457C2×C4—22no—
458C2×C4—34no—
459C2×C4—62no—
460C2×C4—62no—
461C2×C4—62no—
462C2×C4—62no—
463C2×C4—62no—
464C2×C4—92no—
465C2×C4—181no—
466C2×C4—181no—
467C2×C4—182no—
468C2×C4—181no—
469C2×C4C3×#62612yes—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—22no—
474C2×C3—22no—
475C2×C3—22no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—22no—
480C2×C3—22no—
481C2×C3—22no—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—41no—
486C2×C3—42no—
487C2×C3—41no—
488C2×C3—42no—
489C2×C3—41no—
490C2×C3#6384612yes—
491C2×C3#6386611yescharacteristic
492C2×C3#6389912yes—
493C2×C3#6391611yescharacteristic
494C2×C3#6396611yescharacteristic
495C2×C3C3×#86911yescharacteristic
496C2×C3C3×#87812yes—
497C2×C3C3×#88611yescharacteristic
498C2×C3C3×#88912yes—
499C2×C3C3×#89012yes—
500C2×C3C3×#89511yescharacteristic
501C2×C3C3×#89612yes—
502C2×C3C3×#92911yescharacteristic
503C2×C3C3×#93811yescharacteristic
504C2×C3C3×#94411yescharacteristic
505C22C3×D3×Dic612yes—
506C22C3×#379611yescharacteristic
507C22C3×#380211yescharacteristic
508C22C3×#380412yes—
509C22C2×C3×#62611yescharacteristic
510C4—22no—
511C4—64no—
512C4—122no—
513C4—182no—
514C4—361no—
515C3—21no—
516C3—21no—
517C3—21no—
518C3—41no—
519C3#1059135811yescharacteristic, direct product factor
520C3C3×#466111yescharacteristic, complemented
521C3C3×#498311yescharacteristic, complemented
522C2C3×#6384612yes—
523C2C3×#6386611yescharacteristic
524C2C3×#6389912yes—
525C2C3×#6391611yescharacteristic
526C2C3×#6396611yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutator, cosocleC23×C32C23×C3
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socle, prime radicalC23×C33C23
hypercenterC22×C3×C4D32
hypocenterC32C3×#398
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
8. #10591358C3direct (N × H)
282. C33#398semidirect
453. C32#4661semidirect
454. C32#4983semidirect
455. C32C3×#398semidirect
519. C3#10591358direct (N × H)
520. C3C3×#4661semidirect
521. C3C3×#4983semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37502
C2×C32×C4⋊Dic32
C2×C3×Dic3⋊Dic32
C32×C3⋊#952
C3×#635792
C3×#49833
C3×#46613
#105913583