Finite Group Databasebeta

← C3×#10591205 #13158414

Subgroups — C3×#10591205 #13158414

Order 1728.

1146 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#627C211yesmaximal, characteristic, complemented
2C2×C3×#3752C211yesmaximal, characteristic
3C2×C3×#3722C212yesmaximal
4C32×C3⋊#95C212yesmaximal, complemented
5C3×#63579C211yesmaximal, characteristic, complemented
6C3×#4673—32nomaximal
7#10591205C311yesmaximal, characteristic, direct product factor
8C23×C3×C3⋊D3C2211yescharacteristic
9C22×C33×C4C2211yescharacteristic
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C32×Dic3C2212yes—
12C22×C32×Dic3C2212yes—
13C2×C3×#627—22no—
14C2×C3×#627—24no—
15C3×#3752—22no—
16C3×#3722—24no—
17C22×C3×Dic6—32no—
18C22×#627C2×C311yescharacteristic, complemented
19C2×#3752C2×C311yescharacteristic
20C2×C3×(C2×D3)⋊C4—32no—
21C2×C3×(C2×D3)⋊C4—32no—
22C2×C3×(C2×D3)⋊C4—32no—
23C2×C3×(C2×D3)⋊C4—61no—
24C2×#3722C2×C312yes—
25C32×#95—32no—
26C3×C3⋊#95—22no—
27C3×C3⋊#95—32no—
28C3×C3⋊#95D312yescomplemented
29C3×C3⋊#95C2×C312yescomplemented
30#63579C2×C311yescharacteristic, complemented
31C23×C33C2311yescharacteristic
32C22×C3×C3⋊D3—21no—
33C22×C3×C3⋊D3—22no—
34C22×C3×C3⋊D3—21no—
35C22×C3×C3⋊D3—22no—
36C22×C3×C3⋊D3Q812yescomplemented
37C2×C33×C4—21no—
38C2×C33×C4—22no—
39C2×C3×C3⋊Dic3—22no—
40C2×C3×C3⋊Dic3D412yes—
41C2×C32×Dic3—22no—
42C2×C32×Dic3—22no—
43C2×C32×Dic3—22no—
44C2×C32×Dic3—22no—
45C2×C32×Dic3—22no—
46C2×C32×Dic3D414yes—
47C3×#627—28no—
48C3×#393—91no—
49#4673—32no—
50C23×C3⋊D3C22×C311yescharacteristic
51C23×C3×D3—32no—
52C23×C3×D3—61no—
53C22×C32×C4—21no—
54C22×C32×C4—22no—
55C22×C32×C4—32no—
56C22×C32×C4—32no—
57C22×C32×C4—41no—
58C22×C32×C4C2×D312yes—
59C22×C32×C4C22×C311yescharacteristic
60C22×C3⋊Dic3C22×C311yescharacteristic
61C22×C3×Dic3—22no—
62C22×C3×Dic3—22no—
63C22×C3×Dic3—32no—
64C22×C3×Dic3—61no—
65C22×C3×Dic3C2×D312yes—
66C22×C3×Dic3C2×D312yes—
67C22×C3×Dic3C22×C312yes—
68C22×C3×Dic3C22×C312yes—
69C2×C3×Dic6—64no—
70C2×C3×Dic6—64no—
71C2×C3×Dic6—64no—
72C2×#627—22no—
73C2×#627—24no—
74#3752—22no—
75C3×(C2×D3)⋊C4—64no—
76C3×(C2×D3)⋊C4—64no—
77C3×(C2×D3)⋊C4—64no—
78C3×(C2×D3)⋊C4—64no—
79#3722—24no—
80C22×C33C2×D412yes—
81C22×C33C2×D411yescharacteristic
82C22×C33C2×Q811yescharacteristic
83C22×C33#4111yescharacteristic
84C22×C33#4112yes—
85C2×C3×C3⋊D3—22no—
86C2×C3×C3⋊D3—24no—
87C2×C3×C3⋊D3—41no—
88C2×C3×C3⋊D3—42no—
89C2×C3×C3⋊D3—41no—
90C33×C4—41no—
91C3×C3⋊Dic3—22no—
92C32×Dic3—24no—
93C32×Dic3—42no—
94C22×C3×Q8—91no—
95C22×Dic6—32no—
96C2×C3×C22⋊C4—92no—
97C2×C3×C22⋊C4—91no—
98C2×(C2×D3)⋊C4—32no—
99C2×(C2×D3)⋊C4—32no—
100C2×(C2×D3)⋊C4—32no—
101C2×(C2×D3)⋊C4—61no—
102C3×#95—32no—
103C3×#95—32no—
104C3×#95—62no—
105C3×#95—91no—
106C3⋊#95—32no—
107C3⋊#95C3×D312yescomplemented
108C23×C32—21no—
109C23×C32—22no—
110C23×C32—41no—
111C23×C32C22×D312yes—
112C23×C32C23×C311yescharacteristic
113C22×C3⋊D3—21no—
114C22×C3⋊D3—21no—
115C22×C3⋊D3—22no—
116C22×C3⋊D3—22no—
117C22×C3⋊D3C3×Q812yescomplemented
118C22×C3×D3—34no—
119C22×C3×D3—62no—
120C22×C3×D3—62no—
121C22×C3×D3—62no—
122C22×C3×D3—62no—
123C22×C3×D3—62no—
124C22×C3×D3—62no—
125C22×C3×D3—62no—
126C22×C3×D3—62no—
127C22×C3×D3—62no—
128C22×C3×D3—122no—
129C22×C3×D3—122no—
130C2×C32×C4—22no—
131C2×C32×C4—21no—
132C2×C32×C4—22no—
133C2×C32×C4—24no—
134C2×C32×C4—22no—
135C2×C32×C4—22no—
136C2×C32×C4—22no—
137C2×C32×C4—34no—
138C2×C32×C4—42no—
139C2×C32×C4—42no—
140C2×C32×C4—42no—
141C2×C32×C4—42no—
142C2×C32×C4—44no—
143C2×C32×C4—62no—
144C2×C32×C4—62no—
145C2×C32×C4—62no—
146C2×C32×C4—62no—
147C2×C32×C4—62no—
148C2×C3⋊Dic3—22no—
149C2×C3⋊Dic3C3×D412yes—
150C2×C3×Dic3—22no—
151C2×C3×Dic3—22no—
152C2×C3×Dic3—22no—
153C2×C3×Dic3—22no—
154C2×C3×Dic3—24no—
155C2×C3×Dic3—22no—
156C2×C3×Dic3—22no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—22no—
159C2×C3×Dic3—22no—
160C2×C3×Dic3—22no—
161C2×C3×Dic3—24no—
162C2×C3×Dic3—34no—
163C2×C3×Dic3—42no—
164C2×C3×Dic3—42no—
165C2×C3×Dic3—42no—
166C2×C3×Dic3—42no—
167C2×C3×Dic3—62no—
168C2×C3×Dic3—62no—
169C2×C3×Dic3—62no—
170C2×C3×Dic3—122no—
171C2×C3×Dic3D1214yes—
172C2×C3×Dic3C3×D414yes—
173C3×Dic6—616no—
174#627—28no—
175#393—91noSylow
176C2×C33#12011yescharacteristic
177C2×C33C22⋊Q811yescharacteristic
178C2×C33C22⋊Q812yes—
179C2×C33#12412yes—
180C2×C33#12411yescharacteristic
181C3×C3⋊D3—42no—
182C24×C3—91no—
183C23×D3—32no—
184C23×D3—61no—
185C2×C3×Q8—184no—
186C2×C3×Q8—182no—
187C22×C3×C4—22no—
188C22×C3×C4—21no—
189C22×C3×C4—32no—
190C22×C3×C4—32no—
191C22×C3×C4—32no—
192C22×C3×C4—32no—
193C22×C3×C4—41no—
194C22×C3×C4—62no—
195C22×C3×C4—62no—
196C22×C3×C4—91no—
197C22×C3×C4D3211yescharacteristic
198C22×C3×C4C2×C3×D312yes—
199C22×Dic3—32no—
200C22×Dic3—61no—
201C22×Dic3C2×C3×D312yes—
202C22×Dic3C2×C3×D312yes—
203C2×Dic6—64no—
204C2×Dic6—64no—
205C2×Dic6—64no—
206C3×C22⋊C4—182no—
207C3×C22⋊C4—184no—
208(C2×D3)⋊C4—64no—
209(C2×D3)⋊C4—64no—
210(C2×D3)⋊C4—64no—
211(C2×D3)⋊C4—64no—
212C22×C32—21no—
213C22×C32—21no—
214C22×C32—22no—
215C22×C32—22no—
216C22×C32—21no—
217C22×C32—22no—
218C22×C32—22no—
219C22×C32—22no—
220C22×C32—22no—
221C22×C32—22no—
222C22×C32—22no—
223C22×C32—22no—
224C22×C32—41no—
225C22×C32—42no—
226C22×C32—41no—
227C22×C32—42no—
228C22×C32—41no—
229C22×C32C2×D1212yes—
230C22×C32#23512yes—
231C22×C32#23512yes—
232C22×C32D3×D412yes—
233C22×C32D3×D412yes—
234C22×C32#23712yes—
235C22×C32D3×Q812yes—
236C22×C32C2×C3×D412yes—
237C22×C32C2×C3×D411yescharacteristic
238C22×C32C2×C3×Q811yescharacteristic
239C22×C32C3×#4111yescharacteristic
240C22×C32C3×#4112yes—
241C2×C3⋊D3—22no—
242C2×C3⋊D3—24no—
243C2×C3⋊D3—41no—
244C2×C3⋊D3—41no—
245C2×C3⋊D3—42no—
246C2×C3×D3—64no—
247C2×C3×D3—64no—
248C2×C3×D3—64no—
249C2×C3×D3—64no—
250C2×C3×D3—124no—
251C2×C3×D3—124no—
252C2×C3×D3—122no—
253C2×C3×D3—122no—
254C2×C3×D3—122no—
255C2×C3×D3—122no—
256C2×C3×D3—122no—
257C2×C3×D3—122no—
258C32×C4—44no—
259C32×C4—42no—
260C32×C4—44no—
261C32×C4—42no—
262C32×C4—41no—
263C32×C4—64no—
264C32×C4—122no—
265C3⋊Dic3—22no—
266C3×Dic3—24no—
267C3×Dic3—24no—
268C3×Dic3—44no—
269C3×Dic3—44no—
270C3×Dic3—42no—
271C3×Dic3—42no—
272C3×Dic3—64no—
273C3×Dic3—122no—
274C22×Q8—91no—
275C2×C22⋊C4—92no—
276C2×C22⋊C4—91no—
277#95—32no—
278#95—91no—
279C33#39311yescharacteristic, complemented, Sylow
280C23×C3—22no—
281C23×C3—21no—
282C23×C3—41no—
283C23×C3—92no—
284C23×C3—181no—
285C23×C3—182no—
286C23×C3—182no—
287C23×C3—181no—
288C23×C3C2×D3211yescharacteristic
289C23×C3C22×C3×D312yes—
290C22×D3—34no—
291C22×D3—62no—
292C22×D3—62no—
293C22×D3—62no—
294C22×D3—62no—
295C22×D3—62no—
296C22×D3—62no—
297C22×D3—62no—
298C22×D3—62no—
299C22×D3—62no—
300C22×D3—122no—
301C22×D3—122no—
302C3×Q8—188no—
303C2×C3×C4—22no—
304C2×C3×C4—24no—
305C2×C3×C4—22no—
306C2×C3×C4—21no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—34no—
311C2×C3×C4—34no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—44no—
316C2×C3×C4—42no—
317C2×C3×C4—64no—
318C2×C3×C4—62no—
319C2×C3×C4—62no—
320C2×C3×C4—62no—
321C2×C3×C4—62no—
322C2×C3×C4—64no—
323C2×C3×C4—62no—
324C2×C3×C4—62no—
325C2×C3×C4—62no—
326C2×C3×C4—62no—
327C2×C3×C4—62no—
328C2×C3×C4—62no—
329C2×C3×C4—92no—
330C2×C3×C4—122no—
331C2×C3×C4—122no—
332C2×C3×C4—122no—
333C2×C3×C4—122no—
334C2×C3×C4—182no—
335C2×Dic3—22no—
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—34no—
341C2×Dic3—62no—
342C2×Dic3—62no—
343C2×Dic3—62no—
344C2×Dic3—122no—
345C2×Dic3C3×D1214yes—
346Dic6—616no—
347C2×C32—22no—
348C2×C32—22no—
349C2×C32—22no—
350C2×C32—22no—
351C2×C32—22no—
352C2×C32—22no—
353C2×C32—22no—
354C2×C32—21no—
355C2×C32—22no—
356C2×C32—21no—
357C2×C32—21no—
358C2×C32—22no—
359C2×C32—41no—
360C2×C32—41no—
361C2×C32—42no—
362C2×C32—41no—
363C2×C32—42no—
364C2×C32#87512yes—
365C2×C32#88212yes—
366C2×C32#88512yes—
367C2×C32#88512yes—
368C2×C32#89612yes—
369C2×C32#89612yes—
370C2×C32#89712yes—
371C2×C32C3×#12011yescharacteristic
372C2×C32C3×C22⋊Q811yescharacteristic
373C2×C32C3×C22⋊Q812yes—
374C2×C32C3×#12411yescharacteristic
375C2×C32C3×#12412yes—
376C3⋊D3—42no—
377C3×D3—124no—
378C3×D3—124no—
379C24—91no—
380C2×Q8—184no—
381C2×Q8—182no—
382C22×C4—32no—
383C22×C4—32no—
384C22×C4—91no—
385C22×C4C3×D3211yescharacteristic
386C22⋊C4—182no—
387C22⋊C4—184no—
388C22×C3—21no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—21no—
392C22×C3—22no—
393C22×C3—22no—
394C22×C3—22no—
395C22×C3—22no—
396C22×C3—22no—
397C22×C3—22no—
398C22×C3—21no—
399C22×C3—22no—
400C22×C3—41no—
401C22×C3—41no—
402C22×C3—42no—
403C22×C3—42no—
404C22×C3—41no—
405C22×C3—182no—
406C22×C3—184no—
407C22×C3—362no—
408C22×C3—361no—
409C22×C3—361no—
410C22×C3#379711yescharacteristic
411C22×C3#379811yescharacteristic
412C22×C3D3×D1212yes—
413C22×C3#380412yes—
414C22×C3#381111yescharacteristic
415C22×C3C2×C3×D1212yes—
416C22×C3C3×#23512yes—
417C22×C3C3×#23512yes—
418C22×C3C3×D3×D412yes—
419C22×C3C3×D3×D412yes—
420C22×C3C3×#23712yes—
421C22×C3C3×D3×Q812yes—
422C2×D3—64no—
423C2×D3—64no—
424C2×D3—64no—
425C2×D3—64no—
426C2×D3—124no—
427C2×D3—122no—
428C2×D3—124no—
429C2×D3—122no—
430C2×D3—122no—
431C2×D3—122no—
432C2×D3—122no—
433C2×D3—122no—
434C3×C4—42no—
435C3×C4—44no—
436C3×C4—44no—
437C3×C4—42no—
438C3×C4—41no—
439C3×C4—64no—
440C3×C4—64no—
441C3×C4—124no—
442C3×C4—124no—
443C3×C4—122no—
444C3×C4—122no—
445C3×C4—182no—
446Dic3—24no—
447Dic3—42no—
448Dic3—64no—
449Dic3—122no—
450C32—22no—
451C32—21no—
452C32—41no—
453C32#467312yescomplemented
454C32C3×#39311yescharacteristic, complemented
455C23—92no—
456C23—181no—
457C23—182no—
458C23—182no—
459C23—181no—
460C23C2×C3×D3211yescharacteristic
461Q8—188no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—34no—
465C2×C4—62no—
466C2×C4—62no—
467C2×C4—62no—
468C2×C4—62no—
469C2×C4—62no—
470C2×C4—92no—
471C2×C4—182no—
472C2×C3—21no—
473C2×C3—22no—
474C2×C3—22no—
475C2×C3—22no—
476C2×C3—22no—
477C2×C3—21no—
478C2×C3—22no—
479C2×C3—22no—
480C2×C3—22no—
481C2×C3—22no—
482C2×C3—21no—
483C2×C3—22no—
484C2×C3—41no—
485C2×C3—42no—
486C2×C3—41no—
487C2×C3—42no—
488C2×C3—41no—
489C2×C3—362no—
490C2×C3#6385212yes—
491C2×C3#6386511yescharacteristic
492C2×C3#6389512yes—
493C2×C3#6392311yescharacteristic
494C2×C3#6395311yescharacteristic
495C2×C3C3×#87512yes—
496C2×C3C3×#88212yes—
497C2×C3C3×#88512yes—
498C2×C3C3×#88512yes—
499C2×C3C3×#89612yes—
500C2×C3C3×#89612yes—
501C2×C3C3×#89712yes—
502D3—124no—
503D3—124no—
504C22—182no—
505C22—184no—
506C22—361no—
507C22—362no—
508C22—361no—
509C22C3×#379711yescharacteristic
510C22C3×#379811yescharacteristic
511C22C3×D3×D1212yes—
512C22C3×#380412yes—
513C22C3×#381111yescharacteristic
514C4—41no—
515C4—64no—
516C4—122no—
517C4—182no—
518C3—21no—
519C3—22no—
520C3—41no—
521C3#1059120511yescharacteristic, direct product factor
522C3C3×#467312yescomplemented
523C2—362no—
524C2C3×#6385212yes—
525C2C3×#6386511yescharacteristic
526C2C3×#6389512yes—
527C2C3×#6392311yescharacteristic
528C2C3×#6395311yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutator, cosocleC23×C32C23×C3
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socleC23×C33C23
prime radicalC23×C3×C3⋊D3C22
hypercenterC22×C3×C4D32
hypocenterC32C3×#393
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×#627C2semidirect
4. C32×C3⋊#95C2semidirect
5. C3×#63579C2semidirect
7. #10591205C3direct (N × H)
18. C22×#627C2×C3semidirect
28. C3×C3⋊#95D3semidirect
29. C3×C3⋊#95C2×C3semidirect
30. #63579C2×C3semidirect
36. C22×C3×C3⋊D3Q8semidirect
107. C3⋊#95C3×D3semidirect
117. C22×C3⋊D3C3×Q8semidirect
279. C33#393semidirect
453. C32#4673semidirect
454. C32C3×#393semidirect
521. C3#10591205direct (N × H)
522. C3C3×#4673semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#6272
C2×C3×#37522
C2×C3×#37222
C32×C3⋊#952
C3×#635792
C3×#46733
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