Finite Group Databasebeta

← C3×#10591203 #13158228

Subgroups — C3×#10591203 #13158228

Order 1728.

1114 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3752C211yesmaximal, characteristic
2C2×C32×Dic3⋊C4C212yesmaximal, complemented
3C2×C3×#3724C211yesmaximal, characteristic, complemented
4C2×C3×#3722C212yesmaximal
5C3×#63579C211yesmaximal, characteristic, complemented
6C3×#4993—32nomaximal
7#10591203C311yesmaximal, characteristic, direct product factor
8C23×C3×C3⋊D3C2211yescharacteristic
9C22×C33×C4C2211yescharacteristic
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C32×Dic3C2212yes—
12C22×C32×Dic3C2212yes—
13C3×#3752—22no—
14C32×Dic3⋊C4—24no—
15C3×#3724—22no—
16C3×#3722—24no—
17C2×C32×C4⋊C4—32no—
18C2×#3752C2×C311yescharacteristic
19C2×C3×(C2×D3)⋊C4—32no—
20C2×C3×(C2×D3)⋊C4—32no—
21C2×C3×(C2×D3)⋊C4—32no—
22C2×C3×(C2×D3)⋊C4—61no—
23C2×C3×Dic3⋊C4—22no—
24C2×C3×Dic3⋊C4—32no—
25C2×C3×Dic3⋊C4D312yescomplemented
26C2×C3×Dic3⋊C4C2×C312yescomplemented
27C2×#3724C2×C311yescharacteristic, complemented
28C2×#3722C2×C312yes—
29C3×C3⋊#95—32no—
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⋊D3Q812yes—
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×#399—91no—
48#4993—32no—
49C23×C3⋊D3C22×C311yescharacteristic
50C23×C3×D3—32no—
51C23×C3×D3—61no—
52C22×C32×C4—21no—
53C22×C32×C4—22no—
54C22×C32×C4—32no—
55C22×C32×C4—32no—
56C22×C32×C4—41no—
57C22×C32×C4C2×D312yes—
58C22×C32×C4C22×C311yescharacteristic
59C22×C3⋊Dic3C22×C311yescharacteristic
60C22×C3×Dic3—22no—
61C22×C3×Dic3—22no—
62C22×C3×Dic3—32no—
63C22×C3×Dic3—61no—
64C22×C3×Dic3C2×D312yes—
65C22×C3×Dic3C2×D312yes—
66C22×C3×Dic3C22×C312yes—
67C22×C3×Dic3C22×C312yes—
68C32×C4⋊C4—64no—
69#3752—22no—
70C3×(C2×D3)⋊C4—64no—
71C3×(C2×D3)⋊C4—64no—
72C3×(C2×D3)⋊C4—64no—
73C3×(C2×D3)⋊C4—64no—
74C3×Dic3⋊C4—28no—
75C3×Dic3⋊C4—24no—
76C3×Dic3⋊C4—24no—
77C3×Dic3⋊C4—64no—
78#3724—22no—
79#3722—24no—
80C22×C33C2×D411yescharacteristic
81C22×C33C2×D412yes—
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—
94C2×C3×C4⋊C4—32no—
95C2×C3×C4⋊C4—32no—
96C2×C3×C4⋊C4—62no—
97C2×C3×C4⋊C4—91no—
98C2×C3×C22⋊C4—92no—
99C2×C3×C22⋊C4—91no—
100C2×(C2×D3)⋊C4—32no—
101C2×(C2×D3)⋊C4—32no—
102C2×(C2×D3)⋊C4—32no—
103C2×(C2×D3)⋊C4—61no—
104C2×Dic3⋊C4—32no—
105C2×Dic3⋊C4C3×D312yescomplemented
106C3×#95—91no—
107C3⋊#95—32no—
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×Q812yes—
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—24no—
133C2×C32×C4—22no—
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—24no—
151C2×C3×Dic3—24no—
152C2×C3×Dic3—22no—
153C2×C3×Dic3—22no—
154C2×C3×Dic3—22no—
155C2×C3×Dic3—22no—
156C2×C3×Dic3—24no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—22no—
159C2×C3×Dic3—22no—
160C2×C3×Dic3—22no—
161C2×C3×Dic3—22no—
162C2×C3×Dic3—22no—
163C2×C3×Dic3—24no—
164C2×C3×Dic3—34no—
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×Dic3#6714yes—
172C2×C3×Dic3C3×D414yes—
173#399—91noSylow
174C2×C33C22⋊Q811yescharacteristic
175C2×C33C22⋊Q812yes—
176C2×C33#12311yescharacteristic
177C2×C33#12312yes—
178C2×C33#12711yescharacteristic
179C3×C3⋊D3—42no—
180C24×C3—91no—
181C23×D3—32no—
182C23×D3—61no—
183C22×C3×C4—21no—
184C22×C3×C4—22no—
185C22×C3×C4—32no—
186C22×C3×C4—32no—
187C22×C3×C4—32no—
188C22×C3×C4—32no—
189C22×C3×C4—41no—
190C22×C3×C4—62no—
191C22×C3×C4—62no—
192C22×C3×C4—91no—
193C22×C3×C4D3211yescharacteristic
194C22×C3×C4C2×C3×D312yes—
195C22×Dic3—32no—
196C22×Dic3—61no—
197C22×Dic3C2×C3×D312yes—
198C22×Dic3C2×C3×D312yes—
199C3×C4⋊C4—64no—
200C3×C4⋊C4—64no—
201C3×C4⋊C4—68no—
202C3×C4⋊C4—182no—
203C3×C22⋊C4—182no—
204C3×C22⋊C4—184no—
205(C2×D3)⋊C4—64no—
206(C2×D3)⋊C4—64no—
207(C2×D3)⋊C4—64no—
208(C2×D3)⋊C4—64no—
209Dic3⋊C4—24no—
210Dic3⋊C4—64no—
211C22×C32—21no—
212C22×C32—21no—
213C22×C32—22no—
214C22×C32—22no—
215C22×C32—21no—
216C22×C32—22no—
217C22×C32—22no—
218C22×C32—22no—
219C22×C32—22no—
220C22×C32—22no—
221C22×C32—22no—
222C22×C32—22no—
223C22×C32—42no—
224C22×C32—41no—
225C22×C32—41no—
226C22×C32—41no—
227C22×C32—42no—
228C22×C32#23512yes—
229C22×C32#23512yes—
230C22×C32D3×D412yes—
231C22×C32D3×D412yes—
232C22×C32D3×Q812yes—
233C22×C32#23912yes—
234C22×C32C2×#6712yes—
235C22×C32C2×C3×D411yescharacteristic
236C22×C32C2×C3×D412yes—
237C22×C32C2×C3×Q811yescharacteristic
238C22×C32C3×#4111yescharacteristic
239C22×C32C3×#4112yes—
240C2×C3⋊D3—22no—
241C2×C3⋊D3—24no—
242C2×C3⋊D3—41no—
243C2×C3⋊D3—42no—
244C2×C3⋊D3—41no—
245C2×C3×D3—64no—
246C2×C3×D3—64no—
247C2×C3×D3—64no—
248C2×C3×D3—64no—
249C2×C3×D3—124no—
250C2×C3×D3—124no—
251C2×C3×D3—122no—
252C2×C3×D3—122no—
253C2×C3×D3—122no—
254C2×C3×D3—122no—
255C2×C3×D3—122no—
256C2×C3×D3—122no—
257C32×C4—44no—
258C32×C4—42no—
259C32×C4—44no—
260C32×C4—42no—
261C32×C4—41no—
262C32×C4—64no—
263C32×C4—122no—
264C3⋊Dic3—22no—
265C3×Dic3—28no—
266C3×Dic3—24no—
267C3×Dic3—24no—
268C3×Dic3—44no—
269C3×Dic3—42no—
270C3×Dic3—42no—
271C3×Dic3—64no—
272C3×Dic3—122no—
273C2×C4⋊C4—32no—
274C2×C4⋊C4—91no—
275C2×C22⋊C4—92no—
276C2×C22⋊C4—91no—
277#95—91no—
278C33#39911yescharacteristic, complemented, Sylow
279C23×C3—22no—
280C23×C3—21no—
281C23×C3—41no—
282C23×C3—92no—
283C23×C3—181no—
284C23×C3—182no—
285C23×C3—182no—
286C23×C3—181no—
287C23×C3C2×D3211yescharacteristic
288C23×C3C22×C3×D312yes—
289C22×D3—34no—
290C22×D3—62no—
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—122no—
300C22×D3—122no—
301C2×C3×C4—22no—
302C2×C3×C4—24no—
303C2×C3×C4—22no—
304C2×C3×C4—21no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—34no—
309C2×C3×C4—34no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—44no—
314C2×C3×C4—42no—
315C2×C3×C4—64no—
316C2×C3×C4—64no—
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—182no—
333C2×Dic3—22no—
334C2×Dic3—22no—
335C2×Dic3—22no—
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3—34no—
339C2×Dic3—62no—
340C2×Dic3—62no—
341C2×Dic3—62no—
342C2×Dic3—122no—
343C2×Dic3C3×#6714yes—
344C2×C32—22no—
345C2×C32—22no—
346C2×C32—22no—
347C2×C32—22no—
348C2×C32—22no—
349C2×C32—22no—
350C2×C32—22no—
351C2×C32—22no—
352C2×C32—21no—
353C2×C32—22no—
354C2×C32—21no—
355C2×C32—21no—
356C2×C32—41no—
357C2×C32—42no—
358C2×C32—42no—
359C2×C32—41no—
360C2×C32—41no—
361C2×C32#89412yes—
362C2×C32#89412yes—
363C2×C32#89612yes—
364C2×C32#89612yes—
365C2×C32#92912yes—
366C2×C32#94012yes—
367C2×C32#94612yes—
368C2×C32C3×C22⋊Q811yescharacteristic
369C2×C32C3×C22⋊Q812yes—
370C2×C32C3×#12311yescharacteristic
371C2×C32C3×#12312yes—
372C2×C32C3×#12711yescharacteristic
373C3⋊D3—42no—
374C3×D3—124no—
375C3×D3—124no—
376C24—91no—
377C22×C4—32no—
378C22×C4—32no—
379C22×C4—91no—
380C22×C4C3×D3211yescharacteristic
381C4⋊C4—64no—
382C4⋊C4—182no—
383C22⋊C4—182no—
384C22⋊C4—184no—
385C22×C3—21no—
386C22×C3—22no—
387C22×C3—22no—
388C22×C3—21no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—22no—
392C22×C3—22no—
393C22×C3—22no—
394C22×C3—22no—
395C22×C3—21no—
396C22×C3—22no—
397C22×C3—41no—
398C22×C3—41no—
399C22×C3—42no—
400C22×C3—42no—
401C22×C3—41no—
402C22×C3—182no—
403C22×C3—184no—
404C22×C3—362no—
405C22×C3—361no—
406C22×C3—361no—
407C22×C3#379711yescharacteristic
408C22×C3#379811yescharacteristic
409C22×C3#379912yes—
410C22×C3D3×#6712yes—
411C22×C3#381111yescharacteristic
412C22×C3C3×#23512yes—
413C22×C3C3×#23512yes—
414C22×C3C3×D3×D412yes—
415C22×C3C3×D3×D412yes—
416C22×C3C3×D3×Q812yes—
417C22×C3C3×#23912yes—
418C22×C3C2×C3×#6712yes—
419C2×D3—64no—
420C2×D3—64no—
421C2×D3—64no—
422C2×D3—64no—
423C2×D3—124no—
424C2×D3—122no—
425C2×D3—124no—
426C2×D3—122no—
427C2×D3—122no—
428C2×D3—122no—
429C2×D3—122no—
430C2×D3—122no—
431C3×C4—44no—
432C3×C4—42no—
433C3×C4—44no—
434C3×C4—42no—
435C3×C4—41no—
436C3×C4—68no—
437C3×C4—64no—
438C3×C4—64no—
439C3×C4—124no—
440C3×C4—122no—
441C3×C4—122no—
442C3×C4—182no—
443Dic3—24no—
444Dic3—42no—
445Dic3—64no—
446Dic3—122no—
447C32—22no—
448C32—21no—
449C32—41no—
450C32#499312yescomplemented
451C32C3×#39911yescharacteristic, complemented
452C23—92no—
453C23—181no—
454C23—182no—
455C23—182no—
456C23—181no—
457C23C2×C3×D3211yescharacteristic
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—34no—
461C2×C4—62no—
462C2×C4—62no—
463C2×C4—62no—
464C2×C4—62no—
465C2×C4—62no—
466C2×C4—92no—
467C2×C4—182no—
468C2×C3—22no—
469C2×C3—21no—
470C2×C3—22no—
471C2×C3—22no—
472C2×C3—22no—
473C2×C3—21no—
474C2×C3—22no—
475C2×C3—22no—
476C2×C3—22no—
477C2×C3—22no—
478C2×C3—21no—
479C2×C3—22no—
480C2×C3—41no—
481C2×C3—42no—
482C2×C3—41no—
483C2×C3—42no—
484C2×C3—41no—
485C2×C3—362no—
486C2×C3#6385012yes—
487C2×C3#6386511yescharacteristic
488C2×C3#6388812yes—
489C2×C3#6389711yescharacteristic
490C2×C3#6397911yescharacteristic
491C2×C3C3×#89412yes—
492C2×C3C3×#89412yes—
493C2×C3C3×#89612yes—
494C2×C3C3×#89612yes—
495C2×C3C3×#92912yes—
496C2×C3C3×#94012yes—
497C2×C3C3×#94612yes—
498D3—124no—
499D3—124no—
500C22—182no—
501C22—184no—
502C22—361no—
503C22—362no—
504C22—361no—
505C22C3×#379711yescharacteristic
506C22C3×#379811yescharacteristic
507C22C3×#379912yes—
508C22C3×D3×#6712yes—
509C22C3×#381111yescharacteristic
510C4—41no—
511C4—64no—
512C4—122no—
513C4—182no—
514C3—21no—
515C3—22no—
516C3—41no—
517C3#1059120311yescharacteristic, direct product factor
518C3C3×#499312yescomplemented
519C2—362no—
520C2C3×#6385012yes—
521C2C3×#6386511yescharacteristic
522C2C3×#6388812yes—
523C2C3×#6389711yescharacteristic
524C2C3×#6397911yescharacteristic
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×#399
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C32×Dic3⋊C4C2semidirect
3. C2×C3×#3724C2semidirect
5. C3×#63579C2semidirect
7. #10591203C3direct (N × H)
25. C2×C3×Dic3⋊C4D3semidirect
26. C2×C3×Dic3⋊C4C2×C3semidirect
27. C2×#3724C2×C3semidirect
30. #63579C2×C3semidirect
105. C2×Dic3⋊C4C3×D3semidirect
278. C33#399semidirect
450. C32#4993semidirect
451. C32C3×#399semidirect
517. C3#10591203direct (N × H)
518. C3C3×#4993semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37522
C2×C32×Dic3⋊C42
C2×C3×#37242
C2×C3×#37222
C3×#635792
C3×#49933
#105912033