Finite Group Databasebeta

← #13161817

Subgroups — #13161817

Order 1728.

686 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic
2C3×#64133C211yesmaximal, characteristic
3C3×#63657C211yesmaximal, characteristic
4#11722079C212yesmaximal, complemented
5#11722076C212yesmaximal, complemented
6#10591126—31nomaximal
7#10591124—31nomaximal
8#10591101—31nomaximal
9C22×C33×C4C2211yescharacteristic
10C22×C3×C3⋊Dic3—21no—
11C2×C3×#634C2211yescharacteristic
12C3×#3757—21no—
13C3×C4⋊(C3⋊Dic3)—21no—
14C3×C4⋊(C3⋊Dic3)C2212yes—
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×#3750—21no—
17C2×C3×C3⋊(C3⋊C8)C2212yes—
18C2×C4⋊(C3⋊Dic3)D311yescharacteristic
19#64133D311yescharacteristic
20C2×C3×C4⋊Dic3—31no—
21C2×C3×C4⋊Dic3—31no—
22C2×C3×C4⋊Dic3—61no—
23C3×#924—31no—
24C3×#924—31no—
25C3×#924—61no—
26#63657D311yescharacteristic
27C3×#830—31no—
28C3×#830—31no—
29C3×#830—61no—
30#63574—32no—
31#63573—32no—
32#63573—32no—
33#63567—32no—
34#63566—32no—
35#63565—32no—
36C23×C33D411yescharacteristic
37C2×C33×C4—22no—
38C2×C33×C4D411yescharacteristic
39C2×C33×C4C2311yescharacteristic
40C2×C3×C3⋊Dic3—21no—
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3—22no—
43C2×C3×C3⋊Dic3—42no—
44C3×#634—22no—
45C3×C3⋊(C3⋊C8)—22no—
46#5043—91no—
47#4971—91no—
48#4721—91no—
49C22×C32×C4—21no—
50C22×C32×C4—21no—
51C22×C32×C4—21no—
52C22×C32×C4—41no—
53C22×C32×C4C2×D311yescharacteristic
54C22×C32×C4C2×D311yescharacteristic
55C22×C32×C4C2×D311yescharacteristic
56C22×C3⋊Dic3—21no—
57C2×#634C2×D311yescharacteristic
58C22×C3×Dic3—62no—
59C22×C3×Dic3—61no—
60C22×C3×Dic3—61no—
61C2×C3×Dic6—31no—
62C2×C3×Dic6—31no—
63C2×C3×Dic6—61no—
64#3757—21no—
65C4⋊(C3⋊Dic3)—21no—
66C4⋊(C3⋊Dic3)C2×D312yes—
67C4⋊(C3⋊Dic3)C2×D311yescharacteristic
68#3750—21no—
69C2×C3⋊(C3⋊C8)C2×D312yes—
70C3×#217—62no—
71C3×#217—61no—
72C3×#217—61no—
73C3×C4⋊Dic3—31no—
74C3×C4⋊Dic3—31no—
75C3×C4⋊Dic3—32no—
76C3×C4⋊Dic3—32no—
77C3×C4⋊Dic3—61no—
78C3×C4⋊Dic3—62no—
79C3×C4⋊Dic3—62no—
80C3×C4⋊Dic3—61no—
81C3×C4⋊Dic3—61no—
82C3×Dic3⋊C4—62no—
83C3×Dic3⋊C4—61no—
84C3×Dic3⋊C4—61no—
85C2×C3×C3⋊C8—32no—
86C2×C3×C3⋊C8—32no—
87C2×C3×C3⋊C8—62no—
88C22×C33—21no—
89C22×C33—21no—
90C22×C33SD1612yes—
91C22×C33C2×D411yescharacteristic
92C33×C4—21no—
93C33×C4#4112yes—
94C3×C3⋊Dic3—41no—
95C3×C3⋊Dic3—42no—
96C3×C3⋊Dic3—41no—
97C3×C22⋊Q8—91no—
98C2×C3×C4⋊C4—91no—
99C2×C4⋊Dic3—31no—
100C2×C4⋊Dic3—31no—
101C2×C4⋊Dic3—61no—
102#924—31no—
103#924—31no—
104#924—61no—
105C3×C22⋊C8—91no—
106#835—92no—
107#830—31no—
108#830—31no—
109#830—61no—
110#817—92no—
111#816—92no—
112#810—92no—
113#808—92no—
114#808—92no—
115C23×C32—21no—
116C23×C32—21no—
117C23×C32—21no—
118C23×C32—41no—
119C23×C32D1211yescharacteristic
120C23×C32#6711yescharacteristic
121C23×C32#6711yescharacteristic
122C2×C32×C4—22no—
123C2×C32×C4—24no—
124C2×C32×C4—22no—
125C2×C32×C4—22no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—41no—
133C2×C32×C4—41no—
134C2×C32×C4—42no—
135C2×C32×C4—42no—
136C2×C32×C4—44no—
137C2×C32×C4D1211yescharacteristic
138C2×C32×C4#6711yescharacteristic
139C2×C32×C4#6711yescharacteristic
140C2×C32×C4C22×D311yescharacteristic
141C2×C32×C4C22×D311yescharacteristic
142C2×C32×C4C22×D311yescharacteristic
143C2×C3⋊Dic3—21no—
144C2×C3⋊Dic3—21no—
145C2×C3⋊Dic3—22no—
146C2×C3⋊Dic3—42no—
147#634—22no—
148C2×C3×Dic3—62no—
149C2×C3×Dic3—62no—
150C2×C3×Dic3—64no—
151C2×C3×Dic3—61no—
152C2×C3×Dic3—61no—
153C2×C3×Dic3—62no—
154C2×C3×Dic3—61no—
155C2×C3×Dic3—61no—
156C2×C3×Dic3—62no—
157C2×C3×Dic3—124no—
158C2×C3×Dic3—122no—
159C2×C3×Dic3—122no—
160C3×Dic6—62no—
161C3×Dic6—62no—
162C3×Dic6—122no—
163C3⋊(C3⋊C8)—22no—
164C3×C3⋊C8—62no—
165C3×C3⋊C8—62no—
166C3×C3⋊C8—122no—
167#483—271noSylow
168C2×C33—22no—
169C2×C33#12311yescharacteristic
170C2×C33C2×SD1611yescharacteristic
171C2×C33#13711yescharacteristic
172C2×C3×Q8—91no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—41no—
177C22×C3×C4—181no—
178C22×C3×C4D3211yescharacteristic
179C22×C3×C4D3211yescharacteristic
180C22×C3×C4D3211yescharacteristic
181C22×Dic3—62no—
182C22×Dic3—61no—
183C22×Dic3—61no—
184C2×Dic6—31no—
185C2×Dic6—31no—
186C2×Dic6—61no—
187C2×C3×C8—92no—
188C3×C4⋊C4—91no—
189C3×C4⋊C4—92no—
190C3×C4⋊C4—181no—
191C3×C4⋊C4—181no—
192C3×C22⋊C4—181no—
193#217—62no—
194#217—61no—
195#217—61no—
196C4⋊Dic3—31no—
197C4⋊Dic3—32no—
198C4⋊Dic3—32no—
199C4⋊Dic3—31no—
200C4⋊Dic3—61no—
201C4⋊Dic3—62no—
202C4⋊Dic3—62no—
203C4⋊Dic3—61no—
204C4⋊Dic3—61no—
205Dic3⋊C4—62no—
206Dic3⋊C4—61no—
207Dic3⋊C4—61no—
208C2×C3⋊C8—32no—
209C2×C3⋊C8—32no—
210C2×C3⋊C8—62no—
211C22×C32—21no—
212C22×C32—21no—
213C22×C32—22no—
214C22×C32—22no—
215C22×C32—21no—
216C22×C32—21no—
217C22×C32—21no—
218C22×C32—21no—
219C22×C32—22no—
220C22×C32—22no—
221C22×C32—22no—
222C22×C32—21no—
223C22×C32—21no—
224C22×C32—21no—
225C22×C32—42no—
226C22×C32—41no—
227C22×C32—41no—
228C22×C32—41no—
229C22×C32—41no—
230C22×C32—41no—
231C22×C32—42no—
232C22×C32—42no—
233C22×C32#20412yes—
234C22×C32#21412yes—
235C22×C32#21512yes—
236C22×C32C2×D1211yescharacteristic
237C22×C32C2×#6711yescharacteristic
238C22×C32C2×#6711yescharacteristic
239C32×C4—21no—
240C32×C4—22no—
241C32×C4—21no—
242C32×C4—21no—
243C32×C4—22no—
244C32×C4—22no—
245C32×C4—22no—
246C32×C4—42no—
247C32×C4—41no—
248C32×C4—41no—
249C32×C4—42no—
250C32×C4#23512yes—
251C32×C4#23712yes—
252C32×C4#23712yes—
253C3⋊Dic3—42no—
254C3⋊Dic3—41no—
255C3⋊Dic3—41no—
256C3×Dic3—122no—
257C3×Dic3—122no—
258C3×Dic3—124no—
259C3×Dic3—121no—
260C3×Dic3—122no—
261C3×Dic3—121no—
262C3×Dic3—121no—
263C3×Dic3—122no—
264C3×Dic3—121no—
265C22⋊Q8—91no—
266C2×C4⋊C4—91no—
267#106—272no—
268#103—272no—
269C22⋊C8—91no—
270C33#48311yescharacteristic, complemented, Sylow
271C23×C3—21no—
272C23×C3—21no—
273C23×C3—21no—
274C23×C3—41no—
275C23×C3#62511yescharacteristic
276C23×C3#62611yescharacteristic
277C23×C3#62611yescharacteristic
278C3×Q8—182no—
279C2×C3×C4—24no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—44no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—181no—
295C2×C3×C4—181no—
296C2×C3×C4—182no—
297C2×C3×C4—362no—
298C2×C3×C4#62511yescharacteristic
299C2×C3×C4#62611yescharacteristic
300C2×C3×C4#62611yescharacteristic
301C2×C3×C4C2×D3211yescharacteristic
302C2×C3×C4C2×D3211yescharacteristic
303C2×C3×C4C2×D3211yescharacteristic
304C2×Dic3—62no—
305C2×Dic3—62no—
306C2×Dic3—64no—
307C2×Dic3—61no—
308C2×Dic3—61no—
309C2×Dic3—62no—
310C2×Dic3—61no—
311C2×Dic3—61no—
312C2×Dic3—62no—
313C2×Dic3—124no—
314C2×Dic3—122no—
315C2×Dic3—122no—
316Dic6—62no—
317Dic6—62no—
318Dic6—122no—
319C3×C8—182no—
320C3⋊C8—62no—
321C3⋊C8—62no—
322C3⋊C8—122no—
323C2×C32—22no—
324C2×C32—24no—
325C2×C32—22no—
326C2×C32—22no—
327C2×C32—21no—
328C2×C32—21no—
329C2×C32—21no—
330C2×C32—21no—
331C2×C32—21no—
332C2×C32—21no—
333C2×C32—21no—
334C2×C32—21no—
335C2×C32—21no—
336C2×C32—41no—
337C2×C32—41no—
338C2×C32—41no—
339C2×C32—42no—
340C2×C32—42no—
341C2×C32—44no—
342C2×C32#88611yescharacteristic
343C2×C32C2×#20411yescharacteristic
344C2×C32#90911yescharacteristic
345C2×C32#92911yescharacteristic
346C2×C32C2×#21411yescharacteristic
347C2×C32#93511yescharacteristic
348C2×C32C2×#21511yescharacteristic
349C2×C32#94211yescharacteristic
350C2×C32#95111yescharacteristic
351C2×Q8—91no—
352C22×C4—181no—
353C22×C4#111311yescharacteristic
354C2×C8—92no—
355C4⋊C4—92no—
356C4⋊C4—91no—
357C4⋊C4—181no—
358C4⋊C4—181no—
359C22⋊C4—181no—
360C22×C3—22no—
361C22×C3—22no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—22no—
372C22×C3—22no—
373C22×C3—22no—
374C22×C3—41no—
375C22×C3—41no—
376C22×C3—41no—
377C22×C3—41no—
378C22×C3—42no—
379C22×C3—42no—
380C22×C3—41no—
381C22×C3—42no—
382C22×C3#371512yes—
383C22×C3#371612yes—
384C22×C3#371712yes—
385C22×C3C2×#62511yescharacteristic
386C22×C3C2×#62611yescharacteristic
387C22×C3C2×#62611yescharacteristic
388C3×C4—22no—
389C3×C4—21no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—22no—
393C3×C4—22no—
394C3×C4—22no—
395C3×C4—41no—
396C3×C4—41no—
397C3×C4—42no—
398C3×C4—42no—
399C3×C4—362no—
400C3×C4—361no—
401C3×C4—361no—
402C3×C4#380412yes—
403C3×C4#380412yes—
404C3×C4#380512yes—
405Dic3—122no—
406Dic3—124no—
407Dic3—122no—
408Dic3—122no—
409Dic3—121no—
410Dic3—122no—
411Dic3—121no—
412Dic3—121no—
413Dic3—121no—
414C32—21no—
415C32—21no—
416C32—21no—
417C32—41no—
418C32#472111yescharacteristic, complemented
419C32#497111yescharacteristic, complemented
420C32#504311yescharacteristic, complemented
421C23#629511yescharacteristic
422Q8—182no—
423C2×C4—22no—
424C2×C4—181no—
425C2×C4—181no—
426C2×C4—182no—
427C2×C4—362no—
428C2×C4#629511yescharacteristic
429C2×C4C2×#111311yescharacteristic
430C8—182no—
431C2×C3—24no—
432C2×C3—22no—
433C2×C3—22no—
434C2×C3—22no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—42no—
445C2×C3—42no—
446C2×C3—44no—
447C2×C3—41no—
448C2×C3—41no—
449C2×C3—41no—
450C2×C3C2×#371511yescharacteristic
451C2×C3#6382711yescharacteristic
452C2×C3C2×#371611yescharacteristic
453C2×C3#6383111yescharacteristic
454C2×C3C2×#371711yescharacteristic
455C2×C3#6383311yescharacteristic
456C2×C3#6396111yescharacteristic
457C2×C3#6396211yescharacteristic
458C2×C3#6396611yescharacteristic
459C22—21no—
460C22—21no—
461C22#8881112yes—
462C22C2×#629511yescharacteristic
463C4—21no—
464C4—362no—
465C4—361no—
466C4—361no—
467C4#8904612yes—
468C3—21no—
469C3—21no—
470C3—21no—
471C3—41no—
472C3#1059110111yescharacteristic, complemented
473C3#1059112411yescharacteristic, complemented
474C3#1059112611yescharacteristic, complemented
475C2—22no—
476C2C2×#8881111yescharacteristic
477C2#1172366311yescharacteristic
478C2#1172372011yescharacteristic
subgroupG/H
centerC22C2×#6295
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radicalC23×C33D4
hypercenterC22×C4#1113
hypocenterC33#483
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #11722079C2semidirect
5. #11722076C2semidirect
270. C33#483semidirect
418. C32#4721semidirect
419. C32#4971semidirect
420. C32#5043semidirect
472. C3#10591101semidirect
473. C3#10591124semidirect
474. C3#10591126semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊(C3⋊Dic3)2
C3×#641332
C3×#636572
#117220792
#117220762
#105911263
#105911243
#105911013