Finite Group Databasebeta

← #13161810

Subgroups — #13161810

Order 1728.

1194 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊(C3⋊D3)C211yesmaximal, characteristic
2C3×#64135C211yesmaximal, characteristic, complemented
3C2×C3×#3748C211yesmaximal, characteristic, complemented
4#11722075C214yesmaximal, complemented
5#10591096—31nomaximal
6#10591086—31nomaximal
7#10591085—31nomaximal
8C23×C3×C3⋊D3—21no—
9C22×C33×C4C2211yescharacteristic
10C2×C3×C4⋊(C3⋊D3)—21no—
11C2×C3×C4⋊(C3⋊D3)—22no—
12C2×C3×C4⋊(C3⋊D3)C414yescomplemented
13C2×C3×C4⋊(C3⋊D3)C2212yes—
14C3×#3757—21no—
15C3×C4⋊(C3⋊Dic3)C2212yescomplemented
16C3×C4×C3⋊Dic3—21no—
17C3×#3748—22no—
18C2×C3×C3⋊(C3⋊C8)C2212yescomplemented
19C22×C4⋊(C3⋊D3)D311yescharacteristic
20C22×C3×D12—31no—
21C22×C3×D12—31no—
22C22×C3×D12—61no—
23#64135D311yescharacteristic, complemented
24C2×#3748D311yescharacteristic, complemented
25C3×#926—31no—
26C3×#926—31no—
27C3×#926—61no—
28C2×C3×#208—31no—
29C2×C3×#208—31no—
30C2×C3×#208—61no—
31#63564—34no—
32#63563—34no—
33#63562—34no—
34C23×C33D411yescharacteristic
35C22×C3×C3⋊D3—24no—
36C22×C3×C3⋊D3—22no—
37C22×C3×C3⋊D3—44no—
38C2×C33×C4C2×C412yes—
39C2×C33×C4D411yescharacteristic
40C2×C33×C4D412yes—
41C2×C33×C4C2311yescharacteristic
42C2×C3×C3⋊Dic3—22no—
43C3×C4⋊(C3⋊D3)—24no—
44C3×C4⋊(C3⋊D3)—22no—
45C3×C4⋊(C3⋊D3)C2×C414yes—
46C3×C3⋊(C3⋊C8)—22no—
47#5215—91no—
48#5131—91no—
49#5001—91no—
50C23×C3⋊D3—21no—
51C23×C3×D3—62no—
52C23×C3×D3—61no—
53C23×C3×D3—61no—
54C22×C32×C4—21no—
55C22×C32×C4—21no—
56C22×C32×C4—21no—
57C22×C32×C4—41no—
58C22×C32×C4C2×D311yescharacteristic
59C22×C32×C4C2×D311yescharacteristic
60C22×C32×C4C2×D311yescharacteristic
61C2×C4⋊(C3⋊D3)—21no—
62C2×C4⋊(C3⋊D3)—22no—
63C2×C4⋊(C3⋊D3)Dic314yescomplemented
64C2×C4⋊(C3⋊D3)C2×D312yes—
65C2×C3×D12—34no—
66C2×C3×D12—34no—
67C2×C3×D12—32no—
68C2×C3×D12—32no—
69C2×C3×D12—64no—
70C2×C3×D12—62no—
71C2×C3×D12—62no—
72C2×C3×D12—64no—
73C2×C3×D12—61no—
74C2×C3×D12—62no—
75C2×C3×D12—61no—
76C2×C3×D12—62no—
77#3757—21no—
78C4⋊(C3⋊Dic3)C2×D312yescomplemented
79C4×C3⋊Dic3—21no—
80#3748—22no—
81C2×C3⋊(C3⋊C8)C2×D312yescomplemented
82C3×#217—62no—
83C3×#217—61no—
84C3×#217—61no—
85C3×C4⋊Dic3—32no—
86C3×C4⋊Dic3—32no—
87C3×C4⋊Dic3—62no—
88C3×C4×Dic3—62no—
89C3×C4×Dic3—61no—
90C3×C4×Dic3—61no—
91C3×#208—64no—
92C3×#208—62no—
93C3×#208—62no—
94C2×C3×C3⋊C8—32no—
95C2×C3×C3⋊C8—32no—
96C2×C3×C3⋊C8—62no—
97C22×C33—22no—
98C22×C33C22⋊C412yes—
99C22×C33C2×D411yescharacteristic
100C2×C3×C3⋊D3—24no—
101C2×C3×C3⋊D3—48no—
102C2×C3×C3⋊D3—44no—
103C33×C4C22⋊C412yes—
104C33×C4C22×C411yescharacteristic
105C33×C4C2×D411yescharacteristic
106C3×C3⋊Dic3—42no—
107C22×C3×D4—91no—
108C22×D12—31no—
109C22×D12—31no—
110C22×D12—61no—
111C2×C3×M16—91no—
112C3×#117—91no—
113#926—31no—
114#926—31no—
115#926—61no—
116C2×#208—31no—
117C2×#208—31no—
118C2×#208—61no—
119#832—94no—
120#821—94no—
121#809—94no—
122C23×C32—21no—
123C23×C32—21no—
124C23×C32—21no—
125C23×C32—41no—
126C23×C32D1211yescharacteristic
127C23×C32#6711yescharacteristic
128C23×C32#6711yescharacteristic
129C22×C3⋊D3—24no—
130C22×C3⋊D3—22no—
131C22×C3⋊D3—44no—
132C22×C3×D3—68no—
133C22×C3×D3—64no—
134C22×C3×D3—64no—
135C22×C3×D3—64no—
136C22×C3×D3—62no—
137C22×C3×D3—62no—
138C22×C3×D3—128no—
139C22×C3×D3—124no—
140C22×C3×D3—124no—
141C2×C32×C4—22no—
142C2×C32×C4—22no—
143C2×C32×C4—22no—
144C2×C32×C4—21no—
145C2×C32×C4—22no—
146C2×C32×C4—22no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—22no—
150C2×C32×C4—21no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—42no—
154C2×C32×C4—42no—
155C2×C32×C4—41no—
156C2×C32×C4—41no—
157C2×C32×C4C4×D312yes—
158C2×C32×C4C4×D312yes—
159C2×C32×C4D1211yescharacteristic
160C2×C32×C4D1212yes—
161C2×C32×C4C2×Dic312yes—
162C2×C32×C4#6711yescharacteristic
163C2×C32×C4#6712yes—
164C2×C32×C4#6711yescharacteristic
165C2×C32×C4#6712yes—
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C22×D311yescharacteristic
169C2×C3⋊Dic3—22no—
170C4⋊(C3⋊D3)—24no—
171C4⋊(C3⋊D3)—22no—
172C4⋊(C3⋊D3)C2×Dic314yes—
173C2×C3×Dic3—64no—
174C2×C3×Dic3—62no—
175C2×C3×Dic3—62no—
176C3×D12—34no—
177C3×D12—34no—
178C3×D12—64no—
179C3×D12—68no—
180C3×D12—64no—
181C3×D12—64no—
182C3×D12—64no—
183C3×D12—62no—
184C3×D12—62no—
185C3⋊(C3⋊C8)—22no—
186C3×C3⋊C8—64no—
187C3×C3⋊C8—62no—
188C3×C3⋊C8—62no—
189#418—271noSylow
190C2×C33—22no—
191C2×C33C2×C22⋊C411yescharacteristic
192C2×C33AGL(1,ℤ/8)12yes—
193C3×C3⋊D3—44no—
194C24×C3—181no—
195C23×D3—62no—
196C23×D3—61no—
197C23×D3—61no—
198C2×C3×D4—94no—
199C2×C3×D4—92no—
200C2×C3×D4—181no—
201C2×C3×D4—182no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—41no—
206C22×C3×C4D3211yescharacteristic
207C22×C3×C4D3211yescharacteristic
208C22×C3×C4D3211yescharacteristic
209C2×D12—34no—
210C2×D12—34no—
211C2×D12—32no—
212C2×D12—32no—
213C2×D12—64no—
214C2×D12—62no—
215C2×D12—64no—
216C2×D12—62no—
217C2×D12—61no—
218C2×D12—62no—
219C2×D12—61no—
220C2×D12—62no—
221C3×M16—182no—
222C2×C3×C8—92no—
223C3×C4⋊C4—92no—
224C3×C22⋊C4—181no—
225C3×C42—181no—
226#217—62no—
227#217—61no—
228#217—61no—
229C4⋊Dic3—32no—
230C4⋊Dic3—32no—
231C4⋊Dic3—62no—
232C4×Dic3—62no—
233C4×Dic3—61no—
234C4×Dic3—61no—
235#208—64no—
236#208—62no—
237#208—62no—
238C2×C3⋊C8—32no—
239C2×C3⋊C8—32no—
240C2×C3⋊C8—62no—
241C22×C32—22no—
242C22×C32—24no—
243C22×C32—22no—
244C22×C32—22no—
245C22×C32—22no—
246C22×C32—22no—
247C22×C32—22no—
248C22×C32—21no—
249C22×C32—21no—
250C22×C32—21no—
251C22×C32—42no—
252C22×C32—41no—
253C22×C32—42no—
254C22×C32—42no—
255C22×C32—44no—
256C22×C32(C2×D3)⋊C412yes—
257C22×C32(C2×D3)⋊C412yes—
258C22×C32#21712yes—
259C22×C32C2×D1211yescharacteristic
260C22×C32C2×#6711yescharacteristic
261C22×C32C2×#6711yescharacteristic
262C2×C3⋊D3—24no—
263C2×C3⋊D3—48no—
264C2×C3⋊D3—44no—
265C2×C3×D3—68no—
266C2×C3×D3—64no—
267C2×C3×D3—64no—
268C2×C3×D3—1216no—
269C2×C3×D3—128no—
270C2×C3×D3—128no—
271C2×C3×D3—128no—
272C2×C3×D3—124no—
273C2×C3×D3—124no—
274C32×C4—22no—
275C32×C4—22no—
276C32×C4—22no—
277C32×C4—21no—
278C32×C4—21no—
279C32×C4—21no—
280C32×C4—21no—
281C32×C4—21no—
282C32×C4—21no—
283C32×C4—42no—
284C32×C4—41no—
285C32×C4—41no—
286C32×C4(C2×D3)⋊C412yes—
287C32×C4(C2×D3)⋊C412yes—
288C32×C4#21712yes—
289C32×C4C2×C4×D311yescharacteristic
290C32×C4C2×C4×D311yescharacteristic
291C32×C4C2×D1211yescharacteristic
292C32×C4C22×Dic311yescharacteristic
293C32×C4C2×#6711yescharacteristic
294C32×C4C2×#6711yescharacteristic
295C3⋊Dic3—42no—
296C3×Dic3—124no—
297C3×Dic3—122no—
298C3×Dic3—122no—
299C22×D4—91no—
300C2×M16—91no—
301#117—91no—
302#102—274no—
303C33#41811yescharacteristic, complemented, Sylow
304C23×C3—21no—
305C23×C3—21no—
306C23×C3—21no—
307C23×C3—41no—
308C23×C3—184no—
309C23×C3—182no—
310C23×C3—364no—
311C23×C3#62511yescharacteristic
312C23×C3#62611yescharacteristic
313C23×C3#62611yescharacteristic
314C22×D3—64no—
315C22×D3—68no—
316C22×D3—64no—
317C22×D3—64no—
318C22×D3—62no—
319C22×D3—62no—
320C22×D3—128no—
321C22×D3—124no—
322C22×D3—124no—
323C3×D4—94no—
324C3×D4—184no—
325C3×D4—182no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—21no—
331C2×C3×C4—22no—
332C2×C3×C4—21no—
333C2×C3×C4—22no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—182no—
343C2×C3×C4D3×Dic312yes—
344C2×C3×C4D3×Dic312yes—
345C2×C3×C4#62412yes—
346C2×C3×C4#62511yescharacteristic
347C2×C3×C4#62512yes—
348C2×C3×C4#62611yescharacteristic
349C2×C3×C4#62612yes—
350C2×C3×C4#62611yescharacteristic
351C2×C3×C4#62612yes—
352C2×C3×C4C2×D3211yescharacteristic
353C2×C3×C4C2×D3211yescharacteristic
354C2×C3×C4C2×D3211yescharacteristic
355C2×Dic3—64no—
356C2×Dic3—62no—
357C2×Dic3—62no—
358D12—34no—
359D12—34no—
360D12—64no—
361D12—64no—
362D12—68no—
363D12—64no—
364D12—64no—
365D12—62no—
366D12—62no—
367C3×C8—182no—
368C3⋊C8—64no—
369C3⋊C8—62no—
370C3⋊C8—62no—
371C2×C32—22no—
372C2×C32—24no—
373C2×C32—22no—
374C2×C32—22no—
375C2×C32—22no—
376C2×C32—22no—
377C2×C32—22no—
378C2×C32—21no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—41no—
382C2×C32—42no—
383C2×C32—42no—
384C2×C32—42no—
385C2×C32—44no—
386C2×C32#90812yes—
387C2×C32C2×(C2×D3)⋊C411yescharacteristic
388C2×C32C2×(C2×D3)⋊C411yescharacteristic
389C2×C32#93212yes—
390C2×C32#94912yes—
391C2×C32C2×#21711yescharacteristic
392C3⋊D3—44no—
393C3×D3—128no—
394C3×D3—124no—
395C3×D3—124no—
396C24—181no—
397C2×D4—94no—
398C2×D4—92no—
399C2×D4—182no—
400C2×D4—181no—
401C22×C4#111311yescharacteristic
402M16—182no—
403C2×C8—92no—
404C4⋊C4—92no—
405C22⋊C4—181no—
406C42—181no—
407C22×C3—24no—
408C22×C3—22no—
409C22×C3—22no—
410C22×C3—22no—
411C22×C3—21no—
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—22no—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3—42no—
418C22×C3—42no—
419C22×C3—44no—
420C22×C3—41no—
421C22×C3—42no—
422C22×C3—184no—
423C22×C3—368no—
424C22×C3—364no—
425C22×C3(C2×D3)⋊Dic312yes—
426C22×C3(C2×D3)⋊Dic312yes—
427C22×C3#372212yes—
428C22×C3C2×#62511yescharacteristic
429C22×C3C2×#62611yescharacteristic
430C22×C3C2×#62611yescharacteristic
431C2×D3—68no—
432C2×D3—64no—
433C2×D3—64no—
434C2×D3—1216no—
435C2×D3—128no—
436C2×D3—128no—
437C2×D3—128no—
438C2×D3—124no—
439C2×D3—124no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—21no—
446C3×C4—22no—
447C3×C4—22no—
448C3×C4—22no—
449C3×C4—41no—
450C3×C4—41no—
451C3×C4—42no—
452C3×C4—362no—
453C3×C4(C2×D3)⋊Dic312yes—
454C3×C4(C2×D3)⋊Dic312yes—
455C3×C4#372212yes—
456C3×C4C2×D3×Dic311yescharacteristic
457C3×C4C2×D3×Dic311yescharacteristic
458C3×C4C2×#62411yescharacteristic
459C3×C4C2×#62511yescharacteristic
460C3×C4C2×#62611yescharacteristic
461C3×C4C2×#62611yescharacteristic
462Dic3—124no—
463Dic3—122no—
464Dic3—122no—
465C32—21no—
466C32—21no—
467C32—21no—
468C32—41no—
469C32#500111yescharacteristic, complemented
470C32#513111yescharacteristic, complemented
471C32#521511yescharacteristic, complemented
472C23—182no—
473C23—184no—
474C23—364no—
475C23#629511yescharacteristic
476D4—94no—
477D4—182no—
478D4—184no—
479C2×C4—182no—
480C2×C4#629412yes—
481C2×C4#629512yes—
482C2×C4#629511yescharacteristic
483C2×C4C2×#111311yescharacteristic
484C8—182no—
485C2×C3—24no—
486C2×C3—22no—
487C2×C3—22no—
488C2×C3—22no—
489C2×C3—22no—
490C2×C3—22no—
491C2×C3—22no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—42no—
496C2×C3—42no—
497C2×C3—44no—
498C2×C3—42no—
499C2×C3—41no—
500C2×C3—364no—
501C2×C3#6382312yes—
502C2×C3#6382512yes—
503C2×C3#6383012yes—
504C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
505C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
506C2×C3C2×#372211yescharacteristic
507D3—128no—
508D3—124no—
509D3—124no—
510C22—22no—
511C22—184no—
512C22—368no—
513C22—364no—
514C22#8881412yes—
515C22C2×#629511yescharacteristic
516C4—362no—
517C4#8881412yes—
518C4C2×#629411yescharacteristic
519C4C2×#629511yescharacteristic
520C3—21no—
521C3—21no—
522C3—21no—
523C3—41no—
524C3#1059108511yescharacteristic, complemented
525C3#1059108611yescharacteristic, complemented
526C3#1059109611yescharacteristic, complemented
527C2—22no—
528C2—364no—
529C2#1172366112yes—
530C2C2×#8881411yescharacteristic
subgroupG/H
centerC22C2×#6295
commutatorC33×C4C22×C4
FrattiniC2×C4C2×#1113
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radicalC22×C3×C4⋊(C3⋊D3)C2
cosocleC2×C33×C4C23
hypercenterC22×C4#1113
hypocenterC33#418
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C3×#64135C2semidirect
3. C2×C3×#3748C2semidirect
4. #11722075C2semidirect
12. C2×C3×C4⋊(C3⋊D3)C4semidirect
15. C3×C4⋊(C3⋊Dic3)C22semidirect
18. C2×C3×C3⋊(C3⋊C8)C22semidirect
23. #64135D3semidirect
24. C2×#3748D3semidirect
63. C2×C4⋊(C3⋊D3)Dic3semidirect
78. C4⋊(C3⋊Dic3)C2×D3semidirect
81. C2×C3⋊(C3⋊C8)C2×D3semidirect
303. C33#418semidirect
469. C32#5001semidirect
470. C32#5131semidirect
471. C32#5215semidirect
524. C3#10591085semidirect
525. C3#10591086semidirect
526. C3#10591096semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊(C3⋊D3)2
C3×#641352
C2×C3×#37482
#117220752
#105910963
#105910863
#105910853