Finite Group Databasebeta

← #13156701

Subgroups — #13156701

Order 1728.

1666 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6289C211yesmaximal, characteristic
2C2×#88804C212yesmaximal
3C2×C3×Dic32C212yesmaximal, complemented
4#11722082C211yesmaximal, characteristic, complemented
5C3×#63579C211yesmaximal, characteristic, complemented
6C4×#3722—31nomaximal
7#10591233—32nomaximal
8C23×C3⋊(C3⋊D3)C2211yescharacteristic
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C22×C3×C3⋊Dic3C2212yes—
11C22×C32×Dic3C2212yes—
12C22×C32×Dic3C2211yescharacteristic
13C2×#6289—24no—
14C2×#6289C414yescomplemented
15#88804C418yescomplemented
16C3×Dic32—24no—
17C22×C4×C3⋊D3—31no—
18C22×#624—32no—
19C22×#624—61no—
20C2×#3752—32no—
21C2×C3×C4×Dic3—32no—
22C2×C3×C4×Dic3—32no—
23C2×#3722—32no—
24C2×#3722—32no—
25C2×#3722—62no—
26C2×Dic32D312yescomplemented
27C3×C3⋊#95—32no—
28#63579—32no—
29#63579—31no—
30#63579D311yescharacteristic, complemented
31C23×C33C2311yescharacteristic
32C22×C3⋊(C3⋊D3)—22no—
33C22×C3⋊(C3⋊D3)—21no—
34C22×C3⋊(C3⋊D3)—21no—
35C22×C3⋊(C3⋊D3)C2×C412yes—
36C22×C3⋊(C3⋊D3)C2×C414yes—
37C2×C3×C3⋊Dic3—22no—
38C2×C3×C3⋊Dic3—22no—
39C2×C3×C3⋊Dic3—22no—
40C2×C3×C3⋊Dic3C2×C412yes—
41C2×C3×C3⋊Dic3C2×C414yescomplemented
42C2×C32×Dic3—22no—
43C2×C32×Dic3—22no—
44C2×C32×Dic3C2×C412yes—
45C2×C32×Dic3C2×C414yescomplemented
46C2×C32×Dic3D414yes—
47#6289—28no—
48C4×(C2×D3)⋊C4—92no—
49#4666—91no—
50C23×C3⋊D3—32no—
51C23×C3⋊D3—31no—
52C23×C3⋊D3—61no—
53C23×C3⋊D3—62no—
54C23×C3⋊D3—121no—
55C22×C32×C4—31no—
56C22×C32×C4—32no—
57C22×C3⋊Dic3C2×D311yescharacteristic
58C22×C3⋊Dic3C2×D312yes—
59C2×C4×C3⋊D3—34no—
60C2×C4×C3⋊D3—64no—
61C22×C3×Dic3—21no—
62C22×C3×Dic3—22no—
63C22×C3×Dic3—32no—
64C22×C3×Dic3—32no—
65C22×C3×Dic3—32no—
66C22×C3×Dic3—61no—
67C22×C3×Dic3—62no—
68C22×C3×Dic3C2×D312yes—
69C22×C3×Dic3C2×D312yes—
70C22×C3×Dic3C2×D312yes—
71C2×#624—38no—
72C2×#624—64no—
73C2×#624—68no—
74C2×#624—64no—
75C2×#624—64no—
76#3752—38no—
77C3×C4×Dic3—64no—
78C3×C4×Dic3—64no—
79#3722—38no—
80#3722—38no—
81#3722—68no—
82Dic32—24no—
83C22×C33C22×C411yescharacteristic
84C22×C33C22×C412yes—
85C22×C33C2×D412yes—
86C22×C33#4111yescharacteristic
87C22×C33#4111yescharacteristic
88C2×C3⋊(C3⋊D3)—22no—
89C2×C3⋊(C3⋊D3)—22no—
90C2×C3⋊(C3⋊D3)—24no—
91C2×C3⋊(C3⋊D3)—24no—
92C2×C3⋊(C3⋊D3)C4214yescomplemented
93C3×C3⋊Dic3—22no—
94C3×C3⋊Dic3—24no—
95C32×Dic3—24no—
96C32×Dic3C22⋊C414yescomplemented
97C22×C4×D3—92no—
98C22×C4×D3—91no—
99C22×C4×D3—181no—
100C2×C3×C42—92no—
101C2×(C2×D3)⋊C4—92no—
102C2×(C2×D3)⋊C4—92no—
103C2×(C2×D3)⋊C4—92no—
104C2×(C2×D3)⋊C4—182no—
105C2×C4×Dic3—32no—
106C2×C4×Dic3—32no—
107C3×#95—91no—
108C3⋊#95—32no—
109C3⋊#95—91no—
110C3⋊#95—92no—
111C23×C32—21no—
112C23×C32—22no—
113C23×C32—41no—
114C23×C32C22×D312yes—
115C23×C32C22×D311yescharacteristic
116C22×C3⋊D3—34no—
117C22×C3⋊D3—34no—
118C22×C3⋊D3—34no—
119C22×C3⋊D3—34no—
120C22×C3⋊D3—32no—
121C22×C3⋊D3—64no—
122C22×C3⋊D3—62no—
123C22×C3⋊D3—64no—
124C22×C3⋊D3—64no—
125C22×C3⋊D3—64no—
126C22×C3⋊D3—62no—
127C22×C3⋊D3—61no—
128C22×C3⋊D3—61no—
129C22×C3⋊D3—62no—
130C22×C3⋊D3—62no—
131C22×C3⋊D3—64no—
132C22×C3⋊D3—62no—
133C22×C3⋊D3—62no—
134C22×C3⋊D3—62no—
135C22×C3⋊D3—62no—
136C22×C3⋊D3—122no—
137C22×C3⋊D3—124no—
138C22×C3⋊D3—122no—
139C22×C3⋊D3—122no—
140C22×C3⋊D3—122no—
141C22×C3⋊D3—122no—
142C22×C3⋊D3—124no—
143C22×C3⋊D3—122no—
144C22×C3⋊D3—122no—
145C2×C32×C4—32no—
146C2×C32×C4—34no—
147C2×C32×C4—34no—
148C2×C32×C4—62no—
149C2×C32×C4—62no—
150C2×C3⋊Dic3—22no—
151C2×C3⋊Dic3—22no—
152C2×C3⋊Dic3—22no—
153C2×C3⋊Dic3C4×D312yes—
154C2×C3⋊Dic3C4×D314yescomplemented
155C4×C3⋊D3—68no—
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—24no—
163C2×C3×Dic3—34no—
164C2×C3×Dic3—34no—
165C2×C3×Dic3—34no—
166C2×C3×Dic3—42no—
167C2×C3×Dic3—42no—
168C2×C3×Dic3—62no—
169C2×C3×Dic3—64no—
170C2×C3×Dic3—64no—
171C2×C3×Dic3—62no—
172C2×C3×Dic3—62no—
173C2×C3×Dic3—62no—
174C2×C3×Dic3—62no—
175C2×C3×Dic3—62no—
176C2×C3×Dic3—62no—
177C2×C3×Dic3—122no—
178C2×C3×Dic3—122no—
179C2×C3×Dic3C4×D314yes—
180C2×C3×Dic3C4×D314yescomplemented
181C2×C3×Dic3C4×D314yescomplemented
182C2×C3×Dic3D1214yes—
183C2×C3×Dic3#6714yes—
184#624—616no—
185#624—616no—
186C4×C22⋊C4—271noSylow
187C2×C33C2×C4211yescharacteristic
188C2×C33C2×C22⋊C411yescharacteristic
189C2×C33#11711yescharacteristic
190C2×C33C4×D412yes—
191C2×C33C4×D412yes—
192C3⋊(C3⋊D3)—24no—
193C23×D3—91no—
194C23×D3—92no—
195C23×D3—181no—
196C23×D3—182no—
197C23×D3—361no—
198C22×C3×C4—32no—
199C22×C3×C4—32no—
200C22×C3×C4—32no—
201C22×C3×C4—61no—
202C22×C3×C4—62no—
203C22×C3×C4—91no—
204C22×C3×C4—92no—
205C22×Dic3—32no—
206C22×Dic3—32no—
207C22×Dic3—32no—
208C22×Dic3—61no—
209C22×Dic3—62no—
210C22×Dic3D3212yes—
211C22×Dic3D3211yescharacteristic
212C2×C4×D3—98no—
213C2×C4×D3—94no—
214C2×C4×D3—184no—
215C2×C4×D3—184no—
216C2×C4×D3—184no—
217C2×C4×D3—188no—
218C2×C4×D3—184no—
219C3×C42—184no—
220(C2×D3)⋊C4—98no—
221(C2×D3)⋊C4—98no—
222(C2×D3)⋊C4—98no—
223(C2×D3)⋊C4—188no—
224C4×Dic3—64no—
225C4×Dic3—64no—
226C22×C32—21no—
227C22×C32—21no—
228C22×C32—22no—
229C22×C32—22no—
230C22×C32—22no—
231C22×C32—22no—
232C22×C32—22no—
233C22×C32—22no—
234C22×C32—21no—
235C22×C32—22no—
236C22×C32—22no—
237C22×C32—22no—
238C22×C32—41no—
239C22×C32—42no—
240C22×C32—42no—
241C22×C32—41no—
242C22×C32—41no—
243C22×C32C2×C4×D312yes—
244C22×C32C2×C4×D311yescharacteristic
245C22×C32C2×C4×D312yes—
246C22×C32C2×C4×D312yes—
247C22×C32C2×C4×D312yes—
248C22×C32C2×D1212yes—
249C22×C32#23512yes—
250C22×C32#23512yes—
251C22×C32D3×D412yes—
252C22×C32#23711yescharacteristic
253C22×C32#23911yescharacteristic
254C22×C32C2×#6712yes—
255C2×C3⋊D3—34no—
256C2×C3⋊D3—38no—
257C2×C3⋊D3—64no—
258C2×C3⋊D3—68no—
259C2×C3⋊D3—62no—
260C2×C3⋊D3—64no—
261C2×C3⋊D3—62no—
262C2×C3⋊D3—64no—
263C2×C3⋊D3—64no—
264C2×C3⋊D3—64no—
265C2×C3⋊D3—68no—
266C2×C3⋊D3—68no—
267C2×C3⋊D3—64no—
268C2×C3⋊D3—64no—
269C2×C3⋊D3—64no—
270C2×C3⋊D3—64no—
271C2×C3⋊D3—64no—
272C2×C3⋊D3—64no—
273C2×C3⋊D3—124no—
274C2×C3⋊D3—124no—
275C2×C3⋊D3—124no—
276C2×C3⋊D3—124no—
277C2×C3⋊D3—124no—
278C2×C3⋊D3—124no—
279C2×C3⋊D3—124no—
280C2×C3⋊D3—124no—
281C2×C3⋊D3—124no—
282C2×C3⋊D3—128no—
283C2×C3⋊D3—128no—
284C32×C4—34no—
285C32×C4—64no—
286C3⋊Dic3—22no—
287C3⋊Dic3—24no—
288C3×Dic3—24no—
289C3×Dic3—24no—
290C3×Dic3—24no—
291C3×Dic3—44no—
292C3×Dic3—64no—
293C3×Dic3—64no—
294C3×Dic3—64no—
295C3×Dic3—64no—
296C3×Dic3—124no—
297C3×Dic3(C2×D3)⋊C418yescomplemented
298C23×C4—271no—
299C2×C22⋊C4—272no—
300C2×C42—92no—
301#95—91no—
302#95—271no—
303C33C4×C22⋊C411yescharacteristic, complemented, Sylow
304C23×C3—21no—
305C23×C3—22no—
306C23×C3—41no—
307C23×C3C2×D3211yescharacteristic
308C23×C3C2×D3212yes—
309C22×D3—94no—
310C22×D3—94no—
311C22×D3—94no—
312C22×D3—94no—
313C22×D3—92no—
314C22×D3—182no—
315C22×D3—184no—
316C22×D3—184no—
317C22×D3—184no—
318C22×D3—184no—
319C22×D3—184no—
320C22×D3—182no—
321C22×D3—182no—
322C22×D3—181no—
323C22×D3—181no—
324C22×D3—182no—
325C22×D3—182no—
326C22×D3—182no—
327C22×D3—182no—
328C22×D3—182no—
329C22×D3—364no—
330C22×D3—362no—
331C22×D3—362no—
332C22×D3—362no—
333C22×D3—362no—
334C22×D3—362no—
335C22×D3—364no—
336C22×D3—362no—
337C22×D3—362no—
338C2×C3×C4—34no—
339C2×C3×C4—34no—
340C2×C3×C4—34no—
341C2×C3×C4—34no—
342C2×C3×C4—34no—
343C2×C3×C4—62no—
344C2×C3×C4—64no—
345C2×C3×C4—64no—
346C2×C3×C4—62no—
347C2×C3×C4—62no—
348C2×C3×C4—62no—
349C2×C3×C4—62no—
350C2×C3×C4—94no—
351C2×C3×C4—92no—
352C2×C3×C4—122no—
353C2×C3×C4—122no—
354C2×C3×C4—182no—
355C2×C3×C4—182no—
356C2×C3×C4—182no—
357C2×Dic3—22no—
358C2×Dic3—22no—
359C2×Dic3—34no—
360C2×Dic3—34no—
361C2×Dic3—34no—
362C2×Dic3—62no—
363C2×Dic3—64no—
364C2×Dic3—64no—
365C2×Dic3—62no—
366C2×Dic3—62no—
367C2×Dic3—62no—
368C2×Dic3—62no—
369C2×Dic3—62no—
370C2×Dic3—62no—
371C2×Dic3—122no—
372C2×Dic3—122no—
373C2×Dic3#62412yes—
374C2×Dic3#62414yescomplemented
375C2×Dic3#62614yes—
376C4×D3—1816no—
377C4×D3—1816no—
378C4×D3—188no—
379C2×C32—22no—
380C2×C32—21no—
381C2×C32—22no—
382C2×C32—22no—
383C2×C32—21no—
384C2×C32—22no—
385C2×C32—22no—
386C2×C32—22no—
387C2×C32—22no—
388C2×C32—22no—
389C2×C32—22no—
390C2×C32—21no—
391C2×C32—41no—
392C2×C32—41no—
393C2×C32—41no—
394C2×C32—42no—
395C2×C32—42no—
396C2×C32C42×D311yescharacteristic
397C2×C32C42×D312yes—
398C2×C32#87212yes—
399C2×C32C4×D1212yes—
400C2×C32C4×D1212yes—
401C2×C32D3×C22⋊C411yescharacteristic
402C2×C32#67⋊C412yes—
403C2×C32#89211yescharacteristic
404C2×C32#89312yes—
405C2×C32C2×(C2×D3)⋊C412yes—
406C2×C32C4×#6712yes—
407C2×C32C4×#6712yes—
408C3⋊D3—64no—
409C3⋊D3—68no—
410C3⋊D3—68no—
411C3⋊D3—128no—
412C3⋊D3—128no—
413C24—271no—
414C22×C4—31no—
415C22×C4—32no—
416C22×C4—91no—
417C22×C4—92no—
418C22×C4—274no—
419C22×C4—544no—
420C22⋊C4—278no—
421C42—184no—
422C22×C3—21no—
423C22×C3—22no—
424C22×C3—22no—
425C22×C3—22no—
426C22×C3—22no—
427C22×C3—22no—
428C22×C3—22no—
429C22×C3—22no—
430C22×C3—22no—
431C22×C3—21no—
432C22×C3—22no—
433C22×C3—21no—
434C22×C3—41no—
435C22×C3—42no—
436C22×C3—42no—
437C22×C3—41no—
438C22×C3—41no—
439C22×C3#379811yescharacteristic
440C22×C3#379811yescharacteristic
441C22×C3#379912yes—
442C22×C3C4×D3212yes—
443C22×C3C4×D3212yes—
444C22×C3C4×D3212yes—
445C22×C3D3×D1212yes—
446C22×C3#380412yes—
447C22×C3C2×#62411yescharacteristic
448C22×C3C2×#62412yes—
449C22×C3C2×#62612yes—
450C22×C3D3×#6712yes—
451C2×D3—98no—
452C2×D3—94no—
453C2×D3—184no—
454C2×D3—188no—
455C2×D3—184no—
456C2×D3—184no—
457C2×D3—188no—
458C2×D3—188no—
459C2×D3—184no—
460C2×D3—182no—
461C2×D3—184no—
462C2×D3—184no—
463C2×D3—184no—
464C2×D3—184no—
465C2×D3—184no—
466C2×D3—184no—
467C2×D3—184no—
468C2×D3—182no—
469C2×D3—364no—
470C2×D3—364no—
471C2×D3—364no—
472C2×D3—364no—
473C2×D3—364no—
474C2×D3—364no—
475C2×D3—364no—
476C2×D3—364no—
477C2×D3—364no—
478C2×D3—368no—
479C2×D3—368no—
480C3×C4—38no—
481C3×C4—64no—
482C3×C4—64no—
483C3×C4—64no—
484C3×C4—124no—
485C3×C4—184no—
486C3×C4—182no—
487Dic3—24no—
488Dic3—64no—
489Dic3—64no—
490Dic3—64no—
491Dic3—64no—
492Dic3—124no—
493Dic3#372214yescomplemented
494C32—22no—
495C32—21no—
496C32—41no—
497C32#466611yescharacteristic, complemented
498C32C4×(C2×D3)⋊C412yescomplemented
499C23—274no—
500C23—272no—
501C23—542no—
502C23—541no—
503C23—541no—
504C23D3311yescharacteristic
505C2×C4—32no—
506C2×C4—34no—
507C2×C4—34no—
508C2×C4—62no—
509C2×C4—62no—
510C2×C4—94no—
511C2×C4—92no—
512C2×C4—182no—
513C2×C4—182no—
514C2×C4—182no—
515C2×C4—548no—
516C2×C3—21no—
517C2×C3—22no—
518C2×C3—21no—
519C2×C3—22no—
520C2×C3—21no—
521C2×C3—22no—
522C2×C3—22no—
523C2×C3—22no—
524C2×C3—22no—
525C2×C3—22no—
526C2×C3—22no—
527C2×C3—22no—
528C2×C3—41no—
529C2×C3—41no—
530C2×C3—41no—
531C2×C3—42no—
532C2×C3—42no—
533C2×C3#6383612yes—
534C2×C3#6387411yescharacteristic
535C2×C3#6388012yes—
536C2×C3C4×#62412yes—
537C2×C3C4×#62411yescharacteristic
538C2×C3#6389412yes—
539C2×C3C4×#62612yes—
540C2×C3C4×#62612yes—
541C2×C3#6390412yes—
542C2×C3D3×(C2×D3)⋊C412yes—
543C2×C3#6395212yes—
544C2×C3C2×#372211yescharacteristic
545D3—188no—
546D3—188no—
547D3—184no—
548D3—368no—
549D3—368no—
550C22—274no—
551C22—542no—
552C22—542no—
553C22—544no—
554C22—544no—
555C22D3×#62411yescharacteristic
556C22D3×#62412yes—
557C22D3×#62612yes—
558C22#8895911yescharacteristic
559C22#8896511yescharacteristic
560C4—34no—
561C4—64no—
562C4—182no—
563C4—184no—
564C3—21no—
565C3—22no—
566C3—41no—
567C3#1059123312yescomplemented
568C3C4×#372211yescharacteristic, complemented
569C2—544no—
570C2#1172290011yescharacteristic
571C2D3×#372211yescharacteristic
572C2#1172291811yescharacteristic
573C2#1172293212yes—
574C2#1172293312yes—
subgroupG/H
center, Frattini, hypercenterC23D33
commutatorC2×C33C2×C42
Fitting, socle, cosocleC23×C33C23
prime radicalC23×C3⋊(C3⋊D3)C22
hypocenterC33C4×C22⋊C4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
3. C2×C3×Dic32C2semidirect
4. #11722082C2semidirect
5. C3×#63579C2semidirect
14. C2×#6289C4semidirect
15. #88804C4semidirect
26. C2×Dic32D3semidirect
30. #63579D3semidirect
41. C2×C3×C3⋊Dic3C2×C4semidirect
45. C2×C32×Dic3C2×C4semidirect
92. C2×C3⋊(C3⋊D3)C42semidirect
96. C32×Dic3C22⋊C4semidirect
154. C2×C3⋊Dic3C4×D3semidirect
180. C2×C3×Dic3C4×D3semidirect
181. C2×C3×Dic3C4×D3semidirect
297. C3×Dic3(C2×D3)⋊C4semidirect
303. C33C4×C22⋊C4semidirect
374. C2×Dic3#624semidirect
493. Dic3#3722semidirect
497. C32#4666semidirect
498. C32C4×(C2×D3)⋊C4semidirect
567. C3#10591233semidirect
568. C3C4×#3722semidirect

Maximal subgroups

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

subgroupindex
C22×#62892
C2×#888042
C2×C3×Dic322
#117220822
C3×#635792
C4×#37223
#105912333