Finite Group Databasebeta

← #13162057

Subgroups — #13162057

Order 1728.

1210 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3757C212yesmaximal
2C2×C3×#3757C211yesmaximal, characteristic
3C2×#88815C211yesmaximal, characteristic, complemented
4#11722082C212yesmaximal, complemented
5#11722082C211yesmaximal, characteristic, complemented
6#10591717—32nomaximal
7#10591694—31nomaximal
8C24×C33C2211yescharacteristic
9C22×C3×C3⋊Dic3C2212yes—
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C3×C3⋊Dic3C2212yes—
12C22×C3×C3⋊Dic3C2211yescharacteristic
13C3×#3757—24no—
14C3×#3757—22no—
15#88815—22no—
16C2×#3757D311yescharacteristic
17C2×#3757D312yes—
18C2×C3×#217—32no—
19C2×C3×#217—32no—
20C2×C3×#217—32no—
21C2×C3×#217—62no—
22C2×C3×#217—61no—
23C2×#3724—31no—
24C2×Dic3⋊Dic3—32no—
25#63579—31no—
26#63579—32no—
27#63579—32no—
28#63579—32no—
29#63579—32no—
30C23×C33—21no—
31C23×C33—21no—
32C23×C33—22no—
33C23×C33—22no—
34C23×C33D412yes—
35C23×C33C2311yescharacteristic
36C2×C3×C3⋊Dic3—22no—
37C2×C3×C3⋊Dic3—22no—
38C2×C3×C3⋊Dic3—22no—
39C2×C3×C3⋊Dic3—21no—
40C2×C3×C3⋊Dic3—22no—
41C2×C3×C3⋊Dic3—22no—
42C2×C3×C3⋊Dic3—22no—
43C2×C3×C3⋊Dic3—22no—
44C2×C3×C3⋊Dic3—22no—
45C2×C3×C3⋊Dic3D412yes—
46#4953—91no—
47#4952—92no—
48C24×C32—21no—
49C24×C32—22no—
50C24×C32—41no—
51C24×C32C2×D311yescharacteristic
52C24×C32C2×D312yes—
53C22×C3⋊Dic3C2×D311yescharacteristic
54C22×C3⋊Dic3C2×D312yes—
55C22×C3⋊Dic3C2×D312yes—
56C22×C3⋊Dic3C2×D311yescharacteristic
57C22×C3×Dic3—32no—
58C22×C3×Dic3—32no—
59C22×C3×Dic3—32no—
60C22×C3×Dic3—32no—
61C22×C3×Dic3—32no—
62C22×C3×Dic3—32no—
63C22×C3×Dic3—61no—
64C22×C3×Dic3—62no—
65C22×C3×Dic3—61no—
66C22×C3×Dic3—62no—
67#3757—24no—
68#3757—22no—
69C3×#217—64no—
70C3×#217—64no—
71C3×#217—68no—
72C3×#217—64no—
73C3×#217—64no—
74#3724—62no—
75Dic3⋊Dic3—64no—
76C22×C33—22no—
77C22×C33—24no—
78C22×C33—41no—
79C22×C33—41no—
80C22×C33—42no—
81C22×C33C2×D411yescharacteristic
82C22×C33C2×D411yescharacteristic
83C22×C33#4111yescharacteristic
84C22×C33#4112yes—
85C22×C33#4112yes—
86C3×C3⋊Dic3—22no—
87C3×C3⋊Dic3—42no—
88C3×C3⋊Dic3—42no—
89C3×C3⋊Dic3—41no—
90C2×C3×C22⋊C4—92no—
91C2×C3×C22⋊C4—91no—
92C2×#217—32no—
93C2×#217—32no—
94C2×#217—32no—
95C2×#217—61no—
96C2×#217—62no—
97C2×C4⋊Dic3—91no—
98C2×Dic3⋊C4—92no—
99C3⋊#95—92no—
100C3⋊#95—92no—
101C3⋊#95—92no—
102C3⋊#95—91no—
103C3⋊#95—92no—
104C23×C32—22no—
105C23×C32—22no—
106C23×C32—21no—
107C23×C32—21no—
108C23×C32—22no—
109C23×C32—24no—
110C23×C32—24no—
111C23×C32—22no—
112C23×C32—22no—
113C23×C32—22no—
114C23×C32—22no—
115C23×C32—22no—
116C23×C32—22no—
117C23×C32—22no—
118C23×C32—22no—
119C23×C32—24no—
120C23×C32—21no—
121C23×C32—22no—
122C23×C32—42no—
123C23×C32—41no—
124C23×C32—42no—
125C23×C32—42no—
126C23×C32—42no—
127C23×C32—42no—
128C23×C32—42no—
129C23×C32—42no—
130C23×C32—42no—
131C23×C32—42no—
132C23×C32—44no—
133C23×C32—44no—
134C23×C32D1212yes—
135C23×C32#6714yes—
136C23×C32C22×D311yescharacteristic
137C23×C32C22×D312yes—
138C2×C3⋊Dic3—22no—
139C2×C3⋊Dic3—22no—
140C2×C3⋊Dic3—21no—
141C2×C3⋊Dic3—22no—
142C2×C3⋊Dic3—22no—
143C2×C3⋊Dic3—22no—
144C2×C3⋊Dic3—22no—
145C2×C3⋊Dic3—22no—
146C2×C3⋊Dic3—22no—
147C2×C3⋊Dic3#6712yes—
148C2×C3×Dic3—34no—
149C2×C3×Dic3—62no—
150C2×C3×Dic3—62no—
151C2×C3×Dic3—62no—
152C2×C3×Dic3—64no—
153C2×C3×Dic3—64no—
154C2×C3×Dic3—62no—
155C2×C3×Dic3—62no—
156C2×C3×Dic3—62no—
157C2×C3×Dic3—62no—
158C2×C3×Dic3—62no—
159C2×C3×Dic3—62no—
160C2×C3×Dic3—62no—
161C2×C3×Dic3—62no—
162C2×C3×Dic3—62no—
163C2×C3×Dic3—62no—
164C2×C3×Dic3—62no—
165C2×C3×Dic3—62no—
166C2×C3×Dic3—62no—
167C2×C3×Dic3—62no—
168C2×C3×Dic3—62no—
169C2×C3×Dic3—62no—
170C2×C3×Dic3—122no—
171C2×C3×Dic3—122no—
172C2×C3×Dic3—122no—
173C2×C3×Dic3—122no—
174C2×C3×Dic3—122no—
175C2×C3×Dic3—122no—
176#397—271noSylow
177C2×C33—42no—
178C2×C33#12111yescharacteristic
179C2×C33#12311yescharacteristic
180C2×C33#12312yes—
181C2×C33#12411yescharacteristic
182C2×C33#12612yes—
183C24×C3—21no—
184C24×C3—22no—
185C24×C3—41no—
186C24×C3D3211yescharacteristic
187C24×C3D3212yes—
188C22×C3×C4—91no—
189C22×C3×C4—92no—
190C22×C3×C4—91no—
191C22×C3×C4—92no—
192C22×Dic3—32no—
193C22×Dic3—32no—
194C22×Dic3—32no—
195C22×Dic3—32no—
196C22×Dic3—32no—
197C22×Dic3—32no—
198C22×Dic3—61no—
199C22×Dic3—62no—
200C22×Dic3—61no—
201C22×Dic3—62no—
202C3×C22⋊C4—182no—
203C3×C22⋊C4—184no—
204#217—64no—
205#217—64no—
206#217—68no—
207#217—64no—
208#217—64no—
209C4⋊Dic3—182no—
210Dic3⋊C4—184no—
211C22×C32—24no—
212C22×C32—22no—
213C22×C32—28no—
214C22×C32—24no—
215C22×C32—24no—
216C22×C32—24no—
217C22×C32—24no—
218C22×C32—22no—
219C22×C32—22no—
220C22×C32—22no—
221C22×C32—22no—
222C22×C32—22no—
223C22×C32—21no—
224C22×C32—22no—
225C22×C32—22no—
226C22×C32—21no—
227C22×C32—22no—
228C22×C32—22no—
229C22×C32—21no—
230C22×C32—41no—
231C22×C32—41no—
232C22×C32—41no—
233C22×C32—42no—
234C22×C32—42no—
235C22×C32—44no—
236C22×C32—44no—
237C22×C32—44no—
238C22×C32—44no—
239C22×C32—48no—
240C22×C32—42no—
241C22×C32—44no—
242C22×C32—44no—
243C22×C32—44no—
244C22×C32—44no—
245C22×C32—44no—
246C22×C32—42no—
247C22×C32—44no—
248C22×C32—44no—
249C22×C32—48no—
250C22×C32—42no—
251C22×C32—42no—
252C22×C32—42no—
253C22×C32—42no—
254C22×C32—42no—
255C22×C32—41no—
256C22×C32—41no—
257C22×C32C2×D1211yescharacteristic
258C22×C32#23511yescharacteristic
259C22×C32#23512yes—
260C22×C32#23512yes—
261C22×C32D3×D412yes—
262C22×C32#23712yes—
263C22×C32#23712yes—
264C22×C32#23712yes—
265C22×C32#23712yes—
266C22×C32#23712yes—
267C22×C32C2×#6712yes—
268C22×C32C2×#6711yescharacteristic
269C3⋊Dic3—22no—
270C3⋊Dic3—42no—
271C3⋊Dic3—41no—
272C3⋊Dic3—42no—
273C3×Dic3—64no—
274C3×Dic3—122no—
275C3×Dic3—122no—
276C3×Dic3—124no—
277C3×Dic3—124no—
278C3×Dic3—122no—
279C3×Dic3—122no—
280C3×Dic3—122no—
281C3×Dic3—122no—
282C3×Dic3—122no—
283C2×C4⋊C4—271no—
284C2×C22⋊C4—91no—
285C2×C22⋊C4—92no—
286#95—271no—
287#95—272no—
288C33#39711yescharacteristic, complemented, Sylow
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—24no—
292C23×C3—24no—
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—22no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3—21no—
301C23×C3—22no—
302C23×C3—21no—
303C23×C3—22no—
304C23×C3—24no—
305C23×C3—21no—
306C23×C3—22no—
307C23×C3—42no—
308C23×C3—42no—
309C23×C3—42no—
310C23×C3—42no—
311C23×C3—42no—
312C23×C3—42no—
313C23×C3—42no—
314C23×C3—44no—
315C23×C3—44no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3—41no—
319C23×C3#62512yes—
320C23×C3#62614yes—
321C23×C3C2×D3211yescharacteristic
322C23×C3C2×D3212yes—
323C2×C3×C4—92no—
324C2×C3×C4—182no—
325C2×C3×C4—182no—
326C2×C3×C4—182no—
327C2×C3×C4—182no—
328C2×C3×C4—182no—
329C2×C3×C4—182no—
330C2×C3×C4—182no—
331C2×C3×C4—182no—
332C2×C3×C4—181no—
333C2×Dic3—34no—
334C2×Dic3—62no—
335C2×Dic3—62no—
336C2×Dic3—62no—
337C2×Dic3—64no—
338C2×Dic3—64no—
339C2×Dic3—62no—
340C2×Dic3—62no—
341C2×Dic3—62no—
342C2×Dic3—62no—
343C2×Dic3—62no—
344C2×Dic3—62no—
345C2×Dic3—62no—
346C2×Dic3—62no—
347C2×Dic3—62no—
348C2×Dic3—62no—
349C2×Dic3—62no—
350C2×Dic3—62no—
351C2×Dic3—62no—
352C2×Dic3—62no—
353C2×Dic3—62no—
354C2×Dic3—62no—
355C2×Dic3—122no—
356C2×Dic3—122no—
357C2×Dic3—122no—
358C2×Dic3—122no—
359C2×Dic3—122no—
360C2×Dic3—122no—
361C2×C32—21no—
362C2×C32—22no—
363C2×C32—22no—
364C2×C32—22no—
365C2×C32—22no—
366C2×C32—22no—
367C2×C32—22no—
368C2×C32—22no—
369C2×C32—22no—
370C2×C32—21no—
371C2×C32—22no—
372C2×C32—21no—
373C2×C32—41no—
374C2×C32—42no—
375C2×C32—41no—
376C2×C32—41no—
377C2×C32—42no—
378C2×C32—44no—
379C2×C32—48no—
380C2×C32—48no—
381C2×C32—44no—
382C2×C32—42no—
383C2×C32#87912yes—
384C2×C32#87912yes—
385C2×C32#87912yes—
386C2×C32#88312yes—
387C2×C32#88512yes—
388C2×C32#88612yes—
389C2×C32#92912yes—
390C2×C32#93011yescharacteristic
391C2×C32#93512yes—
392C2×C32#93511yescharacteristic
393C2×C32#93611yescharacteristic
394C2×C32#93912yes—
395C24#111311yescharacteristic
396C22×C4—91no—
397C22×C4—92no—
398C22×C4—91no—
399C22×C4—92no—
400C4⋊C4—542no—
401C22⋊C4—182no—
402C22⋊C4—184no—
403C22×C3—24no—
404C22×C3—28no—
405C22×C3—24no—
406C22×C3—24no—
407C22×C3—24no—
408C22×C3—24no—
409C22×C3—22no—
410C22×C3—22no—
411C22×C3—22no—
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—22no—
415C22×C3—21no—
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—22no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—22no—
422C22×C3—44no—
423C22×C3—44no—
424C22×C3—44no—
425C22×C3—44no—
426C22×C3—48no—
427C22×C3—44no—
428C22×C3—44no—
429C22×C3—44no—
430C22×C3—42no—
431C22×C3—44no—
432C22×C3—42no—
433C22×C3—44no—
434C22×C3—48no—
435C22×C3—44no—
436C22×C3—44no—
437C22×C3—41no—
438C22×C3—42no—
439C22×C3—42no—
440C22×C3—42no—
441C22×C3—42no—
442C22×C3—42no—
443C22×C3—41no—
444C22×C3—41no—
445C22×C3—42no—
446C22×C3—42no—
447C22×C3—41no—
448C22×C3—41no—
449C22×C3#380412yes—
450C22×C3#380412yes—
451C22×C3#380412yes—
452C22×C3#380412yes—
453C22×C3#380412yes—
454C22×C3#380511yescharacteristic
455C22×C3#380512yes—
456C22×C3#380512yes—
457C22×C3C2×#62511yescharacteristic
458C22×C3C2×#62612yes—
459C22×C3D3×#6712yes—
460C22×C3#381111yescharacteristic
461C3×C4—182no—
462C3×C4—361no—
463C3×C4—362no—
464C3×C4—362no—
465Dic3—64no—
466Dic3—122no—
467Dic3—124no—
468Dic3—122no—
469Dic3—124no—
470Dic3—122no—
471Dic3—122no—
472Dic3—122no—
473Dic3—122no—
474Dic3—122no—
475C32—21no—
476C32—22no—
477C32—41no—
478C32#495212yescomplemented
479C32#495311yescharacteristic, complemented
480C23—21no—
481C23—22no—
482C23—22no—
483C23—21no—
484C23#629512yes—
485C23C2×#111311yescharacteristic
486C2×C4—92no—
487C2×C4—182no—
488C2×C4—182no—
489C2×C4—182no—
490C2×C4—182no—
491C2×C4—182no—
492C2×C4—182no—
493C2×C4—182no—
494C2×C4—181no—
495C2×C4—182no—
496C2×C3—22no—
497C2×C3—22no—
498C2×C3—22no—
499C2×C3—22no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—22no—
503C2×C3—22no—
504C2×C3—22no—
505C2×C3—22no—
506C2×C3—21no—
507C2×C3—22no—
508C2×C3—48no—
509C2×C3—44no—
510C2×C3—48no—
511C2×C3—42no—
512C2×C3—44no—
513C2×C3—42no—
514C2×C3—41no—
515C2×C3—41no—
516C2×C3—42no—
517C2×C3—41no—
518C2×C3#6395311yescharacteristic
519C2×C3#6395612yes—
520C2×C3#6395612yes—
521C2×C3#6395612yes—
522C2×C3#6395912yes—
523C2×C3#6396112yes—
524C2×C3#6396212yes—
525C2×C3#6396612yes—
526C2×C3#6396912yes—
527C2×C3#6397511yescharacteristic
528C2×C3#6397812yes—
529C2×C3#6398011yescharacteristic
530C22—22no—
531C22—24no—
532C22—42no—
533C22—41no—
534C22—41no—
535C22#8904611yescharacteristic
536C22#8904612yes—
537C22#8904612yes—
538C22C2×#629511yescharacteristic
539C22#8904911yescharacteristic
540C4—182no—
541C4—362no—
542C4—361no—
543C4—362no—
544C3—22no—
545C3—21no—
546C3—41no—
547C3#1059169411yescharacteristic, complemented
548C3#1059171712yescomplemented
549C2—42no—
550C2#1172371612yes—
551C2#1172371711yescharacteristic
552C2#1172371811yescharacteristic
553C2#1172372012yes—
554C2#1172372411yescharacteristic
subgroupG/H
center, FrattiniC23C2×#1113
commutator, socle, cosocleC23×C33C23
Fitting, prime radicalC24×C33C22
hypercenterC24#1113
hypocenterC33#397
LCSG ▷ C23×C33 ▷ C33
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
3. C2×#88815C2semidirect
4. #11722082C2semidirect
5. #11722082C2semidirect
288. C33#397semidirect
478. C32#4952semidirect
479. C32#4953semidirect
547. C3#10591694semidirect
548. C3#10591717semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37572
C2×#888152
#117220822
#105917173
#105916943