Finite Group Databasebeta

← #13178675

Subgroups — #13178675

Order 1728.

1644 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3830C212yesmaximal, complemented
2C3×D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C2×#88815C212yesmaximal, complemented
4#11723720C214yesmaximal, complemented
5#11723717C212yesmaximal, complemented
6C4×#6295C211yesmaximal, characteristic, complemented
7#11723678C211yesmaximal, characteristic, complemented
8#11723674C212yesmaximal, complemented
9#10595252—31nomaximal
10#10595248—32nomaximal
11C2×C33×D4C2211yescharacteristic
12C2×C3×#638C2214yes—
13C22×C3×C3⋊Dic3C2212yes—
14C22×C3×C3⋊Dic3C2214yes—
15C3×#3830—28no—
16C2×C3×C4×C3⋊D3C2212yes—
17C2×C3×#634C2212yes—
18C2×#6295C2211yescharacteristic
19C3×#3757C2212yes—
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C3×C4×C3⋊Dic3C2211yescharacteristic
22#88815C2214yes—
23#88815C2212yes—
24#88815C2212yes—
25#88815C2211yescharacteristic
26#88814C2214yes—
27#88814C2212yes—
28C2×#3830D312yescomplemented
29C2×C3×#237—32no—
30C2×C3×#237—32no—
31C2×C3×#237—62no—
32D4×C3⋊Dic3D311yescharacteristic, complemented
33C3×D4×Dic3—32no—
34C3×D4×Dic3—61no—
35#63980—32no—
36#63978—34no—
37C2×#3724—32no—
38#63966—34no—
39C2×Dic3⋊Dic3—34no—
40#63962—34no—
41#63961—34no—
42C4×#626—32no—
43C4×#625—31no—
44#63873—31no—
45#63871—32no—
46#63866—32no—
47#63857—32no—
48#63856—32no—
49C23×C33C2312yes—
50C22×C3×C3⋊D3C2312yes—
51C33×D4D414yescomplemented
52C2×C33×C4C2311yescharacteristic
53C3×#638—28no—
54C2×C3×C3⋊Dic3—28no—
55C2×C3×C3⋊Dic3—24no—
56C2×C3×C3⋊Dic3C2311yescharacteristic
57C2×C3×C3⋊Dic3C2312yes—
58C2×C3×C3⋊Dic3C2314yes—
59C2×C3×C3⋊Dic3C2312yes—
60C2×C3×C3⋊Dic3C2311yescharacteristic
61C3×C4×C3⋊D3—24no—
62C3×#634—24no—
63#6295—22no—
64#5824—92no—
65#5555—91no—
66C2×C32×D4—21no—
67C2×C32×D4—22no—
68C2×C32×D4—41no—
69C2×C32×D4C2×D311yescharacteristic
70C2×C32×D4C2×D312yes—
71C2×#638C2×D314yes—
72C22×C3⋊Dic3C2×D314yes—
73C22×C3⋊Dic3C2×D312yes—
74#3830—28no—
75C2×C4×C3⋊D3C2×D312yes—
76C2×#634C2×D312yes—
77C2×C3×#67—34no—
78C2×C3×#67—34no—
79C2×C3×#67—64no—
80C22×C3×Dic3—34no—
81C22×C3×Dic3—34no—
82C22×C3×Dic3—34no—
83C22×C3×Dic3—64no—
84C22×C3×Dic3—62no—
85C3×#237—616no—
86C3×#237—68no—
87C3×#237—68no—
88C2×C3×C4×D3—32no—
89C2×C3×C4×D3—32no—
90C2×C3×C4×D3—62no—
91C2×C3×Dic6—32no—
92C2×C3×Dic6—32no—
93C2×C3×Dic6—62no—
94C2×#626—32no—
95C2×#625—31no—
96#3757C2×D312yes—
97C4⋊(C3⋊Dic3)C2×D311yescharacteristic
98C4×C3⋊Dic3C2×D311yescharacteristic
99C3×#217—34no—
100C3×#217—62no—
101C3×C4⋊Dic3—32no—
102C3×C4⋊Dic3—61no—
103C3×C4×Dic3—32no—
104C3×C4×Dic3—61no—
105#3724—34no—
106#3724—32no—
107#3724—32no—
108#3724—31no—
109Dic3⋊Dic3—34no—
110Dic3⋊Dic3—34no—
111Dic3⋊Dic3—34no—
112Dic3⋊Dic3—34no—
113Dic3⋊Dic3—32no—
114#3722—34no—
115#3722—32no—
116(C2×D3)⋊Dic3—34no—
117(C2×D3)⋊Dic3—34no—
118(C2×D3)⋊Dic3—32no—
119(C2×D3)⋊Dic3—32no—
120C22×C33—22no—
121C22×C33—22no—
122C22×C33C2×D414yes—
123C22×C33C2411yescharacteristic
124C2×C3×C3⋊D3—22no—
125C2×C3×C3⋊D3—22no—
126C2×C3×C3⋊D3—22no—
127C33×C4C2×D412yes—
128C3×C3⋊Dic3—24no—
129C3×C3⋊Dic3—22no—
130C3×C3⋊Dic3—22no—
131C3×C3⋊Dic3—21no—
132C3×C3⋊Dic3#4112yescomplemented
133C2×C3×#41—92no—
134C2×#237—32no—
135C2×#237—32no—
136C2×#237—62no—
137C3×C4×D4—91no—
138#946—92no—
139#944—92no—
140#939—94no—
141#935—94no—
142D4×Dic3—32no—
143D4×Dic3—61no—
144#930—92no—
145#929—94no—
146C4×#67—92no—
147C2×C4⋊Dic3—92no—
148#924—92no—
149C2×Dic3⋊C4—94no—
150#897—92no—
151#886—94no—
152C4×D12—91no—
153#869—91no—
154C23×C32—22no—
155C23×C32—24no—
156C23×C32—42no—
157C23×C32C22×D312yes—
158C23×C32C22×D314yes—
159C22×C3⋊D3C22×D312yes—
160C22×C3×D3—32no—
161C22×C3×D3—32no—
162C22×C3×D3—62no—
163C32×D4—24no—
164C32×D4—28no—
165C32×D4—44no—
166C32×D4D1214yescomplemented
167C32×D4#6718yescomplemented
168C2×C32×C4—21no—
169C2×C32×C4—22no—
170C2×C32×C4—41no—
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D312yes—
173#638—28no—
174C2×C3⋊Dic3—28no—
175C2×C3⋊Dic3—24no—
176C2×C3⋊Dic3C22×D311yescharacteristic
177C2×C3⋊Dic3C22×D312yes—
178C2×C3⋊Dic3C22×D314yes—
179C2×C3⋊Dic3C22×D312yes—
180C2×C3⋊Dic3C22×D311yescharacteristic
181C4×C3⋊D3—24no—
182#634—24no—
183C3×#67—616no—
184C3×#67—68no—
185C3×#67—68no—
186C2×C3×Dic3—32no—
187C2×C3×Dic3—32no—
188C2×C3×Dic3—34no—
189C2×C3×Dic3—34no—
190C2×C3×Dic3—32no—
191C2×C3×Dic3—34no—
192C2×C3×Dic3—32no—
193C2×C3×Dic3—61no—
194C2×C3×Dic3—64no—
195C2×C3×Dic3—62no—
196C2×C3×Dic3—61no—
197C2×C3×Dic3—62no—
198C2×C3×Dic3—68no—
199C2×C3×Dic3—616no—
200C2×C3×Dic3—68no—
201C2×C3×Dic3—68no—
202C2×C3×Dic3—124no—
203C3×C4×D3—68no—
204C3×C4×D3—64no—
205C3×C4×D3—64no—
206C3×Dic6—68no—
207C3×Dic6—64no—
208C3×Dic6—64no—
209#626—64no—
210#625—62no—
211#547—271noSylow
212C2×C33—24no—
213C2×C33C22×D411yescharacteristic
214C2×C33C2×#4111yescharacteristic
215C2×C332−1+411yescharacteristic
216C3×C3⋊D3—42no—
217C3×#41—188no—
218C2×C3×Q8—92no—
219C2×C3×D4—21no—
220C2×C3×D4—22no—
221C2×C3×D4—41no—
222C2×C3×D4—94no—
223C2×C3×D4D3211yescharacteristic
224C2×C3×D4D3212yes—
225C22×C3×C4—94no—
226C22×C3×C4—92no—
227C22×C3×C4—92no—
228C2×#67—34no—
229C2×#67—34no—
230C2×#67—64no—
231C2×#67—92no—
232C22×Dic3—34no—
233C22×Dic3—34no—
234C22×Dic3—34no—
235C22×Dic3—64no—
236C22×Dic3—62no—
237#237—616no—
238#237—68no—
239#237—68no—
240C2×D12—91no—
241C2×C4×D3—32no—
242C2×C4×D3—32no—
243C2×C4×D3—62no—
244C2×Dic6—32no—
245C2×Dic6—32no—
246C2×Dic6—62no—
247C3×C4⋊C4—91no—
248C3×C22⋊C4—92no—
249C3×C42—91no—
250#217—34no—
251#217—62no—
252#217—94no—
253#217—92no—
254(C2×D3)⋊C4—92no—
255(C2×D3)⋊C4—94no—
256(C2×D3)⋊C4—92no—
257(C2×D3)⋊C4—94no—
258C4⋊Dic3—32no—
259C4⋊Dic3—61no—
260C4⋊Dic3—92no—
261C4⋊Dic3—92no—
262C4⋊Dic3—91no—
263C4⋊Dic3—94no—
264Dic3⋊C4—92no—
265Dic3⋊C4—94no—
266Dic3⋊C4—94no—
267Dic3⋊C4—94no—
268Dic3⋊C4—94no—
269C4×Dic3—32no—
270C4×Dic3—61no—
271C22×C32—22no—
272C22×C32—24no—
273C22×C32—22no—
274C22×C32—24no—
275C22×C32—21no—
276C22×C32—22no—
277C22×C32—24no—
278C22×C32—28no—
279C22×C32—41no—
280C22×C32—44no—
281C22×C32—42no—
282C22×C32—42no—
283C22×C32—44no—
284C22×C32—44no—
285C22×C32—82no—
286C22×C32—82no—
287C22×C32C2×D1214yes—
288C22×C32C2×#6718yes—
289C22×C32C23×D311yescharacteristic
290C22×C32C23×D312yes—
291C2×C3⋊D3—22no—
292C2×C3⋊D3—22no—
293C2×C3⋊D3—22no—
294C2×C3×D3—64no—
295C2×C3×D3—62no—
296C2×C3×D3—62no—
297C2×C3×D3—62no—
298C2×C3×D3—62no—
299C2×C3×D3—62no—
300C2×C3×D3—62no—
301C2×C3×D3—122no—
302C2×C3×D3—122no—
303C32×C4—22no—
304C32×C4—24no—
305C32×C4—42no—
306C32×C4C2×D1212yes—
307C32×C4C2×#6714yes—
308C3⋊Dic3—24no—
309C3⋊Dic3—22no—
310C3⋊Dic3—22no—
311C3⋊Dic3—21no—
312C3⋊Dic3#23712yescomplemented
313C3×Dic3—34no—
314C3×Dic3—62no—
315C3×Dic3—68no—
316C3×Dic3—64no—
317C3×Dic3—64no—
318C3×Dic3—62no—
319C3×Dic3—64no—
320C3×Dic3—62no—
321C3×Dic3—64no—
322C3×Dic3—62no—
323C3×Dic3—121no—
324C3×Dic3—122no—
325C2×#41—92no—
326C4⋊Q8—271no—
327#123—274no—
328C22⋊Q8—272no—
329#121—272no—
330C4×D4—91no—
331C4×D4—271no—
332C2×C4⋊C4—272no—
333C33#54711yescharacteristic, complemented, Sylow
334C23×C3—22no—
335C23×C3—24no—
336C23×C3—42no—
337C23×C3—92no—
338C23×C3C2×D3212yes—
339C23×C3C2×D3214yes—
340C22×D3—32no—
341C22×D3—32no—
342C22×D3—62no—
343C3×Q8—184no—
344C3×D4—24no—
345C3×D4—28no—
346C3×D4—44no—
347C3×D4—188no—
348C3×D4#62514yescomplemented
349C3×D4#62618yescomplemented
350C2×C3×C4—21no—
351C2×C3×C4—22no—
352C2×C3×C4—41no—
353C2×C3×C4—91no—
354C2×C3×C4—92no—
355C2×C3×C4—94no—
356C2×C3×C4—91no—
357C2×C3×C4—92no—
358C2×C3×C4—184no—
359C2×C3×C4—188no—
360C2×C3×C4—184no—
361C2×C3×C4C2×D3211yescharacteristic
362C2×C3×C4C2×D3212yes—
363#67—616no—
364#67—68no—
365#67—68no—
366#67—184no—
367C2×Dic3—32no—
368C2×Dic3—32no—
369C2×Dic3—34no—
370C2×Dic3—32no—
371C2×Dic3—34no—
372C2×Dic3—34no—
373C2×Dic3—32no—
374C2×Dic3—61no—
375C2×Dic3—64no—
376C2×Dic3—62no—
377C2×Dic3—61no—
378C2×Dic3—62no—
379C2×Dic3—68no—
380C2×Dic3—616no—
381C2×Dic3—68no—
382C2×Dic3—68no—
383C2×Dic3—124no—
384D12—182no—
385C4×D3—68no—
386C4×D3—64no—
387C4×D3—64no—
388Dic6—68no—
389Dic6—64no—
390Dic6—64no—
391C2×C32—24no—
392C2×C32—28no—
393C2×C32—21no—
394C2×C32—22no—
395C2×C32—22no—
396C2×C32—21no—
397C2×C32—21no—
398C2×C32—22no—
399C2×C32—41no—
400C2×C32—41no—
401C2×C32—41no—
402C2×C32—48no—
403C2×C32—44no—
404C2×C32—84no—
405C2×C32C22×D1211yescharacteristic
406C2×C32C2×#23711yescharacteristic
407C2×C32D3×#4112yes—
408C2×C32#101011yescharacteristic
409C2×C32#101012yes—
410C2×C32C22×#6712yes—
411C3⋊D3—42no—
412C3×D3—124no—
413C3×D3—122no—
414C3×D3—122no—
415#41—188no—
416C2×Q8—92no—
417C2×D4—94no—
418C2×D4—271no—
419C2×D4#111311yescharacteristic
420C22×C4—92no—
421C22×C4—94no—
422C22×C4—92no—
423C4⋊C4—91no—
424C4⋊C4—271no—
425C4⋊C4—272no—
426C4⋊C4—274no—
427C4⋊C4—272no—
428C22⋊C4—92no—
429C22⋊C4—272no—
430C22⋊C4—274no—
431C42—91no—
432C22×C3—24no—
433C22×C3—22no—
434C22×C3—24no—
435C22×C3—22no—
436C22×C3—21no—
437C22×C3—22no—
438C22×C3—28no—
439C22×C3—24no—
440C22×C3—42no—
441C22×C3—42no—
442C22×C3—44no—
443C22×C3—44no—
444C22×C3—41no—
445C22×C3—44no—
446C22×C3—82no—
447C22×C3—82no—
448C22×C3—182no—
449C22×C3—182no—
450C22×C3—182no—
451C22×C3C2×#62514yes—
452C22×C3C2×#62618yes—
453C22×C3C22×D3212yes—
454C22×C3C22×D3211yescharacteristic
455C2×D3—64no—
456C2×D3—62no—
457C2×D3—62no—
458C2×D3—62no—
459C2×D3—62no—
460C2×D3—62no—
461C2×D3—62no—
462C2×D3—122no—
463C2×D3—122no—
464C3×C4—24no—
465C3×C4—22no—
466C3×C4—42no—
467C3×C4—92no—
468C3×C4—181no—
469C3×C4—182no—
470C3×C4—182no—
471C3×C4—184no—
472C3×C4C2×#62512yes—
473C3×C4C2×#62614yes—
474Dic3—34no—
475Dic3—62no—
476Dic3—68no—
477Dic3—64no—
478Dic3—62no—
479Dic3—64no—
480Dic3—64no—
481Dic3—62no—
482Dic3—64no—
483Dic3—62no—
484Dic3—122no—
485Dic3—121no—
486C32—22no—
487C32—21no—
488C32—41no—
489C32#555511yescharacteristic, complemented
490C32#582412yescomplemented
491C23—92no—
492C23C2×#111312yes—
493Q8—184no—
494D4—188no—
495D4—542no—
496D4#629514yescomplemented
497C2×C4—91no—
498C2×C4—91no—
499C2×C4—92no—
500C2×C4—92no—
501C2×C4—94no—
502C2×C4—184no—
503C2×C4—188no—
504C2×C4—184no—
505C2×C4C2×#111311yescharacteristic
506C2×C3—28no—
507C2×C3—24no—
508C2×C3—21no—
509C2×C3—22no—
510C2×C3—21no—
511C2×C3—22no—
512C2×C3—22no—
513C2×C3—21no—
514C2×C3—48no—
515C2×C3—44no—
516C2×C3—41no—
517C2×C3—41no—
518C2×C3—41no—
519C2×C3—84no—
520C2×C3—362no—
521C2×C3#6430912yes—
522C2×C3#6430911yescharacteristic
523C2×C3D3×#23712yes—
524C2×C3#6431211yescharacteristic
525C2×C3C22×#62511yescharacteristic
526C2×C3C22×#62612yes—
527D3—124no—
528D3—122no—
529D3—122no—
530C22—22no—
531C22—22no—
532C22—182no—
533C22—182no—
534C22—182no—
535C22C2×#629514yes—
536C22C22×#111311yescharacteristic
537C4—92no—
538C4—181no—
539C4—182no—
540C4—182no—
541C4—184no—
542C4C2×#629512yes—
543C3—22no—
544C3—21no—
545C3—41no—
546C3#1059524812yescomplemented
547C3#1059525211yescharacteristic, complemented
548C2—24no—
549C2—362no—
550C2#1172462011yescharacteristic
551C2#1172462211yescharacteristic
552C2C22×#629511yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4#1113
hypocenterC33#547
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3830C2semidirect
2. C3×D4×C3⋊Dic3C2semidirect
3. C2×#88815C2semidirect
4. #11723720C2semidirect
5. #11723717C2semidirect
6. C4×#6295C2semidirect
7. #11723678C2semidirect
8. #11723674C2semidirect
28. C2×#3830D3semidirect
32. D4×C3⋊Dic3D3semidirect
51. C33×D4D4semidirect
132. C3×C3⋊Dic3#41semidirect
166. C32×D4D12semidirect
167. C32×D4#67semidirect
312. C3⋊Dic3#237semidirect
333. C33#547semidirect
348. C3×D4#625semidirect
349. C3×D4#626semidirect
489. C32#5555semidirect
490. C32#5824semidirect
496. D4#6295semidirect
546. C3#10595248semidirect
547. C3#10595252semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C3×D4×C3⋊Dic32
C2×#888152
#117237202
#117237172
C4×#62952
#117236782
#117236742
#105952523
#105952483