Finite Group Databasebeta

← #13177654

Subgroups — #13177654

Order 1728.

2248 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89029C212yesmaximal, complemented
2D4×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
3C3×#64149C211yesmaximal, characteristic, complemented
4C32×#940C211yesmaximal, characteristic, complemented
5#11723637C212yesmaximal, complemented
6#11723631C212yesmaximal, complemented
7#11723615C212yesmaximal, complemented
8C4×#6293C211yesmaximal, characteristic, complemented
9#11723529C211yesmaximal, characteristic, complemented
10D12×C3⋊Dic3C211yesmaximal, characteristic, complemented
11Dic3×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
12#10596189—31nomaximal
13#10595144—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C22×C3⋊(C3⋊Dic3)C2212yes—
16C2×C3×#638C2212yes—
17C2×C3×#638C2212yes—
18C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
19C2×C32×#67C2212yes—
20C2×C32×#67C2212yes—
21C2×C32×D12C2211yescharacteristic
22C2×#6293C2211yescharacteristic
23C2×#6290C2212yes—
24C2×#6290C2212yes—
25C2×Dic3×C3⋊D3C2212yes—
26#89029—28no—
27C2×D3×C3⋊Dic3C2212yes—
28#88864C2212yes—
29C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
30C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
31C3×C4×C3⋊Dic3C2211yescharacteristic
32C32×C4×Dic3C2211yescharacteristic
33#88807C2211yescharacteristic
34#88806C2211yescharacteristic
35Dic3⋊(C3⋊Dic3)C2211yescharacteristic
36#88803C2212yes—
37(C2×D3)⋊(C3⋊Dic3)C2212yes—
38C2×#3830—32no—
39C2×#3805—38no—
40C32×#127—31no—
41#64149—31no—
42#64149D311yescharacteristic, complemented
43#64148—32no—
44#64148—32no—
45#64147—32no—
46D4×C3⋊Dic3—31no—
47D4×C3⋊Dic3—34no—
48D4×C3⋊Dic3—31no—
49D4×C3⋊Dic3—64no—
50C4×C4⋊(C3⋊D3)—31no—
51C4×#634—31no—
52C3×#940—34no—
53C3×#940D314yescomplemented
54#63980—38no—
55#63970—38no—
56#63970—38no—
57C4×#627—34no—
58#63914—34no—
59Dic3×D12—34no—
60Dic3×D12—34no—
61C23×C33C2312yes—
62C22×C3×C3⋊D3C2312yes—
63C22×C32×D3C2312yes—
64C33×D4—22no—
65C2×C33×C4C2311yescharacteristic
66C2×C3⋊(C3⋊Dic3)—24no—
67C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
68C2×C3⋊(C3⋊Dic3)C2312yes—
69C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
70C3×#638—24no—
71C3×#638—42no—
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C2×C3×C3⋊Dic3C2311yescharacteristic
74C3×C4⋊(C3⋊D3)—22no—
75C32×#67—24no—
76C32×#67—42no—
77C2×C32×Dic3C2311yescharacteristic
78C2×C32×Dic3C2311yescharacteristic
79C32×D12—22no—
80#6293D414yescomplemented
81#6290—24no—
82#6290—42no—
83Dic3×C3⋊D3—24no—
84D3×C3⋊Dic3—24no—
85#5718—94no—
86#5718—91no—
87C2×C32×D4—24no—
88C2×C32×D4—32no—
89C2×C32×D4—32no—
90C2×C32×D4—31no—
91C2×C32×D4C2×D314yes—
92C2×C32×D4C2×D311yescharacteristic
93C2×#638—32no—
94C2×#638—32no—
95C2×#638C2×D312yes—
96C2×#638C2×D312yes—
97C22×C3⋊Dic3—32no—
98C22×C3⋊Dic3—38no—
99C22×C3⋊Dic3—32no—
100C22×C3⋊Dic3—68no—
101#3830—68no—
102C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
103C2×C4×C3⋊D3—32no—
104C2×#634—31no—
105C2×C3×#67—38no—
106C2×C3×#67—38no—
107C2×C3×#67C2×D318yes—
108C2×C3×#67C2×D318yes—
109C2×C3×D12—34no—
110C2×C3×D12C2×D314yes—
111C2×#627—34no—
112C2×#625—38no—
113C2×#625—38no—
114#3805—632no—
115C2×D3×Dic3—38no—
116C2×D3×Dic3—38no—
117C32×C42—31no—
118#3757—32no—
119#3757—38no—
120#3757—32no—
121#3757—68no—
122#3752—32no—
123C4⋊(C3⋊Dic3)—31no—
124C4⋊(C3⋊Dic3)—34no—
125C4⋊(C3⋊Dic3)—31no—
126C4⋊(C3⋊Dic3)—64no—
127#3750—31no—
128#3750—31no—
129C4×C3⋊Dic3—31no—
130C4×C3⋊Dic3—34no—
131C4×C3⋊Dic3—64no—
132C4×C3⋊Dic3C2×D311yescharacteristic
133C3×C4×Dic3—34no—
134C3×C4×Dic3C2×D314yes—
135#3724—34no—
136Dic3⋊Dic3—34no—
137Dic3⋊Dic3—34no—
138(C2×D3)⋊Dic3—38no—
139(C2×D3)⋊Dic3—38no—
140C22×C33—22no—
141C22×C33—22no—
142C22×C33—22no—
143C22×C33C2411yescharacteristic
144C2×C3×C3⋊D3—22no—
145C2×C3×C3⋊D3—22no—
146C2×C3×C3⋊D3—22no—
147C2×C32×D3—22no—
148C2×C32×D3—22no—
149C2×C32×D3—22no—
150C33×C4#4112yescomplemented
151C3⋊(C3⋊Dic3)—22no—
152C3⋊(C3⋊Dic3)—21no—
153C3⋊(C3⋊Dic3)C2×D412yes—
154C3×C3⋊Dic3—21no—
155C3×C3⋊Dic3C2×D412yes—
156C32×Dic3—21no—
157C32×Dic3C2×D412yes—
158C2×#237—98no—
159C2×#237—92no—
160C3×#127—34no—
161C3×#127—91no—
162#940—34no—
163#940—94no—
164#940—91no—
165#940C3⋊D311yescharacteristic, complemented
166#939—98no—
167#939—98no—
168#939—92no—
169#939—92no—
170#938—98no—
171#938—92no—
172D4×Dic3—94no—
173D4×Dic3—94no—
174D4×Dic3—91no—
175D4×Dic3—91no—
176D4×Dic3—184no—
177C4×D12—94no—
178C4×D12—91no—
179C4×Dic6—94no—
180C4×Dic6—91no—
181C23×C32—28no—
182C23×C32—32no—
183C23×C32C22×D318yes—
184C23×C32C22×D312yes—
185C22×C3⋊D3C22×D312yes—
186C22×C3×D3—38no—
187C22×C3×D3C22×D318yes—
188C32×D4—216no—
189C32×D4—28no—
190C32×D4—22no—
191C32×D4—64no—
192C32×D4—62no—
193C32×D4—122no—
194C2×C32×C4—24no—
195C2×C32×C4—31no—
196C2×C32×C4—31no—
197C2×C32×C4C22×D314yes—
198C2×C32×C4C22×D311yescharacteristic
199#638—24no—
200#638—42no—
201#638—64no—
202#638—122no—
203C2×C3⋊Dic3—31no—
204C2×C3⋊Dic3—31no—
205C2×C3⋊Dic3—32no—
206C2×C3⋊Dic3—34no—
207C2×C3⋊Dic3—38no—
208C2×C3⋊Dic3—34no—
209C2×C3⋊Dic3—64no—
210C2×C3⋊Dic3—64no—
211C2×C3⋊Dic3—68no—
212C2×C3⋊Dic3—64no—
213C2×C3⋊Dic3—64no—
214C2×C3⋊Dic3—616no—
215C2×C3⋊Dic3—1216no—
216C2×C3⋊Dic3C22×D311yescharacteristic
217C2×C3⋊Dic3C22×D311yescharacteristic
218C4⋊(C3⋊D3)—22no—
219C4×C3⋊D3—64no—
220#634—34no—
221C3×#67—216no—
222C3×#67—48no—
223C3×#67—616no—
224C3×#67—128no—
225C2×C3×Dic3—34no—
226C2×C3×Dic3—34no—
227C2×C3×Dic3C22×D314yes—
228C2×C3×Dic3C22×D314yes—
229C3×D12—28no—
230C3×D12—68no—
231#627—316no—
232#625—616no—
233#625—128no—
234D3×Dic3—616no—
235D3×Dic3—616no—
236#550—271noSylow
237C2×C33—42no—
238C2×C33C22×D411yescharacteristic
239C2×C33C2×#4111yescharacteristic
240C2×C33H(5,𝔽2)11yescharacteristic
241C3×C3⋊D3—42no—
242C32×D3—42no—
243C2×C3×D4—24no—
244C2×C3×D4—38no—
245C2×C3×D4—38no—
246C2×C3×D4—34no—
247C2×C3×D4—92no—
248C2×C3×D4—92no—
249C2×C3×D4—91no—
250C2×C3×D4D3214yes—
251C2×C3×D4C2×C3⋊D311yescharacteristic
252C2×#67—38no—
253C2×#67—38no—
254C2×#67—98no—
255C2×#67—92no—
256C2×#67—98no—
257C2×#67—92no—
258C2×#67C2×C3⋊D312yes—
259C2×#67C2×C3⋊D312yes—
260C22×Dic3—98no—
261C22×Dic3—98no—
262C22×Dic3—92no—
263C22×Dic3—92no—
264C22×Dic3—188no—
265#237—188no—
266#237—1832no—
267C2×D12—34no—
268C2×D12C2×C3⋊D311yescharacteristic
269C2×C4×D3—98no—
270C2×C4×D3—92no—
271C2×Dic6—94no—
272C2×Dic6—91no—
273C3×C42—34no—
274C3×C42—91no—
275#217—98no—
276#217—98no—
277#217—92no—
278#217—92no—
279#217—188no—
280(C2×D3)⋊C4—98no—
281(C2×D3)⋊C4—92no—
282C4⋊Dic3—94no—
283C4⋊Dic3—94no—
284C4⋊Dic3—91no—
285C4⋊Dic3—91no—
286C4⋊Dic3—184no—
287Dic3⋊C4—94no—
288Dic3⋊C4—94no—
289Dic3⋊C4—91no—
290Dic3⋊C4—91no—
291C4×Dic3—34no—
292C4×Dic3—94no—
293C4×Dic3—91no—
294C4×Dic3—184no—
295C4×Dic3C2×C3⋊D311yescharacteristic
296C22×C32—216no—
297C22×C32—22no—
298C22×C32—28no—
299C22×C32—22no—
300C22×C32—22no—
301C22×C32—24no—
302C22×C32—28no—
303C22×C32—28no—
304C22×C32—48no—
305C22×C32—48no—
306C22×C32—62no—
307C22×C32—62no—
308C22×C32—62no—
309C22×C32C23×D314yes—
310C22×C32C23×D311yescharacteristic
311C2×C3⋊D3—22no—
312C2×C3⋊D3—22no—
313C2×C3⋊D3—22no—
314C2×C3×D3—28no—
315C2×C3×D3—28no—
316C2×C3×D3—28no—
317C2×C3×D3—68no—
318C2×C3×D3—68no—
319C2×C3×D3—68no—
320C32×C4—28no—
321C32×C4—32no—
322C32×C4—61no—
323C32×C4#23718yescomplemented
324C32×C4#23712yescomplemented
325C3⋊Dic3—21no—
326C3⋊Dic3—32no—
327C3⋊Dic3—38no—
328C3⋊Dic3—68no—
329C3⋊Dic3—62no—
330C3⋊Dic3—61no—
331C3⋊Dic3—68no—
332C3⋊Dic3—64no—
333C3⋊Dic3—124no—
334C3⋊Dic3—128no—
335C3⋊Dic3D3×D412yes—
336C3×Dic3—24no—
337C3×Dic3—38no—
338C3×Dic3—64no—
339C3×Dic3D3×D418yes—
340C2×#41—272no—
341#127—31no—
342#127—91no—
343#127—271no—
344#121—272no—
345#121—272no—
346#121—272no—
347C4×Q8—271no—
348C4×D4—271no—
349C4×D4—271no—
350C4×D4—271no—
351C33#55011yescharacteristic, complemented, Sylow
352C23×C3—28no—
353C23×C3—38no—
354C23×C3—92no—
355C23×C3C2×D3218yes—
356C23×C3C22×C3⋊D312yes—
357C22×D3—38no—
358C22×D3C22×C3⋊D312yes—
359C3×D4—216no—
360C3×D4—28no—
361C3×D4—22no—
362C3×D4—616no—
363C3×D4—68no—
364C3×D4—128no—
365C3×D4—184no—
366C3×D4—182no—
367C3×D4—362no—
368C2×C3×C4—24no—
369C2×C3×C4—34no—
370C2×C3×C4—34no—
371C2×C3×C4—91no—
372C2×C3×C4—91no—
373C2×C3×C4C2×D3214yes—
374C2×C3×C4C22×C3⋊D311yescharacteristic
375#67—24no—
376#67—42no—
377#67—616no—
378#67—128no—
379#67—184no—
380#67—1816no—
381#67—362no—
382#67—368no—
383C2×Dic3—34no—
384C2×Dic3—34no—
385C2×Dic3—94no—
386C2×Dic3—91no—
387C2×Dic3—94no—
388C2×Dic3—98no—
389C2×Dic3—91no—
390C2×Dic3—92no—
391C2×Dic3—184no—
392C2×Dic3—184no—
393C2×Dic3—188no—
394C2×Dic3—1816no—
395C2×Dic3—184no—
396C2×Dic3—1816no—
397C2×Dic3—184no—
398C2×Dic3—3616no—
399C2×Dic3C22×C3⋊D311yescharacteristic
400C2×Dic3C22×C3⋊D311yescharacteristic
401D12—22no—
402D12—68no—
403C4×D3—1816no—
404C4×D3—184no—
405Dic6—916no—
406Dic6—94no—
407C2×C32—24no—
408C2×C32—24no—
409C2×C32—24no—
410C2×C32—416no—
411C2×C32—48no—
412C2×C32—42no—
413C2×C32—122no—
414C2×C32C2×D3×D414yes—
415C2×C32C2×D3×D411yescharacteristic
416C2×C32C2×#23714yes—
417C2×C32C2×#23711yescharacteristic
418C2×C32#100414yes—
419C2×C32#100411yescharacteristic
420C3⋊D3—42no—
421C3×D3—48no—
422C3×D3—128no—
423#41—548no—
424C2×Q8—271no—
425C2×D4—32no—
426C2×D4—32no—
427C2×D4—31no—
428C2×D4—92no—
429C2×D4—92no—
430C2×D4—91no—
431C2×D4—272no—
432C2×D4—272no—
433C2×D4D3×C3⋊D311yescharacteristic
434C22×C4—272no—
435C22×C4—272no—
436C22×C4—272no—
437C4⋊C4—271no—
438C4⋊C4—271no—
439C4⋊C4—271no—
440C4⋊C4—271no—
441C22⋊C4—272no—
442C22⋊C4—272no—
443C22⋊C4—272no—
444C42—31no—
445C42—91no—
446C42—271no—
447C22×C3—216no—
448C22×C3—28no—
449C22×C3—28no—
450C22×C3—28no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—22no—
454C22×C3—24no—
455C22×C3—48no—
456C22×C3—48no—
457C22×C3—68no—
458C22×C3—68no—
459C22×C3—68no—
460C22×C3—182no—
461C22×C3—182no—
462C22×C3—182no—
463C22×C3C22×D3214yes—
464C22×C3C23×C3⋊D311yescharacteristic
465C2×D3—22no—
466C2×D3—22no—
467C2×D3—22no—
468C2×D3—68no—
469C2×D3—68no—
470C2×D3—68no—
471C3×C4—28no—
472C3×C4—38no—
473C3×C4—64no—
474C3×C4—92no—
475C3×C4—181no—
476C3×C4#380518yescomplemented
477C3×C4#383012yescomplemented
478Dic3—21no—
479Dic3—38no—
480Dic3—64no—
481Dic3—98no—
482Dic3—92no—
483Dic3—188no—
484Dic3—188no—
485Dic3—184no—
486Dic3—182no—
487Dic3—181no—
488Dic3—368no—
489Dic3—364no—
490Dic3D4×C3⋊D312yes—
491C32—24no—
492C32#571814yescomplemented
493C32#571811yescharacteristic, complemented
494C23—32no—
495C23—92no—
496C23C2×D3×C3⋊D312yes—
497Q8—274no—
498D4—22no—
499D4—64no—
500D4—62no—
501D4—122no—
502D4—184no—
503D4—182no—
504D4—362no—
505D4—544no—
506D4—1082no—
507C2×C4—31no—
508C2×C4—31no—
509C2×C4—91no—
510C2×C4—91no—
511C2×C4—271no—
512C2×C4—271no—
513C2×C4—272no—
514C2×C4—544no—
515C2×C4—544no—
516C2×C4—544no—
517C2×C4C2×D3×C3⋊D311yescharacteristic
518C2×C3—24no—
519C2×C3—24no—
520C2×C3—24no—
521C2×C3—416no—
522C2×C3—48no—
523C2×C3—42no—
524C2×C3—128no—
525C2×C3—362no—
526C2×C3D32×D414yes—
527C2×C3#6431314yes—
528C2×C3C2×#380514yes—
529C2×C3C2×D4×C3⋊D311yescharacteristic
530C2×C3C2×#383011yescharacteristic
531C2×C3#6436111yescharacteristic
532D3—42no—
533D3—128no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—62no—
538C22—62no—
539C22—62no—
540C22—182no—
541C22—182no—
542C22—182no—
543C22C22×D3×C3⋊D311yescharacteristic
544C4—32no—
545C4—61no—
546C4—92no—
547C4—181no—
548C4—272no—
549C4—542no—
550C4—541no—
551C4#8902912yescomplemented
552C3—24no—
553C3#1059514414yescomplemented
554C3#1059618911yescharacteristic, complemented
555C2—42no—
556C2—122no—
557C2—362no—
558C2D3×D4×C3⋊D311yescharacteristic
559C2#1172458711yescharacteristic
560C2C2×#8902911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#550
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#89029C2semidirect
2. D4×C3⋊(C3⋊Dic3)C2semidirect
3. C3×#64149C2semidirect
4. C32×#940C2semidirect
5. #11723637C2semidirect
6. #11723631C2semidirect
7. #11723615C2semidirect
8. C4×#6293C2semidirect
9. #11723529C2semidirect
10. D12×C3⋊Dic3C2semidirect
11. Dic3×C4⋊(C3⋊D3)C2semidirect
42. #64149D3semidirect
53. C3×#940D3semidirect
80. #6293D4semidirect
150. C33×C4#41semidirect
165. #940C3⋊D3semidirect
323. C32×C4#237semidirect
324. C32×C4#237semidirect
351. C33#550semidirect
476. C3×C4#3805semidirect
477. C3×C4#3830semidirect
492. C32#5718semidirect
493. C32#5718semidirect
551. C4#89029semidirect
553. C3#10595144semidirect
554. C3#10596189semidirect

Maximal subgroups

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

subgroupindex
C2×#890292
D4×C3⋊(C3⋊Dic3)2
C3×#641492
C32×#9402
#117236372
#117236312
#117236152
C4×#62932
#117235292
D12×C3⋊Dic32
Dic3×C4⋊(C3⋊D3)2
#105961893
#105951443