Finite Group Databasebeta

← #13156559

Subgroups — #13156559

Order 1728.

622 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723454C211yesmaximal, characteristic, complemented
2C3×#63911C211yesmaximal, characteristic
3#11722075C211yesmaximal, characteristic, complemented
4#11722034C211yesmaximal, characteristic, complemented
5#11722032C211yesmaximal, characteristic, complemented
6C3×#63563C211yesmaximal, characteristic, complemented
7C3×#63554C211yesmaximal, characteristic, complemented
8#10591477—31nomaximal
9#10590882—31nomaximal
10#10590275—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C3×#626—21no—
14C2×C3×#626—21no—
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4×Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88807—21no—
21#88806—21no—
22#64079—31no—
23C3×#940—31no—
24C3×#874—31no—
25#63911D311yescharacteristic
26#63869—31no—
27#63869—31no—
28#63869—61no—
29#63659—31no—
30#63648—31no—
31C3×#832—31no—
32C3×#821—31no—
33C3×#815—31no—
34C3×#803—31no—
35#63564—31no—
36#63563—31no—
37#63563—31no—
38#63563—31no—
39#63563—31no—
40#63563—31no—
41#63563—61no—
42#63563—61no—
43#63563D311yescharacteristic, complemented
44#63562—31no—
45#63554D311yescharacteristic, complemented
46C22×C3×C3⋊D3—21no—
47C22×C32×D3—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)—21no—
50C2×C3×C3⋊Dic3—21no—
51C3×C4⋊(C3⋊D3)—21no—
52C3×C4⋊(C3⋊D3)—21no—
53C2×C32×Dic3D411yescharacteristic
54C2×C32×Dic3D411yescharacteristic
55C32×D12—21no—
56C32×D12—21no—
57C3×#626—42no—
58C3×#626—42no—
59C3×C3⋊(C3⋊C8)—21no—
60C32×C3⋊C8—21no—
61#5071—91no—
62#4764—91no—
63#4713—91no—
64C2×C32×D4—31no—
65C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
66C2×C3×#67—61no—
67C2×C3×#67—61no—
68C2×C3×D12—21no—
69C2×C3×D12—31no—
70C2×C3×D12—31no—
71C2×C3×D12—61no—
72C2×C3×D12—61no—
73C2×C3×D12—61no—
74C2×C3×D12C2×D311yescharacteristic
75C2×C3×D12C2×D311yescharacteristic
76C2×#626—21no—
77C2×#626—21no—
78C2×C32×C8—31no—
79C32×C42—31no—
80C4⋊(C3⋊Dic3)—31no—
81C4⋊(C3⋊Dic3)—31no—
82C4⋊(C3⋊Dic3)—31no—
83C4⋊(C3⋊Dic3)—61no—
84C4⋊(C3⋊Dic3)—61no—
85C4⋊(C3⋊Dic3)—61no—
86C4⋊(C3⋊Dic3)—121no—
87C4⋊(C3⋊Dic3)C2×D311yescharacteristic
88#3750—61no—
89#3750—61no—
90C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
91C3×C4⋊Dic3—31no—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3—61no—
94C3×C4×Dic3—21no—
95C3×C4×Dic3C2×D311yescharacteristic
96C3×C4×Dic3C2×D311yescharacteristic
97C2×C3×C3⋊C8—21no—
98C2×C3×C3⋊C8—31no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—61no—
101C2×C3×C3⋊C8C2×D311yescharacteristic
102C2×C3×C3⋊C8C2×D311yescharacteristic
103#3724—62no—
104#3724—61no—
105#3724—61no—
106Dic3⋊Dic3—62no—
107Dic3⋊Dic3—61no—
108Dic3⋊Dic3—61no—
109C22×C33C2×D411yescharacteristic
110C2×C3×C3⋊D3—41no—
111C2×C3×C3⋊D3—41no—
112C2×C3×C3⋊D3—41no—
113C2×C32×D3—41no—
114C2×C32×D3—41no—
115C2×C32×D3—41no—
116C33×C4#4111yescharacteristic
117C33×C4#4111yescharacteristic
118C3⋊(C3⋊Dic3)—41no—
119C3×C3⋊Dic3—41no—
120C32×Dic3—21no—
121C32×Dic3—21no—
122C3×#127—91no—
123#940—31no—
124#890—91no—
125#874—31no—
126#870—91no—
127#870—91no—
128#870—181no—
129C3×#102—91no—
130C3×#97—91no—
131#832—31no—
132#832—91no—
133#832—91no—
134#832—91no—
135#832—91no—
136#832—181no—
137#821—31no—
138#821—91no—
139#821—91no—
140#821—91no—
141#821—91no—
142#821—181no—
143#815—31no—
144#809—91no—
145#803—31no—
146C23×C32—61no—
147C22×C3⋊D3—21no—
148C22×C3×D3—21no—
149C22×C3×D3—21no—
150C22×C3×D3—41no—
151C22×C3×D3—61no—
152C22×C3×D3—61no—
153C22×C3×D3—121no—
154C32×D4—61no—
155C32×D4—61no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—31no—
160C2×C32×C4—31no—
161C2×C32×C4—41no—
162C2×C32×C4C22×D311yescharacteristic
163C2×C32×C4C22×D311yescharacteristic
164C2×C32×C4C22×D311yescharacteristic
165C2×C3⋊Dic3—21no—
166C2×C3⋊Dic3—61no—
167C2×C3⋊Dic3—62no—
168C2×C3⋊Dic3—61no—
169C2×C3⋊Dic3—61no—
170C2×C3⋊Dic3—121no—
171C2×C3⋊Dic3—121no—
172C2×C3⋊Dic3—122no—
173C4⋊(C3⋊D3)—21no—
174C4⋊(C3⋊D3)—21no—
175C3×#67—122no—
176C3×#67—122no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—62no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3D1211yescharacteristic
183C2×C3×Dic3D1211yescharacteristic
184C2×C3×Dic3#6711yescharacteristic
185C2×C3×Dic3#6711yescharacteristic
186C3×D12—22no—
187C3×D12—21no—
188C3×D12—21no—
189C3×D12—21no—
190C3×D12—21no—
191C3×D12—41no—
192C3×D12—62no—
193C3×D12—61no—
194C3×D12—61no—
195C3×D12—61no—
196C3×D12—61no—
197C3×D12—121no—
198C3×D12—122no—
199C3×D12—122no—
200#626—42no—
201#626—42no—
202C32×C8—61no—
203C3⋊(C3⋊C8)—21no—
204C3×C3⋊C8—21no—
205C3×C3⋊C8—21no—
206C3×C3⋊C8—41no—
207C3×C3⋊C8—61no—
208C3×C3⋊C8—61no—
209C3×C3⋊C8—121no—
210#490—271noSylow
211C2×C33#12411yescharacteristic
212C2×C33AGL(1,ℤ/8)11yescharacteristic
213C2×C33AGL(1,ℤ/8)11yescharacteristic
214C3×C3⋊D3—81no—
215C32×D3—81no—
216C2×C3×D4—31no—
217C2×C3×D4—31no—
218C2×C3×D4—61no—
219C2×C3×D4—91no—
220C2×C3×D4—181no—
221C2×C3×D4—181no—
222C2×#67—61no—
223C2×#67—61no—
224C2×D12—31no—
225C2×D12—31no—
226C2×D12—61no—
227C2×D12—61no—
228C2×D12—61no—
229C2×D12D3211yescharacteristic
230C2×C3×C8—31no—
231C2×C3×C8—31no—
232C2×C3×C8—61no—
233C2×C3×C8—91no—
234C3×C4⋊C4—91no—
235C3×C42—31no—
236C3×C42—31no—
237C3×C42—61no—
238C4⋊Dic3—31no—
239C4⋊Dic3—31no—
240C4⋊Dic3—61no—
241C4⋊Dic3—91no—
242C4⋊Dic3—91no—
243C4⋊Dic3—91no—
244C4⋊Dic3—181no—
245C4⋊Dic3—181no—
246C4⋊Dic3—181no—
247C4⋊Dic3—181no—
248C4⋊Dic3—361no—
249Dic3⋊C4—181no—
250Dic3⋊C4—181no—
251Dic3⋊C4—181no—
252Dic3⋊C4—181no—
253Dic3⋊C4—182no—
254Dic3⋊C4—182no—
255Dic3⋊C4—181no—
256C4×Dic3D3211yescharacteristic
257C2×C3⋊C8—31no—
258C2×C3⋊C8—31no—
259C2×C3⋊C8—61no—
260C2×C3⋊C8D3211yescharacteristic
261C22×C32—21no—
262C22×C32—21no—
263C22×C32—21no—
264C22×C32—41no—
265C22×C32—121no—
266C22×C32—121no—
267C22×C32—121no—
268C22×C32C2×D1211yescharacteristic
269C22×C32D3×D411yescharacteristic
270C22×C32C2×#6711yescharacteristic
271C2×C3⋊D3—41no—
272C2×C3⋊D3—41no—
273C2×C3⋊D3—41no—
274C2×C3×D3—42no—
275C2×C3×D3—41no—
276C2×C3×D3—41no—
277C2×C3×D3—41no—
278C2×C3×D3—41no—
279C2×C3×D3—41no—
280C2×C3×D3—41no—
281C2×C3×D3—81no—
282C2×C3×D3—81no—
283C2×C3×D3—122no—
284C2×C3×D3—121no—
285C2×C3×D3—121no—
286C2×C3×D3—121no—
287C2×C3×D3—121no—
288C2×C3×D3—121no—
289C2×C3×D3—121no—
290C2×C3×D3—241no—
291C2×C3×D3—241no—
292C32×C4—21no—
293C32×C4—21no—
294C32×C4—21no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4—21no—
298C32×C4—41no—
299C32×C4—41no—
300C32×C4—61no—
301C32×C4—61no—
302C32×C4#23511yescharacteristic
303C32×C4#23511yescharacteristic
304C32×C4#23511yescharacteristic
305C32×C4#23711yescharacteristic
306C32×C4#23711yescharacteristic
307C32×C4#23711yescharacteristic
308C3⋊Dic3—41no—
309C3⋊Dic3—121no—
310C3⋊Dic3—122no—
311C3⋊Dic3—121no—
312C3⋊Dic3—121no—
313C3⋊Dic3—241no—
314C3⋊Dic3—241no—
315C3⋊Dic3—242no—
316C3×Dic3—21no—
317C3×Dic3—21no—
318C3×Dic3—21no—
319C3×Dic3—21no—
320C3×Dic3—41no—
321C3×Dic3—41no—
322C3×Dic3—122no—
323C3×Dic3—121no—
324C3×Dic3—121no—
325#127—91no—
326#125—271no—
327#102—91no—
328#102—271no—
329#102—271no—
330#102—271no—
331#97—91no—
332C33#49011yescharacteristic, complemented, Sylow
333C23×C3—61no—
334C23×C3—61no—
335C23×C3—121no—
336C23×C3—181no—
337C22×D3—21no—
338C22×D3—61no—
339C22×D3—61no—
340C22×D3—121no—
341C3×D4—62no—
342C3×D4—61no—
343C3×D4—61no—
344C3×D4—61no—
345C3×D4—61no—
346C3×D4—121no—
347C3×D4—181no—
348C3×D4—181no—
349C3×D4—362no—
350C3×D4—362no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—31no—
355C2×C3×C4—31no—
356C2×C3×C4—31no—
357C2×C3×C4—31no—
358C2×C3×C4—41no—
359C2×C3×C4—61no—
360C2×C3×C4—61no—
361C2×C3×C4—181no—
362C2×C3×C4C2×D3211yescharacteristic
363C2×C3×C4C2×D3211yescharacteristic
364C2×C3×C4C2×D3211yescharacteristic
365#67—122no—
366#67—122no—
367C2×Dic3—62no—
368C2×Dic3—61no—
369C2×Dic3—61no—
370C2×Dic3—182no—
371C2×Dic3—181no—
372C2×Dic3—181no—
373C2×Dic3—181no—
374C2×Dic3—361no—
375C2×Dic3—361no—
376C2×Dic3—362no—
377C2×Dic3#62611yescharacteristic
378C2×Dic3#62611yescharacteristic
379D12—21no—
380D12—21no—
381D12—62no—
382D12—61no—
383D12—61no—
384D12—61no—
385D12—61no—
386D12—121no—
387D12—122no—
388D12—122no—
389C3×C8—61no—
390C3×C8—61no—
391C3×C8—121no—
392C3×C8—181no—
393C3⋊C8—21no—
394C3⋊C8—61no—
395C3⋊C8—61no—
396C3⋊C8—121no—
397C2×C32—21no—
398C2×C32—21no—
399C2×C32—21no—
400C2×C32—21no—
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32—21no—
405C2×C32—21no—
406C2×C32—41no—
407C2×C32—41no—
408C2×C32—41no—
409C2×C32—241no—
410C2×C32#87511yescharacteristic
411C2×C32#88511yescharacteristic
412C2×C32#90811yescharacteristic
413C2×C32#90811yescharacteristic
414C2×C32#91111yescharacteristic
415C2×C32#91411yescharacteristic
416C2×C32#93211yescharacteristic
417C2×C32#93211yescharacteristic
418C2×C32#93611yescharacteristic
419C3⋊D3—81no—
420C3×D3—82no—
421C3×D3—81no—
422C3×D3—81no—
423C3×D3—242no—
424C3×D3—241no—
425C3×D3—241no—
426C2×D4—31no—
427C2×D4—91no—
428C2×D4—181no—
429C2×D4—181no—
430C2×C8—31no—
431C2×C8—91no—
432C4⋊C4—91no—
433C4⋊C4—271no—
434C4⋊C4—541no—
435C4⋊C4—541no—
436C42—31no—
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—21no—
440C22×C3—41no—
441C22×C3—122no—
442C22×C3—121no—
443C22×C3—121no—
444C22×C3—121no—
445C22×C3—121no—
446C22×C3—121no—
447C22×C3—121no—
448C22×C3—241no—
449C22×C3—241no—
450C22×C3—361no—
451C22×C3—361no—
452C22×C3—361no—
453C22×C3D3×D1211yescharacteristic
454C22×C3C2×#62611yescharacteristic
455C22×C3D3×#6711yescharacteristic
456C2×D3—41no—
457C2×D3—41no—
458C2×D3—41no—
459C2×D3—122no—
460C2×D3—121no—
461C2×D3—121no—
462C2×D3—121no—
463C2×D3—121no—
464C2×D3—121no—
465C2×D3—121no—
466C2×D3—241no—
467C2×D3—241no—
468C3×C4—21no—
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—41no—
475C3×C4—41no—
476C3×C4—61no—
477C3×C4—61no—
478C3×C4—61no—
479C3×C4—61no—
480C3×C4—121no—
481C3×C4—121no—
482C3×C4—361no—
483C3×C4#379511yescharacteristic
484C3×C4#379511yescharacteristic
485C3×C4#379811yescharacteristic
486C3×C4#380411yescharacteristic
487C3×C4#380411yescharacteristic
488C3×C4#380511yescharacteristic
489Dic3—21no—
490Dic3—21no—
491Dic3—122no—
492Dic3—121no—
493Dic3—121no—
494Dic3—362no—
495Dic3—361no—
496Dic3—361no—
497Dic3—361no—
498Dic3—721no—
499Dic3—721no—
500Dic3—722no—
501C32—21no—
502C32—21no—
503C32—21no—
504C32—41no—
505C32#471311yescharacteristic, complemented
506C32#476411yescharacteristic, complemented
507C32#507111yescharacteristic, complemented
508C23—61no—
509C23—181no—
510D4—61no—
511D4—61no—
512D4—181no—
513D4—181no—
514D4—362no—
515D4—362no—
516C2×C4—31no—
517C2×C4—31no—
518C2×C4—181no—
519C2×C4—541no—
520C2×C4D3311yescharacteristic
521C8—61no—
522C8—181no—
523C2×C3—21no—
524C2×C3—21no—
525C2×C3—21no—
526C2×C3—21no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—41no—
533C2×C3—41no—
534C2×C3—41no—
535C2×C3—242no—
536C2×C3—241no—
537C2×C3—241no—
538C2×C3—721no—
539C2×C3#6379411yescharacteristic
540C2×C3#6379511yescharacteristic
541C2×C3#6383011yescharacteristic
542C2×C3#6383011yescharacteristic
543C2×C3#6385211yescharacteristic
544C2×C3#6386011yescharacteristic
545C2×C3#6393211yescharacteristic
546C2×C3#6393411yescharacteristic
547C2×C3#6395911yescharacteristic
548D3—81no—
549D3—242no—
550D3—241no—
551D3—241no—
552C22—121no—
553C22—121no—
554C22—121no—
555C22—361no—
556C22—361no—
557C22—361no—
558C22D3×#62611yescharacteristic
559C4—61no—
560C4—61no—
561C4—361no—
562C4—1081no—
563C4#8896011yescharacteristic
564C4#8896111yescharacteristic
565C3—21no—
566C3—21no—
567C3—21no—
568C3—41no—
569C3#1059027511yescharacteristic, complemented
570C3#1059088211yescharacteristic, complemented
571C3#1059147711yescharacteristic, complemented
572C2—241no—
573C2—721no—
574C2#1172286811yescharacteristic
575C2#1172286911yescharacteristic
576C2#1172300211yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC3×#63911C2
hypocenterC33#490
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723454C2semidirect
3. #11722075C2semidirect
4. #11722034C2semidirect
5. #11722032C2semidirect
6. C3×#63563C2semidirect
7. C3×#63554C2semidirect
43. #63563D3semidirect
45. #63554D3semidirect
332. C33#490semidirect
505. C32#4713semidirect
506. C32#4764semidirect
507. C32#5071semidirect
569. C3#10590275semidirect
570. C3#10590882semidirect
571. C3#10591477semidirect

Maximal subgroups

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

subgroupindex
#117234542
C3×#639112
#117220752
#117220342
#117220322
C3×#635632
C3×#635542
#105914773
#105908823
#105902753