Finite Group Databasebeta

← #13156424

Subgroups — #13156424

Order 1728.

612 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723424C211yesmaximal, characteristic, complemented
2C3×#63908C211yesmaximal, characteristic
3#11722080C211yesmaximal, characteristic, complemented
4#11722052C211yesmaximal, characteristic, complemented
5#11722032C211yesmaximal, characteristic, complemented
6C3×#63563C211yesmaximal, characteristic, complemented
7C3×(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10591422—31nomaximal
9#10591093—31nomaximal
10#10590837—31nomaximal
11C2×C32×D12C2211yescharacteristic
12C2×C32×C4×D3C2211yescharacteristic
13C2×C3×#625—21no—
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19(C2×D3)⋊(C3⋊Dic3)—21no—
20Dic3×C3⋊Dic3—21no—
21C3×(C2×D3)⋊Dic3—21no—
22#64135—31no—
23C3×#938—31no—
24C3×#930—31no—
25#63908D311yescharacteristic
26#63841—31no—
27#63841—31no—
28#63841—61no—
29#63659—31no—
30#63644—31no—
31C3×#832—31no—
32C3×#830—31no—
33C3×#821—31no—
34C3×(C2×D3)⋊C8—31no—
35#63576—31no—
36#63575—31no—
37#63575—31no—
38#63573—31no—
39#63573—31no—
40#63573—61no—
41#63563—31no—
42#63563—31no—
43#63563—61no—
44#63563D311yescharacteristic, complemented
45(C2×D3)⋊(C3⋊C8)D311yescharacteristic, complemented
46C22×C32×D3—21no—
47C22×C32×D3D411yescharacteristic
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)—21no—
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3—21no—
52C2×C32×Dic3D411yescharacteristic
53C32×D12—21no—
54C32×D12—21no—
55C32×C4×D3—21no—
56C32×C4×D3—21no—
57C3×#625—42no—
58C3×C3⋊(C3⋊C8)—21no—
59C32×C3⋊C8—21no—
60#5035—91no—
61#4779—91no—
62#4737—91no—
63C2×C32×D4—31no—
64C22×C32×C4—31no—
65C2×C3×#67—61no—
66C2×C3×#67—61no—
67C2×C3×D12—21no—
68C2×C3×D12C2×D311yescharacteristic
69C2×C3×D12C2×D311yescharacteristic
70C2×C3×C4×D3—21no—
71C2×C3×C4×D3C2×D311yescharacteristic
72C2×C3×C4×D3C2×D311yescharacteristic
73C2×#625—21no—
74C2×C32×C8—31no—
75#3757—61no—
76C4⋊(C3⋊Dic3)—31no—
77C4⋊(C3⋊Dic3)—31no—
78C4⋊(C3⋊Dic3)—31no—
79C4⋊(C3⋊Dic3)—61no—
80C4⋊(C3⋊Dic3)—61no—
81C4⋊(C3⋊Dic3)—61no—
82C4⋊(C3⋊Dic3)—121no—
83C4⋊(C3⋊Dic3)C2×D311yescharacteristic
84C4⋊(C3⋊Dic3)C2×D311yescharacteristic
85C4×C3⋊Dic3—61no—
86C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
87C3×#217—61no—
88C3×(C2×D3)⋊C4—61no—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—31no—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3—61no—
94C3×C4⋊Dic3—61no—
95C2×C3×C3⋊C8—21no—
96C2×C3×C3⋊C8—31no—
97C2×C3×C3⋊C8—31no—
98C2×C3×C3⋊C8—61no—
99C2×C3×C3⋊C8C2×D311yescharacteristic
100C2×C3×C3⋊C8C2×D311yescharacteristic
101(C2×D3)⋊Dic3—21no—
102(C2×D3)⋊Dic3—62no—
103(C2×D3)⋊Dic3—61no—
104(C2×D3)⋊Dic3—61no—
105Dic32—62no—
106Dic32—61no—
107Dic32—61no—
108C22×C33C2×D411yescharacteristic
109C2×C32×D3—21no—
110C2×C32×D3—21no—
111C2×C32×D3—21no—
112C2×C32×D3—41no—
113C2×C32×D3—41no—
114C2×C32×D3—41no—
115C33×C4#4111yescharacteristic
116C33×C4#4111yescharacteristic
117C3⋊(C3⋊Dic3)—41no—
118C3×C3⋊Dic3—41no—
119C3×C3⋊Dic3—41no—
120C32×Dic3—21no—
121C3×#121—91no—
122#938—31no—
123#930—31no—
124#926—91no—
125#926—91no—
126#926—181no—
127#892—91no—
128C3×#102—91no—
129C3×C22⋊C8—91no—
130#832—31no—
131#832—91no—
132#832—91no—
133#832—181no—
134#830—31no—
135#821—31no—
136#821—91no—
137(C2×D3)⋊C8—31no—
138#818—91no—
139#817—91no—
140#817—91no—
141#817—181no—
142#808—91no—
143#807—91no—
144#807—91no—
145C23×C32—31no—
146C23×C32—61no—
147C22×C3×D3—21no—
148C22×C3×D3—21no—
149C22×C3×D3—21no—
150C22×C3×D3—41no—
151C22×C3×D3D1211yescharacteristic
152C22×C3×D3#6711yescharacteristic
153C32×D4—61no—
154C32×D4—61no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—31no—
159C2×C32×C4—41no—
160C2×C32×C4—61no—
161C2×C32×C4—61no—
162C2×C32×C4C22×D311yescharacteristic
163C2×C32×C4C22×D311yescharacteristic
164C2×C32×C4C22×D311yescharacteristic
165C2×C3⋊Dic3—21no—
166C2×C3⋊Dic3—21no—
167C2×C3⋊Dic3—61no—
168C2×C3⋊Dic3—62no—
169C2×C3⋊Dic3—61no—
170C2×C3⋊Dic3—61no—
171C2×C3⋊Dic3—121no—
172C2×C3⋊Dic3—121no—
173C2×C3⋊Dic3—122no—
174C3×#67—122no—
175C3×#67—122no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—62no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—121no—
183C2×C3×Dic3D1211yescharacteristic
184C2×C3×Dic3#6711yescharacteristic
185C3×D12—22no—
186C3×D12—21no—
187C3×D12—21no—
188C3×D12—21no—
189C3×D12—21no—
190C3×D12—41no—
191C3×C4×D3—22no—
192C3×C4×D3—22no—
193C3×C4×D3—21no—
194C3×C4×D3—21no—
195C3×C4×D3—21no—
196C3×C4×D3—21no—
197#625—42no—
198C32×C8—61no—
199C3⋊(C3⋊C8)—21no—
200C3×C3⋊C8—22no—
201C3×C3⋊C8—21no—
202C3×C3⋊C8—21no—
203C3×C3⋊C8—61no—
204C3×C3⋊C8—61no—
205C3×C3⋊C8—121no—
206#482—271noSylow
207C2×C33#12311yescharacteristic
208C2×C33#13511yescharacteristic
209C2×C33AGL(1,ℤ/8)11yescharacteristic
210C32×D3—41no—
211C32×D3—81no—
212C2×C3×D4—31no—
213C2×C3×D4—31no—
214C2×C3×D4—61no—
215C2×C3×D4—181no—
216C22×C3×C4—31no—
217C22×C3×C4—31no—
218C22×C3×C4—61no—
219C2×#67—61no—
220C2×#67—61no—
221C2×D12D3211yescharacteristic
222C2×C4×D3D3211yescharacteristic
223C2×C3×C8—31no—
224C2×C3×C8—31no—
225C2×C3×C8—61no—
226C2×C3×C8—91no—
227C3×C4⋊C4—91no—
228C3×C4⋊C4—91no—
229C3×C22⋊C4—181no—
230#217—61no—
231#217—181no—
232#217—181no—
233#217—182no—
234(C2×D3)⋊C4—61no—
235(C2×D3)⋊C4—181no—
236C4⋊Dic3—31no—
237C4⋊Dic3—31no—
238C4⋊Dic3—31no—
239C4⋊Dic3—31no—
240C4⋊Dic3—61no—
241C4⋊Dic3—61no—
242C4⋊Dic3—91no—
243C4⋊Dic3—91no—
244C4⋊Dic3—91no—
245C4⋊Dic3—181no—
246C4⋊Dic3—181no—
247C4⋊Dic3—181no—
248C4⋊Dic3—361no—
249C4×Dic3—181no—
250C4×Dic3—181no—
251C4×Dic3—182no—
252C4×Dic3—181no—
253C2×C3⋊C8—31no—
254C2×C3⋊C8—31no—
255C2×C3⋊C8—61no—
256C2×C3⋊C8D3211yescharacteristic
257C22×C32—21no—
258C22×C32—21no—
259C22×C32—21no—
260C22×C32—41no—
261C22×C32—61no—
262C22×C32—61no—
263C22×C32—61no—
264C22×C32—121no—
265C22×C32—121no—
266C22×C32—121no—
267C22×C32C2×D1211yescharacteristic
268C22×C32D3×D411yescharacteristic
269C22×C32C2×#6711yescharacteristic
270C2×C3×D3—22no—
271C2×C3×D3—21no—
272C2×C3×D3—21no—
273C2×C3×D3—21no—
274C2×C3×D3—21no—
275C2×C3×D3—21no—
276C2×C3×D3—21no—
277C2×C3×D3—41no—
278C2×C3×D3—41no—
279C2×C3×D3—42no—
280C2×C3×D3—41no—
281C2×C3×D3—41no—
282C2×C3×D3—41no—
283C2×C3×D3—41no—
284C2×C3×D3—41no—
285C2×C3×D3—41no—
286C2×C3×D3—81no—
287C2×C3×D3—81no—
288C32×C4—21no—
289C32×C4—21no—
290C32×C4—21no—
291C32×C4—21no—
292C32×C4—21no—
293C32×C4—21no—
294C32×C4—41no—
295C32×C4—41no—
296C32×C4—61no—
297C32×C4#23511yescharacteristic
298C32×C4#23711yescharacteristic
299C32×C4#23711yescharacteristic
300C32×C4#23711yescharacteristic
301C32×C4#23711yescharacteristic
302C32×C4#23711yescharacteristic
303C3⋊Dic3—41no—
304C3⋊Dic3—41no—
305C3⋊Dic3—121no—
306C3⋊Dic3—122no—
307C3⋊Dic3—121no—
308C3⋊Dic3—121no—
309C3⋊Dic3—241no—
310C3⋊Dic3—241no—
311C3⋊Dic3—242no—
312C3×Dic3—22no—
313C3×Dic3—21no—
314C3×Dic3—21no—
315C3×Dic3—122no—
316C3×Dic3—121no—
317C3×Dic3—121no—
318C3×Dic3—121no—
319C3×Dic3—121no—
320C3×Dic3—241no—
321#121—91no—
322#117—271no—
323#107—271no—
324#106—271no—
325#102—91no—
326#102—271no—
327C22⋊C8—91no—
328C33#48211yescharacteristic, complemented, Sylow
329C23×C3—31no—
330C23×C3—31no—
331C23×C3—61no—
332C23×C3—61no—
333C23×C3—61no—
334C23×C3—121no—
335C22×D3—21no—
336C22×D3#62611yescharacteristic
337C3×D4—62no—
338C3×D4—61no—
339C3×D4—61no—
340C3×D4—61no—
341C3×D4—61no—
342C3×D4—121no—
343C3×D4—362no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—31no—
348C2×C3×C4—31no—
349C2×C3×C4—41no—
350C2×C3×C4—61no—
351C2×C3×C4—62no—
352C2×C3×C4—62no—
353C2×C3×C4—61no—
354C2×C3×C4—61no—
355C2×C3×C4—61no—
356C2×C3×C4—61no—
357C2×C3×C4—181no—
358C2×C3×C4—181no—
359C2×C3×C4C2×D3211yescharacteristic
360C2×C3×C4C2×D3211yescharacteristic
361C2×C3×C4C2×D3211yescharacteristic
362#67—122no—
363#67—122no—
364C2×Dic3—62no—
365C2×Dic3—61no—
366C2×Dic3—61no—
367C2×Dic3—61no—
368C2×Dic3—61no—
369C2×Dic3—121no—
370C2×Dic3—182no—
371C2×Dic3—181no—
372C2×Dic3—181no—
373C2×Dic3—181no—
374C2×Dic3—361no—
375C2×Dic3—361no—
376C2×Dic3—362no—
377C2×Dic3#62611yescharacteristic
378D12—21no—
379D12—21no—
380C4×D3—21no—
381C4×D3—21no—
382C3×C8—62no—
383C3×C8—61no—
384C3×C8—61no—
385C3×C8—181no—
386C3⋊C8—21no—
387C3⋊C8—61no—
388C3⋊C8—61no—
389C3⋊C8—121no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32—21no—
396C2×C32—21no—
397C2×C32—21no—
398C2×C32—21no—
399C2×C32—41no—
400C2×C32—41no—
401C2×C32—41no—
402C2×C32—121no—
403C2×C32—241no—
404C2×C32#88311yescharacteristic
405C2×C32#88611yescharacteristic
406C2×C32#90411yescharacteristic
407C2×C32#90811yescharacteristic
408C2×C32#91111yescharacteristic
409C2×C32#91611yescharacteristic
410C2×C32#93211yescharacteristic
411C2×C32#93511yescharacteristic
412C2×C32#95011yescharacteristic
413C3×D3—42no—
414C3×D3—41no—
415C3×D3—41no—
416C3×D3—82no—
417C3×D3—81no—
418C3×D3—81no—
419C2×D4—31no—
420C2×D4—181no—
421C22×C4—31no—
422C2×C8—31no—
423C2×C8—91no—
424C4⋊C4—91no—
425C4⋊C4—91no—
426C4⋊C4—271no—
427C22⋊C4—181no—
428C22⋊C4—541no—
429C42—541no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—41no—
434C22×C3—62no—
435C22×C3—61no—
436C22×C3—61no—
437C22×C3—61no—
438C22×C3—61no—
439C22×C3—61no—
440C22×C3—61no—
441C22×C3—121no—
442C22×C3—121no—
443C22×C3—122no—
444C22×C3—121no—
445C22×C3—121no—
446C22×C3—121no—
447C22×C3—121no—
448C22×C3—121no—
449C22×C3—121no—
450C22×C3—241no—
451C22×C3—241no—
452C22×C3D3×D1211yescharacteristic
453C22×C3C2×#62611yescharacteristic
454C22×C3D3×#6711yescharacteristic
455C2×D3—21no—
456C2×D3—21no—
457C2×D3—21no—
458C2×D3—41no—
459C2×D3—41no—
460C2×D3—41no—
461C3×C4—21no—
462C3×C4—21no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—41no—
468C3×C4—41no—
469C3×C4—62no—
470C3×C4—61no—
471C3×C4—61no—
472C3×C4—361no—
473C3×C4—361no—
474C3×C4#379511yescharacteristic
475C3×C4#380411yescharacteristic
476C3×C4#380511yescharacteristic
477C3×C4#380511yescharacteristic
478C3×C4#380511yescharacteristic
479C3×C4#380511yescharacteristic
480Dic3—21no—
481Dic3—122no—
482Dic3—121no—
483Dic3—121no—
484Dic3—121no—
485Dic3—121no—
486Dic3—241no—
487Dic3—362no—
488Dic3—361no—
489Dic3—361no—
490Dic3—361no—
491Dic3—721no—
492Dic3—721no—
493Dic3—722no—
494C32—21no—
495C32—21no—
496C32—21no—
497C32—41no—
498C32#473711yescharacteristic, complemented
499C32#477911yescharacteristic, complemented
500C32#503511yescharacteristic, complemented
501C23—31no—
502C23—61no—
503D4—61no—
504D4—61no—
505D4—362no—
506C2×C4—31no—
507C2×C4—61no—
508C2×C4—61no—
509C2×C4—181no—
510C2×C4—181no—
511C2×C4—541no—
512C2×C4D3311yescharacteristic
513C8—61no—
514C8—181no—
515C2×C3—21no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—41no—
525C2×C3—41no—
526C2×C3—41no—
527C2×C3—122no—
528C2×C3—121no—
529C2×C3—121no—
530C2×C3—242no—
531C2×C3—241no—
532C2×C3—241no—
533C2×C3#6379511yescharacteristic
534C2×C3#6380611yescharacteristic
535C2×C3#6382911yescharacteristic
536C2×C3#6383011yescharacteristic
537C2×C3#6389111yescharacteristic
538C2×C3#6393211yescharacteristic
539C2×C3#6393511yescharacteristic
540C2×C3#6396211yescharacteristic
541C2×C3#6396911yescharacteristic
542D3—41no—
543D3—81no—
544C22—61no—
545C22—61no—
546C22—61no—
547C22—121no—
548C22—121no—
549C22—121no—
550C22D3×#62611yescharacteristic
551C4—61no—
552C4—361no—
553C4—361no—
554C4—1081no—
555C4#8896111yescharacteristic
556C4#8896311yescharacteristic
557C3—21no—
558C3—21no—
559C3—21no—
560C3—41no—
561C3#1059083711yescharacteristic, complemented
562C3#1059109311yescharacteristic, complemented
563C3#1059142211yescharacteristic, complemented
564C2—121no—
565C2—241no—
566C2#1172285311yescharacteristic
567C2#1172286911yescharacteristic
568C2#1172298911yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC3×#63908C2
hypocenterC33#482
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. #11723424C2semidirect
3. #11722080C2semidirect
4. #11722052C2semidirect
5. #11722032C2semidirect
6. C3×#63563C2semidirect
7. C3×(C2×D3)⋊(C3⋊C8)C2semidirect
44. #63563D3semidirect
45. (C2×D3)⋊(C3⋊C8)D3semidirect
328. C33#482semidirect
498. C32#4737semidirect
499. C32#4779semidirect
500. C32#5035semidirect
561. C3#10590837semidirect
562. C3#10591093semidirect
563. C3#10591422semidirect

Maximal subgroups

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

subgroupindex
#117234242
C3×#639082
#117220802
#117220522
#117220322
C3×#635632
C3×(C2×D3)⋊(C3⋊C8)2
#105914223
#105910933
#105908373