Finite Group Databasebeta

← #13156506

Subgroups — #13156506

Order 1728.

638 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#63918C211yesmaximal, characteristic, complemented
2C3×#63911C211yesmaximal, characteristic
3#11722075C211yesmaximal, characteristic, complemented
4#11722033C211yesmaximal, characteristic, complemented
5#11722031C211yesmaximal, characteristic, complemented
6Dic3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7C3×#63562C211yesmaximal, characteristic, complemented
8#10591487—31nomaximal
9#10590548—31nomaximal
10#10590277—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C3×#627—21no—
14C2×C3×#626—21no—
15C2×C3×#626—21no—
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C32×C4⋊Dic3C2211yescharacteristic
20C32×C4×Dic3C2211yescharacteristic
21C3×Dic3⋊Dic3—21no—
22C3×#940—31no—
23C3×#888—31no—
24C3×#874—31no—
25C3×#869—31no—
26#63918D311yescharacteristic, complemented
27#63911D311yescharacteristic
28#63659—31no—
29#63636—31no—
30C4×C3⋊(C3⋊C8)—31no—
31C3×#832—31no—
32C3×#809—31no—
33#63564—31no—
34#63563—31no—
35#63562—31no—
36#63562—31no—
37#63562—31no—
38#63562—31no—
39#63562—31no—
40#63562—61no—
41#63562—61no—
42#63562D311yescharacteristic, complemented
43Dic3×C3⋊C8—31no—
44Dic3×C3⋊C8—31no—
45Dic3×C3⋊C8—61no—
46C22×C3×C3⋊D3—21no—
47C22×C32×D3—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3×C3⋊Dic3—21no—
50C3×C4⋊(C3⋊D3)—21no—
51C3×C4⋊(C3⋊D3)—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3D411yescharacteristic
54C2×C32×Dic3D411yescharacteristic
55C32×D12—21no—
56C32×D12—21no—
57C3×#627—42no—
58C3×#626—42no—
59C3×#626—42no—
60C3⋊(C3⋊(C3⋊C8))—21no—
61C3×C3⋊(C3⋊C8)—21no—
62#5088—91no—
63#5070—91no—
64#4760—91no—
65C2×C32×D4—31no—
66C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
67C2×C3×#67—61no—
68C2×C3×#67—61no—
69C2×C3×D12—21no—
70C2×C3×D12—31no—
71C2×C3×D12—31no—
72C2×C3×D12—61no—
73C2×C3×D12—61no—
74C2×C3×D12—61no—
75C2×C3×D12C2×D311yescharacteristic
76C2×C3×D12C2×D311yescharacteristic
77C2×C3×Dic6—61no—
78C2×C3×Dic6—61no—
79C2×#627—21no—
80C2×#626—21no—
81C2×#626—21no—
82C32×C4⋊C4—31no—
83C32×C42—31no—
84C4⋊(C3⋊Dic3)C2×D311yescharacteristic
85C2×C3⋊(C3⋊C8)—31no—
86C2×C3⋊(C3⋊C8)—31no—
87C2×C3⋊(C3⋊C8)—31no—
88C2×C3⋊(C3⋊C8)—61no—
89C2×C3⋊(C3⋊C8)—61no—
90C2×C3⋊(C3⋊C8)—61no—
91C2×C3⋊(C3⋊C8)—121no—
92C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
93C3×C4⋊Dic3—21no—
94C3×C4⋊Dic3—31no—
95C3×C4⋊Dic3—31no—
96C3×C4⋊Dic3—61no—
97C3×C4⋊Dic3—61no—
98C3×C4⋊Dic3C2×D311yescharacteristic
99C3×C4⋊Dic3C2×D311yescharacteristic
100C3×Dic3⋊C4—61no—
101C3×C4×Dic3—21no—
102C3×C4×Dic3C2×D311yescharacteristic
103C3×C4×Dic3C2×D311yescharacteristic
104C2×C3×C3⋊C8—31no—
105C2×C3×C3⋊C8—31no—
106C2×C3×C3⋊C8—61no—
107Dic3⋊Dic3—21no—
108C22×C33C2×D411yescharacteristic
109C2×C3×C3⋊D3—41no—
110C2×C3×C3⋊D3—41no—
111C2×C3×C3⋊D3—41no—
112C2×C32×D3—41no—
113C2×C32×D3—41no—
114C2×C32×D3—41no—
115C33×C4#4111yescharacteristic
116C33×C4#4111yescharacteristic
117C3×C3⋊Dic3—41no—
118C32×Dic3—41no—
119C32×Dic3D812yes—
120C32×Dic3SD1612yes—
121C3×C4⋊Q8—91no—
122C3×#127—91no—
123#940—31no—
124#888—31no—
125#874—31no—
126#869—31no—
127C3×#102—91no—
128#832—31no—
129#832—91no—
130#832—91no—
131#832—91no—
132#832—91no—
133#832—181no—
134#821—91no—
135C8×Dic3—91no—
136#809—31no—
137#809—91no—
138#809—91no—
139#809—91no—
140#809—91no—
141#809—181no—
142C4×C3⋊C8—91no—
143C4×C3⋊C8—91no—
144C4×C3⋊C8—181no—
145C23×C32—61no—
146C22×C3⋊D3—21no—
147C22×C3×D3—21no—
148C22×C3×D3—21no—
149C22×C3×D3—41no—
150C22×C3×D3—61no—
151C22×C3×D3—61no—
152C22×C3×D3—121no—
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—31no—
160C2×C32×C4—41no—
161C2×C32×C4—61no—
162C2×C32×C4C22×D311yescharacteristic
163C2×C32×C4C22×D311yescharacteristic
164C2×C32×C4C22×D311yescharacteristic
165C2×C3⋊Dic3—21no—
166C4⋊(C3⋊D3)—21no—
167C4⋊(C3⋊D3)—21no—
168C3×#67—122no—
169C3×#67—122no—
170C2×C3×Dic3—21no—
171C2×C3×Dic3—21no—
172C2×C3×Dic3—21no—
173C2×C3×Dic3—21no—
174C2×C3×Dic3—41no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—121no—
178C2×C3×Dic3#6711yescharacteristic
179C2×C3×Dic3#6711yescharacteristic
180C2×C3×Dic3#6711yescharacteristic
181C2×C3×Dic3#6711yescharacteristic
182C3×D12—22no—
183C3×D12—21no—
184C3×D12—21no—
185C3×D12—21no—
186C3×D12—21no—
187C3×D12—41no—
188C3×D12—62no—
189C3×D12—61no—
190C3×D12—61no—
191C3×D12—61no—
192C3×D12—61no—
193C3×D12—121no—
194C3×D12—122no—
195C3×D12—122no—
196C3×Dic6—122no—
197C3×Dic6—122no—
198#627—42no—
199#626—42no—
200#626—42no—
201C3⋊(C3⋊C8)—21no—
202C3⋊(C3⋊C8)—61no—
203C3⋊(C3⋊C8)—62no—
204C3⋊(C3⋊C8)—61no—
205C3⋊(C3⋊C8)—61no—
206C3⋊(C3⋊C8)—121no—
207C3⋊(C3⋊C8)—121no—
208C3⋊(C3⋊C8)—122no—
209C3×C3⋊C8—62no—
210C3×C3⋊C8—61no—
211C3×C3⋊C8—61no—
212#486—271noSylow
213C2×C33#12411yescharacteristic
214C2×C33C2×D811yescharacteristic
215C2×C33C2×SD1611yescharacteristic
216C3×C3⋊D3—81no—
217C32×D3—81no—
218C2×C3×Q8—181no—
219C2×C3×D4—31no—
220C2×C3×D4—31no—
221C2×C3×D4—61no—
222C2×C3×D4—91no—
223C2×C3×D4—181no—
224C2×C3×D4—181no—
225C2×#67—61no—
226C2×#67—61no—
227C2×D12—31no—
228C2×D12—31no—
229C2×D12—61no—
230C2×D12—61no—
231C2×D12—61no—
232C2×D12D3211yescharacteristic
233C2×Dic6—61no—
234C2×Dic6—61no—
235C2×C3×C8—91no—
236C3×C4⋊C4—31no—
237C3×C4⋊C4—31no—
238C3×C4⋊C4—61no—
239C3×C4⋊C4—91no—
240C3×C4⋊C4—181no—
241C3×C42—31no—
242C3×C42—31no—
243C3×C42—61no—
244C4⋊Dic3—31no—
245C4⋊Dic3—31no—
246C4⋊Dic3—61no—
247C4⋊Dic3—61no—
248C4⋊Dic3D3211yescharacteristic
249Dic3⋊C4—61no—
250C4×Dic3D3211yescharacteristic
251C2×C3⋊C8—31no—
252C2×C3⋊C8—31no—
253C2×C3⋊C8—61no—
254C2×C3⋊C8—91no—
255C2×C3⋊C8—91no—
256C2×C3⋊C8—91no—
257C2×C3⋊C8—181no—
258C2×C3⋊C8—181no—
259C2×C3⋊C8—181no—
260C2×C3⋊C8—361no—
261C22×C32—21no—
262C22×C32—21no—
263C22×C32—21no—
264C22×C32—41no—
265C22×C32—121no—
266C22×C32—121no—
267C22×C32—121no—
268C22×C32D3×D411yescharacteristic
269C22×C32C2×#6711yescharacteristic
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—32no—
299C32×C4—32no—
300C32×C4—41no—
301C32×C4—41no—
302C32×C4—121no—
303C32×C4#23511yescharacteristic
304C32×C4#23711yescharacteristic
305C32×C4#23711yescharacteristic
306C32×C4#23711yescharacteristic
307C32×C4#23911yescharacteristic
308C32×C4#23911yescharacteristic
309C3⋊Dic3—41no—
310C3×Dic3—22no—
311C3×Dic3—22no—
312C3×Dic3—41no—
313C3×Dic3—41no—
314C3×Dic3—81no—
315C3×Dic3—121no—
316C3×Dic3—121no—
317C3×Dic3—241no—
318C3×Dic3#21312yes—
319C3×Dic3#21312yes—
320C3×Dic3#21412yes—
321C3×Dic3#21512yes—
322C4⋊Q8—91no—
323#127—91no—
324#102—91no—
325#102—271no—
326#102—271no—
327#102—271no—
328C4×C8—271no—
329C33#48611yescharacteristic, complemented, Sylow
330C23×C3—61no—
331C23×C3—61no—
332C23×C3—121no—
333C23×C3—181no—
334C22×D3—21no—
335C22×D3—61no—
336C22×D3—61no—
337C22×D3—121no—
338C3×Q8—362no—
339C3×D4—62no—
340C3×D4—61no—
341C3×D4—61no—
342C3×D4—61no—
343C3×D4—61no—
344C3×D4—121no—
345C3×D4—181no—
346C3×D4—181no—
347C3×D4—362no—
348C3×D4—362no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—31no—
353C2×C3×C4—31no—
354C2×C3×C4—31no—
355C2×C3×C4—31no—
356C2×C3×C4—41no—
357C2×C3×C4—61no—
358C2×C3×C4—61no—
359C2×C3×C4—61no—
360C2×C3×C4—61no—
361C2×C3×C4—121no—
362C2×C3×C4—181no—
363C2×C3×C4C2×D3211yescharacteristic
364C2×C3×C4C2×D3211yescharacteristic
365C2×C3×C4C2×D3211yescharacteristic
366#67—122no—
367#67—122no—
368C2×Dic3—21no—
369C2×Dic3—61no—
370C2×Dic3—61no—
371C2×Dic3—121no—
372C2×Dic3#62511yescharacteristic
373C2×Dic3#62511yescharacteristic
374D12—21no—
375D12—21no—
376D12—62no—
377D12—61no—
378D12—61no—
379D12—61no—
380D12—61no—
381D12—121no—
382D12—122no—
383D12—122no—
384Dic6—122no—
385Dic6—122no—
386C3×C8—181no—
387C3⋊C8—62no—
388C3⋊C8—61no—
389C3⋊C8—61no—
390C3⋊C8—182no—
391C3⋊C8—181no—
392C3⋊C8—181no—
393C3⋊C8—181no—
394C3⋊C8—361no—
395C3⋊C8—361no—
396C3⋊C8—362no—
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#88511yescharacteristic
411C2×C32D3×D811yescharacteristic
412C2×C32D3×SD1611yescharacteristic
413C2×C32C2×#21311yescharacteristic
414C2×C32C2×#21311yescharacteristic
415C2×C32C2×#21411yescharacteristic
416C2×C32#93611yescharacteristic
417C2×C32C2×#21511yescharacteristic
418C2×C32#94711yescharacteristic
419C3⋊D3—81no—
420C3×D3—82no—
421C3×D3—81no—
422C3×D3—81no—
423C3×D3—242no—
424C3×D3—241no—
425C3×D3—241no—
426C2×Q8—181no—
427C2×D4—31no—
428C2×D4—91no—
429C2×D4—181no—
430C2×D4—181no—
431C2×C8—91no—
432C2×C8—271no—
433C4⋊C4—31no—
434C4⋊C4—91no—
435C4⋊C4—181no—
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×C3C2×#62511yescharacteristic
454C22×C3D3×#6711yescharacteristic
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—32no—
475C3×C4—32no—
476C3×C4—32no—
477C3×C4—32no—
478C3×C4—41no—
479C3×C4—41no—
480C3×C4—62no—
481C3×C4—62no—
482C3×C4—121no—
483C3×C4—121no—
484C3×C4—241no—
485C3×C4—361no—
486C3×C4#379611yescharacteristic
487C3×C4#379611yescharacteristic
488C3×C4#379611yescharacteristic
489C3×C4#379911yescharacteristic
490C3×C4#380411yescharacteristic
491C3×C4#380511yescharacteristic
492Dic3—41no—
493Dic3—121no—
494Dic3—121no—
495Dic3—241no—
496Dic3#371312yes—
497Dic3#371512yes—
498C32—21no—
499C32—21no—
500C32—21no—
501C32—41no—
502C32#476011yescharacteristic, complemented
503C32#507011yescharacteristic, complemented
504C32#508811yescharacteristic, complemented
505C23—61no—
506C23—181no—
507Q8—362no—
508D4—61no—
509D4—61no—
510D4—181no—
511D4—181no—
512D4—362no—
513D4—362no—
514C2×C4—31no—
515C2×C4—31no—
516C2×C4—61no—
517C2×C4—181no—
518C2×C4D3311yescharacteristic
519C8—181no—
520C8—541no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—21no—
525C2×C3—21no—
526C2×C3—21no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—41no—
531C2×C3—41no—
532C2×C3—41no—
533C2×C3—242no—
534C2×C3—241no—
535C2×C3—241no—
536C2×C3—721no—
537C2×C3C2×#371311yescharacteristic
538C2×C3C2×#371511yescharacteristic
539C2×C3#6386311yescharacteristic
540C2×C3#6390711yescharacteristic
541C2×C3D3×#21311yescharacteristic
542C2×C3D3×#21311yescharacteristic
543C2×C3D3×#21411yescharacteristic
544C2×C3D3×#21511yescharacteristic
545C2×C3#6395911yescharacteristic
546D3—81no—
547D3—242no—
548D3—241no—
549D3—241no—
550C22—121no—
551C22—121no—
552C22—121no—
553C22—361no—
554C22—361no—
555C22—361no—
556C22D3×#62511yescharacteristic
557C4—32no—
558C4—32no—
559C4—121no—
560C4—361no—
561C4#8896211yescharacteristic
562C4#8896411yescharacteristic
563C3—21no—
564C3—21no—
565C3—21no—
566C3—41no—
567C3#1059027711yescharacteristic, complemented
568C3#1059054811yescharacteristic, complemented
569C3#1059148711yescharacteristic, complemented
570C2—241no—
571C2—721no—
572C2D3×#371311yescharacteristic
573C2D3×#371511yescharacteristic
574C2#1172300111yescharacteristic
subgroupG/H
centerC22D3×#625
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC3×#63911C2
hypocenterC33#486
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. C3×#63918C2semidirect
3. #11722075C2semidirect
4. #11722033C2semidirect
5. #11722031C2semidirect
6. Dic3×C3⋊(C3⋊C8)C2semidirect
7. C3×#63562C2semidirect
26. #63918D3semidirect
42. #63562D3semidirect
329. C33#486semidirect
502. C32#4760semidirect
503. C32#5070semidirect
504. C32#5088semidirect
567. C3#10590277semidirect
568. C3#10590548semidirect
569. C3#10591487semidirect

Maximal subgroups

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

subgroupindex
C3×#639182
C3×#639112
#117220752
#117220332
#117220312
Dic3×C3⋊(C3⋊C8)2
C3×#635622
#105914873
#105905483
#105902773