Finite Group Databasebeta

← #13155715

Subgroups — #13155715

Order 1728.

690 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
1#11723691C211yesmaximal, characteristic, complemented
2C3×#63898C211yesmaximal, characteristic, complemented
3C2×C3×#3714C211yesmaximal, characteristic
4#11722033C211yesmaximal, characteristic, complemented
5#11722031C211yesmaximal, characteristic, complemented
6#11722020C211yesmaximal, characteristic, complemented
7C3×#63573C211yesmaximal, characteristic, complemented
8#10591430—31nomaximal
9#10590781—31nomaximal
10#10590589—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×#6294—21no—
14C2×C3×D3×Dic3—21no—
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88814—21no—
21C3×(C2×D3)⋊Dic3—21no—
22C3×#3714C414yescomplemented
23C3×D4×Dic3—31no—
24C2×C3×#213—31no—
25C2×C3×D24—31no—
26C3×#893—31no—
27#63898—31no—
28#63898D311yescharacteristic, complemented
29#63894—31no—
30Dic3×D12—31no—
31C2×#3714D311yescharacteristic
32#63659—31no—
33#63642—31no—
34#63636—31no—
35C3×#817—31no—
36C3×#808—31no—
37#63573D311yescharacteristic, complemented
38#63562—31no—
39#63562—31no—
40#63562—31no—
41#63562—31no—
42#63562—61no—
43#63562—61no—
44#63554—31no—
45#63554—31no—
46#63554—61no—
47C22×C3×C3⋊D3—21no—
48C22×C32×D3—21no—
49C2×C33×C4C2311yescharacteristic
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3D411yescharacteristic
52C2×C3×C3⋊Dic3D411yescharacteristic
53C3×C4⋊(C3⋊D3)—21no—
54C3×C4⋊(C3⋊D3)C2×C412yescomplemented
55C2×C32×Dic3—21no—
56C32×D12—21no—
57C32×D12C2×C412yescomplemented
58#6294—42no—
59C3×D3×Dic3—42no—
60C3⋊(C3⋊(C3⋊C8))—21no—
61C32×C3⋊C8C2×C412yescomplemented
62#5152—91no—
63#5014—91no—
64#4869—91no—
65C2×C32×D4—31no—
66C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
67C22×C3×Dic3—61no—
68C2×C3×D12—21no—
69C2×C3×D12—31no—
70C2×C3×D12—31no—
71C2×C3×D12—61no—
72C2×C3×D12C2×D311yescharacteristic
73C2×C3×D12C2×D311yescharacteristic
74C2×C3×C4×D3—61no—
75C2×#624—61no—
76C2×D3×Dic3—21no—
77C2×D3×Dic3—61no—
78C2×D3×Dic3—61no—
79C2×C32×C8—31no—
80C32×C4⋊C4—31no—
81C4⋊(C3⋊Dic3)C2×D311yescharacteristic
82C4×C3⋊Dic3C2×D311yescharacteristic
83C2×C3⋊(C3⋊C8)—31no—
84C2×C3⋊(C3⋊C8)—31no—
85C2×C3⋊(C3⋊C8)—31no—
86C2×C3⋊(C3⋊C8)—61no—
87C2×C3⋊(C3⋊C8)—61no—
88C2×C3⋊(C3⋊C8)—61no—
89C2×C3⋊(C3⋊C8)—121no—
90C3×#217—61no—
91C3×#213—34no—
92C3×(C2×D3)⋊C4—61no—
93C3×C4⋊Dic3—21no—
94C3×C4⋊Dic3—31no—
95C3×C4⋊Dic3—31no—
96C3×C4⋊Dic3—61no—
97C3×C4⋊Dic3C2×D311yescharacteristic
98C3×C4⋊Dic3C2×D311yescharacteristic
99C3×C4×Dic3—31no—
100C3×C4×Dic3—31no—
101C3×C4×Dic3—61no—
102C2×C3×C3⋊C8—21no—
103C2×C3×C3⋊C8C2×D311yescharacteristic
104C2×C3×C3⋊C8C2×D311yescharacteristic
105C3×D24—34no—
106#3722—61no—
107(C2×D3)⋊Dic3—21no—
108(C2×D3)⋊Dic3—61no—
109(C2×D3)⋊Dic3—61no—
110#3714Dic314yescomplemented
111C22×C33C2×D411yescharacteristic
112C2×C3×C3⋊D3—22no—
113C2×C3×C3⋊D3—41no—
114C2×C3×C3⋊D3—41no—
115C2×C32×D3—22no—
116C2×C32×D3—41no—
117C2×C32×D3—41no—
118C33×C4C22×C411yescharacteristic
119C33×C4#4111yescharacteristic
120C3×C3⋊Dic3—21no—
121C3×C3⋊Dic3—21no—
122C3×C3⋊Dic3—41no—
123C32×Dic3—41no—
124C2×C3×D8—91no—
125C3×C4×D4—91no—
126D4×Dic3—31no—
127D4×Dic3—91no—
128C2×#213—31no—
129C2×D24—31no—
130#893—31no—
131#893—91no—
132C4×D12—91no—
133C3×#106—91no—
134#832—91no—
135#832—91no—
136#832—91no—
137#832—181no—
138#817—31no—
139#815—91no—
140#815—91no—
141#815—181no—
142#809—91no—
143#809—91no—
144#809—91no—
145#809—181no—
146#808—31no—
147#803—91no—
148C23×C32—61no—
149C22×C3⋊D3—21no—
150C22×C3×D3—21no—
151C22×C3×D3—21no—
152C22×C3×D3—41no—
153C22×C3×D3—61no—
154C22×C3×D3—61no—
155C22×C3×D3—121no—
156C32×D4—32no—
157C32×D4—61no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—41no—
162C2×C32×C4—61no—
163C2×C32×C4C22×D311yescharacteristic
164C2×C32×C4C22×D311yescharacteristic
165C2×C32×C4C22×D311yescharacteristic
166C2×C3⋊Dic3—21no—
167C2×C3⋊Dic3#6711yescharacteristic
168C2×C3⋊Dic3#6711yescharacteristic
169C4⋊(C3⋊D3)—21no—
170C4⋊(C3⋊D3)C2×Dic312yescomplemented
171C2×C3×Dic3—21no—
172C2×C3×Dic3—21no—
173C2×C3×Dic3—31no—
174C2×C3×Dic3—31no—
175C2×C3×Dic3—31no—
176C2×C3×Dic3—31no—
177C2×C3×Dic3—41no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—121no—
183C2×C3×Dic3—122no—
184C3×D12—21no—
185C3×D12—21no—
186C3×D12—22no—
187C3×D12—32no—
188C3×D12—32no—
189C3×D12—41no—
190C3×D12—62no—
191C3×D12—61no—
192C3×D12—61no—
193C3×D12—121no—
194C3×D12C4×D312yescomplemented
195C3×D12C2×Dic312yescomplemented
196C3×C4×D3—122no—
197#624—122no—
198D3×Dic3—42no—
199D3×Dic3—122no—
200D3×Dic3—122no—
201C32×C8—32no—
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—22no—
210C3×C3⋊C8C4×D312yescomplemented
211C3×C3⋊C8C2×Dic312yescomplemented
212#442—271noSylow
213C2×C33C4×D411yescharacteristic
214C2×C33AGL(1,ℤ/8)11yescharacteristic
215C2×C33AGL(1,ℤ/8)11yescharacteristic
216C3×C3⋊D3—42no—
217C32×D3—42no—
218C2×C3×D4—31no—
219C2×C3×D4—31no—
220C2×C3×D4—61no—
221C2×C3×D4—91no—
222C22×C3×C4—181no—
223C22×Dic3—61no—
224C22×Dic3—181no—
225C2×D12—31no—
226C2×D12—31no—
227C2×D12—61no—
228C2×D12D3211yescharacteristic
229C2×C4×D3—61no—
230C2×C4×D3—181no—
231C2×C4×D3—181no—
232C3×D8—94no—
233C2×C3×C8—31no—
234C2×C3×C8—31no—
235C2×C3×C8—61no—
236C3×C4⋊C4—31no—
237C3×C4⋊C4—31no—
238C3×C4⋊C4—61no—
239C3×C4⋊C4—91no—
240C3×C22⋊C4—181no—
241C3×C42—91no—
242#217—61no—
243#217—181no—
244#213—34no—
245(C2×D3)⋊C4—61no—
246(C2×D3)⋊C4—181no—
247(C2×D3)⋊C4—181no—
248C4⋊Dic3—31no—
249C4⋊Dic3—31no—
250C4⋊Dic3—61no—
251C4⋊Dic3D3211yescharacteristic
252C4×Dic3—31no—
253C4×Dic3—31no—
254C4×Dic3—61no—
255C2×C3⋊C8—91no—
256C2×C3⋊C8—91no—
257C2×C3⋊C8—91no—
258C2×C3⋊C8—181no—
259C2×C3⋊C8—181no—
260C2×C3⋊C8—181no—
261C2×C3⋊C8—361no—
262C2×C3⋊C8D3211yescharacteristic
263D24—34no—
264C22×C32—21no—
265C22×C32—21no—
266C22×C32—21no—
267C22×C32—41no—
268C22×C32—62no—
269C22×C32—121no—
270C22×C32—121no—
271C22×C32D3×D411yescharacteristic
272C22×C32D3×D411yescharacteristic
273C22×C32C2×#6711yescharacteristic
274C2×C3⋊D3—22no—
275C2×C3⋊D3—41no—
276C2×C3⋊D3—41no—
277C2×C3×D3—22no—
278C2×C3×D3—22no—
279C2×C3×D3—42no—
280C2×C3×D3—41no—
281C2×C3×D3—41no—
282C2×C3×D3—41no—
283C2×C3×D3—41no—
284C2×C3×D3—62no—
285C2×C3×D3—62no—
286C2×C3×D3—81no—
287C2×C3×D3—81no—
288C2×C3×D3—122no—
289C2×C3×D3—121no—
290C2×C3×D3—121no—
291C2×C3×D3—121no—
292C2×C3×D3—121no—
293C2×C3×D3—241no—
294C2×C3×D3—241no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4—21no—
298C32×C4—21no—
299C32×C4—21no—
300C32×C4—21no—
301C32×C4—41no—
302C32×C4—41no—
303C32×C4—121no—
304C32×C4C2×C4×D311yescharacteristic
305C32×C4C2×C4×D311yescharacteristic
306C32×C4#23511yescharacteristic
307C32×C4#23711yescharacteristic
308C32×C4#23911yescharacteristic
309C32×C4C22×Dic311yescharacteristic
310C3⋊Dic3—21no—
311C3⋊Dic3—21no—
312C3⋊Dic3—41no—
313C3×Dic3—41no—
314C3×Dic3—41no—
315C3×Dic3—61no—
316C3×Dic3—61no—
317C3×Dic3—61no—
318C3×Dic3—61no—
319C3×Dic3—81no—
320C3×Dic3—121no—
321C3×Dic3—121no—
322C3×Dic3—121no—
323C3×Dic3—121no—
324C3×Dic3—241no—
325C2×D8—91no—
326C4×D4—91no—
327C4×D4—271no—
328#106—91no—
329#102—271no—
330#102—271no—
331#97—271no—
332C33#44211yescharacteristic, 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—32no—
342C3×D4—32no—
343C3×D4—62no—
344C3×D4—61no—
345C3×D4—61no—
346C3×D4—92no—
347C3×D4—121no—
348C3×D4—181no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—41no—
353C2×C3×C4—61no—
354C2×C3×C4—61no—
355C2×C3×C4—91no—
356C2×C3×C4—91no—
357C2×C3×C4—121no—
358C2×C3×C4—181no—
359C2×C3×C4—362no—
360C2×C3×C4C2×D3211yescharacteristic
361C2×C3×C4C2×D3211yescharacteristic
362C2×C3×C4C2×D3211yescharacteristic
363C2×Dic3—21no—
364C2×Dic3—31no—
365C2×Dic3—31no—
366C2×Dic3—31no—
367C2×Dic3—31no—
368C2×Dic3—61no—
369C2×Dic3—61no—
370C2×Dic3—61no—
371C2×Dic3—61no—
372C2×Dic3—121no—
373C2×Dic3—122no—
374C2×Dic3—362no—
375D12—21no—
376D12—32no—
377D12—32no—
378D12—62no—
379D12—61no—
380D12—61no—
381D12—121no—
382D12D3×Dic312yescomplemented
383C4×D3—122no—
384C4×D3—362no—
385C4×D3—362no—
386C3×C8—32no—
387C3×C8—32no—
388C3×C8—62no—
389C3⋊C8—182no—
390C3⋊C8—181no—
391C3⋊C8—181no—
392C3⋊C8—181no—
393C3⋊C8—361no—
394C3⋊C8—361no—
395C3⋊C8—362no—
396C3⋊C8D3×Dic312yescomplemented
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—122no—
410C2×C32#89311yescharacteristic
411C2×C32#91111yescharacteristic
412C2×C32#91111yescharacteristic
413C2×C32#91411yescharacteristic
414C2×C32#91411yescharacteristic
415C2×C32C4×#6711yescharacteristic
416C2×C32#93211yescharacteristic
417C2×C32D4×Dic311yescharacteristic
418C2×C32#94911yescharacteristic
419C3⋊D3—42no—
420C3×D3—42no—
421C3×D3—42no—
422C3×D3—82no—
423C3×D3—122no—
424C3×D3—122no—
425C3×D3—242no—
426C2×D4—31no—
427C2×D4—91no—
428C22×C4—181no—
429C22×C4—541no—
430D8—94no—
431C2×C8—31no—
432C2×C8—271no—
433C4⋊C4—31no—
434C4⋊C4—91no—
435C22⋊C4—181no—
436C22⋊C4—541no—
437C42—91no—
438C22×C3—21no—
439C22×C3—21no—
440C22×C3—21no—
441C22×C3—41no—
442C22×C3—62no—
443C22×C3—62no—
444C22×C3—122no—
445C22×C3—121no—
446C22×C3—121no—
447C22×C3—121no—
448C22×C3—121no—
449C22×C3—182no—
450C22×C3—241no—
451C22×C3—241no—
452C22×C3—361no—
453C22×C3—361no—
454C22×C3#380211yescharacteristic
455C22×C3D3×#6711yescharacteristic
456C22×C3D3×#6711yescharacteristic
457C2×D3—22no—
458C2×D3—41no—
459C2×D3—41no—
460C2×D3—62no—
461C2×D3—62no—
462C2×D3—122no—
463C2×D3—121no—
464C2×D3—121no—
465C2×D3—121no—
466C2×D3—121no—
467C2×D3—241no—
468C2×D3—241no—
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—41no—
476C3×C4—41no—
477C3×C4—121no—
478C3×C4—121no—
479C3×C4—181no—
480C3×C4—181no—
481C3×C4—241no—
482C3×C4—361no—
483C3×C4#379611yescharacteristic
484C3×C4#379911yescharacteristic
485C3×C4C4×D3211yescharacteristic
486C3×C4C2×D3×Dic311yescharacteristic
487C3×C4C2×D3×Dic311yescharacteristic
488C3×C4#380411yescharacteristic
489Dic3—41no—
490Dic3—61no—
491Dic3—61no—
492Dic3—61no—
493Dic3—61no—
494Dic3—121no—
495Dic3—121no—
496Dic3—121no—
497Dic3—121no—
498Dic3—241no—
499C32—21no—
500C32—21no—
501C32—21no—
502C32—41no—
503C32#486911yescharacteristic, complemented
504C32#501411yescharacteristic, complemented
505C32#515211yescharacteristic, complemented
506C23—61no—
507C23—181no—
508D4—32no—
509D4—61no—
510D4—92no—
511D4—181no—
512C2×C4—61no—
513C2×C4—91no—
514C2×C4—91no—
515C2×C4—181no—
516C2×C4—362no—
517C2×C4—1082no—
518C2×C4D3311yescharacteristic
519C8—32no—
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—122no—
534C2×C3—122no—
535C2×C3—242no—
536C2×C3—362no—
537C2×C3#6379811yescharacteristic
538C2×C3#6379811yescharacteristic
539C2×C3#6389811yescharacteristic
540C2×C3#6390411yescharacteristic
541C2×C3#6392511yescharacteristic
542C2×C3#6393211yescharacteristic
543C2×C3#6393411yescharacteristic
544C2×C3#6393911yescharacteristic
545C2×C3Dic3×#6711yescharacteristic
546D3—42no—
547D3—122no—
548D3—122no—
549D3—242no—
550C22—62no—
551C22—121no—
552C22—121no—
553C22—182no—
554C22—361no—
555C22—361no—
556C22#8895311yescharacteristic
557C4—121no—
558C4—181no—
559C4—181no—
560C4—361no—
561C4D32×Dic311yescharacteristic
562C4#8896211yescharacteristic
563C3—21no—
564C3—21no—
565C3—21no—
566C3—41no—
567C3#1059058911yescharacteristic, complemented
568C3#1059078111yescharacteristic, complemented
569C3#1059143011yescharacteristic, complemented
570C2—122no—
571C2—362no—
572C2#1172287511yescharacteristic
573C2#1172287611yescharacteristic
574C2#1172292711yescharacteristic
subgroupG/H
centerC22#88953
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC2×C3×#3714C2
hypocenterC33#442
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723691C2semidirect
2. C3×#63898C2semidirect
4. #11722033C2semidirect
5. #11722031C2semidirect
6. #11722020C2semidirect
7. C3×#63573C2semidirect
22. C3×#3714C4semidirect
28. #63898D3semidirect
37. #63573D3semidirect
54. C3×C4⋊(C3⋊D3)C2×C4semidirect
57. C32×D12C2×C4semidirect
61. C32×C3⋊C8C2×C4semidirect
110. #3714Dic3semidirect
170. C4⋊(C3⋊D3)C2×Dic3semidirect
194. C3×D12C4×D3semidirect
195. C3×D12C2×Dic3semidirect
210. C3×C3⋊C8C4×D3semidirect
211. C3×C3⋊C8C2×Dic3semidirect
332. C33#442semidirect
382. D12D3×Dic3semidirect
396. C3⋊C8D3×Dic3semidirect
503. C32#4869semidirect
504. C32#5014semidirect
505. C32#5152semidirect
567. C3#10590589semidirect
568. C3#10590781semidirect
569. C3#10591430semidirect

Maximal subgroups

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

subgroupindex
#117236912
C3×#638982
C2×C3×#37142
#117220332
#117220312
#117220202
C3×#635732
#105914303
#105907813
#105905893