Finite Group Databasebeta

← #13155724

Subgroups — #13155724

Order 1728.

702 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#11723691C211yesmaximal, characteristic, complemented
2C2×#88789C211yesmaximal, characteristic
3C3×#63898C211yesmaximal, characteristic, complemented
4#11722053C211yesmaximal, characteristic, complemented
5#11722020C211yesmaximal, characteristic, complemented
6C3×#63564C211yesmaximal, characteristic, complemented
7C3×#63563C211yesmaximal, characteristic, complemented
8#10591404—31nomaximal
9#10590741—31nomaximal
10#10590455—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—
21#88789C414yescomplemented
22C3×(C2×D3)⋊Dic3—21no—
23C2×#3753—31no—
24C3×D4×Dic3—31no—
25C3×#893—31no—
26#63898—31no—
27#63898D311yescharacteristic, complemented
28#63894—31no—
29Dic3×D12—31no—
30C2×#3713—31no—
31C2×#3713—31no—
32C2×#3713—61no—
33#63642—31no—
34#63635—31no—
35C3×#832—31no—
36C3×#821—31no—
37C3×#821—31no—
38C3×#809—31no—
39#63574—31no—
40#63574—31no—
41#63574—61no—
42#63564D311yescharacteristic, complemented
43#63563D311yescharacteristic, complemented
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))C2×C412yescomplemented
61C32×C3⋊C8—21no—
62#5014—91no—
63#4785—91no—
64#4785—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—
81#3753—34no—
82C4⋊(C3⋊Dic3)C2×D311yescharacteristic
83C4×C3⋊Dic3C2×D311yescharacteristic
84C2×C3⋊(C3⋊C8)—31no—
85C2×C3⋊(C3⋊C8)—31no—
86C2×C3⋊(C3⋊C8)—31no—
87C2×C3⋊(C3⋊C8)—61no—
88C2×C3⋊(C3⋊C8)—61no—
89C2×C3⋊(C3⋊C8)—61no—
90C2×C3⋊(C3⋊C8)—121no—
91C3×#217—61no—
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
105#3722—61no—
106(C2×D3)⋊Dic3—21no—
107(C2×D3)⋊Dic3—61no—
108(C2×D3)⋊Dic3—61no—
109#3713—34no—
110#3713—34no—
111#3713—64no—
112C22×C33C2×D411yescharacteristic
113C2×C3×C3⋊D3—22no—
114C2×C3×C3⋊D3—41no—
115C2×C3×C3⋊D3—41no—
116C2×C32×D3—22no—
117C2×C32×D3—41no—
118C2×C32×D3—41no—
119C33×C4C22×C411yescharacteristic
120C33×C4#4111yescharacteristic
121C3×C3⋊Dic3—21no—
122C3×C3⋊Dic3—21no—
123C3×C3⋊Dic3—41no—
124C32×Dic3—41no—
125C3×C4×D4—91no—
126D4×Dic3—31no—
127D4×Dic3—91no—
128C2×#213—91no—
129C2×#213—91no—
130C2×#213—91no—
131C2×#213—181no—
132#893—31no—
133#893—91no—
134C4×D12—91no—
135C3×#102—91no—
136C3×#102—91no—
137#832—31no—
138#821—31no—
139#821—31no—
140#815—91no—
141#815—91no—
142#815—181no—
143#809—31no—
144#808—91no—
145#808—91no—
146#808—91no—
147#808—181no—
148#803—91no—
149C23×C32—61no—
150C22×C3⋊D3—21no—
151C22×C3×D3—21no—
152C22×C3×D3—21no—
153C22×C3×D3—41no—
154C22×C3×D3—61no—
155C22×C3×D3—61no—
156C22×C3×D3—121no—
157C32×D4—32no—
158C32×D4—61no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—41no—
163C2×C32×C4—61no—
164C2×C32×C4C22×D311yescharacteristic
165C2×C32×C4C22×D311yescharacteristic
166C2×C32×C4C22×D311yescharacteristic
167C2×C3⋊Dic3—21no—
168C2×C3⋊Dic3#6711yescharacteristic
169C2×C3⋊Dic3#6711yescharacteristic
170C4⋊(C3⋊D3)—21no—
171C4⋊(C3⋊D3)C4×D312yescomplemented
172C2×C3×Dic3—21no—
173C2×C3×Dic3—21no—
174C2×C3×Dic3—31no—
175C2×C3×Dic3—31no—
176C2×C3×Dic3—31no—
177C2×C3×Dic3—31no—
178C2×C3×Dic3—41no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—61no—
183C2×C3×Dic3—121no—
184C2×C3×Dic3—122no—
185C3×D12—21no—
186C3×D12—21no—
187C3×D12—22no—
188C3×D12—32no—
189C3×D12—32no—
190C3×D12—41no—
191C3×D12—62no—
192C3×D12—61no—
193C3×D12—61no—
194C3×D12—121no—
195C3×D12C4×D312yescomplemented
196C3×D12C4×D312yescomplemented
197C3×C4×D3—122no—
198#624—122no—
199D3×Dic3—42no—
200D3×Dic3—122no—
201D3×Dic3—122no—
202C32×C8—61no—
203C3⋊(C3⋊C8)—32no—
204C3⋊(C3⋊C8)—32no—
205C3⋊(C3⋊C8)—32no—
206C3⋊(C3⋊C8)—62no—
207C3⋊(C3⋊C8)—62no—
208C3⋊(C3⋊C8)—62no—
209C3⋊(C3⋊C8)—122no—
210C3×C3⋊C8—22no—
211C3×C3⋊C8—21no—
212C3×C3⋊C8—21no—
213#442—271noSylow
214C2×C33C4×D411yescharacteristic
215C2×C33AGL(1,ℤ/8)11yescharacteristic
216C2×C33AGL(1,ℤ/8)11yescharacteristic
217C3×C3⋊D3—42no—
218C32×D3—42no—
219C2×C3×D4—31no—
220C2×C3×D4—31no—
221C2×C3×D4—61no—
222C2×C3×D4—91no—
223C22×C3×C4—181no—
224C22×Dic3—61no—
225C22×Dic3—181no—
226C2×D12—31no—
227C2×D12—31no—
228C2×D12—61no—
229C2×D12D3211yescharacteristic
230C2×C4×D3—61no—
231C2×C4×D3—181no—
232C2×C4×D3—181no—
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—94no—
245#213—94no—
246#213—94no—
247#213—184no—
248(C2×D3)⋊C4—61no—
249(C2×D3)⋊C4—181no—
250(C2×D3)⋊C4—181no—
251C4⋊Dic3—31no—
252C4⋊Dic3—31no—
253C4⋊Dic3—61no—
254C4⋊Dic3D3211yescharacteristic
255C4×Dic3—31no—
256C4×Dic3—31no—
257C4×Dic3—61no—
258C2×C3⋊C8—91no—
259C2×C3⋊C8—91no—
260C2×C3⋊C8—91no—
261C2×C3⋊C8—181no—
262C2×C3⋊C8—181no—
263C2×C3⋊C8—181no—
264C2×C3⋊C8—361no—
265C2×C3⋊C8D3211yescharacteristic
266C22×C32—21no—
267C22×C32—21no—
268C22×C32—21no—
269C22×C32—41no—
270C22×C32—62no—
271C22×C32—121no—
272C22×C32—121no—
273C22×C32D3×D411yescharacteristic
274C22×C32D3×D411yescharacteristic
275C22×C32C2×#6711yescharacteristic
276C2×C3⋊D3—22no—
277C2×C3⋊D3—41no—
278C2×C3⋊D3—41no—
279C2×C3×D3—22no—
280C2×C3×D3—22no—
281C2×C3×D3—42no—
282C2×C3×D3—41no—
283C2×C3×D3—41no—
284C2×C3×D3—41no—
285C2×C3×D3—41no—
286C2×C3×D3—62no—
287C2×C3×D3—62no—
288C2×C3×D3—81no—
289C2×C3×D3—81no—
290C2×C3×D3—122no—
291C2×C3×D3—121no—
292C2×C3×D3—121no—
293C2×C3×D3—121no—
294C2×C3×D3—121no—
295C2×C3×D3—241no—
296C2×C3×D3—241no—
297C32×C4—21no—
298C32×C4—21no—
299C32×C4—21no—
300C32×C4—21no—
301C32×C4—21no—
302C32×C4—21no—
303C32×C4—41no—
304C32×C4—41no—
305C32×C4—121no—
306C32×C4C2×C4×D311yescharacteristic
307C32×C4C2×C4×D311yescharacteristic
308C32×C4C2×C4×D311yescharacteristic
309C32×C4#23511yescharacteristic
310C32×C4#23711yescharacteristic
311C32×C4#23711yescharacteristic
312C3⋊Dic3—21no—
313C3⋊Dic3—21no—
314C3⋊Dic3—41no—
315C3×Dic3—41no—
316C3×Dic3—41no—
317C3×Dic3—61no—
318C3×Dic3—61no—
319C3×Dic3—61no—
320C3×Dic3—61no—
321C3×Dic3—81no—
322C3×Dic3—121no—
323C3×Dic3—121no—
324C3×Dic3—121no—
325C3×Dic3—121no—
326C3×Dic3—241no—
327C2×D8—271no—
328C4×D4—91no—
329C4×D4—271no—
330#106—271no—
331#102—91no—
332#102—91no—
333#97—271no—
334C33#44211yescharacteristic, complemented, Sylow
335C23×C3—61no—
336C23×C3—61no—
337C23×C3—121no—
338C23×C3—181no—
339C22×D3—21no—
340C22×D3—61no—
341C22×D3—61no—
342C22×D3—121no—
343C3×D4—32no—
344C3×D4—32no—
345C3×D4—62no—
346C3×D4—61no—
347C3×D4—61no—
348C3×D4—92no—
349C3×D4—121no—
350C3×D4—181no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—41no—
355C2×C3×C4—61no—
356C2×C3×C4—61no—
357C2×C3×C4—91no—
358C2×C3×C4—91no—
359C2×C3×C4—121no—
360C2×C3×C4—181no—
361C2×C3×C4—362no—
362C2×C3×C4C2×D3211yescharacteristic
363C2×C3×C4C2×D3211yescharacteristic
364C2×C3×C4C2×D3211yescharacteristic
365C2×Dic3—21no—
366C2×Dic3—31no—
367C2×Dic3—31no—
368C2×Dic3—31no—
369C2×Dic3—31no—
370C2×Dic3—61no—
371C2×Dic3—61no—
372C2×Dic3—61no—
373C2×Dic3—61no—
374C2×Dic3—121no—
375C2×Dic3—122no—
376C2×Dic3—362no—
377D12—21no—
378D12—32no—
379D12—32no—
380D12—62no—
381D12—61no—
382D12—61no—
383D12—121no—
384D12#62412yescomplemented
385C4×D3—122no—
386C4×D3—362no—
387C4×D3—362no—
388C3×C8—62no—
389C3×C8—61no—
390C3×C8—61no—
391C3⋊C8—21no—
392C3⋊C8—92no—
393C3⋊C8—92no—
394C3⋊C8—92no—
395C3⋊C8—182no—
396C3⋊C8—182no—
397C3⋊C8—182no—
398C3⋊C8—362no—
399C2×C32—21no—
400C2×C32—21no—
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32—21no—
405C2×C32—21no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—41no—
409C2×C32—41no—
410C2×C32—41no—
411C2×C32—122no—
412C2×C32#67⋊C411yescharacteristic
413C2×C32#67⋊C411yescharacteristic
414C2×C32#91111yescharacteristic
415C2×C32#91111yescharacteristic
416C2×C32#91411yescharacteristic
417C2×C32#91411yescharacteristic
418C2×C32C4×#6711yescharacteristic
419C2×C32#93211yescharacteristic
420C2×C32#94911yescharacteristic
421C3⋊D3—42no—
422C3×D3—42no—
423C3×D3—42no—
424C3×D3—82no—
425C3×D3—122no—
426C3×D3—122no—
427C3×D3—242no—
428C2×D4—31no—
429C2×D4—91no—
430C22×C4—181no—
431C22×C4—541no—
432D8—274no—
433C2×C8—31no—
434C2×C8—271no—
435C4⋊C4—31no—
436C4⋊C4—91no—
437C22⋊C4—181no—
438C22⋊C4—541no—
439C42—91no—
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—41no—
444C22×C3—62no—
445C22×C3—62no—
446C22×C3—122no—
447C22×C3—121no—
448C22×C3—121no—
449C22×C3—121no—
450C22×C3—121no—
451C22×C3—182no—
452C22×C3—241no—
453C22×C3—241no—
454C22×C3—361no—
455C22×C3—361no—
456C22×C3#380211yescharacteristic
457C22×C3D3×#6711yescharacteristic
458C22×C3D3×#6711yescharacteristic
459C2×D3—22no—
460C2×D3—41no—
461C2×D3—41no—
462C2×D3—62no—
463C2×D3—62no—
464C2×D3—122no—
465C2×D3—121no—
466C2×D3—121no—
467C2×D3—121no—
468C2×D3—121no—
469C2×D3—241no—
470C2×D3—241no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—21no—
477C3×C4—41no—
478C3×C4—41no—
479C3×C4—121no—
480C3×C4—121no—
481C3×C4—181no—
482C3×C4—181no—
483C3×C4—241no—
484C3×C4—361no—
485C3×C4#379511yescharacteristic
486C3×C4C4×D3211yescharacteristic
487C3×C4C4×D3211yescharacteristic
488C3×C4#380411yescharacteristic
489C3×C4#380511yescharacteristic
490C3×C4C2×#62411yescharacteristic
491Dic3—41no—
492Dic3—61no—
493Dic3—61no—
494Dic3—61no—
495Dic3—61no—
496Dic3—121no—
497Dic3—121no—
498Dic3—121no—
499Dic3—121no—
500Dic3—241no—
501C32—21no—
502C32—21no—
503C32—21no—
504C32—41no—
505C32#478511yescharacteristic, complemented
506C32#478511yescharacteristic, complemented
507C32#501411yescharacteristic, complemented
508C23—61no—
509C23—181no—
510D4—32no—
511D4—61no—
512D4—92no—
513D4—181no—
514C2×C4—61no—
515C2×C4—91no—
516C2×C4—91no—
517C2×C4—181no—
518C2×C4—362no—
519C2×C4—1082no—
520C2×C4D3311yescharacteristic
521C8—61no—
522C8—272no—
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—122no—
536C2×C3—122no—
537C2×C3—242no—
538C2×C3—362no—
539C2×C3#6379811yescharacteristic
540C2×C3#6379811yescharacteristic
541C2×C3#6387911yescharacteristic
542C2×C3#6390211yescharacteristic
543C2×C3#6392511yescharacteristic
544C2×C3#6393211yescharacteristic
545C2×C3#6393411yescharacteristic
546C2×C3#6393911yescharacteristic
547C2×C3#6395211yescharacteristic
548D3—42no—
549D3—122no—
550D3—122no—
551D3—242no—
552C22—62no—
553C22—121no—
554C22—121no—
555C22—182no—
556C22—361no—
557C22—361no—
558C22#8895311yescharacteristic
559C4—121no—
560C4—181no—
561C4—181no—
562C4—361no—
563C4D3×#62411yescharacteristic
564C4#8896111yescharacteristic
565C3—21no—
566C3—21no—
567C3—21no—
568C3—41no—
569C3#1059045511yescharacteristic, complemented
570C3#1059074111yescharacteristic, complemented
571C3#1059140411yescharacteristic, complemented
572C2—122no—
573C2—362no—
574C2#1172287511yescharacteristic
575C2#1172287611yescharacteristic
576C2#1172293611yescharacteristic
subgroupG/H
centerC22#88953
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC2×#88789C2
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
3. C3×#63898C2semidirect
4. #11722053C2semidirect
5. #11722020C2semidirect
6. C3×#63564C2semidirect
7. C3×#63563C2semidirect
21. #88789C4semidirect
27. #63898D3semidirect
42. #63564D3semidirect
43. #63563D3semidirect
54. C3×C4⋊(C3⋊D3)C2×C4semidirect
57. C32×D12C2×C4semidirect
60. C3⋊(C3⋊(C3⋊C8))C2×C4semidirect
171. C4⋊(C3⋊D3)C4×D3semidirect
195. C3×D12C4×D3semidirect
196. C3×D12C4×D3semidirect
334. C33#442semidirect
384. D12#624semidirect
505. C32#4785semidirect
506. C32#4785semidirect
507. C32#5014semidirect
569. C3#10590455semidirect
570. C3#10590741semidirect
571. C3#10591404semidirect

Maximal subgroups

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

subgroupindex
#117236912
C2×#887892
C3×#638982
#117220532
#117220202
C3×#635642
C3×#635632
#105914043
#105907413
#105904553