Finite Group Databasebeta

← #13156392

Subgroups — #13156392

Order 1728.

660 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3⋊D3×C4⋊Dic3C211yesmaximal, characteristic, complemented
2C3×#63916C211yesmaximal, characteristic
3#11722072C211yesmaximal, characteristic, complemented
4#11722057C211yesmaximal, characteristic, complemented
5#11722032C211yesmaximal, characteristic, complemented
6C3×#63576C211yesmaximal, characteristic, complemented
7C3×#63562C211yesmaximal, characteristic, complemented
8#10591413—31nomaximal
9#10590510—31nomaximal
10#10590438—31nomaximal
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×Dic3×C3⋊D3—21no—
14C2×C3×#626—21no—
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C32×C4⋊Dic3C2211yescharacteristic
20#88807—21no—
21C3×(C2×D3)⋊Dic3—21no—
22C4⋊C4×C3⋊D3—31no—
23C3×#938—31no—
24C3×#895—31no—
25#63916D311yescharacteristic
26D3×C4⋊Dic3—31no—
27D3×C4⋊Dic3—31no—
28D3×C4⋊Dic3—61no—
29#63659—31no—
30#63634—31no—
31C3×#832—31no—
32C3×#809—31no—
33C3×#807—31no—
34C3×#807—31no—
35#63576D311yescharacteristic, complemented
36#63575—31no—
37#63575—31no—
38#63575—61no—
39#63563—31no—
40#63563—31no—
41#63563—61no—
42#63562D311yescharacteristic, complemented
43#63557—31no—
44(C2×D3)⋊(C3⋊C8)—31no—
45(C2×D3)⋊(C3⋊C8)—31no—
46C22×C3×C3⋊D3D411yescharacteristic
47C22×C32×D3—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)—21no—
50C2×C3×C3⋊Dic3D411yescharacteristic
51C3×C4×C3⋊D3—21no—
52C3×C4×C3⋊D3—21no—
53C2×C32×Dic3—21no—
54C2×C32×Dic3—21no—
55C32×D12—21no—
56C32×D12—21no—
57Dic3×C3⋊D3—42no—
58C3×#626—42no—
59C3×C3⋊(C3⋊C8)—21no—
60C3×C3⋊(C3⋊C8)—21no—
61#5033—91no—
62#4882—91no—
63#4774—91no—
64C2×C32×D4—31no—
65C2×C4×C3⋊D3—61no—
66C2×C4×C3⋊D3C2×D311yescharacteristic
67C2×C3×#67—61no—
68C2×C3×D12—21no—
69C2×C3×D12—61no—
70C2×C3×D12C2×D311yescharacteristic
71C2×C3×D12C2×D311yescharacteristic
72C2×C3×C4×D3—31no—
73C2×C3×C4×D3—31no—
74C2×C3×C4×D3—61no—
75C2×#626—21no—
76C2×D3×Dic3—61no—
77C2×D3×Dic3—61no—
78C2×D3×Dic3—62no—
79C32×C4⋊C4—31no—
80C32×C4⋊C4—31no—
81C4⋊(C3⋊Dic3)—31no—
82C4⋊(C3⋊Dic3)—31no—
83C4⋊(C3⋊Dic3)—31no—
84C4⋊(C3⋊Dic3)—61no—
85C4⋊(C3⋊Dic3)—61no—
86C4⋊(C3⋊Dic3)—61no—
87C4⋊(C3⋊Dic3)—121no—
88#3750—61no—
89C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
90C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
91C3×#217—61no—
92C3×(C2×D3)⋊C4—61no—
93C3×C4⋊Dic3—21no—
94C3×C4⋊Dic3—21no—
95C3×C4⋊Dic3C2×D311yescharacteristic
96C3×C4⋊Dic3C2×D311yescharacteristic
97C3×C4⋊Dic3C2×D311yescharacteristic
98C3×C4⋊Dic3C2×D311yescharacteristic
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—61no—
104C2×C3×C3⋊C8—61no—
105Dic3⋊Dic3—61no—
106Dic3⋊Dic3—61no—
107Dic3⋊Dic3—62no—
108(C2×D3)⋊Dic3—21no—
109C22×C33C2×D411yescharacteristic
110C2×C3×C3⋊D3—21no—
111C2×C3×C3⋊D3—21no—
112C2×C3×C3⋊D3D812yes—
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—21no—
120C32×Dic3—41no—
121C32×Dic3—41no—
122C3×#121—91no—
123#938—31no—
124C2×C4⋊Dic3—91no—
125#895—31no—
126D3×C4⋊C4—91no—
127D3×C4⋊C4—91no—
128D3×C4⋊C4—181no—
129C3×#107—91no—
130C3×#102—91no—
131#832—31no—
132#832—91no—
133#832—91no—
134#832—181no—
135#830—91no—
136#821—91no—
137(C2×D3)⋊C8—91no—
138(C2×D3)⋊C8—91no—
139#818—91no—
140#809—31no—
141#807—31no—
142#807—31no—
143#807—91no—
144#807—91no—
145#807—181no—
146C23×C32—61no—
147C22×C3⋊D3#6711yescharacteristic
148C22×C3×D3—21no—
149C22×C3×D3—21no—
150C22×C3×D3—31no—
151C22×C3×D3—31no—
152C22×C3×D3—41no—
153C22×C3×D3—61no—
154C32×D4—61no—
155C32×D4—61no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
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—61no—
166C2×C3⋊Dic3—62no—
167C2×C3⋊Dic3—61no—
168C2×C3⋊Dic3—61no—
169C2×C3⋊Dic3—121no—
170C2×C3⋊Dic3—121no—
171C2×C3⋊Dic3—122no—
172C2×C3⋊Dic3#6711yescharacteristic
173C4×C3⋊D3—21no—
174C4×C3⋊D3—21no—
175C4×C3⋊D3—122no—
176C3×#67—122no—
177C2×C3×Dic3—22no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—31no—
183C2×C3×Dic3—31no—
184C2×C3×Dic3—41no—
185C2×C3×Dic3—61no—
186C3×D12—22no—
187C3×D12—21no—
188C3×D12—21no—
189C3×D12—21no—
190C3×D12—21no—
191C3×D12—41no—
192C3×D12—122no—
193C3×C4×D3—62no—
194C3×C4×D3—62no—
195C3×C4×D3—61no—
196C3×C4×D3—61no—
197C3×C4×D3—61no—
198C3×C4×D3—61no—
199#626—42no—
200D3×Dic3—122no—
201D3×Dic3—122no—
202D3×Dic3—124no—
203C3⋊(C3⋊C8)—21no—
204C3⋊(C3⋊C8)—21no—
205C3×C3⋊C8—62no—
206C3×C3⋊C8—61no—
207C3×C3⋊C8—61no—
208C3×C3⋊C8—61no—
209C3×C3⋊C8—61no—
210C3×C3⋊C8—121no—
211#480—271noSylow
212C2×C33#12311yescharacteristic
213C2×C33C2×D811yescharacteristic
214C2×C33#13711yescharacteristic
215C3×C3⋊D3—22no—
216C32×D3—81no—
217C2×C3×D4—31no—
218C2×C3×D4—31no—
219C2×C3×D4—61no—
220C2×C3×D4—181no—
221C22×C3×C4—91no—
222C2×#67—61no—
223C22×Dic3—181no—
224C2×D12—61no—
225C2×D12D3211yescharacteristic
226C2×C4×D3—31no—
227C2×C4×D3—31no—
228C2×C4×D3—61no—
229C2×C4×D3—181no—
230C2×C4×D3—181no—
231C2×C4×D3—182no—
232C2×C3×C8—91no—
233C2×C3×C8—91no—
234C3×C4⋊C4—31no—
235C3×C4⋊C4—31no—
236C3×C4⋊C4—31no—
237C3×C4⋊C4—31no—
238C3×C4⋊C4—61no—
239C3×C4⋊C4—61no—
240C3×C22⋊C4—181no—
241#217—61no—
242(C2×D3)⋊C4—61no—
243C4⋊Dic3—91no—
244C4⋊Dic3—91no—
245C4⋊Dic3—91no—
246C4⋊Dic3—181no—
247C4⋊Dic3—181no—
248C4⋊Dic3—181no—
249C4⋊Dic3—181no—
250C4⋊Dic3—361no—
251C4⋊Dic3D3211yescharacteristic
252C4⋊Dic3D3211yescharacteristic
253Dic3⋊C4—181no—
254Dic3⋊C4—181no—
255Dic3⋊C4—182no—
256C2×C3⋊C8—31no—
257C2×C3⋊C8—31no—
258C2×C3⋊C8—31no—
259C2×C3⋊C8—31no—
260C2×C3⋊C8—61no—
261C2×C3⋊C8—61no—
262C22×C32—21no—
263C22×C32—21no—
264C22×C32—21no—
265C22×C32—41no—
266C22×C32—121no—
267C22×C32—121no—
268C22×C32—121no—
269C22×C32D3×D411yescharacteristic
270C22×C32D3×D411yescharacteristic
271C22×C32C2×#6711yescharacteristic
272C2×C3⋊D3—21no—
273C2×C3⋊D3—21no—
274C2×C3⋊D3#21312yes—
275C2×C3×D3—32no—
276C2×C3×D3—32no—
277C2×C3×D3—42no—
278C2×C3×D3—41no—
279C2×C3×D3—41no—
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—62no—
287C2×C3×D3—61no—
288C2×C3×D3—61no—
289C2×C3×D3—61no—
290C2×C3×D3—61no—
291C2×C3×D3—81no—
292C2×C3×D3—81no—
293C32×C4—21no—
294C32×C4—21no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4—21no—
298C32×C4—21no—
299C32×C4—41no—
300C32×C4—41no—
301C32×C4—121no—
302C32×C4—121no—
303C32×C4#23511yescharacteristic
304C32×C4#23511yescharacteristic
305C32×C4#23711yescharacteristic
306C32×C4#23711yescharacteristic
307C32×C4#23711yescharacteristic
308C32×C4#23911yescharacteristic
309C3⋊Dic3—21no—
310C3⋊Dic3—121no—
311C3⋊Dic3—122no—
312C3⋊Dic3—121no—
313C3⋊Dic3—121no—
314C3⋊Dic3—241no—
315C3⋊Dic3—241no—
316C3⋊Dic3—242no—
317C3×Dic3—42no—
318C3×Dic3—41no—
319C3×Dic3—41no—
320C3×Dic3—41no—
321C3×Dic3—41no—
322C3×Dic3—62no—
323C3×Dic3—61no—
324C3×Dic3—61no—
325C3×Dic3—81no—
326#121—91no—
327C2×C4⋊C4—271no—
328#107—91no—
329#107—271no—
330#102—91no—
331#102—271no—
332C22⋊C8—271no—
333C33#48011yescharacteristic, complemented, Sylow
334C23×C3—61no—
335C23×C3—61no—
336C23×C3—91no—
337C23×C3—121no—
338C22×D3—21no—
339C22×D3—31no—
340C22×D3—31no—
341C22×D3—61no—
342C3×D4—62no—
343C3×D4—61no—
344C3×D4—61no—
345C3×D4—61no—
346C3×D4—61no—
347C3×D4—121no—
348C3×D4—362no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—41no—
353C2×C3×C4—62no—
354C2×C3×C4—61no—
355C2×C3×C4—61no—
356C2×C3×C4—61no—
357C2×C3×C4—61no—
358C2×C3×C4—91no—
359C2×C3×C4—121no—
360C2×C3×C4—181no—
361C2×C3×C4—181no—
362C2×C3×C4C2×D3211yescharacteristic
363C2×C3×C4C2×D3211yescharacteristic
364C2×C3×C4C2×D3211yescharacteristic
365#67—122no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—31no—
369C2×Dic3—31no—
370C2×Dic3—61no—
371C2×Dic3—182no—
372C2×Dic3—181no—
373C2×Dic3—181no—
374C2×Dic3—181no—
375C2×Dic3—361no—
376C2×Dic3—361no—
377C2×Dic3—362no—
378C2×Dic3—362no—
379D12—21no—
380D12—21no—
381D12—122no—
382C4×D3—62no—
383C4×D3—62no—
384C4×D3—61no—
385C4×D3—61no—
386C4×D3—61no—
387C4×D3—61no—
388C4×D3—362no—
389C4×D3—362no—
390C4×D3—364no—
391C3×C8—181no—
392C3×C8—181no—
393C3⋊C8—62no—
394C3⋊C8—61no—
395C3⋊C8—61no—
396C3⋊C8—61no—
397C3⋊C8—61no—
398C3⋊C8—121no—
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—241no—
412C2×C32#88311yescharacteristic
413C2×C32#89411yescharacteristic
414C2×C32D3×D811yescharacteristic
415C2×C32D3×D811yescharacteristic
416C2×C32#91511yescharacteristic
417C2×C32#91811yescharacteristic
418C2×C32C2×#21311yescharacteristic
419C2×C32#93511yescharacteristic
420C2×C32#95111yescharacteristic
421C3⋊D3—22no—
422C3×D3—64no—
423C3×D3—62no—
424C3×D3—62no—
425C3×D3—82no—
426C3×D3—81no—
427C3×D3—81no—
428C2×D4—31no—
429C2×D4—181no—
430C22×C4—91no—
431C22×C4—541no—
432C2×C8—91no—
433C2×C8—91no—
434C4⋊C4—31no—
435C4⋊C4—31no—
436C4⋊C4—271no—
437C4⋊C4—541no—
438C22⋊C4—181no—
439C22×C3—21no—
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—41no—
443C22×C3—92no—
444C22×C3—122no—
445C22×C3—121no—
446C22×C3—121no—
447C22×C3—121no—
448C22×C3—121no—
449C22×C3—121no—
450C22×C3—121no—
451C22×C3—181no—
452C22×C3—181no—
453C22×C3—241no—
454C22×C3—241no—
455C22×C3D3×#6711yescharacteristic
456C22×C3D3×#6711yescharacteristic
457C22×C3#381111yescharacteristic
458C2×D3—32no—
459C2×D3—32no—
460C2×D3—41no—
461C2×D3—41no—
462C2×D3—41no—
463C2×D3—62no—
464C2×D3—62no—
465C2×D3—62no—
466C2×D3—61no—
467C2×D3—61no—
468C2×D3—61no—
469C2×D3—61no—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—41no—
477C3×C4—41no—
478C3×C4—122no—
479C3×C4—121no—
480C3×C4—121no—
481C3×C4—121no—
482C3×C4—121no—
483C3×C4—181no—
484C3×C4—241no—
485C3×C4#379511yescharacteristic
486C3×C4#379611yescharacteristic
487C3×C4#379611yescharacteristic
488C3×C4#379811yescharacteristic
489C3×C4#380411yescharacteristic
490C3×C4#380511yescharacteristic
491Dic3—41no—
492Dic3—41no—
493Dic3—62no—
494Dic3—61no—
495Dic3—61no—
496Dic3—362no—
497Dic3—361no—
498Dic3—361no—
499Dic3—361no—
500Dic3—721no—
501Dic3—721no—
502Dic3—722no—
503C32—21no—
504C32—21no—
505C32—21no—
506C32—41no—
507C32#477411yescharacteristic, complemented
508C32#488211yescharacteristic, complemented
509C32#503311yescharacteristic, complemented
510C23—61no—
511C23—91no—
512D4—61no—
513D4—61no—
514D4—362no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—91no—
518C2×C4—181no—
519C2×C4—181no—
520C2×C4—541no—
521C2×C4—1082no—
522C2×C4D3311yescharacteristic
523C8—181no—
524C8—181no—
525C2×C3—21no—
526C2×C3—21no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—41no—
535C2×C3—41no—
536C2×C3—41no—
537C2×C3—182no—
538C2×C3—242no—
539C2×C3—241no—
540C2×C3—241no—
541C2×C3#6385311yescharacteristic
542C2×C3#6388411yescharacteristic
543C2×C3D3×#21311yescharacteristic
544C2×C3D3×#21311yescharacteristic
545C2×C3#6392611yescharacteristic
546C2×C3#6392911yescharacteristic
547C2×C3#6394311yescharacteristic
548C2×C3#6395111yescharacteristic
549C2×C3#6396911yescharacteristic
550D3—64no—
551D3—62no—
552D3—62no—
553D3—81no—
554C22—92no—
555C22—121no—
556C22—121no—
557C22—121no—
558C22—181no—
559C22—181no—
560C22#8895211yescharacteristic
561C4—121no—
562C4—121no—
563C4—181no—
564C4—1081no—
565C4#8896011yescharacteristic
566C4#8896411yescharacteristic
567C3—21no—
568C3—21no—
569C3—21no—
570C3—41no—
571C3#1059043811yescharacteristic, complemented
572C3#1059051011yescharacteristic, complemented
573C3#1059141311yescharacteristic, complemented
574C2—182no—
575C2—241no—
576C2#1172283011yescharacteristic
577C2#1172288811yescharacteristic
578C2#1172299111yescharacteristic
subgroupG/H
centerC22#88952
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC3×#63916C2
hypocenterC33#480
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⋊D3×C4⋊Dic3C2semidirect
3. #11722072C2semidirect
4. #11722057C2semidirect
5. #11722032C2semidirect
6. C3×#63576C2semidirect
7. C3×#63562C2semidirect
35. #63576D3semidirect
42. #63562D3semidirect
333. C33#480semidirect
507. C32#4774semidirect
508. C32#4882semidirect
509. C32#5033semidirect
571. C3#10590438semidirect
572. C3#10590510semidirect
573. C3#10591413semidirect

Maximal subgroups

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

subgroupindex
C3⋊D3×C4⋊Dic32
C3×#639162
#117220722
#117220572
#117220322
C3×#635762
C3×#635622
#105914133
#105905103
#105904383