Finite Group Databasebeta

← #13156398

Subgroups — #13156398

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
1#11723696C211yesmaximal, characteristic
2C3⋊D3×C4⋊Dic3C211yesmaximal, characteristic, complemented
3#11722055C211yesmaximal, characteristic, complemented
4#11722034C211yesmaximal, characteristic, complemented
5C3×#63575C211yesmaximal, characteristic, complemented
6C3×#63564C211yesmaximal, characteristic, complemented
7C3×#63557C211yesmaximal, characteristic, complemented
8#10590680—31nomaximal
9#10590679—31nomaximal
10#10590510—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×#6295—21no—
14C2×Dic3×C3⋊D3—21no—
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88814—21no—
21#88807—21no—
22C4⋊C4×C3⋊D3—31no—
23#63916—31no—
24#63909—31no—
25#63908—31no—
26D3×C4⋊Dic3—31no—
27D3×C4⋊Dic3—31no—
28D3×C4⋊Dic3—61no—
29#63648—31no—
30#63645—31no—
31C3×#821—31no—
32C3×(C2×D3)⋊C8—31no—
33C3×(C2×D3)⋊C8—31no—
34C3×#818—31no—
35C3×#809—31no—
36C3×#807—31no—
37#63575—31no—
38#63575—31no—
39#63575—61no—
40#63575D311yescharacteristic, complemented
41#63564D311yescharacteristic, complemented
42#63563—31no—
43#63563—31no—
44#63563—61no—
45#63557D311yescharacteristic, complemented
46C22×C3×C3⋊D3—21no—
47C22×C3×C3⋊D3D411yescharacteristic
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)—21no—
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3D411yescharacteristic
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C3×C4×C3⋊D3—21no—
55C3×C4×C3⋊D3—21no—
56C2×C32×Dic3—21no—
57#6295—42no—
58Dic3×C3⋊D3—42no—
59C32×C3⋊C8—21no—
60C32×C3⋊C8—21no—
61#4882—91no—
62#4774—91no—
63#4736—91no—
64C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
65C2×C4×C3⋊D3—61no—
66C2×C4×C3⋊D3C2×D311yescharacteristic
67C2×C3×D12—31no—
68C2×C3×D12—31no—
69C2×C3×D12—61no—
70C2×C3×C4×D3—31no—
71C2×C3×C4×D3—31no—
72C2×C3×C4×D3—61no—
73C2×#626—61no—
74C2×#626—61no—
75C2×#625—61no—
76C2×D3×Dic3—61no—
77C2×D3×Dic3—61no—
78C2×D3×Dic3—62no—
79C2×C32×C8—31no—
80C2×C32×C8—31no—
81C32×C4⋊C4—31no—
82C4⋊(C3⋊Dic3)—31no—
83C4⋊(C3⋊Dic3)—31no—
84C4⋊(C3⋊Dic3)—31no—
85C4⋊(C3⋊Dic3)—61no—
86C4⋊(C3⋊Dic3)—61no—
87C4⋊(C3⋊Dic3)—61no—
88C4⋊(C3⋊Dic3)—121no—
89C4⋊(C3⋊Dic3)C2×D311yescharacteristic
90#3750—61no—
91C3×C4⋊Dic3—21no—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3—31no—
94C3×C4⋊Dic3—61no—
95C3×C4⋊Dic3C2×D311yescharacteristic
96C3×C4⋊Dic3C2×D311yescharacteristic
97C2×C3×C3⋊C8—21no—
98C2×C3×C3⋊C8—21no—
99C2×C3×C3⋊C8C2×D311yescharacteristic
100C2×C3×C3⋊C8C2×D311yescharacteristic
101C2×C3×C3⋊C8C2×D311yescharacteristic
102C2×C3×C3⋊C8C2×D311yescharacteristic
103Dic3⋊Dic3—61no—
104Dic3⋊Dic3—61no—
105Dic3⋊Dic3—62no—
106#3722—61no—
107(C2×D3)⋊Dic3—61no—
108(C2×D3)⋊Dic3—61no—
109C22×C33C2×D411yescharacteristic
110C2×C3×C3⋊D3—21no—
111C2×C3×C3⋊D3—21no—
112C2×C3×C3⋊D3—41no—
113C2×C3×C3⋊D3—41no—
114C2×C3×C3⋊D3—41no—
115C2×C3×C3⋊D3D812yes—
116C33×C4#4111yescharacteristic
117C33×C4#4111yescharacteristic
118C3⋊(C3⋊Dic3)—41no—
119C3×C3⋊Dic3—21no—
120C3×C3⋊Dic3—41no—
121C32×Dic3—41no—
122#938—91no—
123#930—91no—
124C2×C4⋊Dic3—91no—
125#895—91no—
126D3×C4⋊C4—91no—
127D3×C4⋊C4—91no—
128D3×C4⋊C4—181no—
129C3×#107—91no—
130C3×#102—91no—
131C3×C22⋊C8—91no—
132#832—91no—
133#821—31no—
134#821—91no—
135#821—91no—
136#821—181no—
137(C2×D3)⋊C8—31no—
138(C2×D3)⋊C8—31no—
139#818—31no—
140#818—91no—
141#818—91no—
142#818—181no—
143#809—31no—
144#807—31no—
145#807—91no—
146C22×C3⋊D3—21no—
147C22×C3⋊D3D1211yescharacteristic
148C22×C3×D3—31no—
149C22×C3×D3—31no—
150C22×C3×D3—61no—
151C22×C3×D3—61no—
152C22×C3×D3—61no—
153C22×C3×D3—121no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—41no—
158C2×C32×C4—61no—
159C2×C32×C4C22×D311yescharacteristic
160C2×C32×C4C22×D311yescharacteristic
161C2×C32×C4C22×D311yescharacteristic
162C2×C3⋊Dic3—21no—
163C2×C3⋊Dic3—61no—
164C2×C3⋊Dic3—62no—
165C2×C3⋊Dic3—61no—
166C2×C3⋊Dic3—61no—
167C2×C3⋊Dic3—121no—
168C2×C3⋊Dic3—121no—
169C2×C3⋊Dic3—122no—
170C2×C3⋊Dic3D1211yescharacteristic
171C4⋊(C3⋊D3)—21no—
172C4⋊(C3⋊D3)—21no—
173C4×C3⋊D3—21no—
174C4×C3⋊D3—21no—
175C4×C3⋊D3—122no—
176C2×C3×Dic3—22no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3—31no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—61no—
183C2×C3×Dic3—61no—
184C2×C3×Dic3—121no—
185C3×D12—62no—
186C3×D12—61no—
187C3×D12—61no—
188C3×D12—61no—
189C3×D12—61no—
190C3×D12—121no—
191C3×C4×D3—62no—
192C3×C4×D3—62no—
193C3×C4×D3—61no—
194C3×C4×D3—61no—
195C3×C4×D3—61no—
196C3×C4×D3—61no—
197#626—122no—
198#626—122no—
199#625—122no—
200D3×Dic3—122no—
201D3×Dic3—122no—
202D3×Dic3—124no—
203C32×C8—61no—
204C32×C8—61no—
205C3×C3⋊C8—22no—
206C3×C3⋊C8—21no—
207C3×C3⋊C8—21no—
208C3×C3⋊C8—21no—
209C3×C3⋊C8—21no—
210C3×C3⋊C8—41no—
211#480—271noSylow
212C2×C33#12311yescharacteristic
213C2×C33C2×D811yescharacteristic
214C2×C33#13711yescharacteristic
215C3×C3⋊D3—22no—
216C3×C3⋊D3—81no—
217C2×C3×D4—91no—
218C22×C3×C4—91no—
219C2×#67—181no—
220C2×#67—181no—
221C22×Dic3—181no—
222C2×D12—31no—
223C2×D12—31no—
224C2×D12—61no—
225C2×D12—181no—
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—31no—
233C2×C3×C8—31no—
234C2×C3×C8—31no—
235C2×C3×C8—31no—
236C2×C3×C8—61no—
237C2×C3×C8—61no—
238C3×C4⋊C4—31no—
239C3×C4⋊C4—31no—
240C3×C4⋊C4—61no—
241C3×C4⋊C4—91no—
242#217—181no—
243(C2×D3)⋊C4—181no—
244(C2×D3)⋊C4—181no—
245C4⋊Dic3—31no—
246C4⋊Dic3—31no—
247C4⋊Dic3—61no—
248C4⋊Dic3—91no—
249C4⋊Dic3—91no—
250C4⋊Dic3—91no—
251C4⋊Dic3—181no—
252C4⋊Dic3—181no—
253C4⋊Dic3—181no—
254C4⋊Dic3—181no—
255C4⋊Dic3—361no—
256C4⋊Dic3D3211yescharacteristic
257Dic3⋊C4—181no—
258Dic3⋊C4—181no—
259Dic3⋊C4—182no—
260C2×C3⋊C8D3211yescharacteristic
261C2×C3⋊C8D3211yescharacteristic
262C22×C32—21no—
263C22×C32—21no—
264C22×C32—21no—
265C22×C32—41no—
266C22×C32C2×D1211yescharacteristic
267C22×C32D3×D411yescharacteristic
268C22×C32D3×D411yescharacteristic
269C2×C3⋊D3—21no—
270C2×C3⋊D3—21no—
271C2×C3⋊D3—41no—
272C2×C3⋊D3—41no—
273C2×C3⋊D3—41no—
274C2×C3⋊D3D2412yes—
275C2×C3×D3—32no—
276C2×C3×D3—32no—
277C2×C3×D3—62no—
278C2×C3×D3—62no—
279C2×C3×D3—62no—
280C2×C3×D3—61no—
281C2×C3×D3—61no—
282C2×C3×D3—61no—
283C2×C3×D3—61no—
284C2×C3×D3—122no—
285C2×C3×D3—121no—
286C2×C3×D3—121no—
287C2×C3×D3—121no—
288C2×C3×D3—121no—
289C2×C3×D3—121no—
290C2×C3×D3—121no—
291C2×C3×D3—241no—
292C2×C3×D3—241no—
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#23511yescharacteristic
303C32×C4#23511yescharacteristic
304C32×C4#23711yescharacteristic
305C32×C4#23711yescharacteristic
306C32×C4#23711yescharacteristic
307C32×C4#23911yescharacteristic
308C3⋊Dic3—21no—
309C3⋊Dic3—41no—
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—62no—
321C3×Dic3—61no—
322C3×Dic3—61no—
323C3×Dic3—121no—
324C3×Dic3—121no—
325C3×Dic3—241no—
326#121—271no—
327C2×C4⋊C4—271no—
328#107—91no—
329#107—271no—
330#102—91no—
331#102—271no—
332C22⋊C8—91no—
333C33#48011yescharacteristic, complemented, Sylow
334C23×C3—91no—
335C23×C3—181no—
336C22×D3—31no—
337C22×D3—31no—
338C22×D3—61no—
339C22×D3—61no—
340C22×D3—61no—
341C22×D3—121no—
342C3×D4—181no—
343C3×D4—181no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—41no—
348C2×C3×C4—62no—
349C2×C3×C4—61no—
350C2×C3×C4—61no—
351C2×C3×C4—91no—
352C2×C3×C4—181no—
353C2×C3×C4—181no—
354C2×C3×C4—181no—
355C2×C3×C4C2×D3211yescharacteristic
356C2×C3×C4C2×D3211yescharacteristic
357C2×C3×C4C2×D3211yescharacteristic
358#67—362no—
359#67—362no—
360C2×Dic3—21no—
361C2×Dic3—31no—
362C2×Dic3—31no—
363C2×Dic3—61no—
364C2×Dic3—61no—
365C2×Dic3—61no—
366C2×Dic3—121no—
367C2×Dic3—182no—
368C2×Dic3—181no—
369C2×Dic3—181no—
370C2×Dic3—181no—
371C2×Dic3—361no—
372C2×Dic3—361no—
373C2×Dic3—362no—
374C2×Dic3—362no—
375D12—62no—
376D12—61no—
377D12—61no—
378D12—61no—
379D12—61no—
380D12—121no—
381D12—362no—
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—62no—
392C3×C8—61no—
393C3×C8—61no—
394C3×C8—61no—
395C3×C8—61no—
396C3×C8—121no—
397C3⋊C8—21no—
398C3⋊C8—21no—
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#88311yescharacteristic
412C2×C32#88611yescharacteristic
413C2×C32#89411yescharacteristic
414C2×C32C2×D2411yescharacteristic
415C2×C32#90911yescharacteristic
416C2×C32D3×D811yescharacteristic
417C2×C32D3×D811yescharacteristic
418C2×C32#91511yescharacteristic
419C2×C32#91811yescharacteristic
420C3⋊D3—22no—
421C3⋊D3—81no—
422C3×D3—64no—
423C3×D3—62no—
424C3×D3—62no—
425C3×D3—242no—
426C3×D3—241no—
427C3×D3—241no—
428C2×D4—91no—
429C2×D4—541no—
430C22×C4—91no—
431C22×C4—541no—
432C2×C8—31no—
433C2×C8—31no—
434C4⋊C4—31no—
435C4⋊C4—91no—
436C4⋊C4—271no—
437C4⋊C4—541no—
438C22⋊C4—541no—
439C22×C3—21no—
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—41no—
443C22×C3—92no—
444C22×C3—181no—
445C22×C3—181no—
446C22×C3—361no—
447C22×C3—361no—
448C22×C3—361no—
449C22×C3D3×D1211yescharacteristic
450C22×C3D3×D1211yescharacteristic
451C22×C3#381111yescharacteristic
452C2×D3—32no—
453C2×D3—32no—
454C2×D3—62no—
455C2×D3—62no—
456C2×D3—62no—
457C2×D3—61no—
458C2×D3—61no—
459C2×D3—61no—
460C2×D3—61no—
461C2×D3—122no—
462C2×D3—121no—
463C2×D3—121no—
464C2×D3—121no—
465C2×D3—121no—
466C2×D3—121no—
467C2×D3—121no—
468C2×D3—241no—
469C2×D3—241no—
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—181no—
482C3×C4—361no—
483C3×C4#379511yescharacteristic
484C3×C4#379611yescharacteristic
485C3×C4#379611yescharacteristic
486C3×C4#379811yescharacteristic
487C3×C4#380411yescharacteristic
488C3×C4#380511yescharacteristic
489Dic3—41no—
490Dic3—62no—
491Dic3—61no—
492Dic3—61no—
493Dic3—121no—
494Dic3—121no—
495Dic3—241no—
496Dic3—362no—
497Dic3—361no—
498Dic3—361no—
499Dic3—361no—
500Dic3—721no—
501Dic3—721no—
502Dic3—722no—
503C32—21no—
504C32—21no—
505C32—21no—
506C32—41no—
507C32#473611yescharacteristic, complemented
508C32#477411yescharacteristic, complemented
509C32#488211yescharacteristic, complemented
510C23—91no—
511C23—181no—
512D4—181no—
513D4—181no—
514D4—1082no—
515C2×C4—61no—
516C2×C4—91no—
517C2×C4—181no—
518C2×C4—181no—
519C2×C4—181no—
520C2×C4—541no—
521C2×C4—1082no—
522C2×C4D3311yescharacteristic
523C8—61no—
524C8—61no—
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—721no—
539C2×C3D3×D2411yescharacteristic
540C2×C3D3×D2411yescharacteristic
541C2×C3#6380011yescharacteristic
542C2×C3#6380111yescharacteristic
543C2×C3#6385311yescharacteristic
544C2×C3#6389011yescharacteristic
545C2×C3#6389111yescharacteristic
546C2×C3#6392611yescharacteristic
547C2×C3#6395111yescharacteristic
548D3—64no—
549D3—62no—
550D3—62no—
551D3—242no—
552D3—241no—
553D3—241no—
554C22—92no—
555C22—181no—
556C22—181no—
557C22—361no—
558C22—361no—
559C22—361no—
560C22#8895511yescharacteristic
561C4—121no—
562C4—181no—
563C4—361no—
564C4—1081no—
565C4#8896011yescharacteristic
566C4#8896411yescharacteristic
567C3—21no—
568C3—21no—
569C3—21no—
570C3—41no—
571C3#1059051011yescharacteristic, complemented
572C3#1059067911yescharacteristic, complemented
573C3#1059068011yescharacteristic, complemented
574C2—182no—
575C2—721no—
576C2#1172283311yescharacteristic
577C2#1172289611yescharacteristic
578C2#1172299711yescharacteristic
subgroupG/H
centerC22#88955
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radical#11723696C2
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
2. C3⋊D3×C4⋊Dic3C2semidirect
3. #11722055C2semidirect
4. #11722034C2semidirect
5. C3×#63575C2semidirect
6. C3×#63564C2semidirect
7. C3×#63557C2semidirect
40. #63575D3semidirect
41. #63564D3semidirect
45. #63557D3semidirect
333. C33#480semidirect
507. C32#4736semidirect
508. C32#4774semidirect
509. C32#4882semidirect
571. C3#10590510semidirect
572. C3#10590679semidirect
573. C3#10590680semidirect

Maximal subgroups

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

subgroupindex
#117236962
C3⋊D3×C4⋊Dic32
#117220552
#117220342
C3×#635752
C3×#635642
C3×#635572
#105906803
#105906793
#105905103