Finite Group Databasebeta

← #13176470

Subgroups — #13176470

Order 1728.

2248 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89020C212yesmaximal, complemented
2#11723934C211yesmaximal, characteristic, complemented
3C3×#64102C211yesmaximal, characteristic, complemented
4C32×#893C211yesmaximal, characteristic, complemented
5C4×#6293C211yesmaximal, characteristic, complemented
6#11723529C211yesmaximal, characteristic, complemented
7#11723527C212yesmaximal, complemented
8#11723524C211yesmaximal, characteristic, complemented
9#11723520C212yesmaximal, complemented
10#11723517C211yesmaximal, characteristic, complemented
11#11723514C212yesmaximal, complemented
12#10596024—31nomaximal
13#10594520—34nomaximal
14C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
15C2×C4×C3⋊(C3⋊D3)C2212yes—
16C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×C4×C3⋊D3C2212yes—
18C2×C32×D12C2211yescharacteristic
19C2×C32×C4×D3C2212yes—
20C2×#6293C2211yescharacteristic
21C2×#6292C2212yes—
22C2×#6292C2212yes—
23C2×#6291C2212yes—
24C2×#6291C2212yes—
25C2×#6290C2212yes—
26C2×#6290C2212yes—
27#89020—28no—
28C33×C4⋊C4C2211yescharacteristic
29#88859C2212yes—
30C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
31C3×#3752C2212yes—
32C3×C4×C3⋊Dic3C2211yescharacteristic
33C32×(C2×D3)⋊C4C2212yes—
34C32×C4×Dic3C2211yescharacteristic
35#88807C2211yescharacteristic
36#88806C2211yescharacteristic
37Dic3⋊(C3⋊Dic3)C2211yescharacteristic
38C2×#3828—32no—
39C2×#3798—38no—
40C32×C4×D4—31no—
41#64149—31no—
42#64149—31no—
43#64139—32no—
44#64102—31no—
45#64102—34no—
46#64102—64no—
47#64102D311yescharacteristic, complemented
48#64093—32no—
49#64093—32no—
50#64083—31no—
51C4×#634—31no—
52C3×#893—34no—
53C3×#893D314yescomplemented
54C4×#627—34no—
55#63914—34no—
56#63912—38no—
57#63911—34no—
58#63911—34no—
59#63906—38no—
60#63906—38no—
61C22×C3⋊(C3⋊D3)C2312yes—
62C22×C3×C3⋊D3C2312yes—
63C22×C32×D3C2312yes—
64C2×C33×C4C2312yes—
65C2×C33×C4C2311yescharacteristic
66C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
67C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
68C4⋊(C3⋊(C3⋊D3))—22no—
69C4×C3⋊(C3⋊D3)—24no—
70C2×C3×C3⋊Dic3C2311yescharacteristic
71C2×C3×C3⋊Dic3C2311yescharacteristic
72C3×C4⋊(C3⋊D3)—22no—
73C3×C4×C3⋊D3—24no—
74C2×C32×Dic3C2311yescharacteristic
75C2×C32×Dic3C2311yescharacteristic
76C32×D12—22no—
77C32×C4×D3—24no—
78#6293D414yescomplemented
79#6292—24no—
80#6292—42no—
81#6291—24no—
82#6291—42no—
83#6290—24no—
84#6290—42no—
85#5553—94no—
86#5553—91no—
87C2×C32×D4—31no—
88C22×C32×C4—32no—
89C2×#638—32no—
90C2×#638—32no—
91C2×#638—32no—
92C2×#638—32no—
93#3828—68no—
94C2×C4⋊(C3⋊D3)—32no—
95C2×C4⋊(C3⋊D3)—31no—
96C2×C4⋊(C3⋊D3)—34no—
97C2×C4⋊(C3⋊D3)—32no—
98C2×C4⋊(C3⋊D3)—64no—
99C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
100C2×C4×C3⋊D3—32no—
101C2×C4×C3⋊D3—38no—
102C2×C4×C3⋊D3—68no—
103C2×C4×C3⋊D3C2×D312yes—
104C2×#634—31no—
105C2×C3×D12—34no—
106C2×C3×D12C2×D314yes—
107C2×C3×C4×D3—38no—
108C2×C3×C4×D3C2×D318yes—
109C2×#627—34no—
110C2×#626—38no—
111C2×#626—38no—
112C2×#626—38no—
113C2×#626—38no—
114C2×#625—38no—
115C2×#625—38no—
116#3798—632no—
117C32×C4⋊C4—24no—
118C32×C4⋊C4C2×D314yes—
119C32×C4⋊C4C2×D311yescharacteristic
120C32×C22⋊C4—32no—
121C32×C42—31no—
122#3752—32no—
123#3752—38no—
124#3752—68no—
125#3752C2×D312yes—
126C4⋊(C3⋊Dic3)—31no—
127#3750—31no—
128#3750—31no—
129C4×C3⋊Dic3—31no—
130C4×C3⋊Dic3—34no—
131C4×C3⋊Dic3—64no—
132C4×C3⋊Dic3C2×D311yescharacteristic
133C3×(C2×D3)⋊C4—38no—
134C3×(C2×D3)⋊C4C2×D318yes—
135C3×C4×Dic3—34no—
136C3×C4×Dic3C2×D314yes—
137#3724—34no—
138Dic3⋊Dic3—34no—
139Dic3⋊Dic3—34no—
140C22×C33C2411yescharacteristic
141C2×C3⋊(C3⋊D3)—22no—
142C2×C3⋊(C3⋊D3)—22no—
143C2×C3⋊(C3⋊D3)—22no—
144C2×C3×C3⋊D3—22no—
145C2×C3×C3⋊D3—22no—
146C2×C3×C3⋊D3—22no—
147C2×C32×D3—22no—
148C2×C32×D3—22no—
149C2×C32×D3—22no—
150C33×C4—22no—
151C33×C4#4112yescomplemented
152C3⋊(C3⋊Dic3)—21no—
153C3⋊(C3⋊Dic3)C2×D412yes—
154C3×C3⋊Dic3—21no—
155C3×C3⋊Dic3C2×D412yes—
156C32×Dic3—21no—
157C32×Dic3C2×D412yes—
158C2×#235—98no—
159C2×#235—92no—
160C3×C4×D4—34no—
161C3×C4×D4—91no—
162#940—94no—
163#940—94no—
164#940—91no—
165#940—91no—
166#930—98no—
167#930—92no—
168#893—34no—
169#893—94no—
170#893—91no—
171#893—184no—
172#893C3⋊D311yescharacteristic, complemented
173#884—98no—
174#884—98no—
175#884—92no—
176#884—92no—
177#874—94no—
178#874—91no—
179C4×Dic6—94no—
180C4×Dic6—91no—
181C23×C32—32no—
182C22×C3⋊D3—32no—
183C22×C3⋊D3—38no—
184C22×C3⋊D3—68no—
185C22×C3⋊D3C22×D312yes—
186C22×C3×D3—38no—
187C22×C3×D3C22×D318yes—
188C32×D4—62no—
189C2×C32×C4—24no—
190C2×C32×C4—28no—
191C2×C32×C4—31no—
192C2×C32×C4—31no—
193C2×C32×C4—64no—
194C2×C32×C4C22×D318yes—
195C2×C32×C4C22×D314yes—
196C2×C32×C4C22×D312yes—
197C2×C32×C4C22×D311yescharacteristic
198#638—64no—
199#638—64no—
200#638—122no—
201#638—122no—
202C2×C3⋊Dic3—31no—
203C2×C3⋊Dic3—31no—
204C2×C3⋊Dic3—34no—
205C2×C3⋊Dic3—34no—
206C2×C3⋊Dic3—64no—
207C2×C3⋊Dic3—64no—
208C2×C3⋊Dic3C22×D311yescharacteristic
209C2×C3⋊Dic3C22×D311yescharacteristic
210C4⋊(C3⋊D3)—22no—
211C4⋊(C3⋊D3)—616no—
212C4⋊(C3⋊D3)—64no—
213C4⋊(C3⋊D3)—68no—
214C4⋊(C3⋊D3)—62no—
215C4⋊(C3⋊D3)—122no—
216C4×C3⋊D3—24no—
217C4×C3⋊D3—64no—
218C4×C3⋊D3—616no—
219C4×C3⋊D3—1216no—
220#634—34no—
221C2×C3×Dic3—34no—
222C2×C3×Dic3—34no—
223C2×C3×Dic3C22×D314yes—
224C2×C3×Dic3C22×D314yes—
225C3×D12—28no—
226C3×D12—68no—
227C3×C4×D3—216no—
228C3×C4×D3—616no—
229#627—316no—
230#626—616no—
231#626—616no—
232#626—128no—
233#626—128no—
234#625—616no—
235#625—128no—
236#550—271noSylow
237C2×C33C22×D411yescharacteristic
238C2×C33C2×#4111yescharacteristic
239C2×C33H(5,𝔽2)11yescharacteristic
240C3⋊(C3⋊D3)—42no—
241C3×C3⋊D3—42no—
242C32×D3—42no—
243C2×C3×D4—34no—
244C2×C3×D4—91no—
245C22×C3×C4—38no—
246C22×C3×C4—92no—
247C2×#67—98no—
248C2×#67—98no—
249C2×#67—98no—
250C2×#67—92no—
251C2×#67—98no—
252C2×#67—92no—
253C2×#67—92no—
254C2×#67—92no—
255#235—188no—
256#235—1832no—
257C2×D12—34no—
258C2×D12—98no—
259C2×D12—92no—
260C2×D12—94no—
261C2×D12—91no—
262C2×D12—98no—
263C2×D12—92no—
264C2×D12—184no—
265C2×D12C2×C3⋊D311yescharacteristic
266C2×C4×D3—38no—
267C2×C4×D3—98no—
268C2×C4×D3—92no—
269C2×C4×D3—188no—
270C2×C4×D3C2×C3⋊D312yes—
271C2×Dic6—94no—
272C2×Dic6—91no—
273C3×C4⋊C4—24no—
274C3×C4⋊C4D3214yes—
275C3×C4⋊C4C2×C3⋊D311yescharacteristic
276C3×C22⋊C4—38no—
277C3×C22⋊C4—92no—
278C3×C42—34no—
279C3×C42—91no—
280(C2×D3)⋊C4—38no—
281(C2×D3)⋊C4—98no—
282(C2×D3)⋊C4—92no—
283(C2×D3)⋊C4—188no—
284(C2×D3)⋊C4C2×C3⋊D312yes—
285C4⋊Dic3—94no—
286C4⋊Dic3—91no—
287Dic3⋊C4—94no—
288Dic3⋊C4—94no—
289Dic3⋊C4—91no—
290Dic3⋊C4—91no—
291C4×Dic3—34no—
292C4×Dic3—94no—
293C4×Dic3—91no—
294C4×Dic3—184no—
295C4×Dic3C2×C3⋊D311yescharacteristic
296C22×C32—24no—
297C22×C32—62no—
298C22×C32—62no—
299C22×C32—62no—
300C22×C32C23×D314yes—
301C22×C32C23×D311yescharacteristic
302C2×C3⋊D3—22no—
303C2×C3⋊D3—22no—
304C2×C3⋊D3—22no—
305C2×C3⋊D3—616no—
306C2×C3⋊D3—68no—
307C2×C3⋊D3—62no—
308C2×C3⋊D3—62no—
309C2×C3⋊D3—62no—
310C2×C3⋊D3—68no—
311C2×C3⋊D3—68no—
312C2×C3⋊D3—128no—
313C2×C3⋊D3—128no—
314C2×C3×D3—28no—
315C2×C3×D3—28no—
316C2×C3×D3—28no—
317C2×C3×D3—68no—
318C2×C3×D3—68no—
319C2×C3×D3—68no—
320C32×C4—22no—
321C32×C4—28no—
322C32×C4—28no—
323C32×C4—32no—
324C32×C4—48no—
325C32×C4—61no—
326C32×C4#23518yescomplemented
327C32×C4#23512yescomplemented
328C3⋊Dic3—21no—
329C3⋊Dic3—32no—
330C3⋊Dic3—38no—
331C3⋊Dic3—68no—
332C3⋊Dic3—61no—
333C3⋊Dic3—64no—
334C3⋊Dic3—124no—
335C3⋊Dic3D3×D412yes—
336C3×Dic3—24no—
337C3×Dic3—38no—
338C3×Dic3—64no—
339C3×Dic3D3×D418yes—
340C2×#41—272no—
341#127—271no—
342#127—271no—
343#127—271no—
344#121—272no—
345#121—272no—
346#121—272no—
347C4×Q8—271no—
348C4×D4—31no—
349C4×D4—91no—
350C4×D4—271no—
351C33#55011yescharacteristic, complemented, Sylow
352C23×C3—38no—
353C23×C3—92no—
354C22×D3—38no—
355C22×D3—98no—
356C22×D3—92no—
357C22×D3—188no—
358C22×D3C22×C3⋊D312yes—
359C3×D4—68no—
360C3×D4—182no—
361C2×C3×C4—24no—
362C2×C3×C4—28no—
363C2×C3×C4—34no—
364C2×C3×C4—34no—
365C2×C3×C4—616no—
366C2×C3×C4—91no—
367C2×C3×C4—91no—
368C2×C3×C4—184no—
369C2×C3×C4C2×D3218yes—
370C2×C3×C4C2×D3214yes—
371C2×C3×C4C22×C3⋊D312yes—
372C2×C3×C4C22×C3⋊D311yescharacteristic
373#67—1816no—
374#67—184no—
375#67—1816no—
376#67—184no—
377#67—362no—
378#67—368no—
379#67—362no—
380#67—368no—
381C2×Dic3—34no—
382C2×Dic3—34no—
383C2×Dic3—94no—
384C2×Dic3—91no—
385C2×Dic3—94no—
386C2×Dic3—91no—
387C2×Dic3—184no—
388C2×Dic3—184no—
389C2×Dic3C22×C3⋊D311yescharacteristic
390C2×Dic3C22×C3⋊D311yescharacteristic
391D12—22no—
392D12—68no—
393D12—1816no—
394D12—1816no—
395D12—184no—
396D12—188no—
397D12—182no—
398D12—368no—
399D12—362no—
400C4×D3—24no—
401C4×D3—616no—
402C4×D3—184no—
403C4×D3—1816no—
404C4×D3—3616no—
405Dic6—916no—
406Dic6—94no—
407C2×C32—24no—
408C2×C32—24no—
409C2×C32—24no—
410C2×C32—122no—
411C2×C32C2×#23514yes—
412C2×C32C2×#23511yescharacteristic
413C2×C32C2×D3×D414yes—
414C2×C32C2×D3×D411yescharacteristic
415C2×C32#100914yes—
416C2×C32#100911yescharacteristic
417C3⋊D3—42no—
418C3⋊D3—1216no—
419C3⋊D3—128no—
420C3⋊D3—122no—
421C3×D3—48no—
422C3×D3—128no—
423#41—548no—
424C2×Q8—271no—
425C2×D4—31no—
426C2×D4—91no—
427C2×D4—272no—
428C2×D4—272no—
429C2×D4—272no—
430C2×D4—272no—
431C2×D4—272no—
432C2×D4—272no—
433C2×D4—271no—
434C22×C4—32no—
435C22×C4—92no—
436C22×C4—272no—
437C4⋊C4—271no—
438C4⋊C4—271no—
439C4⋊C4—271no—
440C4⋊C4D3×C3⋊D311yescharacteristic
441C22⋊C4—32no—
442C22⋊C4—92no—
443C22⋊C4—272no—
444C42—31no—
445C42—91no—
446C42—271no—
447C22×C3—24no—
448C22×C3—68no—
449C22×C3—68no—
450C22×C3—68no—
451C22×C3—182no—
452C22×C3—182no—
453C22×C3—182no—
454C22×C3C22×D3214yes—
455C22×C3C23×C3⋊D311yescharacteristic
456C2×D3—22no—
457C2×D3—22no—
458C2×D3—22no—
459C2×D3—68no—
460C2×D3—68no—
461C2×D3—68no—
462C2×D3—1816no—
463C2×D3—188no—
464C2×D3—188no—
465C2×D3—188no—
466C2×D3—182no—
467C2×D3—182no—
468C2×D3—182no—
469C2×D3—368no—
470C2×D3—368no—
471C3×C4—28no—
472C3×C4—22no—
473C3×C4—28no—
474C3×C4—38no—
475C3×C4—48no—
476C3×C4—64no—
477C3×C4—92no—
478C3×C4—181no—
479C3×C4#379818yescomplemented
480C3×C4#382812yescomplemented
481Dic3—21no—
482Dic3—38no—
483Dic3—64no—
484Dic3—98no—
485Dic3—92no—
486Dic3—188no—
487Dic3—184no—
488Dic3—181no—
489Dic3—364no—
490Dic3D4×C3⋊D312yes—
491C32—24no—
492C32#555314yescomplemented
493C32#555311yescharacteristic, complemented
494C23—32no—
495C23—92no—
496C23—272no—
497Q8—274no—
498D4—62no—
499D4—182no—
500D4—544no—
501D4—544no—
502D4—544no—
503D4—542no—
504D4—1082no—
505D4—1082no—
506D4—1082no—
507C2×C4—31no—
508C2×C4—31no—
509C2×C4—64no—
510C2×C4—91no—
511C2×C4—91no—
512C2×C4—184no—
513C2×C4—271no—
514C2×C4—271no—
515C2×C4—544no—
516C2×C4C2×D3×C3⋊D312yes—
517C2×C4C2×D3×C3⋊D311yescharacteristic
518C2×C3—24no—
519C2×C3—24no—
520C2×C3—24no—
521C2×C3—128no—
522C2×C3—362no—
523C2×C3C2×#379814yes—
524C2×C3D32×D414yes—
525C2×C3#6432014yes—
526C2×C3C2×#382811yescharacteristic
527C2×C3C2×D4×C3⋊D311yescharacteristic
528C2×C3#6436611yescharacteristic
529D3—42no—
530D3—128no—
531D3—3616no—
532D3—362no—
533D3—368no—
534C22—62no—
535C22—62no—
536C22—62no—
537C22—182no—
538C22—182no—
539C22—182no—
540C22—542no—
541C22—542no—
542C22—542no—
543C22C22×D3×C3⋊D311yescharacteristic
544C4—22no—
545C4—32no—
546C4—61no—
547C4—92no—
548C4—181no—
549C4—272no—
550C4—541no—
551C4#8902012yescomplemented
552C3—24no—
553C3#1059452014yescomplemented
554C3#1059602411yescharacteristic, complemented
555C2—122no—
556C2—362no—
557C2—1082no—
558C2C2×#8902011yescharacteristic
559C2D3×D4×C3⋊D311yescharacteristic
560C2#1172459711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#550
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#89020C2semidirect
2. #11723934C2semidirect
3. C3×#64102C2semidirect
4. C32×#893C2semidirect
5. C4×#6293C2semidirect
6. #11723529C2semidirect
7. #11723527C2semidirect
8. #11723524C2semidirect
9. #11723520C2semidirect
10. #11723517C2semidirect
11. #11723514C2semidirect
47. #64102D3semidirect
53. C3×#893D3semidirect
78. #6293D4semidirect
151. C33×C4#41semidirect
172. #893C3⋊D3semidirect
326. C32×C4#235semidirect
327. C32×C4#235semidirect
351. C33#550semidirect
479. C3×C4#3798semidirect
480. C3×C4#3828semidirect
492. C32#5553semidirect
493. C32#5553semidirect
551. C4#89020semidirect
553. C3#10594520semidirect
554. C3#10596024semidirect

Maximal subgroups

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

subgroupindex
C2×#890202
#117239342
C3×#641022
C32×#8932
C4×#62932
#117235292
#117235272
#117235242
#117235202
#117235172
#117235142
#105960243
#105945203