Finite Group Databasebeta

← #13177251

Subgroups — #13177251

Order 1728.

1804 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723971C211yesmaximal, characteristic, complemented
2C3×C4×#638C211yesmaximal, characteristic, complemented
3C2×C32×C4⋊Dic3C211yesmaximal, characteristic, complemented
4#11723632C212yesmaximal, complemented
5#11723618C212yesmaximal, complemented
6#11723526C211yesmaximal, characteristic, complemented
7#11723504C212yesmaximal, complemented
8#11723494C211yesmaximal, characteristic, complemented
9#11723447C211yesmaximal, characteristic, complemented
10#11723411C212yesmaximal, complemented
11#11723404C211yesmaximal, characteristic, complemented
12#10595985—31nomaximal
13#10594843—34nomaximal
14C22×C33×C4C2211yescharacteristic
15C2×#6312C2212yes—
16C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
17C2×C3×#638C2211yescharacteristic
18C2×C3×C4×C3⋊D3C2211yescharacteristic
19C22×C32×Dic3C2212yes—
20C2×#6292C2212yes—
21C2×#6289C2212yes—
22#88859C2212yes—
23C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
24C3×#3757C2211yescharacteristic
25C3×#3752C2211yescharacteristic
26C3×#3750C2211yescharacteristic
27C3×C4×C3⋊Dic3C2211yescharacteristic
28C32×C4⋊Dic3C2211yescharacteristic
29C32×C4⋊Dic3C2211yescharacteristic
30C32×C4⋊Dic3C2212yes—
31#88806C2212yes—
32Dic3⋊(C3⋊Dic3)C2212yes—
33#88804C2212yes—
34#88804C2212yes—
35#88803C2212yes—
36Dic3×C3⋊Dic3C2212yes—
37C2×C32×C4⋊C4—31no—
38#64139—31no—
39#64139—32no—
40#64139—34no—
41#64139—64no—
42C4×#638—32no—
43C4×#638D311yescharacteristic, complemented
44#64107—32no—
45#64104—31no—
46#64103—32no—
47#64102—31no—
48#64101—31no—
49#64099—31no—
50C3×C4×#67—34no—
51C2×C3×C4⋊Dic3D314yescomplemented
52#63978—38no—
53#63952—38no—
54#63909—34no—
55#63894—34no—
56#63869—34no—
57#63854—38no—
58#63848—38no—
59#63841—34no—
60C23×C33C2311yescharacteristic
61C22×C3⋊(C3⋊D3)C2312yes—
62C22×C3×C3⋊D3C2311yescharacteristic
63C2×C33×C4—22no—
64C2×C33×C4C2311yescharacteristic
65C2×C33×C4C2311yescharacteristic
66#6312—24no—
67C2×C3⋊(C3⋊Dic3)C2312yes—
68C4⋊(C3⋊(C3⋊D3))—22no—
69C3×#638—22no—
70C2×C3×C3⋊Dic3C2311yescharacteristic
71C2×C3×C3⋊Dic3C2311yescharacteristic
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C3×C4×C3⋊D3—22no—
74C2×C32×Dic3—42no—
75C2×C32×Dic3C2312yes—
76C2×C32×Dic3C2312yes—
77#6292—42no—
78#6289—42no—
79#5565—91no—
80#5514—94no—
81C22×C32×C4—24no—
82C22×C32×C4—32no—
83C22×C32×C4C2×D314yes—
84C22×C32×C4C2×D311yescharacteristic
85C2×#638—32no—
86C2×#638—38no—
87C2×#638—68no—
88C2×#638C2×D311yescharacteristic
89C2×C4⋊(C3⋊D3)—31no—
90C2×C4⋊(C3⋊D3)—34no—
91C2×C4⋊(C3⋊D3)—32no—
92C2×C4⋊(C3⋊D3)—64no—
93C2×C4×C3⋊D3—32no—
94C2×C4×C3⋊D3C2×D311yescharacteristic
95C2×C3×#67—34no—
96C22×C3×Dic3C2×D318yes—
97C2×C3×C4×D3—34no—
98C2×#626—38no—
99C2×#624—38no—
100C32×C4⋊C4—31no—
101C32×C4⋊C4—31no—
102C32×C4⋊C4—32no—
103#3757C2×D311yescharacteristic
104#3752—32no—
105#3752—32no—
106#3752—38no—
107#3752—32no—
108#3752—32no—
109#3752—68no—
110#3752C2×D311yescharacteristic
111C4⋊(C3⋊Dic3)—31no—
112C4⋊(C3⋊Dic3)—34no—
113C4⋊(C3⋊Dic3)—32no—
114C4⋊(C3⋊Dic3)—64no—
115#3750—32no—
116#3750C2×D311yescharacteristic
117C4×C3⋊Dic3—32no—
118C4×C3⋊Dic3C2×D311yescharacteristic
119C3×#217—34no—
120C3×(C2×D3)⋊C4—34no—
121C3×C4⋊Dic3C2×D314yes—
122C3×C4⋊Dic3C2×D318yes—
123C3×C4⋊Dic3C2×D314yes—
124C3×Dic3⋊C4—34no—
125C3×C4×Dic3—34no—
126#3724—38no—
127Dic3⋊Dic3—38no—
128#3722—38no—
129#3722—38no—
130(C2×D3)⋊Dic3—38no—
131Dic32—38no—
132C22×C33—21no—
133C22×C33—21no—
134C22×C33#4112yes—
135C22×C33C2411yescharacteristic
136C2×C3⋊(C3⋊D3)—22no—
137C2×C3⋊(C3⋊D3)—22no—
138C2×C3⋊(C3⋊D3)—22no—
139C2×C3×C3⋊D3—21no—
140C2×C3×C3⋊D3—21no—
141C2×C3×C3⋊D3—21no—
142C33×C4—21no—
143C33×C4#4112yes—
144C3⋊(C3⋊Dic3)—22no—
145C3×C3⋊Dic3—21no—
146C3×C3⋊Dic3—21no—
147C3×C3⋊Dic3—21no—
148C32×Dic3—22no—
149C32×Dic3—22no—
150C3×C4×D4—91no—
151C2×C3×C4⋊C4—34no—
152#939—92no—
153#930—98no—
154#930—94no—
155#930—91no—
156#930—91no—
157#930—184no—
158C4×#67—34no—
159C4×#67—98no—
160#926—91no—
161C2×C4⋊Dic3C3⋊D311yescharacteristic, complemented
162#898—98no—
163#898—92no—
164#895—94no—
165#894—98no—
166#893—94no—
167#892—94no—
168#890—94no—
169#883—92no—
170#67⋊C4—92no—
171C4×D12—91no—
172#870—91no—
173C23×C32—24no—
174C23×C32C22×D314yes—
175C23×C32C22×D311yescharacteristic
176C22×C3⋊D3—32no—
177C22×C3⋊D3—38no—
178C22×C3⋊D3—68no—
179C22×C3⋊D3C22×D311yescharacteristic
180C22×C3×D3—34no—
181C2×C32×C4—22no—
182C2×C32×C4—24no—
183C2×C32×C4—24no—
184C2×C32×C4—28no—
185C2×C32×C4—32no—
186C2×C32×C4—32no—
187C2×C32×C4—48no—
188C2×C32×C4—122no—
189C2×C32×C4C22×D314yes—
190C2×C32×C4C22×D314yes—
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C22×D311yescharacteristic
193#638—22no—
194#638—64no—
195#638—616no—
196#638—1216no—
197C2×C3⋊Dic3—32no—
198C2×C3⋊Dic3—38no—
199C2×C3⋊Dic3—68no—
200C2×C3⋊Dic3C22×D311yescharacteristic
201C2×C3⋊Dic3C22×D311yescharacteristic
202C2×C3⋊Dic3C22×D311yescharacteristic
203C4⋊(C3⋊D3)—62no—
204C4⋊(C3⋊D3)—68no—
205C4⋊(C3⋊D3)—128no—
206C4⋊(C3⋊D3)—122no—
207C4×C3⋊D3—22no—
208C4×C3⋊D3—122no—
209C3×#67—68no—
210C2×C3×Dic3—34no—
211C2×C3×Dic3—34no—
212C2×C3×Dic3—34no—
213C2×C3×Dic3—48no—
214C2×C3×Dic3C22×D318yes—
215C2×C3×Dic3C22×D318yes—
216C3×C4×D3—68no—
217#626—128no—
218#624—128no—
219#553—271noSylow
220C2×C33—22no—
221C2×C33C2×#4111yescharacteristic
222C2×C33C2×#4111yescharacteristic
223C2×C33H(5,𝔽2)11yescharacteristic
224C3⋊(C3⋊D3)—42no—
225C3×C3⋊D3—41no—
226C2×C3×D4—91no—
227C22×C3×C4—24no—
228C22×C3×C4—38no—
229C22×C3×C4—91no—
230C22×C3×C4D3214yes—
231C22×C3×C4C2×C3⋊D311yescharacteristic
232C2×#67—34no—
233C2×#67—98no—
234C2×#67—92no—
235C2×#67—92no—
236C2×#67—188no—
237C22×Dic3C2×C3⋊D312yes—
238C2×D12—94no—
239C2×D12—91no—
240C2×D12—98no—
241C2×D12—184no—
242C2×C4×D3—34no—
243C2×C4×D3—98no—
244C2×C4×D3—92no—
245C3×C4⋊C4—34no—
246C3×C4⋊C4—34no—
247C3×C4⋊C4—38no—
248C3×C4⋊C4—91no—
249C3×C22⋊C4—91no—
250C3×C22⋊C4—91no—
251C3×C42—91no—
252#217—34no—
253#217—92no—
254(C2×D3)⋊C4—34no—
255(C2×D3)⋊C4—98no—
256(C2×D3)⋊C4—98no—
257(C2×D3)⋊C4—98no—
258(C2×D3)⋊C4—98no—
259(C2×D3)⋊C4—92no—
260(C2×D3)⋊C4—92no—
261(C2×D3)⋊C4—92no—
262(C2×D3)⋊C4—188no—
263C4⋊Dic3—98no—
264C4⋊Dic3—94no—
265C4⋊Dic3—91no—
266C4⋊Dic3—184no—
267C4⋊Dic3C2×C3⋊D311yescharacteristic
268C4⋊Dic3C2×C3⋊D311yescharacteristic
269C4⋊Dic3C2×C3⋊D312yes—
270Dic3⋊C4—34no—
271Dic3⋊C4—98no—
272Dic3⋊C4—92no—
273Dic3⋊C4—92no—
274C4×Dic3—34no—
275C4×Dic3—98no—
276C4×Dic3—92no—
277C22×C32—21no—
278C22×C32—21no—
279C22×C32—24no—
280C22×C32—28no—
281C22×C32—24no—
282C22×C32—24no—
283C22×C32—44no—
284C22×C32—44no—
285C22×C32#23512yes—
286C22×C32#23918yes—
287C22×C32C23×D314yes—
288C22×C32C23×D311yescharacteristic
289C2×C3⋊D3—21no—
290C2×C3⋊D3—21no—
291C2×C3⋊D3—21no—
292C2×C3⋊D3—62no—
293C2×C3⋊D3—62no—
294C2×C3⋊D3—62no—
295C2×C3⋊D3—68no—
296C2×C3⋊D3—68no—
297C2×C3⋊D3—68no—
298C2×C3⋊D3—128no—
299C2×C3⋊D3—128no—
300C2×C3⋊D3—128no—
301C2×C3×D3—64no—
302C2×C3×D3—64no—
303C2×C3×D3—64no—
304C32×C4—21no—
305C32×C4—28no—
306C32×C4—24no—
307C32×C4—44no—
308C32×C4—62no—
309C32×C4—62no—
310C32×C4#23518yes—
311C32×C4#23712yes—
312C3⋊Dic3—21no—
313C3⋊Dic3—21no—
314C3⋊Dic3—21no—
315C3⋊Dic3—62no—
316C3⋊Dic3—68no—
317C3⋊Dic3—128no—
318C3×Dic3—28no—
319C3×Dic3—28no—
320C3×Dic3—64no—
321C3×Dic3—64no—
322C3×Dic3—64no—
323#126—272no—
324#125—271no—
325#123—272no—
326#121—271no—
327#121—272no—
328#121—271no—
329C4×D4—91no—
330C4×D4—271no—
331C4×D4—272no—
332#117—271no—
333C2×C4⋊C4—31no—
334C33#55311yescharacteristic, complemented, Sylow
335C23×C3—24no—
336C23×C3—91no—
337C23×C3C2×D3214yes—
338C23×C3C22×C3⋊D311yescharacteristic
339C22×D3—34no—
340C22×D3—98no—
341C22×D3—92no—
342C22×D3—188no—
343C3×D4—182no—
344C2×C3×C4—28no—
345C2×C3×C4—22no—
346C2×C3×C4—24no—
347C2×C3×C4—24no—
348C2×C3×C4—38no—
349C2×C3×C4—38no—
350C2×C3×C4—48no—
351C2×C3×C4—91no—
352C2×C3×C4—91no—
353C2×C3×C4—91no—
354C2×C3×C4—128no—
355C2×C3×C4—182no—
356C2×C3×C4C2×D3214yes—
357C2×C3×C4C2×D3214yes—
358C2×C3×C4C22×C3⋊D311yescharacteristic
359C2×C3×C4C22×C3⋊D311yescharacteristic
360#67—68no—
361#67—1816no—
362#67—184no—
363#67—3616no—
364#67—362no—
365C2×Dic3—34no—
366C2×Dic3—34no—
367C2×Dic3—34no—
368C2×Dic3—42no—
369C2×Dic3—98no—
370C2×Dic3—92no—
371C2×Dic3—188no—
372C2×Dic3C22×C3⋊D312yes—
373C2×Dic3C22×C3⋊D312yes—
374D12—188no—
375D12—182no—
376D12—368no—
377D12—368no—
378C4×D3—68no—
379C4×D3—362no—
380C4×D3—368no—
381C2×C32—22no—
382C2×C32—24no—
383C2×C32—24no—
384C2×C32—24no—
385C2×C32—28no—
386C2×C32—48no—
387C2×C32C2×#23514yes—
388C2×C32C2×#23511yescharacteristic
389C2×C32C2×#23711yescharacteristic
390C2×C32#100414yes—
391C2×C32C2×#23914yes—
392C2×C32#100911yescharacteristic
393C3⋊D3—41no—
394C3⋊D3—122no—
395C3⋊D3—128no—
396C3⋊D3—248no—
397C3×D3—124no—
398C2×D4—91no—
399C2×D4—272no—
400C2×D4—272no—
401C2×D4—271no—
402C22×C4—32no—
403C22×C4—91no—
404C22×C4—272no—
405C22×C4D3×C3⋊D311yescharacteristic
406C4⋊C4—31no—
407C4⋊C4—31no—
408C4⋊C4—32no—
409C4⋊C4—91no—
410C4⋊C4—272no—
411C4⋊C4—272no—
412C4⋊C4—271no—
413C22⋊C4—91no—
414C22⋊C4—91no—
415C22⋊C4—272no—
416C22⋊C4—272no—
417C22⋊C4—272no—
418C22⋊C4—272no—
419C42—91no—
420C42—272no—
421C22×C3—24no—
422C22×C3—24no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—24no—
426C22×C3—28no—
427C22×C3—44no—
428C22×C3—44no—
429C22×C3—181no—
430C22×C3—181no—
431C22×C3—181no—
432C22×C3#379918yes—
433C22×C3#383212yes—
434C22×C3C22×D3214yes—
435C22×C3C23×C3⋊D311yescharacteristic
436C2×D3—64no—
437C2×D3—64no—
438C2×D3—64no—
439C2×D3—188no—
440C2×D3—188no—
441C2×D3—188no—
442C2×D3—182no—
443C2×D3—182no—
444C2×D3—182no—
445C2×D3—368no—
446C2×D3—368no—
447C2×D3—368no—
448C3×C4—24no—
449C3×C4—21no—
450C3×C4—28no—
451C3×C4—44no—
452C3×C4—68no—
453C3×C4—68no—
454C3×C4—181no—
455C3×C4—181no—
456C3×C4—181no—
457C3×C4#380418yes—
458C3×C4#382812yes—
459Dic3—22no—
460Dic3—22no—
461Dic3—64no—
462Dic3—64no—
463Dic3—64no—
464Dic3—188no—
465Dic3—182no—
466Dic3—368no—
467C32—24no—
468C32#551414yescomplemented
469C32#556511yescharacteristic, complemented
470C23—91no—
471C23—272no—
472C23C2×D3×C3⋊D311yescharacteristic
473D4—182no—
474D4—544no—
475D4—542no—
476D4—1082no—
477C2×C4—22no—
478C2×C4—32no—
479C2×C4—32no—
480C2×C4—91no—
481C2×C4—91no—
482C2×C4—91no—
483C2×C4—122no—
484C2×C4—182no—
485C2×C4—272no—
486C2×C4—1082no—
487C2×C4C2×D3×C3⋊D311yescharacteristic
488C2×C4C2×D3×C3⋊D311yescharacteristic
489C2×C3—28no—
490C2×C3—22no—
491C2×C3—24no—
492C2×C3—24no—
493C2×C3—24no—
494C2×C3—48no—
495C2×C3—361no—
496C2×C3C2×#379914yes—
497C2×C3#6430714yes—
498C2×C3C2×#380414yes—
499C2×C3C2×#382811yescharacteristic
500C2×C3#6436111yescharacteristic
501C2×C3C2×#383211yescharacteristic
502D3—124no—
503D3—368no—
504D3—362no—
505D3—728no—
506C22—21no—
507C22—21no—
508C22—181no—
509C22—181no—
510C22—181no—
511C22—542no—
512C22—542no—
513C22—542no—
514C22#8901812yes—
515C22C22×D3×C3⋊D311yescharacteristic
516C4—21no—
517C4—62no—
518C4—62no—
519C4—181no—
520C4—181no—
521C4—181no—
522C4—542no—
523C4#8903112yes—
524C3—24no—
525C3#1059484314yescomplemented
526C3#1059598511yescharacteristic, complemented
527C2—22no—
528C2—361no—
529C2—1082no—
530C2C2×#8901811yescharacteristic
531C2#1172457911yescharacteristic
532C2C2×#8903111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC22×C33×C4C22
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#553
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #11723971C2semidirect
2. C3×C4×#638C2semidirect
3. C2×C32×C4⋊Dic3C2semidirect
4. #11723632C2semidirect
5. #11723618C2semidirect
6. #11723526C2semidirect
7. #11723504C2semidirect
8. #11723494C2semidirect
9. #11723447C2semidirect
10. #11723411C2semidirect
11. #11723404C2semidirect
43. C4×#638D3semidirect
51. C2×C3×C4⋊Dic3D3semidirect
161. C2×C4⋊Dic3C3⋊D3semidirect
334. C33#553semidirect
468. C32#5514semidirect
469. C32#5565semidirect
525. C3#10594843semidirect
526. C3#10595985semidirect

Maximal subgroups

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

subgroupindex
#117239712
C3×C4×#6382
C2×C32×C4⋊Dic32
#117236322
#117236182
#117235262
#117235042
#117234942
#117234472
#117234112
#117234042
#105959853
#105948433