Finite Group Databasebeta

← #13154256

Subgroups — #13154256

Order 1728.

1309 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×#126C211yesmaximal, characteristic, complemented
2#11721766C211yesmaximal, characteristic, complemented
3#11721763C211yesmaximal, characteristic, complemented
4#11721762C211yesmaximal, characteristic, complemented
5#11721761C211yesmaximal, characteristic, complemented
6C4⋊C4×#272C211yesmaximal, characteristic, complemented
7#11721757C211yesmaximal, characteristic, complemented
8#11721742C211yesmaximal, characteristic, complemented
9#11721740C211yesmaximal, characteristic, complemented
10#11721740C211yesmaximal, characteristic, complemented
11#11721740C211yesmaximal, characteristic, complemented
12#11721737C211yesmaximal, characteristic, complemented
13#11721737C211yesmaximal, characteristic, complemented
14C4×C4⋊#272C211yesmaximal, characteristic, complemented
15#11721718C211yesmaximal, characteristic, complemented
16C3×#5709—34nomaximal
17C2×#6235C2211yescharacteristic
18C2×#6235C2211yescharacteristic
19C2×#6235C2211yescharacteristic
20C22×#1084C2211yescharacteristic
21C2×C4⋊#272C2211yescharacteristic
22C2×C4⋊#272C2211yescharacteristic
23C2×C4⋊#272C2211yescharacteristic
24C2×C4×#272C2211yescharacteristic
25C2×C4×#272C2211yescharacteristic
26C2×C4×#272C2211yescharacteristic
27C2×C4×#272C2211yescharacteristic
28C2×C4×#272C2211yescharacteristic
29C4⋊C4×H(3,𝔽3)C2211yescharacteristic
30C4⋊C4×H(3,𝔽3)C2211yescharacteristic
31C4⋊C4×H(3,𝔽3)C2211yescharacteristic
32C22⋊C4×H(3,𝔽3)C2211yescharacteristic
33C22⋊C4×H(3,𝔽3)C2211yescharacteristic
34C22⋊C4×H(3,𝔽3)C2211yescharacteristic
35C42×H(3,𝔽3)C2211yescharacteristic
36#88542C2211yescharacteristic
37#88542C2211yescharacteristic
38#88542C2211yescharacteristic
39#88542C2211yescharacteristic
40#88542C2211yescharacteristic
41#88542C2211yescharacteristic
42#88542C2211yescharacteristic
43C4⋊#1084C2211yescharacteristic
44C4⋊#1084C2211yescharacteristic
45#88540C2211yescharacteristic
46#88540C2211yescharacteristic
47#88540C2211yescharacteristic
48#88540C2211yescharacteristic
49#88540C2211yescharacteristic
50C4×#1084C2211yescharacteristic
51C4×#1084C2211yescharacteristic
52C32×#126D314yescomplemented
53C3×#898—34no—
54C3×#895—34no—
55C3×#894—34no—
56C3×#893—34no—
57C3×D3×C4⋊C4—34no—
58C3×#889—34no—
59C3×#886—34no—
60C3×#884—34no—
61C3×#884—34no—
62C3×#884—34no—
63C3×#67⋊C4—34no—
64C3×#67⋊C4—34no—
65C3×C4×D12—34no—
66C3×#872—34no—
67C23×H(3,𝔽3)C2311yescharacteristic
68C22×#272C2311yescharacteristic
69C22×#272C2311yescharacteristic
70C22×#272C2311yescharacteristic
71C2×C4×H(3,𝔽3)C2311yescharacteristic
72C2×C4×H(3,𝔽3)C2311yescharacteristic
73C2×C4×H(3,𝔽3)C2311yescharacteristic
74C2×C4×H(3,𝔽3)C2311yescharacteristic
75C2×C4×H(3,𝔽3)C2311yescharacteristic
76C2×C4×H(3,𝔽3)C2311yescharacteristic
77#6235—22no—
78#6235—22no—
79#6235—41no—
80C2×#1084—41no—
81C2×#1084C2311yescharacteristic
82C2×#1084C2311yescharacteristic
83C2×#1084C2311yescharacteristic
84C2×#1084C2311yescharacteristic
85C2×#1084C2311yescharacteristic
86C4⋊#272—22no—
87C4⋊#272—22no—
88C4⋊#272—41no—
89C4×#272—22no—
90C4×#272—22no—
91C4×#272—41no—
92C4×#272—41no—
93C4×#272—41no—
94C3×#553—91no—
95#5709—34no—
96C2×C3×#67—34no—
97C2×C3×#67—34no—
98C2×C3×#67—34no—
99C22×C3×Dic3—34no—
100C2×C3×D12—34no—
101C2×C3×D12—34no—
102C2×C3×D12—34no—
103C2×C3×C4×D3—34no—
104C2×C3×C4×D3—34no—
105C2×C3×C4×D3—34no—
106C2×C3×C4×D3—34no—
107C2×C3×C4×D3—34no—
108C32×C4⋊C4C2×D314yes—
109C32×C4⋊C4C2×D314yes—
110C32×C4⋊C4C2×D314yes—
111C32×C22⋊C4C2×D314yes—
112C32×C22⋊C4C2×D314yes—
113C32×C22⋊C4C2×D314yes—
114C32×C42C2×D314yes—
115C3×(C2×D3)⋊C4—34no—
116C3×(C2×D3)⋊C4—34no—
117C3×(C2×D3)⋊C4—34no—
118C3×(C2×D3)⋊C4—34no—
119C3×(C2×D3)⋊C4—34no—
120C3×(C2×D3)⋊C4—34no—
121C3×(C2×D3)⋊C4—34no—
122C3×C4⋊Dic3—34no—
123C3×C4⋊Dic3—34no—
124C3×Dic3⋊C4—34no—
125C3×Dic3⋊C4—34no—
126C3×Dic3⋊C4—34no—
127C3×Dic3⋊C4—34no—
128C3×Dic3⋊C4—34no—
129C3×C4×Dic3—34no—
130C3×C4×Dic3—34no—
131C22×H(3,𝔽3)—21no—
132C22×H(3,𝔽3)—21no—
133C22×H(3,𝔽3)—21no—
134C22×H(3,𝔽3)C2411yescharacteristic
135C2×#272—21no—
136C2×#272—21no—
137C2×#272—21no—
138C2×#272—21no—
139C2×#272—21no—
140C2×#272—21no—
141C2×#272—21no—
142C2×#272—21no—
143C2×#272#4112yes—
144C4×H(3,𝔽3)—21no—
145C4×H(3,𝔽3)—21no—
146C4×H(3,𝔽3)—21no—
147C4×H(3,𝔽3)—21no—
148C4×H(3,𝔽3)—21no—
149C4×H(3,𝔽3)—21no—
150#1084—21no—
151#1084—21no—
152#1084—21no—
153#1084—21no—
154#1084#4112yes—
155C3×#126—34no—
156C3×#126—91no—
157C3×#126C3⋊D311yescharacteristic
158C3×#125—91no—
159C3×#123—91no—
160C3×#123—91no—
161C3×#121—91no—
162C3×#121—91no—
163C3×#121—91no—
164C3×#121—91no—
165C3×C4×D4—91no—
166C3×C4×D4—91no—
167C3×C4×D4—91no—
168C3×C4×D4—91no—
169C3×#117—91no—
170C2×C3×C4⋊C4—91no—
171#898—34no—
172#895—34no—
173#894—34no—
174#893—34no—
175D3×C4⋊C4—34no—
176#889—34no—
177#886—34no—
178#884—34no—
179#884—34no—
180#884—34no—
181#67⋊C4—34no—
182#67⋊C4—34no—
183C4×D12—34no—
184#872—34no—
185C23×C32C22×D314yes—
186C22×C3×D3—34no—
187C22×C3×D3—34no—
188C22×C3×D3—34no—
189C2×C32×C4C22×D314yes—
190C2×C32×C4C22×D314yes—
191C2×C32×C4C22×D314yes—
192C2×C32×C4C22×D314yes—
193C2×C32×C4C22×D314yes—
194C2×C32×C4C22×D314yes—
195C3×#67—68no—
196C3×#67—68no—
197C3×#67—124no—
198C2×C3×Dic3—34no—
199C2×C3×Dic3—34no—
200C2×C3×Dic3—34no—
201C2×C3×Dic3—34no—
202C2×C3×Dic3—34no—
203C2×C3×Dic3—124no—
204C3×D12—68no—
205C3×D12—68no—
206C3×D12—124no—
207C3×C4×D3—68no—
208C3×C4×D3—68no—
209C3×C4×D3—124no—
210C3×C4×D3—124no—
211C3×C4×D3—124no—
212#553—91noSylow
213C2×H(3,𝔽3)—41no—
214C2×H(3,𝔽3)C2×#4111yescharacteristic
215C2×H(3,𝔽3)C2×#4111yescharacteristic
216C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
217#272—22no—
218#272—41no—
219#272—41no—
220C2×C3×D4—91no—
221C2×C3×D4—91no—
222C2×C3×D4—91no—
223C2×C3×D4—91no—
224C2×C3×D4—91no—
225C2×C3×D4—91no—
226C22×C3×C4—91no—
227C22×C3×C4—91no—
228C22×C3×C4—91no—
229C22×C3×C4—91no—
230C22×C3×C4—91no—
231C22×C3×C4—91no—
232C2×#67—34no—
233C2×#67—34no—
234C2×#67—34no—
235C22×Dic3—34no—
236C2×D12—34no—
237C2×D12—34no—
238C2×D12—34no—
239C2×C4×D3—34no—
240C2×C4×D3—34no—
241C2×C4×D3—34no—
242C2×C4×D3—34no—
243C2×C4×D3—34no—
244C3×C4⋊C4—34no—
245C3×C4⋊C4—34no—
246C3×C4⋊C4—34no—
247C3×C4⋊C4—91no—
248C3×C4⋊C4—91no—
249C3×C4⋊C4—91no—
250C3×C4⋊C4—91no—
251C3×C4⋊C4—91no—
252C3×C4⋊C4—91no—
253C3×C4⋊C4—91no—
254C3×C4⋊C4C2×C3⋊D311yescharacteristic
255C3×C4⋊C4C2×C3⋊D311yescharacteristic
256C3×C4⋊C4C2×C3⋊D311yescharacteristic
257C3×C22⋊C4—34no—
258C3×C22⋊C4—34no—
259C3×C22⋊C4—34no—
260C3×C22⋊C4—91no—
261C3×C22⋊C4—91no—
262C3×C22⋊C4—91no—
263C3×C22⋊C4—91no—
264C3×C22⋊C4—91no—
265C3×C22⋊C4—91no—
266C3×C22⋊C4—91no—
267C3×C22⋊C4C2×C3⋊D311yescharacteristic
268C3×C22⋊C4C2×C3⋊D311yescharacteristic
269C3×C22⋊C4C2×C3⋊D311yescharacteristic
270C3×C42—34no—
271C3×C42—91no—
272C3×C42—91no—
273C3×C42C2×C3⋊D311yescharacteristic
274(C2×D3)⋊C4—34no—
275(C2×D3)⋊C4—34no—
276(C2×D3)⋊C4—34no—
277(C2×D3)⋊C4—34no—
278(C2×D3)⋊C4—34no—
279(C2×D3)⋊C4—34no—
280(C2×D3)⋊C4—34no—
281C4⋊Dic3—34no—
282C4⋊Dic3—34no—
283Dic3⋊C4—34no—
284Dic3⋊C4—34no—
285Dic3⋊C4—34no—
286Dic3⋊C4—34no—
287Dic3⋊C4—34no—
288C4×Dic3—34no—
289C4×Dic3—34no—
290C22×C32—24no—
291C22×C32—24no—
292C22×C32—24no—
293C22×C32C23×D314yes—
294C2×C3×D3—38no—
295C2×C3×D3—64no—
296C2×C3×D3—64no—
297C2×C3×D3—64no—
298C2×C3×D3—64no—
299C2×C3×D3—64no—
300C2×C3×D3—64no—
301C2×C3×D3—64no—
302C2×C3×D3—64no—
303C32×C4—24no—
304C32×C4—24no—
305C32×C4—24no—
306C32×C4—24no—
307C32×C4—24no—
308C32×C4—24no—
309C3×Dic3—38no—
310C3×Dic3—64no—
311C3×Dic3—64no—
312C3×Dic3—64no—
313C3×Dic3—64no—
314#126—91no—
315#126#27211yescharacteristic, complemented
316#125—91no—
317#123—91no—
318#123—91no—
319#121—91no—
320#121—91no—
321#121—91no—
322#121—91no—
323C4×D4—91no—
324C4×D4—91no—
325C4×D4—91no—
326C4×D4—91no—
327#117—91no—
328C2×C4⋊C4—91no—
329H(3,𝔽3)#55311yescharacteristic, complemented, Sylow
330C23×C3—34no—
331C23×C3—91no—
332C23×C3—91no—
333C23×C3—91no—
334C23×C3C22×C3⋊D311yescharacteristic
335C22×D3—34no—
336C22×D3—34no—
337C22×D3—34no—
338C3×D4—182no—
339C3×D4—182no—
340C3×D4—182no—
341C3×D4—182no—
342C3×D4—361no—
343C3×D4—361no—
344C2×C3×C4—34no—
345C2×C3×C4—34no—
346C2×C3×C4—34no—
347C2×C3×C4—34no—
348C2×C3×C4—34no—
349C2×C3×C4—34no—
350C2×C3×C4—91no—
351C2×C3×C4—91no—
352C2×C3×C4—91no—
353C2×C3×C4—91no—
354C2×C3×C4—91no—
355C2×C3×C4—182no—
356C2×C3×C4—182no—
357C2×C3×C4—361no—
358C2×C3×C4—361no—
359C2×C3×C4—361no—
360C2×C3×C4—361no—
361C2×C3×C4C22×C3⋊D311yescharacteristic
362C2×C3×C4C22×C3⋊D311yescharacteristic
363C2×C3×C4C22×C3⋊D311yescharacteristic
364C2×C3×C4C22×C3⋊D311yescharacteristic
365C2×C3×C4C22×C3⋊D311yescharacteristic
366C2×C3×C4C22×C3⋊D311yescharacteristic
367#67—68no—
368#67—68no—
369#67—124no—
370C2×Dic3—34no—
371C2×Dic3—34no—
372C2×Dic3—34no—
373C2×Dic3—34no—
374C2×Dic3—34no—
375C2×Dic3—124no—
376D12—68no—
377D12—68no—
378D12—124no—
379C4×D3—68no—
380C4×D3—68no—
381C4×D3—124no—
382C4×D3—124no—
383C4×D3—124no—
384C2×C32—44no—
385C2×C32D3×#4114yes—
386C2×C32D3×#4114yes—
387C2×C32#100914yes—
388C3×D3—68no—
389C3×D3—124no—
390C3×D3—124no—
391C2×D4—91no—
392C2×D4—91no—
393C2×D4—91no—
394C2×D4—91no—
395C2×D4—91no—
396C2×D4—91no—
397C22×C4—91no—
398C22×C4—91no—
399C22×C4—91no—
400C22×C4—91no—
401C22×C4—91no—
402C22×C4—91no—
403C4⋊C4—91no—
404C4⋊C4—91no—
405C4⋊C4—91no—
406C4⋊C4—91no—
407C4⋊C4—91no—
408C4⋊C4—91no—
409C4⋊C4—91no—
410C4⋊C4C2×#27211yescharacteristic
411C4⋊C4C2×#27211yescharacteristic
412C4⋊C4C2×#27211yescharacteristic
413C22⋊C4—91no—
414C22⋊C4—91no—
415C22⋊C4—91no—
416C22⋊C4—91no—
417C22⋊C4—91no—
418C22⋊C4—91no—
419C22⋊C4—91no—
420C22⋊C4C2×#27211yescharacteristic
421C22⋊C4C2×#27211yescharacteristic
422C22⋊C4C2×#27211yescharacteristic
423C42—91no—
424C42—91no—
425C42C2×#27211yescharacteristic
426C22×C3—21no—
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—34no—
430C22×C3—64no—
431C22×C3—64no—
432C22×C3—64no—
433C22×C3—92no—
434C22×C3—181no—
435C22×C3—181no—
436C22×C3—181no—
437C22×C3—181no—
438C22×C3—181no—
439C22×C3—181no—
440C22×C3—181no—
441C22×C3—181no—
442C22×C3C23×C3⋊D311yescharacteristic
443C2×D3—38no—
444C2×D3—64no—
445C2×D3—64no—
446C2×D3—64no—
447C2×D3—64no—
448C2×D3—64no—
449C2×D3—64no—
450C2×D3—64no—
451C2×D3—64no—
452C3×C4—21no—
453C3×C4—21no—
454C3×C4—21no—
455C3×C4—21no—
456C3×C4—21no—
457C3×C4—21no—
458C3×C4—64no—
459C3×C4—64no—
460C3×C4—64no—
461C3×C4—64no—
462C3×C4—64no—
463C3×C4—64no—
464C3×C4—92no—
465C3×C4—181no—
466C3×C4—181no—
467C3×C4—181no—
468C3×C4—181no—
469Dic3—38no—
470Dic3—64no—
471Dic3—64no—
472Dic3—64no—
473Dic3—64no—
474C32#570914yescomplemented
475C23—91no—
476C23—91no—
477C23—91no—
478C23C22×#27211yescharacteristic
479D4—182no—
480D4—182no—
481D4—182no—
482D4—182no—
483D4—361no—
484D4—361no—
485C2×C4—91no—
486C2×C4—91no—
487C2×C4—91no—
488C2×C4—91no—
489C2×C4—91no—
490C2×C4—182no—
491C2×C4—182no—
492C2×C4—361no—
493C2×C4—361no—
494C2×C4—361no—
495C2×C4—361no—
496C2×C4C22×#27211yescharacteristic
497C2×C4C22×#27211yescharacteristic
498C2×C4C22×#27211yescharacteristic
499C2×C4C22×#27211yescharacteristic
500C2×C4C22×#27211yescharacteristic
501C2×C4C22×#27211yescharacteristic
502C2×C3—34no—
503C2×C3—34no—
504C2×C3—34no—
505C2×C3—41no—
506C2×C3—124no—
507C2×C3—182no—
508C2×C3—361no—
509C2×C3—361no—
510C2×C3#41×C3⋊D311yescharacteristic
511C2×C3#41×C3⋊D311yescharacteristic
512C2×C3#6436611yescharacteristic
513D3—68no—
514D3—124no—
515D3—124no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—92no—
520C22—181no—
521C22—181no—
522C22—181no—
523C22—181no—
524C22—181no—
525C22—181no—
526C22—181no—
527C22—181no—
528C22C23×#27211yescharacteristic
529C4—21no—
530C4—21no—
531C4—21no—
532C4—21no—
533C4—21no—
534C4—21no—
535C4—92no—
536C4—181no—
537C4—181no—
538C4—181no—
539C4—181no—
540C3—34no—
541C3#1059618011yescharacteristic
542C2—41no—
543C2—182no—
544C2—361no—
545C2—361no—
546C2#41×#27211yescharacteristic
547C2#41×#27211yescharacteristic
548C2#1172267511yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×#126C2
hypercenterC3×#126C3⋊D3
hypocenterH(3,𝔽3)#553
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×#126

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#126C2semidirect
2. #11721766C2semidirect
3. #11721763C2semidirect
4. #11721762C2semidirect
5. #11721761C2semidirect
6. C4⋊C4×#272C2semidirect
7. #11721757C2semidirect
8. #11721742C2semidirect
9. #11721740C2semidirect
10. #11721740C2semidirect
11. #11721740C2semidirect
12. #11721737C2semidirect
13. #11721737C2semidirect
14. C4×C4⋊#272C2semidirect
15. #11721718C2semidirect
52. C32×#126D3semidirect
315. #126#272semidirect
329. H(3,𝔽3)#553semidirect
474. C32#5709semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1262
#117217662
#117217632
#117217622
#117217612
C4⋊C4×#2722
#117217572
#117217422
#117217402
#117217372
C4×C4⋊#2722
#117217182
C3×#57093