Finite Group Databasebeta

← #13154252

Subgroups — #13154252

Order 1728.

1364 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×#126C211yesmaximal, characteristic, complemented
2#11721765C211yesmaximal, characteristic, complemented
3#11721764C211yesmaximal, characteristic, complemented
4#11721762C211yesmaximal, characteristic, complemented
5#11721762C211yesmaximal, characteristic, complemented
6#11721761C211yesmaximal, characteristic, complemented
7#11721760C211yesmaximal, characteristic, complemented
8#11721741C211yesmaximal, characteristic, complemented
9#11721739C211yesmaximal, characteristic, complemented
10#11721738C211yesmaximal, characteristic
11C22⋊C4×#272C211yesmaximal, characteristic, complemented
12C22⋊C4×#272C211yesmaximal, characteristic, complemented
13#11721735C211yesmaximal, characteristic, complemented
14C4×C4⋊#272C211yesmaximal, characteristic, complemented
15#11721718C211yesmaximal, characteristic, complemented
16C3×#5705—34nomaximal
17C23×#272C2211yescharacteristic
18C2×#6235C2211yescharacteristic
19C2×C4⋊#272C2211yescharacteristic
20C2×C4⋊#272C2211yescharacteristic
21C2×C4×#272C2211yescharacteristic
22C2×C4×#272C2211yescharacteristic
23C2×C4×#272C2211yescharacteristic
24C2×C4×#272C2211yescharacteristic
25C2×C4×#272C2211yescharacteristic, complemented
26C2×#6229C2211yescharacteristic, complemented
27C4⋊C4×H(3,𝔽3)C2211yescharacteristic, complemented
28C4⋊C4×H(3,𝔽3)C2211yescharacteristic, complemented
29C4⋊C4×H(3,𝔽3)C2211yescharacteristic, complemented
30C22⋊C4×H(3,𝔽3)C2211yescharacteristic
31C22⋊C4×H(3,𝔽3)C2211yescharacteristic
32C22⋊C4×H(3,𝔽3)C2211yescharacteristic
33C42×H(3,𝔽3)C2211yescharacteristic, complemented
34#88552C2211yescharacteristic
35#88552C2211yescharacteristic
36#88542C2211yescharacteristic
37#88542C2211yescharacteristic
38#88542C2211yescharacteristic
39#88542C2211yescharacteristic
40#88542C2211yescharacteristic
41#88542C2211yescharacteristic
42#88542C2211yescharacteristic, complemented
43#88542C2211yescharacteristic, complemented
44#88542C2211yescharacteristic, complemented
45C4⋊#1084C2211yescharacteristic, complemented
46C4⋊#1084C2211yescharacteristic, complemented
47#88540C2211yescharacteristic, complemented
48#88540C2211yescharacteristic, complemented
49#88540C2211yescharacteristic, complemented
50C4×#1084C2211yescharacteristic, complemented
51C4×#1084C2211yescharacteristic, complemented
52C32×#126D314yescomplemented
53C3×#897—34no—
54C3×#896—34no—
55C3×#894—34no—
56C3×#894—34no—
57C3×#893—34no—
58C3×#892—34no—
59C3×#885—34no—
60C3×#883—34no—
61C3×#882—34no—
62C3×D3×C22⋊C4—34no—
63C3×D3×C22⋊C4—34no—
64C3×#879—34no—
65C3×C4×D12—34no—
66C3×#872—34no—
67C23×H(3,𝔽3)C2311yescharacteristic
68C22×#272—22no—
69C22×#272—22no—
70C22×#272—41no—
71C22×#272C2311yescharacteristic
72C22×#272C2311yescharacteristic
73C22×#272C2311yescharacteristic
74C2×C4×H(3,𝔽3)C2311yescharacteristic
75C2×C4×H(3,𝔽3)C2311yescharacteristic
76C2×C4×H(3,𝔽3)C2311yescharacteristic
77C2×C4×H(3,𝔽3)C2311yescharacteristic
78C2×C4×H(3,𝔽3)C2311yescharacteristic
79C2×C4×H(3,𝔽3)C2311yescharacteristic
80#6235—41no—
81C2×#1084C2311yescharacteristic
82C2×#1084C2311yescharacteristic
83C2×#1084C2311yescharacteristic
84C2×#1084C2311yescharacteristic
85C2×#1084C2311yescharacteristic
86C4⋊#272—22no—
87C4⋊#272—22no—
88C4×#272—22no—
89C4×#272—22no—
90C4×#272—41no—
91C4×#272—41no—
92C4×#272—41no—
93#6229—41no—
94C3×#551—91no—
95#5705—34no—
96C23×C3×D3—34no—
97C2×C3×#67—34no—
98C2×C3×D12—34no—
99C2×C3×D12—34no—
100C2×C3×C4×D3—34no—
101C2×C3×C4×D3—34no—
102C2×C3×C4×D3—34no—
103C2×C3×C4×D3—34no—
104C2×C3×C4×D3—34no—
105C2×C3×Dic6—34no—
106C32×C4⋊C4C2×D314yescomplemented
107C32×C4⋊C4C2×D314yescomplemented
108C32×C4⋊C4C2×D314yescomplemented
109C32×C22⋊C4C2×D314yes—
110C32×C22⋊C4C2×D314yes—
111C32×C22⋊C4C2×D314yes—
112C32×C42C2×D314yescomplemented
113C3×#217—34no—
114C3×#217—34no—
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×(C2×D3)⋊C4—34no—
123C3×(C2×D3)⋊C4—34no—
124C3×C4⋊Dic3—34no—
125C3×C4⋊Dic3—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—44no—
143C2×#272#4112yes—
144C2×#272#4112yes—
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—
150C4×H(3,𝔽3)—21no—
151#1084—21no—
152#1084—21no—
153#1084—21no—
154#1084—21no—
155#1084—21no—
156C3×#126—34no—
157C3×#126—91no—
158C3×#126C3⋊D311yescharacteristic
159C3×#124—91no—
160C3×#123—91no—
161C3×#123—91no—
162C3×#123—91no—
163C3×C22⋊Q8—91no—
164C3×C22⋊Q8—91no—
165C3×#120—91no—
166C3×C4×D4—91no—
167C3×C4×D4—91no—
168C3×#117—91no—
169C3×#117—91no—
170C2×C3×C22⋊C4—91no—
171C2×C3×C22⋊C4—91no—
172#897—34no—
173#896—34no—
174#894—34no—
175#894—34no—
176#893—34no—
177#892—34no—
178#885—34no—
179#883—34no—
180#882—34no—
181D3×C22⋊C4—34no—
182D3×C22⋊C4—34no—
183#879—34no—
184C4×D12—34no—
185#872—34no—
186C23×C32C22×D314yes—
187C22×C3×D3—34no—
188C22×C3×D3—34no—
189C22×C3×D3—34no—
190C22×C3×D3—68no—
191C22×C3×D3—68no—
192C22×C3×D3—124no—
193C2×C32×C4C22×D314yes—
194C2×C32×C4C22×D314yes—
195C2×C32×C4C22×D314yes—
196C2×C32×C4C22×D314yes—
197C2×C32×C4C22×D314yes—
198C2×C32×C4C22×D314yes—
199C3×#67—124no—
200C2×C3×Dic3—34no—
201C2×C3×Dic3—34no—
202C2×C3×Dic3—34no—
203C2×C3×Dic3—34no—
204C2×C3×Dic3—34no—
205C3×D12—68no—
206C3×D12—68no—
207C3×C4×D3—68no—
208C3×C4×D3—68no—
209C3×C4×D3—124no—
210C3×C4×D3—124no—
211C3×C4×D3—124no—
212C3×Dic6—124no—
213#551—91noSylow
214C2×H(3,𝔽3)—41no—
215C2×H(3,𝔽3)C2×#4111yescharacteristic
216C2×H(3,𝔽3)C2×#4111yescharacteristic
217C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
218#272—22no—
219#272—22no—
220#272—41no—
221C24×C3—91no—
222C23×D3—34no—
223C2×C3×Q8—91no—
224C2×C3×D4—91no—
225C2×C3×D4—91no—
226C2×C3×D4—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×D12—34no—
234C2×D12—34no—
235C2×C4×D3—34no—
236C2×C4×D3—34no—
237C2×C4×D3—34no—
238C2×C4×D3—34no—
239C2×C4×D3—34no—
240C2×Dic6—34no—
241C3×C4⋊C4—34no—
242C3×C4⋊C4—34no—
243C3×C4⋊C4—34no—
244C3×C4⋊C4—91no—
245C3×C4⋊C4—91no—
246C3×C4⋊C4—91no—
247C3×C4⋊C4—91no—
248C3×C4⋊C4—91no—
249C3×C4⋊C4C2×C3⋊D311yescharacteristic
250C3×C4⋊C4C2×C3⋊D311yescharacteristic
251C3×C4⋊C4C2×C3⋊D311yescharacteristic
252C3×C22⋊C4—34no—
253C3×C22⋊C4—34no—
254C3×C22⋊C4—34no—
255C3×C22⋊C4—91no—
256C3×C22⋊C4—91no—
257C3×C22⋊C4—91no—
258C3×C22⋊C4—91no—
259C3×C22⋊C4—91no—
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⋊C4C2×C3⋊D311yescharacteristic
267C3×C22⋊C4C2×C3⋊D311yescharacteristic
268C3×C22⋊C4C2×C3⋊D311yescharacteristic
269C3×C42—34no—
270C3×C42—91no—
271C3×C42—91no—
272C3×C42C2×C3⋊D311yescharacteristic
273#217—34no—
274#217—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—
281(C2×D3)⋊C4—34no—
282(C2×D3)⋊C4—34no—
283(C2×D3)⋊C4—34no—
284C4⋊Dic3—34no—
285C4⋊Dic3—34no—
286Dic3⋊C4—34no—
287Dic3⋊C4—34no—
288Dic3⋊C4—34no—
289C4×Dic3—34no—
290C4×Dic3—34no—
291C22×C32—24no—
292C22×C32—24no—
293C22×C32—24no—
294C22×C32C23×D314yes—
295C2×C3×D3—38no—
296C2×C3×D3—38no—
297C2×C3×D3—64no—
298C2×C3×D3—64no—
299C2×C3×D3—64no—
300C2×C3×D3—64no—
301C2×C3×D3—64no—
302C2×C3×D3—64no—
303C2×C3×D3—64no—
304C2×C3×D3—1216no—
305C32×C4—24no—
306C32×C4—24no—
307C32×C4—24no—
308C32×C4—24no—
309C32×C4—24no—
310C32×C4—24no—
311C3×Dic3—64no—
312C3×Dic3—64no—
313C3×Dic3—64no—
314C3×Dic3—64no—
315C3×Dic3—64no—
316#126—91no—
317#126#27211yescharacteristic, complemented
318#124—91no—
319#123—91no—
320#123—91no—
321#123—91no—
322C22⋊Q8—91no—
323C22⋊Q8—91no—
324#120—91no—
325C4×D4—91no—
326C4×D4—91no—
327#117—91no—
328#117—91no—
329C2×C22⋊C4—91no—
330C2×C22⋊C4—91no—
331H(3,𝔽3)#55111yescharacteristic, complemented, Sylow
332C23×C3—34no—
333C23×C3—91no—
334C23×C3—91no—
335C23×C3—91no—
336C23×C3—182no—
337C23×C3—182no—
338C23×C3—361no—
339C23×C3C22×C3⋊D311yescharacteristic
340C22×D3—34no—
341C22×D3—34no—
342C22×D3—34no—
343C22×D3—68no—
344C22×D3—68no—
345C22×D3—124no—
346C3×Q8—361no—
347C3×D4—182no—
348C3×D4—182no—
349C3×D4—361no—
350C2×C3×C4—34no—
351C2×C3×C4—34no—
352C2×C3×C4—34no—
353C2×C3×C4—34no—
354C2×C3×C4—34no—
355C2×C3×C4—34no—
356C2×C3×C4—91no—
357C2×C3×C4—91no—
358C2×C3×C4—91no—
359C2×C3×C4—91no—
360C2×C3×C4—91no—
361C2×C3×C4—182no—
362C2×C3×C4—182no—
363C2×C3×C4—361no—
364C2×C3×C4—361no—
365C2×C3×C4—361no—
366C2×C3×C4C22×C3⋊D311yescharacteristic
367C2×C3×C4C22×C3⋊D311yescharacteristic
368C2×C3×C4C22×C3⋊D311yescharacteristic
369C2×C3×C4C22×C3⋊D311yescharacteristic
370C2×C3×C4C22×C3⋊D311yescharacteristic
371C2×C3×C4C22×C3⋊D311yescharacteristic
372#67—124no—
373C2×Dic3—34no—
374C2×Dic3—34no—
375C2×Dic3—34no—
376C2×Dic3—34no—
377C2×Dic3—34no—
378D12—68no—
379D12—68no—
380C4×D3—68no—
381C4×D3—68no—
382C4×D3—124no—
383C4×D3—124no—
384C4×D3—124no—
385Dic6—124no—
386C2×C32—44no—
387C2×C32D3×#4114yes—
388C2×C32D3×#4114yes—
389C2×C32#100914yes—
390C3×D3—68no—
391C3×D3—68no—
392C3×D3—124no—
393C24—91no—
394C2×Q8—91no—
395C2×D4—91no—
396C2×D4—91no—
397C2×D4—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⋊C4C2×#27211yescharacteristic, complemented
409C4⋊C4C2×#27211yescharacteristic, complemented
410C4⋊C4C2×#27211yescharacteristic, complemented
411C22⋊C4—91no—
412C22⋊C4—91no—
413C22⋊C4—91no—
414C22⋊C4—91no—
415C22⋊C4—91no—
416C22⋊C4—91no—
417C22⋊C4—91no—
418C22⋊C4—91no—
419C22⋊C4—91no—
420C22⋊C4—91no—
421C22⋊C4—91no—
422C22⋊C4C2×#27211yescharacteristic
423C22⋊C4C2×#27211yescharacteristic
424C22⋊C4C2×#27211yescharacteristic
425C42—91no—
426C42—91no—
427C42C2×#27211yescharacteristic, complemented
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—34no—
432C22×C3—64no—
433C22×C3—64no—
434C22×C3—64no—
435C22×C3—92no—
436C22×C3—92no—
437C22×C3—181no—
438C22×C3—181no—
439C22×C3—181no—
440C22×C3—181no—
441C22×C3—181no—
442C22×C3—181no—
443C22×C3—181no—
444C22×C3—364no—
445C22×C3C23×C3⋊D311yescharacteristic
446C2×D3—38no—
447C2×D3—38no—
448C2×D3—64no—
449C2×D3—64no—
450C2×D3—64no—
451C2×D3—64no—
452C2×D3—64no—
453C2×D3—64no—
454C2×D3—64no—
455C2×D3—1216no—
456C3×C4—21no—
457C3×C4—21no—
458C3×C4—21no—
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—64no—
463C3×C4—64no—
464C3×C4—64no—
465C3×C4—64no—
466C3×C4—64no—
467C3×C4—64no—
468C3×C4—181no—
469C3×C4—181no—
470C3×C4—181no—
471C3×C4—181no—
472C3×C4—181no—
473Dic3—64no—
474Dic3—64no—
475Dic3—64no—
476Dic3—64no—
477Dic3—64no—
478C32#570514yescomplemented
479C23—91no—
480C23—91no—
481C23—91no—
482C23—182no—
483C23—182no—
484C23—361no—
485C23C22×#27211yescharacteristic
486Q8—361no—
487D4—182no—
488D4—182no—
489D4—361no—
490C2×C4—91no—
491C2×C4—91no—
492C2×C4—91no—
493C2×C4—91no—
494C2×C4—91no—
495C2×C4—182no—
496C2×C4—182no—
497C2×C4—361no—
498C2×C4—361no—
499C2×C4—361no—
500C2×C4C22×#27211yescharacteristic
501C2×C4C22×#27211yescharacteristic
502C2×C4C22×#27211yescharacteristic
503C2×C4C22×#27211yescharacteristic
504C2×C4C22×#27211yescharacteristic
505C2×C4C22×#27211yescharacteristic
506C2×C3—34no—
507C2×C3—34no—
508C2×C3—34no—
509C2×C3—41no—
510C2×C3—124no—
511C2×C3—182no—
512C2×C3—182no—
513C2×C3—361no—
514C2×C3#41×C3⋊D311yescharacteristic
515C2×C3#41×C3⋊D311yescharacteristic
516C2×C3#6436611yescharacteristic
517D3—68no—
518D3—68no—
519D3—124no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—92no—
524C22—92no—
525C22—181no—
526C22—181no—
527C22—181no—
528C22—181no—
529C22—181no—
530C22—181no—
531C22—181no—
532C22—364no—
533C22C23×#27211yescharacteristic
534C4—21no—
535C4—21no—
536C4—21no—
537C4—21no—
538C4—21no—
539C4—21no—
540C4—181no—
541C4—181no—
542C4—181no—
543C4—181no—
544C4—181no—
545C3—34no—
546C3#1059617611yescharacteristic
547C2—41no—
548C2—182no—
549C2—182no—
550C2—361no—
551C2#41×#27211yescharacteristic
552C2#41×#27211yescharacteristic
553C2#1172267511yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×#126C2
prime radical#11721738C2
hypercenterC3×#126C3⋊D3
hypocenterH(3,𝔽3)#551
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. #11721765C2semidirect
3. #11721764C2semidirect
4. #11721762C2semidirect
5. #11721762C2semidirect
6. #11721761C2semidirect
7. #11721760C2semidirect
8. #11721741C2semidirect
9. #11721739C2semidirect
11. C22⋊C4×#272C2semidirect
12. C22⋊C4×#272C2semidirect
13. #11721735C2semidirect
14. C4×C4⋊#272C2semidirect
15. #11721718C2semidirect
25. C2×C4×#272C22semidirect
26. C2×#6229C22semidirect
27. C4⋊C4×H(3,𝔽3)C22semidirect
28. C4⋊C4×H(3,𝔽3)C22semidirect
29. C4⋊C4×H(3,𝔽3)C22semidirect
33. C42×H(3,𝔽3)C22semidirect
42. #88542C22semidirect
43. #88542C22semidirect
44. #88542C22semidirect
45. C4⋊#1084C22semidirect
46. C4⋊#1084C22semidirect
47. #88540C22semidirect
48. #88540C22semidirect
49. #88540C22semidirect
50. C4×#1084C22semidirect
51. C4×#1084C22semidirect
52. C32×#126D3semidirect
106. C32×C4⋊C4C2×D3semidirect
107. C32×C4⋊C4C2×D3semidirect
108. C32×C4⋊C4C2×D3semidirect
112. C32×C42C2×D3semidirect
317. #126#272semidirect
331. H(3,𝔽3)#551semidirect
408. C4⋊C4C2×#272semidirect
409. C4⋊C4C2×#272semidirect
410. C4⋊C4C2×#272semidirect
427. C42C2×#272semidirect
478. C32#5705semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1262
#117217652
#117217642
#117217622
#117217612
#117217602
#117217412
#117217392
#117217382
C22⋊C4×#2722
#117217352
C4×C4⋊#2722
#117217182
C3×#57053