Finite Group Databasebeta

← #13147863

Subgroups — #13147863

Order 1728.

818 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721721C211yesmaximal, characteristic
2C4×#634⋊C3C211yesmaximal, characteristic, complemented
3C4×#634⋊C3C211yesmaximal, characteristic, complemented
4#11721149C211yesmaximal, characteristic
5#11721148C212yesmaximal, complemented
6#11721114C211yesmaximal, characteristic
7#11721092C212yesmaximal, complemented
8#11721090C211yesmaximal, characteristic, complemented
9#11721088C212yesmaximal, complemented
10#11721083C211yesmaximal, characteristic, complemented
11#11721076C212yesmaximal, complemented
12#10593942—31nomaximal
13#10593932—31nomaximal
14C2×C4⋊#272C2211yescharacteristic
15C2×#6229C2211yescharacteristic
16C2×#634⋊C3C2211yescharacteristic
17C2×#634⋊C3C2211yescharacteristic
18C2×#6199C2212yes—
19C42×H(3,𝔽3)C2211yescharacteristic
20#88542C2212yes—
21#88542C2212yes—
22C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
23C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
24#3750⋊C3C2211yescharacteristic
25#3750⋊C3C2212yes—
26#3750⋊C3C2211yescharacteristic
27C4×(C3⋊Dic3)⋊C3C2212yes—
28C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
29C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
30#88451C2212yes—
31#88451C2212yes—
32#88451C2212yes—
33#88449C2212yes—
34#88448C2212yes—
35#88448C2212yes—
36#88447C2212yes—
37C4×#634D311yescharacteristic, complemented
38C4×#634D311yescharacteristic, complemented
39C3×#875—31no—
40C3×#875—31no—
41C3×#875—61no—
42C3×C4×Dic6—31no—
43C3×C4×Dic6—31no—
44#63898—31no—
45#63896—32no—
46Dic3×D12—31no—
47#63887—32no—
48#63878—32no—
49#63877—32no—
50#63873—31no—
51#63866—31no—
52#63863—31no—
53#63862—32no—
54#63861—32no—
55#63860—31no—
56#63859—32no—
57#63858—32no—
58#63843—31no—
59Dic3×Dic6—31no—
60C22×#272C2312yes—
61C2×C4×H(3,𝔽3)C2311yescharacteristic
62C2×C4×H(3,𝔽3)C2311yescharacteristic
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×#1084C2312yes—
65C4⋊#272—22no—
66#6229—22no—
67C2×(C3⋊Dic3)⋊C3C2312yes—
68C2×(C3⋊Dic3)⋊C3C2312yes—
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
73#634⋊C3—22no—
74#634⋊C3—22no—
75#6199—24no—
76#5671—91no—
77#5584—91no—
78#5568—91no—
79C2×#634C2×D311yescharacteristic
80C2×#634C2×D311yescharacteristic
81C2×C3×D12—31no—
82C2×C3×D12—31no—
83C2×C3×D12—61no—
84C2×C3×Dic6—31no—
85C2×C3×Dic6—31no—
86C2×C3×Dic6—31no—
87C2×C3×Dic6—31no—
88C2×C3×Dic6—61no—
89C2×D3×Dic3—32no—
90C2×D3×Dic3—32no—
91C32×C42—21no—
92C32×C42C2×D311yescharacteristic
93C32×C42C2×D311yescharacteristic
94C4⋊(C3⋊Dic3)C2×D311yescharacteristic
95C4⋊(C3⋊Dic3)C2×D311yescharacteristic
96#3750C2×D312yes—
97#3750C2×D311yescharacteristic
98#3750C2×D311yescharacteristic
99C4×C3⋊Dic3C2×D311yescharacteristic
100C4×C3⋊Dic3C2×D312yes—
101C4×C3⋊Dic3C2×D311yescharacteristic
102C3×(C2×D3)⋊C4—32no—
103C3×(C2×D3)⋊C4—32no—
104C3×(C2×D3)⋊C4—32no—
105C3×(C2×D3)⋊C4—32no—
106C3×(C2×D3)⋊C4—62no—
107C3×(C2×D3)⋊C4—62no—
108C3×C4⋊Dic3—31no—
109C3×C4⋊Dic3—31no—
110C3×Dic3⋊C4—32no—
111C3×Dic3⋊C4—31no—
112C3×Dic3⋊C4—31no—
113C3×C4×Dic3—31no—
114C3×C4×Dic3—32no—
115C3×C4×Dic3—31no—
116#3724—32no—
117#3724—32no—
118Dic3⋊Dic3—32no—
119Dic3⋊Dic3—32no—
120Dic3⋊Dic3—32no—
121Dic3⋊Dic3—32no—
122(C2×D3)⋊Dic3—32no—
123(C2×D3)⋊Dic3—32no—
124(C2×D3)⋊Dic3—32no—
125(C2×D3)⋊Dic3—32no—
126(C2×D3)⋊Dic3—32no—
127(C2×D3)⋊Dic3—32no—
128Dic32—32no—
129Dic32—32no—
130C22×H(3,𝔽3)C2411yescharacteristic
131C2×#272—22no—
132C2×#272—22no—
133C2×#272—22no—
134C4×H(3,𝔽3)—21no—
135C4×H(3,𝔽3)—21no—
136C4×H(3,𝔽3)#4112yes—
137#1084—22no—
138(C3⋊Dic3)⋊C3—22no—
139(C3⋊Dic3)⋊C3—22no—
140(C3⋊Dic3)⋊C3—21no—
141(C3⋊Dic3)⋊C3—21no—
142(C3⋊Dic3)⋊C3—21no—
143(C3⋊Dic3)⋊C3#4112yes—
144C3×#124—91no—
145C3×C4×Q8—91no—
146C3×C4×Q8—91no—
147#947—91no—
148#946—92no—
149Q8×Dic3—91no—
150#944—91no—
151#944—91no—
152#936—91no—
153D4×Dic3—91no—
154#898—92no—
155#898—92no—
156#898—92no—
157#896—92no—
158#893—91no—
159#892—92no—
160#887—91no—
161#879—92no—
162#879—92no—
163#878—92no—
164#877—92no—
165#876—92no—
166#875—31no—
167#875—31no—
168#875—61no—
169#875—91no—
170C4×D12—91no—
171#872—92no—
172#869—91no—
173C4×Dic6—31no—
174C4×Dic6—31no—
175C4×Dic6—91no—
176C4×Dic6—91no—
177C4×Dic6—91no—
178C22×C3×D3—32no—
179C22×C3×D3—32no—
180C22×C3×D3—62no—
181C2×C32×C4—21no—
182C2×C32×C4—21no—
183C2×C32×C4—21no—
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D311yescharacteristic
189C2×C32×C4C22×D311yescharacteristic
190C2×C3⋊Dic3C22×D311yescharacteristic
191C2×C3⋊Dic3C22×D312yes—
192C2×C3⋊Dic3C22×D312yes—
193C2×C3⋊Dic3C22×D311yescharacteristic
194C2×C3⋊Dic3C22×D311yescharacteristic
195C2×C3⋊Dic3C22×D311yescharacteristic
196#634—22no—
197#634—22no—
198C2×C3×Dic3—31no—
199C2×C3×Dic3—31no—
200C2×C3×Dic3—32no—
201C2×C3×Dic3—31no—
202C2×C3×Dic3—31no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3—62no—
207C3×D12—62no—
208C3×D12—62no—
209C3×D12—122no—
210C3×Dic6—62no—
211C3×Dic6—62no—
212C3×Dic6—62no—
213C3×Dic6—62no—
214C3×Dic6—122no—
215D3×Dic3—64no—
216D3×Dic3—64no—
217#556—271noSylow
218C2×H(3,𝔽3)C2×#4111yescharacteristic
219C2×H(3,𝔽3)C2×#4111yescharacteristic
220C2×H(3,𝔽3)2−1+411yescharacteristic
221#272—42no—
222C2×C3×Q8—91no—
223C2×C3×Q8—91no—
224C2×C3×Q8—91no—
225C2×C3×D4—91no—
226C22×Dic3—92no—
227C2×D12—31no—
228C2×D12—31no—
229C2×D12—61no—
230C2×C4×D3—92no—
231C2×C4×D3—92no—
232C2×Dic6—31no—
233C2×Dic6—31no—
234C2×Dic6—31no—
235C2×Dic6—31no—
236C2×Dic6—61no—
237C2×Dic6—91no—
238C2×Dic6—91no—
239C3×C4⋊C4—91no—
240C3×C4⋊C4—91no—
241C3×C4⋊C4—92no—
242C3×C4⋊C4—91no—
243C3×C4⋊C4—91no—
244C3×C22⋊C4—92no—
245C3×C22⋊C4—92no—
246C3×C42—31no—
247C3×C42—31no—
248C3×C42—61no—
249C3×C42—91no—
250C3×C42—91no—
251C3×C42—92no—
252C3×C42D3211yescharacteristic
253#217—92no—
254#217—92no—
255#217—92no—
256(C2×D3)⋊C4—32no—
257(C2×D3)⋊C4—32no—
258(C2×D3)⋊C4—32no—
259(C2×D3)⋊C4—32no—
260(C2×D3)⋊C4—62no—
261(C2×D3)⋊C4—62no—
262(C2×D3)⋊C4—92no—
263(C2×D3)⋊C4—92no—
264(C2×D3)⋊C4—92no—
265(C2×D3)⋊C4—92no—
266(C2×D3)⋊C4—92no—
267(C2×D3)⋊C4—92no—
268C4⋊Dic3—31no—
269C4⋊Dic3—31no—
270C4⋊Dic3—91no—
271C4⋊Dic3—91no—
272C4⋊Dic3—92no—
273C4⋊Dic3—92no—
274C4⋊Dic3—92no—
275Dic3⋊C4—31no—
276Dic3⋊C4—32no—
277Dic3⋊C4—31no—
278Dic3⋊C4—91no—
279Dic3⋊C4—91no—
280Dic3⋊C4—92no—
281Dic3⋊C4—92no—
282Dic3⋊C4—92no—
283Dic3⋊C4—92no—
284Dic3⋊C4—92no—
285Dic3⋊C4—92no—
286Dic3⋊C4—92no—
287C4×Dic3—32no—
288C4×Dic3—31no—
289C4×Dic3—31no—
290C4×Dic3—91no—
291C4×Dic3—92no—
292C4×Dic3—91no—
293C4×Dic3—92no—
294C4×Dic3—92no—
295C4×Dic3—92no—
296C22×C32—21no—
297C22×C32C23×D311yescharacteristic
298C22×C32C23×D311yescharacteristic
299C2×C3×D3—62no—
300C2×C3×D3—62no—
301C2×C3×D3—62no—
302C2×C3×D3—62no—
303C2×C3×D3—62no—
304C2×C3×D3—62no—
305C2×C3×D3—122no—
306C2×C3×D3—122no—
307C2×C3×D3—122no—
308C32×C4—21no—
309C32×C4—21no—
310C32×C4—21no—
311C32×C4—21no—
312C32×C4—22no—
313C32×C4—41no—
314C32×C4—41no—
315C32×C4#23512yes—
316C32×C4#23912yes—
317C3⋊Dic3—22no—
318C3⋊Dic3—21no—
319C3⋊Dic3—22no—
320C3⋊Dic3—21no—
321C3⋊Dic3—21no—
322C3⋊Dic3#23512yes—
323C3×Dic3—32no—
324C3×Dic3—62no—
325C3×Dic3—62no—
326C3×Dic3—62no—
327C3×Dic3—62no—
328C3×Dic3—61no—
329C3×Dic3—61no—
330C3×Dic3—61no—
331C3×Dic3—122no—
332C4⋊Q8—271no—
333#126—272no—
334#126—272no—
335#124—91no—
336#124—271no—
337C22⋊Q8—272no—
338C4×Q8—91no—
339C4×Q8—91no—
340C4×Q8—271no—
341C4×D4—271no—
342#117—272no—
343H(3,𝔽3)#55611yescharacteristic, complemented, Sylow
344C23×C3—92no—
345C22×D3—32no—
346C22×D3—32no—
347C22×D3—62no—
348C3×Q8—182no—
349C3×Q8—182no—
350C3×Q8—182no—
351C3×D4—182no—
352C2×C3×C4—31no—
353C2×C3×C4—31no—
354C2×C3×C4—31no—
355C2×C3×C4—31no—
356C2×C3×C4—31no—
357C2×C3×C4—31no—
358C2×C3×C4—61no—
359C2×C3×C4—61no—
360C2×C3×C4—61no—
361C2×C3×C4—91no—
362C2×C3×C4—91no—
363C2×C3×C4—92no—
364C2×C3×C4—91no—
365C2×C3×C4—92no—
366C2×C3×C4—91no—
367C2×C3×C4—92no—
368C2×C3×C4C2×D3211yescharacteristic
369C2×C3×C4C2×D3211yescharacteristic
370C2×C3×C4C2×D3211yescharacteristic
371C2×Dic3—31no—
372C2×Dic3—32no—
373C2×Dic3—32no—
374C2×Dic3—31no—
375C2×Dic3—32no—
376C2×Dic3—31no—
377C2×Dic3—32no—
378C2×Dic3—31no—
379C2×Dic3—62no—
380C2×Dic3—91no—
381C2×Dic3—91no—
382C2×Dic3—92no—
383C2×Dic3—91no—
384C2×Dic3—91no—
385C2×Dic3—92no—
386C2×Dic3—184no—
387D12—62no—
388D12—62no—
389D12—122no—
390C4×D3—184no—
391C4×D3—184no—
392Dic6—62no—
393Dic6—62no—
394Dic6—62no—
395Dic6—62no—
396Dic6—122no—
397Dic6—182no—
398Dic6—182no—
399C2×C32—21no—
400C2×C32—21no—
401C2×C32—21no—
402C2×C32C2×#23511yescharacteristic
403C2×C32C2×#23511yescharacteristic
404C2×C32C2×#23911yescharacteristic
405C2×C32#100711yescharacteristic
406C2×C32D3×#4111yescharacteristic
407C2×C32#101011yescharacteristic
408C3×D3—122no—
409C3×D3—122no—
410C3×D3—242no—
411C2×Q8—91no—
412C2×Q8—91no—
413C2×Q8—91no—
414C2×D4—91no—
415C22×C4—272no—
416C4⋊C4—91no—
417C4⋊C4—91no—
418C4⋊C4—92no—
419C4⋊C4—91no—
420C4⋊C4—91no—
421C4⋊C4—272no—
422C4⋊C4—272no—
423C4⋊C4—272no—
424C22⋊C4—92no—
425C22⋊C4—92no—
426C22⋊C4—272no—
427C22⋊C4—272no—
428C22⋊C4—272no—
429C42—92no—
430C42—91no—
431C42—91no—
432C42—272no—
433C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
434C22×C3—31no—
435C22×C3—31no—
436C22×C3—61no—
437C22×C3—182no—
438C22×C3—182no—
439C22×C3—182no—
440C22×C3C22×D3211yescharacteristic
441C2×D3—62no—
442C2×D3—62no—
443C2×D3—62no—
444C2×D3—62no—
445C2×D3—62no—
446C2×D3—62no—
447C2×D3—122no—
448C2×D3—122no—
449C2×D3—122no—
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—32no—
453C3×C4—32no—
454C3×C4—61no—
455C3×C4—61no—
456C3×C4—61no—
457C3×C4—61no—
458C3×C4—62no—
459C3×C4—92no—
460C3×C4—121no—
461C3×C4—121no—
462C3×C4—182no—
463C3×C4—182no—
464C3×C4—181no—
465C3×C4—181no—
466C3×C4—181no—
467C3×C4—182no—
468C3×C4#379912yes—
469Dic3—32no—
470Dic3—62no—
471Dic3—62no—
472Dic3—61no—
473Dic3—61no—
474Dic3—62no—
475Dic3—61no—
476Dic3—62no—
477Dic3—92no—
478Dic3—122no—
479Dic3—182no—
480Dic3—181no—
481Dic3—182no—
482Dic3—181no—
483Dic3—181no—
484C32—21no—
485C32#556811yescharacteristic, complemented
486C32#558411yescharacteristic, complemented
487C23—92no—
488Q8—182no—
489Q8—182no—
490Q8—182no—
491D4—182no—
492C2×C4—91no—
493C2×C4—91no—
494C2×C4—92no—
495C2×C4—91no—
496C2×C4—92no—
497C2×C4—91no—
498C2×C4—92no—
499C2×C4—544no—
500C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
501C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
502C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
503C2×C3—31no—
504C2×C3—31no—
505C2×C3—31no—
506C2×C3—31no—
507C2×C3—31no—
508C2×C3—31no—
509C2×C3—61no—
510C2×C3—61no—
511C2×C3—61no—
512C2×C3—362no—
513C2×C3#6429711yescharacteristic
514C2×C3C2×#379911yescharacteristic
515C2×C3D3×#23511yescharacteristic
516D3—122no—
517D3—122no—
518D3—242no—
519C22—182no—
520C22—182no—
521C22—182no—
522C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
523C4—21no—
524C4—21no—
525C4—92no—
526C4—181no—
527C4—181no—
528C4—182no—
529C4—182no—
530C4—182no—
531C4—181no—
532C4#8865212yes—
533C3—31no—
534C3—31no—
535C3—61no—
536C3#1059385011yescharacteristic
537C2—362no—
538C2#1172248211yescharacteristic
539C2C2×#8865211yescharacteristic
540C2#1172249011yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC2×C4⋊#272C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#556
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C4×#634⋊C3C2semidirect
3. C4×#634⋊C3C2semidirect
5. #11721148C2semidirect
7. #11721092C2semidirect
8. #11721090C2semidirect
9. #11721088C2semidirect
10. #11721083C2semidirect
11. #11721076C2semidirect
37. C4×#634D3semidirect
38. C4×#634D3semidirect
343. H(3,𝔽3)#556semidirect
485. C32#5568semidirect
486. C32#5584semidirect

Maximal subgroups

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

subgroupindex
#117217212
C4×#634⋊C32
#117211492
#117211482
#117211142
#117210922
#117210902
#117210882
#117210832
#117210762
#105939423
#105939323