Finite Group Databasebeta

← #13147815

Subgroups — #13147815

Order 1728.

777 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721716C211yesmaximal, characteristic, complemented
2#64081⋊C3C211yesmaximal, characteristic
3#64078⋊C3C211yesmaximal, characteristic, complemented
4#11721134C212yesmaximal
5#11721129C211yesmaximal, characteristic
6#11721128C212yesmaximal
7#11721116C211yesmaximal, characteristic, complemented
8#11721093C211yesmaximal, characteristic, complemented
9#11721087C212yesmaximal, complemented
10#11721081C211yesmaximal, characteristic
11#11721079C212yesmaximal, complemented
12#10593959—31nomaximal
13#10593855—31nomaximal
14C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
15C2×#634⋊C3C2212yes—
16C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
17C2×#6196C2212yes—
18C42×H(3,𝔽3)C2211yescharacteristic
19C4⋊#1084C2211yescharacteristic
20C4⋊#1084C2211yescharacteristic
21#88540C2212yes—
22#88540C2212yes—
23#3752⋊C3C2211yescharacteristic
24#3752⋊C3C2211yescharacteristic
25C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
26C4⋊((C3⋊Dic3)⋊C3)C2212yes—
27C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
28#3750⋊C3C2211yescharacteristic
29#3750⋊C3C2211yescharacteristic
30C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
31#88450C2212yes—
32#88449C2212yes—
33#88448C2212yes—
34#88448C2212yes—
35#88448C2212yes—
36#88447C2212yes—
37#64081D311yescharacteristic
38#64078D311yescharacteristic, complemented
39C3×#872—31no—
40C3×#870—31no—
41C3×#870—31no—
42C3×#870—61no—
43C3×#869—31no—
44#63919—31no—
45#63918—31no—
46#63900—31no—
47#63896—32no—
48#63895—32no—
49#63893—32no—
50D3×C4⋊Dic3—31no—
51#63888—32no—
52#63872—31no—
53#63869—31no—
54#63849—31no—
55#63847—32no—
56#63845—32no—
57#63841—31no—
58#63840—32no—
59#63839—32no—
60C22×(C3⋊D3)⋊C3C2311yescharacteristic
61C2×C4×H(3,𝔽3)C2311yescharacteristic
62C2×C4×H(3,𝔽3)C2311yescharacteristic
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×#1084C2312yes—
65C2×#1084C2312yes—
66C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
67C2×(C3⋊Dic3)⋊C3C2312yes—
68C2×(C3⋊Dic3)⋊C3C2312yes—
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C4×(C3⋊D3)⋊C3Q814yescomplemented
72#634⋊C3—42no—
73((C3⋊D3)⋊C3)⋊C4—42no—
74#6196—42no—
75#5729—91no—
76#5689—91no—
77#5519—91no—
78C2×C4×C3⋊D3C2×D311yescharacteristic
79C2×#634C2×D312yes—
80C2×C3×C4×D3—31no—
81C2×C3×Dic6—32no—
82C2×#627—32no—
83C2×#627—32no—
84C2×#624—32no—
85C2×D3×Dic3—32no—
86C32×C42—21no—
87C32×C42C2×D311yescharacteristic
88C32×C42C2×D311yescharacteristic
89#3752C2×D311yescharacteristic
90#3752C2×D311yescharacteristic
91C4⋊(C3⋊Dic3)C2×D311yescharacteristic
92C4⋊(C3⋊Dic3)C2×D312yes—
93C4⋊(C3⋊Dic3)C2×D311yescharacteristic
94#3750C2×D311yescharacteristic
95#3750C2×D311yescharacteristic
96C4×C3⋊Dic3C2×D311yescharacteristic
97C3×(C2×D3)⋊C4—31no—
98C3×(C2×D3)⋊C4—31no—
99C3×C4⋊Dic3—31no—
100C3×C4⋊Dic3—32no—
101C3×C4⋊Dic3—31no—
102C3×C4⋊Dic3—31no—
103C3×C4⋊Dic3—31no—
104C3×C4⋊Dic3—31no—
105C3×C4⋊Dic3—31no—
106C3×C4⋊Dic3—61no—
107C3×C4⋊Dic3—61no—
108C3×Dic3⋊C4—32no—
109C3×Dic3⋊C4—31no—
110C3×Dic3⋊C4—32no—
111C3×Dic3⋊C4—31no—
112C3×Dic3⋊C4—32no—
113C3×Dic3⋊C4—32no—
114C3×Dic3⋊C4—62no—
115C3×Dic3⋊C4—62no—
116C3×C4×Dic3—31no—
117#3724—32no—
118#3724—32no—
119#3724—32no—
120Dic3⋊Dic3—32no—
121Dic3⋊Dic3—32no—
122Dic3⋊Dic3—32no—
123Dic3⋊Dic3—32no—
124Dic3⋊Dic3—32no—
125#3722—32no—
126(C2×D3)⋊Dic3—32no—
127Dic32—32no—
128Dic32—32no—
129C22×H(3,𝔽3)C2411yescharacteristic
130C2×(C3⋊D3)⋊C3—21no—
131C2×(C3⋊D3)⋊C3—21no—
132C2×(C3⋊D3)⋊C3C2×Q812yes—
133C4×H(3,𝔽3)—21no—
134C4×H(3,𝔽3)—21no—
135C4×H(3,𝔽3)C2×Q812yes—
136#1084—22no—
137#1084—22no—
138(C3⋊Dic3)⋊C3—21no—
139(C3⋊Dic3)⋊C3—22no—
140(C3⋊Dic3)⋊C3—22no—
141(C3⋊Dic3)⋊C3—21no—
142(C3⋊Dic3)⋊C3C2×Q812yes—
143C3×C4⋊Q8—91no—
144C3×#125—91no—
145C3×#117—91no—
146#946—92no—
147#944—92no—
148#926—91no—
149C2×C4⋊Dic3—91no—
150#897—92no—
151#896—92no—
152#896—92no—
153#892—91no—
154#892—91no—
155D3×C4⋊C4—91no—
156D3×C4⋊C4—91no—
157#890—91no—
158#890—92no—
159#890—91no—
160#889—92no—
161#889—92no—
162#888—91no—
163#888—92no—
164#888—92no—
165#888—91no—
166#878—92no—
167#878—92no—
168#872—31no—
169#872—91no—
170#870—31no—
171#870—31no—
172#870—61no—
173#870—91no—
174#869—31no—
175#869—91no—
176#869—91no—
177C22×C3⋊D3C22×D311yescharacteristic
178C22×C3×D3—31no—
179C2×C32×C4—21no—
180C2×C32×C4—21no—
181C2×C32×C4—21no—
182C2×C32×C4C22×D311yescharacteristic
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188C2×C3⋊Dic3C22×D312yes—
189C2×C3⋊Dic3C22×D311yescharacteristic
190C2×C3⋊Dic3C22×D312yes—
191C2×C3⋊Dic3C22×D311yescharacteristic
192C2×C3⋊Dic3C22×D311yescharacteristic
193C4×C3⋊D3Dic614yescomplemented
194#634—42no—
195C2×C3×Dic3—31no—
196C2×C3×Dic3—32no—
197C2×C3×Dic3—31no—
198C2×C3×Dic3—32no—
199C2×C3×Dic3—31no—
200C2×C3×Dic3—32no—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—62no—
205C2×C3×Dic3—62no—
206C3×C4×D3—34no—
207C3×Dic6—122no—
208#627—122no—
209#627—122no—
210#624—122no—
211D3×Dic3—122no—
212#544—271noSylow
213C2×H(3,𝔽3)C22×Q811yescharacteristic
214C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
215C2×H(3,𝔽3)2−1+411yescharacteristic
216(C3⋊D3)⋊C3—22no—
217C2×C3×Q8—92no—
218C22×C3×C4—91no—
219C22×Dic3—92no—
220C2×C4×D3—31no—
221C2×C4×D3—91no—
222C2×C4×D3—92no—
223C2×C4×D3—92no—
224C2×Dic6—32no—
225C2×Dic6—92no—
226C2×Dic6—92no—
227C2×Dic6—92no—
228C2×Dic6—92no—
229C3×C4⋊C4—91no—
230C3×C4⋊C4—91no—
231C3×C4⋊C4—92no—
232C3×C4⋊C4—91no—
233C3×C4⋊C4—91no—
234C3×C4⋊C4—91no—
235C3×C4⋊C4—92no—
236C3×C4⋊C4—92no—
237C3×C4⋊C4—91no—
238C3×C22⋊C4—91no—
239C3×C22⋊C4—91no—
240C3×C42—31no—
241C3×C42—31no—
242C3×C42—61no—
243C3×C42—91no—
244C3×C42D3211yescharacteristic
245#217—92no—
246(C2×D3)⋊C4—31no—
247(C2×D3)⋊C4—31no—
248(C2×D3)⋊C4—91no—
249(C2×D3)⋊C4—91no—
250(C2×D3)⋊C4—92no—
251(C2×D3)⋊C4—92no—
252C4⋊Dic3—31no—
253C4⋊Dic3—31no—
254C4⋊Dic3—31no—
255C4⋊Dic3—31no—
256C4⋊Dic3—31no—
257C4⋊Dic3—32no—
258C4⋊Dic3—31no—
259C4⋊Dic3—61no—
260C4⋊Dic3—61no—
261C4⋊Dic3—91no—
262C4⋊Dic3—91no—
263C4⋊Dic3—92no—
264C4⋊Dic3—92no—
265C4⋊Dic3—92no—
266C4⋊Dic3—92no—
267C4⋊Dic3—92no—
268Dic3⋊C4—32no—
269Dic3⋊C4—32no—
270Dic3⋊C4—32no—
271Dic3⋊C4—32no—
272Dic3⋊C4—31no—
273Dic3⋊C4—31no—
274Dic3⋊C4—62no—
275Dic3⋊C4—62no—
276Dic3⋊C4—91no—
277Dic3⋊C4—91no—
278Dic3⋊C4—92no—
279Dic3⋊C4—92no—
280Dic3⋊C4—92no—
281Dic3⋊C4—92no—
282Dic3⋊C4—92no—
283Dic3⋊C4—92no—
284Dic3⋊C4—92no—
285Dic3⋊C4—92no—
286C4×Dic3—31no—
287C4×Dic3—91no—
288C4×Dic3—92no—
289C4×Dic3—92no—
290C4×Dic3—92no—
291C22×C32—21no—
292C22×C32C23×D311yescharacteristic
293C22×C32C23×D311yescharacteristic
294C2×C3⋊D3—21no—
295C2×C3⋊D3—21no—
296C2×C3⋊D3C2×Dic612yes—
297C2×C3×D3—32no—
298C2×C3×D3—61no—
299C2×C3×D3—61no—
300C32×C4—21no—
301C32×C4—21no—
302C32×C4—21no—
303C32×C4—21no—
304C32×C4—22no—
305C32×C4—41no—
306C32×C4—41no—
307C32×C4C2×Dic612yes—
308C32×C4D3×Q812yes—
309C3⋊Dic3—22no—
310C3⋊Dic3—22no—
311C3⋊Dic3—21no—
312C3⋊Dic3—21no—
313C3⋊Dic3C2×Dic612yes—
314C3×Dic3—32no—
315C3×Dic3—61no—
316C3×Dic3—62no—
317C3×Dic3—61no—
318C3×Dic3—62no—
319C3×Dic3—62no—
320C3×Dic3—62no—
321C3×Dic3—62no—
322C3×Dic3—62no—
323C3×Dic3—122no—
324C3×Dic3—122no—
325C4⋊Q8—91no—
326C4⋊Q8—271no—
327C4⋊Q8—272no—
328#125—91no—
329#125—271no—
330#125—272no—
331C22⋊Q8—272no—
332C22⋊Q8—272no—
333#117—91no—
334#117—271no—
335C2×C4⋊C4—271no—
336H(3,𝔽3)#54411yescharacteristic, complemented, Sylow
337C23×C3—91no—
338C22×D3—31no—
339C22×D3—91no—
340C3×Q8—362no—
341C2×C3×C4—31no—
342C2×C3×C4—31no—
343C2×C3×C4—31no—
344C2×C3×C4—31no—
345C2×C3×C4—31no—
346C2×C3×C4—31no—
347C2×C3×C4—61no—
348C2×C3×C4—61no—
349C2×C3×C4—61no—
350C2×C3×C4—94no—
351C2×C3×C4—91no—
352C2×C3×C4—92no—
353C2×C3×C4—91no—
354C2×C3×C4—92no—
355C2×C3×C4—91no—
356C2×C3×C4—92no—
357C2×C3×C4—92no—
358C2×C3×C4C2×D3211yescharacteristic
359C2×C3×C4C2×D3211yescharacteristic
360C2×C3×C4C2×D3211yescharacteristic
361C2×Dic3—31no—
362C2×Dic3—32no—
363C2×Dic3—32no—
364C2×Dic3—32no—
365C2×Dic3—32no—
366C2×Dic3—31no—
367C2×Dic3—31no—
368C2×Dic3—32no—
369C2×Dic3—32no—
370C2×Dic3—62no—
371C2×Dic3—62no—
372C2×Dic3—91no—
373C2×Dic3—91no—
374C2×Dic3—91no—
375C2×Dic3—92no—
376C2×Dic3—92no—
377C2×Dic3—362no—
378C4×D3—34no—
379C4×D3—94no—
380C4×D3—362no—
381C4×D3—362no—
382Dic6—122no—
383Dic6—362no—
384Dic6—362no—
385Dic6—362no—
386Dic6—362no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32C22×Dic611yescharacteristic
391C2×C32#100411yescharacteristic
392C2×C32C2×D3×Q811yescharacteristic
393C2×C32#100711yescharacteristic
394C2×C32#100911yescharacteristic
395C2×C32#101011yescharacteristic
396C3⋊D3—22no—
397C3×D3—62no—
398C2×Q8—92no—
399C2×Q8—272no—
400C22×C4—91no—
401C22×C4—272no—
402C4⋊C4—91no—
403C4⋊C4—91no—
404C4⋊C4—91no—
405C4⋊C4—91no—
406C4⋊C4—91no—
407C4⋊C4—91no—
408C4⋊C4—92no—
409C4⋊C4—92no—
410C4⋊C4—92no—
411C4⋊C4—272no—
412C4⋊C4—272no—
413C4⋊C4—272no—
414C4⋊C4—272no—
415C22⋊C4—91no—
416C22⋊C4—91no—
417C22⋊C4—272no—
418C42—91no—
419C42—272no—
420C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
421C22×C3—31no—
422C22×C3—31no—
423C22×C3—61no—
424C22×C3—92no—
425C22×C3—181no—
426C22×C3—181no—
427C22×C3C22×D3211yescharacteristic
428C2×D3—32no—
429C2×D3—61no—
430C2×D3—61no—
431C2×D3—92no—
432C2×D3—181no—
433C2×D3—181no—
434C3×C4—21no—
435C3×C4—21no—
436C3×C4—32no—
437C3×C4—32no—
438C3×C4—61no—
439C3×C4—61no—
440C3×C4—61no—
441C3×C4—61no—
442C3×C4—62no—
443C3×C4—92no—
444C3×C4—121no—
445C3×C4—121no—
446C3×C4—181no—
447C3×C4—182no—
448C3×C4—182no—
449C3×C4—181no—
450C3×C4—182no—
451C3×C4—182no—
452C3×C4D3×Dic612yes—
453Dic3—32no—
454Dic3—62no—
455Dic3—62no—
456Dic3—62no—
457Dic3—62no—
458Dic3—61no—
459Dic3—62no—
460Dic3—62no—
461Dic3—61no—
462Dic3—92no—
463Dic3—122no—
464Dic3—122no—
465Dic3—182no—
466Dic3—181no—
467Dic3—182no—
468Dic3—181no—
469C32—21no—
470C32#551911yescharacteristic, complemented
471C32#572911yescharacteristic, complemented
472C23—91no—
473Q8—362no—
474Q8—1082no—
475C2×C4—91no—
476C2×C4—94no—
477C2×C4—91no—
478C2×C4—91no—
479C2×C4—92no—
480C2×C4—92no—
481C2×C4—92no—
482C2×C4—92no—
483C2×C4—1082no—
484C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
485C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
486C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
487C2×C3—31no—
488C2×C3—31no—
489C2×C3—31no—
490C2×C3—31no—
491C2×C3—31no—
492C2×C3—31no—
493C2×C3—61no—
494C2×C3—61no—
495C2×C3—61no—
496C2×C3—182no—
497C2×C3C2×D3×Dic611yescharacteristic
498C2×C3#6429711yescharacteristic
499C2×C3#6430711yescharacteristic
500D3—62no—
501D3—182no—
502C22—92no—
503C22—181no—
504C22—181no—
505C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
506C4—21no—
507C4—21no—
508C4—92no—
509C4—182no—
510C4—181no—
511C4—182no—
512C4—182no—
513C4—182no—
514C4—181no—
515C4((C3⋊D3)⋊C3)⋊Q812yes—
516C3—31no—
517C3—31no—
518C3—61no—
519C3#1059395411yescharacteristic
520C2—182no—
521C2C2×((C3⋊D3)⋊C3)⋊Q811yescharacteristic
522C2#1172248211yescharacteristic
523C2#1172249211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC22×(C3⋊D3)⋊C3C23
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#544
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
1. #11721716C2semidirect
3. #64078⋊C3C2semidirect
7. #11721116C2semidirect
8. #11721093C2semidirect
9. #11721087C2semidirect
11. #11721079C2semidirect
38. #64078D3semidirect
71. C4×(C3⋊D3)⋊C3Q8semidirect
193. C4×C3⋊D3Dic6semidirect
336. H(3,𝔽3)#544semidirect
470. C32#5519semidirect
471. C32#5729semidirect

Maximal subgroups

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

subgroupindex
#117217162
#64081⋊C32
#64078⋊C32
#117211342
#117211292
#117211282
#117211162
#117210932
#117210872
#117210812
#117210792
#105939593
#105938553