Finite Group Databasebeta

← #13147819

Subgroups — #13147819

Order 1728.

789 conjugacy classes of subgroups.

The subgroup lattice has 527 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721715C211yesmaximal, characteristic, complemented
2#64081⋊C3C211yesmaximal, characteristic
3#64079⋊C3C211yesmaximal, characteristic, complemented
4#11721135C212yesmaximal
5#11721133C212yesmaximal
6#11721129C211yesmaximal, characteristic
7#11721116C211yesmaximal, characteristic, complemented
8#11721093C211yesmaximal, characteristic, complemented
9#11721085C212yesmaximal, complemented
10#11721081C211yesmaximal, characteristic
11#11721080C212yesmaximal, complemented
12#10593954—31nomaximal
13#10593855—31nomaximal
14C2×#6229C2212yes—
15C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
16C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
17C2×#6196C2212yes—
18C42×H(3,𝔽3)C2211yescharacteristic
19C4⋊#1084C2211yescharacteristic
20C4⋊#1084C2212yes—
21C4⋊#1084C2211yescharacteristic
22#3752⋊C3C2211yescharacteristic
23#3752⋊C3C2211yescharacteristic
24C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
25C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
26#3750⋊C3C2211yescharacteristic
27#3750⋊C3C2212yes—
28#3750⋊C3C2211yescharacteristic
29#3750⋊C3C2212yes—
30C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
31#88450C2212yes—
32#88449C2212yes—
33#88448C2212yes—
34#88448C2212yes—
35#88448C2212yes—
36#88447C2212yes—
37#64081D311yescharacteristic
38#64079D311yescharacteristic, complemented
39C3×#872—31no—
40C3×#870—31no—
41C3×#869—31no—
42C3×#869—31no—
43C3×#869—61no—
44#63919—31no—
45#63918—31no—
46#63900—31no—
47#63899—32no—
48#63895—32no—
49D3×C4⋊Dic3—31no—
50#63888—32no—
51#63887—32no—
52#63872—31no—
53#63869—31no—
54#63849—31no—
55#63847—32no—
56#63846—32no—
57#63841—31no—
58#63839—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—
66#6229—42no—
67C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
68C2×(C3⋊Dic3)⋊C3C2312yes—
69C2×(C3⋊Dic3)⋊C3C2312yes—
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C4×(C3⋊D3)⋊C3Q814yescomplemented
73((C3⋊D3)⋊C3)⋊C4—42no—
74#6196—42no—
75#5729—91no—
76#5689—91no—
77#5519—91no—
78C2×C4×C3⋊D3C2×D311yescharacteristic
79C2×C3×C4×D3—31no—
80C2×C3×Dic6—32no—
81C2×C3×Dic6—32no—
82C2×C3×Dic6—62no—
83C2×#627—32no—
84C2×#627—32no—
85C2×#624—32no—
86C2×D3×Dic3—32no—
87C32×C42—21no—
88C32×C42C2×D311yescharacteristic
89C32×C42C2×D311yescharacteristic
90#3752C2×D311yescharacteristic
91#3752C2×D311yescharacteristic
92C4⋊(C3⋊Dic3)C2×D311yescharacteristic
93C4⋊(C3⋊Dic3)C2×D311yescharacteristic
94#3750C2×D312yes—
95#3750C2×D312yes—
96#3750C2×D311yescharacteristic
97#3750C2×D311yescharacteristic
98C4×C3⋊Dic3C2×D311yescharacteristic
99C3×(C2×D3)⋊C4—31no—
100C3×(C2×D3)⋊C4—31no—
101C3×C4⋊Dic3—31no—
102C3×C4⋊Dic3—32no—
103C3×C4⋊Dic3—32no—
104C3×C4⋊Dic3—31no—
105C3×C4⋊Dic3—31no—
106C3×C4⋊Dic3—31no—
107C3×C4⋊Dic3—31no—
108C3×C4⋊Dic3—31no—
109C3×C4⋊Dic3—61no—
110C3×C4⋊Dic3—62no—
111C3×C4⋊Dic3—61no—
112C3×Dic3⋊C4—31no—
113C3×Dic3⋊C4—32no—
114C3×Dic3⋊C4—31no—
115C3×Dic3⋊C4—32no—
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—91no—
159#890—92no—
160#889—92no—
161#889—92no—
162#888—91no—
163#888—91no—
164#888—92no—
165#888—92no—
166#878—92no—
167#878—92no—
168#872—31no—
169#872—91no—
170#870—31no—
171#870—91no—
172#870—91no—
173#869—31no—
174#869—31no—
175#869—61no—
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×D311yescharacteristic
191C2×C3⋊Dic3C22×D312yes—
192C2×C3⋊Dic3C22×D311yescharacteristic
193C4×C3⋊D3Dic614yescomplemented
194C2×C3×Dic3—31no—
195C2×C3×Dic3—32no—
196C2×C3×Dic3—31no—
197C2×C3×Dic3—31no—
198C2×C3×Dic3—32no—
199C2×C3×Dic3—32no—
200C2×C3×Dic3—32no—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—62no—
204C2×C3×Dic3—62no—
205C3×C4×D3—34no—
206C3×Dic6—122no—
207C3×Dic6—122no—
208C3×Dic6—124no—
209#627—122no—
210#627—122no—
211#624—122no—
212D3×Dic3—122no—
213#544—271noSylow
214C2×H(3,𝔽3)C22×Q811yescharacteristic
215C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
216C2×H(3,𝔽3)2−1+411yescharacteristic
217(C3⋊D3)⋊C3—22no—
218C2×C3×Q8—92no—
219C22×C3×C4—91no—
220C22×Dic3—92no—
221C2×C4×D3—31no—
222C2×C4×D3—91no—
223C2×C4×D3—92no—
224C2×C4×D3—92no—
225C2×Dic6—32no—
226C2×Dic6—32no—
227C2×Dic6—62no—
228C2×Dic6—92no—
229C2×Dic6—92no—
230C2×Dic6—92no—
231C3×C4⋊C4—91no—
232C3×C4⋊C4—91no—
233C3×C4⋊C4—92no—
234C3×C4⋊C4—92no—
235C3×C4⋊C4—91no—
236C3×C4⋊C4—91no—
237C3×C4⋊C4—91no—
238C3×C4⋊C4—92no—
239C3×C4⋊C4—91no—
240C3×C22⋊C4—91no—
241C3×C22⋊C4—91no—
242C3×C42—31no—
243C3×C42—31no—
244C3×C42—61no—
245C3×C42—91no—
246C3×C42D3211yescharacteristic
247#217—92no—
248(C2×D3)⋊C4—31no—
249(C2×D3)⋊C4—31no—
250(C2×D3)⋊C4—91no—
251(C2×D3)⋊C4—91no—
252(C2×D3)⋊C4—92no—
253(C2×D3)⋊C4—92no—
254C4⋊Dic3—31no—
255C4⋊Dic3—31no—
256C4⋊Dic3—32no—
257C4⋊Dic3—31no—
258C4⋊Dic3—32no—
259C4⋊Dic3—31no—
260C4⋊Dic3—31no—
261C4⋊Dic3—31no—
262C4⋊Dic3—61no—
263C4⋊Dic3—61no—
264C4⋊Dic3—62no—
265C4⋊Dic3—91no—
266C4⋊Dic3—91no—
267C4⋊Dic3—92no—
268C4⋊Dic3—92no—
269C4⋊Dic3—92no—
270C4⋊Dic3—92no—
271Dic3⋊C4—32no—
272Dic3⋊C4—32no—
273Dic3⋊C4—31no—
274Dic3⋊C4—31no—
275Dic3⋊C4—92no—
276Dic3⋊C4—92no—
277Dic3⋊C4—91no—
278Dic3⋊C4—91no—
279Dic3⋊C4—92no—
280Dic3⋊C4—92no—
281Dic3⋊C4—92no—
282Dic3⋊C4—92no—
283Dic3⋊C4—92no—
284Dic3⋊C4—92no—
285Dic3⋊C4—92no—
286Dic3⋊C4—92no—
287C4×Dic3—31no—
288C4×Dic3—91no—
289C4×Dic3—92no—
290C4×Dic3—92no—
291C4×Dic3—92no—
292C22×C32—21no—
293C22×C32C23×D311yescharacteristic
294C22×C32C23×D311yescharacteristic
295C2×C3⋊D3—21no—
296C2×C3⋊D3—21no—
297C2×C3⋊D3C2×Dic612yes—
298C2×C3×D3—32no—
299C2×C3×D3—61no—
300C2×C3×D3—61no—
301C32×C4—22no—
302C32×C4—22no—
303C32×C4—21no—
304C32×C4—21no—
305C32×C4—21no—
306C32×C4—21no—
307C32×C4—22no—
308C32×C4C2×Dic612yes—
309C32×C4D3×Q812yes—
310C3⋊Dic3—22no—
311C3⋊Dic3—21no—
312C3⋊Dic3—22no—
313C3⋊Dic3—21no—
314C3⋊Dic3C2×Dic612yes—
315C3×Dic3—32no—
316C3×Dic3—62no—
317C3×Dic3—61no—
318C3×Dic3—62no—
319C3×Dic3—62no—
320C3×Dic3—62no—
321C3×Dic3—62no—
322C3×Dic3—61no—
323C3×Dic3—62no—
324C3×Dic3—122no—
325C3×Dic3—122no—
326C4⋊Q8—91no—
327C4⋊Q8—271no—
328C4⋊Q8—272no—
329#125—91no—
330#125—271no—
331#125—272no—
332C22⋊Q8—272no—
333C22⋊Q8—272no—
334#117—91no—
335#117—271no—
336C2×C4⋊C4—271no—
337H(3,𝔽3)#54411yescharacteristic, complemented, Sylow
338C23×C3—91no—
339C22×D3—31no—
340C22×D3—91no—
341C3×Q8—362no—
342C2×C3×C4—31no—
343C2×C3×C4—31no—
344C2×C3×C4—31no—
345C2×C3×C4—31no—
346C2×C3×C4—31no—
347C2×C3×C4—31no—
348C2×C3×C4—61no—
349C2×C3×C4—61no—
350C2×C3×C4—61no—
351C2×C3×C4—94no—
352C2×C3×C4—91no—
353C2×C3×C4—92no—
354C2×C3×C4—91no—
355C2×C3×C4—91no—
356C2×C3×C4—92no—
357C2×C3×C4—92no—
358C2×C3×C4—92no—
359C2×C3×C4C2×D3211yescharacteristic
360C2×C3×C4C2×D3211yescharacteristic
361C2×C3×C4C2×D3211yescharacteristic
362C2×Dic3—31no—
363C2×Dic3—32no—
364C2×Dic3—32no—
365C2×Dic3—32no—
366C2×Dic3—32no—
367C2×Dic3—31no—
368C2×Dic3—31no—
369C2×Dic3—32no—
370C2×Dic3—32no—
371C2×Dic3—62no—
372C2×Dic3—62no—
373C2×Dic3—91no—
374C2×Dic3—91no—
375C2×Dic3—92no—
376C2×Dic3—91no—
377C2×Dic3—92no—
378C2×Dic3—362no—
379C4×D3—34no—
380C4×D3—94no—
381C4×D3—362no—
382C4×D3—362no—
383Dic6—122no—
384Dic6—122no—
385Dic6—124no—
386Dic6—362no—
387Dic6—362no—
388Dic6—362no—
389C2×C32—21no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32C22×Dic611yescharacteristic
393C2×C32C2×D3×Q811yescharacteristic
394C2×C32#100911yescharacteristic
395C2×C32#100911yescharacteristic
396C2×C32#101011yescharacteristic
397C2×C32#101011yescharacteristic
398C3⋊D3—22no—
399C3×D3—62no—
400C2×Q8—92no—
401C2×Q8—272no—
402C22×C4—91no—
403C22×C4—272no—
404C4⋊C4—91no—
405C4⋊C4—91no—
406C4⋊C4—91no—
407C4⋊C4—91no—
408C4⋊C4—91no—
409C4⋊C4—91no—
410C4⋊C4—92no—
411C4⋊C4—92no—
412C4⋊C4—92no—
413C4⋊C4—272no—
414C4⋊C4—272no—
415C4⋊C4—272no—
416C4⋊C4—272no—
417C22⋊C4—91no—
418C22⋊C4—91no—
419C22⋊C4—272no—
420C42—91no—
421C42—272no—
422C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
423C22×C3—31no—
424C22×C3—31no—
425C22×C3—61no—
426C22×C3—92no—
427C22×C3—181no—
428C22×C3—181no—
429C22×C3C22×D3211yescharacteristic
430C2×D3—32no—
431C2×D3—61no—
432C2×D3—61no—
433C2×D3—92no—
434C2×D3—181no—
435C2×D3—181no—
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—32no—
439C3×C4—32no—
440C3×C4—61no—
441C3×C4—61no—
442C3×C4—61no—
443C3×C4—61no—
444C3×C4—62no—
445C3×C4—62no—
446C3×C4—62no—
447C3×C4—92no—
448C3×C4—181no—
449C3×C4—182no—
450C3×C4—181no—
451C3×C4—182no—
452C3×C4—182no—
453C3×C4—182no—
454C3×C4D3×Dic612yes—
455Dic3—32no—
456Dic3—62no—
457Dic3—62no—
458Dic3—62no—
459Dic3—62no—
460Dic3—61no—
461Dic3—62no—
462Dic3—61no—
463Dic3—62no—
464Dic3—92no—
465Dic3—122no—
466Dic3—122no—
467Dic3—181no—
468Dic3—182no—
469Dic3—182no—
470Dic3—181no—
471C32—21no—
472C32#551911yescharacteristic, complemented
473C32#568911yescharacteristic, complemented
474C23—91no—
475Q8—362no—
476Q8—1082no—
477C2×C4—91no—
478C2×C4—94no—
479C2×C4—91no—
480C2×C4—91no—
481C2×C4—92no—
482C2×C4—92no—
483C2×C4—92no—
484C2×C4—92no—
485C2×C4—1082no—
486C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
487C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
488C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
489C2×C3—31no—
490C2×C3—31no—
491C2×C3—31no—
492C2×C3—31no—
493C2×C3—31no—
494C2×C3—31no—
495C2×C3—61no—
496C2×C3—61no—
497C2×C3—61no—
498C2×C3—182no—
499C2×C3C2×D3×Dic611yescharacteristic
500C2×C3#6429811yescharacteristic
501C2×C3#6430811yescharacteristic
502D3—62no—
503D3—182no—
504C22—92no—
505C22—181no—
506C22—181no—
507C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
508C4—21no—
509C4—21no—
510C4—92no—
511C4—181no—
512C4—182no—
513C4—182no—
514C4—182no—
515C4—182no—
516C4—181no—
517C4((C3⋊D3)⋊C3)⋊Q812yes—
518C3—31no—
519C3—31no—
520C3—61no—
521C3#1059395911yescharacteristic
522C2—182no—
523C2C2×((C3⋊D3)⋊C3)⋊Q811yescharacteristic
524C2#1172248311yescharacteristic
525C2#1172249311yescharacteristic
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. #11721715C2semidirect
3. #64079⋊C3C2semidirect
7. #11721116C2semidirect
8. #11721093C2semidirect
9. #11721085C2semidirect
11. #11721080C2semidirect
38. #64079D3semidirect
72. C4×(C3⋊D3)⋊C3Q8semidirect
193. C4×C3⋊D3Dic6semidirect
337. H(3,𝔽3)#544semidirect
472. C32#5519semidirect
473. C32#5689semidirect

Maximal subgroups

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

subgroupindex
#117217152
#64081⋊C32
#64079⋊C32
#117211352
#117211332
#117211292
#117211162
#117210932
#117210852
#117210812
#117210802
#105939543
#105938553