Finite Group Databasebeta

← #13149382

Subgroups — #13149382

Order 1728.

746 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721765C211yesmaximal, characteristic
2#64099⋊C3C211yesmaximal, characteristic, complemented
3#64096⋊C3C211yesmaximal, characteristic, complemented
4#11721097C211yesmaximal, characteristic
5#11721094C211yesmaximal, characteristic
6#11721093C212yesmaximal, complemented
7#11721092C212yesmaximal
8#11721090C211yesmaximal, characteristic, complemented
9#11721088C212yesmaximal
10#11721085C212yesmaximal, complemented
11#11721083C211yesmaximal, characteristic, complemented
12#10594346—31nomaximal
13#10594345—31nomaximal
14C2×C4×#272C2211yescharacteristic
15C2×#6229C2211yescharacteristic
16C2×#634⋊C3C2211yescharacteristic
17C4⋊C4×H(3,𝔽3)C2211yescharacteristic
18#88542C2212yes—
19C4⋊#1084C2212yes—
20C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
21C4⋊((C3⋊Dic3)⋊C3)C2212yes—
22#3750⋊C3C2212yes—
23#3750⋊C3C2212yes—
24C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
25C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
26C4×(C3⋊Dic3)⋊C3C2212yes—
27#88451C2211yescharacteristic
28#88451C2212yes—
29#88451C2211yescharacteristic
30#88449C2212yes—
31#88449C2211yescharacteristic
32#88449C2211yescharacteristic
33#88448C2212yes—
34#88448C2212yes—
35#88448C2212yes—
36#88447C2212yes—
37#64099D311yescharacteristic, complemented
38#64096D311yescharacteristic, complemented
39C3×#897—31no—
40C3×#897—31no—
41C3×#897—61no—
42C3×#890—31no—
43C3×#887—31no—
44#63873—31no—
45#63872—32no—
46#63869—32no—
47#63866—31no—
48#63862—32no—
49#63861—32no—
50#63859—32no—
51#63858—32no—
52#63857—31no—
53#63856—31no—
54#63855—31no—
55#63855—31no—
56#63847—32no—
57#63846—32no—
58#63843—31no—
59Dic3×Dic6—31no—
60C22×#272C2311yescharacteristic
61C2×C4×H(3,𝔽3)C2312yes—
62C2×C4×H(3,𝔽3)C2311yescharacteristic
63C2×#1084C2312yes—
64C2×#1084C2311yescharacteristic
65C4×#272—22no—
66#6229—22no—
67C2×(C3⋊Dic3)⋊C3C2312yes—
68C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
69C2×(C3⋊Dic3)⋊C3C2312yes—
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
73#634⋊C3—22no—
74#5698—91no—
75#5625—91no—
76#5583—91no—
77C2×#634C2×D311yescharacteristic
78C2×C3×C4×D3—31no—
79C2×C3×C4×D3—31no—
80C2×C3×C4×D3—61no—
81C2×C3×Dic6—31no—
82C2×C3×Dic6—31no—
83C2×C3×Dic6—31no—
84C2×C3×Dic6—61no—
85C32×C4⋊C4—21no—
86C32×C4⋊C4C2×D311yescharacteristic
87C32×C4⋊C4C2×D311yescharacteristic
88C4⋊(C3⋊Dic3)C2×D311yescharacteristic
89C4⋊(C3⋊Dic3)C2×D312yes—
90#3750C2×D312yes—
91#3750C2×D312yes—
92C4×C3⋊Dic3C2×D311yescharacteristic
93C4×C3⋊Dic3C2×D312yes—
94C4×C3⋊Dic3C2×D311yescharacteristic
95C3×(C2×D3)⋊C4—32no—
96C3×(C2×D3)⋊C4—32no—
97C3×(C2×D3)⋊C4—62no—
98C3×C4⋊Dic3—31no—
99C3×C4⋊Dic3—32no—
100C3×C4⋊Dic3—32no—
101C3×C4⋊Dic3—32no—
102C3×C4⋊Dic3—62no—
103C3×Dic3⋊C4—32no—
104C3×Dic3⋊C4—32no—
105C3×C4×Dic3—31no—
106C3×C4×Dic3—32no—
107C3×C4×Dic3—31no—
108#3724—32no—
109#3724—32no—
110#3724—32no—
111Dic3⋊Dic3—32no—
112Dic3⋊Dic3—32no—
113Dic3⋊Dic3—31no—
114Dic3⋊Dic3—32no—
115Dic3⋊Dic3—32no—
116Dic3⋊Dic3—31no—
117Dic3⋊Dic3—31no—
118Dic3⋊Dic3—31no—
119Dic3⋊Dic3—32no—
120(C2×D3)⋊Dic3—31no—
121(C2×D3)⋊Dic3—31no—
122(C2×D3)⋊Dic3—31no—
123(C2×D3)⋊Dic3—31no—
124(C2×D3)⋊Dic3—32no—
125(C2×D3)⋊Dic3—32no—
126Dic32—32no—
127Dic32—32no—
128C22×H(3,𝔽3)C2411yescharacteristic
129C2×#272—21no—
130C2×#272—21no—
131C2×#272—21no—
132C4×H(3,𝔽3)—22no—
133C4×H(3,𝔽3)#4112yes—
134#1084—22no—
135#1084—21no—
136(C3⋊Dic3)⋊C3—22no—
137(C3⋊Dic3)⋊C3—22no—
138(C3⋊Dic3)⋊C3—21no—
139(C3⋊Dic3)⋊C3—21no—
140(C3⋊Dic3)⋊C3—21no—
141(C3⋊Dic3)⋊C3—21no—
142C3×#125—91no—
143C3×C22⋊Q8—91no—
144C3×C4×Q8—91no—
145#946—91no—
146Q8×Dic3—91no—
147#944—91no—
148#944—91no—
149#926—91no—
150#924—91no—
151#898—92no—
152#898—92no—
153#898—92no—
154#897—31no—
155#897—31no—
156#897—61no—
157#897—91no—
158#890—31no—
159#890—91no—
160#890—92no—
161#890—92no—
162#890—92no—
163#889—92no—
164#889—92no—
165#887—31no—
166#887—91no—
167#887—91no—
168#879—92no—
169#879—92no—
170#876—92no—
171#872—91no—
172#872—91no—
173#870—92no—
174#869—91no—
175C4×Dic6—91no—
176C22×C3×D3—31no—
177C22×C3×D3—31no—
178C22×C3×D3—61no—
179C2×C32×C4—21no—
180C2×C32×C4—22no—
181C2×C32×C4C22×D312yes—
182C2×C32×C4C22×D312yes—
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C22×D311yescharacteristic
185C2×C3⋊Dic3C22×D311yescharacteristic
186C2×C3⋊Dic3C22×D312yes—
187C2×C3⋊Dic3C22×D311yescharacteristic
188C2×C3⋊Dic3C22×D312yes—
189C2×C3⋊Dic3C22×D311yescharacteristic
190C2×C3⋊Dic3C22×D311yescharacteristic
191#634—22no—
192C2×C3×Dic3—31no—
193C2×C3×Dic3—32no—
194C2×C3×Dic3—31no—
195C2×C3×Dic3—31no—
196C2×C3×Dic3—31no—
197C2×C3×Dic3—32no—
198C2×C3×Dic3—32no—
199C2×C3×Dic3—32no—
200C2×C3×Dic3—31no—
201C2×C3×Dic3—31no—
202C2×C3×Dic3—61no—
203C2×C3×Dic3—62no—
204C3×C4×D3—62no—
205C3×C4×D3—62no—
206C3×C4×D3—122no—
207C3×Dic6—62no—
208C3×Dic6—62no—
209C3×Dic6—62no—
210C3×Dic6—122no—
211#541—271noSylow
212C2×H(3,𝔽3)C2×#4111yescharacteristic
213C2×H(3,𝔽3)2−1+411yescharacteristic
214C2×H(3,𝔽3)2−1+411yescharacteristic
215#272—41no—
216C2×C3×Q8—91no—
217C2×C3×Q8—91no—
218C22×C3×C4—91no—
219C2×C4×D3—31no—
220C2×C4×D3—31no—
221C2×C4×D3—61no—
222C2×Dic6—31no—
223C2×Dic6—31no—
224C2×Dic6—31no—
225C2×Dic6—61no—
226C2×Dic6—91no—
227C3×C4⋊C4—31no—
228C3×C4⋊C4—31no—
229C3×C4⋊C4—61no—
230C3×C4⋊C4—91no—
231C3×C4⋊C4—92no—
232C3×C4⋊C4—92no—
233C3×C4⋊C4—92no—
234C3×C4⋊C4—92no—
235C3×C4⋊C4D3211yescharacteristic
236C3×C22⋊C4—92no—
237C3×C42—91no—
238C3×C42—91no—
239C3×C42—92no—
240#217—92no—
241#217—91no—
242#217—91no—
243(C2×D3)⋊C4—32no—
244(C2×D3)⋊C4—32no—
245(C2×D3)⋊C4—62no—
246(C2×D3)⋊C4—91no—
247(C2×D3)⋊C4—91no—
248(C2×D3)⋊C4—91no—
249(C2×D3)⋊C4—91no—
250(C2×D3)⋊C4—92no—
251(C2×D3)⋊C4—92no—
252C4⋊Dic3—32no—
253C4⋊Dic3—32no—
254C4⋊Dic3—32no—
255C4⋊Dic3—31no—
256C4⋊Dic3—62no—
257C4⋊Dic3—91no—
258C4⋊Dic3—92no—
259C4⋊Dic3—92no—
260C4⋊Dic3—91no—
261C4⋊Dic3—91no—
262C4⋊Dic3—92no—
263C4⋊Dic3—92no—
264C4⋊Dic3—92no—
265Dic3⋊C4—32no—
266Dic3⋊C4—32no—
267Dic3⋊C4—92no—
268Dic3⋊C4—92no—
269Dic3⋊C4—92no—
270Dic3⋊C4—92no—
271Dic3⋊C4—92no—
272Dic3⋊C4—92no—
273Dic3⋊C4—91no—
274Dic3⋊C4—92no—
275Dic3⋊C4—92no—
276Dic3⋊C4—91no—
277Dic3⋊C4—91no—
278Dic3⋊C4—91no—
279Dic3⋊C4—92no—
280Dic3⋊C4—92no—
281C4×Dic3—32no—
282C4×Dic3—31no—
283C4×Dic3—31no—
284C4×Dic3—91no—
285C4×Dic3—91no—
286C4×Dic3—92no—
287C4×Dic3—92no—
288C4×Dic3—92no—
289C4×Dic3—92no—
290C22×C32—21no—
291C22×C32C23×D311yescharacteristic
292C22×C32C23×D311yescharacteristic
293C2×C3×D3—62no—
294C2×C3×D3—61no—
295C2×C3×D3—61no—
296C2×C3×D3—61no—
297C2×C3×D3—61no—
298C2×C3×D3—61no—
299C2×C3×D3—61no—
300C2×C3×D3—121no—
301C2×C3×D3—121no—
302C32×C4—22no—
303C32×C4—22no—
304C32×C4—22no—
305C32×C4—42no—
306C32×C4#23512yes—
307C32×C4#23912yes—
308C3⋊Dic3—22no—
309C3⋊Dic3—21no—
310C3⋊Dic3—21no—
311C3⋊Dic3—21no—
312C3⋊Dic3—22no—
313C3⋊Dic3—21no—
314C3×Dic3—62no—
315C3×Dic3—61no—
316C3×Dic3—62no—
317C3×Dic3—62no—
318C3×Dic3—61no—
319C3×Dic3—62no—
320C3×Dic3—61no—
321C3×Dic3—61no—
322C3×Dic3—61no—
323C3×Dic3—61no—
324C3×Dic3—122no—
325C3×Dic3—121no—
326C4⋊Q8—271no—
327#126—272no—
328#126—272no—
329#125—91no—
330#125—272no—
331#125—272no—
332C22⋊Q8—91no—
333C22⋊Q8—271no—
334C4×Q8—91no—
335C4×Q8—271no—
336#117—271no—
337H(3,𝔽3)#54111yescharacteristic, complemented, Sylow
338C23×C3—91no—
339C22×D3—31no—
340C22×D3—31no—
341C22×D3—61no—
342C3×Q8—182no—
343C3×Q8—182no—
344C2×C3×C4—31no—
345C2×C3×C4—31no—
346C2×C3×C4—32no—
347C2×C3×C4—32no—
348C2×C3×C4—61no—
349C2×C3×C4—62no—
350C2×C3×C4—91no—
351C2×C3×C4—92no—
352C2×C3×C4—91no—
353C2×C3×C4—92no—
354C2×C3×C4—91no—
355C2×C3×C4—91no—
356C2×C3×C4—92no—
357C2×C3×C4—91no—
358C2×C3×C4—182no—
359C2×C3×C4C2×D3212yes—
360C2×C3×C4C2×D3211yescharacteristic
361C2×Dic3—32no—
362C2×Dic3—32no—
363C2×Dic3—31no—
364C2×Dic3—31no—
365C2×Dic3—31no—
366C2×Dic3—31no—
367C2×Dic3—31no—
368C2×Dic3—32no—
369C2×Dic3—31no—
370C2×Dic3—32no—
371C2×Dic3—61no—
372C2×Dic3—62no—
373C2×Dic3—91no—
374C2×Dic3—91no—
375C2×Dic3—91no—
376C2×Dic3—91no—
377C2×Dic3—92no—
378C2×Dic3—92no—
379C4×D3—62no—
380C4×D3—62no—
381C4×D3—122no—
382Dic6—62no—
383Dic6—62no—
384Dic6—62no—
385Dic6—122no—
386Dic6—182no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32C2×#23511yescharacteristic
391C2×C32C2×#23911yescharacteristic
392C2×C32#100711yescharacteristic
393C2×C32#101011yescharacteristic
394C2×C32#101011yescharacteristic
395C2×C32#101011yescharacteristic
396C3×D3—122no—
397C3×D3—121no—
398C3×D3—121no—
399C2×Q8—91no—
400C2×Q8—91no—
401C22×C4—91no—
402C4⋊C4—92no—
403C4⋊C4—91no—
404C4⋊C4—92no—
405C4⋊C4—92no—
406C4⋊C4—92no—
407C4⋊C4—271no—
408C4⋊C4—271no—
409C4⋊C4—272no—
410C4⋊C4—272no—
411C4⋊C4—272no—
412C4⋊C4—272no—
413C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
414C22⋊C4—92no—
415C22⋊C4—271no—
416C22⋊C4—271no—
417C22⋊C4—272no—
418C42—91no—
419C42—91no—
420C42—92no—
421C42—272no—
422C22×C3—31no—
423C22×C3—31no—
424C22×C3—61no—
425C22×C3—181no—
426C22×C3—181no—
427C22×C3—181no—
428C22×C3C22×D3211yescharacteristic
429C2×D3—61no—
430C2×D3—61no—
431C2×D3—61no—
432C2×D3—61no—
433C2×D3—61no—
434C2×D3—61no—
435C2×D3—62no—
436C2×D3—121no—
437C2×D3—121no—
438C3×C4—22no—
439C3×C4—32no—
440C3×C4—32no—
441C3×C4—62no—
442C3×C4—62no—
443C3×C4—62no—
444C3×C4—122no—
445C3×C4—182no—
446C3×C4—181no—
447C3×C4—181no—
448C3×C4—181no—
449C3×C4—182no—
450C3×C4—181no—
451C3×C4—181no—
452C3×C4—182no—
453C3×C4#379912yes—
454Dic3—62no—
455Dic3—62no—
456Dic3—61no—
457Dic3—61no—
458Dic3—62no—
459Dic3—61no—
460Dic3—61no—
461Dic3—61no—
462Dic3—61no—
463Dic3—62no—
464Dic3—122no—
465Dic3—121no—
466Dic3—182no—
467Dic3—181no—
468Dic3—181no—
469Dic3—181no—
470Dic3—182no—
471Dic3—181no—
472C32—21no—
473C32#558311yescharacteristic, complemented
474C32#569811yescharacteristic, complemented
475C23—91no—
476Q8—182no—
477Q8—182no—
478C2×C4—91no—
479C2×C4—91no—
480C2×C4—91no—
481C2×C4—91no—
482C2×C4—92no—
483C2×C4—92no—
484C2×C4—92no—
485C2×C4—91no—
486C2×C4—182no—
487C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
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—361no—
499C2×C3C2×#379911yescharacteristic
500C2×C3#6430911yescharacteristic
501C2×C3#6431511yescharacteristic
502D3—121no—
503D3—121no—
504D3—122no—
505C22—181no—
506C22—181no—
507C22—181no—
508C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
509C4—22no—
510C4—181no—
511C4—181no—
512C4—181no—
513C4—181no—
514C4—182no—
515C4—181no—
516C4—182no—
517C4—182no—
518C4#8865212yes—
519C3—31no—
520C3—31no—
521C3—61no—
522C3#1059435511yescharacteristic
523C2—361no—
524C2C2×#8865211yescharacteristic
525C2#1172250211yescharacteristic
526C2#1172251611yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC4⋊C4×H(3,𝔽3)C22
prime radicalC22×#272C23
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#541
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #64099⋊C3C2semidirect
3. #64096⋊C3C2semidirect
6. #11721093C2semidirect
8. #11721090C2semidirect
10. #11721085C2semidirect
11. #11721083C2semidirect
37. #64099D3semidirect
38. #64096D3semidirect
337. H(3,𝔽3)#541semidirect
473. C32#5583semidirect
474. C32#5698semidirect

Maximal subgroups

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

subgroupindex
#117217652
#64099⋊C32
#64096⋊C32
#117210972
#117210942
#117210932
#117210922
#117210902
#117210882
#117210852
#117210832
#105943463
#105943453