Finite Group Databasebeta

← #13149595

Subgroups — #13149595

Order 1728.

853 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721760C211yesmaximal, characteristic
2#64099⋊C3C211yesmaximal, characteristic, complemented
3#64097⋊C3C211yesmaximal, characteristic, complemented
4#11721150C212yesmaximal
5#11721148C212yesmaximal
6C4×#6199C211yesmaximal, characteristic
7#11721116C211yesmaximal, characteristic, complemented
8#11721093C211yesmaximal, characteristic, complemented
9#11721089C211yesmaximal, characteristic
10#11721084C212yesmaximal, complemented
11#11721078C212yesmaximal, complemented
12#10594477—31nomaximal
13#10594445—31nomaximal
14C2×C4×#272C2211yescharacteristic
15C2×#634⋊C3C2212yes—
16C2×#6199C2211yescharacteristic
17C2×#6199C2211yescharacteristic
18C2×#6196C2212yes—
19C4⋊C4×H(3,𝔽3)C2211yescharacteristic
20#88542C2212yes—
21C4⋊#1084C2211yescharacteristic
22C4×#1084C2212yes—
23C4⋊((C3⋊Dic3)⋊C3)C2212yes—
24C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
25C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
26#3750⋊C3C2212yes—
27#3750⋊C3C2212yes—
28C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
29C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
30#88451C2212yes—
31#88449C2212yes—
32#88448C2212yes—
33#88448C2212yes—
34#88447C2211yescharacteristic
35#88447C2212yes—
36#88447C2211yescharacteristic
37#64099D311yescharacteristic, complemented
38#64097D311yescharacteristic, complemented
39C3×#892—31no—
40C3×#892—31no—
41C3×#892—61no—
42C3×#890—31no—
43C3×#888—31no—
44#63919—31no—
45#63918—31no—
46#63899—32no—
47#63896—32no—
48#63893—32no—
49#63887—32no—
50C4×D3×Dic3—31no—
51C4×D3×Dic3—31no—
52#63872—31no—
53#63869—31no—
54#63844—32no—
55Dic3×Dic6—32no—
56#63841—31no—
57#63841—31no—
58#63837—32no—
59#63837—32no—
60C22×#272C2311yescharacteristic
61C2×C4×H(3,𝔽3)C2312yes—
62C2×C4×H(3,𝔽3)C2311yescharacteristic
63C2×#1084C2312yes—
64C2×#1084C2311yescharacteristic
65C4×#272Q814yescomplemented
66C2×(C3⋊Dic3)⋊C3C2312yes—
67C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
68C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
69C2×(C3⋊Dic3)⋊C3C2312yes—
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72#634⋊C3—24no—
73#6199—22no—
74#6199—22no—
75#6196—24no—
76#5728—91no—
77#5688—91no—
78#5517—91no—
79C2×#634C2×D312yes—
80C2×C3×C4×D3—31no—
81C2×C3×C4×D3—31no—
82C2×C3×C4×D3—61no—
83C2×C3×Dic6—32no—
84C2×#627—32no—
85C2×#627—32no—
86C2×D3×Dic3—31no—
87C2×D3×Dic3—31no—
88C2×D3×Dic3—31no—
89C2×D3×Dic3—31no—
90C32×C4⋊C4—21no—
91C32×C4⋊C4C2×D311yescharacteristic
92C32×C4⋊C4C2×D311yescharacteristic
93C4⋊(C3⋊Dic3)C2×D311yescharacteristic
94C4⋊(C3⋊Dic3)C2×D311yescharacteristic
95C4⋊(C3⋊Dic3)C2×D312yes—
96#3750C2×D312yes—
97#3750C2×D312yes—
98C4×C3⋊Dic3C2×D311yescharacteristic
99C4×C3⋊Dic3C2×D311yescharacteristic
100C3×(C2×D3)⋊C4—32no—
101C3×(C2×D3)⋊C4—32no—
102C3×(C2×D3)⋊C4—62no—
103C3×C4⋊Dic3—31no—
104C3×C4⋊Dic3—31no—
105C3×C4⋊Dic3—32no—
106C3×C4⋊Dic3—31no—
107C3×C4⋊Dic3—31no—
108C3×C4⋊Dic3—61no—
109C3×Dic3⋊C4—32no—
110C3×Dic3⋊C4—32no—
111C3×C4×Dic3—31no—
112C3×C4×Dic3—31no—
113C3×C4×Dic3—32no—
114C3×C4×Dic3—32no—
115C3×C4×Dic3—62no—
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—
124Dic32—31no—
125Dic32—31no—
126Dic32—32no—
127Dic32—32no—
128Dic32—31no—
129Dic32—31no—
130C22×H(3,𝔽3)C2411yescharacteristic
131C2×#272—21no—
132C2×#272—21no—
133C2×#272C2×Q812yes—
134C4×H(3,𝔽3)—22no—
135C4×H(3,𝔽3)C2×Q812yes—
136#1084—22no—
137#1084C2×Q812yes—
138(C3⋊Dic3)⋊C3—22no—
139(C3⋊Dic3)⋊C3—21no—
140(C3⋊Dic3)⋊C3—21no—
141(C3⋊Dic3)⋊C3—22no—
142(C3⋊Dic3)⋊C3#4112yes—
143(C3⋊Dic3)⋊C3#4112yes—
144C3×C4⋊Q8—91no—
145C3×#125—91no—
146C3×#117—91no—
147#946—92no—
148Q8×Dic3—92no—
149#926—91no—
150C2×C4×Dic3—91no—
151#897—92no—
152#896—92no—
153#896—92no—
154#892—31no—
155#892—31no—
156#892—61no—
157#892—91no—
158#892—91no—
159#890—31no—
160#890—91no—
161#890—91no—
162#890—91no—
163#888—31no—
164#888—91no—
165#888—91no—
166#888—91no—
167#887—92no—
168#887—92no—
169#887—92no—
170#878—92no—
171#878—92no—
172C42×D3—91no—
173C42×D3—91no—
174#870—91no—
175#869—91no—
176C4×Dic6—92no—
177C4×Dic6—92no—
178C22×C3×D3—31no—
179C22×C3×D3—31no—
180C22×C3×D3—61no—
181C2×C32×C4—21no—
182C2×C32×C4—22no—
183C2×C32×C4C22×D312yes—
184C2×C32×C4C22×D312yes—
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C22×D311yescharacteristic
187C2×C3⋊Dic3C22×D312yes—
188C2×C3⋊Dic3C22×D312yes—
189C2×C3⋊Dic3C22×D311yescharacteristic
190C2×C3⋊Dic3C22×D311yescharacteristic
191C2×C3⋊Dic3C22×D311yescharacteristic
192C2×C3⋊Dic3C22×D311yescharacteristic
193#634—24no—
194C2×C3×Dic3—31no—
195C2×C3×Dic3—31no—
196C2×C3×Dic3—31no—
197C2×C3×Dic3—31no—
198C2×C3×Dic3—32no—
199C2×C3×Dic3—32no—
200C2×C3×Dic3—31no—
201C2×C3×Dic3—31no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—61no—
205C2×C3×Dic3—62no—
206C3×C4×D3—34no—
207C3×C4×D3—34no—
208C3×C4×D3—64no—
209C3×Dic6—64no—
210#627—64no—
211#627—64no—
212D3×Dic3—62no—
213D3×Dic3—62no—
214D3×Dic3—62no—
215D3×Dic3—62no—
216#533—271noSylow
217C2×H(3,𝔽3)C22×Q811yescharacteristic
218C2×H(3,𝔽3)C2×#4111yescharacteristic
219C2×H(3,𝔽3)C2×#4111yescharacteristic
220#272—22no—
221C2×C3×Q8—92no—
222C22×C3×C4—91no—
223C22×Dic3—91no—
224C22×Dic3—91no—
225C2×C4×D3—31no—
226C2×C4×D3—31no—
227C2×C4×D3—61no—
228C2×C4×D3—91no—
229C2×C4×D3—91no—
230C2×C4×D3—91no—
231C2×C4×D3—91no—
232C2×Dic6—32no—
233C2×Dic6—92no—
234C2×Dic6—92no—
235C2×Dic6—92no—
236C2×Dic6—92no—
237C3×C4⋊C4—31no—
238C3×C4⋊C4—31no—
239C3×C4⋊C4—61no—
240C3×C4⋊C4—92no—
241C3×C4⋊C4—92no—
242C3×C4⋊C4—92no—
243C3×C4⋊C4—91no—
244C3×C4⋊C4—91no—
245C3×C4⋊C4—91no—
246C3×C4⋊C4D3211yescharacteristic
247C3×C22⋊C4—92no—
248C3×C42—91no—
249C3×C42—91no—
250C3×C42—92no—
251#217—92no—
252(C2×D3)⋊C4—32no—
253(C2×D3)⋊C4—32no—
254(C2×D3)⋊C4—62no—
255(C2×D3)⋊C4—92no—
256(C2×D3)⋊C4—92no—
257C4⋊Dic3—31no—
258C4⋊Dic3—31no—
259C4⋊Dic3—32no—
260C4⋊Dic3—31no—
261C4⋊Dic3—31no—
262C4⋊Dic3—61no—
263C4⋊Dic3—91no—
264C4⋊Dic3—91no—
265C4⋊Dic3—92no—
266C4⋊Dic3—92no—
267C4⋊Dic3—92no—
268C4⋊Dic3—92no—
269Dic3⋊C4—32no—
270Dic3⋊C4—32no—
271Dic3⋊C4—92no—
272Dic3⋊C4—92no—
273Dic3⋊C4—92no—
274Dic3⋊C4—92no—
275Dic3⋊C4—92no—
276Dic3⋊C4—92no—
277Dic3⋊C4—92no—
278Dic3⋊C4—92no—
279C4×Dic3—32no—
280C4×Dic3—32no—
281C4×Dic3—31no—
282C4×Dic3—31no—
283C4×Dic3—62no—
284C4×Dic3—91no—
285C4×Dic3—91no—
286C4×Dic3—91no—
287C4×Dic3—91no—
288C4×Dic3—92no—
289C4×Dic3—91no—
290C4×Dic3—91no—
291C4×Dic3—91no—
292C4×Dic3—91no—
293C4×Dic3—92no—
294C4×Dic3—92no—
295C22×C32—21no—
296C22×C32C23×D311yescharacteristic
297C22×C32C23×D311yescharacteristic
298C2×C3×D3—32no—
299C2×C3×D3—32no—
300C2×C3×D3—62no—
301C2×C3×D3—61no—
302C2×C3×D3—61no—
303C2×C3×D3—61no—
304C2×C3×D3—61no—
305C2×C3×D3—121no—
306C2×C3×D3—121no—
307C32×C4—22no—
308C32×C4—22no—
309C32×C4—22no—
310C32×C4—42no—
311C32×C4D3×Q812yes—
312C32×C4D3×Q812yes—
313C3⋊Dic3—22no—
314C3⋊Dic3—21no—
315C3⋊Dic3—21no—
316C3⋊Dic3—22no—
317C3⋊Dic3#23712yes—
318C3⋊Dic3#23912yes—
319C3×Dic3—32no—
320C3×Dic3—32no—
321C3×Dic3—32no—
322C3×Dic3—32no—
323C3×Dic3—62no—
324C3×Dic3—61no—
325C3×Dic3—61no—
326C3×Dic3—62no—
327C3×Dic3—62no—
328C3×Dic3—62no—
329C3×Dic3—62no—
330C3×Dic3—122no—
331C4⋊Q8—91no—
332C4⋊Q8—271no—
333#125—91no—
334#125—271no—
335C22⋊Q8—272no—
336C22⋊Q8—272no—
337C4×Q8—272no—
338C4×Q8—272no—
339#117—91no—
340#117—271no—
341C2×C42—271no—
342H(3,𝔽3)#53311yescharacteristic, complemented, Sylow
343C23×C3—91no—
344C22×D3—31no—
345C22×D3—31no—
346C22×D3—61no—
347C3×Q8—184no—
348C2×C3×C4—31no—
349C2×C3×C4—31no—
350C2×C3×C4—32no—
351C2×C3×C4—32no—
352C2×C3×C4—61no—
353C2×C3×C4—62no—
354C2×C3×C4—91no—
355C2×C3×C4—91no—
356C2×C3×C4—91no—
357C2×C3×C4—94no—
358C2×C3×C4—91no—
359C2×C3×C4—91no—
360C2×C3×C4—92no—
361C2×C3×C4—92no—
362C2×C3×C4—92no—
363C2×C3×C4C2×D3212yes—
364C2×C3×C4C2×D3211yescharacteristic
365C2×Dic3—31no—
366C2×Dic3—31no—
367C2×Dic3—31no—
368C2×Dic3—31no—
369C2×Dic3—32no—
370C2×Dic3—32no—
371C2×Dic3—31no—
372C2×Dic3—31no—
373C2×Dic3—32no—
374C2×Dic3—32no—
375C2×Dic3—61no—
376C2×Dic3—62no—
377C2×Dic3—91no—
378C2×Dic3—91no—
379C2×Dic3—91no—
380C2×Dic3—91no—
381C2×Dic3—92no—
382C2×Dic3—92no—
383C2×Dic3—182no—
384C2×Dic3—182no—
385C4×D3—34no—
386C4×D3—34no—
387C4×D3—64no—
388C4×D3—182no—
389C4×D3—182no—
390C4×D3—182no—
391C4×D3—182no—
392Dic6—64no—
393Dic6—184no—
394Dic6—184no—
395Dic6—184no—
396Dic6—184no—
397C2×C32—21no—
398C2×C32—21no—
399C2×C32—21no—
400C2×C32C2×#23711yescharacteristic
401C2×C32C2×D3×Q811yescharacteristic
402C2×C32C2×D3×Q811yescharacteristic
403C2×C32C2×#23911yescharacteristic
404C2×C32D3×#4111yescharacteristic
405C2×C32D3×#4111yescharacteristic
406C3×D3—62no—
407C3×D3—62no—
408C3×D3—122no—
409C2×Q8—92no—
410C2×Q8—272no—
411C22×C4—91no—
412C22×C4—271no—
413C22×C4—271no—
414C4⋊C4—91no—
415C4⋊C4—91no—
416C4⋊C4—91no—
417C4⋊C4—92no—
418C4⋊C4—92no—
419C4⋊C4—92no—
420C4⋊C4—272no—
421C4⋊C4—272no—
422C4⋊C4—272no—
423C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
424C22⋊C4—92no—
425C22⋊C4—272no—
426C42—91no—
427C42—91no—
428C42—92no—
429C42—271no—
430C42—271no—
431C42—272no—
432C22×C3—31no—
433C22×C3—31no—
434C22×C3—61no—
435C22×C3—92no—
436C22×C3—181no—
437C22×C3—181no—
438C22×C3C22×D3211yescharacteristic
439C2×D3—32no—
440C2×D3—32no—
441C2×D3—62no—
442C2×D3—61no—
443C2×D3—61no—
444C2×D3—61no—
445C2×D3—61no—
446C2×D3—121no—
447C2×D3—121no—
448C3×C4—22no—
449C3×C4—32no—
450C3×C4—32no—
451C3×C4—62no—
452C3×C4—62no—
453C3×C4—62no—
454C3×C4—92no—
455C3×C4—92no—
456C3×C4—92no—
457C3×C4—122no—
458C3×C4—182no—
459C3×C4—181no—
460C3×C4—181no—
461C3×C4—182no—
462C3×C4—182no—
463C3×C4#379712yes—
464Dic3—32no—
465Dic3—32no—
466Dic3—32no—
467Dic3—32no—
468Dic3—62no—
469Dic3—62no—
470Dic3—62no—
471Dic3—62no—
472Dic3—61no—
473Dic3—62no—
474Dic3—61no—
475Dic3—92no—
476Dic3—92no—
477Dic3—122no—
478Dic3—182no—
479Dic3—182no—
480Dic3—181no—
481Dic3—181no—
482C32—21no—
483C32#568811yescharacteristic, complemented
484C32#572811yescharacteristic, complemented
485C23—91no—
486Q8—184no—
487Q8—544no—
488C2×C4—91no—
489C2×C4—91no—
490C2×C4—91no—
491C2×C4—94no—
492C2×C4—91no—
493C2×C4—91no—
494C2×C4—92no—
495C2×C4—92no—
496C2×C4—92no—
497C2×C4—542no—
498C2×C4—542no—
499C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
500C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
501C2×C3—31no—
502C2×C3—31no—
503C2×C3—31no—
504C2×C3—31no—
505C2×C3—31no—
506C2×C3—31no—
507C2×C3—61no—
508C2×C3—61no—
509C2×C3—61no—
510C2×C3—182no—
511C2×C3C2×#379711yescharacteristic
512C2×C3D3×#23711yescharacteristic
513C2×C3D3×#23911yescharacteristic
514D3—62no—
515D3—62no—
516D3—122no—
517C22—92no—
518C22—181no—
519C22—181no—
520C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
521C4—22no—
522C4—92no—
523C4—92no—
524C4—92no—
525C4—181no—
526C4—182no—
527C4—181no—
528C4—182no—
529C4—182no—
530C4#8865112yes—
531C3—31no—
532C3—31no—
533C3—61no—
534C3#1059444811yescharacteristic
535C2—182no—
536C2C2×#8865111yescharacteristic
537C2#1172250411yescharacteristic
538C2#1172251911yescharacteristic
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)#533
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. #64097⋊C3C2semidirect
7. #11721116C2semidirect
8. #11721093C2semidirect
10. #11721084C2semidirect
11. #11721078C2semidirect
37. #64099D3semidirect
38. #64097D3semidirect
65. C4×#272Q8semidirect
342. H(3,𝔽3)#533semidirect
483. C32#5688semidirect
484. C32#5728semidirect

Maximal subgroups

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

subgroupindex
#117217602
#64099⋊C32
#64097⋊C32
#117211502
#117211482
C4×#61992
#117211162
#117210932
#117210892
#117210842
#117210782
#105944773
#105944453