Finite Group Databasebeta

← #13149502

Subgroups — #13149502

Order 1728.

833 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721758C211yesmaximal, characteristic, complemented
2#64101⋊C3C211yesmaximal, characteristic
3#64097⋊C3C211yesmaximal, characteristic, complemented
4#11721133C212yesmaximal
5#11721128C212yesmaximal
6C4×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic
7#11721116C211yesmaximal, characteristic, complemented
8#11721095C211yesmaximal, characteristic, complemented
9#11721082C212yesmaximal, complemented
10#11721081C211yesmaximal, characteristic
11#11721077C212yesmaximal, complemented
12#10594477—31nomaximal
13#10594448—31nomaximal
14C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
15C2×#634⋊C3C2212yes—
16C2×((C3⋊D3)⋊C3)⋊C4C2211yescharacteristic
17C2×((C3⋊D3)⋊C3)⋊C4C2211yescharacteristic
18C2×#6196C2212yes—
19C4⋊C4×H(3,𝔽3)C2211yescharacteristic
20C4⋊#1084C2211yescharacteristic
21C4⋊#1084C2212yes—
22#88540C2212yes—
23C4×#1084C2211yescharacteristic
24#3752⋊C3C2212yes—
25C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
26C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
27#3750⋊C3C2212yes—
28C4×(C3⋊Dic3)⋊C3C2212yes—
29C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
30#88450C2212yes—
31#88448C2212yes—
32#88448C2212yes—
33#88448C2212yes—
34#88447C2211yescharacteristic
35#88447C2212yes—
36#88447C2211yescharacteristic
37#64101D311yescharacteristic
38#64097D311yescharacteristic, complemented
39C3×#892—31no—
40C3×#890—31no—
41C3×#890—31no—
42C3×#890—61no—
43C3×#888—31no—
44#63919—31no—
45#63918—31no—
46#63899—32no—
47#63896—32no—
48#63895—32no—
49#63888—32no—
50C4×#624—31no—
51C4×D3×Dic3—31no—
52#63869—31no—
53#63869—31no—
54#63849—31no—
55#63844—32no—
56#63843—32no—
57#63841—31no—
58#63838—32no—
59#63837—32no—
60C22×(C3⋊D3)⋊C3C2311yescharacteristic
61C2×C4×H(3,𝔽3)C2312yes—
62C2×C4×H(3,𝔽3)C2311yescharacteristic
63C2×#1084C2312yes—
64C2×#1084C2311yescharacteristic
65C2×#1084C2311yescharacteristic
66C2×(C3⋊Dic3)⋊C3C2312yes—
67C2×(C3⋊Dic3)⋊C3C2312yes—
68C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C4×(C3⋊D3)⋊C3Q814yescomplemented
72#634⋊C3—24no—
73((C3⋊D3)⋊C3)⋊C4—22no—
74((C3⋊D3)⋊C3)⋊C4—22no—
75#6196—24no—
76#5728—91no—
77#5688—91no—
78#5517—91no—
79C2×C4×C3⋊D3C2×D311yescharacteristic
80C2×#634C2×D312yes—
81C2×C3×C4×D3—31no—
82C2×C3×Dic6—32no—
83C2×#627—32no—
84C2×#627—32no—
85C2×#624—31no—
86C2×#624—31no—
87C2×D3×Dic3—31no—
88C2×D3×Dic3—31no—
89C32×C4⋊C4—21no—
90C32×C4⋊C4C2×D311yescharacteristic
91C32×C4⋊C4C2×D311yescharacteristic
92#3752C2×D312yes—
93C4⋊(C3⋊Dic3)C2×D311yescharacteristic
94C4⋊(C3⋊Dic3)C2×D311yescharacteristic
95#3750C2×D312yes—
96C4×C3⋊Dic3C2×D312yes—
97C4×C3⋊Dic3C2×D311yescharacteristic
98C3×(C2×D3)⋊C4—32no—
99C3×C4⋊Dic3—31no—
100C3×C4⋊Dic3—31no—
101C3×C4⋊Dic3—32no—
102C3×C4⋊Dic3—32no—
103C3×C4⋊Dic3—31no—
104C3×C4⋊Dic3—31no—
105C3×C4⋊Dic3—62no—
106C3×C4⋊Dic3—61no—
107C3×Dic3⋊C4—32no—
108C3×Dic3⋊C4—32no—
109C3×Dic3⋊C4—32no—
110C3×Dic3⋊C4—62no—
111C3×C4×Dic3—31no—
112C3×C4×Dic3—31no—
113C3×C4×Dic3—31no—
114C3×C4×Dic3—32no—
115C3×C4×Dic3—61no—
116#3724—32no—
117#3724—32no—
118#3724—32no—
119Dic3⋊Dic3—32no—
120Dic3⋊Dic3—32no—
121Dic3⋊Dic3—32no—
122#3722—32no—
123(C2×D3)⋊Dic3—32no—
124Dic32—31no—
125Dic32—32no—
126Dic32—31no—
127Dic32—32no—
128Dic32—31no—
129Dic32—31no—
130C22×H(3,𝔽3)C2411yescharacteristic
131C2×(C3⋊D3)⋊C3—21no—
132C2×(C3⋊D3)⋊C3—21no—
133C2×(C3⋊D3)⋊C3C2×Q812yes—
134C4×H(3,𝔽3)—22no—
135C4×H(3,𝔽3)C2×Q812yes—
136#1084—22no—
137#1084—21no—
138#1084—21no—
139(C3⋊Dic3)⋊C3—22no—
140(C3⋊Dic3)⋊C3—22no—
141(C3⋊Dic3)⋊C3C2×Q812yes—
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—91no—
156#892—91no—
157#892—91no—
158#890—31no—
159#890—31no—
160#890—61no—
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⋊D3C22×D311yescharacteristic
179C22×C3×D3—31no—
180C2×C32×C4—21no—
181C2×C32×C4—22no—
182C2×C32×C4C22×D312yes—
183C2×C32×C4C22×D312yes—
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D311yescharacteristic
186C2×C3⋊Dic3C22×D312yes—
187C2×C3⋊Dic3C22×D312yes—
188C2×C3⋊Dic3C22×D311yescharacteristic
189C2×C3⋊Dic3C22×D311yescharacteristic
190C2×C3⋊Dic3C22×D311yescharacteristic
191C4×C3⋊D3Dic614yescomplemented
192#634—24no—
193C2×C3×Dic3—31no—
194C2×C3×Dic3—31no—
195C2×C3×Dic3—31no—
196C2×C3×Dic3—32no—
197C2×C3×Dic3—32no—
198C2×C3×Dic3—32no—
199C2×C3×Dic3—32no—
200C2×C3×Dic3—31no—
201C2×C3×Dic3—31no—
202C2×C3×Dic3—31no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3—61no—
205C2×C3×Dic3—61no—
206C2×C3×Dic3—62no—
207C3×C4×D3—34no—
208C3×Dic6—64no—
209#627—64no—
210#627—64no—
211#624—62no—
212#624—62no—
213D3×Dic3—62no—
214D3×Dic3—62no—
215#533—271noSylow
216C2×H(3,𝔽3)C22×Q811yescharacteristic
217C2×H(3,𝔽3)C2×#4111yescharacteristic
218C2×H(3,𝔽3)C2×#4111yescharacteristic
219(C3⋊D3)⋊C3—22no—
220C2×C3×Q8—92no—
221C22×C3×C4—91no—
222C22×Dic3—91no—
223C22×Dic3—91no—
224C2×C4×D3—31no—
225C2×C4×D3—91no—
226C2×C4×D3—91no—
227C2×C4×D3—91no—
228C2×C4×D3—91no—
229C2×C4×D3—91no—
230C2×Dic6—32no—
231C2×Dic6—92no—
232C2×Dic6—92no—
233C2×Dic6—92no—
234C2×Dic6—92no—
235C3×C4⋊C4—31no—
236C3×C4⋊C4—31no—
237C3×C4⋊C4—61no—
238C3×C4⋊C4—92no—
239C3×C4⋊C4—91no—
240C3×C4⋊C4—91no—
241C3×C4⋊C4—92no—
242C3×C4⋊C4—92no—
243C3×C4⋊C4—91no—
244C3×C4⋊C4D3211yescharacteristic
245C3×C22⋊C4—92no—
246C3×C42—92no—
247C3×C42—91no—
248C3×C42—91no—
249#217—92no—
250(C2×D3)⋊C4—32no—
251(C2×D3)⋊C4—92no—
252(C2×D3)⋊C4—92no—
253(C2×D3)⋊C4—92no—
254C4⋊Dic3—32no—
255C4⋊Dic3—31no—
256C4⋊Dic3—31no—
257C4⋊Dic3—32no—
258C4⋊Dic3—31no—
259C4⋊Dic3—31no—
260C4⋊Dic3—61no—
261C4⋊Dic3—62no—
262C4⋊Dic3—91no—
263C4⋊Dic3—91no—
264C4⋊Dic3—92no—
265C4⋊Dic3—92no—
266C4⋊Dic3—92no—
267Dic3⋊C4—32no—
268Dic3⋊C4—32no—
269Dic3⋊C4—32no—
270Dic3⋊C4—62no—
271Dic3⋊C4—92no—
272Dic3⋊C4—92no—
273Dic3⋊C4—92no—
274Dic3⋊C4—92no—
275Dic3⋊C4—92no—
276Dic3⋊C4—92no—
277Dic3⋊C4—92no—
278C4×Dic3—31no—
279C4×Dic3—31no—
280C4×Dic3—31no—
281C4×Dic3—32no—
282C4×Dic3—61no—
283C4×Dic3—92no—
284C4×Dic3—91no—
285C4×Dic3—91no—
286C4×Dic3—91no—
287C4×Dic3—92no—
288C4×Dic3—91no—
289C4×Dic3—91no—
290C4×Dic3—92no—
291C4×Dic3—91no—
292C4×Dic3—91no—
293C4×Dic3—92no—
294C22×C32—21no—
295C22×C32C23×D311yescharacteristic
296C22×C32C23×D311yescharacteristic
297C2×C3⋊D3—21no—
298C2×C3⋊D3—21no—
299C2×C3⋊D3C2×Dic612yes—
300C2×C3×D3—32no—
301C2×C3×D3—61no—
302C2×C3×D3—61no—
303C32×C4—22no—
304C32×C4—22no—
305C32×C4—22no—
306C32×C4—42no—
307C32×C4C2×Dic612yes—
308C32×C4D3×Q812yes—
309C3⋊Dic3—22no—
310C3⋊Dic3—22no—
311C3⋊Dic3C2×Dic612yes—
312C3⋊Dic3#23712yes—
313C3⋊Dic3#23912yes—
314C3×Dic3—32no—
315C3×Dic3—32no—
316C3×Dic3—32no—
317C3×Dic3—61no—
318C3×Dic3—62no—
319C3×Dic3—61no—
320C3×Dic3—62no—
321C3×Dic3—62no—
322C3×Dic3—61no—
323C3×Dic3—61no—
324C3×Dic3—62no—
325C3×Dic3—121no—
326C3×Dic3—122no—
327C3×Dic3—121no—
328C4⋊Q8—91no—
329C4⋊Q8—271no—
330#125—91no—
331#125—271no—
332C22⋊Q8—272no—
333C22⋊Q8—272no—
334C4×Q8—272no—
335C4×Q8—272no—
336#117—91no—
337#117—271no—
338C2×C42—271no—
339H(3,𝔽3)#53311yescharacteristic, complemented, Sylow
340C23×C3—91no—
341C22×D3—31no—
342C22×D3—91no—
343C3×Q8—184no—
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—94no—
351C2×C3×C4—91no—
352C2×C3×C4—91no—
353C2×C3×C4—91no—
354C2×C3×C4—92no—
355C2×C3×C4—92no—
356C2×C3×C4—91no—
357C2×C3×C4—91no—
358C2×C3×C4—92no—
359C2×C3×C4C2×D3212yes—
360C2×C3×C4C2×D3211yescharacteristic
361C2×Dic3—31no—
362C2×Dic3—31no—
363C2×Dic3—31no—
364C2×Dic3—31no—
365C2×Dic3—31no—
366C2×Dic3—31no—
367C2×Dic3—31no—
368C2×Dic3—32no—
369C2×Dic3—32no—
370C2×Dic3—32no—
371C2×Dic3—32no—
372C2×Dic3—61no—
373C2×Dic3—61no—
374C2×Dic3—62no—
375C2×Dic3—91no—
376C2×Dic3—91no—
377C2×Dic3—91no—
378C2×Dic3—92no—
379C2×Dic3—92no—
380C2×Dic3—182no—
381C2×Dic3—182no—
382C4×D3—34no—
383C4×D3—94no—
384C4×D3—182no—
385C4×D3—182no—
386C4×D3—182no—
387C4×D3—182no—
388Dic6—64no—
389Dic6—184no—
390Dic6—184no—
391Dic6—184no—
392Dic6—184no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32—21no—
396C2×C32C22×Dic611yescharacteristic
397C2×C32C2×#23711yescharacteristic
398C2×C32C2×D3×Q811yescharacteristic
399C2×C32C2×#23911yescharacteristic
400C2×C32D3×#4111yescharacteristic
401C2×C32D3×#4111yescharacteristic
402C3⋊D3—22no—
403C3×D3—62no—
404C2×Q8—92no—
405C2×Q8—272no—
406C22×C4—91no—
407C22×C4—271no—
408C22×C4—271no—
409C4⋊C4—91no—
410C4⋊C4—91no—
411C4⋊C4—91no—
412C4⋊C4—92no—
413C4⋊C4—92no—
414C4⋊C4—92no—
415C4⋊C4—272no—
416C4⋊C4—272no—
417C4⋊C4—272no—
418C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
419C22⋊C4—92no—
420C22⋊C4—272no—
421C42—91no—
422C42—91no—
423C42—92no—
424C42—271no—
425C42—271no—
426C42—272no—
427C22×C3—31no—
428C22×C3—31no—
429C22×C3—61no—
430C22×C3—92no—
431C22×C3—181no—
432C22×C3—181no—
433C22×C3C22×D3211yescharacteristic
434C2×D3—32no—
435C2×D3—61no—
436C2×D3—61no—
437C2×D3—92no—
438C2×D3—181no—
439C2×D3—181no—
440C3×C4—22no—
441C3×C4—32no—
442C3×C4—32no—
443C3×C4—62no—
444C3×C4—62no—
445C3×C4—62no—
446C3×C4—92no—
447C3×C4—92no—
448C3×C4—92no—
449C3×C4—122no—
450C3×C4—182no—
451C3×C4—182no—
452C3×C4—181no—
453C3×C4—182no—
454C3×C4—181no—
455C3×C4D3×Dic612yes—
456Dic3—32no—
457Dic3—32no—
458Dic3—32no—
459Dic3—62no—
460Dic3—62no—
461Dic3—61no—
462Dic3—61no—
463Dic3—61no—
464Dic3—61no—
465Dic3—62no—
466Dic3—62no—
467Dic3—92no—
468Dic3—92no—
469Dic3—92no—
470Dic3—121no—
471Dic3—121no—
472Dic3—122no—
473Dic3—182no—
474Dic3—182no—
475C32—21no—
476C32#551711yescharacteristic, complemented
477C32#572811yescharacteristic, complemented
478C23—91no—
479Q8—184no—
480Q8—544no—
481C2×C4—91no—
482C2×C4—91no—
483C2×C4—91no—
484C2×C4—94no—
485C2×C4—91no—
486C2×C4—91no—
487C2×C4—92no—
488C2×C4—92no—
489C2×C4—92no—
490C2×C4—542no—
491C2×C4—542no—
492C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
493C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
494C2×C3—31no—
495C2×C3—31no—
496C2×C3—31no—
497C2×C3—31no—
498C2×C3—31no—
499C2×C3—31no—
500C2×C3—61no—
501C2×C3—61no—
502C2×C3—61no—
503C2×C3—182no—
504C2×C3C2×D3×Dic611yescharacteristic
505C2×C3D3×#23711yescharacteristic
506C2×C3D3×#23911yescharacteristic
507D3—62no—
508D3—182no—
509C22—92no—
510C22—181no—
511C22—181no—
512C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
513C4—22no—
514C4—92no—
515C4—92no—
516C4—92no—
517C4—181no—
518C4—182no—
519C4—181no—
520C4—182no—
521C4—182no—
522C4((C3⋊D3)⋊C3)⋊Q812yes—
523C3—31no—
524C3—31no—
525C3—61no—
526C3#1059444511yescharacteristic
527C2—182no—
528C2C2×((C3⋊D3)⋊C3)⋊Q811yescharacteristic
529C2#1172250411yescharacteristic
530C2#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×(C3⋊D3)⋊C3C23
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
1. #11721758C2semidirect
3. #64097⋊C3C2semidirect
7. #11721116C2semidirect
8. #11721095C2semidirect
9. #11721082C2semidirect
11. #11721077C2semidirect
38. #64097D3semidirect
71. C4×(C3⋊D3)⋊C3Q8semidirect
191. C4×C3⋊D3Dic6semidirect
339. H(3,𝔽3)#533semidirect
476. C32#5517semidirect
477. C32#5728semidirect

Maximal subgroups

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

subgroupindex
#117217582
#64101⋊C32
#64097⋊C32
#117211332
#117211282
C4×((C3⋊D3)⋊C3)⋊C42
#117211162
#117210952
#117210822
#117210812
#117210772
#105944773
#105944483