Finite Group Databasebeta

← #13175451

Subgroups — #13175451

Order 1728.

2508 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6289C211yesmaximal, characteristic
2C22⋊C4×C3⋊(C3⋊D3)C211yesmaximal, characteristic
3C3×#64086C211yesmaximal, characteristic, complemented
4C32×#877C211yesmaximal, characteristic, complemented
5#11723627C211yesmaximal, characteristic
6#11723595C211yesmaximal, characteristic, complemented
7#11723594C211yesmaximal, characteristic, complemented
8#11723470C212yesmaximal, complemented
9#11723469C212yesmaximal, complemented
10#11723468C212yesmaximal, complemented
11C4×#6289C212yesmaximal, complemented
12C3⋊D3×#117—31nomaximal
13#10594052—34nomaximal
14C23×C3⋊(C3⋊D3)C2211yescharacteristic
15C2×C4×C3⋊(C3⋊D3)C2212yes—
16C22×C3×C3⋊Dic3C2211yescharacteristic
17C22×C32×Dic3C2211yescharacteristic
18C2×#6289C418yescomplemented
19C2×#6289C2212yes—
20C2×#6289C2211yescharacteristic
21C2×#6289C2211yescharacteristic
22C2×#6289C2211yescharacteristic
23C2×#6289C2211yescharacteristic
24C33×C22⋊C4C2211yescharacteristic
25#88864C2211yescharacteristic
26#88859C2212yes—
27C3×#3757C2211yescharacteristic
28C3×#3750C2212yescomplemented
29C3×C4×C3⋊Dic3C2212yescomplemented
30C32×#217C2211yescharacteristic
31C32×Dic3⋊C4C2212yescomplemented
32C32×C4×Dic3C2212yescomplemented
33#88807C2212yescomplemented
34Dic3⋊(C3⋊Dic3)C2212yescomplemented
35#88804C2212yes—
36Dic3×C3⋊Dic3C2212yescomplemented
37Dic3×C3⋊Dic3C2212yescomplemented
38C22×C4×C3⋊D3—31no—
39C22×#624—34no—
40C32×#117—31no—
41#64135—31no—
42#64101—32no—
43C4⋊C4×C3⋊D3—32no—
44C22⋊C4×C3⋊D3—31no—
45C22⋊C4×C3⋊D3—31no—
46C22⋊C4×C3⋊D3—34no—
47C22⋊C4×C3⋊D3—64no—
48#64086—31no—
49#64086D311yescharacteristic, complemented
50#64081—32no—
51C42×C3⋊D3—32no—
52C3×#877—34no—
53C3×#877D314yescomplemented
54#63974—34no—
55#63955—34no—
56#63955—34no—
57#63883—38no—
58C4×#624—38no—
59#63836—38no—
60#63836—38no—
61C23×C33C2311yescharacteristic
62C22×C3⋊(C3⋊D3)—24no—
63C22×C3⋊(C3⋊D3)C2×C414yes—
64C22×C3⋊(C3⋊D3)C2311yescharacteristic
65C22×C3⋊(C3⋊D3)C2311yescharacteristic
66C2×C33×C4C2312yes—
67C2×C3⋊(C3⋊Dic3)C2312yes—
68C4×C3⋊(C3⋊D3)—24no—
69C2×C3×C3⋊Dic3C2×C414yescomplemented
70C2×C3×C3⋊Dic3C2312yes—
71C2×C3×C3⋊Dic3C2311yescharacteristic
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C2×C32×Dic3C2×C414yescomplemented
74C2×C32×Dic3C2312yes—
75C2×C32×Dic3C2311yescharacteristic
76C2×C32×Dic3C2311yescharacteristic
77#6289—24no—
78#6289C2×C414yes—
79#6289C2×C414yes—
80#6289C2×C414yescomplemented
81#6289C2×C414yescomplemented
82D3×#117—94no—
83D3×#117—91no—
84C23×C3⋊D3—31no—
85C23×C3⋊D3—34no—
86C23×C3⋊D3—64no—
87C22×C32×C4—31no—
88C22×C3⋊Dic3C2×D311yescharacteristic
89C2×C4×C3⋊D3—38no—
90C2×C4×C3⋊D3—31no—
91C2×C4×C3⋊D3—31no—
92C2×C4×C3⋊D3—31no—
93C2×C4×C3⋊D3—31no—
94C2×C4×C3⋊D3—32no—
95C2×C4×C3⋊D3—38no—
96C2×C4×C3⋊D3—32no—
97C2×C4×C3⋊D3—68no—
98C22×C3×Dic3—34no—
99C22×C3×Dic3C2×D314yes—
100C2×#624—332no—
101C2×#624—34no—
102C2×#624—34no—
103C2×#624—34no—
104C2×#624—34no—
105C2×#624—38no—
106C32×C4⋊C4—32no—
107C32×C22⋊C4—24no—
108C32×C22⋊C4—31no—
109C32×C22⋊C4C2×D314yes—
110C32×C22⋊C4C2×D311yescharacteristic
111C32×C42—32no—
112#3757—31no—
113#3757—34no—
114#3757—64no—
115#3757C2×D311yescharacteristic
116#3752—32no—
117#3752—38no—
118#3752—32no—
119#3752—68no—
120C4⋊(C3⋊Dic3)—32no—
121#3750—32no—
122#3750C2×D312yescomplemented
123C4×C3⋊Dic3—32no—
124C4×C3⋊Dic3—32no—
125C4×C3⋊Dic3C2×D312yescomplemented
126C3×#217—34no—
127C3×#217C2×D314yes—
128C3×Dic3⋊C4—38no—
129C3×Dic3⋊C4C2×D318yescomplemented
130C3×C4×Dic3—38no—
131C3×C4×Dic3C2×D318yescomplemented
132Dic3⋊Dic3—38no—
133Dic3⋊Dic3—38no—
134#3722—38no—
135Dic32—38no—
136Dic32—38no—
137C22×C33—22no—
138C22×C33C22×C412yes—
139C22×C33C2411yescharacteristic
140C2×C3⋊(C3⋊D3)—22no—
141C2×C3⋊(C3⋊D3)—28no—
142C2×C3⋊(C3⋊D3)C22×C412yes—
143C2×C3⋊(C3⋊D3)C22×C412yes—
144C2×C3⋊(C3⋊D3)#4114yes—
145C33×C4—22no—
146C3⋊(C3⋊Dic3)—22no—
147C3×C3⋊Dic3—22no—
148C3×C3⋊Dic3C22×C412yes—
149C3×C3⋊Dic3C22×C412yes—
150C32×Dic3—22no—
151C32×Dic3C22×C412yes—
152C32×Dic3C22×C412yes—
153C22×C4×D3—94no—
154C22×C4×D3—91no—
155C3×#117—34no—
156C3×#117—91no—
157#926—94no—
158#926—91no—
159#892—98no—
160#892—92no—
161D3×C4⋊C4—98no—
162D3×C4⋊C4—92no—
163D3×C22⋊C4—94no—
164D3×C22⋊C4—94no—
165D3×C22⋊C4—91no—
166D3×C22⋊C4—91no—
167D3×C22⋊C4—184no—
168#877—34no—
169#877—94no—
170#877—91no—
171#877C3⋊D311yescharacteristic, complemented
172#872—98no—
173#872—92no—
174C42×D3—98no—
175C42×D3—92no—
176C23×C32—24no—
177C23×C32C22×D314yes—
178C23×C32C22×D311yescharacteristic
179C22×C3⋊D3—31no—
180C22×C3⋊D3—34no—
181C22×C3⋊D3—31no—
182C22×C3⋊D3—34no—
183C22×C3⋊D3—316no—
184C22×C3⋊D3—34no—
185C22×C3⋊D3—64no—
186C22×C3⋊D3—64no—
187C22×C3⋊D3—616no—
188C22×C3⋊D3—64no—
189C22×C3⋊D3—616no—
190C22×C3⋊D3—1216no—
191C2×C32×C4—28no—
192C2×C32×C4—34no—
193C2×C32×C4—31no—
194C2×C32×C4—31no—
195C2×C32×C4—32no—
196C2×C32×C4C22×D318yes—
197C2×C32×C4C22×D312yes—
198C2×C3⋊Dic3—32no—
199C2×C3⋊Dic3—38no—
200C2×C3⋊Dic3—68no—
201C2×C3⋊Dic3C4×D314yescomplemented
202C2×C3⋊Dic3C22×D312yes—
203C2×C3⋊Dic3C22×D311yescharacteristic
204C2×C3⋊Dic3C22×D311yescharacteristic
205C4×C3⋊D3—34no—
206C4×C3⋊D3—34no—
207C4×C3⋊D3—34no—
208C4×C3⋊D3—34no—
209C4×C3⋊D3—64no—
210C4×C3⋊D3—64no—
211C4×C3⋊D3—616no—
212C4×C3⋊D3—1216no—
213C2×C3×Dic3—316no—
214C2×C3×Dic3—34no—
215C2×C3×Dic3—34no—
216C2×C3×Dic3—38no—
217C2×C3×Dic3C4×D3116yescomplemented
218C2×C3×Dic3C22×D318yes—
219C2×C3×Dic3C22×D314yes—
220C2×C3×Dic3C22×D314yes—
221#624—316no—
222#624—316no—
223#624—316no—
224#624—316no—
225#624—616no—
226C2×#117—271noSylow
227C2×C33—22no—
228C2×C33C23×C411yescharacteristic
229C2×C33C2×#4112yes—
230C3⋊(C3⋊D3)—22no—
231C3⋊(C3⋊D3)#11714yescomplemented
232C23×D3—94no—
233C23×D3—91no—
234C23×D3—184no—
235C22×C3×C4—34no—
236C22×C3×C4—91no—
237C22×Dic3—34no—
238C22×Dic3C2×C3⋊D311yescharacteristic
239C2×C4×D3—94no—
240C2×C4×D3—94no—
241C2×C4×D3—94no—
242C2×C4×D3—94no—
243C2×C4×D3—932no—
244C2×C4×D3—98no—
245C2×C4×D3—91no—
246C2×C4×D3—91no—
247C2×C4×D3—91no—
248C2×C4×D3—91no—
249C2×C4×D3—98no—
250C2×C4×D3—98no—
251C2×C4×D3—92no—
252C2×C4×D3—92no—
253C2×C4×D3—188no—
254C3×C4⋊C4—38no—
255C3×C4⋊C4—92no—
256C3×C22⋊C4—24no—
257C3×C22⋊C4—34no—
258C3×C22⋊C4—91no—
259C3×C22⋊C4D3214yes—
260C3×C22⋊C4C2×C3⋊D311yescharacteristic
261C3×C42—38no—
262C3×C42—92no—
263#217—34no—
264#217—94no—
265#217—91no—
266#217—184no—
267#217C2×C3⋊D311yescharacteristic
268(C2×D3)⋊C4—98no—
269(C2×D3)⋊C4—98no—
270(C2×D3)⋊C4—92no—
271(C2×D3)⋊C4—92no—
272(C2×D3)⋊C4—188no—
273C4⋊Dic3—98no—
274C4⋊Dic3—92no—
275Dic3⋊C4—38no—
276Dic3⋊C4—98no—
277Dic3⋊C4—92no—
278Dic3⋊C4C2×C3⋊D312yescomplemented
279C4×Dic3—38no—
280C4×Dic3—98no—
281C4×Dic3—98no—
282C4×Dic3—92no—
283C4×Dic3—92no—
284C4×Dic3C2×C3⋊D312yescomplemented
285C22×C32—22no—
286C22×C32—24no—
287C22×C32—28no—
288C22×C32—28no—
289C22×C32—48no—
290C22×C32C2×C4×D318yes—
291C22×C32C2×C4×D312yes—
292C22×C32C23×D314yes—
293C22×C32C23×D311yescharacteristic
294C2×C3⋊D3—32no—
295C2×C3⋊D3—34no—
296C2×C3⋊D3—32no—
297C2×C3⋊D3—38no—
298C2×C3⋊D3—316no—
299C2×C3⋊D3—38no—
300C2×C3⋊D3—68no—
301C2×C3⋊D3—616no—
302C2×C3⋊D3—68no—
303C2×C3⋊D3—68no—
304C2×C3⋊D3—62no—
305C2×C3⋊D3—632no—
306C2×C3⋊D3—68no—
307C2×C3⋊D3—1232no—
308C2×C3⋊D3—128no—
309C32×C4—22no—
310C32×C4—28no—
311C32×C4—32no—
312C32×C4—32no—
313C32×C4—48no—
314C32×C4—62no—
315C3⋊Dic3—22no—
316C3⋊Dic3—62no—
317C3⋊Dic3—68no—
318C3⋊Dic3—128no—
319C3⋊Dic3C2×C4×D312yes—
320C3⋊Dic3C2×C4×D312yes—
321C3×Dic3—28no—
322C3×Dic3—38no—
323C3×Dic3—38no—
324C3×Dic3—68no—
325C3×Dic3C2×C4×D318yes—
326C3×Dic3C2×C4×D318yes—
327C23×C4—271no—
328#117—31no—
329#117—91no—
330#117—272no—
331#117—272no—
332#117—271no—
333#117—271no—
334C2×C4⋊C4—272no—
335C2×C22⋊C4—271no—
336C2×C22⋊C4—271no—
337C2×C42—272no—
338C33C2×#11711yescharacteristic, complemented, Sylow
339C23×C3—24no—
340C23×C3C2×D3214yes—
341C23×C3C22×C3⋊D311yescharacteristic
342C22×D3—94no—
343C22×D3—91no—
344C22×D3—94no—
345C22×D3—91no—
346C22×D3—916no—
347C22×D3—94no—
348C22×D3—184no—
349C22×D3—184no—
350C22×D3—1816no—
351C22×D3—184no—
352C22×D3—1816no—
353C22×D3—3616no—
354C2×C3×C4—28no—
355C2×C3×C4—34no—
356C2×C3×C4—34no—
357C2×C3×C4—316no—
358C2×C3×C4—38no—
359C2×C3×C4—91no—
360C2×C3×C4—91no—
361C2×C3×C4—94no—
362C2×C3×C4—92no—
363C2×C3×C4C2×D3218yes—
364C2×C3×C4C22×C3⋊D312yes—
365C2×Dic3—34no—
366C2×Dic3—34no—
367C2×Dic3—316no—
368C2×Dic3—38no—
369C2×Dic3—98no—
370C2×Dic3—92no—
371C2×Dic3—188no—
372C2×Dic3C4×C3⋊D314yescomplemented
373C2×Dic3C22×C3⋊D312yes—
374C2×Dic3C22×C3⋊D311yescharacteristic
375C2×Dic3C22×C3⋊D311yescharacteristic
376C4×D3—916no—
377C4×D3—916no—
378C4×D3—916no—
379C4×D3—916no—
380C4×D3—94no—
381C4×D3—94no—
382C4×D3—94no—
383C4×D3—94no—
384C4×D3—184no—
385C4×D3—1816no—
386C4×D3—184no—
387C4×D3—1816no—
388C4×D3—3616no—
389C2×C32—22no—
390C2×C32—24no—
391C2×C32—28no—
392C2×C32—28no—
393C2×C32—48no—
394C2×C32C22×C4×D314yes—
395C2×C32C22×C4×D311yescharacteristic
396C2×C32D3×#4118yes—
397C2×C32D3×#4112yes—
398C3⋊D3—34no—
399C3⋊D3—316no—
400C3⋊D3—616no—
401C3⋊D3—62no—
402C3⋊D3—68no—
403C3⋊D3—128no—
404C24—271no—
405C22×C4—31no—
406C22×C4—91no—
407C22×C4—271no—
408C22×C4—271no—
409C22×C4—271no—
410C22×C4—271no—
411C22×C4—278no—
412C22×C4—272no—
413C22×C4—272no—
414C4⋊C4—32no—
415C4⋊C4—92no—
416C4⋊C4—272no—
417C4⋊C4—272no—
418C22⋊C4—31no—
419C22⋊C4—91no—
420C22⋊C4—272no—
421C22⋊C4—271no—
422C22⋊C4—272no—
423C22⋊C4D3×C3⋊D311yescharacteristic
424C42—32no—
425C42—92no—
426C42—272no—
427C42—272no—
428C22×C3—28no—
429C22×C3—22no—
430C22×C3—24no—
431C22×C3—28no—
432C22×C3—48no—
433C22×C3C4×D3218yes—
434C22×C3C2×C4×C3⋊D312yes—
435C22×C3C22×D3214yes—
436C22×C3C23×C3⋊D311yescharacteristic
437C2×D3—98no—
438C2×D3—916no—
439C2×D3—92no—
440C2×D3—94no—
441C2×D3—98no—
442C2×D3—92no—
443C2×D3—1816no—
444C2×D3—188no—
445C2×D3—188no—
446C2×D3—188no—
447C2×D3—1832no—
448C2×D3—188no—
449C2×D3—182no—
450C2×D3—3632no—
451C2×D3—368no—
452C3×C4—28no—
453C3×C4—22no—
454C3×C4—38no—
455C3×C4—38no—
456C3×C4—48no—
457C3×C4—68no—
458C3×C4—92no—
459C3×C4—92no—
460C3×C4—182no—
461Dic3—22no—
462Dic3—38no—
463Dic3—38no—
464Dic3—68no—
465Dic3—188no—
466Dic3—182no—
467Dic3—368no—
468Dic3C2×C4×C3⋊D312yes—
469Dic3C2×C4×C3⋊D312yes—
470C32—24no—
471C32D3×#11714yescomplemented
472C32D3×#11711yescharacteristic, complemented
473C23—271no—
474C23—271no—
475C23—274no—
476C23—544no—
477C23C2×D3×C3⋊D311yescharacteristic
478C2×C4—31no—
479C2×C4—31no—
480C2×C4—34no—
481C2×C4—32no—
482C2×C4—91no—
483C2×C4—91no—
484C2×C4—94no—
485C2×C4—92no—
486C2×C4—274no—
487C2×C4—274no—
488C2×C4—274no—
489C2×C4—274no—
490C2×C4—272no—
491C2×C4—544no—
492C2×C4—544no—
493C2×C4C2×D3×C3⋊D312yes—
494C2×C3—28no—
495C2×C3—22no—
496C2×C3—24no—
497C2×C3—28no—
498C2×C3—48no—
499C2×C3C2×C4×D3214yes—
500C2×C3#6431218yes—
501C2×C3C22×C4×C3⋊D311yescharacteristic
502C2×C3#41×C3⋊D312yes—
503D3—916no—
504D3—94no—
505D3—1816no—
506D3—188no—
507D3—182no—
508D3—368no—
509C22—22no—
510C22—272no—
511C22—274no—
512C22—272no—
513C22—548no—
514C22—542no—
515C22C4×D3×C3⋊D312yes—
516C22C22×D3×C3⋊D311yescharacteristic
517C4—22no—
518C4—32no—
519C4—32no—
520C4—62no—
521C4—92no—
522C4—92no—
523C4—182no—
524C4—542no—
525C3—24no—
526C3#1059405214yescomplemented
527C3C3⋊D3×#11711yescharacteristic, complemented
528C2—22no—
529C2—274no—
530C2—542no—
531C2C2×C4×D3×C3⋊D311yescharacteristic
532C2#1172458612yes—
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutatorC2×C33C23×C4
FittingC33×C22⋊C4C22
socle, cosocleC22×C33C24
prime radicalC23×C3⋊(C3⋊D3)C22
hypercenterC22⋊C4D3×C3⋊D3
hypocenterC33C2×#117
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C3×#64086C2semidirect
4. C32×#877C2semidirect
6. #11723595C2semidirect
7. #11723594C2semidirect
8. #11723470C2semidirect
9. #11723469C2semidirect
10. #11723468C2semidirect
11. C4×#6289C2semidirect
18. C2×#6289C4semidirect
28. C3×#3750C22semidirect
29. C3×C4×C3⋊Dic3C22semidirect
31. C32×Dic3⋊C4C22semidirect
32. C32×C4×Dic3C22semidirect
33. #88807C22semidirect
34. Dic3⋊(C3⋊Dic3)C22semidirect
36. Dic3×C3⋊Dic3C22semidirect
37. Dic3×C3⋊Dic3C22semidirect
49. #64086D3semidirect
53. C3×#877D3semidirect
69. C2×C3×C3⋊Dic3C2×C4semidirect
73. C2×C32×Dic3C2×C4semidirect
80. #6289C2×C4semidirect
81. #6289C2×C4semidirect
122. #3750C2×D3semidirect
125. C4×C3⋊Dic3C2×D3semidirect
129. C3×Dic3⋊C4C2×D3semidirect
131. C3×C4×Dic3C2×D3semidirect
171. #877C3⋊D3semidirect
201. C2×C3⋊Dic3C4×D3semidirect
217. C2×C3×Dic3C4×D3semidirect
231. C3⋊(C3⋊D3)#117semidirect
278. Dic3⋊C4C2×C3⋊D3semidirect
284. C4×Dic3C2×C3⋊D3semidirect
338. C33C2×#117semidirect
372. C2×Dic3C4×C3⋊D3semidirect
471. C32D3×#117semidirect
472. C32D3×#117semidirect
526. C3#10594052semidirect
527. C3C3⋊D3×#117semidirect

Maximal subgroups

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

subgroupindex
C22×#62892
C22⋊C4×C3⋊(C3⋊D3)2
C3×#640862
C32×#8772
#117236272
#117235952
#117235942
#117234702
#117234692
#117234682
C4×#62892
C3⋊D3×#1173
#105940523