Finite Group Databasebeta

← #13161439

Subgroups — #13161439

Order 1728.

1668 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88864C211yesmaximal, characteristic
2C2×C3×#3757C211yesmaximal, characteristic
3C2×C32×#217C211yesmaximal, characteristic
4C2×#88806C211yesmaximal, characteristic, complemented
5C2×Dic3×C3⋊Dic3C211yesmaximal, characteristic, complemented
6#11722061C211yesmaximal, characteristic, complemented
7#11722061C211yesmaximal, characteristic, complemented
8#10592643—31nomaximal
9#10591703—34nomaximal
10C24×C33C2211yescharacteristic
11C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
12C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
13C22×C3×C3⋊Dic3C2211yescharacteristic
14C22×C3×C3⋊Dic3C2211yescharacteristic
15C22×C32×Dic3C2211yescharacteristic
16C22×C32×Dic3C2211yescharacteristic
17#88864C414yescomplemented
18C3×#3757—22no—
19C32×#217—22no—
20#88806—22no—
21Dic3×C3⋊Dic3—22no—
22C2×C32×C22⋊C4—31no—
23C2×#3757—31no—
24C2×#3757—34no—
25C2×#3757—64no—
26C2×#3757D311yescharacteristic
27C2×#3750—31no—
28C2×C4×C3⋊Dic3—31no—
29C2×C3×#217—34no—
30C2×C3×#217D314yes—
31C2×#3724—34no—
32C2×Dic32—34no—
33#63658—31no—
34#63658—31no—
35#63579—34no—
36#63579—34no—
37C23×C33—21no—
38C23×C33—21no—
39C23×C33—21no—
40C23×C33—21no—
41C23×C33—21no—
42C23×C33—21no—
43C23×C33C2×C412yes—
44C23×C33C2311yescharacteristic
45C2×C3⋊(C3⋊Dic3)—21no—
46C2×C3⋊(C3⋊Dic3)—21no—
47C2×C3⋊(C3⋊Dic3)—21no—
48C2×C3⋊(C3⋊Dic3)—21no—
49C2×C3⋊(C3⋊Dic3)C2×C412yes—
50C2×C3⋊(C3⋊Dic3)C2×C412yes—
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3—21no—
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3—21no—
55C2×C3×C3⋊Dic3—21no—
56C2×C3×C3⋊Dic3D412yes—
57C2×C32×Dic3—21no—
58C2×C32×Dic3—21no—
59C2×C32×Dic3—21no—
60C2×C32×Dic3—21no—
61C2×C32×Dic3—21no—
62C2×C32×Dic3D412yes—
63#4945—91no—
64#4944—94no—
65C24×C32—24no—
66C24×C32C2×D314yes—
67C24×C32C2×D311yescharacteristic
68C22×C32×C4—31no—
69C22×C32×C4—31no—
70C22×C3⋊Dic3—31no—
71C22×C3⋊Dic3—31no—
72C22×C3⋊Dic3—34no—
73C22×C3⋊Dic3—34no—
74C22×C3⋊Dic3—64no—
75C22×C3⋊Dic3—64no—
76C22×C3⋊Dic3C2×D311yescharacteristic
77C22×C3⋊Dic3C2×D311yescharacteristic
78C22×C3×Dic3—34no—
79C22×C3×Dic3—34no—
80C22×C3×Dic3C2×D314yes—
81C22×C3×Dic3C2×D314yes—
82C32×C22⋊C4—62no—
83#3757—22no—
84#3757—34no—
85#3757—316no—
86#3757—616no—
87#3750—62no—
88C4×C3⋊Dic3—62no—
89C3×#217—28no—
90C3×#217—68no—
91#3724—68no—
92Dic32—68no—
93C22×C33—22no—
94C22×C33—22no—
95C22×C33—22no—
96C22×C33—41no—
97C22×C33—41no—
98C22×C33—41no—
99C22×C33—41no—
100C22×C33C22×C411yescharacteristic
101C22×C33C2×D411yescharacteristic
102C22×C33C2×D411yescharacteristic
103C22×C33#4111yescharacteristic
104C22×C33#4111yescharacteristic
105C22×C33#4111yescharacteristic
106C22×C33#4111yescharacteristic
107C3⋊(C3⋊Dic3)—22no—
108C3⋊(C3⋊Dic3)—22no—
109C3×C3⋊Dic3—22no—
110C3×C3⋊Dic3—41no—
111C32×Dic3—22no—
112C32×Dic3—41no—
113C2×C3×C22⋊C4—34no—
114C2×C3×C22⋊C4—91no—
115C2×#217—34no—
116C2×#217—94no—
117C2×#217—91no—
118C2×#217—184no—
119C2×#217C3⋊D311yescharacteristic
120C2×Dic3⋊C4—94no—
121C2×Dic3⋊C4—91no—
122C2×C4×Dic3—94no—
123C2×C4×Dic3—91no—
124C3⋊#95—94no—
125C3⋊#95—94no—
126C3⋊#95—91no—
127C3⋊#95—91no—
128C23×C32—28no—
129C23×C32—28no—
130C23×C32—21no—
131C23×C32—21no—
132C23×C32—24no—
133C23×C32—24no—
134C23×C32—21no—
135C23×C32—21no—
136C23×C32—21no—
137C23×C32—21no—
138C23×C32—28no—
139C23×C32—24no—
140C23×C32—24no—
141C23×C32—24no—
142C23×C32—24no—
143C23×C32—24no—
144C23×C32—44no—
145C23×C32—44no—
146C23×C32—44no—
147C23×C32—44no—
148C23×C32C4×D318yes—
149C23×C32C4×D312yes—
150C23×C32C22×D314yes—
151C23×C32C22×D311yescharacteristic
152C2×C32×C4—32no—
153C2×C32×C4—61no—
154C2×C32×C4—61no—
155C2×C32×C4—61no—
156C2×C32×C4—61no—
157C2×C32×C4—61no—
158C2×C3⋊Dic3—21no—
159C2×C3⋊Dic3—21no—
160C2×C3⋊Dic3—21no—
161C2×C3⋊Dic3—21no—
162C2×C3⋊Dic3—21no—
163C2×C3⋊Dic3—32no—
164C2×C3⋊Dic3—32no—
165C2×C3⋊Dic3—38no—
166C2×C3⋊Dic3—38no—
167C2×C3⋊Dic3—68no—
168C2×C3⋊Dic3—68no—
169C2×C3⋊Dic3—61no—
170C2×C3⋊Dic3—61no—
171C2×C3⋊Dic3—61no—
172C2×C3⋊Dic3—61no—
173C2×C3⋊Dic3—64no—
174C2×C3⋊Dic3—64no—
175C2×C3⋊Dic3—64no—
176C2×C3⋊Dic3—64no—
177C2×C3⋊Dic3—124no—
178C2×C3⋊Dic3—124no—
179C2×C3⋊Dic3—124no—
180C2×C3⋊Dic3—124no—
181C2×C3⋊Dic3#6712yes—
182C2×C3×Dic3—24no—
183C2×C3×Dic3—24no—
184C2×C3×Dic3—24no—
185C2×C3×Dic3—24no—
186C2×C3×Dic3—24no—
187C2×C3×Dic3—38no—
188C2×C3×Dic3—64no—
189C2×C3×Dic3—64no—
190C2×C3×Dic3—64no—
191C2×C3×Dic3—64no—
192C2×C3×Dic3—64no—
193C2×C3×Dic3#6718yes—
194#388—271noSylow
195C2×C33—42no—
196C2×C33#11711yescharacteristic
197C2×C33C4×D411yescharacteristic
198C2×C33C4×D411yescharacteristic
199C2×C33#12111yescharacteristic
200C2×C33#12311yescharacteristic
201C2×C33#12411yescharacteristic
202C2×C33#12611yescharacteristic
203C24×C3—24no—
204C24×C3D3214yes—
205C24×C3C2×C3⋊D311yescharacteristic
206C22×C3×C4—34no—
207C22×C3×C4—34no—
208C22×C3×C4—91no—
209C22×C3×C4—91no—
210C22×Dic3—34no—
211C22×Dic3—34no—
212C22×Dic3—94no—
213C22×Dic3—94no—
214C22×Dic3—91no—
215C22×Dic3—91no—
216C22×Dic3—184no—
217C22×Dic3—184no—
218C22×Dic3C2×C3⋊D311yescharacteristic
219C22×Dic3C2×C3⋊D311yescharacteristic
220C3×C22⋊C4—68no—
221C3×C22⋊C4—182no—
222#217—22no—
223#217—68no—
224#217—916no—
225#217—94no—
226#217—1816no—
227Dic3⋊C4—182no—
228Dic3⋊C4—188no—
229C4×Dic3—188no—
230C4×Dic3—182no—
231C22×C32—216no—
232C22×C32—28no—
233C22×C32—22no—
234C22×C32—22no—
235C22×C32—22no—
236C22×C32—24no—
237C22×C32—24no—
238C22×C32—24no—
239C22×C32—24no—
240C22×C32—24no—
241C22×C32—24no—
242C22×C32—24no—
243C22×C32—28no—
244C22×C32—28no—
245C22×C32—48no—
246C22×C32—48no—
247C22×C32—48no—
248C22×C32—48no—
249C22×C32—48no—
250C22×C32—48no—
251C22×C32—44no—
252C22×C32—41no—
253C22×C32—41no—
254C22×C32—44no—
255C22×C32—44no—
256C22×C32—41no—
257C22×C32—44no—
258C22×C32—41no—
259C22×C32C2×C4×D314yes—
260C22×C32C2×C4×D311yescharacteristic
261C22×C32#23514yes—
262C22×C32#23511yescharacteristic
263C22×C32D3×D414yes—
264C22×C32D3×D411yescharacteristic
265C22×C32#23714yes—
266C22×C32#23714yes—
267C22×C32#23714yes—
268C22×C32#23711yescharacteristic
269C22×C32#23711yescharacteristic
270C22×C32#23711yescharacteristic
271C22×C32C2×#6714yes—
272C22×C32C2×#6711yescharacteristic
273C32×C4—62no—
274C32×C4—121no—
275C3⋊Dic3—22no—
276C3⋊Dic3—41no—
277C3⋊Dic3—62no—
278C3⋊Dic3—62no—
279C3⋊Dic3—68no—
280C3⋊Dic3—68no—
281C3⋊Dic3—128no—
282C3⋊Dic3—128no—
283C3×Dic3—28no—
284C3×Dic3—44no—
285C3×Dic3—68no—
286C3×Dic3—124no—
287C2×C4⋊C4—271no—
288C2×C22⋊C4—31no—
289C2×C22⋊C4—91no—
290C2×C22⋊C4—271no—
291C2×C42—271no—
292#95—271no—
293#95—271no—
294C33#38811yescharacteristic, complemented, Sylow
295C23×C3—28no—
296C23×C3—28no—
297C23×C3—24no—
298C23×C3—24no—
299C23×C3—24no—
300C23×C3—24no—
301C23×C3—24no—
302C23×C3—24no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—21no—
306C23×C3—21no—
307C23×C3—21no—
308C23×C3—21no—
309C23×C3—28no—
310C23×C3—24no—
311C23×C3—44no—
312C23×C3—44no—
313C23×C3—44no—
314C23×C3—44no—
315C23×C3#62418yes—
316C23×C3C4×C3⋊D312yes—
317C23×C3C2×D3214yes—
318C23×C3C22×C3⋊D311yescharacteristic
319C2×C3×C4—38no—
320C2×C3×C4—64no—
321C2×C3×C4—64no—
322C2×C3×C4—64no—
323C2×C3×C4—64no—
324C2×C3×C4—64no—
325C2×C3×C4—92no—
326C2×C3×C4—181no—
327C2×C3×C4—181no—
328C2×C3×C4—181no—
329C2×C3×C4—181no—
330C2×C3×C4—181no—
331C2×Dic3—21no—
332C2×Dic3—21no—
333C2×Dic3—21no—
334C2×Dic3—21no—
335C2×Dic3—21no—
336C2×Dic3—38no—
337C2×Dic3—64no—
338C2×Dic3—64no—
339C2×Dic3—64no—
340C2×Dic3—64no—
341C2×Dic3—64no—
342C2×Dic3—98no—
343C2×Dic3—98no—
344C2×Dic3—92no—
345C2×Dic3—92no—
346C2×Dic3—188no—
347C2×Dic3—188no—
348C2×Dic3—184no—
349C2×Dic3—184no—
350C2×Dic3—184no—
351C2×Dic3—184no—
352C2×Dic3—181no—
353C2×Dic3—181no—
354C2×Dic3—181no—
355C2×Dic3—181no—
356C2×Dic3—364no—
357C2×Dic3—364no—
358C2×Dic3—364no—
359C2×Dic3—364no—
360C2×Dic3#63812yes—
361C2×C32—24no—
362C2×C32—24no—
363C2×C32—24no—
364C2×C32—24no—
365C2×C32—24no—
366C2×C32—24no—
367C2×C32—24no—
368C2×C32—416no—
369C2×C32—48no—
370C2×C32—42no—
371C2×C32#87714yes—
372C2×C32#87711yescharacteristic
373C2×C32#87914yes—
374C2×C32#87911yescharacteristic
375C2×C32#67⋊C414yes—
376C2×C32#67⋊C411yescharacteristic
377C2×C32#88311yescharacteristic
378C2×C32#88514yes—
379C2×C32C4×#6714yes—
380C2×C32C4×#6711yescharacteristic
381C2×C32#93514yes—
382C2×C32#93611yescharacteristic
383C2×C32#93814yes—
384C2×C32#93911yescharacteristic
385C24D3×C3⋊D311yescharacteristic
386C22×C4—31no—
387C22×C4—31no—
388C22×C4—91no—
389C22×C4—91no—
390C22×C4—271no—
391C22×C4—271no—
392C4⋊C4—542no—
393C22⋊C4—62no—
394C22⋊C4—182no—
395C22⋊C4—274no—
396C42—542no—
397C22×C3—216no—
398C22×C3—22no—
399C22×C3—22no—
400C22×C3—22no—
401C22×C3—28no—
402C22×C3—28no—
403C22×C3—28no—
404C22×C3—24no—
405C22×C3—24no—
406C22×C3—24no—
407C22×C3—24no—
408C22×C3—24no—
409C22×C3—24no—
410C22×C3—24no—
411C22×C3—48no—
412C22×C3—48no—
413C22×C3—48no—
414C22×C3—48no—
415C22×C3—48no—
416C22×C3—48no—
417C22×C3—41no—
418C22×C3—41no—
419C22×C3—41no—
420C22×C3—41no—
421C22×C3—44no—
422C22×C3—44no—
423C22×C3—44no—
424C22×C3—44no—
425C22×C3#380414yes—
426C22×C3#380414yes—
427C22×C3#380514yes—
428C22×C3#380514yes—
429C22×C3C2×#62414yes—
430C22×C3D3×#6714yes—
431C22×C3D3×#6714yes—
432C22×C3C2×C4×C3⋊D311yescharacteristic
433C22×C3#382811yescharacteristic
434C22×C3D4×C3⋊D311yescharacteristic
435C22×C3#383011yescharacteristic
436C22×C3#383011yescharacteristic
437C22×C3#383011yescharacteristic
438C22×C3C2×#63811yescharacteristic
439C3×C4—68no—
440C3×C4—124no—
441C3×C4—182no—
442C3×C4—361no—
443Dic3—22no—
444Dic3—41no—
445Dic3—68no—
446Dic3—124no—
447Dic3—182no—
448Dic3—182no—
449Dic3—188no—
450Dic3—188no—
451Dic3—368no—
452Dic3—368no—
453C32—24no—
454C32#494414yescomplemented
455C32#494511yescharacteristic, complemented
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23#628912yes—
463C23C2×D3×C3⋊D311yescharacteristic
464C2×C4—32no—
465C2×C4—61no—
466C2×C4—61no—
467C2×C4—61no—
468C2×C4—61no—
469C2×C4—61no—
470C2×C4—92no—
471C2×C4—181no—
472C2×C4—181no—
473C2×C4—181no—
474C2×C4—181no—
475C2×C4—181no—
476C2×C4—272no—
477C2×C4—272no—
478C2×C4—541no—
479C2×C4—541no—
480C2×C4—541no—
481C2×C4—541no—
482C2×C3—24no—
483C2×C3—24no—
484C2×C3—24no—
485C2×C3—24no—
486C2×C3—24no—
487C2×C3—24no—
488C2×C3—24no—
489C2×C3—416no—
490C2×C3—42no—
491C2×C3—48no—
492C2×C3#6395214yes—
493C2×C3#6395214yes—
494C2×C3#6395614yes—
495C2×C3#6395714yes—
496C2×C3#6395914yes—
497C2×C3#6396914yes—
498C2×C3#6397014yes—
499C2×C3#6408611yescharacteristic
500C2×C3#6408811yescharacteristic
501C2×C3#6409011yescharacteristic
502C2×C3#6409411yescharacteristic
503C2×C3C4×#63811yescharacteristic
504C2×C3#6414411yescharacteristic
505C2×C3#6414711yescharacteristic
506C22—22no—
507C22—22no—
508C22—22no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22#8902811yescharacteristic
514C22#8902911yescharacteristic
515C22#8902911yescharacteristic
516C22#8903111yescharacteristic
517C22C2×#628911yescharacteristic
518C22D3×#63811yescharacteristic
519C22C3⋊D3×#6711yescharacteristic
520C4—62no—
521C4—121no—
522C4—182no—
523C4—361no—
524C4—542no—
525C4—542no—
526C3—24no—
527C3#1059170314yescomplemented
528C3#1059264311yescharacteristic, complemented
529C2—42no—
530C2#1172359111yescharacteristic
531C2#1172359611yescharacteristic
532C2#1172359811yescharacteristic
533C2#1172361411yescharacteristic
534C2#1172361811yescharacteristic
535C2#1172362211yescharacteristic
536C2#1172363111yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×C3⋊D3
commutatorC22×C33C22×C4
Fitting, prime radicalC24×C33C22
socle, cosocleC23×C33C23
hypercenterC24D3×C3⋊D3
hypocenterC33#388
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
4. C2×#88806C2semidirect
5. C2×Dic3×C3⋊Dic3C2semidirect
6. #11722061C2semidirect
7. #11722061C2semidirect
17. #88864C4semidirect
294. C33#388semidirect
454. C32#4944semidirect
455. C32#4945semidirect
527. C3#10591703semidirect
528. C3#10592643semidirect

Maximal subgroups

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

subgroupindex
C2×#888642
C2×C3×#37572
C2×C32×#2172
C2×#888062
C2×Dic3×C3⋊Dic32
#117220612
#105926433
#105917033