Finite Group Databasebeta

← #13160514

Subgroups — #13160514

Order 1728.

1388 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88857C211yesmaximal, characteristic, complemented
2C2×C3×#3752C211yesmaximal, characteristic
3C2×#88803C211yesmaximal, characteristic
4C2×#88803C211yesmaximal, characteristic
5C32×C3⋊#95C211yesmaximal, characteristic, complemented
6#11722061C211yesmaximal, characteristic, complemented
7#11722061C211yesmaximal, characteristic, complemented
8#10592374—31nomaximal
9#10591298—34nomaximal
10C23×C3×C3⋊D3C2211yescharacteristic
11C22×C33×C4C2211yescharacteristic
12C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
13C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C22×C3×C3⋊Dic3C2211yescharacteristic
15C22×C32×Dic3C2211yescharacteristic
16C22×C32×Dic3C2211yescharacteristic
17#88857—22no—
18C3×#3752—22no—
19#88803—22no—
20#88803—22no—
21C2×#3752—31no—
22C2×#3752—31no—
23C2×#3752D311yescharacteristic
24C2×#3750—31no—
25C2×#3750—34no—
26C2×#3750—64no—
27C2×C3×(C2×D3)⋊C4—34no—
28C2×(C2×D3)⋊Dic3—34no—
29C2×(C2×D3)⋊Dic3—34no—
30C32×#95—31no—
31#63658—31no—
32#63658—31no—
33C3×C3⋊#95D314yescomplemented
34#63579—34no—
35#63579—34no—
36C23×C33C2311yescharacteristic
37C22×C3×C3⋊D3—21no—
38C22×C3×C3⋊D3—21no—
39C22×C3×C3⋊D3—21no—
40C22×C3×C3⋊D3—21no—
41C22×C3×C3⋊D3—21no—
42C22×C3×C3⋊D3—21no—
43C22×C3×C3⋊D3D412yes—
44C2×C33×C4—21no—
45C2×C33×C4—21no—
46C2×C33×C4—21no—
47C2×C3⋊(C3⋊Dic3)—21no—
48C2×C3⋊(C3⋊Dic3)—21no—
49C2×C3⋊(C3⋊Dic3)—21no—
50C2×C3⋊(C3⋊Dic3)—21no—
51C2×C3⋊(C3⋊Dic3)—21no—
52C2×C3⋊(C3⋊Dic3)D412yes—
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3—21no—
55C2×C3×C3⋊Dic3—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3—21no—
58C2×C32×Dic3—21no—
59C2×C32×Dic3—21no—
60C2×C32×Dic3—21no—
61C2×C32×Dic3—21no—
62#4952—91no—
63#4675—94no—
64C23×C3⋊D3C2×D311yescharacteristic
65C23×C3×D3—34no—
66C22×C32×C4—24no—
67C22×C32×C4—31no—
68C22×C32×C4—31no—
69C22×C32×C4C2×D314yes—
70C22×C32×C4C2×D311yescharacteristic
71C22×C3⋊Dic3—31no—
72C22×C3⋊Dic3—34no—
73C22×C3⋊Dic3—31no—
74C22×C3⋊Dic3—34no—
75C22×C3⋊Dic3—64no—
76C22×C3⋊Dic3—64no—
77C22×C3⋊Dic3C2×D311yescharacteristic
78C22×C3×Dic3—34no—
79C22×C3×Dic3C2×D314yes—
80C22×C3×Dic3C2×D314yes—
81#3752—22no—
82#3752—62no—
83#3752—62no—
84#3750—62no—
85#3750—68no—
86#3750—616no—
87C3×(C2×D3)⋊C4—68no—
88(C2×D3)⋊Dic3—68no—
89(C2×D3)⋊Dic3—68no—
90C22×C33C2×D411yescharacteristic
91C22×C33C2×D411yescharacteristic
92C22×C33#4111yescharacteristic
93C22×C33#4111yescharacteristic
94C22×C33#4111yescharacteristic
95C22×C33#4111yescharacteristic
96C22×C33#4111yescharacteristic
97C2×C3×C3⋊D3—22no—
98C2×C3×C3⋊D3—22no—
99C2×C3×C3⋊D3—22no—
100C2×C3×C3⋊D3—41no—
101C2×C3×C3⋊D3—41no—
102C2×C3×C3⋊D3—41no—
103C2×C3×C3⋊D3—41no—
104C33×C4—41no—
105C3⋊(C3⋊Dic3)—22no—
106C3⋊(C3⋊Dic3)—41no—
107C3×C3⋊Dic3—41no—
108C32×Dic3—41no—
109C32×Dic3—41no—
110C2×C3×C22⋊C4—91no—
111C2×#217—91no—
112C2×#217—91no—
113C2×(C2×D3)⋊C4—34no—
114C2×(C2×D3)⋊C4—94no—
115C2×(C2×D3)⋊C4—94no—
116C2×Dic3⋊C4—94no—
117C2×Dic3⋊C4—91no—
118C2×Dic3⋊C4—184no—
119C3×#95—34no—
120C3⋊#95—94no—
121C3⋊#95—94no—
122C3⋊#95—91no—
123C3⋊#95—91no—
124C3⋊#95C3⋊D311yescharacteristic, complemented
125C23×C32—24no—
126C23×C32C22×D314yes—
127C23×C32C22×D311yescharacteristic
128C22×C3⋊D3—21no—
129C22×C3⋊D3—21no—
130C22×C3⋊D3—21no—
131C22×C3⋊D3—21no—
132C22×C3⋊D3—21no—
133C22×C3⋊D3—21no—
134C22×C3⋊D3#6712yes—
135C22×C3×D3—38no—
136C22×C3×D3—64no—
137C22×C3×D3—64no—
138C22×C3×D3—64no—
139C22×C3×D3—64no—
140C22×C3×D3—64no—
141C22×C3×D3—64no—
142C2×C32×C4—28no—
143C2×C32×C4—24no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—24no—
148C2×C32×C4—24no—
149C2×C32×C4—44no—
150C2×C32×C4—44no—
151C2×C32×C4—61no—
152C2×C32×C4—61no—
153C2×C32×C4—61no—
154C2×C32×C4—61no—
155C2×C32×C4—61no—
156C2×C32×C4—61no—
157C2×C3⋊Dic3—21no—
158C2×C3⋊Dic3—21no—
159C2×C3⋊Dic3—21no—
160C2×C3⋊Dic3—32no—
161C2×C3⋊Dic3—38no—
162C2×C3⋊Dic3—68no—
163C2×C3⋊Dic3—68no—
164C2×C3⋊Dic3—68no—
165C2×C3⋊Dic3—64no—
166C2×C3⋊Dic3—61no—
167C2×C3⋊Dic3—61no—
168C2×C3⋊Dic3—64no—
169C2×C3⋊Dic3—68no—
170C2×C3⋊Dic3—64no—
171C2×C3⋊Dic3—61no—
172C2×C3⋊Dic3—61no—
173C2×C3⋊Dic3—61no—
174C2×C3⋊Dic3—64no—
175C2×C3⋊Dic3—64no—
176C2×C3⋊Dic3—124no—
177C2×C3⋊Dic3—124no—
178C2×C3×Dic3—24no—
179C2×C3×Dic3—24no—
180C2×C3×Dic3—24no—
181C2×C3×Dic3—24no—
182C2×C3×Dic3—24no—
183C2×C3×Dic3—24no—
184C2×C3×Dic3—64no—
185C2×C3×Dic3—64no—
186C2×C3×Dic3—64no—
187#397—271noSylow
188C2×C33#12111yescharacteristic
189C2×C33#12311yescharacteristic
190C2×C33#12311yescharacteristic
191C2×C33#12311yescharacteristic
192C2×C33#12411yescharacteristic
193C2×C33#12611yescharacteristic
194C2×C33#12611yescharacteristic
195C3×C3⋊D3—42no—
196C24×C3—91no—
197C23×D3—34no—
198C22×C3×C4—24no—
199C22×C3×C4—34no—
200C22×C3×C4—34no—
201C22×C3×C4—91no—
202C22×C3×C4D3214yes—
203C22×C3×C4C2×C3⋊D311yescharacteristic
204C22×Dic3—34no—
205C22×Dic3—94no—
206C22×Dic3—91no—
207C22×Dic3—94no—
208C22×Dic3—91no—
209C22×Dic3—184no—
210C22×Dic3—184no—
211C22×Dic3C2×C3⋊D311yescharacteristic
212C22×Dic3C2×C3⋊D311yescharacteristic
213C3×C22⋊C4—182no—
214#217—182no—
215#217—182no—
216(C2×D3)⋊C4—68no—
217(C2×D3)⋊C4—188no—
218(C2×D3)⋊C4—188no—
219Dic3⋊C4—188no—
220Dic3⋊C4—182no—
221Dic3⋊C4—1816no—
222C22×C32—24no—
223C22×C32—24no—
224C22×C32—24no—
225C22×C32—24no—
226C22×C32—24no—
227C22×C32—24no—
228C22×C32—24no—
229C22×C32#23514yes—
230C22×C32#23514yes—
231C22×C32#23514yes—
232C22×C32#23511yescharacteristic
233C22×C32#23511yescharacteristic
234C22×C32D3×D414yes—
235C22×C32D3×D414yes—
236C22×C32D3×D411yescharacteristic
237C22×C32#23714yes—
238C22×C32#23711yescharacteristic
239C22×C32#23711yescharacteristic
240C22×C32#23711yescharacteristic
241C22×C32#23914yes—
242C22×C32C2×#6711yescharacteristic
243C2×C3⋊D3—22no—
244C2×C3⋊D3—22no—
245C2×C3⋊D3—22no—
246C2×C3⋊D3—41no—
247C2×C3⋊D3—41no—
248C2×C3⋊D3—41no—
249C2×C3⋊D3—41no—
250C2×C3×D3—68no—
251C2×C3×D3—68no—
252C2×C3×D3—68no—
253C2×C3×D3—124no—
254C2×C3×D3—124no—
255C2×C3×D3—124no—
256C2×C3×D3—124no—
257C32×C4—48no—
258C32×C4—44no—
259C32×C4—41no—
260C32×C4—121no—
261C32×C4—121no—
262C3⋊Dic3—41no—
263C3⋊Dic3—616no—
264C3⋊Dic3—62no—
265C3⋊Dic3—68no—
266C3⋊Dic3—128no—
267C3⋊Dic3—124no—
268C3⋊Dic3—121no—
269C3×Dic3—44no—
270C3×Dic3—44no—
271C3×Dic3—124no—
272C2×C4⋊C4—271no—
273C2×C22⋊C4—91no—
274C2×C22⋊C4—271no—
275C2×C22⋊C4—271no—
276#95—31no—
277#95—271no—
278#95—271no—
279C33#39711yescharacteristic, complemented, Sylow
280C23×C3—24no—
281C23×C3—92no—
282C23×C3—181no—
283C23×C3—181no—
284C23×C3—181no—
285C23×C3—181no—
286C23×C3—181no—
287C23×C3—181no—
288C23×C3C2×D3214yes—
289C23×C3C22×C3⋊D311yescharacteristic
290C22×D3—38no—
291C22×D3—64no—
292C22×D3—64no—
293C22×D3—64no—
294C22×D3—64no—
295C22×D3—64no—
296C22×D3—64no—
297C2×C3×C4—28no—
298C2×C3×C4—24no—
299C2×C3×C4—24no—
300C2×C3×C4—24no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—44no—
305C2×C3×C4—44no—
306C2×C3×C4—64no—
307C2×C3×C4—64no—
308C2×C3×C4—64no—
309C2×C3×C4—64no—
310C2×C3×C4—64no—
311C2×C3×C4—64no—
312C2×C3×C4—181no—
313C2×C3×C4—181no—
314C2×C3×C4—181no—
315C2×Dic3—21no—
316C2×Dic3—21no—
317C2×Dic3—21no—
318C2×Dic3—21no—
319C2×Dic3—21no—
320C2×Dic3—21no—
321C2×Dic3—64no—
322C2×Dic3—64no—
323C2×Dic3—64no—
324C2×Dic3—98no—
325C2×Dic3—92no—
326C2×Dic3—188no—
327C2×Dic3—188no—
328C2×Dic3—188no—
329C2×Dic3—184no—
330C2×Dic3—184no—
331C2×Dic3—181no—
332C2×Dic3—181no—
333C2×Dic3—188no—
334C2×Dic3—184no—
335C2×Dic3—184no—
336C2×Dic3—184no—
337C2×Dic3—181no—
338C2×Dic3—181no—
339C2×Dic3—181no—
340C2×Dic3—364no—
341C2×Dic3—364no—
342C2×C32—24no—
343C2×C32—24no—
344C2×C32—24no—
345C2×C32—24no—
346C2×C32—24no—
347C2×C32—24no—
348C2×C32—24no—
349C2×C32#87614yes—
350C2×C32#87911yescharacteristic
351C2×C32#87911yescharacteristic
352C2×C32#88314yes—
353C2×C32#88311yescharacteristic
354C2×C32#88414yes—
355C2×C32#88514yes—
356C2×C32#88511yescharacteristic
357C2×C32#89414yes—
358C2×C32#89414yes—
359C2×C32#89814yes—
360C2×C32#92911yescharacteristic
361C2×C32#93511yescharacteristic
362C2×C32#93911yescharacteristic
363C3⋊D3—42no—
364C3×D3—128no—
365C24—91no—
366C22×C4—31no—
367C22×C4—31no—
368C22×C4—91no—
369C22×C4—271no—
370C22×C4—271no—
371C22×C4D3×C3⋊D311yescharacteristic
372C4⋊C4—542no—
373C22⋊C4—182no—
374C22⋊C4—542no—
375C22⋊C4—542no—
376C22×C3—24no—
377C22×C3—24no—
378C22×C3—24no—
379C22×C3—24no—
380C22×C3—24no—
381C22×C3—24no—
382C22×C3—24no—
383C22×C3—182no—
384C22×C3—182no—
385C22×C3—182no—
386C22×C3—361no—
387C22×C3—361no—
388C22×C3—361no—
389C22×C3—361no—
390C22×C3#379514yes—
391C22×C3#379514yes—
392C22×C3#379614yes—
393C22×C3#379814yes—
394C22×C3#380414yes—
395C22×C3D3×#6714yes—
396C22×C3#381114yes—
397C22×C3#382811yescharacteristic
398C22×C3#382811yescharacteristic
399C22×C3#382811yescharacteristic
400C22×C3D4×C3⋊D311yescharacteristic
401C22×C3D4×C3⋊D311yescharacteristic
402C22×C3#383011yescharacteristic
403C22×C3#383211yescharacteristic
404C2×D3—68no—
405C2×D3—68no—
406C2×D3—68no—
407C2×D3—124no—
408C2×D3—124no—
409C2×D3—124no—
410C2×D3—124no—
411C3×C4—48no—
412C3×C4—41no—
413C3×C4—44no—
414C3×C4—124no—
415C3×C4—124no—
416C3×C4—361no—
417Dic3—41no—
418Dic3—41no—
419Dic3—124no—
420Dic3—1816no—
421Dic3—182no—
422Dic3—188no—
423Dic3—368no—
424Dic3—361no—
425Dic3—364no—
426C32—24no—
427C32#467514yescomplemented
428C32#495211yescharacteristic, complemented
429C23—92no—
430C23—181no—
431C23—181no—
432C23—181no—
433C23—181no—
434C23—181no—
435C23—181no—
436C23C2×D3×C3⋊D311yescharacteristic
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—61no—
441C2×C4—61no—
442C2×C4—61no—
443C2×C4—61no—
444C2×C4—61no—
445C2×C4—61no—
446C2×C4—181no—
447C2×C4—181no—
448C2×C4—181no—
449C2×C4—272no—
450C2×C4—541no—
451C2×C4—541no—
452C2×C4—541no—
453C2×C4—541no—
454C2×C4—541no—
455C2×C3—24no—
456C2×C3—24no—
457C2×C3—24no—
458C2×C3—24no—
459C2×C3—24no—
460C2×C3—24no—
461C2×C3—24no—
462C2×C3—362no—
463C2×C3#6385314yes—
464C2×C3#6385814yes—
465C2×C3#6385914yes—
466C2×C3#6388414yes—
467C2×C3#6390514yes—
468C2×C3#6391314yes—
469C2×C3#6397114yes—
470C2×C3#6408511yescharacteristic
471C2×C3#6409211yescharacteristic
472C2×C3#6409311yescharacteristic
473C2×C3#6409411yescharacteristic
474C2×C3#6410311yescharacteristic
475C2×C3#6410311yescharacteristic
476C2×C3#6410711yescharacteristic
477D3—128no—
478C22—182no—
479C22—182no—
480C22—182no—
481C22—361no—
482C22—361no—
483C22—361no—
484C22—361no—
485C22#8901411yescharacteristic
486C22#8901711yescharacteristic
487C22#8902011yescharacteristic
488C22#8902111yescharacteristic
489C22#8903111yescharacteristic
490C22C3⋊D3×#6711yescharacteristic
491C22#8903911yescharacteristic
492C4—41no—
493C4—121no—
494C4—121no—
495C4—361no—
496C4—542no—
497C4—1081no—
498C3—24no—
499C3#1059129814yescomplemented
500C3#1059237411yescharacteristic, complemented
501C2—362no—
502C2#1172340711yescharacteristic
503C2#1172342111yescharacteristic
504C2#1172343811yescharacteristic
505C2#1172348811yescharacteristic
506C2#1172351811yescharacteristic
507C2#1172352811yescharacteristic
508C2#1172362411yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×C3⋊D3
commutator, socle, cosocleC23×C33C23
FittingC22×C33×C4C22
prime radicalC23×C3×C3⋊D3C22
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#397
LCSG ▷ C23×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88857C2semidirect
5. C32×C3⋊#95C2semidirect
6. #11722061C2semidirect
7. #11722061C2semidirect
33. C3×C3⋊#95D3semidirect
124. C3⋊#95C3⋊D3semidirect
279. C33#397semidirect
427. C32#4675semidirect
428. C32#4952semidirect
499. C3#10591298semidirect
500. C3#10592374semidirect

Maximal subgroups

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

subgroupindex
C2×#888572
C2×C3×#37522
C2×#888032
C32×C3⋊#952
#117220612
#105923743
#105912983