Finite Group Databasebeta

← #13139698

Subgroups — #13139698

Order 1728.

778 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
1C22×C3×C4×D9C211yesmaximal, characteristic
2C2×C9×Dic3⋊C4C211yesmaximal, characteristic, complemented
3C2×(C2×D9)⋊Dic3C211yesmaximal, characteristic
4C2×(C2×D9)⋊Dic3C211yesmaximal, characteristic
5C2×#88442C211yesmaximal, characteristic, complemented
6#11720572C211yesmaximal, characteristic, complemented
7#11720331C211yesmaximal, characteristic, complemented
8#10591277—31nomaximal
9#10588669—31nomaximal
10C23×C3×D9C2211yescharacteristic
11C22×C3×C4×C9C2211yescharacteristic
12C22×C3⋊Dic9C2211yescharacteristic
13C22×C3⋊Dic9C2211yescharacteristic
14C22×C9×Dic3C2211yescharacteristic
15C22×C9×Dic3C2211yescharacteristic
16C22×C3×Dic9C2211yescharacteristic
17C2×C3×C4×D9—22no—
18C2×C3×C4×D9—22no—
19C2×C3×C4×D9—22no—
20C9×Dic3⋊C4—22no—
21(C2×D9)⋊Dic3—22no—
22(C2×D9)⋊Dic3C414yescomplemented
23#88442—22no—
24C22×C3×C4×D3—31no—
25C2×C3×Dic3⋊C4D311yescharacteristic
26C2×#3724—31no—
27C2×(C2×D3)⋊Dic3—31no—
28C2×(C2×D3)⋊Dic3—31no—
29C22×C4×D9D311yescharacteristic
30#63658—31no—
31#63579—31no—
32C2×C9×C4⋊C4—31no—
33C2×(C2×D9)⋊C4—31no—
34C2×(C2×D9)⋊C4—31no—
35C2×Dic9⋊C4—31no—
36C9⋊#95—31no—
37C9⋊#95—31no—
38C9⋊#95—61no—
39C23×C3×C9C2311yescharacteristic
40C22×C3×D9—21no—
41C22×C3×D9—21no—
42C22×C3×D9—21no—
43C22×C3×D9—21no—
44C22×C3×D9C2×C412yes—
45C22×C3×D9D412yes—
46C22×C3×D9Q812yes—
47C2×C3×C4×C9—21no—
48C2×C3×C4×C9—21no—
49C2×C3×C4×C9D412yes—
50C2×C3⋊Dic9—21no—
51C2×C3⋊Dic9—21no—
52C2×C3⋊Dic9—21no—
53C2×C3⋊Dic9—21no—
54C2×C3⋊Dic9—21no—
55C2×C3⋊Dic9C2×C412yescomplemented
56C2×C9×Dic3—21no—
57C2×C9×Dic3—21no—
58C2×C9×Dic3—21no—
59C2×C9×Dic3—21no—
60C2×C9×Dic3—21no—
61C2×C9×Dic3C2×C412yescomplemented
62C2×C3×Dic9—21no—
63C2×C3×Dic9—21no—
64C2×C3×Dic9D412yes—
65C3×C4×D9—28no—
66#5210—91no—
67#4989—91no—
68C23×C3×D3—31no—
69C22×C32×C4C2×D311yescharacteristic
70C22×C3⋊Dic3—31no—
71C22×C3⋊Dic3—31no—
72C22×C3×Dic3—31no—
73C22×C3×Dic3C2×D311yescharacteristic
74C22×C3×Dic3C2×D311yescharacteristic
75C2×C3×C4×D3—62no—
76C2×C3×C4×D3—62no—
77C2×C3×C4×D3—62no—
78C23×D9C2×D311yescharacteristic
79C3×Dic3⋊C4—22no—
80#3724—62no—
81(C2×D3)⋊Dic3—34no—
82(C2×D3)⋊Dic3—62no—
83C22×C4×C9—21no—
84C22×C4×C9—31no—
85C22×C4×C9—31no—
86C22×C4×C9C2×D311yescharacteristic
87C22×Dic9—31no—
88C22×Dic9—31no—
89C22×Dic9—61no—
90C22×Dic9—61no—
91C22×Dic9C2×D311yescharacteristic
92C2×C4×D9—22no—
93C2×C4×D9—22no—
94C2×C4×D9—22no—
95C9×C4⋊C4—62no—
96(C2×D9)⋊C4—34no—
97(C2×D9)⋊C4—62no—
98Dic9⋊C4—62no—
99C22×C3×C9C22×C411yescharacteristic
100C22×C3×C9C2×D411yescharacteristic
101C22×C3×C9C2×D411yescharacteristic
102C22×C3×C9C2×D411yescharacteristic
103C22×C3×C9C2×Q811yescharacteristic
104C22×C3×C9#4111yescharacteristic
105C22×C3×C9#4111yescharacteristic
106C2×C3×D9—22no—
107C2×C3×D9—22no—
108C2×C3×D9—22no—
109C2×C3×D9—22no—
110C2×C3×D9—22no—
111C2×C3×D9—22no—
112C2×C3×D9C4⋊C414yescomplemented
113C3×C4×C9—22no—
114C3⋊Dic9—22no—
115C3⋊Dic9—41no—
116C9×Dic3—22no—
117C9×Dic3—41no—
118C3×Dic9—22no—
119C23×C3×C4—91no—
120C22×C4×D3—31no—
121C2×C3×C4⋊C4—31no—
122C2×#217—91no—
123C2×#217—91no—
124C2×(C2×D3)⋊C4—91no—
125C2×(C2×D3)⋊C4—91no—
126C2×Dic3⋊C4—91no—
127C2×Dic3⋊C4—91no—
128C2×Dic3⋊C4D911yescharacteristic, complemented
129C3⋊#95—91no—
130C3⋊#95—91no—
131C3⋊#95—91no—
132C3⋊#95—91no—
133C3⋊#95—181no—
134C23×C32C22×D311yescharacteristic
135C22×C3×D3—32no—
136C22×C3×D3—32no—
137C22×C3×D3—32no—
138C22×C3×D3—61no—
139C22×C3×D3—61no—
140C22×C3×D3—61no—
141C22×C3×D3—61no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4#6712yes—
145C2×C3⋊Dic3—32no—
146C2×C3⋊Dic3—61no—
147C2×C3⋊Dic3—61no—
148C2×C3⋊Dic3—61no—
149C2×C3⋊Dic3—61no—
150C2×C3⋊Dic3—61no—
151C2×C3×Dic3—21no—
152C2×C3×Dic3—21no—
153C2×C3×Dic3—21no—
154C2×C3×Dic3—21no—
155C2×C3×Dic3—21no—
156C2×C3×Dic3—32no—
157C2×C3×Dic3—61no—
158C2×C3×Dic3—61no—
159C2×C3×Dic3C4×D312yes—
160C3×C4×D3—68no—
161C23×C9—21no—
162C23×C9C22×D311yescharacteristic
163C22×D9—21no—
164C22×D9—21no—
165C22×D9—21no—
166C22×D9—21no—
167C22×D9Dic612yes—
168C22×D9C4×D312yes—
169C22×D9#6712yes—
170C2×C4×C9—21no—
171C2×C4×C9—21no—
172C2×C4×C9—22no—
173C2×C4×C9—32no—
174C2×C4×C9—41no—
175C2×C4×C9—41no—
176C2×C4×C9—61no—
177C2×C4×C9—61no—
178C2×C4×C9—61no—
179C2×C4×C9—61no—
180C2×C4×C9—61no—
181C2×C4×C9#6712yes—
182C2×Dic9—21no—
183C2×Dic9—21no—
184C2×Dic9—32no—
185C2×Dic9—62no—
186C2×Dic9—61no—
187C2×Dic9—61no—
188C2×Dic9—62no—
189C2×Dic9—61no—
190C2×Dic9—61no—
191C2×Dic9—61no—
192C2×Dic9—121no—
193C2×Dic9—121no—
194C2×Dic9—121no—
195C2×Dic9—121no—
196C2×Dic9#6712yes—
197C4×D9—28no—
198#385—271noSylow
199C2×C3×C9C2×C4⋊C411yescharacteristic
200C2×C3×C9C4×D411yescharacteristic
201C2×C3×C9C4×D411yescharacteristic
202C2×C3×C9#12011yescharacteristic
203C2×C3×C9C22⋊Q811yescharacteristic
204C2×C3×C9C22⋊Q811yescharacteristic
205C2×C3×C9#12311yescharacteristic
206C3×D9—24no—
207C24×C3—91no—
208C23×D3—31no—
209C22×C3×C4—21no—
210C22×C3×C4—31no—
211C22×C3×C4—31no—
212C22×C3×C4—91no—
213C22×C3×C4—182no—
214C22×C3×C4—182no—
215C22×C3×C4—182no—
216C22×C3×C4C2×D911yescharacteristic
217C22×C3×C4D3211yescharacteristic
218C22×Dic3—31no—
219C22×Dic3—91no—
220C22×Dic3—91no—
221C22×Dic3—91no—
222C22×Dic3—91no—
223C22×Dic3—181no—
224C22×Dic3—181no—
225C22×Dic3C2×D911yescharacteristic
226C22×Dic3C2×D911yescharacteristic
227C2×C4×D3—62no—
228C2×C4×D3—62no—
229C2×C4×D3—62no—
230C3×C4⋊C4—62no—
231#217—94no—
232#217—182no—
233(C2×D3)⋊C4—94no—
234(C2×D3)⋊C4—182no—
235Dic3⋊C4—22no—
236Dic3⋊C4—182no—
237Dic3⋊C4—182no—
238C22×C32C2×C4×D311yescharacteristic
239C22×C32#23511yescharacteristic
240C22×C32D3×D411yescharacteristic
241C22×C32D3×D411yescharacteristic
242C22×C32D3×Q811yescharacteristic
243C22×C32#23911yescharacteristic
244C22×C32C2×#6711yescharacteristic
245C2×C3×D3—34no—
246C2×C3×D3—62no—
247C2×C3×D3—62no—
248C2×C3×D3—62no—
249C2×C3×D3—62no—
250C2×C3×D3—62no—
251C2×C3×D3—62no—
252C32×C4—22no—
253C3⋊Dic3—62no—
254C3⋊Dic3—121no—
255C3×Dic3—22no—
256C3×Dic3—41no—
257C3×Dic3—62no—
258C22×C9—21no—
259C22×C9—21no—
260C22×C9—21no—
261C22×C9—21no—
262C22×C9—21no—
263C22×C9—21no—
264C22×C9—21no—
265C22×C9C2×Dic611yescharacteristic
266C22×C9C2×C4×D311yescharacteristic
267C22×C9#23511yescharacteristic
268C22×C9#23511yescharacteristic
269C22×C9C2×#6711yescharacteristic
270C22×C9C2×#6711yescharacteristic
271C22×C9C2×#6711yescharacteristic
272C2×D9—22no—
273C2×D9—22no—
274C2×D9—22no—
275C2×D9—22no—
276C2×D9—22no—
277C2×D9—22no—
278C2×D9Dic3⋊C414yescomplemented
279C4×C9—22no—
280C4×C9—42no—
281C4×C9—62no—
282C4×C9—121no—
283Dic9—22no—
284Dic9—62no—
285Dic9—122no—
286Dic9—122no—
287Dic9—121no—
288C23×C4—91no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—271no—
291C2×C22⋊C4—271no—
292C2×C22⋊C4—271no—
293#95—271no—
294#95—271no—
295C3×C9#38511yescharacteristic, complemented, Sylow
296C23×C3—21no—
297C23×C3—92no—
298C23×C3—92no—
299C23×C3—92no—
300C23×C3—181no—
301C23×C3—181no—
302C23×C3—181no—
303C23×C3—181no—
304C23×C3C22×D911yescharacteristic
305C23×C3C2×D3211yescharacteristic
306C22×D3—32no—
307C22×D3—32no—
308C22×D3—32no—
309C22×D3—61no—
310C22×D3—61no—
311C22×D3—61no—
312C22×D3—61no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—22no—
318C2×C3×C4—32no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—61no—
322C2×C3×C4—61no—
323C2×C3×C4—61no—
324C2×C3×C4—61no—
325C2×C3×C4—61no—
326C2×C3×C4—92no—
327C2×C3×C4—188no—
328C2×C3×C4—181no—
329C2×C3×C4—181no—
330C2×C3×C4#61112yes—
331C2×C3×C4#62512yes—
332C2×Dic3—21no—
333C2×Dic3—21no—
334C2×Dic3—21no—
335C2×Dic3—21no—
336C2×Dic3—21no—
337C2×Dic3—32no—
338C2×Dic3—61no—
339C2×Dic3—61no—
340C2×Dic3—92no—
341C2×Dic3—92no—
342C2×Dic3—182no—
343C2×Dic3—181no—
344C2×Dic3—181no—
345C2×Dic3—181no—
346C2×Dic3—181no—
347C2×Dic3—182no—
348C2×Dic3—181no—
349C2×Dic3—181no—
350C2×Dic3—181no—
351C2×Dic3—181no—
352C2×Dic3—181no—
353C2×Dic3—181no—
354C2×Dic3—361no—
355C2×Dic3—361no—
356C2×Dic3—361no—
357C2×Dic3—361no—
358C2×Dic3C4×D912yescomplemented
359C4×D3—68no—
360C2×C32D3×C4⋊C411yescharacteristic
361C2×C32#89311yescharacteristic
362C2×C32#89411yescharacteristic
363C2×C32#89611yescharacteristic
364C2×C32C4×#6711yescharacteristic
365C2×C32#93711yescharacteristic
366C2×C32#94611yescharacteristic
367C3×D3—64no—
368C2×C9—21no—
369C2×C9—21no—
370C2×C9—21no—
371C2×C9—21no—
372C2×C9—21no—
373C2×C9—21no—
374C2×C9—21no—
375C2×C9C2×Dic3⋊C411yescharacteristic
376C2×C9#92411yescharacteristic
377C2×C9#92411yescharacteristic
378C2×C9C4×#6711yescharacteristic
379C2×C9C4×#6711yescharacteristic
380C2×C9#92911yescharacteristic
381C2×C9#95311yescharacteristic
382D9—24no—
383C24—91no—
384C22×C4—31no—
385C22×C4—31no—
386C22×C4—91no—
387C22×C4—182no—
388C22×C4—182no—
389C22×C4—182no—
390C22×C4—271no—
391C22×C4—271no—
392C22×C4D3×D911yescharacteristic
393C4⋊C4—62no—
394C4⋊C4—542no—
395C22⋊C4—274no—
396C22⋊C4—542no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—94no—
405C22×C3—182no—
406C22×C3—182no—
407C22×C3—182no—
408C22×C3—182no—
409C22×C3—182no—
410C22×C3—182no—
411C22×C3C2×C4×D911yescharacteristic
412C22×C3#369711yescharacteristic
413C22×C3D4×D911yescharacteristic
414C22×C3D4×D911yescharacteristic
415C22×C3Q8×D911yescharacteristic
416C22×C3#370111yescharacteristic
417C22×C3C2×#61111yescharacteristic
418C22×C3D3×Dic611yescharacteristic
419C22×C3#379811yescharacteristic
420C22×C3#379911yescharacteristic
421C22×C3C4×D3211yescharacteristic
422C22×C3C2×#62511yescharacteristic
423C22×C3D3×#6711yescharacteristic
424C22×C3D3×#6711yescharacteristic
425C2×D3—34no—
426C2×D3—62no—
427C2×D3—62no—
428C2×D3—62no—
429C2×D3—62no—
430C2×D3—62no—
431C2×D3—62no—
432C3×C4—22no—
433C3×C4—22no—
434C3×C4—42no—
435C3×C4—62no—
436C3×C4—121no—
437C3×C4—182no—
438Dic3—22no—
439Dic3—41no—
440Dic3—62no—
441Dic3—182no—
442Dic3—182no—
443Dic3—362no—
444Dic3—362no—
445Dic3—361no—
446Dic3—361no—
447C32#498911yescharacteristic
448C9—21no—
449C9#521011yescharacteristic, complemented
450C23—92no—
451C23—92no—
452C23—92no—
453C23—181no—
454C23—181no—
455C23—181no—
456C23—181no—
457C23C2×D3×D911yescharacteristic
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—32no—
461C2×C4—61no—
462C2×C4—61no—
463C2×C4—61no—
464C2×C4—61no—
465C2×C4—61no—
466C2×C4—92no—
467C2×C4—188no—
468C2×C4—181no—
469C2×C4—181no—
470C2×C4—272no—
471C2×C4—541no—
472C2×C4—541no—
473C2×C4—541no—
474C2×C4—541no—
475C2×C4—541no—
476C2×C4#619412yes—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—184no—
485C2×C3C4⋊C4×D911yescharacteristic
486C2×C3#6345511yescharacteristic
487C2×C3#6345611yescharacteristic
488C2×C3#6345811yescharacteristic
489C2×C3C4×#61111yescharacteristic
490C2×C3#6349911yescharacteristic
491C2×C3#6350811yescharacteristic
492C2×C3#6385011yescharacteristic
493C2×C3#6385111yescharacteristic
494C2×C3#6385611yescharacteristic
495C2×C3D3×Dic3⋊C411yescharacteristic
496C2×C3C4×#62511yescharacteristic
497C2×C3#6390411yescharacteristic
498C2×C3#6397611yescharacteristic
499D3—64no—
500C22—94no—
501C22—182no—
502C22—182no—
503C22—182no—
504C22—182no—
505C22—182no—
506C22—182no—
507C22D9×Dic611yescharacteristic
508C22#8863511yescharacteristic
509C22#8864011yescharacteristic
510C22C4×D3×D911yescharacteristic
511C22C2×#619411yescharacteristic
512C22D9×#6711yescharacteristic
513C22D9×#6711yescharacteristic
514C4—22no—
515C4—62no—
516C4—121no—
517C4—182no—
518C4—542no—
519C4—1081no—
520C3—21no—
521C3#1058866911yescharacteristic, complemented
522C3#1059127711yescharacteristic
523C2—184no—
524C2#1172093811yescharacteristic
525C2#1172093911yescharacteristic
526C2#1172094411yescharacteristic
527C2D9×Dic3⋊C411yescharacteristic
528C2#1172100211yescharacteristic
529C2C4×#619411yescharacteristic
530C2#1172128511yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutatorC22×C3×C9C22×C4
FrattiniC23×C3C2×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
prime radicalC23×C3×D9C22
cosocleC23×C3×C9C23
hypercenterC22×C4D3×D9
hypocenterC3×C9#385
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C9×Dic3⋊C4C2semidirect
5. C2×#88442C2semidirect
6. #11720572C2semidirect
7. #11720331C2semidirect
22. (C2×D9)⋊Dic3C4semidirect
55. C2×C3⋊Dic9C2×C4semidirect
61. C2×C9×Dic3C2×C4semidirect
112. C2×C3×D9C4⋊C4semidirect
128. C2×Dic3⋊C4D9semidirect
278. C2×D9Dic3⋊C4semidirect
295. C3×C9#385semidirect
358. C2×Dic3C4×D9semidirect
449. C9#5210semidirect
521. C3#10588669semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×D92
C2×C9×Dic3⋊C42
C2×(C2×D9)⋊Dic32
C2×#884422
#117205722
#117203312
#105912773
#105886693