Finite Group Databasebeta

← #13139813

Subgroups — #13139813

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×D3×Dic9C211yesmaximal, characteristic
2C2×C9×(C2×D3)⋊C4C211yesmaximal, characteristic
3C2×(C2×D3)⋊Dic9C211yesmaximal, characteristic
4C2×#88442C211yesmaximal, characteristic, complemented
5C2×Dic9⋊Dic3C211yesmaximal, characteristic, complemented
6#11720572C211yesmaximal, characteristic, complemented
7C3×C9⋊#95C211yesmaximal, characteristic, complemented
8#10591373—31nomaximal
9#10588626—31nomaximal
10C23×D3×C9C2211yescharacteristic
11C22×C3×C4×C9C2211yescharacteristic
12C22×C3⋊Dic9C2211yescharacteristic
13C22×C3⋊Dic9C2211yescharacteristic
14C22×C9×Dic3C2211yescharacteristic
15C22×C3×Dic9C2211yescharacteristic
16C22×C3×Dic9C2211yescharacteristic
17C2×D3×Dic9—22no—
18C2×D3×Dic9—22no—
19C2×D3×Dic9—22no—
20C9×(C2×D3)⋊C4—22no—
21(C2×D3)⋊Dic9C414yescomplemented
22#88442—22no—
23Dic9⋊Dic3—22no—
24C22×D3×Dic3—31no—
25C2×C3×(C2×D3)⋊C4D311yescharacteristic
26C2×#3724—31no—
27C2×Dic3⋊Dic3—31no—
28C2×(C2×D3)⋊Dic3—31no—
29C23×Dic9—31no—
30#63658—31no—
31C3×C3⋊#95—31no—
32C2×C9×C22⋊C4—31no—
33C2×#3676—31no—
34C2×Dic9⋊C4—31no—
35C2×Dic9⋊C4—31no—
36C9⋊#95—31no—
37C9⋊#95—61no—
38C9⋊#95D311yescharacteristic, complemented
39C23×C3×C9C2311yescharacteristic
40C22×D3×C9—21no—
41C22×D3×C9—21no—
42C22×D3×C9—21no—
43C22×D3×C9—21no—
44C22×D3×C9C2×C412yes—
45C22×D3×C9D412yes—
46C22×D3×C9Q812yes—
47C2×C3×C4×C9—21no—
48C2×C3×C4×C9—21no—
49C2×C3×C4×C9—21no—
50C2×C3⋊Dic9—21no—
51C2×C3⋊Dic9—21no—
52C2×C3⋊Dic9—21no—
53C2×C3⋊Dic9—21no—
54C2×C3⋊Dic9C2×C412yescomplemented
55C2×C3⋊Dic9D412yes—
56C2×C9×Dic3—21no—
57C2×C9×Dic3—21no—
58C2×C9×Dic3—21no—
59C2×C3×Dic9—21no—
60C2×C3×Dic9—21no—
61C2×C3×Dic9—21no—
62C2×C3×Dic9—21no—
63C2×C3×Dic9C2×C412yescomplemented
64C2×C3×Dic9D412yes—
65D3×Dic9—28no—
66#4946—91no—
67#4668—91no—
68C23×C3×D3C2×D311yescharacteristic
69C22×C32×C4C2×D311yescharacteristic
70C22×C3⋊Dic3—31no—
71C22×C3⋊Dic3—31no—
72C22×C3×Dic3—31no—
73C22×C3×Dic3—31no—
74C22×C3×Dic3C2×D311yescharacteristic
75C2×D3×Dic3—62no—
76C2×D3×Dic3—62no—
77C2×D3×Dic3—62no—
78C24×C9—31no—
79C3×(C2×D3)⋊C4—22no—
80#3724—62no—
81Dic3⋊Dic3—62no—
82(C2×D3)⋊Dic3—34no—
83C22×C4×C9—21no—
84C22×C4×C9—31no—
85C22×C4×C9C2×D311yescharacteristic
86C22×Dic9—31no—
87C22×Dic9—31no—
88C22×Dic9—62no—
89C22×Dic9—62no—
90C22×Dic9—62no—
91C22×Dic9—61no—
92C22×Dic9—61no—
93C22×Dic9C2×D311yescharacteristic
94C22×Dic9C2×D311yescharacteristic
95C9×C22⋊C4—62no—
96#3676—34no—
97Dic9⋊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×D3×C9—22no—
107C2×D3×C9—22no—
108C2×D3×C9—22no—
109C2×D3×C9—22no—
110C2×D3×C9—22no—
111C2×D3×C9—22no—
112C2×D3×C9C4⋊C414yescomplemented
113C3×C4×C9—41no—
114C3⋊Dic9—22no—
115C3⋊Dic9—22no—
116C9×Dic3—41no—
117C3×Dic9—22no—
118C3×Dic9—22no—
119C23×Dic3—91no—
120C22×C4×D3—91no—
121C2×C3×C22⋊C4—31no—
122C2×#217—91no—
123C2×(C2×D3)⋊C4—91no—
124C2×(C2×D3)⋊C4D911yescharacteristic
125C2×C4⋊Dic3—91no—
126C2×Dic3⋊C4—91no—
127C2×Dic3⋊C4—91no—
128C2×Dic3⋊C4—91no—
129C3×#95—91no—
130C3⋊#95—31no—
131C3⋊#95—91no—
132C3⋊#95—91no—
133C3⋊#95—181no—
134C23×C32C22×D311yescharacteristic
135C22×C3×D3—21no—
136C22×C3×D3—21no—
137C22×C3×D3—21no—
138C22×C3×D3—21no—
139C22×C3×D3Dic612yes—
140C22×C3×D3C4×D312yes—
141C22×C3×D3#6712yes—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C3⋊Dic3—32no—
146C2×C3⋊Dic3—32no—
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—32no—
155C2×C3×Dic3—32no—
156C2×C3×Dic3—61no—
157C2×C3×Dic3—61no—
158C2×C3×Dic3—61no—
159C2×C3×Dic3—61no—
160D3×Dic3—68no—
161C23×C9—21no—
162C23×C9—32no—
163C23×C9—32no—
164C23×C9—32no—
165C23×C9—61no—
166C23×C9—61no—
167C23×C9—61no—
168C23×C9—61no—
169C23×C9C22×D311yescharacteristic
170C2×C4×C9—22no—
171C2×C4×C9—21no—
172C2×C4×C9—21no—
173C2×C4×C9—21no—
174C2×C4×C9—41no—
175C2×C4×C9—41no—
176C2×C4×C9—61no—
177C2×C4×C9—61no—
178C2×C4×C9—61no—
179C2×Dic9—21no—
180C2×Dic9—21no—
181C2×Dic9—21no—
182C2×Dic9—21no—
183C2×Dic9—32no—
184C2×Dic9—32no—
185C2×Dic9—62no—
186C2×Dic9—62no—
187C2×Dic9—68no—
188C2×Dic9—61no—
189C2×Dic9—61no—
190C2×Dic9—61no—
191C2×Dic9—61no—
192C2×Dic9—121no—
193C2×Dic9—121no—
194C2×Dic9—121no—
195C2×Dic9—121no—
196C2×Dic9C4×D312yescomplemented
197C2×Dic9D1212yes—
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
206D3×C9—24no—
207C24×C3—31no—
208C23×D3C2×D911yescharacteristic
209C22×C3×C4—21no—
210C22×C3×C4—31no—
211C22×C3×C4—91no—
212C22×C3×C4—91no—
213C22×C3×C4C2×D911yescharacteristic
214C22×C3×C4D3211yescharacteristic
215C22×Dic3—31no—
216C22×Dic3—31no—
217C22×Dic3—91no—
218C22×Dic3—91no—
219C22×Dic3—91no—
220C22×Dic3—91no—
221C22×Dic3—181no—
222C22×Dic3—181no—
223C22×Dic3—182no—
224C22×Dic3—182no—
225C22×Dic3—182no—
226C22×Dic3C2×D911yescharacteristic
227C2×C4×D3—182no—
228C2×C4×D3—182no—
229C2×C4×D3—182no—
230C3×C22⋊C4—62no—
231#217—94no—
232(C2×D3)⋊C4—22no—
233(C2×D3)⋊C4—94no—
234C4⋊Dic3—182no—
235Dic3⋊C4—182no—
236Dic3⋊C4—182no—
237Dic3⋊C4—182no—
238C22×C32C2×Dic611yescharacteristic
239C22×C32C2×C4×D311yescharacteristic
240C22×C32D3×D411yescharacteristic
241C22×C32D3×D411yescharacteristic
242C22×C32#23711yescharacteristic
243C22×C32#23711yescharacteristic
244C22×C32C2×#6711yescharacteristic
245C2×C3×D3—22no—
246C2×C3×D3—22no—
247C2×C3×D3—22no—
248C2×C3×D3—22no—
249C2×C3×D3—22no—
250C2×C3×D3—22no—
251C2×C3×D3Dic3⋊C414yes—
252C32×C4—41no—
253C3⋊Dic3—62no—
254C3⋊Dic3—62no—
255C3×Dic3—41no—
256C3×Dic3—62no—
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×C9—34no—
266C22×C9—62no—
267C22×C9—62no—
268C22×C9—62no—
269C22×C9—62no—
270C22×C9—62no—
271C22×C9—62no—
272C22×C9C2×C4×D311yescharacteristic
273C22×C9C2×D1211yescharacteristic
274C22×C9#23511yescharacteristic
275C22×C9D3×D411yescharacteristic
276C22×C9D3×D411yescharacteristic
277C22×C9#23711yescharacteristic
278C22×C9D3×Q811yescharacteristic
279C4×C9—42no—
280C4×C9—41no—
281C4×C9—121no—
282Dic9—22no—
283Dic9—22no—
284Dic9—62no—
285Dic9—62no—
286Dic9—122no—
287Dic9—122no—
288C23×C4—271no—
289C2×C4⋊C4—271no—
290C2×C4⋊C4—271no—
291C2×C22⋊C4—31no—
292C2×C22⋊C4—271no—
293#95—91no—
294#95—271no—
295C3×C9#38511yescharacteristic, complemented, Sylow
296C23×C3—21no—
297C23×C3—32no—
298C23×C3—32no—
299C23×C3—32no—
300C23×C3—61no—
301C23×C3—61no—
302C23×C3—61no—
303C23×C3—61no—
304C23×C3C22×D911yescharacteristic
305C23×C3C2×D3211yescharacteristic
306C22×D3—21no—
307C22×D3—21no—
308C22×D3—21no—
309C22×D3—21no—
310C22×D3Dic1812yes—
311C22×D3C4×D912yes—
312C22×D3#61112yes—
313C2×C3×C4—22no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—61no—
323C2×C3×C4—61no—
324C2×C3×C4—61no—
325C2×C3×C4—92no—
326C2×C3×C4—92no—
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—32no—
335C2×Dic3—32no—
336C2×Dic3—61no—
337C2×Dic3—61no—
338C2×Dic3—61no—
339C2×Dic3—61no—
340C2×Dic3—92no—
341C2×Dic3—92no—
342C2×Dic3—92no—
343C2×Dic3—92no—
344C2×Dic3—182no—
345C2×Dic3—182no—
346C2×Dic3—188no—
347C2×Dic3—181no—
348C2×Dic3—181no—
349C2×Dic3—181no—
350C2×Dic3—181no—
351C2×Dic3—181no—
352C2×Dic3—181no—
353C2×Dic3—181no—
354C2×Dic3—181no—
355C2×Dic3—361no—
356C2×Dic3—361no—
357C2×Dic3—361no—
358C2×Dic3—361no—
359C4×D3—188no—
360C2×C32#87811yescharacteristic
361C2×C32#87811yescharacteristic
362C2×C32#67⋊C411yescharacteristic
363C2×C32#67⋊C411yescharacteristic
364C2×C32C2×Dic3⋊C411yescharacteristic
365C2×C32#93511yescharacteristic
366C2×C32#93711yescharacteristic
367C3×D3—24no—
368C2×C9—21no—
369C2×C9—21no—
370C2×C9—21no—
371C2×C9—21no—
372C2×C9—21no—
373C2×C9—21no—
374C2×C9—21no—
375C2×C9—64no—
376C2×C9C4×D1211yescharacteristic
377C2×C9#67⋊C411yescharacteristic
378C2×C9#88211yescharacteristic
379C2×C9#88311yescharacteristic
380C2×C9D3×C4⋊C411yescharacteristic
381C2×C9#89611yescharacteristic
382C2×C9#89711yescharacteristic
383C24—31no—
384C22×C4—31no—
385C22×C4—91no—
386C22×C4—91no—
387C22×C4—271no—
388C22×C4—271no—
389C22×C4—542no—
390C22×C4—542no—
391C22×C4—542no—
392C22×C4D3×D911yescharacteristic
393C4⋊C4—542no—
394C4⋊C4—542no—
395C22⋊C4—62no—
396C22⋊C4—274no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—34no—
405C22×C3—62no—
406C22×C3—62no—
407C22×C3—62no—
408C22×C3—62no—
409C22×C3—62no—
410C22×C3—62no—
411C22×C3C2×Dic1811yescharacteristic
412C22×C3C2×C4×D911yescharacteristic
413C22×C3D4×D911yescharacteristic
414C22×C3D4×D911yescharacteristic
415C22×C3#369911yescharacteristic
416C22×C3#369911yescharacteristic
417C22×C3C2×#61111yescharacteristic
418C22×C3D3×Dic611yescharacteristic
419C22×C3C4×D3211yescharacteristic
420C22×C3D3×D1211yescharacteristic
421C22×C3#380211yescharacteristic
422C22×C3#380411yescharacteristic
423C22×C3#380511yescharacteristic
424C22×C3D3×#6711yescharacteristic
425C2×D3—22no—
426C2×D3—22no—
427C2×D3—22no—
428C2×D3—22no—
429C2×D3—22no—
430C2×D3—22no—
431C2×D3Dic9⋊C414yescomplemented
432C3×C4—42no—
433C3×C4—41no—
434C3×C4—41no—
435C3×C4—121no—
436C3×C4—182no—
437C3×C4—182no—
438Dic3—41no—
439Dic3—62no—
440Dic3—62no—
441Dic3—182no—
442Dic3—182no—
443Dic3—182no—
444Dic3—182no—
445Dic3—362no—
446Dic3—362no—
447C32#494611yescharacteristic
448C9—21no—
449C9#466811yescharacteristic, complemented
450C23—32no—
451C23—32no—
452C23—32no—
453C23—61no—
454C23—61no—
455C23—61no—
456C23—61no—
457C23C2×D3×D911yescharacteristic
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—61no—
462C2×C4—61no—
463C2×C4—61no—
464C2×C4—92no—
465C2×C4—92no—
466C2×C4—181no—
467C2×C4—181no—
468C2×C4—181no—
469C2×C4—181no—
470C2×C4—272no—
471C2×C4—272no—
472C2×C4—548no—
473C2×C4—541no—
474C2×C4—541no—
475C2×C4—541no—
476C2×C4—541no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—64no—
485C2×C3#6344011yescharacteristic
486C2×C3#6344011yescharacteristic
487C2×C3#611⋊C411yescharacteristic
488C2×C3#611⋊C411yescharacteristic
489C2×C3C2×Dic9⋊C411yescharacteristic
490C2×C3#6349711yescharacteristic
491C2×C3#6349911yescharacteristic
492C2×C3D3×Dic3⋊C411yescharacteristic
493C2×C3#6388011yescharacteristic
494C2×C3#6389311yescharacteristic
495C2×C3#6389911yescharacteristic
496C2×C3#6390211yescharacteristic
497C2×C3#6392211yescharacteristic
498C2×C3#6396911yescharacteristic
499D3—24no—
500C22—34no—
501C22—62no—
502C22—62no—
503C22—62no—
504C22—62no—
505C22—62no—
506C22—62no—
507C22D3×Dic1811yescharacteristic
508C22C4×D3×D911yescharacteristic
509C22D9×D1211yescharacteristic
510C22#8864611yescharacteristic
511C22#8866211yescharacteristic
512C22#8866311yescharacteristic
513C22D3×#61111yescharacteristic
514C4—41no—
515C4—121no—
516C4—182no—
517C4—182no—
518C4—542no—
519C4—542no—
520C3—21no—
521C3#1058862611yescharacteristic, complemented
522C3#1059137311yescharacteristic
523C2—64no—
524C2D3×Dic9⋊C411yescharacteristic
525C2#1172101311yescharacteristic
526C2#1172102311yescharacteristic
527C2#1172102611yescharacteristic
528C2#1172102811yescharacteristic
529C2#1172105011yescharacteristic
530C2#1172128011yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutatorC22×C3×C9C22×C4
FrattiniC23×C3C2×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
prime radicalC23×C3×D3C2×D3
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
4. C2×#88442C2semidirect
5. C2×Dic9⋊Dic3C2semidirect
6. #11720572C2semidirect
7. C3×C9⋊#95C2semidirect
21. (C2×D3)⋊Dic9C4semidirect
38. C9⋊#95D3semidirect
54. C2×C3⋊Dic9C2×C4semidirect
63. C2×C3×Dic9C2×C4semidirect
112. C2×D3×C9C4⋊C4semidirect
196. C2×Dic9C4×D3semidirect
295. C3×C9#385semidirect
431. C2×D3Dic9⋊C4semidirect
449. C9#4668semidirect
521. C3#10588626semidirect

Maximal subgroups

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

subgroupindex
C22×D3×Dic92
C2×C9×(C2×D3)⋊C42
C2×(C2×D3)⋊Dic92
C2×#884422
C2×Dic9⋊Dic32
#117205722
C3×C9⋊#952
#105913733
#105886263