Finite Group Databasebeta

← #13139864

Subgroups — #13139864

Order 1728.

882 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×Dic3×D9C211yesmaximal, characteristic
2C2×C4×C3⋊Dic9C211yesmaximal, characteristic, complemented
3C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic
4C2×(C2×D9)⋊Dic3C211yesmaximal, characteristic
5C2×Dic3×Dic9C211yesmaximal, characteristic, complemented
6C9×C3⋊#95C211yesmaximal, characteristic, complemented
7#11720331C211yesmaximal, characteristic, complemented
8#10591368—31nomaximal
9#10588346—31nomaximal
10C23×C3×D9C2211yescharacteristic
11C22×C3×C4×C9C2211yescharacteristic
12C22×C3⋊Dic9C2211yescharacteristic
13C22×C3⋊Dic9C2211yescharacteristic
14C22×C9×Dic3C2211yescharacteristic
15C22×C9×Dic3C2211yescharacteristic
16C22×C3×Dic9C2211yescharacteristic
17C2×Dic3×D9—22no—
18C2×Dic3×D9—22no—
19C2×Dic3×D9C414yescomplemented
20C4×C3⋊Dic9—22no—
21C3×(C2×D9)⋊C4C414yescomplemented
22(C2×D9)⋊Dic3C414yescomplemented
23Dic3×Dic9—22no—
24C22×D3×Dic3—31no—
25C2×C4×C3⋊Dic3—31no—
26C2×C3×(C2×D3)⋊C4—31no—
27C2×(C2×D3)⋊Dic3—31no—
28C2×Dic32—31no—
29C22×C4×D9—31no—
30C3×C3⋊#95D311yescharacteristic
31#63579—31no—
32C2×(C2×D9)⋊C4—31no—
33C2×(C2×D9)⋊C4D311yescharacteristic
34C2×C4×Dic9—31no—
35C2×C4×Dic9—31no—
36C2×C4×Dic9—61no—
37C9×#95—31no—
38C9⋊#95—31no—
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×D9C2×C412yes—
46C22×C3×D9C2×C412yes—
47C2×C3×C4×C9—21no—
48C2×C3×C4×C9—21no—
49C2×C3×C4×C9C2×C412yescomplemented
50C2×C3⋊Dic9—21no—
51C2×C3⋊Dic9—21no—
52C2×C3⋊Dic9C2×C412yes—
53C2×C3⋊Dic9C2×C412yescomplemented
54C2×C3⋊Dic9D412yes—
55C2×C3⋊Dic9D412yes—
56C2×C9×Dic3—21no—
57C2×C9×Dic3—21no—
58C2×C9×Dic3—21no—
59C2×C9×Dic3—21no—
60C2×C9×Dic3C2×C412yes—
61C2×C9×Dic3C2×C412yescomplemented
62C2×C3×Dic9—21no—
63C2×C3×Dic9—21no—
64C2×C3×Dic9C2×C412yescomplemented
65Dic3×D9—28no—
66Dic3×C22⋊C4—91no—
67#4666—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×D3×Dic3—34no—
76C2×D3×Dic3—62no—
77C2×D3×Dic3—62no—
78C23×D9C2×D311yescharacteristic
79C4×C3⋊Dic3—62no—
80C3×(C2×D3)⋊C4—34no—
81(C2×D3)⋊Dic3—34no—
82Dic32—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—34no—
93C2×C4×D9—62no—
94C2×C4×D9—62no—
95(C2×D9)⋊C4—34no—
96(C2×D9)⋊C4Dic314yescomplemented
97C4×Dic9—62no—
98C4×Dic9—62no—
99C4×Dic9—64no—
100C22×C3×C9C22×C411yescharacteristic
101C22×C3×C9C22×C411yescharacteristic
102C22×C3×C9C22×C411yescharacteristic
103C22×C3×C9C2×D411yescharacteristic
104C22×C3×C9C2×D411yescharacteristic
105C22×C3×C9#4111yescharacteristic
106C22×C3×C9#4111yescharacteristic
107C2×C3×D9—22no—
108C2×C3×D9—22no—
109C2×C3×D9—22no—
110C2×C3×D9—22no—
111C2×C3×D9—22no—
112C2×C3×D9—22no—
113C2×C3×D9C4214yescomplemented
114C3×C4×C9—22no—
115C3⋊Dic9—22no—
116C3⋊Dic9C22⋊C414yescomplemented
117C9×Dic3—22no—
118C9×Dic3—22no—
119C3×Dic9—22no—
120C23×Dic3—91no—
121C22×C4×D3—91no—
122C2×C3×C22⋊C4—91no—
123C2×#217—91no—
124C2×(C2×D3)⋊C4—31no—
125C2×(C2×D3)⋊C4—91no—
126C2×C4×Dic3—91no—
127C2×C4×Dic3—91no—
128C2×C4×Dic3—91no—
129C2×C4×Dic3—91no—
130C2×C4×Dic3—181no—
131C3×#95—31no—
132C3⋊#95—91no—
133C3⋊#95—91no—
134C3⋊#95D911yescharacteristic, complemented
135C23×C32C22×D311yescharacteristic
136C22×C3×D3—32no—
137C22×C3×D3—32no—
138C22×C3×D3—32no—
139C22×C3×D3—61no—
140C22×C3×D3—61no—
141C22×C3×D3—61no—
142C22×C3×D3—61no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4C4×D312yes—
146C2×C3⋊Dic3—32no—
147C2×C3⋊Dic3—32no—
148C2×C3⋊Dic3—32no—
149C2×C3⋊Dic3—32no—
150C2×C3⋊Dic3—61no—
151C2×C3⋊Dic3—61no—
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—
160C2×C3×Dic3C4×D312yes—
161D3×Dic3—68no—
162C23×C9—21no—
163C23×C9C22×D311yescharacteristic
164C22×D9—21no—
165C22×D9—21no—
166C22×D9—21no—
167C22×D9—21no—
168C22×D9C4×D312yes—
169C22×D9C4×D312yes—
170C22×D9C2×Dic312yes—
171C2×C4×C9—22no—
172C2×C4×C9—22no—
173C2×C4×C9—21no—
174C2×C4×C9—21no—
175C2×C4×C9—22no—
176C2×C4×C9—32no—
177C2×C4×C9—32no—
178C2×C4×C9—61no—
179C2×C4×C9—61no—
180C2×C4×C9—61no—
181C2×C4×C9—61no—
182C2×C4×C9C2×Dic312yescomplemented
183C2×Dic9—21no—
184C2×Dic9—21no—
185C2×Dic9—32no—
186C2×Dic9—32no—
187C2×Dic9—32no—
188C2×Dic9—32no—
189C2×Dic9—62no—
190C2×Dic9—62no—
191C2×Dic9—62no—
192C2×Dic9—62no—
193C2×Dic9—62no—
194C2×Dic9—62no—
195C2×Dic9—61no—
196C2×Dic9—61no—
197C2×Dic9C2×Dic312yescomplemented
198C4×D9—68no—
199C4×C22⋊C4—271noSylow
200C2×C3×C9C2×C4211yescharacteristic
201C2×C3×C9C2×C22⋊C411yescharacteristic
202C2×C3×C9#11711yescharacteristic
203C2×C3×C9C4×D411yescharacteristic
204C2×C3×C9C4×D411yescharacteristic
205C2×C3×C9C4×D411yescharacteristic
206C2×C3×C9C4×D411yescharacteristic
207C3×D9—24no—
208C24×C3—91no—
209C23×D3—31no—
210C22×C3×C4—21no—
211C22×C3×C4—31no—
212C22×C3×C4—31no—
213C22×C3×C4—91no—
214C22×C3×C4C2×D911yescharacteristic
215C22×C3×C4D3211yescharacteristic
216C22×Dic3—31no—
217C22×Dic3—91no—
218C22×Dic3—91no—
219C22×Dic3—91no—
220C22×Dic3—91no—
221C22×Dic3—94no—
222C22×Dic3—181no—
223C22×Dic3—181no—
224C22×Dic3—182no—
225C22×Dic3—182no—
226C22×Dic3C2×D911yescharacteristic
227C22×Dic3C2×D911yescharacteristic
228C2×C4×D3—94no—
229C2×C4×D3—182no—
230C2×C4×D3—182no—
231C3×C22⋊C4—94no—
232#217—94no—
233(C2×D3)⋊C4—34no—
234(C2×D3)⋊C4—94no—
235C4×Dic3—182no—
236C4×Dic3—182no—
237C4×Dic3—182no—
238C4×Dic3—182no—
239C4×Dic3—184no—
240C22×C32C2×C4×D311yescharacteristic
241C22×C32C2×C4×D311yescharacteristic
242C22×C32C2×C4×D311yescharacteristic
243C22×C32D3×D411yescharacteristic
244C22×C32D3×D411yescharacteristic
245C22×C32#23711yescharacteristic
246C22×C32#23911yescharacteristic
247C2×C3×D3—34no—
248C2×C3×D3—62no—
249C2×C3×D3—62no—
250C2×C3×D3—62no—
251C2×C3×D3—62no—
252C2×C3×D3—62no—
253C2×C3×D3—62no—
254C32×C4—22no—
255C3⋊Dic3—34no—
256C3⋊Dic3—62no—
257C3×Dic3—22no—
258C3×Dic3—22no—
259C3×Dic3—62no—
260C22×C9—21no—
261C22×C9—21no—
262C22×C9—21no—
263C22×C9—21no—
264C22×C9—21no—
265C22×C9—21no—
266C22×C9—21no—
267C22×C9C2×C4×D311yescharacteristic
268C22×C9C2×C4×D311yescharacteristic
269C22×C9D3×D411yescharacteristic
270C22×C9D3×D411yescharacteristic
271C22×C9#23711yescharacteristic
272C22×C9#23711yescharacteristic
273C22×C9C22×Dic311yescharacteristic
274C2×D9—22no—
275C2×D9—22no—
276C2×D9—22no—
277C2×D9—22no—
278C2×D9—22no—
279C2×D9—22no—
280C2×D9C4×Dic314yescomplemented
281C4×C9—24no—
282C4×C9—22no—
283C4×C9—62no—
284C4×C9—62no—
285Dic9—22no—
286Dic9—34no—
287Dic9—64no—
288Dic9—64no—
289Dic9—62no—
290C23×C4—271no—
291C2×C22⋊C4—91no—
292C2×C22⋊C4—271no—
293C2×C42—271no—
294C2×C42—271no—
295#95—31no—
296#95—271no—
297C3×C9C4×C22⋊C411yescharacteristic, complemented, Sylow
298C23×C3—21no—
299C23×C3—92no—
300C23×C3—92no—
301C23×C3—92no—
302C23×C3—181no—
303C23×C3—181no—
304C23×C3—181no—
305C23×C3—181no—
306C23×C3C22×D911yescharacteristic
307C23×C3C2×D3211yescharacteristic
308C22×D3—32no—
309C22×D3—32no—
310C22×D3—32no—
311C22×D3—61no—
312C22×D3—61no—
313C22×D3—61no—
314C22×D3—61no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—22no—
322C2×C3×C4—32no—
323C2×C3×C4—32no—
324C2×C3×C4—61no—
325C2×C3×C4—61no—
326C2×C3×C4—61no—
327C2×C3×C4—61no—
328C2×C3×C4—92no—
329C2×C3×C4—181no—
330C2×C3×C4—181no—
331C2×C3×C4C4×D912yescomplemented
332C2×C3×C4D3×Dic312yes—
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—92no—
343C2×Dic3—92no—
344C2×Dic3—92no—
345C2×Dic3—92no—
346C2×Dic3—92no—
347C2×Dic3—92no—
348C2×Dic3—182no—
349C2×Dic3—182no—
350C2×Dic3—182no—
351C2×Dic3—182no—
352C2×Dic3—188no—
353C2×Dic3—182no—
354C2×Dic3—182no—
355C2×Dic3—181no—
356C2×Dic3—181no—
357C2×Dic3—181no—
358C2×Dic3—181no—
359C2×Dic3C4×D912yes—
360C2×Dic3C4×D912yescomplemented
361C4×D3—188no—
362C2×C32C42×D311yescharacteristic
363C2×C32D3×C22⋊C411yescharacteristic
364C2×C32#67⋊C411yescharacteristic
365C2×C32#67⋊C411yescharacteristic
366C2×C32#89211yescharacteristic
367C2×C32#89311yescharacteristic
368C2×C32#89311yescharacteristic
369C3×D3—64no—
370C2×C9—21no—
371C2×C9—21no—
372C2×C9—21no—
373C2×C9—21no—
374C2×C9—21no—
375C2×C9—21no—
376C2×C9—21no—
377C2×C9#87711yescharacteristic
378C2×C9D3×C22⋊C411yescharacteristic
379C2×C9#67⋊C411yescharacteristic
380C2×C9#67⋊C411yescharacteristic
381C2×C9C2×C4×Dic311yescharacteristic
382C2×C9D4×Dic311yescharacteristic
383C2×C9D4×Dic311yescharacteristic
384D9—24no—
385C24—91no—
386C22×C4—31no—
387C22×C4—31no—
388C22×C4—91no—
389C22×C4—274no—
390C22×C4—271no—
391C22×C4—271no—
392C22×C4—542no—
393C22×C4—542no—
394C22×C4D3×D911yescharacteristic
395C22⋊C4—94no—
396C22⋊C4—274no—
397C42—542no—
398C42—542no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—94no—
407C22×C3—182no—
408C22×C3—182no—
409C22×C3—182no—
410C22×C3—182no—
411C22×C3—182no—
412C22×C3—182no—
413C22×C3C2×C4×D911yescharacteristic
414C22×C3C2×C4×D911yescharacteristic
415C22×C3C2×C4×D911yescharacteristic
416C22×C3D4×D911yescharacteristic
417C22×C3D4×D911yescharacteristic
418C22×C3#369911yescharacteristic
419C22×C3#370111yescharacteristic
420C22×C3#379611yescharacteristic
421C22×C3C4×D3211yescharacteristic
422C22×C3C4×D3211yescharacteristic
423C22×C3#380211yescharacteristic
424C22×C3C2×D3×Dic311yescharacteristic
425C22×C3#380511yescharacteristic
426C22×C3#381111yescharacteristic
427C2×D3—34no—
428C2×D3—62no—
429C2×D3—62no—
430C2×D3—62no—
431C2×D3—62no—
432C2×D3—62no—
433C2×D3—62no—
434C3×C4—24no—
435C3×C4—22no—
436C3×C4—22no—
437C3×C4—62no—
438C3×C4—62no—
439C3×C4—182no—
440Dic3—22no—
441Dic3—22no—
442Dic3—62no—
443Dic3—94no—
444Dic3—94no—
445Dic3—184no—
446Dic3—184no—
447Dic3—182no—
448Dic3—182no—
449C32#466611yescharacteristic
450C9—21no—
451C9Dic3×C22⋊C411yescharacteristic, complemented
452C23—92no—
453C23—92no—
454C23—92no—
455C23—181no—
456C23—181no—
457C23—181no—
458C23—181no—
459C23C2×D3×D911yescharacteristic
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—32no—
463C2×C4—32no—
464C2×C4—61no—
465C2×C4—61no—
466C2×C4—61no—
467C2×C4—61no—
468C2×C4—92no—
469C2×C4—181no—
470C2×C4—181no—
471C2×C4—272no—
472C2×C4—272no—
473C2×C4—272no—
474C2×C4—272no—
475C2×C4—548no—
476C2×C4—541no—
477C2×C4—541no—
478C2×C4Dic3×D912yescomplemented
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—184no—
487C2×C3C42×D911yescharacteristic
488C2×C3C22⋊C4×D911yescharacteristic
489C2×C3#611⋊C411yescharacteristic
490C2×C3#611⋊C411yescharacteristic
491C2×C3#6345411yescharacteristic
492C2×C3#6345511yescharacteristic
493C2×C3#6345511yescharacteristic
494C2×C3C4×D3×Dic311yescharacteristic
495C2×C3#6387811yescharacteristic
496C2×C3#6388111yescharacteristic
497C2×C3#6389811yescharacteristic
498C2×C3#6390211yescharacteristic
499C2×C3#6392111yescharacteristic
500C2×C3#67⋊Dic311yescharacteristic
501D3—64no—
502C22—94no—
503C22—182no—
504C22—182no—
505C22—182no—
506C22—182no—
507C22—182no—
508C22—182no—
509C22#8863411yescharacteristic
510C22C4×D3×D911yescharacteristic
511C22C4×D3×D911yescharacteristic
512C22#8864611yescharacteristic
513C22C2×Dic3×D911yescharacteristic
514C22#8866311yescharacteristic
515C22#8866811yescharacteristic
516C4—22no—
517C4—62no—
518C4—62no—
519C4—182no—
520C4—274no—
521C4—542no—
522C3—21no—
523C3#1058834611yescharacteristic, complemented
524C3#1059136811yescharacteristic
525C2—184no—
526C2C4×Dic3×D911yescharacteristic
527C2#1172097511yescharacteristic
528C2#1172097811yescharacteristic
529C2#1172099811yescharacteristic
530C2#1172102811yescharacteristic
531C2#1172104811yescharacteristic
532C2#611⋊Dic311yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutatorC2×C3×C9C2×C42
FrattiniC23×C3C2×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
prime radicalC23×C3×D9C22
cosocleC23×C3×C9C23
hypercenterC22×C4D3×D9
hypocenterC3×C9C4×C22⋊C4
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C3⋊Dic9C2semidirect
5. C2×Dic3×Dic9C2semidirect
6. C9×C3⋊#95C2semidirect
7. #11720331C2semidirect
19. C2×Dic3×D9C4semidirect
21. C3×(C2×D9)⋊C4C4semidirect
22. (C2×D9)⋊Dic3C4semidirect
49. C2×C3×C4×C9C2×C4semidirect
53. C2×C3⋊Dic9C2×C4semidirect
61. C2×C9×Dic3C2×C4semidirect
64. C2×C3×Dic9C2×C4semidirect
96. (C2×D9)⋊C4Dic3semidirect
113. C2×C3×D9C42semidirect
116. C3⋊Dic9C22⋊C4semidirect
134. C3⋊#95D9semidirect
182. C2×C4×C9C2×Dic3semidirect
197. C2×Dic9C2×Dic3semidirect
280. C2×D9C4×Dic3semidirect
297. C3×C9C4×C22⋊C4semidirect
331. C2×C3×C4C4×D9semidirect
360. C2×Dic3C4×D9semidirect
451. C9Dic3×C22⋊C4semidirect
478. C2×C4Dic3×D9semidirect
523. C3#10588346semidirect

Maximal subgroups

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

subgroupindex
C22×Dic3×D92
C2×C4×C3⋊Dic92
C2×C3×(C2×D9)⋊C42
C2×(C2×D9)⋊Dic32
C2×Dic3×Dic92
C9×C3⋊#952
#117203312
#105913683
#105883463