Finite Group Databasebeta

← #13144286

Subgroups — #13144286

Order 1728.

1190 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6233C211yesmaximal, characteristic
2C2×C3×C9×C22⋊C4C211yesmaximal, characteristic, complemented
3C2×#88551C211yesmaximal, characteristic, complemented
4C2×#88536C211yesmaximal, characteristic, complemented
5C2×#88536C211yesmaximal, characteristic, complemented
6C2×C4⋊(C3⋊Dic9)C211yesmaximal, characteristic, complemented
7#11720572C211yesmaximal, characteristic, complemented
8#10592660—31nomaximal
9#10588641—33nomaximal
10C24×C3×C9C2211yescharacteristic
11C23×C3⋊D9C2211yescharacteristic
12C22×C3×C4×C9C2211yescharacteristic
13C22×C3×C4×C9C2211yescharacteristic
14C2×#6233—22no—
15C2×#6233—22no—
16C2×#6233—22no—
17C22×C3⋊Dic9C2211yescharacteristic
18C22×C3⋊Dic9C2211yescharacteristic
19C22×C3⋊Dic9C2211yescharacteristic
20C3×C9×C22⋊C4—22no—
21#88551—22no—
22#88536—22no—
23#88536—22no—
24C4⋊(C3⋊Dic9)—22no—
25C22×#638—31no—
26C2×C32×C22⋊C4D311yescharacteristic
27C2×#3757—31no—
28C2×#3752—31no—
29C2×#3752—31no—
30C2×C4⋊(C3⋊Dic3)—31no—
31C22×#611—33no—
32#63658—31no—
33C2×C9×C22⋊C4D313yescomplemented
34C2×#3676—33no—
35C2×(C2×D9)⋊C4—33no—
36C2×(C2×D9)⋊C4—33no—
37C2×C4⋊Dic9—33no—
38C9⋊#95—33no—
39C23×C3×C9—21no—
40C23×C3×C9—21no—
41C23×C3×C9—21no—
42C23×C3×C9—21no—
43C23×C3×C9—21no—
44C23×C3×C9—21no—
45C23×C3×C9D412yes—
46C23×C3×C9C2311yescharacteristic
47C22×C3⋊D9—21no—
48C22×C3⋊D9—21no—
49C22×C3⋊D9—21no—
50C22×C3⋊D9—21no—
51C22×C3⋊D9—21no—
52C22×C3⋊D9—21no—
53C22×C3⋊D9D412yes—
54C2×C3×C4×C9—21no—
55C2×C3×C4×C9—21no—
56C2×C3×C4×C9—21no—
57C2×C3×C4×C9—21no—
58C2×C3×C4×C9—21no—
59C2×C3×C4×C9D412yescomplemented
60#6233—28no—
61C2×C3⋊Dic9—21no—
62C2×C3⋊Dic9—21no—
63C2×C3⋊Dic9—21no—
64C2×C3⋊Dic9—21no—
65C2×C3⋊Dic9—21no—
66C2×C3⋊Dic9—21no—
67C2×C3⋊Dic9—21no—
68C2×C3⋊Dic9—21no—
69C2×C3⋊Dic9D412yes—
70#4961—93no—
71#4961—91no—
72C24×C32C2×D311yescharacteristic
73C23×C3⋊D3—31no—
74C22×C32×C4C2×D311yescharacteristic
75C22×C32×C4C2×D311yescharacteristic
76C2×#638—62no—
77C2×#638—62no—
78C2×#638—62no—
79C22×C3⋊Dic3—31no—
80C22×C3⋊Dic3—31no—
81C22×C3⋊Dic3—31no—
82C24×C9C2×D313yes—
83C23×D9—33no—
84C32×C22⋊C4—22no—
85#3757—62no—
86#3752—62no—
87#3752—62no—
88C4⋊(C3⋊Dic3)—62no—
89C22×C4×C9C2×D313yes—
90C22×C4×C9C2×D313yes—
91C2×#611—66no—
92C2×#611—66no—
93C2×#611—66no—
94C22×Dic9—33no—
95C22×Dic9—33no—
96C22×Dic9—33no—
97C9×C22⋊C4—26no—
98#3676—66no—
99(C2×D9)⋊C4—66no—
100(C2×D9)⋊C4—66no—
101C4⋊Dic9—66no—
102C22×C3×C9—22no—
103C22×C3×C9—22no—
104C22×C3×C9—22no—
105C22×C3×C9—41no—
106C22×C3×C9—41no—
107C22×C3×C9—41no—
108C22×C3×C9—41no—
109C22×C3×C9C2×D411yescharacteristic
110C22×C3×C9C2×D411yescharacteristic
111C22×C3×C9C2×D411yescharacteristic
112C22×C3×C9C2×D411yescharacteristic
113C22×C3×C9#4111yescharacteristic
114C22×C3×C9#4111yescharacteristic
115C22×C3×C9#4111yescharacteristic
116C2×C3⋊D9—22no—
117C2×C3⋊D9—22no—
118C2×C3⋊D9—22no—
119C2×C3⋊D9—41no—
120C2×C3⋊D9—41no—
121C2×C3⋊D9—41no—
122C2×C3⋊D9—41no—
123C3×C4×C9—22no—
124C3×C4×C9—41no—
125C3⋊Dic9—22no—
126C3⋊Dic9—41no—
127C3⋊Dic9—41no—
128C22×#67—93no—
129C22×#67—91no—
130C2×C3×C22⋊C4D913yescomplemented
131C2×C3×C22⋊C4C3⋊D311yescharacteristic
132C2×#217—93no—
133C2×#217—91no—
134C2×(C2×D3)⋊C4—93no—
135C2×(C2×D3)⋊C4—93no—
136C2×(C2×D3)⋊C4—91no—
137C2×(C2×D3)⋊C4—91no—
138C2×C4⋊Dic3—93no—
139C2×C4⋊Dic3—91no—
140C3⋊#95—93no—
141C3⋊#95—91no—
142C23×C32—21no—
143C23×C32—21no—
144C23×C32—21no—
145C23×C32—21no—
146C23×C32—21no—
147C23×C32—21no—
148C23×C32D1212yes—
149C23×C32C22×D311yescharacteristic
150C22×C3⋊D3—32no—
151C22×C3⋊D3—61no—
152C22×C3⋊D3—61no—
153C22×C3⋊D3—61no—
154C22×C3⋊D3—61no—
155C22×C3⋊D3—61no—
156C22×C3⋊D3—61no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4#6712yes—
163#638—68no—
164C2×C3⋊Dic3—32no—
165C2×C3⋊Dic3—61no—
166C2×C3⋊Dic3—61no—
167C2×C3⋊Dic3—61no—
168C2×C3⋊Dic3—61no—
169C2×C3⋊Dic3—61no—
170C2×C3⋊Dic3—61no—
171C2×C3⋊Dic3—61no—
172C2×C3⋊Dic3—61no—
173C23×C9—23no—
174C23×C9—23no—
175C23×C9—23no—
176C23×C9—23no—
177C23×C9—23no—
178C23×C9—23no—
179C23×C9D1216yes—
180C23×C9C22×D313yes—
181C22×D9—36no—
182C22×D9—63no—
183C22×D9—63no—
184C22×D9—63no—
185C22×D9—63no—
186C22×D9—63no—
187C22×D9—63no—
188C2×C4×C9—23no—
189C2×C4×C9—23no—
190C2×C4×C9—23no—
191C2×C4×C9—23no—
192C2×C4×C9—23no—
193C2×C4×C9#6716yescomplemented
194#611—624no—
195C2×Dic9—36no—
196C2×Dic9—63no—
197C2×Dic9—63no—
198C2×Dic9—63no—
199C2×Dic9—63no—
200C2×Dic9—63no—
201C2×Dic9—63no—
202C2×Dic9—63no—
203C2×Dic9—63no—
204#394—271noSylow
205C2×C3×C9—42no—
206C2×C3×C9#12011yescharacteristic
207C2×C3×C9#12111yescharacteristic
208C2×C3×C9#12111yescharacteristic
209C2×C3×C9#12111yescharacteristic
210C2×C3×C9#12311yescharacteristic
211C2×C3×C9#12311yescharacteristic
212C2×C3×C9#12411yescharacteristic
213C3⋊D9—42no—
214C24×C3C2×D913yes—
215C24×C3C2×C3⋊D311yescharacteristic
216C23×D3—93no—
217C23×D3—91no—
218C22×C3×C4C2×D913yes—
219C22×C3×C4C2×D913yes—
220C22×C3×C4C2×C3⋊D311yescharacteristic
221C22×C3×C4C2×C3⋊D311yescharacteristic
222C2×#67—182no—
223C2×#67—186no—
224C2×#67—182no—
225C2×#67—186no—
226C2×#67—186no—
227C2×#67—182no—
228C22×Dic3—93no—
229C22×Dic3—91no—
230C22×Dic3—93no—
231C22×Dic3—93no—
232C22×Dic3—91no—
233C22×Dic3—91no—
234C3×C22⋊C4—26no—
235C3×C22⋊C4—22no—
236#217—186no—
237#217—182no—
238(C2×D3)⋊C4—182no—
239(C2×D3)⋊C4—186no—
240(C2×D3)⋊C4—186no—
241(C2×D3)⋊C4—182no—
242C4⋊Dic3—182no—
243C4⋊Dic3—186no—
244C22×C32—22no—
245C22×C32—22no—
246C22×C32—22no—
247C22×C32—41no—
248C22×C32—41no—
249C22×C32—41no—
250C22×C32—41no—
251C22×C32C2×D1211yescharacteristic
252C22×C32#23511yescharacteristic
253C22×C32D3×D411yescharacteristic
254C22×C32D3×D411yescharacteristic
255C22×C32#23711yescharacteristic
256C22×C32#23711yescharacteristic
257C22×C32C2×#6711yescharacteristic
258C2×C3⋊D3—62no—
259C2×C3⋊D3—62no—
260C2×C3⋊D3—62no—
261C2×C3⋊D3—121no—
262C2×C3⋊D3—121no—
263C2×C3⋊D3—121no—
264C2×C3⋊D3—121no—
265C32×C4—22no—
266C32×C4—41no—
267C3⋊Dic3—62no—
268C3⋊Dic3—121no—
269C3⋊Dic3—121no—
270C22×C9—26no—
271C22×C9—26no—
272C22×C9—26no—
273C22×C9—43no—
274C22×C9—43no—
275C22×C9—43no—
276C22×C9—43no—
277C22×C9C2×D1213yes—
278C22×C9#23513yes—
279C22×C9D3×D413yes—
280C22×C9D3×D413yes—
281C22×C9#23713yes—
282C22×C9#23713yes—
283C22×C9C2×#6713yes—
284C2×D9—66no—
285C2×D9—66no—
286C2×D9—66no—
287C2×D9—123no—
288C2×D9—123no—
289C2×D9—123no—
290C2×D9—123no—
291C4×C9—26no—
292C4×C9—43no—
293Dic9—66no—
294Dic9—123no—
295Dic9—123no—
296C22×D4—271no—
297C2×C4⋊C4—271no—
298C2×C22⋊C4—271no—
299C2×C22⋊C4—271no—
300C2×C22⋊C4—271no—
301C2×C22⋊C4C3⋊D911yescharacteristic, complemented
302#95—271no—
303C3×C9#39411yescharacteristic, complemented, Sylow
304C23×C3—23no—
305C23×C3—23no—
306C23×C3—23no—
307C23×C3—23no—
308C23×C3—23no—
309C23×C3—23no—
310C23×C3—21no—
311C23×C3—21no—
312C23×C3—21no—
313C23×C3—21no—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3D3616yes—
317C23×C3C22×D913yes—
318C23×C3C4⋊(C3⋊D3)12yes—
319C23×C3C22×C3⋊D311yescharacteristic
320C22×D3—96no—
321C22×D3—92no—
322C22×D3—183no—
323C22×D3—183no—
324C22×D3—183no—
325C22×D3—183no—
326C22×D3—183no—
327C22×D3—183no—
328C22×D3—181no—
329C22×D3—181no—
330C22×D3—181no—
331C22×D3—181no—
332C22×D3—181no—
333C22×D3—181no—
334C2×C3×C4—23no—
335C2×C3×C4—23no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—23no—
339C2×C3×C4—23no—
340C2×C3×C4—23no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4#61116yescomplemented
345C2×C3×C4#63812yes—
346#67—188no—
347#67—1824no—
348C2×Dic3—96no—
349C2×Dic3—92no—
350C2×Dic3—183no—
351C2×Dic3—183no—
352C2×Dic3—181no—
353C2×Dic3—181no—
354C2×Dic3—183no—
355C2×Dic3—183no—
356C2×Dic3—183no—
357C2×Dic3—183no—
358C2×Dic3—183no—
359C2×Dic3—183no—
360C2×Dic3—181no—
361C2×Dic3—181no—
362C2×Dic3—181no—
363C2×Dic3—181no—
364C2×Dic3—181no—
365C2×Dic3—181no—
366C2×C32—42no—
367C2×C32#88211yescharacteristic
368C2×C32#88311yescharacteristic
369C2×C32#88511yescharacteristic
370C2×C32#88611yescharacteristic
371C2×C32#93011yescharacteristic
372C2×C32#93811yescharacteristic
373C2×C32#93911yescharacteristic
374C3⋊D3—122no—
375C2×C9—46no—
376C2×C9#88213yes—
377C2×C9#88313yes—
378C2×C9#88513yes—
379C2×C9#88613yes—
380C2×C9#93013yes—
381C2×C9#93813yes—
382C2×C9#93913yes—
383D9—126no—
384C24—271no—
385C24C2×C3⋊D911yescharacteristic
386C2×D4—542no—
387C2×D4—542no—
388C2×D4—542no—
389C22×C4—271no—
390C22×C4—271no—
391C22×C4—271no—
392C22×C4C2×C3⋊D911yescharacteristic
393C22×C4C2×C3⋊D911yescharacteristic
394C4⋊C4—542no—
395C22⋊C4—22no—
396C22⋊C4—542no—
397C22⋊C4—542no—
398C22⋊C4—542no—
399C22×C3—26no—
400C22×C3—22no—
401C22×C3—22no—
402C22×C3—22no—
403C22×C3—26no—
404C22×C3—26no—
405C22×C3—43no—
406C22×C3—43no—
407C22×C3—41no—
408C22×C3—41no—
409C22×C3—41no—
410C22×C3—41no—
411C22×C3—43no—
412C22×C3—43no—
413C22×C3C2×D3613yes—
414C22×C3#369713yes—
415C22×C3D4×D913yes—
416C22×C3D4×D913yes—
417C22×C3#369913yes—
418C22×C3#369913yes—
419C22×C3C2×#61113yes—
420C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
421C22×C3#382811yescharacteristic
422C22×C3D4×C3⋊D311yescharacteristic
423C22×C3D4×C3⋊D311yescharacteristic
424C22×C3#383011yescharacteristic
425C22×C3#383011yescharacteristic
426C22×C3C2×#63811yescharacteristic
427C2×D3—186no—
428C2×D3—182no—
429C2×D3—182no—
430C2×D3—182no—
431C2×D3—186no—
432C2×D3—186no—
433C2×D3—363no—
434C2×D3—363no—
435C2×D3—361no—
436C2×D3—361no—
437C2×D3—361no—
438C2×D3—361no—
439C2×D3—363no—
440C2×D3—363no—
441C3×C4—26no—
442C3×C4—22no—
443C3×C4—43no—
444C3×C4—41no—
445Dic3—182no—
446Dic3—186no—
447Dic3—363no—
448Dic3—361no—
449Dic3—361no—
450Dic3—363no—
451C32#496111yescharacteristic
452C9#496113yescomplemented
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—272no—
460C23—541no—
461C23—541no—
462C23—541no—
463C23—541no—
464C23—541no—
465C23—541no—
466C23C4⋊(C3⋊D9)12yes—
467C23C22×C3⋊D911yescharacteristic
468D4—548no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—272no—
475C2×C4—541no—
476C2×C4—541no—
477C2×C4—541no—
478C2×C4—541no—
479C2×C4—541no—
480C2×C4—541no—
481C2×C4—541no—
482C2×C4—541no—
483C2×C4#623312yescomplemented
484C2×C3—42no—
485C2×C3—46no—
486C2×C3#6344413yes—
487C2×C3#6344513yes—
488C2×C3#6344713yes—
489C2×C3#6344813yes—
490C2×C3#6349213yes—
491C2×C3#6350013yes—
492C2×C3#6350113yes—
493C2×C3#6409111yescharacteristic
494C2×C3#6409211yescharacteristic
495C2×C3#6409411yescharacteristic
496C2×C3#6409511yescharacteristic
497C2×C3#6413911yescharacteristic
498C2×C3#6414711yescharacteristic
499C2×C3#6414811yescharacteristic
500D3—362no—
501D3—366no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—542no—
510C22—542no—
511C22—542no—
512C22—1081no—
513C22—1081no—
514C22—1081no—
515C22—1081no—
516C22C2×C4⋊(C3⋊D9)11yescharacteristic
517C22#8873611yescharacteristic
518C22D4×C3⋊D911yescharacteristic
519C22D4×C3⋊D911yescharacteristic
520C22#8874211yescharacteristic
521C22#8874211yescharacteristic
522C22C2×#623311yescharacteristic
523C4—22no—
524C4—41no—
525C4—542no—
526C4—1081no—
527C4—1081no—
528C3#1058864113yescomplemented
529C3#1059266011yescharacteristic
530C2—42no—
531C2—1082no—
532C2#1172172811yescharacteristic
533C2#1172172911yescharacteristic
534C2#1172173111yescharacteristic
535C2#1172173211yescharacteristic
536C2#1172181911yescharacteristic
537C2#1172183811yescharacteristic
538C2#1172183911yescharacteristic
subgroupG/H
centerC23C22×C3⋊D9
commutator, cosocleC23×C3×C9C23
FrattiniC23×C3C22×C3⋊D3
FittingC2×C3×C9×C22⋊C4C2
socleC23×C32C22×D3
prime radicalC22×#6233C2
hypercenterC2×C22⋊C4C3⋊D9
hypocenterC3×C9#394
LCSG ▷ C23×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C9×C22⋊C4C2semidirect
3. C2×#88551C2semidirect
4. C2×#88536C2semidirect
5. C2×#88536C2semidirect
6. C2×C4⋊(C3⋊Dic9)C2semidirect
7. #11720572C2semidirect
33. C2×C9×C22⋊C4D3semidirect
59. C2×C3×C4×C9D4semidirect
130. C2×C3×C22⋊C4D9semidirect
193. C2×C4×C9#67semidirect
301. C2×C22⋊C4C3⋊D9semidirect
303. C3×C9#394semidirect
344. C2×C3×C4#611semidirect
452. C9#4961semidirect
483. C2×C4#6233semidirect
528. C3#10588641semidirect

Maximal subgroups

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

subgroupindex
C22×#62332
C2×C3×C9×C22⋊C42
C2×#885512
C2×#885362
C2×C4⋊(C3⋊Dic9)2
#117205722
#105926603
#105886413