Finite Group Databasebeta

← #13141115

Subgroups — #13141115

Order 1728.

690 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88551C211yesmaximal, characteristic
2C2×C9×#217C211yesmaximal, characteristic
3C2×C3×#3676C211yesmaximal, characteristic
4C2×Dic3⋊Dic9C211yesmaximal, characteristic, complemented
5C2×#88442C211yesmaximal, characteristic, complemented
6C2×Dic9⋊Dic3C211yesmaximal, characteristic, complemented
7#11720331C211yesmaximal, characteristic, complemented
8#10591691—31nomaximal
9#10588628—31nomaximal
10C24×C3×C9C2211yescharacteristic
11C22×C3⋊Dic9C2211yescharacteristic
12C22×C3⋊Dic9C2211yescharacteristic
13C22×C9×Dic3C2211yescharacteristic
14C22×C9×Dic3C2211yescharacteristic
15C22×C3×Dic9C2211yescharacteristic
16C22×C3×Dic9C2211yescharacteristic
17#88551—22no—
18C9×#217—22no—
19C3×#3676—22no—
20Dic3⋊Dic9—22no—
21#88442—22no—
22Dic9⋊Dic3—22no—
23C2×#3757—31no—
24C2×C3×#217—31no—
25C2×C3×#217D311yescharacteristic
26C2×#3724—31no—
27C2×Dic3⋊Dic3—31no—
28C2×Dic3⋊Dic3—31no—
29#63579—31no—
30C2×C9×C22⋊C4—31no—
31C2×#3676—31no—
32C2×#3676—61no—
33C2×#3676D311yescharacteristic
34C2×C4⋊Dic9—31no—
35C2×Dic9⋊C4—31no—
36C2×Dic9⋊C4—31no—
37C9⋊#95—31no—
38C23×C3×C9—21no—
39C23×C3×C9—21no—
40C23×C3×C9—21no—
41C23×C3×C9—21no—
42C23×C3×C9—21no—
43C23×C3×C9—21no—
44C23×C3×C9Q812yes—
45C23×C3×C9C2311yescharacteristic
46C2×C3⋊Dic9—21no—
47C2×C3⋊Dic9—21no—
48C2×C3⋊Dic9—21no—
49C2×C3⋊Dic9—21no—
50C2×C3⋊Dic9—21no—
51C2×C3⋊Dic9D412yes—
52C2×C9×Dic3—21no—
53C2×C9×Dic3—21no—
54C2×C9×Dic3—21no—
55C2×C9×Dic3—21no—
56C2×C9×Dic3—21no—
57C2×C9×Dic3D412yes—
58C2×C3×Dic9—21no—
59C2×C3×Dic9—21no—
60C2×C3×Dic9—21no—
61C2×C3×Dic9—21no—
62C2×C3×Dic9—21no—
63C2×C3×Dic9D412yes—
64#4948—91no—
65#4948—91no—
66C24×C32C2×D311yescharacteristic
67C22×C3⋊Dic3—31no—
68C22×C3⋊Dic3—31no—
69C22×C3×Dic3—31no—
70C22×C3×Dic3—31no—
71C22×C3×Dic3C2×D311yescharacteristic
72C22×C3×Dic3C2×D311yescharacteristic
73C24×C9—21no—
74C24×C9C2×D311yescharacteristic
75#3757—62no—
76C3×#217—22no—
77C3×#217—62no—
78#3724—62no—
79Dic3⋊Dic3—62no—
80Dic3⋊Dic3—62no—
81C22×C4×C9—31no—
82C22×C4×C9—31no—
83C22×Dic9—31no—
84C22×Dic9—31no—
85C22×Dic9—61no—
86C22×Dic9—61no—
87C22×Dic9C2×D311yescharacteristic
88C22×Dic9C2×D311yescharacteristic
89C9×C22⋊C4—62no—
90#3676—22no—
91#3676—62no—
92#3676—64no—
93C4⋊Dic9—62no—
94Dic9⋊C4—62no—
95Dic9⋊C4—62no—
96C22×C3×C9—22no—
97C22×C3×C9—22no—
98C22×C3×C9—22no—
99C22×C3×C9—41no—
100C22×C3×C9—41no—
101C22×C3×C9—41no—
102C22×C3×C9—41no—
103C22×C3×C9C2×D411yescharacteristic
104C22×C3×C9C2×D411yescharacteristic
105C22×C3×C9C2×D411yescharacteristic
106C22×C3×C9C2×Q811yescharacteristic
107C22×C3×C9#4111yescharacteristic
108C22×C3×C9#4111yescharacteristic
109C22×C3×C9#4111yescharacteristic
110C3⋊Dic9—22no—
111C3⋊Dic9—41no—
112C9×Dic3—22no—
113C9×Dic3—41no—
114C3×Dic9—22no—
115C3×Dic9—41no—
116C2×C3×C22⋊C4—31no—
117C2×C3×C22⋊C4—91no—
118C2×#217—31no—
119C2×#217—91no—
120C2×#217—91no—
121C2×#217—181no—
122C2×#217D911yescharacteristic
123C2×C4⋊Dic3—91no—
124C2×C4⋊Dic3—91no—
125C2×Dic3⋊C4—91no—
126C2×Dic3⋊C4—91no—
127C2×Dic3⋊C4—91no—
128C2×Dic3⋊C4—91no—
129C3⋊#95—91no—
130C3⋊#95—91no—
131C23×C32—21no—
132C23×C32—21no—
133C23×C32—21no—
134C23×C32—21no—
135C23×C32—21no—
136C23×C32—21no—
137C23×C32Dic612yes—
138C23×C32C22×D311yescharacteristic
139C2×C3⋊Dic3—32no—
140C2×C3⋊Dic3—61no—
141C2×C3⋊Dic3—61no—
142C2×C3⋊Dic3—61no—
143C2×C3⋊Dic3—61no—
144C2×C3⋊Dic3—61no—
145C2×C3×Dic3—21no—
146C2×C3×Dic3—21no—
147C2×C3×Dic3—21no—
148C2×C3×Dic3—21no—
149C2×C3×Dic3—21no—
150C2×C3×Dic3—32no—
151C2×C3×Dic3—61no—
152C2×C3×Dic3—61no—
153C2×C3×Dic3—61no—
154C2×C3×Dic3—61no—
155C2×C3×Dic3—61no—
156C2×C3×Dic3#6712yes—
157C23×C9—22no—
158C23×C9—22no—
159C23×C9—21no—
160C23×C9—21no—
161C23×C9—21no—
162C23×C9—21no—
163C23×C9—21no—
164C23×C9—21no—
165C23×C9—22no—
166C23×C9—21no—
167C23×C9—41no—
168C23×C9—41no—
169C23×C9—41no—
170C23×C9—41no—
171C23×C9Dic612yes—
172C23×C9C22×D311yescharacteristic
173C2×C4×C9—32no—
174C2×C4×C9—61no—
175C2×C4×C9—61no—
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—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—
197#396—271noSylow
198C2×C3×C9—42no—
199C2×C3×C9#12111yescharacteristic
200C2×C3×C9#12111yescharacteristic
201C2×C3×C9#12111yescharacteristic
202C2×C3×C9C22⋊Q811yescharacteristic
203C2×C3×C9C22⋊Q811yescharacteristic
204C2×C3×C9C22⋊Q811yescharacteristic
205C2×C3×C9#12611yescharacteristic
206C24×C3—21no—
207C24×C3C2×D911yescharacteristic
208C24×C3D3211yescharacteristic
209C22×C3×C4—31no—
210C22×C3×C4—31no—
211C22×C3×C4—91no—
212C22×C3×C4—91no—
213C22×Dic3—31no—
214C22×Dic3—31no—
215C22×Dic3—91no—
216C22×Dic3—91no—
217C22×Dic3—91no—
218C22×Dic3—91no—
219C22×Dic3—181no—
220C22×Dic3—181no—
221C22×Dic3C2×D911yescharacteristic
222C22×Dic3C2×D911yescharacteristic
223C3×C22⋊C4—62no—
224C3×C22⋊C4—182no—
225#217—22no—
226#217—62no—
227#217—182no—
228#217—182no—
229#217—184no—
230C4⋊Dic3—182no—
231C4⋊Dic3—182no—
232Dic3⋊C4—182no—
233Dic3⋊C4—182no—
234Dic3⋊C4—182no—
235Dic3⋊C4—182no—
236C22×C32—22no—
237C22×C32—22no—
238C22×C32—22no—
239C22×C32—41no—
240C22×C32—41no—
241C22×C32—41no—
242C22×C32—41no—
243C22×C32C2×Dic611yescharacteristic
244C22×C32#23511yescharacteristic
245C22×C32D3×D411yescharacteristic
246C22×C32D3×D411yescharacteristic
247C22×C32#23711yescharacteristic
248C22×C32#23711yescharacteristic
249C22×C32C2×#6711yescharacteristic
250C3⋊Dic3—62no—
251C3⋊Dic3—121no—
252C3×Dic3—22no—
253C3×Dic3—41no—
254C3×Dic3—62no—
255C3×Dic3—121no—
256C22×C9—24no—
257C22×C9—22no—
258C22×C9—22no—
259C22×C9—22no—
260C22×C9—21no—
261C22×C9—21no—
262C22×C9—21no—
263C22×C9—21no—
264C22×C9—21no—
265C22×C9—21no—
266C22×C9—21no—
267C22×C9—42no—
268C22×C9—42no—
269C22×C9—42no—
270C22×C9—42no—
271C22×C9—42no—
272C22×C9—42no—
273C22×C9—41no—
274C22×C9—41no—
275C22×C9—41no—
276C22×C9—41no—
277C22×C9C2×Dic611yescharacteristic
278C22×C9#23511yescharacteristic
279C22×C9D3×D411yescharacteristic
280C22×C9D3×D411yescharacteristic
281C22×C9#23711yescharacteristic
282C22×C9#23711yescharacteristic
283C22×C9C2×#6711yescharacteristic
284C4×C9—62no—
285C4×C9—121no—
286Dic9—22no—
287Dic9—41no—
288Dic9—62no—
289Dic9—122no—
290Dic9—122no—
291Dic9—121no—
292C2×C4⋊C4—271no—
293C2×C4⋊C4—271no—
294C2×C4⋊C4—271no—
295C2×C22⋊C4—31no—
296C2×C22⋊C4—91no—
297C2×C22⋊C4—271no—
298#95—271no—
299C3×C9#39611yescharacteristic, complemented, Sylow
300C23×C3—22no—
301C23×C3—22no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—21no—
306C23×C3—21no—
307C23×C3—21no—
308C23×C3—21no—
309C23×C3—21no—
310C23×C3—21no—
311C23×C3—21no—
312C23×C3—21no—
313C23×C3—21no—
314C23×C3—22no—
315C23×C3—21no—
316C23×C3—41no—
317C23×C3—41no—
318C23×C3—41no—
319C23×C3—41no—
320C23×C3Dic1812yes—
321C23×C3C22×D911yescharacteristic
322C23×C3#62712yes—
323C23×C3C2×D3211yescharacteristic
324C2×C3×C4—32no—
325C2×C3×C4—61no—
326C2×C3×C4—61no—
327C2×C3×C4—61no—
328C2×C3×C4—61no—
329C2×C3×C4—61no—
330C2×C3×C4—92no—
331C2×C3×C4—181no—
332C2×C3×C4—181no—
333C2×C3×C4—181no—
334C2×C3×C4—181no—
335C2×C3×C4—181no—
336C2×Dic3—21no—
337C2×Dic3—21no—
338C2×Dic3—21no—
339C2×Dic3—21no—
340C2×Dic3—21no—
341C2×Dic3—32no—
342C2×Dic3—61no—
343C2×Dic3—61no—
344C2×Dic3—61no—
345C2×Dic3—61no—
346C2×Dic3—61no—
347C2×Dic3—92no—
348C2×Dic3—92no—
349C2×Dic3—182no—
350C2×Dic3—181no—
351C2×Dic3—181no—
352C2×Dic3—181no—
353C2×Dic3—181no—
354C2×Dic3—182no—
355C2×Dic3—181no—
356C2×Dic3—181no—
357C2×Dic3—181no—
358C2×Dic3—181no—
359C2×Dic3—181no—
360C2×Dic3—181no—
361C2×Dic3—361no—
362C2×Dic3—361no—
363C2×Dic3—361no—
364C2×Dic3—361no—
365C2×Dic3#61112yes—
366C2×C32—42no—
367C2×C32#87811yescharacteristic
368C2×C32#87811yescharacteristic
369C2×C32#87911yescharacteristic
370C2×C32#88411yescharacteristic
371C2×C32#92411yescharacteristic
372C2×C32#93811yescharacteristic
373C2×C32#93911yescharacteristic
374C2×C9—21no—
375C2×C9—21no—
376C2×C9—21no—
377C2×C9—21no—
378C2×C9—21no—
379C2×C9—21no—
380C2×C9—21no—
381C2×C9—44no—
382C2×C9—42no—
383C2×C9#87811yescharacteristic
384C2×C9#87811yescharacteristic
385C2×C9#87911yescharacteristic
386C2×C9#88411yescharacteristic
387C2×C9#92411yescharacteristic
388C2×C9#93811yescharacteristic
389C2×C9#93911yescharacteristic
390C24D3×D911yescharacteristic
391C22×C4—31no—
392C22×C4—31no—
393C22×C4—91no—
394C22×C4—91no—
395C22×C4—271no—
396C22×C4—271no—
397C4⋊C4—542no—
398C4⋊C4—542no—
399C4⋊C4—542no—
400C22⋊C4—62no—
401C22⋊C4—182no—
402C22⋊C4—542no—
403C22×C3—24no—
404C22×C3—22no—
405C22×C3—22no—
406C22×C3—22no—
407C22×C3—22no—
408C22×C3—22no—
409C22×C3—22no—
410C22×C3—21no—
411C22×C3—21no—
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—21no—
415C22×C3—21no—
416C22×C3—21no—
417C22×C3—42no—
418C22×C3—42no—
419C22×C3—42no—
420C22×C3—42no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—41no—
424C22×C3—41no—
425C22×C3—41no—
426C22×C3—41no—
427C22×C3—41no—
428C22×C3—41no—
429C22×C3—41no—
430C22×C3—41no—
431C22×C3C2×Dic1811yescharacteristic
432C22×C3#369711yescharacteristic
433C22×C3D4×D911yescharacteristic
434C22×C3D4×D911yescharacteristic
435C22×C3#369911yescharacteristic
436C22×C3#369911yescharacteristic
437C22×C3C2×#61111yescharacteristic
438C22×C3#380411yescharacteristic
439C22×C3#380411yescharacteristic
440C22×C3#380511yescharacteristic
441C22×C3C2×#62711yescharacteristic
442C22×C3D3×#6711yescharacteristic
443C22×C3D3×#6711yescharacteristic
444C22×C3#381111yescharacteristic
445C3×C4—62no—
446C3×C4—121no—
447C3×C4—182no—
448C3×C4—361no—
449Dic3—22no—
450Dic3—41no—
451Dic3—62no—
452Dic3—121no—
453Dic3—182no—
454Dic3—182no—
455Dic3—362no—
456Dic3—362no—
457Dic3—361no—
458Dic3—361no—
459C32#494811yescharacteristic
460C9—21no—
461C9#494811yescharacteristic, complemented
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23#618912yes—
469C23C2×D3×D911yescharacteristic
470C2×C4—32no—
471C2×C4—61no—
472C2×C4—61no—
473C2×C4—61no—
474C2×C4—61no—
475C2×C4—61no—
476C2×C4—92no—
477C2×C4—181no—
478C2×C4—181no—
479C2×C4—181no—
480C2×C4—181no—
481C2×C4—181no—
482C2×C4—272no—
483C2×C4—541no—
484C2×C4—541no—
485C2×C4—541no—
486C2×C4—541no—
487C2×C4—541no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—44no—
496C2×C3—42no—
497C2×C3—42no—
498C2×C3#6344011yescharacteristic
499C2×C3#6344011yescharacteristic
500C2×C3#6344111yescharacteristic
501C2×C3#6344611yescharacteristic
502C2×C3#6348611yescharacteristic
503C2×C3#6350011yescharacteristic
504C2×C3#6350111yescharacteristic
505C2×C3#6395611yescharacteristic
506C2×C3#6395811yescharacteristic
507C2×C3#6396411yescharacteristic
508C2×C3#6396411yescharacteristic
509C2×C3#6397011yescharacteristic
510C2×C3#6397111yescharacteristic
511C2×C3#6398211yescharacteristic
512C22—22no—
513C22—22no—
514C22—22no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22C2×#618911yescharacteristic
520C22#8865811yescharacteristic
521C22#8866211yescharacteristic
522C22#8866311yescharacteristic
523C22D3×#61111yescharacteristic
524C22D9×#6711yescharacteristic
525C22#8866811yescharacteristic
526C4—62no—
527C4—121no—
528C4—182no—
529C4—361no—
530C4—542no—
531C4—1081no—
532C3—21no—
533C3#1058862811yescharacteristic, complemented
534C3#1059169111yescharacteristic
535C2—42no—
536C2#1172124711yescharacteristic
537C2#1172124911yescharacteristic
538C2#1172125311yescharacteristic
539C2#1172125611yescharacteristic
540C2#1172126711yescharacteristic
541C2#1172127011yescharacteristic
542C2#1172128211yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutator, cosocleC23×C3×C9C23
FrattiniC23×C3C2×D32
FittingC24×C3×C9C22
socleC23×C32C22×D3
prime radicalC24×C32C2×D3
hypercenterC24D3×D9
hypocenterC3×C9#396
LCSG ▷ C23×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
4. C2×Dic3⋊Dic9C2semidirect
5. C2×#88442C2semidirect
6. C2×Dic9⋊Dic3C2semidirect
7. #11720331C2semidirect
299. C3×C9#396semidirect
461. C9#4948semidirect
533. C3#10588628semidirect

Maximal subgroups

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

subgroupindex
C2×#885512
C2×C9×#2172
C2×C3×#36762
C2×Dic3⋊Dic92
C2×#884422
C2×Dic9⋊Dic32
#117203312
#105916913
#105886283