Finite Group Databasebeta

← #13141137

Subgroups — #13141137

Order 1728.

738 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6189C211yesmaximal, characteristic, complemented
2C2×#88551C211yesmaximal, characteristic
3C2×C9×#217C211yesmaximal, characteristic
4C2×C3×#3676C211yesmaximal, characteristic
5#11720331C211yesmaximal, characteristic, complemented
6#11720331C211yesmaximal, characteristic, complemented
7#11720331C211yesmaximal, characteristic, complemented
8#10591692—31nomaximal
9#10588627—31nomaximal
10C24×C3×C9C2211yescharacteristic
11C22×C3⋊Dic9C2211yescharacteristic
12C22×C3⋊Dic9C2211yescharacteristic
13C22×C9×Dic3C2211yescharacteristic
14C22×C9×Dic3C2211yescharacteristic
15C22×C3×Dic9C2211yescharacteristic
16C22×C3×Dic9C2211yescharacteristic
17C2×#6189—22no—
18C2×#6189—22no—
19C2×#6189—22no—
20#88551—22no—
21C9×#217—22no—
22C3×#3676—22no—
23C22×#627—31no—
24C2×#3757—31no—
25C2×C3×#217—31no—
26C2×C3×#217D311yescharacteristic
27C22×Dic18—31no—
28#63579—31no—
29#63579—31no—
30#63579—31no—
31C2×C9×C22⋊C4—31no—
32C2×#3676—31no—
33C2×#3676—61no—
34C2×#3676D311yescharacteristic
35C9⋊#95—31no—
36C9⋊#95—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×C9Q812yescomplemented
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#6189—28no—
65#4947—91no—
66#4947—91no—
67C24×C32C2×D311yescharacteristic
68C22×C3⋊Dic3—31no—
69C22×C3⋊Dic3—31no—
70C22×C3×Dic3—31no—
71C22×C3×Dic3—31no—
72C22×C3×Dic3C2×D311yescharacteristic
73C22×C3×Dic3C2×D311yescharacteristic
74C2×#627—62no—
75C2×#627—62no—
76C2×#627—62no—
77C24×C9—21no—
78C24×C9C2×D311yescharacteristic
79#3757—62no—
80C3×#217—22no—
81C3×#217—62no—
82C22×C4×C9—31no—
83C22×C4×C9—31no—
84C22×Dic9—31no—
85C22×Dic9—31no—
86C22×Dic9—61no—
87C22×Dic9—61no—
88C22×Dic9C2×D311yescharacteristic
89C22×Dic9C2×D311yescharacteristic
90C2×Dic18—62no—
91C2×Dic18—62no—
92C2×Dic18—62no—
93C9×C22⋊C4—62no—
94#3676—22no—
95#3676—62no—
96#3676—64no—
97C22×C3×C9—22no—
98C22×C3×C9—22no—
99C22×C3×C9—22no—
100C22×C3×C9—41no—
101C22×C3×C9—41no—
102C22×C3×C9—41no—
103C22×C3×C9—41no—
104C22×C3×C9C2×D411yescharacteristic
105C22×C3×C9C2×D411yescharacteristic
106C22×C3×C9C2×D411yescharacteristic
107C22×C3×C9C2×Q811yescharacteristic
108C22×C3×C9#4111yescharacteristic
109C22×C3×C9#4111yescharacteristic
110C22×C3×C9#4111yescharacteristic
111C3⋊Dic9—22no—
112C3⋊Dic9—41no—
113C9×Dic3—22no—
114C9×Dic3—41no—
115C3×Dic9—22no—
116C3×Dic9—41no—
117C22×Dic6—91no—
118C22×Dic6—91no—
119C2×C3×C22⋊C4—31no—
120C2×C3×C22⋊C4—91no—
121C2×#217—31no—
122C2×#217—91no—
123C2×#217—91no—
124C2×#217—181no—
125C2×#217D911yescharacteristic
126C3⋊#95—91no—
127C3⋊#95—91no—
128C3⋊#95—91no—
129C3⋊#95—91no—
130C3⋊#95—91no—
131C3⋊#95—91no—
132C23×C32—21no—
133C23×C32—21no—
134C23×C32—21no—
135C23×C32—21no—
136C23×C32—21no—
137C23×C32—21no—
138C23×C32Dic612yes—
139C23×C32C22×D311yescharacteristic
140C2×C3⋊Dic3—32no—
141C2×C3⋊Dic3—61no—
142C2×C3⋊Dic3—61no—
143C2×C3⋊Dic3—61no—
144C2×C3⋊Dic3—61no—
145C2×C3⋊Dic3—61no—
146C2×C3×Dic3—21no—
147C2×C3×Dic3—21no—
148C2×C3×Dic3—21no—
149C2×C3×Dic3—21no—
150C2×C3×Dic3—21no—
151C2×C3×Dic3—32no—
152C2×C3×Dic3—61no—
153C2×C3×Dic3—61no—
154C2×C3×Dic3—61no—
155C2×C3×Dic3—61no—
156C2×C3×Dic3—61no—
157C2×C3×Dic3#6712yes—
158#627—68no—
159C23×C9—22no—
160C23×C9—22no—
161C23×C9—21no—
162C23×C9—21no—
163C23×C9—21no—
164C23×C9—21no—
165C23×C9—21no—
166C23×C9—21no—
167C23×C9—22no—
168C23×C9—21no—
169C23×C9—41no—
170C23×C9—41no—
171C23×C9—41no—
172C23×C9—41no—
173C23×C9Dic612yescomplemented
174C23×C9C22×D311yescharacteristic
175C2×C4×C9—32no—
176C2×C4×C9—61no—
177C2×C4×C9—61no—
178C2×C4×C9—61no—
179C2×C4×C9—61no—
180C2×C4×C9—61no—
181C2×Dic9—21no—
182C2×Dic9—21no—
183C2×Dic9—21no—
184C2×Dic9—21no—
185C2×Dic9—21no—
186C2×Dic9—32no—
187C2×Dic9—62no—
188C2×Dic9—61no—
189C2×Dic9—61no—
190C2×Dic9—62no—
191C2×Dic9—61no—
192C2×Dic9—61no—
193C2×Dic9—61no—
194C2×Dic9—121no—
195C2×Dic9—121no—
196C2×Dic9—121no—
197C2×Dic9—121no—
198C2×Dic9#6712yes—
199Dic18—68no—
200#393—271noSylow
201C2×C3×C9—42no—
202C2×C3×C9#12011yescharacteristic
203C2×C3×C9C22⋊Q811yescharacteristic
204C2×C3×C9C22⋊Q811yescharacteristic
205C2×C3×C9C22⋊Q811yescharacteristic
206C2×C3×C9#12411yescharacteristic
207C2×C3×C9#12411yescharacteristic
208C2×C3×C9#12411yescharacteristic
209C24×C3—21no—
210C24×C3C2×D911yescharacteristic
211C24×C3D3211yescharacteristic
212C22×C3×C4—31no—
213C22×C3×C4—31no—
214C22×C3×C4—91no—
215C22×C3×C4—91no—
216C22×Dic3—31no—
217C22×Dic3—31no—
218C22×Dic3—91no—
219C22×Dic3—91no—
220C22×Dic3—91no—
221C22×Dic3—91no—
222C22×Dic3—181no—
223C22×Dic3—181no—
224C22×Dic3C2×D911yescharacteristic
225C22×Dic3C2×D911yescharacteristic
226C2×Dic6—182no—
227C2×Dic6—182no—
228C2×Dic6—182no—
229C2×Dic6—182no—
230C2×Dic6—182no—
231C2×Dic6—182no—
232C3×C22⋊C4—62no—
233C3×C22⋊C4—182no—
234#217—22no—
235#217—62no—
236#217—182no—
237#217—182no—
238#217—184no—
239C22×C32—22no—
240C22×C32—22no—
241C22×C32—22no—
242C22×C32—41no—
243C22×C32—41no—
244C22×C32—41no—
245C22×C32—41no—
246C22×C32C2×Dic611yescharacteristic
247C22×C32#23511yescharacteristic
248C22×C32D3×D411yescharacteristic
249C22×C32D3×D411yescharacteristic
250C22×C32#23711yescharacteristic
251C22×C32#23711yescharacteristic
252C22×C32C2×#6711yescharacteristic
253C3⋊Dic3—62no—
254C3⋊Dic3—121no—
255C3×Dic3—22no—
256C3×Dic3—41no—
257C3×Dic3—62no—
258C3×Dic3—121no—
259C22×C9—24no—
260C22×C9—22no—
261C22×C9—22no—
262C22×C9—22no—
263C22×C9—21no—
264C22×C9—21no—
265C22×C9—21no—
266C22×C9—21no—
267C22×C9—21no—
268C22×C9—21no—
269C22×C9—21no—
270C22×C9—42no—
271C22×C9—42no—
272C22×C9—42no—
273C22×C9—42no—
274C22×C9—42no—
275C22×C9—42no—
276C22×C9—41no—
277C22×C9—41no—
278C22×C9—41no—
279C22×C9—41no—
280C22×C9C2×Dic611yescharacteristic
281C22×C9#23511yescharacteristic
282C22×C9D3×D411yescharacteristic
283C22×C9D3×D411yescharacteristic
284C22×C9#23711yescharacteristic
285C22×C9#23711yescharacteristic
286C22×C9C2×#6711yescharacteristic
287C4×C9—62no—
288C4×C9—121no—
289Dic9—22no—
290Dic9—41no—
291Dic9—62no—
292Dic9—122no—
293Dic9—122no—
294Dic9—121no—
295C22×Q8—271no—
296C2×C22⋊C4—31no—
297C2×C22⋊C4—91no—
298C2×C22⋊C4—271no—
299#95—271no—
300#95—271no—
301#95—271no—
302C3×C9#39311yescharacteristic, complemented, Sylow
303C23×C3—22no—
304C23×C3—22no—
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—21no—
315C23×C3—21no—
316C23×C3—21no—
317C23×C3—22no—
318C23×C3—21no—
319C23×C3—41no—
320C23×C3—41no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3Dic1812yescomplemented
324C23×C3C22×D911yescharacteristic
325C23×C3#62712yes—
326C23×C3C2×D3211yescharacteristic
327C2×C3×C4—32no—
328C2×C3×C4—61no—
329C2×C3×C4—61no—
330C2×C3×C4—61no—
331C2×C3×C4—61no—
332C2×C3×C4—61no—
333C2×C3×C4—92no—
334C2×C3×C4—181no—
335C2×C3×C4—181no—
336C2×C3×C4—181no—
337C2×C3×C4—181no—
338C2×C3×C4—181no—
339C2×Dic3—21no—
340C2×Dic3—21no—
341C2×Dic3—21no—
342C2×Dic3—21no—
343C2×Dic3—21no—
344C2×Dic3—32no—
345C2×Dic3—61no—
346C2×Dic3—61no—
347C2×Dic3—61no—
348C2×Dic3—61no—
349C2×Dic3—61no—
350C2×Dic3—92no—
351C2×Dic3—92no—
352C2×Dic3—182no—
353C2×Dic3—181no—
354C2×Dic3—181no—
355C2×Dic3—181no—
356C2×Dic3—181no—
357C2×Dic3—182no—
358C2×Dic3—181no—
359C2×Dic3—181no—
360C2×Dic3—181no—
361C2×Dic3—181no—
362C2×Dic3—181no—
363C2×Dic3—181no—
364C2×Dic3—361no—
365C2×Dic3—361no—
366C2×Dic3—361no—
367C2×Dic3—361no—
368C2×Dic3#61112yes—
369Dic6—188no—
370Dic6—188no—
371C2×C32—42no—
372C2×C32#87811yescharacteristic
373C2×C32#87811yescharacteristic
374C2×C32#88511yescharacteristic
375C2×C32#88511yescharacteristic
376C2×C32#92411yescharacteristic
377C2×C32#93611yescharacteristic
378C2×C32#93711yescharacteristic
379C2×C9—21no—
380C2×C9—21no—
381C2×C9—21no—
382C2×C9—21no—
383C2×C9—21no—
384C2×C9—21no—
385C2×C9—21no—
386C2×C9—44no—
387C2×C9—42no—
388C2×C9#87811yescharacteristic
389C2×C9#87811yescharacteristic
390C2×C9#88511yescharacteristic
391C2×C9#88511yescharacteristic
392C2×C9#92411yescharacteristic
393C2×C9#93611yescharacteristic
394C2×C9#93711yescharacteristic
395C24D3×D911yescharacteristic
396C2×Q8—542no—
397C2×Q8—542no—
398C2×Q8—542no—
399C22×C4—31no—
400C22×C4—31no—
401C22×C4—91no—
402C22×C4—91no—
403C22×C4—271no—
404C22×C4—271no—
405C22⋊C4—62no—
406C22⋊C4—182no—
407C22⋊C4—542no—
408C22×C3—24no—
409C22×C3—22no—
410C22×C3—22no—
411C22×C3—22no—
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—22no—
415C22×C3—21no—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3—42no—
428C22×C3—41no—
429C22×C3—41no—
430C22×C3—41no—
431C22×C3—41no—
432C22×C3—41no—
433C22×C3—41no—
434C22×C3—41no—
435C22×C3—41no—
436C22×C3C2×Dic1811yescharacteristic
437C22×C3#369711yescharacteristic
438C22×C3D4×D911yescharacteristic
439C22×C3D4×D911yescharacteristic
440C22×C3#369911yescharacteristic
441C22×C3#369911yescharacteristic
442C22×C3C2×#61111yescharacteristic
443C22×C3#380411yescharacteristic
444C22×C3#380411yescharacteristic
445C22×C3#380511yescharacteristic
446C22×C3C2×#62711yescharacteristic
447C22×C3D3×#6711yescharacteristic
448C22×C3D3×#6711yescharacteristic
449C22×C3#381111yescharacteristic
450C3×C4—62no—
451C3×C4—121no—
452C3×C4—182no—
453C3×C4—361no—
454Dic3—22no—
455Dic3—41no—
456Dic3—62no—
457Dic3—121no—
458Dic3—182no—
459Dic3—182no—
460Dic3—362no—
461Dic3—362no—
462Dic3—361no—
463Dic3—361no—
464C32#494711yescharacteristic
465C9—21no—
466C9#494711yescharacteristic, complemented
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23#618912yescomplemented
474C23C2×D3×D911yescharacteristic
475Q8—548no—
476C2×C4—32no—
477C2×C4—61no—
478C2×C4—61no—
479C2×C4—61no—
480C2×C4—61no—
481C2×C4—61no—
482C2×C4—92no—
483C2×C4—181no—
484C2×C4—181no—
485C2×C4—181no—
486C2×C4—181no—
487C2×C4—181no—
488C2×C4—272no—
489C2×C4—541no—
490C2×C4—541no—
491C2×C4—541no—
492C2×C4—541no—
493C2×C4—541no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—44no—
502C2×C3—42no—
503C2×C3—42no—
504C2×C3#6344011yescharacteristic
505C2×C3#6344011yescharacteristic
506C2×C3#6344711yescharacteristic
507C2×C3#6344711yescharacteristic
508C2×C3#6348611yescharacteristic
509C2×C3#6349811yescharacteristic
510C2×C3#6349911yescharacteristic
511C2×C3#6395311yescharacteristic
512C2×C3#6395911yescharacteristic
513C2×C3#6395911yescharacteristic
514C2×C3#6396411yescharacteristic
515C2×C3#6396411yescharacteristic
516C2×C3#6398211yescharacteristic
517C2×C3#6398311yescharacteristic
518C22—22no—
519C22—22no—
520C22—22no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22C2×#618911yescharacteristic
526C22#8865811yescharacteristic
527C22#8866211yescharacteristic
528C22#8866311yescharacteristic
529C22D3×#61111yescharacteristic
530C22D9×#6711yescharacteristic
531C22#8866811yescharacteristic
532C4—62no—
533C4—121no—
534C4—182no—
535C4—361no—
536C4—542no—
537C4—1081no—
538C3—21no—
539C3#1058862711yescharacteristic, complemented
540C3#1059169211yescharacteristic
541C2—42no—
542C2#1172124211yescharacteristic
543C2#1172124911yescharacteristic
544C2#1172125311yescharacteristic
545C2#1172125711yescharacteristic
546C2#1172126311yescharacteristic
547C2#1172126711yescharacteristic
548C2#1172129211yescharacteristic
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#393
LCSG ▷ C23×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C22×#6189C2semidirect
5. #11720331C2semidirect
6. #11720331C2semidirect
7. #11720331C2semidirect
44. C23×C3×C9Q8semidirect
173. C23×C9Dic6semidirect
302. C3×C9#393semidirect
323. C23×C3Dic18semidirect
466. C9#4947semidirect
473. C23#6189semidirect
539. C3#10588627semidirect

Maximal subgroups

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

subgroupindex
C22×#61892
C2×#885512
C2×C9×#2172
C2×C3×#36762
#117203312
#105916923
#105886273