Finite Group Databasebeta

← #13150330

Subgroups — #13150330

Order 1728.

1040 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88736C211yesmaximal, characteristic, complemented
2C2×#88635C211yesmaximal, characteristic, complemented
3C3×C4×#611C211yesmaximal, characteristic, complemented
4C9×#924C211yesmaximal, characteristic, complemented
5#11721275C212yesmaximal, complemented
6#11721256C212yesmaximal, complemented
7#11720992C211yesmaximal, characteristic, complemented
8#11720980C212yesmaximal, complemented
9#11720945C211yesmaximal, characteristic, complemented
10#11720941C212yesmaximal, complemented
11#11720924C211yesmaximal, characteristic, complemented
12#10594834—31nomaximal
13#10589733—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6233C2212yes—
16#88736—24no—
17C2×C4⋊(C3⋊D9)C2211yescharacteristic
18C2×C4×C3⋊D9C2212yes—
19C2×#6226C2211yescharacteristic
20C2×C3×#611C2211yescharacteristic
21C2×C3×C4×D9C2211yescharacteristic
22C2×C9×Dic6C2211yescharacteristic
23C2×#6192C2212yes—
24C2×(C3⋊D9)⋊C4C2212yes—
25C2×(C3⋊D9)⋊C4C2212yes—
26#88635—24no—
27C9×#217C2212yes—
28C3×#3676C2211yescharacteristic
29C3×(C2×D9)⋊C4C2211yescharacteristic
30C9×C4⋊Dic3C2211yescharacteristic
31C9×Dic3⋊C4C2212yes—
32C3×Dic9⋊C4C2211yescharacteristic
33C3×C4×Dic9C2211yescharacteristic
34#88445C2212yes—
35(C2×D9)⋊Dic3C2212yes—
36Dic3⋊Dic9C2212yes—
37Dic9⋊Dic3C2212yes—
38Dic9⋊Dic3C2212yes—
39C2×#3828—31no—
40C2×#3799—31no—
41C3×C4×#67—31no—
42C3×#924D311yescharacteristic
43#63975—32no—
44#63958—32no—
45#63894—31no—
46#63883—32no—
47#63866—31no—
48#63857—31no—
49#63853—32no—
50C2×#3701—31no—
51C2×#3697—31no—
52C2×#3697—61no—
53C9×C22⋊Q8—31no—
54#63506—31no—
55C4×#611D311yescharacteristic, complemented
56#63459—31no—
57#63456—32no—
58#63455—31no—
59C4⋊C4×D9—32no—
60#63446—32no—
61#63445—32no—
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9C2312yes—
64C22×C3×D9C2311yescharacteristic
65C2×C3×C4×C9—22no—
66C2×C3×C4×C9C2311yescharacteristic
67C2×C3×C4×C9C2311yescharacteristic
68#6233—24no—
69C2×C3⋊Dic9C2312yes—
70C4⋊(C3⋊D9)D414yescomplemented
71C4×C3⋊D9—24no—
72#6226—22no—
73C3×#611—22no—
74C2×C9×Dic3C2312yes—
75C2×C9×Dic3C2312yes—
76C2×C3×Dic9C2311yescharacteristic
77C2×C3×Dic9C2311yescharacteristic
78C2×C3×Dic9C2311yescharacteristic
79C3×C4×D9—22no—
80C9×Dic6—22no—
81#6192—24no—
82(C3⋊D9)⋊C4—24no—
83(C3⋊D9)⋊C4—24no—
84#5631—91no—
85#5551—91no—
86C22×C32×C4C2×D311yescharacteristic
87C2×#638—32no—
88#3828—64no—
89C2×C4⋊(C3⋊D3)—31no—
90C2×C4×C3⋊D3—32no—
91C2×#634—31no—
92C2×C3×#67—31no—
93C2×C3×C4×D3—31no—
94C2×C3×Dic6C2×D311yescharacteristic
95C2×#626—32no—
96C2×#624—32no—
97C2×#624—32no—
98#3799—64no—
99C3×#217—31no—
100C3×#217C2×D312yes—
101C3×(C2×D3)⋊C4—31no—
102C3×C4⋊Dic3C2×D311yescharacteristic
103C3×Dic3⋊C4—31no—
104C3×Dic3⋊C4C2×D312yes—
105C3×C4×Dic3—31no—
106Dic3⋊Dic3—32no—
107Dic3⋊Dic3—32no—
108Dic3⋊Dic3—32no—
109#3722—32no—
110(C2×D3)⋊Dic3—32no—
111C2×Q8×C9—31no—
112C22×C4×C9—21no—
113C22×C4×C9C2×D311yescharacteristic
114C2×#611—32no—
115C2×#611—62no—
116C2×#611C2×D311yescharacteristic
117#3701—64no—
118#3697—64no—
119#3697—68no—
120C2×D36—31no—
121C2×D36—32no—
122C2×D36—61no—
123C2×C4×D9—32no—
124C2×C4×D9—32no—
125C2×C4×D9—32no—
126C2×C4×D9—62no—
127C2×C4×D9C2×D311yescharacteristic
128C2×Dic18—31no—
129C2×Dic18—61no—
130C9×C4⋊C4—32no—
131C9×C4⋊C4—31no—
132C9×C22⋊C4—32no—
133#3676C2×D311yescharacteristic
134(C2×D9)⋊C4—32no—
135(C2×D9)⋊C4—32no—
136(C2×D9)⋊C4C2×D311yescharacteristic
137C4⋊Dic9—32no—
138Dic9⋊C4—32no—
139Dic9⋊C4—32no—
140Dic9⋊C4C2×D311yescharacteristic
141C4×Dic9C2×D311yescharacteristic
142C22×C3×C9—21no—
143C22×C3×C9—21no—
144C22×C3×C9—21no—
145C22×C3×C9C2411yescharacteristic
146C2×C3⋊D9—22no—
147C2×C3⋊D9—22no—
148C2×C3⋊D9C2×D414yes—
149C2×C3×D9—21no—
150C2×C3×D9—21no—
151C2×C3×D9—21no—
152C3×C4×C9—21no—
153C3×C4×C9C2×D412yes—
154C3⋊Dic9—22no—
155C9×Dic3—22no—
156C9×Dic3—22no—
157C3×Dic9—21no—
158C3×Dic9—21no—
159C3×Dic9#4112yescomplemented
160C2×#239—91no—
161C2×#235—91no—
162C2×#235—91no—
163C2×#235—91no—
164C2×#235—181no—
165C3×C22⋊Q8—31no—
166C3×C4×D4—91no—
167#944—91no—
168#938—92no—
169C4×#67—31no—
170#924—91no—
171#924D911yescharacteristic, complemented
172#897—91no—
173#894—92no—
174#893—91no—
175D3×C4⋊C4—92no—
176D3×C4⋊C4—92no—
177#884—92no—
178#883—92no—
179#883—92no—
180#883—92no—
181C4×D12—91no—
182#869—91no—
183C23×C32C22×D311yescharacteristic
184C22×C3⋊D3—32no—
185C22×C3×D3—31no—
186C2×C32×C4—22no—
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D311yescharacteristic
189#638—64no—
190C2×C3⋊Dic3—32no—
191C4⋊(C3⋊D3)—34no—
192C4×C3⋊D3—64no—
193#634—62no—
194C3×#67—62no—
195C2×C3×Dic3—31no—
196C2×C3×Dic3—31no—
197C2×C3×Dic3—31no—
198C2×C3×Dic3C22×D312yes—
199C2×C3×Dic3C22×D312yes—
200C3×C4×D3—62no—
201C3×Dic6—22no—
202#626—64no—
203#624—64no—
204#624—64no—
205C23×C9—21no—
206C23×C9C22×D311yescharacteristic
207C22×D9—32no—
208C22×D9—62no—
209C22×D9C22×D311yescharacteristic
210Q8×C9—62no—
211C2×C4×C9—24no—
212C2×C4×C9—22no—
213C2×C4×C9—21no—
214C2×C4×C9—21no—
215C2×C4×C9—32no—
216C2×C4×C9—32no—
217C2×C4×C9C22×D311yescharacteristic
218C2×C4×C9C22×D311yescharacteristic
219#611—22no—
220#611—64no—
221#611—68no—
222C2×Dic9—32no—
223C2×Dic9—62no—
224C2×Dic9C22×D311yescharacteristic
225C2×Dic9C22×D311yescharacteristic
226C2×Dic9C22×D311yescharacteristic
227D36—34no—
228D36—64no—
229D36—64no—
230C4×D9—22no—
231C4×D9—64no—
232C4×D9—64no—
233C4×D9—68no—
234C4×D9—64no—
235Dic18—64no—
236Dic18—62no—
237#547—271noSylow
238C2×C3×C9—41no—
239C2×C3×C9C22×D411yescharacteristic
240C2×C3×C9C2×#4111yescharacteristic
241C2×C3×C92−1+411yescharacteristic
242C3⋊D9—24no—
243C3×D9—41no—
244C2×C3×Q8—31no—
245C2×C3×D4—91no—
246C22×C3×C4—21no—
247C22×C3×C4—91no—
248C22×C3×C4C2×D911yescharacteristic
249C22×C3×C4D3211yescharacteristic
250C2×#67—31no—
251C2×#67—92no—
252C2×#67—92no—
253C2×#67—92no—
254C2×#67—182no—
255#239—184no—
256#235—184no—
257#235—184no—
258#235—184no—
259#235—188no—
260C2×D12—91no—
261C2×D12—91no—
262C2×D12—92no—
263C2×D12—181no—
264C2×C4×D3—31no—
265C2×C4×D3—92no—
266C2×C4×D3—92no—
267C2×C4×D3—92no—
268C2×C4×D3—92no—
269C2×C4×D3—92no—
270C2×C4×D3—92no—
271C2×C4×D3—182no—
272C2×Dic6—91no—
273C2×Dic6—91no—
274C2×Dic6—181no—
275C2×Dic6C2×D911yescharacteristic
276C3×C4⋊C4—32no—
277C3×C4⋊C4—31no—
278C3×C4⋊C4—91no—
279C3×C22⋊C4—32no—
280C3×C22⋊C4—91no—
281C3×C22⋊C4—91no—
282C3×C42—91no—
283#217—31no—
284#217—92no—
285#217C2×D912yes—
286(C2×D3)⋊C4—31no—
287(C2×D3)⋊C4—92no—
288(C2×D3)⋊C4—92no—
289(C2×D3)⋊C4—92no—
290C4⋊Dic3—92no—
291C4⋊Dic3—92no—
292C4⋊Dic3—92no—
293C4⋊Dic3C2×D911yescharacteristic
294Dic3⋊C4—31no—
295Dic3⋊C4—92no—
296Dic3⋊C4—92no—
297Dic3⋊C4—92no—
298Dic3⋊C4C2×D912yes—
299C4×Dic3—31no—
300C22×C32—21no—
301C22×C32—21no—
302C22×C32—21no—
303C22×C32C23×D311yescharacteristic
304C2×C3⋊D3—34no—
305C2×C3⋊D3—62no—
306C2×C3⋊D3—62no—
307C2×C3×D3—61no—
308C2×C3×D3—61no—
309C2×C3×D3—61no—
310C32×C4—21no—
311C32×C4D3×D412yes—
312C3⋊Dic3—62no—
313C3×Dic3—22no—
314C3×Dic3—22no—
315C3×Dic3—32no—
316C3×Dic3—61no—
317C3×Dic3—61no—
318C22×C9—22no—
319C22×C9—21no—
320C22×C9—21no—
321C22×C9—21no—
322C22×C9—21no—
323C22×C9—41no—
324C22×C9—41no—
325C22×C9C23×D311yescharacteristic
326C2×D9—21no—
327C2×D9—21no—
328C2×D9—21no—
329C2×D9—34no—
330C2×D9—64no—
331C2×D9—62no—
332C2×D9—62no—
333C2×D9—122no—
334C2×D9—122no—
335C4×C9—22no—
336C4×C9—21no—
337C4×C9—22no—
338C4×C9—62no—
339C4×C9—62no—
340C4×C9D3×D412yes—
341Dic9—21no—
342Dic9—21no—
343Dic9—64no—
344Dic9—62no—
345Dic9#23512yescomplemented
346C2×#41—271no—
347C2×#41—271no—
348C4⋊Q8—271no—
349#123—272no—
350#123—272no—
351C22⋊Q8—31no—
352C22⋊Q8—271no—
353#121—272no—
354C4×D4—91no—
355C4×D4—271no—
356C2×C4⋊C4—272no—
357C3×C9#54711yescharacteristic, complemented, Sylow
358C23×C3—21no—
359C23×C3—91no—
360C23×C3C22×D911yescharacteristic
361C23×C3C2×D3211yescharacteristic
362C22×D3—31no—
363C22×D3—92no—
364C22×D3—92no—
365C22×D3—182no—
366C3×Q8—62no—
367C3×D4—182no—
368C2×C3×C4—24no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—32no—
374C2×C3×C4—32no—
375C2×C3×C4—91no—
376C2×C3×C4—91no—
377C2×C3×C4—91no—
378C2×C3×C4—182no—
379C2×C3×C4C22×D911yescharacteristic
380C2×C3×C4C22×D911yescharacteristic
381C2×C3×C4C2×D3211yescharacteristic
382C2×C3×C4C2×D3211yescharacteristic
383#67—62no—
384#67—184no—
385#67—184no—
386#67—184no—
387#67—188no—
388C2×Dic3—31no—
389C2×Dic3—31no—
390C2×Dic3—31no—
391C2×Dic3—92no—
392C2×Dic3—92no—
393C2×Dic3—182no—
394C2×Dic3C22×D912yes—
395C2×Dic3C22×D912yes—
396D12—94no—
397D12—94no—
398D12—184no—
399D12—184no—
400C4×D3—62no—
401C4×D3—184no—
402C4×D3—184no—
403C4×D3—184no—
404C4×D3—184no—
405C4×D3—184no—
406C4×D3—184no—
407C4×D3—188no—
408Dic6—22no—
409Dic6—184no—
410Dic6—182no—
411Dic6—182no—
412C2×C32—41no—
413C2×C32C2×D3×D411yescharacteristic
414C2×C32#100711yescharacteristic
415C2×C32D3×#4111yescharacteristic
416C3⋊D3—64no—
417C3×D3—121no—
418C2×C9—21no—
419C2×C9—21no—
420C2×C9—21no—
421C2×C9—42no—
422C2×C9—41no—
423C2×C9C2×#23511yescharacteristic
424C2×C9C2×D3×D411yescharacteristic
425C2×C9#101011yescharacteristic
426D9—41no—
427D9—64no—
428D9—124no—
429#41—544no—
430#41—544no—
431C2×Q8—31no—
432C2×Q8—271no—
433C2×D4—91no—
434C2×D4—272no—
435C2×D4—272no—
436C2×D4—271no—
437C22×C4—91no—
438C22×C4—272no—
439C22×C4—272no—
440C22×C4—272no—
441C22×C4D3×D911yescharacteristic
442C4⋊C4—32no—
443C4⋊C4—31no—
444C4⋊C4—91no—
445C4⋊C4—272no—
446C4⋊C4—272no—
447C4⋊C4—272no—
448C22⋊C4—32no—
449C22⋊C4—91no—
450C22⋊C4—91no—
451C22⋊C4—272no—
452C22⋊C4—272no—
453C42—91no—
454C22×C3—22no—
455C22×C3—21no—
456C22×C3—21no—
457C22×C3—21no—
458C22×C3—21no—
459C22×C3—21no—
460C22×C3—21no—
461C22×C3—21no—
462C22×C3—41no—
463C22×C3—41no—
464C22×C3—181no—
465C22×C3—181no—
466C22×C3—181no—
467C22×C3C23×D911yescharacteristic
468C22×C3C22×D3211yescharacteristic
469C2×D3—61no—
470C2×D3—61no—
471C2×D3—61no—
472C2×D3—94no—
473C2×D3—94no—
474C2×D3—184no—
475C2×D3—182no—
476C2×D3—182no—
477C2×D3—182no—
478C2×D3—182no—
479C2×D3—362no—
480C2×D3—362no—
481C3×C4—22no—
482C3×C4—21no—
483C3×C4—21no—
484C3×C4—22no—
485C3×C4—62no—
486C3×C4—62no—
487C3×C4—92no—
488C3×C4—181no—
489C3×C4—181no—
490C3×C4D4×D912yes—
491C3×C4#381112yes—
492Dic3—22no—
493Dic3—22no—
494Dic3—32no—
495Dic3—61no—
496Dic3—61no—
497Dic3—184no—
498Dic3—182no—
499Dic3—182no—
500C32#563111yescharacteristic
501C9—21no—
502C9#555111yescharacteristic, complemented
503C23—91no—
504C23—272no—
505C23C2×D3×D911yescharacteristic
506Q8—62no—
507Q8—542no—
508D4—182no—
509D4—274no—
510D4—544no—
511D4—544no—
512C2×C4—22no—
513C2×C4—32no—
514C2×C4—32no—
515C2×C4—91no—
516C2×C4—91no—
517C2×C4—91no—
518C2×C4—182no—
519C2×C4—272no—
520C2×C4—544no—
521C2×C4—544no—
522C2×C4—544no—
523C2×C4C2×D3×D911yescharacteristic
524C2×C4C2×D3×D911yescharacteristic
525C2×C3—21no—
526C2×C3—21no—
527C2×C3—21no—
528C2×C3—42no—
529C2×C3—41no—
530C2×C3—41no—
531C2×C3—361no—
532C2×C3C2×D4×D911yescharacteristic
533C2×C3#6371311yescharacteristic
534C2×C3#41×D911yescharacteristic
535C2×C3#6429711yescharacteristic
536C2×C3D3×#23511yescharacteristic
537C2×C3C2×#381111yescharacteristic
538D3—121no—
539D3—184no—
540D3—184no—
541D3—364no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—181no—
546C22—181no—
547C22—181no—
548C22—274no—
549C22—542no—
550C22—542no—
551C22C22×D3×D911yescharacteristic
552C4—21no—
553C4—62no—
554C4—62no—
555C4—92no—
556C4—181no—
557C4—181no—
558C4—542no—
559C4#8866812yes—
560C3—21no—
561C3#1058973311yescharacteristic, complemented
562C3#1059483411yescharacteristic
563C2—41no—
564C2—361no—
565C2—544no—
566C2#1172245811yescharacteristic
567C2D9×#23511yescharacteristic
568C2C2×#8866811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#547
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88736C2semidirect
2. C2×#88635C2semidirect
3. C3×C4×#611C2semidirect
4. C9×#924C2semidirect
5. #11721275C2semidirect
6. #11721256C2semidirect
7. #11720992C2semidirect
8. #11720980C2semidirect
9. #11720945C2semidirect
10. #11720941C2semidirect
11. #11720924C2semidirect
55. C4×#611D3semidirect
70. C4⋊(C3⋊D9)D4semidirect
159. C3×Dic9#41semidirect
171. #924D9semidirect
345. Dic9#235semidirect
357. C3×C9#547semidirect
502. C9#5551semidirect
561. C3#10589733semidirect

Maximal subgroups

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

subgroupindex
C2×#887362
C2×#886352
C3×C4×#6112
C9×#9242
#117212752
#117212562
#117209922
#117209802
#117209452
#117209412
#117209242
#105948343
#105897333