Finite Group Databasebeta

← #13150457

Subgroups — #13150457

Order 1728.

956 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3697C211yesmaximal, characteristic, complemented
2C2×#88638C211yesmaximal, characteristic, complemented
3#11721813C211yesmaximal, characteristic, complemented
4C9×#930C211yesmaximal, characteristic, complemented
5#11721281C212yesmaximal, complemented
6#11721270C212yesmaximal, complemented
7#11721036C211yesmaximal, characteristic, complemented
8#11721018C212yesmaximal, complemented
9D9×C4⋊Dic3C211yesmaximal, characteristic, complemented
10#11720982C212yesmaximal, complemented
11#11720945C211yesmaximal, characteristic, complemented
12#10595004—31nomaximal
13#10589710—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6226C2211yescharacteristic
16C2×C9×#67C2212yes—
17C2×C3×#611C2211yescharacteristic
18C2×C3×#611C2211yescharacteristic
19C2×C9×D12C2211yescharacteristic
20C3×#3697—24no—
21C2×C3×D36C2211yescharacteristic
22C2×C3×C4×D9C2211yescharacteristic
23C2×C3×C4×D9C2211yescharacteristic
24C2×C3×Dic18C2211yescharacteristic
25C2×#6194C2212yes—
26C2×#6194C2212yes—
27C2×D3×Dic9C2212yes—
28C2×Dic3×D9C2212yes—
29#88638—24no—
30#88551C2212yes—
31C4⋊(C3⋊Dic9)C2211yescharacteristic
32#88534C2212yes—
33C9×(C2×D3)⋊C4C2212yes—
34C9×C4⋊Dic3C2211yescharacteristic
35(C2×D3)⋊Dic9C2212yes—
36(C2×D9)⋊Dic3C2212yes—
37Dic3⋊Dic9C2212yes—
38Dic9⋊Dic3C2212yes—
39C2×C3×#235—31no—
40C2×#3795—31no—
41#64133—31no—
42C3×#930D311yescharacteristic
43#63970—32no—
44#63970—32no—
45#63908—31no—
46D3×C4⋊Dic3—31no—
47#63887—32no—
48#63885—32no—
49#63857—31no—
50C2×#3699—31no—
51C2×#3697D311yescharacteristic, complemented
52C9×#121—31no—
53#63501—32no—
54#63500—32no—
55#63500—31no—
56#63486—31no—
57#63486—61no—
58#63459—31no—
59C4⋊C4×D9—31no—
60#63446—32no—
61#63440—32no—
62C23×C3×C9C2311yescharacteristic
63C22×D3×C9C2312yes—
64C22×C3×D9C2311yescharacteristic
65C22×C3×D9C2311yescharacteristic
66C2×C3×C4×C9—22no—
67C2×C3×C4×C9C2311yescharacteristic
68C2×C3×C4×C9C2311yescharacteristic
69C2×C3⋊Dic9C2312yes—
70C2×C3⋊Dic9C2312yes—
71#6226—22no—
72C9×#67—42no—
73C3×#611—22no—
74C3×#611—22no—
75C2×C9×Dic3C2312yes—
76C2×C3×Dic9C2311yescharacteristic
77C2×C3×Dic9C2311yescharacteristic
78C9×D12—22no—
79C3×D36—22no—
80C3×C4×D9—22no—
81C3×C4×D9D414yescomplemented
82C3×Dic18—22no—
83#6194—24no—
84#6194—42no—
85D3×Dic9—24no—
86Dic3×D9—42no—
87#5833—91no—
88#5608—91no—
89C22×C32×C4C2×D311yescharacteristic
90C2×#634—31no—
91C2×C3×#67—31no—
92C2×C3×#67—31no—
93C2×C3×#67C2×D312yes—
94C3×#235—64no—
95C2×C3×D12—31no—
96C2×C3×D12C2×D311yescharacteristic
97C2×C3×C4×D3—31no—
98C2×C3×C4×D3—31no—
99C2×C3×Dic6—31no—
100C2×#625—32no—
101C2×#625—32no—
102C2×D3×Dic3—32no—
103C2×D3×Dic3—32no—
104#3795—64no—
105#3757—32no—
106C4⋊(C3⋊Dic3)—31no—
107#3750—32no—
108C3×(C2×D3)⋊C4C2×D312yes—
109C3×C4⋊Dic3C2×D311yescharacteristic
110Dic3⋊Dic3—32no—
111Dic3⋊Dic3—32no—
112(C2×D3)⋊Dic3—32no—
113(C2×D3)⋊Dic3—32no—
114C2×D4×C9—32no—
115C2×D4×C9—31no—
116C22×C4×C9—21no—
117C22×C4×C9C2×D311yescharacteristic
118C2×#611—32no—
119C2×#611—32no—
120C2×#611C2×D311yescharacteristic
121C2×#611C2×D311yescharacteristic
122C22×Dic9—32no—
123#3699—64no—
124#3697—24no—
125C2×D36C2×D311yescharacteristic
126C2×C4×D9—32no—
127C2×C4×D9C2×D311yescharacteristic
128C2×C4×D9C2×D311yescharacteristic
129C2×Dic18—31no—
130C2×Dic18—61no—
131C2×Dic18C2×D311yescharacteristic
132C9×C4⋊C4—31no—
133C9×C22⋊C4—32no—
134#3676—32no—
135#3676—32no—
136#3676—62no—
137(C2×D9)⋊C4—32no—
138C4⋊Dic9—31no—
139C4⋊Dic9—32no—
140C4⋊Dic9—61no—
141Dic9⋊C4—32no—
142Dic9⋊C4—32no—
143Dic9⋊C4—62no—
144C22×C3×C9—21no—
145C22×C3×C9—21no—
146C22×C3×C9—21no—
147C22×C3×C9C2411yescharacteristic
148C2×D3×C9—22no—
149C2×D3×C9—22no—
150C2×D3×C9—22no—
151C2×C3×D9—21no—
152C2×C3×D9—21no—
153C2×C3×D9—21no—
154C2×C3×D9—21no—
155C2×C3×D9—21no—
156C2×C3×D9C2×D412yes—
157C3×C4×C9—21no—
158C3×C4×C9C2×D412yes—
159C3⋊Dic9—22no—
160C3⋊Dic9—22no—
161C9×Dic3—22no—
162C3×Dic9—21no—
163C3×Dic9C2×D412yes—
164C2×C3×#41—91no—
165C2×#237—91no—
166C2×#235—31no—
167C2×#235—91no—
168C3×#121—31no—
169#946—92no—
170#939—92no—
171#939—92no—
172#939—92no—
173#938—91no—
174#938—92no—
175#938—92no—
176#930—91no—
177#930D911yescharacteristic, complemented
178C2×C4⋊Dic3—91no—
179#924—91no—
180#924—91no—
181#924—91no—
182#924—181no—
183#897—91no—
184D3×C4⋊C4—91no—
185#884—92no—
186#878—92no—
187C23×C32C22×D311yescharacteristic
188C22×C3×D3—31no—
189C22×C3×D3—31no—
190C22×C3×D3C22×D312yes—
191C2×C32×C4—22no—
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C22×D311yescharacteristic
194C2×C3⋊Dic3—32no—
195C2×C3⋊Dic3—32no—
196#634—62no—
197C3×#67—42no—
198C3×#67—62no—
199C3×#67—62no—
200C2×C3×Dic3—31no—
201C2×C3×Dic3—31no—
202C2×C3×Dic3C22×D312yes—
203C3×D12—22no—
204C3×D12—62no—
205C3×C4×D3—34no—
206C3×C4×D3—62no—
207C3×Dic6—62no—
208#625—64no—
209#625—122no—
210D3×Dic3—64no—
211D3×Dic3—122no—
212C23×C9—21no—
213C23×C9—32no—
214C23×C9C22×D311yescharacteristic
215C22×D9C22×D311yescharacteristic
216C22×D9C22×D311yescharacteristic
217D4×C9—62no—
218D4×C9—122no—
219C2×C4×C9—22no—
220C2×C4×C9—21no—
221C2×C4×C9—21no—
222C2×C4×C9—32no—
223C2×C4×C9—42no—
224C2×C4×C9C22×D311yescharacteristic
225C2×C4×C9C22×D311yescharacteristic
226#611—22no—
227#611—22no—
228#611—64no—
229#611—122no—
230C2×Dic9—32no—
231C2×Dic9—32no—
232C2×Dic9—62no—
233C2×Dic9—62no—
234C2×Dic9—64no—
235C2×Dic9C22×D311yescharacteristic
236C2×Dic9C22×D311yescharacteristic
237D36—22no—
238C4×D9—22no—
239C4×D9—122no—
240C4×D9#6714yescomplemented
241Dic18—22no—
242Dic18—62no—
243Dic18—122no—
244#537—271noSylow
245C2×C3×C9—41no—
246C2×C3×C9C22×D411yescharacteristic
247C2×C3×C9H(5,𝔽2)11yescharacteristic
248C2×C3×C92−1+411yescharacteristic
249D3×C9—42no—
250C3×D9—22no—
251C3×D9—41no—
252C3×#41—184no—
253C2×C3×Q8—91no—
254C2×C3×D4—32no—
255C2×C3×D4—31no—
256C2×C3×D4—91no—
257C2×C3×D4—91no—
258C2×C3×D4—91no—
259C22×C3×C4—21no—
260C22×C3×C4—91no—
261C22×C3×C4—91no—
262C22×C3×C4C2×D911yescharacteristic
263C22×C3×C4D3211yescharacteristic
264C2×#67—31no—
265C2×#67—31no—
266C2×#67—92no—
267C2×#67—92no—
268C2×#67—92no—
269C2×#67—92no—
270C2×#67C2×D912yes—
271C22×Dic3—92no—
272C22×Dic3—92no—
273#237—184no—
274#235—64no—
275#235—184no—
276C2×D12—31no—
277C2×D12C2×D911yescharacteristic
278C2×C4×D3—31no—
279C2×C4×D3—31no—
280C2×C4×D3—92no—
281C2×C4×D3—92no—
282C2×Dic6—31no—
283C2×Dic6—91no—
284C2×Dic6—91no—
285C2×Dic6—181no—
286C3×C4⋊C4—31no—
287C3×C22⋊C4—32no—
288#217—92no—
289#217—92no—
290#217—92no—
291#217—92no—
292#217—182no—
293(C2×D3)⋊C4—92no—
294(C2×D3)⋊C4—92no—
295(C2×D3)⋊C4C2×D912yes—
296C4⋊Dic3—92no—
297C4⋊Dic3—91no—
298C4⋊Dic3—91no—
299C4⋊Dic3—92no—
300C4⋊Dic3—181no—
301C4⋊Dic3C2×D911yescharacteristic
302Dic3⋊C4—92no—
303Dic3⋊C4—92no—
304Dic3⋊C4—92no—
305Dic3⋊C4—92no—
306Dic3⋊C4—182no—
307C22×C32—21no—
308C22×C32—21no—
309C22×C32—21no—
310C22×C32C23×D311yescharacteristic
311C2×C3×D3—22no—
312C2×C3×D3—22no—
313C2×C3×D3—22no—
314C2×C3×D3—32no—
315C2×C3×D3—61no—
316C2×C3×D3—61no—
317C2×C3×D3—61no—
318C2×C3×D3—61no—
319C2×C3×D3—61no—
320C32×C4—21no—
321C32×C4D3×D412yes—
322C3⋊Dic3—62no—
323C3⋊Dic3—62no—
324C3×Dic3—22no—
325C3×Dic3—32no—
326C3×Dic3—61no—
327C22×C9—22no—
328C22×C9—21no—
329C22×C9—21no—
330C22×C9—21no—
331C22×C9—21no—
332C22×C9—41no—
333C22×C9—41no—
334C22×C9—62no—
335C22×C9—62no—
336C22×C9—62no—
337C22×C9C23×D311yescharacteristic
338C2×D9—21no—
339C2×D9—21no—
340C2×D9—21no—
341C2×D9—21no—
342C2×D9—21no—
343C2×D9C2×#6712yes—
344C4×C9—21no—
345C4×C9—22no—
346C4×C9—41no—
347C4×C9—62no—
348C4×C9C2×#6712yes—
349Dic9—21no—
350Dic9—62no—
351Dic9—62no—
352Dic9—122no—
353Dic9—122no—
354Dic9C2×#6712yes—
355C2×#41—91no—
356C2×#41—271no—
357C22⋊Q8—272no—
358C22⋊Q8—271no—
359C22⋊Q8—271no—
360#121—31no—
361#121—272no—
362#121—271no—
363#121—272no—
364#121—272no—
365C2×C4⋊C4—271no—
366C3×C9#53711yescharacteristic, complemented, Sylow
367C23×C3—21no—
368C23×C3—32no—
369C23×C3—91no—
370C23×C3—91no—
371C23×C3C22×D911yescharacteristic
372C23×C3C2×D3211yescharacteristic
373C22×D3—31no—
374C22×D3—31no—
375C22×D3C22×D912yes—
376C3×Q8—182no—
377C3×D4—62no—
378C3×D4—122no—
379C3×D4—182no—
380C3×D4—182no—
381C3×D4—182no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—32no—
387C2×C3×C4—42no—
388C2×C3×C4—91no—
389C2×C3×C4—94no—
390C2×C3×C4—91no—
391C2×C3×C4—182no—
392C2×C3×C4C22×D911yescharacteristic
393C2×C3×C4C22×D911yescharacteristic
394C2×C3×C4C2×D3211yescharacteristic
395C2×C3×C4C2×D3211yescharacteristic
396#67—42no—
397#67—62no—
398#67—62no—
399#67—184no—
400#67—184no—
401#67—362no—
402#67—362no—
403C2×Dic3—31no—
404C2×Dic3—31no—
405C2×Dic3—92no—
406C2×Dic3—92no—
407C2×Dic3—92no—
408C2×Dic3—92no—
409C2×Dic3—182no—
410C2×Dic3—182no—
411C2×Dic3—184no—
412C2×Dic3—362no—
413C2×Dic3C22×D912yes—
414D12—22no—
415D12—62no—
416C4×D3—34no—
417C4×D3—62no—
418C4×D3—184no—
419C4×D3—362no—
420Dic6—62no—
421Dic6—182no—
422Dic6—182no—
423Dic6—362no—
424C2×C32—41no—
425C2×C32C2×D3×D411yescharacteristic
426C2×C32#100411yescharacteristic
427C2×C32#101011yescharacteristic
428C3×D3—42no—
429C3×D3—62no—
430C3×D3—121no—
431C2×C9—21no—
432C2×C9—21no—
433C2×C9—21no—
434C2×C9—42no—
435C2×C9—41no—
436C2×C9—122no—
437C2×C9#100911yescharacteristic
438C2×C9#101011yescharacteristic
439C2×C9C22×#6711yescharacteristic
440D9—22no—
441D9—41no—
442#41—184no—
443#41—544no—
444C2×Q8—91no—
445C2×Q8—271no—
446C2×D4—32no—
447C2×D4—31no—
448C2×D4—91no—
449C2×D4—91no—
450C2×D4—91no—
451C2×D4—272no—
452C2×D4—272no—
453C22×C4—91no—
454C22×C4—91no—
455C22×C4—272no—
456C22×C4—272no—
457C22×C4D3×D911yescharacteristic
458C4⋊C4—31no—
459C4⋊C4—272no—
460C4⋊C4—272no—
461C4⋊C4—271no—
462C4⋊C4—272no—
463C22⋊C4—32no—
464C22⋊C4—272no—
465C22⋊C4—272no—
466C22⋊C4—272no—
467C22×C3—22no—
468C22×C3—21no—
469C22×C3—21no—
470C22×C3—21no—
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—21no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3—62no—
478C22×C3—62no—
479C22×C3—62no—
480C22×C3—92no—
481C22×C3—181no—
482C22×C3—181no—
483C22×C3—181no—
484C22×C3—181no—
485C22×C3—181no—
486C22×C3C23×D911yescharacteristic
487C22×C3C22×D3211yescharacteristic
488C2×D3—22no—
489C2×D3—22no—
490C2×D3—22no—
491C2×D3—32no—
492C2×D3—61no—
493C2×D3—61no—
494C2×D3—61no—
495C2×D3—61no—
496C2×D3—61no—
497C3×C4—21no—
498C3×C4—21no—
499C3×C4—22no—
500C3×C4—41no—
501C3×C4—62no—
502C3×C4—92no—
503C3×C4—181no—
504C3×C4D4×D912yes—
505C3×C4D3×#6712yes—
506Dic3—22no—
507Dic3—32no—
508Dic3—61no—
509Dic3—182no—
510Dic3—182no—
511Dic3—182no—
512Dic3—182no—
513Dic3—362no—
514Dic3—362no—
515C32#560811yescharacteristic
516C9—21no—
517C9#583311yescharacteristic, complemented
518C23—32no—
519C23—91no—
520C23—91no—
521C23C2×D3×D911yescharacteristic
522Q8—182no—
523Q8—542no—
524D4—62no—
525D4—122no—
526D4—182no—
527D4—182no—
528D4—182no—
529D4—544no—
530D4—1082no—
531C2×C4—22no—
532C2×C4—32no—
533C2×C4—91no—
534C2×C4—94no—
535C2×C4—91no—
536C2×C4—182no—
537C2×C4—272no—
538C2×C4—272no—
539C2×C4—544no—
540C2×C4—1082no—
541C2×C4C2×D3×D911yescharacteristic
542C2×C4C2×D3×D911yescharacteristic
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—21no—
546C2×C3—42no—
547C2×C3—41no—
548C2×C3—41no—
549C2×C3—122no—
550C2×C3—182no—
551C2×C3—361no—
552C2×C3C2×D4×D911yescharacteristic
553C2×C3#6371011yescharacteristic
554C2×C3#6371611yescharacteristic
555C2×C3#6429811yescharacteristic
556C2×C3#6430711yescharacteristic
557C2×C3C2×D3×#6711yescharacteristic
558D3—42no—
559D3—62no—
560D3—121no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—62no—
565C22—62no—
566C22—62no—
567C22—92no—
568C22—181no—
569C22—181no—
570C22—181no—
571C22—181no—
572C22—181no—
573C22C22×D3×D911yescharacteristic
574C4—21no—
575C4—62no—
576C4—92no—
577C4—181no—
578C4—542no—
579C4—542no—
580C4D9×#6712yes—
581C3—21no—
582C3#1058971011yescharacteristic, complemented
583C3#1059500411yescharacteristic
584C2—41no—
585C2—122no—
586C2—182no—
587C2—361no—
588C2#1172246511yescharacteristic
589C2#1172247611yescharacteristic
590C2C2×D9×#6711yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#537
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×C3×#3697C2semidirect
2. C2×#88638C2semidirect
3. #11721813C2semidirect
4. C9×#930C2semidirect
5. #11721281C2semidirect
6. #11721270C2semidirect
7. #11721036C2semidirect
8. #11721018C2semidirect
9. D9×C4⋊Dic3C2semidirect
10. #11720982C2semidirect
11. #11720945C2semidirect
51. C2×#3697D3semidirect
81. C3×C4×D9D4semidirect
177. #930D9semidirect
240. C4×D9#67semidirect
366. C3×C9#537semidirect
517. C9#5833semidirect
582. C3#10589710semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#36972
C2×#886382
#117218132
C9×#9302
#117212812
#117212702
#117210362
#117210182
D9×C4⋊Dic32
#117209822
#117209452
#105950043
#105897103