Finite Group Databasebeta

← #13149319

Subgroups — #13149319

Order 1728.

1096 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D9×D12C211yesmaximal, characteristic, complemented
2C2×#88639C211yesmaximal, characteristic, complemented
3#11721751C211yesmaximal, characteristic, complemented
4C9×#893C211yesmaximal, characteristic, complemented
5C3×#63459C211yesmaximal, characteristic, complemented
6#11721051C212yesmaximal, complemented
7D3×(C2×D9)⋊C4C212yesmaximal, complemented
8#11721038C212yesmaximal, complemented
9#11721020C212yesmaximal, complemented
10C4×#6192C211yesmaximal, characteristic, complemented
11#11720952C211yesmaximal, characteristic, complemented
12#10594629—31nomaximal
13#10589656—31nomaximal
14C22×D3×D9C2212yes—
15C2×C4⋊(C3⋊D9)C2211yescharacteristic
16C2×C4⋊(C3⋊D9)C2212yescomplemented
17C2×C4×C3⋊D9C2211yescharacteristic, complemented
18C2×C9×D12C2211yescharacteristic
19C2×C4×D3×C9C2212yescomplemented
20C2×C3×C4×D9C2211yescharacteristic
21C2×C3×Dic18C2211yescharacteristic, complemented
22C2×#6195C2212yes—
23C2×#6195C2212yescomplemented
24C2×D3×Dic9C2212yescomplemented
25C2×#6192C2211yescharacteristic
26D9×D12—24no—
27#88639—24no—
28C3×C9×C4⋊C4C2211yescharacteristic, complemented
29#88536C2212yescomplemented
30C9×(C2×D3)⋊C4C2212yescomplemented
31C3×(C2×D9)⋊C4C2212yes—
32C4×C9×Dic3C2211yescharacteristic, complemented
33C3×C4⋊Dic9C2212yescomplemented
34#88445C2211yescharacteristic, complemented
35#88445C2212yescomplemented
36#88445C2212yescomplemented
37(C2×D9)⋊Dic3C2211yescharacteristic
38Dic3⋊Dic9C2211yescharacteristic, complemented
39C2×D3×D12—31no—
40C2×#3796—31no—
41#64104—31no—
42C3×#897—31no—
43C3×#893D311yescharacteristic
44#63923—32no—
45D3×(C2×D3)⋊C4—32no—
46#63909—32no—
47C4×#626—31no—
48#63890—32no—
49#63864—31no—
50C2×#3699—31no—
51C2×D4×D9—31no—
52C4×D4×C9—31no—
53#63492—32no—
54C2×(C2×D9)⋊C4—32no—
55#63459D311yescharacteristic, complemented
56#63457—31no—
57#63457—61no—
58#63448—32no—
59#63444—32no—
60#63437—31no—
61C4×D36—31no—
62C22×C3⋊D9C2312yes—
63C22×C3⋊D9C2311yescharacteristic
64C22×D3×C9C2312yes—
65C22×C3×D9C2311yescharacteristic
66C2×D3×D9—24no—
67C2×D3×D9—24no—
68C2×D3×D9—42no—
69C2×C3×C4×C9C2312yes—
70C2×C3×C4×C9C2311yescharacteristic
71C2×C3⋊Dic9C2311yescharacteristic
72C4⋊(C3⋊D9)—22no—
73C4⋊(C3⋊D9)—42no—
74C4×C3⋊D9—22no—
75C2×C9×Dic3C2311yescharacteristic
76C2×C9×Dic3C2311yescharacteristic
77C2×C3×Dic9C2312yes—
78C2×C3×Dic9C2311yescharacteristic
79C9×D12D414yescomplemented
80C4×D3×C9—42no—
81C3×C4×D9—22no—
82C3×Dic18—22no—
83#6195—24no—
84#6195—24no—
85D3×Dic9—24no—
86#6192—22no—
87#5630—91no—
88#5554—91no—
89C22×D32—32no—
90C2×C4⋊(C3⋊D3)—31no—
91C2×C4⋊(C3⋊D3)—32no—
92C2×C4×C3⋊D3—31no—
93C2×C3×D12C2×D311yescharacteristic
94C2×C3×C4×D3—31no—
95C2×C3×C4×D3C2×D312yes—
96C2×C3×Dic6—31no—
97C2×#626—32no—
98C2×#626—32no—
99C2×#626—31no—
100C2×D3×Dic3—32no—
101D3×D12—64no—
102#3796—64no—
103C23×D9—32no—
104C32×C4⋊C4C2×D311yescharacteristic
105#3752—32no—
106C3×(C2×D3)⋊C4—32no—
107C3×(C2×D3)⋊C4C2×D312yes—
108C3×C4⋊Dic3—32no—
109C3×C4×Dic3C2×D311yescharacteristic
110Dic3⋊Dic3—31no—
111#3722—32no—
112#3722—32no—
113#3722—31no—
114(C2×D3)⋊Dic3—31no—
115C2×D4×C9—31no—
116C22×C4×C9—32no—
117C2×#611—32no—
118C2×#611—32no—
119C22×Dic9—32no—
120#3699—64no—
121D4×D9—64no—
122C2×D36—31no—
123C2×D36—31no—
124C2×D36—32no—
125C2×D36—61no—
126C2×D36—62no—
127C2×C4×D9—31no—
128C2×C4×D9—61no—
129C2×C4×D9C2×D311yescharacteristic
130C2×Dic18C2×D311yescharacteristic, complemented
131C9×C4⋊C4—21no—
132C9×C4⋊C4C2×D311yescharacteristic, complemented
133C9×C22⋊C4—32no—
134C42×C9—31no—
135(C2×D9)⋊C4—31no—
136(C2×D9)⋊C4—32no—
137(C2×D9)⋊C4—32no—
138(C2×D9)⋊C4—31no—
139(C2×D9)⋊C4—32no—
140(C2×D9)⋊C4—62no—
141(C2×D9)⋊C4C2×D312yes—
142C4⋊Dic9—31no—
143C4⋊Dic9C2×D312yescomplemented
144C22×C3×C9C2411yescharacteristic
145C2×C3⋊D9—21no—
146C2×C3⋊D9—22no—
147C2×C3⋊D9—21no—
148C2×C3⋊D9—22no—
149C2×C3⋊D9—21no—
150C2×C3⋊D9—22no—
151C2×D3×C9—22no—
152C2×D3×C9—22no—
153C2×D3×C9C2×D414yes—
154C2×C3×D9—21no—
155C2×C3×D9—21no—
156C2×C3×D9#4112yescomplemented
157D3×D9—48no—
158C3×C4×C9—22no—
159C3×C4×C9C2×D412yes—
160C3⋊Dic9—21no—
161C9×Dic3—21no—
162C9×Dic3—21no—
163C3×Dic9—22no—
164C3×Dic9—21no—
165C2×#239—91no—
166C2×#237—91no—
167C2×D3×D4—91no—
168C22×D12—91no—
169C3×C22⋊Q8—91no—
170C3×C4×D4—31no—
171#947—91no—
172#930—92no—
173C4×#67—91no—
174C2×(C2×D3)⋊C4—92no—
175#897—31no—
176#895—92no—
177#895—91no—
178#895—91no—
179#895—181no—
180#894—92no—
181#893D911yescharacteristic, complemented
182#886—92no—
183#882—92no—
184#882—92no—
185D3×C22⋊C4—92no—
186#875—91no—
187C4×D12—91no—
188C22×C3⋊D3—32no—
189C22×C3⋊D3—31no—
190C22×C3×D3—31no—
191C22×C3×D3C22×D312yes—
192C2×D32—64no—
193C2×D32—64no—
194C2×D32—122no—
195C2×C32×C4C22×D312yes—
196C2×C32×C4C22×D311yescharacteristic
197C2×C3⋊Dic3—31no—
198C4⋊(C3⋊D3)—62no—
199C4⋊(C3⋊D3)—122no—
200C4×C3⋊D3—62no—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—31no—
203C2×C3×Dic3C22×D311yescharacteristic
204C2×C3×Dic3C22×D311yescharacteristic
205C3×D12D1214yes—
206C3×C4×D3—42no—
207C3×C4×D3—62no—
208C3×Dic6—62no—
209#626—62no—
210#626—64no—
211#626—64no—
212D3×Dic3—64no—
213C23×C9—32no—
214C22×D9—32no—
215C22×D9—31no—
216C22×D9—61no—
217C22×D9—62no—
218C22×D9—64no—
219C22×D9—64no—
220C22×D9—122no—
221C22×D9C22×D311yescharacteristic
222D4×C9—34no—
223C2×C4×C9—21no—
224C2×C4×C9—22no—
225C2×C4×C9—31no—
226C2×C4×C9—31no—
227C2×C4×C9—122no—
228C2×C4×C9C22×D312yes—
229C2×C4×C9C22×D311yescharacteristic
230#611—64no—
231#611—64no—
232C2×Dic9—31no—
233C2×Dic9—61no—
234C2×Dic9—64no—
235C2×Dic9C22×D311yescharacteristic
236C2×Dic9C22×D312yes—
237D36—62no—
238D36—62no—
239D36—122no—
240D36—122no—
241D36—124no—
242C4×D9—22no—
243C4×D9—62no—
244C4×D9—122no—
245Dic18—22no—
246#546—271noSylow
247C2×C3×C9C22×D411yescharacteristic
248C2×C3×C9C2×#4111yescharacteristic
249C2×C3×C9H(5,𝔽2)11yescharacteristic
250C3⋊D9—41no—
251C3⋊D9—42no—
252D3×C9—24no—
253C3×D9—22no—
254C23×D3—92no—
255C23×D3—92no—
256C2×C3×Q8—91no—
257C2×C3×D4—31no—
258C22×C3×C4—32no—
259C22×C3×C4—91no—
260C2×#67—92no—
261C2×#67—92no—
262C2×#67—91no—
263C22×Dic3—92no—
264#239—184no—
265#237—184no—
266D3×D4—184no—
267C2×D12—91no—
268C2×D12—92no—
269C2×D12—92no—
270C2×D12—91no—
271C2×D12—91no—
272C2×D12—92no—
273C2×D12—92no—
274C2×D12—181no—
275C2×D12—182no—
276C2×D12—184no—
277C2×D12C2×D911yescharacteristic
278C2×C4×D3—31no—
279C2×C4×D3—92no—
280C2×C4×D3—91no—
281C2×C4×D3—91no—
282C2×C4×D3—181no—
283C2×C4×D3C2×D912yescomplemented
284C2×Dic6—31no—
285C3×C4⋊C4—21no—
286C3×C4⋊C4—92no—
287C3×C4⋊C4C2×D911yescharacteristic, complemented
288C3×C4⋊C4D3211yescharacteristic
289C3×C22⋊C4—32no—
290C3×C22⋊C4—92no—
291C3×C42—31no—
292#217—91no—
293(C2×D3)⋊C4—32no—
294(C2×D3)⋊C4—91no—
295(C2×D3)⋊C4—92no—
296(C2×D3)⋊C4—92no—
297(C2×D3)⋊C4—91no—
298(C2×D3)⋊C4—92no—
299(C2×D3)⋊C4—92no—
300(C2×D3)⋊C4—92no—
301(C2×D3)⋊C4—92no—
302(C2×D3)⋊C4—91no—
303(C2×D3)⋊C4—182no—
304(C2×D3)⋊C4C2×D912yescomplemented
305C4⋊Dic3—32no—
306C4⋊Dic3—91no—
307Dic3⋊C4—91no—
308C4×Dic3C2×D911yescharacteristic, complemented
309C22×C32C23×D311yescharacteristic
310C2×C3⋊D3—61no—
311C2×C3⋊D3—61no—
312C2×C3⋊D3—62no—
313C2×C3⋊D3—62no—
314C2×C3⋊D3—61no—
315C2×C3⋊D3—62no—
316C2×C3×D3—22no—
317C2×C3×D3—22no—
318C2×C3×D3—32no—
319C2×C3×D3—61no—
320C2×C3×D3—61no—
321C2×C3×D3C2×D1214yes—
322D32—128no—
323C32×C4—22no—
324C32×C4C2×D1212yes—
325C3⋊Dic3—61no—
326C3×Dic3—21no—
327C3×Dic3—21no—
328C3×Dic3—62no—
329C3×Dic3—61no—
330C22×C9—21no—
331C22×C9—34no—
332C22×C9—62no—
333C22×C9—62no—
334C22×C9C23×D311yescharacteristic
335C2×D9—21no—
336C2×D9—21no—
337C2×D9—62no—
338C2×D9—61no—
339C2×D9—62no—
340C2×D9—61no—
341C2×D9—62no—
342C2×D9—61no—
343C2×D9—62no—
344C2×D9—121no—
345C2×D9—121no—
346C2×D9—122no—
347C2×D9—122no—
348C2×D9—122no—
349C2×D9—128no—
350C2×D9#23912yescomplemented
351C4×C9—22no—
352C4×C9—22no—
353C4×C9—42no—
354C4×C9—61no—
355C4×C9—61no—
356C4×C9D3×D412yes—
357Dic9—22no—
358Dic9—21no—
359Dic9—61no—
360Dic9—121no—
361C2×#41—271no—
362C22×D4—271no—
363#124—271no—
364#123—272no—
365C22⋊Q8—91no—
366#121—272no—
367#121—271no—
368#120—272no—
369C4×D4—31no—
370C4×D4—271no—
371C2×C22⋊C4—272no—
372C3×C9#54611yescharacteristic, complemented, Sylow
373C23×C3—32no—
374C23×C3—91no—
375C22×D3—31no—
376C22×D3—92no—
377C22×D3—91no—
378C22×D3—92no—
379C22×D3—91no—
380C22×D3—181no—
381C22×D3—182no—
382C22×D3—184no—
383C22×D3—184no—
384C22×D3—184no—
385C22×D3—184no—
386C22×D3—362no—
387C22×D3—362no—
388C22×D3C22×D912yes—
389C3×Q8—182no—
390C3×D4—34no—
391C2×C3×C4—21no—
392C2×C3×C4—22no—
393C2×C3×C4—31no—
394C2×C3×C4—31no—
395C2×C3×C4—91no—
396C2×C3×C4—92no—
397C2×C3×C4—122no—
398C2×C3×C4—182no—
399C2×C3×C4C22×D912yes—
400C2×C3×C4C22×D911yescharacteristic
401C2×C3×C4C2×D3212yes—
402C2×C3×C4C2×D3211yescharacteristic
403#67—182no—
404#67—184no—
405#67—184no—
406C2×Dic3—31no—
407C2×Dic3—32no—
408C2×Dic3—91no—
409C2×Dic3—91no—
410C2×Dic3—181no—
411C2×Dic3—184no—
412C2×Dic3C22×D911yescharacteristic
413C2×Dic3C22×D911yescharacteristic
414D12—182no—
415D12—184no—
416D12—184no—
417D12—182no—
418D12—182no—
419D12—362no—
420D12—362no—
421D12—362no—
422D12—364no—
423D12D3614yescomplemented
424C4×D3—42no—
425C4×D3—62no—
426C4×D3—184no—
427C4×D3—182no—
428C4×D3—182no—
429C4×D3—362no—
430Dic6—62no—
431C2×C32C22×D1211yescharacteristic
432C2×C32#100411yescharacteristic
433C2×C32D3×#4111yescharacteristic
434C3⋊D3—121no—
435C3⋊D3—122no—
436C3×D3—24no—
437C3×D3—62no—
438C2×C9—21no—
439C2×C9—21no—
440C2×C9—21no—
441C2×C9—64no—
442C2×C9C2×D3×D411yescharacteristic
443C2×C9C2×#23911yescharacteristic
444C2×C9#100911yescharacteristic
445D9—22no—
446D9—122no—
447D9—121no—
448D9—122no—
449D9—242no—
450C24—272no—
451#41—544no—
452C2×Q8—91no—
453C2×D4—31no—
454C2×D4—271no—
455C2×D4—272no—
456C2×D4—272no—
457C2×D4—271no—
458C2×D4—272no—
459C2×D4—544no—
460C22×C4—32no—
461C22×C4—91no—
462C22×C4—272no—
463C22×C4—271no—
464C4⋊C4—92no—
465C4⋊C4—271no—
466C4⋊C4D3×D911yescharacteristic, complemented
467C22⋊C4—32no—
468C22⋊C4—92no—
469C22⋊C4—272no—
470C22⋊C4—271no—
471C22⋊C4—272no—
472C22⋊C4—271no—
473C22⋊C4—272no—
474C42—31no—
475C22×C3—21no—
476C22×C3—34no—
477C22×C3—62no—
478C22×C3—62no—
479C22×C3—92no—
480C22×C3—181no—
481C22×C3—181no—
482C22×C3C23×D911yescharacteristic
483C22×C3C22×D3211yescharacteristic
484C2×D3—22no—
485C2×D3—22no—
486C2×D3—32no—
487C2×D3—61no—
488C2×D3—61no—
489C2×D3—182no—
490C2×D3—181no—
491C2×D3—182no—
492C2×D3—181no—
493C2×D3—182no—
494C2×D3—181no—
495C2×D3—182no—
496C2×D3—181no—
497C2×D3—182no—
498C2×D3—181no—
499C2×D3—182no—
500C2×D3—181no—
501C2×D3—182no—
502C2×D3—361no—
503C2×D3—361no—
504C2×D3—362no—
505C2×D3—362no—
506C2×D3—362no—
507C2×D3—368no—
508C2×D3—368no—
509C2×D3C2×D3614yes—
510C3×C4—22no—
511C3×C4—22no—
512C3×C4—22no—
513C3×C4—42no—
514C3×C4—61no—
515C3×C4—61no—
516C3×C4—182no—
517C3×C4—181no—
518C3×C4C2×D3612yes—
519C3×C4D3×D1212yes—
520Dic3—21no—
521Dic3—21no—
522Dic3—62no—
523Dic3—61no—
524Dic3—181no—
525Dic3—181no—
526Dic3—361no—
527C32#555411yescharacteristic
528C9—21no—
529C9#563011yescharacteristic, complemented
530C23—32no—
531C23—91no—
532C23—272no—
533C23—271no—
534C23—544no—
535C23—544no—
536C23—1082no—
537Q8—182no—
538D4—34no—
539D4—542no—
540D4—544no—
541D4—544no—
542D4—542no—
543D4—1082no—
544C2×C4—31no—
545C2×C4—31no—
546C2×C4—92no—
547C2×C4—91no—
548C2×C4—122no—
549C2×C4—182no—
550C2×C4—271no—
551C2×C4—544no—
552C2×C4—542no—
553C2×C4C2×D3×D912yes—
554C2×C4C2×D3×D911yescharacteristic
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—64no—
559C2×C3—182no—
560C2×C3C22×D3611yescharacteristic
561C2×C3#6371011yescharacteristic
562C2×C3#41×D911yescharacteristic
563C2×C3C2×D3×D1211yescharacteristic
564C2×C3#6431411yescharacteristic
565C2×C3D3×#23911yescharacteristic
566D3—24no—
567D3—62no—
568D3—362no—
569D3—362no—
570D3—361no—
571D3—362no—
572D3—361no—
573D3—722no—
574C22—34no—
575C22—62no—
576C22—62no—
577C22—92no—
578C22—181no—
579C22—181no—
580C22—541no—
581C22—542no—
582C22—541no—
583C22—542no—
584C22—541no—
585C22—542no—
586C22—1088no—
587C22C22×D3×D911yescharacteristic
588C4—22no—
589C4—61no—
590C4—61no—
591C4—181no—
592C4—182no—
593C4—541no—
594C4D3×D3612yes—
595C3—21no—
596C3#1058965611yescharacteristic, complemented
597C3#1059462911yescharacteristic
598C2—64no—
599C2—182no—
600C2—1082no—
601C2—1081no—
602C2C2×D3×D3611yescharacteristic
603C2#1172250011yescharacteristic
604C2D9×#23911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#546
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×D9×D12C2semidirect
2. C2×#88639C2semidirect
3. #11721751C2semidirect
4. C9×#893C2semidirect
5. C3×#63459C2semidirect
6. #11721051C2semidirect
7. D3×(C2×D9)⋊C4C2semidirect
8. #11721038C2semidirect
9. #11721020C2semidirect
10. C4×#6192C2semidirect
11. #11720952C2semidirect
16. C2×C4⋊(C3⋊D9)C22semidirect
17. C2×C4×C3⋊D9C22semidirect
19. C2×C4×D3×C9C22semidirect
21. C2×C3×Dic18C22semidirect
23. C2×#6195C22semidirect
24. C2×D3×Dic9C22semidirect
28. C3×C9×C4⋊C4C22semidirect
29. #88536C22semidirect
30. C9×(C2×D3)⋊C4C22semidirect
32. C4×C9×Dic3C22semidirect
33. C3×C4⋊Dic9C22semidirect
34. #88445C22semidirect
35. #88445C22semidirect
36. #88445C22semidirect
38. Dic3⋊Dic9C22semidirect
55. #63459D3semidirect
79. C9×D12D4semidirect
130. C2×Dic18C2×D3semidirect
132. C9×C4⋊C4C2×D3semidirect
143. C4⋊Dic9C2×D3semidirect
156. C2×C3×D9#41semidirect
181. #893D9semidirect
283. C2×C4×D3C2×D9semidirect
287. C3×C4⋊C4C2×D9semidirect
304. (C2×D3)⋊C4C2×D9semidirect
308. C4×Dic3C2×D9semidirect
350. C2×D9#239semidirect
372. C3×C9#546semidirect
423. D12D36semidirect
466. C4⋊C4D3×D9semidirect
529. C9#5630semidirect
596. C3#10589656semidirect

Maximal subgroups

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

subgroupindex
C2×D9×D122
C2×#886392
#117217512
C9×#8932
C3×#634592
#117210512
D3×(C2×D9)⋊C42
#117210382
#117210202
C4×#61922
#117209522
#105946293
#105896563