Finite Group Databasebeta

← #13149325

Subgroups — #13149325

Order 1728.

1096 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D9×D12C211yesmaximal, characteristic, complemented
2C2×#88642C211yesmaximal, characteristic, complemented
3#11721749C211yesmaximal, characteristic, complemented
4C9×#895C211yesmaximal, characteristic, complemented
5C3×#63459C211yesmaximal, characteristic, complemented
6#11721050C212yesmaximal, complemented
7#11721048C212yesmaximal, complemented
8#11721045C212yesmaximal, complemented
9C4×#6194C211yesmaximal, characteristic, complemented
10#11720967C211yesmaximal, characteristic, complemented
11#11720958C212yesmaximal, complemented
12#10594633—31nomaximal
13#10589714—31nomaximal
14C22×D3×D9C2212yes—
15C2×C4⋊(C3⋊D9)C2211yescharacteristic
16C2×C4×C3⋊D9C2212yescomplemented
17C2×C9×D12C2211yescharacteristic
18C2×C9×D12C2212yescomplemented
19C2×C4×D3×C9C2211yescharacteristic, complemented
20C2×C3×C4×D9C2211yescharacteristic
21C2×C3×Dic18C2211yescharacteristic, complemented
22C2×#6195C2212yes—
23C2×#6195C2212yescomplemented
24C2×#6194C2211yescharacteristic
25C2×(C3⋊D9)⋊C4C2212yescomplemented
26D9×D12—24no—
27#88642—24no—
28C3×C9×C4⋊C4C2211yescharacteristic, complemented
29#88536C2212yescomplemented
30C4×C3⋊Dic9C2211yescharacteristic, complemented
31C9×(C2×D3)⋊C4C2212yescomplemented
32C3×(C2×D9)⋊C4C2212yes—
33C3×C4⋊Dic9C2212yescomplemented
34(C2×D3)⋊Dic9C2212yescomplemented
35(C2×D3)⋊Dic9C2212yescomplemented
36(C2×D3)⋊Dic9C2211yescharacteristic, complemented
37(C2×D9)⋊Dic3C2211yescharacteristic
38#88442C2211yescharacteristic, complemented
39C2×D3×D12—31no—
40C2×#3799—31no—
41#64102—31no—
42C3×#897—31no—
43C3×#895D311yescharacteristic
44#63922—32no—
45#63921—32no—
46#63916—32no—
47C4×#625—31no—
48#63863—31no—
49#63854—32no—
50C2×D4×D9—31no—
51C2×#3697—31no—
52C9×#121—31no—
53#63500—32no—
54#63499—32no—
55#63498—31no—
56C4×#611—31no—
57#63459D311yescharacteristic, complemented
58#63455—31no—
59#63455—61no—
60#63445—32no—
61C22⋊C4×D9—32no—
62C22×C3⋊D9C2312yes—
63C22×D3×C9C2312yes—
64C22×D3×C9C2311yescharacteristic
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
72C2×C3⋊Dic9C2311yescharacteristic
73C4⋊(C3⋊D9)D414yescomplemented
74C4×C3⋊D9—42no—
75C2×C9×Dic3C2311yescharacteristic
76C2×C3×Dic9C2312yes—
77C2×C3×Dic9C2311yescharacteristic
78C9×D12—22no—
79C9×D12—42no—
80C4×D3×C9—22no—
81C3×C4×D9—22no—
82C3×Dic18—22no—
83#6195—24no—
84#6195—24no—
85#6194—22no—
86(C3⋊D9)⋊C4—24no—
87#5630—91no—
88#5612—91no—
89C22×D32—32no—
90C2×C4⋊(C3⋊D3)—31no—
91C2×C4×C3⋊D3—32no—
92C2×C3×D12C2×D311yescharacteristic
93C2×C3×D12C2×D312yes—
94C2×C3×C4×D3—31no—
95C2×C3×C4×D3C2×D311yescharacteristic
96C2×C3×Dic6—31no—
97C2×#626—32no—
98C2×#626—32no—
99C2×#625—31no—
100C2×#624—32no—
101D3×D12—64no—
102#3799—64no—
103C23×D9—32no—
104C32×C4⋊C4C2×D311yescharacteristic
105#3752—32no—
106C4×C3⋊Dic3—31no—
107C3×(C2×D3)⋊C4—32no—
108C3×(C2×D3)⋊C4C2×D312yes—
109C3×C4⋊Dic3—32no—
110#3724—31no—
111(C2×D3)⋊Dic3—32no—
112(C2×D3)⋊Dic3—32no—
113(C2×D3)⋊Dic3—31no—
114(C2×D3)⋊Dic3—31no—
115C2×D4×C9—31no—
116C2×D4×C9—32no—
117C22×C4×C9—31no—
118C2×#611—31no—
119C2×#611—32no—
120C2×#611—32no—
121D4×D9—64no—
122#3697—64no—
123C2×D36—31no—
124C2×D36—61no—
125C2×C4×D9—32no—
126C2×C4×D9—32no—
127C2×C4×D9—62no—
128C2×C4×D9C2×D311yescharacteristic
129C2×Dic18C2×D311yescharacteristic, complemented
130C9×C4⋊C4—21no—
131C9×C4⋊C4C2×D311yescharacteristic, complemented
132C9×C22⋊C4—32no—
133#3676—31no—
134#3676—32no—
135#3676—32no—
136(C2×D9)⋊C4—31no—
137(C2×D9)⋊C4—32no—
138(C2×D9)⋊C4—62no—
139(C2×D9)⋊C4C2×D312yes—
140C4⋊Dic9C2×D312yescomplemented
141Dic9⋊C4—31no—
142C4×Dic9—31no—
143C4×Dic9—61no—
144C22×C3×C9C2411yescharacteristic
145C2×C3⋊D9—22no—
146C2×C3⋊D9—22no—
147C2×C3⋊D9C2×D414yes—
148C2×D3×C9—21no—
149C2×D3×C9—22no—
150C2×D3×C9—21no—
151C2×D3×C9—22no—
152C2×D3×C9—21no—
153C2×D3×C9—22no—
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—
161C3⋊Dic9—21no—
162C9×Dic3—21no—
163C3×Dic9—22no—
164C3×Dic9—21no—
165C2×#239—91no—
166C2×D3×D4—91no—
167C2×#235—91no—
168C22×D12—91no—
169C3×C22⋊Q8—91no—
170C3×#121—31no—
171#947—91no—
172#938—92no—
173#937—92no—
174#936—91no—
175C4×#67—91no—
176C4×#67—91no—
177#897—31no—
178#895—92no—
179#895D911yescharacteristic, complemented
180#894—92no—
181#893—91no—
182#893—91no—
183#893—181no—
184#883—92no—
185#882—92no—
186D3×C22⋊C4—92no—
187D3×C22⋊C4—92no—
188C22×C3⋊D3—32no—
189C22×C3×D3—31no—
190C22×C3×D3C22×D312yes—
191C22×C3×D3C22×D311yescharacteristic
192C2×D32—64no—
193C2×D32—64no—
194C2×D32—122no—
195C2×C32×C4C22×D312yes—
196C2×C32×C4C22×D311yescharacteristic
197C2×C3⋊Dic3—31no—
198C2×C3⋊Dic3—31no—
199C4⋊(C3⋊D3)—34no—
200C4×C3⋊D3—122no—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—31no—
203C2×C3×Dic3C22×D311yescharacteristic
204C3×D12—22no—
205C3×D12—42no—
206C3×C4×D3—22no—
207C3×C4×D3—62no—
208C3×Dic6—62no—
209#626—64no—
210#626—64no—
211#625—62no—
212#624—64no—
213C23×C9—32no—
214C23×C9—31no—
215C22×D9—32no—
216C22×D9—62no—
217C22×D9—64no—
218C22×D9—64no—
219C22×D9—122no—
220C22×D9C22×D311yescharacteristic
221D4×C9—62no—
222D4×C9—122no—
223C2×C4×C9—21no—
224C2×C4×C9—22no—
225C2×C4×C9—31no—
226C2×C4×C9—62no—
227C2×C4×C9C22×D312yes—
228C2×C4×C9C22×D311yescharacteristic
229#611—64no—
230#611—64no—
231#611—62no—
232C2×Dic9—31no—
233C2×Dic9—31no—
234C2×Dic9—61no—
235C2×Dic9—61no—
236C2×Dic9C22×D311yescharacteristic
237C2×Dic9C22×D312yes—
238D36—34no—
239D36—64no—
240C4×D9—22no—
241C4×D9—64no—
242C4×D9—122no—
243C4×D9—124no—
244Dic18—22no—
245#546—271noSylow
246C2×C3×C9C22×D411yescharacteristic
247C2×C3×C9C2×#4111yescharacteristic
248C2×C3×C9H(5,𝔽2)11yescharacteristic
249C3⋊D9—24no—
250D3×C9—42no—
251D3×C9—41no—
252C3×D9—22no—
253C23×D3—92no—
254C23×D3—92no—
255C2×C3×Q8—91no—
256C2×C3×D4—31no—
257C2×C3×D4—32no—
258C22×C3×C4—31no—
259C22×C3×C4—91no—
260C2×#67—92no—
261C2×#67—92no—
262C2×#67—91no—
263C2×#67—91no—
264#239—184no—
265D3×D4—184no—
266#235—184no—
267C2×D12—91no—
268C2×D12—91no—
269C2×D12—92no—
270C2×D12—92no—
271C2×D12—181no—
272C2×D12—184no—
273C2×D12C2×D911yescharacteristic
274C2×D12C2×D912yescomplemented
275C2×C4×D3—31no—
276C2×C4×D3—92no—
277C2×C4×D3—92no—
278C2×C4×D3—92no—
279C2×C4×D3—92no—
280C2×C4×D3—182no—
281C2×C4×D3C2×D911yescharacteristic, complemented
282C2×Dic6—31no—
283C3×C4⋊C4—21no—
284C3×C4⋊C4—92no—
285C3×C4⋊C4C2×D911yescharacteristic, complemented
286C3×C4⋊C4D3211yescharacteristic
287C3×C22⋊C4—32no—
288C3×C22⋊C4—92no—
289#217—92no—
290#217—92no—
291#217—91no—
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—182no—
301(C2×D3)⋊C4C2×D912yescomplemented
302C4⋊Dic3—32no—
303Dic3⋊C4—91no—
304Dic3⋊C4—91no—
305C4×Dic3—91no—
306C4×Dic3—91no—
307C4×Dic3—181no—
308C22×C32C23×D311yescharacteristic
309C2×C3⋊D3—34no—
310C2×C3⋊D3—62no—
311C2×C3⋊D3—62no—
312C2×C3×D3—21no—
313C2×C3×D3—22no—
314C2×C3×D3—21no—
315C2×C3×D3—22no—
316C2×C3×D3—21no—
317C2×C3×D3—22no—
318C2×C3×D3—32no—
319C2×C3×D3—61no—
320C2×C3×D3—61no—
321D32—128no—
322C32×C4—22no—
323C32×C4D3×D412yes—
324C3⋊Dic3—61no—
325C3⋊Dic3—61no—
326C3×Dic3—21no—
327C3×Dic3—62no—
328C3×Dic3—61no—
329C22×C9—21no—
330C22×C9—61no—
331C22×C9—62no—
332C22×C9—61no—
333C22×C9—62no—
334C22×C9—61no—
335C22×C9—62no—
336C22×C9C23×D311yescharacteristic
337C2×D9—21no—
338C2×D9—21no—
339C2×D9—34no—
340C2×D9—64no—
341C2×D9—62no—
342C2×D9—62no—
343C2×D9—122no—
344C2×D9—122no—
345C2×D9—128no—
346C2×D9#23912yescomplemented
347C4×C9—22no—
348C4×C9—22no—
349C4×C9—42no—
350C4×C9—61no—
351C4×C9D3×D412yes—
352Dic9—22no—
353Dic9—21no—
354Dic9—62no—
355Dic9—61no—
356Dic9—61no—
357Dic9—121no—
358C2×#41—271no—
359C22×D4—271no—
360#124—271no—
361#123—272no—
362C22⋊Q8—91no—
363#121—31no—
364#121—272no—
365#120—272no—
366C4×D4—271no—
367C4×D4—271no—
368C2×C22⋊C4—272no—
369C3×C9#54611yescharacteristic, complemented, Sylow
370C23×C3—32no—
371C23×C3—31no—
372C23×C3—91no—
373C22×D3—31no—
374C22×D3—92no—
375C22×D3—92no—
376C22×D3—182no—
377C22×D3—184no—
378C22×D3—184no—
379C22×D3—184no—
380C22×D3—184no—
381C22×D3—362no—
382C22×D3—362no—
383C22×D3C22×D912yes—
384C22×D3C22×D911yescharacteristic
385C3×Q8—182no—
386C3×D4—62no—
387C3×D4—122no—
388C2×C3×C4—21no—
389C2×C3×C4—22no—
390C2×C3×C4—31no—
391C2×C3×C4—62no—
392C2×C3×C4—91no—
393C2×C3×C4—92no—
394C2×C3×C4—182no—
395C2×C3×C4C22×D912yes—
396C2×C3×C4C22×D911yescharacteristic
397C2×C3×C4C2×D3212yes—
398C2×C3×C4C2×D3211yescharacteristic
399#67—184no—
400#67—184no—
401#67—182no—
402#67—182no—
403C2×Dic3—31no—
404C2×Dic3—32no—
405C2×Dic3—91no—
406C2×Dic3—91no—
407C2×Dic3—91no—
408C2×Dic3—91no—
409C2×Dic3—181no—
410C2×Dic3—181no—
411C2×Dic3C22×D911yescharacteristic
412D12—22no—
413D12—42no—
414D12—94no—
415D12—94no—
416D12—184no—
417D12—184no—
418D12—184no—
419C4×D3—22no—
420C4×D3—62no—
421C4×D3—184no—
422C4×D3—184no—
423C4×D3—362no—
424C4×D3—362no—
425C4×D3—364no—
426Dic6—62no—
427C2×C32C2×D3×D411yescharacteristic
428C2×C32#100411yescharacteristic
429C2×C32D3×#4111yescharacteristic
430C3⋊D3—64no—
431C3×D3—42no—
432C3×D3—41no—
433C3×D3—62no—
434C2×C9—21no—
435C2×C9—21no—
436C2×C9—21no—
437C2×C9—122no—
438C2×C9—121no—
439C2×C9C2×D3×D411yescharacteristic
440C2×C9C2×#23911yescharacteristic
441C2×C9#100911yescharacteristic
442D9—22no—
443D9—64no—
444D9—124no—
445C24—272no—
446#41—544no—
447C2×Q8—91no—
448C2×D4—31no—
449C2×D4—32no—
450C2×D4—272no—
451C2×D4—272no—
452C2×D4—271no—
453C2×D4—271no—
454C2×D4—544no—
455C22×C4—31no—
456C22×C4—91no—
457C22×C4—272no—
458C22×C4—272no—
459C4⋊C4—92no—
460C4⋊C4—271no—
461C4⋊C4D3×D911yescharacteristic, complemented
462C22⋊C4—32no—
463C22⋊C4—92no—
464C22⋊C4—271no—
465C22⋊C4—272no—
466C22⋊C4—271no—
467C22⋊C4—272no—
468C22⋊C4—272no—
469C42—271no—
470C22×C3—21no—
471C22×C3—61no—
472C22×C3—62no—
473C22×C3—61no—
474C22×C3—62no—
475C22×C3—61no—
476C22×C3—62no—
477C22×C3—92no—
478C22×C3—181no—
479C22×C3—181no—
480C22×C3C23×D911yescharacteristic
481C22×C3C22×D3211yescharacteristic
482C2×D3—21no—
483C2×D3—22no—
484C2×D3—21no—
485C2×D3—22no—
486C2×D3—21no—
487C2×D3—22no—
488C2×D3—32no—
489C2×D3—61no—
490C2×D3—61no—
491C2×D3—94no—
492C2×D3—94no—
493C2×D3—184no—
494C2×D3—182no—
495C2×D3—182no—
496C2×D3—182no—
497C2×D3—182no—
498C2×D3—362no—
499C2×D3—362no—
500C2×D3—368no—
501C2×D3—368no—
502C3×C4—22no—
503C3×C4—22no—
504C3×C4—22no—
505C3×C4—42no—
506C3×C4—61no—
507C3×C4—182no—
508C3×C4—181no—
509C3×C4D4×D912yes—
510C3×C4#380212yes—
511Dic3—21no—
512Dic3—62no—
513Dic3—61no—
514Dic3—182no—
515Dic3—181no—
516Dic3—181no—
517Dic3—181no—
518Dic3—181no—
519Dic3—361no—
520C32#561211yescharacteristic
521C9—21no—
522C9#563011yescharacteristic, complemented
523C23—32no—
524C23—31no—
525C23—91no—
526C23—272no—
527C23—544no—
528C23—544no—
529C23—1082no—
530Q8—182no—
531D4—62no—
532D4—122no—
533D4—274no—
534D4—544no—
535D4—544no—
536D4—542no—
537C2×C4—31no—
538C2×C4—62no—
539C2×C4—92no—
540C2×C4—91no—
541C2×C4—182no—
542C2×C4—271no—
543C2×C4—271no—
544C2×C4—544no—
545C2×C4—1082no—
546C2×C4C2×D3×D912yes—
547C2×C4C2×D3×D911yescharacteristic
548C2×C3—21no—
549C2×C3—21no—
550C2×C3—21no—
551C2×C3—122no—
552C2×C3—121no—
553C2×C3—182no—
554C2×C3C2×D4×D911yescharacteristic
555C2×C3#6371011yescharacteristic
556C2×C3#41×D911yescharacteristic
557C2×C3C2×#380211yescharacteristic
558C2×C3#6431411yescharacteristic
559C2×C3D3×#23911yescharacteristic
560D3—42no—
561D3—41no—
562D3—62no—
563D3—184no—
564D3—184no—
565D3—364no—
566C22—61no—
567C22—62no—
568C22—61no—
569C22—62no—
570C22—61no—
571C22—62no—
572C22—92no—
573C22—181no—
574C22—181no—
575C22—274no—
576C22—542no—
577C22—542no—
578C22—1088no—
579C22C22×D3×D911yescharacteristic
580C4—22no—
581C4—61no—
582C4—181no—
583C4—182no—
584C4—541no—
585C4—541no—
586C4#8864612yes—
587C3—21no—
588C3#1058971411yescharacteristic, complemented
589C3#1059463311yescharacteristic
590C2—122no—
591C2—121no—
592C2—182no—
593C2—544no—
594C2C2×#8864611yescharacteristic
595C2#1172250011yescharacteristic
596C2D9×#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×#88642C2semidirect
3. #11721749C2semidirect
4. C9×#895C2semidirect
5. C3×#63459C2semidirect
6. #11721050C2semidirect
7. #11721048C2semidirect
8. #11721045C2semidirect
9. C4×#6194C2semidirect
10. #11720967C2semidirect
11. #11720958C2semidirect
16. C2×C4×C3⋊D9C22semidirect
18. C2×C9×D12C22semidirect
19. C2×C4×D3×C9C22semidirect
21. C2×C3×Dic18C22semidirect
23. C2×#6195C22semidirect
25. C2×(C3⋊D9)⋊C4C22semidirect
28. C3×C9×C4⋊C4C22semidirect
29. #88536C22semidirect
30. C4×C3⋊Dic9C22semidirect
31. C9×(C2×D3)⋊C4C22semidirect
33. C3×C4⋊Dic9C22semidirect
34. (C2×D3)⋊Dic9C22semidirect
35. (C2×D3)⋊Dic9C22semidirect
36. (C2×D3)⋊Dic9C22semidirect
38. #88442C22semidirect
57. #63459D3semidirect
73. C4⋊(C3⋊D9)D4semidirect
129. C2×Dic18C2×D3semidirect
131. C9×C4⋊C4C2×D3semidirect
140. C4⋊Dic9C2×D3semidirect
156. C2×C3×D9#41semidirect
179. #895D9semidirect
274. C2×D12C2×D9semidirect
281. C2×C4×D3C2×D9semidirect
285. C3×C4⋊C4C2×D9semidirect
301. (C2×D3)⋊C4C2×D9semidirect
346. C2×D9#239semidirect
369. C3×C9#546semidirect
461. C4⋊C4D3×D9semidirect
522. C9#5630semidirect
588. C3#10589714semidirect

Maximal subgroups

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

subgroupindex
C2×D9×D122
C2×#886422
#117217492
C9×#8952
C3×#634592
#117210502
#117210482
#117210452
C4×#61942
#117209672
#117209582
#105946333
#105897143