Finite Group Databasebeta

← #13148877

Subgroups — #13148877

Order 1728.

636 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721754C211yesmaximal, characteristic
2C9×#889C211yesmaximal, characteristic, complemented
3C3×#63449C211yesmaximal, characteristic, complemented
4#11720994C211yesmaximal, characteristic
5#11720985C211yesmaximal, characteristic
6C4×#6189C211yesmaximal, characteristic, complemented
7#11720935C211yesmaximal, characteristic, complemented
8#11720934C211yesmaximal, characteristic
9#11720934C211yesmaximal, characteristic
10#11720933C211yesmaximal, characteristic, complemented
11#11720928C211yesmaximal, characteristic, complemented
12#11720926C211yesmaximal, characteristic
13#11720915C211yesmaximal, characteristic, complemented
14#11720904C211yesmaximal, characteristic, complemented
15#11720899C211yesmaximal, characteristic
16#10594348—31nomaximal
17#10589798—31nomaximal
18C2×C3×Dic18C2211yescharacteristic
19C2×(C3⋊D9)⋊C4C2211yescharacteristic
20C2×#6189C2211yescharacteristic
21C3×C9×C4⋊C4C2211yescharacteristic
22#88536C2211yescharacteristic
23#88536C2211yescharacteristic
24#88536C2211yescharacteristic
25C4⋊(C3⋊Dic9)C2211yescharacteristic
26#88534C2211yescharacteristic
27C4×C3⋊Dic9C2211yescharacteristic
28C9×C4⋊Dic3C2211yescharacteristic
29C9×Dic3⋊C4C2211yescharacteristic
30C9×Dic3⋊C4C2211yescharacteristic
31C9×Dic3⋊C4C2211yescharacteristic
32C9×Dic3⋊C4C2211yescharacteristic
33C4×C9×Dic3C2211yescharacteristic
34C3×Dic9⋊C4C2211yescharacteristic
35C3×Dic9⋊C4C2211yescharacteristic
36C3×C4×Dic9C2211yescharacteristic
37C3×C4×Dic9C2211yescharacteristic
38C3×C4×Dic9C2211yescharacteristic
39#88445C2211yescharacteristic
40#88445C2211yescharacteristic
41#88445C2211yescharacteristic
42Dic3⋊Dic9C2211yescharacteristic
43Dic3⋊Dic9C2211yescharacteristic
44Dic3⋊Dic9C2211yescharacteristic
45#88442C2211yescharacteristic
46#88442C2211yescharacteristic
47#88442C2211yescharacteristic
48#88442C2211yescharacteristic
49Dic9⋊Dic3C2211yescharacteristic
50Dic9⋊Dic3C2211yescharacteristic
51Dic9⋊Dic3C2211yescharacteristic
52Dic3×Dic9C2211yescharacteristic
53#64107—31no—
54C3×#889D311yescharacteristic
55C3×#887—31no—
56C4×#627—31no—
57#63895—31no—
58#63888—31no—
59#63870—31no—
60#63869—31no—
61#63868—31no—
62#63868—31no—
63#63868—31no—
64#63867—31no—
65#63845—31no—
66#63839—31no—
67#63836—31no—
68C9×#125—31no—
69#63460—31no—
70#63460—31no—
71#63460—31no—
72#63460—61no—
73#63458—31no—
74#63458—31no—
75#63454—31no—
76#63452—31no—
77#63451—31no—
78#63451—31no—
79#63451—31no—
80#63450—31no—
81#63449D311yescharacteristic, complemented
82#63438—31no—
83C4×Dic18—31no—
84C22×C3⋊D9C2311yescharacteristic
85C2×C3×C4×C9C2311yescharacteristic
86C2×C3×C4×C9C2311yescharacteristic
87C2×C3×C4×C9C2311yescharacteristic
88C2×C3⋊Dic9C2311yescharacteristic
89C2×C3⋊Dic9C2311yescharacteristic
90C2×C3⋊Dic9C2311yescharacteristic
91C2×C9×Dic3C2311yescharacteristic
92C2×C9×Dic3C2311yescharacteristic
93C2×C9×Dic3C2311yescharacteristic
94C2×C9×Dic3C2311yescharacteristic
95C2×C3×Dic9C2311yescharacteristic
96C2×C3×Dic9C2311yescharacteristic
97C2×C3×Dic9C2311yescharacteristic
98C2×C3×Dic9C2311yescharacteristic
99C3×Dic18—22no—
100(C3⋊D9)⋊C4—22no—
101#6189—22no—
102#5696—91no—
103#5583—91no—
104C2×C3×Dic6—31no—
105C2×#627—31no—
106C2×#624—31no—
107C32×C4⋊C4C2×D311yescharacteristic
108#3752—31no—
109#3752—31no—
110#3752—31no—
111C4⋊(C3⋊Dic3)—31no—
112#3750—31no—
113C4×C3⋊Dic3—31no—
114C3×C4⋊Dic3C2×D311yescharacteristic
115C3×Dic3⋊C4—31no—
116C3×Dic3⋊C4—31no—
117C3×Dic3⋊C4C2×D311yescharacteristic
118C3×Dic3⋊C4C2×D311yescharacteristic
119C3×Dic3⋊C4C2×D311yescharacteristic
120C3×Dic3⋊C4C2×D311yescharacteristic
121C3×C4×Dic3—31no—
122C3×C4×Dic3—31no—
123C3×C4×Dic3—31no—
124C3×C4×Dic3C2×D311yescharacteristic
125#3724—31no—
126#3724—31no—
127#3724—31no—
128#3724—31no—
129Dic3⋊Dic3—31no—
130Dic3⋊Dic3—31no—
131Dic3⋊Dic3—31no—
132Dic3⋊Dic3—31no—
133Dic3⋊Dic3—31no—
134Dic3⋊Dic3—31no—
135#3722—31no—
136#3722—31no—
137#3722—31no—
138Dic32—31no—
139C2×C4×D9—31no—
140C2×Dic18—31no—
141C2×Dic18C2×D311yescharacteristic
142C9×C4⋊C4—21no—
143C9×C4⋊C4—31no—
144C9×C4⋊C4—31no—
145C9×C4⋊C4—31no—
146C9×C4⋊C4—31no—
147C9×C4⋊C4—31no—
148C9×C4⋊C4C2×D311yescharacteristic
149C42×C9—31no—
150(C2×D9)⋊C4—31no—
151(C2×D9)⋊C4—31no—
152(C2×D9)⋊C4—31no—
153(C2×D9)⋊C4—31no—
154(C2×D9)⋊C4—31no—
155(C2×D9)⋊C4—31no—
156(C2×D9)⋊C4—61no—
157(C2×D9)⋊C4—61no—
158(C2×D9)⋊C4—61no—
159C4⋊Dic9—31no—
160C4⋊Dic9—31no—
161C4⋊Dic9—31no—
162C4⋊Dic9—31no—
163C4⋊Dic9—61no—
164Dic9⋊C4—31no—
165Dic9⋊C4—31no—
166Dic9⋊C4—31no—
167Dic9⋊C4—31no—
168Dic9⋊C4—31no—
169Dic9⋊C4—31no—
170Dic9⋊C4—31no—
171Dic9⋊C4—31no—
172Dic9⋊C4—61no—
173Dic9⋊C4C2×D311yescharacteristic
174Dic9⋊C4C2×D311yescharacteristic
175C4×Dic9—31no—
176C4×Dic9—31no—
177C4×Dic9—61no—
178C4×Dic9C2×D311yescharacteristic
179C4×Dic9C2×D311yescharacteristic
180C4×Dic9C2×D311yescharacteristic
181C22×C3×C9C2411yescharacteristic
182C2×C3⋊D9—21no—
183C2×C3⋊D9—21no—
184C2×C3⋊D9—21no—
185C3×C4×C9—21no—
186C3×C4×C9—21no—
187C3×C4×C9—21no—
188C3⋊Dic9—21no—
189C3⋊Dic9—21no—
190C3⋊Dic9—21no—
191C9×Dic3—21no—
192C9×Dic3—21no—
193C9×Dic3—21no—
194C9×Dic3—21no—
195C3×Dic9—21no—
196C3×Dic9—21no—
197C3×Dic9—21no—
198C3×Dic9#4112yes—
199C3×#125—31no—
200C3×C4×Q8—91no—
201#946—91no—
202#944—91no—
203#898—91no—
204#898—91no—
205#898—91no—
206#898—91no—
207#898—91no—
208#898—181no—
209#897—91no—
210#896—91no—
211#896—91no—
212#892—91no—
213#890—91no—
214#890—91no—
215#889—91no—
216#889—91no—
217#889—91no—
218#889—91no—
219#889D911yescharacteristic, complemented
220#888—91no—
221#887—31no—
222#876—91no—
223#876—91no—
224#876—91no—
225#872—91no—
226#870—91no—
227#870—91no—
228C4×Dic6—91no—
229C4×Dic6—91no—
230C22×C3⋊D3—31no—
231C2×C32×C4C22×D311yescharacteristic
232C2×C32×C4C22×D311yescharacteristic
233C2×C32×C4C22×D311yescharacteristic
234C2×C3⋊Dic3—31no—
235C2×C3⋊Dic3—31no—
236C2×C3⋊Dic3—31no—
237C2×C3×Dic3—31no—
238C2×C3×Dic3—31no—
239C2×C3×Dic3—31no—
240C2×C3×Dic3—31no—
241C2×C3×Dic3C22×D311yescharacteristic
242C2×C3×Dic3C22×D311yescharacteristic
243C2×C3×Dic3C22×D311yescharacteristic
244C2×C3×Dic3C22×D311yescharacteristic
245C3×Dic6—62no—
246#627—62no—
247#624—62no—
248C22×D9—31no—
249C22×D9—61no—
250C2×C4×C9—21no—
251C2×C4×C9—21no—
252C2×C4×C9—21no—
253C2×C4×C9—31no—
254C2×C4×C9—31no—
255C2×C4×C9—31no—
256C2×C4×C9—31no—
257C2×C4×C9C22×D311yescharacteristic
258C2×C4×C9C22×D311yescharacteristic
259C2×C4×C9C22×D311yescharacteristic
260C2×Dic9—31no—
261C2×Dic9—31no—
262C2×Dic9—31no—
263C2×Dic9—61no—
264C2×Dic9—61no—
265C2×Dic9—61no—
266C2×Dic9C22×D311yescharacteristic
267C2×Dic9C22×D311yescharacteristic
268C2×Dic9C22×D311yescharacteristic
269C2×Dic9C22×D311yescharacteristic
270C4×D9—62no—
271Dic18—22no—
272Dic18—62no—
273#541—271noSylow
274C2×C3×C9C2×#4111yescharacteristic
275C2×C3×C92−1+411yescharacteristic
276C2×C3×C92−1+411yescharacteristic
277C3⋊D9—41no—
278C2×C3×Q8—91no—
279C2×C4×D3—91no—
280C2×C4×D3—91no—
281C2×Dic6—31no—
282C2×Dic6—91no—
283C2×Dic6—91no—
284C3×C4⋊C4—21no—
285C3×C4⋊C4—31no—
286C3×C4⋊C4—31no—
287C3×C4⋊C4—31no—
288C3×C4⋊C4—31no—
289C3×C4⋊C4—31no—
290C3×C4⋊C4—91no—
291C3×C4⋊C4—91no—
292C3×C4⋊C4C2×D911yescharacteristic
293C3×C4⋊C4D3211yescharacteristic
294C3×C42—31no—
295C3×C42—91no—
296C3×C42—91no—
297C3×C42—91no—
298(C2×D3)⋊C4—91no—
299(C2×D3)⋊C4—91no—
300(C2×D3)⋊C4—91no—
301(C2×D3)⋊C4—91no—
302(C2×D3)⋊C4—91no—
303(C2×D3)⋊C4—91no—
304(C2×D3)⋊C4—91no—
305(C2×D3)⋊C4—91no—
306(C2×D3)⋊C4—91no—
307(C2×D3)⋊C4—91no—
308(C2×D3)⋊C4—91no—
309(C2×D3)⋊C4—91no—
310(C2×D3)⋊C4—181no—
311(C2×D3)⋊C4—181no—
312(C2×D3)⋊C4—181no—
313C4⋊Dic3—91no—
314C4⋊Dic3—91no—
315C4⋊Dic3—91no—
316C4⋊Dic3—91no—
317C4⋊Dic3—91no—
318C4⋊Dic3—91no—
319C4⋊Dic3—91no—
320C4⋊Dic3—91no—
321C4⋊Dic3—181no—
322C4⋊Dic3C2×D911yescharacteristic
323Dic3⋊C4—31no—
324Dic3⋊C4—31no—
325Dic3⋊C4—91no—
326Dic3⋊C4—91no—
327Dic3⋊C4—91no—
328Dic3⋊C4—91no—
329Dic3⋊C4—91no—
330Dic3⋊C4—91no—
331Dic3⋊C4—91no—
332Dic3⋊C4—91no—
333Dic3⋊C4—91no—
334Dic3⋊C4—91no—
335Dic3⋊C4—91no—
336Dic3⋊C4—91no—
337Dic3⋊C4—91no—
338Dic3⋊C4—91no—
339Dic3⋊C4—91no—
340Dic3⋊C4—91no—
341Dic3⋊C4—181no—
342Dic3⋊C4C2×D911yescharacteristic
343Dic3⋊C4C2×D911yescharacteristic
344Dic3⋊C4C2×D911yescharacteristic
345Dic3⋊C4C2×D911yescharacteristic
346C4×Dic3—31no—
347C4×Dic3—31no—
348C4×Dic3—31no—
349C4×Dic3—91no—
350C4×Dic3—91no—
351C4×Dic3—91no—
352C4×Dic3—91no—
353C4×Dic3—181no—
354C4×Dic3C2×D911yescharacteristic
355C22×C32C23×D311yescharacteristic
356C2×C3⋊D3—61no—
357C2×C3⋊D3—61no—
358C2×C3⋊D3—61no—
359C32×C4—21no—
360C32×C4—21no—
361C32×C4—21no—
362C3⋊Dic3—61no—
363C3⋊Dic3—61no—
364C3⋊Dic3—61no—
365C3×Dic3—21no—
366C3×Dic3—21no—
367C3×Dic3—21no—
368C3×Dic3—21no—
369C3×Dic3—32no—
370C3×Dic3—61no—
371C3×Dic3—61no—
372C3×Dic3—61no—
373C22×C9—21no—
374C22×C9C23×D311yescharacteristic
375C2×D9—61no—
376C2×D9—61no—
377C2×D9—61no—
378C2×D9—121no—
379C2×D9—121no—
380C2×D9—121no—
381C4×C9—21no—
382C4×C9—21no—
383C4×C9—21no—
384C4×C9—41no—
385C4×C9—41no—
386C4×C9—41no—
387C4×C9—61no—
388C4×C9—61no—
389C4×C9—61no—
390C4×C9—61no—
391Dic9—21no—
392Dic9—21no—
393Dic9—21no—
394Dic9—61no—
395Dic9—61no—
396Dic9—61no—
397Dic9—121no—
398Dic9—121no—
399Dic9—121no—
400Dic9#23512yes—
401C4⋊Q8—271no—
402#126—271no—
403#126—271no—
404#126—271no—
405#126—271no—
406#125—31no—
407#125—271no—
408#125—271no—
409#125—271no—
410#125—271no—
411C22⋊Q8—271no—
412C22⋊Q8—271no—
413C4×Q8—91no—
414C4×Q8—271no—
415#117—271no—
416C3×C9#54111yescharacteristic, complemented, Sylow
417C22×D3—91no—
418C22×D3—91no—
419C22×D3—181no—
420C3×Q8—182no—
421C2×C3×C4—21no—
422C2×C3×C4—21no—
423C2×C3×C4—21no—
424C2×C3×C4—31no—
425C2×C3×C4—31no—
426C2×C3×C4—31no—
427C2×C3×C4—31no—
428C2×C3×C4—91no—
429C2×C3×C4—91no—
430C2×C3×C4—91no—
431C2×C3×C4—91no—
432C2×C3×C4C22×D911yescharacteristic
433C2×C3×C4C22×D911yescharacteristic
434C2×C3×C4C22×D911yescharacteristic
435C2×C3×C4C2×D3211yescharacteristic
436C2×C3×C4C2×D3211yescharacteristic
437C2×C3×C4C2×D3211yescharacteristic
438C2×Dic3—31no—
439C2×Dic3—31no—
440C2×Dic3—31no—
441C2×Dic3—31no—
442C2×Dic3—91no—
443C2×Dic3—91no—
444C2×Dic3—91no—
445C2×Dic3—91no—
446C2×Dic3—91no—
447C2×Dic3—91no—
448C2×Dic3—181no—
449C2×Dic3—181no—
450C2×Dic3—181no—
451C2×Dic3C22×D911yescharacteristic
452C2×Dic3C22×D911yescharacteristic
453C2×Dic3C22×D911yescharacteristic
454C2×Dic3C22×D911yescharacteristic
455C4×D3—182no—
456C4×D3—182no—
457Dic6—62no—
458Dic6—182no—
459Dic6—182no—
460C2×C32#100711yescharacteristic
461C2×C32D3×#4111yescharacteristic
462C2×C32#101011yescharacteristic
463C3⋊D3—121no—
464C2×C9—21no—
465C2×C9—21no—
466C2×C9—21no—
467C2×C9C2×#23511yescharacteristic
468C2×C9#100711yescharacteristic
469C2×C9#101011yescharacteristic
470D9—121no—
471D9—241no—
472C2×Q8—91no—
473C2×Q8—271no—
474C22×C4—271no—
475C4⋊C4—31no—
476C4⋊C4—31no—
477C4⋊C4—31no—
478C4⋊C4—31no—
479C4⋊C4—31no—
480C4⋊C4—91no—
481C4⋊C4—91no—
482C4⋊C4—271no—
483C4⋊C4—271no—
484C4⋊C4—271no—
485C4⋊C4—271no—
486C4⋊C4—271no—
487C4⋊C4—271no—
488C4⋊C4—271no—
489C4⋊C4—271no—
490C4⋊C4—271no—
491C4⋊C4—271no—
492C4⋊C4—271no—
493C4⋊C4—271no—
494C4⋊C4D3×D911yescharacteristic
495C22⋊C4—271no—
496C22⋊C4—271no—
497C22⋊C4—271no—
498C22⋊C4—271no—
499C22⋊C4—271no—
500C22⋊C4—271no—
501C42—31no—
502C42—91no—
503C42—91no—
504C42—91no—
505C42—271no—
506C42—271no—
507C22×C3—21no—
508C22×C3C23×D911yescharacteristic
509C22×C3C22×D3211yescharacteristic
510C2×D3—181no—
511C2×D3—181no—
512C2×D3—181no—
513C2×D3—181no—
514C2×D3—181no—
515C2×D3—181no—
516C2×D3—361no—
517C2×D3—361no—
518C2×D3—361no—
519C3×C4—21no—
520C3×C4—21no—
521C3×C4—21no—
522C3×C4—21no—
523C3×C4—21no—
524C3×C4—21no—
525C3×C4—41no—
526C3×C4—41no—
527C3×C4—41no—
528C3×C4—61no—
529C3×C4—61no—
530C3×C4—61no—
531C3×C4—61no—
532C3×C4—92no—
533C3×C4—181no—
534C3×C4—181no—
535C3×C4—181no—
536Dic3—21no—
537Dic3—21no—
538Dic3—21no—
539Dic3—21no—
540Dic3—32no—
541Dic3—61no—
542Dic3—61no—
543Dic3—61no—
544Dic3—181no—
545Dic3—181no—
546Dic3—181no—
547Dic3—181no—
548Dic3—181no—
549Dic3—181no—
550Dic3—361no—
551Dic3—361no—
552Dic3—361no—
553C32#569611yescharacteristic
554C9—21no—
555C9#558311yescharacteristic, complemented
556C23—271no—
557Q8—182no—
558Q8—542no—
559C2×C4—31no—
560C2×C4—31no—
561C2×C4—31no—
562C2×C4—31no—
563C2×C4—91no—
564C2×C4—91no—
565C2×C4—91no—
566C2×C4—91no—
567C2×C4—271no—
568C2×C4—271no—
569C2×C4—271no—
570C2×C4—542no—
571C2×C4C2×D3×D911yescharacteristic
572C2×C4C2×D3×D911yescharacteristic
573C2×C4C2×D3×D911yescharacteristic
574C2×C3—21no—
575C2×C3—21no—
576C2×C3—21no—
577C2×C3#6371311yescharacteristic
578C2×C3#41×D911yescharacteristic
579C2×C3#6371611yescharacteristic
580C2×C3D3×#23511yescharacteristic
581C2×C3#6430911yescharacteristic
582C2×C3#6431611yescharacteristic
583D3—361no—
584D3—361no—
585D3—721no—
586C22—541no—
587C22—541no—
588C22—541no—
589C22C22×D3×D911yescharacteristic
590C4—21no—
591C4—21no—
592C4—21no—
593C4—61no—
594C4—61no—
595C4—61no—
596C4—61no—
597C4—92no—
598C4—181no—
599C4—181no—
600C4—181no—
601C4—541no—
602C4—541no—
603C4—541no—
604C3—21no—
605C3#1058979811yescharacteristic, complemented
606C3#1059434811yescharacteristic
607C2—1081no—
608C2D9×#23511yescharacteristic
609C2#1172249411yescharacteristic
610C2#1172250911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC22×C3⋊D9C23
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#541
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C9×#889C2semidirect
3. C3×#63449C2semidirect
6. C4×#6189C2semidirect
7. #11720935C2semidirect
10. #11720933C2semidirect
11. #11720928C2semidirect
13. #11720915C2semidirect
14. #11720904C2semidirect
81. #63449D3semidirect
219. #889D9semidirect
416. C3×C9#541semidirect
555. C9#5583semidirect
605. C3#10589798semidirect

Maximal subgroups

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

subgroupindex
#117217542
C9×#8892
C3×#634492
#117209942
#117209852
C4×#61892
#117209352
#117209342
#117209332
#117209282
#117209262
#117209152
#117209042
#117208992
#105943483
#105897983