Finite Group Databasebeta

← #13151288

Subgroups — #13151288

Order 1728.

1154 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88742C211yesmaximal, characteristic, complemented
2C2×C4×D3×D9C211yesmaximal, characteristic, complemented
3C9×#938C211yesmaximal, characteristic, complemented
4C3×#63500C211yesmaximal, characteristic, complemented
5#11721292C212yesmaximal, complemented
6#67⋊Dic9C212yesmaximal, complemented
7#11721275C212yesmaximal, complemented
8#611⋊Dic3C212yesmaximal, complemented
9#11720961C211yesmaximal, characteristic, complemented
10#11720945C211yesmaximal, characteristic, complemented
11#11720922C211yesmaximal, characteristic, complemented
12#10595306—31nomaximal
13#10589708—31nomaximal
14C22×D3×D9C2211yescharacteristic
15C2×C3×D4×C9C2211yescharacteristic, complemented
16C2×#6233C2212yes—
17C22×C3⋊Dic9C2212yescomplemented
18#88742—24no—
19C2×C4×C3⋊D9C2211yescharacteristic
20C2×#6226C2211yescharacteristic, complemented
21C2×C9×#67C2212yes—
22C2×C3×#611C2212yes—
23C2×C4×D3×C9C2211yescharacteristic
24C2×C3×C4×D9C2211yescharacteristic
25C2×D3×Dic9C2211yescharacteristic
26C2×(C3⋊D9)⋊C4C2211yescharacteristic
27C2×Dic3×D9C2211yescharacteristic
28C4×D3×D9—24no—
29C9×#217C2212yescomplemented
30C3×#3676C2212yescomplemented
31C9×C4⋊Dic3C2211yescharacteristic, complemented
32C3×C4⋊Dic9C2211yescharacteristic, complemented
33(C2×D3)⋊Dic9C2212yes—
34#88445C2212yes—
35(C2×D9)⋊Dic3C2212yes—
36Dic3⋊Dic9C2212yescomplemented
37Dic9⋊Dic3C2212yescomplemented
38Dic3×Dic9C2212yescomplemented
39C2×#3830—31no—
40C2×C4×D32—31no—
41C3×#938—31no—
42C3×#938D311yescharacteristic
43#63983—32no—
44#63975—32no—
45#67⋊Dic3—32no—
46#67⋊Dic3—32no—
47#63857—31no—
48#63857—31no—
49#63849—31no—
50C2×#3699—31no—
51C2×#3699—61no—
52C22×C4×D9—31no—
53C9×#121—31no—
54#63500D311yescharacteristic, complemented
55#63499—32no—
56D4×Dic9—32no—
57#63486—31no—
58#63459—31no—
59#63454—31no—
60#63445—32no—
61#611⋊C4—32no—
62C23×C3×C9C2312yes—
63C22×C3⋊D9C2311yescharacteristic
64C22×D3×C9C2311yescharacteristic
65C22×C3×D9C2311yescharacteristic
66C2×D3×D9—22no—
67C2×D3×D9—22no—
68C2×D3×D9—22no—
69C3×D4×C9—22no—
70C2×C3×C4×C9C2311yescharacteristic
71#6233—24no—
72C2×C3⋊Dic9—24no—
73C2×C3⋊Dic9C2312yes—
74C2×C3⋊Dic9C2311yescharacteristic
75C4×C3⋊D9D414yescomplemented
76#6226—22no—
77C9×#67—24no—
78C3×#611—24no—
79C2×C9×Dic3C2311yescharacteristic
80C2×C9×Dic3C2312yes—
81C2×C3×Dic9C2311yescharacteristic
82C2×C3×Dic9C2312yes—
83C4×D3×C9—22no—
84C3×C4×D9—22no—
85D3×Dic9—22no—
86(C3⋊D9)⋊C4—22no—
87Dic3×D9—22no—
88#5606—91no—
89#5606—91no—
90C22×D32—31no—
91C2×C32×D4C2×D311yescharacteristic
92C2×#638—32no—
93C22×C3⋊Dic3—32no—
94#3830—64no—
95C2×C4×C3⋊D3—31no—
96C2×#634—31no—
97C2×C3×#67—32no—
98C2×C3×#67C2×D312yes—
99C2×C3×C4×D3—31no—
100C2×C3×C4×D3C2×D311yescharacteristic
101C2×#624—31no—
102C2×D3×Dic3—31no—
103C2×D3×Dic3—31no—
104C4×D32—64no—
105C23×D9—31no—
106C3×#217—32no—
107C3×#217C2×D312yes—
108C3×C4⋊Dic3—31no—
109C3×C4⋊Dic3C2×D311yescharacteristic
110Dic3⋊Dic3—32no—
111Dic3⋊Dic3—32no—
112#3722—32no—
113(C2×D3)⋊Dic3—32no—
114(C2×D3)⋊Dic3—32no—
115Dic32—32no—
116C2×D4×C9—21no—
117C2×D4×C9—32no—
118C2×D4×C9C2×D311yescharacteristic, complemented
119C22×C4×C9—31no—
120C2×#611—32no—
121C2×#611—62no—
122C2×#611C2×D312yes—
123C22×Dic9—32no—
124C22×Dic9—31no—
125C22×Dic9—62no—
126#3699—64no—
127#3699—68no—
128C2×C4×D9—31no—
129C2×C4×D9—31no—
130C2×C4×D9—31no—
131C2×C4×D9—64no—
132C2×C4×D9—61no—
133C2×C4×D9C2×D311yescharacteristic
134C2×Dic18—31no—
135C2×Dic18—61no—
136C9×C4⋊C4—31no—
137C9×C22⋊C4—32no—
138#3676—32no—
139#3676C2×D312yescomplemented
140(C2×D9)⋊C4—32no—
141(C2×D9)⋊C4—32no—
142C4⋊Dic9—32no—
143C4⋊Dic9C2×D311yescharacteristic, complemented
144Dic9⋊C4—32no—
145C4×Dic9—32no—
146C22×C3×C9—22no—
147C22×C3×C9—22no—
148C22×C3×C9—22no—
149C22×C3×C9C2411yescharacteristic
150C2×C3⋊D9—21no—
151C2×C3⋊D9—21no—
152C2×C3⋊D9C2×D412yes—
153C2×D3×C9—21no—
154C2×D3×C9—21no—
155C2×D3×C9#4112yescomplemented
156C2×C3×D9—21no—
157C2×C3×D9—21no—
158C2×C3×D9#4112yescomplemented
159D3×D9—28no—
160C3×C4×C9C2×D412yes—
161C3⋊Dic9—22no—
162C3⋊Dic9C2×D412yes—
163C9×Dic3—21no—
164C9×Dic3—22no—
165C3×Dic9—21no—
166C3×Dic9—22no—
167C2×#237—91no—
168C2×#237—91no—
169C2×#237—181no—
170C22×C4×D3—91no—
171C22×C4×D3—91no—
172C3×#121—31no—
173C3×#121—91no—
174#938—31no—
175#938D911yescharacteristic, complemented
176#937—92no—
177#937—92no—
178D4×Dic3—92no—
179D4×Dic3—92no—
180#924—91no—
181#924—91no—
182#897—91no—
183#897—91no—
184#892—91no—
185#892—91no—
186#883—92no—
187#883—92no—
188#67⋊C4—92no—
189#67⋊C4—92no—
190C23×C32C22×D312yes—
191C22×C3⋊D3—31no—
192C22×C3×D3—31no—
193C22×C3×D3C22×D311yescharacteristic
194C2×D32—62no—
195C2×D32—62no—
196C2×D32—62no—
197C32×D4—22no—
198C2×C32×C4C22×D311yescharacteristic
199#638—64no—
200C2×C3⋊Dic3—31no—
201C2×C3⋊Dic3—32no—
202C2×C3⋊Dic3—64no—
203C4×C3⋊D3—34no—
204#634—62no—
205C3×#67—24no—
206C3×#67—64no—
207C2×C3×Dic3—32no—
208C2×C3×Dic3—31no—
209C2×C3×Dic3C22×D312yes—
210C2×C3×Dic3C22×D311yescharacteristic
211C3×C4×D3—22no—
212C3×C4×D3—62no—
213#624—62no—
214D3×Dic3—62no—
215D3×Dic3—62no—
216C23×C9—22no—
217C23×C9—31no—
218C23×C9C22×D312yes—
219C22×D9—31no—
220C22×D9—61no—
221C22×D9—62no—
222C22×D9—62no—
223C22×D9—62no—
224C22×D9C22×D311yescharacteristic
225D4×C9—24no—
226D4×C9—22no—
227D4×C9—64no—
228C2×C4×C9—21no—
229C2×C4×C9—32no—
230C2×C4×C9—31no—
231C2×C4×C9—62no—
232C2×C4×C9C22×D311yescharacteristic
233#611—24no—
234#611—64no—
235#611—68no—
236C2×Dic9—31no—
237C2×Dic9—32no—
238C2×Dic9—62no—
239C2×Dic9—61no—
240C2×Dic9—62no—
241C2×Dic9—64no—
242C2×Dic9—68no—
243C2×Dic9C22×D312yes—
244C2×Dic9C22×D311yescharacteristic
245C4×D9—22no—
246C4×D9—34no—
247C4×D9—64no—
248C4×D9—62no—
249C4×D9—62no—
250Dic18—64no—
251Dic18—62no—
252#525—271noSylow
253C2×C3×C9—42no—
254C2×C3×C9C22×D411yescharacteristic
255C2×C3×C9C2×#4111yescharacteristic
256C2×C3×C9C2×#4111yescharacteristic
257C3⋊D9—22no—
258D3×C9—22no—
259C3×D9—22no—
260C23×D3—91no—
261C23×D3—91no—
262C2×C3×D4—21no—
263C2×C3×D4—32no—
264C2×C3×D4—92no—
265C2×C3×D4C2×D911yescharacteristic, complemented
266C2×C3×D4D3211yescharacteristic
267C22×C3×C4—31no—
268C22×C3×C4—91no—
269C2×#67—32no—
270C2×#67—92no—
271C2×#67—92no—
272C2×#67—182no—
273C2×#67C2×D912yes—
274C22×Dic3—91no—
275C22×Dic3—92no—
276C22×Dic3—92no—
277C22×Dic3—91no—
278C22×Dic3—182no—
279#237—184no—
280#237—184no—
281#237—188no—
282C2×C4×D3—31no—
283C2×C4×D3—91no—
284C2×C4×D3—91no—
285C2×C4×D3—91no—
286C2×C4×D3—91no—
287C2×C4×D3—91no—
288C2×C4×D3—91no—
289C2×C4×D3—181no—
290C2×C4×D3—184no—
291C2×C4×D3—184no—
292C2×C4×D3C2×D911yescharacteristic
293C2×Dic6—91no—
294C2×Dic6—91no—
295C2×Dic6—181no—
296C3×C4⋊C4—31no—
297C3×C4⋊C4—91no—
298C3×C22⋊C4—32no—
299C3×C22⋊C4—92no—
300#217—32no—
301#217—92no—
302#217—92no—
303#217C2×D912yescomplemented
304(C2×D3)⋊C4—92no—
305(C2×D3)⋊C4—92no—
306(C2×D3)⋊C4—92no—
307(C2×D3)⋊C4—92no—
308C4⋊Dic3—31no—
309C4⋊Dic3—92no—
310C4⋊Dic3—92no—
311C4⋊Dic3C2×D911yescharacteristic, complemented
312Dic3⋊C4—92no—
313Dic3⋊C4—92no—
314C4×Dic3—92no—
315C4×Dic3—92no—
316C22×C32—22no—
317C22×C32—22no—
318C22×C32—22no—
319C22×C32C23×D311yescharacteristic
320C2×C3⋊D3—32no—
321C2×C3⋊D3—61no—
322C2×C3⋊D3—61no—
323C2×C3×D3—21no—
324C2×C3×D3—21no—
325C2×C3×D3—32no—
326C2×C3×D3—61no—
327C2×C3×D3—61no—
328C2×C3×D3#23712yes—
329D32—68no—
330C32×C4D3×D412yes—
331C3⋊Dic3—32no—
332C3⋊Dic3—62no—
333C3×Dic3—21no—
334C3×Dic3—22no—
335C3×Dic3—61no—
336C3×Dic3—62no—
337C22×C9—24no—
338C22×C9—22no—
339C22×C9—22no—
340C22×C9—22no—
341C22×C9—21no—
342C22×C9—32no—
343C22×C9—42no—
344C22×C9—42no—
345C22×C9—61no—
346C22×C9—61no—
347C22×C9C23×D311yescharacteristic
348C2×D9—21no—
349C2×D9—21no—
350C2×D9—32no—
351C2×D9—62no—
352C2×D9—68no—
353C2×D9—61no—
354C2×D9—61no—
355C2×D9—121no—
356C2×D9—121no—
357C2×D9#23712yescomplemented
358C4×C9—22no—
359C4×C9—61no—
360C4×C9—62no—
361C4×C9D3×D412yes—
362Dic9—21no—
363Dic9—22no—
364Dic9—32no—
365Dic9—62no—
366Dic9—64no—
367Dic9—62no—
368C2×#41—271no—
369C23×C4—271no—
370#123—272no—
371C22⋊Q8—271no—
372C22⋊Q8—271no—
373#121—31no—
374#121—91no—
375#120—272no—
376C4×D4—272no—
377C4×D4—272no—
378#117—271no—
379C3×C9#52511yescharacteristic, complemented, Sylow
380C23×C3—22no—
381C23×C3—31no—
382C23×C3—91no—
383C23×C3C22×D912yes—
384C23×C3C2×D3212yes—
385C22×D3—31no—
386C22×D3—91no—
387C22×D3—91no—
388C22×D3—181no—
389C22×D3—182no—
390C22×D3—182no—
391C22×D3—182no—
392C22×D3—182no—
393C22×D3—182no—
394C22×D3—182no—
395C22×D3C22×D911yescharacteristic
396C3×D4—24no—
397C3×D4—22no—
398C3×D4—22no—
399C3×D4—64no—
400C3×D4—184no—
401C2×C3×C4—21no—
402C2×C3×C4—32no—
403C2×C3×C4—31no—
404C2×C3×C4—62no—
405C2×C3×C4—92no—
406C2×C3×C4—91no—
407C2×C3×C4—182no—
408C2×C3×C4C22×D911yescharacteristic
409C2×C3×C4C2×D3211yescharacteristic
410#67—24no—
411#67—64no—
412#67—184no—
413#67—184no—
414#67—188no—
415C2×Dic3—32no—
416C2×Dic3—31no—
417C2×Dic3—91no—
418C2×Dic3—91no—
419C2×Dic3—92no—
420C2×Dic3—92no—
421C2×Dic3—182no—
422C2×Dic3—181no—
423C2×Dic3—182no—
424C2×Dic3—182no—
425C2×Dic3—184no—
426C2×Dic3—184no—
427C2×Dic3—188no—
428C2×Dic3C22×D912yes—
429C2×Dic3C22×D911yescharacteristic
430C4×D3—22no—
431C4×D3—62no—
432C4×D3—94no—
433C4×D3—94no—
434C4×D3—184no—
435C4×D3—182no—
436C4×D3—182no—
437C4×D3—182no—
438C4×D3—182no—
439Dic6—184no—
440Dic6—182no—
441Dic6—182no—
442C2×C32—42no—
443C2×C32C2×D3×D411yescharacteristic
444C2×C32C2×#23711yescharacteristic
445C2×C32D3×#4111yescharacteristic
446C3⋊D3—62no—
447C3×D3—22no—
448C3×D3—62no—
449C2×C9—21no—
450C2×C9—21no—
451C2×C9—21no—
452C2×C9—44no—
453C2×C9—42no—
454C2×C9—62no—
455C2×C9C2×D3×D411yescharacteristic
456C2×C9C2×#23711yescharacteristic
457C2×C9D3×#4111yescharacteristic
458D9—22no—
459D9—62no—
460D9—122no—
461C24—271no—
462#41—544no—
463C2×Q8—271no—
464C2×D4—32no—
465C2×D4—92no—
466C2×D4—272no—
467C2×D4D3×D911yescharacteristic, complemented
468C22×C4—31no—
469C22×C4—91no—
470C22×C4—271no—
471C22×C4—271no—
472C22×C4—271no—
473C22×C4—271no—
474C22×C4—272no—
475C22×C4—544no—
476C4⋊C4—31no—
477C4⋊C4—91no—
478C4⋊C4—272no—
479C4⋊C4—272no—
480C22⋊C4—32no—
481C22⋊C4—92no—
482C22⋊C4—272no—
483C22⋊C4—272no—
484C22⋊C4—272no—
485C42—272no—
486C22×C3—24no—
487C22×C3—22no—
488C22×C3—22no—
489C22×C3—22no—
490C22×C3—22no—
491C22×C3—22no—
492C22×C3—22no—
493C22×C3—21no—
494C22×C3—32no—
495C22×C3—42no—
496C22×C3—42no—
497C22×C3—61no—
498C22×C3—61no—
499C22×C3—92no—
500C22×C3—181no—
501C22×C3—181no—
502C22×C3C23×D911yescharacteristic
503C22×C3C22×D3211yescharacteristic
504C2×D3—21no—
505C2×D3—21no—
506C2×D3—32no—
507C2×D3—61no—
508C2×D3—61no—
509C2×D3—92no—
510C2×D3—92no—
511C2×D3—182no—
512C2×D3—188no—
513C2×D3—188no—
514C2×D3—181no—
515C2×D3—181no—
516C2×D3—181no—
517C2×D3—181no—
518C2×D3—361no—
519C2×D3—361no—
520C2×D3#369912yescomplemented
521C3×C4—22no—
522C3×C4—62no—
523C3×C4—61no—
524C3×C4—181no—
525C3×C4—182no—
526C3×C4D4×D912yes—
527C3×C4#381112yes—
528Dic3—21no—
529Dic3—22no—
530Dic3—62no—
531Dic3—61no—
532Dic3—92no—
533Dic3—92no—
534Dic3—182no—
535Dic3—184no—
536Dic3—182no—
537Dic3—182no—
538C32#560611yescharacteristic
539C9—21no—
540C9#560611yescharacteristic, complemented
541C23—31no—
542C23—91no—
543C23—271no—
544C23—542no—
545C23—542no—
546C23—542no—
547C23C2×D3×D912yes—
548Q8—542no—
549D4—22no—
550D4—64no—
551D4—184no—
552D4—544no—
553C2×C4—32no—
554C2×C4—31no—
555C2×C4—62no—
556C2×C4—92no—
557C2×C4—91no—
558C2×C4—182no—
559C2×C4—271no—
560C2×C4—274no—
561C2×C4—272no—
562C2×C4—542no—
563C2×C4—542no—
564C2×C4—542no—
565C2×C4—544no—
566C2×C4C2×D3×D911yescharacteristic
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—44no—
571C2×C3—42no—
572C2×C3—42no—
573C2×C3—62no—
574C2×C3—182no—
575C2×C3C2×D4×D911yescharacteristic
576C2×C3C2×#369911yescharacteristic
577C2×C3#41×D911yescharacteristic
578C2×C3D3×#23711yescharacteristic
579C2×C3D3×#23711yescharacteristic
580C2×C3C2×#381111yescharacteristic
581D3—22no—
582D3—62no—
583D3—182no—
584D3—182no—
585D3—362no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—32no—
590C22—61no—
591C22—61no—
592C22—92no—
593C22—181no—
594C22—181no—
595C22—272no—
596C22—548no—
597C22—541no—
598C22—541no—
599C22C22×D3×D911yescharacteristic
600C4—62no—
601C4—61no—
602C4—182no—
603C4—181no—
604C4—272no—
605C4—542no—
606C4#8866812yes—
607C3—21no—
608C3#1058970811yescharacteristic, complemented
609C3#1059530611yescharacteristic
610C2—42no—
611C2—62no—
612C2—182no—
613C2—542no—
614C2D3×#369911yescharacteristic
615C2D9×#23711yescharacteristic
616C2C2×#8866811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
hypercenterC2×D4D3×D9
hypocenterC3×C9#525
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#88742C2semidirect
2. C2×C4×D3×D9C2semidirect
3. C9×#938C2semidirect
4. C3×#63500C2semidirect
5. #11721292C2semidirect
6. #67⋊Dic9C2semidirect
7. #11721275C2semidirect
8. #611⋊Dic3C2semidirect
9. #11720961C2semidirect
10. #11720945C2semidirect
11. #11720922C2semidirect
15. C2×C3×D4×C9C22semidirect
17. C22×C3⋊Dic9C22semidirect
20. C2×#6226C22semidirect
29. C9×#217C22semidirect
30. C3×#3676C22semidirect
31. C9×C4⋊Dic3C22semidirect
32. C3×C4⋊Dic9C22semidirect
36. Dic3⋊Dic9C22semidirect
37. Dic9⋊Dic3C22semidirect
38. Dic3×Dic9C22semidirect
54. #63500D3semidirect
75. C4×C3⋊D9D4semidirect
118. C2×D4×C9C2×D3semidirect
139. #3676C2×D3semidirect
143. C4⋊Dic9C2×D3semidirect
155. C2×D3×C9#41semidirect
158. C2×C3×D9#41semidirect
175. #938D9semidirect
265. C2×C3×D4C2×D9semidirect
303. #217C2×D9semidirect
311. C4⋊Dic3C2×D9semidirect
357. C2×D9#237semidirect
379. C3×C9#525semidirect
467. C2×D4D3×D9semidirect
520. C2×D3#3699semidirect
540. C9#5606semidirect
608. C3#10589708semidirect

Maximal subgroups

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

subgroupindex
C2×#887422
C2×C4×D3×D92
C9×#9382
C3×#635002
#117212922
#67⋊Dic92
#117212752
#611⋊Dic32
#117209612
#117209452
#117209222
#105953063
#105897083