Finite Group Databasebeta

← #13150226

Subgroups — #13150226

Order 1728.

1202 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C3⋊D9C211yesmaximal, characteristic
2C2×(C3⋊D9)⋊Q8C211yesmaximal, characteristic, complemented
3C9×#924C211yesmaximal, characteristic, complemented
4C3×#63486C211yesmaximal, characteristic, complemented
5#11721274C212yesmaximal
6#11721267C212yesmaximal, complemented
7#11721000C211yesmaximal, characteristic, complemented
8#11720980C212yesmaximal, complemented
9#11720936C211yesmaximal, characteristic, complemented
10#11720929C211yesmaximal, characteristic, complemented
11#11720923C212yesmaximal, complemented
12#10594925—31nomaximal
13D9×C22⋊Q8—31nomaximal
14C23×C3⋊D9C2211yescharacteristic
15C22×C3×C4×C9C2211yescharacteristic
16C22×C3⋊Dic9C2211yescharacteristic
17C2×C4×C3⋊D9—24no—
18C2×C4×C3⋊D9C2211yescharacteristic
19C2×C4×C3⋊D9C2211yescharacteristic
20C2×C4×C3⋊D9C2211yescharacteristic
21C2×C4×C3⋊D9C2211yescharacteristic
22C2×C9×Dic6C2211yescharacteristic, complemented
23C2×C3×Dic18C2211yescharacteristic, complemented
24C2×(C3⋊D9)⋊C4C2212yes—
25C2×(C3⋊D9)⋊C4C2212yes—
26C2×#6189C2212yescomplemented
27(C3⋊D9)⋊Q8—24no—
28C9×#217C2212yes—
29C3×#3676C2212yes—
30C9×C4⋊Dic3C2211yescharacteristic, complemented
31C9×Dic3⋊C4C2212yescomplemented
32C3×C4⋊Dic9C2211yescharacteristic, complemented
33C3×Dic9⋊C4C2212yescomplemented
34#88445C2212yes—
35#88445C2212yes—
36Dic3⋊Dic9C2212yescomplemented
37#88442C2212yescomplemented
38Dic9⋊Dic3C2212yescomplemented
39C22×C4×C3⋊D3—31no—
40C2×#3797—31no—
41C3×#924—31no—
42C3×#924D311yescharacteristic
43#63982—32no—
44#63974—32no—
45#63900—31no—
46#63883—32no—
47#63871—31no—
48#63871—31no—
49#63865—32no—
50C2×Q8×D9—31no—
51C22×C4×D9—31no—
52C22×C4×D9—61no—
53C9×C22⋊Q8—31no—
54#63508—31no—
55#63486D311yescharacteristic, complemented
56#63459—31no—
57#63458—32no—
58C4⋊C4×D9—31no—
59C4⋊C4×D9—32no—
60C22⋊C4×D9—32no—
61#63440—32no—
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9—22no—
64C22×C3⋊D9—22no—
65C22×C3⋊D9Q814yescomplemented
66C22×C3⋊D9C2311yescharacteristic
67C22×C3⋊D9C2311yescharacteristic
68C2×C3×C4×C9—22no—
69C2×C3×C4×C9C2311yescharacteristic
70C2×C3×C4×C9C2311yescharacteristic
71C2×C3⋊Dic9—22no—
72C2×C3⋊Dic9C2311yescharacteristic
73C2×C3⋊Dic9C2311yescharacteristic
74C4×C3⋊D9—22no—
75C4×C3⋊D9—22no—
76C4×C3⋊D9—22no—
77C4×C3⋊D9D414yes—
78C2×C9×Dic3C2312yes—
79C2×C9×Dic3C2312yes—
80C2×C3×Dic9C2312yes—
81C2×C3×Dic9C2312yes—
82C9×Dic6—22no—
83C3×Dic18—22no—
84(C3⋊D9)⋊C4—24no—
85(C3⋊D9)⋊C4—24no—
86#6189—24no—
87D3×C22⋊Q8—91no—
88D3×C22⋊Q8—91no—
89C23×C3⋊D3—31no—
90C22×C32×C4C2×D311yescharacteristic
91C22×C3⋊Dic3—31no—
92C2×C4×C3⋊D3—31no—
93C2×C4×C3⋊D3—31no—
94C2×C4×C3⋊D3—31no—
95C2×C4×C3⋊D3—31no—
96C2×C4×C3⋊D3—64no—
97C2×C3×Dic6—31no—
98C2×C3×Dic6C2×D311yescharacteristic
99C2×#627—32no—
100C2×#624—32no—
101C2×#624—32no—
102#3797—64no—
103C23×D9—31no—
104C23×D9—61no—
105C3×#217—32no—
106C3×#217C2×D312yes—
107C3×C4⋊Dic3—31no—
108C3×C4⋊Dic3C2×D311yescharacteristic
109C3×Dic3⋊C4—32no—
110C3×Dic3⋊C4C2×D312yes—
111#3724—32no—
112Dic3⋊Dic3—32no—
113Dic3⋊Dic3—32no—
114#3722—32no—
115#3722—32no—
116C2×Q8×C9—31no—
117C22×C4×C9—21no—
118C22×C4×C9C2×D311yescharacteristic
119C22×Dic9—31no—
120C22×Dic9—61no—
121Q8×D9—64no—
122C2×C4×D9—31no—
123C2×C4×D9—31no—
124C2×C4×D9—31no—
125C2×C4×D9—31no—
126C2×C4×D9—32no—
127C2×C4×D9—32no—
128C2×C4×D9—64no—
129C2×C4×D9—61no—
130C2×C4×D9—61no—
131C2×C4×D9—61no—
132C2×C4×D9—61no—
133C2×C4×D9—68no—
134C2×Dic18—32no—
135C2×Dic18C2×D311yescharacteristic, complemented
136C9×C4⋊C4—32no—
137C9×C4⋊C4—31no—
138C9×C22⋊C4—32no—
139#3676C2×D312yes—
140(C2×D9)⋊C4—32no—
141(C2×D9)⋊C4—32no—
142C4⋊Dic9—32no—
143C4⋊Dic9C2×D311yescharacteristic, complemented
144Dic9⋊C4—32no—
145Dic9⋊C4—32no—
146Dic9⋊C4C2×D312yescomplemented
147C22×C3×C9—21no—
148C22×C3×C9—21no—
149C22×C3×C9C2×Q812yescomplemented
150C22×C3×C9C2411yescharacteristic
151C2×C3⋊D9—21no—
152C2×C3⋊D9—21no—
153C2×C3⋊D9—28no—
154C2×C3⋊D9C2×D412yes—
155C2×C3⋊D9C2×Q812yes—
156C2×C3⋊D9C2×Q812yescomplemented
157C2×C3⋊D9#4112yes—
158C3×C4×C9—21no—
159C3×C4×C9C2×D412yes—
160C3⋊Dic9—21no—
161C3⋊Dic9C2×D412yes—
162C9×Dic3—22no—
163C9×Dic3—22no—
164C3×Dic9—22no—
165C3×Dic9—22no—
166C2×D3×Q8—91no—
167C2×D3×Q8—91no—
168C22×C4×D3—91no—
169C22×C4×D3—91no—
170C22×C4×D3—181no—
171C3×C22⋊Q8—31no—
172C3×C22⋊Q8—91no—
173#946—91no—
174#946—91no—
175#924—31no—
176#924D911yescharacteristic, complemented
177#897—91no—
178#897—91no—
179#896—92no—
180#896—92no—
181D3×C4⋊C4—92no—
182D3×C4⋊C4—91no—
183D3×C4⋊C4—91no—
184D3×C4⋊C4—92no—
185D3×C22⋊C4—92no—
186D3×C22⋊C4—92no—
187#878—92no—
188#878—92no—
189C23×C32C22×D311yescharacteristic
190C22×C3⋊D3—31no—
191C22×C3⋊D3—34no—
192C22×C3⋊D3—31no—
193C22×C3⋊D3—62no—
194C22×C3⋊D3—62no—
195C2×C32×C4—22no—
196C2×C32×C4C22×D311yescharacteristic
197C2×C32×C4C22×D311yescharacteristic
198C2×C3⋊Dic3—31no—
199C2×C3⋊Dic3—31no—
200C2×C3⋊Dic3—62no—
201C4×C3⋊D3—34no—
202C4×C3⋊D3—62no—
203C4×C3⋊D3—62no—
204C4×C3⋊D3—62no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3—32no—
207C2×C3×Dic3C22×D312yes—
208C2×C3×Dic3C22×D312yes—
209C3×Dic6—22no—
210C3×Dic6—62no—
211#627—64no—
212#624—64no—
213#624—64no—
214C23×C9—21no—
215C23×C9C22×D311yescharacteristic
216C22×D9—31no—
217C22×D9—34no—
218C22×D9—31no—
219C22×D9—61no—
220C22×D9—61no—
221C22×D9—64no—
222C22×D9—62no—
223C22×D9—62no—
224C22×D9—64no—
225C22×D9—64no—
226Q8×C9—62no—
227C2×C4×C9—24no—
228C2×C4×C9—22no—
229C2×C4×C9—21no—
230C2×C4×C9—21no—
231C2×C4×C9—32no—
232C2×C4×C9—32no—
233C2×C4×C9C22×D311yescharacteristic
234C2×C4×C9C22×D311yescharacteristic
235C2×Dic9—31no—
236C2×Dic9—31no—
237C2×Dic9—61no—
238C2×Dic9—61no—
239C2×Dic9—62no—
240C2×Dic9—64no—
241C2×Dic9C22×D312yes—
242C2×Dic9C22×D312yes—
243C4×D9—34no—
244C4×D9—64no—
245C4×D9—64no—
246C4×D9—64no—
247C4×D9—62no—
248C4×D9—62no—
249C4×D9—64no—
250C4×D9—64no—
251C4×D9—64no—
252C4×D9—62no—
253Dic18—22no—
254Dic18—64no—
255C2×C22⋊Q8—271noSylow
256C2×C3×C9—22no—
257C2×C3×C9C22×D411yescharacteristic
258C2×C3×C9C22×Q811yescharacteristic
259C2×C3×C9C2×#4111yescharacteristic
260C3⋊D9—22no—
261C3⋊D9C22⋊Q814yescomplemented
262C23×D3—91no—
263C23×D3—91no—
264C23×D3—181no—
265C2×C3×Q8—31no—
266C2×C3×Q8—91no—
267C22×C3×C4—21no—
268C22×C3×C4C2×D911yescharacteristic
269C22×C3×C4D3211yescharacteristic
270C22×Dic3—91no—
271C22×Dic3—91no—
272C22×Dic3—181no—
273D3×Q8—184no—
274D3×Q8—184no—
275C2×C4×D3—91no—
276C2×C4×D3—91no—
277C2×C4×D3—92no—
278C2×C4×D3—92no—
279C2×C4×D3—91no—
280C2×C4×D3—91no—
281C2×C4×D3—91no—
282C2×C4×D3—91no—
283C2×C4×D3—91no—
284C2×C4×D3—91no—
285C2×C4×D3—92no—
286C2×C4×D3—92no—
287C2×C4×D3—181no—
288C2×C4×D3—181no—
289C2×C4×D3—181no—
290C2×C4×D3—181no—
291C2×C4×D3—184no—
292C2×C4×D3—184no—
293C2×C4×D3—188no—
294C2×Dic6—31no—
295C2×Dic6—92no—
296C2×Dic6—92no—
297C2×Dic6C2×D911yescharacteristic, complemented
298C3×C4⋊C4—32no—
299C3×C4⋊C4—31no—
300C3×C4⋊C4—92no—
301C3×C4⋊C4—91no—
302C3×C22⋊C4—32no—
303C3×C22⋊C4—92no—
304#217—32no—
305#217C2×D912yes—
306(C2×D3)⋊C4—92no—
307(C2×D3)⋊C4—92no—
308(C2×D3)⋊C4—92no—
309(C2×D3)⋊C4—92no—
310C4⋊Dic3—31no—
311C4⋊Dic3—92no—
312C4⋊Dic3—92no—
313C4⋊Dic3C2×D911yescharacteristic, complemented
314Dic3⋊C4—32no—
315Dic3⋊C4—92no—
316Dic3⋊C4—92no—
317Dic3⋊C4—92no—
318Dic3⋊C4—92no—
319Dic3⋊C4C2×D912yescomplemented
320C22×C32—21no—
321C22×C32—21no—
322C22×C32D3×Q812yes—
323C22×C32C23×D311yescharacteristic
324C2×C3⋊D3—32no—
325C2×C3⋊D3—32no—
326C2×C3⋊D3—32no—
327C2×C3⋊D3—32no—
328C2×C3⋊D3—68no—
329C2×C3⋊D3—61no—
330C2×C3⋊D3—61no—
331C32×C4—21no—
332C32×C4D3×D412yes—
333C3⋊Dic3—32no—
334C3⋊Dic3—61no—
335C3×Dic3—22no—
336C3×Dic3—22no—
337C3×Dic3—62no—
338C3×Dic3—62no—
339C22×C9—22no—
340C22×C9—22no—
341C22×C9—21no—
342C22×C9—21no—
343C22×C9—21no—
344C22×C9—22no—
345C22×C9D3×Q812yescomplemented
346C22×C9C23×D311yescharacteristic
347C2×D9—32no—
348C2×D9—32no—
349C2×D9—32no—
350C2×D9—32no—
351C2×D9—62no—
352C2×D9—62no—
353C2×D9—62no—
354C2×D9—62no—
355C2×D9—68no—
356C2×D9—62no—
357C2×D9—616no—
358C2×D9—62no—
359C2×D9—61no—
360C2×D9—61no—
361C4×C9—22no—
362C4×C9—21no—
363C4×C9—22no—
364C4×C9—62no—
365C4×C9—62no—
366C4×C9D3×D412yes—
367Dic9—22no—
368Dic9—22no—
369Dic9—32no—
370Dic9—62no—
371Dic9—62no—
372Dic9—61no—
373C22×Q8—271no—
374C23×C4—271no—
375C22⋊Q8—31no—
376C22⋊Q8—91no—
377C22⋊Q8—272no—
378C22⋊Q8—272no—
379C22⋊Q8—271no—
380C22⋊Q8—271no—
381C2×C4⋊C4—272no—
382C2×C4⋊C4—271no—
383C2×C22⋊C4—272no—
384C3×C9C2×C22⋊Q811yescharacteristic, complemented, Sylow
385C23×C3—21no—
386C23×C3C22×D911yescharacteristic
387C23×C3C2×D3211yescharacteristic
388C22×D3—91no—
389C22×D3—91no—
390C22×D3—91no—
391C22×D3—94no—
392C22×D3—91no—
393C22×D3—94no—
394C22×D3—181no—
395C22×D3—181no—
396C22×D3—184no—
397C22×D3—182no—
398C22×D3—182no—
399C22×D3—182no—
400C22×D3—182no—
401C22×D3—184no—
402C22×D3—184no—
403C3×Q8—62no—
404C3×Q8—182no—
405C2×C3×C4—24no—
406C2×C3×C4—22no—
407C2×C3×C4—22no—
408C2×C3×C4—21no—
409C2×C3×C4—21no—
410C2×C3×C4—32no—
411C2×C3×C4—32no—
412C2×C3×C4—92no—
413C2×C3×C4—92no—
414C2×C3×C4C22×D911yescharacteristic
415C2×C3×C4C22×D911yescharacteristic
416C2×C3×C4C2×D3211yescharacteristic
417C2×C3×C4C2×D3211yescharacteristic
418C2×Dic3—32no—
419C2×Dic3—32no—
420C2×Dic3—91no—
421C2×Dic3—91no—
422C2×Dic3—91no—
423C2×Dic3—91no—
424C2×Dic3—181no—
425C2×Dic3—181no—
426C2×Dic3—182no—
427C2×Dic3—182no—
428C2×Dic3—184no—
429C2×Dic3C22×D912yes—
430C2×Dic3C22×D912yes—
431C4×D3—94no—
432C4×D3—94no—
433C4×D3—184no—
434C4×D3—184no—
435C4×D3—184no—
436C4×D3—184no—
437C4×D3—184no—
438C4×D3—182no—
439C4×D3—182no—
440C4×D3—182no—
441C4×D3—182no—
442C4×D3—184no—
443C4×D3—184no—
444C4×D3—184no—
445C4×D3—182no—
446C4×D3—182no—
447Dic6—22no—
448Dic6—62no—
449Dic6—184no—
450Dic6—184no—
451C2×C32—22no—
452C2×C32C2×D3×D411yescharacteristic
453C2×C32C2×D3×Q811yescharacteristic
454C2×C32D3×#4111yescharacteristic
455C3⋊D3—34no—
456C3⋊D3—62no—
457C2×C9—24no—
458C2×C9—22no—
459C2×C9—21no—
460C2×C9—21no—
461C2×C9—21no—
462C2×C9C2×D3×D411yescharacteristic
463C2×C9C2×D3×Q811yescharacteristic
464C2×C9D3×#4111yescharacteristic
465D9—34no—
466D9—64no—
467D9—64no—
468D9—62no—
469C24—271no—
470C2×Q8—31no—
471C2×Q8—91no—
472C2×Q8—272no—
473C2×Q8—544no—
474C22×C4—272no—
475C22×C4—272no—
476C22×C4—271no—
477C22×C4—271no—
478C22×C4—271no—
479C22×C4—271no—
480C22×C4—271no—
481C22×C4—544no—
482C22×C4D3×D911yescharacteristic
483C4⋊C4—32no—
484C4⋊C4—31no—
485C4⋊C4—92no—
486C4⋊C4—91no—
487C4⋊C4—272no—
488C4⋊C4—272no—
489C4⋊C4—272no—
490C22⋊C4—32no—
491C22⋊C4—92no—
492C22⋊C4—272no—
493C22⋊C4—272no—
494C22×C3—22no—
495C22×C3—22no—
496C22×C3—21no—
497C22×C3—21no—
498C22×C3—21no—
499C22×C3—21no—
500C22×C3—22no—
501C22×C3—21no—
502C22×C3Q8×D912yescomplemented
503C22×C3C23×D911yescharacteristic
504C22×C3#379712yes—
505C22×C3C22×D3211yescharacteristic
506C2×D3—92no—
507C2×D3—92no—
508C2×D3—92no—
509C2×D3—92no—
510C2×D3—92no—
511C2×D3—92no—
512C2×D3—92no—
513C2×D3—92no—
514C2×D3—182no—
515C2×D3—182no—
516C2×D3—182no—
517C2×D3—182no—
518C2×D3—188no—
519C2×D3—188no—
520C2×D3—182no—
521C2×D3—1816no—
522C2×D3—182no—
523C2×D3—181no—
524C2×D3—181no—
525C2×D3—181no—
526C2×D3—181no—
527C3×C4—22no—
528C3×C4—21no—
529C3×C4—21no—
530C3×C4—22no—
531C3×C4—62no—
532C3×C4—62no—
533C3×C4—182no—
534C3×C4—182no—
535C3×C4D4×D912yes—
536C3×C4#381112yes—
537Dic3—22no—
538Dic3—22no—
539Dic3—62no—
540Dic3—62no—
541Dic3—92no—
542Dic3—92no—
543Dic3—182no—
544Dic3—182no—
545Dic3—181no—
546Dic3—181no—
547C32D3×C22⋊Q811yescharacteristic
548C9—21no—
549C9D3×C22⋊Q811yescharacteristic, complemented
550C23—271no—
551C23—274no—
552C23—271no—
553C23—542no—
554C23—542no—
555C23C2×D3×D911yescharacteristic
556Q8—62no—
557Q8—182no—
558Q8—544no—
559C2×C4—22no—
560C2×C4—32no—
561C2×C4—32no—
562C2×C4—92no—
563C2×C4—92no—
564C2×C4—271no—
565C2×C4—274no—
566C2×C4—271no—
567C2×C4—542no—
568C2×C4—542no—
569C2×C4—544no—
570C2×C4—544no—
571C2×C4—542no—
572C2×C4—542no—
573C2×C4C2×D3×D911yescharacteristic
574C2×C4C2×D3×D911yescharacteristic
575C2×C3—24no—
576C2×C3—22no—
577C2×C3—22no—
578C2×C3—21no—
579C2×C3—21no—
580C2×C3—21no—
581C2×C3C2×D4×D911yescharacteristic
582C2×C3C2×Q8×D911yescharacteristic
583C2×C3#41×D911yescharacteristic
584C2×C3C2×#379711yescharacteristic
585C2×C3#6430611yescharacteristic
586C2×C3C2×#381111yescharacteristic
587D3—94no—
588D3—94no—
589D3—184no—
590D3—184no—
591D3—182no—
592D3—182no—
593C22—21no—
594C22—21no—
595C22—272no—
596C22—272no—
597C22—272no—
598C22—272no—
599C22—548no—
600C22—541no—
601C22—541no—
602C22(C3⋊D9)⋊Q812yescomplemented
603C22C22×D3×D911yescharacteristic
604C4—21no—
605C4—62no—
606C4—62no—
607C4—182no—
608C4—182no—
609C4—272no—
610C4—541no—
611C4#8866812yes—
612C3—21no—
613C3D9×C22⋊Q811yescharacteristic, complemented
614C3#1059492511yescharacteristic
615C2—22no—
616C2—274no—
617C2—542no—
618C2C2×(C3⋊D9)⋊Q811yescharacteristic
619C2#1172247511yescharacteristic
620C2C2×#8866811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
prime radicalC23×C3⋊D9C22
hypercenterC22×C4D3×D9
hypocenterC3×C9C2×C22⋊Q8
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×(C3⋊D9)⋊Q8C2semidirect
3. C9×#924C2semidirect
4. C3×#63486C2semidirect
6. #11721267C2semidirect
7. #11721000C2semidirect
8. #11720980C2semidirect
9. #11720936C2semidirect
10. #11720929C2semidirect
11. #11720923C2semidirect
22. C2×C9×Dic6C22semidirect
23. C2×C3×Dic18C22semidirect
26. C2×#6189C22semidirect
30. C9×C4⋊Dic3C22semidirect
31. C9×Dic3⋊C4C22semidirect
32. C3×C4⋊Dic9C22semidirect
33. C3×Dic9⋊C4C22semidirect
36. Dic3⋊Dic9C22semidirect
37. #88442C22semidirect
38. Dic9⋊Dic3C22semidirect
55. #63486D3semidirect
65. C22×C3⋊D9Q8semidirect
135. C2×Dic18C2×D3semidirect
143. C4⋊Dic9C2×D3semidirect
146. Dic9⋊C4C2×D3semidirect
149. C22×C3×C9C2×Q8semidirect
156. C2×C3⋊D9C2×Q8semidirect
176. #924D9semidirect
261. C3⋊D9C22⋊Q8semidirect
297. C2×Dic6C2×D9semidirect
313. C4⋊Dic3C2×D9semidirect
319. Dic3⋊C4C2×D9semidirect
345. C22×C9D3×Q8semidirect
384. C3×C9C2×C22⋊Q8semidirect
502. C22×C3Q8×D9semidirect
549. C9D3×C22⋊Q8semidirect
602. C22(C3⋊D9)⋊Q8semidirect
613. C3D9×C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C3⋊D92
C2×(C3⋊D9)⋊Q82
C9×#9242
C3×#634862
#117212742
#117212672
#117210002
#117209802
#117209362
#117209292
#117209232
#105949253
D9×C22⋊Q83