Finite Group Databasebeta

← #13149324

Subgroups — #13149324

Order 1728.

1058 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D3×D9C211yesmaximal, characteristic
2C2×(C3⋊D9)⋊Q8C211yesmaximal, characteristic, complemented
3C4⋊C4×C3⋊D9C211yesmaximal, characteristic, complemented
4C9×#897C211yesmaximal, characteristic, complemented
5C3×#63459C211yesmaximal, characteristic, complemented
6#11721048C212yesmaximal
7#11721026C212yesmaximal, complemented
8#11721000C212yesmaximal, complemented
9#11720999C212yesmaximal, complemented
10#11720960C211yesmaximal, characteristic, complemented
11#11720944C211yesmaximal, characteristic, complemented
12#10594631—31nomaximal
13D9×C22⋊Q8—31nomaximal
14C22×D3×D9C2211yescharacteristic
15C2×C4×C3⋊D9C2212yes—
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×C4×D3×C9C2211yescharacteristic
18C2×C3×C4×D9C2211yescharacteristic
19C2×C9×Dic6C2211yescharacteristic, complemented
20C2×C3×Dic18C2211yescharacteristic, complemented
21C2×D3×Dic9C2211yescharacteristic
22C2×(C3⋊D9)⋊C4C2211yescharacteristic
23C2×(C3⋊D9)⋊C4C2212yes—
24C2×Dic3×D9C2211yescharacteristic
25C2×#6189C2212yescomplemented
26C4×D3×D9—24no—
27(C3⋊D9)⋊Q8—24no—
28C3×C9×C4⋊C4C2211yescharacteristic, complemented
29C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
30#88534C2212yescomplemented
31C9×(C2×D3)⋊C4C2212yes—
32C3×(C2×D9)⋊C4C2212yes—
33C9×C4⋊Dic3C2212yescomplemented
34C3×C4⋊Dic9C2212yescomplemented
35(C2×D3)⋊Dic9C2212yes—
36(C2×D9)⋊Dic3C2212yes—
37#88442C2212yescomplemented
38#88442C2212yescomplemented
39C2×C4×D32—31no—
40C2×#3797—31no—
41C4⋊C4×C3⋊D3—31no—
42C3×#897—31no—
43C3×#897D311yescharacteristic
44#63921—32no—
45#63900—32no—
46#63899—32no—
47#63899—32no—
48#63856—31no—
49#63856—31no—
50C2×Q8×D9—31no—
51C22×C4×D9—31no—
52C9×C22⋊Q8—31no—
53#63508—31no—
54#63486—31no—
55#63459D311yescharacteristic, complemented
56#63458—32no—
57C4⋊C4×D9—31no—
58C4⋊C4×D9—32no—
59C4⋊C4×D9—61no—
60C22⋊C4×D9—32no—
61#63440—32no—
62C22×C3⋊D9C2311yescharacteristic
63C22×D3×C9C2311yescharacteristic
64C22×C3×D9C2311yescharacteristic
65C2×D3×D9—22no—
66C2×D3×D9—22no—
67C2×D3×D9Q814yescomplemented
68C2×C3×C4×C9C2312yes—
69C2×C3×C4×C9C2311yescharacteristic
70C2×C3⋊Dic9C2312yes—
71C2×C3⋊Dic9C2311yescharacteristic
72C4×C3⋊D9—24no—
73C4×C3⋊D9D414yes—
74C2×C9×Dic3C2312yes—
75C2×C9×Dic3C2311yescharacteristic
76C2×C3×Dic9C2312yes—
77C2×C3×Dic9C2311yescharacteristic
78C4×D3×C9—22no—
79C3×C4×D9—22no—
80C9×Dic6—22no—
81C3×Dic18—22no—
82D3×Dic9—22no—
83(C3⋊D9)⋊C4—24no—
84(C3⋊D9)⋊C4—22no—
85Dic3×D9—22no—
86#6189—24no—
87D3×C22⋊Q8—91no—
88D3×C22⋊Q8—91no—
89C22×D32—31no—
90C2×C4×C3⋊D3—31no—
91C2×C4×C3⋊D3—32no—
92C2×C3×C4×D3—31no—
93C2×C3×C4×D3C2×D311yescharacteristic
94C2×C3×Dic6—31no—
95C2×C3×Dic6C2×D311yescharacteristic
96C2×#627—32no—
97C2×#624—32no—
98C2×#624—31no—
99C2×D3×Dic3—31no—
100C2×D3×Dic3—31no—
101C4×D32—64no—
102#3797—64no—
103C23×D9—31no—
104C32×C4⋊C4C2×D311yescharacteristic
105C4⋊(C3⋊Dic3)—31no—
106#3750—32no—
107C3×(C2×D3)⋊C4—32no—
108C3×(C2×D3)⋊C4C2×D312yes—
109C3×C4⋊Dic3—32no—
110C3×C4⋊Dic3C2×D312yes—
111#3724—32no—
112#3724—32no—
113(C2×D3)⋊Dic3—32no—
114(C2×D3)⋊Dic3—32no—
115C2×Q8×C9—31no—
116C22×C4×C9—31no—
117C22×Dic9—31no—
118Q8×D9—64no—
119C2×C4×D9—31no—
120C2×C4×D9—31no—
121C2×C4×D9—32no—
122C2×C4×D9—32no—
123C2×C4×D9—31no—
124C2×C4×D9—64no—
125C2×C4×D9—61no—
126C2×C4×D9—62no—
127C2×C4×D9C2×D311yescharacteristic
128C2×Dic18—32no—
129C2×Dic18C2×D311yescharacteristic, complemented
130C9×C4⋊C4—21no—
131C9×C4⋊C4—32no—
132C9×C4⋊C4C2×D311yescharacteristic, complemented
133C9×C22⋊C4—32no—
134#3676—32no—
135(C2×D9)⋊C4—32no—
136(C2×D9)⋊C4C2×D312yes—
137C4⋊Dic9—31no—
138C4⋊Dic9—61no—
139C4⋊Dic9C2×D312yescomplemented
140Dic9⋊C4—32no—
141Dic9⋊C4—32no—
142Dic9⋊C4—32no—
143Dic9⋊C4—62no—
144C22×C3×C9C2411yescharacteristic
145C2×C3⋊D9C2×D412yes—
146C2×C3⋊D9C2×Q812yes—
147C2×C3⋊D9#4112yes—
148C2×D3×C9—21no—
149C2×D3×C9—21no—
150C2×D3×C9C2×Q812yescomplemented
151C2×C3×D9—21no—
152C2×C3×D9—21no—
153C2×C3×D9C2×Q812yescomplemented
154D3×D9—28no—
155C3×C4×C9—22no—
156C3×C4×C9C2×D412yes—
157C3⋊Dic9—22no—
158C3⋊Dic9C2×D412yes—
159C9×Dic3—22no—
160C9×Dic3—21no—
161C3×Dic9—22no—
162C3×Dic9—21no—
163C2×D3×Q8—91no—
164C2×D3×Q8—91no—
165C22×C4×D3—91no—
166C22×C4×D3—91no—
167C3×C22⋊Q8—31no—
168C3×C22⋊Q8—91no—
169#946—91no—
170#946—91no—
171#924—91no—
172#924—91no—
173#897—31no—
174#897D911yescharacteristic, complemented
175#896—92no—
176#896—92no—
177D3×C4⋊C4—92no—
178D3×C4⋊C4—91no—
179D3×C4⋊C4—91no—
180D3×C4⋊C4—92no—
181D3×C4⋊C4—181no—
182D3×C22⋊C4—92no—
183D3×C22⋊C4—92no—
184#878—92no—
185#878—92no—
186C22×C3⋊D3—31no—
187C22×C3×D3—31no—
188C22×C3×D3C22×D311yescharacteristic
189C2×D32—34no—
190C2×D32—62no—
191C2×D32—62no—
192C2×C32×C4C22×D312yes—
193C2×C32×C4C22×D311yescharacteristic
194C2×C3⋊Dic3—31no—
195C2×C3⋊Dic3—32no—
196C4×C3⋊D3—34no—
197C4×C3⋊D3—64no—
198C2×C3×Dic3—32no—
199C2×C3×Dic3—31no—
200C2×C3×Dic3C22×D312yes—
201C2×C3×Dic3C22×D311yescharacteristic
202C3×C4×D3—22no—
203C3×C4×D3—62no—
204C3×Dic6—22no—
205C3×Dic6—62no—
206#627—64no—
207#624—64no—
208#624—62no—
209D3×Dic3—62no—
210D3×Dic3—62no—
211C23×C9—31no—
212C22×D9—31no—
213C22×D9—34no—
214C22×D9—61no—
215C22×D9—62no—
216C22×D9—62no—
217C22×D9C22×D311yescharacteristic
218Q8×C9—62no—
219C2×C4×C9—21no—
220C2×C4×C9—22no—
221C2×C4×C9—31no—
222C2×C4×C9—32no—
223C2×C4×C9—62no—
224C2×C4×C9C22×D312yes—
225C2×C4×C9C22×D311yescharacteristic
226C2×Dic9—31no—
227C2×Dic9—32no—
228C2×Dic9—61no—
229C2×Dic9—62no—
230C2×Dic9—62no—
231C2×Dic9C22×D311yescharacteristic
232C2×Dic9C22×D312yes—
233C4×D9—22no—
234C4×D9—34no—
235C4×D9—64no—
236C4×D9—62no—
237C4×D9—64no—
238C4×D9—62no—
239C4×D9—64no—
240C4×D9—124no—
241Dic18—22no—
242Dic18—64no—
243C2×C22⋊Q8—271noSylow
244C2×C3×C9C22×D411yescharacteristic
245C2×C3×C9C22×Q811yescharacteristic
246C2×C3×C9C2×#4111yescharacteristic
247C3⋊D9C22⋊Q814yescomplemented
248D3×C9—22no—
249C3×D9—22no—
250C23×D3—91no—
251C23×D3—91no—
252C2×C3×Q8—31no—
253C2×C3×Q8—91no—
254C22×C3×C4—31no—
255C22×C3×C4—91no—
256C22×Dic3—91no—
257C22×Dic3—91no—
258D3×Q8—184no—
259D3×Q8—184no—
260C2×C4×D3—31no—
261C2×C4×D3—91no—
262C2×C4×D3—91no—
263C2×C4×D3—92no—
264C2×C4×D3—91no—
265C2×C4×D3—91no—
266C2×C4×D3—91no—
267C2×C4×D3—92no—
268C2×C4×D3—92no—
269C2×C4×D3—92no—
270C2×C4×D3—91no—
271C2×C4×D3—181no—
272C2×C4×D3—182no—
273C2×C4×D3—184no—
274C2×C4×D3—184no—
275C2×C4×D3C2×D911yescharacteristic
276C2×Dic6—31no—
277C2×Dic6—92no—
278C2×Dic6—92no—
279C2×Dic6C2×D911yescharacteristic, complemented
280C3×C4⋊C4—21no—
281C3×C4⋊C4—32no—
282C3×C4⋊C4—92no—
283C3×C4⋊C4C2×D911yescharacteristic, complemented
284C3×C4⋊C4D3211yescharacteristic
285C3×C22⋊C4—32no—
286C3×C22⋊C4—92no—
287#217—92no—
288#217—92no—
289(C2×D3)⋊C4—32no—
290(C2×D3)⋊C4—92no—
291(C2×D3)⋊C4—92no—
292(C2×D3)⋊C4C2×D912yes—
293C4⋊Dic3—32no—
294C4⋊Dic3—91no—
295C4⋊Dic3—91no—
296C4⋊Dic3—181no—
297C4⋊Dic3C2×D912yescomplemented
298Dic3⋊C4—92no—
299Dic3⋊C4—92no—
300Dic3⋊C4—92no—
301Dic3⋊C4—92no—
302Dic3⋊C4—92no—
303Dic3⋊C4—92no—
304Dic3⋊C4—182no—
305C22×C32C23×D311yescharacteristic
306C2×C3⋊D3—32no—
307C2×C3⋊D3—32no—
308C2×C3⋊D3—32no—
309C2×C3×D3—21no—
310C2×C3×D3—21no—
311C2×C3×D3—32no—
312C2×C3×D3—61no—
313C2×C3×D3—61no—
314C2×C3×D3D3×Q812yes—
315D32—68no—
316C32×C4—22no—
317C32×C4D3×D412yes—
318C3⋊Dic3—32no—
319C3⋊Dic3—62no—
320C3×Dic3—21no—
321C3×Dic3—22no—
322C3×Dic3—61no—
323C3×Dic3—62no—
324C22×C9—21no—
325C22×C9—32no—
326C22×C9—61no—
327C22×C9—61no—
328C22×C9C23×D311yescharacteristic
329C2×D9—21no—
330C2×D9—21no—
331C2×D9—32no—
332C2×D9—32no—
333C2×D9—32no—
334C2×D9—62no—
335C2×D9—62no—
336C2×D9—62no—
337C2×D9—68no—
338C2×D9D3×Q812yescomplemented
339C4×C9—22no—
340C4×C9—22no—
341C4×C9—42no—
342C4×C9—62no—
343C4×C9—61no—
344C4×C9D3×D412yes—
345Dic9—22no—
346Dic9—21no—
347Dic9—32no—
348Dic9—62no—
349Dic9—62no—
350Dic9—122no—
351C22×Q8—271no—
352C23×C4—271no—
353C22⋊Q8—31no—
354C22⋊Q8—91no—
355C22⋊Q8—272no—
356C22⋊Q8—272no—
357C22⋊Q8—271no—
358C22⋊Q8—271no—
359C2×C4⋊C4—272no—
360C2×C4⋊C4—271no—
361C2×C22⋊C4—272no—
362C3×C9C2×C22⋊Q811yescharacteristic, complemented, Sylow
363C23×C3—31no—
364C23×C3—91no—
365C22×D3—31no—
366C22×D3—94no—
367C22×D3—91no—
368C22×D3—91no—
369C22×D3—94no—
370C22×D3—181no—
371C22×D3—182no—
372C22×D3—182no—
373C22×D3—182no—
374C22×D3—182no—
375C22×D3C22×D911yescharacteristic
376C3×Q8—62no—
377C3×Q8—182no—
378C2×C3×C4—21no—
379C2×C3×C4—22no—
380C2×C3×C4—31no—
381C2×C3×C4—32no—
382C2×C3×C4—62no—
383C2×C3×C4—91no—
384C2×C3×C4—92no—
385C2×C3×C4—182no—
386C2×C3×C4C22×D912yes—
387C2×C3×C4C22×D911yescharacteristic
388C2×C3×C4C2×D3212yes—
389C2×C3×C4C2×D3211yescharacteristic
390C2×Dic3—31no—
391C2×Dic3—32no—
392C2×Dic3—91no—
393C2×Dic3—91no—
394C2×Dic3—92no—
395C2×Dic3—92no—
396C2×Dic3—181no—
397C2×Dic3—182no—
398C2×Dic3—182no—
399C2×Dic3—182no—
400C2×Dic3C22×D911yescharacteristic
401C2×Dic3C22×D912yes—
402C4×D3—22no—
403C4×D3—62no—
404C4×D3—94no—
405C4×D3—94no—
406C4×D3—184no—
407C4×D3—182no—
408C4×D3—184no—
409C4×D3—182no—
410C4×D3—184no—
411C4×D3—182no—
412C4×D3—182no—
413C4×D3—184no—
414C4×D3—184no—
415C4×D3—364no—
416Dic6—22no—
417Dic6—62no—
418Dic6—184no—
419Dic6—184no—
420C2×C32C2×D3×D411yescharacteristic
421C2×C32C2×D3×Q811yescharacteristic
422C2×C32D3×#4111yescharacteristic
423C3⋊D3—34no—
424C3×D3—22no—
425C3×D3—62no—
426C2×C9—21no—
427C2×C9—21no—
428C2×C9—21no—
429C2×C9—62no—
430C2×C9C2×D3×D411yescharacteristic
431C2×C9C2×D3×Q811yescharacteristic
432C2×C9D3×#4111yescharacteristic
433D9—22no—
434D9—34no—
435D9—64no—
436C24—271no—
437C2×Q8—31no—
438C2×Q8—91no—
439C2×Q8—272no—
440C2×Q8—544no—
441C22×C4—31no—
442C22×C4—91no—
443C22×C4—272no—
444C22×C4—271no—
445C22×C4—271no—
446C22×C4—271no—
447C22×C4—271no—
448C22×C4—272no—
449C22×C4—544no—
450C4⋊C4—32no—
451C4⋊C4—92no—
452C4⋊C4—272no—
453C4⋊C4—272no—
454C4⋊C4—272no—
455C4⋊C4—271no—
456C4⋊C4D3×D911yescharacteristic, complemented
457C22⋊C4—32no—
458C22⋊C4—92no—
459C22⋊C4—272no—
460C22⋊C4—272no—
461C22×C3—21no—
462C22×C3—32no—
463C22×C3—61no—
464C22×C3—61no—
465C22×C3—92no—
466C22×C3—181no—
467C22×C3—181no—
468C22×C3C23×D911yescharacteristic
469C22×C3C22×D3211yescharacteristic
470C2×D3—21no—
471C2×D3—21no—
472C2×D3—32no—
473C2×D3—61no—
474C2×D3—61no—
475C2×D3—92no—
476C2×D3—92no—
477C2×D3—92no—
478C2×D3—92no—
479C2×D3—92no—
480C2×D3—92no—
481C2×D3—182no—
482C2×D3—182no—
483C2×D3—182no—
484C2×D3—188no—
485C2×D3—188no—
486C2×D3Q8×D912yescomplemented
487C3×C4—22no—
488C3×C4—22no—
489C3×C4—22no—
490C3×C4—42no—
491C3×C4—62no—
492C3×C4—61no—
493C3×C4—182no—
494C3×C4—181no—
495C3×C4D4×D912yes—
496C3×C4#380212yes—
497Dic3—22no—
498Dic3—21no—
499Dic3—62no—
500Dic3—61no—
501Dic3—92no—
502Dic3—92no—
503Dic3—182no—
504Dic3—182no—
505Dic3—182no—
506Dic3—362no—
507C32D3×C22⋊Q811yescharacteristic
508C9—21no—
509C9D3×C22⋊Q811yescharacteristic, complemented
510C23—31no—
511C23—91no—
512C23—274no—
513C23—271no—
514C23—542no—
515C23—542no—
516Q8—62no—
517Q8—182no—
518Q8—544no—
519C2×C4—32no—
520C2×C4—31no—
521C2×C4—62no—
522C2×C4—92no—
523C2×C4—91no—
524C2×C4—182no—
525C2×C4—271no—
526C2×C4—274no—
527C2×C4—272no—
528C2×C4—544no—
529C2×C4—542no—
530C2×C4—544no—
531C2×C4—542no—
532C2×C4—542no—
533C2×C4C2×D3×D912yes—
534C2×C4C2×D3×D911yescharacteristic
535C2×C3—21no—
536C2×C3—21no—
537C2×C3—21no—
538C2×C3—62no—
539C2×C3—182no—
540C2×C3C2×D4×D911yescharacteristic
541C2×C3C2×Q8×D911yescharacteristic
542C2×C3#41×D911yescharacteristic
543C2×C3C2×#380211yescharacteristic
544C2×C3#6431211yescharacteristic
545C2×C3D32×Q811yescharacteristic
546D3—22no—
547D3—62no—
548D3—94no—
549D3—94no—
550D3—184no—
551C22—32no—
552C22—61no—
553C22—61no—
554C22—92no—
555C22—181no—
556C22—181no—
557C22—272no—
558C22—272no—
559C22—272no—
560C22—548no—
561C22C22×D3×D911yescharacteristic
562C4—22no—
563C4—61no—
564C4—62no—
565C4—181no—
566C4—182no—
567C4—272no—
568C4—542no—
569C4#8864612yes—
570C3—21no—
571C3D9×C22⋊Q811yescharacteristic, complemented
572C3#1059463111yescharacteristic
573C2—62no—
574C2—182no—
575C2—274no—
576C2C2×#8864611yescharacteristic
577C2#1172249811yescharacteristic
578C2D3×Q8×D911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC22×D3×D9C22
hypercenterC4⋊C4D3×D9
hypocenterC3×C9C2×C22⋊Q8
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. C2×(C3⋊D9)⋊Q8C2semidirect
3. C4⋊C4×C3⋊D9C2semidirect
4. C9×#897C2semidirect
5. C3×#63459C2semidirect
7. #11721026C2semidirect
8. #11721000C2semidirect
9. #11720999C2semidirect
10. #11720960C2semidirect
11. #11720944C2semidirect
19. C2×C9×Dic6C22semidirect
20. C2×C3×Dic18C22semidirect
25. C2×#6189C22semidirect
28. C3×C9×C4⋊C4C22semidirect
29. C4⋊(C3⋊Dic9)C22semidirect
30. #88534C22semidirect
33. C9×C4⋊Dic3C22semidirect
34. C3×C4⋊Dic9C22semidirect
37. #88442C22semidirect
38. #88442C22semidirect
55. #63459D3semidirect
67. C2×D3×D9Q8semidirect
129. C2×Dic18C2×D3semidirect
132. C9×C4⋊C4C2×D3semidirect
139. C4⋊Dic9C2×D3semidirect
150. C2×D3×C9C2×Q8semidirect
153. C2×C3×D9C2×Q8semidirect
174. #897D9semidirect
247. C3⋊D9C22⋊Q8semidirect
279. C2×Dic6C2×D9semidirect
283. C3×C4⋊C4C2×D9semidirect
297. C4⋊Dic3C2×D9semidirect
338. C2×D9D3×Q8semidirect
362. C3×C9C2×C22⋊Q8semidirect
456. C4⋊C4D3×D9semidirect
486. C2×D3Q8×D9semidirect
509. C9D3×C22⋊Q8semidirect
571. C3D9×C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D3×D92
C2×(C3⋊D9)⋊Q82
C4⋊C4×C3⋊D92
C9×#8972
C3×#634592
#117210482
#117210262
#117210002
#117209992
#117209602
#117209442
#105946313
D9×C22⋊Q83