Finite Group Databasebeta

← #13150461

Subgroups — #13150461

Order 1728.

1084 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
1C22×C4×D3×C9C211yesmaximal, characteristic, complemented
2C2×#88642C211yesmaximal, characteristic, complemented
3#11721819C211yesmaximal, characteristic, complemented
4C3×#63486C211yesmaximal, characteristic, complemented
5#11721287C212yesmaximal, complemented
6#11721240C212yesmaximal, complemented
7#11721038C211yesmaximal, characteristic, complemented
8#11721030C212yesmaximal, complemented
9#11720960C211yesmaximal, characteristic, complemented
10#11720954C212yesmaximal, complemented
11#11720906C211yesmaximal, characteristic, complemented
12#10595005—31nomaximal
13#10589893—31nomaximal
14C23×D3×C9C2211yescharacteristic
15C22×C3×C4×C9C2211yescharacteristic
16C2×#6233C2212yes—
17C2×C4⋊(C3⋊D9)C2211yescharacteristic, complemented
18C22×C9×Dic3C2211yescharacteristic
19C2×C4×D3×C9—24no—
20C2×C4×D3×C9C2211yescharacteristic
21C2×C4×D3×C9C2211yescharacteristic
22C2×C4×D3×C9C2211yescharacteristic
23C2×C4×D3×C9C2211yescharacteristic
24C2×C3×Dic18C2211yescharacteristic, complemented
25C2×#6195C2212yes—
26C2×#6195C2212yes—
27C2×(C3⋊D9)⋊C4C2212yescomplemented
28#88642—24no—
29#88536C2212yescomplemented
30C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
31C3×#3676C2212yes—
32C3×C4⋊Dic9C2211yescharacteristic, complemented
33C3×Dic9⋊C4C2212yescomplemented
34(C2×D3)⋊Dic9C2212yes—
35(C2×D3)⋊Dic9C2212yes—
36#88445C2212yescomplemented
37#88442C2212yescomplemented
38Dic3×Dic9C2212yescomplemented
39C22×C3×C4×D3D311yescharacteristic
40C2×#3799—31no—
41#64139—31no—
42C3×#924—31no—
43#63977—32no—
44#63952—32no—
45#63909—31no—
46#63904—32no—
47#63856—31no—
48#63850—32no—
49#63841—31no—
50C23×C4×C9—31no—
51C2×#3697—31no—
52#63515—32no—
53#63492—31no—
54#63492—31no—
55#63492—61no—
56#63491—32no—
57C4×#611—32no—
58C4×#611—32no—
59#63488—31no—
60#63486—31no—
61#63486D311yescharacteristic, complemented
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9C2312yes—
64C22×D3×C9—22no—
65C22×D3×C9—22no—
66C22×D3×C9—22no—
67C22×D3×C9C2311yescharacteristic
68C22×D3×C9C2311yescharacteristic
69C2×C3×C4×C9—22no—
70C2×C3×C4×C9C2311yescharacteristic
71C2×C3×C4×C9C2311yescharacteristic
72#6233—24no—
73C2×C3⋊Dic9C2312yes—
74C4⋊(C3⋊D9)—22no—
75C2×C9×Dic3—22no—
76C2×C9×Dic3C2311yescharacteristic
77C2×C9×Dic3C2311yescharacteristic
78C2×C3×Dic9C2312yes—
79C2×C3×Dic9C2312yes—
80C4×D3×C9—22no—
81C4×D3×C9—22no—
82C4×D3×C9—22no—
83C4×D3×C9D414yescomplemented
84C3×Dic18—22no—
85#6195—24no—
86#6195—24no—
87(C3⋊D9)⋊C4—24no—
88#5791—91no—
89#5627—91no—
90C23×C3×D3C2×D311yescharacteristic
91C22×C32×C4C2×D311yescharacteristic
92C2×#638—32no—
93C2×C4⋊(C3⋊D3)—31no—
94C22×C3×Dic3C2×D311yescharacteristic
95C2×C3×C4×D3—24no—
96C2×C3×C4×D3C2×D311yescharacteristic
97C2×C3×C4×D3C2×D311yescharacteristic
98C2×C3×C4×D3C2×D311yescharacteristic
99C2×C3×C4×D3C2×D311yescharacteristic
100C2×C3×Dic6—31no—
101C2×#626—32no—
102C2×#626—32no—
103C2×#624—32no—
104#3799—64no—
105C24×C9—31no—
106#3752—32no—
107C4⋊(C3⋊Dic3)—31no—
108C3×#217—32no—
109C3×C4⋊Dic3—31no—
110C3×Dic3⋊C4—32no—
111#3724—32no—
112#3722—32no—
113(C2×D3)⋊Dic3—32no—
114(C2×D3)⋊Dic3—32no—
115Dic32—32no—
116C22×C4×C9—21no—
117C22×C4×C9—31no—
118C22×C4×C9—31no—
119C22×C4×C9—31no—
120C22×C4×C9—31no—
121C22×C4×C9—31no—
122C22×C4×C9—64no—
123C22×C4×C9C2×D311yescharacteristic
124C2×#611—32no—
125C2×#611—32no—
126C2×#611—32no—
127C2×#611—62no—
128#3697—64no—
129C2×D36—31no—
130C2×D36—61no—
131C2×C4×D9—32no—
132C2×Dic18C2×D311yescharacteristic, complemented
133#3676—32no—
134#3676—32no—
135#3676C2×D312yes—
136(C2×D9)⋊C4—32no—
137(C2×D9)⋊C4—32no—
138(C2×D9)⋊C4—62no—
139C4⋊Dic9—31no—
140C4⋊Dic9—61no—
141C4⋊Dic9C2×D311yescharacteristic, complemented
142Dic9⋊C4—32no—
143Dic9⋊C4C2×D312yescomplemented
144C4×Dic9—32no—
145C22×C3×C9—21no—
146C22×C3×C9—21no—
147C22×C3×C9#4112yescomplemented
148C22×C3×C9C2411yescharacteristic
149C2×C3⋊D9—22no—
150C2×C3⋊D9—22no—
151C2×C3⋊D9—22no—
152C2×D3×C9—28no—
153C2×D3×C9—21no—
154C2×D3×C9—21no—
155C2×D3×C9—21no—
156C2×D3×C9—21no—
157C2×D3×C9C2×D412yes—
158C2×D3×C9#4112yescomplemented
159C3×C4×C9—21no—
160C3×C4×C9C2×D412yes—
161C3⋊Dic9—22no—
162C9×Dic3—21no—
163C9×Dic3C2×D412yes—
164C3×Dic9—22no—
165C3×Dic9—22no—
166C23×C3×C4—31no—
167C2×#239—91no—
168C2×#235—91no—
169C22×C4×D3D911yescharacteristic, complemented
170C3×C22⋊Q8—91no—
171#953—92no—
172#946—91no—
173#930—91no—
174#930—91no—
175#930—91no—
176#930—181no—
177#929—92no—
178C4×#67—92no—
179C4×#67—92no—
180#926—91no—
181#924—31no—
182#924—91no—
183#895—91no—
184#894—92no—
185#893—92no—
186#892—91no—
187#882—92no—
188#67⋊C4—92no—
189C23×C32C22×D311yescharacteristic
190C22×C3⋊D3—32no—
191C22×C3×D3—22no—
192C22×C3×D3—22no—
193C22×C3×D3—22no—
194C22×C3×D3C22×D311yescharacteristic
195C22×C3×D3C22×D311yescharacteristic
196C2×C32×C4—22no—
197C2×C32×C4C22×D311yescharacteristic
198C2×C32×C4C22×D311yescharacteristic
199#638—64no—
200C2×C3⋊Dic3—32no—
201C4⋊(C3⋊D3)—62no—
202C2×C3×Dic3—22no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3C22×D311yescharacteristic
206C2×C3×Dic3C22×D311yescharacteristic
207C3×C4×D3—22no—
208C3×C4×D3—22no—
209C3×C4×D3—22no—
210C3×C4×D3#6714yes—
211C3×Dic6—62no—
212#626—64no—
213#626—64no—
214#624—64no—
215C23×C9—21no—
216C23×C9—31no—
217C23×C9—31no—
218C23×C9—62no—
219C23×C9—62no—
220C23×C9—62no—
221C23×C9C22×D311yescharacteristic
222C22×D9—32no—
223C22×D9—62no—
224C2×C4×C9—22no—
225C2×C4×C9—21no—
226C2×C4×C9—21no—
227C2×C4×C9—31no—
228C2×C4×C9—34no—
229C2×C4×C9—31no—
230C2×C4×C9—42no—
231C2×C4×C9—62no—
232C2×C4×C9—62no—
233C2×C4×C9—62no—
234C2×C4×C9—62no—
235C2×C4×C9C22×D311yescharacteristic
236C2×C4×C9C22×D311yescharacteristic
237#611—64no—
238#611—64no—
239#611—64no—
240#611—124no—
241C2×Dic9—32no—
242C2×Dic9—62no—
243C2×Dic9C22×D312yes—
244C2×Dic9C22×D312yes—
245D36—62no—
246D36—122no—
247C4×D9—64no—
248Dic18—22no—
249#525—271noSylow
250C2×C3×C9—22no—
251C2×C3×C9C22×D411yescharacteristic
252C2×C3×C9C2×#4111yescharacteristic
253C2×C3×C9C2×#4111yescharacteristic
254C3⋊D9—42no—
255D3×C9—22no—
256D3×C9—22no—
257C24×C3—31no—
258C23×D3C2×D911yescharacteristic
259C2×C3×Q8—91no—
260C22×C3×C4—21no—
261C22×C3×C4—31no—
262C22×C3×C4—31no—
263C22×C3×C4—31no—
264C22×C3×C4—31no—
265C22×C3×C4—31no—
266C22×C3×C4—64no—
267C22×C3×C4C2×D911yescharacteristic
268C22×C3×C4D3211yescharacteristic
269C2×#67—92no—
270C2×#67—92no—
271C2×#67—92no—
272C2×#67—92no—
273C2×#67—182no—
274C22×Dic3C2×D911yescharacteristic
275#239—184no—
276#235—184no—
277C2×D12—92no—
278C2×D12—92no—
279C2×D12—91no—
280C2×D12—91no—
281C2×D12—181no—
282C2×C4×D3—24no—
283C2×C4×D3—92no—
284C2×C4×D3—92no—
285C2×C4×D3C2×D911yescharacteristic
286C2×C4×D3C2×D911yescharacteristic
287C2×C4×D3C2×D911yescharacteristic
288C2×C4×D3C2×D911yescharacteristic
289C2×Dic6—31no—
290C3×C4⋊C4—91no—
291C3×C4⋊C4—92no—
292C3×C22⋊C4—92no—
293#217—32no—
294#217—92no—
295#217—92no—
296(C2×D3)⋊C4—92no—
297(C2×D3)⋊C4—92no—
298(C2×D3)⋊C4—92no—
299(C2×D3)⋊C4—92no—
300(C2×D3)⋊C4—92no—
301(C2×D3)⋊C4—92no—
302(C2×D3)⋊C4—182no—
303C4⋊Dic3—31no—
304C4⋊Dic3—91no—
305C4⋊Dic3—91no—
306C4⋊Dic3—181no—
307Dic3⋊C4—32no—
308Dic3⋊C4—92no—
309Dic3⋊C4—92no—
310C4×Dic3—92no—
311C4×Dic3—92no—
312C22×C32—21no—
313C22×C32—21no—
314C22×C32#23512yes—
315C22×C32C23×D311yescharacteristic
316C2×C3⋊D3—62no—
317C2×C3⋊D3—62no—
318C2×C3⋊D3—62no—
319C2×C3×D3—28no—
320C2×C3×D3—21no—
321C2×C3×D3—21no—
322C2×C3×D3—21no—
323C2×C3×D3—21no—
324C2×C3×D3#23512yes—
325C2×C3×D3C2×#6712yes—
326C32×C4—21no—
327C32×C4C2×#6712yes—
328C3⋊Dic3—62no—
329C3×Dic3—21no—
330C3×Dic3—62no—
331C3×Dic3—62no—
332C3×Dic3C2×#6712yes—
333C22×C9—21no—
334C22×C9—21no—
335C22×C9—21no—
336C22×C9—22no—
337C22×C9—32no—
338C22×C9—32no—
339C22×C9—41no—
340C22×C9—41no—
341C22×C9—68no—
342C22×C9—61no—
343C22×C9—61no—
344C22×C9—61no—
345C22×C9—61no—
346C22×C9#23912yescomplemented
347C22×C9C23×D311yescharacteristic
348C2×D9—62no—
349C2×D9—62no—
350C2×D9—62no—
351C2×D9—122no—
352C2×D9—122no—
353C2×D9—122no—
354C4×C9—21no—
355C4×C9—22no—
356C4×C9—32no—
357C4×C9—41no—
358C4×C9—61no—
359C4×C9D3×D412yes—
360Dic9—22no—
361Dic9—22no—
362Dic9—62no—
363Dic9—122no—
364C2×#41—271no—
365C23×C4—31no—
366#123—272no—
367C22⋊Q8—91no—
368C22⋊Q8—271no—
369#121—271no—
370#121—271no—
371#120—272no—
372C4×D4—272no—
373C4×D4—272no—
374#117—271no—
375C3×C9#52511yescharacteristic, complemented, Sylow
376C23×C3—21no—
377C23×C3—31no—
378C23×C3—31no—
379C23×C3—62no—
380C23×C3—62no—
381C23×C3—62no—
382C23×C3C22×D911yescharacteristic
383C23×C3C2×D3211yescharacteristic
384C22×D3—22no—
385C22×D3—22no—
386C22×D3—22no—
387C22×D3—92no—
388C22×D3—92no—
389C22×D3—182no—
390C22×D3C22×D911yescharacteristic
391C22×D3C22×D911yescharacteristic
392C3×Q8—182no—
393C2×C3×C4—22no—
394C2×C3×C4—22no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—31no—
398C2×C3×C4—34no—
399C2×C3×C4—31no—
400C2×C3×C4—42no—
401C2×C3×C4—62no—
402C2×C3×C4—62no—
403C2×C3×C4—62no—
404C2×C3×C4—62no—
405C2×C3×C4—92no—
406C2×C3×C4—92no—
407C2×C3×C4C22×D911yescharacteristic
408C2×C3×C4C22×D911yescharacteristic
409C2×C3×C4C2×D3211yescharacteristic
410C2×C3×C4C2×D3211yescharacteristic
411#67—184no—
412#67—184no—
413#67—184no—
414#67—184no—
415#67—364no—
416C2×Dic3—22no—
417C2×Dic3—32no—
418C2×Dic3—32no—
419C2×Dic3—92no—
420C2×Dic3—92no—
421C2×Dic3—182no—
422C2×Dic3C22×D911yescharacteristic
423C2×Dic3C22×D911yescharacteristic
424D12—184no—
425D12—184no—
426D12—182no—
427D12—182no—
428D12—362no—
429C4×D3—22no—
430C4×D3—22no—
431C4×D3—22no—
432C4×D3—184no—
433C4×D3—184no—
434C4×D3#61114yescomplemented
435Dic6—62no—
436C2×C32—22no—
437C2×C32C2×#23511yescharacteristic
438C2×C32C2×#23511yescharacteristic
439C2×C32C22×#6711yescharacteristic
440C3⋊D3—122no—
441C3×D3—22no—
442C3×D3—22no—
443C2×C9—22no—
444C2×C9—21no—
445C2×C9—21no—
446C2×C9—21no—
447C2×C9—42no—
448C2×C9—62no—
449C2×C9—62no—
450C2×C9C2×D3×D411yescharacteristic
451C2×C9C2×#23911yescharacteristic
452C2×C9D3×#4111yescharacteristic
453D9—122no—
454D9—242no—
455C24—31no—
456#41—544no—
457C2×Q8—91no—
458C2×D4—272no—
459C2×D4—272no—
460C2×D4—272no—
461C2×D4—271no—
462C22×C4—31no—
463C22×C4—31no—
464C22×C4—31no—
465C22×C4—31no—
466C22×C4—31no—
467C22×C4—64no—
468C22×C4—272no—
469C22×C4D3×D911yescharacteristic
470C4⋊C4—91no—
471C4⋊C4—92no—
472C4⋊C4—272no—
473C4⋊C4—271no—
474C22⋊C4—92no—
475C22⋊C4—272no—
476C22⋊C4—272no—
477C22⋊C4—272no—
478C22⋊C4—272no—
479C42—272no—
480C22×C3—21no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—21no—
485C22×C3—22no—
486C22×C3—32no—
487C22×C3—32no—
488C22×C3—41no—
489C22×C3—41no—
490C22×C3—68no—
491C22×C3—61no—
492C22×C3—61no—
493C22×C3—61no—
494C22×C3—61no—
495C22×C3#369712yescomplemented
496C22×C3C23×D911yescharacteristic
497C22×C3#379912yes—
498C22×C3C22×D3211yescharacteristic
499C2×D3—28no—
500C2×D3—21no—
501C2×D3—21no—
502C2×D3—21no—
503C2×D3—21no—
504C2×D3—182no—
505C2×D3—182no—
506C2×D3—182no—
507C2×D3—182no—
508C2×D3—182no—
509C2×D3—182no—
510C2×D3—362no—
511C2×D3—362no—
512C2×D3—362no—
513C2×D3#369712yescomplemented
514C2×D3C2×#61112yes—
515C3×C4—21no—
516C3×C4—21no—
517C3×C4—22no—
518C3×C4—32no—
519C3×C4—41no—
520C3×C4—61no—
521C3×C4—182no—
522C3×C4—182no—
523C3×C4C2×#61112yes—
524C3×C4D3×#6712yes—
525Dic3—21no—
526Dic3—62no—
527Dic3—62no—
528Dic3—182no—
529Dic3—182no—
530Dic3—362no—
531Dic3C2×#61112yes—
532C32#579111yescharacteristic
533C9—21no—
534C9#562711yescharacteristic, complemented
535C23—31no—
536C23—31no—
537C23—62no—
538C23—62no—
539C23—62no—
540C23—272no—
541C23C2×D3×D911yescharacteristic
542Q8—182no—
543D4—544no—
544D4—544no—
545D4—544no—
546D4—542no—
547C2×C4—22no—
548C2×C4—31no—
549C2×C4—34no—
550C2×C4—31no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—62no—
554C2×C4—62no—
555C2×C4—92no—
556C2×C4—92no—
557C2×C4—272no—
558C2×C4—544no—
559C2×C4C2×D3×D911yescharacteristic
560C2×C4C2×D3×D911yescharacteristic
561C2×C3—22no—
562C2×C3—22no—
563C2×C3—21no—
564C2×C3—21no—
565C2×C3—21no—
566C2×C3—42no—
567C2×C3—62no—
568C2×C3—62no—
569C2×C3C2×#369711yescharacteristic
570C2×C3C2×#369711yescharacteristic
571C2×C3C22×#61111yescharacteristic
572C2×C3C2×#379911yescharacteristic
573C2×C3D3×#23511yescharacteristic
574C2×C3C2×D3×#6711yescharacteristic
575D3—22no—
576D3—22no—
577D3—362no—
578D3—362no—
579D3—722no—
580C22—21no—
581C22—21no—
582C22—32no—
583C22—32no—
584C22—68no—
585C22—61no—
586C22—61no—
587C22—61no—
588C22—61no—
589C22—542no—
590C22—542no—
591C22—542no—
592C22#8864212yescomplemented
593C22C22×D3×D911yescharacteristic
594C4—21no—
595C4—32no—
596C4—61no—
597C4—182no—
598C4—182no—
599C4—542no—
600C4D3×#61112yes—
601C3—21no—
602C3#1058989311yescharacteristic, complemented
603C3#1059500511yescharacteristic
604C2—22no—
605C2—62no—
606C2—62no—
607C2—1082no—
608C2C2×#8864211yescharacteristic
609C2D3×#369711yescharacteristic
610C2C2×D3×#61111yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#525
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4×D3×C9C2semidirect
2. C2×#88642C2semidirect
3. #11721819C2semidirect
4. C3×#63486C2semidirect
5. #11721287C2semidirect
6. #11721240C2semidirect
7. #11721038C2semidirect
8. #11721030C2semidirect
9. #11720960C2semidirect
10. #11720954C2semidirect
11. #11720906C2semidirect
17. C2×C4⋊(C3⋊D9)C22semidirect
24. C2×C3×Dic18C22semidirect
27. C2×(C3⋊D9)⋊C4C22semidirect
29. #88536C22semidirect
30. C4⋊(C3⋊Dic9)C22semidirect
32. C3×C4⋊Dic9C22semidirect
33. C3×Dic9⋊C4C22semidirect
36. #88445C22semidirect
37. #88442C22semidirect
38. Dic3×Dic9C22semidirect
61. #63486D3semidirect
83. C4×D3×C9D4semidirect
132. C2×Dic18C2×D3semidirect
141. C4⋊Dic9C2×D3semidirect
143. Dic9⋊C4C2×D3semidirect
147. C22×C3×C9#41semidirect
158. C2×D3×C9#41semidirect
169. C22×C4×D3D9semidirect
346. C22×C9#239semidirect
375. C3×C9#525semidirect
434. C4×D3#611semidirect
495. C22×C3#3697semidirect
513. C2×D3#3697semidirect
534. C9#5627semidirect
592. C22#88642semidirect
602. C3#10589893semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D3×C92
C2×#886422
#117218192
C3×#634862
#117212872
#117212402
#117210382
#117210302
#117209602
#117209542
#117209062
#105950053
#105898933