Finite Group Databasebeta

← #13150321

Subgroups — #13150321

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×C3×C4×D9C211yesmaximal, characteristic, complemented
2C2×#88635C211yesmaximal, characteristic, complemented
3#11721819C211yesmaximal, characteristic, complemented
4C9×#924C211yesmaximal, characteristic, complemented
5#11721276C212yesmaximal, complemented
6#11721241C212yesmaximal, complemented
7#11721004C211yesmaximal, characteristic, complemented
8#11721002C212yesmaximal, complemented
9#11720944C211yesmaximal, characteristic, complemented
10#11720938C212yesmaximal, complemented
11#11720907C211yesmaximal, characteristic, complemented
12#10595005—31nomaximal
13#10589729—31nomaximal
14C23×C3×D9C2211yescharacteristic
15C22×C3×C4×C9C2211yescharacteristic
16C2×#6233C2212yes—
17C2×C4⋊(C3⋊D9)C2211yescharacteristic, complemented
18C22×C3×Dic9C2211yescharacteristic
19C2×C3×C4×D9—24no—
20C2×C3×C4×D9C2211yescharacteristic
21C2×C3×C4×D9C2211yescharacteristic
22C2×C3×C4×D9C2211yescharacteristic
23C2×C3×C4×D9C2211yescharacteristic
24C2×C9×Dic6C2211yescharacteristic, complemented
25C2×#6192C2212yes—
26C2×#6192C2212yes—
27C2×(C3⋊D9)⋊C4C2212yescomplemented
28#88635—24no—
29#88536C2212yescomplemented
30C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
31C9×#217C2212yes—
32C9×C4⋊Dic3C2211yescharacteristic, complemented
33C9×Dic3⋊C4C2212yescomplemented
34#88445C2212yescomplemented
35(C2×D9)⋊Dic3C2212yes—
36(C2×D9)⋊Dic3C2212yes—
37#88442C2212yescomplemented
38Dic3×Dic9C2212yescomplemented
39C22×C3×C4×D3—31no—
40C2×#3799—31no—
41#64139—31no—
42C3×#924D311yescharacteristic
43#63977—32no—
44#63952—32no—
45#63909—31no—
46#63904—32no—
47#63856—31no—
48#63850—32no—
49#63841—31no—
50C2×#3701—31no—
51C22×C4×D9D311yescharacteristic, complemented
52C9×C22⋊Q8—31no—
53#63508—31no—
54#63492—31no—
55#63492—61no—
56#63457—31no—
57#63456—32no—
58#63455—32no—
59#63454—31no—
60#63444—32no—
61#611⋊C4—32no—
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9C2312yes—
64C22×C3×D9—22no—
65C22×C3×D9—22no—
66C22×C3×D9—22no—
67C22×C3×D9C2311yescharacteristic
68C22×C3×D9C2311yescharacteristic
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×Dic3C2312yes—
76C2×C9×Dic3C2312yes—
77C2×C3×Dic9—22no—
78C2×C3×Dic9C2311yescharacteristic
79C2×C3×Dic9C2311yescharacteristic
80C3×C4×D9—22no—
81C3×C4×D9—22no—
82C3×C4×D9—22no—
83C3×C4×D9D414yescomplemented
84C9×Dic6—22no—
85#6192—24no—
86#6192—24no—
87(C3⋊D9)⋊C4—24no—
88#5791—91no—
89#5627—91no—
90C23×C3×D3—31no—
91C22×C32×C4C2×D311yescharacteristic
92C2×#638—32no—
93C2×C4⋊(C3⋊D3)—31no—
94C22×C3×Dic3—31no—
95C2×C3×C4×D3—31no—
96C2×C3×C4×D3—31no—
97C2×C3×C4×D3—31no—
98C2×C3×C4×D3—31no—
99C2×C3×C4×D3—64no—
100C2×C3×Dic6C2×D311yescharacteristic
101C2×#626—32no—
102C2×#626—32no—
103C2×#624—32no—
104#3799—64no—
105C23×D9C2×D311yescharacteristic
106#3752—32no—
107C4⋊(C3⋊Dic3)—31no—
108C3×#217C2×D312yes—
109C3×C4⋊Dic3C2×D311yescharacteristic
110C3×Dic3⋊C4C2×D312yes—
111#3724—32no—
112#3722—32no—
113(C2×D3)⋊Dic3—32no—
114(C2×D3)⋊Dic3—32no—
115Dic32—32no—
116C2×Q8×C9—31no—
117C22×C4×C9—21no—
118C22×C4×C9C2×D311yescharacteristic
119C2×#611—32no—
120C2×#611—62no—
121C22×Dic9C2×D311yescharacteristic
122#3701—64no—
123C2×D36—32no—
124C2×D36—32no—
125C2×D36—31no—
126C2×D36—61no—
127C2×C4×D9—24no—
128C2×C4×D9—32no—
129C2×C4×D9C2×D311yescharacteristic
130C2×C4×D9C2×D311yescharacteristic
131C2×C4×D9C2×D311yescharacteristic
132C2×C4×D9C2×D311yescharacteristic
133C9×C4⋊C4—32no—
134C9×C4⋊C4—31no—
135C9×C22⋊C4—32no—
136(C2×D9)⋊C4—32no—
137(C2×D9)⋊C4—32no—
138(C2×D9)⋊C4—32no—
139(C2×D9)⋊C4—32no—
140(C2×D9)⋊C4—62no—
141C4⋊Dic9—31no—
142C4⋊Dic9—61no—
143Dic9⋊C4—32no—
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×C3×D9—28no—
153C2×C3×D9—21no—
154C2×C3×D9—21no—
155C2×C3×D9—21no—
156C2×C3×D9—21no—
157C2×C3×D9C2×D412yes—
158C2×C3×D9#4112yescomplemented
159C3×C4×C9—21no—
160C3×C4×C9C2×D412yes—
161C3⋊Dic9—22no—
162C9×Dic3—22no—
163C9×Dic3—22no—
164C3×Dic9—21no—
165C3×Dic9C2×D412yes—
166C23×C3×C4—91no—
167C2×#239—91no—
168C2×#235—91no—
169C22×C4×D3—31no—
170C3×C22⋊Q8—31no—
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—91no—
182#924D911yescharacteristic, complemented
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—31no—
192C22×C3×D3—31no—
193C22×C3×D3—62no—
194C22×C3×D3—62no—
195C22×C3×D3—62no—
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—31no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3—62no—
205C2×C3×Dic3C22×D312yes—
206C2×C3×Dic3C22×D312yes—
207C3×C4×D3—34no—
208C3×C4×D3—62no—
209C3×C4×D3—62no—
210C3×C4×D3—62no—
211C3×Dic6—22no—
212#626—64no—
213#626—64no—
214#624—64no—
215C23×C9—21no—
216C23×C9C22×D311yescharacteristic
217C22×D9—22no—
218C22×D9—22no—
219C22×D9—22no—
220C22×D9—32no—
221C22×D9—62no—
222C22×D9C22×D311yescharacteristic
223C22×D9C22×D311yescharacteristic
224Q8×C9—62no—
225C2×C4×C9—22no—
226C2×C4×C9—21no—
227C2×C4×C9—21no—
228C2×C4×C9—32no—
229C2×C4×C9—32no—
230C2×C4×C9—42no—
231C2×C4×C9C22×D311yescharacteristic
232C2×C4×C9C22×D311yescharacteristic
233#611—64no—
234#611—124no—
235C2×Dic9—22no—
236C2×Dic9—32no—
237C2×Dic9—62no—
238C2×Dic9C22×D311yescharacteristic
239C2×Dic9C22×D311yescharacteristic
240D36—64no—
241D36—64no—
242D36—62no—
243D36—122no—
244C4×D9—22no—
245C4×D9—22no—
246C4×D9—22no—
247C4×D9—64no—
248C4×D9#6714yescomplemented
249#525—271noSylow
250C2×C3×C9—22no—
251C2×C3×C9C22×D411yescharacteristic
252C2×C3×C9C2×#4111yescharacteristic
253C2×C3×C9C2×#4111yescharacteristic
254C3⋊D9—42no—
255C3×D9—22no—
256C3×D9—22no—
257C24×C3—91no—
258C23×D3—31no—
259C2×C3×Q8—31no—
260C22×C3×C4—21no—
261C22×C3×C4—91no—
262C22×C3×C4—91no—
263C22×C3×C4—91no—
264C22×C3×C4—91no—
265C22×C3×C4—91no—
266C22×C3×C4—184no—
267C22×C3×C4C2×D911yescharacteristic
268C22×C3×C4D3211yescharacteristic
269C2×#67—92no—
270C2×#67—92no—
271C2×#67—92no—
272C2×#67—92no—
273C2×#67—182no—
274C22×Dic3—31no—
275#239—184no—
276#235—184no—
277C2×D12—92no—
278C2×D12—92no—
279C2×D12—91no—
280C2×D12—91no—
281C2×D12—181no—
282C2×C4×D3—31no—
283C2×C4×D3—31no—
284C2×C4×D3—31no—
285C2×C4×D3—31no—
286C2×C4×D3—64no—
287C2×C4×D3—92no—
288C2×C4×D3—92no—
289C2×Dic6C2×D911yescharacteristic, complemented
290C3×C4⋊C4—31no—
291C3×C4⋊C4—32no—
292C3×C22⋊C4—32no—
293#217—92no—
294#217—92no—
295#217C2×D912yes—
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—91no—
304C4⋊Dic3—91no—
305C4⋊Dic3—181no—
306C4⋊Dic3C2×D911yescharacteristic, complemented
307Dic3⋊C4—92no—
308Dic3⋊C4—92no—
309Dic3⋊C4C2×D912yescomplemented
310C4×Dic3—92no—
311C4×Dic3—92no—
312C22×C32—21no—
313C22×C32—21no—
314C22×C32#23912yes—
315C22×C32C23×D311yescharacteristic
316C2×C3⋊D3—62no—
317C2×C3⋊D3—62no—
318C2×C3⋊D3—62no—
319C2×C3×D3—32no—
320C2×C3×D3—32no—
321C2×C3×D3—68no—
322C2×C3×D3—61no—
323C2×C3×D3—61no—
324C2×C3×D3—61no—
325C2×C3×D3—61no—
326C32×C4—21no—
327C32×C4D3×D412yes—
328C3⋊Dic3—62no—
329C3×Dic3—22no—
330C3×Dic3—22no—
331C3×Dic3—32no—
332C3×Dic3—61no—
333C22×C9—21no—
334C22×C9—21no—
335C22×C9—21no—
336C22×C9—22no—
337C22×C9—41no—
338C22×C9—41no—
339C22×C9#23512yescomplemented
340C22×C9C23×D311yescharacteristic
341C2×D9—28no—
342C2×D9—21no—
343C2×D9—21no—
344C2×D9—21no—
345C2×D9—21no—
346C2×D9—62no—
347C2×D9—62no—
348C2×D9—62no—
349C2×D9—122no—
350C2×D9—122no—
351C2×D9—122no—
352C2×D9#23512yescomplemented
353C2×D9C2×#6712yes—
354C4×C9—21no—
355C4×C9—22no—
356C4×C9—41no—
357C4×C9—62no—
358C4×C9—62no—
359C4×C9C2×#6712yes—
360Dic9—21no—
361Dic9—62no—
362Dic9—122no—
363Dic9C2×#6712yes—
364C2×#41—271no—
365C23×C4—91no—
366#123—272no—
367C22⋊Q8—31no—
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—91no—
378C23×C3—91no—
379C23×C3—182no—
380C23×C3—182no—
381C23×C3—182no—
382C23×C3C22×D911yescharacteristic
383C23×C3C2×D3211yescharacteristic
384C22×D3—31no—
385C22×D3—31no—
386C22×D3—62no—
387C22×D3—62no—
388C22×D3—62no—
389C22×D3—92no—
390C22×D3—92no—
391C22×D3—182no—
392C3×Q8—62no—
393C2×C3×C4—22no—
394C2×C3×C4—22no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—32no—
398C2×C3×C4—32no—
399C2×C3×C4—42no—
400C2×C3×C4—91no—
401C2×C3×C4—94no—
402C2×C3×C4—91no—
403C2×C3×C4—182no—
404C2×C3×C4—182no—
405C2×C3×C4—182no—
406C2×C3×C4—182no—
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—31no—
417C2×Dic3—31no—
418C2×Dic3—62no—
419C2×Dic3—92no—
420C2×Dic3—92no—
421C2×Dic3—182no—
422C2×Dic3C22×D912yes—
423C2×Dic3C22×D912yes—
424D12—184no—
425D12—184no—
426D12—182no—
427D12—182no—
428D12—362no—
429C4×D3—34no—
430C4×D3—62no—
431C4×D3—62no—
432C4×D3—62no—
433C4×D3—184no—
434C4×D3—184no—
435Dic6—22no—
436C2×C32—22no—
437C2×C32C2×D3×D411yescharacteristic
438C2×C32C2×#23911yescharacteristic
439C2×C32D3×#4111yescharacteristic
440C3⋊D3—122no—
441C3×D3—62no—
442C3×D3—62no—
443C2×C9—22no—
444C2×C9—21no—
445C2×C9—21no—
446C2×C9—21no—
447C2×C9—42no—
448C2×C9C2×#23511yescharacteristic
449C2×C9C2×#23511yescharacteristic
450C2×C9C22×#6711yescharacteristic
451D9—22no—
452D9—22no—
453D9—122no—
454D9—242no—
455C24—91no—
456#41—544no—
457C2×Q8—31no—
458C2×D4—272no—
459C2×D4—272no—
460C2×D4—272no—
461C2×D4—271no—
462C22×C4—91no—
463C22×C4—91no—
464C22×C4—91no—
465C22×C4—91no—
466C22×C4—91no—
467C22×C4—184no—
468C22×C4—272no—
469C22×C4D3×D911yescharacteristic
470C4⋊C4—31no—
471C4⋊C4—32no—
472C4⋊C4—272no—
473C4⋊C4—271no—
474C22⋊C4—32no—
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—22no—
485C22×C3—21no—
486C22×C3—41no—
487C22×C3—41no—
488C22×C3—92no—
489C22×C3—92no—
490C22×C3—188no—
491C22×C3—181no—
492C22×C3—181no—
493C22×C3—181no—
494C22×C3—181no—
495C22×C3#370112yescomplemented
496C22×C3C23×D911yescharacteristic
497C22×C3#379912yes—
498C22×C3C22×D3211yescharacteristic
499C2×D3—32no—
500C2×D3—32no—
501C2×D3—68no—
502C2×D3—61no—
503C2×D3—61no—
504C2×D3—61no—
505C2×D3—61no—
506C2×D3—182no—
507C2×D3—182no—
508C2×D3—182no—
509C2×D3—182no—
510C2×D3—182no—
511C2×D3—182no—
512C2×D3—362no—
513C2×D3—362no—
514C2×D3—362no—
515C3×C4—21no—
516C3×C4—21no—
517C3×C4—22no—
518C3×C4—41no—
519C3×C4—62no—
520C3×C4—62no—
521C3×C4—92no—
522C3×C4—181no—
523C3×C4D4×D912yes—
524C3×C4D3×#6712yes—
525Dic3—22no—
526Dic3—22no—
527Dic3—32no—
528Dic3—61no—
529Dic3—182no—
530Dic3—182no—
531Dic3—362no—
532C32#562711yescharacteristic
533C9—21no—
534C9#579111yescharacteristic, complemented
535C23—91no—
536C23—91no—
537C23—182no—
538C23—182no—
539C23—182no—
540C23—272no—
541C23C2×D3×D911yescharacteristic
542Q8—62no—
543D4—544no—
544D4—544no—
545D4—544no—
546D4—542no—
547C2×C4—22no—
548C2×C4—32no—
549C2×C4—32no—
550C2×C4—91no—
551C2×C4—94no—
552C2×C4—91no—
553C2×C4—182no—
554C2×C4—182no—
555C2×C4—182no—
556C2×C4—182no—
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—182no—
568C2×C3—182no—
569C2×C3C2×D4×D911yescharacteristic
570C2×C3C2×#370111yescharacteristic
571C2×C3#41×D911yescharacteristic
572C2×C3C2×#379911yescharacteristic
573C2×C3D3×#23511yescharacteristic
574C2×C3C2×D3×#6711yescharacteristic
575D3—62no—
576D3—62no—
577D3—362no—
578D3—362no—
579D3—722no—
580C22—21no—
581C22—21no—
582C22—92no—
583C22—92no—
584C22—188no—
585C22—181no—
586C22—181no—
587C22—181no—
588C22—181no—
589C22—542no—
590C22—542no—
591C22—542no—
592C22#8863512yescomplemented
593C22C22×D3×D911yescharacteristic
594C4—21no—
595C4—62no—
596C4—62no—
597C4—92no—
598C4—181no—
599C4—542no—
600C4D9×#6712yes—
601C3—21no—
602C3#1058972911yescharacteristic, complemented
603C3#1059500511yescharacteristic
604C2—22no—
605C2—182no—
606C2—182no—
607C2—1082no—
608C2C2×#8863511yescharacteristic
609C2D9×#23511yescharacteristic
610C2C2×D9×#6711yescharacteristic
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×C3×C4×D9C2semidirect
2. C2×#88635C2semidirect
3. #11721819C2semidirect
4. C9×#924C2semidirect
5. #11721276C2semidirect
6. #11721241C2semidirect
7. #11721004C2semidirect
8. #11721002C2semidirect
9. #11720944C2semidirect
10. #11720938C2semidirect
11. #11720907C2semidirect
17. C2×C4⋊(C3⋊D9)C22semidirect
24. C2×C9×Dic6C22semidirect
27. C2×(C3⋊D9)⋊C4C22semidirect
29. #88536C22semidirect
30. C4⋊(C3⋊Dic9)C22semidirect
32. C9×C4⋊Dic3C22semidirect
33. C9×Dic3⋊C4C22semidirect
34. #88445C22semidirect
37. #88442C22semidirect
38. Dic3×Dic9C22semidirect
51. C22×C4×D9D3semidirect
83. C3×C4×D9D4semidirect
147. C22×C3×C9#41semidirect
158. C2×C3×D9#41semidirect
182. #924D9semidirect
248. C4×D9#67semidirect
289. C2×Dic6C2×D9semidirect
306. C4⋊Dic3C2×D9semidirect
309. Dic3⋊C4C2×D9semidirect
339. C22×C9#235semidirect
352. C2×D9#235semidirect
375. C3×C9#525semidirect
495. C22×C3#3701semidirect
534. C9#5791semidirect
592. C22#88635semidirect
602. C3#10589729semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×D92
C2×#886352
#117218192
C9×#9242
#117212762
#117212412
#117210042
#117210022
#117209442
#117209382
#117209072
#105950053
#105897293