Finite Group Databasebeta

← #13150475

Subgroups — #13150475

Order 1728.

1238 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊(C3⋊D9)C211yesmaximal, characteristic, complemented
2C2×#88642C211yesmaximal, characteristic, complemented
3C4×C9×#67C211yesmaximal, characteristic, complemented
4C3×#63486C211yesmaximal, characteristic, complemented
5#11721291C212yesmaximal, complemented
6#11721274C212yesmaximal, complemented
7#11721039C212yesmaximal, complemented
8#11721038C211yesmaximal, characteristic, complemented
9#11720997C212yesmaximal, complemented
10#11720993C211yesmaximal, characteristic, complemented
11#11720952C211yesmaximal, characteristic, complemented
12#10595018—31nomaximal
13#10589652—31nomaximal
14C23×C3⋊D9C2212yes—
15C22×C3×C4×C9C2211yescharacteristic
16C2×C4⋊(C3⋊D9)—24no—
17C2×C4⋊(C3⋊D9)C2212yes—
18C2×C4⋊(C3⋊D9)C2211yescharacteristic
19C2×C4⋊(C3⋊D9)C2211yescharacteristic
20C2×C9×#67C2211yescharacteristic
21C2×C4×D3×C9C2211yescharacteristic, complemented
22C2×C3×Dic18C2211yescharacteristic, complemented
23C2×#6195C2212yescomplemented
24C2×#6195C2212yescomplemented
25C2×(C3⋊D9)⋊C4C2212yescomplemented
26C2×(C3⋊D9)⋊C4C2212yescomplemented
27#88642—24no—
28C9×#217C2211yescharacteristic
29C3×#3676C2212yes—
30C9×(C2×D3)⋊C4C2211yescharacteristic, complemented
31C9×Dic3⋊C4C2211yescharacteristic, complemented
32C4×C9×Dic3C2211yescharacteristic, complemented
33C3×C4⋊Dic9C2211yescharacteristic, complemented
34C3×Dic9⋊C4C2212yescomplemented
35#88445C2212yescomplemented
36#88445C2212yescomplemented
37#88445C2212yescomplemented
38#88445C2212yescomplemented
39C22×C4⋊(C3⋊D3)—31no—
40C2×#3799—31no—
41C3×C4×#67D311yescharacteristic
42C3×#924—31no—
43#63981—32no—
44#63974—32no—
45#63910—32no—
46#63909—31no—
47#63897—32no—
48#63894—31no—
49#63864—31no—
50C2×#3697—31no—
51C22×D36—31no—
52C22×D36—61no—
53C4×D4×C9—31no—
54#63499—32no—
55#63492—31no—
56#63486D311yescharacteristic, complemented
57#63456—32no—
58#63446—32no—
59C22⋊C4×D9—32no—
60#63437—31no—
61C4×D36—31no—
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9—24no—
64C22×C3⋊D9—24no—
65C22×C3⋊D9—42no—
66C22×C3⋊D9C2312yes—
67C22×C3⋊D9C2312yes—
68C22×D3×C9C2311yescharacteristic
69C2×C3×C4×C9—22no—
70C2×C3×C4×C9C2311yescharacteristic
71C2×C3×C4×C9C2311yescharacteristic
72C4⋊(C3⋊D9)—24no—
73C4⋊(C3⋊D9)—22no—
74C4⋊(C3⋊D9)D414yescomplemented
75C9×#67—22no—
76C2×C9×Dic3C2311yescharacteristic
77C2×C9×Dic3C2311yescharacteristic
78C2×C9×Dic3C2311yescharacteristic
79C2×C3×Dic9C2312yes—
80C2×C3×Dic9C2312yes—
81C4×D3×C9—22no—
82C3×Dic18—22no—
83#6195—24no—
84#6195—42no—
85(C3⋊D9)⋊C4—24no—
86(C3⋊D9)⋊C4—42no—
87#5630—91no—
88#5550—91no—
89C23×C3⋊D3—32no—
90C22×C32×C4C2×D311yescharacteristic
91C2×C4⋊(C3⋊D3)—31no—
92C2×C4⋊(C3⋊D3)—31no—
93C2×C4⋊(C3⋊D3)—32no—
94C2×C4⋊(C3⋊D3)—64no—
95C2×C3×#67C2×D311yescharacteristic
96C2×C3×C4×D3C2×D311yescharacteristic
97C2×C3×Dic6—31no—
98C2×#626—32no—
99C2×#626—32no—
100C2×#624—32no—
101C2×#624—32no—
102#3799—64no—
103C23×D9—32no—
104C23×D9—62no—
105C3×#217—32no—
106C3×#217C2×D311yescharacteristic
107C3×(C2×D3)⋊C4C2×D311yescharacteristic
108C3×C4⋊Dic3—31no—
109C3×Dic3⋊C4—32no—
110C3×Dic3⋊C4C2×D311yescharacteristic
111C3×C4×Dic3C2×D311yescharacteristic
112#3722—32no—
113#3722—32no—
114#3722—32no—
115#3722—32no—
116C2×D4×C9—31no—
117C22×C4×C9—21no—
118C22×C4×C9—31no—
119C22×C4×C9C2×D311yescharacteristic
120C2×#611—32no—
121C2×#611—32no—
122#3697—64no—
123C2×D36—31no—
124C2×D36—31no—
125C2×D36—32no—
126C2×D36—64no—
127C2×D36—61no—
128C2×D36—61no—
129C2×D36—62no—
130C2×D36—68no—
131C2×C4×D9—32no—
132C2×C4×D9—32no—
133C2×Dic18C2×D311yescharacteristic, complemented
134C9×C4⋊C4—31no—
135C9×C22⋊C4—31no—
136C9×C22⋊C4—31no—
137C42×C9—31no—
138#3676C2×D312yes—
139(C2×D9)⋊C4—32no—
140(C2×D9)⋊C4—32no—
141(C2×D9)⋊C4—32no—
142(C2×D9)⋊C4—32no—
143C4⋊Dic9C2×D311yescharacteristic, complemented
144Dic9⋊C4C2×D312yescomplemented
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—22no—
153C2×C3⋊D9—22no—
154C2×C3⋊D9—48no—
155C2×C3⋊D9C2×D414yes—
156C2×D3×C9—21no—
157C2×D3×C9—21no—
158C2×D3×C9—21no—
159C3×C4×C9—21no—
160C3×C4×C9C2×D412yes—
161C9×Dic3—21no—
162C9×Dic3—21no—
163C9×Dic3—21no—
164C3×Dic9—22no—
165C3×Dic9—22no—
166C2×#239—91no—
167C2×#235—91no—
168C22×D12—91no—
169C22×D12—91no—
170C22×D12—181no—
171C3×C22⋊Q8—91no—
172C3×C4×D4—31no—
173#947—91no—
174#937—92no—
175#930—91no—
176C4×#67D911yescharacteristic, complemented
177#924—31no—
178#895—91no—
179#895—92no—
180#894—92no—
181#894—92no—
182#893—91no—
183#884—92no—
184#882—92no—
185D3×C22⋊C4—92no—
186D3×C22⋊C4—92no—
187#875—91no—
188C4×D12—91no—
189C23×C32C22×D311yescharacteristic
190C22×C3⋊D3—32no—
191C22×C3⋊D3—32no—
192C22×C3⋊D3—64no—
193C22×C3⋊D3—64no—
194C22×C3⋊D3—122no—
195C22×C3×D3C22×D311yescharacteristic
196C2×C32×C4—22no—
197C2×C32×C4C22×D311yescharacteristic
198C2×C32×C4C22×D311yescharacteristic
199C4⋊(C3⋊D3)—34no—
200C4⋊(C3⋊D3)—64no—
201C4⋊(C3⋊D3)—62no—
202C3×#67—22no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3C22×D311yescharacteristic
206C2×C3×Dic3C22×D311yescharacteristic
207C2×C3×Dic3C22×D311yescharacteristic
208C3×C4×D3—22no—
209C3×Dic6—62no—
210#626—64no—
211#626—122no—
212#624—64no—
213#624—122no—
214C23×C9—21no—
215C23×C9—31no—
216C23×C9C22×D311yescharacteristic
217C22×D9—32no—
218C22×D9—32no—
219C22×D9—62no—
220C22×D9—62no—
221C22×D9—64no—
222C22×D9—68no—
223C22×D9—64no—
224C22×D9—124no—
225C22×D9—122no—
226C22×D9—124no—
227D4×C9—62no—
228C2×C4×C9—24no—
229C2×C4×C9—22no—
230C2×C4×C9—21no—
231C2×C4×C9—21no—
232C2×C4×C9—31no—
233C2×C4×C9—31no—
234C2×C4×C9—31no—
235C2×C4×C9—62no—
236C2×C4×C9C22×D311yescharacteristic
237C2×C4×C9C22×D311yescharacteristic
238#611—64no—
239#611—122no—
240C2×Dic9C22×D312yes—
241C2×Dic9C22×D312yes—
242D36—34no—
243D36—64no—
244D36—64no—
245D36—68no—
246D36—64no—
247D36—62no—
248C4×D9—64no—
249C4×D9—122no—
250Dic18—22no—
251#546—271noSylow
252C2×C3×C9—22no—
253C2×C3×C9C22×D411yescharacteristic
254C2×C3×C9C2×#4111yescharacteristic
255C2×C3×C9H(5,𝔽2)11yescharacteristic
256C3⋊D9—24no—
257C3⋊D9—42no—
258D3×C9—41no—
259C23×D3—92no—
260C23×D3—92no—
261C23×D3—182no—
262C2×C3×Q8—91no—
263C2×C3×D4—31no—
264C22×C3×C4—21no—
265C22×C3×C4—31no—
266C22×C3×C4C2×D911yescharacteristic
267C22×C3×C4D3211yescharacteristic
268C2×#67—92no—
269C2×#67—92no—
270C2×#67C2×D911yescharacteristic
271#239—184no—
272#235—184no—
273C2×D12—91no—
274C2×D12—91no—
275C2×D12—92no—
276C2×D12—92no—
277C2×D12—91no—
278C2×D12—91no—
279C2×D12—92no—
280C2×D12—92no—
281C2×D12—181no—
282C2×D12—182no—
283C2×D12—181no—
284C2×D12—184no—
285C2×D12—184no—
286C2×D12—188no—
287C2×C4×D3—92no—
288C2×C4×D3—92no—
289C2×C4×D3—92no—
290C2×C4×D3—92no—
291C2×C4×D3C2×D911yescharacteristic, complemented
292C2×Dic6—31no—
293C3×C4⋊C4—31no—
294C3×C4⋊C4—91no—
295C3×C4⋊C4—92no—
296C3×C22⋊C4—31no—
297C3×C22⋊C4—31no—
298C3×C22⋊C4—92no—
299C3×C42—31no—
300#217—32no—
301#217C2×D911yescharacteristic
302(C2×D3)⋊C4—92no—
303(C2×D3)⋊C4—92no—
304(C2×D3)⋊C4—92no—
305(C2×D3)⋊C4—92no—
306(C2×D3)⋊C4—92no—
307(C2×D3)⋊C4—92no—
308(C2×D3)⋊C4—92no—
309(C2×D3)⋊C4—92no—
310(C2×D3)⋊C4C2×D911yescharacteristic, complemented
311C4⋊Dic3—31no—
312Dic3⋊C4—32no—
313Dic3⋊C4C2×D911yescharacteristic, complemented
314C4×Dic3C2×D911yescharacteristic, complemented
315C22×C32—21no—
316C22×C32—21no—
317C22×C32#23512yes—
318C22×C32C23×D311yescharacteristic
319C2×C3⋊D3—34no—
320C2×C3⋊D3—62no—
321C2×C3⋊D3—62no—
322C2×C3⋊D3—62no—
323C2×C3⋊D3—62no—
324C2×C3⋊D3—62no—
325C2×C3⋊D3—128no—
326C2×C3×D3—21no—
327C2×C3×D3—21no—
328C2×C3×D3—21no—
329C32×C4—21no—
330C32×C4D3×D412yes—
331C3×Dic3—21no—
332C3×Dic3—21no—
333C3×Dic3—21no—
334C3×Dic3—62no—
335C3×Dic3—62no—
336C22×C9—22no—
337C22×C9—22no—
338C22×C9—21no—
339C22×C9—21no—
340C22×C9—21no—
341C22×C9—22no—
342C22×C9—61no—
343C22×C9—61no—
344C22×C9—61no—
345C22×C9#23912yescomplemented
346C22×C9C23×D311yescharacteristic
347C2×D9—34no—
348C2×D9—64no—
349C2×D9—62no—
350C2×D9—62no—
351C2×D9—64no—
352C2×D9—62no—
353C2×D9—62no—
354C2×D9—62no—
355C2×D9—122no—
356C2×D9—122no—
357C2×D9—122no—
358C2×D9—122no—
359C2×D9—128no—
360C2×D9—1216no—
361C4×C9—22no—
362C4×C9—21no—
363C4×C9—22no—
364C4×C9—61no—
365C4×C9—61no—
366C4×C9—61no—
367C4×C9D3×D412yes—
368Dic9—22no—
369Dic9—22no—
370C2×#41—271no—
371C22×D4—271no—
372#124—271no—
373#123—272no—
374C22⋊Q8—91no—
375#121—272no—
376#121—271no—
377#120—272no—
378C4×D4—31no—
379C4×D4—271no—
380C2×C22⋊C4—272no—
381C3×C9#54611yescharacteristic, complemented, Sylow
382C23×C3—21no—
383C23×C3—31no—
384C23×C3C22×D911yescharacteristic
385C23×C3C2×D3211yescharacteristic
386C22×D3—92no—
387C22×D3—92no—
388C22×D3—92no—
389C22×D3—92no—
390C22×D3—182no—
391C22×D3—182no—
392C22×D3—184no—
393C22×D3—184no—
394C22×D3—188no—
395C22×D3—184no—
396C22×D3—184no—
397C22×D3—364no—
398C22×D3—362no—
399C22×D3—362no—
400C22×D3—364no—
401C22×D3C22×D911yescharacteristic
402C3×Q8—182no—
403C3×D4—62no—
404C2×C3×C4—24no—
405C2×C3×C4—22no—
406C2×C3×C4—22no—
407C2×C3×C4—21no—
408C2×C3×C4—21no—
409C2×C3×C4—31no—
410C2×C3×C4—31no—
411C2×C3×C4—31no—
412C2×C3×C4—62no—
413C2×C3×C4—92no—
414C2×C3×C4—92no—
415C2×C3×C4C22×D911yescharacteristic
416C2×C3×C4C22×D911yescharacteristic
417C2×C3×C4C2×D3211yescharacteristic
418C2×C3×C4C2×D3211yescharacteristic
419#67—22no—
420#67—184no—
421#67—362no—
422C2×Dic3—32no—
423C2×Dic3—32no—
424C2×Dic3C22×D911yescharacteristic
425C2×Dic3C22×D911yescharacteristic
426C2×Dic3C22×D911yescharacteristic
427D12—94no—
428D12—94no—
429D12—184no—
430D12—184no—
431D12—184no—
432D12—188no—
433D12—184no—
434D12—184no—
435D12—182no—
436D12—182no—
437D12—362no—
438C4×D3—22no—
439C4×D3—184no—
440C4×D3—184no—
441C4×D3—362no—
442C4×D3—362no—
443Dic6—62no—
444C2×C32—22no—
445C2×C32C2×#23511yescharacteristic
446C2×C32C2×D3×D411yescharacteristic
447C2×C32#100911yescharacteristic
448C3⋊D3—64no—
449C3⋊D3—122no—
450C3×D3—41no—
451C2×C9—24no—
452C2×C9—22no—
453C2×C9—21no—
454C2×C9—21no—
455C2×C9—21no—
456C2×C9—121no—
457C2×C9C2×D3×D411yescharacteristic
458C2×C9C2×#23911yescharacteristic
459C2×C9#100911yescharacteristic
460D9—64no—
461D9—124no—
462D9—124no—
463D9—122no—
464C24—272no—
465#41—544no—
466C2×Q8—91no—
467C2×D4—31no—
468C2×D4—272no—
469C2×D4—272no—
470C2×D4—271no—
471C2×D4—272no—
472C2×D4—271no—
473C2×D4—544no—
474C22×C4—31no—
475C22×C4—272no—
476C22×C4—272no—
477C22×C4D3×D911yescharacteristic
478C4⋊C4—31no—
479C4⋊C4—91no—
480C4⋊C4—92no—
481C22⋊C4—31no—
482C22⋊C4—31no—
483C22⋊C4—92no—
484C22⋊C4—272no—
485C22⋊C4—272no—
486C22⋊C4—272no—
487C22⋊C4—272no—
488C42—31no—
489C22×C3—22no—
490C22×C3—22no—
491C22×C3—21no—
492C22×C3—21no—
493C22×C3—21no—
494C22×C3—21no—
495C22×C3—21no—
496C22×C3—22no—
497C22×C3—61no—
498C22×C3—61no—
499C22×C3—61no—
500C22×C3#369712yescomplemented
501C22×C3C23×D911yescharacteristic
502C22×C3#379912yes—
503C22×C3C22×D3211yescharacteristic
504C2×D3—21no—
505C2×D3—21no—
506C2×D3—21no—
507C2×D3—94no—
508C2×D3—94no—
509C2×D3—184no—
510C2×D3—182no—
511C2×D3—182no—
512C2×D3—182no—
513C2×D3—182no—
514C2×D3—184no—
515C2×D3—182no—
516C2×D3—182no—
517C2×D3—182no—
518C2×D3—182no—
519C2×D3—182no—
520C2×D3—182no—
521C2×D3—362no—
522C2×D3—362no—
523C2×D3—362no—
524C2×D3—362no—
525C2×D3—368no—
526C2×D3—368no—
527C2×D3—3616no—
528C3×C4—22no—
529C3×C4—21no—
530C3×C4—21no—
531C3×C4—22no—
532C3×C4—61no—
533C3×C4—61no—
534C3×C4—61no—
535C3×C4—182no—
536C3×C4—182no—
537C3×C4D4×D912yes—
538C3×C4#381112yes—
539Dic3—21no—
540Dic3—21no—
541Dic3—21no—
542Dic3—62no—
543Dic3—62no—
544C32#555011yescharacteristic
545C9—21no—
546C9#563011yescharacteristic, complemented
547C23—31no—
548C23—272no—
549C23—272no—
550C23—544no—
551C23—544no—
552C23—1082no—
553C23C2×D3×D911yescharacteristic
554Q8—182no—
555D4—62no—
556D4—274no—
557D4—544no—
558D4—544no—
559D4—542no—
560D4—1082no—
561C2×C4—22no—
562C2×C4—31no—
563C2×C4—31no—
564C2×C4—31no—
565C2×C4—62no—
566C2×C4—92no—
567C2×C4—92no—
568C2×C4—544no—
569C2×C4—1082no—
570C2×C4C2×D3×D911yescharacteristic
571C2×C4C2×D3×D911yescharacteristic
572C2×C3—24no—
573C2×C3—22no—
574C2×C3—22no—
575C2×C3—21no—
576C2×C3—21no—
577C2×C3—21no—
578C2×C3—121no—
579C2×C3C2×#369711yescharacteristic
580C2×C3C2×D4×D911yescharacteristic
581C2×C3#6371511yescharacteristic
582C2×C3C2×#379911yescharacteristic
583C2×C3#6430811yescharacteristic
584C2×C3C2×#381111yescharacteristic
585D3—41no—
586D3—184no—
587D3—184no—
588D3—364no—
589D3—364no—
590D3—362no—
591D3—362no—
592C22—21no—
593C22—21no—
594C22—61no—
595C22—61no—
596C22—61no—
597C22—274no—
598C22—542no—
599C22—542no—
600C22—542no—
601C22—542no—
602C22—542no—
603C22—1088no—
604C22#8864212yescomplemented
605C22C22×D3×D911yescharacteristic
606C4—21no—
607C4—61no—
608C4—61no—
609C4—61no—
610C4—182no—
611C4—182no—
612C4#8866812yes—
613C3—21no—
614C3#1058965211yescharacteristic, complemented
615C3#1059501811yescharacteristic
616C2—22no—
617C2—121no—
618C2—544no—
619C2—1082no—
620C2C2×#8864211yescharacteristic
621C2#1172247811yescharacteristic
622C2C2×#8866811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#546
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⋊(C3⋊D9)C2semidirect
2. C2×#88642C2semidirect
3. C4×C9×#67C2semidirect
4. C3×#63486C2semidirect
5. #11721291C2semidirect
6. #11721274C2semidirect
7. #11721039C2semidirect
8. #11721038C2semidirect
9. #11720997C2semidirect
10. #11720993C2semidirect
11. #11720952C2semidirect
21. C2×C4×D3×C9C22semidirect
22. C2×C3×Dic18C22semidirect
23. C2×#6195C22semidirect
24. C2×#6195C22semidirect
25. C2×(C3⋊D9)⋊C4C22semidirect
26. C2×(C3⋊D9)⋊C4C22semidirect
30. C9×(C2×D3)⋊C4C22semidirect
31. C9×Dic3⋊C4C22semidirect
32. C4×C9×Dic3C22semidirect
33. C3×C4⋊Dic9C22semidirect
34. C3×Dic9⋊C4C22semidirect
35. #88445C22semidirect
36. #88445C22semidirect
37. #88445C22semidirect
38. #88445C22semidirect
56. #63486D3semidirect
74. C4⋊(C3⋊D9)D4semidirect
133. C2×Dic18C2×D3semidirect
143. C4⋊Dic9C2×D3semidirect
144. Dic9⋊C4C2×D3semidirect
147. C22×C3×C9#41semidirect
176. C4×#67D9semidirect
291. C2×C4×D3C2×D9semidirect
310. (C2×D3)⋊C4C2×D9semidirect
313. Dic3⋊C4C2×D9semidirect
314. C4×Dic3C2×D9semidirect
345. C22×C9#239semidirect
381. C3×C9#546semidirect
500. C22×C3#3697semidirect
546. C9#5630semidirect
604. C22#88642semidirect
614. C3#10589652semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊(C3⋊D9)2
C2×#886422
C4×C9×#672
C3×#634862
#117212912
#117212742
#117210392
#117210382
#117209972
#117209932
#117209522
#105950183
#105896523