Finite Group Databasebeta

← #13150329

Subgroups — #13150329

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×#88635C211yesmaximal, characteristic, complemented
3C3×C4×#611C211yesmaximal, characteristic, complemented
4C9×#924C211yesmaximal, characteristic, complemented
5#11721278C212yesmaximal, complemented
6#11721274C212yesmaximal, complemented
7#11721005C212yesmaximal, complemented
8#11721004C211yesmaximal, characteristic, complemented
9#11720997C212yesmaximal, complemented
10#11720992C211yesmaximal, characteristic, complemented
11#11720953C211yesmaximal, characteristic, complemented
12#10595018—31nomaximal
13#10589732—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×C3×#611C2211yescharacteristic
21C2×C3×C4×D9C2211yescharacteristic, complemented
22C2×C9×Dic6C2211yescharacteristic, complemented
23C2×#6192C2212yescomplemented
24C2×#6192C2212yescomplemented
25C2×(C3⋊D9)⋊C4C2212yescomplemented
26C2×(C3⋊D9)⋊C4C2212yescomplemented
27#88635—24no—
28C9×#217C2212yes—
29C3×#3676C2211yescharacteristic
30C3×(C2×D9)⋊C4C2211yescharacteristic, complemented
31C9×C4⋊Dic3C2211yescharacteristic, complemented
32C9×Dic3⋊C4C2212yescomplemented
33C3×Dic9⋊C4C2211yescharacteristic, complemented
34C3×C4×Dic9C2211yescharacteristic, complemented
35#88445C2212yescomplemented
36#88445C2212yescomplemented
37#88445C2212yescomplemented
38#88445C2212yescomplemented
39C22×C4⋊(C3⋊D3)—31no—
40C2×#3799—31no—
41C3×C4×#67—31no—
42C3×#924D311yescharacteristic
43#63981—32no—
44#63974—32no—
45#63910—32no—
46#63909—31no—
47#63897—32no—
48#63894—31no—
49#63864—31no—
50C2×#3701—31no—
51C22×D36—31no—
52C22×D36—61no—
53C9×C22⋊Q8—31no—
54#63509—31no—
55C4×#611D311yescharacteristic, complemented
56#63457—31no—
57#63457—32no—
58#63456—32no—
59#63455—31no—
60#63444—32no—
61C22⋊C4×D9—32no—
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9—24no—
64C22×C3⋊D9—24no—
65C22×C3⋊D9—42no—
66C22×C3⋊D9C2312yes—
67C22×C3⋊D9C2312yes—
68C22×C3×D9C2311yescharacteristic
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
75C3×#611—22no—
76C2×C9×Dic3C2312yes—
77C2×C9×Dic3C2312yes—
78C2×C3×Dic9C2311yescharacteristic
79C2×C3×Dic9C2311yescharacteristic
80C2×C3×Dic9C2311yescharacteristic
81C3×C4×D9—22no—
82C9×Dic6—22no—
83#6192—24no—
84#6192—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×#67—31no—
96C2×C3×C4×D3—31no—
97C2×C3×Dic6C2×D311yescharacteristic
98C2×#626—32no—
99C2×#626—32no—
100C2×#624—32no—
101C2×#624—32no—
102#3799—64no—
103C23×D9—32no—
104C23×D9—62no—
105C3×#217—31no—
106C3×#217C2×D312yes—
107C3×(C2×D3)⋊C4—31no—
108C3×C4⋊Dic3C2×D311yescharacteristic
109C3×Dic3⋊C4—31no—
110C3×Dic3⋊C4C2×D312yes—
111C3×C4×Dic3—31no—
112#3722—32no—
113#3722—32no—
114#3722—32no—
115#3722—32no—
116C2×Q8×C9—31no—
117C22×C4×C9—21no—
118C22×C4×C9C2×D311yescharacteristic
119C2×#611C2×D311yescharacteristic
120#3701—64no—
121C2×D36—31no—
122C2×D36—32no—
123C2×D36—31no—
124C2×D36—32no—
125C2×D36—32no—
126C2×D36—64no—
127C2×D36—61no—
128C2×D36—62no—
129C2×D36—61no—
130C2×D36—68no—
131C2×C4×D9—32no—
132C2×C4×D9—32no—
133C2×C4×D9C2×D311yescharacteristic, complemented
134C9×C4⋊C4—32no—
135C9×C4⋊C4—31no—
136C9×C22⋊C4—32no—
137#3676C2×D311yescharacteristic
138(C2×D9)⋊C4—32no—
139(C2×D9)⋊C4—32no—
140(C2×D9)⋊C4—32no—
141(C2×D9)⋊C4—32no—
142(C2×D9)⋊C4C2×D311yescharacteristic, complemented
143Dic9⋊C4C2×D311yescharacteristic, complemented
144C4×Dic9C2×D311yescharacteristic, complemented
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×C3×D9—21no—
157C2×C3×D9—21no—
158C2×C3×D9—21no—
159C3×C4×C9—21no—
160C3×C4×C9C2×D412yes—
161C9×Dic3—22no—
162C9×Dic3—22no—
163C3×Dic9—21no—
164C3×Dic9—21no—
165C3×Dic9—21no—
166C2×#239—91no—
167C2×#235—91no—
168C22×D12—91no—
169C22×D12—91no—
170C22×D12—181no—
171C3×C22⋊Q8—31no—
172C3×C4×D4—91no—
173#947—91no—
174#937—92no—
175#930—91no—
176C4×#67—31no—
177#924D911yescharacteristic, complemented
178#895—92no—
179#895—91no—
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×D3—31no—
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—62no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3—31no—
205C2×C3×Dic3—31no—
206C2×C3×Dic3C22×D312yes—
207C2×C3×Dic3C22×D312yes—
208C3×C4×D3—62no—
209C3×Dic6—22no—
210#626—64no—
211#626—122no—
212#624—64no—
213#624—122no—
214C23×C9—21no—
215C23×C9C22×D311yescharacteristic
216C22×D9—32no—
217C22×D9—32no—
218C22×D9—62no—
219C22×D9—62no—
220C22×D9—64no—
221C22×D9—68no—
222C22×D9—64no—
223C22×D9—124no—
224C22×D9—122no—
225C22×D9—124no—
226C22×D9C22×D311yescharacteristic
227Q8×C9—62no—
228C2×C4×C9—24no—
229C2×C4×C9—22no—
230C2×C4×C9—21no—
231C2×C4×C9—21no—
232C2×C4×C9—32no—
233C2×C4×C9—32no—
234C2×C4×C9C22×D311yescharacteristic
235C2×C4×C9C22×D311yescharacteristic
236#611—22no—
237C2×Dic9C22×D311yescharacteristic
238C2×Dic9C22×D311yescharacteristic
239C2×Dic9C22×D311yescharacteristic
240D36—34no—
241D36—64no—
242D36—64no—
243D36—68no—
244D36—64no—
245D36—64no—
246D36—62no—
247D36—122no—
248C4×D9—22no—
249C4×D9—64no—
250C4×D9—122no—
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—
258C3×D9—41no—
259C23×D3—92no—
260C23×D3—92no—
261C23×D3—182no—
262C2×C3×Q8—31no—
263C2×C3×D4—91no—
264C22×C3×C4—21no—
265C22×C3×C4—91no—
266C22×C3×C4C2×D911yescharacteristic
267C22×C3×C4D3211yescharacteristic
268C2×#67—31no—
269C2×#67—92no—
270C2×#67—92no—
271#239—184no—
272#235—184no—
273C2×D12—91no—
274C2×D12—91no—
275C2×D12—92no—
276C2×D12—91no—
277C2×D12—92no—
278C2×D12—92no—
279C2×D12—91no—
280C2×D12—92no—
281C2×D12—181no—
282C2×D12—182no—
283C2×D12—181no—
284C2×D12—184no—
285C2×D12—184no—
286C2×D12—188no—
287C2×C4×D3—31no—
288C2×C4×D3—92no—
289C2×C4×D3—92no—
290C2×C4×D3—92no—
291C2×C4×D3—92no—
292C2×Dic6C2×D911yescharacteristic, complemented
293C3×C4⋊C4—31no—
294C3×C4⋊C4—32no—
295C3×C4⋊C4—91no—
296C3×C22⋊C4—32no—
297C3×C22⋊C4—91no—
298C3×C22⋊C4—91no—
299C3×C42—91no—
300#217—31no—
301#217C2×D912yes—
302(C2×D3)⋊C4—31no—
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)⋊C4—92no—
311C4⋊Dic3C2×D911yescharacteristic, complemented
312Dic3⋊C4—31no—
313Dic3⋊C4C2×D912yescomplemented
314C4×Dic3—31no—
315C22×C32—21no—
316C22×C32—21no—
317C22×C32#23912yes—
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—61no—
327C2×C3×D3—61no—
328C2×C3×D3—61no—
329C32×C4—21no—
330C32×C4D3×D412yes—
331C3×Dic3—22no—
332C3×Dic3—22no—
333C3×Dic3—61no—
334C3×Dic3—61no—
335C3×Dic3—61no—
336C22×C9—22no—
337C22×C9—22no—
338C22×C9—21no—
339C22×C9—21no—
340C22×C9—21no—
341C22×C9—22no—
342C22×C9#23512yescomplemented
343C22×C9C23×D311yescharacteristic
344C2×D9—21no—
345C2×D9—21no—
346C2×D9—21no—
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—62no—
365C4×C9—62no—
366C4×C9D3×D412yes—
367Dic9—21no—
368Dic9—21no—
369Dic9—21no—
370C2×#41—271no—
371C22×D4—271no—
372#124—271no—
373#123—272no—
374C22⋊Q8—31no—
375#121—272no—
376#121—271no—
377#120—272no—
378C4×D4—91no—
379C4×D4—271no—
380C2×C22⋊C4—272no—
381C3×C9#54611yescharacteristic, complemented, Sylow
382C23×C3—21no—
383C23×C3—91no—
384C23×C3C22×D911yescharacteristic
385C23×C3C2×D3211yescharacteristic
386C22×D3—31no—
387C22×D3—92no—
388C22×D3—92no—
389C22×D3—92no—
390C22×D3—92no—
391C22×D3—182no—
392C22×D3—182no—
393C22×D3—184no—
394C22×D3—184no—
395C22×D3—188no—
396C22×D3—184no—
397C22×D3—184no—
398C22×D3—364no—
399C22×D3—362no—
400C22×D3—362no—
401C22×D3—364no—
402C3×Q8—62no—
403C3×D4—182no—
404C2×C3×C4—24no—
405C2×C3×C4—22no—
406C2×C3×C4—22no—
407C2×C3×C4—21no—
408C2×C3×C4—21no—
409C2×C3×C4—32no—
410C2×C3×C4—32no—
411C2×C3×C4—91no—
412C2×C3×C4—91no—
413C2×C3×C4—91no—
414C2×C3×C4—182no—
415C2×C3×C4C22×D911yescharacteristic
416C2×C3×C4C22×D911yescharacteristic
417C2×C3×C4C2×D3211yescharacteristic
418C2×C3×C4C2×D3211yescharacteristic
419#67—62no—
420#67—184no—
421#67—362no—
422C2×Dic3—31no—
423C2×Dic3—31no—
424C2×Dic3—31no—
425C2×Dic3C22×D912yes—
426C2×Dic3C22×D912yes—
427D12—94no—
428D12—94no—
429D12—184no—
430D12—184no—
431D12—184no—
432D12—188no—
433D12—184no—
434D12—184no—
435D12—182no—
436D12—182no—
437D12—362no—
438C4×D3—62no—
439C4×D3—184no—
440C4×D3—184no—
441C4×D3—362no—
442C4×D3—362no—
443Dic6—22no—
444C2×C32—22no—
445C2×C32C2×D3×D411yescharacteristic
446C2×C32C2×#23911yescharacteristic
447C2×C32#100911yescharacteristic
448C3⋊D3—64no—
449C3⋊D3—122no—
450C3×D3—121no—
451C2×C9—24no—
452C2×C9—22no—
453C2×C9—21no—
454C2×C9—21no—
455C2×C9—21no—
456C2×C9C2×#23511yescharacteristic
457C2×C9C2×D3×D411yescharacteristic
458C2×C9#100911yescharacteristic
459D9—41no—
460D9—64no—
461D9—124no—
462D9—124no—
463D9—122no—
464C24—272no—
465#41—544no—
466C2×Q8—31no—
467C2×D4—91no—
468C2×D4—272no—
469C2×D4—272no—
470C2×D4—271no—
471C2×D4—272no—
472C2×D4—271no—
473C2×D4—544no—
474C22×C4—91no—
475C22×C4—272no—
476C22×C4—272no—
477C22×C4D3×D911yescharacteristic
478C4⋊C4—31no—
479C4⋊C4—32no—
480C4⋊C4—91no—
481C22⋊C4—32no—
482C22⋊C4—91no—
483C22⋊C4—91no—
484C22⋊C4—272no—
485C22⋊C4—272no—
486C22⋊C4—272no—
487C22⋊C4—272no—
488C42—91no—
489C22×C3—22no—
490C22×C3—22no—
491C22×C3—21no—
492C22×C3—21no—
493C22×C3—21no—
494C22×C3—21no—
495C22×C3—22no—
496C22×C3—21no—
497C22×C3—181no—
498C22×C3—181no—
499C22×C3—181no—
500C22×C3#370112yescomplemented
501C22×C3C23×D911yescharacteristic
502C22×C3#379912yes—
503C22×C3C22×D3211yescharacteristic
504C2×D3—61no—
505C2×D3—61no—
506C2×D3—61no—
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—62no—
533C3×C4—62no—
534C3×C4—181no—
535C3×C4—181no—
536C3×C4—181no—
537C3×C4D4×D912yes—
538C3×C4#381112yes—
539Dic3—22no—
540Dic3—22no—
541Dic3—61no—
542Dic3—61no—
543Dic3—61no—
544C32#563011yescharacteristic
545C9—21no—
546C9#555011yescharacteristic, complemented
547C23—91no—
548C23—272no—
549C23—272no—
550C23—544no—
551C23—544no—
552C23—1082no—
553C23C2×D3×D911yescharacteristic
554Q8—62no—
555D4—182no—
556D4—274no—
557D4—544no—
558D4—544no—
559D4—542no—
560D4—1082no—
561C2×C4—22no—
562C2×C4—32no—
563C2×C4—32no—
564C2×C4—91no—
565C2×C4—91no—
566C2×C4—91no—
567C2×C4—182no—
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—361no—
579C2×C3C2×D4×D911yescharacteristic
580C2×C3C2×#370111yescharacteristic
581C2×C3#6371511yescharacteristic
582C2×C3C2×#379911yescharacteristic
583C2×C3#6430811yescharacteristic
584C2×C3C2×#381111yescharacteristic
585D3—121no—
586D3—184no—
587D3—184no—
588D3—364no—
589D3—364no—
590D3—362no—
591D3—362no—
592C22—21no—
593C22—21no—
594C22—181no—
595C22—181no—
596C22—181no—
597C22—274no—
598C22—542no—
599C22—542no—
600C22—542no—
601C22—542no—
602C22—542no—
603C22—1088no—
604C22#8863512yescomplemented
605C22C22×D3×D911yescharacteristic
606C4—21no—
607C4—62no—
608C4—62no—
609C4—181no—
610C4—181no—
611C4—181no—
612C4#8866812yes—
613C3—21no—
614C3#1058973211yescharacteristic, complemented
615C3#1059501811yescharacteristic
616C2—22no—
617C2—361no—
618C2—544no—
619C2—1082no—
620C2C2×#8863511yescharacteristic
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×#88635C2semidirect
3. C3×C4×#611C2semidirect
4. C9×#924C2semidirect
5. #11721278C2semidirect
6. #11721274C2semidirect
7. #11721005C2semidirect
8. #11721004C2semidirect
9. #11720997C2semidirect
10. #11720992C2semidirect
11. #11720953C2semidirect
21. C2×C3×C4×D9C22semidirect
22. C2×C9×Dic6C22semidirect
23. C2×#6192C22semidirect
24. C2×#6192C22semidirect
25. C2×(C3⋊D9)⋊C4C22semidirect
26. C2×(C3⋊D9)⋊C4C22semidirect
30. C3×(C2×D9)⋊C4C22semidirect
31. C9×C4⋊Dic3C22semidirect
32. C9×Dic3⋊C4C22semidirect
33. C3×Dic9⋊C4C22semidirect
34. C3×C4×Dic9C22semidirect
35. #88445C22semidirect
36. #88445C22semidirect
37. #88445C22semidirect
38. #88445C22semidirect
55. C4×#611D3semidirect
74. C4⋊(C3⋊D9)D4semidirect
133. C2×C4×D9C2×D3semidirect
142. (C2×D9)⋊C4C2×D3semidirect
143. Dic9⋊C4C2×D3semidirect
144. C4×Dic9C2×D3semidirect
147. C22×C3×C9#41semidirect
177. #924D9semidirect
292. C2×Dic6C2×D9semidirect
311. C4⋊Dic3C2×D9semidirect
313. Dic3⋊C4C2×D9semidirect
342. C22×C9#235semidirect
381. C3×C9#546semidirect
500. C22×C3#3701semidirect
546. C9#5550semidirect
604. C22#88635semidirect
614. C3#10589732semidirect

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×#886352
C3×C4×#6112
C9×#9242
#117212782
#117212742
#117210052
#117210042
#117209972
#117209922
#117209532
#105950183
#105897323