Finite Group Databasebeta

← #13147722

Subgroups — #13147722

Order 1728.

1096 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D9×D12C211yesmaximal, characteristic, complemented
2C2×#88635C211yesmaximal, characteristic, complemented
3C4×C4⋊(C3⋊D9)C211yesmaximal, characteristic, complemented
4C9×#875C211yesmaximal, characteristic, complemented
5C3×C4×D36C211yesmaximal, characteristic, complemented
6#11721049C212yesmaximal, complemented
7#11721048C212yesmaximal, complemented
8#11721041C212yesmaximal, complemented
9#11721036C211yesmaximal, characteristic, complemented
10#11720958C212yesmaximal, complemented
11#11720944C211yesmaximal, characteristic, complemented
12#10594007—31nomaximal
13#10589778—31nomaximal
14C22×D3×D9C2212yes—
15C2×C4⋊(C3⋊D9)C2211yescharacteristic
16C2×C4×C3⋊D9C2212yescomplemented
17C2×C9×D12C2211yescharacteristic
18C2×C3×D36C2211yescharacteristic
19C2×C3×C4×D9C2211yescharacteristic
20C2×C3×C4×D9C2211yescharacteristic, complemented
21C2×C9×Dic6C2211yescharacteristic, complemented
22C2×#6195C2212yes—
23C2×#6194C2212yescomplemented
24C2×#6192C2212yescomplemented
25C2×(C3⋊D9)⋊C4C2212yescomplemented
26D9×D12—24no—
27#88635—24no—
28C3×C42×C9C2211yescharacteristic, complemented
29#88536C2212yescomplemented
30C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
31C9×(C2×D3)⋊C4C2212yescomplemented
32C9×(C2×D3)⋊C4C2212yescomplemented
33C3×(C2×D9)⋊C4C2211yescharacteristic
34C3×(C2×D9)⋊C4C2211yescharacteristic, complemented
35C3×C4⋊Dic9C2211yescharacteristic, complemented
36(C2×D3)⋊Dic9C2212yescomplemented
37(C2×D9)⋊Dic3C2212yes—
38#88442C2212yescomplemented
39C2×D3×D12—31no—
40C2×#3799—31no—
41C4×C4⋊(C3⋊D3)—31no—
42C3×#875D311yescharacteristic
43C3×C4×D12—31no—
44#63922—32no—
45#63921—32no—
46#63912—32no—
47#63908—31no—
48#63856—31no—
49#63854—32no—
50C2×#3701—31no—
51C2×D4×D9—31no—
52C9×#124—31no—
53#63508—31no—
54#63500—31no—
55#63446—32no—
56#63445—32no—
57#63444—32no—
58C22⋊C4×D9—32no—
59C4×D36—31no—
60C4×D36—61no—
61C4×D36D311yescharacteristic, complemented
62C22×C3⋊D9C2312yes—
63C22×D3×C9C2312yes—
64C22×C3×D9C2311yescharacteristic
65C22×C3×D9C2311yescharacteristic
66C2×D3×D9—24no—
67C2×D3×D9—24no—
68C2×D3×D9—42no—
69C2×C3×C4×C9C2311yescharacteristic
70C2×C3×C4×C9C2311yescharacteristic
71C2×C3×C4×C9C2311yescharacteristic
72C2×C3⋊Dic9C2312yes—
73C4⋊(C3⋊D9)D414yescomplemented
74C4×C3⋊D9—42no—
75C2×C9×Dic3C2312yes—
76C2×C3×Dic9C2311yescharacteristic
77C2×C3×Dic9C2311yescharacteristic
78C9×D12—22no—
79C3×D36—22no—
80C3×C4×D9—22no—
81C3×C4×D9—22no—
82C9×Dic6—22no—
83#6195—24no—
84#6194—42no—
85#6192—24no—
86(C3⋊D9)⋊C4—24no—
87#5676—91no—
88#5550—91no—
89C22×D32—32no—
90C2×C4⋊(C3⋊D3)—31no—
91C2×C4×C3⋊D3—32no—
92C2×C3×D12—31no—
93C2×C3×D12C2×D311yescharacteristic
94C2×C3×C4×D3—31no—
95C2×C3×C4×D3—31no—
96C2×C3×Dic6C2×D311yescharacteristic
97C2×#626—32no—
98C2×#626—32no—
99C2×#625—32no—
100C2×#624—32no—
101D3×D12—64no—
102#3799—64no—
103C23×D9—32no—
104C32×C42C2×D311yescharacteristic
105#3752—32no—
106C4⋊(C3⋊Dic3)—31no—
107C3×(C2×D3)⋊C4—31no—
108C3×(C2×D3)⋊C4—31no—
109C3×(C2×D3)⋊C4C2×D312yes—
110C3×(C2×D3)⋊C4C2×D312yes—
111C3×C4⋊Dic3—31no—
112#3724—32no—
113(C2×D3)⋊Dic3—32no—
114(C2×D3)⋊Dic3—32no—
115C2×Q8×C9—31no—
116C2×D4×C9—31no—
117C2×#611—32no—
118C2×#611—32no—
119#3701—64no—
120D4×D9—64no—
121C2×D36—31no—
122C2×D36—32no—
123C2×D36—61no—
124C2×D36C2×D311yescharacteristic
125C2×C4×D9—32no—
126C2×C4×D9—32no—
127C2×C4×D9—62no—
128C2×C4×D9C2×D311yescharacteristic
129C2×C4×D9C2×D311yescharacteristic, complemented
130C9×C22⋊C4—32no—
131C9×C22⋊C4—32no—
132C42×C9—21no—
133C42×C9C2×D311yescharacteristic, complemented
134#3676—32no—
135(C2×D9)⋊C4—32no—
136(C2×D9)⋊C4—32no—
137(C2×D9)⋊C4—62no—
138(C2×D9)⋊C4C2×D311yescharacteristic
139(C2×D9)⋊C4C2×D311yescharacteristic, complemented
140C4⋊Dic9—31no—
141C4⋊Dic9—61no—
142C4⋊Dic9C2×D311yescharacteristic, complemented
143Dic9⋊C4—32no—
144C22×C3×C9C2411yescharacteristic
145C2×C3⋊D9—22no—
146C2×C3⋊D9—22no—
147C2×C3⋊D9C2×D414yes—
148C2×D3×C9—22no—
149C2×D3×C9—22no—
150C2×D3×C9—22no—
151C2×C3×D9—21no—
152C2×C3×D9—21no—
153C2×C3×D9—21no—
154C2×C3×D9—21no—
155C2×C3×D9—21no—
156C2×C3×D9#4112yescomplemented
157D3×D9—48no—
158C3×C4×C9—21no—
159C3×C4×C9—21no—
160C3×C4×C9C2×D412yes—
161C3⋊Dic9—22no—
162C9×Dic3—22no—
163C3×Dic9—21no—
164C3×Dic9—21no—
165C2×#239—91no—
166C2×D3×D4—91no—
167C2×#235—91no—
168C22×D12—91no—
169C3×#124—31no—
170C3×C4×D4—91no—
171#946—91no—
172#938—91no—
173#937—92no—
174#930—91no—
175#924—91no—
176#894—92no—
177#884—92no—
178#884—92no—
179#883—92no—
180#882—92no—
181D3×C22⋊C4—92no—
182D3×C22⋊C4—92no—
183#875D911yescharacteristic, complemented
184C4×D12—31no—
185C4×D12—91no—
186C4×D12—91no—
187C4×D12—181no—
188C22×C3⋊D3—32no—
189C22×C3×D3—31no—
190C22×C3×D3—31no—
191C22×C3×D3C22×D312yes—
192C2×D32—64no—
193C2×D32—64no—
194C2×D32—122no—
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C22×D311yescharacteristic
197C2×C32×C4C22×D311yescharacteristic
198C2×C3⋊Dic3—32no—
199C4⋊(C3⋊D3)—34no—
200C4×C3⋊D3—122no—
201C2×C3×Dic3—31no—
202C2×C3×Dic3—31no—
203C2×C3×Dic3C22×D312yes—
204C3×D12—22no—
205C3×D12—62no—
206C3×C4×D3—62no—
207C3×C4×D3—62no—
208C3×Dic6—22no—
209#626—64no—
210#626—64no—
211#625—122no—
212#624—64no—
213C23×C9—32no—
214C22×D9—32no—
215C22×D9—62no—
216C22×D9—64no—
217C22×D9—64no—
218C22×D9—122no—
219C22×D9C22×D311yescharacteristic
220C22×D9C22×D311yescharacteristic
221Q8×C9—62no—
222D4×C9—62no—
223C2×C4×C9—21no—
224C2×C4×C9—21no—
225C2×C4×C9—21no—
226C2×C4×C9—32no—
227C2×C4×C9C22×D311yescharacteristic
228C2×C4×C9C22×D311yescharacteristic
229C2×C4×C9C22×D311yescharacteristic
230#611—64no—
231#611—122no—
232C2×Dic9—32no—
233C2×Dic9—62no—
234C2×Dic9C22×D311yescharacteristic
235C2×Dic9C22×D311yescharacteristic
236D36—22no—
237D36—34no—
238D36—64no—
239D36—64no—
240C4×D9—22no—
241C4×D9—22no—
242C4×D9—64no—
243C4×D9—122no—
244C4×D9—124no—
245#546—271noSylow
246C2×C3×C9C22×D411yescharacteristic
247C2×C3×C9C2×#4111yescharacteristic
248C2×C3×C9H(5,𝔽2)11yescharacteristic
249C3⋊D9—24no—
250D3×C9—42no—
251C3×D9—22no—
252C3×D9—41no—
253C23×D3—92no—
254C23×D3—92no—
255C2×C3×Q8—31no—
256C2×C3×D4—31no—
257C2×C3×D4—91no—
258C22×C3×C4—91no—
259C22×C3×C4—91no—
260C2×#67—92no—
261C2×#67—92no—
262C2×#67—92no—
263C2×#67—92no—
264#239—184no—
265D3×D4—184no—
266#235—184no—
267C2×D12—31no—
268C2×D12—91no—
269C2×D12—91no—
270C2×D12—92no—
271C2×D12—92no—
272C2×D12—181no—
273C2×D12—184no—
274C2×D12C2×D911yescharacteristic
275C2×C4×D3—31no—
276C2×C4×D3—31no—
277C2×C4×D3—92no—
278C2×C4×D3—92no—
279C2×C4×D3—92no—
280C2×C4×D3—92no—
281C2×C4×D3—182no—
282C2×Dic6C2×D911yescharacteristic, complemented
283C3×C4⋊C4—91no—
284C3×C22⋊C4—32no—
285C3×C22⋊C4—32no—
286C3×C22⋊C4—91no—
287C3×C22⋊C4—91no—
288C3×C42—21no—
289C3×C42C2×D911yescharacteristic, complemented
290C3×C42D3211yescharacteristic
291#217—92no—
292#217—92no—
293(C2×D3)⋊C4—31no—
294(C2×D3)⋊C4—31no—
295(C2×D3)⋊C4—92no—
296(C2×D3)⋊C4—92no—
297(C2×D3)⋊C4—92no—
298(C2×D3)⋊C4—92no—
299(C2×D3)⋊C4—182no—
300(C2×D3)⋊C4C2×D912yescomplemented
301(C2×D3)⋊C4C2×D912yescomplemented
302C4⋊Dic3—31no—
303C4⋊Dic3—91no—
304C4⋊Dic3—91no—
305C4⋊Dic3—181no—
306Dic3⋊C4—92no—
307Dic3⋊C4—92no—
308C22×C32C23×D311yescharacteristic
309C2×C3⋊D3—34no—
310C2×C3⋊D3—62no—
311C2×C3⋊D3—62no—
312C2×C3×D3—22no—
313C2×C3×D3—22no—
314C2×C3×D3—22no—
315C2×C3×D3—32no—
316C2×C3×D3—61no—
317C2×C3×D3—61no—
318C2×C3×D3—61no—
319C2×C3×D3—61no—
320C2×C3×D3—61no—
321D32—128no—
322C32×C4—21no—
323C32×C4—21no—
324C32×C4D3×D412yes—
325C3⋊Dic3—62no—
326C3×Dic3—22no—
327C3×Dic3—61no—
328C3×Dic3—61no—
329C22×C9—21no—
330C22×C9—62no—
331C22×C9—62no—
332C22×C9—62no—
333C22×C9C23×D311yescharacteristic
334C2×D9—21no—
335C2×D9—21no—
336C2×D9—21no—
337C2×D9—21no—
338C2×D9—21no—
339C2×D9—34no—
340C2×D9—64no—
341C2×D9—62no—
342C2×D9—62no—
343C2×D9—122no—
344C2×D9—122no—
345C2×D9—128no—
346C2×D9#23512yescomplemented
347C4×C9—22no—
348C4×C9—21no—
349C4×C9—21no—
350C4×C9—22no—
351C4×C9—41no—
352C4×C9—62no—
353C4×C9D3×D412yes—
354Dic9—21no—
355Dic9—21no—
356Dic9—62no—
357Dic9—122no—
358C2×#41—271no—
359C22×D4—271no—
360#124—31no—
361#123—272no—
362C22⋊Q8—271no—
363#121—272no—
364#121—271no—
365#120—272no—
366C4×D4—91no—
367C4×D4—271no—
368C2×C22⋊C4—272no—
369C3×C9#54611yescharacteristic, complemented, Sylow
370C23×C3—32no—
371C23×C3—91no—
372C23×C3—91no—
373C22×D3—31no—
374C22×D3—31no—
375C22×D3—92no—
376C22×D3—92no—
377C22×D3—182no—
378C22×D3—184no—
379C22×D3—184no—
380C22×D3—184no—
381C22×D3—184no—
382C22×D3—362no—
383C22×D3—362no—
384C22×D3C22×D912yes—
385C3×Q8—62no—
386C3×D4—62no—
387C3×D4—182no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—32no—
392C2×C3×C4—91no—
393C2×C3×C4—91no—
394C2×C3×C4—182no—
395C2×C3×C4—182no—
396C2×C3×C4C22×D911yescharacteristic
397C2×C3×C4C22×D911yescharacteristic
398C2×C3×C4C22×D911yescharacteristic
399C2×C3×C4C2×D3211yescharacteristic
400C2×C3×C4C2×D3211yescharacteristic
401C2×C3×C4C2×D3211yescharacteristic
402#67—184no—
403#67—184no—
404#67—362no—
405#67—362no—
406C2×Dic3—31no—
407C2×Dic3—31no—
408C2×Dic3—92no—
409C2×Dic3—92no—
410C2×Dic3—182no—
411C2×Dic3C22×D912yes—
412D12—22no—
413D12—62no—
414D12—94no—
415D12—94no—
416D12—184no—
417D12—184no—
418D12—184no—
419C4×D3—62no—
420C4×D3—62no—
421C4×D3—184no—
422C4×D3—184no—
423C4×D3—362no—
424C4×D3—362no—
425C4×D3—364no—
426Dic6—22no—
427C2×C32C2×D3×D411yescharacteristic
428C2×C32D3×#4111yescharacteristic
429C2×C32#100911yescharacteristic
430C3⋊D3—64no—
431C3×D3—42no—
432C3×D3—62no—
433C3×D3—121no—
434C2×C9—21no—
435C2×C9—21no—
436C2×C9—21no—
437C2×C9—122no—
438C2×C9C2×#23511yescharacteristic
439C2×C9C2×D3×D411yescharacteristic
440C2×C9#100911yescharacteristic
441D9—22no—
442D9—41no—
443D9—64no—
444D9—124no—
445C24—272no—
446#41—544no—
447C2×Q8—31no—
448C2×D4—31no—
449C2×D4—91no—
450C2×D4—272no—
451C2×D4—272no—
452C2×D4—272no—
453C2×D4—271no—
454C2×D4—544no—
455C22×C4—91no—
456C22×C4—91no—
457C22×C4—272no—
458C22×C4—272no—
459C4⋊C4—91no—
460C4⋊C4—272no—
461C4⋊C4—271no—
462C22⋊C4—32no—
463C22⋊C4—32no—
464C22⋊C4—91no—
465C22⋊C4—91no—
466C22⋊C4—272no—
467C22⋊C4—272no—
468C22⋊C4—272no—
469C42D3×D911yescharacteristic, complemented
470C22×C3—21no—
471C22×C3—62no—
472C22×C3—62no—
473C22×C3—62no—
474C22×C3—92no—
475C22×C3—181no—
476C22×C3—181no—
477C22×C3—181no—
478C22×C3—181no—
479C22×C3—181no—
480C22×C3C23×D911yescharacteristic
481C22×C3C22×D3211yescharacteristic
482C2×D3—22no—
483C2×D3—22no—
484C2×D3—22no—
485C2×D3—32no—
486C2×D3—61no—
487C2×D3—61no—
488C2×D3—61no—
489C2×D3—61no—
490C2×D3—61no—
491C2×D3—94no—
492C2×D3—94no—
493C2×D3—184no—
494C2×D3—182no—
495C2×D3—182no—
496C2×D3—182no—
497C2×D3—182no—
498C2×D3—362no—
499C2×D3—362no—
500C2×D3—368no—
501C2×D3—368no—
502C3×C4—22no—
503C3×C4—21no—
504C3×C4—21no—
505C3×C4—21no—
506C3×C4—21no—
507C3×C4—22no—
508C3×C4—41no—
509C3×C4—62no—
510C3×C4—181no—
511C3×C4—181no—
512C3×C4D4×D912yes—
513C3×C4#380212yes—
514Dic3—22no—
515Dic3—61no—
516Dic3—61no—
517Dic3—182no—
518Dic3—182no—
519Dic3—362no—
520C32#567611yescharacteristic
521C9—21no—
522C9#555011yescharacteristic, complemented
523C23—32no—
524C23—91no—
525C23—91no—
526C23—272no—
527C23—544no—
528C23—544no—
529C23—1082no—
530Q8—62no—
531D4—62no—
532D4—182no—
533D4—274no—
534D4—544no—
535D4—544no—
536D4—1082no—
537C2×C4—32no—
538C2×C4—91no—
539C2×C4—91no—
540C2×C4—182no—
541C2×C4—182no—
542C2×C4—272no—
543C2×C4—544no—
544C2×C4—1082no—
545C2×C4C2×D3×D911yescharacteristic
546C2×C4C2×D3×D911yescharacteristic
547C2×C4C2×D3×D911yescharacteristic
548C2×C3—21no—
549C2×C3—21no—
550C2×C3—21no—
551C2×C3—122no—
552C2×C3—182no—
553C2×C3—361no—
554C2×C3C2×D4×D911yescharacteristic
555C2×C3#41×D911yescharacteristic
556C2×C3#6371511yescharacteristic
557C2×C3C2×#380211yescharacteristic
558C2×C3D3×#23511yescharacteristic
559C2×C3#6430811yescharacteristic
560D3—42no—
561D3—62no—
562D3—121no—
563D3—184no—
564D3—184no—
565D3—364no—
566C22—62no—
567C22—62no—
568C22—62no—
569C22—92no—
570C22—181no—
571C22—181no—
572C22—181no—
573C22—181no—
574C22—181no—
575C22—274no—
576C22—542no—
577C22—542no—
578C22—1088no—
579C22C22×D3×D911yescharacteristic
580C4—21no—
581C4—21no—
582C4—62no—
583C4—181no—
584C4—181no—
585C4—542no—
586C4#8864612yes—
587C3—21no—
588C3#1058977811yescharacteristic, complemented
589C3#1059400711yescharacteristic
590C2—122no—
591C2—182no—
592C2—361no—
593C2—544no—
594C2C2×#8864611yescharacteristic
595C2D9×#23511yescharacteristic
596C2#1172247811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
hypercenterC42D3×D9
hypocenterC3×C9#546
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×D9×D12C2semidirect
2. C2×#88635C2semidirect
3. C4×C4⋊(C3⋊D9)C2semidirect
4. C9×#875C2semidirect
5. C3×C4×D36C2semidirect
6. #11721049C2semidirect
7. #11721048C2semidirect
8. #11721041C2semidirect
9. #11721036C2semidirect
10. #11720958C2semidirect
11. #11720944C2semidirect
16. C2×C4×C3⋊D9C22semidirect
20. C2×C3×C4×D9C22semidirect
21. C2×C9×Dic6C22semidirect
23. C2×#6194C22semidirect
24. C2×#6192C22semidirect
25. C2×(C3⋊D9)⋊C4C22semidirect
28. C3×C42×C9C22semidirect
29. #88536C22semidirect
30. C4⋊(C3⋊Dic9)C22semidirect
31. C9×(C2×D3)⋊C4C22semidirect
32. C9×(C2×D3)⋊C4C22semidirect
34. C3×(C2×D9)⋊C4C22semidirect
35. C3×C4⋊Dic9C22semidirect
36. (C2×D3)⋊Dic9C22semidirect
38. #88442C22semidirect
61. C4×D36D3semidirect
73. C4⋊(C3⋊D9)D4semidirect
129. C2×C4×D9C2×D3semidirect
133. C42×C9C2×D3semidirect
139. (C2×D9)⋊C4C2×D3semidirect
142. C4⋊Dic9C2×D3semidirect
156. C2×C3×D9#41semidirect
183. #875D9semidirect
282. C2×Dic6C2×D9semidirect
289. C3×C42C2×D9semidirect
300. (C2×D3)⋊C4C2×D9semidirect
301. (C2×D3)⋊C4C2×D9semidirect
346. C2×D9#235semidirect
369. C3×C9#546semidirect
469. C42D3×D9semidirect
522. C9#5550semidirect
588. C3#10589778semidirect

Maximal subgroups

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

subgroupindex
C2×D9×D122
C2×#886352
C4×C4⋊(C3⋊D9)2
C9×#8752
C3×C4×D362
#117210492
#117210482
#117210412
#117210362
#117209582
#117209442
#105940073
#105897783