Finite Group Databasebeta

← #13139906

Subgroups — #13139906

Order 1728.

990 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6195C211yesmaximal, characteristic
2C22×(C3⋊D9)⋊C4C211yesmaximal, characteristic, complemented
3C2×#88536C211yesmaximal, characteristic, complemented
4C2×C9×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
5C2×(C2×D3)⋊Dic9C211yesmaximal, characteristic, complemented
6C3×C9⋊#95C211yesmaximal, characteristic, complemented
7#11720331C211yesmaximal, characteristic, complemented
8#10591275—31nomaximal
9#10588636—31nomaximal
10C23×C3⋊D9C2211yescharacteristic
11C23×D3×C9C2211yescharacteristic
12C22×C3×C4×C9C2211yescharacteristic
13C22×C3⋊Dic9C2211yescharacteristic
14C22×C9×Dic3C2211yescharacteristic
15C22×C3×Dic9C2211yescharacteristic
16C22×C3×Dic9C2211yescharacteristic
17C2×#6195—22no—
18C2×#6195—22no—
19C2×#6195C414yescomplemented
20C2×(C3⋊D9)⋊C4—22no—
21C2×(C3⋊D9)⋊C4—22no—
22C2×(C3⋊D9)⋊C4—22no—
23#88536—22no—
24C9×(C2×D3)⋊C4—22no—
25(C2×D3)⋊Dic9—22no—
26C22×#626—31no—
27C22×#624—31no—
28C2×#3752—31no—
29C2×C3×(C2×D3)⋊C4D311yescharacteristic
30C2×(C2×D3)⋊Dic3—31no—
31C22×#611—31no—
32C22×C4×D9—31no—
33C3×C3⋊#95—31no—
34#63579—31no—
35C2×C9×C22⋊C4—31no—
36C2×#3676—31no—
37C2×(C2×D9)⋊C4—31no—
38C2×(C2×D9)⋊C4—61no—
39C9⋊#95—31no—
40C9⋊#95D311yescharacteristic, complemented
41C23×C3×C9C2311yescharacteristic
42C22×C3⋊D9—21no—
43C22×C3⋊D9—21no—
44C22×C3⋊D9—21no—
45C22×C3⋊D9—21no—
46C22×C3⋊D9C2×C412yes—
47C22×C3⋊D9D412yes—
48C22×C3⋊D9D412yes—
49C22×D3×C9—21no—
50C22×D3×C9—21no—
51C22×D3×C9—21no—
52C22×D3×C9—21no—
53C22×D3×C9—21no—
54C22×D3×C9—21no—
55C22×D3×C9C2×C412yescomplemented
56C2×C3×C4×C9—21no—
57C2×C3×C4×C9—21no—
58C2×C3×C4×C9—21no—
59C2×C3⋊Dic9—21no—
60C2×C3⋊Dic9—21no—
61C2×C3⋊Dic9—21no—
62C2×C9×Dic3—21no—
63C2×C9×Dic3—21no—
64C2×C9×Dic3D412yescomplemented
65C2×C3×Dic9—21no—
66C2×C3×Dic9—21no—
67C2×C3×Dic9—21no—
68C2×C3×Dic9—21no—
69C2×C3×Dic9C2×C412yescomplemented
70C2×C3×Dic9D412yescomplemented
71#6195—28no—
72(C3⋊D9)⋊C4—28no—
73#4956—91no—
74#4665—91no—
75C23×C3⋊D3—31no—
76C23×C3×D3C2×D311yescharacteristic
77C22×C32×C4C2×D311yescharacteristic
78C22×C3⋊Dic3—31no—
79C22×C3×Dic3—31no—
80C22×C3×Dic3—31no—
81C22×C3×Dic3C2×D311yescharacteristic
82C2×#626—34no—
83C2×#626—62no—
84C2×#626—62no—
85C2×#624—62no—
86C2×#624—62no—
87C2×#624—62no—
88C24×C9—31no—
89C23×D9—31no—
90C23×D9—61no—
91#3752—62no—
92C3×(C2×D3)⋊C4—22no—
93(C2×D3)⋊Dic3—62no—
94C22×C4×C9—21no—
95C22×C4×C9—31no—
96C22×C4×C9C2×D311yescharacteristic
97C2×#611—34no—
98C2×#611—62no—
99C2×#611—62no—
100C22×Dic9—31no—
101C22×Dic9—61no—
102C22×Dic9C2×D311yescharacteristic
103C22×Dic9C2×D311yescharacteristic
104C2×C4×D9—62no—
105C2×C4×D9—62no—
106C2×C4×D9—62no—
107C9×C22⋊C4—62no—
108#3676—62no—
109(C2×D9)⋊C4—62no—
110(C2×D9)⋊C4—64no—
111C22×C3×C9C22×C411yescharacteristic
112C22×C3×C9C2×D411yescharacteristic
113C22×C3×C9C2×D411yescharacteristic
114C22×C3×C9C2×D411yescharacteristic
115C22×C3×C9C2×D411yescharacteristic
116C22×C3×C9#4111yescharacteristic
117C22×C3×C9#4111yescharacteristic
118C2×C3⋊D9—22no—
119C2×C3⋊D9—22no—
120C2×C3⋊D9—22no—
121C2×C3⋊D9—22no—
122C2×C3⋊D9—22no—
123C2×C3⋊D9—22no—
124C2×C3⋊D9C22⋊C414yescomplemented
125C2×D3×C9—22no—
126C2×D3×C9—22no—
127C2×D3×C9—22no—
128C2×D3×C9—41no—
129C2×D3×C9—41no—
130C2×D3×C9—41no—
131C2×D3×C9—41no—
132C3×C4×C9—41no—
133C3⋊Dic9—41no—
134C9×Dic3—22no—
135C3×Dic9—22no—
136C3×Dic9—22no—
137C22×#67—91no—
138C22×D12—91no—
139C22×C4×D3—91no—
140C22×C4×D3—91no—
141C2×C3×C22⋊C4—31no—
142C2×#217—91no—
143C2×(C2×D3)⋊C4—91no—
144C2×(C2×D3)⋊C4—91no—
145C2×(C2×D3)⋊C4—91no—
146C2×(C2×D3)⋊C4—181no—
147C2×(C2×D3)⋊C4D911yescharacteristic, complemented
148C3×#95—91no—
149C3⋊#95—31no—
150C3⋊#95—91no—
151C3⋊#95—91no—
152C23×C32C22×D311yescharacteristic
153C22×C3⋊D3—32no—
154C22×C3⋊D3—32no—
155C22×C3⋊D3—32no—
156C22×C3⋊D3—61no—
157C22×C3⋊D3—61no—
158C22×C3⋊D3—61no—
159C22×C3⋊D3—61no—
160C22×C3×D3—21no—
161C22×C3×D3—21no—
162C22×C3×D3—21no—
163C22×C3×D3—21no—
164C22×C3×D3—21no—
165C22×C3×D3—21no—
166C22×C3×D3C4×D312yes—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C3⋊Dic3—61no—
171C2×C3⋊Dic3—61no—
172C2×C3⋊Dic3—61no—
173C2×C3×Dic3—21no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—32no—
176C2×C3×Dic3—32no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3#6712yes—
182#626—68no—
183#624—68no—
184C23×C9—21no—
185C23×C9—32no—
186C23×C9—61no—
187C23×C9—61no—
188C23×C9—61no—
189C23×C9—61no—
190C23×C9—61no—
191C23×C9—61no—
192C23×C9C22×D311yescharacteristic
193C22×D9—32no—
194C22×D9—32no—
195C22×D9—32no—
196C22×D9—62no—
197C22×D9—62no—
198C22×D9—62no—
199C22×D9—61no—
200C22×D9—61no—
201C22×D9—61no—
202C22×D9—61no—
203C22×D9—121no—
204C22×D9—121no—
205C22×D9—121no—
206C22×D9—121no—
207C2×C4×C9—22no—
208C2×C4×C9—21no—
209C2×C4×C9—21no—
210C2×C4×C9—21no—
211C2×C4×C9—32no—
212C2×C4×C9—41no—
213C2×C4×C9—41no—
214C2×C4×C9—61no—
215C2×C4×C9—61no—
216#611—68no—
217C2×Dic9—21no—
218C2×Dic9—21no—
219C2×Dic9—21no—
220C2×Dic9—21no—
221C2×Dic9—62no—
222C2×Dic9—61no—
223C2×Dic9—61no—
224C2×Dic9—61no—
225C2×Dic9—121no—
226C2×Dic9—121no—
227C2×Dic9C4×D312yescomplemented
228C2×Dic9D1212yescomplemented
229C4×D9—68no—
230#386—271noSylow
231C2×C3×C9C2×C22⋊C411yescharacteristic
232C2×C3×C9C4×D411yescharacteristic
233C2×C3×C9C4×D411yescharacteristic
234C2×C3×C9#12011yescharacteristic
235C2×C3×C9#12111yescharacteristic
236C2×C3×C9#12111yescharacteristic
237C2×C3×C9#12311yescharacteristic
238C3⋊D9—24no—
239D3×C9—42no—
240C24×C3—31no—
241C23×D3—91no—
242C23×D3—91no—
243C23×D3—181no—
244C23×D3C2×D911yescharacteristic
245C22×C3×C4—21no—
246C22×C3×C4—31no—
247C22×C3×C4—91no—
248C22×C3×C4—91no—
249C22×C3×C4C2×D911yescharacteristic
250C22×C3×C4D3211yescharacteristic
251C2×#67—94no—
252C2×#67—182no—
253C2×#67—182no—
254C22×Dic3—31no—
255C22×Dic3—31no—
256C22×Dic3—91no—
257C22×Dic3—91no—
258C22×Dic3—181no—
259C22×Dic3C2×D911yescharacteristic
260C2×D12—94no—
261C2×D12—182no—
262C2×D12—182no—
263C2×C4×D3—182no—
264C2×C4×D3—182no—
265C2×C4×D3—182no—
266C2×C4×D3—182no—
267C2×C4×D3—182no—
268C2×C4×D3—182no—
269C3×C22⋊C4—62no—
270#217—182no—
271(C2×D3)⋊C4—22no—
272(C2×D3)⋊C4—182no—
273(C2×D3)⋊C4—182no—
274(C2×D3)⋊C4—184no—
275(C2×D3)⋊C4—182no—
276C22×C32C2×C4×D311yescharacteristic
277C22×C32#23511yescharacteristic
278C22×C32D3×D411yescharacteristic
279C22×C32D3×D411yescharacteristic
280C22×C32D3×D411yescharacteristic
281C22×C32#23711yescharacteristic
282C22×C32C2×#6711yescharacteristic
283C2×C3⋊D3—34no—
284C2×C3⋊D3—62no—
285C2×C3⋊D3—62no—
286C2×C3⋊D3—62no—
287C2×C3⋊D3—62no—
288C2×C3⋊D3—62no—
289C2×C3⋊D3—62no—
290C2×C3×D3—22no—
291C2×C3×D3—22no—
292C2×C3×D3—22no—
293C2×C3×D3—41no—
294C2×C3×D3—41no—
295C2×C3×D3—41no—
296C2×C3×D3—41no—
297C32×C4—41no—
298C3⋊Dic3—121no—
299C3×Dic3—22no—
300C3×Dic3—62no—
301C3×Dic3—62no—
302C22×C9—21no—
303C22×C9—21no—
304C22×C9—21no—
305C22×C9—21no—
306C22×C9—21no—
307C22×C9—21no—
308C22×C9—21no—
309C22×C9—62no—
310C22×C9—62no—
311C22×C9—62no—
312C22×C9—121no—
313C22×C9—121no—
314C22×C9—121no—
315C22×C9—121no—
316C22×C9C2×C4×D311yescharacteristic
317C22×C9C2×D1211yescharacteristic
318C22×C9#23511yescharacteristic
319C22×C9D3×D411yescharacteristic
320C22×C9D3×D411yescharacteristic
321C22×C9D3×D411yescharacteristic
322C22×C9#23911yescharacteristic
323C2×D9—34no—
324C2×D9—64no—
325C2×D9—62no—
326C2×D9—62no—
327C2×D9—62no—
328C2×D9—62no—
329C2×D9—62no—
330C2×D9—62no—
331C2×D9—122no—
332C2×D9—122no—
333C2×D9—122no—
334C2×D9—122no—
335C2×D9—122no—
336C2×D9—122no—
337C4×C9—42no—
338C4×C9—41no—
339C4×C9—62no—
340Dic9—22no—
341Dic9—22no—
342Dic9—122no—
343Dic9—121no—
344C22×D4—271no—
345C23×C4—271no—
346C2×C22⋊C4—31no—
347C2×C22⋊C4—271no—
348C2×C22⋊C4—271no—
349#95—91no—
350#95—271no—
351C3×C9#38611yescharacteristic, complemented, Sylow
352C23×C3—21no—
353C23×C3—32no—
354C23×C3—61no—
355C23×C3—61no—
356C23×C3—61no—
357C23×C3—61no—
358C23×C3—61no—
359C23×C3—61no—
360C23×C3C22×D911yescharacteristic
361C23×C3C2×D3211yescharacteristic
362C22×D3—21no—
363C22×D3—21no—
364C22×D3—21no—
365C22×D3—21no—
366C22×D3—21no—
367C22×D3—21no—
368C22×D3—92no—
369C22×D3—92no—
370C22×D3—92no—
371C22×D3—92no—
372C22×D3—92no—
373C22×D3—92no—
374C22×D3—182no—
375C22×D3—182no—
376C22×D3—182no—
377C22×D3—181no—
378C22×D3—181no—
379C22×D3—181no—
380C22×D3—181no—
381C22×D3—181no—
382C22×D3—181no—
383C22×D3—181no—
384C22×D3—181no—
385C22×D3—361no—
386C22×D3—361no—
387C22×D3—361no—
388C22×D3—361no—
389C22×D3C4×D912yescomplemented
390C2×C3×C4—22no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—32no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—61no—
401C2×C3×C4—61no—
402C2×C3×C4—92no—
403C2×C3×C4—92no—
404C2×C3×C4—181no—
405C2×C3×C4—181no—
406C2×C3×C4—181no—
407C2×C3×C4—181no—
408#67—188no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—32no—
412C2×Dic3—32no—
413C2×Dic3—61no—
414C2×Dic3—61no—
415C2×Dic3—61no—
416C2×Dic3—61no—
417C2×Dic3—182no—
418C2×Dic3—181no—
419C2×Dic3—181no—
420C2×Dic3—181no—
421C2×Dic3—181no—
422C2×Dic3—181no—
423C2×Dic3—181no—
424C2×Dic3—361no—
425C2×Dic3—361no—
426C2×Dic3#61112yescomplemented
427D12—188no—
428C4×D3—188no—
429C4×D3—188no—
430C2×C32D3×C22⋊C411yescharacteristic
431C2×C32#67⋊C411yescharacteristic
432C2×C32#88311yescharacteristic
433C2×C32#88411yescharacteristic
434C2×C32C4×#6711yescharacteristic
435C2×C32#93711yescharacteristic
436C2×C32#93811yescharacteristic
437C3⋊D3—64no—
438C3×D3—42no—
439C2×C9—21no—
440C2×C9—21no—
441C2×C9—21no—
442C2×C9—21no—
443C2×C9—21no—
444C2×C9—21no—
445C2×C9—21no—
446C2×C9—122no—
447C2×C9C4×D1211yescharacteristic
448C2×C9D3×C22⋊C411yescharacteristic
449C2×C9#88211yescharacteristic
450C2×C9#88411yescharacteristic
451C2×C9#89311yescharacteristic
452C2×C9#89411yescharacteristic
453C2×C9#89511yescharacteristic
454D9—64no—
455D9—124no—
456C24—31no—
457C24—271no—
458C2×D4—274no—
459C2×D4—542no—
460C2×D4—542no—
461C22×C4—31no—
462C22×C4—91no—
463C22×C4—91no—
464C22×C4—271no—
465C22×C4—542no—
466C22×C4—542no—
467C22×C4—542no—
468C22×C4D3×D911yescharacteristic
469C22⋊C4—62no—
470C22⋊C4—542no—
471C22⋊C4—542no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—21no—
475C22×C3—21no—
476C22×C3—21no—
477C22×C3—21no—
478C22×C3—21no—
479C22×C3—62no—
480C22×C3—62no—
481C22×C3—62no—
482C22×C3—121no—
483C22×C3—121no—
484C22×C3—121no—
485C22×C3—121no—
486C22×C3C2×C4×D911yescharacteristic
487C22×C3#369711yescharacteristic
488C22×C3D4×D911yescharacteristic
489C22×C3D4×D911yescharacteristic
490C22×C3D4×D911yescharacteristic
491C22×C3#369911yescharacteristic
492C22×C3C2×#61111yescharacteristic
493C22×C3#379911yescharacteristic
494C22×C3C4×D3211yescharacteristic
495C22×C3D3×D1211yescharacteristic
496C22×C3#380211yescharacteristic
497C22×C3#380411yescharacteristic
498C22×C3D3×#6711yescharacteristic
499C22×C3#381111yescharacteristic
500C2×D3—22no—
501C2×D3—22no—
502C2×D3—22no—
503C2×D3—41no—
504C2×D3—41no—
505C2×D3—41no—
506C2×D3—41no—
507C2×D3—94no—
508C2×D3—94no—
509C2×D3—184no—
510C2×D3—182no—
511C2×D3—182no—
512C2×D3—182no—
513C2×D3—182no—
514C2×D3—182no—
515C2×D3—182no—
516C2×D3—182no—
517C2×D3—182no—
518C2×D3—182no—
519C2×D3—182no—
520C2×D3—182no—
521C2×D3—182no—
522C2×D3—362no—
523C2×D3—362no—
524C2×D3—362no—
525C2×D3—362no—
526C2×D3—362no—
527C2×D3—362no—
528C3×C4—42no—
529C3×C4—41no—
530C3×C4—41no—
531C3×C4—62no—
532C3×C4—182no—
533C3×C4—182no—
534Dic3—22no—
535Dic3—62no—
536Dic3—62no—
537Dic3—362no—
538Dic3—361no—
539Dic3—361no—
540C32#495611yescharacteristic
541C9—21no—
542C9#466511yescharacteristic, complemented
543C23—32no—
544C23—61no—
545C23—61no—
546C23—61no—
547C23—61no—
548C23—61no—
549C23—61no—
550C23—272no—
551C23—272no—
552C23—272no—
553C23—541no—
554C23—541no—
555C23—541no—
556C23—541no—
557C23C2×D3×D911yescharacteristic
558D4—548no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—32no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—92no—
566C2×C4—92no—
567C2×C4—181no—
568C2×C4—181no—
569C2×C4—181no—
570C2×C4—181no—
571C2×C4—548no—
572C2×C4—541no—
573C2×C4—541no—
574C2×C4—541no—
575C2×C3—21no—
576C2×C3—21no—
577C2×C3—21no—
578C2×C3—21no—
579C2×C3—21no—
580C2×C3—21no—
581C2×C3—21no—
582C2×C3—122no—
583C2×C3C22⋊C4×D911yescharacteristic
584C2×C3#611⋊C411yescharacteristic
585C2×C3#6344511yescharacteristic
586C2×C3#6344611yescharacteristic
587C2×C3C4×#61111yescharacteristic
588C2×C3#6349911yescharacteristic
589C2×C3#6350011yescharacteristic
590C2×C3#6385411yescharacteristic
591C2×C3#6388011yescharacteristic
592C2×C3#6390411yescharacteristic
593C2×C3#6391011yescharacteristic
594C2×C3#6392111yescharacteristic
595C2×C3#6392211yescharacteristic
596C2×C3#6395811yescharacteristic
597D3—42no—
598D3—184no—
599D3—184no—
600D3—364no—
601C22—62no—
602C22—62no—
603C22—62no—
604C22—121no—
605C22—121no—
606C22—121no—
607C22—121no—
608C22—274no—
609C22—542no—
610C22—542no—
611C22—542no—
612C22—542no—
613C22—542no—
614C22—542no—
615C22#8864211yescharacteristic
616C22C4×D3×D911yescharacteristic
617C22D9×D1211yescharacteristic
618C22#8864611yescharacteristic
619C22#8866211yescharacteristic
620C22D3×#61111yescharacteristic
621C22#8866811yescharacteristic
622C4—41no—
623C4—62no—
624C4—182no—
625C4—182no—
626C4—1081no—
627C3—21no—
628C3#1058863611yescharacteristic, complemented
629C3#1059127511yescharacteristic
630C2—122no—
631C2—544no—
632C2#1172095811yescharacteristic
633C2#1172101311yescharacteristic
634C2#1172103011yescharacteristic
635C2#1172103911yescharacteristic
636C2#1172104811yescharacteristic
637C2#1172105011yescharacteristic
638C2#1172126211yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutatorC22×C3×C9C22×C4
FrattiniC23×C3C2×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
prime radicalC22×#6195C2
cosocleC23×C3×C9C23
hypercenterC22×C4D3×D9
hypocenterC3×C9#386
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C22×(C3⋊D9)⋊C4C2semidirect
3. C2×#88536C2semidirect
4. C2×C9×(C2×D3)⋊C4C2semidirect
5. C2×(C2×D3)⋊Dic9C2semidirect
6. C3×C9⋊#95C2semidirect
7. #11720331C2semidirect
19. C2×#6195C4semidirect
40. C9⋊#95D3semidirect
55. C22×D3×C9C2×C4semidirect
64. C2×C9×Dic3D4semidirect
69. C2×C3×Dic9C2×C4semidirect
70. C2×C3×Dic9D4semidirect
124. C2×C3⋊D9C22⋊C4semidirect
147. C2×(C2×D3)⋊C4D9semidirect
227. C2×Dic9C4×D3semidirect
228. C2×Dic9D12semidirect
351. C3×C9#386semidirect
389. C22×D3C4×D9semidirect
426. C2×Dic3#611semidirect
542. C9#4665semidirect
628. C3#10588636semidirect

Maximal subgroups

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

subgroupindex
C22×#61952
C22×(C3⋊D9)⋊C42
C2×#885362
C2×C9×(C2×D3)⋊C42
C2×(C2×D3)⋊Dic92
C3×C9⋊#952
#117203312
#105912753
#105886363