Finite Group Databasebeta

← #13139713

Subgroups — #13139713

Order 1728.

938 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C3⋊D9C211yesmaximal, characteristic
2C2×C3×Dic9⋊C4C211yesmaximal, characteristic, complemented
3C2×#88445C211yesmaximal, characteristic
4C2×#88445C211yesmaximal, characteristic
5C2×Dic3⋊Dic9C211yesmaximal, characteristic, complemented
6C9×C3⋊#95C211yesmaximal, characteristic, complemented
7#11720331C211yesmaximal, characteristic, complemented
8#10591290—31nomaximal
9#10588348—31nomaximal
10C23×C3⋊D9C2211yescharacteristic
11C22×C3×C4×C9C2211yescharacteristic
12C22×C3⋊Dic9C2211yescharacteristic
13C2×C4×C3⋊D9—22no—
14C2×C4×C3⋊D9—22no—
15C2×C4×C3⋊D9—22no—
16C22×C9×Dic3C2211yescharacteristic
17C22×C9×Dic3C2211yescharacteristic
18C22×C3×Dic9C2211yescharacteristic
19C22×C3×Dic9C2211yescharacteristic
20C3×Dic9⋊C4—22no—
21#88445—22no—
22#88445C414yescomplemented
23Dic3⋊Dic9—22no—
24C22×C4×C3⋊D3—31no—
25C2×C3×Dic3⋊C4—31no—
26C2×Dic3⋊Dic3—31no—
27C2×#3722—31no—
28C2×#3722—31no—
29C22×C4×D9—31no—
30C22×C4×D9—61no—
31C3×C3⋊#95D311yescharacteristic
32#63579—31no—
33C2×(C2×D9)⋊C4—31no—
34C2×(C2×D9)⋊C4—31no—
35C2×C4⋊Dic9—31no—
36C2×Dic9⋊C4D311yescharacteristic, complemented
37C9×#95—31no—
38C9⋊#95—31no—
39C23×C3×C9C2311yescharacteristic
40C22×C3⋊D9—21no—
41C22×C3⋊D9—21no—
42C22×C3⋊D9—21no—
43C22×C3⋊D9—21no—
44C22×C3⋊D9C2×C412yes—
45C22×C3⋊D9D412yes—
46C22×C3⋊D9Q812yes—
47C2×C3×C4×C9—21no—
48C2×C3×C4×C9—21no—
49C2×C3×C4×C9D412yes—
50C2×C3⋊Dic9—21no—
51C2×C3⋊Dic9—21no—
52C2×C3⋊Dic9D412yes—
53C4×C3⋊D9—28no—
54C2×C9×Dic3—21no—
55C2×C9×Dic3—21no—
56C2×C9×Dic3—21no—
57C2×C9×Dic3—21no—
58C2×C9×Dic3—21no—
59C2×C9×Dic3C2×C412yescomplemented
60C2×C3×Dic9—21no—
61C2×C3×Dic9—21no—
62C2×C3×Dic9—21no—
63C2×C3×Dic9—21no—
64C2×C3×Dic9—21no—
65C2×C3×Dic9C2×C412yescomplemented
66#4989—91no—
67#4668—91no—
68C23×C3⋊D3—31no—
69C22×C32×C4C2×D311yescharacteristic
70C22×C3⋊Dic3—31no—
71C2×C4×C3⋊D3—62no—
72C2×C4×C3⋊D3—62no—
73C2×C4×C3⋊D3—62no—
74C22×C3×Dic3—31no—
75C22×C3×Dic3—31no—
76C22×C3×Dic3C2×D311yescharacteristic
77C22×C3×Dic3C2×D311yescharacteristic
78C23×D9—31no—
79C23×D9—61no—
80C3×Dic3⋊C4—62no—
81Dic3⋊Dic3—62no—
82#3722—34no—
83#3722—62no—
84C22×C4×C9—21no—
85C22×C4×C9—31no—
86C22×C4×C9—31no—
87C22×C4×C9C2×D311yescharacteristic
88C22×Dic9—31no—
89C22×Dic9—61no—
90C22×Dic9C2×D311yescharacteristic
91C22×Dic9C2×D311yescharacteristic
92C2×C4×D9—62no—
93C2×C4×D9—62no—
94C2×C4×D9—64no—
95C2×C4×D9—64no—
96C2×C4×D9—64no—
97C2×C4×D9—62no—
98(C2×D9)⋊C4—34no—
99(C2×D9)⋊C4—62no—
100C4⋊Dic9—62no—
101Dic9⋊C4—22no—
102C22×C3×C9C22×C411yescharacteristic
103C22×C3×C9C2×D411yescharacteristic
104C22×C3×C9C2×D411yescharacteristic
105C22×C3×C9C2×D411yescharacteristic
106C22×C3×C9C2×Q811yescharacteristic
107C22×C3×C9#4111yescharacteristic
108C22×C3×C9#4111yescharacteristic
109C2×C3⋊D9—22no—
110C2×C3⋊D9—22no—
111C2×C3⋊D9—22no—
112C2×C3⋊D9—22no—
113C2×C3⋊D9—22no—
114C2×C3⋊D9—22no—
115C2×C3⋊D9C4⋊C414yescomplemented
116C3×C4×C9—22no—
117C3⋊Dic9—22no—
118C9×Dic3—22no—
119C9×Dic3—41no—
120C3×Dic9—22no—
121C3×Dic9—41no—
122C22×C4×D3—91no—
123C22×C4×D3—91no—
124C22×C4×D3—181no—
125C2×C3×C4⋊C4—91no—
126C2×(C2×D3)⋊C4—91no—
127C2×(C2×D3)⋊C4—91no—
128C2×(C2×D3)⋊C4—91no—
129C2×(C2×D3)⋊C4—91no—
130C2×C4⋊Dic3—91no—
131C2×Dic3⋊C4—31no—
132C2×Dic3⋊C4—91no—
133C3×#95—31no—
134C3⋊#95—91no—
135C3⋊#95—91no—
136C3⋊#95D911yescharacteristic, complemented
137C23×C32C22×D311yescharacteristic
138C22×C3⋊D3—32no—
139C22×C3⋊D3—32no—
140C22×C3⋊D3—32no—
141C22×C3⋊D3—61no—
142C22×C3⋊D3—61no—
143C22×C3⋊D3—61no—
144C22×C3⋊D3—61no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4D1212yes—
148C2×C3⋊Dic3—32no—
149C2×C3⋊Dic3—61no—
150C2×C3⋊Dic3—61no—
151C4×C3⋊D3—68no—
152C2×C3×Dic3—21no—
153C2×C3×Dic3—21no—
154C2×C3×Dic3—21no—
155C2×C3×Dic3—21no—
156C2×C3×Dic3—21no—
157C2×C3×Dic3—32no—
158C2×C3×Dic3—61no—
159C2×C3×Dic3—61no—
160C2×C3×Dic3—61no—
161C2×C3×Dic3—61no—
162C2×C3×Dic3—61no—
163C2×C3×Dic3C4×D312yes—
164C23×C9—21no—
165C23×C9C22×D311yescharacteristic
166C22×D9—32no—
167C22×D9—32no—
168C22×D9—32no—
169C22×D9—62no—
170C22×D9—62no—
171C22×D9—62no—
172C22×D9—62no—
173C22×D9—62no—
174C22×D9—62no—
175C22×D9—62no—
176C22×D9—61no—
177C22×D9—61no—
178C22×D9—61no—
179C22×D9—61no—
180C2×C4×C9—22no—
181C2×C4×C9—22no—
182C2×C4×C9—21no—
183C2×C4×C9—21no—
184C2×C4×C9—22no—
185C2×C4×C9—32no—
186C2×C4×C9—61no—
187C2×C4×C9—61no—
188C2×C4×C9—61no—
189C2×C4×C9—61no—
190C2×C4×C9—61no—
191C2×C4×C9#6712yes—
192C2×Dic9—21no—
193C2×Dic9—21no—
194C2×Dic9—21no—
195C2×Dic9—21no—
196C2×Dic9—21no—
197C2×Dic9—32no—
198C2×Dic9—62no—
199C2×Dic9—62no—
200C2×Dic9—62no—
201C2×Dic9—61no—
202C2×Dic9—61no—
203C2×Dic9C4×D312yescomplemented
204C4×D9—616no—
205C4×D9—68no—
206#385—271noSylow
207C2×C3×C9C2×C4⋊C411yescharacteristic
208C2×C3×C9C4×D411yescharacteristic
209C2×C3×C9C4×D411yescharacteristic
210C2×C3×C9#12011yescharacteristic
211C2×C3×C9C22⋊Q811yescharacteristic
212C2×C3×C9C22⋊Q811yescharacteristic
213C2×C3×C9#12311yescharacteristic
214C3⋊D9—24no—
215C23×D3—91no—
216C23×D3—91no—
217C23×D3—181no—
218C22×C3×C4—21no—
219C22×C3×C4—31no—
220C22×C3×C4—31no—
221C22×C3×C4—91no—
222C22×C3×C4—91no—
223C22×C3×C4C2×D911yescharacteristic
224C22×C3×C4D3211yescharacteristic
225C22×Dic3—31no—
226C22×Dic3—31no—
227C22×Dic3—91no—
228C22×Dic3—91no—
229C22×Dic3—181no—
230C22×Dic3C2×D911yescharacteristic
231C22×Dic3C2×D911yescharacteristic
232C2×C4×D3—182no—
233C2×C4×D3—182no—
234C2×C4×D3—182no—
235C2×C4×D3—182no—
236C2×C4×D3—184no—
237C2×C4×D3—184no—
238C2×C4×D3—184no—
239C2×C4×D3—182no—
240C2×C4×D3—182no—
241C3×C4⋊C4—182no—
242(C2×D3)⋊C4—94no—
243(C2×D3)⋊C4—94no—
244(C2×D3)⋊C4—182no—
245(C2×D3)⋊C4—182no—
246C4⋊Dic3—182no—
247Dic3⋊C4—62no—
248Dic3⋊C4—182no—
249C22×C32C2×C4×D311yescharacteristic
250C22×C32C2×D1211yescharacteristic
251C22×C32#23511yescharacteristic
252C22×C32D3×D411yescharacteristic
253C22×C32D3×D411yescharacteristic
254C22×C32#23711yescharacteristic
255C22×C32D3×Q811yescharacteristic
256C2×C3⋊D3—34no—
257C2×C3⋊D3—62no—
258C2×C3⋊D3—62no—
259C2×C3⋊D3—62no—
260C2×C3⋊D3—62no—
261C2×C3⋊D3—62no—
262C2×C3⋊D3—62no—
263C32×C4—22no—
264C3⋊Dic3—62no—
265C3×Dic3—22no—
266C3×Dic3—41no—
267C3×Dic3—62no—
268C3×Dic3—121no—
269C22×C9—21no—
270C22×C9—21no—
271C22×C9—21no—
272C22×C9—21no—
273C22×C9—21no—
274C22×C9—21no—
275C22×C9—21no—
276C22×C9C2×C4×D311yescharacteristic
277C22×C9#23511yescharacteristic
278C22×C9D3×D411yescharacteristic
279C22×C9D3×D411yescharacteristic
280C22×C9D3×Q811yescharacteristic
281C22×C9#23911yescharacteristic
282C22×C9C2×#6711yescharacteristic
283C2×D9—34no—
284C2×D9—64no—
285C2×D9—64no—
286C2×D9—64no—
287C2×D9—64no—
288C2×D9—64no—
289C2×D9—64no—
290C2×D9—64no—
291C2×D9—62no—
292C2×D9—62no—
293C2×D9—62no—
294C2×D9—62no—
295C2×D9—62no—
296C2×D9—62no—
297C4×C9—24no—
298C4×C9—22no—
299C4×C9—62no—
300C4×C9—121no—
301Dic9—22no—
302Dic9—41no—
303Dic9—64no—
304Dic9—62no—
305C23×C4—271no—
306C2×C4⋊C4—91no—
307C2×C4⋊C4—271no—
308C2×C22⋊C4—271no—
309C2×C22⋊C4—271no—
310#95—31no—
311#95—271no—
312C3×C9#38511yescharacteristic, complemented, Sylow
313C23×C3—21no—
314C23×C3C22×D911yescharacteristic
315C23×C3C2×D3211yescharacteristic
316C22×D3—92no—
317C22×D3—92no—
318C22×D3—92no—
319C22×D3—92no—
320C22×D3—92no—
321C22×D3—92no—
322C22×D3—182no—
323C22×D3—182no—
324C22×D3—182no—
325C22×D3—182no—
326C22×D3—182no—
327C22×D3—182no—
328C22×D3—182no—
329C22×D3—181no—
330C22×D3—181no—
331C22×D3—181no—
332C22×D3—181no—
333C22×D3—181no—
334C22×D3—181no—
335C22×D3—181no—
336C22×D3—181no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—22no—
344C2×C3×C4—32no—
345C2×C3×C4—61no—
346C2×C3×C4—61no—
347C2×C3×C4—61no—
348C2×C3×C4—61no—
349C2×C3×C4—61no—
350C2×C3×C4—92no—
351C2×C3×C4—181no—
352C2×C3×C4—181no—
353C2×C3×C4—181no—
354C2×C3×C4—181no—
355C2×C3×C4—181no—
356C2×C3×C4D3612yes—
357C2×C3×C4#62612yes—
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—32no—
364C2×Dic3—61no—
365C2×Dic3—61no—
366C2×Dic3—61no—
367C2×Dic3—61no—
368C2×Dic3—61no—
369C2×Dic3—92no—
370C2×Dic3—92no—
371C2×Dic3—182no—
372C2×Dic3—182no—
373C2×Dic3—182no—
374C2×Dic3—181no—
375C2×Dic3—181no—
376C2×Dic3—181no—
377C2×Dic3—181no—
378C2×Dic3C4×D912yescomplemented
379C4×D3—188no—
380C4×D3—1816no—
381C4×D3—188no—
382C2×C32C4×D1211yescharacteristic
383C2×C32#67⋊C411yescharacteristic
384C2×C32#88211yescharacteristic
385C2×C32#88311yescharacteristic
386C2×C32D3×C4⋊C411yescharacteristic
387C2×C32#89611yescharacteristic
388C2×C32#89711yescharacteristic
389C3⋊D3—64no—
390C2×C9—21no—
391C2×C9—21no—
392C2×C9—21no—
393C2×C9—21no—
394C2×C9—21no—
395C2×C9—21no—
396C2×C9—21no—
397C2×C9D3×C4⋊C411yescharacteristic
398C2×C9#89311yescharacteristic
399C2×C9#89411yescharacteristic
400C2×C9#89611yescharacteristic
401C2×C9C4×#6711yescharacteristic
402C2×C9#93711yescharacteristic
403C2×C9#94611yescharacteristic
404D9—68no—
405D9—64no—
406C24—271no—
407C22×C4—31no—
408C22×C4—31no—
409C22×C4—91no—
410C22×C4—91no—
411C22×C4—271no—
412C22×C4—542no—
413C22×C4—542no—
414C22×C4—542no—
415C22×C4D3×D911yescharacteristic
416C4⋊C4—182no—
417C4⋊C4—542no—
418C22⋊C4—274no—
419C22⋊C4—542no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—21no—
427C22×C3C2×C4×D911yescharacteristic
428C22×C3C2×D3611yescharacteristic
429C22×C3#369711yescharacteristic
430C22×C3D4×D911yescharacteristic
431C22×C3D4×D911yescharacteristic
432C22×C3#369911yescharacteristic
433C22×C3Q8×D911yescharacteristic
434C22×C3#379611yescharacteristic
435C22×C3#379711yescharacteristic
436C22×C3#379811yescharacteristic
437C22×C3C4×D3211yescharacteristic
438C22×C3C2×#62611yescharacteristic
439C22×C3#381111yescharacteristic
440C22×C3#381111yescharacteristic
441C2×D3—94no—
442C2×D3—94no—
443C2×D3—184no—
444C2×D3—184no—
445C2×D3—184no—
446C2×D3—184no—
447C2×D3—184no—
448C2×D3—184no—
449C2×D3—184no—
450C2×D3—182no—
451C2×D3—182no—
452C2×D3—182no—
453C2×D3—182no—
454C2×D3—182no—
455C2×D3—182no—
456C2×D3—182no—
457C2×D3—182no—
458C2×D3—182no—
459C2×D3—182no—
460C2×D3—182no—
461C2×D3—182no—
462C3×C4—24no—
463C3×C4—22no—
464C3×C4—22no—
465C3×C4—62no—
466C3×C4—121no—
467C3×C4—182no—
468C3×C4—361no—
469Dic3—22no—
470Dic3—41no—
471Dic3—62no—
472Dic3—121no—
473Dic3—184no—
474Dic3—182no—
475Dic3—182no—
476C32#466811yescharacteristic
477C9—21no—
478C9#498911yescharacteristic, complemented
479C23—272no—
480C23—272no—
481C23—272no—
482C23—541no—
483C23—541no—
484C23—541no—
485C23—541no—
486C23C2×D3×D911yescharacteristic
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—32no—
490C2×C4—61no—
491C2×C4—61no—
492C2×C4—61no—
493C2×C4—61no—
494C2×C4—61no—
495C2×C4—92no—
496C2×C4—181no—
497C2×C4—181no—
498C2×C4—181no—
499C2×C4—181no—
500C2×C4—181no—
501C2×C4—272no—
502C2×C4—548no—
503C2×C4—541no—
504C2×C4—541no—
505C2×C4#619212yes—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—21no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3C4×D3611yescharacteristic
514C2×C3#611⋊C411yescharacteristic
515C2×C3#6344411yescharacteristic
516C2×C3#6344511yescharacteristic
517C2×C3C4⋊C4×D911yescharacteristic
518C2×C3#6345811yescharacteristic
519C2×C3#6345911yescharacteristic
520C2×C3#6385311yescharacteristic
521C2×C3#6386511yescharacteristic
522C2×C3#6387111yescharacteristic
523C2×C3#6388111yescharacteristic
524C2×C3#6388311yescharacteristic
525C2×C3C4×#62611yescharacteristic
526C2×C3#6398111yescharacteristic
527D3—188no—
528D3—184no—
529D3—184no—
530C22—274no—
531C22—542no—
532C22—542no—
533C22—542no—
534C22—542no—
535C22—542no—
536C22—542no—
537C22(C3⋊D9)⋊Q811yescharacteristic
538C22#8863911yescharacteristic
539C22#8864011yescharacteristic
540C22C4×D3×D911yescharacteristic
541C22C2×#619211yescharacteristic
542C22#8866811yescharacteristic
543C22#8866811yescharacteristic
544C4—22no—
545C4—62no—
546C4—121no—
547C4—182no—
548C4—361no—
549C4—542no—
550C3—21no—
551C3#1058834811yescharacteristic, complemented
552C3#1059129011yescharacteristic
553C2—544no—
554C2#1172092311yescharacteristic
555C2#1172093611yescharacteristic
556C2#1172095711yescharacteristic
557C2#1172098011yescharacteristic
558C2C4×#619211yescharacteristic
559C2#1172101411yescharacteristic
560C2#1172127811yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutatorC22×C3×C9C22×C4
FrattiniC23×C3C2×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
prime radicalC23×C3⋊D9C22
cosocleC23×C3×C9C23
hypercenterC22×C4D3×D9
hypocenterC3×C9#385
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. C2×C3×Dic9⋊C4C2semidirect
5. C2×Dic3⋊Dic9C2semidirect
6. C9×C3⋊#95C2semidirect
7. #11720331C2semidirect
22. #88445C4semidirect
36. C2×Dic9⋊C4D3semidirect
59. C2×C9×Dic3C2×C4semidirect
65. C2×C3×Dic9C2×C4semidirect
115. C2×C3⋊D9C4⋊C4semidirect
136. C3⋊#95D9semidirect
203. C2×Dic9C4×D3semidirect
312. C3×C9#385semidirect
378. C2×Dic3C4×D9semidirect
478. C9#4989semidirect
551. C3#10588348semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C3⋊D92
C2×C3×Dic9⋊C42
C2×#884452
C2×Dic3⋊Dic92
C9×C3⋊#952
#117203312
#105912903
#105883483