Finite Group Databasebeta

← #13143699

Subgroups — #13143699

Order 1728.

1430 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊(C3⋊D9)C211yesmaximal, characteristic
2C22×C4×C3⋊D9C211yesmaximal, characteristic, complemented
3C2×#88536C211yesmaximal, characteristic, complemented
4C2×#88536C211yesmaximal, characteristic, complemented
5C2×#88536C211yesmaximal, characteristic, complemented
6C3×C9×#95C211yesmaximal, characteristic, complemented
7#11720572C211yesmaximal, characteristic, complemented
8#10592364—31nomaximal
9#10588345—33nomaximal
10C23×C3⋊D9C2211yescharacteristic
11C23×C3⋊D9C2211yescharacteristic
12C22×C3×C4×C9C2211yescharacteristic
13C22×C3×C4×C9C2211yescharacteristic
14C22×C3×C4×C9C2211yescharacteristic
15C22×C3⋊Dic9C2211yescharacteristic
16C22×C3⋊Dic9C2211yescharacteristic
17C2×C4⋊(C3⋊D9)—22no—
18C2×C4⋊(C3⋊D9)—22no—
19C2×C4⋊(C3⋊D9)C414yescomplemented
20C2×C4×C3⋊D9—22no—
21C2×C4×C3⋊D9—22no—
22C2×C4×C3⋊D9—22no—
23#88536—22no—
24#88536—22no—
25#88536—22no—
26C22×C4⋊(C3⋊D3)—31no—
27C22×C4×C3⋊D3—31no—
28C2×#3752—31no—
29C2×#3752—31no—
30C2×#3752—31no—
31C22×D36—33no—
32C22×C4×D9—33no—
33C32×#95D311yescharacteristic
34#63658—31no—
35C2×(C2×D9)⋊C4—33no—
36C2×(C2×D9)⋊C4—33no—
37C2×(C2×D9)⋊C4—33no—
38C9×#95D313yescomplemented
39C9⋊#95—33no—
40C23×C3×C9C2311yescharacteristic
41C22×C3⋊D9—21no—
42C22×C3⋊D9—21no—
43C22×C3⋊D9—21no—
44C22×C3⋊D9—21no—
45C22×C3⋊D9—21no—
46C22×C3⋊D9—21no—
47C22×C3⋊D9—21no—
48C22×C3⋊D9—21no—
49C22×C3⋊D9—21no—
50C22×C3⋊D9—21no—
51C22×C3⋊D9C2×C412yes—
52C22×C3⋊D9C2×C412yescomplemented
53C22×C3⋊D9D412yes—
54C22×C3⋊D9D412yes—
55C2×C3×C4×C9—21no—
56C2×C3×C4×C9—21no—
57C2×C3×C4×C9—21no—
58C2×C3×C4×C9—21no—
59C2×C3×C4×C9—21no—
60C2×C3×C4×C9—21no—
61C2×C3×C4×C9—21no—
62C2×C3×C4×C9C2×C412yescomplemented
63C2×C3×C4×C9D412yescomplemented
64C2×C3⋊Dic9—21no—
65C2×C3⋊Dic9—21no—
66C2×C3⋊Dic9—21no—
67C2×C3⋊Dic9—21no—
68C2×C3⋊Dic9—21no—
69C2×C3⋊Dic9D412yescomplemented
70C4⋊(C3⋊D9)—28no—
71C4×C3⋊D9—28no—
72#4665—93no—
73#4665—91no—
74C23×C3⋊D3—31no—
75C23×C3⋊D3—31no—
76C22×C32×C4C2×D311yescharacteristic
77C22×C32×C4C2×D311yescharacteristic
78C22×C32×C4C2×D311yescharacteristic
79C22×C3⋊Dic3—31no—
80C22×C3⋊Dic3—31no—
81C2×C4⋊(C3⋊D3)—34no—
82C2×C4⋊(C3⋊D3)—62no—
83C2×C4⋊(C3⋊D3)—62no—
84C2×C4×C3⋊D3—62no—
85C2×C4×C3⋊D3—62no—
86C2×C4×C3⋊D3—62no—
87C23×D9—33no—
88C23×D9—33no—
89#3752—62no—
90#3752—62no—
91#3752—62no—
92C22×C4×C9C2×D313yes—
93C22×C4×C9C2×D313yes—
94C22×C4×C9C2×D313yes—
95C22×Dic9—33no—
96C22×Dic9—33no—
97C2×D36—312no—
98C2×D36—66no—
99C2×D36—66no—
100C2×C4×D9—66no—
101C2×C4×D9—66no—
102C2×C4×D9—66no—
103(C2×D9)⋊C4—66no—
104(C2×D9)⋊C4—66no—
105(C2×D9)⋊C4—66no—
106C22×C3×C9C22×C411yescharacteristic
107C22×C3×C9C2×D411yescharacteristic
108C22×C3×C9C2×D411yescharacteristic
109C22×C3×C9C2×D411yescharacteristic
110C22×C3×C9C2×D411yescharacteristic
111C22×C3×C9#4111yescharacteristic
112C22×C3×C9#4111yescharacteristic
113C2×C3⋊D9—22no—
114C2×C3⋊D9—22no—
115C2×C3⋊D9—22no—
116C2×C3⋊D9—22no—
117C2×C3⋊D9—22no—
118C2×C3⋊D9—22no—
119C2×C3⋊D9—22no—
120C2×C3⋊D9—22no—
121C2×C3⋊D9—22no—
122C2×C3⋊D9—41no—
123C2×C3⋊D9—41no—
124C2×C3⋊D9—41no—
125C2×C3⋊D9—41no—
126C2×C3⋊D9C22⋊C414yescomplemented
127C3×C4×C9—22no—
128C3×C4×C9—22no—
129C3×C4×C9—41no—
130C3⋊Dic9—22no—
131C3⋊Dic9—41no—
132C22×D12—93no—
133C22×D12—91no—
134C22×C4×D3—93no—
135C22×C4×D3—91no—
136C2×(C2×D3)⋊C4—93no—
137C2×(C2×D3)⋊C4—93no—
138C2×(C2×D3)⋊C4—93no—
139C2×(C2×D3)⋊C4—91no—
140C2×(C2×D3)⋊C4—91no—
141C2×(C2×D3)⋊C4—91no—
142C3×#95D913yescomplemented
143C3×#95C3⋊D311yescharacteristic
144C3⋊#95—93no—
145C3⋊#95—91no—
146C23×C32C22×D311yescharacteristic
147C22×C3⋊D3—32no—
148C22×C3⋊D3—32no—
149C22×C3⋊D3—32no—
150C22×C3⋊D3—32no—
151C22×C3⋊D3—61no—
152C22×C3⋊D3—61no—
153C22×C3⋊D3—61no—
154C22×C3⋊D3—61no—
155C22×C3⋊D3—61no—
156C22×C3⋊D3—61no—
157C22×C3⋊D3—61no—
158C22×C3⋊D3—61no—
159C22×C3⋊D3—61no—
160C22×C3⋊D3—61no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—21no—
168C2×C32×C4C4×D312yes—
169C2×C32×C4D1212yes—
170C2×C3⋊Dic3—32no—
171C2×C3⋊Dic3—61no—
172C2×C3⋊Dic3—61no—
173C2×C3⋊Dic3—61no—
174C2×C3⋊Dic3—61no—
175C2×C3⋊Dic3—61no—
176C4⋊(C3⋊D3)—68no—
177C4×C3⋊D3—68no—
178C23×C9C22×D313yes—
179C22×D9—36no—
180C22×D9—36no—
181C22×D9—36no—
182C22×D9—36no—
183C22×D9—63no—
184C22×D9—63no—
185C22×D9—63no—
186C22×D9—63no—
187C22×D9—63no—
188C22×D9—63no—
189C22×D9—63no—
190C22×D9—63no—
191C22×D9—63no—
192C22×D9—63no—
193C2×C4×C9—23no—
194C2×C4×C9—23no—
195C2×C4×C9—23no—
196C2×C4×C9—23no—
197C2×C4×C9—23no—
198C2×C4×C9—23no—
199C2×C4×C9—23no—
200C2×C4×C9C4×D316yescomplemented
201C2×C4×C9D1216yescomplemented
202C2×Dic9—36no—
203C2×Dic9—63no—
204C2×Dic9—63no—
205C2×Dic9—63no—
206C2×Dic9—63no—
207C2×Dic9—63no—
208D36—624no—
209C4×D9—624no—
210#386—271noSylow
211C2×C3×C9C2×C22⋊C411yescharacteristic
212C2×C3×C9C4×D411yescharacteristic
213C2×C3×C9C4×D411yescharacteristic
214C2×C3×C9#12011yescharacteristic
215C2×C3×C9#12111yescharacteristic
216C2×C3×C9#12111yescharacteristic
217C2×C3×C9#12311yescharacteristic
218C3⋊D9—24no—
219C3⋊D9—42no—
220C23×D3—93no—
221C23×D3—91no—
222C23×D3—93no—
223C23×D3—91no—
224C22×C3×C4C2×D913yes—
225C22×C3×C4C2×D913yes—
226C22×C3×C4C2×D913yes—
227C22×C3×C4C2×C3⋊D311yescharacteristic
228C22×C3×C4C2×C3⋊D311yescharacteristic
229C22×C3×C4C2×C3⋊D311yescharacteristic
230C22×Dic3—93no—
231C22×Dic3—91no—
232C22×Dic3—93no—
233C22×Dic3—91no—
234C2×D12—912no—
235C2×D12—94no—
236C2×D12—182no—
237C2×D12—182no—
238C2×D12—186no—
239C2×D12—186no—
240C2×C4×D3—182no—
241C2×C4×D3—186no—
242C2×C4×D3—182no—
243C2×C4×D3—186no—
244C2×C4×D3—186no—
245C2×C4×D3—182no—
246(C2×D3)⋊C4—182no—
247(C2×D3)⋊C4—186no—
248(C2×D3)⋊C4—182no—
249(C2×D3)⋊C4—186no—
250(C2×D3)⋊C4—186no—
251(C2×D3)⋊C4—182no—
252C22×C32C2×C4×D311yescharacteristic
253C22×C32C2×D1211yescharacteristic
254C22×C32#23511yescharacteristic
255C22×C32D3×D411yescharacteristic
256C22×C32D3×D411yescharacteristic
257C22×C32D3×D411yescharacteristic
258C22×C32#23911yescharacteristic
259C2×C3⋊D3—34no—
260C2×C3⋊D3—62no—
261C2×C3⋊D3—62no—
262C2×C3⋊D3—62no—
263C2×C3⋊D3—62no—
264C2×C3⋊D3—62no—
265C2×C3⋊D3—62no—
266C2×C3⋊D3—62no—
267C2×C3⋊D3—62no—
268C2×C3⋊D3—62no—
269C2×C3⋊D3—121no—
270C2×C3⋊D3—121no—
271C2×C3⋊D3—121no—
272C2×C3⋊D3—121no—
273C32×C4—22no—
274C32×C4—22no—
275C32×C4—41no—
276C3⋊Dic3—62no—
277C3⋊Dic3—121no—
278C22×C9C2×C4×D313yes—
279C22×C9C2×D1213yes—
280C22×C9#23513yes—
281C22×C9D3×D413yes—
282C22×C9D3×D413yes—
283C22×C9D3×D413yes—
284C22×C9#23913yes—
285C2×D9—312no—
286C2×D9—66no—
287C2×D9—66no—
288C2×D9—66no—
289C2×D9—66no—
290C2×D9—66no—
291C2×D9—66no—
292C2×D9—66no—
293C2×D9—66no—
294C2×D9—66no—
295C2×D9—123no—
296C2×D9—123no—
297C2×D9—123no—
298C2×D9—123no—
299C4×C9—26no—
300C4×C9—26no—
301C4×C9—43no—
302Dic9—66no—
303Dic9—123no—
304C22×D4—271no—
305C23×C4—271no—
306C2×C22⋊C4—271no—
307C2×C22⋊C4—271no—
308C2×C22⋊C4—271no—
309#95—271no—
310#95C3⋊D911yescharacteristic, complemented
311C3×C9#38611yescharacteristic, complemented, Sylow
312C23×C3C22×D913yes—
313C23×C3C22×C3⋊D311yescharacteristic
314C22×D3—96no—
315C22×D3—96no—
316C22×D3—96no—
317C22×D3—92no—
318C22×D3—92no—
319C22×D3—92no—
320C22×D3—96no—
321C22×D3—92no—
322C22×D3—183no—
323C22×D3—183no—
324C22×D3—183no—
325C22×D3—183no—
326C22×D3—181no—
327C22×D3—181no—
328C22×D3—181no—
329C22×D3—181no—
330C22×D3—183no—
331C22×D3—183no—
332C22×D3—183no—
333C22×D3—183no—
334C22×D3—183no—
335C22×D3—183no—
336C22×D3—181no—
337C22×D3—181no—
338C22×D3—181no—
339C22×D3—181no—
340C22×D3—181no—
341C22×D3—181no—
342C2×C3×C4—23no—
343C2×C3×C4—23no—
344C2×C3×C4—23no—
345C2×C3×C4—23no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—23no—
351C2×C3×C4—23no—
352C2×C3×C4—23no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4C4×D916yescomplemented
357C2×C3×C4D3616yescomplemented
358C2×C3×C4C4×C3⋊D312yes—
359C2×C3×C4C4⋊(C3⋊D3)12yes—
360C2×Dic3—96no—
361C2×Dic3—92no—
362C2×Dic3—183no—
363C2×Dic3—183no—
364C2×Dic3—181no—
365C2×Dic3—181no—
366C2×Dic3—183no—
367C2×Dic3—183no—
368C2×Dic3—183no—
369C2×Dic3—181no—
370C2×Dic3—181no—
371C2×Dic3—181no—
372D12—188no—
373D12—1824no—
374C4×D3—188no—
375C4×D3—1824no—
376C2×C32C4×D1211yescharacteristic
377C2×C32D3×C22⋊C411yescharacteristic
378C2×C32#88211yescharacteristic
379C2×C32#88411yescharacteristic
380C2×C32#89311yescharacteristic
381C2×C32#89411yescharacteristic
382C2×C32#89511yescharacteristic
383C3⋊D3—64no—
384C3⋊D3—122no—
385C2×C9C4×D1213yes—
386C2×C9D3×C22⋊C413yes—
387C2×C9#88213yes—
388C2×C9#88413yes—
389C2×C9#89313yes—
390C2×C9#89413yes—
391C2×C9#89513yes—
392D9—612no—
393D9—126no—
394C24—271no—
395C24—271no—
396C2×D4—274no—
397C2×D4—542no—
398C2×D4—542no—
399C22×C4—271no—
400C22×C4—271no—
401C22×C4—542no—
402C22×C4—542no—
403C22×C4—542no—
404C22×C4C2×C3⋊D911yescharacteristic
405C22×C4C2×C3⋊D911yescharacteristic
406C22×C4C2×C3⋊D911yescharacteristic
407C22⋊C4—542no—
408C22⋊C4—542no—
409C22⋊C4—542no—
410C22×C3C2×C4×D913yes—
411C22×C3C2×D3613yes—
412C22×C3#369713yes—
413C22×C3D4×D913yes—
414C22×C3D4×D913yes—
415C22×C3D4×D913yes—
416C22×C3#370113yes—
417C22×C3C2×C4×C3⋊D311yescharacteristic
418C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
419C22×C3#382811yescharacteristic
420C22×C3D4×C3⋊D311yescharacteristic
421C22×C3D4×C3⋊D311yescharacteristic
422C22×C3D4×C3⋊D311yescharacteristic
423C22×C3#383211yescharacteristic
424C2×D3—94no—
425C2×D3—912no—
426C2×D3—182no—
427C2×D3—182no—
428C2×D3—182no—
429C2×D3—182no—
430C2×D3—182no—
431C2×D3—182no—
432C2×D3—186no—
433C2×D3—186no—
434C2×D3—186no—
435C2×D3—186no—
436C2×D3—186no—
437C2×D3—186no—
438C2×D3—182no—
439C2×D3—182no—
440C2×D3—182no—
441C2×D3—186no—
442C2×D3—186no—
443C2×D3—186no—
444C2×D3—361no—
445C2×D3—361no—
446C2×D3—361no—
447C2×D3—361no—
448C2×D3—363no—
449C2×D3—363no—
450C2×D3—363no—
451C2×D3—363no—
452C3×C4—22no—
453C3×C4—22no—
454C3×C4—26no—
455C3×C4—26no—
456C3×C4—41no—
457C3×C4—43no—
458Dic3—182no—
459Dic3—186no—
460Dic3—361no—
461Dic3—363no—
462C32#466511yescharacteristic
463C9#466513yescomplemented
464C23—272no—
465C23—272no—
466C23—272no—
467C23—272no—
468C23—541no—
469C23—541no—
470C23—541no—
471C23—541no—
472C23—541no—
473C23—541no—
474C23—541no—
475C23—541no—
476C23—541no—
477C23—541no—
478C23C22×C3⋊D911yescharacteristic
479D4—548no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—272no—
488C2×C4—548no—
489C2×C4—541no—
490C2×C4—541no—
491C2×C4—541no—
492C2×C4—541no—
493C2×C4—541no—
494C2×C4C4×C3⋊D912yescomplemented
495C2×C4C4⋊(C3⋊D9)12yescomplemented
496C2×C3C4×D3613yes—
497C2×C3C22⋊C4×D913yes—
498C2×C3#6344413yes—
499C2×C3#6344613yes—
500C2×C3#6345513yes—
501C2×C3#6345613yes—
502C2×C3#6345713yes—
503C2×C3C4×C4⋊(C3⋊D3)11yescharacteristic
504C2×C3C22⋊C4×C3⋊D311yescharacteristic
505C2×C3#6409111yescharacteristic
506C2×C3#6409311yescharacteristic
507C2×C3#6410211yescharacteristic
508C2×C3#6410311yescharacteristic
509C2×C3#6410411yescharacteristic
510D3—184no—
511D3—1812no—
512D3—362no—
513D3—366no—
514C22—274no—
515C22—542no—
516C22—542no—
517C22—542no—
518C22—542no—
519C22—542no—
520C22—542no—
521C22—542no—
522C22—542no—
523C22—542no—
524C22—1081no—
525C22—1081no—
526C22—1081no—
527C22—1081no—
528C22C2×C4×C3⋊D911yescharacteristic
529C22C2×C4⋊(C3⋊D9)11yescharacteristic
530C22#8873611yescharacteristic
531C22D4×C3⋊D911yescharacteristic
532C22D4×C3⋊D911yescharacteristic
533C22D4×C3⋊D911yescharacteristic
534C22#8874611yescharacteristic
535C4—22no—
536C4—22no—
537C4—41no—
538C4—542no—
539C4—1081no—
540C3#1058834513yescomplemented
541C3#1059236411yescharacteristic
542C2—544no—
543C2—1082no—
544C2C4×C4⋊(C3⋊D9)11yescharacteristic
545C2C22⋊C4×C3⋊D911yescharacteristic
546C2#1172172811yescharacteristic
547C2#1172173011yescharacteristic
548C2#1172174911yescharacteristic
549C2#1172175011yescharacteristic
550C2#1172175111yescharacteristic
subgroupG/H
centerC23C22×C3⋊D9
commutatorC22×C3×C9C22×C4
FrattiniC23×C3C22×C3⋊D3
FittingC3×C9×#95C2
socleC23×C32C22×D3
prime radicalC22×C4⋊(C3⋊D9)C2
cosocleC23×C3×C9C23
hypercenter#95C3⋊D9
hypocenterC3×C9#386
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
2. C22×C4×C3⋊D9C2semidirect
3. C2×#88536C2semidirect
4. C2×#88536C2semidirect
5. C2×#88536C2semidirect
6. C3×C9×#95C2semidirect
7. #11720572C2semidirect
19. C2×C4⋊(C3⋊D9)C4semidirect
38. C9×#95D3semidirect
52. C22×C3⋊D9C2×C4semidirect
62. C2×C3×C4×C9C2×C4semidirect
63. C2×C3×C4×C9D4semidirect
69. C2×C3⋊Dic9D4semidirect
126. C2×C3⋊D9C22⋊C4semidirect
142. C3×#95D9semidirect
200. C2×C4×C9C4×D3semidirect
201. C2×C4×C9D12semidirect
310. #95C3⋊D9semidirect
311. C3×C9#386semidirect
356. C2×C3×C4C4×D9semidirect
357. C2×C3×C4D36semidirect
463. C9#4665semidirect
494. C2×C4C4×C3⋊D9semidirect
495. C2×C4C4⋊(C3⋊D9)semidirect
540. C3#10588345semidirect

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
C22×C4×C3⋊D92
C2×#885362
C3×C9×#952
#117205722
#105923643
#105883453