Finite Group Databasebeta

← #72254708

Subgroups — #72254708

Order 5184.

1482 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72254699C212yesmaximal, complemented
2#72254696C212yesmaximal, complemented
3C2×#13047909C211yesmaximal, characteristic
4C3×3−1+2×#106C211yesmaximal, characteristic, complemented
5#72254601C211yesmaximal, characteristic, complemented
6C3×#10592568—31nomaximal
7#13144106C311yesmaximal, characteristic, complemented
8#10588549⋊C3—33nomaximal
9C2×C4⋊(C3⋊AGL(1,ℤ/9))C2212yes—
10C2×C4×C3⋊AGL(1,ℤ/9)—22no—
11C2×C3×C8×3−1+2C2211yescharacteristic
12C3×C4⋊C4×3−1+2C2212yes—
13#13047949—22no—
14C4×C3⋊(3−1+2⋊C4)C2211yescharacteristic
15C2×C3⋊(3−1+2⋊C8)C2211yescharacteristic
16#13047909C414yescomplemented
17C2×C3×#3745—31no—
18C3×#64102—32no—
19C33×#106D311yescharacteristic
20C3×#63642—31no—
21C3×#63636—32no—
22C2×#88525C2×C311yescharacteristic
23#11721749C2×C312yescomplemented
24C2×D72⋊C3—33no—
25#63455⋊C3—36no—
263−1+2×#106—23no—
273−1+2×#106D313yescomplemented
28C3×C9×#106—21no—
29C3×C9×#106C2×C311yescharacteristic, complemented
30#11720538C2×C311yescharacteristic, complemented
31#11720528C2×C312yescomplemented
32#63375⋊C3—33no—
33#63369⋊C3—36no—
34C22×C3⋊AGL(1,ℤ/9)—22no—
35C2×C3×C4×3−1+2—22no—
36C2×C3×C4×3−1+2C2311yescharacteristic
37C2×C3⋊(3−1+2⋊C4)D412yes—
38C4⋊(C3⋊AGL(1,ℤ/9))—22no—
39C4⋊(C3⋊AGL(1,ℤ/9))C2×C414yescomplemented
40C4×C3⋊AGL(1,ℤ/9)—44no—
41C3×C8×3−1+2C2×C412yescomplemented
42C3⋊(3−1+2⋊C8)—21no—
43#10592568—31no—
44C3×#4869—91no—
45C3×#4869—92no—
46C3×#4869—91no—
47#10588549—33no—
48C2×C3×C4⋊(C3⋊D3)—32no—
49C2×C3×C4×C3⋊D3—62no—
50C2×C33×C8C2×D311yescharacteristic
51C33×C4⋊C4C2×D312yes—
52C3×#3752—62no—
53C3×C4×C3⋊Dic3—31no—
54C2×C3×C3⋊(C3⋊C8)—31no—
55C3×#3745—34no—
56C2×C4⋊(C3⋊D9)C22×C312yes—
57C2×C4×C3⋊D9—22no—
58C2×C4⋊AGL(1,ℤ/9)—36no—
59C2×C4×AGL(1,ℤ/9)—66no—
60C2×C8×3−1+2—23no—
61C2×C8×3−1+2C2×D313yes—
62C2×C3×C8×C9—21no—
63C2×C3×C8×C9C22×C311yescharacteristic
64C4⋊C4×3−1+2—26no—
65C4⋊C4×3−1+2C2×D316yes—
66C3×C9×C4⋊C4—22no—
67C3×C9×C4⋊C4C22×C312yes—
68#88536—22no—
69C4×C3⋊Dic9C22×C311yescharacteristic
70C2×C3⋊(C9⋊C8)C22×C311yescharacteristic
71#88525C3×C414yescomplemented
72(C2×AGL(1,ℤ/9))⋊C4—66no—
73C4×3−1+2⋊C4—33no—
74C2×3−1+2⋊C8—33no—
75D72⋊C3—312no—
76C22×C3×3−1+2C2×D411yescharacteristic
77C2×C3⋊AGL(1,ℤ/9)—24no—
78C2×C3⋊AGL(1,ℤ/9)—42no—
79C2×C3⋊AGL(1,ℤ/9)—42no—
80C3×C4×3−1+2—42no—
81C3×C4×3−1+2C22×C411yescharacteristic
82C3×C4×3−1+2#4111yescharacteristic
83C3⋊(3−1+2⋊C4)—22no—
84C2×#3745—31no—
85#64102—32no—
86C2×C3×D24—91no—
87C2×C3×D24—92no—
88C2×C3×D24—91no—
89C3×#893—92no—
90C3×#893—94no—
91C3×#893—92no—
92C32×#106—21no—
93C32×#106—31no—
94C32×#106—62no—
95C32×#106C3×D311yescharacteristic
96C32×#106C3⋊D311yescharacteristic
97#63642—31no—
98#63636—32no—
99C3×#815—91no—
100C3×#815—92no—
101C3×#815—91no—
102C3×#809—92no—
103C3×#809—94no—
104C3×#809—92no—
105C2×D72—33no—
106#63455—36no—
107C9×#106—23no—
108C9×#106C3×D313yescomplemented
109#63375—33no—
110#63369—36no—
111C22×C3×C3⋊D3—62no—
112C2×C33×C4—22no—
113C2×C33×C4C22×D311yescharacteristic
114C2×C3×C3⋊Dic3—32no—
115C3×C4⋊(C3⋊D3)—34no—
116C3×C4⋊(C3⋊D3)—62no—
117C3×C4×C3⋊D3—124no—
118C22×C3⋊D9—22no—
119C22×AGL(1,ℤ/9)—66no—
120C33×C8C4×D312yes—
121C3×C3⋊(C3⋊C8)—61no—
122C2×C4×3−1+2—23no—
123C2×C4×3−1+2—26no—
124C2×C4×3−1+2—46no—
125C2×C4×3−1+2C22×D313yes—
126C2×C3×C4×C9—21no—
127C2×C3×C4×C9—22no—
128C2×C3×C4×C9—42no—
129C2×C3×C4×C9C23×C311yescharacteristic
130C2×C3⋊Dic9C3×D412yes—
131C4⋊(C3⋊D9)—22no—
132C4⋊(C3⋊D9)C2×C3×C414yescomplemented
133C4×C3⋊D9—44no—
134C2×3−1+2⋊C4—36no—
135C4⋊AGL(1,ℤ/9)—312no—
136C4⋊AGL(1,ℤ/9)—66no—
137C4×AGL(1,ℤ/9)—1212no—
138C8×3−1+2—26no—
139C8×3−1+2C4×D316yescomplemented
140C3×C8×C9—22no—
141C3×C8×C9C2×C3×C412yescomplemented
142C3⋊(C9⋊C8)—21no—
1433−1+2⋊C8—63no—
144C3×#442—271no—
145#4869—91no—
146#4869—92no—
147#4869—91no—
148C2×C3×3−1+2C4×D411yescharacteristic
149C2×C3×3−1+2AGL(1,ℤ/8)12yes—
150C3⋊AGL(1,ℤ/9)—44no—
151C2×C4⋊(C3⋊D3)—32no—
152C2×C4×C3⋊D3—62no—
153C2×C3×D12—92no—
154C2×C3×D12—94no—
155C2×C3×D12—92no—
156C2×C3×C4×D3—182no—
157C2×C3×C4×D3—182no—
158C2×C3×C4×D3—184no—
159C2×C32×C8—21no—
160C2×C32×C8—31no—
161C2×C32×C8—62no—
162C2×C32×C8C2×C3×D311yescharacteristic
163C2×C32×C8C2×C3⋊D311yescharacteristic
164C32×C4⋊C4—22no—
165C32×C4⋊C4—32no—
166C32×C4⋊C4—64no—
167C32×C4⋊C4C2×C3×D312yes—
168C32×C4⋊C4C2×C3⋊D312yes—
169#3752—62no—
170C4×C3⋊Dic3—31no—
171C2×C3⋊(C3⋊C8)—31no—
172#3745—34no—
173C3×(C2×D3)⋊C4—182no—
174C3×(C2×D3)⋊C4—182no—
175C3×(C2×D3)⋊C4—184no—
176C3×C4×Dic3—91no—
177C3×C4×Dic3—92no—
178C3×C4×Dic3—91no—
179C2×C3×C3⋊C8—91no—
180C2×C3×C3⋊C8—92no—
181C2×C3×C3⋊C8—91no—
182C3×D24—94no—
183C3×D24—98no—
184C3×D24—94no—
185C2×D36—36no—
186C2×C4×D9—66no—
187C2×C8×C9—23no—
188C2×C8×C9C2×C3×D313yes—
189C9×C4⋊C4—26no—
190C9×C4⋊C4C2×C3×D316yes—
191(C2×D9)⋊C4—66no—
192C4×Dic9—33no—
193C2×C9⋊C8—33no—
194D72—312no—
195C22×C33D3×D411yescharacteristic
196C2×C3×C3⋊D3—64no—
197C2×C3×C3⋊D3—122no—
198C2×C3×C3⋊D3—122no—
199C33×C4—42no—
200C33×C4C2×C4×D311yescharacteristic
201C33×C4#23911yescharacteristic
202C3×C3⋊Dic3—62no—
203C22×3−1+2—23no—
204C22×3−1+2D3×D413yes—
205C22×C3×C9—21no—
206C22×C3×C9C2×C3×D411yescharacteristic
207C2×C3⋊D9—24no—
208C2×C3⋊D9—42no—
209C2×C3⋊D9—42no—
210C2×AGL(1,ℤ/9)—612no—
211C2×AGL(1,ℤ/9)—126no—
212C2×AGL(1,ℤ/9)—126no—
213C4×3−1+2—23no—
214C4×3−1+2—23no—
215C4×3−1+2—46no—
216C4×3−1+2—86no—
217C4×3−1+2C2×C4×D313yes—
218C4×3−1+2#23913yes—
219C3×C4×C9—21no—
220C3×C4×C9—21no—
221C3×C4×C9—42no—
222C3×C4×C9—82no—
223C3×C4×C9C22×C3×C411yescharacteristic
224C3×C4×C9C3×#4111yescharacteristic
225C3⋊Dic9—22no—
2263−1+2⋊C4—66no—
227C2×C3×D8—271no—
228C3×C4×D4—272no—
229C2×D24—91no—
230C2×D24—92no—
231C2×D24—91no—
232#893—92no—
233#893—94no—
234#893—92no—
235C3×#106—61no—
236C3×#106—61no—
237C3×#106AGL(1,ℤ/9)11yescharacteristic, complemented
238C3×#106AGL(1,ℤ/9)12yescomplemented
239C3×#106C3×C3⋊D311yescharacteristic
240C3×#102—272no—
241C3×#97—271no—
242#815—91no—
243#815—92no—
244#815—91no—
245#809—92no—
246#809—94no—
247#809—92no—
248C3×3−1+2#44211yescharacteristic, complemented, Sylow
249C22×C3⋊D3—62no—
250C22×C3×D3—182no—
251C22×C3×D3—184no—
252C22×C3×D3—182no—
253C2×C32×C4—22no—
254C2×C32×C4—22no—
255C2×C32×C4—21no—
256C2×C32×C4—31no—
257C2×C32×C4—42no—
258C2×C32×C4—62no—
259C2×C32×C4—62no—
260C2×C32×C4—124no—
261C2×C32×C4C22×C3×D311yescharacteristic
262C2×C32×C4C22×C3⋊D311yescharacteristic
263C2×C3⋊Dic3—32no—
264C4⋊(C3⋊D3)—34no—
265C4⋊(C3⋊D3)—62no—
266C4×C3⋊D3—124no—
267C2×C3×Dic3—92no—
268C2×C3×Dic3—94no—
269C2×C3×Dic3—92no—
270C3×D12—94no—
271C3×D12—98no—
272C3×D12—94no—
273C3×D12—182no—
274C3×D12—184no—
275C3×D12—182no—
276C3×C4×D3—364no—
277C3×C4×D3—364no—
278C3×C4×D3—368no—
279C22×D9—66no—
280C32×C8—22no—
281C32×C8—32no—
282C32×C8—64no—
283C32×C8C3×C4×D312yes—
284C32×C8C4×C3⋊D312yes—
285C3⋊(C3⋊C8)—61no—
286C3×C3⋊C8—181no—
287C3×C3⋊C8—182no—
288C3×C3⋊C8—181no—
289C2×C4×C9—26no—
290C2×C4×C9—23no—
291C2×C4×C9—46no—
292C2×C4×C9C22×C3×D313yes—
293C2×Dic9—36no—
294D36—312no—
295D36—66no—
296C4×D9—1212no—
297C8×C9—26no—
298C8×C9C3×C4×D316yescomplemented
299C9⋊C8—63no—
300#442—271noSylow
301C2×C33#89311yescharacteristic
302C2×C33#91412yes—
303C3×C3⋊D3—124no—
304C2×3−1+2—23no—
305C2×3−1+2—26no—
306C2×3−1+2#89313yes—
307C2×3−1+2#91416yes—
308C2×C3×C9—21no—
309C2×C3×C9—22no—
310C2×C3×C9C3×C4×D411yescharacteristic
311C2×C3×C9C3×AGL(1,ℤ/8)12yes—
312C3⋊D9—44no—
313AGL(1,ℤ/9)—1212no—
314C2×C3×D4—272no—
315C22×C3×C4—542no—
316C2×D12—92no—
317C2×D12—94no—
318C2×D12—92no—
319C2×C4×D3—182no—
320C2×C4×D3—182no—
321C2×C4×D3—184no—
322C3×D8—274no—
323C2×C3×C8—61no—
324C2×C3×C8—61no—
325C2×C3×C8—271no—
326C2×C3×C8C2×AGL(1,ℤ/9)11yescharacteristic
327C2×C3×C8C2×AGL(1,ℤ/9)12yes—
328C2×C3×C8C2×C3×C3⋊D311yescharacteristic
329C3×C4⋊C4—62no—
330C3×C4⋊C4—62no—
331C3×C4⋊C4C2×AGL(1,ℤ/9)12yes—
332C3×C4⋊C4C2×AGL(1,ℤ/9)14yes—
333C3×C4⋊C4C2×C3×C3⋊D312yes—
334C3×C22⋊C4—542no—
335C3×C42—271no—
336(C2×D3)⋊C4—182no—
337(C2×D3)⋊C4—182no—
338(C2×D3)⋊C4—184no—
339C4×Dic3—91no—
340C4×Dic3—92no—
341C4×Dic3—91no—
342C2×C3⋊C8—91no—
343C2×C3⋊C8—92no—
344C2×C3⋊C8—91no—
345D24—94no—
346D24—98no—
347D24—94no—
348C22×C32—21no—
349C22×C32—31no—
350C22×C32—62no—
351C22×C32C3×D3×D411yescharacteristic
352C22×C32D4×C3⋊D311yescharacteristic
353C2×C3⋊D3—64no—
354C2×C3⋊D3—122no—
355C2×C3⋊D3—122no—
356C2×C3×D3—184no—
357C2×C3×D3—188no—
358C2×C3×D3—184no—
359C2×C3×D3—362no—
360C2×C3×D3—362no—
361C2×C3×D3—364no—
362C2×C3×D3—364no—
363C2×C3×D3—362no—
364C2×C3×D3—362no—
365C32×C4—21no—
366C32×C4—21no—
367C32×C4—31no—
368C32×C4—31no—
369C32×C4—42no—
370C32×C4—42no—
371C32×C4—62no—
372C32×C4—62no—
373C32×C4—82no—
374C32×C4—122no—
375C32×C4—244no—
376C32×C4C2×C3×C4×D311yescharacteristic
377C32×C4C3×#23911yescharacteristic
378C32×C4C2×C4×C3⋊D311yescharacteristic
379C32×C4#383211yescharacteristic
380C3⋊Dic3—62no—
381C3×Dic3—182no—
382C3×Dic3—184no—
383C3×Dic3—182no—
384C22×C9—23no—
385C22×C9C3×D3×D413yes—
386C2×D9—612no—
387C2×D9—126no—
388C2×D9—126no—
389C4×C9—23no—
390C4×C9—23no—
391C4×C9—46no—
392C4×C9—86no—
393C4×C9C2×C3×C4×D313yes—
394C4×C9C3×#23913yes—
395Dic9—66no—
396C2×D8—271no—
397C4×D4—272no—
398#106C3⋊AGL(1,ℤ/9)11yescharacteristic, complemented
399#102—272no—
400#97—271no—
401C33#486911yescharacteristic
4023−1+2—23no—
4033−1+2#486913yescomplemented
404C3×C9—21no—
405C3×C9C3×#44211yescharacteristic, complemented
406C23×C3—542no—
407C22×D3—182no—
408C22×D3—184no—
409C22×D3—182no—
410C3×D4—274no—
411C3×D4—542no—
412C2×C3×C4—22no—
413C2×C3×C4—24no—
414C2×C3×C4—22no—
415C2×C3×C4—61no—
416C2×C3×C4—61no—
417C2×C3×C4—122no—
418C2×C3×C4—122no—
419C2×C3×C4—272no—
420C2×C3×C4—1084no—
421C2×C3×C4C22×AGL(1,ℤ/9)11yescharacteristic
422C2×C3×C4C22×AGL(1,ℤ/9)12yes—
423C2×C3×C4C22×C3×C3⋊D311yescharacteristic
424C2×Dic3—92no—
425C2×Dic3—94no—
426C2×Dic3—92no—
427D12—94no—
428D12—98no—
429D12—94no—
430D12—182no—
431D12—184no—
432D12—182no—
433C4×D3—364no—
434C4×D3—364no—
435C4×D3—368no—
436C3×C8—62no—
437C3×C8—62no—
438C3×C8—541no—
439C3×C8C4×AGL(1,ℤ/9)12yescomplemented
440C3×C8C4×AGL(1,ℤ/9)14yescomplemented
441C3×C8C3×C4×C3⋊D312yes—
442C3⋊C8—181no—
443C3⋊C8—182no—
444C3⋊C8—181no—
445C2×C32—21no—
446C2×C32—22no—
447C2×C32—31no—
448C2×C32—32no—
449C2×C32—62no—
450C2×C32—64no—
451C2×C32C3×#89311yescharacteristic
452C2×C32C3×#91412yes—
453C2×C32#6410211yescharacteristic
454C2×C32#6412312yes—
455C3⋊D3—124no—
456C3×D3—364no—
457C3×D3—368no—
458C3×D3—364no—
459C2×C9—23no—
460C2×C9—26no—
461C2×C9C3×#89313yes—
462C2×C9C3×#91416yes—
463D9—1212no—
464C2×D4—272no—
465C22×C4—542no—
466D8—274no—
467C2×C8—271no—
468C2×C8C2×C3⋊AGL(1,ℤ/9)11yescharacteristic
469C4⋊C4C2×C3⋊AGL(1,ℤ/9)12yes—
470C22⋊C4—542no—
471C42—271no—
472C22×C3—61no—
473C22×C3—61no—
474C22×C3—544no—
475C22×C3—1082no—
476C22×C3—1082no—
477C22×C3D4×AGL(1,ℤ/9)11yescharacteristic
478C22×C3D4×AGL(1,ℤ/9)12yes—
479C22×C3C3×D4×C3⋊D311yescharacteristic
480C2×D3—184no—
481C2×D3—188no—
482C2×D3—184no—
483C2×D3—362no—
484C2×D3—364no—
485C2×D3—362no—
486C2×D3—364no—
487C2×D3—362no—
488C2×D3—362no—
489C3×C4—42no—
490C3×C4—44no—
491C3×C4—42no—
492C3×C4—61no—
493C3×C4—61no—
494C3×C4—61no—
495C3×C4—61no—
496C3×C4—242no—
497C3×C4—242no—
498C3×C4—542no—
499C3×C4C2×C4×AGL(1,ℤ/9)11yescharacteristic
500C3×C4C2×C4×AGL(1,ℤ/9)12yes—
501C3×C4#8872411yescharacteristic
502C3×C4#8872412yes—
503C3×C4C2×C3×C4×C3⋊D311yescharacteristic
504C3×C4C3×#383211yescharacteristic
505Dic3—182no—
506Dic3—184no—
507Dic3—182no—
508C32—21no—
509C32—31no—
510C32—62no—
511C32C3×#486911yescharacteristic
512C32#1059256811yescharacteristic
513C9—23no—
514C9C3×#486913yescomplemented
515C23—542no—
516D4—274no—
517D4—542no—
518C2×C4—22no—
519C2×C4—272no—
520C2×C4—1084no—
521C2×C4C22×C3⋊AGL(1,ℤ/9)11yescharacteristic
522C8—541no—
523C8C4×C3⋊AGL(1,ℤ/9)12yescomplemented
524C2×C3—61no—
525C2×C3—61no—
526C2×C3—62no—
527C2×C3—62no—
528C2×C3—1084no—
529C2×C3#63455⋊C311yescharacteristic
530C2×C3#63455⋊C312yes—
531C2×C3#63476⋊C312yes—
532C2×C3#63476⋊C314yes—
533C2×C3C3×#6410211yescharacteristic
534C2×C3C3×#6412312yes—
535D3—364no—
536D3—368no—
537D3—364no—
538C22—544no—
539C22—1082no—
540C22—1082no—
541C22D4×C3⋊AGL(1,ℤ/9)11yescharacteristic
542C4—42no—
543C4—542no—
544C4C2×C4×C3⋊AGL(1,ℤ/9)11yescharacteristic
545C4#1304893211yescharacteristic
546C3—61no—
547C3—61no—
548C3#10588549⋊C311yescharacteristic, complemented
549C3#10588549⋊C312yescomplemented
550C3C3×#1059256811yescharacteristic
551C2—1084no—
552C2#7225469611yescharacteristic
553C2#7225470412yes—
subgroupG/H
centerC22D4×C3⋊AGL(1,ℤ/9)
commutatorC3×C4×C9C22×C3×C4
FrattiniC2×C3×C4C22×C3×C3⋊D3
FittingC3×3−1+2×#106C2
socleC22×C32C3×D3×D4
prime radicalC2×#13047909C2
cosocleC2×C3×C4×C9C23×C3
hypercenter#106C3⋊AGL(1,ℤ/9)
hypocenterC3×C9C3×#442
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #106

Complements

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

NH ≅ G/Ntype
1. #72254699C2semidirect
2. #72254696C2semidirect
4. C3×3−1+2×#106C2semidirect
5. #72254601C2semidirect
7. #13144106C3semidirect
16. #13047909C4semidirect
23. #11721749C2×C3semidirect
27. 3−1+2×#106D3semidirect
29. C3×C9×#106C2×C3semidirect
30. #11720538C2×C3semidirect
31. #11720528C2×C3semidirect
39. C4⋊(C3⋊AGL(1,ℤ/9))C2×C4semidirect
41. C3×C8×3−1+2C2×C4semidirect
71. #88525C3×C4semidirect
108. C9×#106C3×D3semidirect
132. C4⋊(C3⋊D9)C2×C3×C4semidirect
139. C8×3−1+2C4×D3semidirect
141. C3×C8×C9C2×C3×C4semidirect
237. C3×#106AGL(1,ℤ/9)semidirect
238. C3×#106AGL(1,ℤ/9)semidirect
248. C3×3−1+2#442semidirect
298. C8×C9C3×C4×D3semidirect
398. #106C3⋊AGL(1,ℤ/9)semidirect
403. 3−1+2#4869semidirect
405. C3×C9C3×#442semidirect
439. C3×C8C4×AGL(1,ℤ/9)semidirect
440. C3×C8C4×AGL(1,ℤ/9)semidirect
514. C9C3×#4869semidirect
523. C8C4×C3⋊AGL(1,ℤ/9)semidirect
548. C3#10588549⋊C3semidirect
549. C3#10588549⋊C3semidirect

Maximal subgroups

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

subgroupindex
#722546992
#722546962
C2×#130479092
C3×3−1+2×#1062
#722546012
C3×#105925683
#131441063
#10588549⋊C33