Finite Group Databasebeta

← #13143700

Subgroups — #13143700

Order 1728.

1170 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C3⋊D9C211yesmaximal, characteristic
2C2×#88536C211yesmaximal, characteristic
3C2×#88536C211yesmaximal, characteristic
4C2×C4⋊(C3⋊Dic9)C211yesmaximal, characteristic, complemented
5C2×#88534C211yesmaximal, characteristic, complemented
6C3×C9×#95C211yesmaximal, characteristic, complemented
7#11720572C211yesmaximal, characteristic, complemented
8#10592367—31nomaximal
9#10588348—33nomaximal
10C23×C3⋊D9C2211yescharacteristic
11C22×C3×C4×C9C2211yescharacteristic
12C22×C3×C4×C9C2211yescharacteristic
13C22×C3×C4×C9C2211yescharacteristic
14C22×C3⋊Dic9C2211yescharacteristic
15C22×C3⋊Dic9C2211yescharacteristic
16C22×C3⋊Dic9C2211yescharacteristic
17C2×C4×C3⋊D9—22no—
18C2×C4×C3⋊D9—22no—
19C2×C4×C3⋊D9—22no—
20#88536—22no—
21#88536C414yescomplemented
22C4⋊(C3⋊Dic9)—22no—
23#88534—22no—
24C22×C4×C3⋊D3—31no—
25C2×#3752—31no—
26C2×#3752—31no—
27C2×C4⋊(C3⋊Dic3)—31no—
28C2×#3750—31no—
29C22×C4×D9—33no—
30C32×#95D311yescharacteristic
31#63658—31no—
32C2×(C2×D9)⋊C4—33no—
33C2×(C2×D9)⋊C4—33no—
34C2×C4⋊Dic9—33no—
35C2×Dic9⋊C4—33no—
36C9×#95D313yescomplemented
37C9⋊#95—33no—
38C23×C3×C9C2311yescharacteristic
39C22×C3⋊D9—21no—
40C22×C3⋊D9—21no—
41C22×C3⋊D9—21no—
42C22×C3⋊D9—21no—
43C22×C3⋊D9C2×C412yes—
44C22×C3⋊D9D412yes—
45C22×C3⋊D9Q812yes—
46C2×C3×C4×C9—21no—
47C2×C3×C4×C9—21no—
48C2×C3×C4×C9—21no—
49C2×C3×C4×C9—21no—
50C2×C3×C4×C9—21no—
51C2×C3×C4×C9—21no—
52C2×C3×C4×C9—21no—
53C2×C3×C4×C9C2×C412yescomplemented
54C2×C3×C4×C9D412yes—
55C2×C3⋊Dic9—21no—
56C2×C3⋊Dic9—21no—
57C2×C3⋊Dic9—21no—
58C2×C3⋊Dic9—21no—
59C2×C3⋊Dic9—21no—
60C2×C3⋊Dic9—21no—
61C2×C3⋊Dic9—21no—
62C2×C3⋊Dic9C2×C412yescomplemented
63C2×C3⋊Dic9D412yes—
64C4×C3⋊D9—28no—
65#4668—93no—
66#4668—91no—
67C23×C3⋊D3—31no—
68C22×C32×C4C2×D311yescharacteristic
69C22×C32×C4C2×D311yescharacteristic
70C22×C32×C4C2×D311yescharacteristic
71C22×C3⋊Dic3—31no—
72C22×C3⋊Dic3—31no—
73C22×C3⋊Dic3—31no—
74C2×C4×C3⋊D3—62no—
75C2×C4×C3⋊D3—62no—
76C2×C4×C3⋊D3—62no—
77C23×D9—33no—
78#3752—34no—
79#3752—62no—
80C4⋊(C3⋊Dic3)—62no—
81#3750—62no—
82C22×C4×C9C2×D313yes—
83C22×C4×C9C2×D313yes—
84C22×C4×C9C2×D313yes—
85C22×Dic9—33no—
86C22×Dic9—33no—
87C22×Dic9—33no—
88C2×C4×D9—66no—
89C2×C4×D9—66no—
90C2×C4×D9—66no—
91(C2×D9)⋊C4—312no—
92(C2×D9)⋊C4—66no—
93C4⋊Dic9—66no—
94Dic9⋊C4—66no—
95C22×C3×C9C22×C411yescharacteristic
96C22×C3×C9C2×D411yescharacteristic
97C22×C3×C9C2×D411yescharacteristic
98C22×C3×C9C2×D411yescharacteristic
99C22×C3×C9C2×Q811yescharacteristic
100C22×C3×C9#4111yescharacteristic
101C22×C3×C9#4111yescharacteristic
102C2×C3⋊D9—22no—
103C2×C3⋊D9—22no—
104C2×C3⋊D9—22no—
105C2×C3⋊D9—22no—
106C2×C3⋊D9—22no—
107C2×C3⋊D9—22no—
108C2×C3⋊D9C4⋊C414yescomplemented
109C3×C4×C9—22no—
110C3×C4×C9—22no—
111C3×C4×C9—41no—
112C3⋊Dic9—22no—
113C3⋊Dic9—22no—
114C3⋊Dic9—41no—
115C22×C4×D3—93no—
116C22×C4×D3—91no—
117C2×(C2×D3)⋊C4—93no—
118C2×(C2×D3)⋊C4—91no—
119C2×(C2×D3)⋊C4—93no—
120C2×(C2×D3)⋊C4—91no—
121C2×C4⋊Dic3—93no—
122C2×C4⋊Dic3—91no—
123C2×Dic3⋊C4—93no—
124C2×Dic3⋊C4—91no—
125C3×#95D913yescomplemented
126C3×#95C3⋊D311yescharacteristic
127C3⋊#95—93no—
128C3⋊#95—91no—
129C23×C32C22×D311yescharacteristic
130C22×C3⋊D3—32no—
131C22×C3⋊D3—32no—
132C22×C3⋊D3—32no—
133C22×C3⋊D3—61no—
134C22×C3⋊D3—61no—
135C22×C3⋊D3—61no—
136C22×C3⋊D3—61no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4C4×D312yes—
145C2×C32×C4D1212yes—
146C2×C3⋊Dic3—32no—
147C2×C3⋊Dic3—32no—
148C2×C3⋊Dic3—61no—
149C2×C3⋊Dic3—61no—
150C2×C3⋊Dic3—61no—
151C2×C3⋊Dic3—61no—
152C2×C3⋊Dic3—61no—
153C2×C3⋊Dic3—61no—
154C2×C3⋊Dic3—61no—
155C4×C3⋊D3—68no—
156C23×C9C22×D313yes—
157C22×D9—36no—
158C22×D9—36no—
159C22×D9—36no—
160C22×D9—63no—
161C22×D9—63no—
162C22×D9—63no—
163C22×D9—63no—
164C2×C4×C9—23no—
165C2×C4×C9—23no—
166C2×C4×C9—23no—
167C2×C4×C9—23no—
168C2×C4×C9—23no—
169C2×C4×C9—23no—
170C2×C4×C9—23no—
171C2×C4×C9C4×D316yescomplemented
172C2×C4×C9D1216yes—
173C2×Dic9—36no—
174C2×Dic9—36no—
175C2×Dic9—63no—
176C2×Dic9—63no—
177C2×Dic9—63no—
178C2×Dic9—63no—
179C2×Dic9—63no—
180C2×Dic9—63no—
181C2×Dic9—63no—
182C4×D9—624no—
183#385—271noSylow
184C2×C3×C9C2×C4⋊C411yescharacteristic
185C2×C3×C9C4×D411yescharacteristic
186C2×C3×C9C4×D411yescharacteristic
187C2×C3×C9#12011yescharacteristic
188C2×C3×C9C22⋊Q811yescharacteristic
189C2×C3×C9C22⋊Q811yescharacteristic
190C2×C3×C9#12311yescharacteristic
191C3⋊D9—24no—
192C23×D3—93no—
193C23×D3—91no—
194C22×C3×C4C2×D913yes—
195C22×C3×C4C2×D913yes—
196C22×C3×C4C2×D913yes—
197C22×C3×C4C2×C3⋊D311yescharacteristic
198C22×C3×C4C2×C3⋊D311yescharacteristic
199C22×C3×C4C2×C3⋊D311yescharacteristic
200C22×Dic3—93no—
201C22×Dic3—93no—
202C22×Dic3—91no—
203C22×Dic3—91no—
204C22×Dic3—93no—
205C22×Dic3—91no—
206C2×C4×D3—182no—
207C2×C4×D3—186no—
208C2×C4×D3—182no—
209C2×C4×D3—186no—
210C2×C4×D3—186no—
211C2×C4×D3—182no—
212(C2×D3)⋊C4—912no—
213(C2×D3)⋊C4—94no—
214(C2×D3)⋊C4—182no—
215(C2×D3)⋊C4—186no—
216C4⋊Dic3—182no—
217C4⋊Dic3—186no—
218Dic3⋊C4—186no—
219Dic3⋊C4—182no—
220C22×C32C2×C4×D311yescharacteristic
221C22×C32C2×D1211yescharacteristic
222C22×C32#23511yescharacteristic
223C22×C32D3×D411yescharacteristic
224C22×C32D3×D411yescharacteristic
225C22×C32#23711yescharacteristic
226C22×C32D3×Q811yescharacteristic
227C2×C3⋊D3—34no—
228C2×C3⋊D3—62no—
229C2×C3⋊D3—62no—
230C2×C3⋊D3—62no—
231C2×C3⋊D3—62no—
232C2×C3⋊D3—62no—
233C2×C3⋊D3—62no—
234C32×C4—22no—
235C32×C4—22no—
236C32×C4—41no—
237C3⋊Dic3—62no—
238C3⋊Dic3—62no—
239C3⋊Dic3—121no—
240C22×C9C2×C4×D313yes—
241C22×C9C2×D1213yes—
242C22×C9#23513yes—
243C22×C9D3×D413yes—
244C22×C9D3×D413yes—
245C22×C9#23713yes—
246C22×C9D3×Q813yes—
247C2×D9—312no—
248C2×D9—66no—
249C2×D9—66no—
250C2×D9—66no—
251C2×D9—66no—
252C2×D9—66no—
253C2×D9—66no—
254C4×C9—26no—
255C4×C9—26no—
256C4×C9—43no—
257Dic9—66no—
258Dic9—66no—
259Dic9—123no—
260C23×C4—271no—
261C2×C4⋊C4—271no—
262C2×C4⋊C4—271no—
263C2×C22⋊C4—271no—
264C2×C22⋊C4—271no—
265#95—271no—
266#95C3⋊D911yescharacteristic, complemented
267C3×C9#38511yescharacteristic, complemented, Sylow
268C23×C3C22×D913yes—
269C23×C3C22×C3⋊D311yescharacteristic
270C22×D3—96no—
271C22×D3—96no—
272C22×D3—96no—
273C22×D3—92no—
274C22×D3—92no—
275C22×D3—92no—
276C22×D3—183no—
277C22×D3—183no—
278C22×D3—183no—
279C22×D3—183no—
280C22×D3—181no—
281C22×D3—181no—
282C22×D3—181no—
283C22×D3—181no—
284C2×C3×C4—23no—
285C2×C3×C4—23no—
286C2×C3×C4—23no—
287C2×C3×C4—23no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—23no—
293C2×C3×C4—23no—
294C2×C3×C4—23no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4C4×D916yescomplemented
299C2×C3×C4D3616yes—
300C2×C3×C4C4×C3⋊D312yes—
301C2×C3×C4C4⋊(C3⋊D3)12yes—
302C2×Dic3—96no—
303C2×Dic3—96no—
304C2×Dic3—92no—
305C2×Dic3—92no—
306C2×Dic3—183no—
307C2×Dic3—183no—
308C2×Dic3—183no—
309C2×Dic3—183no—
310C2×Dic3—181no—
311C2×Dic3—181no—
312C2×Dic3—181no—
313C2×Dic3—181no—
314C2×Dic3—183no—
315C2×Dic3—183no—
316C2×Dic3—183no—
317C2×Dic3—181no—
318C2×Dic3—181no—
319C2×Dic3—181no—
320C4×D3—188no—
321C4×D3—1824no—
322C2×C32C4×D1211yescharacteristic
323C2×C32#67⋊C411yescharacteristic
324C2×C32#88211yescharacteristic
325C2×C32#88311yescharacteristic
326C2×C32D3×C4⋊C411yescharacteristic
327C2×C32#89611yescharacteristic
328C2×C32#89711yescharacteristic
329C3⋊D3—64no—
330C2×C9C4×D1213yes—
331C2×C9#67⋊C413yes—
332C2×C9#88213yes—
333C2×C9#88313yes—
334C2×C9D3×C4⋊C413yes—
335C2×C9#89613yes—
336C2×C9#89713yes—
337D9—612no—
338C24—271no—
339C22×C4—271no—
340C22×C4—271no—
341C22×C4—271no—
342C22×C4—542no—
343C22×C4—542no—
344C22×C4—542no—
345C22×C4C2×C3⋊D911yescharacteristic
346C22×C4C2×C3⋊D911yescharacteristic
347C22×C4C2×C3⋊D911yescharacteristic
348C4⋊C4—542no—
349C4⋊C4—542no—
350C22⋊C4—274no—
351C22⋊C4—542no—
352C22×C3C2×C4×D913yes—
353C22×C3C2×D3613yes—
354C22×C3#369713yes—
355C22×C3D4×D913yes—
356C22×C3D4×D913yes—
357C22×C3#369913yes—
358C22×C3Q8×D913yes—
359C22×C3C2×C4×C3⋊D311yescharacteristic
360C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
361C22×C3#382811yescharacteristic
362C22×C3D4×C3⋊D311yescharacteristic
363C22×C3D4×C3⋊D311yescharacteristic
364C22×C3#383011yescharacteristic
365C22×C3Q8×C3⋊D311yescharacteristic
366C2×D3—94no—
367C2×D3—912no—
368C2×D3—182no—
369C2×D3—182no—
370C2×D3—182no—
371C2×D3—182no—
372C2×D3—182no—
373C2×D3—182no—
374C2×D3—186no—
375C2×D3—186no—
376C2×D3—186no—
377C2×D3—186no—
378C2×D3—186no—
379C2×D3—186no—
380C3×C4—22no—
381C3×C4—22no—
382C3×C4—26no—
383C3×C4—26no—
384C3×C4—41no—
385C3×C4—43no—
386Dic3—182no—
387Dic3—182no—
388Dic3—186no—
389Dic3—186no—
390Dic3—361no—
391Dic3—363no—
392C32#466811yescharacteristic
393C9#466813yescomplemented
394C23—272no—
395C23—272no—
396C23—272no—
397C23—541no—
398C23—541no—
399C23—541no—
400C23—541no—
401C23C22×C3⋊D911yescharacteristic
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—272no—
410C2×C4—272no—
411C2×C4—548no—
412C2×C4—541no—
413C2×C4—541no—
414C2×C4—541no—
415C2×C4—541no—
416C2×C4—541no—
417C2×C4—541no—
418C2×C4—541no—
419C2×C4C4×C3⋊D912yescomplemented
420C2×C4C4⋊(C3⋊D9)12yes—
421C2×C3C4×D3613yes—
422C2×C3#611⋊C413yes—
423C2×C3#6344413yes—
424C2×C3#6344513yes—
425C2×C3C4⋊C4×D913yes—
426C2×C3#6345813yes—
427C2×C3#6345913yes—
428C2×C3C4×C4⋊(C3⋊D3)11yescharacteristic
429C2×C3#6409011yescharacteristic
430C2×C3#6409111yescharacteristic
431C2×C3#6409211yescharacteristic
432C2×C3C4⋊C4×C3⋊D311yescharacteristic
433C2×C3#6410511yescharacteristic
434C2×C3#6410611yescharacteristic
435D3—184no—
436D3—1812no—
437C22—274no—
438C22—542no—
439C22—542no—
440C22—542no—
441C22—542no—
442C22—542no—
443C22—542no—
444C22C2×C4×C3⋊D911yescharacteristic
445C22C2×C4⋊(C3⋊D9)11yescharacteristic
446C22#8873611yescharacteristic
447C22D4×C3⋊D911yescharacteristic
448C22D4×C3⋊D911yescharacteristic
449C22#8874211yescharacteristic
450C22Q8×C3⋊D911yescharacteristic
451C4—22no—
452C4—22no—
453C4—41no—
454C4—542no—
455C4—542no—
456C4—1081no—
457C3#1058834813yescomplemented
458C3#1059236711yescharacteristic
459C2—544no—
460C2C4×C4⋊(C3⋊D9)11yescharacteristic
461C2#1172172711yescharacteristic
462C2#1172172811yescharacteristic
463C2#1172172911yescharacteristic
464C2C4⋊C4×C3⋊D911yescharacteristic
465C2#1172175211yescharacteristic
466C2#1172175311yescharacteristic
subgroupG/H
centerC23C22×C3⋊D9
commutatorC22×C3×C9C22×C4
FrattiniC23×C3C22×C3⋊D3
FittingC3×C9×#95C2
socleC23×C32C22×D3
prime radicalC23×C3⋊D9C22
cosocleC23×C3×C9C23
hypercenter#95C3⋊D9
hypocenterC3×C9#385
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
4. C2×C4⋊(C3⋊Dic9)C2semidirect
5. C2×#88534C2semidirect
6. C3×C9×#95C2semidirect
7. #11720572C2semidirect
21. #88536C4semidirect
36. C9×#95D3semidirect
53. C2×C3×C4×C9C2×C4semidirect
62. C2×C3⋊Dic9C2×C4semidirect
108. C2×C3⋊D9C4⋊C4semidirect
125. C3×#95D9semidirect
171. C2×C4×C9C4×D3semidirect
266. #95C3⋊D9semidirect
267. C3×C9#385semidirect
298. C2×C3×C4C4×D9semidirect
393. C9#4668semidirect
419. C2×C4C4×C3⋊D9semidirect
457. 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×#885362
C2×C4⋊(C3⋊Dic9)2
C2×#885342
C3×C9×#952
#117205722
#105923673
#105883483