Finite Group Databasebeta

← #13148920

Subgroups — #13148920

Order 1728.

796 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721751C211yesmaximal, characteristic
2C9×#887C211yesmaximal, characteristic, complemented
3C3×#63452C211yesmaximal, characteristic, complemented
4#11720997C212yesmaximal, complemented
5#11720993C212yesmaximal, complemented
6#11720993C211yesmaximal, characteristic, complemented
7C4×(C3⋊D9)⋊C4C211yesmaximal, characteristic, complemented
8#11720953C211yesmaximal, characteristic, complemented
9#11720929C211yesmaximal, characteristic, complemented
10#11720921C212yesmaximal, complemented
11#11720898C212yesmaximal, complemented
12#10594464—31nomaximal
13#10589682—31nomaximal
14C2×C4⋊(C3⋊D9)C2212yes—
15C2×C4⋊(C3⋊D9)C2211yescharacteristic
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×C9×Dic6C2211yescharacteristic
18C2×(C3⋊D9)⋊C4C2211yescharacteristic
19C2×(C3⋊D9)⋊C4C2211yescharacteristic
20C2×(C3⋊D9)⋊C4C2212yes—
21C3×C9×C4⋊C4C2211yescharacteristic
22#88536C2212yes—
23C9×Dic3⋊C4C2212yes—
24C4×C9×Dic3C2211yescharacteristic
25C4×C9×Dic3C2212yes—
26C3×C4⋊Dic9C2212yes—
27C3×C4⋊Dic9C2211yescharacteristic
28C3×Dic9⋊C4C2212yes—
29C3×C4×Dic9C2211yescharacteristic
30#88445C2212yes—
31#88445C2212yes—
32#88445C2212yes—
33#88445C2212yes—
34Dic9⋊Dic3C2212yes—
35Dic3×Dic9C2211yescharacteristic
36Dic3×Dic9C2211yescharacteristic
37#64104—31no—
38C3×#890—31no—
39C3×#887D311yescharacteristic
40#63897—32no—
41#63894—32no—
42#63894—31no—
43C4×#624—31no—
44#63871—31no—
45#63864—31no—
46#63848—32no—
47#63836—32no—
48C4×Q8×C9—31no—
49#63509—31no—
50#63508—31no—
51#63460—32no—
52#63457—31no—
53#63457—61no—
54#63456—32no—
55#63452D311yescharacteristic, complemented
56C4×D36—31no—
57C4×D36—32no—
58#63434—32no—
59C42×D9—31no—
60C22×C3⋊D9C2312yes—
61C22×C3⋊D9C2311yescharacteristic
62C2×C3×C4×C9C2312yes—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2311yescharacteristic
65C4⋊(C3⋊D9)—24no—
66C4⋊(C3⋊D9)—22no—
67C4×C3⋊D9—22no—
68C2×C9×Dic3C2311yescharacteristic
69C2×C9×Dic3C2312yes—
70C2×C9×Dic3C2311yescharacteristic
71C2×C3×Dic9C2311yescharacteristic
72C2×C3×Dic9C2311yescharacteristic
73C2×C3×Dic9C2312yes—
74C9×Dic6—22no—
75(C3⋊D9)⋊C4—22no—
76(C3⋊D9)⋊C4—22no—
77(C3⋊D9)⋊C4—24no—
78#5691—91no—
79#5580—91no—
80C2×C4⋊(C3⋊D3)—31no—
81C2×C4⋊(C3⋊D3)—32no—
82C2×C4×C3⋊D3—31no—
83C2×C3×Dic6C2×D311yescharacteristic
84C2×#624—32no—
85C2×#624—31no—
86C2×#624—31no—
87C32×C4⋊C4C2×D311yescharacteristic
88#3752—32no—
89C3×C4⋊Dic3—32no—
90C3×C4⋊Dic3—31no—
91C3×Dic3⋊C4—32no—
92C3×Dic3⋊C4C2×D312yes—
93C3×C4×Dic3—31no—
94C3×C4×Dic3C2×D312yes—
95C3×C4×Dic3C2×D311yescharacteristic
96Dic3⋊Dic3—32no—
97#3722—32no—
98#3722—32no—
99#3722—32no—
100#3722—32no—
101Dic32—31no—
102Dic32—31no—
103C2×Q8×C9—31no—
104C2×D36—31no—
105C2×D36—32no—
106C2×D36—61no—
107C2×D36—62no—
108C2×C4×D9—31no—
109C2×C4×D9—31no—
110C2×C4×D9—31no—
111C2×C4×D9—32no—
112C2×C4×D9—61no—
113C9×C4⋊C4—21no—
114C9×C4⋊C4—32no—
115C9×C4⋊C4C2×D311yescharacteristic
116C42×C9—31no—
117C42×C9—32no—
118(C2×D9)⋊C4—32no—
119(C2×D9)⋊C4—32no—
120(C2×D9)⋊C4—32no—
121(C2×D9)⋊C4—32no—
122(C2×D9)⋊C4—32no—
123(C2×D9)⋊C4—62no—
124C4⋊Dic9C2×D311yescharacteristic
125C4⋊Dic9C2×D312yes—
126Dic9⋊C4—32no—
127Dic9⋊C4C2×D312yes—
128C4×Dic9—31no—
129C4×Dic9—31no—
130C4×Dic9C2×D311yescharacteristic
131C22×C3×C9C2411yescharacteristic
132C2×C3⋊D9—21no—
133C2×C3⋊D9—21no—
134C2×C3⋊D9—22no—
135C2×C3⋊D9—22no—
136C2×C3⋊D9—22no—
137C2×C3⋊D9#4112yes—
138C3×C4×C9—22no—
139C3×C4×C9#4112yes—
140C3⋊Dic9—21no—
141C9×Dic3—21no—
142C9×Dic3—22no—
143C9×Dic3#4112yes—
144C3×Dic9—21no—
145C3×Dic9—22no—
146C3×Dic9—21no—
147C3×#125—91no—
148C3×C4×Q8—31no—
149#947—91no—
150#946—91no—
151#898—92no—
152#898—92no—
153#897—91no—
154#895—91no—
155#895—91no—
156#895—181no—
157#894—92no—
158#894—92no—
159#893—91no—
160#893—92no—
161#892—92no—
162#890—31no—
163#887D911yescharacteristic, complemented
164#875—91no—
165C4×D12—92no—
166C4×D12—91no—
167#872—92no—
168C42×D3—91no—
169C42×D3—91no—
170C22×C3⋊D3—31no—
171C22×C3⋊D3—32no—
172C2×C32×C4C22×D312yes—
173C2×C32×C4C22×D311yescharacteristic
174C2×C3⋊Dic3—31no—
175C4⋊(C3⋊D3)—64no—
176C4⋊(C3⋊D3)—62no—
177C4×C3⋊D3—62no—
178C2×C3×Dic3—31no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3C22×D311yescharacteristic
182C2×C3×Dic3C22×D312yes—
183C2×C3×Dic3C22×D311yescharacteristic
184C3×Dic6—22no—
185#624—62no—
186#624—62no—
187#624—64no—
188C22×D9—31no—
189C22×D9—32no—
190C22×D9—61no—
191C22×D9—62no—
192Q8×C9—62no—
193C2×C4×C9—21no—
194C2×C4×C9—22no—
195C2×C4×C9—31no—
196C2×C4×C9—31no—
197C2×C4×C9—32no—
198C2×C4×C9C22×D312yes—
199C2×C4×C9C22×D311yescharacteristic
200C2×Dic9—31no—
201C2×Dic9—61no—
202C2×Dic9C22×D311yescharacteristic
203C2×Dic9C22×D311yescharacteristic
204C2×Dic9C22×D312yes—
205D36—64no—
206D36—62no—
207D36—122no—
208D36—124no—
209C4×D9—62no—
210C4×D9—64no—
211C4×D9—62no—
212C4×D9—62no—
213C4×D9—122no—
214#529—271noSylow
215C2×C3×C9C2×#4111yescharacteristic
216C2×C3×C9C2×#4111yescharacteristic
217C2×C3×C9C2×#4111yescharacteristic
218C3⋊D9—22no—
219C3⋊D9—42no—
220C2×C3×Q8—31no—
221C2×D12—91no—
222C2×D12—92no—
223C2×D12—92no—
224C2×D12—91no—
225C2×D12—182no—
226C2×D12—181no—
227C2×C4×D3—91no—
228C2×C4×D3—91no—
229C2×C4×D3—92no—
230C2×C4×D3—91no—
231C2×C4×D3—91no—
232C2×C4×D3—91no—
233C2×C4×D3—91no—
234C2×C4×D3—92no—
235C2×C4×D3—181no—
236C2×Dic6C2×D911yescharacteristic
237C3×C4⋊C4—21no—
238C3×C4⋊C4—32no—
239C3×C4⋊C4—92no—
240C3×C4⋊C4—91no—
241C3×C4⋊C4—92no—
242C3×C4⋊C4C2×D911yescharacteristic
243C3×C4⋊C4D3211yescharacteristic
244C3×C42—31no—
245C3×C42—32no—
246C3×C42—91no—
247(C2×D3)⋊C4—92no—
248(C2×D3)⋊C4—92no—
249(C2×D3)⋊C4—92no—
250(C2×D3)⋊C4—92no—
251(C2×D3)⋊C4—92no—
252(C2×D3)⋊C4—92no—
253(C2×D3)⋊C4—92no—
254(C2×D3)⋊C4—92no—
255(C2×D3)⋊C4—92no—
256(C2×D3)⋊C4—92no—
257(C2×D3)⋊C4—182no—
258C4⋊Dic3—32no—
259C4⋊Dic3—31no—
260C4⋊Dic3—92no—
261Dic3⋊C4—32no—
262Dic3⋊C4—92no—
263Dic3⋊C4C2×D912yes—
264C4×Dic3—31no—
265C4×Dic3—91no—
266C4×Dic3—91no—
267C4×Dic3—91no—
268C4×Dic3—91no—
269C4×Dic3C2×D911yescharacteristic
270C4×Dic3C2×D912yes—
271C22×C32C23×D311yescharacteristic
272C2×C3⋊D3—32no—
273C2×C3⋊D3—61no—
274C2×C3⋊D3—61no—
275C2×C3⋊D3—62no—
276C2×C3⋊D3—62no—
277C2×C3⋊D3—62no—
278C32×C4—22no—
279C32×C4#23512yes—
280C3⋊Dic3—61no—
281C3×Dic3—22no—
282C3×Dic3—21no—
283C3×Dic3—61no—
284C3×Dic3—62no—
285C3×Dic3—61no—
286C3×Dic3#23912yes—
287C22×C9—21no—
288C22×C9C23×D311yescharacteristic
289C2×D9—32no—
290C2×D9—62no—
291C2×D9—61no—
292C2×D9—61no—
293C2×D9—62no—
294C2×D9—62no—
295C2×D9—62no—
296C2×D9—121no—
297C2×D9—121no—
298C2×D9—122no—
299C2×D9—122no—
300C2×D9—122no—
301C4×C9—22no—
302C4×C9—22no—
303C4×C9—32no—
304C4×C9—42no—
305C4×C9—62no—
306C4×C9—61no—
307C4×C9#23912yes—
308Dic9—21no—
309Dic9—21no—
310Dic9—22no—
311Dic9—61no—
312Dic9—121no—
313#126—272no—
314#125—91no—
315#124—271no—
316#123—272no—
317C22⋊Q8—271no—
318#121—271no—
319C4×Q8—31no—
320C4×D4—271no—
321C4×D4—272no—
322#117—272no—
323C2×C42—271no—
324C3×C9#52911yescharacteristic, complemented, Sylow
325C22×D3—91no—
326C22×D3—91no—
327C22×D3—92no—
328C22×D3—92no—
329C22×D3—181no—
330C22×D3—182no—
331C3×Q8—62no—
332C2×C3×C4—21no—
333C2×C3×C4—22no—
334C2×C3×C4—31no—
335C2×C3×C4—31no—
336C2×C3×C4—32no—
337C2×C3×C4—91no—
338C2×C3×C4—91no—
339C2×C3×C4—92no—
340C2×C3×C4C22×D912yes—
341C2×C3×C4C22×D911yescharacteristic
342C2×C3×C4C2×D3212yes—
343C2×C3×C4C2×D3211yescharacteristic
344C2×Dic3—31no—
345C2×Dic3—31no—
346C2×Dic3—32no—
347C2×Dic3—91no—
348C2×Dic3—91no—
349C2×Dic3—181no—
350C2×Dic3C22×D911yescharacteristic
351C2×Dic3C22×D912yes—
352C2×Dic3C22×D911yescharacteristic
353D12—184no—
354D12—184no—
355D12—182no—
356D12—182no—
357D12—362no—
358D12—364no—
359C4×D3—182no—
360C4×D3—182no—
361C4×D3—184no—
362C4×D3—182no—
363C4×D3—182no—
364C4×D3—182no—
365C4×D3—182no—
366C4×D3—184no—
367C4×D3—362no—
368Dic6—22no—
369C2×C32C2×#23511yescharacteristic
370C2×C32C2×#23911yescharacteristic
371C2×C32D3×#4111yescharacteristic
372C3⋊D3—62no—
373C3⋊D3—122no—
374C2×C9—21no—
375C2×C9—21no—
376C2×C9—21no—
377C2×C9C2×#23911yescharacteristic
378C2×C9D3×#4111yescharacteristic
379C2×C9D3×#4111yescharacteristic
380D9—62no—
381D9—122no—
382D9—122no—
383D9—242no—
384C2×Q8—31no—
385C2×D4—272no—
386C2×D4—271no—
387C22×C4—271no—
388C22×C4—271no—
389C22×C4—272no—
390C22×C4—271no—
391C4⋊C4—32no—
392C4⋊C4—92no—
393C4⋊C4—91no—
394C4⋊C4—92no—
395C4⋊C4—272no—
396C4⋊C4D3×D911yescharacteristic
397C22⋊C4—272no—
398C22⋊C4—272no—
399C22⋊C4—272no—
400C22⋊C4—272no—
401C22⋊C4—272no—
402C42—31no—
403C42—32no—
404C42—91no—
405C42—271no—
406C42—271no—
407C22×C3—21no—
408C22×C3C23×D911yescharacteristic
409C22×C3C22×D3211yescharacteristic
410C2×D3—92no—
411C2×D3—92no—
412C2×D3—182no—
413C2×D3—181no—
414C2×D3—181no—
415C2×D3—181no—
416C2×D3—181no—
417C2×D3—182no—
418C2×D3—182no—
419C2×D3—182no—
420C2×D3—182no—
421C2×D3—182no—
422C2×D3—182no—
423C2×D3—361no—
424C2×D3—361no—
425C2×D3—362no—
426C2×D3—362no—
427C2×D3—362no—
428C3×C4—22no—
429C3×C4—22no—
430C3×C4—22no—
431C3×C4—32no—
432C3×C4—42no—
433C3×C4—62no—
434C3×C4—61no—
435C3×C4—181no—
436C3×C4—182no—
437C3×C4—181no—
438C3×C4#369712yes—
439C3×C4#379912yes—
440Dic3—22no—
441Dic3—21no—
442Dic3—61no—
443Dic3—61no—
444Dic3—62no—
445Dic3—181no—
446Dic3—181no—
447Dic3—361no—
448Dic3#370112yes—
449C32#558011yescharacteristic
450C9—21no—
451C9#569111yescharacteristic, complemented
452C23—271no—
453C23—272no—
454Q8—62no—
455D4—544no—
456D4—542no—
457C2×C4—31no—
458C2×C4—31no—
459C2×C4—32no—
460C2×C4—91no—
461C2×C4—91no—
462C2×C4—92no—
463C2×C4—271no—
464C2×C4—542no—
465C2×C4—544no—
466C2×C4—542no—
467C2×C4—542no—
468C2×C4C2×D3×D912yes—
469C2×C4C2×D3×D911yescharacteristic
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3C2×#369711yescharacteristic
474C2×C3C2×#370111yescharacteristic
475C2×C3#41×D911yescharacteristic
476C2×C3C2×#379911yescharacteristic
477C2×C3#6431211yescharacteristic
478C2×C3D3×#23911yescharacteristic
479D3—182no—
480D3—182no—
481D3—362no—
482D3—362no—
483D3—362no—
484D3—722no—
485C22—272no—
486C22—541no—
487C22—541no—
488C22—542no—
489C22—542no—
490C22—542no—
491C22C22×D3×D911yescharacteristic
492C4—22no—
493C4—32no—
494C4—61no—
495C4—62no—
496C4—181no—
497C4—181no—
498C4—182no—
499C4—541no—
500C4#8864212yes—
501C3—21no—
502C3#1058968211yescharacteristic, complemented
503C3#1059446411yescharacteristic
504C2—542no—
505C2—1082no—
506C2C2×#8864211yescharacteristic
507C2#1172249811yescharacteristic
508C2D3×#370111yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radical#11721751C2
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#529
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C9×#887C2semidirect
3. C3×#63452C2semidirect
4. #11720997C2semidirect
5. #11720993C2semidirect
6. #11720993C2semidirect
7. C4×(C3⋊D9)⋊C4C2semidirect
8. #11720953C2semidirect
9. #11720929C2semidirect
10. #11720921C2semidirect
11. #11720898C2semidirect
55. #63452D3semidirect
163. #887D9semidirect
324. C3×C9#529semidirect
451. C9#5691semidirect
502. C3#10589682semidirect

Maximal subgroups

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

subgroupindex
#117217512
C9×#8872
C3×#634522
#117209972
#117209932
C4×(C3⋊D9)⋊C42
#117209532
#117209292
#117209212
#117208982
#105944643
#105896823