Finite Group Databasebeta

← #13148960

Subgroups — #13148960

Order 1728.

742 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721751C211yesmaximal, characteristic
2C9×#888C211yesmaximal, characteristic, complemented
3C3×#63449C211yesmaximal, characteristic, complemented
4#11720997C212yesmaximal, complemented
5#11720992C211yesmaximal, characteristic, complemented
6#11720986C212yesmaximal, complemented
7#11720953C212yesmaximal, complemented
8#11720952C211yesmaximal, characteristic, complemented
9#11720936C211yesmaximal, characteristic, complemented
10#11720934C212yesmaximal, complemented
11#11720931C211yesmaximal, characteristic, complemented
12#10594352—31nomaximal
13#10589835—31nomaximal
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×C4⋊(C3⋊D9)C2212yes—
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×C9×Dic6C2212yes—
18C2×C3×Dic18C2211yescharacteristic
19C2×(C3⋊D9)⋊C4C2212yes—
20C3×C9×C4⋊C4C2211yescharacteristic
21#88536C2212yes—
22C9×C4⋊Dic3C2211yescharacteristic
23C9×Dic3⋊C4C2212yes—
24C4×C9×Dic3C2211yescharacteristic
25C3×Dic9⋊C4C2212yes—
26C3×C4×Dic9C2211yescharacteristic
27C3×C4×Dic9C2212yes—
28#88445C2212yes—
29#88445C2212yes—
30#88445C2211yescharacteristic
31#88445C2212yes—
32#88445C2212yes—
33#88445C2211yescharacteristic
34Dic3⋊Dic9C2212yes—
35#88442C2211yescharacteristic
36#88442C2211yescharacteristic
37#64104—31no—
38C3×#888D311yescharacteristic
39C3×#887—31no—
40#63897—32no—
41#63894—31no—
42#63888—32no—
43#63874—31no—
44#63871—31no—
45#63868—32no—
46#63864—32no—
47#63864—31no—
48C9×C4⋊Q8—31no—
49#63509—32no—
50#63508—32no—
51#63460—32no—
52#63459—31no—
53#63457—31no—
54#63457—61no—
55#63456—32no—
56#63455—31no—
57#63449D311yescharacteristic, complemented
58#63437—31no—
59#63434—31no—
60C22×C3⋊D9C2312yes—
61C22×C3⋊D9C2311yescharacteristic
62C2×C3×C4×C9C2312yes—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2311yescharacteristic
65C4⋊(C3⋊D9)—22no—
66C4⋊(C3⋊D9)—42no—
67C4×C3⋊D9—22no—
68C2×C9×Dic3C2312yes—
69C2×C9×Dic3C2311yescharacteristic
70C2×C9×Dic3C2311yescharacteristic
71C2×C3×Dic9C2311yescharacteristic
72C2×C3×Dic9C2312yes—
73C2×C3×Dic9C2311yescharacteristic
74C9×Dic6—42no—
75C3×Dic18—22no—
76(C3⋊D9)⋊C4—42no—
77#5733—91no—
78#5582—91no—
79C2×C4⋊(C3⋊D3)—31no—
80C2×C4⋊(C3⋊D3)—32no—
81C2×C4×C3⋊D3—31no—
82C2×C3×Dic6—31no—
83C2×C3×Dic6C2×D312yes—
84C2×#624—32no—
85C32×C4⋊C4C2×D311yescharacteristic
86#3752—32no—
87C3×C4⋊Dic3C2×D311yescharacteristic
88C3×Dic3⋊C4—32no—
89C3×Dic3⋊C4C2×D312yes—
90C3×C4×Dic3—32no—
91C3×C4×Dic3—31no—
92C3×C4×Dic3C2×D311yescharacteristic
93#3724—31no—
94#3724—31no—
95Dic3⋊Dic3—32no—
96#3722—31no—
97#3722—32no—
98#3722—32no—
99#3722—32no—
100#3722—32no—
101#3722—31no—
102C2×Q8×C9—32no—
103C2×D36—31no—
104C2×D36—32no—
105C2×D36—61no—
106C2×D36—62no—
107C2×C4×D9—31no—
108C2×C4×D9—32no—
109C2×C4×D9—61no—
110C2×Dic18C2×D311yescharacteristic
111C9×C4⋊C4—21no—
112C9×C4⋊C4—32no—
113C9×C4⋊C4—31no—
114C9×C4⋊C4C2×D311yescharacteristic
115C42×C9—31no—
116(C2×D9)⋊C4—32no—
117(C2×D9)⋊C4—32no—
118(C2×D9)⋊C4—31no—
119(C2×D9)⋊C4—32no—
120(C2×D9)⋊C4—32no—
121(C2×D9)⋊C4—31no—
122(C2×D9)⋊C4—32no—
123(C2×D9)⋊C4—62no—
124C4⋊Dic9—32no—
125Dic9⋊C4—31no—
126Dic9⋊C4—31no—
127Dic9⋊C4C2×D312yes—
128C4×Dic9C2×D312yes—
129C4×Dic9C2×D311yescharacteristic
130C22×C3×C9C2411yescharacteristic
131C2×C3⋊D9—21no—
132C2×C3⋊D9—22no—
133C2×C3⋊D9—21no—
134C2×C3⋊D9—22no—
135C2×C3⋊D9—21no—
136C2×C3⋊D9—22no—
137C3×C4×C9—22no—
138C3×C4×C9#4112yes—
139C3⋊Dic9—21no—
140C9×Dic3—22no—
141C9×Dic3—21no—
142C9×Dic3—21no—
143C3×Dic9—22no—
144C3×Dic9—21no—
145C3×Dic9—21no—
146C3×C4⋊Q8—31no—
147C3×C4×Q8—91no—
148#947—92no—
149#947—91no—
150#946—92no—
151#946—91no—
152#898—92no—
153#897—91no—
154#896—92no—
155#895—91no—
156#895—91no—
157#895—181no—
158#894—92no—
159#894—92no—
160#893—91no—
161#888D911yescharacteristic, complemented
162#887—31no—
163#876—92no—
164#875—91no—
165#875—92no—
166C4×D12—91no—
167#872—91no—
168#872—91no—
169C22×C3⋊D3—31no—
170C22×C3⋊D3—32no—
171C2×C32×C4C22×D312yes—
172C2×C32×C4C22×D311yescharacteristic
173C2×C3⋊Dic3—31no—
174C4⋊(C3⋊D3)—62no—
175C4⋊(C3⋊D3)—122no—
176C4×C3⋊D3—62no—
177C2×C3×Dic3—32no—
178C2×C3×Dic3—31no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3C22×D311yescharacteristic
181C2×C3×Dic3C22×D311yescharacteristic
182C2×C3×Dic3C22×D312yes—
183C3×Dic6—42no—
184C3×Dic6—62no—
185#624—122no—
186C22×D9—31no—
187C22×D9—32no—
188C22×D9—61no—
189C22×D9—62no—
190Q8×C9—122no—
191C2×C4×C9—21no—
192C2×C4×C9—22no—
193C2×C4×C9—31no—
194C2×C4×C9—31no—
195C2×C4×C9—32no—
196C2×C4×C9C22×D312yes—
197C2×C4×C9C22×D311yescharacteristic
198C2×Dic9—31no—
199C2×Dic9—61no—
200C2×Dic9C22×D311yescharacteristic
201C2×Dic9C22×D311yescharacteristic
202C2×Dic9C22×D312yes—
203D36—62no—
204D36—122no—
205D36—122no—
206D36—124no—
207C4×D9—62no—
208C4×D9—122no—
209C4×D9—122no—
210Dic18—22no—
211#542—271noSylow
212C2×C3×C9C2×#4111yescharacteristic
213C2×C3×C9H(5,𝔽2)11yescharacteristic
214C2×C3×C92−1+411yescharacteristic
215C3⋊D9—41no—
216C3⋊D9—42no—
217C2×C3×Q8—32no—
218C2×C3×Q8—91no—
219C2×D12—91no—
220C2×D12—91no—
221C2×D12—92no—
222C2×D12—92no—
223C2×D12—181no—
224C2×D12—182no—
225C2×C4×D3—91no—
226C2×C4×D3—91no—
227C2×C4×D3—92no—
228C2×C4×D3—92no—
229C2×C4×D3—181no—
230C2×Dic6—31no—
231C2×Dic6C2×D912yes—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—31no—
234C3×C4⋊C4—32no—
235C3×C4⋊C4—92no—
236C3×C4⋊C4C2×D911yescharacteristic
237C3×C4⋊C4D3211yescharacteristic
238C3×C42—31no—
239C3×C42—91no—
240C3×C42—92no—
241(C2×D3)⋊C4—91no—
242(C2×D3)⋊C4—91no—
243(C2×D3)⋊C4—92no—
244(C2×D3)⋊C4—92no—
245(C2×D3)⋊C4—92no—
246(C2×D3)⋊C4—92no—
247(C2×D3)⋊C4—92no—
248(C2×D3)⋊C4—92no—
249(C2×D3)⋊C4—91no—
250(C2×D3)⋊C4—91no—
251(C2×D3)⋊C4—92no—
252(C2×D3)⋊C4—92no—
253(C2×D3)⋊C4—92no—
254(C2×D3)⋊C4—92no—
255(C2×D3)⋊C4—182no—
256C4⋊Dic3—92no—
257C4⋊Dic3C2×D911yescharacteristic
258Dic3⋊C4—32no—
259Dic3⋊C4—91no—
260Dic3⋊C4—91no—
261Dic3⋊C4—91no—
262Dic3⋊C4—91no—
263Dic3⋊C4—92no—
264Dic3⋊C4C2×D912yes—
265C4×Dic3—31no—
266C4×Dic3—32no—
267C4×Dic3C2×D911yescharacteristic
268C22×C32C23×D311yescharacteristic
269C2×C3⋊D3—61no—
270C2×C3⋊D3—62no—
271C2×C3⋊D3—61no—
272C2×C3⋊D3—62no—
273C2×C3⋊D3—61no—
274C2×C3⋊D3—62no—
275C32×C4—22no—
276C32×C4#23912yes—
277C3⋊Dic3—61no—
278C3×Dic3—22no—
279C3×Dic3—21no—
280C3×Dic3—21no—
281C3×Dic3—61no—
282C3×Dic3—62no—
283C3×Dic3—61no—
284C22×C9—21no—
285C22×C9C23×D311yescharacteristic
286C2×D9—62no—
287C2×D9—61no—
288C2×D9—62no—
289C2×D9—61no—
290C2×D9—62no—
291C2×D9—62no—
292C2×D9—61no—
293C2×D9—121no—
294C2×D9—121no—
295C2×D9—122no—
296C2×D9—122no—
297C2×D9—122no—
298C4×C9—22no—
299C4×C9—22no—
300C4×C9—42no—
301C4×C9—61no—
302C4×C9—62no—
303C4×C9—61no—
304C4×C9#23512yes—
305Dic9—22no—
306Dic9—21no—
307Dic9—21no—
308Dic9—61no—
309Dic9—121no—
310C4⋊Q8—31no—
311#126—272no—
312#124—271no—
313#124—272no—
314#123—272no—
315C22⋊Q8—271no—
316C22⋊Q8—272no—
317#121—271no—
318C4×Q8—91no—
319C4×D4—271no—
320#117—271no—
321C3×C9#54211yescharacteristic, complemented, Sylow
322C22×D3—91no—
323C22×D3—92no—
324C22×D3—92no—
325C22×D3—91no—
326C22×D3—181no—
327C22×D3—182no—
328C3×Q8—122no—
329C3×Q8—182no—
330C2×C3×C4—21no—
331C2×C3×C4—22no—
332C2×C3×C4—31no—
333C2×C3×C4—31no—
334C2×C3×C4—32no—
335C2×C3×C4—91no—
336C2×C3×C4—92no—
337C2×C3×C4—91no—
338C2×C3×C4C22×D912yes—
339C2×C3×C4C22×D911yescharacteristic
340C2×C3×C4C2×D3212yes—
341C2×C3×C4C2×D3211yescharacteristic
342C2×Dic3—31no—
343C2×Dic3—31no—
344C2×Dic3—32no—
345C2×Dic3—91no—
346C2×Dic3—91no—
347C2×Dic3—181no—
348C2×Dic3C22×D911yescharacteristic
349C2×Dic3C22×D911yescharacteristic
350C2×Dic3C22×D912yes—
351D12—182no—
352D12—182no—
353D12—362no—
354D12—362no—
355D12—362no—
356D12—364no—
357C4×D3—182no—
358C4×D3—182no—
359C4×D3—362no—
360C4×D3—362no—
361C4×D3—362no—
362Dic6—42no—
363Dic6—62no—
364C2×C32#100411yescharacteristic
365C2×C32C2×#23911yescharacteristic
366C2×C32#100711yescharacteristic
367C3⋊D3—121no—
368C3⋊D3—122no—
369C2×C9—21no—
370C2×C9—21no—
371C2×C9—21no—
372C2×C9C2×#23511yescharacteristic
373C2×C9#100711yescharacteristic
374C2×C9#100911yescharacteristic
375D9—122no—
376D9—121no—
377D9—122no—
378D9—242no—
379C2×Q8—32no—
380C2×Q8—91no—
381C2×D4—271no—
382C2×D4—272no—
383C22×C4—272no—
384C22×C4—271no—
385C4⋊C4—31no—
386C4⋊C4—32no—
387C4⋊C4—92no—
388C4⋊C4—271no—
389C4⋊C4—271no—
390C4⋊C4—272no—
391C4⋊C4D3×D911yescharacteristic
392C22⋊C4—271no—
393C22⋊C4—271no—
394C22⋊C4—272no—
395C22⋊C4—272no—
396C22⋊C4—272no—
397C22⋊C4—272no—
398C22⋊C4—272no—
399C42—31no—
400C42—91no—
401C42—92no—
402C22×C3—21no—
403C22×C3C23×D911yescharacteristic
404C22×C3C22×D3211yescharacteristic
405C2×D3—182no—
406C2×D3—181no—
407C2×D3—182no—
408C2×D3—182no—
409C2×D3—181no—
410C2×D3—181no—
411C2×D3—182no—
412C2×D3—182no—
413C2×D3—181no—
414C2×D3—181no—
415C2×D3—182no—
416C2×D3—181no—
417C2×D3—182no—
418C2×D3—361no—
419C2×D3—361no—
420C2×D3—362no—
421C2×D3—362no—
422C2×D3—362no—
423C3×C4—22no—
424C3×C4—22no—
425C3×C4—22no—
426C3×C4—42no—
427C3×C4—62no—
428C3×C4—61no—
429C3×C4—61no—
430C3×C4—181no—
431C3×C4—182no—
432C3×C4—181no—
433C3×C4#370112yes—
434C3×C4#379912yes—
435Dic3—21no—
436Dic3—22no—
437Dic3—21no—
438Dic3—62no—
439Dic3—61no—
440Dic3—61no—
441Dic3—181no—
442Dic3—181no—
443Dic3—361no—
444C32#573311yescharacteristic
445C9—21no—
446C9#558211yescharacteristic, complemented
447C23—272no—
448C23—271no—
449Q8—122no—
450Q8—182no—
451D4—542no—
452D4—1082no—
453C2×C4—31no—
454C2×C4—31no—
455C2×C4—32no—
456C2×C4—91no—
457C2×C4—92no—
458C2×C4—91no—
459C2×C4—271no—
460C2×C4—542no—
461C2×C4—1082no—
462C2×C4C2×D3×D912yes—
463C2×C4C2×D3×D911yescharacteristic
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3#6371011yescharacteristic
468C2×C3C2×#370111yescharacteristic
469C2×C3#6371311yescharacteristic
470C2×C3C2×#379911yescharacteristic
471C2×C3#6431411yescharacteristic
472C2×C3#6431611yescharacteristic
473D3—362no—
474D3—361no—
475D3—362no—
476D3—361no—
477D3—362no—
478D3—722no—
479C22—542no—
480C22—541no—
481C22—541no—
482C22—542no—
483C22—541no—
484C22—542no—
485C22C22×D3×D911yescharacteristic
486C4—22no—
487C4—62no—
488C4—61no—
489C4—61no—
490C4—181no—
491C4—181no—
492C4—182no—
493C4—541no—
494C4#8863512yes—
495C3—21no—
496C3#1058983511yescharacteristic, complemented
497C3#1059435211yescharacteristic
498C2—1081no—
499C2—1082no—
500C2C2×#8863511yescharacteristic
501C2#1172250011yescharacteristic
502C2#1172250911yescharacteristic
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#542
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×#888C2semidirect
3. C3×#63449C2semidirect
4. #11720997C2semidirect
5. #11720992C2semidirect
6. #11720986C2semidirect
7. #11720953C2semidirect
8. #11720952C2semidirect
9. #11720936C2semidirect
10. #11720934C2semidirect
11. #11720931C2semidirect
57. #63449D3semidirect
161. #888D9semidirect
321. C3×C9#542semidirect
446. C9#5582semidirect
496. C3#10589835semidirect

Maximal subgroups

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

subgroupindex
#117217512
C9×#8882
C3×#634492
#117209972
#117209922
#117209862
#117209532
#117209522
#117209362
#117209342
#117209312
#105943523
#105898353