Finite Group Databasebeta

← #13150153

Subgroups — #13150153

Order 1728.

730 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
1C4×#6233C211yesmaximal, characteristic
2C9×#924C211yesmaximal, characteristic, complemented
3C3×#63488C211yesmaximal, characteristic, complemented
4#11721247C212yesmaximal, complemented
5#11721242C212yesmaximal
6#11720971C211yesmaximal, characteristic, complemented
7#11720934C212yesmaximal, complemented
8#11720929C211yesmaximal, characteristic, complemented
9#11720922C211yesmaximal, characteristic, complemented
10#11720911C211yesmaximal, characteristic, complemented
11#11720901C212yesmaximal, complemented
12#10594809—31nomaximal
13#10589743—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6233C2211yescharacteristic
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×C9×Dic6C2211yescharacteristic
18C2×#6189C2212yes—
19#88551C2211yescharacteristic
20#88536C2211yescharacteristic
21#88534C2211yescharacteristic
22C4×C3⋊Dic9C2211yescharacteristic
23C9×#217C2212yes—
24C3×#3676C2212yes—
25C9×C4⋊Dic3C2211yescharacteristic
26C9×Dic3⋊C4C2212yes—
27C3×C4⋊Dic9C2211yescharacteristic
28C3×C4⋊Dic9C2211yescharacteristic
29C3×C4×Dic9C2212yes—
30#88445C2212yes—
31#88445C2212yes—
32Dic3⋊Dic9C2212yes—
33#88442C2212yes—
34Dic9⋊Dic3C2212yes—
35Dic3×Dic9C2212yes—
36Dic3×Dic9C2212yes—
37C4×#638—31no—
38C3×#926—31no—
39C3×#924D311yescharacteristic
40#63956—32no—
41#63953—32no—
42#63919—31no—
43#63871—31no—
44#63868—32no—
45#63849—31no—
46#63843—31no—
47#63837—32no—
48C9×C22⋊Q8—31no—
49#63508—31no—
50Q8×Dic9—31no—
51C4×#611—31no—
52C4×#611—61no—
53#63488D311yescharacteristic, complemented
54#63460—32no—
55#63454—31no—
56#63450—31no—
57#63449—32no—
58#63447—32no—
59#63441—32no—
60C23×C3×C9C2311yescharacteristic
61C22×C3⋊D9C2311yescharacteristic
62C2×C3×C4×C9—22no—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65#6233—22no—
66C2×C3⋊Dic9C2311yescharacteristic
67C2×C3⋊Dic9C2311yescharacteristic
68C2×C3⋊Dic9C2311yescharacteristic
69C4×C3⋊D9—22no—
70C2×C9×Dic3C2312yes—
71C2×C9×Dic3C2312yes—
72C2×C3×Dic9C2312yes—
73C2×C3×Dic9C2312yes—
74C9×Dic6—22no—
75#6189—24no—
76#5641—91no—
77#5533—91no—
78C22×C32×C4C2×D311yescharacteristic
79C2×#638—31no—
80C2×C4×C3⋊D3—31no—
81C2×C3×Dic6C2×D311yescharacteristic
82C2×#627—32no—
83#3757—31no—
84#3752—31no—
85#3750—31no—
86C4×C3⋊Dic3—31no—
87C3×#217—32no—
88C3×#217C2×D312yes—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×Dic3⋊C4C2×D312yes—
93C3×C4×Dic3—32no—
94#3724—32no—
95Dic3⋊Dic3—32no—
96Dic3⋊Dic3—32no—
97#3722—32no—
98#3722—32no—
99Dic32—32no—
100Dic32—32no—
101C2×Q8×C9—31no—
102C22×C4×C9—21no—
103C22×C4×C9C2×D311yescharacteristic
104C2×#611—31no—
105C2×#611—61no—
106C2×C4×D9—31no—
107C2×C4×D9—61no—
108C2×Dic18—32no—
109C9×C4⋊C4—32no—
110C9×C4⋊C4—31no—
111C9×C22⋊C4—32no—
112#3676—31no—
113#3676—61no—
114#3676C2×D312yes—
115(C2×D9)⋊C4—31no—
116(C2×D9)⋊C4—32no—
117(C2×D9)⋊C4—32no—
118(C2×D9)⋊C4—61no—
119C4⋊Dic9—32no—
120C4⋊Dic9C2×D311yescharacteristic
121C4⋊Dic9C2×D311yescharacteristic
122Dic9⋊C4—31no—
123Dic9⋊C4—32no—
124Dic9⋊C4—32no—
125Dic9⋊C4—61no—
126C4×Dic9—31no—
127C4×Dic9—32no—
128C4×Dic9—32no—
129C4×Dic9—61no—
130C4×Dic9C2×D312yes—
131C22×C3×C9—21no—
132C22×C3×C9—21no—
133C22×C3×C9—21no—
134C22×C3×C9C2411yescharacteristic
135C2×C3⋊D9—21no—
136C2×C3⋊D9—21no—
137C2×C3⋊D9—21no—
138C3×C4×C9—21no—
139C3×C4×C9#4112yes—
140C3⋊Dic9—21no—
141C3⋊Dic9—21no—
142C3⋊Dic9#4112yes—
143C9×Dic3—22no—
144C9×Dic3—22no—
145C3×Dic9—22no—
146C3×Dic9—22no—
147C3×C22⋊Q8—31no—
148C3×#117—91no—
149#946—91no—
150Q8×Dic3—91no—
151C4×#67—91no—
152C4×#67—91no—
153C4×#67—181no—
154#926—31no—
155#924D911yescharacteristic, complemented
156#898—92no—
157#897—91no—
158#892—91no—
159#892—91no—
160#888—91no—
161#888—91no—
162#887—92no—
163#887—91no—
164#885—92no—
165#885—92no—
166#879—92no—
167#879—92no—
168#876—92no—
169C4×Dic6—92no—
170C23×C32C22×D311yescharacteristic
171C22×C3⋊D3—31no—
172C2×C32×C4—22no—
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175#638—62no—
176C2×C3⋊Dic3—31no—
177C2×C3⋊Dic3—31no—
178C2×C3⋊Dic3—31no—
179C4×C3⋊D3—62no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3—32no—
182C2×C3×Dic3C22×D312yes—
183C2×C3×Dic3C22×D312yes—
184C3×Dic6—22no—
185#627—64no—
186C23×C9—21no—
187C23×C9C22×D311yescharacteristic
188C22×D9—31no—
189C22×D9—61no—
190Q8×C9—62no—
191C2×C4×C9—22no—
192C2×C4×C9—21no—
193C2×C4×C9—21no—
194C2×C4×C9—32no—
195C2×C4×C9—32no—
196C2×C4×C9—42no—
197C2×C4×C9C22×D311yescharacteristic
198C2×C4×C9C22×D311yescharacteristic
199#611—62no—
200#611—64no—
201C2×Dic9—31no—
202C2×Dic9—31no—
203C2×Dic9—31no—
204C2×Dic9—61no—
205C2×Dic9—61no—
206C2×Dic9—61no—
207C2×Dic9C22×D312yes—
208C2×Dic9C22×D312yes—
209C4×D9—62no—
210C4×D9—122no—
211Dic18—64no—
212#556—271noSylow
213C2×C3×C9—41no—
214C2×C3×C9C2×#4111yescharacteristic
215C2×C3×C9C2×#4111yescharacteristic
216C2×C3×C92−1+411yescharacteristic
217C3⋊D9—41no—
218C2×C3×Q8—31no—
219C22×C3×C4—21no—
220C22×C3×C4C2×D911yescharacteristic
221C22×C3×C4D3211yescharacteristic
222C2×#67—91no—
223C2×#67—91no—
224C2×#67—181no—
225C2×C4×D3—91no—
226C2×C4×D3—91no—
227C2×C4×D3—181no—
228C2×Dic6—92no—
229C2×Dic6—92no—
230C2×Dic6C2×D911yescharacteristic
231C3×C4⋊C4—32no—
232C3×C4⋊C4—31no—
233C3×C4⋊C4—91no—
234C3×C4⋊C4—91no—
235C3×C22⋊C4—32no—
236C3×C22⋊C4—92no—
237C3×C42—92no—
238#217—32no—
239#217—91no—
240#217—91no—
241#217—181no—
242#217C2×D912yes—
243(C2×D3)⋊C4—92no—
244(C2×D3)⋊C4—92no—
245(C2×D3)⋊C4—91no—
246(C2×D3)⋊C4—91no—
247(C2×D3)⋊C4—92no—
248(C2×D3)⋊C4—92no—
249(C2×D3)⋊C4—181no—
250C4⋊Dic3—31no—
251C4⋊Dic3—31no—
252C4⋊Dic3—92no—
253C4⋊Dic3—92no—
254C4⋊Dic3C2×D911yescharacteristic
255Dic3⋊C4—92no—
256Dic3⋊C4—92no—
257Dic3⋊C4—91no—
258Dic3⋊C4—91no—
259Dic3⋊C4—92no—
260Dic3⋊C4—92no—
261Dic3⋊C4—181no—
262Dic3⋊C4C2×D912yes—
263C4×Dic3—32no—
264C4×Dic3—92no—
265C4×Dic3—92no—
266C4×Dic3—91no—
267C4×Dic3—91no—
268C4×Dic3—92no—
269C4×Dic3—92no—
270C4×Dic3—181no—
271C22×C32—21no—
272C22×C32—21no—
273C22×C32—21no—
274C22×C32C23×D311yescharacteristic
275C2×C3⋊D3—61no—
276C2×C3⋊D3—61no—
277C2×C3⋊D3—61no—
278C32×C4—21no—
279C32×C4#23712yes—
280C3⋊Dic3—32no—
281C3⋊Dic3—61no—
282C3⋊Dic3—61no—
283C3×Dic3—22no—
284C3×Dic3—22no—
285C3×Dic3—62no—
286C3×Dic3—62no—
287C22×C9—22no—
288C22×C9—21no—
289C22×C9—21no—
290C22×C9—21no—
291C22×C9—21no—
292C22×C9—41no—
293C22×C9—41no—
294C22×C9C23×D311yescharacteristic
295C2×D9—62no—
296C2×D9—61no—
297C2×D9—61no—
298C2×D9—61no—
299C2×D9—121no—
300C2×D9—121no—
301C4×C9—21no—
302C4×C9—22no—
303C4×C9—41no—
304C4×C9—62no—
305C4×C9—62no—
306C4×C9#23512yes—
307Dic9—22no—
308Dic9—22no—
309Dic9—32no—
310Dic9—62no—
311Dic9—61no—
312Dic9—61no—
313Dic9—121no—
314Dic9—121no—
315C4⋊Q8—271no—
316#126—272no—
317#126—272no—
318#124—272no—
319C22⋊Q8—31no—
320C22⋊Q8—271no—
321C4×Q8—272no—
322C4×Q8—271no—
323C4×D4—271no—
324#117—91no—
325#117—271no—
326C3×C9#55611yescharacteristic, complemented, Sylow
327C23×C3—21no—
328C23×C3C22×D911yescharacteristic
329C23×C3C2×D3211yescharacteristic
330C22×D3—91no—
331C22×D3—91no—
332C22×D3—181no—
333C3×Q8—62no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—32no—
339C2×C3×C4—32no—
340C2×C3×C4—42no—
341C2×C3×C4—92no—
342C2×C3×C4—92no—
343C2×C3×C4C22×D911yescharacteristic
344C2×C3×C4C22×D911yescharacteristic
345C2×C3×C4C2×D3211yescharacteristic
346C2×C3×C4C2×D3211yescharacteristic
347#67—182no—
348#67—182no—
349#67—184no—
350C2×Dic3—32no—
351C2×Dic3—32no—
352C2×Dic3—91no—
353C2×Dic3—91no—
354C2×Dic3—91no—
355C2×Dic3—91no—
356C2×Dic3—91no—
357C2×Dic3—91no—
358C2×Dic3—181no—
359C2×Dic3—181no—
360C2×Dic3—181no—
361C2×Dic3C22×D912yes—
362C2×Dic3C22×D912yes—
363C4×D3—182no—
364C4×D3—182no—
365C4×D3—362no—
366Dic6—22no—
367Dic6—184no—
368Dic6—184no—
369C2×C32—41no—
370C2×C32C2×#23711yescharacteristic
371C2×C32#100711yescharacteristic
372C2×C32D3×#4111yescharacteristic
373C3⋊D3—121no—
374C2×C9—21no—
375C2×C9—21no—
376C2×C9—21no—
377C2×C9—42no—
378C2×C9—41no—
379C2×C9C2×#23511yescharacteristic
380C2×C9D3×#4111yescharacteristic
381C2×C9#101011yescharacteristic
382D9—122no—
383D9—121no—
384C2×Q8—31no—
385C2×Q8—272no—
386C2×D4—271no—
387C22×C4—271no—
388C22×C4D3×D911yescharacteristic
389C4⋊C4—31no—
390C4⋊C4—32no—
391C4⋊C4—91no—
392C4⋊C4—91no—
393C4⋊C4—272no—
394C4⋊C4—272no—
395C4⋊C4—272no—
396C4⋊C4—271no—
397C22⋊C4—32no—
398C22⋊C4—92no—
399C22⋊C4—272no—
400C22⋊C4—272no—
401C22⋊C4—271no—
402C22⋊C4—271no—
403C42—92no—
404C42—272no—
405C42—272no—
406C42—271no—
407C22×C3—22no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—21no—
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—21no—
415C22×C3—41no—
416C22×C3—41no—
417C22×C3C23×D911yescharacteristic
418C22×C3C22×D3211yescharacteristic
419C2×D3—182no—
420C2×D3—181no—
421C2×D3—181no—
422C2×D3—181no—
423C2×D3—181no—
424C2×D3—181no—
425C2×D3—181no—
426C2×D3—361no—
427C2×D3—361no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—22no—
431C3×C4—41no—
432C3×C4—62no—
433C3×C4—62no—
434C3×C4—182no—
435C3×C4—182no—
436C3×C4#369912yes—
437C3×C4#380412yes—
438Dic3—22no—
439Dic3—22no—
440Dic3—62no—
441Dic3—62no—
442Dic3—92no—
443Dic3—92no—
444Dic3—182no—
445Dic3—181no—
446Dic3—181no—
447Dic3—181no—
448Dic3—181no—
449Dic3—361no—
450Dic3—361no—
451C32#564111yescharacteristic
452C9—21no—
453C9#553311yescharacteristic, complemented
454C23—271no—
455C23C2×D3×D911yescharacteristic
456Q8—62no—
457Q8—544no—
458D4—542no—
459C2×C4—22no—
460C2×C4—32no—
461C2×C4—32no—
462C2×C4—92no—
463C2×C4—92no—
464C2×C4—271no—
465C2×C4—271no—
466C2×C4—271no—
467C2×C4—542no—
468C2×C4C2×D3×D911yescharacteristic
469C2×C4C2×D3×D911yescharacteristic
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—42no—
474C2×C3—41no—
475C2×C3—41no—
476C2×C3C2×#369911yescharacteristic
477C2×C3#6371311yescharacteristic
478C2×C3#41×D911yescharacteristic
479C2×C3#6429711yescharacteristic
480C2×C3#6430611yescharacteristic
481C2×C3C2×#380411yescharacteristic
482D3—362no—
483D3—361no—
484D3—361no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—541no—
489C22—541no—
490C22—541no—
491C22C22×D3×D911yescharacteristic
492C4—21no—
493C4—62no—
494C4—62no—
495C4—182no—
496C4—182no—
497C4—272no—
498C4—541no—
499C4—541no—
500C4#8866212yes—
501C3—21no—
502C3#1058974311yescharacteristic, complemented
503C3#1059480911yescharacteristic
504C2—41no—
505C2—1081no—
506C2#1172245811yescharacteristic
507C2#1172247511yescharacteristic
508C2C2×#8866211yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
prime radicalC2×#6233C22
hypercenterC22×C4D3×D9
hypocenterC3×C9#556
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C9×#924C2semidirect
3. C3×#63488C2semidirect
4. #11721247C2semidirect
6. #11720971C2semidirect
7. #11720934C2semidirect
8. #11720929C2semidirect
9. #11720922C2semidirect
10. #11720911C2semidirect
11. #11720901C2semidirect
53. #63488D3semidirect
155. #924D9semidirect
326. C3×C9#556semidirect
453. C9#5533semidirect
502. C3#10589743semidirect

Maximal subgroups

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

subgroupindex
C4×#62332
C9×#9242
C3×#634882
#117212472
#117212422
#117209712
#117209342
#117209292
#117209222
#117209112
#117209012
#105948093
#105897433