Finite Group Databasebeta

← #13147688

Subgroups — #13147688

Order 1728.

724 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721709C211yesmaximal, characteristic, complemented
2C4×C9×D12C211yesmaximal, characteristic, complemented
3C3×C4×Dic18C211yesmaximal, characteristic, complemented
4#11721045C211yesmaximal, characteristic
5#11720971C211yesmaximal, characteristic, complemented
6#11720967C211yesmaximal, characteristic, complemented
7#11720963C212yesmaximal, complemented
8#11720956C212yesmaximal, complemented
9#11720955C212yesmaximal, complemented
10#11720954C212yesmaximal, complemented
11#11720936C211yesmaximal, characteristic, complemented
12#10593876—31nomaximal
13#10589663—31nomaximal
14C2×C4×C3⋊D9C2211yescharacteristic
15C2×C9×D12C2211yescharacteristic
16C2×C4×D3×C9C2212yes—
17C2×C3×Dic18C2211yescharacteristic
18C2×#6195C2212yes—
19C2×#6189C2212yes—
20C3×C42×C9C2211yescharacteristic
21#88536C2211yescharacteristic
22#88536C2211yescharacteristic
23#88534C2211yescharacteristic
24#88534C2211yescharacteristic
25C4×C3⋊Dic9C2211yescharacteristic
26C9×(C2×D3)⋊C4C2212yes—
27C9×C4⋊Dic3C2211yescharacteristic
28C3×C4⋊Dic9C2211yescharacteristic
29C3×Dic9⋊C4C2212yes—
30C3×C4×Dic9C2212yes—
31(C2×D3)⋊Dic9C2212yes—
32(C2×D3)⋊Dic9C2212yes—
33(C2×D3)⋊Dic9C2212yes—
34#88445C2212yes—
35Dic3⋊Dic9C2212yes—
36#88442C2212yes—
37#64081—31no—
38C3×C4×D12D311yescharacteristic
39C3×C4×Dic6—31no—
40#63919—31no—
41#63916—31no—
42#63871—31no—
43#63863—31no—
44#63859—32no—
45#63852—32no—
46#63851—32no—
47#63850—32no—
48C4×D4×C9—31no—
49#63500—31no—
50#63498—31no—
51#63491—32no—
52#63486—32no—
53#63459—31no—
54#63450—31no—
55#63447—32no—
56#63441—32no—
57#63434—31no—
58#63434—61no—
59C4×Dic18D311yescharacteristic, complemented
60C22×C3⋊D9C2311yescharacteristic
61C22×D3×C9C2312yes—
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9C2311yescharacteristic
66C2×C3⋊Dic9C2311yescharacteristic
67C2×C3⋊Dic9C2311yescharacteristic
68C4×C3⋊D9—22no—
69C2×C9×Dic3C2312yes—
70C2×C3×Dic9C2312yes—
71C2×C3×Dic9C2312yes—
72C9×D12—22no—
73C4×D3×C9—42no—
74C3×Dic18—22no—
75#6195—42no—
76#6189—42no—
77#5582—91no—
78#5561—91no—
79C2×C4×C3⋊D3—31no—
80C2×C3×D12C2×D311yescharacteristic
81C2×C3×C4×D3C2×D312yes—
82C2×C3×Dic6—31no—
83C2×#627—32no—
84C2×#626—32no—
85C32×C42C2×D311yescharacteristic
86#3752—31no—
87#3752—31no—
88#3750—31no—
89#3750—31no—
90C4×C3⋊Dic3—31no—
91C3×(C2×D3)⋊C4C2×D312yes—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×Dic3⋊C4—32no—
95C3×C4×Dic3—32no—
96#3724—32no—
97Dic3⋊Dic3—32no—
98#3722—32no—
99(C2×D3)⋊Dic3—32no—
100(C2×D3)⋊Dic3—32no—
101(C2×D3)⋊Dic3—32no—
102C2×D4×C9—31no—
103C22×C4×C9—32no—
104C2×#611—32no—
105C2×C4×D9—31no—
106C2×C4×D9—61no—
107C2×Dic18—32no—
108C2×Dic18C2×D311yescharacteristic
109C9×C4⋊C4—31no—
110C9×C22⋊C4—32no—
111C42×C9—21no—
112C42×C9C2×D311yescharacteristic
113#3676—32no—
114#3676—32no—
115#3676—32no—
116(C2×D9)⋊C4—31no—
117(C2×D9)⋊C4—31no—
118(C2×D9)⋊C4—32no—
119(C2×D9)⋊C4—61no—
120(C2×D9)⋊C4—61no—
121C4⋊Dic9—32no—
122C4⋊Dic9C2×D311yescharacteristic
123Dic9⋊C4—31no—
124Dic9⋊C4—31no—
125Dic9⋊C4—32no—
126Dic9⋊C4—61no—
127Dic9⋊C4—61no—
128Dic9⋊C4C2×D312yes—
129C4×Dic9—31no—
130C4×Dic9—61no—
131C4×Dic9C2×D312yes—
132C22×C3×C9C2411yescharacteristic
133C2×C3⋊D9—21no—
134C2×C3⋊D9—21no—
135C2×C3⋊D9—21no—
136C2×D3×C9—22no—
137C2×D3×C9—22no—
138C2×D3×C9—22no—
139C3×C4×C9—21no—
140C3×C4×C9—21no—
141C3×C4×C9#4112yes—
142C3⋊Dic9—21no—
143C3⋊Dic9—21no—
144C3⋊Dic9—21no—
145C9×Dic3—22no—
146C3×Dic9—22no—
147C3×Dic9—22no—
148C3×C4×Q8—91no—
149C3×C4×D4—31no—
150#947—91no—
151#946—91no—
152#938—91no—
153#936—91no—
154#929—92no—
155#924—92no—
156#897—91no—
157#896—92no—
158#895—91no—
159#894—92no—
160#888—91no—
161#888—91no—
162#885—92no—
163#879—92no—
164#876—92no—
165#875—92no—
166C4×D12D911yescharacteristic, complemented
167#872—91no—
168#872—91no—
169#872—181no—
170C4×Dic6—31no—
171C22×C3⋊D3—31no—
172C22×C3×D3C22×D312yes—
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175C2×C32×C4C22×D311yescharacteristic
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—
183C3×D12—22no—
184C3×C4×D3—42no—
185C3×Dic6—62no—
186#627—122no—
187#626—122no—
188C23×C9—32no—
189C22×D9—31no—
190C22×D9—61no—
191D4×C9—62no—
192C2×C4×C9—21no—
193C2×C4×C9—21no—
194C2×C4×C9—21no—
195C2×C4×C9—32no—
196C2×C4×C9—122no—
197C2×C4×C9C22×D311yescharacteristic
198C2×C4×C9C22×D311yescharacteristic
199C2×C4×C9C22×D311yescharacteristic
200#611—122no—
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—64no—
210C4×D9—62no—
211Dic18—22no—
212Dic18—122no—
213#542—271noSylow
214C2×C3×C9C2×#4111yescharacteristic
215C2×C3×C9H(5,𝔽2)11yescharacteristic
216C2×C3×C92−1+411yescharacteristic
217C3⋊D9—41no—
218D3×C9—42no—
219C2×C3×Q8—91no—
220C2×C3×D4—31no—
221C22×C3×C4—32no—
222C2×#67—92no—
223C2×D12—92no—
224C2×D12C2×D911yescharacteristic
225C2×C4×D3—91no—
226C2×C4×D3—91no—
227C2×C4×D3—181no—
228C2×C4×D3C2×D912yes—
229C2×Dic6—31no—
230C2×Dic6—92no—
231C2×Dic6—92no—
232C3×C4⋊C4—31no—
233C3×C4⋊C4—91no—
234C3×C4⋊C4—92no—
235C3×C22⋊C4—32no—
236C3×C42—21no—
237C3×C42—92no—
238C3×C42C2×D911yescharacteristic
239C3×C42D3211yescharacteristic
240#217—92no—
241#217—92no—
242#217—92no—
243(C2×D3)⋊C4—92no—
244(C2×D3)⋊C4—92no—
245(C2×D3)⋊C4—92no—
246(C2×D3)⋊C4—92no—
247(C2×D3)⋊C4—91no—
248(C2×D3)⋊C4—91no—
249(C2×D3)⋊C4—91no—
250(C2×D3)⋊C4—91no—
251(C2×D3)⋊C4—92no—
252(C2×D3)⋊C4—181no—
253(C2×D3)⋊C4—181no—
254(C2×D3)⋊C4C2×D912yes—
255C4⋊Dic3—31no—
256C4⋊Dic3—92no—
257C4⋊Dic3C2×D911yescharacteristic
258Dic3⋊C4—32no—
259Dic3⋊C4—92no—
260Dic3⋊C4—91no—
261Dic3⋊C4—91no—
262Dic3⋊C4—91no—
263Dic3⋊C4—91no—
264Dic3⋊C4—92no—
265Dic3⋊C4—92no—
266Dic3⋊C4—181no—
267Dic3⋊C4—181no—
268C4×Dic3—32no—
269C4×Dic3—91no—
270C4×Dic3—91no—
271C4×Dic3—181no—
272C22×C32C23×D311yescharacteristic
273C2×C3⋊D3—61no—
274C2×C3⋊D3—61no—
275C2×C3⋊D3—61no—
276C2×C3×D3—22no—
277C2×C3×D3—22no—
278C2×C3×D3—22no—
279C32×C4—21no—
280C32×C4—21no—
281C32×C4#23512yes—
282C3⋊Dic3—61no—
283C3⋊Dic3—61no—
284C3⋊Dic3—61no—
285C3×Dic3—22no—
286C3×Dic3—62no—
287C3×Dic3—62no—
288C22×C9—21no—
289C22×C9—62no—
290C22×C9—62no—
291C22×C9—62no—
292C22×C9C23×D311yescharacteristic
293C2×D9—62no—
294C2×D9—61no—
295C2×D9—61no—
296C2×D9—61no—
297C2×D9—121no—
298C2×D9—121no—
299C4×C9—21no—
300C4×C9—21no—
301C4×C9—22no—
302C4×C9—41no—
303C4×C9—41no—
304C4×C9—62no—
305C4×C9#23512yes—
306Dic9—22no—
307Dic9—22no—
308Dic9—62no—
309Dic9—61no—
310Dic9—61no—
311Dic9—61no—
312Dic9—121no—
313Dic9—121no—
314C4⋊Q8—271no—
315#126—272no—
316#124—272no—
317#124—271no—
318#123—272no—
319C22⋊Q8—271no—
320C22⋊Q8—272no—
321#121—271no—
322C4×Q8—91no—
323C4×D4—31no—
324#117—271no—
325C3×C9#54211yescharacteristic, complemented, Sylow
326C23×C3—32no—
327C22×D3—91no—
328C22×D3—91no—
329C22×D3—181no—
330C22×D3C22×D912yes—
331C3×Q8—182no—
332C3×D4—62no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—32no—
337C2×C3×C4—92no—
338C2×C3×C4—92no—
339C2×C3×C4—122no—
340C2×C3×C4C22×D911yescharacteristic
341C2×C3×C4C22×D911yescharacteristic
342C2×C3×C4C22×D911yescharacteristic
343C2×C3×C4C2×D3211yescharacteristic
344C2×C3×C4C2×D3211yescharacteristic
345C2×C3×C4C2×D3211yescharacteristic
346#67—362no—
347C2×Dic3—32no—
348C2×Dic3—32no—
349C2×Dic3—91no—
350C2×Dic3—91no—
351C2×Dic3—91no—
352C2×Dic3—91no—
353C2×Dic3—91no—
354C2×Dic3—91no—
355C2×Dic3—181no—
356C2×Dic3—181no—
357C2×Dic3—181no—
358C2×Dic3C22×D912yes—
359D12—22no—
360D12—362no—
361C4×D3—42no—
362C4×D3—184no—
363C4×D3—182no—
364C4×D3—182no—
365Dic6—62no—
366Dic6—362no—
367Dic6—362no—
368C2×C32C2×#23511yescharacteristic
369C2×C32#100411yescharacteristic
370C2×C32#101011yescharacteristic
371C3⋊D3—121no—
372C3×D3—42no—
373C2×C9—21no—
374C2×C9—21no—
375C2×C9—21no—
376C2×C9—122no—
377C2×C9C2×#23511yescharacteristic
378C2×C9#100711yescharacteristic
379C2×C9#100911yescharacteristic
380D9—122no—
381D9—121no—
382C2×Q8—91no—
383C2×Q8—272no—
384C2×D4—31no—
385C2×D4—272no—
386C22×C4—32no—
387C22×C4—271no—
388C4⋊C4—31no—
389C4⋊C4—91no—
390C4⋊C4—92no—
391C4⋊C4—272no—
392C4⋊C4—272no—
393C4⋊C4—271no—
394C4⋊C4—271no—
395C22⋊C4—32no—
396C22⋊C4—272no—
397C22⋊C4—272no—
398C22⋊C4—272no—
399C22⋊C4—272no—
400C22⋊C4—271no—
401C22⋊C4—271no—
402C42—92no—
403C42—271no—
404C42D3×D911yescharacteristic
405C22×C3—21no—
406C22×C3—62no—
407C22×C3—62no—
408C22×C3—62no—
409C22×C3C23×D911yescharacteristic
410C22×C3C22×D3211yescharacteristic
411C2×D3—22no—
412C2×D3—22no—
413C2×D3—22no—
414C2×D3—182no—
415C2×D3—181no—
416C2×D3—181no—
417C2×D3—181no—
418C2×D3—181no—
419C2×D3—181no—
420C2×D3—181no—
421C2×D3—361no—
422C2×D3—361no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—22no—
428C3×C4—41no—
429C3×C4—41no—
430C3×C4—62no—
431C3×C4—182no—
432C3×C4—182no—
433C3×C4#369712yes—
434C3×C4#379812yes—
435Dic3—22no—
436Dic3—62no—
437Dic3—62no—
438Dic3—182no—
439Dic3—181no—
440Dic3—181no—
441Dic3—181no—
442Dic3—181no—
443Dic3—181no—
444Dic3—181no—
445Dic3—361no—
446Dic3—361no—
447C32#556111yescharacteristic
448C9—21no—
449C9#558211yescharacteristic, complemented
450C23—32no—
451C23—271no—
452Q8—182no—
453Q8—1082no—
454D4—62no—
455D4—1082no—
456C2×C4—32no—
457C2×C4—92no—
458C2×C4—92no—
459C2×C4—122no—
460C2×C4—271no—
461C2×C4—271no—
462C2×C4—271no—
463C2×C4—542no—
464C2×C4C2×D3×D911yescharacteristic
465C2×C4C2×D3×D911yescharacteristic
466C2×C4C2×D3×D911yescharacteristic
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—122no—
471C2×C3C2×#369711yescharacteristic
472C2×C3#6371011yescharacteristic
473C2×C3#6371611yescharacteristic
474C2×C3#6429711yescharacteristic
475C2×C3C2×#379811yescharacteristic
476C2×C3#6430711yescharacteristic
477D3—42no—
478D3—362no—
479D3—361no—
480D3—361no—
481C22—62no—
482C22—62no—
483C22—62no—
484C22—541no—
485C22—541no—
486C22—541no—
487C22C22×D3×D911yescharacteristic
488C4—21no—
489C4—21no—
490C4—62no—
491C4—182no—
492C4—182no—
493C4—541no—
494C4—541no—
495C4—541no—
496C4#8864012yes—
497C3—21no—
498C3#1058966311yescharacteristic, complemented
499C3#1059387611yescharacteristic
500C2—122no—
501C2—1081no—
502C2#1172246411yescharacteristic
503C2C2×#8864011yescharacteristic
504C2#1172247611yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radical#11721045C2
hypercenterC42D3×D9
hypocenterC3×C9#542
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721709C2semidirect
2. C4×C9×D12C2semidirect
3. C3×C4×Dic18C2semidirect
5. #11720971C2semidirect
6. #11720967C2semidirect
7. #11720963C2semidirect
8. #11720956C2semidirect
9. #11720955C2semidirect
10. #11720954C2semidirect
11. #11720936C2semidirect
59. C4×Dic18D3semidirect
166. C4×D12D9semidirect
325. C3×C9#542semidirect
449. C9#5582semidirect
498. C3#10589663semidirect

Maximal subgroups

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

subgroupindex
#117217092
C4×C9×D122
C3×C4×Dic182
#117210452
#117209712
#117209672
#117209632
#117209562
#117209552
#117209542
#117209362
#105938763
#105896633