Finite Group Databasebeta

← #13150431

Subgroups — #13150431

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#11721815C211yesmaximal, characteristic, complemented
2C9×#930C211yesmaximal, characteristic
3C3×#63486C211yesmaximal, characteristic, complemented
4#11721280C212yesmaximal, complemented
5#11721263C212yesmaximal, complemented
6#11721025C211yesmaximal, characteristic, complemented
7#11721023C212yesmaximal, complemented
8#11720971C211yesmaximal, characteristic, complemented
9#11720967C211yesmaximal, characteristic, complemented
10#11720965C212yesmaximal, complemented
11#11720910C211yesmaximal, characteristic, complemented
12#10594984—31nomaximal
13#10589705—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×C9×#67C2212yes—
16C2×C9×D12C2211yescharacteristic
17C2×C3×Dic18C2211yescharacteristic
18C2×D3×Dic9C2212yes—
19C2×#6189C2212yes—
20#88551C2211yescharacteristic
21#88551C2211yescharacteristic
22C4⋊(C3⋊Dic9)C2211yescharacteristic
23C4⋊(C3⋊Dic9)C2211yescharacteristic
24C4×C3⋊Dic9C2211yescharacteristic
25C4×C3⋊Dic9C2211yescharacteristic
26C3×#3676C2212yes—
27C9×(C2×D3)⋊C4C2212yes—
28C9×C4⋊Dic3C2211yescharacteristic
29C3×C4⋊Dic9C2211yescharacteristic
30C3×Dic9⋊C4C2212yes—
31(C2×D3)⋊Dic9C2212yes—
32(C2×D3)⋊Dic9C2212yes—
33(C2×D3)⋊Dic9C2212yes—
34#88442C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic3×Dic9C2212yes—
37#64135—31no—
38C3×#930D311yescharacteristic
39C3×#924—31no—
40#63969—32no—
41#63959—32no—
42#63919—31no—
43#63898—31no—
44#63893—32no—
45#63863—31no—
46#63861—32no—
47#63843—31no—
48C9×#121—31no—
49#63498—31no—
50#63498—32no—
51#63497—32no—
52D4×Dic9—31no—
53#63488—31no—
54#63488—61no—
55#63486D311yescharacteristic, complemented
56#63450—31no—
57#63449—31no—
58#63441—32no—
59#63440—32no—
60C23×C3×C9C2311yescharacteristic
61C22×D3×C9C2312yes—
62C2×C3×C4×C9—22no—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9C2311yescharacteristic
66C2×C3⋊Dic9C2311yescharacteristic
67C2×C3⋊Dic9C2311yescharacteristic
68C2×C3⋊Dic9C2311yescharacteristic
69C9×#67—42no—
70C2×C9×Dic3C2312yes—
71C2×C3×Dic9C2312yes—
72C2×C3×Dic9C2312yes—
73C9×D12—22no—
74C3×Dic18—22no—
75D3×Dic9—42no—
76#6189—42no—
77#5643—91no—
78#5603—91no—
79C22×C32×C4C2×D311yescharacteristic
80C2×C3×#67C2×D312yes—
81C2×C3×D12C2×D311yescharacteristic
82C2×C3×Dic6—31no—
83C2×#627—32no—
84C2×D3×Dic3—32no—
85#3757—31no—
86#3757—31no—
87C4⋊(C3⋊Dic3)—31no—
88C4⋊(C3⋊Dic3)—31no—
89C4×C3⋊Dic3—31no—
90C4×C3⋊Dic3—31no—
91C3×#217—32no—
92C3×(C2×D3)⋊C4C2×D312yes—
93C3×C4⋊Dic3—31no—
94C3×C4⋊Dic3C2×D311yescharacteristic
95C3×Dic3⋊C4—32no—
96#3724—32no—
97Dic3⋊Dic3—32no—
98(C2×D3)⋊Dic3—32no—
99(C2×D3)⋊Dic3—32no—
100(C2×D3)⋊Dic3—32no—
101Dic32—32no—
102C2×D4×C9—32no—
103C2×D4×C9—31no—
104C22×C4×C9—21no—
105C22×C4×C9C2×D311yescharacteristic
106C22×Dic9—32no—
107C2×Dic18—32no—
108C2×Dic18C2×D311yescharacteristic
109C9×C4⋊C4—31no—
110C9×C22⋊C4—32no—
111#3676—31no—
112#3676—31no—
113#3676—32no—
114#3676—32no—
115#3676—32no—
116#3676—61no—
117#3676—61no—
118#3676C2×D312yes—
119C4⋊Dic9—31no—
120C4⋊Dic9—31no—
121C4⋊Dic9—61no—
122C4⋊Dic9—61no—
123C4⋊Dic9C2×D311yescharacteristic
124Dic9⋊C4—32no—
125Dic9⋊C4—32no—
126Dic9⋊C4C2×D312yes—
127C4×Dic9—31no—
128C4×Dic9—31no—
129C4×Dic9—32no—
130C4×Dic9—61no—
131C4×Dic9—61no—
132C22×C3×C9—21no—
133C22×C3×C9—21no—
134C22×C3×C9—21no—
135C22×C3×C9C2411yescharacteristic
136C2×D3×C9—22no—
137C2×D3×C9—22no—
138C2×D3×C9—22no—
139C3×C4×C9—21no—
140C3×C4×C9#4112yes—
141C3⋊Dic9—21no—
142C3⋊Dic9—21no—
143C3⋊Dic9—21no—
144C3⋊Dic9—21no—
145C9×Dic3—22no—
146C3×Dic9—22no—
147C3×Dic9—22no—
148C3×C22⋊Q8—91no—
149C3×#121—31no—
150#947—91no—
151Q8×Dic3—91no—
152#936—92no—
153#936—91no—
154#935—92no—
155D4×Dic3—91no—
156#930D911yescharacteristic
157#926—91no—
158#926—91no—
159#926—181no—
160#924—31no—
161#898—92no—
162#897—92no—
163#893—91no—
164#888—91no—
165#888—91no—
166#887—91no—
167#885—92no—
168#883—92no—
169#879—92no—
170#878—92no—
171C23×C32C22×D311yescharacteristic
172C22×C3×D3C22×D312yes—
173C2×C32×C4—22no—
174C2×C32×C4C22×D311yescharacteristic
175C2×C32×C4C22×D311yescharacteristic
176C2×C3⋊Dic3—31no—
177C2×C3⋊Dic3—31no—
178C2×C3⋊Dic3—31no—
179C2×C3⋊Dic3—31no—
180C3×#67—42no—
181C2×C3×Dic3—32no—
182C2×C3×Dic3—32no—
183C2×C3×Dic3C22×D312yes—
184C3×D12—22no—
185C3×Dic6—62no—
186#627—122no—
187D3×Dic3—122no—
188C23×C9—21no—
189C23×C9—32no—
190C23×C9C22×D311yescharacteristic
191D4×C9—62no—
192D4×C9—122no—
193C2×C4×C9—24no—
194C2×C4×C9—22no—
195C2×C4×C9—21no—
196C2×C4×C9—21no—
197C2×C4×C9—32no—
198C2×C4×C9C22×D311yescharacteristic
199C2×C4×C9C22×D311yescharacteristic
200C2×Dic9—31no—
201C2×Dic9—31no—
202C2×Dic9—31no—
203C2×Dic9—31no—
204C2×Dic9—61no—
205C2×Dic9—61no—
206C2×Dic9—61no—
207C2×Dic9—61no—
208C2×Dic9—122no—
209C2×Dic9C22×D312yes—
210C2×Dic9C22×D312yes—
211Dic18—22no—
212Dic18—122no—
213#542—271noSylow
214C2×C3×C9—41no—
215C2×C3×C9C2×#4111yescharacteristic
216C2×C3×C9H(5,𝔽2)11yescharacteristic
217C2×C3×C92−1+411yescharacteristic
218D3×C9—42no—
219C2×C3×Q8—91no—
220C2×C3×D4—32no—
221C2×C3×D4—31no—
222C22×C3×C4—21no—
223C22×C3×C4C2×D911yescharacteristic
224C22×C3×C4D3211yescharacteristic
225C2×#67C2×D912yes—
226C22×Dic3—92no—
227C2×D12C2×D911yescharacteristic
228C2×C4×D3—92no—
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×C22⋊C4—92no—
237#217—32no—
238#217—92no—
239#217—92no—
240#217—92no—
241#217—91no—
242#217—91no—
243#217—91no—
244#217—91no—
245#217—181no—
246#217—181no—
247(C2×D3)⋊C4—92no—
248(C2×D3)⋊C4—92no—
249(C2×D3)⋊C4—92no—
250(C2×D3)⋊C4C2×D912yes—
251C4⋊Dic3—31no—
252C4⋊Dic3—91no—
253C4⋊Dic3—91no—
254C4⋊Dic3—91no—
255C4⋊Dic3—91no—
256C4⋊Dic3—92no—
257C4⋊Dic3—181no—
258C4⋊Dic3—181no—
259C4⋊Dic3C2×D911yescharacteristic
260Dic3⋊C4—32no—
261Dic3⋊C4—92no—
262Dic3⋊C4—92no—
263Dic3⋊C4—92no—
264C4×Dic3—92no—
265C4×Dic3—91no—
266C4×Dic3—91no—
267C4×Dic3—91no—
268C4×Dic3—91no—
269C4×Dic3—92no—
270C4×Dic3—181no—
271C4×Dic3—181no—
272C22×C32—21no—
273C22×C32—21no—
274C22×C32—21no—
275C22×C32C23×D311yescharacteristic
276C2×C3×D3—22no—
277C2×C3×D3—22no—
278C2×C3×D3—22no—
279C32×C4—21no—
280C32×C4#23712yes—
281C3⋊Dic3—61no—
282C3⋊Dic3—61no—
283C3⋊Dic3—61no—
284C3⋊Dic3—61no—
285C3×Dic3—22no—
286C3×Dic3—62no—
287C3×Dic3—62no—
288C22×C9—22no—
289C22×C9—21no—
290C22×C9—21no—
291C22×C9—21no—
292C22×C9—21no—
293C22×C9—41no—
294C22×C9—41no—
295C22×C9—62no—
296C22×C9—62no—
297C22×C9—62no—
298C22×C9C23×D311yescharacteristic
299C4×C9—22no—
300C4×C9—21no—
301C4×C9—22no—
302C4×C9—62no—
303C4×C9#23712yes—
304Dic9—22no—
305Dic9—22no—
306Dic9—61no—
307Dic9—61no—
308Dic9—61no—
309Dic9—61no—
310Dic9—121no—
311Dic9—121no—
312Dic9—121no—
313Dic9—121no—
314C4⋊Q8—271no—
315#126—272no—
316#124—272no—
317#124—271no—
318#123—272no—
319C22⋊Q8—91no—
320C22⋊Q8—272no—
321#121—31no—
322C4×Q8—271no—
323C4×D4—271no—
324#117—271no—
325C3×C9#54211yescharacteristic, complemented, Sylow
326C23×C3—21no—
327C23×C3—32no—
328C23×C3C22×D911yescharacteristic
329C23×C3C2×D3211yescharacteristic
330C22×D3C22×D912yes—
331C3×Q8—182no—
332C3×D4—62no—
333C3×D4—122no—
334C2×C3×C4—24no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—32no—
340C2×C3×C4—92no—
341C2×C3×C4—92no—
342C2×C3×C4C22×D911yescharacteristic
343C2×C3×C4C22×D911yescharacteristic
344C2×C3×C4C2×D3211yescharacteristic
345C2×C3×C4C2×D3211yescharacteristic
346#67—42no—
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—91no—
356C2×Dic3—91no—
357C2×Dic3—181no—
358C2×Dic3—181no—
359C2×Dic3—181no—
360C2×Dic3—181no—
361C2×Dic3—362no—
362C2×Dic3C22×D912yes—
363D12—22no—
364C4×D3—362no—
365Dic6—62no—
366Dic6—362no—
367Dic6—362no—
368C2×C32—41no—
369C2×C32C2×#23711yescharacteristic
370C2×C32#100411yescharacteristic
371C2×C32#101011yescharacteristic
372C3×D3—42no—
373C2×C9—21no—
374C2×C9—21no—
375C2×C9—21no—
376C2×C9—42no—
377C2×C9—41no—
378C2×C9—122no—
379C2×C9C2×#23711yescharacteristic
380C2×C9#100711yescharacteristic
381C2×C9#100911yescharacteristic
382C2×Q8—91no—
383C2×Q8—272no—
384C2×D4—32no—
385C2×D4—31no—
386C22×C4—272no—
387C22×C4D3×D911yescharacteristic
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—92no—
397C22⋊C4—272no—
398C22⋊C4—272no—
399C22⋊C4—272no—
400C22⋊C4—271no—
401C22⋊C4—271no—
402C42—272no—
403C42—271no—
404C42—271no—
405C22×C3—22no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—21no—
412C22×C3—21no—
413C22×C3—41no—
414C22×C3—41no—
415C22×C3—62no—
416C22×C3—62no—
417C22×C3—62no—
418C22×C3C23×D911yescharacteristic
419C22×C3C22×D3211yescharacteristic
420C2×D3—22no—
421C2×D3—22no—
422C2×D3—22no—
423C3×C4—22no—
424C3×C4—21no—
425C3×C4—21no—
426C3×C4—22no—
427C3×C4—62no—
428C3×C4—182no—
429C3×C4—182no—
430C3×C4#369912yes—
431C3×C4#380512yes—
432Dic3—22no—
433Dic3—62no—
434Dic3—62no—
435Dic3—181no—
436Dic3—181no—
437Dic3—181no—
438Dic3—181no—
439Dic3—181no—
440Dic3—181no—
441Dic3—181no—
442Dic3—181no—
443Dic3—361no—
444Dic3—361no—
445Dic3—361no—
446Dic3—361no—
447C32#560311yescharacteristic
448C9—21no—
449C9#564311yescharacteristic, complemented
450C23—32no—
451C23C2×D3×D911yescharacteristic
452Q8—182no—
453Q8—1082no—
454D4—62no—
455D4—122no—
456C2×C4—22no—
457C2×C4—32no—
458C2×C4—92no—
459C2×C4—92no—
460C2×C4—271no—
461C2×C4—271no—
462C2×C4—271no—
463C2×C4—271no—
464C2×C4—1082no—
465C2×C4C2×D3×D911yescharacteristic
466C2×C4C2×D3×D911yescharacteristic
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—42no—
471C2×C3—41no—
472C2×C3—41no—
473C2×C3—122no—
474C2×C3C2×#369911yescharacteristic
475C2×C3#6371011yescharacteristic
476C2×C3#6371611yescharacteristic
477C2×C3#6429711yescharacteristic
478C2×C3#6430711yescharacteristic
479C2×C3C2×#380511yescharacteristic
480D3—42no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—62no—
485C22—62no—
486C22—62no—
487C22C22×D3×D911yescharacteristic
488C4—21no—
489C4—62no—
490C4—182no—
491C4—182no—
492C4—541no—
493C4—541no—
494C4—541no—
495C4—541no—
496C4#8866312yes—
497C3—21no—
498C3#1058970511yescharacteristic, complemented
499C3#1059498411yescharacteristic
500C2—41no—
501C2—122no—
502C2#1172246411yescharacteristic
503C2#1172247611yescharacteristic
504C2C2×#8866311yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
prime radicalC3×#930D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#542
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #11721815C2semidirect
3. C3×#63486C2semidirect
4. #11721280C2semidirect
5. #11721263C2semidirect
6. #11721025C2semidirect
7. #11721023C2semidirect
8. #11720971C2semidirect
9. #11720967C2semidirect
10. #11720965C2semidirect
11. #11720910C2semidirect
55. #63486D3semidirect
325. C3×C9#542semidirect
449. C9#5643semidirect
498. C3#10589705semidirect

Maximal subgroups

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

subgroupindex
#117218152
C9×#9302
C3×#634862
#117212802
#117212632
#117210252
#117210232
#117209712
#117209672
#117209652
#117209102
#105949843
#105897053