Finite Group Databasebeta

← #13147588

Subgroups — #13147588

Order 1728.

868 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×Dic18C211yesmaximal, characteristic
2C4×#6226C211yesmaximal, characteristic, complemented
3C9×#872C211yesmaximal, characteristic
4C3×C4×Dic18C211yesmaximal, characteristic, complemented
5#11721011C212yesmaximal
6#11721010C212yesmaximal
7#11720912C211yesmaximal, characteristic, complemented
8Dic3×Dic18C211yesmaximal, characteristic, complemented
9#11720906C211yesmaximal, characteristic
10#11720902C212yesmaximal, complemented
11#11720901C212yesmaximal, complemented
12#10593885—31nomaximal
13#10589618—31nomaximal
14C2×#6226C2211yescharacteristic
15C2×C4×D3×C9C2211yescharacteristic
16C2×C3×Dic18C2211yescharacteristic
17C2×D3×Dic9C2212yes—
18C2×#6189C2212yes—
19D3×Dic18C418yescomplemented
20C3×C42×C9C2211yescharacteristic
21C4⋊(C3⋊Dic9)C2211yescharacteristic
22#88534C2212yes—
23C4×C3⋊Dic9C2212yes—
24C9×(C2×D3)⋊C4C2211yescharacteristic
25C9×(C2×D3)⋊C4C2211yescharacteristic
26C9×Dic3⋊C4C2211yescharacteristic
27C9×Dic3⋊C4C2211yescharacteristic
28C4×C9×Dic3C2211yescharacteristic
29C3×C4⋊Dic9C2211yescharacteristic
30C3×Dic9⋊C4C2212yes—
31C3×C4×Dic9C2212yes—
32(C2×D3)⋊Dic9C2212yes—
33#88442C2212yes—
34Dic9⋊Dic3C2212yes—
35Dic3×Dic9C2212yes—
36Dic3×Dic9C2212yes—
37Dic3×Dic9C2212yes—
38C2×D3×Dic6—31no—
39C4×#634—31no—
40C3×#872D311yescharacteristic
41C3×C4×Dic6—31no—
42#63878—32no—
43#63877—32no—
44#63844—31no—
45Dic3×Dic6—31no—
46#63841—31no—
47#63838—32no—
48#63837—32no—
49C22×Dic18—31no—
50C9×#117—31no—
51#63488—31no—
52#63449—32no—
53#63449—32no—
54#63439—32no—
55#63439—32no—
56C4×Dic18—31no—
57C4×Dic18—31no—
58C4×Dic18—31no—
59C4×Dic18—61no—
60C4×Dic18D311yescharacteristic, complemented
61C22×D3×C9C2311yescharacteristic
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9C2312yes—
66C2×C3⋊Dic9C2312yes—
67#6226C2×C414yes—
68C2×C9×Dic3C2311yescharacteristic
69C2×C9×Dic3C2311yescharacteristic
70C2×C9×Dic3C2311yescharacteristic
71C2×C3×Dic9C2312yes—
72C2×C3×Dic9C2312yes—
73C4×D3×C9C2×C414yes—
74C3×Dic18C2×C414yes—
75D3×Dic9C2×C418yes—
76#6189C2×C418yes—
77#5572—91no—
78#5516—91no—
79C2×#634—31no—
80C2×C3×C4×D3C2×D311yescharacteristic
81C2×C3×Dic6—31no—
82C2×#627—32no—
83C2×D3×Dic3—32no—
84D3×Dic6—38no—
85C32×C42C2×D311yescharacteristic
86C4⋊(C3⋊Dic3)—31no—
87#3750—32no—
88C4×C3⋊Dic3—32no—
89C3×(C2×D3)⋊C4C2×D311yescharacteristic
90C3×(C2×D3)⋊C4C2×D311yescharacteristic
91C3×C4⋊Dic3—31no—
92C3×Dic3⋊C4—32no—
93C3×Dic3⋊C4C2×D311yescharacteristic
94C3×Dic3⋊C4C2×D311yescharacteristic
95C3×C4×Dic3—32no—
96C3×C4×Dic3C2×D311yescharacteristic
97#3724—32no—
98Dic3⋊Dic3—32no—
99(C2×D3)⋊Dic3—32no—
100Dic32—32no—
101Dic32—32no—
102Dic32—32no—
103C22×C4×C9—31no—
104C22×Dic9—32no—
105C2×Dic18—31no—
106C2×Dic18—32no—
107C2×Dic18—38no—
108C2×Dic18—61no—
109C2×Dic18C2×D311yescharacteristic
110C9×C4⋊C4—31no—
111C9×C4⋊C4—31no—
112C9×C22⋊C4—31no—
113C9×C22⋊C4—31no—
114C42×C9—21no—
115C42×C9—31no—
116C42×C9C2×D311yescharacteristic
117#3676—32no—
118C4⋊Dic9—31no—
119C4⋊Dic9—61no—
120C4⋊Dic9C2×D311yescharacteristic
121Dic9⋊C4—32no—
122Dic9⋊C4—32no—
123Dic9⋊C4—32no—
124Dic9⋊C4—62no—
125Dic9⋊C4C2×D312yes—
126C4×Dic9—32no—
127C4×Dic9—32no—
128C4×Dic9—32no—
129C4×Dic9—32no—
130C4×Dic9—62no—
131C4×Dic9C2×D312yes—
132C22×C3×C9C2411yescharacteristic
133C2×D3×C9—21no—
134C2×D3×C9—21no—
135C2×D3×C9C22×C412yes—
136C3×C4×C9—21no—
137C3×C4×C9—21no—
138C3×C4×C9C22×C412yes—
139C3⋊Dic9—22no—
140C3⋊Dic9C22×C414yes—
141C9×Dic3—21no—
142C9×Dic3—21no—
143C9×Dic3C22×C412yes—
144C3×Dic9—22no—
145C3×Dic9C22×C414yes—
146C2×D3×Q8—91no—
147C22×Dic6—91no—
148C3×C4×Q8—91no—
149C3×#117—31no—
150Q8×Dic3—91no—
151#926—91no—
152#892—91no—
153#892—92no—
154#887—92no—
155#887—92no—
156#887—92no—
157#887—91no—
158#877—92no—
159#877—92no—
160#872—92no—
161#872D911yescharacteristic
162C4×Dic6—31no—
163C4×Dic6—91no—
164C4×Dic6—91no—
165C4×Dic6—91no—
166C4×Dic6—91no—
167C4×Dic6—92no—
168C4×Dic6—181no—
169C22×C3×D3C22×D311yescharacteristic
170C2×C32×C4C22×D311yescharacteristic
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C3⋊Dic3—32no—
174C2×C3⋊Dic3—32no—
175#634—34no—
176C2×C3×Dic3—32no—
177C2×C3×Dic3—32no—
178C2×C3×Dic3C22×D311yescharacteristic
179C2×C3×Dic3C22×D311yescharacteristic
180C2×C3×Dic3C22×D311yescharacteristic
181C3×C4×D3C4×D314yes—
182C3×Dic6—34no—
183#627—38no—
184D3×Dic3—38no—
185C23×C9—31no—
186C2×C4×C9—21no—
187C2×C4×C9—21no—
188C2×C4×C9—21no—
189C2×C4×C9—31no—
190C2×C4×C9—34no—
191C2×C4×C9—31no—
192C2×C4×C9—31no—
193C2×C4×C9C22×D311yescharacteristic
194C2×C4×C9C22×D311yescharacteristic
195C2×C4×C9C22×D311yescharacteristic
196C2×Dic9—32no—
197C2×Dic9—38no—
198C2×Dic9—32no—
199C2×Dic9—62no—
200C2×Dic9—62no—
201C2×Dic9C22×D312yes—
202C2×Dic9C22×D312yes—
203Dic18—34no—
204Dic18—38no—
205Dic18—64no—
206Dic18C4×D314yes—
207#519—271noSylow
208C2×C3×C9C23×C411yescharacteristic
209C2×C3×C92−1+411yescharacteristic
210C2×C3×C92−1+411yescharacteristic
211D3×C9—22no—
212C2×C3×Q8—91no—
213C22×C3×C4—31no—
214C22×Dic3—92no—
215D3×Q8—98no—
216C2×C4×D3—92no—
217C2×C4×D3C2×D911yescharacteristic
218C2×Dic6—31no—
219C2×Dic6—98no—
220C2×Dic6—92no—
221C2×Dic6—91no—
222C2×Dic6—91no—
223C2×Dic6—92no—
224C2×Dic6—181no—
225C3×C4⋊C4—31no—
226C3×C4⋊C4—31no—
227C3×C4⋊C4—92no—
228C3×C4⋊C4—91no—
229C3×C22⋊C4—31no—
230C3×C22⋊C4—31no—
231C3×C42—21no—
232C3×C42—31no—
233C3×C42—92no—
234C3×C42C2×D911yescharacteristic
235C3×C42D3211yescharacteristic
236#217—92no—
237(C2×D3)⋊C4—92no—
238(C2×D3)⋊C4C2×D911yescharacteristic
239(C2×D3)⋊C4C2×D911yescharacteristic
240C4⋊Dic3—31no—
241C4⋊Dic3—91no—
242C4⋊Dic3—91no—
243C4⋊Dic3—92no—
244C4⋊Dic3—181no—
245Dic3⋊C4—32no—
246Dic3⋊C4—92no—
247Dic3⋊C4—92no—
248Dic3⋊C4—92no—
249Dic3⋊C4—92no—
250Dic3⋊C4—92no—
251Dic3⋊C4—182no—
252Dic3⋊C4C2×D911yescharacteristic
253Dic3⋊C4C2×D911yescharacteristic
254C4×Dic3—32no—
255C4×Dic3—92no—
256C4×Dic3—92no—
257C4×Dic3—92no—
258C4×Dic3—92no—
259C4×Dic3—92no—
260C4×Dic3—92no—
261C4×Dic3—92no—
262C4×Dic3—92no—
263C4×Dic3—182no—
264C4×Dic3C2×D911yescharacteristic
265C22×C32C23×D311yescharacteristic
266C2×C3×D3—21no—
267C2×C3×D3—21no—
268C2×C3×D3C2×C4×D312yes—
269C32×C4—21no—
270C32×C4—21no—
271C32×C4C2×C4×D312yes—
272C3⋊Dic3—34no—
273C3⋊Dic3—62no—
274C3×Dic3—21no—
275C3×Dic3—21no—
276C3×Dic3—34no—
277C3×Dic3—62no—
278C3×Dic3C2×C4×D312yes—
279C22×C9—21no—
280C22×C9—32no—
281C22×C9—61no—
282C22×C9—61no—
283C22×C9C23×D311yescharacteristic
284C4×C9—22no—
285C4×C9—21no—
286C4×C9—21no—
287C4×C9—22no—
288C4×C9—32no—
289C4×C9—41no—
290C4×C9—61no—
291C4×C9—61no—
292C4×C9C2×C4×D312yes—
293Dic9—22no—
294Dic9—34no—
295Dic9—64no—
296Dic9—62no—
297Dic9—122no—
298Dic9C2×C4×D314yes—
299C22×Q8—271no—
300C4×Q8—91no—
301C4×Q8—271no—
302C4×Q8—271no—
303C4×Q8—272no—
304C4×Q8—272no—
305C4×Q8—271no—
306#117—31no—
307#117—272no—
308#117—272no—
309#117—271no—
310C3×C9#51911yescharacteristic, complemented, Sylow
311C23×C3—31no—
312C22×D3C22×D911yescharacteristic
313C3×Q8—94no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—31no—
318C2×C3×C4—34no—
319C2×C3×C4—31no—
320C2×C3×C4—31no—
321C2×C3×C4—92no—
322C2×C3×C4—92no—
323C2×C3×C4C22×D911yescharacteristic
324C2×C3×C4C22×D911yescharacteristic
325C2×C3×C4C22×D911yescharacteristic
326C2×C3×C4C2×D3211yescharacteristic
327C2×C3×C4C2×D3211yescharacteristic
328C2×C3×C4C2×D3211yescharacteristic
329C2×Dic3—32no—
330C2×Dic3—32no—
331C2×Dic3—98no—
332C2×Dic3—92no—
333C2×Dic3—92no—
334C2×Dic3—92no—
335C2×Dic3—92no—
336C2×Dic3—182no—
337C2×Dic3—182no—
338C2×Dic3C22×D911yescharacteristic
339C2×Dic3C22×D911yescharacteristic
340C2×Dic3C22×D911yescharacteristic
341C4×D3—98no—
342C4×D3C4×D914yes—
343Dic6—34no—
344Dic6—98no—
345Dic6—94no—
346Dic6—94no—
347Dic6—98no—
348Dic6—184no—
349C2×C32C22×C4×D311yescharacteristic
350C2×C32#101011yescharacteristic
351C2×C32#101011yescharacteristic
352C3×D3—22no—
353C2×C9—21no—
354C2×C9—21no—
355C2×C9—21no—
356C2×C9—62no—
357C2×C9C22×C4×D311yescharacteristic
358C2×C9#100711yescharacteristic
359C2×C9#101011yescharacteristic
360C2×Q8—91no—
361C2×Q8—272no—
362C2×Q8—278no—
363C2×Q8—271no—
364C22×C4—31no—
365C22×C4—272no—
366C4⋊C4—31no—
367C4⋊C4—31no—
368C4⋊C4—92no—
369C4⋊C4—91no—
370C4⋊C4—272no—
371C4⋊C4—272no—
372C4⋊C4—272no—
373C4⋊C4—271no—
374C22⋊C4—31no—
375C22⋊C4—31no—
376C22⋊C4—272no—
377C42—31no—
378C42—92no—
379C42—272no—
380C42—272no—
381C42—272no—
382C42—272no—
383C42D3×D911yescharacteristic
384C22×C3—21no—
385C22×C3—32no—
386C22×C3—61no—
387C22×C3—61no—
388C22×C3C23×D911yescharacteristic
389C22×C3C22×D3211yescharacteristic
390C2×D3—21no—
391C2×D3—21no—
392C2×D3C2×C4×D912yes—
393C3×C4—22no—
394C3×C4—21no—
395C3×C4—21no—
396C3×C4—21no—
397C3×C4—21no—
398C3×C4—22no—
399C3×C4—32no—
400C3×C4—41no—
401C3×C4—61no—
402C3×C4—61no—
403C3×C4—94no—
404C3×C4—182no—
405C3×C4C2×C4×D912yes—
406C3×C4C4×D3212yes—
407Dic3—21no—
408Dic3—21no—
409Dic3—34no—
410Dic3—62no—
411Dic3—94no—
412Dic3—94no—
413Dic3—184no—
414Dic3—182no—
415Dic3—182no—
416Dic3—362no—
417Dic3C2×C4×D912yes—
418C32#551611yescharacteristic
419C9—21no—
420C9#557211yescharacteristic, complemented
421C23—31no—
422Q8—94no—
423Q8—278no—
424Q8—274no—
425C2×C4—31no—
426C2×C4—34no—
427C2×C4—31no—
428C2×C4—31no—
429C2×C4—92no—
430C2×C4—92no—
431C2×C4—278no—
432C2×C4—272no—
433C2×C4—272no—
434C2×C4C2×D3×D911yescharacteristic
435C2×C4C2×D3×D911yescharacteristic
436C2×C4C2×D3×D911yescharacteristic
437C2×C3—21no—
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—62no—
441C2×C3C22×C4×D911yescharacteristic
442C2×C3#6371611yescharacteristic
443C2×C3#6371611yescharacteristic
444C2×C3#6429711yescharacteristic
445C2×C3#6429811yescharacteristic
446C2×C3C2×C4×D3211yescharacteristic
447D3—22no—
448C22—32no—
449C22—61no—
450C22—61no—
451C22C22×D3×D911yescharacteristic
452C4—21no—
453C4—21no—
454C4—32no—
455C4—61no—
456C4—61no—
457C4—94no—
458C4—182no—
459C4—274no—
460C4—542no—
461C4C4×D3×D912yes—
462C3—21no—
463C3#1058961811yescharacteristic, complemented
464C3#1059388511yescharacteristic
465C2—62no—
466C2#1172246411yescharacteristic
467C2#1172246511yescharacteristic
468C2C2×C4×D3×D911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC22×C3×D3C22×D3
cosocleC22×C3×C9C24
hypercenterC42D3×D9
hypocenterC3×C9#519
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C4×#6226C2semidirect
4. C3×C4×Dic18C2semidirect
7. #11720912C2semidirect
8. Dic3×Dic18C2semidirect
10. #11720902C2semidirect
11. #11720901C2semidirect
19. D3×Dic18C4semidirect
60. C4×Dic18D3semidirect
310. C3×C9#519semidirect
420. C9#5572semidirect
463. C3#10589618semidirect

Maximal subgroups

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

subgroupindex
C2×D3×Dic182
C4×#62262
C9×#8722
C3×C4×Dic182
#117210112
#117210102
#117209122
Dic3×Dic182
#117209062
#117209022
#117209012
#105938853
#105896183