Finite Group Databasebeta

← #13148892

Subgroups — #13148892

Order 1728.

722 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721746C211yesmaximal, characteristic, complemented
2C3×#63454C211yesmaximal, characteristic
3C9×#888C211yesmaximal, characteristic, complemented
4#11720991C212yesmaximal
5#11720984C212yesmaximal
6C4×Dic3×D9C211yesmaximal, characteristic
7#11720970C211yesmaximal, characteristic, complemented
8#11720937C211yesmaximal, characteristic, complemented
9Dic6×Dic9C212yesmaximal, complemented
10#11720907C211yesmaximal, characteristic
11#11720901C212yesmaximal, complemented
12#10594445—31nomaximal
13#10589830—31nomaximal
14C2×C3×C4×D9C2211yescharacteristic
15C2×C9×Dic6C2212yes—
16C2×Dic3×D9C2211yescharacteristic
17C2×Dic3×D9C2211yescharacteristic
18C2×#6189C2212yes—
19C3×C9×C4⋊C4C2211yescharacteristic
20C4⋊(C3⋊Dic9)C2211yescharacteristic
21C4⋊(C3⋊Dic9)C2212yes—
22#88534C2212yes—
23C4×C3⋊Dic9C2211yescharacteristic
24C3×(C2×D9)⋊C4C2212yes—
25C9×C4⋊Dic3C2211yescharacteristic
26C9×Dic3⋊C4C2212yes—
27C4×C9×Dic3C2211yescharacteristic
28C3×C4⋊Dic9C2211yescharacteristic
29C3×C4×Dic9C2212yes—
30(C2×D9)⋊Dic3C2212yes—
31Dic3⋊Dic9C2212yes—
32Dic3⋊Dic9C2212yes—
33Dic9⋊Dic3C2212yes—
34Dic3×Dic9C2211yescharacteristic
35Dic3×Dic9C2211yescharacteristic
36Dic3×Dic9C2212yes—
37#64099—31no—
38C3×#892—31no—
39C3×#888D311yescharacteristic
40#63918—31no—
41#63893—32no—
42#63887—32no—
43C4×D3×Dic3—31no—
44#63872—31no—
45Dic3×Dic6—32no—
46#63841—31no—
47#63837—32no—
48C9×C4⋊Q8—31no—
49#63508—32no—
50Q8×Dic9—32no—
51#63459—32no—
52#63454—31no—
53#63454D311yescharacteristic
54#63452—31no—
55#63452—31no—
56#63452—61no—
57#63449—32no—
58C42×D9—31no—
59#63431—31no—
60C22×C3×D9C2311yescharacteristic
61C2×C3×C4×C9C2312yes—
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3⋊Dic9C2312yes—
64C2×C3⋊Dic9C2311yescharacteristic
65C2×C3⋊Dic9C2311yescharacteristic
66C2×C9×Dic3C2312yes—
67C2×C9×Dic3C2311yescharacteristic
68C2×C9×Dic3C2311yescharacteristic
69C2×C3×Dic9C2312yes—
70C2×C3×Dic9C2311yescharacteristic
71C3×C4×D9Q814yescomplemented
72C9×Dic6—24no—
73Dic3×D9—22no—
74Dic3×D9—22no—
75#6189—24no—
76#5728—91no—
77#5517—91no—
78C2×C3×C4×D3—31no—
79C2×C3×Dic6C2×D312yes—
80C2×#627—32no—
81C2×D3×Dic3—31no—
82C2×D3×Dic3—31no—
83C32×C4⋊C4C2×D311yescharacteristic
84C4⋊(C3⋊Dic3)—31no—
85C4⋊(C3⋊Dic3)—32no—
86#3750—32no—
87C4×C3⋊Dic3—31no—
88C3×(C2×D3)⋊C4—32no—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×Dic3⋊C4C2×D312yes—
92C3×C4×Dic3—32no—
93C3×C4×Dic3C2×D311yescharacteristic
94Dic3⋊Dic3—32no—
95Dic3⋊Dic3—32no—
96Dic3⋊Dic3—32no—
97(C2×D3)⋊Dic3—32no—
98Dic32—31no—
99Dic32—32no—
100Dic32—31no—
101C2×Q8×C9—32no—
102C2×C4×D9—31no—
103C2×C4×D9—31no—
104C2×C4×D9C2×D311yescharacteristic
105C2×Dic18—32no—
106C9×C4⋊C4—21no—
107C9×C4⋊C4—32no—
108C9×C4⋊C4—31no—
109C9×C4⋊C4C2×D311yescharacteristic
110C42×C9—31no—
111(C2×D9)⋊C4—32no—
112(C2×D9)⋊C4C2×D312yes—
113C4⋊Dic9—32no—
114C4⋊Dic9—31no—
115C4⋊Dic9—32no—
116C4⋊Dic9—32no—
117C4⋊Dic9—62no—
118C4⋊Dic9—61no—
119C4⋊Dic9C2×D311yescharacteristic
120Dic9⋊C4—32no—
121Dic9⋊C4—32no—
122Dic9⋊C4—62no—
123C4×Dic9—31no—
124C4×Dic9—31no—
125C4×Dic9—32no—
126C4×Dic9—31no—
127C4×Dic9—61no—
128C4×Dic9C2×D312yes—
129C22×C3×C9C2411yescharacteristic
130C2×C3×D9—21no—
131C2×C3×D9—21no—
132C2×C3×D9C2×Q812yes—
133C3×C4×C9—22no—
134C3×C4×C9C2×Q812yes—
135C3⋊Dic9—22no—
136C3⋊Dic9—21no—
137C3⋊Dic9—21no—
138C9×Dic3—22no—
139C9×Dic3#4112yes—
140C9×Dic3#4112yes—
141C3×Dic9—22no—
142C3×Dic9C2×Q812yes—
143C3×C4⋊Q8—31no—
144C3×#117—91no—
145#946—92no—
146Q8×Dic3—92no—
147#926—91no—
148C2×C4×Dic3—91no—
149#897—92no—
150#892—31no—
151#892—91no—
152#890—91no—
153#890—91no—
154#890—91no—
155#890—91no—
156#890—181no—
157#888—91no—
158#888D911yescharacteristic, complemented
159#887—92no—
160#878—92no—
161#878—92no—
162C42×D3—91no—
163#869—91no—
164C4×Dic6—92no—
165C4×Dic6—92no—
166C22×C3×D3—31no—
167C2×C32×C4C22×D312yes—
168C2×C32×C4C22×D311yescharacteristic
169C2×C3⋊Dic3—32no—
170C2×C3⋊Dic3—31no—
171C2×C3⋊Dic3—31no—
172C2×C3×Dic3—31no—
173C2×C3×Dic3—32no—
174C2×C3×Dic3C22×D312yes—
175C2×C3×Dic3C22×D311yescharacteristic
176C2×C3×Dic3C22×D311yescharacteristic
177C3×C4×D3—34no—
178C3×Dic6—24no—
179#627—64no—
180D3×Dic3—62no—
181D3×Dic3—62no—
182C22×D9C22×D311yescharacteristic
183Q8×C9—64no—
184C2×C4×C9—21no—
185C2×C4×C9—22no—
186C2×C4×C9—31no—
187C2×C4×C9—31no—
188C2×C4×C9—32no—
189C2×C4×C9C22×D312yes—
190C2×C4×C9C22×D311yescharacteristic
191C2×Dic9—31no—
192C2×Dic9—31no—
193C2×Dic9—32no—
194C2×Dic9—61no—
195C2×Dic9—61no—
196C2×Dic9—62no—
197C2×Dic9C22×D312yes—
198C2×Dic9C22×D311yescharacteristic
199C4×D9—62no—
200C4×D9—62no—
201C4×D9Dic614yescomplemented
202Dic18—64no—
203#533—271noSylow
204C2×C3×C9C22×Q811yescharacteristic
205C2×C3×C9C2×#4111yescharacteristic
206C2×C3×C9C2×#4111yescharacteristic
207C3×D9—22no—
208C2×C3×Q8—32no—
209C22×C3×C4—91no—
210C22×Dic3—91no—
211C22×Dic3—91no—
212C2×C4×D3—31no—
213C2×C4×D3—91no—
214C2×C4×D3—91no—
215C2×Dic6—92no—
216C2×Dic6—92no—
217C2×Dic6C2×D912yes—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—32no—
220C3×C4⋊C4—31no—
221C3×C4⋊C4—91no—
222C3×C4⋊C4C2×D911yescharacteristic
223C3×C4⋊C4D3211yescharacteristic
224C3×C22⋊C4—92no—
225C3×C42—31no—
226C3×C42—92no—
227#217—92no—
228(C2×D3)⋊C4—32no—
229(C2×D3)⋊C4—92no—
230C4⋊Dic3—31no—
231C4⋊Dic3—92no—
232C4⋊Dic3—92no—
233C4⋊Dic3—92no—
234C4⋊Dic3—91no—
235C4⋊Dic3—91no—
236C4⋊Dic3—92no—
237C4⋊Dic3—92no—
238C4⋊Dic3—181no—
239C4⋊Dic3—182no—
240C4⋊Dic3C2×D911yescharacteristic
241Dic3⋊C4—92no—
242Dic3⋊C4—92no—
243Dic3⋊C4—92no—
244Dic3⋊C4—92no—
245Dic3⋊C4—92no—
246Dic3⋊C4—182no—
247Dic3⋊C4C2×D912yes—
248C4×Dic3—32no—
249C4×Dic3—91no—
250C4×Dic3—91no—
251C4×Dic3—92no—
252C4×Dic3—91no—
253C4×Dic3—91no—
254C4×Dic3—91no—
255C4×Dic3—91no—
256C4×Dic3—92no—
257C4×Dic3—181no—
258C4×Dic3C2×D911yescharacteristic
259C22×C32C23×D311yescharacteristic
260C2×C3×D3—32no—
261C2×C3×D3—61no—
262C2×C3×D3—61no—
263C32×C4—22no—
264C32×C4D3×Q812yes—
265C3⋊Dic3—61no—
266C3⋊Dic3—61no—
267C3⋊Dic3—62no—
268C3×Dic3—22no—
269C3×Dic3—32no—
270C3×Dic3—62no—
271C3×Dic3#23712yes—
272C3×Dic3#23912yes—
273C22×C9—21no—
274C22×C9C23×D311yescharacteristic
275C2×D9—21no—
276C2×D9—21no—
277C2×D9C2×Dic612yes—
278C4×C9—22no—
279C4×C9—22no—
280C4×C9—32no—
281C4×C9—32no—
282C4×C9—42no—
283C4×C9—62no—
284C4×C9C2×Dic612yes—
285Dic9—22no—
286Dic9—62no—
287Dic9—61no—
288Dic9—61no—
289Dic9—121no—
290Dic9—122no—
291Dic9—121no—
292Dic9C2×Dic612yes—
293C4⋊Q8—31no—
294C4⋊Q8—271no—
295#125—271no—
296#125—271no—
297C22⋊Q8—272no—
298C22⋊Q8—272no—
299C4×Q8—272no—
300C4×Q8—272no—
301#117—91no—
302#117—271no—
303C2×C42—271no—
304C3×C9#53311yescharacteristic, complemented, Sylow
305C23×C3—91no—
306C22×D3—31no—
307C3×Q8—64no—
308C2×C3×C4—21no—
309C2×C3×C4—22no—
310C2×C3×C4—31no—
311C2×C3×C4—31no—
312C2×C3×C4—32no—
313C2×C3×C4—91no—
314C2×C3×C4—94no—
315C2×C3×C4—92no—
316C2×C3×C4C22×D912yes—
317C2×C3×C4C22×D911yescharacteristic
318C2×C3×C4C2×D3212yes—
319C2×C3×C4C2×D3211yescharacteristic
320C2×Dic3—31no—
321C2×Dic3—32no—
322C2×Dic3—91no—
323C2×Dic3—91no—
324C2×Dic3—92no—
325C2×Dic3—91no—
326C2×Dic3—91no—
327C2×Dic3—92no—
328C2×Dic3—181no—
329C2×Dic3—181no—
330C2×Dic3—182no—
331C2×Dic3—182no—
332C2×Dic3—182no—
333C2×Dic3C22×D912yes—
334C2×Dic3C22×D911yescharacteristic
335C2×Dic3C22×D911yescharacteristic
336C4×D3—34no—
337C4×D3—182no—
338C4×D3—182no—
339Dic6—24no—
340Dic6—184no—
341Dic6—184no—
342C2×C32C2×#23711yescharacteristic
343C2×C32C2×D3×Q811yescharacteristic
344C2×C32C2×#23911yescharacteristic
345C3×D3—62no—
346C2×C9—21no—
347C2×C9—21no—
348C2×C9—21no—
349C2×C9C22×Dic611yescharacteristic
350C2×C9D3×#4111yescharacteristic
351C2×C9D3×#4111yescharacteristic
352D9—22no—
353C2×Q8—32no—
354C2×Q8—272no—
355C22×C4—91no—
356C22×C4—271no—
357C22×C4—271no—
358C4⋊C4—31no—
359C4⋊C4—32no—
360C4⋊C4—91no—
361C4⋊C4—272no—
362C4⋊C4—272no—
363C4⋊C4—272no—
364C4⋊C4—271no—
365C4⋊C4—272no—
366C4⋊C4—272no—
367C4⋊C4D3×D911yescharacteristic
368C22⋊C4—92no—
369C22⋊C4—272no—
370C42—31no—
371C42—92no—
372C42—271no—
373C42—271no—
374C42—272no—
375C42—271no—
376C22×C3—21no—
377C22×C3—92no—
378C22×C3—181no—
379C22×C3—181no—
380C22×C3C23×D911yescharacteristic
381C22×C3C22×D3211yescharacteristic
382C2×D3—32no—
383C2×D3—61no—
384C2×D3—61no—
385C3×C4—22no—
386C3×C4—22no—
387C3×C4—22no—
388C3×C4—32no—
389C3×C4—32no—
390C3×C4—42no—
391C3×C4—62no—
392C3×C4—92no—
393C3×C4—182no—
394C3×C4Q8×D912yes—
395C3×C4D3×Dic612yes—
396Dic3—22no—
397Dic3—32no—
398Dic3—62no—
399Dic3—181no—
400Dic3—182no—
401Dic3—181no—
402Dic3—181no—
403Dic3—182no—
404Dic3—181no—
405Dic3—361no—
406Dic3—361no—
407Dic3—362no—
408Dic3#369912yes—
409Dic3#370112yes—
410C32#572811yescharacteristic
411C9—21no—
412C9#551711yescharacteristic, complemented
413C23—91no—
414Q8—64no—
415Q8—544no—
416C2×C4—31no—
417C2×C4—31no—
418C2×C4—32no—
419C2×C4—91no—
420C2×C4—94no—
421C2×C4—92no—
422C2×C4—271no—
423C2×C4—271no—
424C2×C4—272no—
425C2×C4—542no—
426C2×C4—542no—
427C2×C4C2×D3×D912yes—
428C2×C4C2×D3×D911yescharacteristic
429C2×C3—21no—
430C2×C3—21no—
431C2×C3—21no—
432C2×C3—182no—
433C2×C3C2×#369911yescharacteristic
434C2×C3C2×Q8×D911yescharacteristic
435C2×C3C2×#370111yescharacteristic
436C2×C3C2×D3×Dic611yescharacteristic
437C2×C3D3×#23711yescharacteristic
438C2×C3D3×#23911yescharacteristic
439D3—62no—
440C22—92no—
441C22—181no—
442C22—181no—
443C22C22×D3×D911yescharacteristic
444C4—22no—
445C4—32no—
446C4—32no—
447C4—62no—
448C4—92no—
449C4—182no—
450C4—541no—
451C4—541no—
452C4—542no—
453C4D9×Dic612yes—
454C3—21no—
455C3#1058983011yescharacteristic, complemented
456C3#1059444511yescharacteristic
457C2—182no—
458C2C2×D9×Dic611yescharacteristic
459C2D3×#369911yescharacteristic
460C2D3×#370111yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC22×C3×D9C23
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#533
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11721746C2semidirect
3. C9×#888C2semidirect
7. #11720970C2semidirect
8. #11720937C2semidirect
9. Dic6×Dic9C2semidirect
11. #11720901C2semidirect
71. C3×C4×D9Q8semidirect
158. #888D9semidirect
201. C4×D9Dic6semidirect
304. C3×C9#533semidirect
412. C9#5517semidirect
455. C3#10589830semidirect

Maximal subgroups

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

subgroupindex
#117217462
C3×#634542
C9×#8882
#117209912
#117209842
C4×Dic3×D92
#117209702
#117209372
Dic6×Dic92
#117209072
#117209012
#105944453
#105898303