Finite Group Databasebeta

← #13148842

Subgroups — #13148842

Order 1728.

736 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
1#11721743C211yesmaximal, characteristic, complemented
2C3×#63459C211yesmaximal, characteristic
3C9×#888C211yesmaximal, characteristic, complemented
4#11720984C212yesmaximal
5#11720975C212yesmaximal
6C4×Dic3×D9C211yesmaximal, characteristic
7#11720945C211yesmaximal, characteristic
8#11720932C211yesmaximal, characteristic, complemented
9#11720916C212yesmaximal, complemented
10#11720911C212yesmaximal, complemented
11#11720910C211yesmaximal, characteristic, complemented
12#10594387—31nomaximal
13#10589830—31nomaximal
14C2×#6226C2211yescharacteristic
15C2×C3×C4×D9C2211yescharacteristic
16C2×C9×Dic6C2212yes—
17C2×C3×Dic18C2211yescharacteristic
18C2×Dic3×D9C2211yescharacteristic
19C2×Dic3×D9C2211yescharacteristic
20C3×C9×C4⋊C4C2211yescharacteristic
21#88534C2212yes—
22C4×C3⋊Dic9C2211yescharacteristic
23C4×C3⋊Dic9C2212yes—
24C3×(C2×D9)⋊C4C2212yes—
25C9×C4⋊Dic3C2211yescharacteristic
26C9×Dic3⋊C4C2212yes—
27C4×C9×Dic3C2211yescharacteristic
28C3×C4⋊Dic9C2212yes—
29(C2×D9)⋊Dic3C2212yes—
30Dic3⋊Dic9C2212yes—
31Dic3⋊Dic9C2212yes—
32Dic3⋊Dic9C2212yes—
33Dic9⋊Dic3C2212yes—
34Dic3×Dic9C2211yescharacteristic
35Dic3×Dic9C2211yescharacteristic
36Dic3×Dic9C2212yes—
37#64096—31no—
38C3×#897—31no—
39C3×#888D311yescharacteristic
40#63887—32no—
41#63878—32no—
42C4×D3×Dic3—31no—
43#63866—31no—
44#63857—31no—
45#63846—32no—
46#63843—32no—
47#63843—31no—
48C9×C4⋊Q8—31no—
49#63508—32no—
50Q8×Dic9—32no—
51#63459—31no—
52#63459D311yescharacteristic
53#63454—32no—
54#63452—32no—
55#63449—31no—
56#63449—31no—
57#63449—61no—
58C42×D9—31no—
59#63431—31no—
60C22×C3×D9C2311yescharacteristic
61C2×C3×C4×C9C2312yes—
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3⋊Dic9C2311yescharacteristic
64C2×C3⋊Dic9C2312yes—
65C2×C3⋊Dic9C2311yescharacteristic
66#6226—22no—
67C2×C9×Dic3C2312yes—
68C2×C9×Dic3C2311yescharacteristic
69C2×C9×Dic3C2311yescharacteristic
70C2×C3×Dic9C2312yes—
71C2×C3×Dic9C2311yescharacteristic
72C3×C4×D9—22no—
73C9×Dic6—24no—
74C3×Dic18—22no—
75Dic3×D9—22no—
76Dic3×D9Q814yescomplemented
77#5728—91no—
78#5622—91no—
79C2×#634—31no—
80C2×C3×C4×D3—31no—
81C2×C3×Dic6—31no—
82C2×C3×Dic6C2×D312yes—
83C2×D3×Dic3—31no—
84C2×D3×Dic3—31no—
85C32×C4⋊C4C2×D311yescharacteristic
86#3750—32no—
87C4×C3⋊Dic3—31no—
88C4×C3⋊Dic3—32no—
89C3×(C2×D3)⋊C4—32no—
90C3×C4⋊Dic3—32no—
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×Dic3⋊C4C2×D312yes—
93C3×C4×Dic3C2×D311yescharacteristic
94Dic3⋊Dic3—32no—
95Dic3⋊Dic3—32no—
96Dic3⋊Dic3—32no—
97Dic3⋊Dic3—32no—
98(C2×D3)⋊Dic3—32no—
99Dic32—31no—
100Dic32—32no—
101Dic32—31no—
102C2×Q8×C9—32no—
103C2×C4×D9—31no—
104C2×C4×D9—31no—
105C2×C4×D9C2×D311yescharacteristic
106C2×Dic18—31no—
107C2×Dic18—61no—
108C2×Dic18C2×D311yescharacteristic
109C9×C4⋊C4—21no—
110C9×C4⋊C4—32no—
111C9×C4⋊C4—31no—
112C9×C4⋊C4C2×D311yescharacteristic
113C42×C9—31no—
114(C2×D9)⋊C4—32no—
115(C2×D9)⋊C4C2×D312yes—
116C4⋊Dic9—32no—
117C4⋊Dic9—32no—
118C4⋊Dic9—32no—
119C4⋊Dic9C2×D312yes—
120Dic9⋊C4—32no—
121Dic9⋊C4—32no—
122Dic9⋊C4—62no—
123C4×Dic9—32no—
124C4×Dic9—31no—
125C4×Dic9—31no—
126C4×Dic9—32no—
127C4×Dic9—31no—
128C4×Dic9—62no—
129C4×Dic9—61no—
130C22×C3×C9C2411yescharacteristic
131C2×C3×D9—21no—
132C2×C3×D9—21no—
133C2×C3×D9C2×Q812yes—
134C3×C4×C9—22no—
135C3×C4×C9#4112yes—
136C3⋊Dic9—22no—
137C3⋊Dic9—21no—
138C3⋊Dic9C2×Q812yes—
139C9×Dic3—22no—
140C9×Dic3C2×Q812yes—
141C9×Dic3#4112yes—
142C3×Dic9—22no—
143C3×Dic9—21no—
144C3×C4⋊Q8—31no—
145C3×C22⋊Q8—91no—
146#946—92no—
147Q8×Dic3—92no—
148Q8×Dic3—91no—
149#944—91no—
150#924—91no—
151C2×C4×Dic3—91no—
152#897—31no—
153#897—91no—
154#892—92no—
155#890—92no—
156#889—92no—
157#888D911yescharacteristic, complemented
158#887—91no—
159#887—92no—
160#887—91no—
161#887—91no—
162#887—181no—
163#878—92no—
164#877—92no—
165C42×D3—91no—
166#869—91no—
167C22×C3×D3—31no—
168C2×C32×C4C22×D312yes—
169C2×C32×C4C22×D311yescharacteristic
170C2×C3⋊Dic3—31no—
171C2×C3⋊Dic3—31no—
172C2×C3⋊Dic3—32no—
173#634—62no—
174C2×C3×Dic3—31no—
175C2×C3×Dic3—32no—
176C2×C3×Dic3C22×D312yes—
177C2×C3×Dic3C22×D311yescharacteristic
178C2×C3×Dic3C22×D311yescharacteristic
179C3×C4×D3—62no—
180C3×Dic6—24no—
181C3×Dic6—62no—
182D3×Dic3—34no—
183D3×Dic3—62no—
184C22×D9C22×D311yescharacteristic
185Q8×C9—64no—
186C2×C4×C9—21no—
187C2×C4×C9—22no—
188C2×C4×C9—31no—
189C2×C4×C9—31no—
190C2×C4×C9—32no—
191C2×C4×C9C22×D312yes—
192C2×C4×C9C22×D311yescharacteristic
193C2×Dic9—31no—
194C2×Dic9—31no—
195C2×Dic9—32no—
196C2×Dic9—61no—
197C2×Dic9—62no—
198C2×Dic9—61no—
199C2×Dic9C22×D311yescharacteristic
200C2×Dic9C22×D312yes—
201C4×D9—22no—
202C4×D9—34no—
203C4×D9—62no—
204Dic18—22no—
205Dic18—64no—
206Dic18—62no—
207#533—271noSylow
208C2×C3×C9C22×Q811yescharacteristic
209C2×C3×C9C2×#4111yescharacteristic
210C2×C3×C9C2×#4111yescharacteristic
211C3×D9—22no—
212C2×C3×Q8—32no—
213C2×C3×Q8—91no—
214C22×C3×C4—91no—
215C22×Dic3—91no—
216C22×Dic3—91no—
217C2×C4×D3—31no—
218C2×C4×D3—91no—
219C2×C4×D3—91no—
220C2×Dic6—31no—
221C2×Dic6—91no—
222C2×Dic6—91no—
223C2×Dic6—181no—
224C2×Dic6C2×D912yes—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—31no—
227C3×C4⋊C4—32no—
228C3×C4⋊C4—92no—
229C3×C4⋊C4C2×D911yescharacteristic
230C3×C4⋊C4D3211yescharacteristic
231C3×C22⋊C4—92no—
232C3×C42—31no—
233#217—92no—
234(C2×D3)⋊C4—32no—
235(C2×D3)⋊C4—92no—
236C4⋊Dic3—32no—
237C4⋊Dic3—92no—
238C4⋊Dic3—92no—
239C4⋊Dic3—92no—
240C4⋊Dic3—92no—
241C4⋊Dic3C2×D911yescharacteristic
242Dic3⋊C4—92no—
243Dic3⋊C4—92no—
244Dic3⋊C4—92no—
245Dic3⋊C4—92no—
246Dic3⋊C4—92no—
247Dic3⋊C4—92no—
248Dic3⋊C4—182no—
249Dic3⋊C4C2×D912yes—
250C4×Dic3—91no—
251C4×Dic3—91no—
252C4×Dic3—92no—
253C4×Dic3—92no—
254C4×Dic3—91no—
255C4×Dic3—91no—
256C4×Dic3—92no—
257C4×Dic3—92no—
258C4×Dic3—91no—
259C4×Dic3—91no—
260C4×Dic3—181no—
261C4×Dic3—182no—
262C4×Dic3C2×D911yescharacteristic
263C22×C32C23×D311yescharacteristic
264C2×C3×D3—32no—
265C2×C3×D3—61no—
266C2×C3×D3—61no—
267C32×C4—22no—
268C32×C4#23912yes—
269C3⋊Dic3—32no—
270C3⋊Dic3—62no—
271C3⋊Dic3—61no—
272C3×Dic3—22no—
273C3×Dic3—62no—
274C3×Dic3—61no—
275C3×Dic3#23712yes—
276C3×Dic3D3×Q812yes—
277C22×C9—21no—
278C22×C9C23×D311yescharacteristic
279C2×D9—21no—
280C2×D9—21no—
281C2×D9D3×Q812yes—
282C4×C9—22no—
283C4×C9—22no—
284C4×C9—32no—
285C4×C9—32no—
286C4×C9—42no—
287C4×C9—62no—
288C4×C9#23712yes—
289Dic9—22no—
290Dic9—21no—
291Dic9—32no—
292Dic9—62no—
293Dic9—64no—
294Dic9—62no—
295Dic9—61no—
296Dic9—121no—
297C4⋊Q8—31no—
298C4⋊Q8—271no—
299#125—272no—
300C22⋊Q8—91no—
301C22⋊Q8—272no—
302C22⋊Q8—271no—
303C4×Q8—271no—
304C4×Q8—272no—
305C4×Q8—271no—
306#117—272no—
307C2×C42—271no—
308C3×C9#53311yescharacteristic, complemented, Sylow
309C23×C3—91no—
310C22×D3—31no—
311C3×Q8—64no—
312C3×Q8—182no—
313C2×C3×C4—21no—
314C2×C3×C4—22no—
315C2×C3×C4—31no—
316C2×C3×C4—31no—
317C2×C3×C4—32no—
318C2×C3×C4—91no—
319C2×C3×C4—92no—
320C2×C3×C4—182no—
321C2×C3×C4C22×D912yes—
322C2×C3×C4C22×D911yescharacteristic
323C2×C3×C4C2×D3212yes—
324C2×C3×C4C2×D3211yescharacteristic
325C2×Dic3—31no—
326C2×Dic3—32no—
327C2×Dic3—91no—
328C2×Dic3—91no—
329C2×Dic3—94no—
330C2×Dic3—91no—
331C2×Dic3—92no—
332C2×Dic3—91no—
333C2×Dic3—92no—
334C2×Dic3—181no—
335C2×Dic3—182no—
336C2×Dic3—181no—
337C2×Dic3—182no—
338C2×Dic3C22×D912yes—
339C2×Dic3C22×D911yescharacteristic
340C2×Dic3C22×D911yescharacteristic
341C4×D3—62no—
342C4×D3—94no—
343C4×D3—182no—
344Dic6—24no—
345Dic6—62no—
346Dic6—184no—
347Dic6—182no—
348Dic6—182no—
349C2×C32C2×#23711yescharacteristic
350C2×C32C2×D3×Q811yescharacteristic
351C2×C32C2×#23911yescharacteristic
352C3×D3—62no—
353C2×C9—21no—
354C2×C9—21no—
355C2×C9—21no—
356C2×C9C2×#23711yescharacteristic
357C2×C9C2×D3×Q811yescharacteristic
358C2×C9D3×#4111yescharacteristic
359D9—22no—
360C2×Q8—32no—
361C2×Q8—91no—
362C2×Q8—271no—
363C22×C4—91no—
364C22×C4—271no—
365C22×C4—271no—
366C4⋊C4—32no—
367C4⋊C4—31no—
368C4⋊C4—92no—
369C4⋊C4—272no—
370C4⋊C4—272no—
371C4⋊C4—272no—
372C4⋊C4—272no—
373C4⋊C4—272no—
374C4⋊C4D3×D911yescharacteristic
375C22⋊C4—92no—
376C22⋊C4—272no—
377C42—31no—
378C42—271no—
379C42—271no—
380C42—272no—
381C42—271no—
382C42—272no—
383C22×C3—21no—
384C22×C3—92no—
385C22×C3—181no—
386C22×C3—181no—
387C22×C3C23×D911yescharacteristic
388C22×C3C22×D3211yescharacteristic
389C2×D3—32no—
390C2×D3—61no—
391C2×D3—61no—
392C3×C4—22no—
393C3×C4—22no—
394C3×C4—22no—
395C3×C4—32no—
396C3×C4—32no—
397C3×C4—42no—
398C3×C4—62no—
399C3×C4—181no—
400C3×C4—182no—
401C3×C4#370112yes—
402C3×C4#379612yes—
403Dic3—22no—
404Dic3—62no—
405Dic3—61no—
406Dic3—92no—
407Dic3—92no—
408Dic3—182no—
409Dic3—184no—
410Dic3—182no—
411Dic3—181no—
412Dic3—181no—
413Dic3—182no—
414Dic3—361no—
415Dic3#369912yes—
416Dic3Q8×D912yes—
417C32#572811yescharacteristic
418C9—21no—
419C9#562211yescharacteristic, complemented
420C23—91no—
421Q8—64no—
422Q8—182no—
423Q8—542no—
424C2×C4—31no—
425C2×C4—31no—
426C2×C4—32no—
427C2×C4—91no—
428C2×C4—92no—
429C2×C4—182no—
430C2×C4—274no—
431C2×C4—271no—
432C2×C4—271no—
433C2×C4—272no—
434C2×C4—542no—
435C2×C4C2×D3×D912yes—
436C2×C4C2×D3×D911yescharacteristic
437C2×C3—21no—
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—182no—
441C2×C3C2×#369911yescharacteristic
442C2×C3C2×Q8×D911yescharacteristic
443C2×C3C2×#370111yescharacteristic
444C2×C3C2×#379611yescharacteristic
445C2×C3D3×#23711yescharacteristic
446C2×C3D32×Q811yescharacteristic
447D3—62no—
448C22—92no—
449C22—181no—
450C22—181no—
451C22C22×D3×D911yescharacteristic
452C4—22no—
453C4—32no—
454C4—32no—
455C4—62no—
456C4—181no—
457C4—182no—
458C4—272no—
459C4—541no—
460C4—542no—
461C4#8863412yes—
462C3—21no—
463C3#1058983011yescharacteristic, complemented
464C3#1059438711yescharacteristic
465C2—182no—
466C2C2×#8863411yescharacteristic
467C2D3×#369911yescharacteristic
468C2D3×Q8×D911yescharacteristic
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. #11721743C2semidirect
3. C9×#888C2semidirect
8. #11720932C2semidirect
9. #11720916C2semidirect
10. #11720911C2semidirect
11. #11720910C2semidirect
76. Dic3×D9Q8semidirect
157. #888D9semidirect
308. C3×C9#533semidirect
419. C9#5622semidirect
463. C3#10589830semidirect

Maximal subgroups

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

subgroupindex
#117217432
C3×#634592
C9×#8882
#117209842
#117209752
C4×Dic3×D92
#117209452
#117209322
#117209162
#117209112
#117209102
#105943873
#105898303