Finite Group Databasebeta

← #13148941

Subgroups — #13148941

Order 1728.

734 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721744C211yesmaximal, characteristic, complemented
2C3×#63454C211yesmaximal, characteristic
3C9×#890C211yesmaximal, characteristic, complemented
4#11720999C212yesmaximal
5#11720996C212yesmaximal
6C4×Dic3×D9C211yesmaximal, characteristic
7#11720971C211yesmaximal, characteristic, complemented
8#11720927C211yesmaximal, characteristic, complemented
9#11720913C212yesmaximal, complemented
10#11720907C211yesmaximal, characteristic
11#11720901C212yesmaximal, complemented
12#10594477—31nomaximal
13#10589790—31nomaximal
14C2×#6226C2212yes—
15C2×C3×C4×D9C2211yescharacteristic
16C2×Dic3×D9C2211yescharacteristic
17C2×Dic3×D9C2211yescharacteristic
18C2×#6189C2212yes—
19C3×C9×C4⋊C4C2211yescharacteristic
20C4⋊(C3⋊Dic9)C2211yescharacteristic
21#88534C2212yes—
22C4×C3⋊Dic9C2211yescharacteristic
23C3×(C2×D9)⋊C4C2212yes—
24C9×C4⋊Dic3C2211yescharacteristic
25C9×C4⋊Dic3C2212yes—
26C9×Dic3⋊C4C2212yes—
27C4×C9×Dic3C2211yescharacteristic
28C3×C4⋊Dic9C2211yescharacteristic
29C3×C4×Dic9C2212yes—
30(C2×D9)⋊Dic3C2212yes—
31#88442C2212yes—
32#88442C2212yes—
33Dic9⋊Dic3C2212yes—
34Dic3×Dic9C2211yescharacteristic
35Dic3×Dic9C2211yescharacteristic
36Dic3×Dic9C2212yes—
37#64097—31no—
38C3×#892—31no—
39C3×#890D311yescharacteristic
40#63919—31no—
41#63899—32no—
42#63896—32no—
43C4×D3×Dic3—31no—
44#63869—31no—
45#63844—32no—
46#63841—31no—
47#63837—32no—
48C9×#125—31no—
49#63458—32no—
50#63458—32no—
51#63454—31no—
52#63454D311yescharacteristic
53#63450—31no—
54#63450—31no—
55#63450—61no—
56#63449—32no—
57#63449—32no—
58C42×D9—31no—
59#63432—31no—
60C22×C3×D9C2311yescharacteristic
61C2×C3×C4×C9C2312yes—
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3⋊Dic9C2312yes—
64C2×C3⋊Dic9C2311yescharacteristic
65C2×C3⋊Dic9C2311yescharacteristic
66#6226—24no—
67C2×C9×Dic3C2312yes—
68C2×C9×Dic3C2311yescharacteristic
69C2×C9×Dic3C2311yescharacteristic
70C2×C3×Dic9C2312yes—
71C2×C3×Dic9C2311yescharacteristic
72C3×C4×D9Q814yescomplemented
73Dic3×D9—22no—
74Dic3×D9—22no—
75#6189—24no—
76#5688—91no—
77#5517—91no—
78C2×#634—32no—
79C2×C3×C4×D3—31no—
80C2×#627—32no—
81C2×D3×Dic3—31no—
82C2×D3×Dic3—31no—
83C32×C4⋊C4C2×D311yescharacteristic
84C4⋊(C3⋊Dic3)—31no—
85#3750—32no—
86C4×C3⋊Dic3—31no—
87C3×(C2×D3)⋊C4—32no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3C2×D311yescharacteristic
90C3×C4⋊Dic3C2×D312yes—
91C3×Dic3⋊C4C2×D312yes—
92C3×C4×Dic3—32no—
93C3×C4×Dic3C2×D311yescharacteristic
94#3724—32no—
95#3724—32no—
96Dic3⋊Dic3—32no—
97(C2×D3)⋊Dic3—32no—
98Dic32—31no—
99Dic32—32no—
100Dic32—31no—
101C2×C4×D9—31no—
102C2×C4×D9—31no—
103C2×C4×D9C2×D311yescharacteristic
104C2×Dic18—32no—
105C2×Dic18—32no—
106C2×Dic18—62no—
107C9×C4⋊C4—21no—
108C9×C4⋊C4—32no—
109C9×C4⋊C4—32no—
110C9×C4⋊C4—31no—
111C9×C4⋊C4C2×D311yescharacteristic
112C42×C9—31no—
113(C2×D9)⋊C4—32no—
114(C2×D9)⋊C4C2×D312yes—
115C4⋊Dic9—31no—
116C4⋊Dic9—61no—
117C4⋊Dic9C2×D311yescharacteristic
118Dic9⋊C4—32no—
119Dic9⋊C4—32no—
120Dic9⋊C4—32no—
121Dic9⋊C4—32no—
122Dic9⋊C4—62no—
123C4×Dic9—31no—
124C4×Dic9—31no—
125C4×Dic9—31no—
126C4×Dic9—32no—
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#4112yes—
137C3⋊Dic9#4112yes—
138C9×Dic3—22no—
139C9×Dic3—21no—
140C9×Dic3—21no—
141C3×Dic9—22no—
142C3×Dic9C2×Q812yes—
143C3×#125—31no—
144C3×#117—91no—
145#926—91no—
146C2×C4×Dic3—91no—
147#896—92no—
148#896—92no—
149#892—31no—
150#892—91no—
151#890—91no—
152#890D911yescharacteristic, complemented
153#888—91no—
154#888—91no—
155#888—91no—
156#888—91no—
157#888—181no—
158#887—92no—
159#887—92no—
160#878—92no—
161#878—92no—
162C42×D3—91no—
163#870—91no—
164C4×Dic6—92no—
165C4×Dic6—92no—
166C22×C3×D3—31no—
167C2×C32×C4C22×D312yes—
168C2×C32×C4C22×D311yescharacteristic
169C2×C3⋊Dic3—31no—
170C2×C3⋊Dic3—31no—
171C2×C3⋊Dic3—32no—
172#634—64no—
173C2×C3×Dic3—31no—
174C2×C3×Dic3—32no—
175C2×C3×Dic3C22×D312yes—
176C2×C3×Dic3C22×D311yescharacteristic
177C2×C3×Dic3C22×D311yescharacteristic
178C3×C4×D3—34no—
179#627—64no—
180D3×Dic3—62no—
181D3×Dic3—62no—
182C22×D9C22×D311yescharacteristic
183C2×C4×C9—21no—
184C2×C4×C9—22no—
185C2×C4×C9—31no—
186C2×C4×C9—31no—
187C2×C4×C9—32no—
188C2×C4×C9C22×D312yes—
189C2×C4×C9C22×D311yescharacteristic
190C2×Dic9—31no—
191C2×Dic9—31no—
192C2×Dic9—32no—
193C2×Dic9—61no—
194C2×Dic9—61no—
195C2×Dic9—62no—
196C2×Dic9C22×D312yes—
197C2×Dic9C22×D311yescharacteristic
198C4×D9—62no—
199C4×D9—62no—
200C4×D9Dic614yescomplemented
201Dic18—64no—
202Dic18—64no—
203Dic18—124no—
204#533—271noSylow
205C2×C3×C9C22×Q811yescharacteristic
206C2×C3×C9C2×#4111yescharacteristic
207C2×C3×C9C2×#4111yescharacteristic
208C3×D9—22no—
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×Dic6—92no—
218C2×Dic6—92no—
219C2×Dic6—182no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—32no—
222C3×C4⋊C4—32no—
223C3×C4⋊C4—31no—
224C3×C4⋊C4—91no—
225C3×C4⋊C4C2×D911yescharacteristic
226C3×C4⋊C4D3211yescharacteristic
227C3×C22⋊C4—92no—
228C3×C42—31no—
229C3×C42—92no—
230#217—92no—
231(C2×D3)⋊C4—32no—
232(C2×D3)⋊C4—92no—
233C4⋊Dic3—31no—
234C4⋊Dic3—91no—
235C4⋊Dic3—91no—
236C4⋊Dic3—92no—
237C4⋊Dic3—181no—
238C4⋊Dic3C2×D912yes—
239C4⋊Dic3C2×D911yescharacteristic
240Dic3⋊C4—92no—
241Dic3⋊C4—92no—
242Dic3⋊C4—92no—
243Dic3⋊C4—92no—
244Dic3⋊C4—92no—
245Dic3⋊C4—92no—
246Dic3⋊C4—92no—
247Dic3⋊C4—182no—
248Dic3⋊C4C2×D912yes—
249C4×Dic3—32no—
250C4×Dic3—91no—
251C4×Dic3—91no—
252C4×Dic3—92no—
253C4×Dic3—91no—
254C4×Dic3—91no—
255C4×Dic3—91no—
256C4×Dic3—91no—
257C4×Dic3—92no—
258C4×Dic3—181no—
259C4×Dic3C2×D911yescharacteristic
260C22×C32C23×D311yescharacteristic
261C2×C3×D3—32no—
262C2×C3×D3—61no—
263C2×C3×D3—61no—
264C32×C4—22no—
265C32×C4D3×Q812yes—
266C3⋊Dic3—32no—
267C3⋊Dic3—32no—
268C3⋊Dic3—62no—
269C3×Dic3—21no—
270C3×Dic3—21no—
271C3×Dic3—22no—
272C3×Dic3—32no—
273C3×Dic3—62no—
274C22×C9—21no—
275C22×C9C23×D311yescharacteristic
276C2×D9—21no—
277C2×D9—21no—
278C2×D9C2×Dic612yes—
279C4×C9—22no—
280C4×C9—22no—
281C4×C9—42no—
282C4×C9—62no—
283C4×C9—61no—
284C4×C9—61no—
285C4×C9C2×Dic612yes—
286Dic9—22no—
287Dic9—32no—
288Dic9—32no—
289Dic9—62no—
290Dic9—62no—
291Dic9—62no—
292Dic9—122no—
293Dic9C2×Dic612yes—
294C4⋊Q8—271no—
295C4⋊Q8—271no—
296#125—31no—
297#125—271no—
298C22⋊Q8—272no—
299C22⋊Q8—272no—
300C4×Q8—272no—
301C4×Q8—272no—
302#117—91no—
303#117—271no—
304C2×C42—271no—
305C3×C9#53311yescharacteristic, complemented, Sylow
306C23×C3—91no—
307C22×D3—31no—
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—91no—
325C2×Dic3—91no—
326C2×Dic3—92no—
327C2×Dic3—92no—
328C2×Dic3—181no—
329C2×Dic3—181no—
330C2×Dic3—182no—
331C2×Dic3—182no—
332C2×Dic3—182no—
333C2×Dic3C22×D911yescharacteristic
334C2×Dic3C22×D911yescharacteristic
335C2×Dic3C22×D912yes—
336C4×D3—34no—
337C4×D3—182no—
338C4×D3—182no—
339Dic6—184no—
340Dic6—184no—
341Dic6—184no—
342Dic6—184no—
343Dic6—364no—
344C2×C32C2×D3×Q811yescharacteristic
345C2×C32D3×#4111yescharacteristic
346C2×C32D3×#4111yescharacteristic
347C3×D3—62no—
348C2×C9—21no—
349C2×C9—21no—
350C2×C9—21no—
351C2×C9C22×Dic611yescharacteristic
352C2×C9D3×#4111yescharacteristic
353C2×C9D3×#4111yescharacteristic
354D9—22no—
355C2×Q8—272no—
356C2×Q8—272no—
357C22×C4—91no—
358C22×C4—271no—
359C22×C4—271no—
360C4⋊C4—31no—
361C4⋊C4—32no—
362C4⋊C4—32no—
363C4⋊C4—91no—
364C4⋊C4—272no—
365C4⋊C4—272no—
366C4⋊C4—272no—
367C4⋊C4—271no—
368C4⋊C4—272no—
369C4⋊C4D3×D911yescharacteristic
370C22⋊C4—92no—
371C22⋊C4—272no—
372C42—31no—
373C42—92no—
374C42—271no—
375C42—271no—
376C42—272no—
377C42—271no—
378C22×C3—21no—
379C22×C3—92no—
380C22×C3—181no—
381C22×C3—181no—
382C22×C3C23×D911yescharacteristic
383C22×C3C22×D3211yescharacteristic
384C2×D3—32no—
385C2×D3—61no—
386C2×D3—61no—
387C3×C4—22no—
388C3×C4—22no—
389C3×C4—22no—
390C3×C4—42no—
391C3×C4—61no—
392C3×C4—62no—
393C3×C4—61no—
394C3×C4—92no—
395C3×C4—182no—
396C3×C4Q8×D912yes—
397C3×C4D3×Dic612yes—
398Dic3—22no—
399Dic3—21no—
400Dic3—21no—
401Dic3—32no—
402Dic3—62no—
403Dic3—92no—
404Dic3—92no—
405Dic3—92no—
406Dic3—92no—
407Dic3—182no—
408Dic3—182no—
409Dic3—182no—
410Dic3—182no—
411Dic3—362no—
412C32#568811yescharacteristic
413C9—21no—
414C9#551711yescharacteristic, complemented
415C23—91no—
416Q8—544no—
417Q8—544no—
418C2×C4—31no—
419C2×C4—31no—
420C2×C4—32no—
421C2×C4—91no—
422C2×C4—94no—
423C2×C4—92no—
424C2×C4—271no—
425C2×C4—271no—
426C2×C4—272no—
427C2×C4—542no—
428C2×C4—542no—
429C2×C4C2×D3×D912yes—
430C2×C4C2×D3×D911yescharacteristic
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—182no—
435C2×C3C2×Q8×D911yescharacteristic
436C2×C3#41×D911yescharacteristic
437C2×C3#41×D911yescharacteristic
438C2×C3C2×D3×Dic611yescharacteristic
439C2×C3#6431211yescharacteristic
440C2×C3#6431911yescharacteristic
441D3—62no—
442C22—92no—
443C22—181no—
444C22—181no—
445C22C22×D3×D911yescharacteristic
446C4—22no—
447C4—61no—
448C4—61no—
449C4—62no—
450C4—92no—
451C4—182no—
452C4—272no—
453C4—272no—
454C4—542no—
455C4D9×Dic612yes—
456C3—21no—
457C3#1058979011yescharacteristic, complemented
458C3#1059447711yescharacteristic
459C2—182no—
460C2C2×D9×Dic611yescharacteristic
461C2#1172249811yescharacteristic
462C2#1172251411yescharacteristic
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. #11721744C2semidirect
3. C9×#890C2semidirect
7. #11720971C2semidirect
8. #11720927C2semidirect
9. #11720913C2semidirect
11. #11720901C2semidirect
72. C3×C4×D9Q8semidirect
152. #890D9semidirect
200. C4×D9Dic6semidirect
305. C3×C9#533semidirect
414. C9#5517semidirect
457. C3#10589790semidirect

Maximal subgroups

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

subgroupindex
#117217442
C3×#634542
C9×#8902
#117209992
#117209962
C4×Dic3×D92
#117209712
#117209272
#117209132
#117209072
#117209012
#105944773
#105897903