Finite Group Databasebeta

← #13153918

Subgroups — #13153918

Order 1728.

1470 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
1C2×#88742C211yesmaximal, characteristic
2C3×C4×D4×C9C211yesmaximal, characteristic, complemented
3D4×C3⋊Dic9C211yesmaximal, characteristic, complemented
4C4×#6233C212yesmaximal, complemented
5C2×#88534C212yesmaximal, complemented
6C4⋊C4×C3⋊D9C211yesmaximal, characteristic, complemented
7#11721743C211yesmaximal, characteristic, complemented
8#11721727C212yesmaximal, complemented
9#11721723C212yesmaximal, complemented
10#11721709C211yesmaximal, characteristic, complemented
11C4×#6226C211yesmaximal, characteristic, complemented
12#10596017—31nomaximal
13#10589648—33nomaximal
14C2×C3×D4×C9C2211yescharacteristic
15C22×C3×C4×C9C2212yes—
16C2×#6233C2212yes—
17C22×C3⋊Dic9C2212yes—
18C22×C3⋊Dic9C2212yes—
19#88742C418yescomplemented
20C2×C4×C3⋊D9C2212yes—
21C2×C4×C3⋊D9C2211yescharacteristic
22C2×#6226C2211yescharacteristic
23C3×C9×C4⋊C4C2211yescharacteristic
24C3×C9×C22⋊C4C2212yes—
25C3×C42×C9C2211yescharacteristic
26#88551C2212yes—
27#88536C2211yescharacteristic
28#88536C2211yescharacteristic
29C4⋊(C3⋊Dic9)C2211yescharacteristic
30#88534C2212yes—
31#88534C2212yes—
32#88534C2211yescharacteristic
33#88534C2212yes—
34#88534C2211yescharacteristic
35C4×C3⋊Dic9C2212yes—
36C4×C3⋊Dic9C2211yescharacteristic
37C4×C3⋊Dic9C2212yes—
38C2×#3830—31no—
39C32×C4×D4D311yescharacteristic
40D4×C3⋊Dic3—31no—
41C4×#638—32no—
42C2×#3750—32no—
43C4⋊C4×C3⋊D3—31no—
44#64096—31no—
45#64090—32no—
46#64086—32no—
47#64081—31no—
48C4×#634—31no—
49C2×#3699—33no—
50C4×D4×C9D313yescomplemented
51D4×Dic9—33no—
52C4×#611—36no—
53C2×Dic9⋊C4—36no—
54C4⋊C4×D9—33no—
55#63449—33no—
56#611⋊C4—36no—
57#63439—36no—
58#63434—33no—
59C4×Dic18—33no—
60C23×C3×C9C2312yes—
61C22×C3⋊D9C2311yescharacteristic
62C3×D4×C9C2×C414yescomplemented
63C2×C3×C4×C9—42no—
64C2×C3×C4×C9C2312yes—
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3×C4×C9C2311yescharacteristic
67C2×C3×C4×C9C2311yescharacteristic
68#6233C2×C418yescomplemented
69C2×C3⋊Dic9—42no—
70C2×C3⋊Dic9C2×C418yescomplemented
71C2×C3⋊Dic9C2312yes—
72C2×C3⋊Dic9C2311yescharacteristic
73C2×C3⋊Dic9C2311yescharacteristic
74C2×C3⋊Dic9C2312yes—
75C2×C3⋊Dic9C2311yescharacteristic
76C4×C3⋊D9—42no—
77C4×C3⋊D9C2×C414yescomplemented
78#6226C2×C414yescomplemented
79#5546—93no—
80#5546—91no—
81C2×C32×D4C2×D311yescharacteristic
82C22×C32×C4C2×D312yes—
83C2×#638—32no—
84C22×C3⋊Dic3—32no—
85C22×C3⋊Dic3—32no—
86#3830—38no—
87C2×C4×C3⋊D3—31no—
88C2×C4×C3⋊D3—32no—
89C2×#634—31no—
90C32×C4⋊C4C2×D311yescharacteristic
91C32×C22⋊C4C2×D312yes—
92C32×C42C2×D311yescharacteristic
93#3757—32no—
94#3752—31no—
95#3752—31no—
96C4⋊(C3⋊Dic3)—31no—
97#3750—32no—
98#3750—31no—
99#3750—31no—
100#3750—32no—
101#3750—32no—
102C4×C3⋊Dic3—32no—
103C4×C3⋊Dic3—31no—
104C4×C3⋊Dic3—32no—
105C2×D4×C9C2×D313yes—
106C22×C4×C9C2×D316yes—
107C2×#611—36no—
108C22×Dic9—36no—
109C22×Dic9—36no—
110#3699—324no—
111C2×C4×D9—33no—
112C2×C4×D9—36no—
113C2×Dic18—33no—
114C9×C4⋊C4C2×D313yes—
115C9×C22⋊C4C2×D316yes—
116C42×C9C2×D313yes—
117#3676—36no—
118(C2×D9)⋊C4—33no—
119(C2×D9)⋊C4—33no—
120C4⋊Dic9—33no—
121Dic9⋊C4—36no—
122Dic9⋊C4—33no—
123Dic9⋊C4—33no—
124Dic9⋊C4—36no—
125Dic9⋊C4—36no—
126C4×Dic9—36no—
127C4×Dic9—33no—
128C4×Dic9—36no—
129C22×C3×C9—22no—
130C22×C3×C9—22no—
131C22×C3×C9C22×C414yes—
132C22×C3×C9C2411yescharacteristic
133C2×C3⋊D9—21no—
134C2×C3⋊D9—21no—
135C2×C3⋊D9C22×C412yes—
136C3×C4×C9—22no—
137C3×C4×C9—21no—
138C3×C4×C9—21no—
139C3×C4×C9C22×C412yes—
140C3⋊Dic9—22no—
141C3⋊Dic9—21no—
142C3⋊Dic9—21no—
143C3⋊Dic9C22×C414yes—
144C3⋊Dic9C22×C412yes—
145C2×#237—93no—
146C2×#237—91no—
147C3×C4×D4D913yescomplemented
148C3×C4×D4C3⋊D311yescharacteristic
149D4×Dic3—93no—
150D4×Dic3—91no—
151C4×#67—96no—
152C4×#67—92no—
153C2×Dic3⋊C4—96no—
154C2×Dic3⋊C4—92no—
155D3×C4⋊C4—93no—
156D3×C4⋊C4—91no—
157#887—93no—
158#887—91no—
159#67⋊C4—96no—
160#67⋊C4—92no—
161#877—96no—
162#877—92no—
163#872—93no—
164#872—91no—
165C4×Dic6—93no—
166C4×Dic6—91no—
167C23×C32C22×D312yes—
168C22×C3⋊D3—31no—
169C32×D4C4×D314yes—
170C2×C32×C4—42no—
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D312yes—
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175#638—38no—
176C2×C3⋊Dic3—38no—
177C2×C3⋊Dic3—32no—
178C2×C3⋊Dic3—31no—
179C2×C3⋊Dic3—31no—
180C2×C3⋊Dic3—31no—
181C2×C3⋊Dic3—32no—
182C2×C3⋊Dic3—122no—
183C4×C3⋊D3—34no—
184C4×C3⋊D3—122no—
185#634—34no—
186C23×C9C22×D316yes—
187C22×D9—33no—
188D4×C9C4×D3112yescomplemented
189C2×C4×C9—46no—
190C2×C4×C9C22×D313yes—
191C2×C4×C9C22×D316yes—
192C2×C4×C9C22×D313yes—
193C2×C4×C9C22×D313yes—
194#611—324no—
195C2×Dic9—324no—
196C2×Dic9—36no—
197C2×Dic9—33no—
198C2×Dic9—33no—
199C2×Dic9—33no—
200C2×Dic9—36no—
201C2×Dic9—126no—
202C4×D9—312no—
203C4×D9—126no—
204Dic18—312no—
205#520—271noSylow
206C2×C3×C9—24no—
207C2×C3×C9C23×C411yescharacteristic
208C2×C3×C9H(5,𝔽2)11yescharacteristic
209C2×C3×C92−1+411yescharacteristic
210C3⋊D9—22no—
211C2×C3×D4C2×D913yes—
212C2×C3×D4C2×C3⋊D311yescharacteristic
213C22×C3×C4C2×D916yes—
214C22×C3×C4C2×C3⋊D312yes—
215C2×#67—96no—
216C2×#67—92no—
217C22×Dic3—96no—
218C22×Dic3—92no—
219C22×Dic3—96no—
220C22×Dic3—92no—
221#237—924no—
222#237—98no—
223C2×C4×D3—93no—
224C2×C4×D3—91no—
225C2×C4×D3—96no—
226C2×C4×D3—92no—
227C2×Dic6—93no—
228C2×Dic6—91no—
229C3×C4⋊C4C2×D913yes—
230C3×C4⋊C4C2×C3⋊D311yescharacteristic
231C3×C22⋊C4C2×D916yes—
232C3×C22⋊C4C2×C3⋊D312yes—
233C3×C42C2×D913yes—
234C3×C42C2×C3⋊D311yescharacteristic
235#217—96no—
236#217—92no—
237(C2×D3)⋊C4—93no—
238(C2×D3)⋊C4—93no—
239(C2×D3)⋊C4—91no—
240(C2×D3)⋊C4—91no—
241C4⋊Dic3—93no—
242C4⋊Dic3—91no—
243Dic3⋊C4—96no—
244Dic3⋊C4—93no—
245Dic3⋊C4—93no—
246Dic3⋊C4—96no—
247Dic3⋊C4—96no—
248Dic3⋊C4—92no—
249Dic3⋊C4—91no—
250Dic3⋊C4—91no—
251Dic3⋊C4—92no—
252Dic3⋊C4—92no—
253C4×Dic3—96no—
254C4×Dic3—93no—
255C4×Dic3—96no—
256C4×Dic3—92no—
257C4×Dic3—91no—
258C4×Dic3—92no—
259C22×C32—22no—
260C22×C32—22no—
261C22×C32C2×C4×D314yes—
262C22×C32C23×D311yescharacteristic
263C2×C3⋊D3—32no—
264C2×C3⋊D3—61no—
265C2×C3⋊D3—61no—
266C32×C4—22no—
267C32×C4—21no—
268C32×C4—21no—
269C32×C4C2×C4×D312yes—
270C3⋊Dic3—34no—
271C3⋊Dic3—32no—
272C3⋊Dic3—62no—
273C3⋊Dic3—61no—
274C3⋊Dic3—61no—
275C22×C9—26no—
276C22×C9—26no—
277C22×C9C2×C4×D3112yes—
278C22×C9C23×D313yes—
279C2×D9—36no—
280C2×D9—63no—
281C2×D9—63no—
282C4×C9—26no—
283C4×C9—23no—
284C4×C9—23no—
285C4×C9C2×C4×D316yes—
286Dic9—312no—
287Dic9—36no—
288Dic9—66no—
289Dic9—63no—
290Dic9—63no—
291C2×#41—271no—
292C4×Q8—271no—
293C4×Q8—271no—
294C4×D4—271no—
295C4×D4—272no—
296C4×D4—272no—
297C4×D4C3⋊D911yescharacteristic, complemented
298#117—272no—
299#117—271no—
300C2×C4⋊C4—272no—
301C2×C4⋊C4—271no—
302C3×C9#52011yescharacteristic, complemented, Sylow
303C23×C3C22×D916yes—
304C23×C3C22×C3⋊D312yes—
305C22×D3—91no—
306C22×D3—93no—
307C3×D4C4×D9112yescomplemented
308C3×D4C4×C3⋊D314yes—
309C2×C3×C4—42no—
310C2×C3×C4—46no—
311C2×C3×C4C22×D913yes—
312C2×C3×C4C22×D916yes—
313C2×C3×C4C22×D913yes—
314C2×C3×C4C22×D913yes—
315C2×C3×C4C22×C3⋊D311yescharacteristic
316C2×C3×C4C22×C3⋊D312yes—
317C2×C3×C4C22×C3⋊D311yescharacteristic
318C2×C3×C4C22×C3⋊D311yescharacteristic
319#67—924no—
320#67—98no—
321C2×Dic3—92no—
322C2×Dic3—91no—
323C2×Dic3—924no—
324C2×Dic3—96no—
325C2×Dic3—93no—
326C2×Dic3—98no—
327C2×Dic3—91no—
328C2×Dic3—91no—
329C2×Dic3—92no—
330C2×Dic3—93no—
331C2×Dic3—93no—
332C2×Dic3—96no—
333C2×Dic3—362no—
334C2×Dic3—366no—
335C4×D3—912no—
336C4×D3—94no—
337C4×D3—362no—
338C4×D3—366no—
339Dic6—912no—
340Dic6—94no—
341C2×C32—24no—
342C2×C32C22×C4×D311yescharacteristic
343C2×C32#100411yescharacteristic
344C2×C32#101011yescharacteristic
345C3⋊D3—62no—
346C2×C9—212no—
347C2×C9C22×C4×D313yes—
348C2×C9#100413yes—
349C2×C9#101013yes—
350D9—66no—
351#41—278no—
352C2×Q8—271no—
353C2×D4—272no—
354C2×D4C2×C3⋊D911yescharacteristic
355C22×C4—272no—
356C22×C4—271no—
357C22×C4—272no—
358C22×C4—272no—
359C22×C4C2×C3⋊D912yes—
360C4⋊C4—272no—
361C4⋊C4—271no—
362C4⋊C4—272no—
363C4⋊C4—272no—
364C4⋊C4—271no—
365C4⋊C4—271no—
366C4⋊C4C2×C3⋊D911yescharacteristic
367C22⋊C4—272no—
368C22⋊C4—271no—
369C22⋊C4—271no—
370C22⋊C4C2×C3⋊D912yes—
371C42—272no—
372C42—272no—
373C42—271no—
374C42C2×C3⋊D911yescharacteristic
375C22×C3—22no—
376C22×C3—22no—
377C22×C3—26no—
378C22×C3—26no—
379C22×C3C2×C4×D9112yes—
380C22×C3C23×D913yes—
381C22×C3C2×C4×C3⋊D314yes—
382C22×C3C23×C3⋊D311yescharacteristic
383C2×D3—92no—
384C2×D3—96no—
385C2×D3—181no—
386C2×D3—181no—
387C2×D3—183no—
388C2×D3—183no—
389C3×C4—22no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—26no—
393C3×C4—23no—
394C3×C4—23no—
395C3×C4C2×C4×D916yes—
396C3×C4C2×C4×C3⋊D312yes—
397Dic3—94no—
398Dic3—92no—
399Dic3—912no—
400Dic3—96no—
401Dic3—182no—
402Dic3—181no—
403Dic3—181no—
404Dic3—186no—
405Dic3—183no—
406Dic3—183no—
407C32#554611yescharacteristic
408C9#554613yescomplemented
409C23—271no—
410C23C22×C3⋊D912yes—
411Q8—274no—
412D4—278no—
413D4C4×C3⋊D914yescomplemented
414C2×C4—42no—
415C2×C4—272no—
416C2×C4—271no—
417C2×C4—278no—
418C2×C4—274no—
419C2×C4—271no—
420C2×C4—272no—
421C2×C4—271no—
422C2×C4—1082no—
423C2×C4—1082no—
424C2×C4C22×C3⋊D911yescharacteristic
425C2×C4C22×C3⋊D912yes—
426C2×C4C22×C3⋊D911yescharacteristic
427C2×C4C22×C3⋊D911yescharacteristic
428C2×C3—24no—
429C2×C3—212no—
430C2×C3C22×C4×D913yes—
431C2×C3#6371013yes—
432C2×C3#6371613yes—
433C2×C3C22×C4×C3⋊D311yescharacteristic
434C2×C3#6436111yescharacteristic
435C2×C3#6436711yescharacteristic
436D3—182no—
437D3—186no—
438C22—22no—
439C22—22no—
440C22—272no—
441C22—541no—
442C22—541no—
443C22C2×C4×C3⋊D914yes—
444C22C23×C3⋊D911yescharacteristic
445C4—21no—
446C4—22no—
447C4—21no—
448C4—272no—
449C4—274no—
450C4—542no—
451C4—541no—
452C4—541no—
453C4C2×C4×C3⋊D912yes—
454C3#1058964813yescomplemented
455C3#1059601711yescharacteristic
456C2—24no—
457C2—542no—
458C2C22×C4×C3⋊D911yescharacteristic
459C2#1172266111yescharacteristic
460C2#1172267311yescharacteristic
subgroupG/H
centerC22C23×C3⋊D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C23×C3⋊D3
FittingC3×C4×D4×C9C2
socleC22×C32C23×D3
prime radicalC2×#88742C2
cosocleC22×C3×C9C24
hypercenterC4×D4C3⋊D9
hypocenterC3×C9#520
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. C3×C4×D4×C9C2semidirect
3. D4×C3⋊Dic9C2semidirect
4. C4×#6233C2semidirect
5. C2×#88534C2semidirect
6. C4⋊C4×C3⋊D9C2semidirect
7. #11721743C2semidirect
8. #11721727C2semidirect
9. #11721723C2semidirect
10. #11721709C2semidirect
11. C4×#6226C2semidirect
19. #88742C4semidirect
50. C4×D4×C9D3semidirect
62. C3×D4×C9C2×C4semidirect
68. #6233C2×C4semidirect
70. C2×C3⋊Dic9C2×C4semidirect
77. C4×C3⋊D9C2×C4semidirect
78. #6226C2×C4semidirect
147. C3×C4×D4D9semidirect
188. D4×C9C4×D3semidirect
297. C4×D4C3⋊D9semidirect
302. C3×C9#520semidirect
307. C3×D4C4×D9semidirect
408. C9#5546semidirect
413. D4C4×C3⋊D9semidirect
454. C3#10589648semidirect

Maximal subgroups

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

subgroupindex
C2×#887422
C3×C4×D4×C92
D4×C3⋊Dic92
C4×#62332
C2×#885342
C4⋊C4×C3⋊D92
#117217432
#117217272
#117217232
#117217092
C4×#62262
#105960173
#105896483