Finite Group Databasebeta

← #13150333

Subgroups — #13150333

Order 1728.

1154 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88736C211yesmaximal, characteristic, complemented
2C22×C3×C4×D9C211yesmaximal, characteristic, complemented
3C9×#926C211yesmaximal, characteristic, complemented
4#11721276C212yesmaximal, complemented
5#11721255C212yesmaximal, complemented
6#11721004C212yesmaximal, complemented
7C4×#6192C214yesmaximal, complemented
8#11720945C212yesmaximal, complemented
9#10595023—31nomaximal
10#10589626—31nomaximal
11C23×C3×D9C2211yescharacteristic
12C22×C3×C4×C9C2211yescharacteristic
13C2×#6233C2212yes—
14#88736—24no—
15C2×C4⋊(C3⋊D9)C2211yescharacteristic, complemented
16C2×C4×C3⋊D9C2212yescomplemented
17C2×#6226C2211yescharacteristic, complemented
18C22×C3×Dic9C2211yescharacteristic
19C2×C3×C4×D9—24no—
20C2×C3×C4×D9C2212yes—
21C2×C3×C4×D9C2212yes—
22C2×#6192C2214yes—
23C9×#217C2212yes—
24C9×C4⋊Dic3C2212yescomplemented
25C4×C9×Dic3C2212yescomplemented
26#88445C2214yescomplemented
27(C2×D9)⋊Dic3C2214yes—
28Dic3⋊Dic9C2214yescomplemented
29C2×#3828—31no—
30C22×C3×C4×D3—31no—
31C3×#926D311yescharacteristic
32#63977—32no—
33#63966—32no—
34#63909—32no—
35C4×#626—34no—
36#63857—32no—
37C2×#3697—31no—
38C2×#3697—61no—
39C22×C4×D9D311yescharacteristic, complemented
40C9×#117—31no—
41#63459—32no—
42#63457—32no—
43#63448—32no—
44#63444—32no—
45C4×D36—34no—
46C23×C3×C9C2311yescharacteristic
47C22×C3⋊D9C2312yes—
48C22×C3×D9—22no—
49C22×C3×D9—24no—
50C22×C3×D9C2312yes—
51C2×C3×C4×C9D414yescomplemented
52C2×C3×C4×C9C2311yescharacteristic
53C2×C3×C4×C9C2311yescharacteristic
54#6233—24no—
55C2×C3⋊Dic9C2312yes—
56C4⋊(C3⋊D9)—22no—
57C4×C3⋊D9—24no—
58#6226—22no—
59C2×C9×Dic3C2314yes—
60C2×C3×Dic9—22no—
61C2×C3×Dic9C2312yes—
62C3×C4×D9—24no—
63C3×C4×D9—24no—
64#6192—28no—
65#5791—91no—
66#5524—91no—
67C23×C3×D3—31no—
68C22×C32×C4C2×D311yescharacteristic
69C2×#638—32no—
70#3828—64no—
71C2×C4⋊(C3⋊D3)—31no—
72C2×C4×C3⋊D3—32no—
73C2×#634—31no—
74C22×C3×Dic3—31no—
75C2×C3×C4×D3—32no—
76C2×C3×C4×D3—32no—
77C2×C3×C4×D3—64no—
78C2×#626—34no—
79C23×D9C2×D311yescharacteristic
80C3×#217C2×D312yes—
81C3×C4⋊Dic3C2×D312yes—
82C3×C4×Dic3C2×D312yes—
83Dic3⋊Dic3—34no—
84#3722—34no—
85(C2×D3)⋊Dic3—34no—
86C22×C4×C9—21no—
87C22×C4×C9C2×D311yescharacteristic
88C2×#611—32no—
89C2×#611—62no—
90C22×Dic9C2×D311yescharacteristic
91#3697—68no—
92#3697—64no—
93C2×D36—34no—
94C2×D36—31no—
95C2×D36—61no—
96C2×C4×D9—24no—
97C2×C4×D9—32no—
98C2×C4×D9—62no—
99C2×C4×D9C2×D312yes—
100C2×C4×D9C2×D312yes—
101C2×Dic18—31no—
102C2×Dic18—61no—
103C9×C4⋊C4—32no—
104C9×C22⋊C4—32no—
105C42×C9—32no—
106(C2×D9)⋊C4—34no—
107(C2×D9)⋊C4—34no—
108C4⋊Dic9—34no—
109C22×C3×C9—22no—
110C22×C3×C9C2×D412yes—
111C22×C3×C9C2411yescharacteristic
112C2×C3⋊D9—22no—
113C2×C3⋊D9—24no—
114C2×C3×D9—28no—
115C2×C3×D9—22no—
116C2×C3×D9—22no—
117C2×C3×D9#4114yescomplemented
118C3×C4×C9C2×D412yes—
119C3×C4×C9C2×D412yes—
120C3⋊Dic9—22no—
121C9×Dic3—24no—
122C3×Dic9—22no—
123C23×C3×C4—91no—
124C2×#235—91no—
125C2×#235—91no—
126C2×#235—181no—
127C22×C4×D3—31no—
128C3×#117—31no—
129#953—92no—
130#930—92no—
131#929—92no—
132C4×#67—94no—
133#926D911yescharacteristic, complemented
134#924—92no—
135#897—92no—
136#895—92no—
137#886—92no—
138#882—92no—
139C4×D12—94no—
140C23×C32C22×D311yescharacteristic
141C22×C3⋊D3—32no—
142C22×C3×D3—32no—
143C22×C3×D3—64no—
144C22×C3×D3—62no—
145C2×C32×C4D1214yes—
146C2×C32×C4C22×D311yescharacteristic
147C2×C32×C4C22×D311yescharacteristic
148#638—64no—
149C2×C3⋊Dic3—32no—
150C4⋊(C3⋊D3)—62no—
151C4×C3⋊D3—64no—
152#634—62no—
153C2×C3×Dic3—32no—
154C2×C3×Dic3—62no—
155C2×C3×Dic3C22×D314yes—
156C3×C4×D3—64no—
157C3×C4×D3—64no—
158#626—68no—
159C23×C9—21no—
160C23×C9C22×D311yescharacteristic
161C22×D9—22no—
162C22×D9—24no—
163C22×D9—32no—
164C22×D9—62no—
165C22×D9C22×D312yes—
166C2×C4×C9—24no—
167C2×C4×C9—21no—
168C2×C4×C9—21no—
169C2×C4×C9—34no—
170C2×C4×C9#6714yescomplemented
171C2×C4×C9C22×D311yescharacteristic
172C2×C4×C9C22×D311yescharacteristic
173#611—68no—
174#611—64no—
175C2×Dic9—22no—
176C2×Dic9—32no—
177C2×Dic9—62no—
178C2×Dic9C22×D312yes—
179D36—64no—
180D36—68no—
181D36—62no—
182C4×D9—24no—
183C4×D9—24no—
184C4×D9—68no—
185C4×D9—64no—
186Dic18—64no—
187Dic18—62no—
188#525—271noSylow
189C2×C3×C9—22no—
190C2×C3×C9C22×D411yescharacteristic
191C2×C3×C9C2×#4112yes—
192C3⋊D9—42no—
193C3×D9—24no—
194C24×C3—91no—
195C23×D3—31no—
196C22×C3×C4—21no—
197C22×C3×C4—91no—
198C22×C3×C4—92no—
199C22×C3×C4—92no—
200C22×C3×C4—184no—
201C22×C3×C4C2×D911yescharacteristic
202C22×C3×C4D3211yescharacteristic
203C2×#67—92no—
204C2×#67—92no—
205C2×#67—94no—
206C2×#67—182no—
207C22×Dic3—31no—
208#235—188no—
209#235—184no—
210#235—184no—
211C2×D12—94no—
212C2×D12—91no—
213C2×D12—91no—
214C2×D12—181no—
215C2×C4×D3—32no—
216C2×C4×D3—32no—
217C2×C4×D3—64no—
218C2×C4×D3—92no—
219C2×C4×D3—92no—
220C2×C4×D3—182no—
221C2×Dic6—91no—
222C2×Dic6—91no—
223C2×Dic6—181no—
224C3×C4⋊C4—32no—
225C3×C22⋊C4—32no—
226C3×C42—32no—
227#217—94no—
228#217C2×D912yes—
229(C2×D3)⋊C4—94no—
230(C2×D3)⋊C4—94no—
231(C2×D3)⋊C4—94no—
232C4⋊Dic3—94no—
233C4⋊Dic3C2×D912yescomplemented
234Dic3⋊C4—94no—
235C4×Dic3C2×D912yescomplemented
236C22×C32—22no—
237C22×C32C2×D1212yes—
238C22×C32C23×D311yescharacteristic
239C2×C3⋊D3—62no—
240C2×C3⋊D3—64no—
241C2×C3×D3—34no—
242C2×C3×D3—68no—
243C2×C3×D3—62no—
244C2×C3×D3—62no—
245C32×C4C2×D1212yes—
246C32×C4C2×D1212yes—
247C3⋊Dic3—62no—
248C3×Dic3—24no—
249C3×Dic3—62no—
250C22×C9—22no—
251C22×C9—21no—
252C22×C9—22no—
253C22×C9—42no—
254C22×C9C2×#6712yes—
255C22×C9C23×D311yescharacteristic
256C2×D9—28no—
257C2×D9—22no—
258C2×D9—22no—
259C2×D9—64no—
260C2×D9—64no—
261C2×D9—62no—
262C2×D9—124no—
263C2×D9#23514yescomplemented
264C4×C9—22no—
265C4×C9—22no—
266C4×C9—64no—
267C4×C9C2×#6712yes—
268C4×C9C2×#6712yes—
269Dic9—22no—
270Dic9—64no—
271Dic9—62no—
272C2×#41—271no—
273C23×C4—91no—
274#123—272no—
275C22⋊Q8—272no—
276#121—272no—
277#120—272no—
278C4×D4—274no—
279#117—31no—
280C3×C9#52511yescharacteristic, complemented, Sylow
281C23×C3—21no—
282C23×C3—92no—
283C23×C3—184no—
284C23×C3—182no—
285C23×C3C22×D911yescharacteristic
286C23×C3C2×D3211yescharacteristic
287C22×D3—32no—
288C22×D3—64no—
289C22×D3—62no—
290C22×D3—92no—
291C22×D3—92no—
292C22×D3—182no—
293C2×C3×C4—24no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—34no—
297C2×C3×C4—92no—
298C2×C3×C4—182no—
299C2×C3×C4—184no—
300C2×C3×C4—184no—
301C2×C3×C4D3614yescomplemented
302C2×C3×C4C22×D911yescharacteristic
303C2×C3×C4C22×D911yescharacteristic
304C2×C3×C4#62614yes—
305C2×C3×C4C2×D3211yescharacteristic
306C2×C3×C4C2×D3211yescharacteristic
307#67—188no—
308#67—188no—
309#67—184no—
310#67—184no—
311C2×Dic3—32no—
312C2×Dic3—62no—
313C2×Dic3—92no—
314C2×Dic3—92no—
315C2×Dic3—182no—
316C2×Dic3C22×D914yes—
317D12—184no—
318D12—188no—
319D12—182no—
320D12—182no—
321C4×D3—64no—
322C4×D3—64no—
323C4×D3—188no—
324C4×D3—184no—
325C4×D3—184no—
326Dic6—184no—
327Dic6—182no—
328Dic6—182no—
329C2×C32—22no—
330C2×C32C22×D1211yescharacteristic
331C2×C32D3×#4112yes—
332C3⋊D3—122no—
333C3×D3—64no—
334C2×C9—22no—
335C2×C9—22no—
336C2×C9—21no—
337C2×C9—42no—
338C2×C9C2×#23512yes—
339C2×C9C22×#6711yescharacteristic
340D9—24no—
341D9—124no—
342D9—122no—
343C24—91no—
344#41—544no—
345C2×Q8—271no—
346C2×D4—274no—
347C2×D4—272no—
348C2×D4—271no—
349C22×C4—91no—
350C22×C4—92no—
351C22×C4—92no—
352C22×C4—184no—
353C22×C4—272no—
354C22×C4D3×D911yescharacteristic
355C4⋊C4—32no—
356C4⋊C4—274no—
357C22⋊C4—32no—
358C22⋊C4—274no—
359C22⋊C4—274no—
360C42—32no—
361C22×C3—22no—
362C22×C3—22no—
363C22×C3—21no—
364C22×C3—22no—
365C22×C3—42no—
366C22×C3—94no—
367C22×C3—188no—
368C22×C3—182no—
369C22×C3—182no—
370C22×C3C2×D3612yes—
371C22×C3C23×D911yescharacteristic
372C22×C3C2×#62612yes—
373C22×C3C22×D3211yescharacteristic
374C2×D3—34no—
375C2×D3—68no—
376C2×D3—62no—
377C2×D3—62no—
378C2×D3—184no—
379C2×D3—184no—
380C2×D3—182no—
381C2×D3—182no—
382C2×D3—184no—
383C2×D3—364no—
384C3×C4—22no—
385C3×C4—22no—
386C3×C4—64no—
387C3×C4—182no—
388C3×C4C2×D3612yes—
389C3×C4C2×D3612yes—
390C3×C4C2×#62612yes—
391C3×C4C2×#62612yes—
392Dic3—24no—
393Dic3—62no—
394Dic3—184no—
395Dic3—182no—
396Dic3—182no—
397C32#552411yescharacteristic
398C9—21no—
399C9#579111yescharacteristic, complemented
400C23—92no—
401C23—184no—
402C23—182no—
403C23—272no—
404C23C2×D3×D911yescharacteristic
405Q8—542no—
406D4—548no—
407D4—544no—
408D4—542no—
409C2×C4—34no—
410C2×C4—92no—
411C2×C4—182no—
412C2×C4—184no—
413C2×C4—184no—
414C2×C4—272no—
415C2×C4—544no—
416C2×C4#619214yescomplemented
417C2×C4C2×D3×D911yescharacteristic
418C2×C4C2×D3×D911yescharacteristic
419C2×C3—22no—
420C2×C3—22no—
421C2×C3—21no—
422C2×C3—22no—
423C2×C3—42no—
424C2×C3—184no—
425C2×C3C22×D3611yescharacteristic
426C2×C3#41×D912yes—
427C2×C3D3×#23512yes—
428C2×C3C22×#62611yescharacteristic
429D3—64no—
430D3—364no—
431D3—362no—
432D3—362no—
433C22—22no—
434C22—94no—
435C22—188no—
436C22—182no—
437C22—182no—
438C22—542no—
439C22—544no—
440C22C2×#619212yes—
441C22C22×D3×D911yescharacteristic
442C4—64no—
443C4—182no—
444C4—542no—
445C4C2×#619212yes—
446C4C2×#619212yes—
447C3—21no—
448C3#1058962611yescharacteristic, complemented
449C3#1059502311yescharacteristic
450C2—22no—
451C2—184no—
452C2—1082no—
453C2D9×#23512yes—
454C2C22×#619211yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#525
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88736C2semidirect
2. C22×C3×C4×D9C2semidirect
3. C9×#926C2semidirect
4. #11721276C2semidirect
5. #11721255C2semidirect
6. #11721004C2semidirect
7. C4×#6192C2semidirect
8. #11720945C2semidirect
15. C2×C4⋊(C3⋊D9)C22semidirect
16. C2×C4×C3⋊D9C22semidirect
17. C2×#6226C22semidirect
24. C9×C4⋊Dic3C22semidirect
25. C4×C9×Dic3C22semidirect
26. #88445C22semidirect
28. Dic3⋊Dic9C22semidirect
39. C22×C4×D9D3semidirect
51. C2×C3×C4×C9D4semidirect
117. C2×C3×D9#41semidirect
133. #926D9semidirect
170. C2×C4×C9#67semidirect
233. C4⋊Dic3C2×D9semidirect
235. C4×Dic3C2×D9semidirect
263. C2×D9#235semidirect
280. C3×C9#525semidirect
301. C2×C3×C4D36semidirect
399. C9#5791semidirect
416. C2×C4#6192semidirect
448. C3#10589626semidirect

Maximal subgroups

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

subgroupindex
C2×#887362
C22×C3×C4×D92
C9×#9262
#117212762
#117212552
#117210042
C4×#61922
#117209452
#105950233
#105896263