Finite Group Databasebeta

← #13150338

Subgroups — #13150338

Order 1728.

1226 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
1C22×C4×C3⋊D9C211yesmaximal, characteristic, complemented
2C2×C3×#3697C211yesmaximal, characteristic, complemented
3C9×#926C211yesmaximal, characteristic, complemented
4#11721278C212yesmaximal, complemented
5#11721260C212yesmaximal, complemented
6#11721007C212yesmaximal, complemented
7C4×#6192C214yesmaximal, complemented
8#11720936C212yesmaximal, complemented
9#10595034—31nomaximal
10#10589626—31nomaximal
11C23×C3⋊D9C2211yescharacteristic
12C22×C3×C4×C9C2211yescharacteristic
13C22×C3⋊Dic9C2211yescharacteristic
14C2×C4×C3⋊D9—24no—
15C2×C4×C3⋊D9C2212yes—
16C2×C4×C3⋊D9C2212yes—
17C2×C3×#611C2212yes—
18C3×#3697—24no—
19C2×C3×D36C2211yescharacteristic, complemented
20C2×C3×C4×D9C2212yescomplemented
21C2×C3×Dic18C2211yescharacteristic, complemented
22C2×#6192C2214yes—
23C9×#217C2212yes—
24C9×C4⋊Dic3C2212yescomplemented
25C4×C9×Dic3C2212yescomplemented
26#88445C2214yes—
27(C2×D9)⋊Dic3C2214yescomplemented
28Dic3⋊Dic9C2214yescomplemented
29C22×C4×C3⋊D3—31no—
30C2×C3×#235—31no—
31C3×#926D311yescharacteristic
32#63981—32no—
33#63962—32no—
34#63916—32no—
35C4×#626—34no—
36#63871—32no—
37C2×#3697D311yescharacteristic, complemented
38C22×C4×D9—31no—
39C22×C4×D9—61no—
40C9×#117—31no—
41#63459—32no—
42#63457—32no—
43#63448—32no—
44#63444—32no—
45C4×D36—34no—
46C23×C3×C9C2311yescharacteristic
47C22×C3⋊D9—22no—
48C22×C3⋊D9—24no—
49C22×C3⋊D9C2312yes—
50C22×C3×D9C2312yes—
51C2×C3×C4×C9D414yescomplemented
52C2×C3×C4×C9C2311yescharacteristic
53C2×C3×C4×C9C2311yescharacteristic
54C2×C3⋊Dic9—22no—
55C2×C3⋊Dic9C2312yes—
56C4×C3⋊D9—24no—
57C4×C3⋊D9—24no—
58C3×#611—24no—
59C2×C9×Dic3C2314yes—
60C2×C3×Dic9C2312yes—
61C3×D36—22no—
62C3×C4×D9—24no—
63C3×Dic18—22no—
64#6192—28no—
65#5828—91no—
66#5524—91no—
67C23×C3⋊D3—31no—
68C22×C32×C4C2×D311yescharacteristic
69C22×C3⋊Dic3—31no—
70C2×C4×C3⋊D3—32no—
71C2×C4×C3⋊D3—32no—
72C2×C4×C3⋊D3—64no—
73C2×C3×#67—32no—
74C3×#235—64no—
75C2×C3×D12—31no—
76C2×C3×C4×D3—32no—
77C2×C3×Dic6—31no—
78C2×#626—34no—
79C23×D9—31no—
80C23×D9—61no—
81C3×#217C2×D312yes—
82C3×C4⋊Dic3C2×D312yes—
83C3×C4×Dic3C2×D312yes—
84Dic3⋊Dic3—34no—
85#3722—34no—
86(C2×D3)⋊Dic3—34no—
87C22×C4×C9—21no—
88C22×C4×C9C2×D311yescharacteristic
89C2×#611C2×D312yes—
90C22×Dic9—31no—
91C22×Dic9—61no—
92#3697—24no—
93C2×D36—34no—
94C2×D36C2×D311yescharacteristic, complemented
95C2×C4×D9—32no—
96C2×C4×D9—32no—
97C2×C4×D9—62no—
98C2×C4×D9—62no—
99C2×C4×D9—68no—
100C2×C4×D9—64no—
101C2×C4×D9C2×D312yescomplemented
102C2×Dic18C2×D311yescharacteristic, complemented
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—28no—
113C2×C3⋊D9—22no—
114C2×C3⋊D9—22no—
115C2×C3⋊D9#4114yescomplemented
116C2×C3×D9—22no—
117C2×C3×D9—24no—
118C3×C4×C9C2×D412yes—
119C3×C4×C9C2×D412yes—
120C3⋊Dic9—22no—
121C9×Dic3—24no—
122C3×Dic9—22no—
123C2×C3×#41—91no—
124C2×#235—31no—
125C22×C4×D3—91no—
126C22×C4×D3—91no—
127C22×C4×D3—181no—
128C3×#117—31no—
129#946—92no—
130#938—92no—
131#937—92no—
132#935—92no—
133C4×#67—94no—
134#926D911yescharacteristic, complemented
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—64no—
143C22×C3⋊D3—62no—
144C22×C3×D3—32no—
145C2×C32×C4D1214yes—
146C2×C32×C4C22×D311yescharacteristic
147C2×C32×C4C22×D311yescharacteristic
148C2×C3⋊Dic3—32no—
149C2×C3⋊Dic3—62no—
150C4×C3⋊D3—64no—
151C4×C3⋊D3—64no—
152C3×#67—64no—
153C2×C3×Dic3—32no—
154C2×C3×Dic3C22×D314yes—
155C3×D12—62no—
156C3×C4×D3—64no—
157C3×Dic6—62no—
158#626—68no—
159C23×C9—21no—
160C23×C9C22×D311yescharacteristic
161C22×D9—32no—
162C22×D9—62no—
163C22×D9—64no—
164C22×D9—64no—
165C22×D9—68no—
166C22×D9—62no—
167C22×D9C22×D312yes—
168C2×C4×C9—24no—
169C2×C4×C9—21no—
170C2×C4×C9—21no—
171C2×C4×C9—34no—
172C2×C4×C9#6714yescomplemented
173C2×C4×C9C22×D311yescharacteristic
174C2×C4×C9C22×D311yescharacteristic
175#611—24no—
176C2×Dic9—32no—
177C2×Dic9—62no—
178C2×Dic9—64no—
179C2×Dic9—62no—
180C2×Dic9C22×D312yes—
181D36—22no—
182D36—68no—
183C4×D9—24no—
184C4×D9—68no—
185C4×D9—68no—
186C4×D9—64no—
187C4×D9—64no—
188Dic18—22no—
189#525—271noSylow
190C2×C3×C9—22no—
191C2×C3×C9C22×D411yescharacteristic
192C2×C3×C9C2×#4112yes—
193C3⋊D9—24no—
194C3×D9—42no—
195C23×D3—91no—
196C23×D3—91no—
197C23×D3—181no—
198C3×#41—184no—
199C2×C3×Q8—91no—
200C2×C3×D4—92no—
201C2×C3×D4—91no—
202C22×C3×C4—21no—
203C22×C3×C4—92no—
204C22×C3×C4C2×D911yescharacteristic
205C22×C3×C4D3211yescharacteristic
206C2×#67—32no—
207C2×#67—94no—
208C22×Dic3—91no—
209C22×Dic3—91no—
210C22×Dic3—181no—
211#235—64no—
212C2×D12—31no—
213C2×D12—94no—
214C2×C4×D3—32no—
215C2×C4×D3—92no—
216C2×C4×D3—92no—
217C2×C4×D3—92no—
218C2×C4×D3—92no—
219C2×C4×D3—182no—
220C2×C4×D3—182no—
221C2×C4×D3—188no—
222C2×C4×D3—184no—
223C2×C4×D3—184no—
224C2×Dic6—31no—
225C3×C4⋊C4—32no—
226C3×C22⋊C4—32no—
227C3×C42—32no—
228#217—94no—
229#217C2×D912yes—
230(C2×D3)⋊C4—94no—
231(C2×D3)⋊C4—94no—
232(C2×D3)⋊C4—94no—
233C4⋊Dic3—94no—
234C4⋊Dic3C2×D912yescomplemented
235Dic3⋊C4—94no—
236C4×Dic3C2×D912yescomplemented
237C22×C32—22no—
238C22×C32C2×D1212yes—
239C22×C32C23×D311yescharacteristic
240C2×C3⋊D3—34no—
241C2×C3⋊D3—68no—
242C2×C3⋊D3—62no—
243C2×C3⋊D3—62no—
244C2×C3×D3—62no—
245C2×C3×D3—64no—
246C32×C4C2×D1212yes—
247C32×C4C2×D1212yes—
248C3⋊Dic3—62no—
249C3×Dic3—24no—
250C3×Dic3—62no—
251C22×C9—24no—
252C22×C9—22no—
253C22×C9—21no—
254C22×C9—22no—
255C22×C9C2×#6712yes—
256C22×C9C23×D311yescharacteristic
257C2×D9—24no—
258C2×D9—22no—
259C2×D9—34no—
260C2×D9—64no—
261C2×D9—64no—
262C2×D9—616no—
263C2×D9—68no—
264C2×D9—64no—
265C2×D9—62no—
266C2×D9—62no—
267C4×C9—22no—
268C4×C9—22no—
269C4×C9—64no—
270C4×C9C2×#6712yes—
271C4×C9C2×#6712yes—
272Dic9—22no—
273Dic9—64no—
274Dic9—62no—
275C2×#41—91no—
276C23×C4—271no—
277#123—272no—
278C22⋊Q8—272no—
279#121—272no—
280#120—272no—
281C4×D4—274no—
282#117—31no—
283C3×C9#52511yescharacteristic, complemented, Sylow
284C23×C3—21no—
285C23×C3—92no—
286C23×C3C22×D911yescharacteristic
287C23×C3C2×D3211yescharacteristic
288C22×D3—32no—
289C22×D3—92no—
290C22×D3—92no—
291C22×D3—182no—
292C22×D3—184no—
293C22×D3—184no—
294C22×D3—184no—
295C22×D3—188no—
296C22×D3—182no—
297C22×D3—182no—
298C3×Q8—182no—
299C3×D4—184no—
300C3×D4—182no—
301C2×C3×C4—24no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—34no—
305C2×C3×C4—92no—
306C2×C3×C4—184no—
307C2×C3×C4D3614yescomplemented
308C2×C3×C4C22×D911yescharacteristic
309C2×C3×C4C22×D911yescharacteristic
310C2×C3×C4#62614yes—
311C2×C3×C4C2×D3211yescharacteristic
312C2×C3×C4C2×D3211yescharacteristic
313#67—64no—
314#67—188no—
315C2×Dic3—32no—
316C2×Dic3—92no—
317C2×Dic3—92no—
318C2×Dic3—182no—
319C2×Dic3—184no—
320C2×Dic3—182no—
321C2×Dic3—182no—
322C2×Dic3C22×D914yes—
323D12—62no—
324D12—188no—
325C4×D3—64no—
326C4×D3—188no—
327C4×D3—188no—
328C4×D3—184no—
329C4×D3—184no—
330C4×D3—184no—
331C4×D3—184no—
332Dic6—62no—
333C2×C32—22no—
334C2×C32C22×D1211yescharacteristic
335C2×C32D3×#4112yes—
336C3⋊D3—64no—
337C3×D3—122no—
338C2×C9—24no—
339C2×C9—22no—
340C2×C9—22no—
341C2×C9—21no—
342C2×C9D3×#4112yes—
343C2×C9C22×#6711yescharacteristic
344D9—42no—
345D9—68no—
346D9—64no—
347C24—271no—
348#41—184no—
349C2×Q8—91no—
350C2×D4—92no—
351C2×D4—91no—
352C2×D4—274no—
353C22×C4—92no—
354C22×C4—271no—
355C22×C4—272no—
356C22×C4—272no—
357C22×C4—544no—
358C22×C4D3×D911yescharacteristic
359C4⋊C4—32no—
360C4⋊C4—274no—
361C22⋊C4—32no—
362C22⋊C4—274no—
363C22⋊C4—274no—
364C42—32no—
365C22×C3—24no—
366C22×C3—22no—
367C22×C3—22no—
368C22×C3—21no—
369C22×C3—22no—
370C22×C3—182no—
371C22×C3—184no—
372C22×C3C2×D3612yes—
373C22×C3C23×D911yescharacteristic
374C22×C3C2×#62612yes—
375C22×C3C22×D3211yescharacteristic
376C2×D3—64no—
377C2×D3—62no—
378C2×D3—94no—
379C2×D3—94no—
380C2×D3—184no—
381C2×D3—184no—
382C2×D3—188no—
383C2×D3—1816no—
384C2×D3—188no—
385C2×D3—184no—
386C2×D3—182no—
387C2×D3—182no—
388C2×D3—182no—
389C2×D3—182no—
390C3×C4—22no—
391C3×C4—22no—
392C3×C4—64no—
393C3×C4—182no—
394C3×C4C2×D3612yes—
395C3×C4C2×D3612yes—
396C3×C4C2×#62612yes—
397C3×C4C2×#62612yes—
398Dic3—24no—
399Dic3—62no—
400Dic3—184no—
401Dic3—182no—
402Dic3—182no—
403C32#552411yescharacteristic
404C9—21no—
405C9#582811yescharacteristic, complemented
406C23—92no—
407C23—272no—
408C23—544no—
409C23—542no—
410C23C2×D3×D911yescharacteristic
411Q8—182no—
412D4—184no—
413D4—182no—
414D4—548no—
415C2×C4—34no—
416C2×C4—92no—
417C2×C4—184no—
418C2×C4—272no—
419C2×C4—542no—
420C2×C4—544no—
421C2×C4—544no—
422C2×C4#619214yescomplemented
423C2×C4C2×D3×D911yescharacteristic
424C2×C4C2×D3×D911yescharacteristic
425C2×C3—24no—
426C2×C3—22no—
427C2×C3—22no—
428C2×C3—21no—
429C2×C3—22no—
430C2×C3—362no—
431C2×C3C22×D3611yescharacteristic
432C2×C3#41×D912yes—
433C2×C3#6430612yes—
434C2×C3C22×#62611yescharacteristic
435D3—122no—
436D3—188no—
437D3—184no—
438D3—184no—
439C22—22no—
440C22—182no—
441C22—184no—
442C22—274no—
443C22—548no—
444C22—542no—
445C22—542no—
446C22C2×#619212yes—
447C22C22×D3×D911yescharacteristic
448C4—64no—
449C4—182no—
450C4—542no—
451C4C2×#619212yes—
452C4C2×#619212yes—
453C3—21no—
454C3#1058962611yescharacteristic, complemented
455C3#1059503411yescharacteristic
456C2—22no—
457C2—362no—
458C2—544no—
459C2#1172247512yes—
460C2C22×#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. C22×C4×C3⋊D9C2semidirect
2. C2×C3×#3697C2semidirect
3. C9×#926C2semidirect
4. #11721278C2semidirect
5. #11721260C2semidirect
6. #11721007C2semidirect
7. C4×#6192C2semidirect
8. #11720936C2semidirect
19. C2×C3×D36C22semidirect
20. C2×C3×C4×D9C22semidirect
21. C2×C3×Dic18C22semidirect
24. C9×C4⋊Dic3C22semidirect
25. C4×C9×Dic3C22semidirect
27. (C2×D9)⋊Dic3C22semidirect
28. Dic3⋊Dic9C22semidirect
37. C2×#3697D3semidirect
51. C2×C3×C4×C9D4semidirect
94. C2×D36C2×D3semidirect
101. C2×C4×D9C2×D3semidirect
102. C2×Dic18C2×D3semidirect
115. C2×C3⋊D9#41semidirect
134. #926D9semidirect
172. C2×C4×C9#67semidirect
234. C4⋊Dic3C2×D9semidirect
236. C4×Dic3C2×D9semidirect
283. C3×C9#525semidirect
307. C2×C3×C4D36semidirect
405. C9#5828semidirect
422. C2×C4#6192semidirect
454. C3#10589626semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C3⋊D92
C2×C3×#36972
C9×#9262
#117212782
#117212602
#117210072
C4×#61922
#117209362
#105950343
#105896263