Finite Group Databasebeta

← #13150350

Subgroups — #13150350

Order 1728.

1046 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88736C211yesmaximal, characteristic, complemented
2C2×C3×#3697C211yesmaximal, characteristic, complemented
3C2×C9×C4⋊Dic3C211yesmaximal, characteristic, complemented
4#11721277C214yesmaximal, complemented
5#11721007C212yesmaximal, complemented
6#11721004C212yesmaximal, complemented
7#11720945C212yesmaximal, complemented
8#11720936C212yesmaximal, complemented
9#10595043—31nomaximal
10#10589612—31nomaximal
11C22×C3×C4×C9C2211yescharacteristic
12C2×#6233C2212yes—
13#88736—24no—
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×C4×C3⋊D9C2212yes—
16C2×#6226C2211yescharacteristic
17C2×C3×#611C2212yes—
18C22×C9×Dic3C2212yes—
19C3×#3697—24no—
20C2×C3×D36C2211yescharacteristic
21C2×C3×C4×D9C2212yes—
22C2×C3×Dic18C2211yescharacteristic
23C2×#6192C2214yes—
24C9×C4⋊Dic3C2212yes—
25C9×C4⋊Dic3C2212yes—
26#88445C2214yes—
27(C2×D9)⋊Dic3C2214yes—
28Dic3⋊Dic9C2214yes—
29C2×#3828—31no—
30C2×C3×#235—31no—
31C2×C3×C4⋊Dic3D311yescharacteristic
32#63978—34no—
33#63916—32no—
34#63909—32no—
35#63871—32no—
36#63857—32no—
37C2×#3697—31no—
38C2×#3697—61no—
39C2×#3697D311yescharacteristic, complemented
40C2×C9×C4⋊C4—31no—
41#63492—34no—
42#63459—32no—
43#63459—32no—
44#63457—32no—
45#63457—32no—
46C23×C3×C9C2311yescharacteristic
47C22×C3⋊D9C2312yes—
48C22×C3×D9C2312yes—
49C2×C3×C4×C9D414yescomplemented
50C2×C3×C4×C9C2311yescharacteristic
51C2×C3×C4×C9C2311yescharacteristic
52#6233—24no—
53C2×C3⋊Dic9C2312yes—
54C4⋊(C3⋊D9)—22no—
55C4×C3⋊D9—24no—
56#6226—22no—
57C3×#611—24no—
58C2×C9×Dic3—42no—
59C2×C9×Dic3C2314yes—
60C2×C3×Dic9C2312yes—
61C3×D36—22no—
62C3×C4×D9—24no—
63C3×Dic18—22no—
64#6192—44no—
65#5833—91no—
66#5510—91no—
67C22×C32×C4C2×D311yescharacteristic
68C2×#638—32no—
69#3828—64no—
70C2×C4⋊(C3⋊D3)—31no—
71C2×C4×C3⋊D3—32no—
72C2×#634—31no—
73C2×C3×#67—32no—
74C22×C3×Dic3C2×D312yes—
75C3×#235—64no—
76C2×C3×D12—31no—
77C2×C3×C4×D3—32no—
78C2×C3×Dic6—31no—
79C2×#626—34no—
80C3×C4⋊Dic3C2×D312yes—
81C3×C4⋊Dic3C2×D312yes—
82Dic3⋊Dic3—34no—
83#3722—34no—
84(C2×D3)⋊Dic3—34no—
85C22×C4×C9—21no—
86C22×C4×C9—32no—
87C22×C4×C9C2×D311yescharacteristic
88C2×#611—32no—
89C2×#611—62no—
90C2×#611C2×D312yes—
91#3697—24no—
92#3697—68no—
93#3697—64no—
94C2×D36—31no—
95C2×D36—34no—
96C2×D36—61no—
97C2×D36C2×D311yescharacteristic
98C2×C4×D9—32no—
99C2×C4×D9—62no—
100C2×C4×D9C2×D312yes—
101C2×Dic18—31no—
102C2×Dic18—61no—
103C2×Dic18C2×D311yescharacteristic
104C9×C4⋊C4—32no—
105C9×C4⋊C4—32no—
106(C2×D9)⋊C4—34no—
107(C2×D9)⋊C4—34no—
108C4⋊Dic9—34no—
109C22×C3×C9—21no—
110C22×C3×C9—21no—
111C22×C3×C9C2×D412yes—
112C22×C3×C9C2411yescharacteristic
113C2×C3⋊D9—22no—
114C2×C3⋊D9—22no—
115C2×C3⋊D9—22no—
116C2×C3×D9—22no—
117C2×C3×D9—22no—
118C2×C3×D9—22no—
119C3×C4×C9C2×D412yes—
120C3×C4×C9C2×D412yes—
121C3⋊Dic9—22no—
122C9×Dic3—24no—
123C3×Dic9—22no—
124C2×C3×#41—91no—
125C2×#235—31no—
126C2×#235—91no—
127C2×#235—91no—
128C2×#235—181no—
129C2×C3×C4⋊C4—31no—
130#946—92no—
131#939—94no—
132#938—92no—
133#930—94no—
134#930—92no—
135C2×C4⋊Dic3D911yescharacteristic, complemented
136#924—92no—
137#897—92no—
138#897—92no—
139#895—92no—
140#895—92no—
141C23×C32C22×D311yescharacteristic
142C22×C3⋊D3—32no—
143C22×C3×D3—32no—
144C2×C32×C4D1214yes—
145C2×C32×C4C22×D311yescharacteristic
146C2×C32×C4C22×D311yescharacteristic
147#638—64no—
148C2×C3⋊Dic3—32no—
149C4⋊(C3⋊D3)—62no—
150C4×C3⋊D3—64no—
151#634—62no—
152C3×#67—64no—
153C2×C3×Dic3—32no—
154C2×C3×Dic3—42no—
155C2×C3×Dic3C22×D314yes—
156C3×D12—62no—
157C3×C4×D3—64no—
158C3×Dic6—62no—
159#626—124no—
160C23×C9—21no—
161C23×C9C22×D311yescharacteristic
162C22×D9—32no—
163C22×D9—62no—
164C22×D9C22×D312yes—
165C2×C4×C9—24no—
166C2×C4×C9—21no—
167C2×C4×C9—21no—
168C2×C4×C9—34no—
169C2×C4×C9—122no—
170C2×C4×C9#6714yescomplemented
171C2×C4×C9C22×D311yescharacteristic
172C2×C4×C9C22×D311yescharacteristic
173#611—24no—
174#611—68no—
175#611—64no—
176C2×Dic9—32no—
177C2×Dic9—62no—
178C2×Dic9C22×D312yes—
179D36—22no—
180D36—64no—
181D36—62no—
182D36—124no—
183C4×D9—24no—
184C4×D9—68no—
185C4×D9—64no—
186Dic18—22no—
187Dic18—64no—
188Dic18—62no—
189#537—271noSylow
190C2×C3×C9—22no—
191C2×C3×C9C22×D411yescharacteristic
192C2×C3×C9H(5,𝔽2)11yescharacteristic
193C2×C3×C92−1+411yescharacteristic
194C3⋊D9—42no—
195C3×D9—42no—
196C3×#41—184no—
197C2×C3×Q8—91no—
198C2×C3×D4—92no—
199C2×C3×D4—91no—
200C22×C3×C4—21no—
201C22×C3×C4—32no—
202C22×C3×C4—92no—
203C22×C3×C4C2×D911yescharacteristic
204C22×C3×C4D3211yescharacteristic
205C2×#67—32no—
206C2×#67—92no—
207C2×#67—92no—
208C2×#67—94no—
209C2×#67—182no—
210C22×Dic3C2×D912yes—
211#235—64no—
212#235—188no—
213#235—184no—
214#235—184no—
215C2×D12—31no—
216C2×D12—91no—
217C2×D12—91no—
218C2×D12—94no—
219C2×D12—181no—
220C2×C4×D3—32no—
221C2×C4×D3—92no—
222C2×C4×D3—92no—
223C2×C4×D3—182no—
224C2×Dic6—31no—
225C2×Dic6—91no—
226C2×Dic6—91no—
227C2×Dic6—181no—
228C3×C4⋊C4—32no—
229C3×C4⋊C4—32no—
230#217—94no—
231(C2×D3)⋊C4—94no—
232(C2×D3)⋊C4—94no—
233(C2×D3)⋊C4—94no—
234C4⋊Dic3—94no—
235C4⋊Dic3C2×D912yes—
236C4⋊Dic3C2×D912yes—
237Dic3⋊C4—94no—
238C22×C32—21no—
239C22×C32—21no—
240C22×C32C2×D1212yes—
241C22×C32C23×D311yescharacteristic
242C2×C3⋊D3—62no—
243C2×C3⋊D3—62no—
244C2×C3⋊D3—62no—
245C2×C3×D3—62no—
246C2×C3×D3—62no—
247C2×C3×D3—62no—
248C32×C4C2×D1212yes—
249C32×C4C2×D1212yes—
250C3⋊Dic3—62no—
251C3×Dic3—24no—
252C3×Dic3—62no—
253C22×C9—21no—
254C22×C9—21no—
255C22×C9—21no—
256C22×C9—22no—
257C22×C9—41no—
258C22×C9—41no—
259C22×C9C2×#6712yes—
260C22×C9C23×D311yescharacteristic
261C2×D9—22no—
262C2×D9—22no—
263C2×D9—22no—
264C2×D9—64no—
265C2×D9—62no—
266C2×D9—62no—
267C2×D9—62no—
268C2×D9—122no—
269C2×D9—122no—
270C4×C9—22no—
271C4×C9—22no—
272C4×C9—64no—
273C4×C9C2×#6712yes—
274C4×C9C2×#6712yes—
275Dic9—22no—
276Dic9—64no—
277Dic9—62no—
278C2×#41—91no—
279C2×#41—271no—
280C22⋊Q8—272no—
281C22⋊Q8—272no—
282#121—272no—
283#121—272no—
284#121—274no—
285C2×C4⋊C4—31no—
286C3×C9#53711yescharacteristic, complemented, Sylow
287C23×C3—21no—
288C23×C3—92no—
289C23×C3C22×D911yescharacteristic
290C23×C3C2×D3211yescharacteristic
291C22×D3—32no—
292C22×D3—92no—
293C22×D3—92no—
294C22×D3—182no—
295C3×Q8—182no—
296C3×D4—184no—
297C3×D4—182no—
298C2×C3×C4—24no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—34no—
302C2×C3×C4—92no—
303C2×C3×C4—122no—
304C2×C3×C4—184no—
305C2×C3×C4D3614yescomplemented
306C2×C3×C4C22×D911yescharacteristic
307C2×C3×C4C22×D911yescharacteristic
308C2×C3×C4#62614yes—
309C2×C3×C4C2×D3211yescharacteristic
310C2×C3×C4C2×D3211yescharacteristic
311#67—64no—
312#67—188no—
313#67—184no—
314#67—184no—
315#67—364no—
316C2×Dic3—32no—
317C2×Dic3—42no—
318C2×Dic3—92no—
319C2×Dic3—92no—
320C2×Dic3—182no—
321C2×Dic3C22×D914yes—
322D12—62no—
323D12—184no—
324D12—182no—
325D12—182no—
326D12—364no—
327C4×D3—64no—
328C4×D3—188no—
329C4×D3—184no—
330C4×D3—184no—
331Dic6—62no—
332Dic6—184no—
333Dic6—182no—
334Dic6—182no—
335C2×C32—22no—
336C2×C32C22×D1211yescharacteristic
337C2×C32#100411yescharacteristic
338C2×C32#100711yescharacteristic
339C3⋊D3—122no—
340C3×D3—122no—
341C2×C9—22no—
342C2×C9—21no—
343C2×C9—21no—
344C2×C9—21no—
345C2×C9—42no—
346C2×C9#100911yescharacteristic
347C2×C9#101011yescharacteristic
348C2×C9C22×#6711yescharacteristic
349D9—42no—
350D9—124no—
351D9—122no—
352#41—184no—
353#41—544no—
354C2×Q8—91no—
355C2×Q8—271no—
356C2×D4—92no—
357C2×D4—91no—
358C2×D4—274no—
359C2×D4—272no—
360C2×D4—271no—
361C22×C4—32no—
362C22×C4—92no—
363C22×C4—272no—
364C22×C4D3×D911yescharacteristic
365C4⋊C4—32no—
366C4⋊C4—32no—
367C4⋊C4—274no—
368C22⋊C4—274no—
369C22⋊C4—274no—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—22no—
376C22×C3—41no—
377C22×C3—41no—
378C22×C3—182no—
379C22×C3—182no—
380C22×C3—182no—
381C22×C3C2×D3612yes—
382C22×C3C23×D911yescharacteristic
383C22×C3C2×#62612yes—
384C22×C3C22×D3211yescharacteristic
385C2×D3—62no—
386C2×D3—62no—
387C2×D3—62no—
388C2×D3—184no—
389C2×D3—182no—
390C2×D3—182no—
391C2×D3—182no—
392C2×D3—182no—
393C2×D3—182no—
394C2×D3—182no—
395C2×D3—362no—
396C2×D3—362no—
397C3×C4—22no—
398C3×C4—22no—
399C3×C4—64no—
400C3×C4—182no—
401C3×C4C2×D3612yes—
402C3×C4C2×D3612yes—
403C3×C4C2×#62612yes—
404C3×C4C2×#62612yes—
405Dic3—24no—
406Dic3—62no—
407Dic3—184no—
408Dic3—182no—
409Dic3—182no—
410C32#551011yescharacteristic
411C9—21no—
412C9#583311yescharacteristic, complemented
413C23—92no—
414C23—272no—
415C23C2×D3×D911yescharacteristic
416Q8—182no—
417Q8—542no—
418D4—184no—
419D4—182no—
420D4—544no—
421D4—542no—
422D4—1084no—
423C2×C4—34no—
424C2×C4—92no—
425C2×C4—122no—
426C2×C4—184no—
427C2×C4—272no—
428C2×C4—544no—
429C2×C4#619214yescomplemented
430C2×C4C2×D3×D911yescharacteristic
431C2×C4C2×D3×D911yescharacteristic
432C2×C3—22no—
433C2×C3—22no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—42no—
438C2×C3—362no—
439C2×C3C22×D3611yescharacteristic
440C2×C3#6371011yescharacteristic
441C2×C3#6371311yescharacteristic
442C2×C3#6429711yescharacteristic
443C2×C3#6430711yescharacteristic
444C2×C3C22×#62611yescharacteristic
445D3—122no—
446D3—364no—
447D3—362no—
448D3—362no—
449C22—21no—
450C22—21no—
451C22—182no—
452C22—182no—
453C22—182no—
454C22—542no—
455C22—542no—
456C22—542no—
457C22C2×#619212yes—
458C22C22×D3×D911yescharacteristic
459C4—64no—
460C4—182no—
461C4—542no—
462C4C2×#619212yes—
463C4C2×#619212yes—
464C3—21no—
465C3#1058961211yescharacteristic, complemented
466C3#1059504311yescharacteristic
467C2—22no—
468C2—362no—
469C2—1082no—
470C2#1172245811yescharacteristic
471C2#1172247611yescharacteristic
472C2C22×#619211yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#537
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88736C2semidirect
2. C2×C3×#3697C2semidirect
3. C2×C9×C4⋊Dic3C2semidirect
4. #11721277C2semidirect
5. #11721007C2semidirect
6. #11721004C2semidirect
7. #11720945C2semidirect
8. #11720936C2semidirect
39. C2×#3697D3semidirect
49. C2×C3×C4×C9D4semidirect
135. C2×C4⋊Dic3D9semidirect
170. C2×C4×C9#67semidirect
286. C3×C9#537semidirect
305. C2×C3×C4D36semidirect
412. C9#5833semidirect
429. C2×C4#6192semidirect
465. C3#10589612semidirect

Maximal subgroups

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

subgroupindex
C2×#887362
C2×C3×#36972
C2×C9×C4⋊Dic32
#117212772
#117210072
#117210042
#117209452
#117209362
#105950433
#105896123