Finite Group Databasebeta

← #13150208

Subgroups — #13150208

Order 1728.

976 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88658C212yesmaximal, complemented
2#11721819C211yesmaximal, characteristic, complemented
3C3×#63492C211yesmaximal, characteristic, complemented
4C2×C9×Dic3⋊C4C211yesmaximal, characteristic, complemented
5#11721277C212yesmaximal, complemented
6#11721005C212yesmaximal, complemented
7#11720995C212yesmaximal, complemented
8#11720991C212yesmaximal, complemented
9#11720983C212yesmaximal, complemented
10#10594911—31nomaximal
11#10589612—31nomaximal
12C22×C3×C4×C9C2211yescharacteristic
13C2×#6233C2212yes—
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×C3×#611C2212yes—
16C22×C9×Dic3C2211yescharacteristic
17C22×C9×Dic3C2211yescharacteristic
18C2×C3×D36C2211yescharacteristic
19C2×#6192C2212yes—
20C2×#6192C2212yes—
21C2×(C3⋊D9)⋊C4C2212yes—
22#88658—28no—
23C2×Dic3×D9C2212yes—
24C2×#6189C2212yes—
25#88536C2212yes—
26C4⋊(C3⋊Dic9)C2211yescharacteristic
27C3×(C2×D9)⋊C4C2212yes—
28C9×Dic3⋊C4C2212yes—
29C9×Dic3⋊C4C2212yes—
30C3×C4⋊Dic9C2211yescharacteristic
31#88445C2212yes—
32(C2×D9)⋊Dic3C2212yes—
33Dic3⋊Dic9C2212yes—
34C2×#3804—32no—
35#64139—31no—
36C3×#930—31no—
37C2×C3×Dic3⋊C4D311yescharacteristic
38#63978—32no—
39#63910—32no—
40#63895—32no—
41#63893—32no—
42#63886—32no—
43C2×#3697—32no—
44C2×C9×C4⋊C4—31no—
45#63492—31no—
46#63492—32no—
47#63492—61no—
48#63492D311yescharacteristic, complemented
49#63459—32no—
50#63459—32no—
51#63457—32no—
52#63457—32no—
53C23×C3×C9C2311yescharacteristic
54C22×C3⋊D9C2312yes—
55C22×C3×D9C2312yes—
56C2×C3×C4×C9—41no—
57C2×C3×C4×C9C2311yescharacteristic
58C2×C3×C4×C9C2311yescharacteristic
59#6233—24no—
60C2×C3⋊Dic9C2312yes—
61C4⋊(C3⋊D9)—41no—
62C3×#611—24no—
63C2×C9×Dic3—41no—
64C2×C9×Dic3D414yescomplemented
65C2×C9×Dic3C2312yes—
66C2×C9×Dic3C2312yes—
67C2×C3×Dic9C2312yes—
68C3×D36—41no—
69#6192—24no—
70#6192—42no—
71(C3⋊D9)⋊C4—24no—
72Dic3×D9—24no—
73#6189—24no—
74#5597—91no—
75#5510—91no—
76C22×C32×C4C2×D311yescharacteristic
77C2×#638—32no—
78C2×C4⋊(C3⋊D3)—31no—
79C2×C3×#67—32no—
80C22×C3×Dic3C2×D311yescharacteristic
81C22×C3×Dic3C2×D311yescharacteristic
82C2×C3×D12—31no—
83C2×#627—32no—
84C2×#626—32no—
85C2×#626—32no—
86C2×#624—32no—
87#3804—68no—
88C2×D3×Dic3—32no—
89#3752—32no—
90C4⋊(C3⋊Dic3)—31no—
91C3×(C2×D3)⋊C4—32no—
92C3×C4⋊Dic3—31no—
93C3×Dic3⋊C4C2×D312yes—
94C3×Dic3⋊C4C2×D312yes—
95Dic3⋊Dic3—32no—
96#3722—32no—
97(C2×D3)⋊Dic3—32no—
98C22×C4×C9—21no—
99C22×C4×C9—31no—
100C22×C4×C9—31no—
101C22×C4×C9C2×D311yescharacteristic
102C2×#611—32no—
103C2×#611—62no—
104C2×#611C2×D312yes—
105#3697—68no—
106C2×D36—32no—
107C2×D36—32no—
108C2×D36—31no—
109C2×D36—61no—
110C2×D36C2×D311yescharacteristic
111C2×C4×D9—32no—
112C2×C4×D9—32no—
113C2×Dic18—32no—
114C9×C4⋊C4—32no—
115C9×C4⋊C4—32no—
116(C2×D9)⋊C4—32no—
117(C2×D9)⋊C4—32no—
118(C2×D9)⋊C4—32no—
119(C2×D9)⋊C4—62no—
120(C2×D9)⋊C4C2×D312yes—
121C4⋊Dic9—31no—
122C4⋊Dic9—32no—
123C4⋊Dic9—61no—
124C4⋊Dic9C2×D311yescharacteristic
125C22×C3×C9—21no—
126C22×C3×C9—21no—
127C22×C3×C9C2×D412yes—
128C22×C3×C9C2411yescharacteristic
129C2×C3⋊D9—22no—
130C2×C3⋊D9—22no—
131C2×C3⋊D9—22no—
132C2×C3×D9—22no—
133C2×C3×D9—22no—
134C2×C3×D9—22no—
135C3×C4×C9—21no—
136C3×C4×C9—21no—
137C3⋊Dic9—22no—
138C9×Dic3—22no—
139C9×Dic3C2×D414yes—
140C3×Dic9—22no—
141C2×#237—92no—
142C2×#235—92no—
143C3×#121—91no—
144C2×C3×C4⋊C4—31no—
145#939—92no—
146#939—92no—
147#930—31no—
148#930—92no—
149#930—91no—
150#930—91no—
151#930—181no—
152C2×Dic3⋊C4D911yescharacteristic, complemented
153#897—92no—
154#897—92no—
155#896—92no—
156#895—92no—
157#895—92no—
158#884—92no—
159#878—92no—
160C23×C32C22×D311yescharacteristic
161C22×C3⋊D3—32no—
162C22×C3×D3—32no—
163C2×C32×C4—41no—
164C2×C32×C4C22×D311yescharacteristic
165C2×C32×C4C22×D311yescharacteristic
166#638—64no—
167C2×C3⋊Dic3—32no—
168C4⋊(C3⋊D3)—121no—
169C3×#67—64no—
170C2×C3×Dic3—32no—
171C2×C3×Dic3—41no—
172C2×C3×Dic3D1214yes—
173C2×C3×Dic3C22×D312yes—
174C2×C3×Dic3C22×D312yes—
175C3×D12—121no—
176#627—64no—
177#626—64no—
178#626—122no—
179#624—64no—
180D3×Dic3—64no—
181C23×C9—21no—
182C23×C9C22×D311yescharacteristic
183C22×D9—32no—
184C22×D9—62no—
185C22×D9C22×D312yes—
186C2×C4×C9—21no—
187C2×C4×C9—21no—
188C2×C4×C9—34no—
189C2×C4×C9—32no—
190C2×C4×C9—32no—
191C2×C4×C9—41no—
192C2×C4×C9—42no—
193C2×C4×C9—121no—
194C2×C4×C9C22×D311yescharacteristic
195C2×C4×C9C22×D311yescharacteristic
196#611—24no—
197#611—64no—
198#611—124no—
199C2×Dic9—32no—
200C2×Dic9—62no—
201C2×Dic9C22×D312yes—
202D36—41no—
203D36—64no—
204D36—121no—
205D36—122no—
206D36—122no—
207C4×D9—64no—
208C4×D9—64no—
209Dic18—64no—
210#537—271noSylow
211C2×C3×C9—22no—
212C2×C3×C9C22×D411yescharacteristic
213C2×C3×C9H(5,𝔽2)11yescharacteristic
214C2×C3×C92−1+411yescharacteristic
215C3⋊D9—42no—
216C3×D9—42no—
217C2×C3×D4—92no—
218C2×C3×D4—91no—
219C22×C3×C4—21no—
220C22×C3×C4—31no—
221C22×C3×C4—31no—
222C22×C3×C4C2×D911yescharacteristic
223C22×C3×C4D3211yescharacteristic
224C2×#67—32no—
225C2×#67—92no—
226C2×#67—92no—
227C2×#67—92no—
228C2×#67—92no—
229C2×#67—182no—
230C22×Dic3—92no—
231C22×Dic3C2×D911yescharacteristic
232C22×Dic3C2×D911yescharacteristic
233#237—188no—
234#235—188no—
235C2×D12—31no—
236C2×D12—92no—
237C2×D12—92no—
238C2×D12—91no—
239C2×D12—91no—
240C2×D12—181no—
241C2×C4×D3—92no—
242C2×C4×D3—92no—
243C2×C4×D3—92no—
244C2×Dic6—92no—
245C2×Dic6—92no—
246C3×C4⋊C4—32no—
247C3×C4⋊C4—32no—
248C3×C4⋊C4—91no—
249C3×C22⋊C4—92no—
250#217—92no—
251(C2×D3)⋊C4—32no—
252(C2×D3)⋊C4—92no—
253(C2×D3)⋊C4—92no—
254(C2×D3)⋊C4—92no—
255(C2×D3)⋊C4—92no—
256(C2×D3)⋊C4—92no—
257(C2×D3)⋊C4—182no—
258C4⋊Dic3—31no—
259C4⋊Dic3—92no—
260C4⋊Dic3—91no—
261C4⋊Dic3—91no—
262C4⋊Dic3—181no—
263Dic3⋊C4—92no—
264Dic3⋊C4C2×D912yes—
265Dic3⋊C4C2×D912yes—
266C22×C32—21no—
267C22×C32—21no—
268C22×C32C2×D1212yes—
269C22×C32C23×D311yescharacteristic
270C2×C3⋊D3—62no—
271C2×C3⋊D3—62no—
272C2×C3⋊D3—62no—
273C2×C3×D3—62no—
274C2×C3×D3—62no—
275C2×C3×D3—62no—
276C32×C4—21no—
277C32×C4—21no—
278C3⋊Dic3—62no—
279C3×Dic3—22no—
280C3×Dic3—62no—
281C3×Dic3C2×D1214yes—
282C22×C9—21no—
283C22×C9—21no—
284C22×C9—21no—
285C22×C9—22no—
286C22×C9—41no—
287C22×C9—41no—
288C22×C9D3×D412yes—
289C22×C9C23×D311yescharacteristic
290C2×D9—22no—
291C2×D9—22no—
292C2×D9—22no—
293C2×D9—62no—
294C2×D9—62no—
295C2×D9—62no—
296C2×D9—122no—
297C2×D9—122no—
298C2×D9—122no—
299C4×C9—22no—
300C4×C9—21no—
301C4×C9—21no—
302C4×C9—34no—
303C4×C9—41no—
304C4×C9—62no—
305Dic9—22no—
306Dic9—62no—
307Dic9—122no—
308C2×#41—272no—
309C22⋊Q8—272no—
310C22⋊Q8—272no—
311#121—91no—
312#121—272no—
313#121—272no—
314#121—272no—
315#121—271no—
316C2×C4⋊C4—31no—
317C3×C9#53711yescharacteristic, complemented, Sylow
318C23×C3—21no—
319C23×C3—92no—
320C23×C3C22×D911yescharacteristic
321C23×C3C2×D3211yescharacteristic
322C22×D3—32no—
323C22×D3—92no—
324C22×D3—92no—
325C22×D3—182no—
326C3×D4—184no—
327C3×D4—361no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—32no—
331C2×C3×C4—34no—
332C2×C3×C4—32no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—42no—
336C2×C3×C4—92no—
337C2×C3×C4—121no—
338C2×C3×C4C22×D911yescharacteristic
339C2×C3×C4C22×D911yescharacteristic
340C2×C3×C4C2×D3211yescharacteristic
341C2×C3×C4C2×D3211yescharacteristic
342#67—64no—
343#67—184no—
344#67—184no—
345#67—184no—
346#67—364no—
347#67—362no—
348C2×Dic3—32no—
349C2×Dic3—41no—
350C2×Dic3—92no—
351C2×Dic3—92no—
352C2×Dic3—182no—
353C2×Dic3—184no—
354C2×Dic3D3614yescomplemented
355C2×Dic3C22×D912yes—
356C2×Dic3C22×D912yes—
357D12—121no—
358D12—184no—
359D12—361no—
360D12—361no—
361D12—362no—
362D12—362no—
363C4×D3—184no—
364C4×D3—184no—
365C4×D3—184no—
366Dic6—184no—
367Dic6—184no—
368C2×C32—22no—
369C2×C32C22×D1211yescharacteristic
370C2×C32#100411yescharacteristic
371C2×C32#100711yescharacteristic
372C3⋊D3—122no—
373C3×D3—122no—
374C2×C9—22no—
375C2×C9—21no—
376C2×C9—21no—
377C2×C9—21no—
378C2×C9—42no—
379C2×C9C2×D3×D411yescharacteristic
380C2×C9#100411yescharacteristic
381C2×C9#101011yescharacteristic
382D9—42no—
383D9—122no—
384D9—242no—
385#41—548no—
386C2×Q8—272no—
387C2×D4—92no—
388C2×D4—91no—
389C2×D4—272no—
390C2×D4—272no—
391C2×D4—272no—
392C2×D4—271no—
393C22×C4—31no—
394C22×C4—31no—
395C22×C4—272no—
396C22×C4—272no—
397C22×C4D3×D911yescharacteristic
398C4⋊C4—32no—
399C4⋊C4—32no—
400C4⋊C4—91no—
401C4⋊C4—272no—
402C4⋊C4—271no—
403C22⋊C4—92no—
404C22⋊C4—272no—
405C22⋊C4—272no—
406C22⋊C4—272no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—21no—
412C22×C3—22no—
413C22×C3—41no—
414C22×C3—41no—
415C22×C3—182no—
416C22×C3—182no—
417C22×C3—182no—
418C22×C3C2×D3612yes—
419C22×C3C23×D911yescharacteristic
420C22×C3D3×D1212yes—
421C22×C3C22×D3211yescharacteristic
422C2×D3—62no—
423C2×D3—62no—
424C2×D3—62no—
425C2×D3—182no—
426C2×D3—182no—
427C2×D3—182no—
428C2×D3—182no—
429C2×D3—182no—
430C2×D3—182no—
431C2×D3—362no—
432C2×D3—362no—
433C2×D3—362no—
434C3×C4—22no—
435C3×C4—21no—
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—34no—
440C3×C4—41no—
441C3×C4—62no—
442C3×C4—182no—
443Dic3—22no—
444Dic3—62no—
445Dic3—182no—
446Dic3—182no—
447Dic3—362no—
448Dic3C2×D3614yes—
449C32#551011yescharacteristic
450C9—21no—
451C9#559711yescharacteristic, complemented
452C23—92no—
453C23—272no—
454C23C2×D3×D911yescharacteristic
455Q8—544no—
456D4—184no—
457D4—361no—
458D4—544no—
459D4—544no—
460D4—1081no—
461D4—1082no—
462C2×C4—32no—
463C2×C4—34no—
464C2×C4—32no—
465C2×C4—41no—
466C2×C4—92no—
467C2×C4—121no—
468C2×C4—272no—
469C2×C4—544no—
470C2×C4—544no—
471C2×C4C2×D3×D911yescharacteristic
472C2×C4C2×D3×D911yescharacteristic
473C2×C3—22no—
474C2×C3—22no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—42no—
479C2×C3—362no—
480C2×C3C22×D3611yescharacteristic
481C2×C3#6371011yescharacteristic
482C2×C3#6371311yescharacteristic
483C2×C3#6429711yescharacteristic
484C2×C3C2×D3×D1211yescharacteristic
485C2×C3#6433011yescharacteristic
486D3—122no—
487D3—362no—
488D3—362no—
489D3—722no—
490C22—21no—
491C22—21no—
492C22—182no—
493C22—182no—
494C22—182no—
495C22—542no—
496C22—542no—
497C22—542no—
498C22D3×D3612yes—
499C22C22×D3×D911yescharacteristic
500C4—21no—
501C4—21no—
502C4—34no—
503C4—62no—
504C4—182no—
505C4—542no—
506C3—21no—
507C3#1058961211yescharacteristic, complemented
508C3#1059491111yescharacteristic
509C2—22no—
510C2—362no—
511C2—1082no—
512C2#1172245811yescharacteristic
513C2C2×D3×D3611yescharacteristic
514C2#1172253511yescharacteristic
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×#88658C2semidirect
2. #11721819C2semidirect
3. C3×#63492C2semidirect
4. C2×C9×Dic3⋊C4C2semidirect
5. #11721277C2semidirect
6. #11721005C2semidirect
7. #11720995C2semidirect
8. #11720991C2semidirect
9. #11720983C2semidirect
48. #63492D3semidirect
64. C2×C9×Dic3D4semidirect
152. C2×Dic3⋊C4D9semidirect
317. C3×C9#537semidirect
354. C2×Dic3D36semidirect
451. C9#5597semidirect
507. C3#10589612semidirect

Maximal subgroups

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

subgroupindex
C2×#886582
#117218192
C3×#634922
C2×C9×Dic3⋊C42
#117212772
#117210052
#117209952
#117209912
#117209832
#105949113
#105896123