Finite Group Databasebeta

← #13177239

Subgroups — #13177239

Order 1728.

2028 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×#88859C211yesmaximal, characteristic, complemented
2C3×#64139C211yesmaximal, characteristic, complemented
3C32×#924C211yesmaximal, characteristic, complemented
4#11723639C212yesmaximal, complemented
5#11723626C212yesmaximal, complemented
6#11723493C212yesmaximal, complemented
7#11723482C212yesmaximal, complemented
8#11723403C212yesmaximal, complemented
9#11723392C212yesmaximal, complemented
10#10596115—31nomaximal
11#10594979—34nomaximal
12C23×C3⋊(C3⋊D3)C2211yescharacteristic
13C22×C33×C4C2211yescharacteristic
14C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×#638C2212yes—
16C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
17C2×C32×Dic6C2211yescharacteristic, complemented
18C2×#6292C2212yes—
19C2×#6292C2212yes—
20C2×Dic3×C3⋊D3C2212yescomplemented
21#88859C2212yes—
22#88859C2212yes—
23C3×#3752C2212yescomplemented
24C3×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
25C32×#217C2212yes—
26C32×C4⋊Dic3C2211yescharacteristic, complemented
27C32×Dic3⋊C4C2212yescomplemented
28#88807C2212yescomplemented
29#88804C2212yes—
30#88804C2212yes—
31#88803C2212yescomplemented
32Dic3×C3⋊Dic3C2212yescomplemented
33C32×C22⋊Q8—31no—
34#64156—32no—
35#64139D311yescharacteristic, complemented
36C2×#3752—31no—
37C2×#3752—34no—
38C2×#3752—64no—
39#64107—32no—
40#64104—32no—
41#64103—32no—
42#64091—32no—
43#64090—32no—
44C3×#930—34no—
45C3×#924D314yescomplemented
46#63981—38no—
47#67⋊Dic3—38no—
48#63907—38no—
49#63890—38no—
50#63886—38no—
51#63848—38no—
52C23×C33C2311yescharacteristic
53C22×C3⋊(C3⋊D3)—22no—
54C22×C3⋊(C3⋊D3)—41no—
55C22×C3⋊(C3⋊D3)—41no—
56C22×C3⋊(C3⋊D3)C2312yes—
57C22×C3×C3⋊D3C2312yes—
58C2×C33×C4—41no—
59C2×C33×C4C2311yescharacteristic
60C2×C33×C4C2311yescharacteristic
61C2×C3⋊(C3⋊Dic3)—41no—
62C2×C3⋊(C3⋊Dic3)C2312yes—
63C3×#638—24no—
64C2×C3×C3⋊Dic3C2312yes—
65C3×C4⋊(C3⋊D3)—41no—
66C2×C32×Dic3C2312yes—
67C2×C32×Dic3C2312yes—
68C32×Dic6—41no—
69#6292—42no—
70#6292—42no—
71Dic3×C3⋊D3—42no—
72#5644—94no—
73#5617—91no—
74C23×C3⋊D3—31no—
75C23×C3⋊D3—34no—
76C23×C3⋊D3—64no—
77C2×C32×Q8—31no—
78C22×C32×C4—24no—
79C22×C32×C4C2×D314yes—
80C22×C32×C4C2×D311yescharacteristic
81C2×#638C2×D312yes—
82C22×C3⋊Dic3—31no—
83C22×C3⋊Dic3—34no—
84C22×C3⋊Dic3—64no—
85C2×C4⋊(C3⋊D3)—32no—
86C2×C4⋊(C3⋊D3)—32no—
87C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
88C2×C4×C3⋊D3—32no—
89C2×C3×#67—38no—
90C2×C3×D12—34no—
91C2×C3×Dic6C2×D314yescomplemented
92C2×#626—38no—
93C2×#626—38no—
94C2×D3×Dic3—38no—
95C32×C4⋊C4—31no—
96C32×C4⋊C4—32no—
97C32×C22⋊C4—32no—
98#3752—32no—
99#3752—32no—
100#3752—32no—
101#3752—38no—
102#3752—38no—
103#3752—32no—
104#3752—32no—
105#3752—68no—
106#3752—68no—
107#3752C2×D312yescomplemented
108C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
109#3750—32no—
110C4×C3⋊Dic3—32no—
111C3×#217C2×D318yes—
112C3×(C2×D3)⋊C4—38no—
113C3×C4⋊Dic3—34no—
114C3×C4⋊Dic3C2×D314yescomplemented
115C3×Dic3⋊C4C2×D318yescomplemented
116Dic3⋊Dic3—38no—
117#3722—38no—
118#3722—38no—
119(C2×D3)⋊Dic3—38no—
120Dic32—38no—
121C22×C33—21no—
122C22×C33—21no—
123C22×C33#4112yes—
124C22×C33C2411yescharacteristic
125C2×C3⋊(C3⋊D3)—22no—
126C2×C3⋊(C3⋊D3)—22no—
127C2×C3⋊(C3⋊D3)—22no—
128C2×C3⋊(C3⋊D3)—44no—
129C2×C3×C3⋊D3—22no—
130C2×C3×C3⋊D3—22no—
131C2×C3×C3⋊D3—22no—
132C33×C4—21no—
133C33×C4—21no—
134C3⋊(C3⋊Dic3)—22no—
135C3×C3⋊Dic3—22no—
136C32×Dic3—22no—
137C32×Dic3—22no—
138C3×C22⋊Q8—34no—
139C3×#121—91no—
140#947—98no—
141#939—92no—
142#937—92no—
143D4×Dic3—92no—
144#930—34no—
145C2×(C2×D3)⋊C4—94no—
146C2×(C2×D3)⋊C4—91no—
147C2×(C2×D3)⋊C4—184no—
148#924C3⋊D311yescharacteristic, complemented
149#898—98no—
150#898—92no—
151#895—98no—
152#894—98no—
153#886—92no—
154#885—92no—
155#882—98no—
156#67⋊C4—98no—
157C23×C32—24no—
158C23×C32C22×D314yes—
159C23×C32C22×D311yescharacteristic
160C22×C3⋊D3—32no—
161C22×C3⋊D3—38no—
162C22×C3⋊D3—68no—
163C22×C3⋊D3—616no—
164C22×C3⋊D3—62no—
165C22×C3⋊D3—68no—
166C22×C3⋊D3—121no—
167C22×C3⋊D3—121no—
168C22×C3⋊D3—124no—
169C22×C3⋊D3—124no—
170C22×C3⋊D3—128no—
171C22×C3⋊D3—128no—
172C22×C3⋊D3C22×D312yes—
173C22×C3×D3—38no—
174C32×Q8—121no—
175C2×C32×C4—24no—
176C2×C32×C4—24no—
177C2×C32×C4—32no—
178C2×C32×C4—32no—
179C2×C32×C4—41no—
180C2×C32×C4—44no—
181C2×C32×C4—48no—
182C2×C32×C4C22×D314yes—
183C2×C32×C4C22×D314yes—
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D311yescharacteristic
186#638—24no—
187C2×C3⋊Dic3—32no—
188C2×C3⋊Dic3—38no—
189C2×C3⋊Dic3—68no—
190C2×C3⋊Dic3—121no—
191C2×C3⋊Dic3—124no—
192C2×C3⋊Dic3—128no—
193C2×C3⋊Dic3C22×D312yes—
194C4⋊(C3⋊D3)—41no—
195C4⋊(C3⋊D3)—122no—
196C4⋊(C3⋊D3)—122no—
197C4×C3⋊D3—122no—
198C3×#67—616no—
199C2×C3×Dic3—38no—
200C2×C3×Dic3C22×D318yes—
201C2×C3×Dic3C22×D318yes—
202C3×D12—124no—
203C3×Dic6—44no—
204#626—128no—
205#626—128no—
206D3×Dic3—128no—
207#538—271noSylow
208C2×C33—22no—
209C2×C33C2×#4111yescharacteristic
210C2×C33H(5,𝔽2)11yescharacteristic
211C2×C33H(5,𝔽2)11yescharacteristic
212C3⋊(C3⋊D3)—42no—
213C3×C3⋊D3—42no—
214C23×D3—94no—
215C23×D3—91no—
216C23×D3—184no—
217C2×C3×Q8—34no—
218C2×C3×D4—92no—
219C2×C3×D4—91no—
220C22×C3×C4—24no—
221C22×C3×C4D3214yes—
222C22×C3×C4C2×C3⋊D311yescharacteristic
223C2×#67—38no—
224C2×#67—92no—
225C2×#67—92no—
226C22×Dic3—94no—
227C22×Dic3—91no—
228C22×Dic3—92no—
229C22×Dic3—184no—
230C2×D12—34no—
231C2×D12—98no—
232C2×D12—98no—
233C2×C4×D3—98no—
234C2×Dic6C2×C3⋊D311yescharacteristic, complemented
235C3×C4⋊C4—34no—
236C3×C4⋊C4—38no—
237C3×C4⋊C4—91no—
238C3×C22⋊C4—38no—
239C3×C22⋊C4—92no—
240#217—92no—
241#217C2×C3⋊D312yes—
242(C2×D3)⋊C4—38no—
243(C2×D3)⋊C4—98no—
244(C2×D3)⋊C4—98no—
245(C2×D3)⋊C4—98no—
246(C2×D3)⋊C4—98no—
247(C2×D3)⋊C4—98no—
248(C2×D3)⋊C4—92no—
249(C2×D3)⋊C4—92no—
250(C2×D3)⋊C4—92no—
251(C2×D3)⋊C4—92no—
252(C2×D3)⋊C4—188no—
253(C2×D3)⋊C4—188no—
254C4⋊Dic3—34no—
255C4⋊Dic3—92no—
256C4⋊Dic3C2×C3⋊D311yescharacteristic, complemented
257Dic3⋊C4—98no—
258Dic3⋊C4C2×C3⋊D312yescomplemented
259C4×Dic3—98no—
260C4×Dic3—92no—
261C22×C32—28no—
262C22×C32—28no—
263C22×C32—24no—
264C22×C32—21no—
265C22×C32—24no—
266C22×C32—21no—
267C22×C32—24no—
268C22×C32—28no—
269C22×C32#23712yes—
270C22×C32#23918yes—
271C22×C32C23×D314yes—
272C22×C32C23×D311yescharacteristic
273C2×C3⋊D3—22no—
274C2×C3⋊D3—22no—
275C2×C3⋊D3—22no—
276C2×C3⋊D3—616no—
277C2×C3⋊D3—62no—
278C2×C3⋊D3—62no—
279C2×C3⋊D3—68no—
280C2×C3⋊D3—62no—
281C2×C3⋊D3—68no—
282C2×C3⋊D3—68no—
283C2×C3⋊D3—128no—
284C2×C3⋊D3—128no—
285C2×C3⋊D3—1232no—
286C2×C3⋊D3—1216no—
287C2×C3⋊D3—124no—
288C2×C3×D3—68no—
289C2×C3×D3—68no—
290C2×C3×D3—68no—
291C32×C4—21no—
292C32×C4—21no—
293C32×C4—24no—
294C32×C4—24no—
295C32×C4—44no—
296C32×C4—44no—
297C32×C4—62no—
298C32×C4—62no—
299C3⋊Dic3—22no—
300C3⋊Dic3—62no—
301C3⋊Dic3—68no—
302C3⋊Dic3—128no—
303C3×Dic3—28no—
304C3×Dic3—28no—
305C3×Dic3—68no—
306#126—272no—
307#124—272no—
308#123—272no—
309C22⋊Q8—31no—
310#121—91no—
311#121—272no—
312#120—272no—
313C4×D4—272no—
314C2×C22⋊C4—271no—
315C33#53811yescharacteristic, complemented, Sylow
316C23×C3—24no—
317C23×C3—92no—
318C23×C3C2×D3214yes—
319C23×C3C22×C3⋊D311yescharacteristic
320C22×D3—38no—
321C22×D3—98no—
322C22×D3—92no—
323C22×D3—188no—
324C22×D3—1816no—
325C22×D3—188no—
326C22×D3—182no—
327C22×D3—364no—
328C22×D3—364no—
329C22×D3—361no—
330C22×D3—361no—
331C22×D3—368no—
332C22×D3—368no—
333C3×Q8—124no—
334C3×D4—184no—
335C3×D4—361no—
336C2×C3×C4—24no—
337C2×C3×C4—24no—
338C2×C3×C4—38no—
339C2×C3×C4—38no—
340C2×C3×C4—44no—
341C2×C3×C4—41no—
342C2×C3×C4—48no—
343C2×C3×C4—92no—
344C2×C3×C4C2×D3214yes—
345C2×C3×C4C2×D3214yes—
346C2×C3×C4C22×C3⋊D311yescharacteristic
347C2×C3×C4C22×C3⋊D311yescharacteristic
348#67—616no—
349#67—362no—
350#67—362no—
351C2×Dic3—38no—
352C2×Dic3—98no—
353C2×Dic3—92no—
354C2×Dic3—188no—
355C2×Dic3—362no—
356C2×Dic3—364no—
357C2×Dic3—361no—
358C2×Dic3—368no—
359C2×Dic3C22×C3⋊D312yes—
360C2×Dic3C22×C3⋊D312yes—
361D12—124no—
362D12—368no—
363D12—368no—
364C4×D3—368no—
365Dic6—41no—
366C2×C32—216no—
367C2×C32—28no—
368C2×C32—22no—
369C2×C32—24no—
370C2×C32—24no—
371C2×C32—24no—
372C2×C32C2×#23711yescharacteristic
373C2×C32#100414yes—
374C2×C32#100411yescharacteristic
375C2×C32C2×#23914yes—
376C2×C32#100914yes—
377C2×C32#100911yescharacteristic
378C3⋊D3—42no—
379C3⋊D3—1216no—
380C3⋊D3—128no—
381C3⋊D3—122no—
382C3×D3—128no—
383C24—271no—
384C2×Q8—31no—
385C2×D4—92no—
386C2×D4—91no—
387C2×D4—272no—
388C2×D4—272no—
389C22×C4—272no—
390C22×C4—271no—
391C22×C4D3×C3⋊D311yescharacteristic
392C4⋊C4—32no—
393C4⋊C4—31no—
394C4⋊C4—91no—
395C4⋊C4—272no—
396C22⋊C4—32no—
397C22⋊C4—92no—
398C22⋊C4—272no—
399C22⋊C4—272no—
400C22⋊C4—272no—
401C22⋊C4—272no—
402C22⋊C4—272no—
403C42—272no—
404C22×C3—28no—
405C22×C3—28no—
406C22×C3—24no—
407C22×C3—24no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—24no—
411C22×C3—28no—
412C22×C3—182no—
413C22×C3—182no—
414C22×C3—182no—
415C22×C3#379618yes—
416C22×C3#383212yes—
417C22×C3C22×D3214yes—
418C22×C3C23×C3⋊D311yescharacteristic
419C2×D3—68no—
420C2×D3—68no—
421C2×D3—68no—
422C2×D3—1816no—
423C2×D3—188no—
424C2×D3—188no—
425C2×D3—188no—
426C2×D3—182no—
427C2×D3—182no—
428C2×D3—182no—
429C2×D3—368no—
430C2×D3—368no—
431C2×D3—3632no—
432C2×D3—364no—
433C2×D3—3616no—
434C3×C4—24no—
435C3×C4—24no—
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—44no—
439C3×C4—44no—
440C3×C4—68no—
441C3×C4—68no—
442C3×C4—182no—
443Dic3—22no—
444Dic3—22no—
445Dic3—68no—
446Dic3—188no—
447Dic3—182no—
448Dic3—368no—
449C32—24no—
450C32#561711yescharacteristic, complemented
451C32#564414yescomplemented
452C23—92no—
453C23—272no—
454C23—542no—
455C23—1081no—
456C23—1081no—
457C23C2×D3×C3⋊D311yescharacteristic
458Q8—121no—
459D4—184no—
460D4—361no—
461D4—1082no—
462D4—1082no—
463C2×C4—32no—
464C2×C4—32no—
465C2×C4—41no—
466C2×C4—92no—
467C2×C4—272no—
468C2×C4—1082no—
469C2×C4—1081no—
470C2×C4C2×D3×C3⋊D311yescharacteristic
471C2×C4C2×D3×C3⋊D311yescharacteristic
472C2×C3—216no—
473C2×C3—22no—
474C2×C3—28no—
475C2×C3—24no—
476C2×C3—24no—
477C2×C3—24no—
478C2×C3—362no—
479C2×C3C2×#379614yes—
480C2×C3#6430814yes—
481C2×C3#6433014yes—
482C2×C3#6436111yescharacteristic
483C2×C3C2×#383211yescharacteristic
484C2×C3#6436611yescharacteristic
485D3—128no—
486D3—3616no—
487D3—362no—
488D3—368no—
489C22—21no—
490C22—21no—
491C22—182no—
492C22—182no—
493C22—182no—
494C22—542no—
495C22—542no—
496C22—542no—
497C22—1084no—
498C22#8901712yes—
499C22C22×D3×C3⋊D311yescharacteristic
500C4—21no—
501C4—21no—
502C4—62no—
503C4—62no—
504C4—182no—
505C4—542no—
506C3—24no—
507C3#1059497914yescomplemented
508C3#1059611511yescharacteristic, complemented
509C2—22no—
510C2—362no—
511C2—1082no—
512C2C2×#8901711yescharacteristic
513C2#1172458111yescharacteristic
514C2#1172461111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC22×C33×C4C22
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#538
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88859C2semidirect
2. C3×#64139C2semidirect
3. C32×#924C2semidirect
4. #11723639C2semidirect
5. #11723626C2semidirect
6. #11723493C2semidirect
7. #11723482C2semidirect
8. #11723403C2semidirect
9. #11723392C2semidirect
16. C2×C3×C4⋊(C3⋊D3)C22semidirect
17. C2×C32×Dic6C22semidirect
20. C2×Dic3×C3⋊D3C22semidirect
23. C3×#3752C22semidirect
24. C3×C4⋊(C3⋊Dic3)C22semidirect
26. C32×C4⋊Dic3C22semidirect
27. C32×Dic3⋊C4C22semidirect
28. #88807C22semidirect
31. #88803C22semidirect
32. Dic3×C3⋊Dic3C22semidirect
35. #64139D3semidirect
45. C3×#924D3semidirect
87. C2×C4⋊(C3⋊D3)C2×D3semidirect
91. C2×C3×Dic6C2×D3semidirect
107. #3752C2×D3semidirect
108. C4⋊(C3⋊Dic3)C2×D3semidirect
114. C3×C4⋊Dic3C2×D3semidirect
115. C3×Dic3⋊C4C2×D3semidirect
148. #924C3⋊D3semidirect
234. C2×Dic6C2×C3⋊D3semidirect
256. C4⋊Dic3C2×C3⋊D3semidirect
258. Dic3⋊C4C2×C3⋊D3semidirect
315. C33#538semidirect
450. C32#5617semidirect
451. C32#5644semidirect
507. C3#10594979semidirect
508. C3#10596115semidirect

Maximal subgroups

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

subgroupindex
C2×#888592
C3×#641392
C32×#9242
#117236392
#117236262
#117234932
#117234822
#117234032
#117233922
#105961153
#105949793