Finite Group Databasebeta

← #13177637

Subgroups — #13177637

Order 1728.

2080 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723978C211yesmaximal, characteristic, complemented
2C3×#64145C211yesmaximal, characteristic, complemented
3C32×#937C211yesmaximal, characteristic, complemented
4#11723638C212yesmaximal, complemented
5#11723630C212yesmaximal, complemented
6#11723600C212yesmaximal, complemented
7#11723592C212yesmaximal, complemented
8#11723521C212yesmaximal, complemented
9#11723419C212yesmaximal, complemented
10#10596065—31nomaximal
11#10595262—34nomaximal
12C23×C3⋊(C3⋊D3)C2211yescharacteristic, complemented
13C23×C32×D3C2211yescharacteristic, complemented
14C2×C33×D4C2211yescharacteristic
15C2×#6312C2212yes—
16C2×C3×#634C2211yescharacteristic, complemented
17C2×C32×#67C2212yes—
18C2×#6293C2212yescomplemented
19C2×#6291C2212yes—
20C2×#6291C2212yes—
21#88864C2211yescharacteristic, complemented
22#88859C2212yescomplemented
23C3×#3757C2212yescomplemented
24C3×#3757C2212yescomplemented
25C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
26C32×#217C2211yescharacteristic, complemented
27C32×(C2×D3)⋊C4C2212yescomplemented
28#88804C2212yescomplemented
29#88804C2212yescomplemented
30(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
31(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
32Dic3×C3⋊Dic3C2212yescomplemented
33C32×#120—31no—
34#64162—32no—
35#64146—31no—
36#64146—32no—
37#64146—34no—
38#64146—64no—
39#64145—32no—
40#64145D311yescharacteristic, complemented
41#64094—32no—
42#64094—32no—
43#64094—32no—
44C3×#937D314yescomplemented
45C3×#936—34no—
46#63981—38no—
47#63977—38no—
48#63959—38no—
49#63953—38no—
50#63907—38no—
51#63852—38no—
52C23×C33C2311yescharacteristic
53C23×C33C2311yescharacteristic
54C22×C3⋊(C3⋊D3)—42no—
55C22×C3⋊(C3⋊D3)—41no—
56C22×C3⋊(C3⋊D3)C2312yes—
57C22×C32×D3—41no—
58C22×C32×D3—42no—
59C22×C32×D3C2312yes—
60C33×D4—41no—
61C2×C33×C4C2311yescharacteristic
62#6312—42no—
63C2×C3⋊(C3⋊Dic3)C2312yes—
64C2×C3×C3⋊Dic3C2312yes—
65C2×C3×C3⋊Dic3C2312yes—
66C3×#634—41no—
67C32×#67—42no—
68C2×C32×Dic3C2312yes—
69#6293—42no—
70#6291—42no—
71#6291—42no—
72#5683—91no—
73#5594—94no—
74C24×C32—31no—
75C23×C3⋊D3—31no—
76C23×C3⋊D3—34no—
77C23×C3⋊D3—64no—
78C23×C3×D3C2×D314yescomplemented
79C2×C32×D4—24no—
80C2×C32×D4—32no—
81C2×C32×D4C2×D314yes—
82C2×C32×D4C2×D311yescharacteristic
83C2×#638—32no—
84C2×#638—38no—
85C2×#638—32no—
86C2×#638—32no—
87C2×#638—68no—
88C2×#634—32no—
89C2×#634C2×D311yescharacteristic, complemented
90C2×C3×#67C2×D318yes—
91C2×C3×Dic6—34no—
92C2×#627—38no—
93C2×#626—38no—
94C2×#626—38no—
95C32×C22⋊C4—31no—
96C32×C22⋊C4—32no—
97#3757—31no—
98#3757—34no—
99#3757—32no—
100#3757—32no—
101#3757—64no—
102#3757C2×D312yescomplemented
103#3757C2×D312yescomplemented
104#3752—32no—
105#3752—38no—
106#3752—32no—
107#3752—32no—
108#3752—68no—
109C4×C3⋊Dic3—32no—
110C4×C3⋊Dic3C2×D311yescharacteristic, complemented
111C3×#217—38no—
112C3×#217—38no—
113C3×#217C2×D314yescomplemented
114C3×(C2×D3)⋊C4C2×D318yescomplemented
115C3×C4×Dic3—34no—
116#3722—38no—
117#3722—38no—
118(C2×D3)⋊Dic3—38no—
119(C2×D3)⋊Dic3—38no—
120Dic32—38no—
121C22×C33—21no—
122C22×C33—22no—
123C22×C33—22no—
124C22×C33—21no—
125C22×C33C2411yescharacteristic
126C2×C3⋊(C3⋊D3)—22no—
127C2×C3⋊(C3⋊D3)—22no—
128C2×C3⋊(C3⋊D3)—22no—
129C2×C3⋊(C3⋊D3)—44no—
130C2×C32×D3—22no—
131C2×C32×D3—22no—
132C2×C32×D3—22no—
133C2×C32×D3—44no—
134C33×C4—21no—
135C3⋊(C3⋊Dic3)—22no—
136C3×C3⋊Dic3—22no—
137C3×C3⋊Dic3—22no—
138C32×Dic3—22no—
139C3×#124—91no—
140C3×#120—34no—
141#953—98no—
142#947—92no—
143#937—98no—
144#937—94no—
145#937—91no—
146#937—184no—
147#937C3⋊D311yescharacteristic, complemented
148#936—34no—
149#936—98no—
150#885—98no—
151#885—98no—
152#885—98no—
153#885—92no—
154#885—92no—
155#882—92no—
156#882—92no—
157#875—92no—
158C23×C32—24no—
159C23×C32—24no—
160C23×C32—32no—
161C23×C32—121no—
162C23×C32—122no—
163C23×C32C22×D314yes—
164C23×C32C22×D314yes—
165C23×C32C22×D311yescharacteristic
166C23×C32C22×D311yescharacteristic
167C22×C3⋊D3—32no—
168C22×C3⋊D3—38no—
169C22×C3⋊D3—68no—
170C22×C3⋊D3—121no—
171C22×C3⋊D3—122no—
172C22×C3⋊D3—128no—
173C22×C3⋊D3—124no—
174C22×C3⋊D3—1216no—
175C22×C3⋊D3—128no—
176C22×C3×D3—44no—
177C22×C3×D3—48no—
178C22×C3×D3C22×D318yes—
179C32×D4—41no—
180C32×D4—44no—
181C32×D4—48no—
182C32×D4—122no—
183C2×C32×C4—24no—
184C2×C32×C4—32no—
185C2×C32×C4C22×D314yes—
186C2×C32×C4C22×D311yescharacteristic
187#638—122no—
188#638—122no—
189#638—122no—
190#638—128no—
191#638—1216no—
192C2×C3⋊Dic3—32no—
193C2×C3⋊Dic3—38no—
194C2×C3⋊Dic3—68no—
195C2×C3⋊Dic3C22×D312yes—
196C2×C3⋊Dic3C22×D312yes—
197#634—41no—
198#634—122no—
199C3×#67—48no—
200C2×C3×Dic3—38no—
201C2×C3×Dic3—38no—
202C2×C3×Dic3C22×D318yes—
203C3×Dic6—124no—
204#627—128no—
205#626—128no—
206#626—128no—
207H(3,𝔽4)—271noSylow
208C2×C33—41no—
209C2×C33—41no—
210C2×C33H(5,𝔽2)11yescharacteristic
211C2×C33H(5,𝔽2)12yes—
212C3⋊(C3⋊D3)—42no—
213C32×D3—42no—
214C24×C3—34no—
215C23×D3—94no—
216C23×D3—91no—
217C23×D3—184no—
218C23×D3C2×C3⋊D311yescharacteristic, complemented
219C2×C3×Q8—91no—
220C2×C3×D4—24no—
221C2×C3×D4—38no—
222C2×C3×D4D3214yes—
223C2×C3×D4C2×C3⋊D311yescharacteristic
224C2×#67—98no—
225C2×#67—98no—
226C2×#67—98no—
227C2×#67—92no—
228C2×#67—188no—
229C2×#67C2×C3⋊D312yes—
230C2×D12—92no—
231C2×D12—92no—
232C2×Dic6—34no—
233C2×Dic6—98no—
234C2×Dic6—92no—
235C3×C22⋊C4—34no—
236C3×C22⋊C4—38no—
237C3×C22⋊C4—92no—
238C3×C22⋊C4—92no—
239C3×C42—91no—
240#217—38no—
241#217—38no—
242#217—98no—
243#217—98no—
244#217—94no—
245#217—91no—
246#217—184no—
247#217C2×C3⋊D311yescharacteristic, complemented
248(C2×D3)⋊C4—98no—
249(C2×D3)⋊C4—98no—
250(C2×D3)⋊C4—92no—
251(C2×D3)⋊C4—92no—
252(C2×D3)⋊C4—98no—
253(C2×D3)⋊C4—92no—
254(C2×D3)⋊C4—92no—
255(C2×D3)⋊C4—92no—
256(C2×D3)⋊C4—188no—
257(C2×D3)⋊C4C2×C3⋊D312yescomplemented
258C4×Dic3—34no—
259C4×Dic3—98no—
260C4×Dic3—92no—
261C22×C32—28no—
262C22×C32—24no—
263C22×C32—21no—
264C22×C32—21no—
265C22×C32—22no—
266C22×C32—22no—
267C22×C32—24no—
268C22×C32—28no—
269C22×C32—28no—
270C22×C32—24no—
271C22×C32—48no—
272C22×C32—48no—
273C22×C32—44no—
274C22×C32—62no—
275C22×C32—62no—
276C22×C32—62no—
277C22×C32—124no—
278C22×C32C23×D314yes—
279C22×C32C23×D311yescharacteristic
280C2×C3⋊D3—616no—
281C2×C3⋊D3—68no—
282C2×C3⋊D3—62no—
283C2×C3⋊D3—62no—
284C2×C3⋊D3—62no—
285C2×C3⋊D3—68no—
286C2×C3⋊D3—68no—
287C2×C3⋊D3—128no—
288C2×C3⋊D3—128no—
289C2×C3⋊D3—1232no—
290C2×C3⋊D3—124no—
291C2×C3⋊D3—1216no—
292C2×C3×D3—28no—
293C2×C3×D3—28no—
294C2×C3×D3—28no—
295C2×C3×D3—416no—
296C32×C4—21no—
297C32×C4—24no—
298C32×C4—44no—
299C32×C4—62no—
300C3⋊Dic3—22no—
301C3⋊Dic3—22no—
302C3⋊Dic3—62no—
303C3⋊Dic3—68no—
304C3⋊Dic3—128no—
305C3×Dic3—28no—
306C3×Dic3—68no—
307C3×Dic3—68no—
308#124—91no—
309#124—272no—
310#124—272no—
311#124—272no—
312#124—272no—
313#120—31no—
314#120—272no—
315#120—272no—
316#120—271no—
317C33H(3,𝔽4)11yescharacteristic, complemented, Sylow
318C23×C3—24no—
319C23×C3—24no—
320C23×C3—38no—
321C23×C3—128no—
322C23×C3—124no—
323C23×C3C2×D3214yes—
324C23×C3C2×D3214yes—
325C23×C3C22×C3⋊D311yescharacteristic
326C23×C3C22×C3⋊D311yescharacteristic
327C22×D3—41no—
328C22×D3—42no—
329C22×D3—98no—
330C22×D3—92no—
331C22×D3—188no—
332C22×D3—368no—
333C22×D3—364no—
334C22×D3—361no—
335C22×D3—362no—
336C22×D3—3616no—
337C22×D3—368no—
338C22×D3C22×C3⋊D312yes—
339C3×Q8—361no—
340C3×D4—44no—
341C3×D4—41no—
342C3×D4—48no—
343C3×D4—128no—
344C2×C3×C4—24no—
345C2×C3×C4—38no—
346C2×C3×C4—92no—
347C2×C3×C4—92no—
348C2×C3×C4C2×D3214yes—
349C2×C3×C4C22×C3⋊D311yescharacteristic
350#67—42no—
351#67—368no—
352#67—368no—
353#67—368no—
354#67—362no—
355#67—3616no—
356C2×Dic3—38no—
357C2×Dic3—38no—
358C2×Dic3—98no—
359C2×Dic3—92no—
360C2×Dic3—188no—
361C2×Dic3C22×C3⋊D312yes—
362D12—362no—
363D12—362no—
364Dic6—124no—
365Dic6—362no—
366Dic6—368no—
367C2×C32—28no—
368C2×C32—24no—
369C2×C32—48no—
370C2×C32—41no—
371C2×C32—41no—
372C2×C32—44no—
373C2×C32—44no—
374C2×C32—84no—
375C2×C32—122no—
376C2×C32#100418yes—
377C2×C32#100414yes—
378C2×C32#100411yescharacteristic
379C2×C32#100912yes—
380C3⋊D3—1216no—
381C3⋊D3—122no—
382C3⋊D3—128no—
383C3×D3—48no—
384C24—31no—
385C24—271no—
386C2×Q8—91no—
387C2×Q8—272no—
388C2×D4—32no—
389C2×D4—272no—
390C2×D4—272no—
391C2×D4—272no—
392C2×D4D3×C3⋊D311yescharacteristic
393C22⋊C4—31no—
394C22⋊C4—32no—
395C22⋊C4—92no—
396C22⋊C4—92no—
397C22⋊C4—272no—
398C22⋊C4—272no—
399C22⋊C4—272no—
400C22⋊C4—272no—
401C22⋊C4—271no—
402C22⋊C4—272no—
403C42—91no—
404C42—272no—
405C22×C3—28no—
406C22×C3—28no—
407C22×C3—28no—
408C22×C3—24no—
409C22×C3—24no—
410C22×C3—22no—
411C22×C3—22no—
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—24no—
415C22×C3—48no—
416C22×C3—48no—
417C22×C3—44no—
418C22×C3—68no—
419C22×C3—68no—
420C22×C3—68no—
421C22×C3—1216no—
422C22×C3C22×D3214yes—
423C22×C3C23×C3⋊D311yescharacteristic
424C2×D3—22no—
425C2×D3—22no—
426C2×D3—22no—
427C2×D3—44no—
428C2×D3—1816no—
429C2×D3—188no—
430C2×D3—188no—
431C2×D3—188no—
432C2×D3—182no—
433C2×D3—182no—
434C2×D3—182no—
435C2×D3—368no—
436C2×D3—368no—
437C2×D3—3632no—
438C2×D3—3616no—
439C2×D3—364no—
440C3×C4—24no—
441C3×C4—21no—
442C3×C4—44no—
443C3×C4—68no—
444C3×C4—182no—
445C3×C4—182no—
446Dic3—22no—
447Dic3—68no—
448Dic3—68no—
449Dic3—188no—
450Dic3—182no—
451Dic3—368no—
452C32—24no—
453C32#559414yescomplemented
454C32#568311yescharacteristic, complemented
455C23—32no—
456C23—122no—
457C23—121no—
458C23—272no—
459C23—1082no—
460C23—1081no—
461C23C2×D3×C3⋊D311yescharacteristic
462C23C2×D3×C3⋊D311yescharacteristic
463Q8—361no—
464Q8—1082no—
465D4—41no—
466D4—122no—
467D4—1082no—
468D4—1082no—
469D4—1082no—
470C2×C4—32no—
471C2×C4—92no—
472C2×C4—92no—
473C2×C4—272no—
474C2×C4C2×D3×C3⋊D311yescharacteristic
475C2×C3—28no—
476C2×C3—24no—
477C2×C3—48no—
478C2×C3—44no—
479C2×C3—44no—
480C2×C3—41no—
481C2×C3—41no—
482C2×C3—84no—
483C2×C3—128no—
484C2×C3#6431418yes—
485C2×C3#6433014yes—
486C2×C3#6436112yes—
487C2×C3#6436111yescharacteristic
488D3—42no—
489D3—3616no—
490D3—368no—
491D3—362no—
492C22—22no—
493C22—22no—
494C22—21no—
495C22—21no—
496C22—62no—
497C22—62no—
498C22—62no—
499C22—124no—
500C22—542no—
501C22—542no—
502C22—542no—
503C22—1084no—
504C22C22×D3×C3⋊D311yescharacteristic
505C4—21no—
506C4—62no—
507C4—182no—
508C4—182no—
509C4—542no—
510C3—24no—
511C3#1059526214yescomplemented
512C3#1059606511yescharacteristic, complemented
513C2—41no—
514C2—41no—
515C2—122no—
516C2—1082no—
517C2#1172458812yes—
518C2#1172461111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33H(3,𝔽4)
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #11723978C2semidirect
2. C3×#64145C2semidirect
3. C32×#937C2semidirect
4. #11723638C2semidirect
5. #11723630C2semidirect
6. #11723600C2semidirect
7. #11723592C2semidirect
8. #11723521C2semidirect
9. #11723419C2semidirect
12. C23×C3⋊(C3⋊D3)C22semidirect
13. C23×C32×D3C22semidirect
16. C2×C3×#634C22semidirect
18. C2×#6293C22semidirect
21. #88864C22semidirect
22. #88859C22semidirect
23. C3×#3757C22semidirect
24. C3×#3757C22semidirect
25. C3×C4×C3⋊Dic3C22semidirect
26. C32×#217C22semidirect
27. C32×(C2×D3)⋊C4C22semidirect
28. #88804C22semidirect
29. #88804C22semidirect
30. (C2×D3)⋊(C3⋊Dic3)C22semidirect
31. (C2×D3)⋊(C3⋊Dic3)C22semidirect
32. Dic3×C3⋊Dic3C22semidirect
40. #64145D3semidirect
44. C3×#937D3semidirect
78. C23×C3×D3C2×D3semidirect
89. C2×#634C2×D3semidirect
102. #3757C2×D3semidirect
103. #3757C2×D3semidirect
110. C4×C3⋊Dic3C2×D3semidirect
113. C3×#217C2×D3semidirect
114. C3×(C2×D3)⋊C4C2×D3semidirect
147. #937C3⋊D3semidirect
218. C23×D3C2×C3⋊D3semidirect
247. #217C2×C3⋊D3semidirect
257. (C2×D3)⋊C4C2×C3⋊D3semidirect
317. C33H(3,𝔽4)semidirect
453. C32#5594semidirect
454. C32#5683semidirect
511. C3#10595262semidirect
512. C3#10596065semidirect

Maximal subgroups

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

subgroupindex
#117239782
C3×#641452
C32×#9372
#117236382
#117236302
#117236002
#117235922
#117235212
#117234192
#105960653
#105952623