Finite Group Databasebeta

← #13177961

Subgroups — #13177961

Order 1728.

1864 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723988C211yesmaximal, characteristic, complemented
2C3×#64156C211yesmaximal, characteristic, complemented
3C32×#947C211yesmaximal, characteristic, complemented
4#11723542C212yesmaximal, complemented
5#11723541C212yesmaximal, complemented
6#11723540C212yesmaximal, complemented
7#11723530C212yesmaximal, complemented
8#11723419C212yesmaximal, complemented
9#11723416C212yesmaximal, complemented
10#10596154—31nomaximal
11#10595439—34nomaximal
12C22×D3×C3⋊D3C2212yescomplemented
13C2×C33×Q8C2211yescharacteristic, complemented
14C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
16C2×C32×D12C2211yescharacteristic
17C2×#6293C2212yescomplemented
18C2×#6292C2212yes—
19C2×#6291C2212yes—
20C2×#6290C2212yes—
21#88859C2212yescomplemented
22#88859C2212yescomplemented
23C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
24C3×#3752C2212yescomplemented
25C3×#3752C2212yescomplemented
26C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
27C32×(C2×D3)⋊C4C2212yescomplemented
28C32×(C2×D3)⋊C4C2212yescomplemented
29C32×C4×Dic3C2211yescharacteristic, complemented
30#88804C2212yescomplemented
31#88803C2212yescomplemented
32(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
33C32×#124—31no—
34#64156—31no—
35#64156—34no—
36#64156—64no—
37#64156D311yescharacteristic, complemented
38#64146—32no—
39#64094—32no—
40#64094—32no—
41#64091—32no—
42#64091—32no—
43#64084—32no—
44C3×#947—34no—
45C3×#947D314yescomplemented
46#63923—38no—
47#63922—38no—
48#63922—38no—
49#63915—38no—
50#63852—38no—
51#63852—38no—
52C22×C3⋊(C3⋊D3)C2312yes—
53C22×C3×C3⋊D3C2312yes—
54C22×C32×D3C2312yes—
55C2×D3×C3⋊D3—42no—
56C2×D3×C3⋊D3—42no—
57C2×D3×C3⋊D3—42no—
58C33×Q8—41no—
59C2×C33×C4C2311yescharacteristic
60C2×C33×C4C2311yescharacteristic
61C2×C33×C4C2311yescharacteristic
62C2×C3⋊(C3⋊Dic3)C2312yes—
63C4⋊(C3⋊(C3⋊D3))—41no—
64C2×C3×C3⋊Dic3C2312yes—
65C3×C4⋊(C3⋊D3)—41no—
66C2×C32×Dic3C2312yes—
67C32×D12—41no—
68#6293—42no—
69#6292—42no—
70#6291—42no—
71#6290—42no—
72#5683—94no—
73#5683—91no—
74C23×C3⋊D3—32no—
75C22×D32—38no—
76C2×C32×Q8—24no—
77C2×C32×Q8C2×D314yescomplemented
78C2×C32×Q8C2×D311yescharacteristic, complemented
79C2×C32×D4—31no—
80C2×#638—32no—
81C2×#638—32no—
82C2×C4⋊(C3⋊D3)—32no—
83C2×C4⋊(C3⋊D3)—31no—
84C2×C4⋊(C3⋊D3)—34no—
85C2×C4⋊(C3⋊D3)—64no—
86C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
87C2×#634—32no—
88C2×C3×D12—34no—
89C2×C3×D12C2×D314yes—
90C2×#627—38no—
91C2×#626—38no—
92C2×#626—38no—
93C2×#625—38no—
94C32×C22⋊C4—32no—
95C32×C22⋊C4—32no—
96C32×C42—31no—
97#3757—32no—
98#3752—32no—
99#3752—32no—
100#3752—32no—
101#3752—38no—
102#3752—38no—
103#3752—32no—
104#3752—68no—
105#3752—68no—
106#3752C2×D312yescomplemented
107#3752C2×D312yescomplemented
108C4×C3⋊Dic3—31no—
109C4×C3⋊Dic3—34no—
110C4×C3⋊Dic3—64no—
111C4×C3⋊Dic3C2×D311yescharacteristic, complemented
112C3×(C2×D3)⋊C4—38no—
113C3×(C2×D3)⋊C4—38no—
114C3×(C2×D3)⋊C4C2×D318yescomplemented
115C3×(C2×D3)⋊C4C2×D318yescomplemented
116C3×C4×Dic3—34no—
117C3×C4×Dic3C2×D314yescomplemented
118#3722—38no—
119(C2×D3)⋊Dic3—38no—
120(C2×D3)⋊Dic3—38no—
121C22×C33C2411yescharacteristic
122C2×C3⋊(C3⋊D3)—22no—
123C2×C3⋊(C3⋊D3)—22no—
124C2×C3⋊(C3⋊D3)—22no—
125C2×C3×C3⋊D3—22no—
126C2×C3×C3⋊D3—22no—
127C2×C3×C3⋊D3—22no—
128C2×C32×D3—22no—
129C2×C32×D3—22no—
130C2×C32×D3—22no—
131D3×C3⋊D3—48no—
132C33×C4—21no—
133C33×C4—21no—
134C33×C4—21no—
135C3⋊(C3⋊Dic3)—22no—
136C3×C3⋊Dic3—22no—
137C32×Dic3—22no—
138C3×#124—34no—
139C3×#124—91no—
140#947—34no—
141#947—94no—
142#947—91no—
143#947—184no—
144#947C3⋊D311yescharacteristic, complemented
145#937—98no—
146#937—92no—
147#885—98no—
148#885—98no—
149#885—92no—
150#885—92no—
151#882—98no—
152#882—98no—
153#882—92no—
154#882—92no—
155#875—98no—
156#875—92no—
157C23×C32—32no—
158C22×C3⋊D3—32no—
159C22×C3⋊D3—38no—
160C22×C3⋊D3—68no—
161C22×C3⋊D3—122no—
162C22×C3⋊D3—122no—
163C22×C3⋊D3—122no—
164C22×C3⋊D3C22×D312yes—
165C22×C3×D3—38no—
166C22×C3×D3C22×D318yes—
167C2×D32—128no—
168C2×D32—128no—
169C2×D32—128no—
170C32×Q8—41no—
171C32×Q8—44no—
172C32×Q8—48no—
173C32×D4—121no—
174C2×C32×C4—24no—
175C2×C32×C4—24no—
176C2×C32×C4—24no—
177C2×C32×C4—32no—
178C2×C32×C4C22×D314yes—
179C2×C32×C4C22×D314yes—
180C2×C32×C4C22×D314yes—
181C2×C32×C4C22×D311yescharacteristic
182C2×C32×C4C22×D311yescharacteristic
183C2×C32×C4C22×D311yescharacteristic
184#638—122no—
185#638—122no—
186C2×C3⋊Dic3—32no—
187C2×C3⋊Dic3—38no—
188C2×C3⋊Dic3—68no—
189C2×C3⋊Dic3C22×D312yes—
190C4⋊(C3⋊D3)—41no—
191C4⋊(C3⋊D3)—121no—
192C4⋊(C3⋊D3)—124no—
193C4⋊(C3⋊D3)—128no—
194C4⋊(C3⋊D3)—122no—
195#634—122no—
196C2×C3×Dic3—38no—
197C2×C3×Dic3C22×D318yes—
198C3×D12—44no—
199C3×D12—124no—
200#627—128no—
201#626—128no—
202#626—128no—
203#625—128no—
204H(3,𝔽4)—271noSylow
205C2×C33H(5,𝔽2)11yescharacteristic
206C2×C33H(5,𝔽2)12yes—
207C3⋊(C3⋊D3)—42no—
208C3×C3⋊D3—42no—
209C32×D3—42no—
210C23×D3—98no—
211C23×D3—92no—
212C2×C3×Q8—24no—
213C2×C3×Q8D3214yescomplemented
214C2×C3×Q8C2×C3⋊D311yescharacteristic, complemented
215C2×C3×D4—34no—
216C2×C3×D4—91no—
217C2×#67—98no—
218C2×#67—98no—
219C2×#67—92no—
220C2×#67—92no—
221C2×D12—34no—
222C2×D12—98no—
223C2×D12—92no—
224C2×D12—94no—
225C2×D12—91no—
226C2×D12—184no—
227C2×D12C2×C3⋊D311yescharacteristic
228C2×Dic6—98no—
229C2×Dic6—92no—
230C3×C22⋊C4—38no—
231C3×C22⋊C4—38no—
232C3×C22⋊C4—92no—
233C3×C22⋊C4—92no—
234C3×C42—34no—
235C3×C42—91no—
236#217—98no—
237#217—92no—
238(C2×D3)⋊C4—38no—
239(C2×D3)⋊C4—38no—
240(C2×D3)⋊C4—98no—
241(C2×D3)⋊C4—98no—
242(C2×D3)⋊C4—92no—
243(C2×D3)⋊C4—98no—
244(C2×D3)⋊C4—98no—
245(C2×D3)⋊C4—92no—
246(C2×D3)⋊C4—92no—
247(C2×D3)⋊C4—92no—
248(C2×D3)⋊C4—188no—
249(C2×D3)⋊C4—188no—
250(C2×D3)⋊C4C2×C3⋊D312yescomplemented
251(C2×D3)⋊C4C2×C3⋊D312yescomplemented
252C4×Dic3—34no—
253C4×Dic3—94no—
254C4×Dic3—91no—
255C4×Dic3—184no—
256C4×Dic3C2×C3⋊D311yescharacteristic, complemented
257C22×C32—24no—
258C22×C32—62no—
259C22×C32—62no—
260C22×C32—62no—
261C22×C32C23×D314yes—
262C22×C32C23×D311yescharacteristic
263C2×C3⋊D3—22no—
264C2×C3⋊D3—22no—
265C2×C3⋊D3—22no—
266C2×C3⋊D3—62no—
267C2×C3⋊D3—62no—
268C2×C3⋊D3—62no—
269C2×C3⋊D3—68no—
270C2×C3⋊D3—68no—
271C2×C3⋊D3—68no—
272C2×C3⋊D3—128no—
273C2×C3⋊D3—128no—
274C2×C3⋊D3—128no—
275C2×C3⋊D3—128no—
276C2×C3×D3—28no—
277C2×C3×D3—28no—
278C2×C3×D3—28no—
279C2×C3×D3—68no—
280C2×C3×D3—68no—
281C2×C3×D3—68no—
282D32—1232no—
283C32×C4—28no—
284C32×C4—21no—
285C32×C4—24no—
286C32×C4—21no—
287C32×C4—21no—
288C32×C4—24no—
289C32×C4—24no—
290C32×C4—44no—
291C32×C4—44no—
292C32×C4—62no—
293C3⋊Dic3—22no—
294C3⋊Dic3—62no—
295C3⋊Dic3—68no—
296C3⋊Dic3—128no—
297C3×Dic3—28no—
298C3×Dic3—68no—
299#124—31no—
300#124—91no—
301#124—272no—
302#124—272no—
303#124—272no—
304#124—271no—
305#120—272no—
306#120—272no—
307#120—272no—
308C33H(3,𝔽4)11yescharacteristic, complemented, Sylow
309C23×C3—38no—
310C23×C3—92no—
311C22×D3—38no—
312C22×D3—98no—
313C22×D3—92no—
314C22×D3—188no—
315C22×D3—362no—
316C22×D3—362no—
317C22×D3—362no—
318C22×D3—368no—
319C22×D3—368no—
320C22×D3—368no—
321C22×D3C22×C3⋊D312yes—
322C3×Q8—44no—
323C3×Q8—41no—
324C3×Q8—48no—
325C3×D4—124no—
326C3×D4—361no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—24no—
330C2×C3×C4—38no—
331C2×C3×C4—92no—
332C2×C3×C4C2×D3214yes—
333C2×C3×C4C2×D3214yes—
334C2×C3×C4C2×D3214yes—
335C2×C3×C4C22×C3⋊D311yescharacteristic
336C2×C3×C4C22×C3⋊D311yescharacteristic
337C2×C3×C4C22×C3⋊D311yescharacteristic
338#67—362no—
339#67—368no—
340#67—362no—
341#67—368no—
342C2×Dic3—38no—
343C2×Dic3—98no—
344C2×Dic3—92no—
345C2×Dic3—188no—
346C2×Dic3C22×C3⋊D312yes—
347D12—41no—
348D12—124no—
349D12—364no—
350D12—361no—
351D12—368no—
352D12—368no—
353D12—362no—
354Dic6—362no—
355Dic6—368no—
356C2×C32—28no—
357C2×C32—24no—
358C2×C32—122no—
359C2×C32#100414yes—
360C2×C32#100411yescharacteristic
361C2×C32#100918yes—
362C2×C32#100912yes—
363C3⋊D3—42no—
364C3⋊D3—122no—
365C3⋊D3—128no—
366C3⋊D3—248no—
367C3×D3—48no—
368C3×D3—128no—
369C24—272no—
370C2×Q8—272no—
371C2×Q8D3×C3⋊D311yescharacteristic, complemented
372C2×D4—31no—
373C2×D4—91no—
374C2×D4—272no—
375C2×D4—272no—
376C2×D4—272no—
377C2×D4—271no—
378C22⋊C4—32no—
379C22⋊C4—32no—
380C22⋊C4—92no—
381C22⋊C4—92no—
382C22⋊C4—272no—
383C22⋊C4—272no—
384C22⋊C4—272no—
385C22⋊C4—272no—
386C22⋊C4—272no—
387C42—31no—
388C42—91no—
389C42—271no—
390C22×C3—24no—
391C22×C3—68no—
392C22×C3—68no—
393C22×C3—68no—
394C22×C3—182no—
395C22×C3—182no—
396C22×C3—182no—
397C22×C3C22×D3214yes—
398C22×C3C23×C3⋊D311yescharacteristic
399C2×D3—22no—
400C2×D3—22no—
401C2×D3—22no—
402C2×D3—68no—
403C2×D3—68no—
404C2×D3—68no—
405C2×D3—188no—
406C2×D3—188no—
407C2×D3—188no—
408C2×D3—182no—
409C2×D3—182no—
410C2×D3—182no—
411C2×D3—368no—
412C2×D3—368no—
413C2×D3—368no—
414C2×D3—368no—
415C2×D3—3632no—
416C3×C4—28no—
417C3×C4—24no—
418C3×C4—24no—
419C3×C4—24no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—44no—
424C3×C4—44no—
425C3×C4—68no—
426C3×C4—182no—
427Dic3—22no—
428Dic3—68no—
429Dic3—188no—
430Dic3—182no—
431Dic3—368no—
432C32—24no—
433C32#568314yescomplemented
434C32#568311yescharacteristic, complemented
435C23—32no—
436C23—92no—
437C23—272no—
438C23—1082no—
439C23—1082no—
440C23—1082no—
441Q8—41no—
442Q8—1082no—
443D4—121no—
444D4—361no—
445D4—1081no—
446D4—1082no—
447D4—1082no—
448D4—1082no—
449C2×C4—32no—
450C2×C4—92no—
451C2×C4—272no—
452C2×C4C2×D3×C3⋊D311yescharacteristic
453C2×C4C2×D3×C3⋊D311yescharacteristic
454C2×C4C2×D3×C3⋊D311yescharacteristic
455C2×C3—28no—
456C2×C3—24no—
457C2×C3—128no—
458C2×C3—362no—
459C2×C3#6432018yes—
460C2×C3#6433014yes—
461C2×C3#6436111yescharacteristic
462C2×C3#6436612yes—
463D3—42no—
464D3—128no—
465D3—362no—
466D3—368no—
467D3—728no—
468C22—62no—
469C22—62no—
470C22—62no—
471C22—182no—
472C22—182no—
473C22—182no—
474C22—542no—
475C22—542no—
476C22—542no—
477C22—1088no—
478C22C22×D3×C3⋊D311yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—62no—
483C4—182no—
484C4—542no—
485C3—24no—
486C3#1059543914yescomplemented
487C3#1059615411yescharacteristic, complemented
488C2—122no—
489C2—362no—
490C2—1082no—
491C2#1172459712yes—
492C2#1172461111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×Q8C22
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33H(3,𝔽4)
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. #11723988C2semidirect
2. C3×#64156C2semidirect
3. C32×#947C2semidirect
4. #11723542C2semidirect
5. #11723541C2semidirect
6. #11723540C2semidirect
7. #11723530C2semidirect
8. #11723419C2semidirect
9. #11723416C2semidirect
12. C22×D3×C3⋊D3C22semidirect
13. C2×C33×Q8C22semidirect
17. C2×#6293C22semidirect
21. #88859C22semidirect
22. #88859C22semidirect
23. C4×C3⋊(C3⋊Dic3)C22semidirect
24. C3×#3752C22semidirect
25. C3×#3752C22semidirect
26. C3×C4×C3⋊Dic3C22semidirect
27. C32×(C2×D3)⋊C4C22semidirect
28. C32×(C2×D3)⋊C4C22semidirect
29. C32×C4×Dic3C22semidirect
30. #88804C22semidirect
31. #88803C22semidirect
32. (C2×D3)⋊(C3⋊Dic3)C22semidirect
37. #64156D3semidirect
45. C3×#947D3semidirect
77. C2×C32×Q8C2×D3semidirect
78. C2×C32×Q8C2×D3semidirect
106. #3752C2×D3semidirect
107. #3752C2×D3semidirect
111. C4×C3⋊Dic3C2×D3semidirect
114. C3×(C2×D3)⋊C4C2×D3semidirect
115. C3×(C2×D3)⋊C4C2×D3semidirect
117. C3×C4×Dic3C2×D3semidirect
144. #947C3⋊D3semidirect
213. C2×C3×Q8D32semidirect
214. C2×C3×Q8C2×C3⋊D3semidirect
250. (C2×D3)⋊C4C2×C3⋊D3semidirect
251. (C2×D3)⋊C4C2×C3⋊D3semidirect
256. C4×Dic3C2×C3⋊D3semidirect
308. C33H(3,𝔽4)semidirect
371. C2×Q8D3×C3⋊D3semidirect
433. C32#5683semidirect
434. C32#5683semidirect
486. C3#10595439semidirect
487. C3#10596154semidirect

Maximal subgroups

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

subgroupindex
#117239882
C3×#641562
C32×#9472
#117235422
#117235412
#117235402
#117235302
#117234192
#117234162
#105961543
#105954393