Finite Group Databasebeta

← #13178091

Subgroups — #13178091

Order 1728.

3020 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
1C22×#6312C211yesmaximal, characteristic, complemented
2C22×C3×#638C211yesmaximal, characteristic, complemented
3C2×C32×#217C211yesmaximal, characteristic, complemented
4#11723639C212yesmaximal, complemented
5#11723636C212yesmaximal, complemented
6#11723632C214yesmaximal, complemented
7#11723609C212yesmaximal, complemented
8#11723608C212yesmaximal, complemented
9#10595964—31nomaximal
10#10595519—34nomaximal
11C24×C33C2211yescharacteristic
12C23×C3⋊(C3⋊D3)C2211yescharacteristic, complemented
13C23×C3×C3⋊D3C2211yescharacteristic, complemented
14C2×#6312—24no—
15C2×#6312C2212yes—
16C2×#6312C2212yescomplemented
17C22×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
18C2×C3×#638—24no—
19C2×C3×#638C2212yes—
20C2×C3×#638C2212yescomplemented
21C22×C3×C3⋊Dic3C2211yescharacteristic, complemented
22C22×C32×Dic3C2212yescomplemented
23C2×#6292C2214yescomplemented
24C32×#217C2212yescomplemented
25C32×#217C2212yescomplemented
26Dic3⋊(C3⋊Dic3)C2214yescomplemented
27#88804C2214yescomplemented
28#88803C2214yescomplemented
29C22×#638—31no—
30C22×#638—34no—
31C22×#638—64no—
32C22×#638D311yescharacteristic, complemented
33C22×C3×#67—34no—
34C2×C32×C22⋊C4—31no—
35#64139—34no—
36#64095—32no—
37#64095—32no—
38#64091—32no—
39#64091—32no—
40C2×C3×#217D314yescomplemented
41#63981—38no—
42#63978—316no—
43#63977—38no—
44#63966—38no—
45#63962—38no—
46C23×C33—42no—
47C23×C33D414yescomplemented
48C23×C33C2311yescharacteristic
49C23×C33C2311yescharacteristic
50C23×C33C2311yescharacteristic
51C22×C3⋊(C3⋊D3)—22no—
52C22×C3⋊(C3⋊D3)—42no—
53C22×C3⋊(C3⋊D3)C2312yes—
54C22×C3×C3⋊D3—22no—
55C22×C3×C3⋊D3—42no—
56C22×C3×C3⋊D3C2312yes—
57#6312—24no—
58#6312—24no—
59C2×C3⋊(C3⋊Dic3)—22no—
60C2×C3⋊(C3⋊Dic3)C2312yes—
61C3×#638—24no—
62C3×#638—24no—
63C2×C3×C3⋊Dic3—22no—
64C2×C3×C3⋊Dic3C2312yes—
65C2×C32×Dic3—42no—
66C2×C32×Dic3C2314yes—
67#6292—44no—
68#5804—91no—
69#5493—94no—
70C24×C32—24no—
71C24×C32C2×D314yes—
72C24×C32C2×D311yescharacteristic
73C23×C3⋊D3—31no—
74C23×C3⋊D3—34no—
75C23×C3⋊D3—64no—
76C23×C3⋊D3C2×D311yescharacteristic, complemented
77C23×C3×D3—34no—
78C22×C32×C4—32no—
79C2×#638—24no—
80C2×#638—32no—
81C2×#638—32no—
82C2×#638—38no—
83C2×#638—38no—
84C2×#638—68no—
85C2×#638—68no—
86C2×#638—632no—
87C2×#638—616no—
88C2×#638—64no—
89C2×#638C2×D312yes—
90C2×#638C2×D312yescomplemented
91C22×C3⋊Dic3—31no—
92C22×C3⋊Dic3—34no—
93C22×C3⋊Dic3—64no—
94C22×C3⋊Dic3C2×D311yescharacteristic, complemented
95C2×C4⋊(C3⋊D3)—34no—
96C2×C3×#67—38no—
97C2×C3×#67—38no—
98C2×C3×#67—616no—
99C22×C3×Dic3—34no—
100C22×C3×Dic3C2×D318yescomplemented
101C2×#626—316no—
102C32×C22⋊C4—32no—
103C32×C22⋊C4—32no—
104#3752—34no—
105#3752—34no—
106C4⋊(C3⋊Dic3)—34no—
107C3×#217C2×D318yescomplemented
108C3×#217C2×D318yescomplemented
109Dic3⋊Dic3—316no—
110#3722—316no—
111(C2×D3)⋊Dic3—316no—
112C22×C33—22no—
113C22×C33—22no—
114C22×C33—22no—
115C22×C33—44no—
116C22×C33C2×D412yes—
117C22×C33C2×D412yes—
118C22×C33C2×D412yes—
119C22×C33C2411yescharacteristic
120C2×C3⋊(C3⋊D3)—22no—
121C2×C3⋊(C3⋊D3)—24no—
122C2×C3⋊(C3⋊D3)—44no—
123C2×C3×C3⋊D3—22no—
124C2×C3×C3⋊D3—24no—
125C2×C3×C3⋊D3—44no—
126C3⋊(C3⋊Dic3)—22no—
127C3×C3⋊Dic3—22no—
128C32×Dic3—24no—
129C22×C3×D4—91no—
130C22×#67—34no—
131C22×#67—94no—
132C22×#67—91no—
133C22×#67—184no—
134C2×C3×C22⋊C4—34no—
135#953—92no—
136C2×#217C3⋊D311yescharacteristic, complemented
137#939—94no—
138#937—92no—
139#935—92no—
140#930—916no—
141#929—92no—
142#886—98no—
143#886—98no—
144#882—98no—
145#882—98no—
146C23×C32—216no—
147C23×C32—24no—
148C23×C32—24no—
149C23×C32—24no—
150C23×C32—42no—
151C23×C32—48no—
152C23×C32—416no—
153C23×C32D12116yescomplemented
154C23×C32#6714yescomplemented
155C23×C32C22×D314yes—
156C23×C32C22×D314yes—
157C23×C32C22×D314yes—
158C23×C32C22×D311yescharacteristic
159C23×C32C22×D311yescharacteristic
160C23×C32C22×D311yescharacteristic
161C22×C3⋊D3—22no—
162C22×C3⋊D3—32no—
163C22×C3⋊D3—38no—
164C22×C3⋊D3—42no—
165C22×C3⋊D3—68no—
166C22×C3⋊D3—616no—
167C22×C3⋊D3—68no—
168C22×C3⋊D3—62no—
169C22×C3⋊D3—122no—
170C22×C3⋊D3—128no—
171C22×C3⋊D3—1216no—
172C22×C3⋊D3C22×D312yes—
173C22×C3×D3—38no—
174C22×C3×D3—68no—
175C22×C3×D3—128no—
176C2×C32×C4—34no—
177C2×C32×C4—122no—
178#638—24no—
179#638—24no—
180#638—632no—
181#638—632no—
182#638—616no—
183#638—616no—
184#638—64no—
185#638—64no—
186C2×C3⋊Dic3—22no—
187C2×C3⋊Dic3—32no—
188C2×C3⋊Dic3—38no—
189C2×C3⋊Dic3—68no—
190C2×C3⋊Dic3—616no—
191C2×C3⋊Dic3—68no—
192C2×C3⋊Dic3—62no—
193C2×C3⋊Dic3C22×D312yes—
194C4⋊(C3⋊D3)—124no—
195C3×#67—616no—
196C3×#67—616no—
197C2×C3×Dic3—38no—
198C2×C3×Dic3—48no—
199C2×C3×Dic3—68no—
200C2×C3×Dic3C22×D3116yes—
201#626—1216no—
202#534—271noSylow
203C2×C33—22no—
204C2×C33—22no—
205C2×C33—22no—
206C2×C33C22×D411yescharacteristic
207C2×C33H(5,𝔽2)12yes—
208C3⋊(C3⋊D3)—42no—
209C3×C3⋊D3—42no—
210C24×C3—24no—
211C24×C3—91no—
212C24×C3D3214yes—
213C24×C3C2×C3⋊D311yescharacteristic
214C23×D3—34no—
215C23×D3—94no—
216C23×D3—91no—
217C23×D3—184no—
218C2×C3×D4—92no—
219C2×C3×D4—92no—
220C2×C3×D4—184no—
221C22×C3×C4—38no—
222C22×C3×C4—91no—
223C2×#67—38no—
224C2×#67—38no—
225C2×#67—616no—
226C2×#67—98no—
227C2×#67—98no—
228C2×#67—92no—
229C2×#67—92no—
230C2×#67—94no—
231C2×#67—188no—
232C2×#67—188no—
233C2×#67—1832no—
234C2×#67—1816no—
235C2×#67—184no—
236C22×Dic3—34no—
237C22×Dic3—94no—
238C22×Dic3—91no—
239C22×Dic3—184no—
240C22×Dic3C2×C3⋊D312yescomplemented
241C2×D12—916no—
242C3×C22⋊C4—38no—
243C3×C22⋊C4—38no—
244#217—94no—
245#217C2×C3⋊D312yescomplemented
246#217C2×C3⋊D312yescomplemented
247(C2×D3)⋊C4—916no—
248(C2×D3)⋊C4—916no—
249(C2×D3)⋊C4—94no—
250C4⋊Dic3—916no—
251Dic3⋊C4—94no—
252C22×C32—216no—
253C22×C32—22no—
254C22×C32—22no—
255C22×C32—28no—
256C22×C32—22no—
257C22×C32—24no—
258C22×C32—28no—
259C22×C32—28no—
260C22×C32—28no—
261C22×C32—28no—
262C22×C32—28no—
263C22×C32—48no—
264C22×C32—48no—
265C22×C32—416no—
266C22×C32—44no—
267C22×C32—432no—
268C22×C32C2×D1218yes—
269C22×C32C2×D1218yes—
270C22×C32C2×D1218yes—
271C22×C32C2×#6712yes—
272C22×C32C2×#6712yes—
273C22×C32C2×#6712yes—
274C22×C32C23×D314yes—
275C22×C32C23×D311yescharacteristic
276C2×C3⋊D3—24no—
277C2×C3⋊D3—22no—
278C2×C3⋊D3—44no—
279C2×C3⋊D3—616no—
280C2×C3⋊D3—68no—
281C2×C3⋊D3—64no—
282C2×C3⋊D3—62no—
283C2×C3⋊D3—616no—
284C2×C3⋊D3—1216no—
285C2×C3⋊D3—1232no—
286C2×C3⋊D3—1216no—
287C2×C3⋊D3—124no—
288C2×C3×D3—68no—
289C2×C3×D3—616no—
290C2×C3×D3—1216no—
291C32×C4—64no—
292C3⋊Dic3—22no—
293C3⋊Dic3—616no—
294C3⋊Dic3—68no—
295C3⋊Dic3—62no—
296C3×Dic3—216no—
297C3×Dic3—68no—
298C22×D4—91no—
299C22×D4—271no—
300#123—272no—
301#123—272no—
302#121—274no—
303#120—272no—
304#120—272no—
305C2×C22⋊C4—31no—
306C33#53411yescharacteristic, complemented, Sylow
307C23×C3—216no—
308C23×C3—24no—
309C23×C3—24no—
310C23×C3—24no—
311C23×C3—48no—
312C23×C3—42no—
313C23×C3—416no—
314C23×C3—92no—
315C23×C3—182no—
316C23×C3—362no—
317C23×C3#626116yescomplemented
318C23×C3C4⋊(C3⋊D3)14yescomplemented
319C23×C3C2×D3214yes—
320C23×C3C2×D3214yes—
321C23×C3C2×D3214yes—
322C23×C3C22×C3⋊D311yescharacteristic
323C23×C3C22×C3⋊D311yescharacteristic
324C23×C3C22×C3⋊D311yescharacteristic
325C22×D3—38no—
326C22×D3—68no—
327C22×D3—98no—
328C22×D3—92no—
329C22×D3—128no—
330C22×D3—188no—
331C22×D3—1816no—
332C22×D3—188no—
333C22×D3—182no—
334C22×D3—368no—
335C22×D3—362no—
336C22×D3—3616no—
337C3×D4—184no—
338C3×D4—184no—
339C2×C3×C4—316no—
340C2×C3×C4—92no—
341C2×C3×C4—128no—
342C2×C3×C4—182no—
343#67—616no—
344#67—616no—
345#67—1832no—
346#67—1832no—
347#67—1816no—
348#67—1816no—
349#67—184no—
350#67—184no—
351#67—364no—
352C2×Dic3—38no—
353C2×Dic3—42no—
354C2×Dic3—68no—
355C2×Dic3—98no—
356C2×Dic3—92no—
357C2×Dic3—188no—
358C2×Dic3—1816no—
359C2×Dic3—188no—
360C2×Dic3—182no—
361C2×Dic3C22×C3⋊D314yes—
362D12—3616no—
363C2×C32—216no—
364C2×C32—28no—
365C2×C32—22no—
366C2×C32—22no—
367C2×C32—22no—
368C2×C32—24no—
369C2×C32—28no—
370C2×C32—28no—
371C2×C32—28no—
372C2×C32—48no—
373C2×C32—48no—
374C2×C32C22×D1214yes—
375C2×C32#100418yes—
376C2×C32#100412yes—
377C2×C32C22×#6711yescharacteristic
378C3⋊D3—42no—
379C3⋊D3—1216no—
380C3⋊D3—128no—
381C3⋊D3—122no—
382C3×D3—128no—
383C24—91no—
384C24—271no—
385C24D3×C3⋊D311yescharacteristic
386C2×D4—92no—
387C2×D4—92no—
388C2×D4—184no—
389C2×D4—274no—
390C2×D4—272no—
391C2×D4—272no—
392C2×D4—544no—
393C22×C4—32no—
394C22×C4—91no—
395C22×C4—271no—
396C4⋊C4—274no—
397C22⋊C4—32no—
398C22⋊C4—32no—
399C22⋊C4—274no—
400C22⋊C4—274no—
401C22×C3—216no—
402C22×C3—28no—
403C22×C3—28no—
404C22×C3—28no—
405C22×C3—22no—
406C22×C3—22no—
407C22×C3—22no—
408C22×C3—24no—
409C22×C3—28no—
410C22×C3—28no—
411C22×C3—28no—
412C22×C3—48no—
413C22×C3—48no—
414C22×C3—416no—
415C22×C3—44no—
416C22×C3—432no—
417C22×C3—182no—
418C22×C3—184no—
419C22×C3—364no—
420C22×C3C2×#62618yes—
421C22×C3C2×#62618yes—
422C22×C3C2×#62618yes—
423C22×C3C2×C4⋊(C3⋊D3)12yes—
424C22×C3C2×C4⋊(C3⋊D3)12yes—
425C22×C3C2×C4⋊(C3⋊D3)12yes—
426C22×C3C22×D3214yes—
427C22×C3C23×C3⋊D311yescharacteristic
428C2×D3—68no—
429C2×D3—616no—
430C2×D3—1216no—
431C2×D3—1816no—
432C2×D3—188no—
433C2×D3—1816no—
434C2×D3—182no—
435C2×D3—184no—
436C2×D3—3616no—
437C2×D3—3632no—
438C2×D3—364no—
439C2×D3—3616no—
440C3×C4—616no—
441C3×C4—182no—
442Dic3—24no—
443Dic3—68no—
444Dic3—1816no—
445Dic3—188no—
446Dic3—182no—
447C32—24no—
448C32#549314yescomplemented
449C32#580411yescharacteristic, complemented
450C23—42no—
451C23—92no—
452C23—182no—
453C23—272no—
454C23—362no—
455C23—542no—
456C23—1082no—
457C23#629214yescomplemented
458C23C2×D3×C3⋊D311yescharacteristic
459C23C2×D3×C3⋊D311yescharacteristic
460C23C2×D3×C3⋊D311yescharacteristic
461D4—184no—
462D4—184no—
463D4—544no—
464D4—544no—
465D4—1084no—
466C2×C4—34no—
467C2×C4—92no—
468C2×C4—122no—
469C2×C4—182no—
470C2×C4—272no—
471C2×C4—542no—
472C2×C3—216no—
473C2×C3—22no—
474C2×C3—22no—
475C2×C3—22no—
476C2×C3—28no—
477C2×C3—28no—
478C2×C3—28no—
479C2×C3—24no—
480C2×C3—28no—
481C2×C3—48no—
482C2×C3—48no—
483C2×C3—362no—
484C2×C3C22×#62614yes—
485C2×C3#6433018yes—
486C2×C3C22×C4⋊(C3⋊D3)11yescharacteristic
487C2×C3#6436112yes—
488D3—128no—
489D3—3616no—
490D3—362no—
491D3—368no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—44no—
496C22—182no—
497C22—184no—
498C22—364no—
499C22—542no—
500C22—544no—
501C22—1084no—
502C22C2×#629212yes—
503C22C2×#629212yes—
504C22C2×#629212yes—
505C22C22×D3×C3⋊D311yescharacteristic
506C4—64no—
507C4—182no—
508C4—542no—
509C3—24no—
510C3#1059551914yescomplemented
511C3#1059596411yescharacteristic, complemented
512C2—22no—
513C2—22no—
514C2—22no—
515C2—362no—
516C2—1082no—
517C2C22×#629211yescharacteristic
518C2#1172461112yes—
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC24×C33C22
hypercenterC24D3×C3⋊D3
hypocenterC33#534
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C22×#6312C2semidirect
2. C22×C3×#638C2semidirect
3. C2×C32×#217C2semidirect
4. #11723639C2semidirect
5. #11723636C2semidirect
6. #11723632C2semidirect
7. #11723609C2semidirect
8. #11723608C2semidirect
12. C23×C3⋊(C3⋊D3)C22semidirect
13. C23×C3×C3⋊D3C22semidirect
16. C2×#6312C22semidirect
17. C22×C3⋊(C3⋊Dic3)C22semidirect
20. C2×C3×#638C22semidirect
21. C22×C3×C3⋊Dic3C22semidirect
22. C22×C32×Dic3C22semidirect
23. C2×#6292C22semidirect
24. C32×#217C22semidirect
25. C32×#217C22semidirect
26. Dic3⋊(C3⋊Dic3)C22semidirect
27. #88804C22semidirect
28. #88803C22semidirect
32. C22×#638D3semidirect
40. C2×C3×#217D3semidirect
47. C23×C33D4semidirect
76. C23×C3⋊D3C2×D3semidirect
90. C2×#638C2×D3semidirect
94. C22×C3⋊Dic3C2×D3semidirect
100. C22×C3×Dic3C2×D3semidirect
107. C3×#217C2×D3semidirect
108. C3×#217C2×D3semidirect
136. C2×#217C3⋊D3semidirect
153. C23×C32D12semidirect
154. C23×C32#67semidirect
240. C22×Dic3C2×C3⋊D3semidirect
245. #217C2×C3⋊D3semidirect
246. #217C2×C3⋊D3semidirect
306. C33#534semidirect
317. C23×C3#626semidirect
318. C23×C3C4⋊(C3⋊D3)semidirect
448. C32#5493semidirect
449. C32#5804semidirect
457. C23#6292semidirect
510. C3#10595519semidirect
511. C3#10595964semidirect

Maximal subgroups

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

subgroupindex
C22×#63122
C22×C3×#6382
C2×C32×#2172
#117236392
#117236362
#117236322
#117236092
#117236082
#105959643
#105955193