Finite Group Databasebeta

← #13172166

Subgroups — #13172166

Order 1728.

1726 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6295C211yesmaximal, characteristic
2C3⋊D3×#217C212yesmaximal, complemented
3#11723595C211yesmaximal, characteristic, complemented
4C3×#63974C212yesmaximal, complemented
5C3×#63957C211yesmaximal, characteristic, complemented
6#11722938C212yesmaximal, complemented
7#11722934C214yesmaximal, complemented
8#11722931C212yesmaximal, complemented
9#10594182—32nomaximal
10#10594181—31nomaximal
11C23×C3×C3⋊D3C2212yes—
12C22×C3×C3⋊Dic3C2211yescharacteristic
13C2×#6295C418yescomplemented
14C2×#6295C2211yescharacteristic
15C2×#6295C2212yes—
16C2×#6295C2211yescharacteristic
17C2×Dic3×C3⋊D3C2214yescomplemented
18C2×C3×#624C2214yescomplemented
19#88864C2211yescharacteristic
20C32×#217C2212yes—
21C32×#217C2211yescharacteristic
22#88807C2212yescomplemented
23#88803C2214yescomplemented
24Dic3×C3⋊Dic3C2212yescomplemented
25C3×#3724C2212yescomplemented
26C3×#3722C2214yescomplemented
27C3×Dic32C2212yescomplemented
28C22×#626—32no—
29C22×#625—31no—
30C22⋊C4×C3⋊D3—32no—
31#64086—31no—
32C3×D3×C22⋊C4—32no—
33C3×D3×C22⋊C4—32no—
34C3×#877—32no—
35#63974D312yescomplemented
36D3×#217—32no—
37D3×#217—32no—
38D3×#217—62no—
39#63957D311yescharacteristic, complemented
40#63955—32no—
41#63955—61no—
42#63904—34no—
43C4×#626—34no—
44#63902—32no—
45C4×#625—32no—
46#63881—34no—
47#63880—34no—
48#63879—34no—
49C23×C33C2311yescharacteristic
50C22×C3×C3⋊D3—44no—
51C22×C3×C3⋊D3C2×C418yescomplemented
52C22×C3×C3⋊D3C2312yes—
53C22×C3×C3⋊D3C2312yes—
54C2×C3⋊(C3⋊Dic3)C2312yes—
55C2×C3×C3⋊Dic3C2×C414yescomplemented
56C2×C3×C3⋊Dic3C2311yescharacteristic
57C2×C3×C3⋊Dic3C2311yescharacteristic
58C2×C32×Dic3C2312yes—
59C2×C32×Dic3C2314yes—
60#6295C2×C414yescomplemented
61#6295C2×C418yescomplemented
62#6295C2×C414yescomplemented
63Dic3×C3⋊D3—44no—
64C3×#624—44no—
65#5521—91no—
66#5486—92no—
67C23×C3⋊D3C2×D312yes—
68C23×C3×D3—32no—
69C23×C3×D3—32no—
70C23×C3×D3—62no—
71C22×C3⋊Dic3C2×D311yescharacteristic
72C2×C4×C3⋊D3—34no—
73C22×C3×Dic3—32no—
74C22×C3×Dic3—61no—
75C2×C3×C4×D3—34no—
76C2×C3×C4×D3—34no—
77C2×#626—316no—
78C2×#626—32no—
79C2×#626—32no—
80C2×#626—32no—
81C2×#626—32no—
82C2×#625—38no—
83C2×#625—32no—
84C2×#625—31no—
85C2×#625—31no—
86C2×#624C2×D314yescomplemented
87C2×D3×Dic3—34no—
88C2×D3×Dic3—34no—
89C2×D3×Dic3—64no—
90C32×C22⋊C4—32no—
91C32×C22⋊C4—31no—
92#3757—31no—
93#3757—32no—
94#3757—61no—
95#3757—62no—
96#3757—121no—
97#3752—34no—
98#3750—32no—
99C4×C3⋊Dic3—32no—
100C3×#217—21no—
101C3×#217—22no—
102C3×#217C2×D312yes—
103C3×#217C2×D312yes—
104C3×#217C2×D312yes—
105C3×(C2×D3)⋊C4—34no—
106C3×(C2×D3)⋊C4—34no—
107C3×Dic3⋊C4—34no—
108C3×C4×Dic3—34no—
109#3724C2×D312yescomplemented
110Dic3⋊Dic3—34no—
111Dic3⋊Dic3—62no—
112#3722C2×D314yescomplemented
113(C2×D3)⋊Dic3—34no—
114(C2×D3)⋊Dic3—34no—
115(C2×D3)⋊Dic3—64no—
116Dic32—34no—
117Dic32—62no—
118Dic32C2×D312yescomplemented
119C22×C33—22no—
120C22×C33C22×C412yes—
121C22×C33C2411yescharacteristic
122C2×C3×C3⋊D3—24no—
123C2×C3×C3⋊D3—24no—
124C2×C3×C3⋊D3—48no—
125C2×C3×C3⋊D3C22×C414yes—
126C2×C3×C3⋊D3C22×C414yes—
127C3⋊(C3⋊Dic3)—22no—
128C3×C3⋊Dic3C22×C412yes—
129C3×C3⋊Dic3C22×C412yes—
130C32×Dic3—22no—
131C32×Dic3—24no—
132C22×#67—92no—
133C22×D12—91no—
134C3×#117—91no—
135C2×C3×C22⋊C4—92no—
136C2×#217—92no—
137C4×#67—94no—
138C4×#67—94no—
139#926—91no—
140#893—92no—
141#893—92no—
142#67⋊C4—94no—
143#67⋊C4—94no—
144D3×C22⋊C4—32no—
145D3×C22⋊C4—32no—
146D3×C22⋊C4—92no—
147D3×C22⋊C4—92no—
148D3×C22⋊C4—182no—
149#877—32no—
150#877—92no—
151#877—181no—
152C4×D12—94no—
153C23×C32—22no—
154C23×C32—21no—
155C23×C32—41no—
156C23×C32C22×D311yescharacteristic
157C23×C32C22×D312yes—
158C22×C3⋊D3—44no—
159C22×C3⋊D3C4×D318yescomplemented
160C22×C3⋊D3C22×D312yes—
161C22×C3⋊D3C22×D312yes—
162C22×C3×D3—32no—
163C22×C3×D3—32no—
164C22×C3×D3—32no—
165C22×C3×D3—32no—
166C22×C3×D3—38no—
167C22×C3×D3—38no—
168C22×C3×D3—62no—
169C22×C3×D3—62no—
170C22×C3×D3—68no—
171C22×C3×D3—124no—
172C22×C3×D3—124no—
173C22×C3×D3—128no—
174C2×C32×C4—34no—
175C2×C32×C4—32no—
176C2×C3⋊Dic3—34no—
177C2×C3⋊Dic3—32no—
178C2×C3⋊Dic3—62no—
179C2×C3⋊Dic3—64no—
180C2×C3⋊Dic3—122no—
181C2×C3⋊Dic3C4×D314yescomplemented
182C2×C3⋊Dic3C22×D311yescharacteristic
183C2×C3⋊Dic3C22×D311yescharacteristic
184C4×C3⋊D3—124no—
185C2×C3×Dic3—22no—
186C2×C3×Dic3—24no—
187C2×C3×Dic3—38no—
188C2×C3×Dic3—32no—
189C2×C3×Dic3—32no—
190C2×C3×Dic3—61no—
191C2×C3×Dic3—61no—
192C2×C3×Dic3—64no—
193C2×C3×Dic3C22×D314yes—
194C2×C3×Dic3C22×D314yes—
195C2×C3×Dic3C22×D314yes—
196C3×C4×D3—124no—
197C3×C4×D3—124no—
198#626—38no—
199#626—38no—
200#626—38no—
201#626—38no—
202#625—38no—
203#625—34no—
204#625—34no—
205#624—44no—
206D3×Dic3—124no—
207D3×Dic3—124no—
208D3×Dic3—128no—
209#518—271noSylow
210C2×C33—22no—
211C2×C33C23×C411yescharacteristic
212C2×C33H(5,𝔽2)12yes—
213C3×C3⋊D3—24no—
214C3×C3⋊D3—24no—
215C24×C3—92no—
216C23×D3—32no—
217C23×D3—32no—
218C23×D3—62no—
219C22×C3×C4—94no—
220C22×C3×C4—91no—
221C2×#67—92no—
222C2×#67—92no—
223C2×#67—92no—
224C2×#67—92no—
225C2×#67—916no—
226C22×Dic3—32no—
227C22×Dic3—61no—
228C22×Dic3—94no—
229C2×D12—92no—
230C2×D12—91no—
231C2×D12—91no—
232C2×D12—98no—
233C2×C4×D3—34no—
234C2×C4×D3—34no—
235C2×C4×D3—94no—
236C2×C4×D3—94no—
237C2×C4×D3—184no—
238C3×C4⋊C4—92no—
239C3×C22⋊C4—32no—
240C3×C22⋊C4—32no—
241C3×C22⋊C4—32no—
242C3×C22⋊C4—61no—
243C3×C22⋊C4—62no—
244C3×C22⋊C4—94no—
245C3×C42—92no—
246#217—92no—
247#217—91no—
248#217—94no—
249#217—181no—
250#217—182no—
251#217—361no—
252#217D3212yes—
253#217D3211yescharacteristic
254(C2×D3)⋊C4—34no—
255(C2×D3)⋊C4—34no—
256(C2×D3)⋊C4—94no—
257(C2×D3)⋊C4—94no—
258(C2×D3)⋊C4—184no—
259C4⋊Dic3—92no—
260Dic3⋊C4—34no—
261Dic3⋊C4—94no—
262Dic3⋊C4—182no—
263C4×Dic3—34no—
264C4×Dic3—94no—
265C4×Dic3—92no—
266C4×Dic3—182no—
267C22×C32—24no—
268C22×C32—22no—
269C22×C32—21no—
270C22×C32—22no—
271C22×C32—22no—
272C22×C32—24no—
273C22×C32—41no—
274C22×C32—42no—
275C22×C32—44no—
276C22×C32—42no—
277C22×C32—82no—
278C22×C32C2×C4×D314yes—
279C22×C32C2×C4×D312yes—
280C22×C32C23×D312yes—
281C22×C32C23×D311yescharacteristic
282C2×C3⋊D3—24no—
283C2×C3⋊D3—24no—
284C2×C3⋊D3—48no—
285C2×C3⋊D3C2×C4×D314yes—
286C2×C3⋊D3C2×C4×D314yes—
287C2×C3×D3—34no—
288C2×C3×D3—34no—
289C2×C3×D3—34no—
290C2×C3×D3—34no—
291C2×C3×D3—64no—
292C2×C3×D3—64no—
293C2×C3×D3—68no—
294C2×C3×D3—64no—
295C2×C3×D3—64no—
296C2×C3×D3—64no—
297C2×C3×D3—64no—
298C2×C3×D3—124no—
299C2×C3×D3—128no—
300C2×C3×D3—128no—
301C2×C3×D3—1216no—
302C32×C4—64no—
303C32×C4—62no—
304C3⋊Dic3—64no—
305C3⋊Dic3—62no—
306C3⋊Dic3—124no—
307C3⋊Dic3—122no—
308C3⋊Dic3—242no—
309C3⋊Dic3C2×C4×D312yes—
310C3⋊Dic3C2×C4×D312yes—
311C3×Dic3—24no—
312C3×Dic3—24no—
313C3×Dic3—24no—
314C3×Dic3—34no—
315C3×Dic3—34no—
316C3×Dic3—44no—
317C3×Dic3—42no—
318C3×Dic3—62no—
319C3×Dic3—62no—
320C22×D4—271no—
321C4×D4—274no—
322C4×D4—272no—
323C4×D4—272no—
324#117—91no—
325#117—271no—
326C2×C22⋊C4—92no—
327C2×C22⋊C4—272no—
328C33#51811yescharacteristic, complemented, Sylow
329C23×C3—21no—
330C23×C3—22no—
331C23×C3—41no—
332C23×C3—92no—
333C23×C3—92no—
334C23×C3—98no—
335C23×C3—364no—
336C23×C3C2×D3212yes—
337C23×C3C2×D3211yescharacteristic
338C22×D3—32no—
339C22×D3—32no—
340C22×D3—32no—
341C22×D3—32no—
342C22×D3—38no—
343C22×D3—38no—
344C22×D3—62no—
345C22×D3—68no—
346C22×D3—62no—
347C22×D3—124no—
348C22×D3—124no—
349C22×D3—128no—
350C2×C3×C4—34no—
351C2×C3×C4—34no—
352C2×C3×C4—34no—
353C2×C3×C4—62no—
354C2×C3×C4—64no—
355C2×C3×C4—91no—
356C2×C3×C4—91no—
357C2×C3×C4—94no—
358C2×C3×C4—364no—
359#67—98no—
360#67—98no—
361#67—98no—
362#67—98no—
363C2×Dic3—32no—
364C2×Dic3—32no—
365C2×Dic3—38no—
366C2×Dic3—64no—
367C2×Dic3—61no—
368C2×Dic3—61no—
369C2×Dic3—94no—
370C2×Dic3—92no—
371C2×Dic3—182no—
372C2×Dic3—184no—
373C2×Dic3—362no—
374C2×Dic3—364no—
375C2×Dic3C2×D3214yes—
376C2×Dic3C2×D3212yes—
377D12—98no—
378D12—94no—
379D12—94no—
380C4×D3—124no—
381C4×D3—124no—
382C4×D3—364no—
383C4×D3—364no—
384C4×D3—368no—
385C2×C32—24no—
386C2×C32—22no—
387C2×C32—21no—
388C2×C32—22no—
389C2×C32—22no—
390C2×C32—24no—
391C2×C32—41no—
392C2×C32—42no—
393C2×C32—44no—
394C2×C32—42no—
395C2×C32—82no—
396C2×C32C22×C4×D312yes—
397C2×C32C22×C4×D311yescharacteristic
398C2×C32#100414yes—
399C2×C32#100912yes—
400C3⋊D3—24no—
401C3⋊D3—24no—
402C3×D3—68no—
403C3×D3—64no—
404C3×D3—64no—
405C3×D3—64no—
406C3×D3—64no—
407C3×D3—124no—
408C24—92no—
409C2×D4—272no—
410C2×D4—271no—
411C2×D4—271no—
412C2×D4—278no—
413C22×C4—94no—
414C22×C4—91no—
415C22×C4—274no—
416C4⋊C4—92no—
417C4⋊C4—272no—
418C22⋊C4—31no—
419C22⋊C4—32no—
420C22⋊C4—94no—
421C22⋊C4—274no—
422C22⋊C4—271no—
423C42—92no—
424C42—272no—
425C22×C3—22no—
426C22×C3—24no—
427C22×C3—21no—
428C22×C3—22no—
429C22×C3—22no—
430C22×C3—24no—
431C22×C3—44no—
432C22×C3—42no—
433C22×C3—41no—
434C22×C3—42no—
435C22×C3—82no—
436C22×C3—94no—
437C22×C3—94no—
438C22×C3—184no—
439C22×C3—184no—
440C22×C3—368no—
441C22×C3C4×D3214yes—
442C22×C3C4×D3212yes—
443C22×C3C22×D3211yescharacteristic
444C22×C3C22×D3212yes—
445C2×D3—34no—
446C2×D3—34no—
447C2×D3—34no—
448C2×D3—34no—
449C2×D3—64no—
450C2×D3—64no—
451C2×D3—68no—
452C2×D3—64no—
453C2×D3—64no—
454C2×D3—64no—
455C2×D3—64no—
456C2×D3—124no—
457C2×D3—128no—
458C2×D3—128no—
459C2×D3—1216no—
460C3×C4—64no—
461C3×C4—64no—
462C3×C4—64no—
463C3×C4—92no—
464C3×C4—92no—
465C3×C4—124no—
466C3×C4—122no—
467Dic3—22no—
468Dic3—24no—
469Dic3—34no—
470Dic3—34no—
471Dic3—62no—
472Dic3—62no—
473Dic3—182no—
474Dic3—184no—
475Dic3—364no—
476Dic3—362no—
477Dic3—722no—
478C32—21no—
479C32—22no—
480C32—41no—
481C32#548612yescomplemented
482C32#552111yescharacteristic, complemented
483C23—92no—
484C23—92no—
485C23—98no—
486C23—364no—
487C23D3311yescharacteristic
488D4—278no—
489D4—274no—
490D4—274no—
491C2×C4—34no—
492C2×C4—32no—
493C2×C4—91no—
494C2×C4—91no—
495C2×C4—94no—
496C2×C4—272no—
497C2×C4—364no—
498C2×C4—1084no—
499C2×C3—22no—
500C2×C3—24no—
501C2×C3—21no—
502C2×C3—22no—
503C2×C3—24no—
504C2×C3—22no—
505C2×C3—44no—
506C2×C3—42no—
507C2×C3—41no—
508C2×C3—42no—
509C2×C3—82no—
510C2×C3—184no—
511C2×C3—184no—
512C2×C3C2×C4×D3211yescharacteristic
513C2×C3C2×C4×D3212yes—
514C2×C3#6431312yes—
515C2×C3#6431414yes—
516D3—68no—
517D3—64no—
518D3—64no—
519D3—64no—
520D3—64no—
521D3—124no—
522C22—22no—
523C22—94no—
524C22—94no—
525C22—184no—
526C22—184no—
527C22—368no—
528C22#8894912yes—
529C22C2×D3311yescharacteristic
530C4—62no—
531C4—64no—
532C4—92no—
533C4—92no—
534C4—542no—
535C3—22no—
536C3—21no—
537C3—41no—
538C3#1059418111yescharacteristic, complemented
539C3#1059418212yescomplemented
540C2—22no—
541C2—184no—
542C2—184no—
543C2C2×#8894911yescharacteristic
544C2#1172441612yes—
subgroupG/H
center, FrattiniC22C2×D33
commutatorC2×C33C23×C4
FittingC23×C33C23
socle, cosocleC22×C33C24
prime radicalC22×#6295C2
hypercenterC23D33
hypocenterC33#518
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C3⋊D3×#217C2semidirect
3. #11723595C2semidirect
4. C3×#63974C2semidirect
5. C3×#63957C2semidirect
6. #11722938C2semidirect
7. #11722934C2semidirect
8. #11722931C2semidirect
13. C2×#6295C4semidirect
17. C2×Dic3×C3⋊D3C22semidirect
18. C2×C3×#624C22semidirect
22. #88807C22semidirect
23. #88803C22semidirect
24. Dic3×C3⋊Dic3C22semidirect
25. C3×#3724C22semidirect
26. C3×#3722C22semidirect
27. C3×Dic32C22semidirect
35. #63974D3semidirect
39. #63957D3semidirect
51. C22×C3×C3⋊D3C2×C4semidirect
55. C2×C3×C3⋊Dic3C2×C4semidirect
60. #6295C2×C4semidirect
61. #6295C2×C4semidirect
62. #6295C2×C4semidirect
86. C2×#624C2×D3semidirect
109. #3724C2×D3semidirect
112. #3722C2×D3semidirect
118. Dic32C2×D3semidirect
159. C22×C3⋊D3C4×D3semidirect
181. C2×C3⋊Dic3C4×D3semidirect
328. C33#518semidirect
481. C32#5486semidirect
482. C32#5521semidirect
538. C3#10594181semidirect
539. C3#10594182semidirect

Maximal subgroups

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

subgroupindex
C22×#62952
C3⋊D3×#2172
#117235952
C3×#639742
C3×#639572
#117229382
#117229342
#117229312
#105941823
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