Finite Group Databasebeta

← C2×#11722953 #13172320

Subgroups — C2×#11722953 #13172320

Order 1728.

2258 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D3×C3⋊Dic3C211yesmaximal, characteristic, complemented
2C22×C3×#625C211yesmaximal, characteristic, complemented
3C22×C3×#625C212yesmaximal, complemented
4C2×#88815C211yesmaximal, characteristic, complemented
5C2×(C2×D3)⋊(C3⋊Dic3)C212yesmaximal, complemented
6#11722953C218yesmaximal, direct product factor
7C2×#63980—31nomaximal
8C2×#63970—32nomaximal
9C23×C32×D3C2212yes—
10C23×C32×D3C2211yescharacteristic
11C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
12C22×C3×C3⋊Dic3C2212yes—
13C22×C3×C3⋊Dic3C2211yescharacteristic
14C2×D3×C3⋊Dic3—24no—
15C2×D3×C3⋊Dic3C2214yescomplemented
16C2×C3×#625—28no—
17C2×C3×#625—24no—
18C2×C3×#625C2218yescomplemented
19C2×C3×#625C2214yescomplemented
20#88815C2214yescomplemented
21(C2×D3)⋊(C3⋊Dic3)C2218yescomplemented
22C23×C3⋊Dic3—31no—
23C22×C3×#67—32no—
24C22×C3×#67—32no—
25C22×C3×#67—32no—
26C22×#625D312yescomplemented
27C22×#625D311yescharacteristic, complemented
28C22×D3×Dic3—32no—
29C22×D3×Dic3—61no—
30C2×#3757—32no—
31#63980—38no—
32#63970—316no—
33C2×#3724—31no—
34C2×Dic3⋊Dic3—32no—
35C2×(C2×D3)⋊Dic3—32no—
36C2×(C2×D3)⋊Dic3—32no—
37C2×(C2×D3)⋊Dic3—62no—
38C23×C33C2311yescharacteristic
39C22×C32×D3—22no—
40C22×C32×D3—22no—
41C22×C32×D3—24no—
42C22×C32×D3—24no—
43C22×C32×D3—24no—
44C22×C32×D3D414yescomplemented
45C22×C32×D3C2314yes—
46C22×C32×D3C2312yes—
47C2×C3⋊(C3⋊Dic3)—22no—
48C2×C3⋊(C3⋊Dic3)C2312yes—
49C2×C3×C3⋊Dic3—24no—
50C2×C3×C3⋊Dic3D414yescomplemented
51C2×C3×C3⋊Dic3C2314yes—
52C2×C3×C3⋊Dic3C2312yes—
53D3×C3⋊Dic3—28no—
54C3×#625—216no—
55C3×#625—28no—
56C2×#939—92no—
57C2×#938—91no—
58C24×C32—31no—
59C24×C32—32no—
60C23×C3×D3—21no—
61C23×C3×D3—22no—
62C23×C3×D3C2×D312yes—
63C23×C3×D3C2×D312yes—
64C23×C3×D3C2×D312yes—
65C22×C3⋊Dic3—32no—
66C22×C3⋊Dic3—31no—
67C22×C3⋊Dic3—34no—
68C22×C3⋊Dic3—61no—
69C22×C3⋊Dic3—64no—
70C22×C3⋊Dic3—62no—
71C22×C3⋊Dic3—121no—
72C22×C3⋊Dic3C2×D312yes—
73C22×C3⋊Dic3C2×D311yescharacteristic
74C2×C3×#67—38no—
75C2×C3×#67—38no—
76C2×C3×#67—38no—
77C2×C3×#67—68no—
78C2×C3×#67—68no—
79C2×C3×#67—68no—
80C22×C3×Dic3—32no—
81C22×C3×Dic3—32no—
82C22×C3×Dic3—32no—
83C22×C3×Dic3—61no—
84C22×C3×Dic3—62no—
85C2×#625—28no—
86C2×#625—24no—
87C2×#625C2×D318yescomplemented
88C2×#625C2×D314yescomplemented
89C2×D3×Dic3—38no—
90C2×D3×Dic3—68no—
91C2×D3×Dic3—64no—
92C2×D3×Dic3—68no—
93#3757—38no—
94#3724—34no—
95Dic3⋊Dic3—38no—
96(C2×D3)⋊Dic3—38no—
97(C2×D3)⋊Dic3—38no—
98(C2×D3)⋊Dic3—68no—
99C22×C33C2×D412yes—
100C22×C33C2×D412yes—
101C22×C33#4112yes—
102C22×C33C2411yescharacteristic
103C2×C32×D3—28no—
104C2×C32×D3—22no—
105C2×C32×D3—22no—
106C2×C32×D3—24no—
107C2×C32×D3—24no—
108C2×C32×D3—24no—
109C2×C32×D3—48no—
110C2×C32×D3C2×D414yescomplemented
111C3⋊(C3⋊Dic3)—22no—
112C3×C3⋊Dic3—24no—
113C3×C3⋊Dic3C2×D414yescomplemented
114C22×C3×D4—91no—
115C22×C3×D4—92no—
116C22×#67—32no—
117C22×#67—32no—
118C22×#67—32no—
119C23×Dic3—92no—
120C23×Dic3—181no—
121C22×C4×D3—91no—
122C2×#217—92no—
123C2×#217—92no—
124C2×#217—182no—
125#939—916no—
126#938—98no—
127C2×(C2×D3)⋊C4—92no—
128C2×C4⋊Dic3—91no—
129C2×Dic3⋊C4—92no—
130C23×C32—21no—
131C23×C32—22no—
132C23×C32—34no—
133C23×C32—32no—
134C23×C32—34no—
135C23×C32—41no—
136C23×C32—62no—
137C23×C32—62no—
138C23×C32—64no—
139C23×C32—64no—
140C23×C32—64no—
141C23×C32C22×D312yes—
142C23×C32C22×D311yescharacteristic
143C22×C3×D3—24no—
144C22×C3×D3—24no—
145C22×C3×D3—24no—
146C22×C3×D3—24no—
147C22×C3×D3—28no—
148C22×C3×D3—24no—
149C22×C3×D3—24no—
150C22×C3×D3—24no—
151C22×C3×D3—24no—
152C22×C3×D3—24no—
153C22×C3×D3—24no—
154C22×C3×D3—24no—
155C22×C3×D3—22no—
156C22×C3×D3—24no—
157C22×C3×D3—44no—
158C22×C3×D3—44no—
159C22×C3×D3#6718yescomplemented
160C22×C3×D3C22×D314yes—
161C22×C3×D3C22×D314yes—
162C22×C3×D3C22×D314yes—
163C2×C3⋊Dic3—24no—
164C2×C3⋊Dic3—32no—
165C2×C3⋊Dic3—34no—
166C2×C3⋊Dic3—62no—
167C2×C3⋊Dic3—64no—
168C2×C3⋊Dic3—62no—
169C2×C3⋊Dic3—68no—
170C2×C3⋊Dic3—64no—
171C2×C3⋊Dic3—64no—
172C2×C3⋊Dic3—122no—
173C2×C3⋊Dic3—124no—
174C2×C3⋊Dic3—124no—
175C2×C3⋊Dic3#6714yescomplemented
176C2×C3⋊Dic3C22×D314yes—
177C2×C3⋊Dic3C22×D312yes—
178C3×#67—616no—
179C3×#67—616no—
180C3×#67—616no—
181C2×C3×Dic3—38no—
182C2×C3×Dic3—34no—
183C2×C3×Dic3—34no—
184C2×C3×Dic3—34no—
185C2×C3×Dic3—64no—
186C2×C3×Dic3—62no—
187C2×C3×Dic3—64no—
188C2×C3×Dic3—64no—
189C2×C3×Dic3—64no—
190C2×C3×Dic3—124no—
191#625—216no—
192#625—28no—
193D3×Dic3—616no—
194D3×Dic3—616no—
195C2×#121—271noSylow
196C2×C33#12114yescomplemented
197C2×C33C22×D411yescharacteristic
198C2×C33C22×D411yescharacteristic
199C2×C33C2×#4111yescharacteristic
200C32×D3—24no—
201C32×D3—44no—
202C24×C3—32no—
203C24×C3—32no—
204C24×C3—32no—
205C24×C3—61no—
206C24×C3—62no—
207C23×D3D3212yes—
208C23×D3D3211yescharacteristic
209C2×C3×D4—94no—
210C2×C3×D4—98no—
211C2×C3×D4—184no—
212C2×C3×D4—188no—
213C22×C3×C4—91no—
214C22×C3×C4—92no—
215C2×#67—38no—
216C2×#67—38no—
217C2×#67—38no—
218C2×#67—68no—
219C2×#67—68no—
220C2×#67—68no—
221C22×Dic3—32no—
222C22×Dic3—32no—
223C22×Dic3—32no—
224C22×Dic3—61no—
225C22×Dic3—62no—
226C22×Dic3—98no—
227C22×Dic3—92no—
228C22×Dic3—91no—
229C22×Dic3—184no—
230C22×Dic3—181no—
231C22×Dic3—182no—
232C22×Dic3—188no—
233C22×Dic3—188no—
234C22×Dic3—361no—
235C2×C4×D3—94no—
236C2×C4×D3—184no—
237#217—98no—
238#217—98no—
239#217—188no—
240(C2×D3)⋊C4—98no—
241C4⋊Dic3—94no—
242Dic3⋊C4—98no—
243C22×C32—21no—
244C22×C32—22no—
245C22×C32—24no—
246C22×C32—22no—
247C22×C32—22no—
248C22×C32—24no—
249C22×C32—24no—
250C22×C32—22no—
251C22×C32—34no—
252C22×C32—42no—
253C22×C32—42no—
254C22×C32—42no—
255C22×C32—41no—
256C22×C32—68no—
257C22×C32—62no—
258C22×C32—62no—
259C22×C32—64no—
260C22×C32—64no—
261C22×C32—64no—
262C22×C32—128no—
263C22×C32D3×D412yes—
264C22×C32D3×D414yes—
265C22×C32#23712yes—
266C22×C32#23714yes—
267C22×C32C2×#6714yes—
268C22×C32C2×#6712yes—
269C22×C32C23×D312yes—
270C22×C32C23×D311yescharacteristic
271C2×C3×D3—24no—
272C2×C3×D3—24no—
273C2×C3×D3—216no—
274C2×C3×D3—216no—
275C2×C3×D3—24no—
276C2×C3×D3—24no—
277C2×C3×D3—28no—
278C2×C3×D3—24no—
279C2×C3×D3—24no—
280C2×C3×D3—24no—
281C2×C3×D3—24no—
282C2×C3×D3—24no—
283C2×C3×D3—24no—
284C2×C3×D3—24no—
285C2×C3×D3—44no—
286C2×C3×D3—44no—
287C2×C3×D3—416no—
288C2×C3×D3—48no—
289C2×C3×D3—48no—
290C2×C3×D3C2×#6718yescomplemented
291C3⋊Dic3—24no—
292C3⋊Dic3—62no—
293C3⋊Dic3—64no—
294C3⋊Dic3—64no—
295C3⋊Dic3—124no—
296C3⋊Dic3—124no—
297C3⋊Dic3C2×#6714yescomplemented
298C3×Dic3—38no—
299C3×Dic3—64no—
300C3×Dic3—64no—
301C3×Dic3—64no—
302C3×Dic3—124no—
303C22×D4—91no—
304C22×D4—92no—
305C23×C4—271no—
306#121—278no—
307C2×C4⋊C4—271no—
308C2×C22⋊C4—272no—
309C33C2×#12111yescharacteristic, complemented, Sylow
310C23×C3—22no—
311C23×C3—21no—
312C23×C3—34no—
313C23×C3—38no—
314C23×C3—34no—
315C23×C3—34no—
316C23×C3—41no—
317C23×C3—62no—
318C23×C3—64no—
319C23×C3—64no—
320C23×C3—64no—
321C23×C3—64no—
322C23×C3—68no—
323C23×C3—64no—
324C23×C3—64no—
325C23×C3—64no—
326C23×C3—64no—
327C23×C3—64no—
328C23×C3—64no—
329C23×C3—64no—
330C23×C3—64no—
331C23×C3—124no—
332C23×C3—124no—
333C23×C3C2×D3211yescharacteristic
334C23×C3C2×D3212yes—
335C22×D3—22no—
336C22×D3—22no—
337C22×D3—24no—
338C22×D3—24no—
339C22×D3—24no—
340C22×D3#62514yescomplemented
341C22×D3C2×D3214yes—
342C22×D3C2×D3212yes—
343C3×D4—188no—
344C3×D4—1816no—
345C2×C3×C4—92no—
346C2×C3×C4—94no—
347C2×C3×C4—94no—
348C2×C3×C4—184no—
349#67—616no—
350#67—616no—
351#67—616no—
352C2×Dic3—34no—
353C2×Dic3—38no—
354C2×Dic3—34no—
355C2×Dic3—34no—
356C2×Dic3—64no—
357C2×Dic3—62no—
358C2×Dic3—64no—
359C2×Dic3—64no—
360C2×Dic3—64no—
361C2×Dic3—94no—
362C2×Dic3—92no—
363C2×Dic3—124no—
364C2×Dic3—182no—
365C2×Dic3—184no—
366C2×Dic3—1816no—
367C2×Dic3—1816no—
368C2×Dic3—184no—
369C2×Dic3—184no—
370C2×Dic3—182no—
371C2×Dic3—362no—
372C2×Dic3—364no—
373C2×Dic3—364no—
374C4×D3—188no—
375C2×C32—22no—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32—22no—
379C2×C32—22no—
380C2×C32—21no—
381C2×C32—28no—
382C2×C32—24no—
383C2×C32—44no—
384C2×C32—41no—
385C2×C32—41no—
386C2×C32—41no—
387C2×C32—64no—
388C2×C32—124no—
389C2×C32#93814yescomplemented
390C2×C32#93918yescomplemented
391C2×C32C2×D3×D411yescharacteristic
392C2×C32C2×D3×D412yes—
393C2×C32C2×#23711yescharacteristic
394C2×C32C2×#23712yes—
395C2×C32C22×#6712yes—
396C2×C32C22×#6711yescharacteristic
397C3×D3—28no—
398C3×D3—28no—
399C3×D3—48no—
400C3×D3—44no—
401C3×D3—44no—
402C24—31no—
403C24—32no—
404C2×D4—94no—
405C2×D4—98no—
406C2×D4—184no—
407C2×D4—188no—
408C22×C4—91no—
409C22×C4—92no—
410C22×C4—274no—
411C22×C4—271no—
412C22×C4—544no—
413C4⋊C4—274no—
414C22⋊C4—278no—
415C22×C3—22no—
416C22×C3—21no—
417C22×C3—24no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—24no—
421C22×C3—22no—
422C22×C3—24no—
423C22×C3—38no—
424C22×C3—41no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3—42no—
428C22×C3—64no—
429C22×C3—64no—
430C22×C3—64no—
431C22×C3—616no—
432C22×C3—68no—
433C22×C3—64no—
434C22×C3—64no—
435C22×C3—616no—
436C22×C3—64no—
437C22×C3—64no—
438C22×C3—64no—
439C22×C3—64no—
440C22×C3—64no—
441C22×C3—64no—
442C22×C3—124no—
443C22×C3—124no—
444C22×C3—1216no—
445C22×C3—128no—
446C22×C3—128no—
447C22×C3#380514yes—
448C22×C3#380512yes—
449C22×C3C2×#62512yes—
450C22×C3D3×#6714yes—
451C22×C3D3×#6714yes—
452C22×C3#381112yes—
453C22×C3C22×D3211yescharacteristic
454C22×C3C22×D3212yes—
455C2×D3—22no—
456C2×D3—22no—
457C2×D3—28no—
458C2×D3—24no—
459C2×D3—24no—
460C2×D3—24no—
461C2×D3—48no—
462C2×D3C2×#62514yescomplemented
463C3×C4—94no—
464C3×C4—184no—
465Dic3—38no—
466Dic3—64no—
467Dic3—64no—
468Dic3—64no—
469Dic3—124no—
470Dic3—184no—
471Dic3—184no—
472Dic3—182no—
473Dic3—364no—
474Dic3—364no—
475C32—22no—
476C32—21no—
477C32—41no—
478C32C2×#93811yescharacteristic, complemented
479C32C2×#93912yescomplemented
480C23—32no—
481C23—34no—
482C23—34no—
483C23—62no—
484C23—62no—
485C23—64no—
486C23—64no—
487C23—64no—
488C23D3311yescharacteristic
489D4—188no—
490D4—1816no—
491C2×C4—92no—
492C2×C4—94no—
493C2×C4—94no—
494C2×C4—184no—
495C2×C4—272no—
496C2×C4—548no—
497C2×C4—542no—
498C2×C3—22no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—22no—
502C2×C3—21no—
503C2×C3—22no—
504C2×C3—24no—
505C2×C3—28no—
506C2×C3—41no—
507C2×C3—41no—
508C2×C3—41no—
509C2×C3—44no—
510C2×C3—68no—
511C2×C3—68no—
512C2×C3—128no—
513C2×C3—124no—
514C2×C3—124no—
515C2×C3#6397018yescomplemented
516C2×C3#6398014yescomplemented
517C2×C3C2×#380512yes—
518C2×C3C2×#380511yescharacteristic
519C2×C3C22×#62511yescharacteristic
520C2×C3C2×D3×#6712yes—
521C2×C3C2×D3×#6712yes—
522C2×C3C2×#381111yescharacteristic
523D3—24no—
524D3—44no—
525C22—34no—
526C22—62no—
527C22—62no—
528C22—68no—
529C22—64no—
530C22—64no—
531C22—64no—
532C22—128no—
533C22D3×#62512yes—
534C22#8895212yes—
535C22#8896312yes—
536C22C2×D3311yescharacteristic
537C4—94no—
538C4—184no—
539C4—542no—
540C3—21no—
541C3—22no—
542C3—41no—
543C3C2×#6397012yescomplemented
544C3C2×#6398011yescharacteristic, complemented
545C2—64no—
546C2—124no—
547C2#1172295314yesdirect product factor
548C2C2×D3×#62511yescharacteristic
549C2C2×#8895211yescharacteristic
550C2C2×#8896311yescharacteristic
subgroupG/H
center, hypercenterC23D33
commutator, cosocleC22×C33C24
FrattiniC22C2×D33
Fitting, socleC23×C33C23
hypocenterC33C2×#121
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×D3×C3⋊Dic3C2semidirect
2. C22×C3×#625C2semidirect
3. C22×C3×#625C2semidirect
4. C2×#88815C2semidirect
5. C2×(C2×D3)⋊(C3⋊Dic3)C2semidirect
6. #11722953C2direct (N × H)
15. C2×D3×C3⋊Dic3C22semidirect
18. C2×C3×#625C22semidirect
19. C2×C3×#625C22semidirect
20. #88815C22semidirect
21. (C2×D3)⋊(C3⋊Dic3)C22semidirect
26. C22×#625D3semidirect
27. C22×#625D3semidirect
44. C22×C32×D3D4semidirect
50. C2×C3×C3⋊Dic3D4semidirect
87. C2×#625C2×D3semidirect
88. C2×#625C2×D3semidirect
110. C2×C32×D3C2×D4semidirect
113. C3×C3⋊Dic3C2×D4semidirect
159. C22×C3×D3#67semidirect
175. C2×C3⋊Dic3#67semidirect
196. C2×C33#121semidirect
290. C2×C3×D3C2×#67semidirect
297. C3⋊Dic3C2×#67semidirect
309. C33C2×#121semidirect
340. C22×D3#625semidirect
389. C2×C32#938semidirect
390. C2×C32#939semidirect
462. C2×D3C2×#625semidirect
478. C32C2×#938semidirect
479. C32C2×#939semidirect
515. C2×C3#63970semidirect
516. C2×C3#63980semidirect
543. C3C2×#63970semidirect
544. C3C2×#63980semidirect
547. C2#11722953direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D3×C3⋊Dic32
C22×C3×#6252
C2×#888152
C2×(C2×D3)⋊(C3⋊Dic3)2
#117229532
C2×#639803
C2×#639703