Finite Group Databasebeta

← C2×#11723615 #13178045

Subgroups — C2×#11723615 #13178045

Order 1728.

3084 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#638C211yesmaximal, characteristic, complemented
2C22×C32×#67C211yesmaximal, characteristic, complemented
3C22×#6290C211yesmaximal, characteristic, complemented
4C22×D3×C3⋊Dic3C211yesmaximal, characteristic, complemented
5C2×#88864C211yesmaximal, characteristic, complemented
6#11723615C218yesmaximal, direct product factor
7C2×#88807C211yesmaximal, characteristic, complemented
8C2×#88803C211yesmaximal, characteristic, complemented
9C2×#64148—31nomaximal
10C2×#63970—34nomaximal
11C24×C33C2211yescharacteristic
12C23×C3×C3⋊D3C2211yescharacteristic
13C23×C32×D3C2211yescharacteristic
14C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×#638—24no—
17C2×C3×#638C2214yescomplemented
18C22×C3×C3⋊Dic3C2211yescharacteristic
19C2×C32×#67—24no—
20C2×C32×#67C2214yescomplemented
21C22×C32×Dic3C2211yescharacteristic
22C2×#6290—24no—
23C2×#6290C2214yescomplemented
24C2×D3×C3⋊Dic3—24no—
25C2×D3×C3⋊Dic3C2214yescomplemented
26#88864C2214yescomplemented
27#88807C2214yescomplemented
28#88803C2214yescomplemented
29C22×C32×D4—31no—
30C22×#638—31no—
31C22×#638D311yescharacteristic, complemented
32C23×C3⋊Dic3—31no—
33C22×C3×#67—34no—
34C22×C3×#67D314yescomplemented
35C22×#625—34no—
36C22×D3×Dic3—34no—
37C2×#3757—31no—
38C2×#3757—34no—
39C2×#3757—64no—
40#64148—38no—
41C2×#3752—31no—
42C2×#3750—31no—
43#63970—332no—
44C2×Dic3⋊Dic3—34no—
45C2×(C2×D3)⋊Dic3—34no—
46C23×C33—22no—
47C23×C33—22no—
48C23×C33—22no—
49C23×C33C2312yes—
50C23×C33C2311yescharacteristic
51C22×C3×C3⋊D3—22no—
52C22×C3×C3⋊D3—22no—
53C22×C3×C3⋊D3—22no—
54C22×C3×C3⋊D3C2312yes—
55C22×C32×D3—22no—
56C22×C32×D3—22no—
57C22×C32×D3D414yescomplemented
58C22×C32×D3C2312yes—
59C2×C3⋊(C3⋊Dic3)—22no—
60C2×C3⋊(C3⋊Dic3)—22no—
61C2×C3⋊(C3⋊Dic3)C2312yes—
62C2×C3⋊(C3⋊Dic3)C2312yes—
63C3×#638—28no—
64C2×C3×C3⋊Dic3D414yescomplemented
65C2×C3×C3⋊Dic3C2312yes—
66C32×#67—28no—
67C2×C32×Dic3—22no—
68C2×C32×Dic3C2312yes—
69#6290—28no—
70D3×C3⋊Dic3—28no—
71C2×#939—94no—
72C2×#938—91no—
73C24×C32—24no—
74C24×C32—31no—
75C24×C32C2×D314yes—
76C24×C32C2×D311yescharacteristic
77C23×C3⋊D3C2×D311yescharacteristic
78C23×C3×D3—34no—
79C23×C3×D3C2×D314yes—
80C2×C32×D4—34no—
81C2×C32×D4—64no—
82C22×C32×C4—31no—
83C2×#638—24no—
84C2×#638—34no—
85C2×#638—64no—
86C2×#638C2×D314yescomplemented
87C22×C3⋊Dic3—31no—
88C22×C3⋊Dic3—31no—
89C22×C3⋊Dic3—34no—
90C22×C3⋊Dic3—34no—
91C22×C3⋊Dic3—34no—
92C22×C3⋊Dic3—64no—
93C22×C3⋊Dic3—64no—
94C22×C3⋊Dic3—64no—
95C22×C3⋊Dic3C2×D311yescharacteristic
96C2×C3×#67—216no—
97C2×C3×#67—316no—
98C2×C3×#67—616no—
99C2×C3×#67C2×D3116yescomplemented
100C22×C3×Dic3—34no—
101C22×C3×Dic3C2×D314yes—
102C2×#625—316no—
103C2×#625—616no—
104C2×D3×Dic3—316no—
105C2×D3×Dic3—616no—
106#3757—34no—
107#3757—316no—
108#3757—616no—
109#3752—34no—
110#3750—34no—
111Dic3⋊Dic3—316no—
112(C2×D3)⋊Dic3—316no—
113C22×C33—22no—
114C22×C33—22no—
115C22×C33—22no—
116C22×C33—44no—
117C22×C33C2×D412yes—
118C22×C33C2×D412yes—
119C22×C33#4112yes—
120C22×C33C2411yescharacteristic
121C2×C3×C3⋊D3—22no—
122C2×C3×C3⋊D3—22no—
123C2×C3×C3⋊D3—22no—
124C2×C3×C3⋊D3—44no—
125C2×C32×D3—28no—
126C2×C32×D3—22no—
127C2×C32×D3—22no—
128C2×C32×D3C2×D414yescomplemented
129C3⋊(C3⋊Dic3)—22no—
130C3⋊(C3⋊Dic3)—22no—
131C3×C3⋊Dic3C2×D414yescomplemented
132C32×Dic3—22no—
133C22×C3×D4—34no—
134C22×C3×D4—91no—
135C22×#67—34no—
136C22×#67—94no—
137C22×#67—91no—
138C22×#67C3⋊D311yescharacteristic, complemented
139C23×Dic3—94no—
140C22×C4×D3—91no—
141C2×#217—94no—
142C2×#217—91no—
143C2×#217—91no—
144C2×#217—184no—
145#939—932no—
146#938—98no—
147C2×(C2×D3)⋊C4—94no—
148C2×C4⋊Dic3—91no—
149C2×Dic3⋊C4—94no—
150C23×C32—216no—
151C23×C32—22no—
152C23×C32—28no—
153C23×C32—22no—
154C23×C32—22no—
155C23×C32—24no—
156C23×C32—28no—
157C23×C32—28no—
158C23×C32—28no—
159C23×C32—34no—
160C23×C32—32no—
161C23×C32—48no—
162C23×C32—48no—
163C23×C32—62no—
164C23×C32—62no—
165C23×C32C22×D318yes—
166C23×C32C22×D314yes—
167C23×C32C22×D312yes—
168C23×C32C22×D311yescharacteristic
169C22×C3⋊D3—22no—
170C22×C3⋊D3—22no—
171C22×C3⋊D3—22no—
172C22×C3⋊D3C22×D312yes—
173C22×C3×D3—28no—
174C22×C3×D3—28no—
175C22×C3×D3—38no—
176C22×C3×D3—68no—
177C22×C3×D3—68no—
178C22×C3×D3—68no—
179C22×C3×D3#67116yescomplemented
180C22×C3×D3C22×D318yes—
181C32×D4—68no—
182C2×C32×C4—32no—
183C2×C32×C4—62no—
184#638—28no—
185#638—68no—
186C2×C3⋊Dic3—32no—
187C2×C3⋊Dic3—32no—
188C2×C3⋊Dic3—38no—
189C2×C3⋊Dic3—38no—
190C2×C3⋊Dic3—68no—
191C2×C3⋊Dic3—68no—
192C2×C3⋊Dic3—62no—
193C2×C3⋊Dic3—62no—
194C2×C3⋊Dic3—68no—
195C2×C3⋊Dic3—68no—
196C2×C3⋊Dic3—68no—
197C2×C3⋊Dic3—128no—
198C2×C3⋊Dic3—128no—
199C2×C3⋊Dic3#6714yescomplemented
200C2×C3⋊Dic3C22×D312yes—
201C3×#67—232no—
202C3×#67—632no—
203C2×C3×Dic3—28no—
204C2×C3×Dic3—316no—
205C2×C3×Dic3—38no—
206C2×C3×Dic3C22×D318yes—
207#625—632no—
208D3×Dic3—632no—
209C2×#121—271noSylow
210C2×C33—42no—
211C2×C33#12114yescomplemented
212C2×C33C22×D411yescharacteristic
213C2×C33C22×D411yescharacteristic
214C2×C33C2×#4111yescharacteristic
215C3×C3⋊D3—42no—
216C32×D3—24no—
217C24×C3—24no—
218C24×C3—34no—
219C24×C3—91no—
220C24×C3D3214yes—
221C24×C3C2×C3⋊D311yescharacteristic
222C23×D3—34no—
223C23×D3C2×C3⋊D311yescharacteristic
224C2×C3×D4—316no—
225C2×C3×D4—616no—
226C2×C3×D4—94no—
227C2×C3×D4—184no—
228C22×C3×C4—34no—
229C22×C3×C4—91no—
230C2×#67—24no—
231C2×#67—316no—
232C2×#67—616no—
233C2×#67—916no—
234C2×#67—94no—
235C2×#67—1816no—
236C2×#67—184no—
237C2×#67C2×C3⋊D314yescomplemented
238C22×Dic3—34no—
239C22×Dic3—916no—
240C22×Dic3—94no—
241C22×Dic3—94no—
242C22×Dic3—91no—
243C22×Dic3—91no—
244C22×Dic3—184no—
245C22×Dic3—184no—
246C22×Dic3—1816no—
247C22×Dic3C2×C3⋊D311yescharacteristic
248C2×C4×D3—94no—
249C2×C4×D3—184no—
250#217—916no—
251#217—94no—
252#217—94no—
253#217—1816no—
254(C2×D3)⋊C4—916no—
255C4⋊Dic3—94no—
256Dic3⋊C4—916no—
257C22×C32—216no—
258C22×C32—22no—
259C22×C32—28no—
260C22×C32—22no—
261C22×C32—22no—
262C22×C32—24no—
263C22×C32—28no—
264C22×C32—28no—
265C22×C32—28no—
266C22×C32—28no—
267C22×C32—28no—
268C22×C32—34no—
269C22×C32—48no—
270C22×C32—48no—
271C22×C32—432no—
272C22×C32—44no—
273C22×C32—416no—
274C22×C32—68no—
275C22×C32—62no—
276C22×C32—62no—
277C22×C32D3×D418yes—
278C22×C32D3×D412yes—
279C22×C32#23718yes—
280C22×C32#23712yes—
281C22×C32C2×#6718yes—
282C22×C32C2×#6712yes—
283C22×C32C23×D314yes—
284C22×C32C23×D311yescharacteristic
285C2×C3⋊D3—22no—
286C2×C3⋊D3—22no—
287C2×C3⋊D3—22no—
288C2×C3⋊D3—44no—
289C2×C3×D3—232no—
290C2×C3×D3—28no—
291C2×C3×D3—28no—
292C2×C3×D3—68no—
293C2×C3×D3—68no—
294C2×C3×D3—68no—
295C2×C3×D3—1216no—
296C2×C3×D3C2×#67116yescomplemented
297C32×C4—62no—
298C3⋊Dic3—62no—
299C3⋊Dic3—62no—
300C3⋊Dic3—68no—
301C3⋊Dic3—68no—
302C3⋊Dic3—128no—
303C3⋊Dic3—128no—
304C3⋊Dic3C2×#6714yescomplemented
305C3×Dic3—28no—
306C3×Dic3—316no—
307C22×D4—31no—
308C22×D4—91no—
309C22×D4—271no—
310C23×C4—271no—
311#121—278no—
312C2×C4⋊C4—271no—
313C2×C22⋊C4—271no—
314C2×C22⋊C4—271no—
315C33C2×#12111yescharacteristic, complemented, Sylow
316C23×C3—216no—
317C23×C3—28no—
318C23×C3—28no—
319C23×C3—28no—
320C23×C3—22no—
321C23×C3—22no—
322C23×C3—22no—
323C23×C3—28no—
324C23×C3—24no—
325C23×C3—38no—
326C23×C3—316no—
327C23×C3—48no—
328C23×C3—48no—
329C23×C3—68no—
330C23×C3—68no—
331C23×C3—92no—
332C23×C3—182no—
333C23×C3—182no—
334C23×C3—182no—
335C23×C3C2×D3218yes—
336C23×C3C2×D3214yes—
337C23×C3C22×C3⋊D312yes—
338C23×C3C22×C3⋊D311yescharacteristic
339C22×D3—22no—
340C22×D3—22no—
341C22×D3—38no—
342C22×D3—68no—
343C22×D3—68no—
344C22×D3—68no—
345C22×D3#63814yescomplemented
346C22×D3C22×C3⋊D312yes—
347C3×D4—632no—
348C3×D4—188no—
349C2×C3×C4—38no—
350C2×C3×C4—68no—
351C2×C3×C4—92no—
352C2×C3×C4—94no—
353#67—28no—
354#67—632no—
355#67—1832no—
356#67—188no—
357C2×Dic3—22no—
358C2×Dic3—38no—
359C2×Dic3—316no—
360C2×Dic3—98no—
361C2×Dic3—98no—
362C2×Dic3—92no—
363C2×Dic3—92no—
364C2×Dic3—188no—
365C2×Dic3—188no—
366C2×Dic3—1832no—
367C2×Dic3—188no—
368C2×Dic3—188no—
369C2×Dic3—182no—
370C2×Dic3—182no—
371C2×Dic3—368no—
372C2×Dic3—368no—
373C2×Dic3C22×C3⋊D312yes—
374C4×D3—188no—
375C2×C32—24no—
376C2×C32—24no—
377C2×C32—24no—
378C2×C32—216no—
379C2×C32—416no—
380C2×C32—42no—
381C2×C32—48no—
382C2×C32—64no—
383C2×C32#93814yescomplemented
384C2×C32#939116yescomplemented
385C2×C32C2×D3×D414yes—
386C2×C32C2×D3×D411yescharacteristic
387C2×C32C2×#23714yes—
388C2×C32C2×#23711yescharacteristic
389C2×C32C22×#6714yes—
390C2×C32C22×#6711yescharacteristic
391C3⋊D3—42no—
392C3×D3—216no—
393C3×D3—128no—
394C24—31no—
395C24—91no—
396C24D3×C3⋊D311yescharacteristic
397C2×D4—34no—
398C2×D4—64no—
399C2×D4—94no—
400C2×D4—184no—
401C2×D4—274no—
402C2×D4—544no—
403C22×C4—31no—
404C22×C4—91no—
405C22×C4—274no—
406C22×C4—271no—
407C22×C4—271no—
408C22×C4—544no—
409C4⋊C4—274no—
410C22⋊C4—274no—
411C22⋊C4—274no—
412C22×C3—216no—
413C22×C3—28no—
414C22×C3—28no—
415C22×C3—28no—
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—22no—
419C22×C3—24no—
420C22×C3—28no—
421C22×C3—28no—
422C22×C3—28no—
423C22×C3—316no—
424C22×C3—48no—
425C22×C3—48no—
426C22×C3—432no—
427C22×C3—44no—
428C22×C3—416no—
429C22×C3—68no—
430C22×C3—68no—
431C22×C3—632no—
432C22×C3—182no—
433C22×C3—182no—
434C22×C3—182no—
435C22×C3—364no—
436C22×C3#380518yes—
437C22×C3D3×#6718yes—
438C22×C3D3×#6718yes—
439C22×C3D4×C3⋊D312yes—
440C22×C3#383012yes—
441C22×C3C2×#63812yes—
442C22×C3C22×D3214yes—
443C22×C3C23×C3⋊D311yescharacteristic
444C2×D3—22no—
445C2×D3—22no—
446C2×D3—28no—
447C2×D3—68no—
448C2×D3—68no—
449C2×D3—68no—
450C2×D3—1216no—
451C2×D3C2×#63814yescomplemented
452C3×C4—68no—
453C3×C4—94no—
454Dic3—22no—
455Dic3—316no—
456Dic3—188no—
457Dic3—188no—
458Dic3—182no—
459Dic3—182no—
460Dic3—368no—
461Dic3—368no—
462C32—24no—
463C32C2×#93811yescharacteristic, complemented
464C32C2×#93914yescomplemented
465C23—22no—
466C23—22no—
467C23—22no—
468C23—32no—
469C23—34no—
470C23—62no—
471C23—62no—
472C23—92no—
473C23—182no—
474C23—182no—
475C23—182no—
476C23C2×D3×C3⋊D312yes—
477C23C2×D3×C3⋊D311yescharacteristic
478D4—68no—
479D4—188no—
480D4—548no—
481C2×C4—32no—
482C2×C4—62no—
483C2×C4—92no—
484C2×C4—94no—
485C2×C4—272no—
486C2×C4—272no—
487C2×C4—548no—
488C2×C4—542no—
489C2×C4—542no—
490C2×C3—24no—
491C2×C3—24no—
492C2×C3—24no—
493C2×C3—216no—
494C2×C3—416no—
495C2×C3—48no—
496C2×C3—42no—
497C2×C3—616no—
498C2×C3—362no—
499C2×C3#63970116yescomplemented
500C2×C3#6414814yescomplemented
501C2×C3C2×#380514yes—
502C2×C3C2×D3×#6714yes—
503C2×C3C2×D3×#6714yes—
504C2×C3C2×D4×C3⋊D311yescharacteristic
505C2×C3C2×#383011yescharacteristic
506C2×C3C22×#63811yescharacteristic
507D3—24no—
508D3—128no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—34no—
513C22—44no—
514C22—62no—
515C22—62no—
516C22—68no—
517C22—182no—
518C22—182no—
519C22—182no—
520C22—364no—
521C22#8902912yes—
522C22D3×#63812yes—
523C22C3⋊D3×#6712yes—
524C22C22×D3×C3⋊D311yescharacteristic
525C4—62no—
526C4—94no—
527C4—542no—
528C4—542no—
529C3—24no—
530C3C2×#6397014yescomplemented
531C3C2×#6414811yescharacteristic, complemented
532C2—42no—
533C2—64no—
534C2—362no—
535C2#1172361514yesdirect product factor
536C2C2×#8902911yescharacteristic
537C2C2×D3×#63811yescharacteristic
538C2C2×C3⋊D3×#6711yescharacteristic
subgroupG/H
centerC23C2×D3×C3⋊D3
commutator, cosocleC22×C33C24
FrattiniC22C22×D3×C3⋊D3
FittingC24×C33C22
socleC23×C33C23
hypercenterC24D3×C3⋊D3
hypocenterC33C2×#121
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C22×C3×#638C2semidirect
2. C22×C32×#67C2semidirect
3. C22×#6290C2semidirect
4. C22×D3×C3⋊Dic3C2semidirect
5. C2×#88864C2semidirect
6. #11723615C2direct (N × H)
7. C2×#88807C2semidirect
8. C2×#88803C2semidirect
17. C2×C3×#638C22semidirect
20. C2×C32×#67C22semidirect
23. C2×#6290C22semidirect
25. C2×D3×C3⋊Dic3C22semidirect
26. #88864C22semidirect
27. #88807C22semidirect
28. #88803C22semidirect
31. C22×#638D3semidirect
34. C22×C3×#67D3semidirect
57. C22×C32×D3D4semidirect
64. C2×C3×C3⋊Dic3D4semidirect
86. C2×#638C2×D3semidirect
99. C2×C3×#67C2×D3semidirect
128. C2×C32×D3C2×D4semidirect
131. C3×C3⋊Dic3C2×D4semidirect
138. C22×#67C3⋊D3semidirect
179. C22×C3×D3#67semidirect
199. C2×C3⋊Dic3#67semidirect
211. C2×C33#121semidirect
237. C2×#67C2×C3⋊D3semidirect
296. C2×C3×D3C2×#67semidirect
304. C3⋊Dic3C2×#67semidirect
315. C33C2×#121semidirect
345. C22×D3#638semidirect
383. C2×C32#938semidirect
384. C2×C32#939semidirect
451. C2×D3C2×#638semidirect
463. C32C2×#938semidirect
464. C32C2×#939semidirect
499. C2×C3#63970semidirect
500. C2×C3#64148semidirect
530. C3C2×#63970semidirect
531. C3C2×#64148semidirect
535. C2#11723615direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6382
C22×C32×#672
C22×#62902
C22×D3×C3⋊Dic32
C2×#888642
#117236152
C2×#888072
C2×#888032
C2×#641483
C2×#639703