Finite Group Databasebeta

← C2×#11723635 #13178081

Subgroups — C2×#11723635 #13178081

Order 1728.

3500 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6312C211yesmaximal, characteristic, complemented
2C23×C3×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C22×C32×#67C211yesmaximal, characteristic, complemented
4C22×#6291C211yesmaximal, characteristic, complemented
5#11723635C218yesmaximal, direct product factor
6C2×#88807C211yesmaximal, characteristic, complemented
7C2×#88804C211yesmaximal, characteristic, complemented
8C2×(C2×D3)⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
9C2×#64148—31nomaximal
10C2×#63978—34nomaximal
11C24×C33C2211yescharacteristic
12C23×C3⋊(C3⋊D3)C2211yescharacteristic
13C23×C32×D3C2211yescharacteristic
14C2×#6312—24no—
15C2×#6312C2214yescomplemented
16C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
17C22×C3×C3⋊Dic3—24no—
18C22×C3×C3⋊Dic3C2211yescharacteristic
19C22×C3×C3⋊Dic3C2211yescharacteristic
20C22×C3×C3⋊Dic3C2214yescomplemented
21C2×C32×#67—24no—
22C2×C32×#67C2214yescomplemented
23C22×C32×Dic3C2211yescharacteristic
24C2×#6291—24no—
25C2×#6291C2214yescomplemented
26#88807C2214yescomplemented
27#88804C2214yescomplemented
28(C2×D3)⋊(C3⋊Dic3)C2214yescomplemented
29C22×C32×D4—31no—
30C22×#638—31no—
31C22×#638—31no—
32C22×#638—34no—
33C22×#638—64no—
34C23×C3⋊Dic3D311yescharacteristic, complemented
35C22×C3×#67D314yescomplemented
36C23×C3×Dic3—34no—
37C22×#626—34no—
38C2×#3757—31no—
39#64148—38no—
40C2×#3752—31no—
41C2×#3750—31no—
42#63978—332no—
43C2×Dic3⋊Dic3—34no—
44C2×#3722—34no—
45C2×(C2×D3)⋊Dic3—34no—
46C23×C33—22no—
47C23×C33—22no—
48C23×C33D414yescomplemented
49C23×C33C2312yes—
50C23×C33C2311yescharacteristic
51C22×C3⋊(C3⋊D3)—22no—
52C22×C3⋊(C3⋊D3)—22no—
53C22×C3⋊(C3⋊D3)—22no—
54C22×C3⋊(C3⋊D3)C2312yes—
55C22×C32×D3—22no—
56C22×C32×D3—22no—
57C22×C32×D3—22no—
58C22×C32×D3C2312yes—
59#6312—28no—
60C2×C3⋊(C3⋊Dic3)—22no—
61C2×C3⋊(C3⋊Dic3)C2312yes—
62C2×C3×C3⋊Dic3—22no—
63C2×C3×C3⋊Dic3—28no—
64C2×C3×C3⋊Dic3D414yescomplemented
65C2×C3×C3⋊Dic3C2312yes—
66C2×C3×C3⋊Dic3C2312yes—
67C32×#67—28no—
68C2×C32×Dic3—22no—
69C2×C32×Dic3C2312yes—
70#6291—28no—
71C2×#939—94no—
72C2×#930—91no—
73C24×C32—24no—
74C24×C32—31no—
75C24×C32C2×D311yescharacteristic
76C24×C32C2×D314yes—
77C23×C3⋊D3—31no—
78C23×C3⋊D3—34no—
79C23×C3⋊D3—64no—
80C23×C3×D3C2×D314yes—
81C2×C32×D4—34no—
82C2×C32×D4—64no—
83C22×C32×C4—31no—
84C2×#638—34no—
85C2×#638—316no—
86C2×#638—34no—
87C2×#638—616no—
88C2×#638—632no—
89C2×#638—64no—
90C2×#638—616no—
91C2×#638—64no—
92C22×C3⋊Dic3—24no—
93C22×C3⋊Dic3—31no—
94C22×C3⋊Dic3—34no—
95C22×C3⋊Dic3—64no—
96C22×C3⋊Dic3C2×D311yescharacteristic
97C22×C3⋊Dic3C2×D311yescharacteristic
98C22×C3⋊Dic3C2×D314yescomplemented
99C2×C3×#67—216no—
100C2×C3×#67C2×D3116yescomplemented
101C22×C3×Dic3—34no—
102C22×C3×Dic3—34no—
103C22×C3×Dic3—316no—
104C22×C3×Dic3—616no—
105C22×C3×Dic3C2×D314yes—
106C2×#626—316no—
107C2×#626—616no—
108#3757—34no—
109#3752—34no—
110#3750—34no—
111Dic3⋊Dic3—316no—
112#3722—316no—
113(C2×D3)⋊Dic3—316no—
114C22×C33—28no—
115C22×C33—22no—
116C22×C33—22no—
117C22×C33C2×D412yes—
118C22×C33C2×D412yes—
119C22×C33C2×D414yescomplemented
120C22×C33#4112yes—
121C22×C33C2411yescharacteristic
122C2×C3⋊(C3⋊D3)—22no—
123C2×C3⋊(C3⋊D3)—22no—
124C2×C3⋊(C3⋊D3)—22no—
125C2×C3⋊(C3⋊D3)—44no—
126C2×C32×D3—22no—
127C2×C32×D3—22no—
128C2×C32×D3—22no—
129C2×C32×D3—44no—
130C3⋊(C3⋊Dic3)—22no—
131C3×C3⋊Dic3—22no—
132C3×C3⋊Dic3C2×D414yescomplemented
133C32×Dic3—22no—
134C22×C3×D4—34no—
135C23×C3×C4—91no—
136C22×#67—94no—
137C22×#67—94no—
138C22×#67—91no—
139C22×#67—184no—
140C22×#67C3⋊D311yescharacteristic, complemented
141C23×Dic3—34no—
142C22×D12—91no—
143C2×#217—94no—
144#939—932no—
145#930—98no—
146C2×(C2×D3)⋊C4—94no—
147C2×(C2×D3)⋊C4—91no—
148C2×(C2×D3)⋊C4—91no—
149C2×C4⋊Dic3—91no—
150C2×Dic3⋊C4—94no—
151C23×C32—22no—
152C23×C32—22no—
153C23×C32—216no—
154C23×C32—24no—
155C23×C32—28no—
156C23×C32—28no—
157C23×C32—28no—
158C23×C32—32no—
159C23×C32—48no—
160C23×C32—48no—
161C23×C32—62no—
162C23×C32—62no—
163C23×C32—62no—
164C23×C32D1214yescomplemented
165C23×C32#67116yescomplemented
166C23×C32C22×D318yes—
167C23×C32C22×D314yes—
168C23×C32C22×D312yes—
169C23×C32C22×D311yescharacteristic
170C22×C3⋊D3—32no—
171C22×C3⋊D3—38no—
172C22×C3⋊D3—68no—
173C22×C3⋊D3—616no—
174C22×C3⋊D3—62no—
175C22×C3⋊D3—62no—
176C22×C3⋊D3—68no—
177C22×C3⋊D3—62no—
178C22×C3⋊D3—68no—
179C22×C3⋊D3—68no—
180C22×C3⋊D3—128no—
181C22×C3⋊D3—128no—
182C22×C3×D3—28no—
183C22×C3×D3—28no—
184C22×C3×D3—28no—
185C22×C3×D3C22×D318yes—
186C32×D4—68no—
187C2×C32×C4—32no—
188C2×C32×C4—62no—
189#638—664no—
190#638—68no—
191#638—632no—
192#638—68no—
193C2×C3⋊Dic3—22no—
194C2×C3⋊Dic3—28no—
195C2×C3⋊Dic3—32no—
196C2×C3⋊Dic3—38no—
197C2×C3⋊Dic3—68no—
198C2×C3⋊Dic3—616no—
199C2×C3⋊Dic3—62no—
200C2×C3⋊Dic3—68no—
201C2×C3⋊Dic3#6714yescomplemented
202C2×C3⋊Dic3C22×D312yes—
203C2×C3⋊Dic3C22×D312yes—
204C3×#67—232no—
205C2×C3×Dic3—28no—
206C2×C3×Dic3—316no—
207C2×C3×Dic3—38no—
208C2×C3×Dic3—38no—
209C2×C3×Dic3—632no—
210C2×C3×Dic3—68no—
211C2×C3×Dic3C22×D318yes—
212#626—632no—
213C2×#121—271noSylow
214C2×C33—24no—
215C2×C33#12114yescomplemented
216C2×C33C22×D411yescharacteristic
217C2×C33C22×D411yescharacteristic
218C2×C33C2×#4111yescharacteristic
219C3⋊(C3⋊D3)—42no—
220C32×D3—42no—
221C24×C3—24no—
222C24×C3—34no—
223C24×C3D3214yes—
224C24×C3C2×C3⋊D311yescharacteristic
225C23×D3—94no—
226C23×D3—91no—
227C23×D3—184no—
228C23×D3C2×C3⋊D311yescharacteristic
229C2×C3×D4—316no—
230C2×C3×D4—616no—
231C22×C3×C4—34no—
232C22×C3×C4—91no—
233C22×C3×C4—91no—
234C22×C3×C4—94no—
235C22×C3×C4—184no—
236C2×#67—24no—
237C2×#67—916no—
238C2×#67—916no—
239C2×#67—94no—
240C2×#67—1816no—
241C2×#67—1816no—
242C2×#67—1832no—
243C2×#67—1816no—
244C2×#67—184no—
245C2×#67C2×C3⋊D314yescomplemented
246C22×Dic3—34no—
247C22×Dic3—34no—
248C22×Dic3—316no—
249C22×Dic3—616no—
250C22×Dic3—94no—
251C22×Dic3—91no—
252C22×Dic3—184no—
253C22×Dic3C2×C3⋊D311yescharacteristic
254C2×D12—94no—
255C2×D12—184no—
256#217—916no—
257(C2×D3)⋊C4—916no—
258(C2×D3)⋊C4—94no—
259(C2×D3)⋊C4—94no—
260C4⋊Dic3—94no—
261Dic3⋊C4—916no—
262C22×C32—28no—
263C22×C32—22no—
264C22×C32—22no—
265C22×C32—24no—
266C22×C32—28no—
267C22×C32—28no—
268C22×C32—216no—
269C22×C32—28no—
270C22×C32—232no—
271C22×C32—28no—
272C22×C32—28no—
273C22×C32—48no—
274C22×C32—48no—
275C22×C32—432no—
276C22×C32—62no—
277C22×C32—62no—
278C22×C32—62no—
279C22×C32—124no—
280C22×C32C2×D1212yes—
281C22×C32C2×D1214yescomplemented
282C22×C32#23512yes—
283C22×C32D3×D418yes—
284C22×C32#23718yes—
285C22×C32C2×#6718yes—
286C22×C32C2×#6712yes—
287C22×C32C2×#67116yescomplemented
288C22×C32C23×D314yes—
289C22×C32C23×D311yescharacteristic
290C2×C3⋊D3—616no—
291C2×C3⋊D3—62no—
292C2×C3⋊D3—62no—
293C2×C3⋊D3—68no—
294C2×C3⋊D3—62no—
295C2×C3⋊D3—68no—
296C2×C3⋊D3—68no—
297C2×C3⋊D3—128no—
298C2×C3⋊D3—128no—
299C2×C3⋊D3—1232no—
300C2×C3⋊D3—124no—
301C2×C3⋊D3—1216no—
302C2×C3×D3—28no—
303C2×C3×D3—28no—
304C2×C3×D3—28no—
305C2×C3×D3—416no—
306C32×C4—62no—
307C3⋊Dic3—22no—
308C3⋊Dic3—616no—
309C3⋊Dic3—62no—
310C3⋊Dic3—68no—
311C3⋊Dic3C2×#6714yescomplemented
312C3×Dic3—28no—
313C3×Dic3—316no—
314C3×Dic3—68no—
315C22×D4—31no—
316C22×D4—271no—
317C22×D4—271no—
318C23×C4—91no—
319#121—278no—
320C2×C4⋊C4—271no—
321C2×C22⋊C4—271no—
322C2×C22⋊C4—271no—
323C33C2×#12111yescharacteristic, complemented, Sylow
324C23×C3—28no—
325C23×C3—28no—
326C23×C3—22no—
327C23×C3—22no—
328C23×C3—216no—
329C23×C3—28no—
330C23×C3—24no—
331C23×C3—38no—
332C23×C3—48no—
333C23×C3—48no—
334C23×C3—68no—
335C23×C3—68no—
336C23×C3—68no—
337C23×C3#626116yescomplemented
338C23×C3#63814yescomplemented
339C23×C3C2×D3218yes—
340C23×C3C2×D3214yes—
341C23×C3C22×C3⋊D312yes—
342C23×C3C22×C3⋊D311yescharacteristic
343C22×D3—22no—
344C22×D3—22no—
345C22×D3—22no—
346C22×D3—98no—
347C22×D3—92no—
348C22×D3—188no—
349C22×D3—1816no—
350C22×D3—188no—
351C22×D3—188no—
352C22×D3—188no—
353C22×D3—182no—
354C22×D3—182no—
355C22×D3—182no—
356C22×D3—368no—
357C22×D3—368no—
358C22×D3C22×C3⋊D312yes—
359C3×D4—632no—
360C2×C3×C4—38no—
361C2×C3×C4—68no—
362C2×C3×C4—92no—
363C2×C3×C4—94no—
364C2×C3×C4—92no—
365C2×C3×C4—188no—
366C2×C3×C4—182no—
367#67—28no—
368#67—1832no—
369#67—1864no—
370#67—1832no—
371#67—188no—
372C2×Dic3—22no—
373C2×Dic3—38no—
374C2×Dic3—316no—
375C2×Dic3—38no—
376C2×Dic3—632no—
377C2×Dic3—68no—
378C2×Dic3—98no—
379C2×Dic3—92no—
380C2×Dic3—188no—
381C2×Dic3—1816no—
382C2×Dic3—188no—
383C2×Dic3—182no—
384C2×Dic3C22×C3⋊D312yes—
385D12—188no—
386C2×C32—24no—
387C2×C32—24no—
388C2×C32—24no—
389C2×C32—24no—
390C2×C32—216no—
391C2×C32—216no—
392C2×C32—416no—
393C2×C32—122no—
394C2×C32#93014yescomplemented
395C2×C32#939116yescomplemented
396C2×C32C22×D1211yescharacteristic
397C2×C32C2×#23511yescharacteristic
398C2×C32C2×D3×D414yes—
399C2×C32C2×#23714yes—
400C2×C32C22×#6714yes—
401C2×C32C22×#6711yescharacteristic
402C3⋊D3—1216no—
403C3⋊D3—122no—
404C3⋊D3—128no—
405C3×D3—48no—
406C24—31no—
407C24—271no—
408C24D3×C3⋊D311yescharacteristic
409C2×D4—34no—
410C2×D4—64no—
411C2×D4—274no—
412C2×D4—274no—
413C2×D4—544no—
414C2×D4—544no—
415C22×C4—31no—
416C22×C4—91no—
417C22×C4—94no—
418C22×C4—91no—
419C22×C4—184no—
420C22×C4—271no—
421C4⋊C4—274no—
422C22⋊C4—274no—
423C22⋊C4—274no—
424C22×C3—28no—
425C22×C3—28no—
426C22×C3—22no—
427C22×C3—22no—
428C22×C3—28no—
429C22×C3—232no—
430C22×C3—24no—
431C22×C3—28no—
432C22×C3—28no—
433C22×C3—216no—
434C22×C3—28no—
435C22×C3—48no—
436C22×C3—48no—
437C22×C3—432no—
438C22×C3—68no—
439C22×C3—68no—
440C22×C3—68no—
441C22×C3—1216no—
442C22×C3#380418yes—
443C22×C3C2×#62618yes—
444C22×C3C2×#626116yescomplemented
445C22×C3D3×#6718yes—
446C22×C3D4×C3⋊D312yes—
447C22×C3#383012yes—
448C22×C3C2×#63812yes—
449C22×C3C2×#63814yescomplemented
450C22×C3C22×D3214yes—
451C22×C3C23×C3⋊D311yescharacteristic
452C2×D3—22no—
453C2×D3—22no—
454C2×D3—22no—
455C2×D3—44no—
456C2×D3—1816no—
457C2×D3—188no—
458C2×D3—188no—
459C2×D3—188no—
460C2×D3—182no—
461C2×D3—182no—
462C2×D3—182no—
463C2×D3—368no—
464C2×D3—368no—
465C2×D3—3632no—
466C2×D3—364no—
467C2×D3—3616no—
468C3×C4—68no—
469C3×C4—94no—
470C3×C4—182no—
471Dic3—22no—
472Dic3—316no—
473Dic3—68no—
474Dic3—1816no—
475Dic3—188no—
476Dic3—182no—
477C32—24no—
478C32C2×#93011yescharacteristic, complemented
479C32C2×#93914yescomplemented
480C23—22no—
481C23—22no—
482C23—32no—
483C23—62no—
484C23—62no—
485C23—62no—
486C23—272no—
487C23—542no—
488C23—542no—
489C23—542no—
490C23#629114yescomplemented
491C23C2×D3×C3⋊D312yes—
492C23C2×D3×C3⋊D311yescharacteristic
493D4—68no—
494D4—548no—
495D4—548no—
496C2×C4—32no—
497C2×C4—62no—
498C2×C4—92no—
499C2×C4—94no—
500C2×C4—92no—
501C2×C4—188no—
502C2×C4—182no—
503C2×C4—272no—
504C2×C4—542no—
505C2×C3—24no—
506C2×C3—216no—
507C2×C3—24no—
508C2×C3—24no—
509C2×C3—24no—
510C2×C3—216no—
511C2×C3—416no—
512C2×C3—128no—
513C2×C3#63978116yescomplemented
514C2×C3#6414814yescomplemented
515C2×C3C2×#380414yes—
516C2×C3C22×#62614yes—
517C2×C3C2×D3×#6714yes—
518C2×C3C2×D4×C3⋊D311yescharacteristic
519C2×C3C2×#383011yescharacteristic
520C2×C3C22×#63811yescharacteristic
521D3—42no—
522D3—3616no—
523D3—362no—
524D3—368no—
525C22—22no—
526C22—22no—
527C22—28no—
528C22—62no—
529C22—62no—
530C22—62no—
531C22—124no—
532C22—542no—
533C22—542no—
534C22—542no—
535C22—1084no—
536C22#8902812yes—
537C22C2×#629112yes—
538C22C2×#629114yescomplemented
539C22C3⋊D3×#6712yes—
540C22C22×D3×C3⋊D311yescharacteristic
541C4—62no—
542C4—94no—
543C4—182no—
544C4—542no—
545C3—24no—
546C3C2×#6397814yescomplemented
547C3C2×#6414811yescharacteristic, complemented
548C2—24no—
549C2—122no—
550C2—1082no—
551C2#1172363514yesdirect product factor
552C2C2×#8902811yescharacteristic
553C2C22×#629111yescharacteristic
554C2C2×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×#6312C2semidirect
2. C23×C3×C3⋊Dic3C2semidirect
3. C22×C32×#67C2semidirect
4. C22×#6291C2semidirect
5. #11723635C2direct (N × H)
6. C2×#88807C2semidirect
7. C2×#88804C2semidirect
8. C2×(C2×D3)⋊(C3⋊Dic3)C2semidirect
15. C2×#6312C22semidirect
20. C22×C3×C3⋊Dic3C22semidirect
22. C2×C32×#67C22semidirect
25. C2×#6291C22semidirect
26. #88807C22semidirect
27. #88804C22semidirect
28. (C2×D3)⋊(C3⋊Dic3)C22semidirect
34. C23×C3⋊Dic3D3semidirect
35. C22×C3×#67D3semidirect
48. C23×C33D4semidirect
64. C2×C3×C3⋊Dic3D4semidirect
98. C22×C3⋊Dic3C2×D3semidirect
100. C2×C3×#67C2×D3semidirect
119. C22×C33C2×D4semidirect
132. C3×C3⋊Dic3C2×D4semidirect
140. C22×#67C3⋊D3semidirect
164. C23×C32D12semidirect
165. C23×C32#67semidirect
201. C2×C3⋊Dic3#67semidirect
215. C2×C33#121semidirect
245. C2×#67C2×C3⋊D3semidirect
281. C22×C32C2×D12semidirect
287. C22×C32C2×#67semidirect
311. C3⋊Dic3C2×#67semidirect
323. C33C2×#121semidirect
337. C23×C3#626semidirect
338. C23×C3#638semidirect
394. C2×C32#930semidirect
395. C2×C32#939semidirect
444. C22×C3C2×#626semidirect
449. C22×C3C2×#638semidirect
478. C32C2×#930semidirect
479. C32C2×#939semidirect
490. C23#6291semidirect
513. C2×C3#63978semidirect
514. C2×C3#64148semidirect
538. C22C2×#6291semidirect
546. C3C2×#63978semidirect
547. C3C2×#64148semidirect
551. C2#11723635direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#63122
C23×C3×C3⋊Dic32
C22×C32×#672
C22×#62912
#117236352
C2×#888072
C2×#888042
C2×(C2×D3)⋊(C3⋊Dic3)2
C2×#641483
C2×#639783