Finite Group Databasebeta

← C2×#11721005 #13150214

Subgroups — C2×#11721005 #13150214

Order 1728.

1514 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×C4⋊(C3⋊D9)C211yesmaximal, characteristic, complemented
2C22×#6192C211yesmaximal, characteristic, complemented
3C22×#6192C211yesmaximal, characteristic, complemented
4C22×(C3⋊D9)⋊C4C211yesmaximal, characteristic, complemented
5C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic, complemented
6C2×C9×Dic3⋊C4C211yesmaximal, characteristic, complemented
7C2×#88445C211yesmaximal, characteristic, complemented
8#11721005C218yesmaximal, direct product factor
9C2×#63910—31nomaximal
10C2×#63457—31nomaximal
11C23×C3⋊D9C2211yescharacteristic
12C23×C3⋊D9C2211yescharacteristic
13C23×C3×D9C2211yescharacteristic
14C22×C3×C4×C9C2211yescharacteristic
15C2×C4⋊(C3⋊D9)—24no—
16C2×C4⋊(C3⋊D9)C2214yescomplemented
17C22×C9×Dic3C2211yescharacteristic
18C22×C9×Dic3C2211yescharacteristic
19C22×C3×Dic9C2211yescharacteristic
20C2×#6192—24no—
21C2×#6192—24no—
22C2×#6192C2214yescomplemented
23C2×#6192C2214yescomplemented
24C2×(C3⋊D9)⋊C4—24no—
25C2×(C3⋊D9)⋊C4C2214yescomplemented
26C3×(C2×D9)⋊C4C2214yescomplemented
27C9×Dic3⋊C4C2214yescomplemented
28#88445C2214yescomplemented
29C22×C4⋊(C3⋊D3)—31no—
30C22×#626—31no—
31C22×#626—31no—
32C22×#624—31no—
33C2×C3×(C2×D3)⋊C4—31no—
34C2×C3×Dic3⋊C4D311yescharacteristic
35C2×#3722—31no—
36#63910—38no—
37C22×D36—31no—
38C22×D36—31no—
39C22×D36—31no—
40C22×D36—61no—
41C22×C4×D9—31no—
42C2×C9×C4⋊C4—31no—
43C2×(C2×D9)⋊C4—31no—
44C2×(C2×D9)⋊C4D311yescharacteristic, complemented
45#63457—38no—
46C23×C3×C9C2311yescharacteristic
47C22×C3⋊D9—22no—
48C22×C3⋊D9—22no—
49C22×C3⋊D9—22no—
50C22×C3⋊D9—22no—
51C22×C3⋊D9—22no—
52C22×C3⋊D9D414yescomplemented
53C22×C3⋊D9C2312yes—
54C22×C3⋊D9C2312yes—
55C22×C3×D9—22no—
56C22×C3×D9—22no—
57C22×C3×D9—22no—
58C22×C3×D9C2312yes—
59C2×C3×C4×C9—22no—
60C2×C3×C4×C9C2312yes—
61C4⋊(C3⋊D9)—28no—
62C2×C9×Dic3—22no—
63C2×C9×Dic3D414yescomplemented
64C2×C9×Dic3C2312yes—
65C2×C9×Dic3C2312yes—
66C2×C3×Dic9—22no—
67C2×C3×Dic9C2312yes—
68#6192—28no—
69#6192—28no—
70(C3⋊D9)⋊C4—28no—
71C2×#895—91no—
72C2×#884—91no—
73C23×C3⋊D3—31no—
74C23×C3⋊D3—31no—
75C23×C3×D3—31no—
76C22×C32×C4C2×D311yescharacteristic
77C2×C4⋊(C3⋊D3)—34no—
78C2×C4⋊(C3⋊D3)—64no—
79C22×C3×Dic3—31no—
80C22×C3×Dic3C2×D311yescharacteristic
81C22×C3×Dic3C2×D311yescharacteristic
82C2×#626—34no—
83C2×#626—34no—
84C2×#626—64no—
85C2×#626—64no—
86C2×#624—34no—
87C2×#624—64no—
88C23×D9—31no—
89C23×D9—31no—
90C23×D9—61no—
91C23×D9—61no—
92C23×D9C2×D311yescharacteristic
93C3×(C2×D3)⋊C4—34no—
94C3×Dic3⋊C4C2×D314yes—
95#3722—34no—
96C22×C4×C9—21no—
97C22×C4×C9—31no—
98C22×C4×C9—31no—
99C22×C4×C9C2×D311yescharacteristic
100C22×Dic9C2×D311yescharacteristic
101C2×D36—34no—
102C2×D36—34no—
103C2×D36—34no—
104C2×D36—64no—
105C2×D36—64no—
106C2×D36—68no—
107C2×D36—64no—
108C2×D36—64no—
109C2×C4×D9—34no—
110C2×C4×D9—64no—
111C9×C4⋊C4—34no—
112(C2×D9)⋊C4—34no—
113(C2×D9)⋊C4C2×D314yescomplemented
114C22×C3×C9C2×D412yes—
115C22×C3×C9C2×D412yes—
116C22×C3×C9#4112yes—
117C22×C3×C9C2411yescharacteristic
118C2×C3⋊D9—28no—
119C2×C3⋊D9—22no—
120C2×C3⋊D9—22no—
121C2×C3⋊D9—22no—
122C2×C3⋊D9—22no—
123C2×C3⋊D9—22no—
124C2×C3⋊D9—44no—
125C2×C3⋊D9C2×D414yescomplemented
126C2×C3×D9—22no—
127C2×C3×D9—22no—
128C2×C3×D9—22no—
129C2×C3×D9—44no—
130C3×C4×C9—22no—
131C9×Dic3—22no—
132C9×Dic3C2×D414yescomplemented
133C3×Dic9—22no—
134C22×#67—91no—
135C22×#67—91no—
136C22×D12—91no—
137C22×D12—91no—
138C22×D12—91no—
139C22×D12—91no—
140C22×D12—181no—
141C22×C4×D3—91no—
142C22×C4×D3—91no—
143C2×C3×C4⋊C4—31no—
144C2×C3×C22⋊C4—91no—
145C2×(C2×D3)⋊C4—31no—
146C2×(C2×D3)⋊C4—91no—
147C2×(C2×D3)⋊C4—91no—
148C2×Dic3⋊C4D911yescharacteristic, complemented
149#895—98no—
150#884—98no—
151C23×C32C22×D311yescharacteristic
152C22×C3⋊D3—34no—
153C22×C3⋊D3—32no—
154C22×C3⋊D3—32no—
155C22×C3⋊D3—62no—
156C22×C3⋊D3—62no—
157C22×C3⋊D3—62no—
158C22×C3⋊D3—62no—
159C22×C3⋊D3—62no—
160C22×C3×D3—32no—
161C22×C3×D3—62no—
162C22×C3×D3—62no—
163C22×C3×D3—62no—
164C2×C32×C4—22no—
165C2×C32×C4C22×D312yes—
166C4⋊(C3⋊D3)—68no—
167C2×C3×Dic3—22no—
168C2×C3×Dic3—32no—
169C2×C3×Dic3—62no—
170C2×C3×Dic3D1214yes—
171C2×C3×Dic3C22×D312yes—
172C2×C3×Dic3C22×D312yes—
173#626—68no—
174#626—68no—
175#624—68no—
176C23×C9—21no—
177C23×C9C22×D311yescharacteristic
178C22×D9—22no—
179C22×D9—22no—
180C22×D9—22no—
181C22×D9—34no—
182C22×D9—32no—
183C22×D9—32no—
184C22×D9—64no—
185C22×D9—62no—
186C22×D9—62no—
187C22×D9—62no—
188C22×D9—62no—
189C22×D9—64no—
190C22×D9—62no—
191C22×D9—62no—
192C22×D9—62no—
193C22×D9—122no—
194C22×D9—122no—
195C22×D9—122no—
196C22×D9—122no—
197C22×D9C22×D312yes—
198C2×C4×C9—24no—
199C2×C4×C9—22no—
200C2×C4×C9—22no—
201C2×C4×C9—34no—
202C2×C4×C9—32no—
203C2×C4×C9—32no—
204C2×C4×C9—62no—
205C2×C4×C9C22×D312yes—
206C2×Dic9—22no—
207C2×Dic9C22×D312yes—
208D36—68no—
209D36—616no—
210D36—68no—
211D36—68no—
212C4×D9—68no—
213C2×#121—271noSylow
214C2×C3×C9#12114yescomplemented
215C2×C3×C9C22×D411yescharacteristic
216C2×C3×C9C22×D411yescharacteristic
217C2×C3×C9C2×#4111yescharacteristic
218C3⋊D9—24no—
219C3⋊D9—42no—
220C3×D9—42no—
221C24×C3—91no—
222C23×D3—31no—
223C23×D3—91no—
224C23×D3—91no—
225C23×D3—91no—
226C23×D3—91no—
227C23×D3—181no—
228C23×D3—181no—
229C22×C3×C4—21no—
230C22×C3×C4—31no—
231C22×C3×C4—31no—
232C22×C3×C4—91no—
233C22×C3×C4C2×D911yescharacteristic
234C22×C3×C4D3211yescharacteristic
235C2×#67—94no—
236C2×#67—94no—
237C2×#67—184no—
238C2×#67—184no—
239C22×Dic3—31no—
240C22×Dic3C2×D911yescharacteristic
241C22×Dic3C2×D911yescharacteristic
242C2×D12—94no—
243C2×D12—94no—
244C2×D12—94no—
245C2×D12—94no—
246C2×D12—184no—
247C2×D12—184no—
248C2×D12—184no—
249C2×D12—188no—
250C2×D12—184no—
251C2×D12—184no—
252C2×C4×D3—94no—
253C2×C4×D3—94no—
254C2×C4×D3—184no—
255C2×C4×D3—184no—
256C3×C4⋊C4—34no—
257C3×C22⋊C4—94no—
258(C2×D3)⋊C4—34no—
259(C2×D3)⋊C4—94no—
260(C2×D3)⋊C4—94no—
261Dic3⋊C4C2×D914yescomplemented
262C22×C32C2×D1212yes—
263C22×C32D3×D412yes—
264C22×C32#23912yes—
265C22×C32C23×D311yescharacteristic
266C2×C3⋊D3—34no—
267C2×C3⋊D3—62no—
268C2×C3⋊D3—68no—
269C2×C3⋊D3—62no—
270C2×C3⋊D3—62no—
271C2×C3⋊D3—62no—
272C2×C3⋊D3—62no—
273C2×C3⋊D3—124no—
274C2×C3×D3—62no—
275C2×C3×D3—62no—
276C2×C3×D3—62no—
277C2×C3×D3—124no—
278C32×C4—22no—
279C3×Dic3—22no—
280C3×Dic3—62no—
281C3×Dic3C2×D1214yes—
282C22×C9—21no—
283C22×C9—22no—
284C22×C9—22no—
285C22×C9—22no—
286C22×C9#23512yes—
287C22×C9D3×D412yes—
288C22×C9D3×D412yes—
289C22×C9C23×D311yescharacteristic
290C2×D9—22no—
291C2×D9—22no—
292C2×D9—22no—
293C2×D9—34no—
294C2×D9—44no—
295C2×D9—64no—
296C2×D9—68no—
297C2×D9—62no—
298C2×D9—62no—
299C2×D9—64no—
300C2×D9—62no—
301C2×D9—62no—
302C2×D9—62no—
303C2×D9—122no—
304C2×D9—122no—
305C2×D9—128no—
306C2×D9—122no—
307C2×D9—122no—
308C2×D9—128no—
309C2×D9—124no—
310C4×C9—24no—
311C4×C9—22no—
312C4×C9—34no—
313C4×C9—62no—
314Dic9—22no—
315C22×D4—271no—
316C22×D4—271no—
317C22×D4—271no—
318C23×C4—271no—
319#121—278no—
320C2×C4⋊C4—31no—
321C2×C22⋊C4—91no—
322C2×C22⋊C4—271no—
323C3×C9C2×#12111yescharacteristic, complemented, Sylow
324C23×C3—21no—
325C23×C3—92no—
326C23×C3—182no—
327C23×C3—182no—
328C23×C3—182no—
329C23×C3C22×D911yescharacteristic
330C23×C3C2×D3211yescharacteristic
331C22×D3—32no—
332C22×D3—62no—
333C22×D3—62no—
334C22×D3—62no—
335C22×D3—92no—
336C22×D3—94no—
337C22×D3—92no—
338C22×D3—94no—
339C22×D3—92no—
340C22×D3—92no—
341C22×D3—182no—
342C22×D3—184no—
343C22×D3—182no—
344C22×D3—182no—
345C22×D3—182no—
346C22×D3—182no—
347C22×D3—182no—
348C22×D3—184no—
349C22×D3—182no—
350C22×D3—182no—
351C22×D3—182no—
352C22×D3—182no—
353C22×D3—182no—
354C22×D3—182no—
355C22×D3—362no—
356C22×D3—362no—
357C22×D3—362no—
358C22×D3—362no—
359C2×C3×C4—24no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—32no—
364C2×C3×C4—34no—
365C2×C3×C4—32no—
366C2×C3×C4—62no—
367C2×C3×C4—92no—
368C2×C3×C4—182no—
369C2×C3×C4C22×D912yes—
370C2×C3×C4C2×D3212yes—
371#67—188no—
372#67—188no—
373C2×Dic3—22no—
374C2×Dic3—32no—
375C2×Dic3—62no—
376C2×Dic3D3614yescomplemented
377C2×Dic3C22×D912yes—
378C2×Dic3C22×D912yes—
379D12—188no—
380D12—188no—
381D12—1816no—
382D12—188no—
383D12—188no—
384C4×D3—188no—
385C4×D3—188no—
386C2×C32#89514yes—
387C2×C32C22×D1211yescharacteristic
388C2×C32C2×D3×D411yescharacteristic
389C2×C32C2×#23911yescharacteristic
390C3⋊D3—64no—
391C3⋊D3—122no—
392C3×D3—122no—
393C2×C9—21no—
394C2×C9—21no—
395C2×C9—21no—
396C2×C9—24no—
397C2×C9#88414yescomplemented
398C2×C9C2×#23511yescharacteristic
399C2×C9C2×D3×D411yescharacteristic
400C2×C9C2×D3×D411yescharacteristic
401D9—42no—
402D9—64no—
403D9—124no—
404D9—124no—
405D9—122no—
406C24—91no—
407C24—271no—
408C24—271no—
409C2×D4—274no—
410C2×D4—274no—
411C2×D4—274no—
412C2×D4—544no—
413C2×D4—544no—
414C2×D4—544no—
415C22×C4—31no—
416C22×C4—31no—
417C22×C4—91no—
418C22×C4—274no—
419C22×C4—544no—
420C22×C4D3×D911yescharacteristic
421C4⋊C4—34no—
422C22⋊C4—94no—
423C22⋊C4—274no—
424C22×C3—22no—
425C22×C3—21no—
426C22×C3—22no—
427C22×C3—22no—
428C22×C3—182no—
429C22×C3—182no—
430C22×C3—182no—
431C22×C3—364no—
432C22×C3C2×D3612yes—
433C22×C3D4×D912yes—
434C22×C3#370112yes—
435C22×C3C23×D911yescharacteristic
436C22×C3#379912yes—
437C22×C3D3×D1212yes—
438C22×C3#381112yes—
439C22×C3C22×D3211yescharacteristic
440C2×D3—62no—
441C2×D3—62no—
442C2×D3—62no—
443C2×D3—94no—
444C2×D3—94no—
445C2×D3—124no—
446C2×D3—184no—
447C2×D3—182no—
448C2×D3—182no—
449C2×D3—188no—
450C2×D3—182no—
451C2×D3—182no—
452C2×D3—188no—
453C2×D3—184no—
454C2×D3—182no—
455C2×D3—182no—
456C2×D3—182no—
457C2×D3—182no—
458C2×D3—182no—
459C2×D3—182no—
460C2×D3—368no—
461C2×D3—362no—
462C2×D3—362no—
463C2×D3—362no—
464C2×D3—362no—
465C2×D3—368no—
466C2×D3—364no—
467C2×D3—364no—
468C3×C4—24no—
469C3×C4—22no—
470C3×C4—22no—
471C3×C4—34no—
472C3×C4—62no—
473C3×C4—182no—
474Dic3—22no—
475Dic3—62no—
476Dic3C2×D3614yescomplemented
477C32C2×#89511yescharacteristic
478C9—21no—
479C9C2×#88411yescharacteristic, complemented
480C23—92no—
481C23—182no—
482C23—182no—
483C23—182no—
484C23—272no—
485C23—274no—
486C23—272no—
487C23—542no—
488C23—542no—
489C23—542no—
490C23—542no—
491C23—542no—
492C23C2×D3×D911yescharacteristic
493D4—548no—
494D4—548no—
495D4—548no—
496C2×C4—22no—
497C2×C4—32no—
498C2×C4—34no—
499C2×C4—32no—
500C2×C4—62no—
501C2×C4—92no—
502C2×C4—182no—
503C2×C4—548no—
504C2×C4C2×D3×D912yes—
505C2×C3—21no—
506C2×C3—24no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—362no—
510C2×C3#6345714yescomplemented
511C2×C3C22×D3611yescharacteristic
512C2×C3C2×D4×D911yescharacteristic
513C2×C3C2×#370111yescharacteristic
514C2×C3#6391014yes—
515C2×C3C2×#379911yescharacteristic
516C2×C3C2×D3×D1211yescharacteristic
517C2×C3C2×#381111yescharacteristic
518D3—122no—
519D3—184no—
520D3—184no—
521D3—364no—
522D3—364no—
523D3—362no—
524D3—362no—
525C22—182no—
526C22—182no—
527C22—182no—
528C22—274no—
529C22—364no—
530C22—548no—
531C22—542no—
532C22—542no—
533C22—542no—
534C22—542no—
535C22—542no—
536C22—1084no—
537C22#8863512yes—
538C22D3×D3612yes—
539C22#8866812yes—
540C22C22×D3×D911yescharacteristic
541C4—22no—
542C4—34no—
543C4—62no—
544C4—182no—
545C3—21no—
546C3C2×#6345711yescharacteristic, complemented
547C3C2×#6391011yescharacteristic
548C2—362no—
549C2—544no—
550C2—1082no—
551C2#1172100514yesdirect product factor
552C2C2×#8863511yescharacteristic
553C2C2×D3×D3611yescharacteristic
554C2C2×#8866811yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9C2×#121
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4⋊(C3⋊D9)C2semidirect
2. C22×#6192C2semidirect
3. C22×#6192C2semidirect
4. C22×(C3⋊D9)⋊C4C2semidirect
5. C2×C3×(C2×D9)⋊C4C2semidirect
6. C2×C9×Dic3⋊C4C2semidirect
7. C2×#88445C2semidirect
8. #11721005C2direct (N × H)
16. C2×C4⋊(C3⋊D9)C22semidirect
22. C2×#6192C22semidirect
23. C2×#6192C22semidirect
25. C2×(C3⋊D9)⋊C4C22semidirect
26. C3×(C2×D9)⋊C4C22semidirect
27. C9×Dic3⋊C4C22semidirect
28. #88445C22semidirect
44. C2×(C2×D9)⋊C4D3semidirect
52. C22×C3⋊D9D4semidirect
63. C2×C9×Dic3D4semidirect
113. (C2×D9)⋊C4C2×D3semidirect
125. C2×C3⋊D9C2×D4semidirect
132. C9×Dic3C2×D4semidirect
148. C2×Dic3⋊C4D9semidirect
214. C2×C3×C9#121semidirect
261. Dic3⋊C4C2×D9semidirect
323. C3×C9C2×#121semidirect
376. C2×Dic3D36semidirect
397. C2×C9#884semidirect
476. Dic3C2×D36semidirect
479. C9C2×#884semidirect
510. C2×C3#63457semidirect
546. C3C2×#63457semidirect
551. C2#11721005direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4⋊(C3⋊D9)2
C22×#61922
C22×(C3⋊D9)⋊C42
C2×C3×(C2×D9)⋊C42
C2×C9×Dic3⋊C42
C2×#884452
#117210052
C2×#639103
C2×#634573