Finite Group Databasebeta

← C2×#11722478 #13164708

Subgroups — C2×#11722478 #13164708

Order 1728.

2930 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊(C3⋊D9)C211yesmaximal, characteristic, complemented
2C2×C9×#235C211yesmaximal, characteristic, complemented
3C2×C3×#3697C211yesmaximal, characteristic, complemented
4C2×#88668C214yesmaximal, complemented
5#11722478C2116yesmaximal, direct product factor
6C2×D9×D12C212yesmaximal, complemented
7C2×D3×D36C212yesmaximal, complemented
8C2×#88642C212yesmaximal, complemented
9C2×#88635C212yesmaximal, complemented
10C2×#64308—31nomaximal
11C2×#63715—31nomaximal
12C23×C3⋊D9C2212yescomplemented
13C22×D3×D9C2214yescomplemented
14C22×C3×C4×C9C2211yescharacteristic, complemented
15C2×C4⋊(C3⋊D9)C2212yescomplemented
16C2×C4⋊(C3⋊D9)C2212yescomplemented
17C2×C4⋊(C3⋊D9)C2218yescomplemented
18C2×C9×#67C2212yescomplemented
19C2×C3×#611C2212yescomplemented
20C9×#235C2218yescomplemented
21C2×C9×D12C2211yescharacteristic, complemented
22C2×C4×D3×C9C2212yescomplemented
23C3×#3697C2218yescomplemented
24C2×C3×D36C2211yescharacteristic, complemented
25C2×C3×C4×D9C2212yescomplemented
26C2×C9×Dic6C2211yescharacteristic, complemented
27C2×C3×Dic18C2211yescharacteristic, complemented
28#88668C22132yescomplemented
29C2×#6195C2214yescomplemented
30C2×#6192C2214yescomplemented
31C2×(C3⋊D9)⋊C4C2214yescomplemented
32D9×D12C22116yescomplemented
33D3×D36C22116yescomplemented
34#88642C22116yescomplemented
35#88635C22116yescomplemented
36C22×C4⋊(C3⋊D3)—31no—
37C2×C3×#235—31no—
38C2×C3×#235D311yescharacteristic
39C2×#3811—34no—
40#64308—316no—
41C2×D3×D12—32no—
42C2×D3×D12—32no—
43C2×#3799—32no—
44C2×#3799—32no—
45C2×C9×#41—31no—
46#63715—316no—
47C2×#3701—32no—
48C2×D4×D9—34no—
49C2×D4×D9—32no—
50C2×#3697—32no—
51C2×#3697D311yescharacteristic, complemented
52C22×D36—31no—
53C22×D36—32no—
54C22×D36—61no—
55C23×C3×C9C2311yescharacteristic
56C22×C3⋊D9—44no—
57C22×C3⋊D9C2314yes—
58C22×C3⋊D9C2318yescomplemented
59C22×D3×C9C2312yes—
60C22×C3×D9C2312yes—
61C2×D3×D9—48no—
62C2×D3×D9C23116yescomplemented
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3×C4×C9C2314yescomplemented
66C4⋊(C3⋊D9)C2318yescomplemented
67C4⋊(C3⋊D9)C2318yescomplemented
68C9×#67C2318yescomplemented
69C3×#611C2318yescomplemented
70C2×C9×Dic3C2312yes—
71C2×C3×Dic9C2312yes—
72C9×D12C2314yescomplemented
73C4×D3×C9C2318yescomplemented
74C3×D36C2314yescomplemented
75C3×C4×D9C2318yescomplemented
76C9×Dic6C2314yescomplemented
77C3×Dic18C2314yescomplemented
78#6195C23116yescomplemented
79#6192C23116yescomplemented
80(C3⋊D9)⋊C4C23116yescomplemented
81C2×#1009—91no—
82C2×#1009—91no—
83C23×C3⋊D3—32no—
84C22×D32—34no—
85C22×C32×C4C2×D311yescharacteristic
86C2×C4⋊(C3⋊D3)—32no—
87C2×C4⋊(C3⋊D3)—32no—
88C2×C4⋊(C3⋊D3)—38no—
89C2×C3×#67—32no—
90C2×C3×#67C2×D312yes—
91C3×#235—38no—
92C3×#235C2×D318yes—
93C2×C3×D12—31no—
94C2×C3×D12C2×D311yescharacteristic
95C2×C3×C4×D3—32no—
96C2×C3×C4×D3C2×D312yes—
97C2×C3×Dic6—31no—
98C2×C3×Dic6C2×D311yescharacteristic
99#3811—332no—
100C2×#626—34no—
101C2×#626—34no—
102C2×#624—34no—
103D3×D12—316no—
104D3×D12—316no—
105#3799—316no—
106#3799—316no—
107C23×D9—32no—
108C23×D9—34no—
109C23×D9—62no—
110C9×#41—38no—
111C2×Q8×C9—31no—
112C2×D4×C9—32no—
113C2×D4×C9—31no—
114C22×C4×C9—21no—
115C22×C4×C9—32no—
116C22×C4×C9C2×D311yescharacteristic, complemented
117C2×#611—34no—
118C2×#611C2×D312yescomplemented
119#3701—316no—
120D4×D9—332no—
121D4×D9—316no—
122#3697—316no—
123#3697C2×D318yescomplemented
124C2×D36—34no—
125C2×D36—32no—
126C2×D36—32no—
127C2×D36—38no—
128C2×D36—316no—
129C2×D36—62no—
130C2×D36—62no—
131C2×D36—68no—
132C2×D36C2×D311yescharacteristic, complemented
133C2×C4×D9—34no—
134C2×C4×D9C2×D312yescomplemented
135C2×Dic18C2×D311yescharacteristic, complemented
136C22×C3×C9—22no—
137C22×C3×C9C2412yes—
138C22×C3×C9C2411yescharacteristic
139C2×C3⋊D9—28no—
140C2×C3⋊D9—48no—
141C2×C3⋊D9C2418yes—
142C2×D3×C9—24no—
143C2×D3×C9C2414yes—
144C2×C3×D9—24no—
145C2×C3×D9C2414yes—
146D3×D9—416no—
147C3×C4×C9C2412yes—
148C3×C4×C9C2412yes—
149C9×Dic3C2414yes—
150C3×Dic9C2414yes—
151C2×C3×#41—31no—
152C2×C3×#41—91no—
153#1009—916no—
154#1009—916no—
155C2×#239—92no—
156C2×#239—92no—
157C2×D3×D4—92no—
158C2×D3×D4—94no—
159C2×D3×D4—94no—
160C2×D3×D4—92no—
161C2×#235—31no—
162C2×#235—92no—
163C2×#235—92no—
164C2×#235D911yescharacteristic, complemented
165C22×D12—92no—
166C22×D12—91no—
167C22×D12—91no—
168C22×D12—92no—
169C22×D12—181no—
170C23×C32C22×D311yescharacteristic
171C22×C3⋊D3—34no—
172C22×C3⋊D3—38no—
173C22×C3⋊D3—124no—
174C22×C3×D3—32no—
175C22×C3×D3C22×D312yes—
176C2×D32—316no—
177C2×D32—128no—
178C2×C32×C4C22×D311yescharacteristic
179C2×C32×C4C22×D311yescharacteristic
180C2×C32×C4C22×D314yes—
181C4⋊(C3⋊D3)—38no—
182C4⋊(C3⋊D3)—38no—
183C3×#67—38no—
184C3×#67C22×D318yes—
185C2×C3×Dic3—32no—
186C2×C3×Dic3C22×D312yes—
187C3×D12—34no—
188C3×D12C22×D314yes—
189C3×C4×D3—38no—
190C3×C4×D3C22×D318yes—
191C3×Dic6—34no—
192C3×Dic6C22×D314yes—
193#626—316no—
194#626—316no—
195#624—316no—
196C23×C9—21no—
197C23×C9—32no—
198C23×C9C22×D311yescharacteristic
199C22×D9—34no—
200C22×D9—38no—
201C22×D9—316no—
202C22×D9—64no—
203C22×D9—68no—
204C22×D9—128no—
205C22×D9—124no—
206C22×D9—128no—
207C22×D9C22×D312yes—
208Q8×C9—34no—
209D4×C9—38no—
210D4×C9—34no—
211C2×C4×C9—21no—
212C2×C4×C9—21no—
213C2×C4×C9—24no—
214C2×C4×C9—32no—
215C2×C4×C9—38no—
216C2×C4×C9C22×D311yescharacteristic
217C2×C4×C9C22×D311yescharacteristic
218C2×C4×C9C22×D314yescomplemented
219#611—316no—
220#611C22×D318yescomplemented
221C2×Dic9C22×D312yes—
222D36—316no—
223D36—38no—
224D36—38no—
225D36—68no—
226D36—68no—
227D36C22×D314yescomplemented
228C4×D9—316no—
229C4×D9C22×D318yescomplemented
230Dic18C22×D314yescomplemented
231C2×H(5,𝔽2)—271noSylow
232C2×C3×C9—22no—
233C2×C3×C9H(5,𝔽2)12yescomplemented
234C2×C3×C9C2511yescharacteristic
235C3⋊D9—28no—
236D3×C9—24no—
237C3×D9—24no—
238C23×D3—94no—
239C23×D3—92no—
240C23×D3—92no—
241C23×D3—94no—
242C23×D3—182no—
243C3×#41—38no—
244C3×#41—98no—
245C2×C3×Q8—31no—
246C2×C3×Q8—91no—
247C2×C3×D4—32no—
248C2×C3×D4—31no—
249C2×C3×D4—92no—
250C2×C3×D4—91no—
251C22×C3×C4—21no—
252C22×C3×C4—32no—
253C22×C3×C4—92no—
254C22×C3×C4C2×D911yescharacteristic, complemented
255C22×C3×C4D3211yescharacteristic
256C2×#67—32no—
257C2×#67—94no—
258C2×#67—94no—
259C2×#67C2×D912yescomplemented
260#239—916no—
261#239—916no—
262D3×D4—916no—
263D3×D4—932no—
264D3×D4—932no—
265D3×D4—916no—
266#235—38no—
267#235—916no—
268#235—916no—
269#235C2×D918yescomplemented
270C2×D12—31no—
271C2×D12—916no—
272C2×D12—94no—
273C2×D12—94no—
274C2×D12—92no—
275C2×D12—92no—
276C2×D12—98no—
277C2×D12—92no—
278C2×D12—92no—
279C2×D12—98no—
280C2×D12—916no—
281C2×D12—182no—
282C2×D12—182no—
283C2×D12—188no—
284C2×D12C2×D911yescharacteristic, complemented
285C2×C4×D3—32no—
286C2×C4×D3—94no—
287C2×C4×D3—94no—
288C2×C4×D3C2×D912yescomplemented
289C2×Dic6—31no—
290C2×Dic6C2×D911yescharacteristic, complemented
291C22×C32—22no—
292C22×C32C23×D312yes—
293C22×C32C23×D311yescharacteristic
294C2×C3⋊D3—38no—
295C2×C3⋊D3—68no—
296C2×C3⋊D3—128no—
297C2×C3×D3—24no—
298C2×C3×D3—34no—
299C2×C3×D3—64no—
300C2×C3×D3C23×D314yes—
301D32—1216no—
302C32×C4C23×D312yes—
303C32×C4C23×D312yes—
304C3×Dic3—34no—
305C3×Dic3C23×D314yes—
306C22×C9—24no—
307C22×C9—22no—
308C22×C9—22no—
309C22×C9—21no—
310C22×C9—34no—
311C22×C9—64no—
312C22×C9C23×D312yes—
313C22×C9C23×D311yescharacteristic
314C2×D9—24no—
315C2×D9—38no—
316C2×D9—68no—
317C2×D9—68no—
318C2×D9—128no—
319C2×D9—1216no—
320C2×D9—128no—
321C2×D9—1216no—
322C2×D9C23×D314yes—
323C4×C9—22no—
324C4×C9—22no—
325C4×C9—34no—
326C4×C9C23×D312yes—
327C4×C9C23×D312yes—
328Dic9C23×D314yes—
329H(5,𝔽2)—2716no—
330C2×#41—31no—
331C2×#41—91no—
332C2×#41—272no—
333C2×#41—272no—
334C22×D4—272no—
335C22×D4—272no—
336C22×D4—274no—
337C22×D4—271no—
338C3×C9C2×H(5,𝔽2)11yescharacteristic, complemented, Sylow
339C23×C3—21no—
340C23×C3—32no—
341C23×C3—92no—
342C23×C3C22×D911yescharacteristic
343C23×C3C2×D3211yescharacteristic
344C22×D3—32no—
345C22×D3—916no—
346C22×D3—94no—
347C22×D3—94no—
348C22×D3—98no—
349C22×D3—98no—
350C22×D3—916no—
351C22×D3—184no—
352C22×D3—188no—
353C22×D3—368no—
354C22×D3—368no—
355C22×D3—364no—
356C22×D3—364no—
357C22×D3—368no—
358C22×D3C22×D912yes—
359C3×Q8—34no—
360C3×Q8—94no—
361C3×D4—38no—
362C3×D4—34no—
363C3×D4—98no—
364C3×D4—94no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—24no—
368C2×C3×C4—32no—
369C2×C3×C4—38no—
370C2×C3×C4—92no—
371C2×C3×C4—98no—
372C2×C3×C4C22×D911yescharacteristic
373C2×C3×C4C22×D911yescharacteristic
374C2×C3×C4C22×D914yescomplemented
375C2×C3×C4C2×D3211yescharacteristic
376C2×C3×C4C2×D3211yescharacteristic
377C2×C3×C4C2×D3214yes—
378#67—38no—
379#67—916no—
380#67—916no—
381#67C22×D918yescomplemented
382C2×Dic3—32no—
383C2×Dic3C22×D912yes—
384D12—34no—
385D12—916no—
386D12—916no—
387D12—98no—
388D12—98no—
389D12—98no—
390D12—98no—
391D12—188no—
392D12—188no—
393D12C22×D914yescomplemented
394C4×D3—38no—
395C4×D3—916no—
396C4×D3—916no—
397C4×D3C22×D918yescomplemented
398Dic6—34no—
399Dic6C22×D914yescomplemented
400C2×C32—22no—
401C2×C32#100912yes—
402C2×C32C24×D311yescharacteristic
403C3⋊D3—68no—
404C3×D3—24no—
405C3×D3—64no—
406C2×C9—24no—
407C2×C9—22no—
408C2×C9—22no—
409C2×C9—21no—
410C2×C9—64no—
411C2×C9#100912yescomplemented
412C2×C9C24×D311yescharacteristic
413D9—24no—
414D9—68no—
415D9—128no—
416C24—274no—
417C24—272no—
418#41—38no—
419#41—98no—
420#41—2716no—
421#41—2716no—
422C2×Q8—31no—
423C2×Q8—91no—
424C2×D4—32no—
425C2×D4—31no—
426C2×D4—92no—
427C2×D4—91no—
428C2×D4—2716no—
429C2×D4—274no—
430C2×D4—2716no—
431C2×D4—274no—
432C2×D4—2732no—
433C2×D4—272no—
434C2×D4—272no—
435C2×D4—278no—
436C22×C4—32no—
437C22×C4—92no—
438C22×C4—274no—
439C22×C4D3×D911yescharacteristic, complemented
440C22×C3—24no—
441C22×C3—22no—
442C22×C3—22no—
443C22×C3—22no—
444C22×C3—21no—
445C22×C3—34no—
446C22×C3—64no—
447C22×C3—94no—
448C22×C3—184no—
449C22×C3C23×D912yes—
450C22×C3C23×D911yescharacteristic
451C22×C3C22×D3212yes—
452C22×C3C22×D3211yescharacteristic
453C2×D3—24no—
454C2×D3—34no—
455C2×D3—64no—
456C2×D3—98no—
457C2×D3—98no—
458C2×D3—188no—
459C2×D3—188no—
460C2×D3—188no—
461C2×D3—368no—
462C2×D3—3616no—
463C2×D3—3616no—
464C2×D3—368no—
465C2×D3—368no—
466C2×D3—3616no—
467C2×D3C23×D914yes—
468C3×C4—22no—
469C3×C4—22no—
470C3×C4—34no—
471C3×C4—94no—
472C3×C4C23×D912yes—
473C3×C4C23×D912yes—
474C3×C4C22×D3212yes—
475C3×C4C22×D3212yes—
476Dic3—34no—
477Dic3C23×D914yes—
478C32C2×#100911yescharacteristic
479C9—21no—
480C9C2×#100911yescharacteristic, complemented
481C23—32no—
482C23—92no—
483C23—2716no—
484C23—274no—
485C23—278no—
486C23—1084no—
487C23—1088no—
488C23C2×D3×D911yescharacteristic
489Q8—34no—
490Q8—94no—
491D4—38no—
492D4—34no—
493D4—98no—
494D4—94no—
495D4—2716no—
496D4—2716no—
497D4—278no—
498D4—278no—
499C2×C4—32no—
500C2×C4—38no—
501C2×C4—92no—
502C2×C4—98no—
503C2×C4—2716no—
504C2×C4C2×D3×D911yescharacteristic
505C2×C4C2×D3×D911yescharacteristic
506C2×C4C2×D3×D914yescomplemented
507C2×C3—24no—
508C2×C3—22no—
509C2×C3—22no—
510C2×C3—21no—
511C2×C3—22no—
512C2×C3—64no—
513C2×C3—184no—
514C2×C3#6371512yescomplemented
515C2×C3C24×D911yescharacteristic
516C2×C3#6430812yes—
517C2×C3C23×D3211yescharacteristic
518D3—24no—
519D3—64no—
520D3—188no—
521D3—188no—
522D3—368no—
523C22—22no—
524C22—34no—
525C22—64no—
526C22—94no—
527C22—184no—
528C22—278no—
529C22—548no—
530C22—1088no—
531C22—10816no—
532C22C22×D3×D912yes—
533C22C22×D3×D911yescharacteristic
534C4—34no—
535C4—94no—
536C4C22×D3×D912yes—
537C4C22×D3×D912yes—
538C3—21no—
539C3C2×#6371511yescharacteristic, complemented
540C3C2×#6430811yescharacteristic
541C2—22no—
542C2—64no—
543C2—184no—
544C2—548no—
545C2#1172247812yesdirect product factor
546C2C23×D3×D911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC2×C3×C9C25
FrattiniC2×C3C23×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9C2×H(5,𝔽2)
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4⋊(C3⋊D9)C2semidirect
2. C2×C9×#235C2semidirect
3. C2×C3×#3697C2semidirect
4. C2×#88668C2semidirect
5. #11722478C2direct (N × H)
6. C2×D9×D12C2semidirect
7. C2×D3×D36C2semidirect
8. C2×#88642C2semidirect
9. C2×#88635C2semidirect
12. C23×C3⋊D9C22semidirect
13. C22×D3×D9C22semidirect
14. C22×C3×C4×C9C22semidirect
15. C2×C4⋊(C3⋊D9)C22semidirect
16. C2×C4⋊(C3⋊D9)C22semidirect
17. C2×C4⋊(C3⋊D9)C22semidirect
18. C2×C9×#67C22semidirect
19. C2×C3×#611C22semidirect
20. C9×#235C22semidirect
21. C2×C9×D12C22semidirect
22. C2×C4×D3×C9C22semidirect
23. C3×#3697C22semidirect
24. C2×C3×D36C22semidirect
25. C2×C3×C4×D9C22semidirect
26. C2×C9×Dic6C22semidirect
27. C2×C3×Dic18C22semidirect
28. #88668C22semidirect
29. C2×#6195C22semidirect
30. C2×#6192C22semidirect
31. C2×(C3⋊D9)⋊C4C22semidirect
32. D9×D12C22semidirect
33. D3×D36C22semidirect
34. #88642C22semidirect
35. #88635C22semidirect
51. C2×#3697D3semidirect
58. C22×C3⋊D9C23semidirect
62. C2×D3×D9C23semidirect
65. C2×C3×C4×C9C23semidirect
66. C4⋊(C3⋊D9)C23semidirect
67. C4⋊(C3⋊D9)C23semidirect
68. C9×#67C23semidirect
69. C3×#611C23semidirect
72. C9×D12C23semidirect
73. C4×D3×C9C23semidirect
74. C3×D36C23semidirect
75. C3×C4×D9C23semidirect
76. C9×Dic6C23semidirect
77. C3×Dic18C23semidirect
78. #6195C23semidirect
79. #6192C23semidirect
80. (C3⋊D9)⋊C4C23semidirect
116. C22×C4×C9C2×D3semidirect
118. C2×#611C2×D3semidirect
123. #3697C2×D3semidirect
132. C2×D36C2×D3semidirect
134. C2×C4×D9C2×D3semidirect
135. C2×Dic18C2×D3semidirect
164. C2×#235D9semidirect
218. C2×C4×C9C22×D3semidirect
220. #611C22×D3semidirect
227. D36C22×D3semidirect
229. C4×D9C22×D3semidirect
230. Dic18C22×D3semidirect
233. C2×C3×C9H(5,𝔽2)semidirect
254. C22×C3×C4C2×D9semidirect
259. C2×#67C2×D9semidirect
269. #235C2×D9semidirect
284. C2×D12C2×D9semidirect
288. C2×C4×D3C2×D9semidirect
290. C2×Dic6C2×D9semidirect
338. C3×C9C2×H(5,𝔽2)semidirect
374. C2×C3×C4C22×D9semidirect
381. #67C22×D9semidirect
393. D12C22×D9semidirect
397. C4×D3C22×D9semidirect
399. Dic6C22×D9semidirect
411. C2×C9#1009semidirect
439. C22×C4D3×D9semidirect
480. C9C2×#1009semidirect
506. C2×C4C2×D3×D9semidirect
514. C2×C3#63715semidirect
539. C3C2×#63715semidirect
545. C2#11722478direct (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
C2×C9×#2352
C2×C3×#36972
C2×#886682
#117224782
C2×D9×D122
C2×D3×D362
C2×#886422
C2×#886352
C2×#643083
C2×#637153