Finite Group Databasebeta

← C2×#11722458 #13164689

Subgroups — C2×#11722458 #13164689

Order 1728.

2546 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88736C211yesmaximal, characteristic, complemented
2C2×C3×#3697C211yesmaximal, characteristic, complemented
3C22×C9×Dic6C211yesmaximal, characteristic, complemented
4C2×#88658C214yesmaximal, complemented
5C2×(C3⋊D9)⋊Q8C212yesmaximal, complemented
6C2×#88635C212yesmaximal, complemented
7#11722458C2116yesmaximal, direct product factor
8C2×#88634C212yesmaximal, complemented
9C2×D9×Dic6C212yesmaximal, complemented
10C2×#64297—31nomaximal
11C2×#63713—31nomaximal
12C22×C3×C4×C9C2211yescharacteristic
13C2×#6233C2212yes—
14#88736C2218yescomplemented
15C2×C4⋊(C3⋊D9)C2211yescharacteristic
16C2×C4×C3⋊D9C2212yes—
17C2×#6226C2211yescharacteristic
18C2×C3×#611C2212yes—
19C22×C9×Dic3C2212yes—
20C3×#3697C2218yescomplemented
21C2×C3×D36C2211yescharacteristic
22C2×C3×C4×D9C2212yes—
23C2×C9×Dic6C2212yes—
24C2×C9×Dic6C2212yes—
25C2×C9×Dic6C2218yescomplemented
26C2×C3×Dic18C2211yescharacteristic
27C2×#6192C2214yes—
28C2×(C3⋊D9)⋊C4C2214yes—
29#88658C22132yescomplemented
30C2×Dic3×D9C2214yes—
31C2×#6189C2214yes—
32(C3⋊D9)⋊Q8C22116yescomplemented
33#88635C22116yescomplemented
34#88634C22116yescomplemented
35D9×Dic6C22116yescomplemented
36C2×#3828—31no—
37C2×C3×#235—31no—
38C22×C3×Dic6D311yescharacteristic
39C2×#3804—34no—
40C2×#3799—32no—
41C2×#3797—32no—
42#64297—316no—
43C2×#3796—32no—
44C2×D3×Dic6—32no—
45C22×Q8×C9—31no—
46#63713—316no—
47C2×#3701—32no—
48C2×#3701—32no—
49C2×Q8×D9—32no—
50C2×Q8×D9—32no—
51C2×#3697—31no—
52C2×#3697—34no—
53C2×#3697—61no—
54C2×#3697D311yescharacteristic, complemented
55C23×C3×C9C2311yescharacteristic
56C22×C3⋊D9C2312yes—
57C22×C3×D9C2312yes—
58C2×C3×C4×C9C2311yescharacteristic
59C2×C3×C4×C9C2311yescharacteristic
60C2×C3×C4×C9C2314yes—
61#6233C2318yes—
62C2×C3⋊Dic9C2312yes—
63C4⋊(C3⋊D9)C2314yes—
64C4×C3⋊D9C2318yes—
65#6226C2314yes—
66C3×#611C2318yes—
67C2×C9×Dic3C2314yes—
68C2×C9×Dic3C2318yes—
69C2×C3×Dic9C2312yes—
70C3×D36C2314yes—
71C3×C4×D9C2318yes—
72C9×Dic6C2318yes—
73C9×Dic6C2318yes—
74C3×Dic18C2314yes—
75#6192C23116yes—
76(C3⋊D9)⋊C4C23116yes—
77Dic3×D9C23116yes—
78#6189C23116yes—
79C2×#1010—91no—
80C2×#1007—91no—
81C22×C32×C4C2×D311yescharacteristic
82C2×#638—32no—
83#3828—38no—
84C2×C4⋊(C3⋊D3)—31no—
85C2×C4×C3⋊D3—32no—
86C2×#634—31no—
87C2×C3×#67—32no—
88C22×C3×Dic3C2×D312yes—
89C3×#235—38no—
90C2×C3×D12—31no—
91C2×C3×C4×D3—32no—
92C2×C3×Dic6—31no—
93C2×C3×Dic6C2×D312yes—
94C2×C3×Dic6C2×D312yes—
95C2×C3×Dic6C2×D318yes—
96C2×#627—34no—
97C2×#626—34no—
98C2×#624—34no—
99#3804—332no—
100C2×D3×Dic3—34no—
101#3799—316no—
102#3797—316no—
103#3796—316no—
104D3×Dic6—316no—
105C2×Q8×C9—32no—
106C2×Q8×C9—32no—
107C2×Q8×C9—38no—
108C22×C4×C9—21no—
109C22×C4×C9—32no—
110C22×C4×C9C2×D311yescharacteristic
111C2×#611—32no—
112C2×#611—62no—
113C2×#611C2×D312yes—
114#3701—316no—
115#3701—316no—
116Q8×D9—316no—
117Q8×D9—316no—
118#3697—38no—
119#3697—332no—
120#3697—68no—
121#3697C2×D318yescomplemented
122C2×D36—34no—
123C2×D36—31no—
124C2×D36—61no—
125C2×D36C2×D311yescharacteristic
126C2×C4×D9—34no—
127C2×C4×D9—32no—
128C2×C4×D9—34no—
129C2×C4×D9—62no—
130C2×C4×D9C2×D312yes—
131C2×Dic18—31no—
132C2×Dic18—34no—
133C2×Dic18—61no—
134C2×Dic18C2×D311yescharacteristic
135C22×C3×C9—22no—
136C22×C3×C9C2412yes—
137C22×C3×C9C2411yescharacteristic
138C2×C3⋊D9—24no—
139C2×C3⋊D9C2414yes—
140C2×C3×D9—24no—
141C2×C3×D9C2414yes—
142C3×C4×C9C2412yes—
143C3×C4×C9C2412yes—
144C3⋊Dic9C2414yes—
145C9×Dic3C2418yes—
146C3×Dic9C2414yes—
147C2×C3×#41—91no—
148C22×C3×Q8—31no—
149#1010—916no—
150#1007—916no—
151C2×#239—92no—
152C2×#239—92no—
153C2×D3×Q8—92no—
154C2×D3×Q8—92no—
155C2×D3×Q8—92no—
156C2×#237—94no—
157C2×#237—92no—
158C2×#235—31no—
159C2×#235—94no—
160C2×#235—91no—
161C2×#235—91no—
162C2×#235—92no—
163C2×#235—181no—
164C22×Dic6—92no—
165C22×Dic6D911yescharacteristic, complemented
166C23×C32C22×D311yescharacteristic
167C22×C3⋊D3—32no—
168C22×C3×D3—32no—
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C22×D311yescharacteristic
171C2×C32×C4C22×D314yes—
172#638—38no—
173C2×C3⋊Dic3—32no—
174C4⋊(C3⋊D3)—34no—
175C4×C3⋊D3—38no—
176#634—34no—
177C3×#67—38no—
178C2×C3×Dic3—32no—
179C2×C3×Dic3C22×D314yes—
180C2×C3×Dic3C22×D318yes—
181C3×D12—34no—
182C3×C4×D3—38no—
183C3×Dic6—34no—
184C3×Dic6C22×D318yes—
185C3×Dic6C22×D318yes—
186#627—316no—
187#626—316no—
188#624—316no—
189D3×Dic3—316no—
190C23×C9—21no—
191C23×C9C22×D311yescharacteristic
192C22×D9—32no—
193C22×D9—62no—
194C22×D9C22×D312yes—
195Q8×C9—38no—
196Q8×C9—38no—
197C2×C4×C9—21no—
198C2×C4×C9—21no—
199C2×C4×C9—24no—
200C2×C4×C9—34no—
201C2×C4×C9—38no—
202C2×C4×C9C22×D311yescharacteristic
203C2×C4×C9C22×D311yescharacteristic
204C2×C4×C9C22×D314yes—
205#611—38no—
206#611—68no—
207#611C22×D318yes—
208C2×Dic9—32no—
209C2×Dic9—62no—
210C2×Dic9C22×D312yes—
211D36—316no—
212D36—34no—
213D36—64no—
214D36C22×D314yes—
215C4×D9—316no—
216C4×D9—38no—
217C4×D9—316no—
218C4×D9—68no—
219C4×D9C22×D318yes—
220Dic18—34no—
221Dic18—316no—
222Dic18—64no—
223Dic18C22×D314yes—
224C2×2−1+4—271noSylow
225C2×C3×C9—22no—
226C2×C3×C92−1+412yescomplemented
227C2×C3×C9C2511yescharacteristic
228C3⋊D9—24no—
229C3×D9—24no—
230C3×#41—98no—
231C2×C3×Q8—32no—
232C2×C3×Q8—32no—
233C2×C3×Q8—38no—
234C2×C3×Q8—91no—
235C2×C3×D4—92no—
236C2×C3×D4—91no—
237C22×C3×C4—21no—
238C22×C3×C4—32no—
239C22×C3×C4—92no—
240C22×C3×C4C2×D911yescharacteristic
241C22×C3×C4D3211yescharacteristic
242C2×#67—32no—
243C2×#67—92no—
244C2×#67—92no—
245C2×#67—94no—
246C2×#67—182no—
247C22×Dic3—94no—
248C22×Dic3C2×D912yes—
249#239—916no—
250#239—916no—
251D3×Q8—916no—
252D3×Q8—916no—
253D3×Q8—916no—
254#237—932no—
255#237—916no—
256#235—38no—
257#235—932no—
258#235—98no—
259#235—98no—
260#235—916no—
261#235—188no—
262C2×D12—31no—
263C2×D12—94no—
264C2×D12—91no—
265C2×D12—91no—
266C2×D12—181no—
267C2×C4×D3—32no—
268C2×C4×D3—94no—
269C2×C4×D3—94no—
270C2×C4×D3—92no—
271C2×C4×D3—92no—
272C2×C4×D3—94no—
273C2×C4×D3—182no—
274C2×Dic6—31no—
275C2×Dic6—94no—
276C2×Dic6—91no—
277C2×Dic6—91no—
278C2×Dic6—94no—
279C2×Dic6—916no—
280C2×Dic6—181no—
281C2×Dic6C2×D912yes—
282C2×Dic6C2×D912yes—
283C2×Dic6C2×D918yescomplemented
284C22×C32—22no—
285C22×C32C23×D312yes—
286C22×C32C23×D311yescharacteristic
287C2×C3⋊D3—34no—
288C2×C3⋊D3—64no—
289C2×C3×D3—34no—
290C2×C3×D3—64no—
291C32×C4C23×D312yes—
292C32×C4C23×D312yes—
293C3⋊Dic3—34no—
294C3×Dic3—34no—
295C3×Dic3C23×D318yes—
296C22×C9—22no—
297C22×C9—21no—
298C22×C9—22no—
299C22×C9—42no—
300C22×C9C23×D312yes—
301C22×C9C23×D311yescharacteristic
302C2×D9—24no—
303C2×D9—34no—
304C2×D9—64no—
305C2×D9—64no—
306C2×D9—124no—
307C2×D9C23×D314yes—
308C4×C9—22no—
309C4×C9—22no—
310C4×C9—38no—
311C4×C9C23×D312yes—
312C4×C9C23×D312yes—
313Dic9—34no—
314Dic9—64no—
315Dic9C23×D314yes—
3162−1+4—2716no—
317C2×#41—91no—
318C2×#41—274no—
319C2×#41—272no—
320C2×#41—272no—
321C2×#41—271no—
322C22×Q8—31no—
323C22×Q8—272no—
324C22×Q8—272no—
325C3×C9C2×2−1+411yescharacteristic, complemented, Sylow
326C23×C3—21no—
327C23×C3—92no—
328C23×C3C22×D911yescharacteristic
329C23×C3C2×D3211yescharacteristic
330C22×D3—32no—
331C22×D3—92no—
332C22×D3—92no—
333C22×D3—182no—
334C3×Q8—38no—
335C3×Q8—38no—
336C3×Q8—94no—
337C3×D4—98no—
338C3×D4—94no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—24no—
342C2×C3×C4—34no—
343C2×C3×C4—38no—
344C2×C3×C4—92no—
345C2×C3×C4—98no—
346C2×C3×C4C22×D911yescharacteristic
347C2×C3×C4C22×D911yescharacteristic
348C2×C3×C4C22×D914yes—
349C2×C3×C4C2×D3211yescharacteristic
350C2×C3×C4C2×D3211yescharacteristic
351C2×C3×C4C2×D3214yes—
352#67—38no—
353#67—98no—
354#67—98no—
355#67—916no—
356#67—188no—
357C2×Dic3—32no—
358C2×Dic3—92no—
359C2×Dic3—92no—
360C2×Dic3—916no—
361C2×Dic3—182no—
362C2×Dic3C22×D914yes—
363C2×Dic3C22×D918yes—
364D12—34no—
365D12—916no—
366D12—94no—
367D12—94no—
368D12—184no—
369C4×D3—38no—
370C4×D3—916no—
371C4×D3—916no—
372C4×D3—98no—
373C4×D3—98no—
374C4×D3—916no—
375C4×D3—188no—
376Dic6—34no—
377Dic6—916no—
378Dic6—94no—
379Dic6—94no—
380Dic6—916no—
381Dic6—184no—
382Dic6C22×D918yes—
383Dic6C22×D918yes—
384C2×C32—22no—
385C2×C32#100712yes—
386C2×C32C24×D311yescharacteristic
387C3⋊D3—64no—
388C3×D3—64no—
389C2×C9—22no—
390C2×C9—22no—
391C2×C9—21no—
392C2×C9—42no—
393C2×C9#101012yescomplemented
394C2×C9C24×D311yescharacteristic
395D9—24no—
396D9—64no—
397D9—124no—
398#41—98no—
399#41—2732no—
400#41—2716no—
401#41—2716no—
402#41—278no—
403C2×Q8—32no—
404C2×Q8—32no—
405C2×Q8—38no—
406C2×Q8—91no—
407C2×Q8—274no—
408C2×Q8—2716no—
409C2×Q8—2716no—
410C2×Q8—271no—
411C2×D4—92no—
412C2×D4—91no—
413C2×D4—274no—
414C2×D4—272no—
415C2×D4—271no—
416C22×C4—32no—
417C22×C4—92no—
418C22×C4—274no—
419C22×C4—274no—
420C22×C4—272no—
421C22×C4D3×D911yescharacteristic
422C22×C3—22no—
423C22×C3—22no—
424C22×C3—22no—
425C22×C3—21no—
426C22×C3—42no—
427C22×C3—94no—
428C22×C3—184no—
429C22×C3C23×D912yes—
430C22×C3C23×D911yescharacteristic
431C22×C3C22×D3212yes—
432C22×C3C22×D3211yescharacteristic
433C2×D3—34no—
434C2×D3—64no—
435C2×D3—94no—
436C2×D3—94no—
437C2×D3—184no—
438C2×D3—184no—
439C2×D3—184no—
440C2×D3—364no—
441C3×C4—22no—
442C3×C4—22no—
443C3×C4—38no—
444C3×C4—94no—
445C3×C4C23×D912yes—
446C3×C4C23×D912yes—
447C3×C4C22×D3212yes—
448C3×C4C22×D3212yes—
449Dic3—34no—
450Dic3—94no—
451Dic3—94no—
452Dic3—184no—
453Dic3C23×D918yes—
454C32C2×#100711yescharacteristic
455C9—21no—
456C9C2×#101011yescharacteristic, complemented
457C23—92no—
458C23—272no—
459C23C2×D3×D911yescharacteristic
460Q8—38no—
461Q8—38no—
462Q8—94no—
463Q8—2716no—
464Q8—274no—
465D4—98no—
466D4—94no—
467D4—2716no—
468D4—278no—
469D4—274no—
470C2×C4—38no—
471C2×C4—34no—
472C2×C4—92no—
473C2×C4—98no—
474C2×C4—2716no—
475C2×C4—2716no—
476C2×C4—272no—
477C2×C4—278no—
478C2×C4C2×D3×D914yes—
479C2×C4C2×D3×D911yescharacteristic
480C2×C4C2×D3×D911yescharacteristic
481C2×C3—22no—
482C2×C3—22no—
483C2×C3—21no—
484C2×C3—22no—
485C2×C3—42no—
486C2×C3—184no—
487C2×C3#6371312yescomplemented
488C2×C3C24×D911yescharacteristic
489C2×C3#6429712yes—
490C2×C3C23×D3211yescharacteristic
491D3—64no—
492D3—184no—
493D3—184no—
494D3—364no—
495C22—22no—
496C22—94no—
497C22—184no—
498C22—274no—
499C22—544no—
500C22C22×D3×D912yes—
501C22C22×D3×D911yescharacteristic
502C4—38no—
503C4—94no—
504C4—274no—
505C4C22×D3×D912yes—
506C4C22×D3×D912yes—
507C3—21no—
508C3C2×#6371311yescharacteristic, complemented
509C3C2×#6429711yescharacteristic
510C2—22no—
511C2—184no—
512C2—544no—
513C2#1172245812yesdirect product factor
514C2C23×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×C9C2×2−1+4
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. C2×#88736C2semidirect
2. C2×C3×#3697C2semidirect
3. C22×C9×Dic6C2semidirect
4. C2×#88658C2semidirect
5. C2×(C3⋊D9)⋊Q8C2semidirect
6. C2×#88635C2semidirect
7. #11722458C2direct (N × H)
8. C2×#88634C2semidirect
9. C2×D9×Dic6C2semidirect
14. #88736C22semidirect
20. C3×#3697C22semidirect
25. C2×C9×Dic6C22semidirect
29. #88658C22semidirect
32. (C3⋊D9)⋊Q8C22semidirect
33. #88635C22semidirect
34. #88634C22semidirect
35. D9×Dic6C22semidirect
54. C2×#3697D3semidirect
121. #3697C2×D3semidirect
165. C22×Dic6D9semidirect
226. C2×C3×C92−1+4semidirect
283. C2×Dic6C2×D9semidirect
325. C3×C9C2×2−1+4semidirect
393. C2×C9#1010semidirect
456. C9C2×#1010semidirect
487. C2×C3#63713semidirect
508. C3C2×#63713semidirect
513. C2#11722458direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#887362
C2×C3×#36972
C22×C9×Dic62
C2×#886582
C2×(C3⋊D9)⋊Q82
C2×#886352
#117224582
C2×#886342
C2×D9×Dic62
C2×#642973
C2×#637133