Finite Group Databasebeta

← C2×#11722465 #13164690

Subgroups — C2×#11722465 #13164690

Order 1728.

2538 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6226C211yesmaximal, characteristic, complemented
2C2×C9×#235C211yesmaximal, characteristic, complemented
3C2×C3×#3697C211yesmaximal, characteristic, complemented
4C2×#88663C214yesmaximal, complemented
5C2×#88641C212yesmaximal, complemented
6#11722465C2116yesmaximal, direct product factor
7C2×#88638C212yesmaximal, complemented
8C2×D3×Dic18C212yesmaximal, complemented
9C2×D9×Dic6C212yesmaximal, complemented
10C2×#64298—31nomaximal
11C2×#63716—31nomaximal
12C22×C3×C4×C9C2211yescharacteristic
13C22×C3⋊Dic9C2212yes—
14C2×#6226C2212yes—
15C2×#6226C2212yes—
16C2×#6226C2218yescomplemented
17C2×C9×#67C2212yes—
18C2×C3×#611C2212yes—
19C9×#235C2218yescomplemented
20C2×C9×D12C2211yescharacteristic
21C2×C4×D3×C9C2212yes—
22C3×#3697C2218yescomplemented
23C2×C3×D36C2211yescharacteristic
24C2×C3×C4×D9C2212yes—
25C2×C9×Dic6C2211yescharacteristic
26C2×C3×Dic18C2211yescharacteristic
27C2×#6194C2214yes—
28#88663C22132yescomplemented
29C2×D3×Dic9C2214yes—
30C2×Dic3×D9C2214yes—
31C2×#6189C2214yes—
32#88641C22116yescomplemented
33#88638C22116yescomplemented
34D3×Dic18C22116yescomplemented
35D9×Dic6C22116yescomplemented
36C22×#634—31no—
37C2×C3×#235—31no—
38C2×C3×#235D311yescharacteristic
39C2×#3805—34no—
40#64298—316no—
41C2×#3795—32no—
42C2×#3795—32no—
43C2×D3×Dic6—32no—
44C2×D3×Dic6—32no—
45C2×C9×#41—31no—
46#63716—316no—
47C2×Q8×D9—32no—
48C2×#3699—34no—
49C2×#3699—32no—
50C2×#3697—32no—
51C2×#3697D311yescharacteristic, complemented
52C22×Dic18—31no—
53C22×Dic18—32no—
54C22×Dic18—61no—
55C23×C3×C9C2311yescharacteristic
56C22×D3×C9C2312yes—
57C22×C3×D9C2312yes—
58C2×C3×C4×C9C2311yescharacteristic
59C2×C3×C4×C9C2311yescharacteristic
60C2×C3×C4×C9C2314yes—
61C2×C3⋊Dic9C2314yes—
62C2×C3⋊Dic9C2318yes—
63#6226C2318yes—
64#6226C2318yes—
65C9×#67C2318yes—
66C3×#611C2318yes—
67C2×C9×Dic3C2312yes—
68C2×C3×Dic9C2312yes—
69C9×D12C2314yes—
70C4×D3×C9C2318yes—
71C3×D36C2314yes—
72C3×C4×D9C2318yes—
73C9×Dic6C2314yes—
74C3×Dic18C2314yes—
75#6194C23116yes—
76D3×Dic9C23116yes—
77Dic3×D9C23116yes—
78#6189C23116yes—
79C2×#1010—91no—
80C2×#1010—91no—
81C22×C32×C4C2×D311yescharacteristic
82C22×C3⋊Dic3—32no—
83C2×#634—32no—
84C2×#634—32no—
85C2×#634—38no—
86C2×C3×#67—32no—
87C2×C3×#67C2×D312yes—
88C3×#235—38no—
89C3×#235C2×D318yes—
90C2×C3×D12—31no—
91C2×C3×D12C2×D311yescharacteristic
92C2×C3×C4×D3—32no—
93C2×C3×C4×D3C2×D312yes—
94C2×C3×Dic6—31no—
95C2×C3×Dic6C2×D311yescharacteristic
96C2×#627—34no—
97C2×#625—34no—
98#3805—332no—
99C2×D3×Dic3—34no—
100C2×D3×Dic3—34no—
101#3795—316no—
102#3795—316no—
103D3×Dic6—316no—
104D3×Dic6—316no—
105C9×#41—38no—
106C2×Q8×C9—31no—
107C2×D4×C9—32no—
108C2×D4×C9—31no—
109C22×C4×C9—21no—
110C22×C4×C9—32no—
111C22×C4×C9C2×D311yescharacteristic
112C2×#611—34no—
113C2×#611C2×D312yes—
114C22×Dic9—32no—
115C22×Dic9—34no—
116C22×Dic9—62no—
117Q8×D9—316no—
118#3699—332no—
119#3699—316no—
120#3697—316no—
121#3697C2×D318yescomplemented
122C2×D36C2×D311yescharacteristic
123C2×C4×D9—34no—
124C2×C4×D9C2×D312yes—
125C2×Dic18—32no—
126C2×Dic18—32no—
127C2×Dic18—38no—
128C2×Dic18—34no—
129C2×Dic18—316no—
130C2×Dic18—62no—
131C2×Dic18—62no—
132C2×Dic18—68no—
133C2×Dic18C2×D311yescharacteristic
134C22×C3×C9—22no—
135C22×C3×C9C2412yes—
136C22×C3×C9C2411yescharacteristic
137C2×D3×C9—24no—
138C2×D3×C9C2414yes—
139C2×C3×D9—24no—
140C2×C3×D9C2414yes—
141C3×C4×C9C2412yes—
142C3×C4×C9C2412yes—
143C3⋊Dic9C2418yes—
144C9×Dic3C2414yes—
145C3×Dic9C2414yes—
146C2×C3×#41—31no—
147C2×C3×#41—91no—
148#1010—916no—
149#1010—916no—
150C2×D3×Q8—92no—
151C2×D3×Q8—92no—
152C2×#237—92no—
153C2×#237—94no—
154C2×#237—94no—
155C2×#237—92no—
156C2×#235—31no—
157C2×#235—92no—
158C2×#235—92no—
159C2×#235D911yescharacteristic, complemented
160C22×Dic6—92no—
161C22×Dic6—91no—
162C22×Dic6—91no—
163C22×Dic6—92no—
164C22×Dic6—181no—
165C23×C32C22×D311yescharacteristic
166C22×C3×D3—32no—
167C22×C3×D3C22×D312yes—
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C22×D314yes—
171C2×C3⋊Dic3—34no—
172C2×C3⋊Dic3—38no—
173#634—38no—
174#634—38no—
175C3×#67—38no—
176C3×#67C22×D318yes—
177C2×C3×Dic3—32no—
178C2×C3×Dic3C22×D312yes—
179C3×D12—34no—
180C3×D12C22×D314yes—
181C3×C4×D3—38no—
182C3×C4×D3C22×D318yes—
183C3×Dic6—34no—
184C3×Dic6C22×D314yes—
185#627—316no—
186#625—316no—
187D3×Dic3—316no—
188D3×Dic3—316no—
189C23×C9—21no—
190C23×C9—32no—
191C23×C9C22×D311yescharacteristic
192C22×D9C22×D312yes—
193Q8×C9—34no—
194D4×C9—38no—
195D4×C9—34no—
196C2×C4×C9—21no—
197C2×C4×C9—21no—
198C2×C4×C9—24no—
199C2×C4×C9—32no—
200C2×C4×C9—38no—
201C2×C4×C9C22×D311yescharacteristic
202C2×C4×C9C22×D311yescharacteristic
203C2×C4×C9C22×D314yes—
204#611—316no—
205#611C22×D318yes—
206C2×Dic9—34no—
207C2×Dic9—38no—
208C2×Dic9—316no—
209C2×Dic9—64no—
210C2×Dic9—68no—
211C2×Dic9C22×D312yes—
212D36C22×D314yes—
213C4×D9—316no—
214C4×D9C22×D318yes—
215Dic18—38no—
216Dic18—38no—
217Dic18—316no—
218Dic18—68no—
219Dic18—68no—
220Dic18C22×D314yes—
221C2×2−1+4—271noSylow
222C2×C3×C9—22no—
223C2×C3×C92−1+412yescomplemented
224C2×C3×C9C2511yescharacteristic
225D3×C9—24no—
226C3×D9—24no—
227C3×#41—38no—
228C3×#41—98no—
229C2×C3×Q8—31no—
230C2×C3×Q8—91no—
231C2×C3×D4—32no—
232C2×C3×D4—31no—
233C2×C3×D4—92no—
234C2×C3×D4—91no—
235C22×C3×C4—21no—
236C22×C3×C4—32no—
237C22×C3×C4—92no—
238C22×C3×C4C2×D911yescharacteristic
239C22×C3×C4D3211yescharacteristic
240C2×#67—32no—
241C2×#67—94no—
242C2×#67—94no—
243C2×#67C2×D912yes—
244C22×Dic3—94no—
245C22×Dic3—92no—
246C22×Dic3—92no—
247C22×Dic3—94no—
248C22×Dic3—182no—
249D3×Q8—916no—
250D3×Q8—916no—
251#237—916no—
252#237—932no—
253#237—932no—
254#237—916no—
255#235—38no—
256#235—916no—
257#235—916no—
258#235C2×D918yescomplemented
259C2×D12—31no—
260C2×D12C2×D911yescharacteristic
261C2×C4×D3—32no—
262C2×C4×D3—94no—
263C2×C4×D3—94no—
264C2×C4×D3C2×D912yes—
265C2×Dic6—31no—
266C2×Dic6—916no—
267C2×Dic6—94no—
268C2×Dic6—92no—
269C2×Dic6—92no—
270C2×Dic6—98no—
271C2×Dic6—92no—
272C2×Dic6—92no—
273C2×Dic6—98no—
274C2×Dic6—94no—
275C2×Dic6—916no—
276C2×Dic6—182no—
277C2×Dic6—182no—
278C2×Dic6—188no—
279C2×Dic6C2×D911yescharacteristic
280C22×C32—22no—
281C22×C32C23×D312yes—
282C22×C32C23×D311yescharacteristic
283C2×C3×D3—24no—
284C2×C3×D3—34no—
285C2×C3×D3—64no—
286C2×C3×D3C23×D314yes—
287C32×C4C23×D312yes—
288C32×C4C23×D312yes—
289C3⋊Dic3—38no—
290C3×Dic3—34no—
291C3×Dic3C23×D314yes—
292C22×C9—24no—
293C22×C9—22no—
294C22×C9—22no—
295C22×C9—21no—
296C22×C9—34no—
297C22×C9—64no—
298C22×C9C23×D312yes—
299C22×C9C23×D311yescharacteristic
300C2×D9—24no—
301C2×D9C23×D314yes—
302C4×C9—22no—
303C4×C9—22no—
304C4×C9—34no—
305C4×C9C23×D312yes—
306C4×C9C23×D312yes—
307Dic9—38no—
308Dic9—68no—
309Dic9C23×D314yes—
3102−1+4—2716no—
311C2×#41—31no—
312C2×#41—91no—
313C2×#41—272no—
314C2×#41—272no—
315C2×#41—274no—
316C22×Q8—272no—
317C22×Q8—272no—
318C22×Q8—271no—
319C3×C9C2×2−1+411yescharacteristic, complemented, Sylow
320C23×C3—21no—
321C23×C3—32no—
322C23×C3—92no—
323C23×C3C22×D911yescharacteristic
324C23×C3C2×D3211yescharacteristic
325C22×D3—32no—
326C22×D3C22×D912yes—
327C3×Q8—34no—
328C3×Q8—94no—
329C3×D4—38no—
330C3×D4—34no—
331C3×D4—98no—
332C3×D4—94no—
333C2×C3×C4—24no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—32no—
337C2×C3×C4—38no—
338C2×C3×C4—92no—
339C2×C3×C4—98no—
340C2×C3×C4C22×D911yescharacteristic
341C2×C3×C4C22×D911yescharacteristic
342C2×C3×C4C22×D914yes—
343C2×C3×C4C2×D3211yescharacteristic
344C2×C3×C4C2×D3211yescharacteristic
345C2×C3×C4C2×D3214yes—
346#67—38no—
347#67—916no—
348#67—916no—
349#67C22×D918yes—
350C2×Dic3—32no—
351C2×Dic3—916no—
352C2×Dic3—94no—
353C2×Dic3—98no—
354C2×Dic3—94no—
355C2×Dic3—98no—
356C2×Dic3—916no—
357C2×Dic3—188no—
358C2×Dic3—184no—
359C2×Dic3C22×D912yes—
360D12—34no—
361D12C22×D914yes—
362C4×D3—38no—
363C4×D3—916no—
364C4×D3—916no—
365C4×D3C22×D918yes—
366Dic6—34no—
367Dic6—916no—
368Dic6—98no—
369Dic6—98no—
370Dic6—98no—
371Dic6—98no—
372Dic6—916no—
373Dic6—188no—
374Dic6—188no—
375Dic6C22×D914yes—
376C2×C32—22no—
377C2×C32#101012yes—
378C2×C32C24×D311yescharacteristic
379C3×D3—24no—
380C3×D3—64no—
381C2×C9—24no—
382C2×C9—22no—
383C2×C9—22no—
384C2×C9—21no—
385C2×C9—64no—
386C2×C9#101012yescomplemented
387C2×C9C24×D311yescharacteristic
388D9—24no—
389#41—38no—
390#41—98no—
391#41—2716no—
392#41—2716no—
393#41—2732no—
394C2×Q8—31no—
395C2×Q8—91no—
396C2×Q8—274no—
397C2×Q8—2716no—
398C2×Q8—2716no—
399C2×Q8—272no—
400C2×Q8—272no—
401C2×Q8—278no—
402C2×D4—32no—
403C2×D4—31no—
404C2×D4—92no—
405C2×D4—91no—
406C2×D4—274no—
407C22×C4—32no—
408C22×C4—92no—
409C22×C4—274no—
410C22×C4—274no—
411C22×C4—272no—
412C22×C4D3×D911yescharacteristic
413C22×C3—24no—
414C22×C3—22no—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3—21no—
418C22×C3—34no—
419C22×C3—64no—
420C22×C3—94no—
421C22×C3—184no—
422C22×C3C23×D912yes—
423C22×C3C23×D911yescharacteristic
424C22×C3C22×D3212yes—
425C22×C3C22×D3211yescharacteristic
426C2×D3—24no—
427C2×D3—34no—
428C2×D3—64no—
429C2×D3C23×D914yes—
430C3×C4—22no—
431C3×C4—22no—
432C3×C4—34no—
433C3×C4—94no—
434C3×C4C23×D912yes—
435C3×C4C23×D912yes—
436C3×C4C22×D3212yes—
437C3×C4C22×D3212yes—
438Dic3—34no—
439Dic3—98no—
440Dic3—98no—
441Dic3—188no—
442Dic3C23×D914yes—
443C32C2×#101011yescharacteristic
444C9—21no—
445C9C2×#101011yescharacteristic, complemented
446C23—32no—
447C23—92no—
448C23C2×D3×D911yescharacteristic
449Q8—34no—
450Q8—94no—
451Q8—2716no—
452Q8—278no—
453Q8—278no—
454D4—38no—
455D4—34no—
456D4—98no—
457D4—94no—
458D4—2716no—
459C2×C4—32no—
460C2×C4—38no—
461C2×C4—92no—
462C2×C4—98no—
463C2×C4—2716no—
464C2×C4—2716no—
465C2×C4—278no—
466C2×C4—274no—
467C2×C4C2×D3×D914yes—
468C2×C4C2×D3×D911yescharacteristic
469C2×C4C2×D3×D911yescharacteristic
470C2×C3—24no—
471C2×C3—22no—
472C2×C3—22no—
473C2×C3—21no—
474C2×C3—22no—
475C2×C3—64no—
476C2×C3—184no—
477C2×C3#6371612yescomplemented
478C2×C3C24×D911yescharacteristic
479C2×C3#6429812yes—
480C2×C3C23×D3211yescharacteristic
481D3—24no—
482D3—64no—
483C22—22no—
484C22—34no—
485C22—64no—
486C22—94no—
487C22—184no—
488C22C22×D3×D912yes—
489C22C22×D3×D911yescharacteristic
490C4—34no—
491C4—94no—
492C4—278no—
493C4C22×D3×D912yes—
494C4C22×D3×D912yes—
495C3—21no—
496C3C2×#6371611yescharacteristic, complemented
497C3C2×#6429811yescharacteristic
498C2—22no—
499C2—64no—
500C2—184no—
501C2#1172246512yesdirect product factor
502C2C23×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. C22×#6226C2semidirect
2. C2×C9×#235C2semidirect
3. C2×C3×#3697C2semidirect
4. C2×#88663C2semidirect
5. C2×#88641C2semidirect
6. #11722465C2direct (N × H)
7. C2×#88638C2semidirect
8. C2×D3×Dic18C2semidirect
9. C2×D9×Dic6C2semidirect
16. C2×#6226C22semidirect
19. C9×#235C22semidirect
22. C3×#3697C22semidirect
28. #88663C22semidirect
32. #88641C22semidirect
33. #88638C22semidirect
34. D3×Dic18C22semidirect
35. D9×Dic6C22semidirect
51. C2×#3697D3semidirect
121. #3697C2×D3semidirect
159. C2×#235D9semidirect
223. C2×C3×C92−1+4semidirect
258. #235C2×D9semidirect
319. C3×C9C2×2−1+4semidirect
386. C2×C9#1010semidirect
445. C9C2×#1010semidirect
477. C2×C3#63716semidirect
496. C3C2×#63716semidirect
501. C2#11722465direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62262
C2×C9×#2352
C2×C3×#36972
C2×#886632
C2×#886412
#117224652
C2×#886382
C2×D3×Dic182
C2×D9×Dic62
C2×#642983
C2×#637163