Finite Group Databasebeta

← C2×#11721014 #13150446

Subgroups — C2×#11721014 #13150446

Order 1728.

1514 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C3⋊D9C211yesmaximal, characteristic, complemented
2C22×#6195C211yesmaximal, characteristic
3C22×D3×Dic9C211yesmaximal, characteristic, complemented
4C2×C9×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
5C2×C3×Dic9⋊C4C211yesmaximal, characteristic, complemented
6C2×#88445C211yesmaximal, characteristic, complemented
7C2×Dic3×Dic9C211yesmaximal, characteristic, complemented
8#11721014C218yesmaximal, direct product factor
9C2×#63881—31nomaximal
10C2×#611⋊C4—31nomaximal
11C23×C3⋊D9C2211yescharacteristic
12C23×D3×C9C2211yescharacteristic
13C22×C3×C4×C9C2211yescharacteristic
14C22×C3⋊Dic9C2211yescharacteristic
15C2×C4×C3⋊D9—24no—
16C2×C4×C3⋊D9C2214yescomplemented
17C22×C9×Dic3C2211yescharacteristic
18C22×C3×Dic9C2211yescharacteristic
19C22×C3×Dic9C2211yescharacteristic
20C2×#6195C418yescomplemented
21C2×#6195C2214yes—
22C2×D3×Dic9—24no—
23C2×D3×Dic9C2214yescomplemented
24C9×(C2×D3)⋊C4C2214yescomplemented
25C3×Dic9⋊C4C2214yescomplemented
26#88445C2214yescomplemented
27Dic3×Dic9C2214yescomplemented
28C22×C4×C3⋊D3—31no—
29C22×#626—31no—
30C22×D3×Dic3—31no—
31C2×C3×(C2×D3)⋊C4D311yescharacteristic
32C2×C3×Dic3⋊C4—31no—
33C2×#3722—31no—
34C2×Dic32—31no—
35#63881—38no—
36C22×#611—31no—
37C23×Dic9—31no—
38C22×C4×D9—31no—
39C22×C4×D9—61no—
40C2×C9×C22⋊C4—31no—
41C2×(C2×D9)⋊C4—31no—
42C2×Dic9⋊C4D311yescharacteristic, complemented
43C2×C4×Dic9—31no—
44#611⋊C4—38no—
45C23×C3×C9C2311yescharacteristic
46C22×C3⋊D9—22no—
47C22×C3⋊D9—22no—
48C22×C3⋊D9C2×C414yescomplemented
49C22×C3⋊D9C2312yes—
50C22×D3×C9—22no—
51C22×D3×C9—22no—
52C22×D3×C9C2×C414yescomplemented
53C22×D3×C9C2312yes—
54C2×C3×C4×C9—22no—
55C2×C3×C4×C9C2312yes—
56C2×C3⋊Dic9D414yescomplemented
57C2×C3⋊Dic9C2312yes—
58C4×C3⋊D9—28no—
59C2×C9×Dic3—22no—
60C2×C9×Dic3C2312yes—
61C2×C3×Dic9—22no—
62C2×C3×Dic9C2×C414yescomplemented
63C2×C3×Dic9C2312yes—
64C2×C3×Dic9C2312yes—
65#6195C2×C4116yescomplemented
66D3×Dic9—28no—
67C2×#893—91no—
68C2×#67⋊C4—91no—
69C23×C3⋊D3—31no—
70C23×C3×D3C2×D311yescharacteristic
71C22×C32×C4C2×D311yescharacteristic
72C22×C3⋊Dic3—31no—
73C2×C4×C3⋊D3—34no—
74C2×C4×C3⋊D3—64no—
75C22×C3×Dic3—31no—
76C22×C3×Dic3—31no—
77C22×C3×Dic3C2×D311yescharacteristic
78C2×#626—38no—
79C2×#626—34no—
80C2×D3×Dic3—34no—
81C2×D3×Dic3—64no—
82C24×C9—31no—
83C23×D9—31no—
84C23×D9—61no—
85C3×(C2×D3)⋊C4C2×D314yes—
86C3×Dic3⋊C4—34no—
87#3722—34no—
88Dic32—34no—
89C22×C4×C9—21no—
90C22×C4×C9—31no—
91C22×C4×C9C2×D311yescharacteristic
92C2×#611—38no—
93C2×#611—34no—
94C22×Dic9—31no—
95C22×Dic9—34no—
96C22×Dic9—64no—
97C22×Dic9—61no—
98C22×Dic9C2×D311yescharacteristic
99C22×Dic9C2×D311yescharacteristic
100C2×C4×D9—34no—
101C2×C4×D9—64no—
102C2×C4×D9—64no—
103C2×C4×D9—68no—
104C9×C22⋊C4—34no—
105(C2×D9)⋊C4—34no—
106Dic9⋊C4C2×D314yescomplemented
107C4×Dic9—34no—
108C22×C3×C9C22×C412yes—
109C22×C3×C9C2×D412yes—
110C22×C3×C9#4112yes—
111C22×C3×C9C2411yescharacteristic
112C2×C3⋊D9—28no—
113C2×C3⋊D9—22no—
114C2×C3⋊D9—22no—
115C2×C3⋊D9C22×C414yescomplemented
116C2×D3×C9—28no—
117C2×D3×C9—22no—
118C2×D3×C9—22no—
119C2×D3×C9C22×C414yescomplemented
120C3×C4×C9—22no—
121C3⋊Dic9C2×D414yescomplemented
122C9×Dic3—22no—
123C3×Dic9—22no—
124C3×Dic9C22×C414yescomplemented
125C22×#67—91no—
126C23×Dic3—91no—
127C22×D12—91no—
128C22×C4×D3—91no—
129C22×C4×D3—91no—
130C22×C4×D3—91no—
131C22×C4×D3—181no—
132C2×C3×C4⋊C4—91no—
133C2×C3×C22⋊C4—31no—
134C2×(C2×D3)⋊C4—91no—
135C2×(C2×D3)⋊C4—91no—
136C2×(C2×D3)⋊C4D911yescharacteristic, complemented
137C2×Dic3⋊C4—31no—
138C2×C4×Dic3—91no—
139C2×C4×Dic3—91no—
140#893—98no—
141#67⋊C4—98no—
142C23×C32C22×D311yescharacteristic
143C22×C3⋊D3—34no—
144C22×C3⋊D3—32no—
145C22×C3⋊D3—62no—
146C22×C3⋊D3—62no—
147C22×C3×D3—22no—
148C22×C3×D3—22no—
149C22×C3×D3C4×D314yes—
150C22×C3×D3C22×D312yes—
151C2×C32×C4—22no—
152C2×C32×C4C22×D312yes—
153C2×C3⋊Dic3—34no—
154C2×C3⋊Dic3—32no—
155C4×C3⋊D3—68no—
156C2×C3×Dic3—22no—
157C2×C3×Dic3—34no—
158C2×C3×Dic3—32no—
159C2×C3×Dic3—32no—
160C2×C3×Dic3—62no—
161C2×C3×Dic3C22×D312yes—
162#626—316no—
163D3×Dic3—68no—
164C23×C9—21no—
165C23×C9—34no—
166C23×C9—32no—
167C23×C9—62no—
168C23×C9—62no—
169C23×C9C22×D311yescharacteristic
170C22×D9—34no—
171C22×D9—32no—
172C22×D9—62no—
173C22×D9—64no—
174C22×D9—64no—
175C22×D9—64no—
176C22×D9—62no—
177C22×D9—62no—
178C2×C4×C9—24no—
179C2×C4×C9—22no—
180C2×C4×C9—22no—
181C2×C4×C9—32no—
182C2×C4×C9—62no—
183C2×C4×C9C22×D312yes—
184#611—316no—
185C2×Dic9—22no—
186C2×Dic9—34no—
187C2×Dic9—32no—
188C2×Dic9—64no—
189C2×Dic9—62no—
190C2×Dic9—68no—
191C2×Dic9C4×D314yescomplemented
192C2×Dic9C22×D312yes—
193C2×Dic9C22×D312yes—
194C4×D9—68no—
195C4×D9—616no—
196C2×C4×D4—271noSylow
197C2×C3×C9C4×D414yescomplemented
198C2×C3×C9C23×C411yescharacteristic
199C2×C3×C9C22×D411yescharacteristic
200C2×C3×C9C2×#4111yescharacteristic
201C3⋊D9—24no—
202D3×C9—24no—
203C24×C3—31no—
204C23×D3—91no—
205C23×D3—91no—
206C23×D3—181no—
207C23×D3C2×D911yescharacteristic
208C22×C3×C4—21no—
209C22×C3×C4—31no—
210C22×C3×C4—91no—
211C22×C3×C4—91no—
212C22×C3×C4C2×D911yescharacteristic
213C22×C3×C4D3211yescharacteristic
214C2×#67—94no—
215C2×#67—98no—
216C22×Dic3—31no—
217C22×Dic3—31no—
218C22×Dic3—94no—
219C22×Dic3—91no—
220C22×Dic3—91no—
221C22×Dic3—181no—
222C22×Dic3—184no—
223C22×Dic3C2×D911yescharacteristic
224C2×D12—98no—
225C2×D12—94no—
226C2×C4×D3—94no—
227C2×C4×D3—94no—
228C2×C4×D3—94no—
229C2×C4×D3—184no—
230C2×C4×D3—184no—
231C2×C4×D3—184no—
232C2×C4×D3—184no—
233C2×C4×D3—188no—
234C3×C4⋊C4—94no—
235C3×C22⋊C4—34no—
236(C2×D3)⋊C4—94no—
237(C2×D3)⋊C4—94no—
238(C2×D3)⋊C4C2×D914yescomplemented
239Dic3⋊C4—34no—
240C4×Dic3—94no—
241C4×Dic3—94no—
242C22×C32C2×C4×D312yes—
243C22×C32D3×D412yes—
244C22×C32#23712yes—
245C22×C32C23×D311yescharacteristic
246C2×C3⋊D3—34no—
247C2×C3⋊D3—68no—
248C2×C3⋊D3—62no—
249C2×C3⋊D3—62no—
250C2×C3×D3—28no—
251C2×C3×D3—22no—
252C2×C3×D3—22no—
253C2×C3×D3C2×C4×D314yes—
254C32×C4—22no—
255C3⋊Dic3—34no—
256C3×Dic3—22no—
257C3×Dic3—34no—
258C3×Dic3—62no—
259C22×C9—21no—
260C22×C9—22no—
261C22×C9—22no—
262C22×C9—22no—
263C22×C9—34no—
264C22×C9—68no—
265C22×C9—62no—
266C22×C9—62no—
267C22×C9C2×C4×D312yes—
268C22×C9D3×D412yes—
269C22×C9#23912yes—
270C22×C9C23×D311yescharacteristic
271C2×D9—34no—
272C2×D9—64no—
273C2×D9—64no—
274C2×D9—64no—
275C2×D9—616no—
276C2×D9—68no—
277C2×D9—62no—
278C2×D9—62no—
279C4×C9—24no—
280C4×C9—22no—
281C4×C9—62no—
282Dic9—22no—
283Dic9—34no—
284Dic9—64no—
285Dic9C2×C4×D314yescomplemented
286C22×D4—271no—
287C23×C4—271no—
288C23×C4—271no—
289C4×D4—278no—
290C2×C4⋊C4—91no—
291C2×C22⋊C4—31no—
292C2×C22⋊C4—271no—
293C2×C42—271no—
294C3×C9C2×C4×D411yescharacteristic, complemented, Sylow
295C23×C3—21no—
296C23×C3—32no—
297C23×C3—34no—
298C23×C3—62no—
299C23×C3—62no—
300C23×C3C22×D911yescharacteristic
301C23×C3C2×D3211yescharacteristic
302C22×D3—22no—
303C22×D3—22no—
304C22×D3—92no—
305C22×D3—94no—
306C22×D3—92no—
307C22×D3—94no—
308C22×D3—182no—
309C22×D3—184no—
310C22×D3—184no—
311C22×D3—184no—
312C22×D3—182no—
313C22×D3—182no—
314C22×D3—182no—
315C22×D3—182no—
316C22×D3C4×D914yescomplemented
317C22×D3C22×D912yes—
318C2×C3×C4—24no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—32no—
323C2×C3×C4—62no—
324C2×C3×C4—94no—
325C2×C3×C4—92no—
326C2×C3×C4—92no—
327C2×C3×C4—182no—
328C2×C3×C4C22×D912yes—
329C2×C3×C4C2×D3212yes—
330#67—916no—
331C2×Dic3—22no—
332C2×Dic3—32no—
333C2×Dic3—34no—
334C2×Dic3—32no—
335C2×Dic3—62no—
336C2×Dic3—92no—
337C2×Dic3—94no—
338C2×Dic3—92no—
339C2×Dic3—94no—
340C2×Dic3—182no—
341C2×Dic3—184no—
342C2×Dic3—188no—
343C2×Dic3C22×D912yes—
344D12—916no—
345C4×D3—188no—
346C4×D3—188no—
347C4×D3—188no—
348C4×D3—1816no—
349C2×C32#67⋊C414yes—
350C2×C32C22×C4×D311yescharacteristic
351C2×C32C2×D3×D411yescharacteristic
352C2×C32C2×#23711yescharacteristic
353C3⋊D3—64no—
354C3×D3—24no—
355C2×C9—21no—
356C2×C9—21no—
357C2×C9—21no—
358C2×C9—24no—
359C2×C9—64no—
360C2×C9#89314yescomplemented
361C2×C9C22×C4×D311yescharacteristic
362C2×C9C2×D3×D411yescharacteristic
363C2×C9C2×#23911yescharacteristic
364D9—68no—
365D9—64no—
366C24—31no—
367C24—271no—
368C2×D4—274no—
369C2×D4—278no—
370C22×C4—31no—
371C22×C4—91no—
372C22×C4—91no—
373C22×C4—274no—
374C22×C4—271no—
375C22×C4—274no—
376C22×C4—544no—
377C22×C4—544no—
378C22×C4D3×D911yescharacteristic
379C4⋊C4—94no—
380C22⋊C4—34no—
381C22⋊C4—274no—
382C42—274no—
383C22×C3—22no—
384C22×C3—21no—
385C22×C3—22no—
386C22×C3—22no—
387C22×C3—34no—
388C22×C3—62no—
389C22×C3—62no—
390C22×C3—68no—
391C22×C3C2×C4×D912yes—
392C22×C3D4×D912yes—
393C22×C3#369912yes—
394C22×C3C23×D911yescharacteristic
395C22×C3#379612yes—
396C22×C3C4×D3212yes—
397C22×C3#381112yes—
398C22×C3C22×D3211yescharacteristic
399C2×D3—28no—
400C2×D3—22no—
401C2×D3—22no—
402C2×D3—94no—
403C2×D3—94no—
404C2×D3—184no—
405C2×D3—184no—
406C2×D3—1816no—
407C2×D3—184no—
408C2×D3—182no—
409C2×D3—182no—
410C2×D3—188no—
411C2×D3—188no—
412C2×D3—182no—
413C2×D3—182no—
414C2×D3C2×C4×D914yescomplemented
415C3×C4—24no—
416C3×C4—22no—
417C3×C4—22no—
418C3×C4—62no—
419C3×C4—94no—
420C3×C4—182no—
421Dic3—22no—
422Dic3—34no—
423Dic3—62no—
424Dic3—94no—
425Dic3—94no—
426Dic3—184no—
427C32C2×#67⋊C411yescharacteristic
428C9—21no—
429C9C2×#89311yescharacteristic, complemented
430C23—32no—
431C23—34no—
432C23—62no—
433C23—62no—
434C23—272no—
435C23—274no—
436C23—542no—
437C23—542no—
438C23C2×D3×D911yescharacteristic
439D4—2716no—
440C2×C4—22no—
441C2×C4—32no—
442C2×C4—62no—
443C2×C4—92no—
444C2×C4—94no—
445C2×C4—92no—
446C2×C4—182no—
447C2×C4—272no—
448C2×C4—274no—
449C2×C4—548no—
450C2×C4—548no—
451C2×C4C2×D3×D912yes—
452C2×C3—21no—
453C2×C3—24no—
454C2×C3—21no—
455C2×C3—21no—
456C2×C3—64no—
457C2×C3#611⋊C414yescomplemented
458C2×C3C22×C4×D911yescharacteristic
459C2×C3C2×D4×D911yescharacteristic
460C2×C3C2×#369911yescharacteristic
461C2×C3#6388114yes—
462C2×C3C2×#379611yescharacteristic
463C2×C3C2×C4×D3211yescharacteristic
464C2×C3C2×#381111yescharacteristic
465D3—24no—
466D3—188no—
467D3—184no—
468D3—184no—
469C22—34no—
470C22—62no—
471C22—68no—
472C22—62no—
473C22—274no—
474C22—542no—
475C22—548no—
476C22—542no—
477C22#8863912yes—
478C22C4×D3×D912yes—
479C22#8866812yes—
480C22C22×D3×D911yescharacteristic
481C4—22no—
482C4—62no—
483C4—94no—
484C4—182no—
485C4—274no—
486C3—21no—
487C3C2×#611⋊C411yescharacteristic, complemented
488C3C2×#6388111yescharacteristic
489C2—64no—
490C2—544no—
491C2#1172101414yesdirect product factor
492C2C2×#8863911yescharacteristic
493C2C2×C4×D3×D911yescharacteristic
494C2C2×#8866811yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
prime radicalC22×#6195C2
cosocleC22×C3×C9C24
hypercenterC22×C4D3×D9
hypocenterC3×C9C2×C4×D4
LCSG ▷ C2×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⋊D9C2semidirect
3. C22×D3×Dic9C2semidirect
4. C2×C9×(C2×D3)⋊C4C2semidirect
5. C2×C3×Dic9⋊C4C2semidirect
6. C2×#88445C2semidirect
7. C2×Dic3×Dic9C2semidirect
8. #11721014C2direct (N × H)
16. C2×C4×C3⋊D9C22semidirect
20. C2×#6195C4semidirect
23. C2×D3×Dic9C22semidirect
24. C9×(C2×D3)⋊C4C22semidirect
25. C3×Dic9⋊C4C22semidirect
26. #88445C22semidirect
27. Dic3×Dic9C22semidirect
42. C2×Dic9⋊C4D3semidirect
48. C22×C3⋊D9C2×C4semidirect
52. C22×D3×C9C2×C4semidirect
56. C2×C3⋊Dic9D4semidirect
62. C2×C3×Dic9C2×C4semidirect
65. #6195C2×C4semidirect
106. Dic9⋊C4C2×D3semidirect
115. C2×C3⋊D9C22×C4semidirect
119. C2×D3×C9C22×C4semidirect
121. C3⋊Dic9C2×D4semidirect
124. C3×Dic9C22×C4semidirect
136. C2×(C2×D3)⋊C4D9semidirect
191. C2×Dic9C4×D3semidirect
197. C2×C3×C9C4×D4semidirect
238. (C2×D3)⋊C4C2×D9semidirect
285. Dic9C2×C4×D3semidirect
294. C3×C9C2×C4×D4semidirect
316. C22×D3C4×D9semidirect
360. C2×C9#893semidirect
414. C2×D3C2×C4×D9semidirect
429. C9C2×#893semidirect
457. C2×C3#611⋊C4semidirect
487. C3C2×#611⋊C4semidirect
491. C2#11721014direct (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⋊D92
C22×#61952
C22×D3×Dic92
C2×C9×(C2×D3)⋊C42
C2×C3×Dic9⋊C42
C2×#884452
C2×Dic3×Dic92
#117210142
C2×#638813
C2×#611⋊C43