Finite Group Databasebeta

← C2×#11721042 #13150471

Subgroups — C2×#11721042 #13150471

Order 1728.

1146 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6226C211yesmaximal, characteristic, complemented
2C22×#6194C211yesmaximal, characteristic
3C2×C9×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
4C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic, complemented
5C2×(C2×D3)⋊Dic9C211yesmaximal, characteristic, complemented
6C2×(C2×D9)⋊Dic3C211yesmaximal, characteristic, complemented
7C2×Dic3×Dic9C211yesmaximal, characteristic, complemented
8#11721042C218yesmaximal, direct product factor
9C2×#63913—31nomaximal
10C2×#63447—31nomaximal
11C23×D3×C9C2211yescharacteristic
12C23×C3×D9C2211yescharacteristic
13C22×C3×C4×C9C2211yescharacteristic
14C22×C3⋊Dic9C2211yescharacteristic
15C22×C3⋊Dic9C2211yescharacteristic
16C2×#6226—24no—
17C2×#6226C2214yescomplemented
18C22×C9×Dic3C2211yescharacteristic
19C22×C3×Dic9C2211yescharacteristic
20C2×#6194—24no—
21C2×#6194C2214yes—
22C9×(C2×D3)⋊C4C2214yescomplemented
23C3×(C2×D9)⋊C4C2214yescomplemented
24(C2×D3)⋊Dic9C2214yescomplemented
25(C2×D9)⋊Dic3C2214yescomplemented
26Dic3×Dic9C2214yescomplemented
27C22×#634—31no—
28C22×#625—31no—
29C2×C3×(C2×D3)⋊C4—31no—
30C2×C3×(C2×D3)⋊C4D311yescharacteristic
31C2×(C2×D3)⋊Dic3—31no—
32C2×(C2×D3)⋊Dic3—31no—
33C2×Dic32—31no—
34#63913—38no—
35C22×#611—31no—
36C22×Dic18—31no—
37C22×Dic18—61no—
38C2×C9×C22⋊C4—31no—
39C2×#3676—31no—
40C2×(C2×D9)⋊C4—31no—
41C2×(C2×D9)⋊C4D311yescharacteristic, complemented
42C2×C4×Dic9—31no—
43#63447—38no—
44C23×C3×C9C2311yescharacteristic
45C22×D3×C9—22no—
46C22×D3×C9—22no—
47C22×D3×C9—22no—
48C22×D3×C9C2312yes—
49C22×C3×D9—22no—
50C22×C3×D9—22no—
51C22×C3×D9—22no—
52C22×C3×D9C2312yes—
53C2×C3×C4×C9—22no—
54C2×C3×C4×C9C2312yes—
55C2×C3⋊Dic9—22no—
56C2×C3⋊Dic9D414yes—
57C2×C3⋊Dic9C2312yes—
58C2×C3⋊Dic9C2312yes—
59#6226—28no—
60C2×C9×Dic3—22no—
61C2×C9×Dic3C2312yes—
62C2×C3×Dic9—22no—
63C2×C3×Dic9C2312yes—
64#6194—28no—
65C2×#885—91no—
66C2×#885—91no—
67C23×C3×D3—31no—
68C23×C3×D3C2×D311yescharacteristic
69C22×C32×C4C2×D311yescharacteristic
70C22×C3⋊Dic3—31no—
71C22×C3⋊Dic3—31no—
72C2×#634—34no—
73C2×#634—64no—
74C22×C3×Dic3—31no—
75C22×C3×Dic3C2×D311yescharacteristic
76C2×#625—34no—
77C2×#625—64no—
78C24×C9—31no—
79C23×D9C2×D311yescharacteristic
80C3×(C2×D3)⋊C4—34no—
81C3×(C2×D3)⋊C4C2×D314yes—
82(C2×D3)⋊Dic3—34no—
83(C2×D3)⋊Dic3—34no—
84Dic32—34no—
85C22×C4×C9—21no—
86C22×C4×C9—31no—
87C22×C4×C9C2×D311yescharacteristic
88C2×#611—34no—
89C2×#611—64no—
90C22×Dic9—31no—
91C22×Dic9—31no—
92C22×Dic9—61no—
93C22×Dic9—61no—
94C22×Dic9C2×D311yescharacteristic
95C2×Dic18—34no—
96C2×Dic18—64no—
97C2×Dic18—64no—
98C2×Dic18—68no—
99C9×C22⋊C4—34no—
100#3676—34no—
101(C2×D9)⋊C4—34no—
102(C2×D9)⋊C4C2×D314yescomplemented
103C4×Dic9—34no—
104C22×C3×C9C2×D412yes—
105C22×C3×C9#4112yes—
106C22×C3×C9#4112yes—
107C22×C3×C9C2411yescharacteristic
108C2×D3×C9—22no—
109C2×D3×C9—22no—
110C2×D3×C9—22no—
111C2×D3×C9—44no—
112C2×C3×D9—22no—
113C2×C3×D9—22no—
114C2×C3×D9—22no—
115C2×C3×D9—44no—
116C3×C4×C9—22no—
117C3⋊Dic9—22no—
118C3⋊Dic9C2×D414yes—
119C9×Dic3—22no—
120C3×Dic9—22no—
121C22×#67—91no—
122C22×#67—91no—
123C22×Dic6—91no—
124C22×Dic6—91no—
125C22×Dic6—181no—
126C2×C3×C22⋊C4—31no—
127C2×C3×C22⋊C4—91no—
128C2×#217—91no—
129C2×#217—91no—
130C2×(C2×D3)⋊C4—31no—
131C2×(C2×D3)⋊C4—91no—
132C2×(C2×D3)⋊C4—91no—
133C2×(C2×D3)⋊C4D911yescharacteristic, complemented
134C2×C4×Dic3—91no—
135C2×C4×Dic3—91no—
136#885—98no—
137#885—98no—
138C23×C32C22×D311yescharacteristic
139C22×C3×D3—22no—
140C22×C3×D3—22no—
141C22×C3×D3—22no—
142C22×C3×D3—32no—
143C22×C3×D3—62no—
144C22×C3×D3—62no—
145C22×C3×D3—62no—
146C22×C3×D3C22×D312yes—
147C2×C32×C4—22no—
148C2×C32×C4C22×D312yes—
149C2×C3⋊Dic3—34no—
150C2×C3⋊Dic3—32no—
151C2×C3⋊Dic3—32no—
152C2×C3⋊Dic3—62no—
153#634—68no—
154C2×C3×Dic3—22no—
155C2×C3×Dic3—32no—
156C2×C3×Dic3—62no—
157C2×C3×Dic3C22×D312yes—
158#625—68no—
159C23×C9—21no—
160C23×C9—32no—
161C23×C9—62no—
162C23×C9—62no—
163C23×C9—62no—
164C23×C9C22×D311yescharacteristic
165C22×D9—22no—
166C22×D9—22no—
167C22×D9—22no—
168C22×D9C22×D312yes—
169C2×C4×C9—24no—
170C2×C4×C9—22no—
171C2×C4×C9—22no—
172C2×C4×C9—32no—
173C2×C4×C9—62no—
174C2×C4×C9C22×D312yes—
175#611—68no—
176C2×Dic9—22no—
177C2×Dic9—34no—
178C2×Dic9—32no—
179C2×Dic9—32no—
180C2×Dic9—64no—
181C2×Dic9—62no—
182C2×Dic9—62no—
183C2×Dic9—64no—
184C2×Dic9—62no—
185C2×Dic9C22×D312yes—
186Dic18—68no—
187Dic18—616no—
188C2×#124—271noSylow
189C2×C3×C9#12414yescomplemented
190C2×C3×C9C22×D411yescharacteristic
191C2×C3×C9C2×#4111yescharacteristic
192C2×C3×C9C2×#4111yescharacteristic
193D3×C9—42no—
194C3×D9—42no—
195C24×C3—31no—
196C24×C3—91no—
197C23×D3—31no—
198C23×D3C2×D911yescharacteristic
199C22×C3×C4—21no—
200C22×C3×C4—31no—
201C22×C3×C4—91no—
202C22×C3×C4C2×D911yescharacteristic
203C22×C3×C4D3211yescharacteristic
204C2×#67—94no—
205C2×#67—94no—
206C2×#67—184no—
207C2×#67—184no—
208C22×Dic3—31no—
209C22×Dic3—91no—
210C22×Dic3—91no—
211C22×Dic3—91no—
212C22×Dic3—91no—
213C22×Dic3—181no—
214C22×Dic3—181no—
215C22×Dic3C2×D911yescharacteristic
216C2×Dic6—94no—
217C2×Dic6—94no—
218C2×Dic6—184no—
219C2×Dic6—184no—
220C2×Dic6—184no—
221C2×Dic6—188no—
222C3×C22⋊C4—34no—
223C3×C22⋊C4—94no—
224#217—94no—
225#217—94no—
226(C2×D3)⋊C4—34no—
227(C2×D3)⋊C4—94no—
228(C2×D3)⋊C4—94no—
229(C2×D3)⋊C4C2×D914yescomplemented
230C4×Dic3—94no—
231C4×Dic3—94no—
232C22×C32#23512yes—
233C22×C32D3×D412yes—
234C22×C32#23712yes—
235C22×C32C23×D311yescharacteristic
236C2×C3×D3—22no—
237C2×C3×D3—22no—
238C2×C3×D3—22no—
239C2×C3×D3—44no—
240C2×C3×D3—62no—
241C2×C3×D3—62no—
242C2×C3×D3—62no—
243C2×C3×D3—124no—
244C32×C4—22no—
245C3⋊Dic3—34no—
246C3⋊Dic3—62no—
247C3×Dic3—22no—
248C3×Dic3—62no—
249C22×C9—21no—
250C22×C9—22no—
251C22×C9—22no—
252C22×C9—22no—
253C22×C9—62no—
254C22×C9—62no—
255C22×C9—62no—
256C22×C9—124no—
257C22×C9#23512yes—
258C22×C9D3×D412yes—
259C22×C9#23712yes—
260C22×C9C23×D311yescharacteristic
261C2×D9—22no—
262C2×D9—22no—
263C2×D9—22no—
264C2×D9—44no—
265C4×C9—24no—
266C4×C9—22no—
267C4×C9—62no—
268Dic9—22no—
269Dic9—34no—
270Dic9—64no—
271Dic9—64no—
272Dic9—62no—
273C22×Q8—271no—
274C22×D4—271no—
275#124—278no—
276C2×C22⋊C4—31no—
277C2×C22⋊C4—91no—
278C2×C22⋊C4—271no—
279C2×C22⋊C4—271no—
280C2×C42—271no—
281C3×C9C2×#12411yescharacteristic, complemented, Sylow
282C23×C3—21no—
283C23×C3—32no—
284C23×C3—62no—
285C23×C3—62no—
286C23×C3—62no—
287C23×C3—92no—
288C23×C3—182no—
289C23×C3—182no—
290C23×C3—182no—
291C23×C3C22×D911yescharacteristic
292C23×C3C2×D3211yescharacteristic
293C22×D3—22no—
294C22×D3—22no—
295C22×D3—22no—
296C22×D3—32no—
297C22×D3—62no—
298C22×D3—62no—
299C22×D3—62no—
300C22×D3C22×D912yes—
301C2×C3×C4—24no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—32no—
306C2×C3×C4—62no—
307C2×C3×C4—92no—
308C2×C3×C4—182no—
309C2×C3×C4C22×D912yes—
310C2×C3×C4C2×D3212yes—
311#67—188no—
312#67—188no—
313C2×Dic3—22no—
314C2×Dic3—32no—
315C2×Dic3—62no—
316C2×Dic3—92no—
317C2×Dic3—94no—
318C2×Dic3—92no—
319C2×Dic3—94no—
320C2×Dic3—92no—
321C2×Dic3—92no—
322C2×Dic3—182no—
323C2×Dic3—184no—
324C2×Dic3—182no—
325C2×Dic3—184no—
326C2×Dic3—182no—
327C2×Dic3—182no—
328C2×Dic3C22×D912yes—
329Dic6—188no—
330Dic6—188no—
331Dic6—1816no—
332C2×C32#88514yes—
333C2×C32C2×#23511yescharacteristic
334C2×C32C2×D3×D411yescharacteristic
335C2×C32C2×#23711yescharacteristic
336C3×D3—42no—
337C3×D3—122no—
338C2×C9—21no—
339C2×C9—21no—
340C2×C9—21no—
341C2×C9—24no—
342C2×C9—122no—
343C2×C9#88514yescomplemented
344C2×C9C2×#23511yescharacteristic
345C2×C9C2×D3×D411yescharacteristic
346C2×C9C2×#23711yescharacteristic
347D9—42no—
348C24—31no—
349C24—91no—
350C2×Q8—274no—
351C2×Q8—544no—
352C2×D4—274no—
353C2×D4—544no—
354C22×C4—31no—
355C22×C4—91no—
356C22×C4—271no—
357C22×C4—271no—
358C22×C4D3×D911yescharacteristic
359C22⋊C4—34no—
360C22⋊C4—94no—
361C22⋊C4—274no—
362C22⋊C4—274no—
363C42—274no—
364C22×C3—22no—
365C22×C3—22no—
366C22×C3—21no—
367C22×C3—22no—
368C22×C3—62no—
369C22×C3—62no—
370C22×C3—62no—
371C22×C3—124no—
372C22×C3—182no—
373C22×C3—182no—
374C22×C3—182no—
375C22×C3—364no—
376C22×C3#369712yes—
377C22×C3D4×D912yes—
378C22×C3#369912yes—
379C22×C3C23×D911yescharacteristic
380C22×C3#379512yes—
381C22×C3#379512yes—
382C22×C3#381112yes—
383C22×C3C22×D3211yescharacteristic
384C2×D3—22no—
385C2×D3—22no—
386C2×D3—22no—
387C2×D3—44no—
388C2×D3—62no—
389C2×D3—62no—
390C2×D3—62no—
391C2×D3—124no—
392C3×C4—24no—
393C3×C4—22no—
394C3×C4—22no—
395C3×C4—62no—
396C3×C4—182no—
397Dic3—22no—
398Dic3—62no—
399Dic3—94no—
400Dic3—94no—
401Dic3—184no—
402Dic3—184no—
403Dic3—182no—
404Dic3—182no—
405C32C2×#88511yescharacteristic
406C9—21no—
407C9C2×#88511yescharacteristic, complemented
408C23—32no—
409C23—62no—
410C23—62no—
411C23—62no—
412C23—92no—
413C23—182no—
414C23—182no—
415C23—182no—
416C23C2×D3×D911yescharacteristic
417Q8—548no—
418D4—548no—
419C2×C4—22no—
420C2×C4—32no—
421C2×C4—62no—
422C2×C4—92no—
423C2×C4—182no—
424C2×C4—272no—
425C2×C4—274no—
426C2×C4—272no—
427C2×C4—542no—
428C2×C4C2×D3×D912yes—
429C2×C3—21no—
430C2×C3—24no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—122no—
434C2×C3—362no—
435C2×C3#6344714yescomplemented
436C2×C3C2×#369711yescharacteristic
437C2×C3C2×D4×D911yescharacteristic
438C2×C3C2×#369911yescharacteristic
439C2×C3#6391314yes—
440C2×C3C2×#379511yescharacteristic
441C2×C3C2×#379511yescharacteristic
442C2×C3C2×#381111yescharacteristic
443D3—42no—
444D3—122no—
445C22—62no—
446C22—62no—
447C22—62no—
448C22—124no—
449C22—182no—
450C22—182no—
451C22—182no—
452C22—364no—
453C22#8863812yes—
454C22#8864112yes—
455C22#8866812yes—
456C22C22×D3×D911yescharacteristic
457C4—22no—
458C4—62no—
459C4—182no—
460C4—274no—
461C4—542no—
462C3—21no—
463C3C2×#6344711yescharacteristic, complemented
464C3C2×#6391311yescharacteristic
465C2—122no—
466C2—362no—
467C2#1172104214yesdirect product factor
468C2C2×#8863811yescharacteristic
469C2C2×#8864111yescharacteristic
470C2C2×#8866811yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
prime radicalC22×#6194C2
hypercenterC22×C4D3×D9
hypocenterC3×C9C2×#124
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×#6226C2semidirect
3. C2×C9×(C2×D3)⋊C4C2semidirect
4. C2×C3×(C2×D9)⋊C4C2semidirect
5. C2×(C2×D3)⋊Dic9C2semidirect
6. C2×(C2×D9)⋊Dic3C2semidirect
7. C2×Dic3×Dic9C2semidirect
8. #11721042C2direct (N × H)
17. C2×#6226C22semidirect
22. C9×(C2×D3)⋊C4C22semidirect
23. C3×(C2×D9)⋊C4C22semidirect
24. (C2×D3)⋊Dic9C22semidirect
25. (C2×D9)⋊Dic3C22semidirect
26. Dic3×Dic9C22semidirect
41. C2×(C2×D9)⋊C4D3semidirect
102. (C2×D9)⋊C4C2×D3semidirect
133. C2×(C2×D3)⋊C4D9semidirect
189. C2×C3×C9#124semidirect
229. (C2×D3)⋊C4C2×D9semidirect
281. C3×C9C2×#124semidirect
343. C2×C9#885semidirect
407. C9C2×#885semidirect
435. C2×C3#63447semidirect
463. C3C2×#63447semidirect
467. C2#11721042direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62262
C22×#61942
C2×C9×(C2×D3)⋊C42
C2×C3×(C2×D9)⋊C42
C2×(C2×D3)⋊Dic92
C2×(C2×D9)⋊Dic32
C2×Dic3×Dic92
#117210422
C2×#639133
C2×#634473