Finite Group Databasebeta

← C2×#11720988 #13150201

Subgroups — C2×#11720988 #13150201

Order 1728.

1158 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6192C211yesmaximal, characteristic
2C22×Dic3×D9C211yesmaximal, characteristic, complemented
3C2×#88536C211yesmaximal, characteristic, complemented
4C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic, complemented
5C2×C9×C4⋊Dic3C211yesmaximal, characteristic, complemented
6C2×#88445C211yesmaximal, characteristic, complemented
7C2×Dic9⋊Dic3C211yesmaximal, characteristic, complemented
8#11720988C218yesmaximal, direct product factor
9C2×#63890—31nomaximal
10C2×#63456—31nomaximal
11C23×C3⋊D9C2211yescharacteristic
12C23×C3×D9C2211yescharacteristic
13C22×C3×C4×C9C2211yescharacteristic
14C22×C3⋊Dic9C2211yescharacteristic
15C22×C9×Dic3C2211yescharacteristic
16C22×C9×Dic3C2211yescharacteristic
17C22×C3×Dic9C2211yescharacteristic
18C2×#6192—24no—
19C2×#6192C2214yes—
20C2×Dic3×D9—24no—
21C2×Dic3×D9C2214yescomplemented
22#88536C2214yescomplemented
23C3×(C2×D9)⋊C4C2214yescomplemented
24C9×C4⋊Dic3C2214yescomplemented
25#88445C2214yescomplemented
26Dic9⋊Dic3C2214yescomplemented
27C22×#626—31no—
28C22×D3×Dic3—31no—
29C2×#3752—31no—
30C2×C3×(C2×D3)⋊C4—31no—
31C2×C3×C4⋊Dic3D311yescharacteristic
32C2×Dic3⋊Dic3—31no—
33C2×#3722—31no—
34#63890—38no—
35C22×D36—31no—
36C22×C4×D9—31no—
37C2×C9×C4⋊C4—31no—
38C2×(C2×D9)⋊C4—31no—
39C2×(C2×D9)⋊C4—31no—
40C2×(C2×D9)⋊C4—61no—
41C2×(C2×D9)⋊C4D311yescharacteristic, complemented
42C2×Dic9⋊C4—31no—
43#63456—38no—
44C23×C3×C9C2311yescharacteristic
45C22×C3⋊D9—22no—
46C22×C3⋊D9—22no—
47C22×C3⋊D9—22no—
48C22×C3⋊D9C2312yes—
49C22×C3×D9—22no—
50C22×C3×D9—22no—
51C22×C3×D9D414yes—
52C22×C3×D9C2312yes—
53C2×C3×C4×C9—22no—
54C2×C3×C4×C9C2312yes—
55C2×C3⋊Dic9—22no—
56C2×C3⋊Dic9C2312yes—
57C2×C9×Dic3—22no—
58C2×C9×Dic3—22no—
59C2×C9×Dic3C2312yes—
60C2×C9×Dic3C2312yes—
61C2×C3×Dic9—22no—
62C2×C3×Dic9C2312yes—
63#6192—28no—
64Dic3×D9—28no—
65C2×#894—91no—
66C2×#886—91no—
67C23×C3⋊D3—31no—
68C23×C3×D3—31no—
69C22×C32×C4C2×D311yescharacteristic
70C22×C3⋊Dic3—31no—
71C22×C3×Dic3—31no—
72C22×C3×Dic3C2×D311yescharacteristic
73C22×C3×Dic3C2×D311yescharacteristic
74C2×#626—34no—
75C2×#626—64no—
76C2×D3×Dic3—34no—
77C2×D3×Dic3—64no—
78C23×D9—31no—
79C23×D9—61no—
80C23×D9C2×D311yescharacteristic
81#3752—34no—
82C3×(C2×D3)⋊C4—34no—
83C3×C4⋊Dic3C2×D314yes—
84Dic3⋊Dic3—34no—
85#3722—34no—
86C22×C4×C9—21no—
87C22×C4×C9—31no—
88C22×C4×C9—31no—
89C22×C4×C9C2×D311yescharacteristic
90C22×Dic9—31no—
91C22×Dic9—61no—
92C22×Dic9C2×D311yescharacteristic
93C2×D36—34no—
94C2×D36—64no—
95C2×C4×D9—34no—
96C2×C4×D9—64no—
97C9×C4⋊C4—34no—
98(C2×D9)⋊C4—34no—
99(C2×D9)⋊C4—34no—
100(C2×D9)⋊C4—64no—
101(C2×D9)⋊C4C2×D314yescomplemented
102Dic9⋊C4—34no—
103C22×C3×C9C2×D412yes—
104C22×C3×C9#4112yes—
105C22×C3×C9#4112yes—
106C22×C3×C9C2411yescharacteristic
107C2×C3⋊D9—22no—
108C2×C3⋊D9—22no—
109C2×C3⋊D9—22no—
110C2×C3⋊D9—44no—
111C2×C3×D9—28no—
112C2×C3×D9—22no—
113C2×C3×D9—22no—
114C2×C3×D9C2×D414yes—
115C3×C4×C9—22no—
116C3⋊Dic9—22no—
117C9×Dic3—22no—
118C9×Dic3—22no—
119C3×Dic9—22no—
120C22×#67—91no—
121C23×Dic3—91no—
122C22×D12—91no—
123C22×C4×D3—91no—
124C2×C3×C4⋊C4—31no—
125C2×C3×C22⋊C4—91no—
126C2×(C2×D3)⋊C4—31no—
127C2×(C2×D3)⋊C4—91no—
128C2×(C2×D3)⋊C4—91no—
129C2×(C2×D3)⋊C4—91no—
130C2×(C2×D3)⋊C4—91no—
131C2×(C2×D3)⋊C4—181no—
132C2×C4⋊Dic3—91no—
133C2×C4⋊Dic3D911yescharacteristic, complemented
134C2×Dic3⋊C4—91no—
135#894—98no—
136#886—98no—
137C23×C32C22×D311yescharacteristic
138C22×C3⋊D3—32no—
139C22×C3⋊D3—62no—
140C22×C3⋊D3—62no—
141C22×C3⋊D3—62no—
142C22×C3×D3—34no—
143C22×C3×D3—32no—
144C22×C3×D3—62no—
145C22×C3×D3—62no—
146C2×C32×C4—22no—
147C2×C32×C4C22×D312yes—
148C2×C3⋊Dic3—32no—
149C2×C3⋊Dic3—62no—
150C2×C3×Dic3—22no—
151C2×C3×Dic3—22no—
152C2×C3×Dic3—32no—
153C2×C3×Dic3—62no—
154C2×C3×Dic3C22×D312yes—
155C2×C3×Dic3C22×D312yes—
156#626—68no—
157D3×Dic3—68no—
158C23×C9—21no—
159C23×C9C22×D311yescharacteristic
160C22×D9—22no—
161C22×D9—22no—
162C22×D9—32no—
163C22×D9—62no—
164C22×D9—64no—
165C22×D9—62no—
166C22×D9—62no—
167C22×D9—62no—
168C22×D9—122no—
169C22×D9—122no—
170C22×D9D1214yes—
171C22×D9C22×D312yes—
172C2×C4×C9—22no—
173C2×C4×C9—22no—
174C2×C4×C9—32no—
175C2×C4×C9—32no—
176C2×C4×C9—42no—
177C2×C4×C9—62no—
178C2×C4×C9—62no—
179C2×C4×C9C22×D312yes—
180C2×Dic9—22no—
181C2×Dic9—32no—
182C2×Dic9—62no—
183C2×Dic9—62no—
184C2×Dic9—122no—
185C2×Dic9C22×D312yes—
186D36—68no—
187C4×D9—68no—
188C2×#123—271noSylow
189C2×C3×C9#12314yescomplemented
190C2×C3×C9C22×D411yescharacteristic
191C2×C3×C9C2×#4111yescharacteristic
192C2×C3×C9C2×#4111yescharacteristic
193C3⋊D9—42no—
194C3×D9—24no—
195C24×C3—91no—
196C23×D3—31no—
197C23×D3—91no—
198C23×D3—91no—
199C23×D3—181no—
200C22×C3×C4—21no—
201C22×C3×C4—31no—
202C22×C3×C4—31no—
203C22×C3×C4—91no—
204C22×C3×C4C2×D911yescharacteristic
205C22×C3×C4D3211yescharacteristic
206C2×#67—94no—
207C2×#67—184no—
208C22×Dic3—31no—
209C22×Dic3—91no—
210C22×Dic3—91no—
211C22×Dic3—94no—
212C22×Dic3—181no—
213C22×Dic3—184no—
214C22×Dic3C2×D911yescharacteristic
215C22×Dic3C2×D911yescharacteristic
216C2×D12—94no—
217C2×D12—184no—
218C2×C4×D3—94no—
219C2×C4×D3—184no—
220C3×C4⋊C4—34no—
221C3×C22⋊C4—94no—
222(C2×D3)⋊C4—34no—
223(C2×D3)⋊C4—94no—
224(C2×D3)⋊C4—94no—
225(C2×D3)⋊C4—94no—
226(C2×D3)⋊C4—94no—
227(C2×D3)⋊C4—184no—
228C4⋊Dic3—94no—
229C4⋊Dic3C2×D914yescomplemented
230Dic3⋊C4—94no—
231C22×C32#23512yes—
232C22×C32D3×D412yes—
233C22×C32#23912yes—
234C22×C32C23×D311yescharacteristic
235C2×C3⋊D3—62no—
236C2×C3⋊D3—62no—
237C2×C3⋊D3—62no—
238C2×C3⋊D3—124no—
239C2×C3×D3—34no—
240C2×C3×D3—68no—
241C2×C3×D3—62no—
242C2×C3×D3—62no—
243C32×C4—22no—
244C3⋊Dic3—62no—
245C3×Dic3—22no—
246C3×Dic3—22no—
247C3×Dic3—62no—
248C22×C9—21no—
249C22×C9—22no—
250C22×C9—22no—
251C22×C9—22no—
252C22×C9C2×D1212yes—
253C22×C9#23712yes—
254C22×C9#23712yes—
255C22×C9C23×D311yescharacteristic
256C2×D9—28no—
257C2×D9—22no—
258C2×D9—22no—
259C2×D9—64no—
260C2×D9—62no—
261C2×D9—62no—
262C2×D9—62no—
263C2×D9—122no—
264C2×D9—122no—
265C2×D9—128no—
266C2×D9—124no—
267C2×D9C2×D1214yes—
268C4×C9—22no—
269C4×C9—42no—
270C4×C9—62no—
271C4×C9—62no—
272Dic9—22no—
273Dic9—62no—
274Dic9—122no—
275C22×D4—271no—
276C23×C4—271no—
277#123—278no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—271no—
280C2×C22⋊C4—91no—
281C2×C22⋊C4—271no—
282C2×C22⋊C4—271no—
283C3×C9C2×#12311yescharacteristic, complemented, Sylow
284C23×C3—21no—
285C23×C3—94no—
286C23×C3—92no—
287C23×C3—182no—
288C23×C3—182no—
289C23×C3C22×D911yescharacteristic
290C23×C3C2×D3211yescharacteristic
291C22×D3—32no—
292C22×D3—34no—
293C22×D3—62no—
294C22×D3—62no—
295C22×D3—92no—
296C22×D3—92no—
297C22×D3—182no—
298C22×D3—184no—
299C22×D3—182no—
300C22×D3—182no—
301C22×D3—182no—
302C22×D3—182no—
303C22×D3—182no—
304C22×D3—182no—
305C22×D3—362no—
306C22×D3—362no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—32no—
311C2×C3×C4—32no—
312C2×C3×C4—42no—
313C2×C3×C4—62no—
314C2×C3×C4—62no—
315C2×C3×C4—92no—
316C2×C3×C4—182no—
317C2×C3×C4C22×D912yes—
318C2×C3×C4C2×D3212yes—
319#67—188no—
320C2×Dic3—22no—
321C2×Dic3—22no—
322C2×Dic3—32no—
323C2×Dic3—62no—
324C2×Dic3—92no—
325C2×Dic3—92no—
326C2×Dic3—182no—
327C2×Dic3—188no—
328C2×Dic3—182no—
329C2×Dic3—182no—
330C2×Dic3—362no—
331C2×Dic3C22×D912yes—
332C2×Dic3C22×D912yes—
333D12—188no—
334C4×D3—188no—
335C2×C32#89414yes—
336C2×C32C2×#23511yescharacteristic
337C2×C32C2×D3×D411yescharacteristic
338C2×C32C2×#23911yescharacteristic
339C3⋊D3—122no—
340C3×D3—64no—
341C2×C9—21no—
342C2×C9—21no—
343C2×C9—21no—
344C2×C9—24no—
345C2×C9#88614yescomplemented
346C2×C9C22×D1211yescharacteristic
347C2×C9C2×#23711yescharacteristic
348C2×C9C2×#23711yescharacteristic
349D9—24no—
350D9—124no—
351D9—122no—
352C24—91no—
353C24—271no—
354C2×D4—274no—
355C2×D4—544no—
356C22×C4—31no—
357C22×C4—31no—
358C22×C4—91no—
359C22×C4—274no—
360C22×C4—271no—
361C22×C4—544no—
362C22×C4D3×D911yescharacteristic
363C4⋊C4—34no—
364C4⋊C4—274no—
365C22⋊C4—94no—
366C22⋊C4—274no—
367C22⋊C4—274no—
368C22×C3—22no—
369C22×C3—21no—
370C22×C3—22no—
371C22×C3—22no—
372C22×C3—94no—
373C22×C3—182no—
374C22×C3—188no—
375C22×C3—182no—
376C22×C3#369712yes—
377C22×C3D4×D912yes—
378C22×C3#370112yes—
379C22×C3C23×D911yescharacteristic
380C22×C3#379612yes—
381C22×C3D3×D1212yes—
382C22×C3#380412yes—
383C22×C3C22×D3211yescharacteristic
384C2×D3—34no—
385C2×D3—62no—
386C2×D3—62no—
387C2×D3—68no—
388C2×D3—184no—
389C2×D3—182no—
390C2×D3—182no—
391C2×D3—182no—
392C2×D3—182no—
393C2×D3—182no—
394C2×D3—182no—
395C2×D3—362no—
396C2×D3—362no—
397C2×D3—368no—
398C2×D3—364no—
399C2×D3—364no—
400C3×C4—22no—
401C3×C4—22no—
402C3×C4—42no—
403C3×C4—62no—
404C3×C4—62no—
405C3×C4—182no—
406Dic3—22no—
407Dic3—22no—
408Dic3—62no—
409Dic3—182no—
410Dic3—182no—
411Dic3—362no—
412C32C2×#89411yescharacteristic
413C9—21no—
414C9C2×#88611yescharacteristic, complemented
415C23—92no—
416C23—94no—
417C23—182no—
418C23—182no—
419C23—272no—
420C23—542no—
421C23—542no—
422C23—542no—
423C23C2×D3×D911yescharacteristic
424D4—548no—
425C2×C4—22no—
426C2×C4—32no—
427C2×C4—32no—
428C2×C4—62no—
429C2×C4—62no—
430C2×C4—92no—
431C2×C4—182no—
432C2×C4—272no—
433C2×C4—548no—
434C2×C4—542no—
435C2×C4C2×D3×D912yes—
436C2×C3—24no—
437C2×C3—21no—
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—184no—
441C2×C3#6345614yescomplemented
442C2×C3C2×#369711yescharacteristic
443C2×C3C2×D4×D911yescharacteristic
444C2×C3C2×#370111yescharacteristic
445C2×C3#6389014yes—
446C2×C3C2×#379611yescharacteristic
447C2×C3C2×D3×D1211yescharacteristic
448C2×C3C2×#380411yescharacteristic
449D3—64no—
450D3—364no—
451D3—362no—
452D3—362no—
453C22—94no—
454C22—188no—
455C22—182no—
456C22—182no—
457C22—542no—
458C22—542no—
459C22—542no—
460C22—1084no—
461C22#8863412yes—
462C22D9×D1212yes—
463C22#8865812yes—
464C22C22×D3×D911yescharacteristic
465C4—22no—
466C4—62no—
467C4—62no—
468C4—182no—
469C4—542no—
470C3—21no—
471C3C2×#6345611yescharacteristic, complemented
472C3C2×#6389011yescharacteristic
473C2—184no—
474C2—1082no—
475C2#1172098814yesdirect product factor
476C2C2×#8863411yescharacteristic
477C2C2×D9×D1211yescharacteristic
478C2C2×#8865811yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
prime radicalC22×#6192C2
hypercenterC22×C4D3×D9
hypocenterC3×C9C2×#123
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C22×Dic3×D9C2semidirect
3. C2×#88536C2semidirect
4. C2×C3×(C2×D9)⋊C4C2semidirect
5. C2×C9×C4⋊Dic3C2semidirect
6. C2×#88445C2semidirect
7. C2×Dic9⋊Dic3C2semidirect
8. #11720988C2direct (N × H)
21. C2×Dic3×D9C22semidirect
22. #88536C22semidirect
23. C3×(C2×D9)⋊C4C22semidirect
24. C9×C4⋊Dic3C22semidirect
25. #88445C22semidirect
26. Dic9⋊Dic3C22semidirect
41. C2×(C2×D9)⋊C4D3semidirect
101. (C2×D9)⋊C4C2×D3semidirect
133. C2×C4⋊Dic3D9semidirect
189. C2×C3×C9#123semidirect
229. C4⋊Dic3C2×D9semidirect
283. C3×C9C2×#123semidirect
345. C2×C9#886semidirect
414. C9C2×#886semidirect
441. C2×C3#63456semidirect
471. C3C2×#63456semidirect
475. C2#11720988direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#61922
C22×Dic3×D92
C2×#885362
C2×C3×(C2×D9)⋊C42
C2×C9×C4⋊Dic32
C2×#884452
C2×Dic9⋊Dic32
#117209882
C2×#638903
C2×#634563