Finite Group Databasebeta

← #13150495

Subgroups — #13150495

Order 1728.

1038 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C9×#235C211yesmaximal, characteristic, complemented
2C2×C3×#3697C211yesmaximal, characteristic, complemented
3C2×C4×C3⋊Dic9C211yesmaximal, characteristic, complemented
4#11721290C214yesmaximal, complemented
5#11721043C211yesmaximal, characteristic, complemented
6C4×#6194C214yesmaximal, complemented
7#11720967C211yesmaximal, characteristic, complemented
8#11720951C211yesmaximal, characteristic, complemented
9#11720930C211yesmaximal, characteristic, complemented
10#10595032—31nomaximal
11#10589934—31nomaximal
12C22×C3×C4×C9C2211yescharacteristic
13C22×C3⋊Dic9C2212yes—
14C2×C9×#67C2212yes—
15C2×C3×#611C2212yes—
16C9×#235—24no—
17C2×C9×D12C2211yescharacteristic
18C2×C4×D3×C9C2212yes—
19C3×#3697—24no—
20C2×C3×D36C2211yescharacteristic
21C2×C3×C4×D9C2212yes—
22C2×C9×Dic6C2211yescharacteristic
23C2×C3×Dic18C2211yescharacteristic
24C2×#6194C2214yes—
25C4×C3⋊Dic9C2211yescharacteristic
26C4×C3⋊Dic9C2212yes—
27C4×C3⋊Dic9C2211yescharacteristic
28(C2×D3)⋊Dic9C2214yes—
29(C2×D9)⋊Dic3C2214yes—
30#88442C2214yes—
31C2×C3×#235—31no—
32C2×C3×#235D311yescharacteristic
33C2×C4×C3⋊Dic3—31no—
34#63980—34no—
35#63914—31no—
36C4×#625—34no—
37#63873—31no—
38#63863—31no—
39#63863—31no—
40C2×C9×#41—31no—
41C2×#3697D311yescharacteristic, complemented
42#63509—31no—
43#63506—31no—
44#63502—31no—
45#63501—34no—
46#63498—31no—
47C4×#611—34no—
48C2×C4×Dic9—31no—
49C2×C4×Dic9—61no—
50C23×C3×C9C2311yescharacteristic
51C22×D3×C9C2312yes—
52C22×C3×D9C2312yes—
53C2×C3×C4×C9D414yescomplemented
54C2×C3×C4×C9C2311yescharacteristic
55C2×C3×C4×C9C2311yescharacteristic
56C2×C3⋊Dic9—24no—
57C2×C3⋊Dic9C2312yes—
58C2×C3⋊Dic9C2312yes—
59C9×#67—24no—
60C3×#611—24no—
61C2×C9×Dic3C2312yes—
62C2×C3×Dic9C2312yes—
63C9×D12—22no—
64C4×D3×C9—24no—
65C3×D36—22no—
66C3×C4×D9—24no—
67C9×Dic6—22no—
68C3×Dic18—22no—
69#6194—28no—
70#5832—91no—
71#5832—91no—
72C22×C32×C4C2×D311yescharacteristic
73C22×C3⋊Dic3—32no—
74C2×C3×#67—32no—
75C2×C3×#67C2×D312yes—
76C3×#235—24no—
77C3×#235—64no—
78C2×C3×D12—31no—
79C2×C3×D12C2×D311yescharacteristic
80C2×C3×C4×D3—32no—
81C2×C3×C4×D3C2×D312yes—
82C2×C3×Dic6—31no—
83C2×C3×Dic6C2×D311yescharacteristic
84C2×#625—34no—
85C4×C3⋊Dic3—31no—
86C4×C3⋊Dic3—31no—
87C4×C3⋊Dic3—32no—
88#3724—34no—
89(C2×D3)⋊Dic3—34no—
90(C2×D3)⋊Dic3—34no—
91C9×#41—64no—
92C2×Q8×C9—31no—
93C2×D4×C9—32no—
94C2×D4×C9—31no—
95C22×C4×C9—21no—
96C22×C4×C9—32no—
97C22×C4×C9C2×D311yescharacteristic
98C2×#611—34no—
99C2×#611C2×D312yes—
100C22×Dic9—32no—
101C22×Dic9—62no—
102#3697—24no—
103C2×D36C2×D311yescharacteristic
104C2×C4×D9C2×D312yes—
105C2×Dic18C2×D311yescharacteristic
106#3676—34no—
107(C2×D9)⋊C4—34no—
108Dic9⋊C4—34no—
109C4×Dic9—32no—
110C4×Dic9—31no—
111C4×Dic9—31no—
112C4×Dic9—62no—
113C4×Dic9—61no—
114C4×Dic9—61no—
115C22×C3×C9—22no—
116C22×C3×C9C2×D412yes—
117C22×C3×C9C2411yescharacteristic
118C2×D3×C9—22no—
119C2×D3×C9—24no—
120C2×C3×D9—22no—
121C2×C3×D9—24no—
122C3×C4×C9C2×D412yes—
123C3×C4×C9C2×D412yes—
124C3⋊Dic9—22no—
125C3⋊Dic9#4114yescomplemented
126C9×Dic3—22no—
127C3×Dic9—22no—
128C2×C3×#41—31no—
129C2×C3×#41—91no—
130C2×#235—31no—
131C2×#235D911yescharacteristic, complemented
132#947—91no—
133#947—91no—
134#944—91no—
135#944—91no—
136#940—91no—
137#940—91no—
138#939—94no—
139#939—94no—
140#936—91no—
141#936—91no—
142C4×#67—94no—
143C4×#67—94no—
144C2×C4×Dic3—91no—
145C2×C4×Dic3—91no—
146C2×C4×Dic3—181no—
147C23×C32C22×D311yescharacteristic
148C22×C3×D3—32no—
149C22×C3×D3C22×D312yes—
150C2×C32×C4#6714yes—
151C2×C32×C4C22×D311yescharacteristic
152C2×C32×C4C22×D311yescharacteristic
153C2×C3⋊Dic3—32no—
154C2×C3⋊Dic3—32no—
155C2×C3⋊Dic3—64no—
156C3×#67—24no—
157C3×#67—64no—
158C2×C3×Dic3—32no—
159C2×C3×Dic3C22×D312yes—
160C3×D12—22no—
161C3×D12—62no—
162C3×C4×D3—24no—
163C3×C4×D3—64no—
164C3×Dic6—22no—
165C3×Dic6—62no—
166#625—68no—
167C23×C9—21no—
168C23×C9—32no—
169C23×C9C22×D311yescharacteristic
170C22×D9C22×D312yes—
171Q8×C9—62no—
172D4×C9—64no—
173D4×C9—62no—
174C2×C4×C9—24no—
175C2×C4×C9—21no—
176C2×C4×C9—21no—
177C2×C4×C9—32no—
178C2×C4×C9—64no—
179C2×C4×C9#6714yescomplemented
180C2×C4×C9C22×D311yescharacteristic
181C2×C4×C9C22×D311yescharacteristic
182#611—24no—
183#611—68no—
184C2×Dic9—32no—
185C2×Dic9—32no—
186C2×Dic9—62no—
187C2×Dic9—62no—
188C2×Dic9—64no—
189C2×Dic9—68no—
190C2×Dic9C22×D312yes—
191D36—22no—
192C4×D9—24no—
193Dic18—22no—
194#532—271noSylow
195C2×C3×C9—22no—
196C2×C3×C9C22×D411yescharacteristic
197C2×C3×C9C2×#4112yes—
198D3×C9—42no—
199C3×D9—42no—
200C3×#41—64no—
201C3×#41—184no—
202C2×C3×Q8—31no—
203C2×C3×Q8—91no—
204C2×C3×D4—32no—
205C2×C3×D4—31no—
206C2×C3×D4—92no—
207C2×C3×D4—91no—
208C22×C3×C4—21no—
209C22×C3×C4—32no—
210C22×C3×C4—92no—
211C22×C3×C4C2×D911yescharacteristic
212C22×C3×C4D3211yescharacteristic
213C2×#67—32no—
214C2×#67—94no—
215C2×#67—94no—
216C2×#67C2×D912yes—
217C22×Dic3—92no—
218C22×Dic3—92no—
219C22×Dic3—182no—
220#235—24no—
221#235—64no—
222C2×D12—31no—
223C2×D12C2×D911yescharacteristic
224C2×C4×D3—32no—
225C2×C4×D3C2×D912yes—
226C2×Dic6—31no—
227C2×Dic6C2×D911yescharacteristic
228#217—94no—
229#217—94no—
230(C2×D3)⋊C4—94no—
231(C2×D3)⋊C4—94no—
232Dic3⋊C4—94no—
233Dic3⋊C4—94no—
234C4×Dic3—92no—
235C4×Dic3—91no—
236C4×Dic3—91no—
237C4×Dic3—92no—
238C4×Dic3—91no—
239C4×Dic3—91no—
240C4×Dic3—181no—
241C4×Dic3—181no—
242C4×Dic3—182no—
243C22×C32—22no—
244C22×C32C2×#6712yes—
245C22×C32C23×D311yescharacteristic
246C2×C3×D3—24no—
247C2×C3×D3—22no—
248C2×C3×D3—64no—
249C2×C3×D3—62no—
250C32×C4C2×#6712yes—
251C32×C4C2×#6712yes—
252C3⋊Dic3—34no—
253C3⋊Dic3—62no—
254C3×Dic3—22no—
255C3×Dic3—62no—
256C22×C9—24no—
257C22×C9—22no—
258C22×C9—21no—
259C22×C9—22no—
260C22×C9—64no—
261C22×C9—62no—
262C22×C9C2×#6712yes—
263C22×C9C23×D311yescharacteristic
264C2×D9—24no—
265C2×D9—22no—
266C4×C9—22no—
267C4×C9—22no—
268C4×C9—62no—
269C4×C9C2×#6712yes—
270C4×C9C2×#6712yes—
271Dic9—22no—
272Dic9—34no—
273Dic9—64no—
274Dic9—64no—
275Dic9—62no—
276C2×#41—31no—
277C2×#41—91no—
278C4⋊Q8—271no—
279#127—271no—
280#124—271no—
281#124—271no—
282#121—274no—
283C4×D4—274no—
284C2×C42—271no—
285C3×C9#53211yescharacteristic, complemented, Sylow
286C23×C3—21no—
287C23×C3—32no—
288C23×C3—92no—
289C23×C3C22×D911yescharacteristic
290C23×C3C2×D3211yescharacteristic
291C22×D3—32no—
292C22×D3C22×D912yes—
293C3×Q8—62no—
294C3×Q8—182no—
295C3×D4—64no—
296C3×D4—62no—
297C3×D4—184no—
298C3×D4—182no—
299C2×C3×C4—24no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—32no—
303C2×C3×C4—64no—
304C2×C3×C4—92no—
305C2×C3×C4—184no—
306C2×C3×C4#61114yescomplemented
307C2×C3×C4C22×D911yescharacteristic
308C2×C3×C4C22×D911yescharacteristic
309C2×C3×C4#62514yes—
310C2×C3×C4C2×D3211yescharacteristic
311C2×C3×C4C2×D3211yescharacteristic
312#67—24no—
313#67—64no—
314#67—188no—
315#67—188no—
316C2×Dic3—32no—
317C2×Dic3—92no—
318C2×Dic3—92no—
319C2×Dic3—92no—
320C2×Dic3—92no—
321C2×Dic3—182no—
322C2×Dic3—182no—
323C2×Dic3—184no—
324C2×Dic3—184no—
325C2×Dic3—188no—
326C2×Dic3C22×D912yes—
327D12—22no—
328D12—62no—
329C4×D3—24no—
330C4×D3—64no—
331Dic6—22no—
332Dic6—62no—
333C2×C32—22no—
334C2×C32D3×#4112yes—
335C2×C32C22×#6711yescharacteristic
336C3×D3—42no—
337C3×D3—122no—
338C2×C9—24no—
339C2×C9—22no—
340C2×C9—22no—
341C2×C9—21no—
342C2×C9—122no—
343C2×C9D3×#4112yes—
344C2×C9C22×#6711yescharacteristic
345D9—42no—
346#41—64no—
347#41—184no—
348C2×Q8—31no—
349C2×Q8—91no—
350C2×D4—32no—
351C2×D4—31no—
352C2×D4—92no—
353C2×D4—91no—
354C2×D4—274no—
355C22×C4—32no—
356C22×C4—92no—
357C22×C4—272no—
358C22×C4D3×D911yescharacteristic
359C4⋊C4—274no—
360C22⋊C4—274no—
361C22⋊C4—274no—
362C42—271no—
363C42—271no—
364C42—272no—
365C22×C3—24no—
366C22×C3—22no—
367C22×C3—22no—
368C22×C3—21no—
369C22×C3—22no—
370C22×C3—64no—
371C22×C3—62no—
372C22×C3—184no—
373C22×C3—182no—
374C22×C3C2×#61112yes—
375C22×C3C23×D911yescharacteristic
376C22×C3C2×#62512yes—
377C22×C3C22×D3211yescharacteristic
378C2×D3—24no—
379C2×D3—22no—
380C2×D3—64no—
381C2×D3—62no—
382C3×C4—22no—
383C3×C4—22no—
384C3×C4—62no—
385C3×C4—182no—
386C3×C4C2×#61112yes—
387C3×C4C2×#61112yes—
388C3×C4C2×#62512yes—
389C3×C4C2×#62512yes—
390Dic3—22no—
391Dic3—62no—
392Dic3—94no—
393Dic3—94no—
394Dic3—184no—
395Dic3—184no—
396Dic3—182no—
397Dic3—182no—
398C32#583211yescharacteristic
399C9—21no—
400C9#583211yescharacteristic, complemented
401C23—32no—
402C23—92no—
403C23C2×D3×D911yescharacteristic
404Q8—62no—
405Q8—182no—
406D4—64no—
407D4—62no—
408D4—184no—
409D4—182no—
410D4—548no—
411C2×C4—32no—
412C2×C4—64no—
413C2×C4—92no—
414C2×C4—184no—
415C2×C4—272no—
416C2×C4—272no—
417C2×C4—544no—
418C2×C4#619414yescomplemented
419C2×C4C2×D3×D911yescharacteristic
420C2×C4C2×D3×D911yescharacteristic
421C2×C3—24no—
422C2×C3—22no—
423C2×C3—22no—
424C2×C3—21no—
425C2×C3—22no—
426C2×C3—122no—
427C2×C3—362no—
428C2×C3#41×D912yes—
429C2×C3C22×#61111yescharacteristic
430C2×C3#6430612yes—
431C2×C3C22×#62511yescharacteristic
432D3—42no—
433D3—122no—
434C22—22no—
435C22—62no—
436C22—64no—
437C22—182no—
438C22—184no—
439C22C2×#619412yes—
440C22C22×D3×D911yescharacteristic
441C4—62no—
442C4—182no—
443C4—274no—
444C4—542no—
445C4C2×#619412yes—
446C4C2×#619412yes—
447C3—21no—
448C3#1058993411yescharacteristic, complemented
449C3#1059503211yescharacteristic
450C2—22no—
451C2—122no—
452C2—362no—
453C2#1172247512yes—
454C2C22×#619411yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#532
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C9×#235C2semidirect
2. C2×C3×#3697C2semidirect
3. C2×C4×C3⋊Dic9C2semidirect
4. #11721290C2semidirect
5. #11721043C2semidirect
6. C4×#6194C2semidirect
7. #11720967C2semidirect
8. #11720951C2semidirect
9. #11720930C2semidirect
41. C2×#3697D3semidirect
53. C2×C3×C4×C9D4semidirect
125. C3⋊Dic9#41semidirect
131. C2×#235D9semidirect
179. C2×C4×C9#67semidirect
285. C3×C9#532semidirect
306. C2×C3×C4#611semidirect
400. C9#5832semidirect
418. C2×C4#6194semidirect
448. C3#10589934semidirect

Maximal subgroups

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

subgroupindex
C2×C9×#2352
C2×C3×#36972
C2×C4×C3⋊Dic92
#117212902
#117210432
C4×#61942
#117209672
#117209512
#117209302
#105950323
#105899343