Finite Group Databasebeta

← #13147500

Subgroups — #13147500

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×#88639C211yesmaximal, characteristic, complemented
2C2×#88634C211yesmaximal, characteristic, complemented
3C42×C3⋊D9C211yesmaximal, characteristic, complemented
4C9×#875C211yesmaximal, characteristic, complemented
5C3×#63437C211yesmaximal, characteristic, complemented
6#11721043C211yesmaximal, characteristic, complemented
7#11721041C214yesmaximal, complemented
8#11721028C214yesmaximal, complemented
9#11720930C211yesmaximal, characteristic, complemented
10#10593971—31nomaximal
11#10589770—31nomaximal
12C2×C4×C3⋊D9C2212yes—
13C2×C4×C3⋊D9C2211yescharacteristic
14C2×C9×D12C2211yescharacteristic
15C2×C3×D36C2211yescharacteristic
16C2×C9×Dic6C2211yescharacteristic
17C2×C3×Dic18C2211yescharacteristic
18C2×#6195C2212yes—
19C2×#6194C2214yes—
20C2×D3×Dic9C2212yes—
21C2×#6192C2212yes—
22C2×Dic3×D9C2212yes—
23#88639—24no—
24#88634—24no—
25C3×C42×C9C2211yescharacteristic
26C4×C3⋊Dic9C2212yes—
27C4×C3⋊Dic9C2211yescharacteristic
28C9×(C2×D3)⋊C4C2214yes—
29C3×(C2×D9)⋊C4C2214yes—
30#88442C2214yes—
31C2×#3796—31no—
32C2×#3796—31no—
33C42×C3⋊D3—31no—
34C3×#875—31no—
35C3×#875D311yescharacteristic
36#63914—31no—
37#63912—34no—
38#63902—34no—
39#63873—31no—
40C2×#3701—31no—
41C2×#3699—31no—
42C9×#124—31no—
43#63506—31no—
44#63502—31no—
45#63446—34no—
46#611⋊C4—34no—
47#63437D311yescharacteristic, complemented
48C42×D9—31no—
49C42×D9—61no—
50C22×C3⋊D9C2311yescharacteristic
51C22×D3×C9C2312yes—
52C22×C3×D9C2312yes—
53C2×C3×C4×C9C2312yes—
54C2×C3×C4×C9C2311yescharacteristic
55C2×C3⋊Dic9C2311yescharacteristic
56C2×C3⋊Dic9C2312yes—
57C4×C3⋊D9—24no—
58C4×C3⋊D9D414yescomplemented
59C2×C9×Dic3C2312yes—
60C2×C3×Dic9C2312yes—
61C9×D12—22no—
62C3×D36—22no—
63C9×Dic6—22no—
64C3×Dic18—22no—
65#6195—24no—
66#6194—28no—
67D3×Dic9—24no—
68#6192—24no—
69Dic3×D9—24no—
70#5668—91no—
71#5668—91no—
72C2×C4×C3⋊D3—31no—
73C2×C4×C3⋊D3—32no—
74C2×C3×D12—31no—
75C2×C3×D12C2×D311yescharacteristic
76C2×C3×Dic6—31no—
77C2×C3×Dic6C2×D311yescharacteristic
78C2×#626—32no—
79C2×#626—32no—
80C2×#625—34no—
81C2×D3×Dic3—32no—
82C2×D3×Dic3—32no—
83#3796—64no—
84#3796—64no—
85C32×C42C2×D311yescharacteristic
86C4×C3⋊Dic3—31no—
87C4×C3⋊Dic3—32no—
88C3×(C2×D3)⋊C4—34no—
89C3×(C2×D3)⋊C4C2×D314yes—
90#3724—34no—
91C2×Q8×C9—31no—
92C2×D4×C9—31no—
93C2×#611—34no—
94C2×#611—32no—
95C22×Dic9—32no—
96#3701—64no—
97#3699—64no—
98C2×D36—32no—
99C2×D36C2×D311yescharacteristic
100C2×C4×D9—31no—
101C2×C4×D9—32no—
102C2×C4×D9—32no—
103C2×C4×D9—61no—
104C2×C4×D9—62no—
105C2×Dic18C2×D311yescharacteristic
106C9×C22⋊C4—34no—
107C42×C9—21no—
108C42×C9C2×D311yescharacteristic
109(C2×D9)⋊C4C2×D314yes—
110Dic9⋊C4—34no—
111C4×Dic9—31no—
112C4×Dic9—32no—
113C4×Dic9—62no—
114C4×Dic9—61no—
115C22×C3×C9C2411yescharacteristic
116C2×C3⋊D9—22no—
117C2×C3⋊D9C2×D412yes—
118C2×D3×C9—22no—
119C2×D3×C9—24no—
120C2×C3×D9—22no—
121C2×C3×D9—24no—
122C3×C4×C9—22no—
123C3×C4×C9C2×D412yes—
124C3⋊Dic9C2×D412yes—
125C3⋊Dic9#4114yescomplemented
126C9×Dic3—22no—
127C3×Dic9—22no—
128C2×#239—91no—
129C2×#239—91no—
130C2×#237—91no—
131C2×#237—91no—
132C3×#124—31no—
133C3×#124—91no—
134#944—91no—
135#944—91no—
136#940—91no—
137#940—91no—
138#884—94no—
139#884—94no—
140#67⋊C4—94no—
141#67⋊C4—94no—
142#875—31no—
143#875D911yescharacteristic, complemented
144C42×D3—91no—
145C42×D3—91no—
146C42×D3—181no—
147C22×C3⋊D3—31no—
148C22×C3×D3—32no—
149C22×C3×D3C22×D312yes—
150C2×C32×C4C22×D312yes—
151C2×C32×C4C22×D311yescharacteristic
152C2×C3⋊Dic3—32no—
153C2×C3⋊Dic3—31no—
154C4×C3⋊D3—34no—
155C4×C3⋊D3—64no—
156C2×C3×Dic3—32no—
157C2×C3×Dic3C22×D312yes—
158C3×D12—22no—
159C3×D12—62no—
160C3×Dic6—22no—
161C3×Dic6—62no—
162#626—64no—
163#626—64no—
164#625—68no—
165D3×Dic3—64no—
166D3×Dic3—64no—
167C23×C9—32no—
168C22×D9—31no—
169C22×D9—61no—
170C22×D9C22×D312yes—
171Q8×C9—62no—
172D4×C9—62no—
173C2×C4×C9—22no—
174C2×C4×C9—21no—
175C2×C4×C9—32no—
176C2×C4×C9C22×D312yes—
177C2×C4×C9C22×D311yescharacteristic
178#611—64no—
179#611—68no—
180C2×Dic9—32no—
181C2×Dic9—31no—
182C2×Dic9—62no—
183C2×Dic9—61no—
184C2×Dic9—64no—
185C2×Dic9C22×D312yes—
186D36—22no—
187D36—64no—
188C4×D9—34no—
189C4×D9—64no—
190C4×D9—64no—
191C4×D9—64no—
192C4×D9—68no—
193Dic18—22no—
194#532—271noSylow
195C2×C3×C9C22×D411yescharacteristic
196C2×C3×C9C2×#4112yes—
197C3⋊D9—22no—
198D3×C9—42no—
199C3×D9—42no—
200C2×C3×Q8—31no—
201C2×C3×Q8—91no—
202C2×C3×D4—31no—
203C2×C3×D4—91no—
204C2×#67—92no—
205C2×#67—94no—
206C2×#67—94no—
207C2×#67—92no—
208C22×Dic3—92no—
209C22×Dic3—92no—
210#239—184no—
211#239—184no—
212#237—184no—
213#237—184no—
214C2×D12—31no—
215C2×D12—92no—
216C2×D12—92no—
217C2×D12C2×D911yescharacteristic
218C2×C4×D3—91no—
219C2×C4×D3—91no—
220C2×C4×D3—92no—
221C2×C4×D3—92no—
222C2×C4×D3—92no—
223C2×C4×D3—92no—
224C2×C4×D3—181no—
225C2×C4×D3—182no—
226C2×Dic6—31no—
227C2×Dic6C2×D911yescharacteristic
228C3×C22⋊C4—34no—
229C3×C22⋊C4—94no—
230C3×C42—21no—
231C3×C42C2×D911yescharacteristic
232C3×C42D3211yescharacteristic
233(C2×D3)⋊C4—34no—
234(C2×D3)⋊C4C2×D914yes—
235Dic3⋊C4—94no—
236Dic3⋊C4—94no—
237C4×Dic3—92no—
238C4×Dic3—91no—
239C4×Dic3—92no—
240C4×Dic3—91no—
241C4×Dic3—182no—
242C4×Dic3—181no—
243C22×C32C23×D311yescharacteristic
244C2×C3⋊D3—32no—
245C2×C3⋊D3—62no—
246C2×C3×D3—22no—
247C2×C3×D3—24no—
248C2×C3×D3—62no—
249C2×C3×D3—64no—
250C32×C4—22no—
251C32×C4D3×D412yes—
252C3⋊Dic3—32no—
253C3⋊Dic3—34no—
254C3×Dic3—22no—
255C3×Dic3—62no—
256C22×C9—21no—
257C22×C9—62no—
258C22×C9—64no—
259C22×C9C23×D311yescharacteristic
260C2×D9—22no—
261C2×D9—24no—
262C2×D9—32no—
263C2×D9—62no—
264C2×D9—64no—
265C2×D9—62no—
266C4×C9—24no—
267C4×C9—22no—
268C4×C9—22no—
269C4×C9—62no—
270C4×C9D3×D412yes—
271Dic9—22no—
272Dic9—32no—
273Dic9—34no—
274Dic9—62no—
275Dic9—64no—
276C2×#41—271no—
277C2×#41—271no—
278C4⋊Q8—271no—
279#127—271no—
280#124—31no—
281#124—91no—
282#121—274no—
283C4×D4—274no—
284C2×C42—271no—
285C3×C9#53211yescharacteristic, complemented, Sylow
286C23×C3—32no—
287C23×C3—92no—
288C22×D3—32no—
289C22×D3—91no—
290C22×D3—91no—
291C22×D3—181no—
292C22×D3C22×D912yes—
293C3×Q8—62no—
294C3×Q8—182no—
295C3×D4—62no—
296C3×D4—182no—
297C2×C3×C4—22no—
298C2×C3×C4—21no—
299C2×C3×C4—32no—
300C2×C3×C4—92no—
301C2×C3×C4C22×D912yes—
302C2×C3×C4C22×D911yescharacteristic
303C2×C3×C4C2×D3212yes—
304C2×C3×C4C2×D3211yescharacteristic
305#67—184no—
306#67—184no—
307#67—188no—
308#67—188no—
309C2×Dic3—32no—
310C2×Dic3—92no—
311C2×Dic3—91no—
312C2×Dic3—92no—
313C2×Dic3—91no—
314C2×Dic3—182no—
315C2×Dic3—181no—
316C2×Dic3—184no—
317C2×Dic3—184no—
318C2×Dic3C22×D912yes—
319D12—22no—
320D12—62no—
321D12—184no—
322D12—184no—
323C4×D3—94no—
324C4×D3—94no—
325C4×D3—184no—
326C4×D3—184no—
327C4×D3—184no—
328C4×D3—184no—
329C4×D3—184no—
330C4×D3—188no—
331Dic6—22no—
332Dic6—62no—
333C2×C32C2×D3×D411yescharacteristic
334C2×C32D3×#4112yes—
335C3⋊D3—62no—
336C3×D3—42no—
337C3×D3—122no—
338C2×C9—21no—
339C2×C9—22no—
340C2×C9—122no—
341C2×C9C2×D3×D411yescharacteristic
342C2×C9D3×#4112yes—
343D9—42no—
344D9—64no—
345D9—62no—
346#41—544no—
347#41—544no—
348C2×Q8—31no—
349C2×Q8—91no—
350C2×D4—31no—
351C2×D4—91no—
352C2×D4—272no—
353C2×D4—272no—
354C2×D4—274no—
355C22×C4—272no—
356C22×C4—272no—
357C22×C4—271no—
358C22×C4—272no—
359C4⋊C4—274no—
360C22⋊C4—34no—
361C22⋊C4—94no—
362C42—272no—
363C42—271no—
364C42D3×D911yescharacteristic
365C22×C3—21no—
366C22×C3—62no—
367C22×C3—64no—
368C22×C3—182no—
369C22×C3—184no—
370C22×C3C23×D911yescharacteristic
371C22×C3C22×D3211yescharacteristic
372C2×D3—22no—
373C2×D3—24no—
374C2×D3—62no—
375C2×D3—64no—
376C2×D3—92no—
377C2×D3—92no—
378C2×D3—182no—
379C2×D3—184no—
380C2×D3—182no—
381C2×D3—182no—
382C3×C4—24no—
383C3×C4—22no—
384C3×C4—22no—
385C3×C4—22no—
386C3×C4—62no—
387C3×C4—182no—
388C3×C4D4×D912yes—
389C3×C4#380212yes—
390Dic3—22no—
391Dic3—62no—
392Dic3—92no—
393Dic3—94no—
394Dic3—92no—
395Dic3—94no—
396Dic3—182no—
397Dic3—184no—
398C32#566811yescharacteristic
399C9—21no—
400C9#566811yescharacteristic, complemented
401C23—32no—
402C23—92no—
403C23—271no—
404Q8—62no—
405Q8—182no—
406D4—62no—
407D4—182no—
408D4—544no—
409D4—544no—
410D4—548no—
411C2×C4—32no—
412C2×C4—92no—
413C2×C4—272no—
414C2×C4—271no—
415C2×C4—274no—
416C2×C4—544no—
417C2×C4—544no—
418C2×C4—544no—
419C2×C4C2×D3×D912yes—
420C2×C4C2×D3×D911yescharacteristic
421C2×C3—21no—
422C2×C3—22no—
423C2×C3—122no—
424C2×C3—362no—
425C2×C3C2×D4×D911yescharacteristic
426C2×C3#41×D912yes—
427C2×C3C2×#380211yescharacteristic
428C2×C3#6430612yes—
429D3—42no—
430D3—122no—
431D3—184no—
432D3—182no—
433D3—182no—
434C22—62no—
435C22—64no—
436C22—182no—
437C22—184no—
438C22—272no—
439C22—542no—
440C22C22×D3×D911yescharacteristic
441C4—22no—
442C4—62no—
443C4—182no—
444C4—272no—
445C4—274no—
446C4#8864612yes—
447C3—21no—
448C3#1058977011yescharacteristic, complemented
449C3#1059397111yescharacteristic
450C2—122no—
451C2—362no—
452C2—542no—
453C2C2×#8864611yescharacteristic
454C2#1172247512yes—
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
hypercenterC42D3×D9
hypocenterC3×C9#532
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#88639C2semidirect
2. C2×#88634C2semidirect
3. C42×C3⋊D9C2semidirect
4. C9×#875C2semidirect
5. C3×#63437C2semidirect
6. #11721043C2semidirect
7. #11721041C2semidirect
8. #11721028C2semidirect
9. #11720930C2semidirect
47. #63437D3semidirect
58. C4×C3⋊D9D4semidirect
125. C3⋊Dic9#41semidirect
143. #875D9semidirect
285. C3×C9#532semidirect
400. C9#5668semidirect
448. C3#10589770semidirect

Maximal subgroups

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

subgroupindex
C2×#886392
C2×#886342
C42×C3⋊D92
C9×#8752
C3×#634372
#117210432
#117210412
#117210282
#117209302
#105939713
#105897703