Finite Group Databasebeta

← #13150335

Subgroups — #13150335

Order 1728.

1094 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88736C211yesmaximal, characteristic, complemented
2C2×C3×#3697C211yesmaximal, characteristic, complemented
3C2×C4×C9×Dic3C211yesmaximal, characteristic, complemented
4#11721277C214yesmaximal, complemented
5#11721006C211yesmaximal, characteristic, complemented
6C4×#6192C214yesmaximal, complemented
7#11720952C211yesmaximal, characteristic, complemented
8#11720948C211yesmaximal, characteristic, complemented
9#11720932C211yesmaximal, characteristic, complemented
10#10595029—31nomaximal
11#10589579—31nomaximal
12C22×C3×C4×C9C2211yescharacteristic
13C2×#6233C2212yes—
14#88736—24no—
15C2×C4⋊(C3⋊D9)C2211yescharacteristic
16C2×C4×C3⋊D9C2212yes—
17C2×#6226C2211yescharacteristic
18C2×C3×#611C2212yes—
19C22×C9×Dic3C2212yes—
20C3×#3697—24no—
21C2×C3×D36C2211yescharacteristic
22C2×C3×C4×D9C2212yes—
23C2×C3×Dic18C2211yescharacteristic
24C2×#6192C2214yes—
25C4×C9×Dic3C2212yes—
26C4×C9×Dic3C2211yescharacteristic
27C4×C9×Dic3C2211yescharacteristic
28#88445C2214yes—
29(C2×D9)⋊Dic3C2214yes—
30Dic3⋊Dic9C2214yes—
31C2×#3828—31no—
32C2×C3×#235—31no—
33C2×C3×C4×Dic3D311yescharacteristic
34#63978—34no—
35#63911—31no—
36C4×#626—34no—
37#63866—31no—
38#63864—31no—
39#63860—31no—
40C2×#3697—31no—
41C2×#3697—61no—
42C2×#3697D311yescharacteristic, complemented
43C2×C42×C9—31no—
44#63492—34no—
45#63437—31no—
46#63437—31no—
47#63436—31no—
48C4×D36—34no—
49#63431—31no—
50C23×C3×C9C2311yescharacteristic
51C22×C3⋊D9C2312yes—
52C22×C3×D9C2312yes—
53C2×C3×C4×C9D414yescomplemented
54C2×C3×C4×C9C2311yescharacteristic
55C2×C3×C4×C9C2311yescharacteristic
56#6233—24no—
57C2×C3⋊Dic9C2312yes—
58C4⋊(C3⋊D9)—22no—
59C4×C3⋊D9—24no—
60#6226—22no—
61C3×#611—24no—
62C2×C9×Dic3—24no—
63C2×C9×Dic3C2312yes—
64C2×C9×Dic3C2312yes—
65C2×C3×Dic9C2312yes—
66C3×D36—22no—
67C3×C4×D9—24no—
68C3×Dic18—22no—
69#6192—28no—
70#5832—91no—
71#5477—91no—
72C22×C32×C4C2×D311yescharacteristic
73C2×#638—32no—
74#3828—64no—
75C2×C4⋊(C3⋊D3)—31no—
76C2×C4×C3⋊D3—32no—
77C2×#634—31no—
78C2×C3×#67—32no—
79C22×C3×Dic3C2×D312yes—
80C3×#235—64no—
81C2×C3×D12—31no—
82C2×C3×C4×D3—32no—
83C2×C3×Dic6—31no—
84C2×#626—34no—
85C3×C4×Dic3C2×D311yescharacteristic
86C3×C4×Dic3C2×D311yescharacteristic
87C3×C4×Dic3C2×D312yes—
88Dic3⋊Dic3—34no—
89#3722—34no—
90(C2×D3)⋊Dic3—34no—
91C22×C4×C9—21no—
92C22×C4×C9—32no—
93C22×C4×C9C2×D311yescharacteristic
94C2×#611—32no—
95C2×#611—62no—
96C2×#611C2×D312yes—
97#3697—24no—
98#3697—68no—
99#3697—64no—
100C2×D36—34no—
101C2×D36—31no—
102C2×D36—61no—
103C2×D36C2×D311yescharacteristic
104C2×C4×D9—32no—
105C2×C4×D9—62no—
106C2×C4×D9C2×D312yes—
107C2×Dic18—31no—
108C2×Dic18—61no—
109C2×Dic18C2×D311yescharacteristic
110C42×C9—31no—
111C42×C9—32no—
112C42×C9—31no—
113(C2×D9)⋊C4—34no—
114(C2×D9)⋊C4—34no—
115C4⋊Dic9—34no—
116C22×C3×C9—22no—
117C22×C3×C9C2×D412yes—
118C22×C3×C9C2411yescharacteristic
119C2×C3⋊D9—22no—
120C2×C3⋊D9—24no—
121C2×C3×D9—22no—
122C2×C3×D9—24no—
123C3×C4×C9C2×D412yes—
124C3×C4×C9C2×D412yes—
125C3⋊Dic9—22no—
126C9×Dic3—22no—
127C9×Dic3#4114yescomplemented
128C3×Dic9—22no—
129C2×C3×#41—91no—
130C2×#235—31no—
131C2×#235—91no—
132C2×#235—91no—
133C2×#235—181no—
134C2×C3×C42—31no—
135#947—91no—
136#944—91no—
137#940—91no—
138#939—94no—
139#936—91no—
140#930—94no—
141C4×#67—94no—
142C2×C4×Dic3D911yescharacteristic, complemented
143#875—91no—
144#875—91no—
145#874—91no—
146C4×D12—94no—
147#869—91no—
148C23×C32C22×D311yescharacteristic
149C22×C3⋊D3—32no—
150C22×C3×D3—32no—
151C2×C32×C4D1214yes—
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C22×D311yescharacteristic
154#638—64no—
155C2×C3⋊Dic3—32no—
156C4⋊(C3⋊D3)—62no—
157C4×C3⋊D3—64no—
158#634—62no—
159C3×#67—64no—
160C2×C3×Dic3—24no—
161C2×C3×Dic3—32no—
162C2×C3×Dic3C22×D312yes—
163C2×C3×Dic3C22×D312yes—
164C3×D12—62no—
165C3×C4×D3—64no—
166C3×Dic6—62no—
167#626—68no—
168C23×C9—21no—
169C23×C9C22×D311yescharacteristic
170C22×D9—32no—
171C22×D9—62no—
172C22×D9C22×D312yes—
173C2×C4×C9—24no—
174C2×C4×C9—21no—
175C2×C4×C9—21no—
176C2×C4×C9—32no—
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—
184#611—64no—
185C2×Dic9—32no—
186C2×Dic9—62no—
187C2×Dic9C22×D312yes—
188D36—22no—
189D36—64no—
190D36—68no—
191D36—62no—
192C4×D9—24no—
193C4×D9—68no—
194C4×D9—64no—
195Dic18—22no—
196Dic18—64no—
197Dic18—62no—
198#532—271noSylow
199C2×C3×C9—22no—
200C2×C3×C9C22×D411yescharacteristic
201C2×C3×C9C2×#4112yes—
202C3⋊D9—42no—
203C3×D9—42no—
204C3×#41—184no—
205C2×C3×Q8—91no—
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—92no—
215C2×#67—92no—
216C2×#67—94no—
217C2×#67—182no—
218C22×Dic3C2×D912yes—
219#235—64no—
220#235—188no—
221#235—184no—
222#235—184no—
223C2×D12—31no—
224C2×D12—94no—
225C2×D12—91no—
226C2×D12—91no—
227C2×D12—181no—
228C2×C4×D3—32no—
229C2×C4×D3—92no—
230C2×C4×D3—92no—
231C2×C4×D3—182no—
232C2×Dic6—31no—
233C2×Dic6—91no—
234C2×Dic6—91no—
235C2×Dic6—181no—
236C3×C42—31no—
237C3×C42—32no—
238C3×C42—31no—
239#217—94no—
240(C2×D3)⋊C4—94no—
241(C2×D3)⋊C4—94no—
242(C2×D3)⋊C4—94no—
243C4⋊Dic3—94no—
244Dic3⋊C4—94no—
245C4×Dic3C2×D912yes—
246C4×Dic3C2×D911yescharacteristic
247C4×Dic3C2×D911yescharacteristic
248C22×C32—22no—
249C22×C32C2×D1212yes—
250C22×C32C23×D311yescharacteristic
251C2×C3⋊D3—62no—
252C2×C3⋊D3—64no—
253C2×C3×D3—62no—
254C2×C3×D3—64no—
255C32×C4C2×D1212yes—
256C32×C4C2×D1212yes—
257C3⋊Dic3—62no—
258C3×Dic3—22no—
259C3×Dic3—62no—
260C3×Dic3#23514yes—
261C22×C9—22no—
262C22×C9—21no—
263C22×C9—22no—
264C22×C9—42no—
265C22×C9C2×#6712yes—
266C22×C9C23×D311yescharacteristic
267C2×D9—24no—
268C2×D9—22no—
269C2×D9—64no—
270C2×D9—64no—
271C2×D9—62no—
272C2×D9—124no—
273C4×C9—22no—
274C4×C9—22no—
275C4×C9—34no—
276C4×C9—62no—
277C4×C9C2×#6712yes—
278C4×C9C2×#6712yes—
279Dic9—22no—
280Dic9—64no—
281Dic9—62no—
282C2×#41—91no—
283C2×#41—271no—
284C4⋊Q8—271no—
285#127—271no—
286#124—271no—
287#124—271no—
288#121—274no—
289C4×D4—274no—
290C2×C42—31no—
291C3×C9#53211yescharacteristic, complemented, Sylow
292C23×C3—21no—
293C23×C3—92no—
294C23×C3C22×D911yescharacteristic
295C23×C3C2×D3211yescharacteristic
296C22×D3—32no—
297C22×D3—92no—
298C22×D3—92no—
299C22×D3—182no—
300C3×Q8—182no—
301C3×D4—184no—
302C3×D4—182no—
303C2×C3×C4—24no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—32no—
307C2×C3×C4—32no—
308C2×C3×C4—64no—
309C2×C3×C4—92no—
310C2×C3×C4—184no—
311C2×C3×C4D3614yescomplemented
312C2×C3×C4C22×D911yescharacteristic
313C2×C3×C4C22×D911yescharacteristic
314C2×C3×C4#62614yes—
315C2×C3×C4C2×D3211yescharacteristic
316C2×C3×C4C2×D3211yescharacteristic
317#67—64no—
318#67—188no—
319#67—184no—
320#67—184no—
321#67—188no—
322C2×Dic3—24no—
323C2×Dic3—32no—
324C2×Dic3—92no—
325C2×Dic3—92no—
326C2×Dic3—182no—
327C2×Dic3C22×D912yes—
328C2×Dic3C22×D912yes—
329D12—62no—
330D12—184no—
331D12—188no—
332D12—182no—
333D12—182no—
334C4×D3—64no—
335C4×D3—188no—
336C4×D3—184no—
337C4×D3—184no—
338Dic6—62no—
339Dic6—184no—
340Dic6—182no—
341Dic6—182no—
342C2×C32—22no—
343C2×C32C22×D1211yescharacteristic
344C2×C32C2×#23512yes—
345C3⋊D3—122no—
346C3×D3—122no—
347C2×C9—22no—
348C2×C9—22no—
349C2×C9—21no—
350C2×C9—42no—
351C2×C9D3×#4112yes—
352C2×C9C22×#6711yescharacteristic
353D9—42no—
354D9—124no—
355D9—122no—
356#41—184no—
357#41—544no—
358C2×Q8—91no—
359C2×Q8—271no—
360C2×D4—92no—
361C2×D4—91no—
362C2×D4—274no—
363C2×D4—272no—
364C2×D4—271no—
365C22×C4—32no—
366C22×C4—92no—
367C22×C4—272no—
368C22×C4D3×D911yescharacteristic
369C4⋊C4—274no—
370C22⋊C4—274no—
371C22⋊C4—274no—
372C42—31no—
373C42—31no—
374C42—32no—
375C22×C3—22no—
376C22×C3—22no—
377C22×C3—21no—
378C22×C3—22no—
379C22×C3—42no—
380C22×C3—182no—
381C22×C3—184no—
382C22×C3C2×D3612yes—
383C22×C3C23×D911yescharacteristic
384C22×C3C2×#62612yes—
385C22×C3C22×D3211yescharacteristic
386C2×D3—64no—
387C2×D3—62no—
388C2×D3—184no—
389C2×D3—184no—
390C2×D3—182no—
391C2×D3—182no—
392C2×D3—184no—
393C2×D3—364no—
394C3×C4—22no—
395C3×C4—22no—
396C3×C4—34no—
397C3×C4—62no—
398C3×C4—182no—
399C3×C4C2×D3612yes—
400C3×C4C2×D3612yes—
401C3×C4C2×#62612yes—
402C3×C4C2×#62612yes—
403Dic3—22no—
404Dic3—62no—
405Dic3—184no—
406Dic3—182no—
407Dic3—182no—
408Dic3#369714yescomplemented
409C32#547711yescharacteristic
410C9—21no—
411C9#583211yescharacteristic, complemented
412C23—92no—
413C23—272no—
414C23C2×D3×D911yescharacteristic
415Q8—182no—
416Q8—542no—
417D4—184no—
418D4—182no—
419D4—548no—
420D4—544no—
421D4—542no—
422C2×C4—32no—
423C2×C4—32no—
424C2×C4—64no—
425C2×C4—92no—
426C2×C4—184no—
427C2×C4—272no—
428C2×C4—544no—
429C2×C4#619214yescomplemented
430C2×C4C2×D3×D911yescharacteristic
431C2×C4C2×D3×D911yescharacteristic
432C2×C3—22no—
433C2×C3—22no—
434C2×C3—21no—
435C2×C3—22no—
436C2×C3—42no—
437C2×C3—362no—
438C2×C3C22×D3611yescharacteristic
439C2×C3C2×#369712yes—
440C2×C3D3×#23512yes—
441C2×C3C22×#62611yescharacteristic
442D3—122no—
443D3—364no—
444D3—362no—
445D3—362no—
446C22—22no—
447C22—182no—
448C22—184no—
449C22—542no—
450C22—544no—
451C22C2×#619212yes—
452C22C22×D3×D911yescharacteristic
453C4—34no—
454C4—62no—
455C4—182no—
456C4—542no—
457C4C2×#619212yes—
458C4C2×#619212yes—
459C3—21no—
460C3#1058957911yescharacteristic, complemented
461C3#1059502911yescharacteristic
462C2—22no—
463C2—362no—
464C2—1082no—
465C2D3×#369712yes—
466C2C22×#619211yescharacteristic
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×#88736C2semidirect
2. C2×C3×#3697C2semidirect
3. C2×C4×C9×Dic3C2semidirect
4. #11721277C2semidirect
5. #11721006C2semidirect
6. C4×#6192C2semidirect
7. #11720952C2semidirect
8. #11720948C2semidirect
9. #11720932C2semidirect
42. C2×#3697D3semidirect
53. C2×C3×C4×C9D4semidirect
127. C9×Dic3#41semidirect
142. C2×C4×Dic3D9semidirect
179. C2×C4×C9#67semidirect
291. C3×C9#532semidirect
311. C2×C3×C4D36semidirect
408. Dic3#3697semidirect
411. C9#5832semidirect
429. C2×C4#6192semidirect
460. C3#10589579semidirect

Maximal subgroups

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

subgroupindex
C2×#887362
C2×C3×#36972
C2×C4×C9×Dic32
#117212772
#117210062
C4×#61922
#117209522
#117209482
#117209322
#105950293
#105895793