Finite Group Databasebeta

← #13147727

Subgroups — #13147727

Order 1728.

1098 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88646C211yesmaximal, characteristic
2C2×(C3⋊D9)⋊Q8C211yesmaximal, characteristic, complemented
3C42×C3⋊D9C211yesmaximal, characteristic, complemented
4C9×#875C211yesmaximal, characteristic, complemented
5C3×#63437C211yesmaximal, characteristic, complemented
6#11721048C214yesmaximal, complemented
7#11721044C214yesmaximal, complemented
8#11720967C211yesmaximal, characteristic, complemented
9#11720951C211yesmaximal, characteristic, complemented
10#10594009—31nomaximal
11D9×#124—31nomaximal
12C22×D3×D9C2212yes—
13C2×C4×C3⋊D9C2212yes—
14C2×C4×C3⋊D9C2211yescharacteristic
15C2×C9×D12C2211yescharacteristic
16C2×C3×D36C2211yescharacteristic
17C2×C9×Dic6C2211yescharacteristic, complemented
18C2×C3×Dic18C2211yescharacteristic, complemented
19C2×#6194C2212yes—
20C2×(C3⋊D9)⋊C4C2212yes—
21C2×#6189C2212yescomplemented
22#88646—24no—
23(C3⋊D9)⋊Q8—24no—
24C3×C42×C9C2211yescharacteristic, complemented
25C4×C3⋊Dic9C2211yescharacteristic, complemented
26C4×C3⋊Dic9C2212yescomplemented
27C9×(C2×D3)⋊C4C2214yescomplemented
28C3×(C2×D9)⋊C4C2214yescomplemented
29(C2×D3)⋊Dic9C2214yescomplemented
30(C2×D9)⋊Dic3C2214yescomplemented
31C2×#3802—31no—
32C2×#3797—31no—
33C42×C3⋊D3—31no—
34C3×#875—31no—
35C3×#875D311yescharacteristic
36#63921—34no—
37#63915—34no—
38#63863—31no—
39#63863—31no—
40C2×Q8×D9—31no—
41C2×D4×D9—31no—
42C9×#124—31no—
43#63509—31no—
44#63498—31no—
45#63447—34no—
46C22⋊C4×D9—34no—
47#63437D311yescharacteristic, complemented
48C42×D9—31no—
49C42×D9—61no—
50C22×C3⋊D9C2311yescharacteristic
51C22×D3×C9C2312yes—
52C22×C3×D9C2312yes—
53C2×D3×D9—24no—
54C2×D3×D9—28no—
55C2×C3×C4×C9C2312yes—
56C2×C3×C4×C9C2311yescharacteristic
57C2×C3⋊Dic9C2312yes—
58C2×C3⋊Dic9C2311yescharacteristic
59C4×C3⋊D9—24no—
60C4×C3⋊D9D414yes—
61C2×C9×Dic3C2312yes—
62C2×C3×Dic9C2312yes—
63C9×D12—22no—
64C3×D36—22no—
65C9×Dic6—22no—
66C3×Dic18—22no—
67#6194—24no—
68(C3⋊D9)⋊C4—24no—
69#6189—24no—
70D3×#124—91no—
71D3×#124—91no—
72C22×D32—32no—
73C2×C4×C3⋊D3—31no—
74C2×C4×C3⋊D3—32no—
75C2×C3×D12—31no—
76C2×C3×D12C2×D311yescharacteristic
77C2×C3×Dic6—31no—
78C2×C3×Dic6C2×D311yescharacteristic
79C2×#627—32no—
80C2×#625—32no—
81C2×#624—32no—
82#3802—64no—
83#3797—64no—
84C23×D9—32no—
85C32×C42C2×D311yescharacteristic
86C4×C3⋊Dic3—31no—
87C4×C3⋊Dic3—32no—
88C3×(C2×D3)⋊C4—34no—
89C3×(C2×D3)⋊C4C2×D314yes—
90(C2×D3)⋊Dic3—34no—
91(C2×D3)⋊Dic3—34no—
92C2×Q8×C9—31no—
93C2×D4×C9—31no—
94C2×#611—32no—
95Q8×D9—64no—
96D4×D9—64no—
97C2×D36C2×D311yescharacteristic
98C2×C4×D9—31no—
99C2×C4×D9—32no—
100C2×C4×D9—32no—
101C2×C4×D9—61no—
102C2×C4×D9—62no—
103C2×Dic18—32no—
104C2×Dic18C2×D311yescharacteristic, complemented
105C9×C22⋊C4—34no—
106C42×C9—21no—
107C42×C9C2×D311yescharacteristic, complemented
108#3676—34no—
109(C2×D9)⋊C4—34no—
110(C2×D9)⋊C4C2×D314yescomplemented
111C4×Dic9—31no—
112C4×Dic9—32no—
113C4×Dic9—62no—
114C4×Dic9—61no—
115C22×C3×C9C2411yescharacteristic
116C2×C3⋊D9C2×D412yes—
117C2×C3⋊D9#4114yes—
118C2×D3×C9—22no—
119C2×D3×C9—24no—
120C2×C3×D9—22no—
121C2×C3×D9—24no—
122D3×D9—48no—
123C3×C4×C9—22no—
124C3×C4×C9C2×D412yes—
125C3⋊Dic9—22no—
126C3⋊Dic9C2×D412yes—
127C9×Dic3—22no—
128C3×Dic9—22no—
129C2×D3×Q8—91no—
130C2×D3×Q8—91no—
131C2×D3×D4—91no—
132C2×D3×D4—91no—
133C3×#124—31no—
134C3×#124—91no—
135#947—91no—
136#947—91no—
137#936—91no—
138#936—91no—
139#885—94no—
140#885—94no—
141D3×C22⋊C4—94no—
142D3×C22⋊C4—94no—
143#875—31no—
144#875D911yescharacteristic, complemented
145C42×D3—91no—
146C42×D3—91no—
147C42×D3—181no—
148C22×C3⋊D3—31no—
149C22×C3×D3—32no—
150C22×C3×D3C22×D312yes—
151C2×D32—68no—
152C2×D32—64no—
153C2×C32×C4C22×D312yes—
154C2×C32×C4C22×D311yescharacteristic
155C2×C3⋊Dic3—31no—
156C2×C3⋊Dic3—32no—
157C4×C3⋊D3—34no—
158C4×C3⋊D3—64no—
159C2×C3×Dic3—32no—
160C2×C3×Dic3C22×D312yes—
161C3×D12—22no—
162C3×D12—62no—
163C3×Dic6—22no—
164C3×Dic6—62no—
165#627—64no—
166#625—64no—
167#624—64no—
168C23×C9—32no—
169C22×D9—31no—
170C22×D9—61no—
171C22×D9—64no—
172C22×D9—68no—
173C22×D9C22×D312yes—
174Q8×C9—62no—
175D4×C9—62no—
176C2×C4×C9—22no—
177C2×C4×C9—21no—
178C2×C4×C9—32no—
179C2×C4×C9C22×D312yes—
180C2×C4×C9C22×D311yescharacteristic
181#611—64no—
182C2×Dic9—31no—
183C2×Dic9—32no—
184C2×Dic9—61no—
185C2×Dic9—62no—
186C2×Dic9C22×D312yes—
187D36—22no—
188C4×D9—34no—
189C4×D9—64no—
190C4×D9—64no—
191C4×D9—64no—
192C4×D9—68no—
193Dic18—22no—
194Dic18—64no—
195C2×#124—271noSylow
196C2×C3×C9C22×D411yescharacteristic
197C2×C3×C9C2×#4112yes—
198C3⋊D9#12414yescomplemented
199D3×C9—42no—
200C3×D9—42no—
201C23×D3—92no—
202C23×D3—92no—
203C2×C3×Q8—31no—
204C2×C3×Q8—91no—
205C2×C3×D4—31no—
206C2×C3×D4—91no—
207C2×#67—92no—
208C2×#67—92no—
209D3×Q8—184no—
210D3×Q8—184no—
211D3×D4—184no—
212D3×D4—184no—
213C2×D12—31no—
214C2×D12C2×D911yescharacteristic
215C2×C4×D3—91no—
216C2×C4×D3—91no—
217C2×C4×D3—92no—
218C2×C4×D3—92no—
219C2×C4×D3—92no—
220C2×C4×D3—92no—
221C2×C4×D3—181no—
222C2×C4×D3—182no—
223C2×Dic6—31no—
224C2×Dic6—92no—
225C2×Dic6—92no—
226C2×Dic6C2×D911yescharacteristic, complemented
227C3×C22⋊C4—34no—
228C3×C22⋊C4—94no—
229C3×C42—21no—
230C3×C42C2×D911yescharacteristic, complemented
231C3×C42D3211yescharacteristic
232#217—94no—
233#217—94no—
234(C2×D3)⋊C4—34no—
235(C2×D3)⋊C4—94no—
236(C2×D3)⋊C4—94no—
237(C2×D3)⋊C4C2×D914yescomplemented
238C4×Dic3—92no—
239C4×Dic3—91no—
240C4×Dic3—92no—
241C4×Dic3—91no—
242C4×Dic3—181no—
243C4×Dic3—182no—
244C22×C32C23×D311yescharacteristic
245C2×C3⋊D3—32no—
246C2×C3⋊D3—34no—
247C2×C3×D3—22no—
248C2×C3×D3—24no—
249C2×C3×D3—62no—
250C2×C3×D3—64no—
251D32—128no—
252C32×C4—22no—
253C32×C4D3×D412yes—
254C3⋊Dic3—32no—
255C3⋊Dic3—62no—
256C3×Dic3—22no—
257C3×Dic3—62no—
258C22×C9—21no—
259C22×C9—62no—
260C22×C9—64no—
261C22×C9C23×D311yescharacteristic
262C2×D9—22no—
263C2×D9—24no—
264C2×D9—32no—
265C2×D9—34no—
266C2×D9—62no—
267C2×D9—64no—
268C2×D9—128no—
269C4×C9—24no—
270C4×C9—22no—
271C4×C9—22no—
272C4×C9—62no—
273C4×C9D3×D412yes—
274Dic9—22no—
275Dic9—32no—
276Dic9—62no—
277Dic9—64no—
278Dic9—62no—
279C22×Q8—271no—
280C22×D4—271no—
281#124—31no—
282#124—91no—
283#124—274no—
284#124—271no—
285#124—271no—
286C2×C22⋊C4—274no—
287C2×C42—271no—
288C3×C9C2×#12411yescharacteristic, complemented, Sylow
289C23×C3—32no—
290C23×C3—92no—
291C22×D3—32no—
292C22×D3—91no—
293C22×D3—91no—
294C22×D3—181no—
295C22×D3—184no—
296C22×D3—188no—
297C22×D3—184no—
298C22×D3—188no—
299C22×D3C22×D912yes—
300C3×Q8—62no—
301C3×Q8—182no—
302C3×D4—62no—
303C3×D4—182no—
304C2×C3×C4—22no—
305C2×C3×C4—21no—
306C2×C3×C4—32no—
307C2×C3×C4—92no—
308C2×C3×C4C22×D912yes—
309C2×C3×C4C22×D911yescharacteristic
310C2×C3×C4C2×D3212yes—
311C2×C3×C4C2×D3211yescharacteristic
312#67—184no—
313#67—184no—
314C2×Dic3—32no—
315C2×Dic3—91no—
316C2×Dic3—91no—
317C2×Dic3—92no—
318C2×Dic3—92no—
319C2×Dic3—181no—
320C2×Dic3—182no—
321C2×Dic3C22×D912yes—
322D12—22no—
323D12—62no—
324C4×D3—94no—
325C4×D3—94no—
326C4×D3—184no—
327C4×D3—184no—
328C4×D3—184no—
329C4×D3—184no—
330C4×D3—184no—
331C4×D3—188no—
332Dic6—22no—
333Dic6—62no—
334Dic6—184no—
335Dic6—184no—
336C2×C32C2×D3×D411yescharacteristic
337C2×C32D3×#4112yes—
338C3⋊D3—34no—
339C3×D3—42no—
340C3×D3—122no—
341C2×C9—21no—
342C2×C9—22no—
343C2×C9—122no—
344C2×C9C2×D3×D411yescharacteristic
345C2×C9D3×#4112yes—
346D9—34no—
347D9—42no—
348D9—64no—
349C24—272no—
350C2×Q8—31no—
351C2×Q8—91no—
352C2×Q8—272no—
353C2×Q8—544no—
354C2×D4—31no—
355C2×D4—91no—
356C2×D4—272no—
357C2×D4—544no—
358C22×C4—272no—
359C22×C4—271no—
360C22×C4—272no—
361C22⋊C4—34no—
362C22⋊C4—94no—
363C22⋊C4—274no—
364C22⋊C4—274no—
365C42—272no—
366C42—271no—
367C42D3×D911yescharacteristic, complemented
368C22×C3—21no—
369C22×C3—62no—
370C22×C3—64no—
371C22×C3—182no—
372C22×C3—184no—
373C22×C3C23×D911yescharacteristic
374C22×C3C22×D3211yescharacteristic
375C2×D3—22no—
376C2×D3—24no—
377C2×D3—62no—
378C2×D3—64no—
379C2×D3—92no—
380C2×D3—94no—
381C2×D3—92no—
382C2×D3—94no—
383C2×D3—182no—
384C2×D3—184no—
385C2×D3—368no—
386C2×D3—368no—
387C3×C4—24no—
388C3×C4—22no—
389C3×C4—22no—
390C3×C4—22no—
391C3×C4—62no—
392C3×C4—182no—
393C3×C4D4×D912yes—
394C3×C4#380212yes—
395Dic3—22no—
396Dic3—62no—
397Dic3—92no—
398Dic3—92no—
399Dic3—182no—
400Dic3—184no—
401Dic3—182no—
402Dic3—182no—
403C32D3×#12411yescharacteristic
404C9—21no—
405C9D3×#12411yescharacteristic, complemented
406C23—32no—
407C23—92no—
408C23—271no—
409C23—544no—
410C23—548no—
411Q8—62no—
412Q8—182no—
413Q8—544no—
414D4—62no—
415D4—182no—
416D4—544no—
417C2×C4—32no—
418C2×C4—92no—
419C2×C4—271no—
420C2×C4—274no—
421C2×C4—272no—
422C2×C4—544no—
423C2×C4—544no—
424C2×C4C2×D3×D912yes—
425C2×C4C2×D3×D911yescharacteristic
426C2×C3—21no—
427C2×C3—22no—
428C2×C3—122no—
429C2×C3—362no—
430C2×C3C2×D4×D911yescharacteristic
431C2×C3#41×D912yes—
432C2×C3C2×#380211yescharacteristic
433C2×C3#6430612yes—
434D3—42no—
435D3—94no—
436D3—94no—
437D3—122no—
438D3—184no—
439C22—62no—
440C22—64no—
441C22—182no—
442C22—184no—
443C22—272no—
444C22—274no—
445C22—1088no—
446C22C22×D3×D911yescharacteristic
447C4—22no—
448C4—62no—
449C4—182no—
450C4—272no—
451C4—542no—
452C4#8864612yes—
453C3—21no—
454C3D9×#12411yescharacteristic, complemented
455C3#1059400911yescharacteristic
456C2—122no—
457C2—274no—
458C2—362no—
459C2C2×#8864611yescharacteristic
460C2#1172247512yes—
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC2×#88646C2
hypercenterC42D3×D9
hypocenterC3×C9C2×#124
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C2×(C3⋊D9)⋊Q8C2semidirect
3. C42×C3⋊D9C2semidirect
4. C9×#875C2semidirect
5. C3×#63437C2semidirect
6. #11721048C2semidirect
7. #11721044C2semidirect
8. #11720967C2semidirect
9. #11720951C2semidirect
17. C2×C9×Dic6C22semidirect
18. C2×C3×Dic18C22semidirect
21. C2×#6189C22semidirect
24. C3×C42×C9C22semidirect
25. C4×C3⋊Dic9C22semidirect
26. C4×C3⋊Dic9C22semidirect
27. C9×(C2×D3)⋊C4C22semidirect
28. C3×(C2×D9)⋊C4C22semidirect
29. (C2×D3)⋊Dic9C22semidirect
30. (C2×D9)⋊Dic3C22semidirect
47. #63437D3semidirect
104. C2×Dic18C2×D3semidirect
107. C42×C9C2×D3semidirect
110. (C2×D9)⋊C4C2×D3semidirect
144. #875D9semidirect
198. C3⋊D9#124semidirect
226. C2×Dic6C2×D9semidirect
230. C3×C42C2×D9semidirect
237. (C2×D3)⋊C4C2×D9semidirect
288. C3×C9C2×#124semidirect
367. C42D3×D9semidirect
405. C9D3×#124semidirect
454. C3D9×#124semidirect

Maximal subgroups

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

subgroupindex
C2×#886462
C2×(C3⋊D9)⋊Q82
C42×C3⋊D92
C9×#8752
C3×#634372
#117210482
#117210442
#117209672
#117209512
#105940093
D9×#1243