Finite Group Databasebeta

← #13151165

Subgroups — #13151165

Order 1728.

822 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721835C211yesmaximal, characteristic, complemented
2C9×#937C211yesmaximal, characteristic
3C3×#63498C211yesmaximal, characteristic, complemented
4#67⋊Dic9C212yesmaximal, complemented
5#11721280C212yesmaximal, complemented
6D3×#3676C212yesmaximal, complemented
7#11721246C212yesmaximal, complemented
8#11721018C212yesmaximal, complemented
9#11721010C212yesmaximal, complemented
10#10595176—31nomaximal
11#10589689—31nomaximal
12C23×D3×C9C2211yescharacteristic
13C2×C3×D4×C9C2211yescharacteristic
14C22×C3⋊Dic9C2211yescharacteristic, complemented
15C2×C9×#67C2212yes—
16C2×C3×Dic18C2211yescharacteristic, complemented
17C2×D3×Dic9C2212yes—
18C2×D3×Dic9C2212yes—
19#88551C2212yes—
20#88551C2211yescharacteristic, complemented
21#88534C2212yescomplemented
22C9×#217C2211yescharacteristic
23C3×#3676C2212yes—
24C3×#3676C2212yescomplemented
25C9×(C2×D3)⋊C4C2212yes—
26C3×C4×Dic9C2211yescharacteristic, complemented
27(C2×D3)⋊Dic9C2212yes—
28(C2×D3)⋊Dic9C2212yes—
29Dic3⋊Dic9C2212yescomplemented
30#88442C2212yescomplemented
31Dic9⋊Dic3C2212yescomplemented
32Dic3×Dic9C2212yescomplemented
33#64144—31no—
34C3×#937D311yescharacteristic
35C3×#936—31no—
36#67⋊Dic3—32no—
37#63969—32no—
38D3×#217—32no—
39#63956—32no—
40#63887—32no—
41#63877—32no—
42C9×#120—31no—
43C2×#3676—32no—
44#63498D311yescharacteristic, complemented
45#63497—31no—
46#63497—32no—
47#63497—61no—
48D4×Dic9—32no—
49#63441—32no—
50#63440—32no—
51#63439—32no—
52C23×C3×C9C2311yescharacteristic
53C23×C3×C9C2311yescharacteristic
54C22×D3×C9—24no—
55C22×D3×C9—41no—
56C22×D3×C9C2312yes—
57C3×D4×C9—41no—
58C2×C3×C4×C9C2311yescharacteristic
59C2×C3⋊Dic9—41no—
60C2×C3⋊Dic9C2312yes—
61C2×C3⋊Dic9C2312yes—
62C9×#67—24no—
63C2×C9×Dic3C2312yes—
64C2×C3×Dic9C2312yes—
65C2×C3×Dic9C2312yes—
66C3×Dic18—41no—
67D3×Dic9—24no—
68D3×Dic9—42no—
69#5679—91no—
70#5587—91no—
71C23×C3×D3C2×D311yescharacteristic
72C2×C32×D4C2×D311yescharacteristic
73C22×C3⋊Dic3—31no—
74C2×C3×#67C2×D312yes—
75C2×C3×Dic6—31no—
76C2×D3×Dic3—32no—
77C2×D3×Dic3—32no—
78C24×C9—31no—
79#3757—32no—
80#3757—31no—
81#3750—32no—
82C3×#217—32no—
83C3×#217—32no—
84C3×#217C2×D311yescharacteristic
85C3×(C2×D3)⋊C4C2×D312yes—
86C3×C4×Dic3—31no—
87#3724—32no—
88Dic3⋊Dic3—32no—
89Dic3⋊Dic3—32no—
90(C2×D3)⋊Dic3—32no—
91(C2×D3)⋊Dic3—32no—
92Dic32—32no—
93C2×D4×C9—21no—
94C2×D4×C9—32no—
95C2×D4×C9C2×D311yescharacteristic
96C22×Dic9—31no—
97C22×Dic9—32no—
98C22×Dic9—32no—
99C22×Dic9—61no—
100C2×Dic18C2×D311yescharacteristic, complemented
101C9×C22⋊C4—31no—
102C9×C22⋊C4—32no—
103#3676—31no—
104#3676—32no—
105#3676—32no—
106#3676—32no—
107#3676—62no—
108#3676—61no—
109#3676C2×D312yes—
110#3676C2×D312yescomplemented
111C4⋊Dic9—32no—
112Dic9⋊C4—32no—
113Dic9⋊C4—32no—
114Dic9⋊C4—32no—
115Dic9⋊C4—62no—
116C4×Dic9—32no—
117C4×Dic9C2×D311yescharacteristic, complemented
118C22×C3×C9—22no—
119C22×C3×C9—22no—
120C22×C3×C9—21no—
121C22×C3×C9—21no—
122C22×C3×C9C2411yescharacteristic
123C2×D3×C9—22no—
124C2×D3×C9—22no—
125C2×D3×C9—44no—
126C2×D3×C9#4114yes—
127C3×C4×C9—21no—
128C3⋊Dic9—22no—
129C3⋊Dic9—22no—
130C9×Dic3—22no—
131C3×Dic9—22no—
132C3×Dic9—22no—
133C3×#124—91no—
134C3×#120—31no—
135C2×#217—92no—
136#946—92no—
137#937D911yescharacteristic
138#936—31no—
139#935—92no—
140#935—91no—
141#935—91no—
142#935—181no—
143D4×Dic3—92no—
144#883—92no—
145#67⋊C4—92no—
146D3×C22⋊C4—92no—
147#879—92no—
148#879—92no—
149#878—92no—
150#877—92no—
151#872—92no—
152C23×C32C22×D311yescharacteristic
153C23×C32C22×D311yescharacteristic
154C22×C3×D3—24no—
155C22×C3×D3—41no—
156C22×C3×D3C22×D312yes—
157C32×D4—41no—
158C2×C32×C4C22×D311yescharacteristic
159C2×C3⋊Dic3—32no—
160C2×C3⋊Dic3—32no—
161C2×C3⋊Dic3—121no—
162C3×#67—24no—
163C2×C3×Dic3—32no—
164C2×C3×Dic3—32no—
165C2×C3×Dic3C22×D312yes—
166C3×Dic6—121no—
167D3×Dic3—64no—
168D3×Dic3—122no—
169C23×C9—21no—
170C23×C9—21no—
171C23×C9—32no—
172C23×C9—64no—
173C23×C9—121no—
174C23×C9C22×D311yescharacteristic
175C23×C9C22×D311yescharacteristic
176D4×C9—41no—
177D4×C9—42no—
178D4×C9—64no—
179C2×C4×C9—21no—
180C2×C4×C9—32no—
181C2×C4×C9C22×D311yescharacteristic
182C2×Dic9—32no—
183C2×Dic9—32no—
184C2×Dic9—62no—
185C2×Dic9—62no—
186C2×Dic9—64no—
187C2×Dic9—122no—
188C2×Dic9—121no—
189C2×Dic9—122no—
190C2×Dic9C22×D312yes—
191C2×Dic9C22×D312yes—
192Dic18—41no—
193#551—271noSylow
194C2×C3×C9—41no—
195C2×C3×C9—41no—
196C2×C3×C9C2×#4112yes—
197C2×C3×C9H(5,𝔽2)11yescharacteristic
198D3×C9—24no—
199C24×C3—31no—
200C23×D3C2×D911yescharacteristic
201C2×C3×Q8—91no—
202C2×C3×D4—21no—
203C2×C3×D4—32no—
204C2×C3×D4C2×D911yescharacteristic
205C2×C3×D4D3211yescharacteristic
206C2×#67C2×D912yes—
207C22×Dic3—92no—
208C22×Dic3—92no—
209C22×Dic3—91no—
210C22×Dic3—91no—
211C22×Dic3—181no—
212C2×C4×D3—92no—
213C2×C4×D3—92no—
214C2×Dic6—31no—
215C3×C22⋊C4—31no—
216C3×C22⋊C4—32no—
217C3×C22⋊C4—92no—
218C3×C22⋊C4—92no—
219C3×C42—91no—
220#217—32no—
221#217—32no—
222#217—92no—
223#217—92no—
224#217—92no—
225#217—91no—
226#217—92no—
227#217—91no—
228#217—182no—
229#217—181no—
230#217C2×D911yescharacteristic
231(C2×D3)⋊C4—92no—
232(C2×D3)⋊C4—92no—
233(C2×D3)⋊C4C2×D912yes—
234C4⋊Dic3—92no—
235C4⋊Dic3—92no—
236Dic3⋊C4—92no—
237Dic3⋊C4—92no—
238Dic3⋊C4—92no—
239Dic3⋊C4—92no—
240Dic3⋊C4—92no—
241Dic3⋊C4—92no—
242Dic3⋊C4—182no—
243C4×Dic3—31no—
244C4×Dic3—92no—
245C4×Dic3—92no—
246C22×C32—21no—
247C22×C32—22no—
248C22×C32—22no—
249C22×C32—21no—
250C22×C32C23×D311yescharacteristic
251C2×C3×D3—22no—
252C2×C3×D3—22no—
253C2×C3×D3—44no—
254C2×C3×D3#23714yes—
255C32×C4—21no—
256C3⋊Dic3—62no—
257C3⋊Dic3—62no—
258C3×Dic3—22no—
259C3×Dic3—62no—
260C3×Dic3—62no—
261C22×C9—22no—
262C22×C9—21no—
263C22×C9—21no—
264C22×C9—22no—
265C22×C9—22no—
266C22×C9—21no—
267C22×C9—34no—
268C22×C9—42no—
269C22×C9—41no—
270C22×C9—42no—
271C22×C9—62no—
272C22×C9—62no—
273C22×C9—124no—
274C22×C9C23×D311yescharacteristic
275C4×C9—21no—
276C4×C9—41no—
277C4×C9—62no—
278Dic9—22no—
279Dic9—22no—
280Dic9—62no—
281Dic9—62no—
282Dic9—122no—
283Dic9—122no—
284#126—272no—
285#124—91no—
286#123—272no—
287#123—271no—
288C22⋊Q8—272no—
289#120—31no—
290C4×D4—272no—
291#117—272no—
292C2×C22⋊C4—272no—
293C3×C9#55111yescharacteristic, complemented, Sylow
294C23×C3—21no—
295C23×C3—21no—
296C23×C3—32no—
297C23×C3—64no—
298C23×C3—121no—
299C23×C3C22×D911yescharacteristic
300C23×C3C22×D911yescharacteristic
301C23×C3C2×D3211yescharacteristic
302C23×C3C2×D3211yescharacteristic
303C22×D3—24no—
304C22×D3—41no—
305C22×D3C22×D912yes—
306C3×Q8—361no—
307C3×D4—41no—
308C3×D4—41no—
309C3×D4—42no—
310C3×D4—64no—
311C2×C3×C4—21no—
312C2×C3×C4—32no—
313C2×C3×C4—92no—
314C2×C3×C4—92no—
315C2×C3×C4C22×D911yescharacteristic
316C2×C3×C4C2×D3211yescharacteristic
317#67—24no—
318C2×Dic3—32no—
319C2×Dic3—32no—
320C2×Dic3—92no—
321C2×Dic3—92no—
322C2×Dic3—92no—
323C2×Dic3—92no—
324C2×Dic3—182no—
325C2×Dic3—182no—
326C2×Dic3—184no—
327C2×Dic3—362no—
328C2×Dic3—361no—
329C2×Dic3—361no—
330C2×Dic3—362no—
331C2×Dic3C22×D912yes—
332C4×D3—184no—
333C4×D3—362no—
334Dic6—121no—
335C2×C32—41no—
336C2×C32—41no—
337C2×C32C2×#23712yes—
338C2×C32#100411yescharacteristic
339C3×D3—24no—
340C2×C9—22no—
341C2×C9—21no—
342C2×C9—42no—
343C2×C9—41no—
344C2×C9—41no—
345C2×C9—64no—
346C2×C9—81no—
347C2×C9#100411yescharacteristic
348C2×C9D3×#4112yes—
349C24—31no—
350C2×Q8—91no—
351C2×D4—32no—
352C2×D4D3×D911yescharacteristic
353C22×C4—272no—
354C22×C4—272no—
355C22×C4—271no—
356C4⋊C4—272no—
357C4⋊C4—272no—
358C4⋊C4—272no—
359C4⋊C4—272no—
360C22⋊C4—31no—
361C22⋊C4—32no—
362C22⋊C4—92no—
363C22⋊C4—92no—
364C22⋊C4—272no—
365C22⋊C4—272no—
366C22⋊C4—272no—
367C22⋊C4—271no—
368C42—91no—
369C42—272no—
370C22×C3—22no—
371C22×C3—22no—
372C22×C3—22no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—22no—
376C22×C3—22no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—34no—
381C22×C3—42no—
382C22×C3—41no—
383C22×C3—42no—
384C22×C3—62no—
385C22×C3—62no—
386C22×C3—124no—
387C22×C3C23×D911yescharacteristic
388C22×C3C22×D3211yescharacteristic
389C2×D3—22no—
390C2×D3—22no—
391C2×D3—44no—
392C2×D3#369914yes—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—41no—
396C3×C4—62no—
397C3×C4—182no—
398C3×C4—182no—
399Dic3—22no—
400Dic3—62no—
401Dic3—62no—
402Dic3—182no—
403Dic3—182no—
404Dic3—182no—
405Dic3—182no—
406Dic3—362no—
407Dic3—362no—
408C32#558711yescharacteristic
409C9—21no—
410C9#567911yescharacteristic, complemented
411C23—32no—
412C23—64no—
413C23—121no—
414C23C2×D3×D911yescharacteristic
415C23C2×D3×D911yescharacteristic
416Q8—361no—
417D4—41no—
418D4—64no—
419C2×C4—32no—
420C2×C4—92no—
421C2×C4—92no—
422C2×C4—272no—
423C2×C4—272no—
424C2×C4—544no—
425C2×C4—1082no—
426C2×C4—1081no—
427C2×C4C2×D3×D911yescharacteristic
428C2×C3—22no—
429C2×C3—21no—
430C2×C3—42no—
431C2×C3—41no—
432C2×C3—41no—
433C2×C3—41no—
434C2×C3—41no—
435C2×C3—64no—
436C2×C3—81no—
437C2×C3C2×#369912yes—
438C2×C3#6371011yescharacteristic
439C2×C3D3×#23712yes—
440C2×C3#6433011yescharacteristic
441D3—24no—
442C22—22no—
443C22—22no—
444C22—21no—
445C22—21no—
446C22—34no—
447C22—62no—
448C22—62no—
449C22—124no—
450C22C22×D3×D911yescharacteristic
451C4—21no—
452C4—62no—
453C4—182no—
454C4—182no—
455C4—542no—
456C4—542no—
457C3—21no—
458C3#1058968911yescharacteristic, complemented
459C3#1059517611yescharacteristic
460C2—41no—
461C2—41no—
462C2—64no—
463C2D3×#369912yes—
464C2#1172253511yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
prime radicalC3×#937D3
hypercenterC2×D4D3×D9
hypocenterC3×C9#551
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #11721835C2semidirect
3. C3×#63498C2semidirect
4. #67⋊Dic9C2semidirect
5. #11721280C2semidirect
6. D3×#3676C2semidirect
7. #11721246C2semidirect
8. #11721018C2semidirect
9. #11721010C2semidirect
14. C22×C3⋊Dic9C22semidirect
16. C2×C3×Dic18C22semidirect
20. #88551C22semidirect
21. #88534C22semidirect
24. C3×#3676C22semidirect
26. C3×C4×Dic9C22semidirect
29. Dic3⋊Dic9C22semidirect
30. #88442C22semidirect
31. Dic9⋊Dic3C22semidirect
32. Dic3×Dic9C22semidirect
44. #63498D3semidirect
100. C2×Dic18C2×D3semidirect
110. #3676C2×D3semidirect
117. C4×Dic9C2×D3semidirect
293. C3×C9#551semidirect
410. C9#5679semidirect
458. C3#10589689semidirect

Maximal subgroups

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

subgroupindex
#117218352
C9×#9372
C3×#634982
#67⋊Dic92
#117212802
D3×#36762
#117212462
#117210182
#117210102
#105951763
#105896893