Finite Group Databasebeta

← #13148774

Subgroups — #13148774

Order 1728.

706 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721743C211yesmaximal, characteristic, complemented
2C9×#893C211yesmaximal, characteristic
3C3×#63449C211yesmaximal, characteristic, complemented
4C4×#6189C211yesmaximal, characteristic, complemented
5#11720967C211yesmaximal, characteristic
6#11720966C212yesmaximal, complemented
7#11720964C211yesmaximal, characteristic
8#11720963C212yesmaximal, complemented
9#11720959C212yesmaximal, complemented
10#11720955C212yesmaximal, complemented
11#11720932C211yesmaximal, characteristic, complemented
12#10594335—31nomaximal
13#10589644—31nomaximal
14C2×#6226C2211yescharacteristic
15C2×C9×D12C2211yescharacteristic
16C2×C4×D3×C9C2212yes—
17C2×C3×Dic18C2211yescharacteristic
18C2×#6189C2211yescharacteristic
19C3×C9×C4⋊C4C2211yescharacteristic
20#88534C2212yes—
21C4×C3⋊Dic9C2212yes—
22C4×C3⋊Dic9C2211yescharacteristic
23C9×(C2×D3)⋊C4C2212yes—
24C4×C9×Dic3C2211yescharacteristic
25C3×Dic9⋊C4C2212yes—
26C3×C4×Dic9C2212yes—
27C3×C4×Dic9C2211yescharacteristic
28(C2×D3)⋊Dic9C2212yes—
29(C2×D3)⋊Dic9C2212yes—
30(C2×D3)⋊Dic9C2212yes—
31(C2×D3)⋊Dic9C2212yes—
32Dic3⋊Dic9C2212yes—
33Dic3⋊Dic9C2211yescharacteristic
34Dic3⋊Dic9C2212yes—
35#88442C2211yescharacteristic
36Dic9⋊Dic3C2211yescharacteristic
37#64096—31no—
38C3×#893D311yescharacteristic
39C3×#887—31no—
40C4×#627—31no—
41#63866—31no—
42#63863—31no—
43#63862—32no—
44#63860—31no—
45#63859—32no—
46#63855—32no—
47#63851—32no—
48C4×D4×C9—31no—
49#63498—31no—
50#63498—31no—
51#63488—32no—
52#63486—32no—
53#63449—31no—
54#63449—61no—
55#63449D311yescharacteristic, complemented
56#63441—32no—
57#63441—32no—
58#63431—31no—
59C4×Dic18—31no—
60C22×D3×C9C2312yes—
61C2×C3×C4×C9C2312yes—
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3⋊Dic9C2311yescharacteristic
64C2×C3⋊Dic9C2311yescharacteristic
65C2×C3⋊Dic9C2312yes—
66#6226—22no—
67C2×C9×Dic3C2311yescharacteristic
68C2×C9×Dic3C2311yescharacteristic
69C2×C3×Dic9C2311yescharacteristic
70C2×C3×Dic9C2312yes—
71C2×C3×Dic9C2311yescharacteristic
72C9×D12—22no—
73C4×D3×C9—24no—
74C3×Dic18—22no—
75#6189—22no—
76#5568—91no—
77#5542—91no—
78C2×#634—31no—
79C2×C3×D12C2×D311yescharacteristic
80C2×C3×C4×D3C2×D312yes—
81C2×C3×Dic6—31no—
82C2×#627—31no—
83C32×C4⋊C4C2×D311yescharacteristic
84#3750—32no—
85C4×C3⋊Dic3—32no—
86C4×C3⋊Dic3—31no—
87C3×(C2×D3)⋊C4C2×D312yes—
88C3×Dic3⋊C4—32no—
89C3×C4×Dic3—32no—
90C3×C4×Dic3—31no—
91C3×C4×Dic3C2×D311yescharacteristic
92#3724—31no—
93Dic3⋊Dic3—32no—
94Dic3⋊Dic3—31no—
95Dic3⋊Dic3—32no—
96Dic3⋊Dic3—31no—
97(C2×D3)⋊Dic3—32no—
98(C2×D3)⋊Dic3—32no—
99(C2×D3)⋊Dic3—32no—
100(C2×D3)⋊Dic3—32no—
101C2×D4×C9—31no—
102C22×C4×C9—32no—
103C2×Dic18—31no—
104C2×Dic18—31no—
105C2×Dic18—61no—
106C2×Dic18C2×D311yescharacteristic
107C9×C4⋊C4—21no—
108C9×C4⋊C4C2×D311yescharacteristic
109C9×C22⋊C4—32no—
110C42×C9—31no—
111#3676—32no—
112#3676—32no—
113#3676—32no—
114#3676—32no—
115C4⋊Dic9—31no—
116C4⋊Dic9—32no—
117C4⋊Dic9—32no—
118Dic9⋊C4—32no—
119Dic9⋊C4—31no—
120Dic9⋊C4—31no—
121Dic9⋊C4—62no—
122Dic9⋊C4C2×D312yes—
123C4×Dic9—32no—
124C4×Dic9—31no—
125C4×Dic9—62no—
126C4×Dic9—61no—
127C4×Dic9C2×D312yes—
128C4×Dic9C2×D311yescharacteristic
129C22×C3×C9C2411yescharacteristic
130C2×D3×C9—22no—
131C2×D3×C9—22no—
132C2×D3×C9—22no—
133C3×C4×C9—22no—
134C3×C4×C9#4112yes—
135C3⋊Dic9—22no—
136C3⋊Dic9—21no—
137C3⋊Dic9—21no—
138C9×Dic3—21no—
139C9×Dic3#4112yes—
140C3×Dic9—21no—
141C3×Dic9—22no—
142C3×Dic9—21no—
143C3×C4×Q8—91no—
144C3×C4×D4—31no—
145#947—91no—
146#944—91no—
147#936—91no—
148#936—91no—
149#926—92no—
150#924—92no—
151#898—92no—
152#896—92no—
153#893D911yescharacteristic
154#887—31no—
155#887—91no—
156#887—91no—
157#887—181no—
158#879—92no—
159#879—92no—
160#876—92no—
161#875—91no—
162#872—92no—
163#869—91no—
164C4×Dic6—91no—
165C4×Dic6—91no—
166C22×C3×D3C22×D312yes—
167C2×C32×C4C22×D312yes—
168C2×C32×C4C22×D311yescharacteristic
169C2×C3⋊Dic3—31no—
170C2×C3⋊Dic3—32no—
171C2×C3⋊Dic3—31no—
172#634—62no—
173C2×C3×Dic3—32no—
174C2×C3×Dic3—31no—
175C2×C3×Dic3—31no—
176C2×C3×Dic3C22×D311yescharacteristic
177C2×C3×Dic3C22×D311yescharacteristic
178C3×D12—22no—
179C3×C4×D3—24no—
180C3×Dic6—62no—
181#627—62no—
182C23×C9—32no—
183D4×C9—62no—
184C2×C4×C9—21no—
185C2×C4×C9—22no—
186C2×C4×C9—31no—
187C2×C4×C9—31no—
188C2×C4×C9—64no—
189C2×C4×C9C22×D312yes—
190C2×C4×C9C22×D311yescharacteristic
191C2×Dic9—31no—
192C2×Dic9—31no—
193C2×Dic9—32no—
194C2×Dic9—61no—
195C2×Dic9—62no—
196C2×Dic9—61no—
197C2×Dic9C22×D311yescharacteristic
198C2×Dic9C22×D311yescharacteristic
199C2×Dic9C22×D312yes—
200Dic18—22no—
201Dic18—62no—
202Dic18—64no—
203Dic18—62no—
204#556—271noSylow
205C2×C3×C9C2×#4111yescharacteristic
206C2×C3×C9C2×#4111yescharacteristic
207C2×C3×C92−1+411yescharacteristic
208D3×C9—42no—
209C2×C3×Q8—91no—
210C2×C3×D4—31no—
211C22×C3×C4—32no—
212C2×D12C2×D911yescharacteristic
213C2×C4×D3C2×D912yes—
214C2×Dic6—31no—
215C2×Dic6—91no—
216C2×Dic6—91no—
217C2×Dic6—91no—
218C2×Dic6—91no—
219C2×Dic6—181no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—92no—
222C3×C4⋊C4C2×D911yescharacteristic
223C3×C4⋊C4D3211yescharacteristic
224C3×C22⋊C4—32no—
225C3×C42—31no—
226C3×C42—91no—
227C3×C42—92no—
228#217—92no—
229#217—92no—
230#217—92no—
231#217—92no—
232(C2×D3)⋊C4—92no—
233(C2×D3)⋊C4—92no—
234(C2×D3)⋊C4—92no—
235(C2×D3)⋊C4—92no—
236(C2×D3)⋊C4C2×D912yes—
237C4⋊Dic3—91no—
238C4⋊Dic3—92no—
239C4⋊Dic3—92no—
240C4⋊Dic3—91no—
241Dic3⋊C4—32no—
242Dic3⋊C4—91no—
243Dic3⋊C4—91no—
244Dic3⋊C4—92no—
245Dic3⋊C4—92no—
246Dic3⋊C4—91no—
247Dic3⋊C4—91no—
248Dic3⋊C4—92no—
249Dic3⋊C4—92no—
250Dic3⋊C4—182no—
251C4×Dic3—31no—
252C4×Dic3—32no—
253C4×Dic3—91no—
254C4×Dic3—92no—
255C4×Dic3—91no—
256C4×Dic3—92no—
257C4×Dic3—182no—
258C4×Dic3—181no—
259C4×Dic3C2×D911yescharacteristic
260C22×C32C23×D311yescharacteristic
261C2×C3×D3—22no—
262C2×C3×D3—22no—
263C2×C3×D3—22no—
264C32×C4—22no—
265C32×C4#23512yes—
266C3⋊Dic3—62no—
267C3⋊Dic3—61no—
268C3⋊Dic3—61no—
269C3×Dic3—21no—
270C3×Dic3—62no—
271C3×Dic3—61no—
272C3×Dic3—61no—
273C3×Dic3#23712yes—
274C22×C9—21no—
275C22×C9—62no—
276C22×C9—62no—
277C22×C9—62no—
278C22×C9C23×D311yescharacteristic
279C4×C9—22no—
280C4×C9—22no—
281C4×C9—32no—
282C4×C9—42no—
283C4×C9—61no—
284C4×C9#23512yes—
285Dic9—22no—
286Dic9—21no—
287Dic9—21no—
288Dic9—64no—
289Dic9—62no—
290Dic9—62no—
291Dic9—61no—
292Dic9—61no—
293Dic9—121no—
294C4⋊Q8—271no—
295#126—272no—
296#126—272no—
297#124—271no—
298#124—271no—
299C22⋊Q8—272no—
300C4×Q8—91no—
301C4×Q8—271no—
302C4×Q8—271no—
303C4×D4—31no—
304#117—272no—
305C3×C9#55611yescharacteristic, complemented, Sylow
306C23×C3—32no—
307C22×D3C22×D912yes—
308C3×Q8—182no—
309C3×D4—62no—
310C2×C3×C4—21no—
311C2×C3×C4—22no—
312C2×C3×C4—31no—
313C2×C3×C4—31no—
314C2×C3×C4—64no—
315C2×C3×C4—91no—
316C2×C3×C4—92no—
317C2×C3×C4—91no—
318C2×C3×C4C22×D912yes—
319C2×C3×C4C22×D911yescharacteristic
320C2×C3×C4C2×D3212yes—
321C2×C3×C4C2×D3211yescharacteristic
322C2×Dic3—31no—
323C2×Dic3—31no—
324C2×Dic3—32no—
325C2×Dic3—91no—
326C2×Dic3—91no—
327C2×Dic3—92no—
328C2×Dic3—91no—
329C2×Dic3—92no—
330C2×Dic3—91no—
331C2×Dic3—181no—
332C2×Dic3—182no—
333C2×Dic3—181no—
334C2×Dic3C22×D911yescharacteristic
335C2×Dic3C22×D911yescharacteristic
336D12—22no—
337C4×D3—24no—
338Dic6—62no—
339Dic6—182no—
340Dic6—184no—
341Dic6—182no—
342Dic6—182no—
343Dic6—182no—
344C2×C32C2×#23511yescharacteristic
345C2×C32C2×#23711yescharacteristic
346C2×C32#101011yescharacteristic
347C3×D3—42no—
348C2×C9—21no—
349C2×C9—21no—
350C2×C9—21no—
351C2×C9—122no—
352C2×C9C2×#23511yescharacteristic
353C2×C9#100711yescharacteristic
354C2×C9D3×#4111yescharacteristic
355C2×Q8—91no—
356C2×Q8—271no—
357C2×Q8—271no—
358C2×D4—31no—
359C22×C4—32no—
360C4⋊C4—92no—
361C4⋊C4—271no—
362C4⋊C4—272no—
363C4⋊C4—271no—
364C4⋊C4—271no—
365C4⋊C4—272no—
366C4⋊C4—272no—
367C4⋊C4D3×D911yescharacteristic
368C22⋊C4—32no—
369C22⋊C4—272no—
370C22⋊C4—272no—
371C22⋊C4—272no—
372C22⋊C4—272no—
373C42—31no—
374C42—92no—
375C42—91no—
376C42—272no—
377C42—271no—
378C22×C3—21no—
379C22×C3—62no—
380C22×C3—62no—
381C22×C3—62no—
382C22×C3C23×D911yescharacteristic
383C22×C3C22×D3211yescharacteristic
384C2×D3—22no—
385C2×D3—22no—
386C2×D3—22no—
387C3×C4—22no—
388C3×C4—22no—
389C3×C4—22no—
390C3×C4—32no—
391C3×C4—42no—
392C3×C4—61no—
393C3×C4—182no—
394C3×C4—181no—
395C3×C4—181no—
396C3×C4#369712yes—
397C3×C4#379812yes—
398Dic3—21no—
399Dic3—62no—
400Dic3—61no—
401Dic3—61no—
402Dic3—184no—
403Dic3—182no—
404Dic3—182no—
405Dic3—181no—
406Dic3—181no—
407Dic3—182no—
408Dic3—181no—
409Dic3—181no—
410Dic3—361no—
411Dic3#369912yes—
412C32#554211yescharacteristic
413C9—21no—
414C9#556811yescharacteristic, complemented
415C23—32no—
416Q8—182no—
417Q8—542no—
418Q8—542no—
419D4—62no—
420C2×C4—31no—
421C2×C4—31no—
422C2×C4—64no—
423C2×C4—91no—
424C2×C4—91no—
425C2×C4—92no—
426C2×C4—271no—
427C2×C4—271no—
428C2×C4—272no—
429C2×C4C2×D3×D912yes—
430C2×C4C2×D3×D911yescharacteristic
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—122no—
435C2×C3C2×#369711yescharacteristic
436C2×C3C2×#369911yescharacteristic
437C2×C3#6371611yescharacteristic
438C2×C3C2×#379811yescharacteristic
439C2×C3D3×#23711yescharacteristic
440C2×C3#6431511yescharacteristic
441D3—42no—
442C22—62no—
443C22—62no—
444C22—62no—
445C22C22×D3×D911yescharacteristic
446C4—22no—
447C4—32no—
448C4—61no—
449C4—181no—
450C4—181no—
451C4—182no—
452C4—541no—
453C4—541no—
454C4—542no—
455C4#8864012yes—
456C3—21no—
457C3#1058964411yescharacteristic, complemented
458C3#1059433511yescharacteristic
459C2—122no—
460C2C2×#8864011yescharacteristic
461C2D3×#369911yescharacteristic
462C2#1172250811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC2×C3×D12C2×D3
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#556
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11721743C2semidirect
3. C3×#63449C2semidirect
4. C4×#6189C2semidirect
6. #11720966C2semidirect
8. #11720963C2semidirect
9. #11720959C2semidirect
10. #11720955C2semidirect
11. #11720932C2semidirect
55. #63449D3semidirect
305. C3×C9#556semidirect
414. C9#5568semidirect
457. C3#10589644semidirect

Maximal subgroups

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

subgroupindex
#117217432
C9×#8932
C3×#634492
C4×#61892
#117209672
#117209662
#117209642
#117209632
#117209592
#117209552
#117209322
#105943353
#105896443