Finite Group Databasebeta

← #13147532

Subgroups — #13147532

Order 1728.

712 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721712C211yesmaximal, characteristic
2C4×C9×Dic6C211yesmaximal, characteristic, complemented
3C3×C4×Dic18C211yesmaximal, characteristic, complemented
4#11720993C211yesmaximal, characteristic
5#11720986C212yesmaximal, complemented
6#11720952C211yesmaximal, characteristic
7#11720934C212yesmaximal, complemented
8#11720932C211yesmaximal, characteristic, complemented
9#11720920C212yesmaximal, complemented
10#11720912C211yesmaximal, characteristic, complemented
11#11720899C212yesmaximal, complemented
12#10593850—31nomaximal
13#10589686—31nomaximal
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×#6226C2211yescharacteristic
16C2×C9×Dic6C2211yescharacteristic
17C2×C3×Dic18C2211yescharacteristic
18C2×(C3⋊D9)⋊C4C2212yes—
19C3×C42×C9C2211yescharacteristic
20#88536C2212yes—
21#88536C2212yes—
22C9×C4⋊Dic3C2211yescharacteristic
23C9×Dic3⋊C4C2211yescharacteristic
24C9×Dic3⋊C4C2211yescharacteristic
25C4×C9×Dic3C2211yescharacteristic
26C4×C9×Dic3C2211yescharacteristic
27C3×C4⋊Dic9C2211yescharacteristic
28C3×Dic9⋊C4C2212yes—
29C3×C4×Dic9C2212yes—
30#88445C2212yes—
31#88445C2212yes—
32#88445C2212yes—
33Dic3⋊Dic9C2212yes—
34Dic3⋊Dic9C2212yes—
35#88442C2212yes—
36Dic3×Dic9C2212yes—
37#64084—31no—
38C3×C4×Dic6—31no—
39C3×C4×Dic6D311yescharacteristic
40#63894—31no—
41#63888—32no—
42#63868—32no—
43#63866—31no—
44#63864—31no—
45#63848—32no—
46#63844—31no—
47#63836—32no—
48C4×Q8×C9—31no—
49#63508—32no—
50#63460—32no—
51#63460—32no—
52#63454—32no—
53#63437—31no—
54#63437—31no—
55#63437—61no—
56C4×D36—31no—
57#63431—31no—
58C4×Dic18—31no—
59C4×Dic18D311yescharacteristic, complemented
60C22×C3⋊D9C2312yes—
61C2×C3×C4×C9C2311yescharacteristic
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2312yes—
65C4⋊(C3⋊D9)—22no—
66#6226—22no—
67C2×C9×Dic3C2311yescharacteristic
68C2×C9×Dic3C2311yescharacteristic
69C2×C9×Dic3C2311yescharacteristic
70C2×C9×Dic3C2311yescharacteristic
71C2×C3×Dic9C2312yes—
72C2×C3×Dic9C2312yes—
73C9×Dic6—22no—
74C3×Dic18—22no—
75(C3⋊D9)⋊C4—24no—
76#5584—91no—
77#5568—91no—
78C2×C4⋊(C3⋊D3)—31no—
79C2×#634—31no—
80C2×C3×Dic6—31no—
81C2×C3×Dic6C2×D311yescharacteristic
82C2×#624—32no—
83C32×C42C2×D311yescharacteristic
84#3752—32no—
85#3752—32no—
86C3×C4⋊Dic3—31no—
87C3×C4⋊Dic3C2×D311yescharacteristic
88C3×Dic3⋊C4—32no—
89C3×Dic3⋊C4C2×D311yescharacteristic
90C3×Dic3⋊C4C2×D311yescharacteristic
91C3×C4×Dic3—32no—
92C3×C4×Dic3C2×D311yescharacteristic
93C3×C4×Dic3C2×D311yescharacteristic
94#3724—32no—
95Dic3⋊Dic3—32no—
96Dic3⋊Dic3—32no—
97#3722—32no—
98#3722—32no—
99#3722—32no—
100Dic32—32no—
101C2×Q8×C9—31no—
102C2×D36—31no—
103C2×D36—61no—
104C2×C4×D9—32no—
105C2×Dic18—31no—
106C2×Dic18—61no—
107C2×Dic18C2×D311yescharacteristic
108C9×C4⋊C4—31no—
109C9×C4⋊C4—31no—
110C9×C4⋊C4—31no—
111C42×C9—21no—
112C42×C9—31no—
113C42×C9—31no—
114C42×C9C2×D311yescharacteristic
115(C2×D9)⋊C4—32no—
116(C2×D9)⋊C4—32no—
117(C2×D9)⋊C4—32no—
118(C2×D9)⋊C4—32no—
119(C2×D9)⋊C4—32no—
120(C2×D9)⋊C4—62no—
121(C2×D9)⋊C4—62no—
122C4⋊Dic9—32no—
123C4⋊Dic9—32no—
124C4⋊Dic9C2×D311yescharacteristic
125Dic9⋊C4—32no—
126Dic9⋊C4C2×D312yes—
127C4×Dic9—32no—
128C4×Dic9C2×D312yes—
129C22×C3×C9C2411yescharacteristic
130C2×C3⋊D9—22no—
131C2×C3⋊D9—22no—
132C2×C3⋊D9—22no—
133C3×C4×C9—21no—
134C3×C4×C9—21no—
135C3×C4×C9#4112yes—
136C3⋊Dic9—22no—
137C9×Dic3—21no—
138C9×Dic3—21no—
139C9×Dic3—21no—
140C9×Dic3#4112yes—
141C3×Dic9—22no—
142C3×Dic9—22no—
143C3×C4×Q8—31no—
144C3×C4×Q8—91no—
145#947—91no—
146#946—92no—
147#944—91no—
148#898—92no—
149#898—92no—
150#898—92no—
151#896—92no—
152#893—91no—
153#892—92no—
154#887—91no—
155#876—92no—
156#875—91no—
157#875—91no—
158#875—91no—
159#875—181no—
160C4×D12—91no—
161#872—92no—
162#869—91no—
163C4×Dic6—31no—
164C4×Dic6—91no—
165C4×Dic6D911yescharacteristic, complemented
166C22×C3⋊D3—32no—
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C22×D311yescharacteristic
170C2×C3⋊Dic3—32no—
171C4⋊(C3⋊D3)—62no—
172#634—62no—
173C2×C3×Dic3—32no—
174C2×C3×Dic3—32no—
175C2×C3×Dic3C22×D311yescharacteristic
176C2×C3×Dic3C22×D311yescharacteristic
177C2×C3×Dic3C22×D311yescharacteristic
178C2×C3×Dic3C22×D311yescharacteristic
179C3×Dic6—22no—
180C3×Dic6—62no—
181#624—64no—
182C22×D9—32no—
183C22×D9—62no—
184Q8×C9—62no—
185C2×C4×C9—21no—
186C2×C4×C9—21no—
187C2×C4×C9—21no—
188C2×C4×C9—31no—
189C2×C4×C9—31no—
190C2×C4×C9—31no—
191C2×C4×C9—31no—
192C2×C4×C9C22×D311yescharacteristic
193C2×C4×C9C22×D311yescharacteristic
194C2×C4×C9C22×D311yescharacteristic
195C2×Dic9—32no—
196C2×Dic9—62no—
197C2×Dic9C22×D312yes—
198C2×Dic9C22×D312yes—
199D36—62no—
200D36—122no—
201C4×D9—64no—
202Dic18—22no—
203Dic18—62no—
204Dic18—122no—
205#556—271noSylow
206C2×C3×C9C2×#4111yescharacteristic
207C2×C3×C9C2×#4111yescharacteristic
208C2×C3×C92−1+411yescharacteristic
209C3⋊D9—42no—
210C2×C3×Q8—31no—
211C2×C3×Q8—91no—
212C2×D12—91no—
213C2×D12—91no—
214C2×D12—181no—
215C2×C4×D3—92no—
216C2×C4×D3—92no—
217C2×Dic6—31no—
218C2×Dic6—91no—
219C2×Dic6—91no—
220C2×Dic6—181no—
221C2×Dic6C2×D911yescharacteristic
222C3×C4⋊C4—31no—
223C3×C4⋊C4—31no—
224C3×C4⋊C4—31no—
225C3×C4⋊C4—91no—
226C3×C4⋊C4—92no—
227C3×C42—21no—
228C3×C42—31no—
229C3×C42—31no—
230C3×C42—92no—
231C3×C42C2×D911yescharacteristic
232C3×C42D3211yescharacteristic
233(C2×D3)⋊C4—92no—
234(C2×D3)⋊C4—92no—
235(C2×D3)⋊C4—92no—
236(C2×D3)⋊C4—92no—
237(C2×D3)⋊C4—92no—
238(C2×D3)⋊C4—92no—
239(C2×D3)⋊C4—92no—
240(C2×D3)⋊C4—92no—
241(C2×D3)⋊C4—92no—
242(C2×D3)⋊C4—92no—
243(C2×D3)⋊C4—182no—
244(C2×D3)⋊C4—182no—
245C4⋊Dic3—31no—
246C4⋊Dic3—92no—
247C4⋊Dic3—92no—
248C4⋊Dic3C2×D911yescharacteristic
249Dic3⋊C4—32no—
250Dic3⋊C4—92no—
251Dic3⋊C4—92no—
252Dic3⋊C4—92no—
253Dic3⋊C4—92no—
254Dic3⋊C4C2×D911yescharacteristic
255Dic3⋊C4C2×D911yescharacteristic
256C4×Dic3—32no—
257C4×Dic3—92no—
258C4×Dic3—92no—
259C4×Dic3C2×D911yescharacteristic
260C4×Dic3C2×D911yescharacteristic
261C22×C32C23×D311yescharacteristic
262C2×C3⋊D3—62no—
263C2×C3⋊D3—62no—
264C2×C3⋊D3—62no—
265C32×C4—21no—
266C32×C4—21no—
267C32×C4#23912yes—
268C3⋊Dic3—62no—
269C3×Dic3—21no—
270C3×Dic3—21no—
271C3×Dic3—21no—
272C3×Dic3—62no—
273C3×Dic3—62no—
274C3×Dic3#23512yes—
275C22×C9—21no—
276C22×C9C23×D311yescharacteristic
277C2×D9—62no—
278C2×D9—62no—
279C2×D9—62no—
280C2×D9—122no—
281C2×D9—122no—
282C2×D9—122no—
283C4×C9—21no—
284C4×C9—21no—
285C4×C9—22no—
286C4×C9—32no—
287C4×C9—41no—
288C4×C9—41no—
289C4×C9—61no—
290C4×C9—61no—
291C4×C9—61no—
292C4×C9#23512yes—
293Dic9—22no—
294Dic9—22no—
295Dic9—62no—
296Dic9—122no—
297C4⋊Q8—271no—
298#126—272no—
299#126—272no—
300#124—271no—
301#124—271no—
302C22⋊Q8—272no—
303C4×Q8—31no—
304C4×Q8—91no—
305C4×Q8—271no—
306C4×D4—271no—
307#117—272no—
308C3×C9#55611yescharacteristic, complemented, Sylow
309C22×D3—92no—
310C22×D3—92no—
311C22×D3—182no—
312C3×Q8—62no—
313C3×Q8—182no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—31no—
318C2×C3×C4—31no—
319C2×C3×C4—31no—
320C2×C3×C4—31no—
321C2×C3×C4—92no—
322C2×C3×C4—92no—
323C2×C3×C4C22×D911yescharacteristic
324C2×C3×C4C22×D911yescharacteristic
325C2×C3×C4C22×D911yescharacteristic
326C2×C3×C4C2×D3211yescharacteristic
327C2×C3×C4C2×D3211yescharacteristic
328C2×C3×C4C2×D3211yescharacteristic
329C2×Dic3—32no—
330C2×Dic3—32no—
331C2×Dic3—92no—
332C2×Dic3—92no—
333C2×Dic3—182no—
334C2×Dic3C22×D911yescharacteristic
335C2×Dic3C22×D911yescharacteristic
336C2×Dic3C22×D911yescharacteristic
337C2×Dic3C22×D911yescharacteristic
338D12—182no—
339D12—182no—
340D12—362no—
341C4×D3—184no—
342C4×D3—184no—
343Dic6—22no—
344Dic6—62no—
345Dic6—182no—
346Dic6—182no—
347Dic6—362no—
348C2×C32C2×#23511yescharacteristic
349C2×C32C2×#23911yescharacteristic
350C2×C32#101011yescharacteristic
351C3⋊D3—122no—
352C2×C9—21no—
353C2×C9—21no—
354C2×C9—21no—
355C2×C9C2×#23511yescharacteristic
356C2×C9#100711yescharacteristic
357C2×C9D3×#4111yescharacteristic
358D9—122no—
359D9—242no—
360C2×Q8—31no—
361C2×Q8—91no—
362C2×Q8—271no—
363C2×D4—271no—
364C22×C4—272no—
365C4⋊C4—31no—
366C4⋊C4—31no—
367C4⋊C4—31no—
368C4⋊C4—91no—
369C4⋊C4—92no—
370C4⋊C4—272no—
371C4⋊C4—272no—
372C4⋊C4—272no—
373C22⋊C4—272no—
374C22⋊C4—272no—
375C22⋊C4—272no—
376C22⋊C4—272no—
377C22⋊C4—272no—
378C42—31no—
379C42—31no—
380C42—92no—
381C42—272no—
382C42D3×D911yescharacteristic
383C22×C3—21no—
384C22×C3C23×D911yescharacteristic
385C22×C3C22×D3211yescharacteristic
386C2×D3—182no—
387C2×D3—182no—
388C2×D3—182no—
389C2×D3—182no—
390C2×D3—182no—
391C2×D3—182no—
392C2×D3—362no—
393C2×D3—362no—
394C2×D3—362no—
395C3×C4—21no—
396C3×C4—21no—
397C3×C4—21no—
398C3×C4—21no—
399C3×C4—22no—
400C3×C4—32no—
401C3×C4—41no—
402C3×C4—41no—
403C3×C4—61no—
404C3×C4—61no—
405C3×C4—61no—
406C3×C4—182no—
407C3×C4—182no—
408C3×C4#370112yes—
409C3×C4#379912yes—
410Dic3—21no—
411Dic3—21no—
412Dic3—21no—
413Dic3—62no—
414Dic3—62no—
415Dic3—182no—
416Dic3—182no—
417Dic3—362no—
418Dic3#369712yes—
419C32#558411yescharacteristic
420C9—21no—
421C9#556811yescharacteristic, complemented
422C23—272no—
423Q8—62no—
424Q8—182no—
425Q8—542no—
426D4—542no—
427C2×C4—31no—
428C2×C4—31no—
429C2×C4—31no—
430C2×C4—31no—
431C2×C4—92no—
432C2×C4—92no—
433C2×C4—272no—
434C2×C4—544no—
435C2×C4C2×D3×D911yescharacteristic
436C2×C4C2×D3×D911yescharacteristic
437C2×C4C2×D3×D911yescharacteristic
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3C2×#369711yescharacteristic
442C2×C3C2×#370111yescharacteristic
443C2×C3#6371611yescharacteristic
444C2×C3#6429711yescharacteristic
445C2×C3C2×#379911yescharacteristic
446C2×C3D3×#23511yescharacteristic
447D3—362no—
448D3—362no—
449D3—722no—
450C22—542no—
451C22—542no—
452C22—542no—
453C22C22×D3×D911yescharacteristic
454C4—21no—
455C4—21no—
456C4—32no—
457C4—61no—
458C4—61no—
459C4—61no—
460C4—182no—
461C4—182no—
462C4—542no—
463C4#8863512yes—
464C3—21no—
465C3#1058968611yescharacteristic, complemented
466C3#1059385011yescharacteristic
467C2—1082no—
468C2C2×#8863511yescharacteristic
469C2#1172246411yescharacteristic
470C2D3×#369711yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC2×C4⋊(C3⋊D9)C22
hypercenterC42D3×D9
hypocenterC3×C9#556
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C4×C9×Dic6C2semidirect
3. C3×C4×Dic18C2semidirect
5. #11720986C2semidirect
7. #11720934C2semidirect
8. #11720932C2semidirect
9. #11720920C2semidirect
10. #11720912C2semidirect
11. #11720899C2semidirect
59. C4×Dic18D3semidirect
165. C4×Dic6D9semidirect
308. C3×C9#556semidirect
421. C9#5568semidirect
465. C3#10589686semidirect

Maximal subgroups

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

subgroupindex
#117217122
C4×C9×Dic62
C3×C4×Dic182
#117209932
#117209862
#117209522
#117209342
#117209322
#117209202
#117209122
#117208992
#105938503
#105896863