Finite Group Databasebeta

← #13148788

Subgroups — #13148788

Order 1728.

730 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#11721749C211yesmaximal, characteristic
2C9×#887C211yesmaximal, characteristic, complemented
3C3×#63449C211yesmaximal, characteristic, complemented
4C4×#6189C211yesmaximal, characteristic, complemented
5#11720953C211yesmaximal, characteristic
6#11720952C211yesmaximal, characteristic
7#11720934C212yesmaximal, complemented
8#11720931C212yesmaximal, complemented
9#11720930C211yesmaximal, characteristic, complemented
10#11720926C212yesmaximal, complemented
11#11720923C212yesmaximal, complemented
12#10594350—31nomaximal
13#10589670—31nomaximal
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×C4×C3⋊D9C2212yes—
16C2×C9×Dic6C2211yescharacteristic
17C2×C3×Dic18C2211yescharacteristic
18C2×#6189C2211yescharacteristic
19C3×C9×C4⋊C4C2211yescharacteristic
20#88536C2212yes—
21C4×C3⋊Dic9C2211yescharacteristic
22C9×Dic3⋊C4C2212yes—
23C4×C9×Dic3C2211yescharacteristic
24C4×C9×Dic3C2212yes—
25C3×Dic9⋊C4C2212yes—
26C3×C4×Dic9C2211yescharacteristic
27C3×C4×Dic9C2212yes—
28#88445C2212yes—
29#88445C2212yes—
30#88445C2212yes—
31#88445C2212yes—
32Dic3⋊Dic9C2211yescharacteristic
33#88442C2211yescharacteristic
34#88442C2212yes—
35#88442C2212yes—
36Dic9⋊Dic3C2211yescharacteristic
37#64102—31no—
38C3×#887—31no—
39C3×#887D311yescharacteristic
40C4×#627—31no—
41#63874—32no—
42#63873—31no—
43#63868—32no—
44#63868—32no—
45#63865—32no—
46#63864—31no—
47#63864—31no—
48C4×Q8×C9—31no—
49#63509—31no—
50#63506—31no—
51#63460—32no—
52#63458—32no—
53#63455—31no—
54#63455—61no—
55#63449D311yescharacteristic, complemented
56#63438—32no—
57#63437—31no—
58#63434—32no—
59C4×Dic18—31no—
60C22×C3⋊D9C2312yes—
61C2×C3×C4×C9C2312yes—
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3⋊Dic9C2311yescharacteristic
64C2×C3⋊Dic9C2311yescharacteristic
65C4⋊(C3⋊D9)—22no—
66C4×C3⋊D9—24no—
67C2×C9×Dic3C2311yescharacteristic
68C2×C9×Dic3C2312yes—
69C2×C9×Dic3C2311yescharacteristic
70C2×C3×Dic9C2311yescharacteristic
71C2×C3×Dic9C2312yes—
72C2×C3×Dic9C2311yescharacteristic
73C9×Dic6—22no—
74C3×Dic18—22no—
75#6189—22no—
76#5568—91no—
77#5568—91no—
78C2×C4⋊(C3⋊D3)—31no—
79C2×C4×C3⋊D3—32no—
80C2×C3×Dic6—31no—
81C2×C3×Dic6C2×D311yescharacteristic
82C2×#627—31no—
83C32×C4⋊C4C2×D311yescharacteristic
84#3752—32no—
85C4×C3⋊Dic3—31no—
86C3×Dic3⋊C4—32no—
87C3×Dic3⋊C4C2×D312yes—
88C3×C4×Dic3—31no—
89C3×C4×Dic3—32no—
90C3×C4×Dic3C2×D311yescharacteristic
91C3×C4×Dic3C2×D312yes—
92#3724—32no—
93#3724—32no—
94#3724—31no—
95Dic3⋊Dic3—31no—
96Dic3⋊Dic3—31no—
97#3722—32no—
98#3722—32no—
99#3722—32no—
100#3722—32no—
101C2×Q8×C9—31no—
102C2×D36—31no—
103C2×D36—61no—
104C2×C4×D9—32no—
105C2×C4×D9—62no—
106C2×Dic18—31no—
107C2×Dic18C2×D311yescharacteristic
108C9×C4⋊C4—21no—
109C9×C4⋊C4—32no—
110C9×C4⋊C4C2×D311yescharacteristic
111C42×C9—31no—
112C42×C9—32no—
113(C2×D9)⋊C4—32no—
114(C2×D9)⋊C4—32no—
115(C2×D9)⋊C4—32no—
116(C2×D9)⋊C4—32no—
117(C2×D9)⋊C4—32no—
118(C2×D9)⋊C4—62no—
119C4⋊Dic9—31no—
120Dic9⋊C4—32no—
121Dic9⋊C4—32no—
122Dic9⋊C4—31no—
123Dic9⋊C4—31no—
124Dic9⋊C4C2×D312yes—
125C4×Dic9—31no—
126C4×Dic9—61no—
127C4×Dic9C2×D311yescharacteristic
128C4×Dic9C2×D312yes—
129C22×C3×C9C2411yescharacteristic
130C2×C3⋊D9—22no—
131C2×C3⋊D9—22no—
132C2×C3⋊D9—22no—
133C3×C4×C9—22no—
134C3×C4×C9#4112yes—
135C3⋊Dic9—21no—
136C3⋊Dic9#4112yes—
137C9×Dic3—21no—
138C9×Dic3—22no—
139C9×Dic3—21no—
140C3×Dic9—21no—
141C3×Dic9—22no—
142C3×Dic9—21no—
143C3×C4×Q8—31no—
144C3×C4×Q8—91no—
145#947—91no—
146#947—91no—
147#944—91no—
148#944—91no—
149#898—92no—
150#898—92no—
151#896—92no—
152#896—92no—
153#893—91no—
154#893—91no—
155#893—181no—
156#887—31no—
157#887D911yescharacteristic, complemented
158#876—92no—
159#876—92no—
160#875—91no—
161#875—91no—
162#872—92no—
163#872—92no—
164C4×Dic6—91no—
165C4×Dic6—91no—
166C22×C3⋊D3—32no—
167C2×C32×C4C22×D312yes—
168C2×C32×C4C22×D311yescharacteristic
169C2×C3⋊Dic3—31no—
170C2×C3⋊Dic3—31no—
171C4⋊(C3⋊D3)—62no—
172C4×C3⋊D3—64no—
173C2×C3×Dic3—31no—
174C2×C3×Dic3—31no—
175C2×C3×Dic3—32no—
176C2×C3×Dic3C22×D311yescharacteristic
177C2×C3×Dic3C22×D311yescharacteristic
178C2×C3×Dic3C22×D312yes—
179C3×Dic6—22no—
180C3×Dic6—62no—
181#627—62no—
182C22×D9—32no—
183C22×D9—62no—
184Q8×C9—62no—
185C2×C4×C9—21no—
186C2×C4×C9—22no—
187C2×C4×C9—31no—
188C2×C4×C9—32no—
189C2×C4×C9—31no—
190C2×C4×C9C22×D312yes—
191C2×C4×C9C22×D311yescharacteristic
192C2×Dic9—31no—
193C2×Dic9—31no—
194C2×Dic9—61no—
195C2×Dic9—61no—
196C2×Dic9C22×D311yescharacteristic
197C2×Dic9C22×D312yes—
198C2×Dic9C22×D311yescharacteristic
199D36—64no—
200D36—62no—
201C4×D9—64no—
202C4×D9—124no—
203Dic18—22no—
204Dic18—62no—
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×C4×D3—182no—
218C2×Dic6—31no—
219C2×Dic6—91no—
220C2×Dic6—91no—
221C2×Dic6C2×D911yescharacteristic
222C3×C4⋊C4—21no—
223C3×C4⋊C4—32no—
224C3×C4⋊C4—92no—
225C3×C4⋊C4C2×D911yescharacteristic
226C3×C4⋊C4D3211yescharacteristic
227C3×C42—31no—
228C3×C42—32no—
229C3×C42—91no—
230C3×C42—92no—
231(C2×D3)⋊C4—92no—
232(C2×D3)⋊C4—92no—
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—182no—
242C4⋊Dic3—91no—
243C4⋊Dic3—91no—
244Dic3⋊C4—32no—
245Dic3⋊C4—91no—
246Dic3⋊C4—91no—
247Dic3⋊C4—92no—
248Dic3⋊C4—92no—
249Dic3⋊C4—91no—
250Dic3⋊C4—91no—
251Dic3⋊C4—92no—
252Dic3⋊C4—92no—
253Dic3⋊C4C2×D912yes—
254C4×Dic3—31no—
255C4×Dic3—32no—
256C4×Dic3—91no—
257C4×Dic3—91no—
258C4×Dic3—181no—
259C4×Dic3C2×D911yescharacteristic
260C4×Dic3C2×D912yes—
261C22×C32C23×D311yescharacteristic
262C2×C3⋊D3—62no—
263C2×C3⋊D3—62no—
264C2×C3⋊D3—62no—
265C32×C4—22no—
266C32×C4#23512yes—
267C3⋊Dic3—32no—
268C3⋊Dic3—61no—
269C3×Dic3—22no—
270C3×Dic3—21no—
271C3×Dic3—21no—
272C3×Dic3—62no—
273C3×Dic3—61no—
274C3×Dic3—61no—
275C22×C9—21no—
276C22×C9C23×D311yescharacteristic
277C2×D9—64no—
278C2×D9—62no—
279C2×D9—62no—
280C2×D9—62no—
281C2×D9—122no—
282C2×D9—122no—
283C4×C9—22no—
284C4×C9—22no—
285C4×C9—42no—
286C4×C9—62no—
287C4×C9—61no—
288C4×C9—61no—
289C4×C9#23512yes—
290Dic9—22no—
291Dic9—21no—
292Dic9—21no—
293Dic9—32no—
294Dic9—62no—
295Dic9—61no—
296Dic9—121no—
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—22no—
316C2×C3×C4—31no—
317C2×C3×C4—32no—
318C2×C3×C4—31no—
319C2×C3×C4—91no—
320C2×C3×C4—92no—
321C2×C3×C4—91no—
322C2×C3×C4C22×D912yes—
323C2×C3×C4C22×D911yescharacteristic
324C2×C3×C4C2×D3212yes—
325C2×C3×C4C2×D3211yescharacteristic
326C2×Dic3—31no—
327C2×Dic3—31no—
328C2×Dic3—32no—
329C2×Dic3—91no—
330C2×Dic3—91no—
331C2×Dic3—91no—
332C2×Dic3—91no—
333C2×Dic3—181no—
334C2×Dic3—181no—
335C2×Dic3C22×D911yescharacteristic
336C2×Dic3C22×D912yes—
337C2×Dic3C22×D911yescharacteristic
338D12—184no—
339D12—182no—
340D12—182no—
341C4×D3—184no—
342C4×D3—184no—
343C4×D3—364no—
344Dic6—22no—
345Dic6—62no—
346Dic6—182no—
347Dic6—182no—
348C2×C32C2×#23511yescharacteristic
349C2×C32#100711yescharacteristic
350C2×C32D3×#4111yescharacteristic
351C3⋊D3—122no—
352C2×C9—21no—
353C2×C9—21no—
354C2×C9—21no—
355C2×C9C2×#23511yescharacteristic
356C2×C9#100711yescharacteristic
357C2×C9D3×#4111yescharacteristic
358D9—124no—
359D9—122no—
360C2×Q8—31no—
361C2×Q8—91no—
362C2×Q8—271no—
363C2×D4—271no—
364C22×C4—272no—
365C4⋊C4—32no—
366C4⋊C4—92no—
367C4⋊C4—271no—
368C4⋊C4—272no—
369C4⋊C4—271no—
370C4⋊C4—271no—
371C4⋊C4—272no—
372C4⋊C4D3×D911yescharacteristic
373C22⋊C4—272no—
374C22⋊C4—272no—
375C22⋊C4—272no—
376C22⋊C4—272no—
377C22⋊C4—272no—
378C42—32no—
379C42—31no—
380C42—92no—
381C42—91no—
382C42—271no—
383C22×C3—21no—
384C22×C3C23×D911yescharacteristic
385C22×C3C22×D3211yescharacteristic
386C2×D3—184no—
387C2×D3—182no—
388C2×D3—182no—
389C2×D3—182no—
390C2×D3—182no—
391C2×D3—182no—
392C2×D3—182no—
393C2×D3—362no—
394C2×D3—362no—
395C3×C4—22no—
396C3×C4—22no—
397C3×C4—22no—
398C3×C4—42no—
399C3×C4—62no—
400C3×C4—61no—
401C3×C4—61no—
402C3×C4—182no—
403C3×C4—181no—
404C3×C4—181no—
405C3×C4#369712yes—
406C3×C4#379812yes—
407Dic3—22no—
408Dic3—21no—
409Dic3—21no—
410Dic3—62no—
411Dic3—61no—
412Dic3—61no—
413Dic3—92no—
414Dic3—92no—
415Dic3—182no—
416Dic3—181no—
417Dic3—181no—
418Dic3—361no—
419C32#556811yescharacteristic
420C9—21no—
421C9#556811yescharacteristic, complemented
422C23—272no—
423Q8—62no—
424Q8—182no—
425Q8—542no—
426D4—542no—
427C2×C4—31no—
428C2×C4—31no—
429C2×C4—32no—
430C2×C4—91no—
431C2×C4—91no—
432C2×C4—92no—
433C2×C4—271no—
434C2×C4—271no—
435C2×C4—544no—
436C2×C4C2×D3×D912yes—
437C2×C4C2×D3×D911yescharacteristic
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3C2×#369711yescharacteristic
442C2×C3#6371311yescharacteristic
443C2×C3#41×D911yescharacteristic
444C2×C3C2×#379811yescharacteristic
445C2×C3#6431211yescharacteristic
446C2×C3#6431611yescharacteristic
447D3—364no—
448D3—362no—
449D3—362no—
450C22—542no—
451C22—542no—
452C22—542no—
453C22C22×D3×D911yescharacteristic
454C4—22no—
455C4—61no—
456C4—61no—
457C4—62no—
458C4—181no—
459C4—181no—
460C4—182no—
461C4—272no—
462C4—541no—
463C4#8864012yes—
464C3—21no—
465C3#1058967011yescharacteristic, complemented
466C3#1059435011yescharacteristic
467C2—1082no—
468C2C2×#8864011yescharacteristic
469C2#1172249811yescharacteristic
470C2#1172250911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC2×C4⋊(C3⋊D9)C22
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
2. C9×#887C2semidirect
3. C3×#63449C2semidirect
4. C4×#6189C2semidirect
7. #11720934C2semidirect
8. #11720931C2semidirect
9. #11720930C2semidirect
10. #11720926C2semidirect
11. #11720923C2semidirect
55. #63449D3semidirect
157. #887D9semidirect
308. C3×C9#556semidirect
421. C9#5568semidirect
465. C3#10589670semidirect

Maximal subgroups

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

subgroupindex
#117217492
C9×#8872
C3×#634492
C4×#61892
#117209532
#117209522
#117209342
#117209312
#117209302
#117209262
#117209232
#105943503
#105896703