Finite Group Databasebeta

← #13148946

Subgroups — #13148946

Order 1728.

752 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721744C211yesmaximal, characteristic, complemented
2C3×#63455C211yesmaximal, characteristic
3C9×#890C211yesmaximal, characteristic, complemented
4#11720999C212yesmaximal, complemented
5#11720998C211yesmaximal, characteristic
6#11720996C212yesmaximal, complemented
7Dic3×D36C211yesmaximal, characteristic
8D9×Dic3⋊C4C212yesmaximal, complemented
9C4×#6189C211yesmaximal, characteristic, complemented
10#11720945C212yesmaximal, complemented
11#11720935C211yesmaximal, characteristic, complemented
12#10594485—31nomaximal
13#10589792—31nomaximal
14C2×#6226C2212yes—
15C2×C3×D36C2211yescharacteristic
16C2×C3×C4×D9C2212yes—
17C2×Dic3×D9C2212yes—
18C2×Dic3×D9C2212yes—
19C2×#6189C2211yescharacteristic
20C3×C9×C4⋊C4C2211yescharacteristic
21C4⋊(C3⋊Dic9)C2211yescharacteristic
22#88534C2212yes—
23C4×C3⋊Dic9C2211yescharacteristic
24C3×(C2×D9)⋊C4C2212yes—
25C9×C4⋊Dic3C2211yescharacteristic
26C9×C4⋊Dic3C2212yes—
27C9×Dic3⋊C4C2212yes—
28C4×C9×Dic3C2211yescharacteristic
29C3×C4×Dic9C2211yescharacteristic
30(C2×D9)⋊Dic3C2212yes—
31(C2×D9)⋊Dic3C2212yes—
32Dic3⋊Dic9C2211yescharacteristic
33Dic3⋊Dic9C2212yes—
34#88442C2211yescharacteristic
35#88442C2212yes—
36Dic9⋊Dic3C2211yescharacteristic
37#64097—31no—
38C3×#893—31no—
39C3×#890D311yescharacteristic
40C4×#627—31no—
41#63899—32no—
42#63898—31no—
43#63896—32no—
44Dic3×D12—31no—
45D3×Dic3⋊C4—32no—
46#63869—31no—
47#63857—32no—
48C9×#125—31no—
49#63459—32no—
50#63458—32no—
51#63458—32no—
52#63455—31no—
53#63455D311yescharacteristic
54C4⋊C4×D9—32no—
55#63452—31no—
56#63450—31no—
57#63450—61no—
58C4×D36—31no—
59C4×Dic18—31no—
60C22×C3×D9C2312yes—
61C2×C3×C4×C9C2312yes—
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3⋊Dic9C2312yes—
64C2×C3⋊Dic9C2311yescharacteristic
65C2×C3⋊Dic9C2311yescharacteristic
66#6226—42no—
67C2×C9×Dic3C2311yescharacteristic
68C2×C9×Dic3C2311yescharacteristic
69C2×C9×Dic3C2312yes—
70C2×C3×Dic9C2311yescharacteristic
71C2×C3×Dic9C2311yescharacteristic
72C3×D36Q814yescomplemented
73C3×C4×D9—42no—
74Dic3×D9—24no—
75Dic3×D9—42no—
76#6189—22no—
77#5690—91no—
78#5539—91no—
79C2×#634—32no—
80C2×C3×D12—31no—
81C2×C3×C4×D3—32no—
82C2×#627—31no—
83C2×D3×Dic3—32no—
84C2×D3×Dic3—32no—
85C32×C4⋊C4C2×D311yescharacteristic
86C4⋊(C3⋊Dic3)—31no—
87#3750—32no—
88C4×C3⋊Dic3—31no—
89C3×(C2×D3)⋊C4—32no—
90C3×C4⋊Dic3C2×D312yes—
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×Dic3⋊C4C2×D312yes—
93C3×C4×Dic3—31no—
94C3×C4×Dic3C2×D311yescharacteristic
95#3724—32no—
96#3724—31no—
97Dic3⋊Dic3—31no—
98Dic3⋊Dic3—32no—
99Dic3⋊Dic3—31no—
100(C2×D3)⋊Dic3—32no—
101(C2×D3)⋊Dic3—32no—
102C2×D36C2×D311yescharacteristic
103C2×C4×D9—32no—
104C2×C4×D9—32no—
105C2×C4×D9C2×D312yes—
106C2×Dic18—31no—
107C2×Dic18—32no—
108C2×Dic18—62no—
109C9×C4⋊C4—21no—
110C9×C4⋊C4—32no—
111C9×C4⋊C4—31no—
112C9×C4⋊C4—32no—
113C9×C4⋊C4C2×D311yescharacteristic
114C42×C9—31no—
115(C2×D9)⋊C4—32no—
116(C2×D9)⋊C4—32no—
117(C2×D9)⋊C4C2×D312yes—
118C4⋊Dic9—31no—
119C4⋊Dic9—31no—
120C4⋊Dic9—32no—
121C4⋊Dic9—61no—
122Dic9⋊C4—32no—
123Dic9⋊C4—31no—
124Dic9⋊C4—31no—
125Dic9⋊C4—32no—
126Dic9⋊C4—62no—
127C4×Dic9—31no—
128C4×Dic9—61no—
129C4×Dic9C2×D311yescharacteristic
130C22×C3×C9C2411yescharacteristic
131C2×C3×D9—22no—
132C2×C3×D9—22no—
133C2×C3×D9C2×Q814yes—
134C3×C4×C9—22no—
135C3×C4×C9C2×Q812yes—
136C3⋊Dic9—21no—
137C3⋊Dic9—22no—
138C3⋊Dic9#4112yes—
139C9×Dic3—21no—
140C9×Dic3—22no—
141C9×Dic3—21no—
142C3×Dic9—21no—
143C3×Dic9—21no—
144C3×#125—31no—
145C3×C4×D4—91no—
146D4×Dic3—91no—
147D4×Dic3—91no—
148#924—92no—
149C2×Dic3⋊C4—92no—
150#897—92no—
151#896—92no—
152#896—92no—
153#893—31no—
154#893—91no—
155D3×C4⋊C4—92no—
156#890—91no—
157#890D911yescharacteristic, complemented
158#888—91no—
159#888—91no—
160#888—181no—
161#878—92no—
162#878—92no—
163C4×D12—91no—
164#870—91no—
165C4×Dic6—91no—
166C4×Dic6—91no—
167C22×C3×D3—32no—
168C2×C32×C4C22×D312yes—
169C2×C32×C4C22×D311yescharacteristic
170C2×C3⋊Dic3—31no—
171C2×C3⋊Dic3—31no—
172C2×C3⋊Dic3—32no—
173#634—122no—
174C2×C3×Dic3—31no—
175C2×C3×Dic3—31no—
176C2×C3×Dic3C22×D311yescharacteristic
177C2×C3×Dic3C22×D312yes—
178C2×C3×Dic3C22×D311yescharacteristic
179C3×D12—34no—
180C3×C4×D3—122no—
181#627—62no—
182D3×Dic3—64no—
183D3×Dic3—122no—
184C22×D9C22×D312yes—
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—32no—
195C2×Dic9—61no—
196C2×Dic9—61no—
197C2×Dic9—62no—
198C2×Dic9C22×D311yescharacteristic
199C2×Dic9C22×D311yescharacteristic
200D36Dic614yescomplemented
201C4×D9—42no—
202C4×D9—64no—
203C4×D9—122no—
204Dic18—62no—
205Dic18—122no—
206Dic18—124no—
207#554—271noSylow
208C2×C3×C9C22×Q811yescharacteristic
209C2×C3×C9C2×#4111yescharacteristic
210C2×C3×C9H(5,𝔽2)11yescharacteristic
211C3×D9—24no—
212C2×C3×D4—91no—
213C22×C3×C4—92no—
214C22×Dic3—92no—
215C22×Dic3—92no—
216C2×D12—31no—
217C2×C4×D3—32no—
218C2×C4×D3—92no—
219C2×C4×D3—92no—
220C2×Dic6—91no—
221C2×Dic6—91no—
222C2×Dic6—92no—
223C2×Dic6—92no—
224C2×Dic6—182no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—31no—
227C3×C4⋊C4—32no—
228C3×C4⋊C4—32no—
229C3×C4⋊C4C2×D911yescharacteristic
230C3×C4⋊C4D3211yescharacteristic
231C3×C22⋊C4—92no—
232C3×C42—31no—
233C3×C42—91no—
234#217—92no—
235#217—92no—
236(C2×D3)⋊C4—32no—
237(C2×D3)⋊C4—92no—
238(C2×D3)⋊C4—92no—
239C4⋊Dic3—91no—
240C4⋊Dic3—92no—
241C4⋊Dic3—91no—
242C4⋊Dic3—91no—
243C4⋊Dic3—91no—
244C4⋊Dic3—181no—
245C4⋊Dic3C2×D911yescharacteristic
246C4⋊Dic3C2×D912yes—
247Dic3⋊C4—91no—
248Dic3⋊C4—91no—
249Dic3⋊C4—92no—
250Dic3⋊C4—92no—
251Dic3⋊C4—92no—
252Dic3⋊C4—91no—
253Dic3⋊C4—91no—
254Dic3⋊C4—92no—
255Dic3⋊C4—92no—
256Dic3⋊C4—182no—
257Dic3⋊C4C2×D912yes—
258C4×Dic3—31no—
259C4×Dic3—91no—
260C4×Dic3—91no—
261C4×Dic3—181no—
262C4×Dic3C2×D911yescharacteristic
263C22×C32C23×D311yescharacteristic
264C2×C3×D3—34no—
265C2×C3×D3—62no—
266C2×C3×D3—62no—
267C32×C4—22no—
268C32×C4D3×Q812yes—
269C3⋊Dic3—32no—
270C3⋊Dic3—61no—
271C3⋊Dic3—62no—
272C3×Dic3—22no—
273C3×Dic3—21no—
274C3×Dic3—21no—
275C3×Dic3—61no—
276C3×Dic3—61no—
277C22×C9—21no—
278C22×C9C23×D311yescharacteristic
279C2×D9—22no—
280C2×D9—22no—
281C2×D9C2×Dic614yes—
282C4×C9—22no—
283C4×C9—22no—
284C4×C9—42no—
285C4×C9—61no—
286C4×C9—61no—
287C4×C9—62no—
288C4×C9C2×Dic612yes—
289Dic9—21no—
290Dic9—21no—
291Dic9—32no—
292Dic9—62no—
293Dic9—62no—
294Dic9—61no—
295Dic9—62no—
296Dic9—122no—
297C4⋊Q8—271no—
298#125—31no—
299#125—271no—
300C22⋊Q8—272no—
301C22⋊Q8—272no—
302C22⋊Q8—272no—
303C4×Q8—271no—
304C4×D4—91no—
305C4×D4—271no—
306C4×D4—271no—
307C2×C4⋊C4—272no—
308C3×C9#55411yescharacteristic, complemented, Sylow
309C23×C3—92no—
310C22×D3—32no—
311C3×D4—94no—
312C2×C3×C4—21no—
313C2×C3×C4—22no—
314C2×C3×C4—31no—
315C2×C3×C4—32no—
316C2×C3×C4—31no—
317C2×C3×C4—91no—
318C2×C3×C4—91no—
319C2×C3×C4—362no—
320C2×C3×C4C22×D912yes—
321C2×C3×C4C22×D911yescharacteristic
322C2×C3×C4C2×D3212yes—
323C2×C3×C4C2×D3211yescharacteristic
324C2×Dic3—31no—
325C2×Dic3—31no—
326C2×Dic3—91no—
327C2×Dic3—91no—
328C2×Dic3—91no—
329C2×Dic3—92no—
330C2×Dic3—91no—
331C2×Dic3—92no—
332C2×Dic3—181no—
333C2×Dic3—181no—
334C2×Dic3—182no—
335C2×Dic3—184no—
336C2×Dic3—362no—
337C2×Dic3C22×D911yescharacteristic
338C2×Dic3C22×D911yescharacteristic
339C2×Dic3C22×D912yes—
340D12—34no—
341C4×D3—122no—
342C4×D3—184no—
343C4×D3—362no—
344Dic6—182no—
345Dic6—182no—
346Dic6—362no—
347Dic6—362no—
348Dic6—364no—
349C2×C32C2×D3×Q811yescharacteristic
350C2×C32D3×#4111yescharacteristic
351C2×C32#100911yescharacteristic
352C3×D3—64no—
353C2×C9—21no—
354C2×C9—21no—
355C2×C9—21no—
356C2×C9C22×Dic611yescharacteristic
357C2×C9#100411yescharacteristic
358C2×C9D3×#4111yescharacteristic
359D9—24no—
360C2×Q8—271no—
361C2×Q8—272no—
362C2×D4—91no—
363C22×C4—92no—
364C22×C4—272no—
365C22×C4—272no—
366C4⋊C4—32no—
367C4⋊C4—31no—
368C4⋊C4—32no—
369C4⋊C4—271no—
370C4⋊C4—272no—
371C4⋊C4—271no—
372C4⋊C4—271no—
373C4⋊C4—272no—
374C4⋊C4—272no—
375C4⋊C4—271no—
376C4⋊C4D3×D911yescharacteristic
377C22⋊C4—92no—
378C22⋊C4—272no—
379C22⋊C4—272no—
380C42—31no—
381C42—91no—
382C42—271no—
383C22×C3—21no—
384C22×C3—94no—
385C22×C3—182no—
386C22×C3—182no—
387C22×C3C23×D911yescharacteristic
388C22×C3C22×D3211yescharacteristic
389C2×D3—34no—
390C2×D3—62no—
391C2×D3—62no—
392C3×C4—22no—
393C3×C4—22no—
394C3×C4—22no—
395C3×C4—42no—
396C3×C4—61no—
397C3×C4—61no—
398C3×C4—62no—
399C3×C4—181no—
400C3×C4—181no—
401C3×C4Q8×D912yes—
402C3×C4D3×Dic612yes—
403Dic3—21no—
404Dic3—22no—
405Dic3—21no—
406Dic3—61no—
407Dic3—61no—
408Dic3—92no—
409Dic3—92no—
410Dic3—182no—
411Dic3—182no—
412Dic3—181no—
413Dic3—182no—
414Dic3—181no—
415Dic3—182no—
416Dic3—362no—
417C32#569011yescharacteristic
418C9—21no—
419C9#553911yescharacteristic, complemented
420C23—92no—
421Q8—542no—
422Q8—1082no—
423D4—94no—
424C2×C4—31no—
425C2×C4—31no—
426C2×C4—32no—
427C2×C4—91no—
428C2×C4—91no—
429C2×C4—271no—
430C2×C4—271no—
431C2×C4—272no—
432C2×C4—362no—
433C2×C4—544no—
434C2×C4—1082no—
435C2×C4C2×D3×D912yes—
436C2×C4C2×D3×D911yescharacteristic
437C2×C3—21no—
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—184no—
441C2×C3C2×Q8×D911yescharacteristic
442C2×C3#41×D911yescharacteristic
443C2×C3#6371511yescharacteristic
444C2×C3C2×D3×Dic611yescharacteristic
445C2×C3#6431411yescharacteristic
446C2×C3#6431911yescharacteristic
447D3—64no—
448C22—94no—
449C22—182no—
450C22—182no—
451C22C22×D3×D911yescharacteristic
452C4—22no—
453C4—61no—
454C4—62no—
455C4—61no—
456C4—181no—
457C4—181no—
458C4—272no—
459C4—542no—
460C4—541no—
461C4D9×Dic612yes—
462C3—21no—
463C3#1058979211yescharacteristic, complemented
464C3#1059448511yescharacteristic
465C2—184no—
466C2C2×D9×Dic611yescharacteristic
467C2#1172250111yescharacteristic
468C2#1172251411yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC2×C3×D36C22
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#554
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. #11721744C2semidirect
3. C9×#890C2semidirect
4. #11720999C2semidirect
6. #11720996C2semidirect
8. D9×Dic3⋊C4C2semidirect
9. C4×#6189C2semidirect
10. #11720945C2semidirect
11. #11720935C2semidirect
72. C3×D36Q8semidirect
157. #890D9semidirect
200. D36Dic6semidirect
308. C3×C9#554semidirect
419. C9#5539semidirect
463. C3#10589792semidirect

Maximal subgroups

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

subgroupindex
#117217442
C3×#634552
C9×#8902
#117209992
#117209982
#117209962
Dic3×D362
D9×Dic3⋊C42
C4×#61892
#117209452
#117209352
#105944853
#105897923