Finite Group Databasebeta

← #13150947

Subgroups — #13150947

Order 1728.

816 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#11721840C211yesmaximal, characteristic
2C9×#935C211yesmaximal, characteristic, complemented
3C3×#63497C211yesmaximal, characteristic, complemented
4#11721242C212yesmaximal, complemented
5#11721242C212yesmaximal, complemented
6#11721241C212yesmaximal, complemented
7#11721240C212yesmaximal, complemented
8#11720997C212yesmaximal, complemented
9#11720902C212yesmaximal, complemented
10#10595212—31nomaximal
11#10589766—31nomaximal
12C2×C3×D4×C9C2211yescharacteristic
13C2×#6233C2212yes—
14C2×#6233C2212yes—
15C2×C4⋊(C3⋊D9)C2211yescharacteristic
16C22×C9×Dic3C2211yescharacteristic
17C22×C3×Dic9C2211yescharacteristic
18C2×(C3⋊D9)⋊C4C2212yes—
19C2×#6189C2212yes—
20C4×C3⋊Dic9C2211yescharacteristic
21C9×#217C2212yes—
22C9×#217C2211yescharacteristic
23C3×#3676C2212yes—
24C3×#3676C2211yescharacteristic
25C9×Dic3⋊C4C2212yes—
26C3×Dic9⋊C4C2212yes—
27#88445C2212yes—
28#88445C2212yes—
29#88445C2212yes—
30#88442C2212yes—
31Dic3×Dic9C2212yes—
32Dic3×Dic9C2212yes—
33#64149—31no—
34C3×#935—31no—
35C3×#935D311yescharacteristic
36#63953—32no—
37#63953—32no—
38#63952—32no—
39#63952—32no—
40#63897—32no—
41#63838—32no—
42C9×#123—31no—
43#63502—31no—
44#63502—61no—
45#63497D311yescharacteristic, complemented
46C4×#611—32no—
47#63456—32no—
48#63449—32no—
49#63447—32no—
50#63447—32no—
51#611⋊C4—32no—
52C23×C3×C9C2311yescharacteristic
53C23×C3×C9C2311yescharacteristic
54C22×C3⋊D9C2312yes—
55C3×D4×C9—41no—
56C2×C3×C4×C9C2311yescharacteristic
57#6233—24no—
58#6233—24no—
59C2×C3⋊Dic9C2312yes—
60C4⋊(C3⋊D9)—41no—
61C2×C9×Dic3—41no—
62C2×C9×Dic3C2312yes—
63C2×C9×Dic3C2312yes—
64C2×C3×Dic9—41no—
65C2×C3×Dic9C2312yes—
66C2×C3×Dic9C2312yes—
67(C3⋊D9)⋊C4—42no—
68#6189—24no—
69#5664—91no—
70#5664—91no—
71C2×C32×D4C2×D311yescharacteristic
72C2×#638—32no—
73C2×#638—32no—
74C2×C4⋊(C3⋊D3)—31no—
75C22×C3×Dic3—31no—
76C22×C3×Dic3C2×D311yescharacteristic
77C2×#627—32no—
78C2×#624—32no—
79C4×C3⋊Dic3—31no—
80C3×#217—32no—
81C3×#217—31no—
82C3×#217C2×D312yes—
83C3×#217C2×D311yescharacteristic
84C3×Dic3⋊C4—32no—
85C3×Dic3⋊C4C2×D312yes—
86#3724—32no—
87#3722—32no—
88#3722—32no—
89#3722—32no—
90Dic32—32no—
91Dic32—32no—
92C2×D4×C9—21no—
93C2×D4×C9C2×D311yescharacteristic
94C22×C4×C9—31no—
95C2×#611—32no—
96C2×#611—32no—
97C2×#611—62no—
98C2×#611—62no—
99C22×Dic9C2×D311yescharacteristic
100C2×D36—31no—
101C2×D36—61no—
102C2×C4×D9—32no—
103C2×Dic18—32no—
104C9×C4⋊C4—32no—
105C9×C22⋊C4—32no—
106C9×C22⋊C4—31no—
107#3676C2×D312yes—
108#3676C2×D311yescharacteristic
109(C2×D9)⋊C4—32no—
110(C2×D9)⋊C4—32no—
111(C2×D9)⋊C4—32no—
112Dic9⋊C4—32no—
113Dic9⋊C4C2×D312yes—
114C4×Dic9—31no—
115C4×Dic9—32no—
116C4×Dic9—32no—
117C4×Dic9—61no—
118C22×C3×C9—22no—
119C22×C3×C9—22no—
120C22×C3×C9—21no—
121C22×C3×C9—21no—
122C22×C3×C9C2411yescharacteristic
123C2×C3⋊D9—22no—
124C2×C3⋊D9—22no—
125C2×C3⋊D9—22no—
126C3×C4×C9—21no—
127C3⋊Dic9#4114yes—
128C9×Dic3—22no—
129C9×Dic3—22no—
130C3×Dic9—22no—
131C3×Dic9—22no—
132C3×#123—31no—
133C3×#123—91no—
134#940—91no—
135#940—91no—
136#940—181no—
137#935—31no—
138#935D911yescharacteristic, complemented
139C4×#67—92no—
140C4×#67—92no—
141#894—92no—
142#894—92no—
143#887—92no—
144#887—92no—
145#885—92no—
146#885—92no—
147#885—92no—
148#885—92no—
149#67⋊C4—92no—
150#67⋊C4—92no—
151C23×C32C22×D311yescharacteristic
152C23×C32C22×D311yescharacteristic
153C22×C3⋊D3—32no—
154C32×D4—41no—
155C2×C32×C4C22×D311yescharacteristic
156#638—64no—
157#638—64no—
158C2×C3⋊Dic3—32no—
159C4⋊(C3⋊D3)—121no—
160C2×C3×Dic3—32no—
161C2×C3×Dic3—32no—
162C2×C3×Dic3—41no—
163C2×C3×Dic3—121no—
164C2×C3×Dic3C22×D312yes—
165C2×C3×Dic3C22×D312yes—
166#627—64no—
167#624—122no—
168C23×C9—21no—
169C23×C9—21no—
170C23×C9C22×D311yescharacteristic
171C23×C9C22×D311yescharacteristic
172C22×D9—32no—
173C22×D9—62no—
174D4×C9—41no—
175D4×C9—42no—
176C2×C4×C9—21no—
177C2×C4×C9—32no—
178C2×C4×C9—32no—
179C2×C4×C9—121no—
180C2×C4×C9C22×D311yescharacteristic
181#611—64no—
182#611—64no—
183#611—124no—
184#611—124no—
185C2×Dic9—32no—
186C2×Dic9—41no—
187C2×Dic9—62no—
188C2×Dic9C22×D312yes—
189C2×Dic9C22×D312yes—
190D36—121no—
191D36—122no—
192C4×D9—122no—
193Dic18—64no—
194#559—271noSylow
195C2×C3×C9—41no—
196C2×C3×C9—41no—
197C2×C3×C9C2×#4112yes—
198C2×C3×C9H(5,𝔽2)11yescharacteristic
199C3⋊D9—42no—
200C2×C3×D4—21no—
201C2×C3×D4C2×D911yescharacteristic
202C2×C3×D4D3211yescharacteristic
203C22×C3×C4—31no—
204C22×C3×C4—91no—
205C2×#67—92no—
206C2×#67—92no—
207C2×#67—92no—
208C2×#67—92no—
209C2×#67—182no—
210C2×#67—182no—
211C22×Dic3—31no—
212C22×Dic3C2×D911yescharacteristic
213C2×D12—91no—
214C2×D12—91no—
215C2×D12—181no—
216C2×C4×D3—92no—
217C2×C4×D3—92no—
218C2×Dic6—92no—
219C2×Dic6—92no—
220C3×C4⋊C4—32no—
221C3×C4⋊C4—92no—
222C3×C22⋊C4—32no—
223C3×C22⋊C4—31no—
224C3×C22⋊C4—92no—
225C3×C22⋊C4—91no—
226#217—31no—
227#217—32no—
228#217C2×D911yescharacteristic
229#217C2×D912yes—
230(C2×D3)⋊C4—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—
236Dic3⋊C4—32no—
237Dic3⋊C4—92no—
238Dic3⋊C4—92no—
239Dic3⋊C4C2×D912yes—
240C4×Dic3—92no—
241C4×Dic3—92no—
242C4×Dic3—91no—
243C4×Dic3—91no—
244C4×Dic3—92no—
245C4×Dic3—92no—
246C4×Dic3—181no—
247C22×C32—21no—
248C22×C32—21no—
249C22×C32—22no—
250C22×C32—22no—
251C22×C32C23×D311yescharacteristic
252C2×C3⋊D3—62no—
253C2×C3⋊D3—62no—
254C2×C3⋊D3—62no—
255C32×C4—21no—
256C3⋊Dic3—34no—
257C3×Dic3—22no—
258C3×Dic3—22no—
259C3×Dic3—62no—
260C3×Dic3—62no—
261C22×C9—21no—
262C22×C9—21no—
263C22×C9—22no—
264C22×C9—22no—
265C22×C9—21no—
266C22×C9—42no—
267C22×C9—42no—
268C22×C9—41no—
269C22×C9—41no—
270C22×C9C23×D311yescharacteristic
271C2×D9—62no—
272C2×D9—62no—
273C2×D9—62no—
274C2×D9—122no—
275C2×D9—122no—
276C2×D9—122no—
277C4×C9—22no—
278C4×C9—21no—
279C4×C9—62no—
280C4×C9—62no—
281Dic9—22no—
282Dic9—22no—
283Dic9—34no—
284Dic9—64no—
285#127—271no—
286#124—272no—
287#124—272no—
288#123—31no—
289#123—91no—
290#123—272no—
291C4×Q8—272no—
292C4×D4—272no—
293C4×D4—272no—
294C3×C9#55911yescharacteristic, complemented, Sylow
295C23×C3—21no—
296C23×C3—21no—
297C23×C3C22×D911yescharacteristic
298C23×C3C22×D911yescharacteristic
299C23×C3C2×D3211yescharacteristic
300C23×C3C2×D3211yescharacteristic
301C22×D3—92no—
302C22×D3—92no—
303C22×D3—182no—
304C3×D4—41no—
305C3×D4—41no—
306C3×D4—42no—
307C2×C3×C4—21no—
308C2×C3×C4—32no—
309C2×C3×C4—32no—
310C2×C3×C4—92no—
311C2×C3×C4—92no—
312C2×C3×C4—121no—
313C2×C3×C4—361no—
314C2×C3×C4C22×D911yescharacteristic
315C2×C3×C4C2×D3211yescharacteristic
316#67—184no—
317#67—184no—
318#67—184no—
319#67—184no—
320#67—364no—
321#67—364no—
322C2×Dic3—32no—
323C2×Dic3—32no—
324C2×Dic3—41no—
325C2×Dic3—92no—
326C2×Dic3—92no—
327C2×Dic3—121no—
328C2×Dic3—182no—
329C2×Dic3C22×D912yes—
330C2×Dic3C22×D912yes—
331D12—361no—
332D12—361no—
333D12—362no—
334C4×D3—362no—
335C4×D3—362no—
336Dic6—184no—
337Dic6—184no—
338C2×C32—41no—
339C2×C32—41no—
340C2×C32#100411yescharacteristic
341C2×C32D3×#4112yes—
342C3⋊D3—122no—
343C2×C9—22no—
344C2×C9—21no—
345C2×C9—41no—
346C2×C9—41no—
347C2×C9—81no—
348C2×C9—81no—
349C2×C9#100411yescharacteristic
350C2×C9D3×#4112yes—
351D9—122no—
352D9—242no—
353C2×Q8—272no—
354C2×D4—272no—
355C2×D4—272no—
356C2×D4—271no—
357C2×D4D3×D911yescharacteristic
358C22×C4—31no—
359C22×C4—91no—
360C22×C4—272no—
361C4⋊C4—32no—
362C4⋊C4—92no—
363C4⋊C4—272no—
364C22⋊C4—32no—
365C22⋊C4—31no—
366C22⋊C4—92no—
367C22⋊C4—91no—
368C22⋊C4—272no—
369C22⋊C4—272no—
370C22⋊C4—272no—
371C42—272no—
372C42—272no—
373C42—271no—
374C22×C3—22no—
375C22×C3—22no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—22no—
381C22×C3—22no—
382C22×C3—21no—
383C22×C3—42no—
384C22×C3—42no—
385C22×C3—41no—
386C22×C3—41no—
387C22×C3C23×D911yescharacteristic
388C22×C3C22×D3211yescharacteristic
389C2×D3—182no—
390C2×D3—182no—
391C2×D3—182no—
392C2×D3—182no—
393C2×D3—182no—
394C2×D3—182no—
395C2×D3—362no—
396C2×D3—362no—
397C2×D3—362no—
398C3×C4—22no—
399C3×C4—21no—
400C3×C4—21no—
401C3×C4—62no—
402C3×C4—62no—
403C3×C4—182no—
404C3×C4—182no—
405Dic3—22no—
406Dic3—22no—
407Dic3—62no—
408Dic3—62no—
409Dic3—94no—
410Dic3—94no—
411Dic3—184no—
412C32#566411yescharacteristic
413C9—21no—
414C9#566411yescharacteristic, complemented
415C23—272no—
416C23C2×D3×D911yescharacteristic
417C23C2×D3×D911yescharacteristic
418Q8—544no—
419D4—41no—
420D4—544no—
421D4—544no—
422D4—1081no—
423C2×C4—32no—
424C2×C4—32no—
425C2×C4—92no—
426C2×C4—92no—
427C2×C4—121no—
428C2×C4—272no—
429C2×C4—361no—
430C2×C4—1082no—
431C2×C4C2×D3×D911yescharacteristic
432C2×C3—22no—
433C2×C3—21no—
434C2×C3—41no—
435C2×C3—41no—
436C2×C3—41no—
437C2×C3—41no—
438C2×C3—81no—
439C2×C3—81no—
440C2×C3#6371011yescharacteristic
441C2×C3#41×D912yes—
442C2×C3#6431212yes—
443C2×C3#6433011yescharacteristic
444D3—362no—
445D3—362no—
446D3—722no—
447C22—22no—
448C22—22no—
449C22—21no—
450C22—21no—
451C22—542no—
452C22—542no—
453C22—542no—
454C22C22×D3×D911yescharacteristic
455C4—21no—
456C4—62no—
457C4—62no—
458C4—182no—
459C4—182no—
460C4—274no—
461C3—21no—
462C3#1058976611yescharacteristic, complemented
463C3#1059521211yescharacteristic
464C2—41no—
465C2—41no—
466C2—1082no—
467C2#1172249812yes—
468C2#1172253511yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
prime radical#11721840C2
hypercenterC2×D4D3×D9
hypocenterC3×C9#559
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C9×#935C2semidirect
3. C3×#63497C2semidirect
4. #11721242C2semidirect
5. #11721242C2semidirect
6. #11721241C2semidirect
7. #11721240C2semidirect
8. #11720997C2semidirect
9. #11720902C2semidirect
45. #63497D3semidirect
138. #935D9semidirect
294. C3×C9#559semidirect
414. C9#5664semidirect
462. C3#10589766semidirect

Maximal subgroups

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

subgroupindex
#117218402
C9×#9352
C3×#634972
#117212422
#117212412
#117212402
#117209972
#117209022
#105952123
#105897663