Finite Group Databasebeta

← #13153882

Subgroups — #13153882

Order 1728.

1190 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×#117C211yesmaximal, characteristic, complemented
2#11721819C211yesmaximal, characteristic, complemented
3C4×#6233C211yesmaximal, characteristic, complemented
4#11721751C211yesmaximal, characteristic, complemented
5#11721749C211yesmaximal, characteristic, complemented
6C4⋊C4×C3⋊D9C211yesmaximal, characteristic, complemented
7#11721746C211yesmaximal, characteristic, complemented
8#11721730C212yesmaximal, complemented
9#11721729C212yesmaximal, complemented
10#11721713C212yesmaximal, complemented
11C4×C4⋊(C3⋊D9)C212yesmaximal, complemented
12#10596001—31nomaximal
13#10589632—33nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6233C2211yescharacteristic
16C2×#6233C2212yes—
17C2×C4⋊(C3⋊D9)C2212yes—
18C2×C4⋊(C3⋊D9)C2211yescharacteristic
19C2×C4×C3⋊D9C2211yescharacteristic
20C2×C4×C3⋊D9C2212yes—
21C2×C4×C3⋊D9C2212yes—
22C3×C9×C4⋊C4C2211yescharacteristic
23C3×C9×C4⋊C4C2211yescharacteristic
24C3×C9×C22⋊C4C2212yes—
25C3×C42×C9C2212yes—
26#88551C2211yescharacteristic
27#88536C2212yes—
28#88536C2212yes—
29#88536C2211yescharacteristic
30#88536C2212yes—
31C4⋊(C3⋊Dic9)C2212yes—
32C4⋊(C3⋊Dic9)C2211yescharacteristic
33#88534C2212yes—
34#88534C2212yes—
35#88534C2211yescharacteristic
36C4×C3⋊Dic9C2211yescharacteristic
37C32×#117D311yescharacteristic
38#64139—31no—
39C4×#638—31no—
40#64104—31no—
41#64102—31no—
42C4⋊C4×C3⋊D3—31no—
43#64099—31no—
44#64093—32no—
45#64092—32no—
46#64085—32no—
47C4×C4⋊(C3⋊D3)—32no—
48C9×#117D313yescomplemented
49#63492—33no—
50C4×#611—33no—
51#63457—33no—
52#63455—33no—
53C4⋊C4×D9—33no—
54#63452—33no—
55#63446—36no—
56#63445—36no—
57#63438—36no—
58C4×D36—36no—
59C23×C3×C9C2311yescharacteristic
60C22×C3⋊D9C2312yes—
61C22×C3⋊D9C2311yescharacteristic
62C2×C3×C4×C9—22no—
63C2×C3×C4×C9C2312yes—
64C2×C3×C4×C9C2312yes—
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3×C4×C9C2311yescharacteristic
67#6233—22no—
68#6233—42no—
69C2×C3⋊Dic9C2311yescharacteristic
70C2×C3⋊Dic9C2312yes—
71C2×C3⋊Dic9C2311yescharacteristic
72C2×C3⋊Dic9C2311yescharacteristic
73C4⋊(C3⋊D9)—24no—
74C4⋊(C3⋊D9)—22no—
75C4×C3⋊D9—22no—
76C4×C3⋊D9—42no—
77C4×C3⋊D9—42no—
78#5530—93no—
79#5530—91no—
80C22×C32×C4C2×D311yescharacteristic
81C2×#638—31no—
82C2×#638—32no—
83C2×C4⋊(C3⋊D3)—32no—
84C2×C4⋊(C3⋊D3)—31no—
85C2×C4×C3⋊D3—31no—
86C2×C4×C3⋊D3—32no—
87C2×C4×C3⋊D3—32no—
88C32×C4⋊C4C2×D311yescharacteristic
89C32×C4⋊C4C2×D311yescharacteristic
90C32×C22⋊C4C2×D312yes—
91C32×C42C2×D312yes—
92#3757—31no—
93#3752—31no—
94#3752—32no—
95#3752—32no—
96#3752—32no—
97C4⋊(C3⋊Dic3)—32no—
98C4⋊(C3⋊Dic3)—31no—
99#3750—31no—
100#3750—32no—
101#3750—32no—
102C4×C3⋊Dic3—31no—
103C22×C4×C9C2×D313yes—
104C2×#611—33no—
105C2×#611—36no—
106C2×D36—33no—
107C2×D36—36no—
108C2×C4×D9—33no—
109C2×C4×D9—36no—
110C2×C4×D9—36no—
111C9×C4⋊C4C2×D313yes—
112C9×C4⋊C4C2×D313yes—
113C9×C22⋊C4C2×D316yes—
114C42×C9C2×D316yes—
115#3676—33no—
116(C2×D9)⋊C4—33no—
117(C2×D9)⋊C4—36no—
118(C2×D9)⋊C4—36no—
119(C2×D9)⋊C4—36no—
120C4⋊Dic9—33no—
121C4⋊Dic9—36no—
122Dic9⋊C4—33no—
123Dic9⋊C4—36no—
124Dic9⋊C4—36no—
125C4×Dic9—33no—
126C22×C3×C9—21no—
127C22×C3×C9—21no—
128C22×C3×C9—21no—
129C22×C3×C9C2411yescharacteristic
130C2×C3⋊D9—21no—
131C2×C3⋊D9—21no—
132C2×C3⋊D9—22no—
133C2×C3⋊D9—22no—
134C2×C3⋊D9—22no—
135C2×C3⋊D9#4112yes—
136C3×C4×C9—22no—
137C3×C4×C9—21no—
138C3×C4×C9—22no—
139C3×C4×C9#4112yes—
140C3⋊Dic9—21no—
141C3⋊Dic9—21no—
142C3⋊Dic9—21no—
143C3⋊Dic9—22no—
144C3×#117D913yescomplemented
145C3×#117C3⋊D311yescharacteristic
146#930—93no—
147#930—91no—
148C4×#67—93no—
149C4×#67—91no—
150#895—93no—
151#895—91no—
152#893—93no—
153#893—91no—
154D3×C4⋊C4—93no—
155D3×C4⋊C4—91no—
156#890—93no—
157#890—91no—
158#884—96no—
159#884—92no—
160#883—96no—
161#883—92no—
162#876—96no—
163#876—92no—
164C4×D12—96no—
165C4×D12—92no—
166C23×C32C22×D311yescharacteristic
167C22×C3⋊D3—31no—
168C22×C3⋊D3—32no—
169C2×C32×C4—22no—
170C2×C32×C4C22×D312yes—
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D312yes—
173C2×C32×C4C22×D311yescharacteristic
174#638—62no—
175#638—122no—
176C2×C3⋊Dic3—31no—
177C2×C3⋊Dic3—31no—
178C2×C3⋊Dic3—31no—
179C2×C3⋊Dic3—32no—
180C4⋊(C3⋊D3)—64no—
181C4⋊(C3⋊D3)—62no—
182C4×C3⋊D3—62no—
183C4×C3⋊D3—122no—
184C4×C3⋊D3—122no—
185C23×C9C22×D313yes—
186C22×D9—33no—
187C22×D9—36no—
188C2×C4×C9—26no—
189C2×C4×C9C22×D316yes—
190C2×C4×C9C22×D313yes—
191C2×C4×C9C22×D316yes—
192C2×C4×C9C22×D313yes—
193#611—66no—
194#611—126no—
195C2×Dic9—33no—
196C2×Dic9—33no—
197C2×Dic9—33no—
198C2×Dic9—36no—
199D36—612no—
200D36—66no—
201C4×D9—66no—
202C4×D9—126no—
203C4×D9—126no—
204#553—271noSylow
205C2×C3×C9—41no—
206C2×C3×C9C2×#4111yescharacteristic
207C2×C3×C9C2×#4111yescharacteristic
208C2×C3×C9H(5,𝔽2)11yescharacteristic
209C3⋊D9—22no—
210C3⋊D9—42no—
211C22×C3×C4C2×D913yes—
212C22×C3×C4C2×C3⋊D311yescharacteristic
213C2×#67—93no—
214C2×#67—91no—
215C2×#67—96no—
216C2×#67—92no—
217C2×D12—96no—
218C2×D12—93no—
219C2×D12—92no—
220C2×D12—91no—
221C2×C4×D3—93no—
222C2×C4×D3—91no—
223C2×C4×D3—96no—
224C2×C4×D3—96no—
225C2×C4×D3—92no—
226C2×C4×D3—92no—
227C3×C4⋊C4C2×D913yes—
228C3×C4⋊C4C2×D913yes—
229C3×C4⋊C4C2×C3⋊D311yescharacteristic
230C3×C4⋊C4C2×C3⋊D311yescharacteristic
231C3×C22⋊C4C2×D916yes—
232C3×C22⋊C4C2×C3⋊D312yes—
233C3×C42C2×D916yes—
234C3×C42C2×C3⋊D312yes—
235#217—93no—
236#217—91no—
237(C2×D3)⋊C4—93no—
238(C2×D3)⋊C4—96no—
239(C2×D3)⋊C4—96no—
240(C2×D3)⋊C4—96no—
241(C2×D3)⋊C4—91no—
242(C2×D3)⋊C4—92no—
243(C2×D3)⋊C4—92no—
244(C2×D3)⋊C4—92no—
245C4⋊Dic3—96no—
246C4⋊Dic3—93no—
247C4⋊Dic3—92no—
248C4⋊Dic3—91no—
249Dic3⋊C4—93no—
250Dic3⋊C4—96no—
251Dic3⋊C4—96no—
252Dic3⋊C4—91no—
253Dic3⋊C4—92no—
254Dic3⋊C4—92no—
255C4×Dic3—93no—
256C4×Dic3—91no—
257C22×C32—21no—
258C22×C32—21no—
259C22×C32—21no—
260C22×C32C23×D311yescharacteristic
261C2×C3⋊D3—32no—
262C2×C3⋊D3—61no—
263C2×C3⋊D3—61no—
264C2×C3⋊D3—62no—
265C2×C3⋊D3—62no—
266C2×C3⋊D3—62no—
267C32×C4—21no—
268C32×C4—22no—
269C32×C4—22no—
270C32×C4#23512yes—
271C3⋊Dic3—61no—
272C3⋊Dic3—61no—
273C3⋊Dic3—61no—
274C3⋊Dic3—62no—
275C22×C9—23no—
276C22×C9—23no—
277C22×C9—23no—
278C22×C9C23×D313yes—
279C2×D9—36no—
280C2×D9—63no—
281C2×D9—63no—
282C2×D9—66no—
283C2×D9—66no—
284C2×D9—66no—
285C4×C9—23no—
286C4×C9—26no—
287C4×C9—26no—
288C4×C9#23516yes—
289Dic9—63no—
290Dic9—63no—
291Dic9—63no—
292Dic9—66no—
293#126—272no—
294#125—271no—
295#123—272no—
296#121—271no—
297#121—271no—
298#121—272no—
299C4×D4—271no—
300C4×D4—271no—
301C4×D4—272no—
302#117C3⋊D911yescharacteristic, complemented
303C2×C4⋊C4—271no—
304C3×C9#55311yescharacteristic, complemented, Sylow
305C23×C3C22×D913yes—
306C23×C3C22×C3⋊D311yescharacteristic
307C22×D3—91no—
308C22×D3—93no—
309C22×D3—92no—
310C22×D3—96no—
311C2×C3×C4—26no—
312C2×C3×C4—22no—
313C2×C3×C4C22×D916yes—
314C2×C3×C4C22×D913yes—
315C2×C3×C4C22×D916yes—
316C2×C3×C4C22×D913yes—
317C2×C3×C4C22×C3⋊D312yes—
318C2×C3×C4C22×C3⋊D311yescharacteristic
319C2×C3×C4C22×C3⋊D312yes—
320C2×C3×C4C22×C3⋊D311yescharacteristic
321#67—182no—
322#67—186no—
323#67—362no—
324#67—366no—
325C2×Dic3—91no—
326C2×Dic3—91no—
327C2×Dic3—91no—
328C2×Dic3—92no—
329C2×Dic3—93no—
330C2×Dic3—93no—
331C2×Dic3—93no—
332C2×Dic3—96no—
333D12—184no—
334D12—1812no—
335D12—186no—
336D12—182no—
337C4×D3—186no—
338C4×D3—182no—
339C4×D3—362no—
340C4×D3—362no—
341C4×D3—366no—
342C4×D3—366no—
343C2×C32—41no—
344C2×C32C2×#23511yescharacteristic
345C2×C32D3×#4111yescharacteristic
346C2×C32#100911yescharacteristic
347C3⋊D3—62no—
348C3⋊D3—122no—
349C2×C9—43no—
350C2×C9C2×#23513yes—
351C2×C9D3×#4113yes—
352C2×C9#100913yes—
353D9—66no—
354D9—126no—
355C2×D4—271no—
356C2×D4—272no—
357C2×D4—272no—
358C2×D4—271no—
359C22×C4—271no—
360C22×C4—272no—
361C22×C4—272no—
362C22×C4C2×C3⋊D911yescharacteristic
363C4⋊C4—271no—
364C4⋊C4—272no—
365C4⋊C4—272no—
366C4⋊C4—271no—
367C4⋊C4—272no—
368C4⋊C4C2×C3⋊D911yescharacteristic
369C4⋊C4C2×C3⋊D911yescharacteristic
370C22⋊C4—271no—
371C22⋊C4—271no—
372C22⋊C4—272no—
373C22⋊C4—272no—
374C22⋊C4—272no—
375C22⋊C4C2×C3⋊D912yes—
376C42—271no—
377C42C2×C3⋊D912yes—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—23no—
382C22×C3—23no—
383C22×C3—23no—
384C22×C3C23×D913yes—
385C22×C3C23×C3⋊D311yescharacteristic
386C2×D3—92no—
387C2×D3—96no—
388C2×D3—181no—
389C2×D3—181no—
390C2×D3—183no—
391C2×D3—183no—
392C2×D3—182no—
393C2×D3—182no—
394C2×D3—182no—
395C2×D3—186no—
396C2×D3—186no—
397C2×D3—186no—
398C3×C4—21no—
399C3×C4—22no—
400C3×C4—22no—
401C3×C4—23no—
402C3×C4—26no—
403C3×C4—26no—
404C3×C4#369716yes—
405C3×C4#382812yes—
406Dic3—181no—
407Dic3—181no—
408Dic3—181no—
409Dic3—182no—
410Dic3—183no—
411Dic3—183no—
412Dic3—183no—
413Dic3—186no—
414C32#553011yescharacteristic
415C9#553013yescomplemented
416C23—271no—
417C23—272no—
418C23C22×C3⋊D911yescharacteristic
419D4—544no—
420D4—542no—
421D4—542no—
422D4—1082no—
423C2×C4—22no—
424C2×C4—271no—
425C2×C4—271no—
426C2×C4—271no—
427C2×C4—272no—
428C2×C4—542no—
429C2×C4—1082no—
430C2×C4—1082no—
431C2×C4C22×C3⋊D911yescharacteristic
432C2×C4C22×C3⋊D912yes—
433C2×C4C22×C3⋊D912yes—
434C2×C4C22×C3⋊D911yescharacteristic
435C2×C3—41no—
436C2×C3—43no—
437C2×C3C2×#369713yes—
438C2×C3#41×D913yes—
439C2×C3#6371513yes—
440C2×C3C2×#382811yescharacteristic
441C2×C3#41×C3⋊D311yescharacteristic
442C2×C3#6436611yescharacteristic
443D3—182no—
444D3—186no—
445D3—362no—
446D3—366no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—272no—
451C22—541no—
452C22—541no—
453C22—542no—
454C22—542no—
455C22—542no—
456C22C23×C3⋊D911yescharacteristic
457C4—21no—
458C4—22no—
459C4—22no—
460C4—541no—
461C4—541no—
462C4—541no—
463C4—542no—
464C4#8873612yes—
465C3#1058963213yescomplemented
466C3#1059600111yescharacteristic
467C2—41no—
468C2—542no—
469C2—1082no—
470C2C2×#8873611yescharacteristic
471C2#41×C3⋊D911yescharacteristic
472C2#1172267211yescharacteristic
subgroupG/H
centerC22C23×C3⋊D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C23×C3⋊D3
FittingC3×C9×#117C2
socleC22×C32C23×D3
hypercenter#117C3⋊D9
hypocenterC3×C9#553
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ #117

Complements

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

NH ≅ G/Ntype
1. C3×C9×#117C2semidirect
2. #11721819C2semidirect
3. C4×#6233C2semidirect
4. #11721751C2semidirect
5. #11721749C2semidirect
6. C4⋊C4×C3⋊D9C2semidirect
7. #11721746C2semidirect
8. #11721730C2semidirect
9. #11721729C2semidirect
10. #11721713C2semidirect
11. C4×C4⋊(C3⋊D9)C2semidirect
48. C9×#117D3semidirect
144. C3×#117D9semidirect
302. #117C3⋊D9semidirect
304. C3×C9#553semidirect
415. C9#5530semidirect
465. C3#10589632semidirect

Maximal subgroups

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

subgroupindex
C3×C9×#1172
#117218192
C4×#62332
#117217512
#117217492
C4⋊C4×C3⋊D92
#117217462
#117217302
#117217292
#117217132
C4×C4⋊(C3⋊D9)2
#105960013
#105896323