Finite Group Databasebeta

← #13144338

Subgroups — #13144338

Order 1728.

950 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C9×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#88536C211yesmaximal, characteristic
3C2×#88536C211yesmaximal, characteristic
4C2×#88536C211yesmaximal, characteristic
5C2×C4×C3⋊Dic9C211yesmaximal, characteristic, complemented
6#11720572C211yesmaximal, characteristic, complemented
7#11720572C211yesmaximal, characteristic, complemented
8#10592689—31nomaximal
9#10588670—33nomaximal
10C23×C3⋊D9C2211yescharacteristic
11C22×C3×C4×C9C2211yescharacteristic
12C22×C3×C4×C9C2211yescharacteristic
13C22×C3×C4×C9C2211yescharacteristic
14C22×C3⋊Dic9C2211yescharacteristic
15C22×C3⋊Dic9C2211yescharacteristic
16C22×C3⋊Dic9C2211yescharacteristic
17C3×C9×C4⋊C4—22no—
18#88536—22no—
19#88536—22no—
20#88536C414yescomplemented
21C4×C3⋊Dic9—22no—
22C2×C32×C4⋊C4D311yescharacteristic
23C2×#3752—31no—
24C2×#3752—31no—
25C2×#3752—31no—
26C2×C4×C3⋊Dic3—31no—
27#63658—31no—
28#63658—31no—
29C2×C9×C4⋊C4D313yescomplemented
30C2×(C2×D9)⋊C4—33no—
31C2×(C2×D9)⋊C4—33no—
32C2×(C2×D9)⋊C4—33no—
33C2×C4×Dic9—33no—
34C9⋊#95—33no—
35C9⋊#95—33no—
36C23×C3×C9C2311yescharacteristic
37C22×C3⋊D9—21no—
38C22×C3⋊D9—21no—
39C22×C3⋊D9—21no—
40C22×C3⋊D9—21no—
41C22×C3⋊D9—21no—
42C22×C3⋊D9—21no—
43C22×C3⋊D9C2×C412yes—
44C2×C3×C4×C9—21no—
45C2×C3×C4×C9—21no—
46C2×C3×C4×C9—21no—
47C2×C3×C4×C9—21no—
48C2×C3×C4×C9—21no—
49C2×C3×C4×C9—21no—
50C2×C3×C4×C9—21no—
51C2×C3×C4×C9C2×C412yes—
52C2×C3×C4×C9D412yes—
53C2×C3⋊Dic9—21no—
54C2×C3⋊Dic9—21no—
55C2×C3⋊Dic9—21no—
56C2×C3⋊Dic9—21no—
57C2×C3⋊Dic9—21no—
58C2×C3⋊Dic9—21no—
59C2×C3⋊Dic9—21no—
60C2×C3⋊Dic9C2×C412yes—
61C2×C3⋊Dic9D412yes—
62#4990—93no—
63#4990—91no—
64C23×C3⋊D3—31no—
65C22×C32×C4C2×D311yescharacteristic
66C22×C32×C4C2×D311yescharacteristic
67C22×C32×C4C2×D311yescharacteristic
68C22×C3⋊Dic3—31no—
69C22×C3⋊Dic3—31no—
70C22×C3⋊Dic3—31no—
71C23×D9—33no—
72C32×C4⋊C4—22no—
73#3752—34no—
74#3752—62no—
75#3752—62no—
76C4×C3⋊Dic3—62no—
77C22×C4×C9C2×D313yes—
78C22×C4×C9C2×D313yes—
79C22×C4×C9C2×D313yes—
80C22×Dic9—33no—
81C22×Dic9—33no—
82C22×Dic9—33no—
83C9×C4⋊C4—26no—
84(C2×D9)⋊C4—312no—
85(C2×D9)⋊C4—66no—
86(C2×D9)⋊C4—66no—
87C4×Dic9—66no—
88C22×C3×C9C22×C411yescharacteristic
89C22×C3×C9C2×D411yescharacteristic
90C22×C3×C9C2×D411yescharacteristic
91C22×C3×C9#4111yescharacteristic
92C22×C3×C9#4111yescharacteristic
93C22×C3×C9#4111yescharacteristic
94C22×C3×C9#4111yescharacteristic
95C2×C3⋊D9—22no—
96C2×C3⋊D9—22no—
97C2×C3⋊D9—22no—
98C2×C3⋊D9—41no—
99C2×C3⋊D9—41no—
100C2×C3⋊D9—41no—
101C2×C3⋊D9—41no—
102C3×C4×C9—22no—
103C3×C4×C9—22no—
104C3×C4×C9—41no—
105C3⋊Dic9—22no—
106C3⋊Dic9—22no—
107C3⋊Dic9—41no—
108C2×C3×C4⋊C4D913yescomplemented
109C2×C3×C4⋊C4C3⋊D311yescharacteristic
110C2×(C2×D3)⋊C4—93no—
111C2×(C2×D3)⋊C4—91no—
112C2×(C2×D3)⋊C4—93no—
113C2×(C2×D3)⋊C4—93no—
114C2×(C2×D3)⋊C4—91no—
115C2×(C2×D3)⋊C4—91no—
116C2×C4×Dic3—93no—
117C2×C4×Dic3—91no—
118C3⋊#95—93no—
119C3⋊#95—93no—
120C3⋊#95—91no—
121C3⋊#95—91no—
122C23×C32C22×D311yescharacteristic
123C22×C3⋊D3—32no—
124C22×C3⋊D3—61no—
125C22×C3⋊D3—61no—
126C22×C3⋊D3—61no—
127C22×C3⋊D3—61no—
128C22×C3⋊D3—61no—
129C22×C3⋊D3—61no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4C4×D312yes—
138C2×C32×C4#6712yes—
139C2×C3⋊Dic3—32no—
140C2×C3⋊Dic3—32no—
141C2×C3⋊Dic3—61no—
142C2×C3⋊Dic3—61no—
143C2×C3⋊Dic3—61no—
144C2×C3⋊Dic3—61no—
145C2×C3⋊Dic3—61no—
146C2×C3⋊Dic3—61no—
147C2×C3⋊Dic3—61no—
148C23×C9C22×D313yes—
149C22×D9—36no—
150C22×D9—63no—
151C22×D9—63no—
152C22×D9—63no—
153C22×D9—63no—
154C22×D9—63no—
155C22×D9—63no—
156C2×C4×C9—23no—
157C2×C4×C9—23no—
158C2×C4×C9—23no—
159C2×C4×C9—23no—
160C2×C4×C9—23no—
161C2×C4×C9—23no—
162C2×C4×C9—23no—
163C2×C4×C9C4×D316yes—
164C2×C4×C9#6716yes—
165C2×Dic9—36no—
166C2×Dic9—36no—
167C2×Dic9—63no—
168C2×Dic9—63no—
169C2×Dic9—63no—
170C2×Dic9—63no—
171C2×Dic9—63no—
172C2×Dic9—63no—
173C2×Dic9—63no—
174#388—271noSylow
175C2×C3×C9#11711yescharacteristic
176C2×C3×C9C4×D411yescharacteristic
177C2×C3×C9C4×D411yescharacteristic
178C2×C3×C9#12111yescharacteristic
179C2×C3×C9#12311yescharacteristic
180C2×C3×C9#12411yescharacteristic
181C2×C3×C9#12611yescharacteristic
182C3⋊D9—42no—
183C23×D3—93no—
184C23×D3—91no—
185C22×C3×C4C2×D913yes—
186C22×C3×C4C2×D913yes—
187C22×C3×C4C2×D913yes—
188C22×C3×C4C2×C3⋊D311yescharacteristic
189C22×C3×C4C2×C3⋊D311yescharacteristic
190C22×C3×C4C2×C3⋊D311yescharacteristic
191C22×Dic3—93no—
192C22×Dic3—93no—
193C22×Dic3—91no—
194C22×Dic3—91no—
195C22×Dic3—93no—
196C22×Dic3—91no—
197C3×C4⋊C4—26no—
198C3×C4⋊C4—22no—
199(C2×D3)⋊C4—912no—
200(C2×D3)⋊C4—94no—
201(C2×D3)⋊C4—182no—
202(C2×D3)⋊C4—186no—
203(C2×D3)⋊C4—186no—
204(C2×D3)⋊C4—182no—
205C4×Dic3—182no—
206C4×Dic3—186no—
207C22×C32C2×C4×D311yescharacteristic
208C22×C32#23511yescharacteristic
209C22×C32D3×D411yescharacteristic
210C22×C32#23711yescharacteristic
211C22×C32#23911yescharacteristic
212C22×C32#23911yescharacteristic
213C22×C32C2×#6711yescharacteristic
214C2×C3⋊D3—62no—
215C2×C3⋊D3—62no—
216C2×C3⋊D3—62no—
217C2×C3⋊D3—121no—
218C2×C3⋊D3—121no—
219C2×C3⋊D3—121no—
220C2×C3⋊D3—121no—
221C32×C4—22no—
222C32×C4—22no—
223C32×C4—41no—
224C3⋊Dic3—62no—
225C3⋊Dic3—62no—
226C3⋊Dic3—121no—
227C22×C9C2×C4×D313yes—
228C22×C9#23513yes—
229C22×C9D3×D413yes—
230C22×C9#23713yes—
231C22×C9#23913yes—
232C22×C9#23913yes—
233C22×C9C2×#6713yes—
234C2×D9—66no—
235C2×D9—66no—
236C2×D9—66no—
237C2×D9—123no—
238C2×D9—123no—
239C2×D9—123no—
240C2×D9—123no—
241C4×C9—26no—
242C4×C9—26no—
243C4×C9—43no—
244Dic9—66no—
245Dic9—66no—
246Dic9—123no—
247C2×C4⋊C4C3⋊D911yescharacteristic, complemented
248C2×C22⋊C4—271no—
249C2×C22⋊C4—271no—
250C2×C22⋊C4—271no—
251C2×C42—271no—
252#95—271no—
253#95—271no—
254C3×C9#38811yescharacteristic, complemented, Sylow
255C23×C3C22×D913yes—
256C23×C3C22×C3⋊D311yescharacteristic
257C22×D3—96no—
258C22×D3—92no—
259C22×D3—183no—
260C22×D3—183no—
261C22×D3—183no—
262C22×D3—183no—
263C22×D3—183no—
264C22×D3—183no—
265C22×D3—181no—
266C22×D3—181no—
267C22×D3—181no—
268C22×D3—181no—
269C22×D3—181no—
270C22×D3—181no—
271C2×C3×C4—23no—
272C2×C3×C4—23no—
273C2×C3×C4—23no—
274C2×C3×C4—23no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—23no—
280C2×C3×C4—23no—
281C2×C3×C4—23no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4C4×D916yes—
286C2×C3×C4#61116yes—
287C2×C3×C4C4×C3⋊D312yes—
288C2×C3×C4#63812yes—
289C2×Dic3—96no—
290C2×Dic3—96no—
291C2×Dic3—92no—
292C2×Dic3—92no—
293C2×Dic3—183no—
294C2×Dic3—183no—
295C2×Dic3—183no—
296C2×Dic3—183no—
297C2×Dic3—181no—
298C2×Dic3—181no—
299C2×Dic3—181no—
300C2×Dic3—181no—
301C2×Dic3—183no—
302C2×Dic3—183no—
303C2×Dic3—183no—
304C2×Dic3—181no—
305C2×Dic3—181no—
306C2×Dic3—181no—
307C2×C32#89211yescharacteristic
308C2×C32#89311yescharacteristic
309C2×C32#89411yescharacteristic
310C2×C32#89811yescharacteristic
311C2×C32C4×#6711yescharacteristic
312C2×C32#93911yescharacteristic
313C2×C32#94711yescharacteristic
314C3⋊D3—122no—
315C2×C9#89213yes—
316C2×C9#89313yes—
317C2×C9#89413yes—
318C2×C9#89813yes—
319C2×C9C4×#6713yes—
320C2×C9#93913yes—
321C2×C9#94713yes—
322D9—126no—
323C24—271no—
324C22×C4—271no—
325C22×C4—271no—
326C22×C4—271no—
327C22×C4C2×C3⋊D911yescharacteristic
328C22×C4C2×C3⋊D911yescharacteristic
329C22×C4C2×C3⋊D911yescharacteristic
330C4⋊C4—22no—
331C22⋊C4—274no—
332C22⋊C4—542no—
333C22⋊C4—542no—
334C42—542no—
335C22×C3C2×C4×D913yes—
336C22×C3#369713yes—
337C22×C3D4×D913yes—
338C22×C3#369913yes—
339C22×C3#370113yes—
340C22×C3#370113yes—
341C22×C3C2×#61113yes—
342C22×C3C2×C4×C3⋊D311yescharacteristic
343C22×C3#382811yescharacteristic
344C22×C3D4×C3⋊D311yescharacteristic
345C22×C3#383011yescharacteristic
346C22×C3#383211yescharacteristic
347C22×C3#383211yescharacteristic
348C22×C3C2×#63811yescharacteristic
349C2×D3—186no—
350C2×D3—182no—
351C2×D3—182no—
352C2×D3—182no—
353C2×D3—186no—
354C2×D3—186no—
355C2×D3—363no—
356C2×D3—363no—
357C2×D3—361no—
358C2×D3—361no—
359C2×D3—361no—
360C2×D3—361no—
361C2×D3—363no—
362C2×D3—363no—
363C3×C4—26no—
364C3×C4—26no—
365C3×C4—22no—
366C3×C4—22no—
367C3×C4—43no—
368C3×C4—41no—
369Dic3—182no—
370Dic3—182no—
371Dic3—186no—
372Dic3—186no—
373Dic3—363no—
374Dic3—361no—
375C32#499011yescharacteristic
376C9#499013yescomplemented
377C23—272no—
378C23—541no—
379C23—541no—
380C23—541no—
381C23—541no—
382C23—541no—
383C23—541no—
384C23C22×C3⋊D911yescharacteristic
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—272no—
393C2×C4—272no—
394C2×C4—541no—
395C2×C4—541no—
396C2×C4—541no—
397C2×C4—541no—
398C2×C4—541no—
399C2×C4—541no—
400C2×C4—541no—
401C2×C4C4×C3⋊D912yes—
402C2×C4#623312yes—
403C2×C3#6345413yes—
404C2×C3#6345513yes—
405C2×C3#6345613yes—
406C2×C3#6346013yes—
407C2×C3C4×#61113yes—
408C2×C3#6350113yes—
409C2×C3#6350913yes—
410C2×C3#6410111yescharacteristic
411C2×C3#6410211yescharacteristic
412C2×C3#6410311yescharacteristic
413C2×C3#6410711yescharacteristic
414C2×C3C4×#63811yescharacteristic
415C2×C3#6414811yescharacteristic
416C2×C3#6415611yescharacteristic
417D3—362no—
418D3—366no—
419C22—542no—
420C22—542no—
421C22—542no—
422C22—1081no—
423C22—1081no—
424C22—1081no—
425C22—1081no—
426C22C2×C4×C3⋊D911yescharacteristic
427C22#8873611yescharacteristic
428C22D4×C3⋊D911yescharacteristic
429C22#8874211yescharacteristic
430C22#8874611yescharacteristic
431C22#8874611yescharacteristic
432C22C2×#623311yescharacteristic
433C4—22no—
434C4—22no—
435C4—41no—
436C4—542no—
437C4—542no—
438C4—1081no—
439C3#1058867013yescomplemented
440C3#1059268911yescharacteristic
441C2—1082no—
442C2#1172174811yescharacteristic
443C2#1172174911yescharacteristic
444C2#1172175011yescharacteristic
445C2#1172175411yescharacteristic
446C2C4×#623311yescharacteristic
447C2#1172183911yescharacteristic
448C2#1172185711yescharacteristic
subgroupG/H
centerC23C22×C3⋊D9
commutatorC22×C3×C9C22×C4
FrattiniC23×C3C22×C3⋊D3
FittingC2×C3×C9×C4⋊C4C2
socleC23×C32C22×D3
prime radicalC23×C3⋊D9C22
cosocleC23×C3×C9C23
hypercenterC2×C4⋊C4C3⋊D9
hypocenterC3×C9#388
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C9×C4⋊C4C2semidirect
5. C2×C4×C3⋊Dic9C2semidirect
6. #11720572C2semidirect
7. #11720572C2semidirect
20. #88536C4semidirect
29. C2×C9×C4⋊C4D3semidirect
108. C2×C3×C4⋊C4D9semidirect
247. C2×C4⋊C4C3⋊D9semidirect
254. C3×C9#388semidirect
376. C9#4990semidirect
439. C3#10588670semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C9×C4⋊C42
C2×#885362
C2×C4×C3⋊Dic92
#117205722
#105926893
#105886703