Finite Group Databasebeta

← #99×C3⋊D9 #13143867

Subgroups — #99×C3⋊D9 #13143867

Order 1728.

1050 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 1050 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C3⋊D9C211yesmaximal, characteristic
2C22⋊C4×C3⋊D9C212yesmaximal, complemented
3C3×C9×#99C211yesmaximal, characteristic, complemented
4#11720577C211yesmaximal, characteristic, complemented
5#11720523C212yesmaximal, complemented
6C3⋊D3×#99—31nomaximal
7D9×#99—33nomaximal
8C23×C3⋊D9C411yescharacteristic, complemented
9C23×C3⋊D9C2211yescharacteristic
10C2×C3×D4×C9C2211yescharacteristic
11C2×#6233C411yescharacteristic, complemented
12C2×#6233C2211yescharacteristic
13D4×C3⋊D9—22no—
14D4×C3⋊D9—22no—
15C2×C4⋊(C3⋊D9)C411yescharacteristic, complemented
16C2×C4×C3⋊D9—22no—
17C2×C4×C3⋊D9C411yescharacteristic, complemented
18C3×C9×C22⋊C4C2212yescomplemented
19#88551C2212yescomplemented
20#88536—22no—
21C2×D4×C3⋊D3—31no—
22C22⋊C4×C3⋊D3—32no—
23C2×D4×D9—33no—
24C32×#99D311yescharacteristic
25#63661—31no—
26#63633—32no—
27C22⋊C4×D9—36no—
28C9×#99D313yescomplemented
29#63393—33no—
30#63365—36no—
31C23×C3×C9C2×C411yescharacteristic, complemented
32C23×C3×C9C2311yescharacteristic
33C22×C3⋊D9—21no—
34C22×C3⋊D9—21no—
35C22×C3⋊D9—42no—
36C22×C3⋊D9—42no—
37C22×C3⋊D9C2×C411yescharacteristic
38C22×C3⋊D9C2×C411yescharacteristic
39C22×C3⋊D9C2×C411yescharacteristic, complemented
40C22×C3⋊D9D412yes—
41C22×C3⋊D9D412yes—
42C3×D4×C9—22no—
43C2×C3×C4×C9—22no—
44C2×C3×C4×C9C2×C411yescharacteristic, complemented
45#6233—22no—
46#6233—22no—
47C2×C3⋊Dic9—22no—
48C2×C3⋊Dic9C2×C411yescharacteristic, complemented
49C4⋊(C3⋊D9)—22no—
50C4×C3⋊D9—21no—
51C4×C3⋊D9—21no—
52C4×C3⋊D9—44no—
53D3×#99—93no—
54D3×#99—91no—
55C23×C3⋊D3—31no—
56C23×C3⋊D3—31no—
57C2×C32×D4C2×D311yescharacteristic
58C2×#638—31no—
59C2×#638—31no—
60D4×C3⋊D3—62no—
61D4×C3⋊D3—62no—
62C2×C4⋊(C3⋊D3)—31no—
63C2×C4×C3⋊D3—31no—
64C2×C4×C3⋊D3—62no—
65C23×D9—33no—
66C23×D9—33no—
67C32×C22⋊C4C2×D312yes—
68#3757—32no—
69#3752—62no—
70C2×D4×C9C2×D313yes—
71C2×#611—33no—
72C2×#611—33no—
73D4×D9—66no—
74D4×D9—66no—
75C2×D36—33no—
76C2×C4×D9—33no—
77C2×C4×D9—66no—
78C9×C22⋊C4C2×D316yescomplemented
79#3676—36no—
80(C2×D9)⋊C4—66no—
81C22×C3×C9—21no—
82C22×C3×C9—41no—
83C22×C3×C9—41no—
84C22×C3×C9C22×C411yescharacteristic
85C22×C3×C9C2×D412yes—
86C2×C3⋊D9—24no—
87C2×C3⋊D9—22no—
88C2×C3⋊D9—21no—
89C2×C3⋊D9—41no—
90C2×C3⋊D9—42no—
91C2×C3⋊D9—42no—
92C2×C3⋊D9—42no—
93C2×C3⋊D9—41no—
94C2×C3⋊D9C22⋊C411yescharacteristic
95C2×C3⋊D9C22⋊C412yes—
96C2×C3⋊D9C22⋊C411yescharacteristic
97C3×C4×C9—21no—
98C3×C4×C9—42no—
99C3⋊Dic9—21no—
100C3⋊Dic9—42no—
101C2×D3×D4—93no—
102C2×D3×D4—91no—
103D3×C22⋊C4—96no—
104D3×C22⋊C4—92no—
105C3×#99D913yescomplemented
106C3×#99C3⋊D311yescharacteristic
107#834—93no—
108#834—91no—
109#806—96no—
110#806—92no—
111C23×C32C4×D311yescharacteristic
112C23×C32C22×D311yescharacteristic
113C22×C3⋊D3—31no—
114C22×C3⋊D3—32no—
115C22×C3⋊D3—31no—
116C22×C3⋊D3—32no—
117C22×C3⋊D3—31no—
118C22×C3⋊D3—61no—
119C22×C3⋊D3—61no—
120C22×C3⋊D3—122no—
121C22×C3⋊D3—122no—
122C32×D4—22no—
123C2×C32×C4—22no—
124C2×C32×C4C4×D311yescharacteristic
125#638—62no—
126#638—62no—
127C2×C3⋊Dic3—31no—
128C2×C3⋊Dic3—62no—
129C4⋊(C3⋊D3)—62no—
130C4×C3⋊D3—61no—
131C4×C3⋊D3—61no—
132C4×C3⋊D3—124no—
133C23×C9C4×D313yescomplemented
134C23×C9C22×D313yes—
135C22×D9—33no—
136C22×D9—36no—
137C22×D9—33no—
138C22×D9—36no—
139C22×D9—33no—
140C22×D9—63no—
141C22×D9—63no—
142C22×D9—126no—
143C22×D9—126no—
144D4×C9—26no—
145C2×C4×C9—26no—
146C2×C4×C9C4×D313yescomplemented
147#611—66no—
148#611—66no—
149C2×Dic9—33no—
150C2×Dic9—66no—
151D36—66no—
152C4×D9—63no—
153C4×D9—63no—
154C4×D9—1212no—
155C2×#99—271noSylow
156C2×C3×C9—22no—
157C2×C3×C9—21no—
158C2×C3×C9—41no—
159C2×C3×C9C2×C22⋊C411yescharacteristic
160C3⋊D9—21no—
161C3⋊D9—22no—
162C3⋊D9—41no—
163C3⋊D9#9912yesdirect product factor
164C23×D3—93no—
165C23×D3—91no—
166C23×D3—93no—
167C23×D3—91no—
168C2×C3×D4C2×D913yes—
169C2×C3×D4C2×C3⋊D311yescharacteristic
170C2×#67—93no—
171C2×#67—93no—
172C2×#67—91no—
173C2×#67—91no—
174D3×D4—182no—
175D3×D4—182no—
176D3×D4—186no—
177D3×D4—186no—
178C2×D12—93no—
179C2×D12—91no—
180C2×C4×D3—93no—
181C2×C4×D3—91no—
182C2×C4×D3—182no—
183C2×C4×D3—186no—
184C3×C22⋊C4C2×D916yescomplemented
185C3×C22⋊C4C2×C3⋊D312yes—
186#217—96no—
187#217—92no—
188(C2×D3)⋊C4—182no—
189(C2×D3)⋊C4—186no—
190C22×C32—21no—
191C22×C32—41no—
192C22×C32—41no—
193C22×C32C2×C4×D311yescharacteristic
194C22×C32D3×D412yes—
195C2×C3⋊D3—31no—
196C2×C3⋊D3—31no—
197C2×C3⋊D3—32no—
198C2×C3⋊D3—64no—
199C2×C3⋊D3—62no—
200C2×C3⋊D3—61no—
201C2×C3⋊D3—122no—
202C2×C3⋊D3—121no—
203C2×C3⋊D3—122no—
204C2×C3⋊D3—122no—
205C2×C3⋊D3—121no—
206C32×C4—21no—
207C32×C4—42no—
208C3⋊Dic3—61no—
209C3⋊Dic3—122no—
210C22×C9—23no—
211C22×C9—43no—
212C22×C9—43no—
213C22×C9C2×C4×D313yes—
214C22×C9D3×D416yes—
215C2×D9—33no—
216C2×D9—33no—
217C2×D9—36no—
218C2×D9—612no—
219C2×D9—66no—
220C2×D9—63no—
221C2×D9—126no—
222C2×D9—123no—
223C2×D9—126no—
224C2×D9—126no—
225C2×D9—123no—
226C4×C9—23no—
227C4×C9—46no—
228Dic9—63no—
229Dic9—126no—
230C22×D4—271no—
231C2×C22⋊C4—272no—
232#99—271no—
233#99—272no—
234#99C3⋊D911yescharacteristic, direct product factor
235C3×C9C2×#9911yescharacteristic, complemented, Sylow
236C23×C3C4×D913yescomplemented
237C23×C3C22×D913yes—
238C23×C3C4×C3⋊D311yescharacteristic
239C23×C3C22×C3⋊D311yescharacteristic
240C22×D3—93no—
241C22×D3—96no—
242C22×D3—91no—
243C22×D3—92no—
244C22×D3—93no—
245C22×D3—96no—
246C22×D3—91no—
247C22×D3—92no—
248C22×D3—93no—
249C22×D3—91no—
250C22×D3—181no—
251C22×D3—181no—
252C22×D3—183no—
253C22×D3—183no—
254C22×D3—362no—
255C22×D3—366no—
256C22×D3—362no—
257C22×D3—366no—
258C3×D4—26no—
259C3×D4—22no—
260C2×C3×C4—26no—
261C2×C3×C4—22no—
262C2×C3×C4C4×D913yescomplemented
263C2×C3×C4C4×C3⋊D311yescharacteristic
264#67—182no—
265#67—186no—
266#67—186no—
267#67—182no—
268C2×Dic3—93no—
269C2×Dic3—91no—
270C2×Dic3—186no—
271C2×Dic3—182no—
272D12—182no—
273D12—186no—
274C4×D3—181no—
275C4×D3—181no—
276C4×D3—183no—
277C4×D3—183no—
278C4×D3—364no—
279C4×D3—3612no—
280C2×C32—22no—
281C2×C32—21no—
282C2×C32—41no—
283C2×C32D3×C22⋊C411yescharacteristic
284C3⋊D3—32no—
285C3⋊D3—62no—
286C3⋊D3—61no—
287C3⋊D3—121no—
288C2×C9—26no—
289C2×C9—23no—
290C2×C9—43no—
291C2×C9D3×C22⋊C413yes—
292D9—36no—
293D9—66no—
294D9—63no—
295D9—123no—
296C24—271no—
297C24—271no—
298C2×D4—271no—
299C2×D4—271no—
300C2×D4—271no—
301C2×D4—542no—
302C2×D4—542no—
303C2×D4C2×C3⋊D911yescharacteristic
304C22×C4—271no—
305C22×C4—542no—
306C22⋊C4—272no—
307C22⋊C4—542no—
308C22⋊C4C2×C3⋊D912yescomplemented
309C22×C3—21no—
310C22×C3—23no—
311C22×C3—43no—
312C22×C3—41no—
313C22×C3—43no—
314C22×C3—41no—
315C22×C3C2×C4×D913yes—
316C22×C3D4×D916yes—
317C22×C3C2×C4×C3⋊D311yescharacteristic
318C22×C3D4×C3⋊D312yes—
319C2×D3—91no—
320C2×D3—93no—
321C2×D3—91no—
322C2×D3—92no—
323C2×D3—93no—
324C2×D3—96no—
325C2×D3—184no—
326C2×D3—1812no—
327C2×D3—182no—
328C2×D3—186no—
329C2×D3—181no—
330C2×D3—183no—
331C2×D3—362no—
332C2×D3—366no—
333C2×D3—362no—
334C2×D3—361no—
335C2×D3—366no—
336C2×D3—363no—
337C2×D3—362no—
338C2×D3—366no—
339C2×D3—361no—
340C2×D3—363no—
341C3×C4—21no—
342C3×C4—23no—
343C3×C4—42no—
344C3×C4—46no—
345Dic3—181no—
346Dic3—183no—
347Dic3—362no—
348Dic3—366no—
349C32D3×#9911yescharacteristic
350C9D3×#9913yescomplemented
351C23—271no—
352C23—272no—
353C23—271no—
354C23—272no—
355C23—271no—
356C23—541no—
357C23—541no—
358C23—1082no—
359C23—1082no—
360C23C4×C3⋊D911yescharacteristic, complemented
361C23C22×C3⋊D911yescharacteristic
362D4—22no—
363D4—542no—
364D4—542no—
365D4—542no—
366C2×C4—22no—
367C2×C4—271no—
368C2×C4—541no—
369C2×C4—541no—
370C2×C4—542no—
371C2×C4—1084no—
372C2×C4C4×C3⋊D911yescharacteristic, complemented
373C2×C3—26no—
374C2×C3—23no—
375C2×C3—22no—
376C2×C3—21no—
377C2×C3—43no—
378C2×C3—41no—
379C2×C3C22⋊C4×D913yes—
380C2×C3C22⋊C4×C3⋊D311yescharacteristic
381D3—96no—
382D3—92no—
383D3—183no—
384D3—186no—
385D3—181no—
386D3—182no—
387D3—363no—
388D3—361no—
389C22—21no—
390C22—41no—
391C22—41no—
392C22—271no—
393C22—271no—
394C22—272no—
395C22—544no—
396C22—542no—
397C22—541no—
398C22—1082no—
399C22—1081no—
400C22—1081no—
401C22—1082no—
402C22—1082no—
403C22C2×C4×C3⋊D911yescharacteristic
404C22D4×C3⋊D912yes—
405C4—21no—
406C4—42no—
407C4—541no—
408C4—1082no—
409C3D9×#9913yescomplemented
410C3C3⋊D3×#9911yescharacteristic
411C2—22no—
412C2—21no—
413C2—41no—
414C2—272no—
415C2—541no—
416C2—542no—
417C2—1081no—
418C2C22⋊C4×C3⋊D911yescharacteristic
subgroupG/H
centerC2C22⋊C4×C3⋊D9
commutatorC22×C3×C9C22×C4
FrattiniC23×C3C22×C3⋊D3
FittingC3×C9×#99C2
socleC2×C32D3×C22⋊C4
prime radicalC2×D4×C3⋊D9C2
cosocleC23×C3×C9C23
hypercenter#99C3⋊D9
hypocenterC3×C9C2×#99
LCSG ▷ C22×C3×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C23 ◁ #99

Complements

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

NH ≅ G/Ntype
2. C22⋊C4×C3⋊D9C2semidirect
3. C3×C9×#99C2semidirect
4. #11720577C2semidirect
5. #11720523C2semidirect
8. C23×C3⋊D9C4semidirect
11. C2×#6233C4semidirect
15. C2×C4⋊(C3⋊D9)C4semidirect
17. C2×C4×C3⋊D9C4semidirect
18. C3×C9×C22⋊C4C22semidirect
19. #88551C22semidirect
28. C9×#99D3semidirect
31. C23×C3×C9C2×C4semidirect
39. C22×C3⋊D9C2×C4semidirect
44. C2×C3×C4×C9C2×C4semidirect
48. C2×C3⋊Dic9C2×C4semidirect
78. C9×C22⋊C4C2×D3semidirect
105. C3×#99D9semidirect
133. C23×C9C4×D3semidirect
146. C2×C4×C9C4×D3semidirect
163. C3⋊D9#99direct (N × H)
184. C3×C22⋊C4C2×D9semidirect
234. #99C3⋊D9direct (N × H)
235. C3×C9C2×#99semidirect
236. C23×C3C4×D9semidirect
262. C2×C3×C4C4×D9semidirect
308. C22⋊C4C2×C3⋊D9semidirect
350. C9D3×#99semidirect
360. C23C4×C3⋊D9semidirect
372. C2×C4C4×C3⋊D9semidirect
409. C3D9×#99semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C3⋊D92
C22⋊C4×C3⋊D92
C3×C9×#992
#117205772
#117205232
C3⋊D3×#993
D9×#993