Finite Group Databasebeta

← #13143914

Subgroups — #13143914

Order 1728.

990 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 990 conjugacy classes of subgroups — too many to draw. Only its 157 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, complemented
2C2×#88544C211yesmaximal, characteristic, complemented
3C2×#88525C211yesmaximal, characteristic, complemented
4#11721751C211yesmaximal, characteristic, complemented
5C3×C9×#102C211yesmaximal, characteristic, complemented
6#11720542C211yesmaximal, characteristic, complemented
7#11720528C211yesmaximal, characteristic, complemented
8#10592472—31nomaximal
9#10588453—33nomaximal
10C23×C3⋊D9—21no—
11C2×C3×D4×C9C2211yescharacteristic
12C2×#6233—21no—
13D4×C3⋊D9—22no—
14D4×C3⋊D9—22no—
15C2×C4⋊(C3⋊D9)—21no—
16C2×C4⋊(C3⋊D9)C2211yescharacteristic
17C2×C4⋊(C3⋊D9)C2211yescharacteristic, complemented
18C2×C4×C3⋊D9C2211yescharacteristic
19C2×C3×C8×C9C2211yescharacteristic, complemented
20C3×C9×C4⋊C4C2211yescharacteristic, complemented
21#88544—22no—
22#88536—21no—
23C2×C3⋊(C9⋊C8)C2211yescharacteristic, complemented
24#88525—22no—
25C2×D4×C3⋊D3—31no—
26C2×#3753—31no—
27C2×#3745—31no—
28#64104—31no—
29C2×D4×D9—33no—
30C32×#102D311yescharacteristic
31#63647—31no—
32#63636—31no—
33C2×#3673—33no—
34C2×D72—33no—
35#63457—33no—
36C9×#102D313yescomplemented
37(C2×D9)⋊C8—33no—
38#63369—33no—
39C23×C3×C9—21no—
40C22×C3⋊D9—24no—
41C22×C3⋊D9—21no—
42C22×C3⋊D9—21no—
43C22×C3⋊D9—42no—
44C22×C3⋊D9—42no—
45C22×C3⋊D9D411yescharacteristic
46C3×D4×C9—21no—
47C3×D4×C9D412yescomplemented
48C2×C3×C4×C9—21no—
49C2×C3×C4×C9C2311yescharacteristic
50#6233—24no—
51C2×C3⋊Dic9D411yescharacteristic
52C4⋊(C3⋊D9)—21no—
53C4⋊(C3⋊D9)—21no—
54C4⋊(C3⋊D9)—21no—
55C4⋊(C3⋊D9)—42no—
56C4⋊(C3⋊D9)D412yescomplemented
57C4×C3⋊D9—21no—
58C4×C3⋊D9—21no—
59C3×C8×C9—21no—
60C3⋊(C9⋊C8)—21no—
61#4773—93no—
62#4773—91no—
63C23×C3⋊D3—61no—
64C2×C32×D4C2×D311yescharacteristic
65C2×#638—61no—
66D4×C3⋊D3—62no—
67D4×C3⋊D3—62no—
68C2×C4⋊(C3⋊D3)—31no—
69C2×C4⋊(C3⋊D3)—31no—
70C2×C4⋊(C3⋊D3)—61no—
71C2×C4×C3⋊D3—31no—
72C23×D9—63no—
73C2×C32×C8C2×D311yescharacteristic
74C32×C4⋊C4C2×D311yescharacteristic
75#3753—62no—
76#3752—61no—
77C2×C3⋊(C3⋊C8)—31no—
78#3745—62no—
79C2×D4×C9C2×D313yes—
80C2×#611—63no—
81D4×D9—66no—
82D4×D9—66no—
83C2×D36—33no—
84C2×D36—33no—
85C2×D36—63no—
86C2×C4×D9—33no—
87C2×C8×C9C2×D313yescomplemented
88C9×C4⋊C4C2×D313yescomplemented
89#3673—66no—
90(C2×D9)⋊C4—63no—
91C2×C9⋊C8—33no—
92D72—66no—
93C22×C3×C9—22no—
94C22×C3×C9—41no—
95C22×C3×C9—41no—
96C22×C3×C9C2×D411yescharacteristic
97C2×C3⋊D9—21no—
98C2×C3⋊D9—21no—
99C2×C3⋊D9—22no—
100C2×C3⋊D9—48no—
101C2×C3⋊D9—41no—
102C2×C3⋊D9—41no—
103C2×C3⋊D9—41no—
104C2×C3⋊D9—41no—
105C2×C3⋊D9—41no—
106C2×C3⋊D9D812yescomplemented
107C3×C4×C9—41no—
108C3×C4×C9C2×D411yescharacteristic
109C3×C4×C9C2×D411yescharacteristic
110C3⋊Dic9—21no—
111C2×D3×D4—93no—
112C2×D3×D4—91no—
113C2×#213—93no—
114C2×#213—91no—
115C2×D24—93no—
116C2×D24—91no—
117#895—93no—
118#895—91no—
119C3×#102D913yescomplemented
120C3×#102C3⋊D311yescharacteristic
121(C2×D3)⋊C8—93no—
122(C2×D3)⋊C8—91no—
123#809—93no—
124#809—91no—
125C23×C32—21no—
126C22×C3⋊D3—31no—
127C22×C3⋊D3—64no—
128C22×C3⋊D3—61no—
129C22×C3⋊D3—61no—
130C22×C3⋊D3—122no—
131C22×C3⋊D3—122no—
132C32×D4—21no—
133C32×D4D1212yes—
134C2×C32×C4—21no—
135C2×C32×C4C22×D311yescharacteristic
136#638—64no—
137C2×C3⋊Dic3—31no—
138C4⋊(C3⋊D3)—32no—
139C4⋊(C3⋊D3)—61no—
140C4⋊(C3⋊D3)—61no—
141C4⋊(C3⋊D3)—61no—
142C4⋊(C3⋊D3)—122no—
143C4×C3⋊D3—61no—
144C4×C3⋊D3—61no—
145C23×C9—23no—
146C22×D9—33no—
147C22×D9—612no—
148C22×D9—63no—
149C22×D9—63no—
150C22×D9—126no—
151C22×D9—126no—
152C32×C8—21no—
153C3⋊(C3⋊C8)—61no—
154D4×C9—23no—
155D4×C9D1216yescomplemented
156C2×C4×C9—23no—
157C2×C4×C9C22×D313yes—
158#611—612no—
159C2×Dic9—33no—
160D36—36no—
161D36—63no—
162D36—63no—
163D36—63no—
164D36—126no—
165C4×D9—63no—
166C4×D9—63no—
167C8×C9—23no—
168C9⋊C8—63no—
169#447—271noSylow
170C2×C3×C9—42no—
171C2×C3×C9#12011yescharacteristic
172C2×C3×C9C2×D811yescharacteristic
173C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
174C3⋊D9—22no—
175C3⋊D9—42no—
176C3⋊D9—81no—
177C23×D3—181no—
178C23×D3—183no—
179C2×C3×D4C2×D913yes—
180C2×C3×D4C2×C3⋊D311yescharacteristic
181C2×#67—181no—
182C2×#67—183no—
183D3×D4—182no—
184D3×D4—186no—
185D3×D4—186no—
186D3×D4—182no—
187C2×D12—93no—
188C2×D12—91no—
189C2×D12—93no—
190C2×D12—91no—
191C2×D12—181no—
192C2×D12—183no—
193C2×C4×D3—93no—
194C2×C4×D3—91no—
195C2×C3×C8C2×D913yescomplemented
196C2×C3×C8C2×C3⋊D311yescharacteristic
197C3×C4⋊C4C2×D913yescomplemented
198C3×C4⋊C4C2×C3⋊D311yescharacteristic
199#213—186no—
200#213—182no—
201(C2×D3)⋊C4—181no—
202(C2×D3)⋊C4—183no—
203C2×C3⋊C8—93no—
204C2×C3⋊C8—91no—
205D24—182no—
206D24—186no—
207C22×C32—22no—
208C22×C32—41no—
209C22×C32—41no—
210C22×C32D3×D411yescharacteristic
211C2×C3⋊D3—32no—
212C2×C3⋊D3—61no—
213C2×C3⋊D3—61no—
214C2×C3⋊D3—62no—
215C2×C3⋊D3—128no—
216C2×C3⋊D3—121no—
217C2×C3⋊D3—121no—
218C2×C3⋊D3—121no—
219C2×C3⋊D3—121no—
220C2×C3⋊D3—121no—
221C32×C4—41no—
222C32×C4C2×D1211yescharacteristic
223C32×C4D3×D411yescharacteristic
224C3⋊Dic3—61no—
225C22×C9—26no—
226C22×C9—43no—
227C22×C9—43no—
228C22×C9D3×D413yes—
229C2×D9—36no—
230C2×D9—63no—
231C2×D9—63no—
232C2×D9—66no—
233C2×D9—1224no—
234C2×D9—123no—
235C2×D9—123no—
236C2×D9—123no—
237C2×D9—123no—
238C2×D9—123no—
239C4×C9—43no—
240C4×C9C2×D1213yes—
241C4×C9D3×D413yes—
242Dic9—63no—
243C22×D4—271no—
244C2×D8—271no—
245C2×D8—271no—
246#121—271no—
247#102—271no—
248#102C3⋊D911yescharacteristic, complemented
249C22⋊C8—271no—
250C3×C9#44711yescharacteristic, complemented, Sylow
251C23×C3—23no—
252C23×C3—21no—
253C22×D3—93no—
254C22×D3—91no—
255C22×D3—184no—
256C22×D3—1812no—
257C22×D3—183no—
258C22×D3—181no—
259C22×D3—183no—
260C22×D3—181no—
261C22×D3—362no—
262C22×D3—366no—
263C22×D3—362no—
264C22×D3—366no—
265C3×D4—23no—
266C3×D4—21no—
267C3×D4D3616yescomplemented
268C3×D4C4⋊(C3⋊D3)12yes—
269C2×C3×C4—23no—
270C2×C3×C4—21no—
271C2×C3×C4C22×D913yes—
272C2×C3×C4C22×C3⋊D311yescharacteristic
273#67—184no—
274#67—1812no—
275C2×Dic3—93no—
276C2×Dic3—91no—
277D12—96no—
278D12—92no—
279D12—183no—
280D12—183no—
281D12—183no—
282D12—181no—
283D12—181no—
284D12—181no—
285D12—362no—
286D12—366no—
287C4×D3—183no—
288C4×D3—183no—
289C4×D3—181no—
290C4×D3—181no—
291C3×C8—23no—
292C3×C8—21no—
293C3⋊C8—183no—
294C3⋊C8—181no—
295C2×C32—42no—
296C2×C32#88211yescharacteristic
297C2×C32D3×D811yescharacteristic
298C2×C32#91411yescharacteristic
299C3⋊D3—62no—
300C3⋊D3—122no—
301C3⋊D3—241no—
302C2×C9—46no—
303C2×C9#88213yes—
304C2×C9D3×D813yes—
305C2×C9#91413yes—
306D9—66no—
307D9—126no—
308D9—243no—
309C24—541no—
310C2×D4—271no—
311C2×D4—271no—
312C2×D4—541no—
313C2×D4—541no—
314C2×D4—542no—
315C2×D4—542no—
316C2×D4C2×C3⋊D911yescharacteristic
317C22×C4—271no—
318D8—542no—
319D8—542no—
320C2×C8—271no—
321C2×C8C2×C3⋊D911yescharacteristic, complemented
322C4⋊C4C2×C3⋊D911yescharacteristic, complemented
323C22⋊C4—541no—
324C22×C3—22no—
325C22×C3—26no—
326C22×C3—41no—
327C22×C3—41no—
328C22×C3—43no—
329C22×C3—43no—
330C22×C3D4×D913yes—
331C22×C3D4×C3⋊D311yescharacteristic
332C2×D3—92no—
333C2×D3—96no—
334C2×D3—181no—
335C2×D3—181no—
336C2×D3—183no—
337C2×D3—183no—
338C2×D3—182no—
339C2×D3—186no—
340C2×D3—368no—
341C2×D3—3624no—
342C2×D3—361no—
343C2×D3—361no—
344C2×D3—363no—
345C2×D3—363no—
346C2×D3—361no—
347C2×D3—361no—
348C2×D3—361no—
349C2×D3—363no—
350C2×D3—363no—
351C2×D3—363no—
352C3×C4—41no—
353C3×C4—43no—
354C3×C4C2×D3613yes—
355C3×C4D4×D913yes—
356C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
357C3×C4D4×C3⋊D311yescharacteristic
358Dic3—181no—
359Dic3—183no—
360C32#477311yescharacteristic
361C9#477313yescomplemented
362C23—21no—
363C23—271no—
364C23—544no—
365C23—541no—
366C23—541no—
367C23—1082no—
368C23—1082no—
369D4—21no—
370D4—272no—
371D4—544no—
372D4—541no—
373D4—541no—
374D4—541no—
375D4—1082no—
376D4C4⋊(C3⋊D9)12yescomplemented
377C2×C4—21no—
378C2×C4—271no—
379C2×C4—541no—
380C2×C4—541no—
381C2×C4C22×C3⋊D911yescharacteristic
382C8—21no—
383C8—541no—
384C2×C3—46no—
385C2×C3—42no—
386C2×C3#6344413yes—
387C2×C3D8×D913yes—
388C2×C3#6347613yes—
389C2×C3#6409111yescharacteristic
390C2×C3D8×C3⋊D311yescharacteristic
391C2×C3#6412311yescharacteristic
392D3—186no—
393D3—182no—
394D3—366no—
395D3—362no—
396D3—723no—
397D3—721no—
398C22—22no—
399C22—41no—
400C22—41no—
401C22—272no—
402C22—541no—
403C22—541no—
404C22—542no—
405C22—1088no—
406C22—1081no—
407C22—1081no—
408C22—1081no—
409C22—1081no—
410C22—1081no—
411C22D4×C3⋊D911yescharacteristic
412C4—41no—
413C4—541no—
414C4C2×C4⋊(C3⋊D9)11yescharacteristic
415C4D4×C3⋊D911yescharacteristic
416C3#1058845313yescomplemented
417C3#1059247211yescharacteristic
418C2—42no—
419C2—542no—
420C2—1082no—
421C2—2161no—
422C2#1172172811yescharacteristic
423C2D8×C3⋊D911yescharacteristic
424C2#1172179611yescharacteristic
subgroupG/H
centerC22D4×C3⋊D9
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#102C2
socleC22×C32D3×D4
hypercenter#102C3⋊D9
hypocenterC3×C9#447
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
1. C2×D4×C3⋊D9C2semidirect
2. C2×#88544C2semidirect
3. C2×#88525C2semidirect
4. #11721751C2semidirect
5. C3×C9×#102C2semidirect
6. #11720542C2semidirect
7. #11720528C2semidirect
17. C2×C4⋊(C3⋊D9)C22semidirect
19. C2×C3×C8×C9C22semidirect
20. C3×C9×C4⋊C4C22semidirect
23. C2×C3⋊(C9⋊C8)C22semidirect
36. C9×#102D3semidirect
47. C3×D4×C9D4semidirect
56. C4⋊(C3⋊D9)D4semidirect
87. C2×C8×C9C2×D3semidirect
88. C9×C4⋊C4C2×D3semidirect
106. C2×C3⋊D9D8semidirect
119. C3×#102D9semidirect
155. D4×C9D12semidirect
195. C2×C3×C8C2×D9semidirect
197. C3×C4⋊C4C2×D9semidirect
248. #102C3⋊D9semidirect
250. C3×C9#447semidirect
267. C3×D4D36semidirect
321. C2×C8C2×C3⋊D9semidirect
322. C4⋊C4C2×C3⋊D9semidirect
361. C9#4773semidirect
376. D4C4⋊(C3⋊D9)semidirect
416. C3#10588453semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C3⋊D92
C2×#885442
C2×#885252
#117217512
C3×C9×#1022
#117205422
#117205282
#105924723
#105884533