Finite Group Databasebeta

← #13140339

Subgroups — #13140339

Order 1728.

842 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 842 conjugacy classes of subgroups — too many to draw. Only its 113 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
2#11721000C211yesmaximal, characteristic, complemented
3C2×(C3⋊D9)⋊C8C211yesmaximal, characteristic, complemented
4C9×#832C211yesmaximal, characteristic, complemented
5C3×#63392C211yesmaximal, characteristic, complemented
6#11720284C211yesmaximal, characteristic, complemented
7#11720267C211yesmaximal, characteristic, complemented
8#10591416—31nomaximal
9D9×#102—31nomaximal
10C23×C3⋊D9—21no—
11C2×C3×D4×C9C2211yescharacteristic
12C2×#6233—21no—
13D4×C3⋊D9—22no—
14D4×C3⋊D9C414yescomplemented
15C2×C4⋊(C3⋊D9)C2211yescharacteristic
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×(C3⋊D9)⋊C4—21no—
18C9×C4⋊Dic3C2211yescharacteristic, complemented
19C3×C4⋊Dic9C2211yescharacteristic, complemented
20C2×C9×C3⋊C8C2211yescharacteristic, complemented
21C2×C3×C9⋊C8C2211yescharacteristic, complemented
22#88442—21no—
23(C3⋊D9)⋊C8—22no—
24C2×D4×C3⋊D3—31no—
25#63900—31no—
26C2×#3710—31no—
27C2×D4×D9—31no—
28C2×D4×D9—61no—
29C3×#832—31no—
30C3×#832D311yescharacteristic
31#63564—31no—
32#63564—31no—
33C2×C8×D9—31no—
34C4⋊C4×D9—31no—
35C9×#102—31no—
36#63392D311yescharacteristic, complemented
37#63380—31no—
38#63369—31no—
39C23×C3×C9—21no—
40C22×C3⋊D9—21no—
41C22×C3⋊D9—24no—
42C22×C3⋊D9—42no—
43C22×C3⋊D9—42no—
44C22×C3⋊D9D411yescharacteristic
45C3×D4×C9—21no—
46C3×D4×C9C2×C412yescomplemented
47C2×C3×C4×C9C2311yescharacteristic
48#6233—24no—
49C2×C3⋊Dic9D411yescharacteristic
50C4⋊(C3⋊D9)—21no—
51C4⋊(C3⋊D9)C2×C412yescomplemented
52C4×C3⋊D9C2×C412yes—
53C4×C3⋊D9D412yes—
54C2×C9×Dic3—21no—
55C2×C3×Dic9—21no—
56(C3⋊D9)⋊C4—42no—
57C9×C3⋊C8—21no—
58C3×C9⋊C8—21no—
59D3×#102—91no—
60D3×#102—91no—
61C23×C3⋊D3—61no—
62C2×C32×D4C2×D311yescharacteristic
63C2×#638—61no—
64D4×C3⋊D3—34no—
65D4×C3⋊D3—62no—
66C2×C4⋊(C3⋊D3)—31no—
67C2×C4×C3⋊D3—31no—
68C2×#624—61no—
69C23×D9—61no—
70C23×D9—62no—
71C3×C4⋊Dic3—31no—
72C3×C4⋊Dic3C2×D311yescharacteristic
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8C2×D311yescharacteristic
75#3724—61no—
76#3710—62no—
77C2×D4×C9—21no—
78C2×D4×C9C2×D311yescharacteristic
79C2×#611—61no—
80C2×#611—62no—
81D4×D9—34no—
82D4×D9—62no—
83D4×D9—64no—
84D4×D9—64no—
85C2×D36—31no—
86C2×D36—61no—
87C2×C4×D9—31no—
88C2×C4×D9—61no—
89C2×C4×D9—61no—
90C2×C8×C9—31no—
91C9×C4⋊C4—31no—
92C4⋊Dic9C2×D311yescharacteristic, complemented
93Dic9⋊C4—61no—
94C2×C9⋊C8C2×D311yescharacteristic, complemented
95C8×D9—62no—
96C22×C3×C9—22no—
97C22×C3×C9—41no—
98C22×C3×C9—41no—
99C22×C3×C9C2×D411yescharacteristic
100C2×C3⋊D9—22no—
101C2×C3⋊D9—41no—
102C2×C3⋊D9—41no—
103C2×C3⋊D9—48no—
104C2×C3⋊D9C22⋊C412yes—
105C2×C3⋊D9D812yes—
106C2×C3⋊D9SD1612yes—
107C3×C4×C9C22×C411yescharacteristic
108C3×C4×C9C2×D411yescharacteristic
109C3⋊Dic9C22⋊C412yes—
110C9×Dic3—41no—
111C3×Dic9—41no—
112C2×D3×D4—91no—
113C2×D3×D4—91no—
114C2×D3×D4—181no—
115C2×D3×C8—91no—
116C2×D3×C8—91no—
117D3×C4⋊C4—91no—
118D3×C4⋊C4—91no—
119C3×#102—31no—
120C3×#102—91no—
121#832—31no—
122#832D911yescharacteristic, complemented
123#821—91no—
124#821—91no—
125#809—91no—
126#809—91no—
127C23×C32—21no—
128C22×C3⋊D3—31no—
129C22×C3⋊D3—64no—
130C22×C3⋊D3—61no—
131C22×C3⋊D3—122no—
132C22×C3⋊D3—122no—
133C32×D4—21no—
134C32×D4C4×D312yes—
135C2×C32×C4C22×D311yescharacteristic
136#638—64no—
137C2×C3⋊Dic3—31no—
138C4⋊(C3⋊D3)—32no—
139C4⋊(C3⋊D3)—61no—
140C4×C3⋊D3—32no—
141C4×C3⋊D3—32no—
142C2×C3×Dic3—21no—
143C2×C3×Dic3—61no—
144#624—122no—
145C23×C9—22no—
146C23×C9—21no—
147C22×D9—31no—
148C22×D9—61no—
149C22×D9—64no—
150C22×D9—62no—
151C22×D9—68no—
152C22×D9—61no—
153C22×D9—122no—
154C22×D9—122no—
155C22×D9—124no—
156C22×D9—124no—
157C3×C3⋊C8—21no—
158C3×C3⋊C8—61no—
159D4×C9—22no—
160D4×C9—21no—
161D4×C9—22no—
162D4×C9C4×D312yescomplemented
163C2×C4×C9—21no—
164C2×C4×C9—61no—
165C2×C4×C9C22×D311yescharacteristic
166#611—64no—
167#611—68no—
168C2×Dic9—21no—
169C2×Dic9—31no—
170C2×Dic9—61no—
171D36—32no—
172D36—62no—
173D36—62no—
174D36—61no—
175C4×D9—32no—
176C4×D9—32no—
177C4×D9—62no—
178C4×D9—62no—
179C4×D9—122no—
180C8×C9—61no—
181C9⋊C8—21no—
182C2×#102—271noSylow
183C2×C3×C9—42no—
184C2×C3×C9C2×C22⋊C411yescharacteristic
185C2×C3×C9C2×D811yescharacteristic
186C2×C3×C9C2×SD1611yescharacteristic
187C3⋊D9—42no—
188C3⋊D9#10214yescomplemented
189C23×D3—181no—
190C23×D3—181no—
191C23×D3—182no—
192C2×C3×D4—21no—
193C2×C3×D4C2×D911yescharacteristic
194C2×C3×D4D3211yescharacteristic
195C2×#67—181no—
196C2×#67—181no—
197C2×#67—182no—
198D3×D4—94no—
199D3×D4—94no—
200D3×D4—184no—
201D3×D4—182no—
202D3×D4—182no—
203D3×D4—184no—
204C2×D12—91no—
205C2×D12—91no—
206C2×D12—181no—
207C2×C4×D3—91no—
208C2×C4×D3—91no—
209C2×C4×D3—181no—
210C2×C4×D3—181no—
211C2×C4×D3—181no—
212C2×C3×C8—31no—
213C2×C3×C8—91no—
214C3×C4⋊C4—31no—
215C3×C4⋊C4—91no—
216C4⋊Dic3—31no—
217C4⋊Dic3C2×D911yescharacteristic, complemented
218Dic3⋊C4—181no—
219Dic3⋊C4—181no—
220C2×C3⋊C8—31no—
221C2×C3⋊C8C2×D911yescharacteristic, complemented
222D3×C8—182no—
223D3×C8—182no—
224C22×C32—22no—
225C22×C32—41no—
226C22×C32—41no—
227C22×C32D3×D411yescharacteristic
228C2×C3⋊D3—32no—
229C2×C3⋊D3—32no—
230C2×C3⋊D3—32no—
231C2×C3⋊D3—62no—
232C2×C3⋊D3—128no—
233C2×C3⋊D3—121no—
234C2×C3⋊D3—121no—
235C32×C4C2×C4×D311yescharacteristic
236C32×C4D3×D411yescharacteristic
237C3⋊Dic3—32no—
238C3×Dic3—41no—
239C3×Dic3—121no—
240C22×C9—24no—
241C22×C9—22no—
242C22×C9—21no—
243C22×C9—42no—
244C22×C9—42no—
245C22×C9—41no—
246C22×C9—41no—
247C22×C9D3×D411yescharacteristic
248C2×D9—32no—
249C2×D9—32no—
250C2×D9—32no—
251C2×D9—62no—
252C2×D9—62no—
253C2×D9—62no—
254C2×D9—64no—
255C2×D9—62no—
256C2×D9—128no—
257C2×D9—1216no—
258C2×D9—122no—
259C2×D9—122no—
260C2×D9—121no—
261C2×D9—121no—
262C4×C9—21no—
263C4×C9—21no—
264C4×C9—121no—
265C4×C9C2×C4×D311yescharacteristic
266C4×C9D3×D411yescharacteristic
267Dic9—32no—
268Dic9—41no—
269Dic9—62no—
270C22×D4—271no—
271C22×C8—271no—
272C2×C4⋊C4—271no—
273#102—31no—
274#102—91no—
275#102—271no—
276#102—271no—
277C3×C9C2×#10211yescharacteristic, complemented, Sylow
278C23×C3—22no—
279C23×C3—21no—
280C23×C3—21no—
281C22×D3—91no—
282C22×D3—91no—
283C22×D3—181no—
284C22×D3—184no—
285C22×D3—184no—
286C22×D3—182no—
287C22×D3—188no—
288C22×D3—181no—
289C22×D3—181no—
290C22×D3—362no—
291C22×D3—362no—
292C22×D3—362no—
293C22×D3—362no—
294C22×D3—364no—
295C22×D3—364no—
296C3×D4—22no—
297C3×D4—21no—
298C3×D4—21no—
299C3×D4—22no—
300C3×D4C4×D912yescomplemented
301C3×D4#62412yes—
302C2×C3×C4—21no—
303C2×C3×C4—61no—
304C2×C3×C4—181no—
305C2×C3×C4C22×D911yescharacteristic
306C2×C3×C4C2×D3211yescharacteristic
307#67—184no—
308#67—184no—
309#67—188no—
310C2×Dic3—21no—
311C2×Dic3—61no—
312C2×Dic3—91no—
313C2×Dic3—91no—
314C2×Dic3—181no—
315D12—92no—
316D12—92no—
317D12—182no—
318D12—182no—
319D12—181no—
320D12—181no—
321C4×D3—92no—
322C4×D3—92no—
323C4×D3—92no—
324C4×D3—92no—
325C4×D3—182no—
326C4×D3—182no—
327C4×D3—362no—
328C4×D3—362no—
329C3×C8—61no—
330C3×C8—181no—
331C3⋊C8—21no—
332C3⋊C8—61no—
333C2×C32—42no—
334C2×C32D3×C22⋊C411yescharacteristic
335C2×C32D3×D811yescharacteristic
336C2×C32D3×SD1611yescharacteristic
337C3⋊D3—34no—
338C3⋊D3—122no—
339C2×C9—21no—
340C2×C9—21no—
341C2×C9—21no—
342C2×C9—44no—
343C2×C9—42no—
344C2×C9D3×C22⋊C411yescharacteristic
345C2×C9D3×D811yescharacteristic
346C2×C9D3×SD1611yescharacteristic
347D9—34no—
348D9—64no—
349D9—124no—
350D9—122no—
351C24—541no—
352C2×D4—274no—
353C2×D4—271no—
354C2×D4—542no—
355C2×D4—541no—
356C2×D4D3×D911yescharacteristic
357C22×C4—271no—
358C22×C4—541no—
359C2×C8—31no—
360C2×C8—91no—
361C2×C8—542no—
362C4⋊C4—31no—
363C4⋊C4—91no—
364C4⋊C4—541no—
365C22×C3—24no—
366C22×C3—22no—
367C22×C3—22no—
368C22×C3—21no—
369C22×C3—42no—
370C22×C3—42no—
371C22×C3—41no—
372C22×C3—41no—
373C22×C3—41no—
374C22×C3—41no—
375C22×C3D4×D911yescharacteristic
376C22×C3#381111yescharacteristic
377C2×D3—92no—
378C2×D3—92no—
379C2×D3—92no—
380C2×D3—92no—
381C2×D3—92no—
382C2×D3—92no—
383C2×D3—182no—
384C2×D3—182no—
385C2×D3—182no—
386C2×D3—184no—
387C2×D3—182no—
388C2×D3—182no—
389C2×D3—368no—
390C2×D3—368no—
391C2×D3—3616no—
392C2×D3—362no—
393C2×D3—362no—
394C2×D3—361no—
395C2×D3—361no—
396C2×D3—361no—
397C2×D3—361no—
398C3×C4—21no—
399C3×C4—21no—
400C3×C4—121no—
401C3×C4—361no—
402C3×C4C2×C4×D911yescharacteristic
403C3×C4D4×D911yescharacteristic
404C3×C4C2×#62411yescharacteristic
405C3×C4#381111yescharacteristic
406Dic3—41no—
407Dic3—92no—
408Dic3—92no—
409Dic3—121no—
410Dic3—182no—
411C32D3×#10211yescharacteristic
412C9—21no—
413C9D3×#10211yescharacteristic, complemented
414C23—21no—
415C23—271no—
416C23—544no—
417C23—541no—
418C23—1082no—
419C23—1082no—
420D4—21no—
421D4—272no—
422D4—544no—
423D4—541no—
424D4(C3⋊D9)⋊C412yescomplemented
425C2×C4—61no—
426C2×C4—181no—
427C2×C4—271no—
428C2×C4—272no—
429C2×C4—272no—
430C2×C4—1082no—
431C2×C4C2×D3×D911yescharacteristic
432C8—61no—
433C8—181no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—44no—
438C2×C3—42no—
439C2×C3—42no—
440C2×C3C22⋊C4×D911yescharacteristic
441C2×C3D8×D911yescharacteristic
442C2×C3SD16×D911yescharacteristic
443C2×C3#6392611yescharacteristic
444C2×C3#6393011yescharacteristic
445C2×C3#6397411yescharacteristic
446D3—94no—
447D3—94no—
448D3—184no—
449D3—364no—
450D3—362no—
451D3—362no—
452C22—22no—
453C22—41no—
454C22—41no—
455C22—272no—
456C22—272no—
457C22—272no—
458C22—542no—
459C22—1088no—
460C22—1081no—
461C22—1081no—
462C22#8866811yescharacteristic
463C4—121no—
464C4—272no—
465C4—361no—
466C4C2×(C3⋊D9)⋊C411yescharacteristic
467C4#8866811yescharacteristic
468C3—21no—
469C3D9×#10211yescharacteristic, complemented
470C3#1059141611yescharacteristic
471C2—42no—
472C2—274no—
473C2—1082no—
474C2#1172117411yescharacteristic
475C2#1172118211yescharacteristic
476C2#1172127411yescharacteristic
subgroupG/H
centerC22#88668
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC22×C32D3×D4
prime radicalC2×D4×C3⋊D9C2
cosocleC2×C3×C4×C9C23
hypercenterC2×D4D3×D9
hypocenterC3×C9C2×#102
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11721000C2semidirect
3. C2×(C3⋊D9)⋊C8C2semidirect
4. C9×#832C2semidirect
5. C3×#63392C2semidirect
6. #11720284C2semidirect
7. #11720267C2semidirect
14. D4×C3⋊D9C4semidirect
18. C9×C4⋊Dic3C22semidirect
19. C3×C4⋊Dic9C22semidirect
20. C2×C9×C3⋊C8C22semidirect
21. C2×C3×C9⋊C8C22semidirect
36. #63392D3semidirect
46. C3×D4×C9C2×C4semidirect
51. C4⋊(C3⋊D9)C2×C4semidirect
92. C4⋊Dic9C2×D3semidirect
94. C2×C9⋊C8C2×D3semidirect
122. #832D9semidirect
162. D4×C9C4×D3semidirect
188. C3⋊D9#102semidirect
217. C4⋊Dic3C2×D9semidirect
221. C2×C3⋊C8C2×D9semidirect
277. C3×C9C2×#102semidirect
300. C3×D4C4×D9semidirect
413. C9D3×#102semidirect
424. D4(C3⋊D9)⋊C4semidirect
469. C3D9×#102semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C3⋊D92
#117210002
C2×(C3⋊D9)⋊C82
C9×#8322
C3×#633922
#117202842
#117202672
#105914163
D9×#1023