Finite Group Databasebeta

← #13140348

Subgroups — #13140348

Order 1728.

806 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 105 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#11720922C211yesmaximal, characteristic, complemented
3C2×#88414C211yesmaximal, characteristic, complemented
4C9×#832C211yesmaximal, characteristic, complemented
5C3×#63392C211yesmaximal, characteristic, complemented
6#11720284C211yesmaximal, characteristic, complemented
7#11720267C211yesmaximal, characteristic, complemented
8#10591426—31nomaximal
9#10588450—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
17C9×C4⋊Dic3C2211yescharacteristic, complemented
18C3×C4⋊Dic9C2211yescharacteristic, complemented
19C2×C9×C3⋊C8C2211yescharacteristic, complemented
20C2×C3×C9⋊C8C2211yescharacteristic, complemented
21#88445—21no—
22Dic3×Dic9—21no—
23#88414—22no—
24C2×D4×C3⋊D3—31no—
25#63849—31no—
26C2×#3712—31no—
27C2×D4×D9—31no—
28C2×D4×D9—61no—
29C3×#832—31no—
30C3×#832D311yescharacteristic
31#63564—31no—
32#63564—31no—
33C2×#3663—31no—
34#63454—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×C412yes—
47C2×C3×C4×C9C2311yescharacteristic
48#6233—24no—
49C2×C3⋊Dic9D411yescharacteristic
50C4⋊(C3⋊D9)—21no—
51C4⋊(C3⋊D9)C2×C412yes—
52C4×C3⋊D9C2×C412yes—
53C4×C3⋊D9D412yes—
54C2×C9×Dic3—21no—
55C2×C3×Dic9—21no—
56C9×C3⋊C8—21no—
57C3×C9⋊C8—21no—
58#4770—91no—
59#4770—91no—
60C23×C3⋊D3—61no—
61C2×C32×D4C2×D311yescharacteristic
62C2×#638—61no—
63D4×C3⋊D3—34no—
64D4×C3⋊D3—62no—
65C2×C4⋊(C3⋊D3)—31no—
66C2×C4×C3⋊D3—31no—
67C23×D9—61no—
68C23×D9—62no—
69C3×C4⋊Dic3—31no—
70C3×C4⋊Dic3C2×D311yescharacteristic
71C2×C3×C3⋊C8—31no—
72C2×C3×C3⋊C8C2×D311yescharacteristic
73#3722—61no—
74Dic32—61no—
75#3712—62no—
76C2×D4×C9—21no—
77C2×D4×C9C2×D311yescharacteristic
78C2×#611—61no—
79C2×#611—62no—
80D4×D9—34no—
81D4×D9—62no—
82D4×D9—64no—
83D4×D9—64no—
84C2×D36—31no—
85C2×D36—61no—
86C2×C4×D9—31no—
87C2×C4×D9—61no—
88C2×C8×C9—31no—
89C9×C4⋊C4—31no—
90(C2×D9)⋊C4—61no—
91C4⋊Dic9C2×D311yescharacteristic, complemented
92C4×Dic9—61no—
93C2×C9⋊C8C2×D311yescharacteristic, complemented
94#3663—62no—
95C22×C3×C9—22no—
96C22×C3×C9—41no—
97C22×C3×C9—41no—
98C22×C3×C9C2×D411yescharacteristic
99C2×C3⋊D9—21no—
100C2×C3⋊D9—21no—
101C2×C3⋊D9—22no—
102C2×C3⋊D9—48no—
103C2×C3⋊D9—41no—
104C2×C3⋊D9—41no—
105C2×C3⋊D9C22⋊C412yes—
106C3×C4×C9C22×C411yescharacteristic
107C3×C4×C9C2×D411yescharacteristic
108C3⋊Dic9C22⋊C412yes—
109C9×Dic3—41no—
110C3×Dic9—41no—
111C2×D3×D4—91no—
112C2×D3×D4—91no—
113C2×D3×D4—181no—
114C2×ΓL(1,𝔽25)—91no—
115C2×ΓL(1,𝔽25)—91no—
116#892—91no—
117#892—91no—
118C3×#102—31no—
119C3×#102—91no—
120#832—31no—
121#832D911yescharacteristic, complemented
122#821—91no—
123#821—91no—
124#809—91no—
125#809—91no—
126C23×C32—21no—
127C22×C3⋊D3—31no—
128C22×C3⋊D3—64no—
129C22×C3⋊D3—61no—
130C22×C3⋊D3—122no—
131C22×C3⋊D3—122no—
132C32×D4—21no—
133C32×D4C4×D312yes—
134C2×C32×C4C22×D311yescharacteristic
135#638—64no—
136C2×C3⋊Dic3—31no—
137C4⋊(C3⋊D3)—32no—
138C4⋊(C3⋊D3)—61no—
139C4×C3⋊D3—32no—
140C4×C3⋊D3—32no—
141C2×C3×Dic3—21no—
142C2×C3×Dic3—61no—
143C23×C9—22no—
144C23×C9—21no—
145C22×D9—31no—
146C22×D9—61no—
147C22×D9—64no—
148C22×D9—62no—
149C22×D9—68no—
150C22×D9—61no—
151C22×D9—124no—
152C22×D9—124no—
153C22×D9—122no—
154C22×D9—122no—
155C3×C3⋊C8—21no—
156C3×C3⋊C8—61no—
157D4×C9—22no—
158D4×C9—21no—
159D4×C9—22no—
160D4×C9C4×D312yes—
161C2×C4×C9—21no—
162C2×C4×C9—61no—
163C2×C4×C9C22×D311yescharacteristic
164#611—64no—
165#611—68no—
166C2×Dic9—21no—
167C2×Dic9—31no—
168C2×Dic9—61no—
169D36—32no—
170D36—62no—
171D36—62no—
172D36—61no—
173C4×D9—32no—
174C4×D9—32no—
175C4×D9—62no—
176C4×D9—62no—
177C8×C9—61no—
178C9⋊C8—21no—
179#418—271noSylow
180C2×C3×C9—42no—
181C2×C3×C9C2×C22⋊C411yescharacteristic
182C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
183C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
184C3⋊D9—22no—
185C3⋊D9—42no—
186C23×D3—181no—
187C23×D3—181no—
188C23×D3—182no—
189C2×C3×D4—21no—
190C2×C3×D4C2×D911yescharacteristic
191C2×C3×D4D3211yescharacteristic
192C2×#67—181no—
193C2×#67—181no—
194C2×#67—182no—
195D3×D4—94no—
196D3×D4—94no—
197D3×D4—184no—
198D3×D4—182no—
199D3×D4—182no—
200D3×D4—184no—
201C2×D12—91no—
202C2×D12—91no—
203C2×D12—181no—
204C2×C4×D3—91no—
205C2×C4×D3—91no—
206C2×C4×D3—181no—
207C2×C3×C8—31no—
208C2×C3×C8—91no—
209C3×C4⋊C4—31no—
210C3×C4⋊C4—91no—
211(C2×D3)⋊C4—181no—
212(C2×D3)⋊C4—181no—
213C4⋊Dic3—31no—
214C4⋊Dic3C2×D911yescharacteristic, complemented
215C4×Dic3—181no—
216C4×Dic3—181no—
217C2×C3⋊C8—31no—
218C2×C3⋊C8C2×D911yescharacteristic, complemented
219ΓL(1,𝔽25)—182no—
220ΓL(1,𝔽25)—182no—
221C22×C32—22no—
222C22×C32—41no—
223C22×C32—41no—
224C22×C32D3×D411yescharacteristic
225C2×C3⋊D3—32no—
226C2×C3⋊D3—61no—
227C2×C3⋊D3—61no—
228C2×C3⋊D3—62no—
229C2×C3⋊D3—128no—
230C2×C3⋊D3—121no—
231C2×C3⋊D3—121no—
232C32×C4C2×C4×D311yescharacteristic
233C32×C4D3×D411yescharacteristic
234C3⋊Dic3—32no—
235C3×Dic3—41no—
236C3×Dic3—121no—
237C22×C9—24no—
238C22×C9—22no—
239C22×C9—21no—
240C22×C9—42no—
241C22×C9—42no—
242C22×C9—41no—
243C22×C9—41no—
244C22×C9D3×D411yescharacteristic
245C2×D9—32no—
246C2×D9—62no—
247C2×D9—62no—
248C2×D9—62no—
249C2×D9—61no—
250C2×D9—61no—
251C2×D9—64no—
252C2×D9—62no—
253C2×D9—1216no—
254C2×D9—128no—
255C2×D9—122no—
256C2×D9—122no—
257C2×D9—121no—
258C2×D9—121no—
259C4×C9—21no—
260C4×C9—21no—
261C4×C9—121no—
262C4×C9C2×C4×D311yescharacteristic
263C4×C9D3×D411yescharacteristic
264Dic9—32no—
265Dic9—41no—
266Dic9—62no—
267C22×D4—271no—
268C2×M16—271no—
269#117—271no—
270#102—31no—
271#102—91no—
272#102—271no—
273#102—271no—
274C3×C9#41811yescharacteristic, complemented, Sylow
275C23×C3—22no—
276C23×C3—21no—
277C23×C3—21no—
278C22×D3—91no—
279C22×D3—91no—
280C22×D3—181no—
281C22×D3—184no—
282C22×D3—184no—
283C22×D3—182no—
284C22×D3—188no—
285C22×D3—181no—
286C22×D3—181no—
287C22×D3—362no—
288C22×D3—362no—
289C22×D3—364no—
290C22×D3—364no—
291C22×D3—362no—
292C22×D3—362no—
293C3×D4—22no—
294C3×D4—21no—
295C3×D4—21no—
296C3×D4—22no—
297C3×D4C4×D912yes—
298C3×D4#62412yes—
299C2×C3×C4—21no—
300C2×C3×C4—61no—
301C2×C3×C4—181no—
302C2×C3×C4C22×D911yescharacteristic
303C2×C3×C4C2×D3211yescharacteristic
304#67—184no—
305#67—184no—
306#67—188no—
307C2×Dic3—21no—
308C2×Dic3—61no—
309C2×Dic3—91no—
310C2×Dic3—91no—
311C2×Dic3—181no—
312D12—92no—
313D12—92no—
314D12—182no—
315D12—182no—
316D12—181no—
317D12—181no—
318C4×D3—92no—
319C4×D3—92no—
320C4×D3—92no—
321C4×D3—92no—
322C4×D3—182no—
323C4×D3—182no—
324C3×C8—61no—
325C3×C8—181no—
326C3⋊C8—21no—
327C3⋊C8—61no—
328C2×C32—42no—
329C2×C32D3×C22⋊C411yescharacteristic
330C2×C32#91111yescharacteristic
331C2×C32#91411yescharacteristic
332C3⋊D3—62no—
333C3⋊D3—122no—
334C2×C9—21no—
335C2×C9—21no—
336C2×C9—21no—
337C2×C9—44no—
338C2×C9—42no—
339C2×C9D3×C22⋊C411yescharacteristic
340C2×C9#91111yescharacteristic
341C2×C9#91411yescharacteristic
342D9—64no—
343D9—62no—
344D9—124no—
345D9—122no—
346C24—541no—
347C2×D4—274no—
348C2×D4—271no—
349C2×D4—542no—
350C2×D4—541no—
351C2×D4D3×D911yescharacteristic
352C22×C4—271no—
353M16—542no—
354C2×C8—31no—
355C2×C8—91no—
356C4⋊C4—31no—
357C4⋊C4—91no—
358C22⋊C4—541no—
359C42—541no—
360C22×C3—24no—
361C22×C3—22no—
362C22×C3—22no—
363C22×C3—21no—
364C22×C3—42no—
365C22×C3—42no—
366C22×C3—41no—
367C22×C3—41no—
368C22×C3—41no—
369C22×C3—41no—
370C22×C3D4×D911yescharacteristic
371C22×C3#381111yescharacteristic
372C2×D3—92no—
373C2×D3—92no—
374C2×D3—182no—
375C2×D3—182no—
376C2×D3—182no—
377C2×D3—181no—
378C2×D3—181no—
379C2×D3—181no—
380C2×D3—181no—
381C2×D3—184no—
382C2×D3—182no—
383C2×D3—182no—
384C2×D3—368no—
385C2×D3—3616no—
386C2×D3—368no—
387C2×D3—362no—
388C2×D3—362no—
389C2×D3—361no—
390C2×D3—361no—
391C2×D3—361no—
392C2×D3—361no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—121no—
396C3×C4—361no—
397C3×C4C2×C4×D911yescharacteristic
398C3×C4D4×D911yescharacteristic
399C3×C4C2×#62411yescharacteristic
400C3×C4#381111yescharacteristic
401Dic3—41no—
402Dic3—92no—
403Dic3—92no—
404Dic3—121no—
405Dic3—182no—
406C32#477011yescharacteristic
407C9—21no—
408C9#477011yescharacteristic, complemented
409C23—21no—
410C23—271no—
411C23—544no—
412C23—541no—
413C23—1082no—
414C23—1082no—
415D4—21no—
416D4—272no—
417D4—544no—
418D4—541no—
419D4(C3⋊D9)⋊C412yes—
420C2×C4—61no—
421C2×C4—181no—
422C2×C4—271no—
423C2×C4—272no—
424C2×C4—272no—
425C2×C4C2×D3×D911yescharacteristic
426C8—61no—
427C8—181no—
428C2×C3—21no—
429C2×C3—21no—
430C2×C3—21no—
431C2×C3—44no—
432C2×C3—42no—
433C2×C3—42no—
434C2×C3C22⋊C4×D911yescharacteristic
435C2×C3#6347311yescharacteristic
436C2×C3#6347611yescharacteristic
437C2×C3#6393711yescharacteristic
438C2×C3#6393711yescharacteristic
439C2×C3#6397411yescharacteristic
440D3—184no—
441D3—182no—
442D3—182no—
443D3—364no—
444D3—362no—
445D3—362no—
446C22—22no—
447C22—41no—
448C22—41no—
449C22—272no—
450C22—541no—
451C22—541no—
452C22—542no—
453C22—1088no—
454C22—1081no—
455C22—1081no—
456C22#8866811yescharacteristic
457C4—121no—
458C4—272no—
459C4—361no—
460C4C2×(C3⋊D9)⋊C411yescharacteristic
461C4#8866811yescharacteristic
462C3—21no—
463C3#1058845011yescharacteristic, complemented
464C3#1059142611yescharacteristic
465C2—42no—
466C2—542no—
467C2—1082no—
468C2#1172117911yescharacteristic
469C2#1172118711yescharacteristic
470C2#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×C9#418
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. #11720922C2semidirect
3. C2×#88414C2semidirect
4. C9×#832C2semidirect
5. C3×#63392C2semidirect
6. #11720284C2semidirect
7. #11720267C2semidirect
14. D4×C3⋊D9C4semidirect
17. C9×C4⋊Dic3C22semidirect
18. C3×C4⋊Dic9C22semidirect
19. C2×C9×C3⋊C8C22semidirect
20. C2×C3×C9⋊C8C22semidirect
36. #63392D3semidirect
91. C4⋊Dic9C2×D3semidirect
93. C2×C9⋊C8C2×D3semidirect
121. #832D9semidirect
214. C4⋊Dic3C2×D9semidirect
218. C2×C3⋊C8C2×D9semidirect
274. C3×C9#418semidirect
408. C9#4770semidirect
463. C3#10588450semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C3⋊D92
#117209222
C2×#884142
C9×#8322
C3×#633922
#117202842
#117202672
#105914263
#105884503