Finite Group Databasebeta

← #13139124

Subgroups — #13139124

Order 1728.

506 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#3697C211yesmaximal, characteristic
2D3×C9×M16C211yesmaximal, characteristic, complemented
3#11720959C211yesmaximal, characteristic, complemented
4#11720558C211yesmaximal, characteristic, complemented
5C3×#63384C211yesmaximal, characteristic, complemented
6#11720292C211yesmaximal, characteristic, complemented
7#11720291C211yesmaximal, characteristic, complemented
8#10591005—31nomaximal
9#10588819—31nomaximal
10#88736C2211yescharacteristic
11C2×C4×D3×C9C2211yescharacteristic
12C3×#3697C2211yescharacteristic
13D3×#611—21no—
14#88658—21no—
15D3×D36C411yescharacteristic, complemented
16C4×D3×D9—21no—
17#88642C411yescharacteristic, complemented
18#88641C411yescharacteristic, complemented
19#88640—21no—
20D3×Dic18C411yescharacteristic, complemented
21C3×C9×M16C2211yescharacteristic, complemented
22C4×C3⋊Dic9C2211yescharacteristic, complemented
23C3×C4×Dic9C2211yescharacteristic, complemented
24C9×#208C2211yescharacteristic, complemented
25C9×ΓL(1,𝔽25)—21no—
26D3×C8×C9—21no—
27(C2×D3)⋊Dic9—21no—
28Dic3⋊Dic9—21no—
29D3×#235—31no—
30C3×D3×M16D311yescharacteristic
31#63855—31no—
32C2×#3697—31no—
33#63652—31no—
34C3×#825—31no—
35#63570—31no—
36#63568—31no—
37C2×C9×M16—31no—
38#63488—31no—
39#63384—31no—
40#63384—31no—
41#63384—31no—
42#63384—61no—
43#63384D311yescharacteristic, complemented
44C22×D3×C9D411yescharacteristic
45C2×D3×D9—21no—
46C2×C3×C4×C9C2311yescharacteristic
47#6233—21no—
48C2×C3⋊Dic9—21no—
49C4⋊(C3⋊D9)C2×C411yescharacteristic
50C4×C3⋊D9—21no—
51#6226C2×C411yescharacteristic
52C3×#611—21no—
53C2×C9×Dic3D411yescharacteristic
54C2×C3×Dic9—21no—
55C4×D3×C9C2×C411yescharacteristic
56C4×D3×C9C2×C411yescharacteristic
57C4×D3×C9D411yescharacteristic
58C4×D3×C9D411yescharacteristic
59C3×D36C2×C411yescharacteristic
60C3×C4×D9—21no—
61C3×Dic18C2×C411yescharacteristic
62#6195—21no—
63#6194—21no—
64D3×Dic9—21no—
65#6192—21no—
66(C3⋊D9)⋊C4—21no—
67Dic3×D9—21no—
68#6189—21no—
69C3×C8×C9—21no—
70C9×C3⋊C8—21no—
71#5139—91no—
72#4821—91no—
73#3828—31no—
74C3×#235—31no—
75C2×C3×C4×D3C2×D311yescharacteristic
76D3×#67—61no—
77#3804—61no—
78D3×D12—31no—
79C4×D32—61no—
80#3799—31no—
81#3798—61no—
82#3795—31no—
83D3×Dic6—31no—
84C32×M16C2×D311yescharacteristic
85C4×C3⋊Dic3—31no—
86C3×C4×Dic3—31no—
87C3×#208C2×D311yescharacteristic
88C3×ΓL(1,𝔽25)—21no—
89C3×D3×C8—21no—
90Dic3⋊Dic3—61no—
91(C2×D3)⋊Dic3—61no—
92C22×C4×C9—31no—
93C2×#611—61no—
94#3697—31no—
95#3697—31no—
96#3697—31no—
97#3697—61no—
98#3697—61no—
99#3697—61no—
100#3697C2×D311yescharacteristic
101C2×D36—31no—
102C2×C4×D9—61no—
103C2×Dic18—31no—
104C9×M16—21no—
105C9×M16—31no—
106C9×M16—61no—
107C9×M16C2×D311yescharacteristic, complemented
108C2×C8×C9—61no—
109#3676—61no—
110C4⋊Dic9—61no—
111C4×Dic9—31no—
112C4×Dic9—61no—
113C4×Dic9C2×D311yescharacteristic, complemented
114C22×C3×C9C2×D411yescharacteristic
115C2×C3⋊D9—21no—
116C2×D3×C9—41no—
117C2×D3×C9C22⋊C411yescharacteristic
118C2×D3×C9C22⋊C411yescharacteristic
119C2×C3×D9—21no—
120D3×D9—42no—
121C3×C4×C9C22×C411yescharacteristic
122C3×C4×C9C2×D411yescharacteristic
123C3⋊Dic9—21no—
124C3⋊Dic9—41no—
125C9×Dic3C22⋊C411yescharacteristic
126C9×Dic3C22⋊C411yescharacteristic
127C3×Dic9—21no—
128C3×Dic9—41no—
129D3×#41—91no—
130C2×#235—91no—
131C2×C3×M16—31no—
132#926—91no—
133D3×M16D911yescharacteristic, complemented
134#872—91no—
135C3×#104—91no—
136#837—91no—
137#825—31no—
138#825—91no—
139#825—91no—
140#825—91no—
141#825—91no—
142#825—181no—
143#805—91no—
144C22×C3×D3D1211yescharacteristic
145C2×D32—61no—
146C2×C32×C4C22×D311yescharacteristic
147#638—61no—
148C2×C3⋊Dic3—61no—
149C4⋊(C3⋊D3)—31no—
150C4×C3⋊D3—61no—
151#634—31no—
152C3×#67—61no—
153C2×C3×Dic3—61no—
154C2×C3×Dic3D1211yescharacteristic
155C3×D12—31no—
156C3×C4×D3—61no—
157C3×C4×D3C4×D311yescharacteristic
158C3×C4×D3C4×D311yescharacteristic
159C3×C4×D3#6711yescharacteristic
160C3×C4×D3#6711yescharacteristic
161C3×Dic6—31no—
162#627—61no—
163#626—61no—
164#626—61no—
165#625—61no—
166#624—61no—
167D3×Dic3—61no—
168D3×Dic3—61no—
169C23×C9—31no—
170C22×D9—61no—
171C32×C8—21no—
172C3×C3⋊C8—21no—
173C2×C4×C9—21no—
174C2×C4×C9—31no—
175C2×C4×C9—31no—
176C2×C4×C9—31no—
177C2×C4×C9—31no—
178C2×C4×C9—31no—
179C2×C4×C9C22×D311yescharacteristic
180#611—21no—
181#611—61no—
182#611—61no—
183#611—61no—
184#611—121no—
185C2×Dic9—21no—
186C2×Dic9—61no—
187C2×Dic9—61no—
188C2×Dic9—121no—
189D36—31no—
190D36—61no—
191D36—61no—
192D36C4×D311yescharacteristic
193C4×D9—21no—
194C4×D9—61no—
195C4×D9—61no—
196C4×D9—61no—
197C4×D9—121no—
198Dic18—31no—
199Dic18—61no—
200Dic18—61no—
201Dic18C4×D311yescharacteristic
202C8×C9—21no—
203C8×C9—41no—
204C8×C9—61no—
205#421—271noSylow
206C2×C3×C9—21no—
207C2×C3×C9C2×C22⋊C411yescharacteristic
208C3⋊D9—41no—
209D3×C9—21no—
210D3×C9—21no—
211C3×D9—41no—
212C3×#41—91no—
213C22×C3×C4—31no—
214C2×#67—181no—
215#239—91no—
216D3×Q8—91no—
217#237—91no—
218#237—181no—
219D3×D4—91no—
220D3×D4—181no—
221#235—31no—
222#235—91no—
223#235—91no—
224#235—91no—
225#235—91no—
226#235—181no—
227#235—181no—
228#235—181no—
229#235—181no—
230C2×D12—91no—
231C2×C4×D3—181no—
232C2×C4×D3—181no—
233C2×C4×D3C2×D911yescharacteristic
234C2×Dic6—91no—
235C3×M16—21no—
236C3×M16—31no—
237C3×M16—61no—
238C3×M16C2×D911yescharacteristic, complemented
239C3×M16D3211yescharacteristic
240C2×C3×C8—61no—
241C3×C42—91no—
242#217—181no—
243(C2×D3)⋊C4—181no—
244C4⋊Dic3—181no—
245Dic3⋊C4—181no—
246C4×Dic3—31no—
247C4×Dic3—91no—
248C4×Dic3—91no—
249C4×Dic3—181no—
250#208C2×D911yescharacteristic, complemented
251ΓL(1,𝔽25)—21no—
252D3×C8—21no—
253C22×C32C2×D1211yescharacteristic
254C2×C3⋊D3—61no—
255C2×C3×D3—41no—
256C2×C3×D3—61no—
257C2×C3×D3(C2×D3)⋊C411yescharacteristic
258C2×C3×D3(C2×D3)⋊C411yescharacteristic
259D32—122no—
260C32×C4C2×C4×D311yescharacteristic
261C32×C4C2×#6711yescharacteristic
262C3⋊Dic3—61no—
263C3⋊Dic3—121no—
264C3×Dic3—61no—
265C3×Dic3—121no—
266C3×Dic3(C2×D3)⋊C411yescharacteristic
267C3×Dic3(C2×D3)⋊C411yescharacteristic
268C22×C9—21no—
269C22×C9—31no—
270C22×C9—31no—
271C22×C9—121no—
272C22×C9D3×D411yescharacteristic
273C2×D9—21no—
274C2×D9—61no—
275C2×D9—121no—
276C2×D9—122no—
277C4×C9—21no—
278C4×C9—21no—
279C4×C9—31no—
280C4×C9—31no—
281C4×C9C2×C4×D311yescharacteristic
282C4×C9D3×D411yescharacteristic
283Dic9—21no—
284Dic9—41no—
285Dic9—61no—
286Dic9—121no—
287Dic9—121no—
288Dic9—122no—
289C2×#41—271no—
290C2×M16—31no—
291#117—271no—
292#104—91no—
293#104—271no—
294#104—271no—
295#104—271no—
296C3×C9#42111yescharacteristic, complemented, Sylow
297C23×C3—31no—
298C22×D3—181no—
299C22×D3—181no—
300C22×D3D3611yescharacteristic
301C3×Q8—91no—
302C3×D4—91no—
303C3×D4—181no—
304C2×C3×C4—21no—
305C2×C3×C4—31no—
306C2×C3×C4—31no—
307C2×C3×C4—31no—
308C2×C3×C4—31no—
309C2×C3×C4—31no—
310C2×C3×C4—181no—
311C2×C3×C4—181no—
312C2×C3×C4C22×D911yescharacteristic
313C2×C3×C4C2×D3211yescharacteristic
314#67—61no—
315#67—181no—
316#67—181no—
317#67—181no—
318#67—181no—
319#67—181no—
320#67—181no—
321#67—361no—
322C2×Dic3—61no—
323C2×Dic3—181no—
324C2×Dic3—181no—
325C2×Dic3—181no—
326C2×Dic3—181no—
327C2×Dic3—361no—
328C2×Dic3D3611yescharacteristic
329D12—31no—
330D12—91no—
331D12—91no—
332D12—181no—
333D12—181no—
334D12—181no—
335C4×D3—61no—
336C4×D3—181no—
337C4×D3—181no—
338C4×D3—181no—
339C4×D3—181no—
340C4×D3—181no—
341C4×D3—181no—
342C4×D3—361no—
343C4×D3C4×D911yescharacteristic
344C4×D3C4×D911yescharacteristic
345C4×D3#61111yescharacteristic
346C4×D3#61111yescharacteristic
347Dic6—31no—
348Dic6—91no—
349Dic6—91no—
350Dic6—181no—
351Dic6—181no—
352Dic6—181no—
353C3×C8—21no—
354C3×C8—21no—
355C3×C8—41no—
356C3×C8—61no—
357C3⋊C8—21no—
358C2×C32—21no—
359C2×C32C2×(C2×D3)⋊C411yescharacteristic
360C3⋊D3—121no—
361C3×D3—21no—
362C3×D3—21no—
363C3×D3—121no—
364C2×C9—21no—
365C2×C9—21no—
366C2×C9—41no—
367C2×C9—61no—
368C2×C9—61no—
369C2×C9D3×C22⋊C411yescharacteristic
370D9—41no—
371D9—121no—
372D9—241no—
373#41—91no—
374#41—271no—
375#41—271no—
376#41—271no—
377#41—541no—
378#41—541no—
379C2×Q8—271no—
380C2×D4—271no—
381C2×D4—541no—
382C22×C4—31no—
383C22×C4—541no—
384M16—31no—
385M16—61no—
386M16D3×D911yescharacteristic, complemented
387C2×C8—61no—
388C4⋊C4—541no—
389C22⋊C4—541no—
390C42—91no—
391C42—271no—
392C22×C3—21no—
393C22×C3—31no—
394C22×C3—31no—
395C22×C3—121no—
396C22×C3—181no—
397C22×C3C2×D3611yescharacteristic
398C22×C3D3×D1211yescharacteristic
399C2×D3—41no—
400C2×D3—61no—
401C2×D3—181no—
402C2×D3—181no—
403C2×D3—361no—
404C2×D3—362no—
405C2×D3—362no—
406C2×D3(C2×D9)⋊C411yescharacteristic
407C2×D3(C2×D9)⋊C411yescharacteristic
408C3×C4—21no—
409C3×C4—21no—
410C3×C4—31no—
411C3×C4—31no—
412C3×C4—181no—
413C3×C4—361no—
414C3×C4C2×C4×D911yescharacteristic
415C3×C4C2×#61111yescharacteristic
416C3×C4C4×D3211yescharacteristic
417C3×C4D3×#6711yescharacteristic
418Dic3—61no—
419Dic3—121no—
420Dic3—181no—
421Dic3—181no—
422Dic3—361no—
423Dic3—361no—
424Dic3—361no—
425Dic3—362no—
426Dic3(C2×D9)⋊C411yescharacteristic
427Dic3(C2×D9)⋊C411yescharacteristic
428C32#513911yescharacteristic
429C9—21no—
430C9#482111yescharacteristic, complemented
431C23—31no—
432C23—541no—
433Q8—91no—
434Q8—271no—
435Q8—541no—
436D4—91no—
437D4—181no—
438D4—271no—
439D4—541no—
440D4—541no—
441D4—541no—
442D4—541no—
443C2×C4—31no—
444C2×C4—31no—
445C2×C4—31no—
446C2×C4—31no—
447C2×C4—31no—
448C2×C4—181no—
449C2×C4—181no—
450C2×C4—541no—
451C2×C4—541no—
452C2×C4—541no—
453C2×C4—541no—
454C2×C4—541no—
455C2×C4C2×D3×D911yescharacteristic
456C8—21no—
457C8—61no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—41no—
462C2×C3—61no—
463C2×C3—61no—
464C2×C3—361no—
465C2×C3C2×(C2×D9)⋊C411yescharacteristic
466C2×C3D3×(C2×D3)⋊C411yescharacteristic
467D3—21no—
468D3—21no—
469D3—121no—
470D3—361no—
471D3—361no—
472D3—721no—
473C22—31no—
474C22—31no—
475C22—121no—
476C22—181no—
477C22—541no—
478C22—1082no—
479C22D3×D3611yescharacteristic
480C4—31no—
481C4—31no—
482C4—181no—
483C4—361no—
484C4—541no—
485C4—1081no—
486C4C4×D3×D911yescharacteristic
487C4D3×#61111yescharacteristic
488C3—21no—
489C3#1058881911yescharacteristic, complemented
490C3#1059100511yescharacteristic
491C2—21no—
492C2—61no—
493C2—61no—
494C2—361no—
495C2—1081no—
496C2D3×(C2×D9)⋊C411yescharacteristic
subgroupG/H
centerC4D3×#611
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC3×C9×M16C22
socleC2×C32C2×(C2×D3)⋊C4
prime radicalD3×#3697C2
cosocleC2×C3×C4×C9C23
hypercenterM16D3×D9
hypocenterC3×C9#421
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C4 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
2. D3×C9×M16C2semidirect
3. #11720959C2semidirect
4. #11720558C2semidirect
5. C3×#63384C2semidirect
6. #11720292C2semidirect
7. #11720291C2semidirect
15. D3×D36C4semidirect
17. #88642C4semidirect
18. #88641C4semidirect
20. D3×Dic18C4semidirect
21. C3×C9×M16C22semidirect
22. C4×C3⋊Dic9C22semidirect
23. C3×C4×Dic9C22semidirect
24. C9×#208C22semidirect
43. #63384D3semidirect
107. C9×M16C2×D3semidirect
113. C4×Dic9C2×D3semidirect
133. D3×M16D9semidirect
238. C3×M16C2×D9semidirect
250. #208C2×D9semidirect
296. C3×C9#421semidirect
386. M16D3×D9semidirect
430. C9#4821semidirect
489. C3#10588819semidirect

Maximal subgroups

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

subgroupindex
D3×#36972
D3×C9×M162
#117209592
#117205582
C3×#633842
#117202922
#117202912
#105910053
#105888193