Finite Group Databasebeta

← #13161362

Subgroups — #13161362

Order 1728.

1100 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
1C3×#41×C3⋊D3C211yesmaximal, characteristic
2#11723437C211yesmaximal, characteristic, complemented
3C3⋊D3×#208C211yesmaximal, characteristic, complemented
4#11722207C211yesmaximal, characteristic, complemented
5C32×#837C211yesmaximal, characteristic, complemented
6#11722046C211yesmaximal, characteristic, complemented
7#11722045C211yesmaximal, characteristic, complemented
8#10592520—31nomaximal
9#10591670—34nomaximal
10C33×#41C2211yescharacteristic
11C3×#3832C411yescharacteristic, complemented
12C3×Q8×C3⋊D3C411yescharacteristic, complemented
13C3×#3830—21no—
14C3×#3830C411yescharacteristic, complemented
15C3×D4×C3⋊D3—21no—
16C3×D4×C3⋊D3C411yescharacteristic, complemented
17C3×#3828—21no—
18C3×#3828C2211yescharacteristic
19C2×C3×C4×C3⋊D3—21no—
20C2×C3×C4×C3⋊D3C2211yescharacteristic
21C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
22#88855C2211yescharacteristic, complemented
23C32×C4×Dic3C2211yescharacteristic, complemented
24C32×#208C2211yescharacteristic, complemented
25#88807—21no—
26#88803—21no—
27#88787—21no—
28C3⋊D3×C3⋊C8—21no—
29#41×C3⋊D3D311yescharacteristic
30C3×D3×#41—34no—
31M16×C3⋊D3—31no—
32#64081—31no—
33#63855—34no—
34D3×#208—34no—
35C32×#104—31no—
36#63664—31no—
37#63664—34no—
38#63664—64no—
39#63652—31no—
40#63632—31no—
41C3×#837D314yescomplemented
42#63569—34no—
43#63568—34no—
44C22×C3×C3⋊D3—21no—
45C22×C3×C3⋊D3D411yescharacteristic
46C33×Q8C2×C411yescharacteristic
47C33×D4—21no—
48C33×D4C2×C411yescharacteristic
49C2×C33×C4—21no—
50C2×C33×C4C2311yescharacteristic
51C2×C3⋊(C3⋊Dic3)—21no—
52C3×#638—21no—
53C3×#638—21no—
54C3×#638—21no—
55C2×C3×C3⋊Dic3—21no—
56C2×C3×C3⋊Dic3D411yescharacteristic
57C3×C4⋊(C3⋊D3)—21no—
58C3×C4⋊(C3⋊D3)C2×C411yescharacteristic
59C3×C4×C3⋊D3—21no—
60C3×C4×C3⋊D3—21no—
61C3×C4×C3⋊D3—21no—
62C3×C4×C3⋊D3C2×C411yescharacteristic
63C3×C4×C3⋊D3C2×C411yescharacteristic
64C3×C4×C3⋊D3D411yescharacteristic
65C3×C4×C3⋊D3D411yescharacteristic
66C3×#634—21no—
67C3×#634C2×C411yescharacteristic
68C2×C32×Dic3—21no—
69C3⋊(C3⋊(C3⋊C8))—21no—
70C32×C3⋊C8—21no—
71#5236—91no—
72#4821—94no—
73C32×#41—24no—
74C32×#41C2×D314yes—
75C32×#41C2×D311yescharacteristic
76#3832Dic311yescharacteristic, complemented
77Q8×C3⋊D3Dic311yescharacteristic, complemented
78#3830—21no—
79#3830Dic311yescharacteristic, complemented
80D4×C3⋊D3—21no—
81D4×C3⋊D3Dic311yescharacteristic, complemented
82#3828—21no—
83#3828C2×D311yescharacteristic
84C2×C4×C3⋊D3—21no—
85C2×C4×C3⋊D3C2×D311yescharacteristic
86C3×#239—34no—
87C3×D3×Q8—34no—
88C3×#237—34no—
89C3×#237—64no—
90C3×D3×D4—34no—
91C3×D3×D4—64no—
92C3×#235—34no—
93C3×#235—64no—
94C2×C3×C4×D3—34no—
95C2×C3×C4×D3—64no—
96C32×M16—31no—
97C32×C42—31no—
98#3752—61no—
99#3750—61no—
100C4×C3⋊Dic3—31no—
101C4×C3⋊Dic3—34no—
102C4×C3⋊Dic3—64no—
103#3748—31no—
104#3748—34no—
105#3748—64no—
106#3743—61no—
107C8×C3⋊D3—61no—
108C3×C4×Dic3C2×D314yescomplemented
109C3×#208C2×D314yescomplemented
110Dic3⋊Dic3—64no—
111(C2×D3)⋊Dic3—64no—
112#3711—64no—
113D3×C3⋊C8—64no—
114C22×C33—21no—
115C22×C33C2×D411yescharacteristic
116C2×C3×C3⋊D3—21no—
117C2×C3×C3⋊D3—41no—
118C2×C3×C3⋊D3—42no—
119C2×C3×C3⋊D3C22⋊C411yescharacteristic
120C2×C3×C3⋊D3C22⋊C411yescharacteristic
121C33×C4—21no—
122C33×C4C22×C411yescharacteristic
123C33×C4C2×D411yescharacteristic
124C3⋊(C3⋊Dic3)—41no—
125C3×C3⋊Dic3—21no—
126C3×C3⋊Dic3C22⋊C411yescharacteristic
127C3×C3⋊Dic3C22⋊C411yescharacteristic
128C32×Dic3—41no—
129C2×C3×#41—91no—
130D3×#41—34no—
131#926—91no—
132C2×#208—91no—
133D3×M16—94no—
134#872—94no—
135C3×#104—34no—
136#837—94no—
137#837—91no—
138#837—91no—
139#837—91no—
140#837—184no—
141#837C3⋊D311yescharacteristic, complemented
142#825—94no—
143#805—94no—
144C22×C3⋊D3—21no—
145C22×C3⋊D3#6711yescharacteristic
146C22×C3×D3—34no—
147C22×C3×D3—64no—
148C32×Q8—24no—
149C32×Q8C4×D314yes—
150C32×Q8C2×Dic311yescharacteristic
151C32×D4—21no—
152C32×D4—24no—
153C32×D4—24no—
154C32×D4—44no—
155C32×D4C4×D314yes—
156C32×D4C2×Dic311yescharacteristic
157C2×C32×C4—21no—
158C2×C32×C4—24no—
159C2×C32×C4—24no—
160C2×C32×C4—44no—
161C2×C32×C4—61no—
162C2×C32×C4C22×D314yes—
163C2×C32×C4C22×D311yescharacteristic
164#638—21no—
165#638—21no—
166#638—21no—
167C2×C3⋊Dic3—21no—
168C2×C3⋊Dic3—61no—
169C2×C3⋊Dic3—64no—
170C2×C3⋊Dic3—124no—
171C2×C3⋊Dic3#6711yescharacteristic
172C4⋊(C3⋊D3)—21no—
173C4⋊(C3⋊D3)C2×Dic311yescharacteristic
174C4×C3⋊D3—21no—
175C4×C3⋊D3—21no—
176C4×C3⋊D3—21no—
177C4×C3⋊D3C2×Dic311yescharacteristic
178C4×C3⋊D3C2×Dic311yescharacteristic
179C4×C3⋊D3#6711yescharacteristic
180C4×C3⋊D3#6711yescharacteristic
181#634—21no—
182#634C2×Dic311yescharacteristic
183C3×#67—64no—
184C3×#67—64no—
185C3×#67—64no—
186C2×C3×Dic3—24no—
187C2×C3×Dic3—34no—
188C2×C3×Dic3—64no—
189C3×D12—34no—
190C3×D12—64no—
191C3×C4×D3—34no—
192C3×C4×D3—34no—
193C3×C4×D3—34no—
194C3×C4×D3—34no—
195C3×C4×D3—64no—
196C3×C4×D3—64no—
197C3×C4×D3—64no—
198C3×Dic6—34no—
199C3×Dic6—64no—
200C32×C8—61no—
201C3⋊(C3⋊C8)—61no—
202C3⋊(C3⋊C8)—64no—
203C3⋊(C3⋊C8)—124no—
204C3×C3⋊C8—24no—
205#421—271noSylow
206C2×C33—21no—
207C2×C33—41no—
208C2×C33C2×C22⋊C411yescharacteristic
209C3×C3⋊D3—21no—
210C3×C3⋊D3—21no—
211C3×C3⋊D3—41no—
212C3×#41—24no—
213C3×#41—91no—
214C3×#41—91no—
215C3×#41—91no—
216C3×#41—181no—
217C3×#41—181no—
218C3×#41D3214yes—
219C3×#41C2×C3⋊D311yescharacteristic
220C2×C3×Q8—91no—
221C2×C3×D4—91no—
222C2×C3×D4—181no—
223C22×C3×C4—91no—
224C22×C3×C4—181no—
225#239—34no—
226D3×Q8—34no—
227#237—34no—
228#237—64no—
229D3×D4—34no—
230D3×D4—64no—
231#235—34no—
232#235—64no—
233C2×C4×D3—34no—
234C2×C4×D3—64no—
235C3×M16—34no—
236C3×C42—34no—
237#217—181no—
238(C2×D3)⋊C4—184no—
239C4⋊Dic3—181no—
240Dic3⋊C4—184no—
241C4×Dic3—94no—
242C4×Dic3—91no—
243C4×Dic3—184no—
244C4×Dic3C2×C3⋊D311yescharacteristic, complemented
245#208—94no—
246#208—91no—
247#208—184no—
248#208—181no—
249#208C2×C3⋊D311yescharacteristic, complemented
250C2×C3⋊C8—181no—
251ΓL(1,𝔽25)—184no—
252D3×C8—184no—
253C22×C32—21no—
254C22×C32—24no—
255C22×C32—24no—
256C22×C32—44no—
257C22×C32D3×D414yes—
258C22×C32C2×#6711yescharacteristic
259C2×C3⋊D3—21no—
260C2×C3⋊D3—41no—
261C2×C3⋊D3—42no—
262C2×C3⋊D3#21711yescharacteristic
263C2×C3⋊D3#21711yescharacteristic
264C2×C3×D3—34no—
265C2×C3×D3—34no—
266C2×C3×D3—64no—
267C2×C3×D3—124no—
268C2×C3×D3—128no—
269C32×C4—21no—
270C32×C4—24no—
271C32×C4—24no—
272C32×C4—24no—
273C32×C4—44no—
274C32×C4—121no—
275C32×C4C2×C4×D314yes—
276C32×C4D3×D414yes—
277C32×C4C22×Dic311yescharacteristic
278C32×C4C2×#6711yescharacteristic
279C3⋊Dic3—21no—
280C3⋊Dic3—121no—
281C3⋊Dic3—124no—
282C3⋊Dic3—128no—
283C3⋊Dic3#21711yescharacteristic
284C3⋊Dic3#21711yescharacteristic
285C3×Dic3—34no—
286C3×Dic3—34no—
287C3×Dic3—44no—
288C3×Dic3—64no—
289C2×#41—91no—
290C2×M16—271no—
291#117—271no—
292#104—31no—
293#104—271no—
294#104—271no—
295#104—271no—
296C33#42111yescharacteristic, complemented, Sylow
297C23×C3—91no—
298C23×C3—181no—
299C22×D3—34no—
300C22×D3—64no—
301C3×Q8—24no—
302C3×Q8—91no—
303C3×Q8—181no—
304C3×Q8D3×Dic314yes—
305C3×Q8C4×C3⋊D311yescharacteristic
306C3×D4—24no—
307C3×D4—21no—
308C3×D4—24no—
309C3×D4—44no—
310C3×D4—91no—
311C3×D4—181no—
312C3×D4—181no—
313C3×D4—181no—
314C3×D4—181no—
315C3×D4D3×Dic314yes—
316C3×D4C4×C3⋊D311yescharacteristic
317C2×C3×C4—24no—
318C2×C3×C4—21no—
319C2×C3×C4—24no—
320C2×C3×C4—44no—
321C2×C3×C4—64no—
322C2×C3×C4—91no—
323C2×C3×C4—91no—
324C2×C3×C4—91no—
325C2×C3×C4—91no—
326C2×C3×C4—91no—
327C2×C3×C4—181no—
328C2×C3×C4—181no—
329C2×C3×C4—181no—
330C2×C3×C4—181no—
331C2×C3×C4C2×D3214yes—
332C2×C3×C4C22×C3⋊D311yescharacteristic
333#67—64no—
334#67—64no—
335#67—64no—
336C2×Dic3—21no—
337C2×Dic3—34no—
338C2×Dic3—64no—
339C2×Dic3—184no—
340C2×Dic3—181no—
341C2×Dic3—364no—
342D12—34no—
343D12—64no—
344C4×D3—34no—
345C4×D3—34no—
346C4×D3—34no—
347C4×D3—34no—
348C4×D3—64no—
349C4×D3—64no—
350C4×D3—64no—
351Dic6—34no—
352Dic6—64no—
353C3×C8—64no—
354C3⋊C8—21no—
355C3⋊C8—184no—
356C3⋊C8—181no—
357C3⋊C8—364no—
358C2×C32—21no—
359C2×C32—24no—
360C2×C32—24no—
361C2×C32—44no—
362C2×C32—41no—
363C2×C32—44no—
364C2×C32—84no—
365C2×C32D3×C22⋊C414yes—
366C2×C32C2×#21711yescharacteristic
367C3⋊D3—21no—
368C3⋊D3—21no—
369C3⋊D3—41no—
370C3×D3—64no—
371C3×D3—64no—
372C3×D3—124no—
373#41—91no—
374#41—91no—
375#41—91no—
376#41—181no—
377#41—181no—
378#41D3×C3⋊D311yescharacteristic
379C2×Q8—91no—
380C2×D4—91no—
381C2×D4—181no—
382C22×C4—91no—
383C22×C4—181no—
384M16—31no—
385M16—271no—
386M16—541no—
387C2×C8—541no—
388C4⋊C4—541no—
389C22⋊C4—541no—
390C42—31no—
391C42—271no—
392C22×C3—21no—
393C22×C3—24no—
394C22×C3—24no—
395C22×C3—44no—
396C22×C3—91no—
397C22×C3—91no—
398C22×C3—181no—
399C22×C3—361no—
400C22×C3—362no—
401C22×C3D3×#6714yes—
402C22×C3D4×C3⋊D311yescharacteristic
403C2×D3—34no—
404C2×D3—34no—
405C2×D3—64no—
406C2×D3—124no—
407C2×D3—128no—
408C3×C4—21no—
409C3×C4—24no—
410C3×C4—24no—
411C3×C4—24no—
412C3×C4—44no—
413C3×C4—91no—
414C3×C4—91no—
415C3×C4—124no—
416C3×C4—181no—
417C3×C4C2×D3×Dic314yes—
418C3×C4D3×#6714yes—
419C3×C4C2×C4×C3⋊D311yescharacteristic
420C3×C4D4×C3⋊D311yescharacteristic
421Dic3—34no—
422Dic3—34no—
423Dic3—41no—
424Dic3—64no—
425Dic3—361no—
426Dic3—364no—
427Dic3—368no—
428C32—24no—
429C32#482114yescomplemented
430C32#523611yescharacteristic, complemented
431C23—91no—
432C23—181no—
433Q8—91no—
434Q8—181no—
435Q8Dic3×C3⋊D311yescharacteristic
436D4—21no—
437D4—91no—
438D4—181no—
439D4—181no—
440D4—181no—
441D4—181no—
442D4Dic3×C3⋊D311yescharacteristic
443C2×C4—21no—
444C2×C4—61no—
445C2×C4—91no—
446C2×C4—91no—
447C2×C4—91no—
448C2×C4—91no—
449C2×C4—91no—
450C2×C4—181no—
451C2×C4—181no—
452C2×C4—181no—
453C2×C4—181no—
454C2×C4—541no—
455C2×C4C2×D3×C3⋊D311yescharacteristic
456C8—61no—
457C8—541no—
458C2×C3—21no—
459C2×C3—24no—
460C2×C3—24no—
461C2×C3—44no—
462C2×C3—41no—
463C2×C3—44no—
464C2×C3—84no—
465C2×C3—181no—
466C2×C3—181no—
467C2×C3—361no—
468C2×C3D3×#21714yes—
469C2×C3C22⋊C4×C3⋊D311yescharacteristic
470D3—64no—
471D3—64no—
472D3—124no—
473C22—21no—
474C22—91no—
475C22—91no—
476C22—181no—
477C22—361no—
478C22—362no—
479C22C3⋊D3×#6711yescharacteristic
480C4—21no—
481C4—91no—
482C4—91no—
483C4—121no—
484C4—181no—
485C4—1081no—
486C4C2×Dic3×C3⋊D311yescharacteristic
487C4C3⋊D3×#6711yescharacteristic
488C3—24no—
489C3#1059167014yescomplemented
490C3#1059252011yescharacteristic, complemented
491C2—21no—
492C2—41no—
493C2—181no—
494C2—181no—
495C2—361no—
496C2C3⋊D3×#21711yescharacteristic
subgroupG/H
centerC4C3⋊D3×#67
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×#41C22
socleC2×C33C2×C22⋊C4
prime radicalC3×#41×C3⋊D3C2
cosocleC2×C33×C4C23
hypercenter#41D3×C3⋊D3
hypocenterC33#421
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
2. #11723437C2semidirect
3. C3⋊D3×#208C2semidirect
4. #11722207C2semidirect
5. C32×#837C2semidirect
6. #11722046C2semidirect
7. #11722045C2semidirect
11. C3×#3832C4semidirect
12. C3×Q8×C3⋊D3C4semidirect
14. C3×#3830C4semidirect
16. C3×D4×C3⋊D3C4semidirect
21. C4×C3⋊(C3⋊Dic3)C22semidirect
22. #88855C22semidirect
23. C32×C4×Dic3C22semidirect
24. C32×#208C22semidirect
41. C3×#837D3semidirect
76. #3832Dic3semidirect
77. Q8×C3⋊D3Dic3semidirect
79. #3830Dic3semidirect
81. D4×C3⋊D3Dic3semidirect
108. C3×C4×Dic3C2×D3semidirect
109. C3×#208C2×D3semidirect
141. #837C3⋊D3semidirect
244. C4×Dic3C2×C3⋊D3semidirect
249. #208C2×C3⋊D3semidirect
296. C33#421semidirect
429. C32#4821semidirect
430. C32#5236semidirect
489. C3#10591670semidirect
490. C3#10592520semidirect

Maximal subgroups

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

subgroupindex
C3×#41×C3⋊D32
#117234372
C3⋊D3×#2082
#117222072
C32×#8372
#117220462
#117220452
#105925203
#105916703