Finite Group Databasebeta

← #13155505

Subgroups — #13155505

Order 1728.

556 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#3748C211yesmaximal, characteristic
2C2×C3×#3711C211yesmaximal, characteristic
3C3×D3×#208C211yesmaximal, characteristic
4#11722081C211yesmaximal, characteristic, complemented
5#11722060C211yesmaximal, characteristic, complemented
6#11722058C211yesmaximal, characteristic, complemented
7C3×#63577C211yesmaximal, characteristic, complemented
8#10591141—31nomaximal
9#10590973—31nomaximal
10#10590912—31nomaximal
11C2×C32×C4×D3C2211yescharacteristic
12#88855C2211yescharacteristic
13C3×#3748C2211yescharacteristic
14C3×#3748C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×#208C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88786—21no—
19D3×C3⋊(C3⋊C8)—21no—
20C3×#3711—21no—
21C3×#3711C412yes—
22C3×#3711C412yes—
23C3×D3×C3⋊C8—21no—
24C2×#3748—31no—
25C2×C3×#208—31no—
26C2×C3×#208—31no—
27C3×D3×M16—31no—
28C2×C3×ΓL(1,𝔽25)—31no—
29C2×#3711D311yescharacteristic
30D3×#208—31no—
31D3×#208—31no—
32D3×#208—61no—
33D3×#208D311yescharacteristic
34#63649—31no—
35#63646—31no—
36C3×#822—31no—
37C3×#819—31no—
38#63578—31no—
39#63577—31no—
40#63577—31no—
41#63577—31no—
42#63577—31no—
43#63577—31no—
44#63577—31no—
45#63577—61no—
46#63577—61no—
47#63577D311yescharacteristic, complemented
48C22×C32×D3Q811yescharacteristic
49C2×C33×C4C2311yescharacteristic
50C2×C32×Dic3Q811yescharacteristic
51C32×C4×D3C2×C411yescharacteristic
52C32×C4×D3C2×C411yescharacteristic
53C32×C4×D3D411yescharacteristic
54C32×C4×D3D411yescharacteristic
55C3⋊(C3⋊(C3⋊C8))—21no—
56C3×C3⋊(C3⋊C8)—21no—
57C3×C3⋊(C3⋊C8)—21no—
58C3×C3⋊(C3⋊C8)C2×C412yes—
59C32×C3⋊C8—21no—
60C32×C3⋊C8C2×C412yes—
61#5125—91no—
62#5124—91no—
63#4893—91no—
64C22×C32×C4—31no—
65C2×C3×C4×D3—21no—
66C2×C3×C4×D3C2×D311yescharacteristic
67C2×C3×C4×D3C2×D311yescharacteristic
68C32×M16—31no—
69C2×C32×C8—31no—
70#3748—31no—
71#3748—31no—
72#3748—31no—
73#3748—61no—
74#3748—61no—
75#3748—61no—
76#3748—61no—
77#3748—121no—
78#3748C2×D311yescharacteristic
79#3748C2×D311yescharacteristic
80C2×C3⋊(C3⋊C8)—61no—
81C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
82C3×#208—21no—
83C3×#208—32no—
84C3×#208—32no—
85C3×#208—31no—
86C3×#208—31no—
87C3×#208—31no—
88C3×#208—31no—
89C3×#208—61no—
90C3×#208—61no—
91C3×#208—61no—
92C3×#208C2×D311yescharacteristic
93C3×#208C2×D311yescharacteristic
94C2×C3×C3⋊C8—21no—
95C2×C3×C3⋊C8—31no—
96C2×C3×C3⋊C8—31no—
97C2×C3×C3⋊C8—61no—
98C2×C3×C3⋊C8—61no—
99C2×C3×C3⋊C8C2×D311yescharacteristic
100C2×C3×C3⋊C8C2×D311yescharacteristic
101C3×ΓL(1,𝔽25)—32no—
102C3×ΓL(1,𝔽25)—32no—
103C3×ΓL(1,𝔽25)—61no—
104C3×D3×C8—61no—
105#3711—21no—
106#3711—62no—
107#3711—61no—
108#3711—61no—
109#3711Dic312yes—
110#3711Dic312yes—
111D3×C3⋊C8—21no—
112D3×C3⋊C8—62no—
113D3×C3⋊C8—61no—
114D3×C3⋊C8—61no—
115C22×C33C2×Q811yescharacteristic
116C2×C32×D3—41no—
117C2×C32×D3C4⋊C411yescharacteristic
118C2×C32×D3C4⋊C411yescharacteristic
119C33×C4C22×C411yescharacteristic
120C33×C4C2×D411yescharacteristic
121C32×Dic3C4⋊C411yescharacteristic
122C32×Dic3C4⋊C411yescharacteristic
123C2×C3×M16—91no—
124C2×C3×M16—91no—
125C2×#208—31no—
126C2×#208—31no—
127C2×#208—91no—
128C2×#208—91no—
129C2×#208—181no—
130D3×M16—31no—
131D3×M16—91no—
132C2×ΓL(1,𝔽25)—31no—
133C3×#108—91no—
134#822—31no—
135#822—91no—
136#822—91no—
137#822—91no—
138#822—91no—
139#822—91no—
140#822—181no—
141#819—31no—
142#819—91no—
143#819—91no—
144#819—91no—
145#819—91no—
146#819—181no—
147C23×C32—31no—
148C22×C3×D3—21no—
149C22×C3×D3Dic611yescharacteristic
150C22×C3×D3Dic611yescharacteristic
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—31no—
155C2×C32×C4—31no—
156C2×C32×C4—31no—
157C2×C32×C4—31no—
158C2×C32×C4—31no—
159C2×C32×C4—41no—
160C2×C32×C4C22×D311yescharacteristic
161C2×C32×C4C22×D311yescharacteristic
162C2×C32×C4C22×D311yescharacteristic
163C2×C3×Dic3—21no—
164C2×C3×Dic3Dic611yescharacteristic
165C2×C3×Dic3Dic611yescharacteristic
166C3×C4×D3—21no—
167C3×C4×D3—21no—
168C3×C4×D3—21no—
169C3×C4×D3—21no—
170C3×C4×D3C4×D311yescharacteristic
171C3×C4×D3C4×D311yescharacteristic
172C3×C4×D3D1211yescharacteristic
173C3×C4×D3D1211yescharacteristic
174C3×C4×D3C2×Dic311yescharacteristic
175C3×C4×D3C2×Dic311yescharacteristic
176C3×C4×D3#6711yescharacteristic
177C3×C4×D3#6711yescharacteristic
178C32×C8—32no—
179C32×C8—61no—
180C3⋊(C3⋊C8)—21no—
181C3⋊(C3⋊C8)—21no—
182C3⋊(C3⋊C8)—61no—
183C3⋊(C3⋊C8)—62no—
184C3⋊(C3⋊C8)—61no—
185C3⋊(C3⋊C8)—61no—
186C3⋊(C3⋊C8)—121no—
187C3⋊(C3⋊C8)—121no—
188C3⋊(C3⋊C8)—122no—
189C3⋊(C3⋊C8)C2×Dic312yes—
190C3×C3⋊C8—21no—
191C3×C3⋊C8—21no—
192C3×C3⋊C8—22no—
193C3×C3⋊C8—32no—
194C3×C3⋊C8—32no—
195C3×C3⋊C8—41no—
196C3×C3⋊C8—62no—
197C3×C3⋊C8—62no—
198C3×C3⋊C8—61no—
199C3×C3⋊C8—61no—
200C3×C3⋊C8—61no—
201C3×C3⋊C8—61no—
202C3×C3⋊C8—121no—
203C3×C3⋊C8C4×D312yes—
204C3×C3⋊C8C2×Dic312yes—
205#430—271noSylow
206C2×C33—21no—
207C2×C33C2×C4⋊C411yescharacteristic
208C32×D3—21no—
209C32×D3—21no—
210C22×C3×C4—31no—
211C22×C3×C4—31no—
212C22×C3×C4—61no—
213C2×C4×D3D3211yescharacteristic
214C3×M16—31no—
215C3×M16—31no—
216C3×M16—61no—
217C3×M16—92no—
218C3×M16—92no—
219C3×M16—91no—
220C3×M16—91no—
221C3×M16—181no—
222C2×C3×C8—31no—
223C2×C3×C8—31no—
224C2×C3×C8—61no—
225C2×C3×C8—91no—
226C2×C3×C8—181no—
227#208—32no—
228#208—32no—
229#208—31no—
230#208—31no—
231#208—31no—
232#208—31no—
233#208—61no—
234#208—61no—
235#208—61no—
236#208—91no—
237#208—91no—
238#208—91no—
239#208—181no—
240#208—181no—
241#208—181no—
242#208—181no—
243#208—181no—
244#208—182no—
245#208—361no—
246#208D3211yescharacteristic
247C2×C3⋊C8—31no—
248C2×C3⋊C8—31no—
249C2×C3⋊C8—61no—
250C2×C3⋊C8—61no—
251C2×C3⋊C8—181no—
252C2×C3⋊C8—181no—
253C2×C3⋊C8—182no—
254C2×C3⋊C8D3211yescharacteristic
255ΓL(1,𝔽25)—32no—
256ΓL(1,𝔽25)—32no—
257ΓL(1,𝔽25)—61no—
258ΓL(1,𝔽25)—181no—
259D3×C8—61no—
260D3×C8—181no—
261C22×C32—21no—
262C22×C32—21no—
263C22×C32—21no—
264C22×C32—31no—
265C22×C32—31no—
266C22×C32—41no—
267C22×C32—121no—
268C22×C32C2×Dic611yescharacteristic
269C22×C32C2×Dic611yescharacteristic
270C22×C32D3×Q811yescharacteristic
271C2×C3×D3—21no—
272C2×C3×D3—21no—
273C2×C3×D3—41no—
274C2×C3×D3—41no—
275C2×C3×D3—42no—
276C2×C3×D3Dic3⋊C411yescharacteristic
277C2×C3×D3Dic3⋊C411yescharacteristic
278C2×C3×D3C4⋊Dic311yescharacteristic
279C2×C3×D3C4⋊Dic311yescharacteristic
280C32×C4—21no—
281C32×C4—21no—
282C32×C4—21no—
283C32×C4—21no—
284C32×C4—21no—
285C32×C4—21no—
286C32×C4—31no—
287C32×C4—31no—
288C32×C4—41no—
289C32×C4—41no—
290C32×C4C2×C4×D311yescharacteristic
291C32×C4C2×C4×D311yescharacteristic
292C32×C4C2×D1211yescharacteristic
293C32×C4D3×D411yescharacteristic
294C32×C4C22×Dic311yescharacteristic
295C32×C4C2×#6711yescharacteristic
296C3×Dic3—21no—
297C3×Dic3—21no—
298C3×Dic3Dic3⋊C411yescharacteristic
299C3×Dic3Dic3⋊C411yescharacteristic
300C3×Dic3C4⋊Dic311yescharacteristic
301C3×Dic3C4⋊Dic311yescharacteristic
302C2×M16—91no—
303C2×M16—91no—
304C2×M16—271no—
305#108—91no—
306#108—271no—
307#108—271no—
308#108—271no—
309C33#43011yescharacteristic, complemented, Sylow
310C23×C3—31no—
311C23×C3—31no—
312C23×C3—61no—
313C22×D3#62711yescharacteristic
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—31no—
318C2×C3×C4—31no—
319C2×C3×C4—31no—
320C2×C3×C4—31no—
321C2×C3×C4—31no—
322C2×C3×C4—31no—
323C2×C3×C4—31no—
324C2×C3×C4—31no—
325C2×C3×C4—31no—
326C2×C3×C4—31no—
327C2×C3×C4—41no—
328C2×C3×C4—61no—
329C2×C3×C4—61no—
330C2×C3×C4—61no—
331C2×C3×C4—61no—
332C2×C3×C4—61no—
333C2×C3×C4C2×D3211yescharacteristic
334C2×C3×C4C2×D3211yescharacteristic
335C2×C3×C4C2×D3211yescharacteristic
336C2×Dic3#62711yescharacteristic
337C4×D3D3×Dic311yescharacteristic
338C4×D3D3×Dic311yescharacteristic
339C4×D3#62611yescharacteristic
340C4×D3#62611yescharacteristic
341C3×C8—32no—
342C3×C8—32no—
343C3×C8—62no—
344C3×C8—61no—
345C3×C8—61no—
346C3×C8—92no—
347C3×C8—121no—
348C3×C8—181no—
349C3×C8—181no—
350C3⋊C8—21no—
351C3⋊C8—32no—
352C3⋊C8—32no—
353C3⋊C8—62no—
354C3⋊C8—62no—
355C3⋊C8—61no—
356C3⋊C8—61no—
357C3⋊C8—61no—
358C3⋊C8—61no—
359C3⋊C8—121no—
360C3⋊C8—182no—
361C3⋊C8—181no—
362C3⋊C8—181no—
363C3⋊C8—181no—
364C3⋊C8—361no—
365C3⋊C8—361no—
366C3⋊C8—362no—
367C3⋊C8D3×Dic312yes—
368C2×C32—21no—
369C2×C32—21no—
370C2×C32—21no—
371C2×C32—21no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32—41no—
375C2×C32—41no—
376C2×C32—41no—
377C2×C32—41no—
378C2×C32—61no—
379C2×C32—61no—
380C2×C32—81no—
381C2×C32D3×C4⋊C411yescharacteristic
382C2×C32C2×Dic3⋊C411yescharacteristic
383C2×C32C2×C4⋊Dic311yescharacteristic
384C3×D3—22no—
385C3×D3—21no—
386C3×D3—21no—
387C3×D3—21no—
388C3×D3—21no—
389C3×D3—41no—
390C22×C4—31no—
391M16—31no—
392M16—92no—
393M16—92no—
394M16—91no—
395M16—91no—
396M16—181no—
397M16—271no—
398M16—541no—
399C2×C8—31no—
400C2×C8—91no—
401C2×C8—181no—
402C2×C8—541no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—31no—
407C22×C3—31no—
408C22×C3—31no—
409C22×C3—31no—
410C22×C3—41no—
411C22×C3—61no—
412C22×C3—61no—
413C22×C3—121no—
414C22×C3—121no—
415C22×C3—122no—
416C22×C3D3×Dic611yescharacteristic
417C22×C3D3×Dic611yescharacteristic
418C22×C3C2×#62711yescharacteristic
419C2×D3—41no—
420C2×D3Dic3⋊Dic311yescharacteristic
421C2×D3Dic3⋊Dic311yescharacteristic
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—31no—
429C3×C4—31no—
430C3×C4—31no—
431C3×C4—31no—
432C3×C4—41no—
433C3×C4—41no—
434C3×C4—61no—
435C3×C4—61no—
436C3×C4C4×D3211yescharacteristic
437C3×C4D3×D1211yescharacteristic
438C3×C4C2×D3×Dic311yescharacteristic
439C3×C4C2×D3×Dic311yescharacteristic
440C3×C4C2×#62611yescharacteristic
441C3×C4D3×#6711yescharacteristic
442Dic3Dic3⋊Dic311yescharacteristic
443Dic3Dic3⋊Dic311yescharacteristic
444C32—21no—
445C32—21no—
446C32—21no—
447C32—41no—
448C32#489311yescharacteristic, complemented
449C32#512411yescharacteristic, complemented
450C32#512511yescharacteristic, complemented
451C23—31no—
452C2×C4—31no—
453C2×C4—31no—
454C2×C4—31no—
455C2×C4—31no—
456C2×C4—31no—
457C2×C4D3311yescharacteristic
458C8—32no—
459C8—61no—
460C8—92no—
461C8—181no—
462C8—181no—
463C8—541no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—41no—
471C2×C3—41no—
472C2×C3—41no—
473C2×C3—41no—
474C2×C3—62no—
475C2×C3—61no—
476C2×C3—61no—
477C2×C3—61no—
478C2×C3—61no—
479C2×C3—81no—
480C2×C3—121no—
481C2×C3D3×Dic3⋊C411yescharacteristic
482C2×C3D3×C4⋊Dic311yescharacteristic
483C2×C3C2×Dic3⋊Dic311yescharacteristic
484D3—21no—
485D3—21no—
486C22—31no—
487C22—31no—
488C22—121no—
489C22D3×#62711yescharacteristic
490C4—31no—
491C4—31no—
492C4D32×Dic311yescharacteristic
493C4D3×#62611yescharacteristic
494C3—21no—
495C3—21no—
496C3—21no—
497C3—41no—
498C3#1059091211yescharacteristic, complemented
499C3#1059097311yescharacteristic, complemented
500C3#1059114111yescharacteristic, complemented
501C2—21no—
502C2—61no—
503C2—61no—
504C2D3×Dic3⋊Dic311yescharacteristic
subgroupG/H
centerC4D3×#626
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC2×C33C2×C4⋊C4
prime radicalC22×C32×D3Q8
hypocenterC33#430
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722081C2semidirect
5. #11722060C2semidirect
6. #11722058C2semidirect
7. C3×#63577C2semidirect
47. #63577D3semidirect
309. C33#430semidirect
448. C32#4893semidirect
449. C32#5124semidirect
450. C32#5125semidirect
498. C3#10590912semidirect
499. C3#10590973semidirect
500. C3#10591141semidirect

Maximal subgroups

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

subgroupindex
D3×#37482
C2×C3×#37112
C3×D3×#2082
#117220812
#117220602
#117220582
C3×#635772
#105911413
#105909733
#105909123