Finite Group Databasebeta

← #11786695

Subgroups — #11786695

Order 1152.

663 conjugacy classes of subgroups.

The subgroup lattice has 499 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10594986C211yesmaximal, characteristic
2#10592820C211yesmaximal, characteristic, complemented
3C2×C3×#825C211yesmaximal, characteristic
4#10591349C211yesmaximal, characteristic, complemented
5#10591088C211yesmaximal, characteristic
6C3×#4554C211yesmaximal, characteristic, complemented
7#10589162C211yesmaximal, characteristic, complemented
8#70363—31nomaximal
9#69002—31nomaximal
10C2×C3×#235C2211yescharacteristic
11C2×C32×M16C2211yescharacteristic
12#64135C2211yescharacteristic
13C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3×C4⋊Dic3C2211yescharacteristic
15C2×C3×C4×Dic3C2211yescharacteristic
16C2×C3×#208C2211yescharacteristic
17Dic3×#67—22no—
18C2×Dic3⋊Dic3—22no—
19#63898—21no—
20D3×C4⋊Dic3—21no—
21#63843—21no—
22#63841—21no—
23#63645—21no—
24#63644—21no—
25C3×#825C414yescomplemented
26#63566—21no—
27#63563—21no—
28#5827—31no—
29#5504—31no—
30C2×C3×#104—31no—
31#5232—31no—
32C2×#825D311yescharacteristic
33#5133—31no—
34#5121—31no—
35#5121—31no—
36#5121—61no—
37#4975—31no—
38#4935—31no—
39C3×#343—31no—
40#4555—31no—
41#4554D311yescharacteristic, complemented
42#4523—31no—
43C22×C32×C4C2311yescharacteristic
44C22×C3⋊Dic3—21no—
45C2×C3×#67—21no—
46C22×C3×Dic3—21no—
47C22×C3×Dic3D412yes—
48C3×#235—21no—
49C3×#235—21no—
50C3×#235C2×C412yes—
51C3×#235Q812yes—
52C2×C3×D12D411yescharacteristic
53C2×C3×C4×D3—21no—
54C2×C3×Dic6D411yescharacteristic
55C2×D3×Dic3—22no—
56C32×M16—21no—
57C32×M16C2×C412yescomplemented
58C2×C32×C8—21no—
59#3757—21no—
60C4⋊(C3⋊Dic3)—21no—
61C4⋊(C3⋊Dic3)—21no—
62C4⋊(C3⋊Dic3)D411yescharacteristic
63C4⋊(C3⋊Dic3)D411yescharacteristic
64C4×C3⋊Dic3—21no—
65C3×C4⋊Dic3—21no—
66C3×C4⋊Dic3—21no—
67C3×C4×Dic3—21no—
68C3×C4×Dic3C2×C412yescomplemented
69C3×#208—21no—
70C3×#208—21no—
71C2×C3×C3⋊C8—21no—
72Dic3⋊Dic3—22no—
73Dic3⋊Dic3—44no—
74(C2×D3)⋊Dic3—22no—
75Dic32—22no—
76#1859—91noSylow
77C2×C3×#41—31no—
78C2×#235C2×D311yescharacteristic
79C2×C3×M16—21no—
80C2×C3×M16—31no—
81C2×C3×M16C2×D311yescharacteristic
82C2×C3×M16C2×D311yescharacteristic
83C2×C3×C4⋊C4—31no—
84C2×C3×C42—31no—
85Q8×Dic3—61no—
86D4×Dic3—61no—
87D4×Dic3—62no—
88C4×#67—62no—
89#926—31no—
90#926—31no—
91#926—61no—
92#926—61no—
93C2×C4⋊Dic3—31no—
94C2×C4⋊Dic3—31no—
95C2×C4⋊Dic3—61no—
96C2×C4⋊Dic3—61no—
97C2×C4⋊Dic3—62no—
98C2×C4⋊Dic3C2×D311yescharacteristic
99C2×Dic3⋊C4—62no—
100C2×C4×Dic3C2×D311yescharacteristic
101C2×#208C2×D311yescharacteristic
102#893—61no—
103#892—61no—
104D3×C4⋊C4—61no—
105#887—61no—
106C3×#104—34no—
107#835—61no—
108#832—61no—
109#825Dic314yescomplemented
110#821—61no—
111#818—61no—
112#818—61no—
113#818—62no—
114#817—61no—
115#817—61no—
116#817—62no—
117#816—61no—
118C23×C32C2×D411yescharacteristic
119C22×C3×D3—21no—
120C2×C32×C4C22×C411yescharacteristic
121C2×C32×C4C2×D411yescharacteristic
122C2×C32×C4C2×D411yescharacteristic
123C2×C32×C4C2×Q811yescharacteristic
124C2×C32×C4#4111yescharacteristic
125C2×C32×C4#4111yescharacteristic
126C2×C3⋊Dic3—21no—
127C2×C3⋊Dic3—21no—
128C2×C3⋊Dic3—41no—
129C2×C3⋊Dic3—42no—
130C3×#67—22no—
131C3×#67—22no—
132C2×C3×Dic3—22no—
133C2×C3×Dic3—22no—
134C2×C3×Dic3—22no—
135C2×C3×Dic3—21no—
136C2×C3×Dic3—22no—
137C2×C3×Dic3—42no—
138C3×D12—21no—
139C3×D12C4⋊C412yescomplemented
140C3×C4×D3—22no—
141C3×C4×D3—22no—
142C3×Dic6—21no—
143C3×Dic6C4⋊C412yescomplemented
144D3×Dic3—82no—
145C32×C8—22no—
146C3×C3⋊C8—41no—
147#520—91no—
148#428—91no—
149C2×#104—31no—
150#417—91no—
151#384—91no—
152#343—31no—
153#339—91no—
154C3×#41—32no—
155C3×#41—32no—
156C3×#41—61no—
157C3×#41—61no—
158C2×C3×Q8—31no—
159C2×C3×D4—31no—
160C2×C3×D4—61no—
161C22×C3×C4—21no—
162C22×C3×C4—32no—
163C22×C3×C4—61no—
164C22×C3×C4—61no—
165C22×C3×C4C22×D311yescharacteristic
166C22×C3×C4C22×D311yescharacteristic
167C2×#67—21no—
168C22×Dic3—21no—
169C22×Dic3—62no—
170C22×Dic3—61no—
171C22×Dic3—61no—
172C22×Dic3—121no—
173C22×Dic3D1212yes—
174#235—21no—
175#235—21no—
176#235Dic612yes—
177#235C2×Dic312yes—
178C2×D12D1211yescharacteristic
179C2×C4×D3—21no—
180C2×C4×D3—62no—
181C2×Dic6D1211yescharacteristic
182C3×M16—22no—
183C3×M16—21no—
184C3×M16—21no—
185C3×M16—22no—
186C3×M16—61no—
187C3×M16—61no—
188C3×M16C4×D312yescomplemented
189C3×M16C2×Dic312yescomplemented
190C2×C3×C8—22no—
191C2×C3×C8—21no—
192C2×C3×C8—21no—
193C2×C3×C8—61no—
194C3×C4⋊C4—61no—
195C3×C4⋊C4—61no—
196C3×C42—32no—
197C3×C42—61no—
198#217—62no—
199#217—61no—
200#217—61no—
201#217—121no—
202(C2×D3)⋊C4—62no—
203C4⋊Dic3—21no—
204C4⋊Dic3—21no—
205C4⋊Dic3—31no—
206C4⋊Dic3—31no—
207C4⋊Dic3—31no—
208C4⋊Dic3—31no—
209C4⋊Dic3—61no—
210C4⋊Dic3—61no—
211C4⋊Dic3—62no—
212C4⋊Dic3—62no—
213C4⋊Dic3—61no—
214C4⋊Dic3—61no—
215C4⋊Dic3—61no—
216C4⋊Dic3—61no—
217C4⋊Dic3—121no—
218C4⋊Dic3—124no—
219Dic3⋊C4—62no—
220Dic3⋊C4—124no—
221C4×Dic3—21no—
222C4×Dic3—62no—
223C4×Dic3—62no—
224C4×Dic3—61no—
225C4×Dic3—61no—
226C4×Dic3—121no—
227C4×Dic3C2×Dic312yescomplemented
228#208—21no—
229#208—21no—
230C2×C3⋊C8—21no—
231C22×C32—21no—
232C22×C32—21no—
233C22×C32C4×D411yescharacteristic
234C22×C32#12011yescharacteristic
235C22×C32C22⋊Q811yescharacteristic
236C2×C3×D3—22no—
237C2×C3×D3—41no—
238C2×C3×D3—41no—
239C32×C4C2×C4⋊C411yescharacteristic
240C32×C4C4×D411yescharacteristic
241C32×C4C22⋊Q811yescharacteristic
242C32×C4#12311yescharacteristic
243C3⋊Dic3—41no—
244C3⋊Dic3—41no—
245C3⋊Dic3—81no—
246C3×Dic3—24no—
247C3×Dic3—22no—
248C3×Dic3—42no—
249C2×#41—31no—
250C2×M16—31no—
251C2×M16D3211yescharacteristic
252C4×Q8—181no—
253C4×D4—181no—
254C4×D4—182no—
255#117—91no—
256#117—181no—
257C2×C4⋊C4—31no—
258C2×C4⋊C4—91no—
259C2×C4⋊C4—181no—
260C2×C4⋊C4—182no—
261C2×C42—31no—
262#107—181no—
263#106—181no—
264#104—34no—
265#103—181no—
266#102—181no—
267C23×C3—21no—
268C23×C3—61no—
269C23×C3C2×D1211yescharacteristic
270C23×C3D3×D411yescharacteristic
271C22×D3—21no—
272C3×Q8—32no—
273C3×Q8—61no—
274C3×D4—32no—
275C3×D4—62no—
276C3×D4—62no—
277C3×D4—61no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—62no—
285C2×C3×C4—62no—
286C2×C3×C4—62no—
287C2×C3×C4—61no—
288C2×C3×C4—62no—
289C2×C3×C4—62no—
290C2×C3×C4—62no—
291C2×C3×C4—122no—
292C2×C3×C4C2×Dic611yescharacteristic
293C2×C3×C4C2×C4×D311yescharacteristic
294C2×C3×C4C2×D1211yescharacteristic
295C2×C3×C4#23511yescharacteristic
296C2×C3×C4D3×D411yescharacteristic
297C2×C3×C4D3×D411yescharacteristic
298C2×C3×C4D3×D411yescharacteristic
299C2×C3×C4#23711yescharacteristic
300C2×C3×C4#23711yescharacteristic
301C2×C3×C4#23711yescharacteristic
302C2×C3×C4D3×Q811yescharacteristic
303C2×C3×C4C22×Dic311yescharacteristic
304#67—22no—
305#67—22no—
306C2×Dic3—22no—
307C2×Dic3—22no—
308C2×Dic3—21no—
309C2×Dic3—22no—
310C2×Dic3—22no—
311C2×Dic3—42no—
312C2×Dic3—61no—
313C2×Dic3—61no—
314C2×Dic3—61no—
315C2×Dic3—61no—
316C2×Dic3—121no—
317C2×Dic3—121no—
318C2×Dic3—122no—
319C2×Dic3—122no—
320C2×Dic3—122no—
321C2×Dic3—121no—
322C2×Dic3—121no—
323C2×Dic3—242no—
324C2×Dic3—242no—
325D12—21no—
326D12C4⋊Dic312yescomplemented
327C4×D3—22no—
328C4×D3—22no—
329C4×D3—242no—
330Dic6—21no—
331Dic6C4⋊Dic312yescomplemented
332C3×C8—24no—
333C3×C8—22no—
334C3×C8—22no—
335C3×C8—121no—
336C3⋊C8—41no—
337C2×C32—21no—
338C2×C32—21no—
339C2×C32#38511yescharacteristic
340C2×C32#45311yescharacteristic
341C2×C32#45611yescharacteristic
342C3×D3—42no—
343#41—32no—
344#41—32no—
345#41—61no—
346#41—61no—
347C2×Q8—31no—
348C2×D4—31no—
349C2×D4—61no—
350C22×C4—32no—
351C22×C4—61no—
352C22×C4—61no—
353C22×C4—182no—
354C22×C4—181no—
355C22×C4C2×D3211yescharacteristic
356M16—21no—
357M16—61no—
358M16—61no—
359M16D3×Dic312yescomplemented
360C2×C8—21no—
361C2×C8—61no—
362C4⋊C4—61no—
363C4⋊C4—61no—
364C4⋊C4—91no—
365C4⋊C4—91no—
366C4⋊C4—182no—
367C4⋊C4—181no—
368C4⋊C4—181no—
369C4⋊C4—364no—
370C22⋊C4—182no—
371C22⋊C4—181no—
372C42—32no—
373C42—61no—
374C42—182no—
375C42—181no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—41no—
384C22×C3—41no—
385C22×C3—62no—
386C22×C3—121no—
387C22×C3—121no—
388C22×C3C4×D1211yescharacteristic
389C22×C3#87811yescharacteristic
390C22×C3#88211yescharacteristic
391C22×C3#88211yescharacteristic
392C22×C3#89711yescharacteristic
393C22×C3D4×Dic311yescharacteristic
394C2×D3—22no—
395C2×D3—41no—
396C2×D3—41no—
397C3×C4—21no—
398C3×C4—21no—
399C3×C4—21no—
400C3×C4—21no—
401C3×C4—64no—
402C3×C4—62no—
403C3×C4—122no—
404C3×C4#87811yescharacteristic
405C3×C4#67⋊C411yescharacteristic
406C3×C4#88311yescharacteristic
407C3×C4#88611yescharacteristic
408C3×C4D3×C4⋊C411yescharacteristic
409C3×C4#89611yescharacteristic
410C3×C4C2×C4⋊Dic311yescharacteristic
411C3×C4D4×Dic311yescharacteristic
412Dic3—24no—
413Dic3—22no—
414Dic3—42no—
415Dic3—121no—
416Dic3—121no—
417Dic3—121no—
418Dic3—121no—
419Dic3—241no—
420Dic3—241no—
421Dic3—242no—
422Dic3—241no—
423Dic3—241no—
424C32#185911yescharacteristic, complemented, Sylow
425C23—61no—
426C23D3×D1211yescharacteristic
427Q8—32no—
428Q8—61no—
429D4—32no—
430D4—62no—
431D4—62no—
432D4—61no—
433C2×C4—62no—
434C2×C4—62no—
435C2×C4—62no—
436C2×C4—61no—
437C2×C4—62no—
438C2×C4—62no—
439C2×C4—62no—
440C2×C4—122no—
441C2×C4—181no—
442C2×C4—181no—
443C2×C4—362no—
444C2×C4—361no—
445C2×C4—722no—
446C2×C4D3×Dic611yescharacteristic
447C2×C4#379511yescharacteristic
448C2×C4D3×D1211yescharacteristic
449C2×C4C2×D3×Dic311yescharacteristic
450C2×C4#380511yescharacteristic
451C2×C4#381111yescharacteristic
452C8—22no—
453C8—121no—
454C2×C3—21no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—41no—
462C2×C3—41no—
463C2×C3—122no—
464C2×C3#466811yescharacteristic
465C2×C3#474711yescharacteristic
466C2×C3#475211yescharacteristic
467C2×C3#482211yescharacteristic
468C2×C3#482611yescharacteristic
469C2×C3#495011yescharacteristic
470D3—42no—
471C22—21no—
472C22—21no—
473C22—62no—
474C22—121no—
475C22—121no—
476C22Dic3×D1211yescharacteristic
477C22#6389311yescharacteristic
478C22#6392311yescharacteristic
479C4—64no—
480C4—62no—
481C4—122no—
482C4—361no—
483C4—361no—
484C4—721no—
485C4D3×C4⋊Dic311yescharacteristic
486C4#6389111yescharacteristic
487C4#6389611yescharacteristic
488C4#67⋊Dic311yescharacteristic
489C3—21no—
490C3#6900211yescharacteristic, complemented
491C3#7036311yescharacteristic, complemented
492C2—21no—
493C2—21no—
494C2—122no—
495C2#1059101511yescharacteristic
496C2#1059102111yescharacteristic
497C2#1059137911yescharacteristic
subgroupG/H
centerC22#63923
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×M16C22
socleC22×C32#120
prime radicalC2×C3×#235C22
cosocleC22×C32×C4C23
hypercenterC2×M16D32
hypocenterC32#1859
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. #10592820C2semidirect
4. #10591349C2semidirect
6. C3×#4554C2semidirect
7. #10589162C2semidirect
25. C3×#825C4semidirect
41. #4554D3semidirect
57. C32×M16C2×C4semidirect
68. C3×C4×Dic3C2×C4semidirect
109. #825Dic3semidirect
139. C3×D12C4⋊C4semidirect
143. C3×Dic6C4⋊C4semidirect
188. C3×M16C4×D3semidirect
189. C3×M16C2×Dic3semidirect
227. C4×Dic3C2×Dic3semidirect
326. D12C4⋊Dic3semidirect
331. Dic6C4⋊Dic3semidirect
359. M16D3×Dic3semidirect
424. C32#1859semidirect
490. C3#69002semidirect
491. C3#70363semidirect

Maximal subgroups

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

subgroupindex
#105949862
#105928202
C2×C3×#8252
#105913492
#105910882
C3×#45542
#105891622
#703633
#690023