Finite Group Databasebeta

← #14736640

Subgroups — #14736640

Order 2816.

682 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13079075C211yesmaximal, characteristic
2#13078647C211yesmaximal, characteristic
3C2×#10622633C211yesmaximal, characteristic, complemented
4#13068110C212yesmaximal, complemented
5#13068088C212yesmaximal, complemented
6#13067986C212yesmaximal, complemented
7#13067964C212yesmaximal, complemented
8#13067368C211yesmaximal, characteristic, complemented
9#13067294C211yesmaximal, characteristic
10#13067289C211yesmaximal, characteristic, complemented
11Q8×C11×M16C211yesmaximal, characteristic, complemented
12C2×C4×Q8×C11C2211yescharacteristic
13C2×#66962—21no—
14C2×#66962—21no—
15#10623008C2211yescharacteristic
16C2×#66922C2211yescharacteristic
17C11×#446—22no—
18C11×#446C2212yes—
19C8×Q8×C11C2212yes—
20C11×#423—21no—
21C11×#423C2211yescharacteristic
22C4×C11×M16—21no—
23C4×C11×M16C2211yescharacteristic
24#10622840—21no—
25C2×#66899—21no—
26#10622640C2211yescharacteristic
27#10622640C2211yescharacteristic
28C4×#4325—21no—
29C4×#4325—21no—
30#10622633—21no—
31#10622633—21no—
32#10622633C2212yes—
33#10622633C2211yescharacteristic
34#10622633C2212yes—
35#10622633C2211yescharacteristic
36#10622581—21no—
37C2×#66872—21no—
38#10622538C2211yescharacteristic
39C2×C4⋊(C11⋊C8)C2211yescharacteristic
40#10622328—22no—
41#10622312C2212yes—
42#10622306C2212yes—
43#10622123C2212yes—
44#10622120C2212yes—
45#10622120C2212yes—
46#10622120C2212yes—
47#10622096C2212yes—
48#10622095C2212yes—
49C22×Q8×C11D411yescharacteristic
50C2×#4339—21no—
51C22×Dic22—21no—
52C2×C11×M16—21no—
53C2×C11×M16D411yescharacteristic
54C2×C11×M16D411yescharacteristic
55C4×Q8×C11—21no—
56C4×Q8×C11—21no—
57C4×Q8×C11D411yescharacteristic
58C4×Q8×C11D411yescharacteristic
59C4×Q8×C11C2311yescharacteristic
60C4×Q8×C11C2311yescharacteristic
61C2×C11×C4⋊C4—21no—
62C2×C11×C4⋊C4D411yescharacteristic
63C2×C42×C11—21no—
64C2×C42×C11C2311yescharacteristic
65C2×#4325—21no—
66C2×#4325—21no—
67#66983—21no—
68#66983—21no—
69C2×C4⋊Dic11—21no—
70C2×C4⋊Dic11—21no—
71C22×C11⋊C8—21no—
72#66962—44no—
73#66962—44no—
74C2×Dic44—22no—
75C2×Dic44—22no—
76C2×#4314—22no—
77C2×#4314—22no—
78#66928—21no—
79#66927C2311yescharacteristic
80C4×D44—21no—
81C4×D44—21no—
82#66922—21no—
83#66922—21no—
84#66922—21no—
85#66922C2312yes—
86#66922C2311yescharacteristic
87C11×C4⋊C8—22no—
88C11×C4⋊C8—22no—
89C11×C4⋊C8—21no—
90C11×C4⋊C8C2312yes—
91C11×#97—22no—
92C11×#97D412yes—
93C4×C8×C11—22no—
94C4×C8×C11C2312yes—
95#66899—22no—
96#66899—22no—
97#66899—21no—
98#66899—21no—
99C22⋊(C11⋊C8)—21no—
100#66873—21no—
101#66873—21no—
102#66872—22no—
103#66872—22no—
104#66872—21no—
105#66872—21no—
106C4⋊(C11⋊C8)—21no—
107C4⋊(C11⋊C8)C2312yes—
108C4⋊(C11⋊C8)C2311yescharacteristic
109C4⋊(C11⋊C8)C2311yescharacteristic
110C4×C11⋊C8—21no—
111#19171—111nomaximal, Sylow
112C2×Q8×C11—21no—
113C2×Q8×C11—22no—
114C2×Q8×C11—41no—
115C2×Q8×C11C2×D411yescharacteristic
116C2×Q8×C11C2×D411yescharacteristic
117C22×C4×C11—21no—
118C22×C4×C11—21no—
119C22×C4×C11C2×D411yescharacteristic
120C22×C4×C11C2×D411yescharacteristic
121C22×C4×C11#4111yescharacteristic
122C2×C22⋊D11—41no—
123C22×Dic11—41no—
124#4339—42no—
125#4339—42no—
126C2×D44—21no—
127C2×D44—21no—
128C2×D44—21no—
129C2×C4×D11—41no—
130C2×Dic22—22no—
131C2×Dic22—21no—
132C2×Dic22—41no—
133C2×Dic22—42no—
134C2×Dic22—42no—
135C11×M16—22no—
136C11×M16—42no—
137C11×M16SD1614yescomplemented
138C2×C8×C11—22no—
139C2×C8×C11C2×D412yes—
140C2×C8×C11C2×D412yes—
141C11×C4⋊C4—21no—
142C11×C4⋊C4—21no—
143C11×C4⋊C4—21no—
144C11×C4⋊C4—21no—
145C11×C4⋊C4—21no—
146C11×C4⋊C4C2×D411yescharacteristic
147C11×C4⋊C4C2×D411yescharacteristic
148C42×C11—21no—
149C42×C11—21no—
150C42×C11—21no—
151C42×C11—21no—
152C42×C11C2×D411yescharacteristic
153C42×C11C2×D411yescharacteristic
154C42×C11#4111yescharacteristic
155C42×C11C2411yescharacteristic
156#4325—42no—
157#4325—42no—
158(C2×D11)⋊C4—41no—
159(C2×D11)⋊C4—41no—
160C4⋊Dic11—22no—
161C4⋊Dic11—21no—
162C4⋊Dic11—22no—
163C4⋊Dic11—21no—
164C4⋊Dic11—41no—
165C4⋊Dic11—41no—
166C2×C11⋊C8—21no—
167C2×C11⋊C8—21no—
168C2×C11⋊C8—22no—
169C2×C11⋊C8—42no—
170Dic44—44no—
171Dic44—44no—
172#4314—44no—
173#4314—44no—
174#3152—111no—
175#3103—111no—
176C2×#475—111no—
177Q8×M16D1111yescharacteristic, complemented
178#1690—112no—
179#1686—112no—
180#1670—112no—
181#1666—112no—
182#1565—111no—
183#1547—111no—
184#1546—111no—
185C23×C11#12111yescharacteristic
186C22×D11—41no—
187Q8×C11—21no—
188Q8×C11—21no—
189Q8×C11—21no—
190Q8×C11—42no—
191Q8×C11#13712yescomplemented
192C2×C4×C11—21no—
193C2×C4×C11—21no—
194C2×C4×C11—21no—
195C2×C4×C11—21no—
196C2×C4×C11—21no—
197C2×C4×C11—21no—
198C2×C4×C11—41no—
199C2×C4×C11—42no—
200C2×C4×C11—81no—
201C2×C4×C11#12111yescharacteristic
202C2×C4×C11#12111yescharacteristic
203C2×C4×C11#12111yescharacteristic
204C2×C4×C11C2×SD1612yes—
205C2×C4×C11AGL(1,ℤ/8)12yes—
206C2×C4×C11C22×D411yescharacteristic
207C2×C4×C11C22×D411yescharacteristic
208C2×C4×C11C2×#4111yescharacteristic
209C22⋊D11—82no—
210C2×Dic11—42no—
211C2×Dic11—41no—
212C2×Dic11—82no—
213D44—41no—
214D44—41no—
215D44—41no—
216D44—41no—
217D44—81no—
218C4×D11—82no—
219Dic22—42no—
220Dic22—41no—
221Dic22—42no—
222Dic22—41no—
223Dic22—82no—
224C8×C11—22no—
225C8×C11—42no—
226C8×C11C2×SD1614yes—
227C11⋊C8—41no—
228C11⋊C8—42no—
229C11⋊C8—41no—
230C2×#137—221no—
231C2×#137—221no—
232#532—111no—
233C2×C4⋊Q8—111no—
234C2×C4×Q8C2×D1111yescharacteristic
235#497—222no—
236#494—112no—
237#492—112no—
238#481—221no—
239#475—112no—
240#475—112no—
241#475—111no—
242#475—111no—
243#475—221no—
244#475—221no—
245#460—111no—
246#460—111no—
247#448—221no—
248#446—22no—
249#446C2×D1112yes—
250C8×Q8C2×D1112yes—
251C4×SD16—221no—
252C4×SD16—221no—
253#431—111no—
254C2×#106—221no—
255#423—21no—
256#423C2×D1111yescharacteristic
257C2×C4⋊C8—111no—
258C2×#103—221no—
259C4×M16—21no—
260C4×M16C2×D1111yescharacteristic
261#335—112no—
262#334—112no—
263#332—112no—
264#332—112no—
265#332—112no—
266#331—112no—
267C22×C11—21no—
268C22×C11—21no—
269C22×C11#46012yes—
270C22×C11C2×#12111yescharacteristic
271C2×D11—81no—
272C2×D11—81no—
273C2×D11—81no—
274C4×C11—21no—
275C4×C11—22no—
276C4×C11—21no—
277C4×C11—21no—
278C4×C11—41no—
279C4×C11—41no—
280C4×C11—41no—
281C4×C11#46012yes—
282C4×C11C22×SD1611yescharacteristic
283C4×C11C2×AGL(1,ℤ/8)11yescharacteristic
284C4×C11C2×#13711yescharacteristic
285C4×C11#57511yescharacteristic
286Dic11—82no—
287Dic11—81no—
288C2×#41—221no—
289C22×Q8—221no—
290C22×Q8D4411yescharacteristic
291#137—444no—
292#137—444no—
293C2×Q16—222no—
294C2×Q16—222no—
295C2×SD16—221no—
296C2×SD16—221no—
297C2×SD16—222no—
298C2×SD16—222no—
299C2×M16—21no—
300C2×M16C22⋊D1111yescharacteristic
301C2×M16C22⋊D1111yescharacteristic
302C22×C8—221no—
303C4⋊Q8—112no—
304C4⋊Q8—111no—
305C4⋊Q8—221no—
306C4⋊Q8—221no—
307C4⋊Q8—221no—
308#127—111no—
309#124—221no—
310#121—221no—
311#121—221no—
312C4×Q8—21no—
313C4×Q8—21no—
314C4×Q8D4411yescharacteristic
315C4×Q8D4411yescharacteristic
316C4×Q8C22×D1111yescharacteristic
317C4×Q8C22×D1111yescharacteristic
318C4×D4—221no—
319C4×D4—221no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—221no—
322C2×C4⋊C4—221no—
323C2×C4⋊C4D4411yescharacteristic
324C2×C42—21no—
325C2×C42C22×D1111yescharacteristic
326#106—222no—
327#106—222no—
328#106—221no—
329#106—221no—
330C4⋊C8—22no—
331C4⋊C8—22no—
332C4⋊C8—21no—
333C4⋊C8—112no—
334C4⋊C8—111no—
335C4⋊C8—111no—
336C4⋊C8—221no—
337C4⋊C8C22×D1112yes—
338#103—222no—
339#103—222no—
340#103—221no—
341#103—221no—
342#102—221no—
343#102—221no—
344C22⋊C8—221no—
345#97—22no—
346#97C22⋊D1112yes—
347C4×C8—22no—
348C4×C8—221no—
349C4×C8C22×D1112yes—
350C2×C11—22no—
351C2×C11C2×#46011yescharacteristic
352C2×C11#305611yescharacteristic
353C2×C11#306911yescharacteristic
354D11—161no—
355#41—442no—
356#41—442no—
357C2×Q8—21no—
358C2×Q8—22no—
359C2×Q8—41no—
360C2×Q8—221no—
361C2×Q8—222no—
362C2×Q8—441no—
363C2×Q8—442no—
364C2×Q8—442no—
365C2×Q8C2×D4411yescharacteristic
366C2×Q8C2×D4411yescharacteristic
367C2×D4—221no—
368C2×D4—221no—
369C2×D4—221no—
370C2×D4—441no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—441no—
374C22×C4—441no—
375C22×C4C2×D4411yescharacteristic
376C22×C4#433911yescharacteristic
377C22×C4C2×C22⋊D1111yescharacteristic
378Q16—444no—
379Q16—444no—
380SD16—444no—
381SD16—442no—
382SD16—442no—
383SD16—444no—
384M16—22no—
385M16—42no—
386M16#432514yescomplemented
387C2×C8—22no—
388C2×C8—221no—
389C2×C8—221no—
390C2×C8—222no—
391C2×C8—442no—
392C2×C8C2×C22⋊D1112yes—
393C2×C8C2×C22⋊D1112yes—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—222no—
400C4⋊C4—221no—
401C4⋊C4—222no—
402C4⋊C4—221no—
403C4⋊C4—441no—
404C4⋊C4—441no—
405C4⋊C4C2×D4411yescharacteristic
406C4⋊C4C2×D4411yescharacteristic
407C22⋊C4—441no—
408C22⋊C4—441no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42C2×D4411yescharacteristic
414C42#433911yescharacteristic
415C42C2×C22⋊D1111yescharacteristic
416C42C23×D1111yescharacteristic
417C11#1917111yescharacteristic, complemented, Sylow
418C23—441no—
419C23#6698311yescharacteristic
420Q8—21no—
421Q8—21no—
422Q8—21no—
423Q8—42no—
424Q8—442no—
425Q8—441no—
426Q8—442no—
427Q8—441no—
428Q8—882no—
429Q8#6696212yescomplemented
430D4—441no—
431D4—441no—
432D4—441no—
433D4—441no—
434D4—881no—
435D4—882no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—81no—
445C2×C4—441no—
446C2×C4—442no—
447C2×C4—882no—
448C2×C4—882no—
449C2×C4#6698311yescharacteristic
450C2×C4#6698311yescharacteristic
451C2×C4#6698311yescharacteristic
452C2×C4C2×#432512yes—
453C2×C4#6700212yes—
454C2×C4C22×D4411yescharacteristic
455C2×C4C2×#433911yescharacteristic
456C2×C4C22×C22⋊D1111yescharacteristic
457C8—22no—
458C8—42no—
459C8—441no—
460C8—442no—
461C8—441no—
462C8C2×#432514yes—
463C22—21no—
464C22—21no—
465C22—881no—
466C22—881no—
467C22—881no—
468C22#1062264012yes—
469C22C2×#6698311yescharacteristic
470C4—21no—
471C4—21no—
472C4—21no—
473C4—22no—
474C4—41no—
475C4—41no—
476C4—41no—
477C4—881no—
478C4—882no—
479C4#1062264012yes—
480C4C2×#6696211yescharacteristic
481C4#1062327911yescharacteristic
482C4C22×#432511yescharacteristic
483C4C2×#6700211yescharacteristic
484C2—22no—
485C2—1761no—
486C2C2×#1062264011yescharacteristic
487C2#1307824611yescharacteristic
488C2#1307835311yescharacteristic
subgroupG/H
centerC22C2×#66983
commutator, cosocleC42×C11C24
FrattiniC42C23×D11
FittingQ8×C11×M16C2
socleC22×C11C2×#121
prime radical#10623008C22
hypercenterQ8×M16D11
hypocenterC11#19171
LCSG ▷ C42×C11 ▷ C22×C11 ▷ C11
UCSC1 ◁ C22 ◁ C2×C42 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
3. C2×#10622633C2semidirect
4. #13068110C2semidirect
5. #13068088C2semidirect
6. #13067986C2semidirect
7. #13067964C2semidirect
8. #13067368C2semidirect
10. #13067289C2semidirect
11. Q8×C11×M16C2semidirect
137. C11×M16SD16semidirect
177. Q8×M16D11semidirect
191. Q8×C11#137semidirect
386. M16#4325semidirect
417. C11#19171semidirect
429. Q8#66962semidirect

Maximal subgroups

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

subgroupindex
#130790752
#130786472
C2×#106226332
#130681102
#130680882
#130679862
#130679642
#130673682
#130672942
#130672892
Q8×C11×M162
#1917111