Finite Group Databasebeta

← #14763710

Subgroups — #14763710

Order 2816.

1086 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623153C211yesmaximal, characteristic
2C2×#10622731C211yesmaximal, characteristic
3#13071011C211yesmaximal, characteristic
4#13069562C212yesmaximal, complemented
5#13069537C212yesmaximal, complemented
6#13069050C211yesmaximal, characteristic
7#13069049C211yesmaximal, characteristic
8#13069047C211yesmaximal, characteristic
9#13068630C212yesmaximal, complemented
10#13068629C212yesmaximal, complemented
11C11×#1894C211yesmaximal, characteristic
12C22×C11×M16C2211yescharacteristic
13C2×C11×C22⋊Q8C2211yescharacteristic
14C2×Q8×Dic11C2211yescharacteristic
15C2×#66997C2211yescharacteristic
16C2×#66977C2211yescharacteristic
17C22×C4×Dic11C2211yescharacteristic
18C22×#4318C2211yescharacteristic
19#10623153—24no—
20#10623153C418yescomplemented
21C2×#66942—21no—
22C2×#66941C2211yescharacteristic
23C2×#66940—21no—
24C2×#66940C2211yescharacteristic
25C2×#66931—21no—
26C2×#66930—21no—
27C2×C11×#103C2212yes—
28C11×#380C2211yescharacteristic
29C2×#66899C2212yes—
30#10622825C2211yescharacteristic
31#10622731—24no—
32C2×#66880C2212yes—
33C2×Dic11⋊C8C2212yes—
34C2×Dic11⋊C8C2212yes—
35#10622590C2212yes—
36C4⋊C4×Dic11C2212yes—
37C2×#66874C2212yes—
38#10622556—21no—
39C22⋊C4×Dic11—21no—
40#10622264—21no—
41#10622261—21no—
42C11×#340C2212yes—
43#10622189C2212yes—
44#10622167C2212yes—
45#10622167C2212yes—
46C22×Q8×C11C2311yescharacteristic
47C23×C4×C11C2311yescharacteristic
48C23×Dic11D411yescharacteristic
49C23×Dic11D411yescharacteristic
50C22×Dic22—21no—
51C22×Dic22C2311yescharacteristic
52C2×C11×M16—22no—
53C2×C11×M16—42no—
54C22×C8×C11C2312yes—
55C11×C22⋊Q8—22no—
56C11×C22⋊Q8C2×C414yes—
57C2×C11×C4⋊C4—21no—
58C2×C11×C4⋊C4C2312yes—
59C2×C11×C4⋊C4C2311yescharacteristic
60C2×C11×C22⋊C4—21no—
61C2×C11×C22⋊C4—21no—
62C2×C22⋊Dic11—21no—
63C2×C22⋊Dic11—21no—
64Q8×Dic11—22no—
65Q8×Dic11C2×C414yes—
66#66997—22no—
67#66997C2×C414yes—
68C2×C4⋊Dic11—21no—
69C2×C4⋊Dic11C2312yes—
70C2×C4⋊Dic11C2311yescharacteristic
71#66977—22no—
72#66977C2×C414yes—
73C2×Dic11⋊C4—21no—
74C2×Dic11⋊C4—21no—
75C2×Dic11⋊C4—21no—
76C2×Dic11⋊C4—21no—
77C2×Dic11⋊C4—21no—
78C2×C4×Dic11—21no—
79C2×C4×Dic11—21no—
80C2×C4×Dic11—22no—
81C2×C4×Dic11C2×C414yes—
82C2×C4×Dic11D411yescharacteristic
83C2×C4×Dic11D411yescharacteristic
84C2×C4×Dic11D414yes—
85C2×C4×Dic11C2311yescharacteristic
86C2×C4×Dic11C2311yescharacteristic
87C2×#4318—22no—
88C2×#4318—42no—
89C22×C11⋊C8C2312yes—
90#66942—28no—
91#66941—22no—
92#66941C2×C414yes—
93#66940—28no—
94#66940—22no—
95#66940C2×C414yes—
96#66931—28no—
97#66930—28no—
98C11×#103—42no—
99C11×#95—21no—
100#66899—42no—
101C11⋊#95—21no—
102C11⋊#95—22no—
103C11⋊#95—22no—
104C11⋊#95—22no—
105#66880—42no—
106Dic11⋊C8—24no—
107Dic11⋊C8—24no—
108#66874—42no—
109#21892—111nomaximal, Sylow
110C24×C11C2×D411yescharacteristic
111C2×Q8×C11—21no—
112C2×Q8×C11—22no—
113C2×Q8×C11—22no—
114C2×Q8×C11C22×C412yes—
115C22×C4×C11—22no—
116C22×C4×C11—22no—
117C22×C4×C11—21no—
118C22×C4×C11—21no—
119C22×C4×C11—22no—
120C22×C4×C11C22×C412yes—
121C22×C4×C11C2×D411yescharacteristic
122C22×C4×C11C2×D412yes—
123C22×C4×C11C2411yescharacteristic
124C22×Dic11—24no—
125C22×Dic11—21no—
126C22×Dic11—21no—
127C22×Dic11—22no—
128C22×Dic11—41no—
129C22×Dic11—41no—
130C22×Dic11C22⋊C414yes—
131C22×Dic11C22⋊C414yes—
132C22×Dic11C2×D411yescharacteristic
133C22×Dic11C2×D411yescharacteristic
134C22×Dic11C2×D411yescharacteristic
135C22×Dic11C2×D411yescharacteristic
136C2×Dic22—24no—
137C2×Dic22—21no—
138C2×Dic22—22no—
139C2×Dic22—22no—
140C2×Dic22—42no—
141C2×Dic22—42no—
142C2×Dic22C22×C412yes—
143C11×M16—44no—
144C2×C8×C11—22no—
145C2×C8×C11—22no—
146C2×C8×C11—22no—
147C11×C4⋊C4—24no—
148C11×C4⋊C4—21no—
149C11×C4⋊C4—22no—
150C11×C4⋊C4—22no—
151C11×C4⋊C4C22×C412yes—
152C11×C22⋊C4—24no—
153C11×C22⋊C4—42no—
154C22⋊Dic11—24no—
155C22⋊Dic11—42no—
156C4⋊Dic11—24no—
157C4⋊Dic11—21no—
158C4⋊Dic11—22no—
159C4⋊Dic11—22no—
160C4⋊Dic11C22×C412yes—
161Dic11⋊C4—24no—
162Dic11⋊C4—24no—
163Dic11⋊C4—24no—
164Dic11⋊C4—24no—
165Dic11⋊C4—24no—
166C4×Dic11—24no—
167C4×Dic11—24no—
168C4×Dic11—44no—
169C4×Dic11C22⋊C414yes—
170C4×Dic11C22⋊C414yes—
171C4×Dic11C22×C412yes—
172C4×Dic11C22×C412yes—
173C4×Dic11C2×D412yes—
174C4×Dic11C2×D412yes—
175#4318—44no—
176C2×C11⋊C8—22no—
177C2×C11⋊C8—22no—
178C2×C11⋊C8—22no—
179C2×#533—111no—
180C2×#423—111no—
181#2283—111no—
182#1987—112no—
183#1981—112no—
184#1894—111no—
185#1894—111no—
186#1894—111no—
187#1894D1111yescharacteristic
188#1805—112no—
189#1805—112no—
190C23×C11—22no—
191C23×C11—22no—
192C23×C11—21no—
193C23×C11—21no—
194C23×C11C2×C22⋊C412yes—
195C23×C11C22×D411yescharacteristic
196Q8×C11—24no—
197Q8×C11—41no—
198Q8×C11—41no—
199C2×C4×C11—22no—
200C2×C4×C11—24no—
201C2×C4×C11—22no—
202C2×C4×C11—22no—
203C2×C4×C11—22no—
204C2×C4×C11—22no—
205C2×C4×C11—42no—
206C2×C4×C11—41no—
207C2×C4×C11—41no—
208C2×C4×C11—42no—
209C2×C4×C11—42no—
210C2×C4×C11—42no—
211C2×C4×C11C2×C22⋊C412yes—
212C2×C4×C11C23×C411yescharacteristic
213C2×C4×C11C22×D411yescharacteristic
214C2×C4×C11H(5,𝔽2)12yes—
215C2×C4×C112−1+412yes—
216C2×Dic11—22no—
217C2×Dic11—21no—
218C2×Dic11—21no—
219C2×Dic11—21no—
220C2×Dic11—21no—
221C2×Dic11—28no—
222C2×Dic11—22no—
223C2×Dic11—22no—
224C2×Dic11—44no—
225C2×Dic11—42no—
226C2×Dic11—41no—
227C2×Dic11—41no—
228C2×Dic11—42no—
229C2×Dic11—42no—
230C2×Dic11—42no—
231C2×Dic11C2×C22⋊C412yes—
232C2×Dic11C2×C22⋊C412yes—
233C2×Dic11C2×C22⋊C412yes—
234C2×Dic11C2×C22⋊C412yes—
235C2×Dic11#13714yes—
236Dic22—24no—
237Dic22—48no—
238Dic22—41no—
239Dic22—41no—
240C8×C11—42no—
241C11⋊C8—42no—
242C22×M16—111no—
243C22×M16C2×D1111yescharacteristic
244#533—118no—
245#533—224no—
246C2×C4⋊Q8—111no—
247C2×C4⋊Q8—111no—
248C2×#125—221no—
249C2×C22⋊Q8—111no—
250C2×C22⋊Q8—221no—
251C2×C22⋊Q8C2×D1111yescharacteristic
252C2×C4×Q8—111no—
253C2×C4×Q8—111no—
254C2×C4×Q8—221no—
255C2×#117—221no—
256C22×C42—111no—
257#423—224no—
258C2×C4⋊C8—112no—
259C2×C4⋊C8—112no—
260C2×#103—112no—
261C2×#103—112no—
262C2×#103—112no—
263C2×#103C2×D1112yes—
264#383—221no—
265#382—112no—
266#381—221no—
267#380—111no—
268#380C2×D1111yescharacteristic
269C4×C4⋊C4—112no—
270C4×C22⋊C4—221no—
271#376—221no—
272#340—112no—
273#340—112no—
274#340—112no—
275#340C2×D1112yes—
276C22×C11—22no—
277C22×C11—22no—
278C22×C11—22no—
279C22×C11—42no—
280C22×C11—42no—
281C22×C11C22×C22⋊C411yescharacteristic
282C22×C11#53712yes—
283C22×C11C2×#13712yes—
284C22×C11#57511yescharacteristic
285C22×C11#57511yescharacteristic
286C4×C11—22no—
287C4×C11—42no—
288C4×C11—42no—
289C4×C11—82no—
290C4×C11#52012yes—
291C4×C11#53511yescharacteristic
292C4×C11#53611yescharacteristic
293Dic11—22no—
294Dic11—22no—
295Dic11—22no—
296Dic11—42no—
297Dic11—42no—
298Dic11—82no—
299Dic11#41914yes—
300C22×Q8—111no—
301C22×Q8—221no—
302C22×Q8C22×D1111yescharacteristic
303C23×C4—111no—
304C23×C4—111no—
305C23×C4C22×D1111yescharacteristic
306C2×M16—22no—
307C2×M16—42no—
308C2×M16—222no—
309C2×M16—442no—
310C22×C8—112no—
311C22×C8C22×D1112yes—
312C4⋊Q8—114no—
313C4⋊Q8—114no—
314C4⋊Q8—222no—
315C4⋊Q8—222no—
316#125—228no—
317C22⋊Q8—22no—
318C22⋊Q8—114no—
319C22⋊Q8—228no—
320C22⋊Q8—222no—
321C22⋊Q8C4×D1114yes—
322C4×Q8—114no—
323C4×Q8—114no—
324C4×Q8—222no—
325C4×Q8—228no—
326C4×Q8—222no—
327#117—228no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—111no—
330C2×C4⋊C4—112no—
331C2×C4⋊C4—221no—
332C2×C4⋊C4—221no—
333C2×C4⋊C4—221no—
334C2×C4⋊C4—221no—
335C2×C4⋊C4—221no—
336C2×C4⋊C4—221no—
337C2×C4⋊C4C22×D1112yes—
338C2×C4⋊C4C22×D1111yescharacteristic
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—221no—
342C2×C22⋊C4—221no—
343C2×C42—114no—
344C2×C42—111no—
345C2×C42—111no—
346C2×C42—114no—
347C2×C42—111no—
348C2×C42—111no—
349C2×C42—221no—
350C2×C42—221no—
351C2×C42—222no—
352C4⋊C8—224no—
353C4⋊C8—224no—
354#103—42no—
355#103—442no—
356#103—442no—
357#103—442no—
358#95—21no—
359#95—222no—
360#95—222no—
361#95—222no—
362#95—221no—
363C2×C11—42no—
364C2×C11#231111yescharacteristic
365C2×C11C2×#41911yescharacteristic
366C2×C11#289611yescharacteristic
367C2×C11#325212yes—
368C2×C11#325212yes—
369C24D4×D1111yescharacteristic
370C2×Q8—21no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—112no—
374C2×Q8—224no—
375C2×Q8—221no—
376C2×Q8—222no—
377C2×Q8—222no—
378C2×Q8—442no—
379C2×Q8—442no—
380C2×Q8C2×C4×D1112yes—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—111no—
387C22×C4—114no—
388C22×C4—111no—
389C22×C4—111no—
390C22×C4—111no—
391C22×C4—114no—
392C22×C4—224no—
393C22×C4—221no—
394C22×C4—221no—
395C22×C4—222no—
396C22×C4—441no—
397C22×C4—441no—
398C22×C4C2×C4×D1112yes—
399C22×C4D4×D1111yescharacteristic
400C22×C4D4×D1112yes—
401C22×C4C23×D1111yescharacteristic
402M16—44no—
403M16—444no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—222no—
408C2×C8—222no—
409C2×C8—222no—
410C4⋊C4—24no—
411C4⋊C4—21no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—112no—
415C4⋊C4—224no—
416C4⋊C4—224no—
417C4⋊C4—224no—
418C4⋊C4—224no—
419C4⋊C4—224no—
420C4⋊C4—224no—
421C4⋊C4—221no—
422C4⋊C4—222no—
423C4⋊C4—222no—
424C4⋊C4C2×C4×D1112yes—
425C22⋊C4—24no—
426C22⋊C4—42no—
427C22⋊C4—224no—
428C22⋊C4—442no—
429C42—114no—
430C42—114no—
431C42—112no—
432C42—112no—
433C42—112no—
434C42—112no—
435C42—224no—
436C42—224no—
437C42—444no—
438C11#2189211yescharacteristic, complemented, Sylow
439C23—21no—
440C23—22no—
441C23—21no—
442C23—22no—
443C23C22⋊C4×D1112yes—
444C23C2×D4×D1111yescharacteristic
445Q8—24no—
446Q8—41no—
447Q8—41no—
448Q8—224no—
449Q8—448no—
450Q8—441no—
451Q8—441no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—112no—
465C2×C4—114no—
466C2×C4—112no—
467C2×C4—112no—
468C2×C4—112no—
469C2×C4—228no—
470C2×C4—221no—
471C2×C4—221no—
472C2×C4—222no—
473C2×C4—221no—
474C2×C4—221no—
475C2×C4—222no—
476C2×C4—222no—
477C2×C4—444no—
478C2×C4—441no—
479C2×C4—441no—
480C2×C4—442no—
481C2×C4—442no—
482C2×C4—442no—
483C2×C4—442no—
484C2×C4C22⋊C4×D1112yes—
485C2×C4C22×C4×D1111yescharacteristic
486C2×C4C2×D4×D1111yescharacteristic
487C2×C4#6703712yes—
488C2×C4#6704312yes—
489C8—42no—
490C8—442no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—42no—
495C22—42no—
496C22C2×C22⋊C4×D1111yescharacteristic
497C22#1062312812yes—
498C22#1062330411yescharacteristic
499C22D11×#13712yes—
500C22#1062330811yescharacteristic
501C4—22no—
502C4—42no—
503C4—42no—
504C4—82no—
505C4—114no—
506C4—222no—
507C4—222no—
508C4—222no—
509C4—442no—
510C4—442no—
511C4—882no—
512C4#1062307712yes—
513C4#1062312311yescharacteristic
514C4#1062317911yescharacteristic
515C2—42no—
516C2#1307120311yescharacteristic
517C2D11×#41911yescharacteristic
518C2#1307713211yescharacteristic
519C2#1307998212yes—
520C2#1307998412yes—
subgroupG/H
centerC23C2×D4×D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC11×#1894C2
socleC23×C11C22×D4
prime radicalC24×C11C2×D4
cosocleC22×C4×C11C24
hypercenter#1894D11
hypocenterC11#21892
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1894

Complements

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

NH ≅ G/Ntype
4. #13069562C2semidirect
5. #13069537C2semidirect
9. #13068630C2semidirect
10. #13068629C2semidirect
20. #10623153C4semidirect
438. C11#21892semidirect

Maximal subgroups

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

subgroupindex
C2×#106231532
C2×#106227312
#130710112
#130695622
#130695372
#130690502
#130690492
#130690472
#130686302
#130686292
C11×#18942
#2189211