Finite Group Databasebeta

← #14809909

Subgroups — #14809909

Order 2816.

1076 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623070C211yesmaximal, characteristic
2C2×#10622315C211yesmaximal, characteristic, complemented
3C2×#10622621C211yesmaximal, characteristic
4#13073604C211yesmaximal, characteristic, complemented
5#13069957C212yesmaximal, complemented
6#13069948C212yesmaximal, complemented
7#13069573C211yesmaximal, characteristic, complemented
8#13069556C211yesmaximal, characteristic, complemented
9#13068879C212yesmaximal
10#13068865C212yesmaximal
11C11×#1762C211yesmaximal, characteristic
12C2×C4×D4×C11C2211yescharacteristic
13C2×D4×Dic11C2211yescharacteristic
14C2×D4×Dic11C2211yescharacteristic
15C2×#66977—21no—
16C22×Dic11⋊C4—21no—
17#10623070—24no—
18#10623070—24no—
19C2×#66943C2211yescharacteristic
20C2×#66941C2211yescharacteristic
21C2×#66931—21no—
22C2×#66931—21no—
23C2×#66923C2211yescharacteristic
24C2×#66922—21no—
25C2×#66922C2211yescharacteristic
26C2×C4×Dic22C2211yescharacteristic
27C2×C11×#102C2212yes—
28C2×C11×#97C2211yescharacteristic
29C4×C11×C4⋊C4C2211yescharacteristic
30C2×#66896C2212yes—
31C2×#66882C2212yes—
32C2×#66881C2212yes—
33#10622621—24no—
34#10622598C2212yes—
35#10622598C2212yes—
36C2×#66872C2212yes—
37#10622542—21no—
38C2×C4⋊(C11⋊C8)C2211yescharacteristic
39#10622315—24no—
40#10622274—21no—
41#10622274—21no—
42C11×#340C2212yes—
43#10622188C2212yes—
44#10622188C2212yes—
45#10622167C2212yes—
46#10622167C2212yes—
47C22×D4×C11C2311yescharacteristic
48C23×C4×C11—21no—
49C23×Dic11—21no—
50C23×Dic11—21no—
51C22×Dic22—21no—
52C22×Dic22—21no—
53C22×Dic22D411yescharacteristic
54C22×C8×C11C2312yes—
55C4×D4×C11—22no—
56C4×D4×C11D414yes—
57C2×C11×C4⋊C4C2311yescharacteristic
58C2×C11×C4⋊C4C2312yes—
59C2×C11×C22⋊C4—21no—
60C2×C42×C11—21no—
61C2×C42×C11C2311yescharacteristic
62C2×C22⋊Dic11—21no—
63C2×C22⋊Dic11—21no—
64D4×Dic11—22no—
65D4×Dic11—22no—
66D4×Dic11—22no—
67D4×Dic11—22no—
68C2×C4⋊Dic11—21no—
69C2×C4⋊Dic11—21no—
70C2×C4⋊Dic11—22no—
71C2×C4⋊Dic11—22no—
72C2×C4⋊Dic11D411yescharacteristic
73C2×C4⋊Dic11C2312yes—
74C2×C4⋊Dic11C2311yescharacteristic
75C2×C4⋊Dic11C2311yescharacteristic
76#66977—28no—
77C2×Dic11⋊C4—28no—
78C2×Dic11⋊C4—21no—
79C2×Dic11⋊C4—21no—
80C2×Dic11⋊C4—21no—
81C2×Dic11⋊C4—21no—
82C2×Dic11⋊C4D411yescharacteristic
83C2×Dic11⋊C4D411yescharacteristic
84C2×C4×Dic11C2311yescharacteristic
85C2×C4×Dic11C2311yescharacteristic
86C22×C11⋊C8C2312yes—
87#66943—22no—
88#66943—22no—
89#66941—22no—
90#66941—22no—
91#66931—28no—
92#66931—28no—
93#66923—22no—
94#66923D414yes—
95#66922—22no—
96#66922—22no—
97#66922—44no—
98C4×Dic22—22no—
99C4×Dic22—22no—
100C11×#102Q818yescomplemented
101C11×#97—22no—
102C11×#95—21no—
103#66896—24no—
104C11⋊#95—22no—
105C11⋊#95—22no—
106#66882—24no—
107#66881—24no—
108#66872—24no—
109C4⋊(C11⋊C8)—22no—
110#25851—111nomaximal, Sylow
111C24×C11—21no—
112C2×D4×C11—21no—
113C2×D4×C11—22no—
114C2×D4×C11C2×D412yes—
115C2×D4×C11C2×Q814yes—
116C22×C4×C11—22no—
117C22×C4×C11—21no—
118C22×C4×C11—24no—
119C22×C4×C11—82no—
120C22×C4×C11C2×D411yescharacteristic
121C22×C4×C11C2×D411yescharacteristic
122C22×C4×C11C2411yescharacteristic
123C22×Dic11—24no—
124C22×Dic11—24no—
125C22×Dic11—21no—
126C22×Dic11—21no—
127C22×Dic11—22no—
128C22×Dic11—42no—
129C22×Dic11—42no—
130C22×Dic11—82no—
131C22×Dic11C2×D411yescharacteristic
132C22×Dic11C2×D411yescharacteristic
133C22×Dic11C2×D411yescharacteristic
134C22×Dic11C2×D411yescharacteristic
135C2×Dic22—22no—
136C2×Dic22—24no—
137C2×Dic22—22no—
138C2×Dic22—41no—
139C2×Dic22—42no—
140C2×Dic22—42no—
141C2×Dic22—42no—
142C2×Dic22—82no—
143C2×C8×C11—22no—
144C2×C8×C11—22no—
145C2×C8×C11C2×Q814yes—
146C11×C4⋊C4—21no—
147C11×C4⋊C4—22no—
148C11×C4⋊C4C2×D412yes—
149C11×C4⋊C4C2×Q814yes—
150C11×C22⋊C4—24no—
151C42×C11—42no—
152C42×C11C2×D412yes—
153C42×C11C2×D412yes—
154C22⋊Dic11—24no—
155C22⋊Dic11—24no—
156C4⋊Dic11—24no—
157C4⋊Dic11—22no—
158C4⋊Dic11—21no—
159C4⋊Dic11—21no—
160C4⋊Dic11—22no—
161C4⋊Dic11—22no—
162C4⋊Dic11—44no—
163C4⋊Dic11—44no—
164C4⋊Dic11—42no—
165C4⋊Dic11C2×D412yes—
166C4⋊Dic11C2×D412yes—
167Dic11⋊C4—24no—
168Dic11⋊C4—24no—
169Dic11⋊C4—22no—
170Dic11⋊C4—22no—
171Dic11⋊C4—24no—
172Dic11⋊C4—24no—
173C4×Dic11—21no—
174C4×Dic11—21no—
175C4×Dic11—21no—
176C4×Dic11—21no—
177C2×C11⋊C8—22no—
178C2×C11⋊C8—24no—
179C2×#554—111no—
180C2×#501—111no—
181C2×#476—111no—
182#2546—111no—
183#2059—112no—
184#2058—112no—
185#1989—111no—
186#1985—111no—
187#1864—112no—
188#1862—112no—
189#1762D1111yescharacteristic
190C23×C11—22no—
191C23×C11—24no—
192C23×C11—41no—
193C23×C11—41no—
194C23×C11—42no—
195C23×C11C22×D411yescharacteristic
196D4×C11—22no—
197D4×C11—24no—
198D4×C11C4⋊Q814yescomplemented
199C2×C4×C11—21no—
200C2×C4×C11—21no—
201C2×C4×C11—22no—
202C2×C4×C11—22no—
203C2×C4×C11—24no—
204C2×C4×C11—42no—
205C2×C4×C11—42no—
206C2×C4×C11—42no—
207C2×C4×C11—84no—
208C2×C4×C11#12012yes—
209C2×C4×C11#12012yes—
210C2×C4×C11C22×D411yescharacteristic
211C2×C4×C11C22×D411yescharacteristic
212C2×C4×C11C22×Q812yes—
213C2×C4×C112−1+412yes—
214C2×Dic11—21no—
215C2×Dic11—21no—
216C2×Dic11—21no—
217C2×Dic11—21no—
218C2×Dic11—21no—
219C2×Dic11—22no—
220C2×Dic11—22no—
221C2×Dic11—21no—
222C2×Dic11—21no—
223C2×Dic11—22no—
224C2×Dic11—22no—
225C2×Dic11—48no—
226C2×Dic11—41no—
227C2×Dic11—41no—
228C2×Dic11—42no—
229C2×Dic11—42no—
230C2×Dic11—42no—
231C2×Dic11—42no—
232C2×Dic11—84no—
233C2×Dic11AGL(1,ℤ/8)12yes—
234Dic22—44no—
235Dic22—84no—
236Dic22—84no—
237C8×C11—24no—
238C11⋊C8—42no—
239#554—224no—
240#554—224no—
241C2×C4⋊Q8—111no—
242C2×C4⋊Q8—111no—
243C2×C4⋊Q8—221no—
244C2×#125—111no—
245C2×#125—111no—
246C2×C22⋊Q8—221no—
247C2×C22⋊Q8—221no—
248C2×C22⋊Q8—221no—
249C2×C4×Q8—111no—
250C2×C4×D4—111no—
251C2×C4×D4—111no—
252C2×C4×D4C2×D1111yescharacteristic
253C22×C4⋊C4—221no—
254#501—224no—
255#476—224no—
256C2×#107—112no—
257C2×#106—112no—
258C2×#106—112no—
259C2×C4⋊C8—111no—
260C2×#102—112no—
261C2×#102C2×D1112yes—
262C2×#97C2×D1111yescharacteristic
263#398—221no—
264#398—221no—
265#391—221no—
266#389—112no—
267#389—112no—
268C4×C4⋊C4C2×D1111yescharacteristic
269#340—112no—
270#340—112no—
271#340C2×D1112yes—
272#336—112no—
273#336—112no—
274C22×C11—24no—
275C22×C11—44no—
276C22×C11—44no—
277C22×C11—42no—
278C22×C11—42no—
279C22×C11C2×#12011yescharacteristic
280C22×C11D4×Q812yes—
281C22×C11C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C11#57511yescharacteristic
283C22×C11#57712yes—
284C4×C11—22no—
285C4×C11—22no—
286C4×C11—44no—
287C4×C11—42no—
288C4×C11C2×C4⋊Q811yescharacteristic
289C4×C11#53611yescharacteristic
290C4×C11D4×Q812yes—
291Dic11—22no—
292Dic11—41no—
293Dic11—42no—
294Dic11—41no—
295Dic11—42no—
296Dic11—41no—
297Dic11—82no—
298C22×Q8—111no—
299C22×Q8—221no—
300C22×Q8—221no—
301C22×D4C22×D1111yescharacteristic
302C23×C4—21no—
303C23×C4—221no—
304C23×C4—221no—
305C22×C8—112no—
306C22×C8C22×D1112yes—
307C4⋊Q8—222no—
308C4⋊Q8—222no—
309C4⋊Q8—222no—
310C4⋊Q8—222no—
311C4⋊Q8—444no—
312#125—114no—
313#125—222no—
314#125—222no—
315#125—222no—
316C22⋊Q8—228no—
317C22⋊Q8—228no—
318C22⋊Q8—228no—
319C4×Q8—222no—
320C4×Q8—222no—
321C4×D4—22no—
322C4×D4—222no—
323C4×D4—222no—
324C4×D4—222no—
325C4×D4—222no—
326C4×D4D4414yes—
327C2×C4⋊C4—111no—
328C2×C4⋊C4—111no—
329C2×C4⋊C4—111no—
330C2×C4⋊C4—111no—
331C2×C4⋊C4—111no—
332C2×C4⋊C4—112no—
333C2×C4⋊C4—221no—
334C2×C4⋊C4—221no—
335C2×C4⋊C4—221no—
336C2×C4⋊C4—228no—
337C2×C4⋊C4—221no—
338C2×C4⋊C4—221no—
339C2×C4⋊C4—222no—
340C2×C4⋊C4—222no—
341C2×C4⋊C4—221no—
342C2×C4⋊C4C22×D1112yes—
343C2×C4⋊C4C22×D1111yescharacteristic
344C2×C22⋊C4—21no—
345C2×C22⋊C4—221no—
346C2×C22⋊C4—221no—
347C2×C42—21no—
348C2×C42—111no—
349C2×C42—111no—
350C2×C42C22×D1111yescharacteristic
351#107—224no—
352#106—224no—
353#106—224no—
354C4⋊C8—222no—
355#102—224no—
356#102Dic2218yescomplemented
357#97—22no—
358#95—21no—
359#95—222no—
360#95—222no—
361C2×C11—44no—
362C2×C11#243411yescharacteristic
363C2×C11#300611yescharacteristic
364C2×C11#300911yescharacteristic
365C2×C11#338412yes—
366C2×C11#338812yes—
367C24—21no—
368C2×Q8—222no—
369C2×Q8—224no—
370C2×Q8—222no—
371C2×Q8—441no—
372C2×Q8—442no—
373C2×Q8—442no—
374C2×Q8—442no—
375C2×Q8—882no—
376C2×D4—21no—
377C2×D4—22no—
378C2×D4C2×Dic2214yes—
379C2×D4C2×D4412yes—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—24no—
383C22×C4—82no—
384C22×C4—111no—
385C22×C4—111no—
386C22×C4—111no—
387C22×C4—111no—
388C22×C4—224no—
389C22×C4—221no—
390C22×C4—221no—
391C22×C4—224no—
392C22×C4—222no—
393C22×C4—442no—
394C22×C4—442no—
395C22×C4—882no—
396C22×C4D4×D1111yescharacteristic
397C22×C4D4×D1111yescharacteristic
398C22×C4C23×D1111yescharacteristic
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—222no—
402C2×C8—224no—
403C2×C8C2×Dic2214yes—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—112no—
407C4⋊C4—112no—
408C4⋊C4—224no—
409C4⋊C4—224no—
410C4⋊C4—224no—
411C4⋊C4—222no—
412C4⋊C4—222no—
413C4⋊C4—222no—
414C4⋊C4—224no—
415C4⋊C4—224no—
416C4⋊C4—221no—
417C4⋊C4—221no—
418C4⋊C4—222no—
419C4⋊C4—222no—
420C4⋊C4—444no—
421C4⋊C4—444no—
422C4⋊C4—442no—
423C4⋊C4C2×Dic2214yes—
424C4⋊C4C2×D4412yes—
425C22⋊C4—24no—
426C22⋊C4—224no—
427C22⋊C4—224no—
428C42—42no—
429C42—221no—
430C42—221no—
431C42—221no—
432C42—221no—
433C42C2×D4412yes—
434C42D4×D1112yes—
435C11#2585111yescharacteristic, complemented, Sylow
436C23—22no—
437C23—24no—
438C23—41no—
439C23—42no—
440C23—41no—
441C23C2×D4×D1111yescharacteristic
442Q8—444no—
443Q8—884no—
444Q8—884no—
445D4—22no—
446D4—24no—
447D4#6692214yescomplemented
448C2×C4—21no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—84no—
457C2×C4—112no—
458C2×C4—221no—
459C2×C4—221no—
460C2×C4—221no—
461C2×C4—221no—
462C2×C4—221no—
463C2×C4—222no—
464C2×C4—222no—
465C2×C4—221no—
466C2×C4—222no—
467C2×C4—222no—
468C2×C4—221no—
469C2×C4—448no—
470C2×C4—441no—
471C2×C4—442no—
472C2×C4—441no—
473C2×C4—442no—
474C2×C4—442no—
475C2×C4—442no—
476C2×C4—884no—
477C2×C4#6693512yes—
478C2×C4#6693512yes—
479C2×C4C22×Dic2212yes—
480C2×C4C22×D4411yescharacteristic
481C2×C4C2×D4×D1111yescharacteristic
482C2×C4#6704312yes—
483C8—24no—
484C8—442no—
485C22—24no—
486C22—42no—
487C22—42no—
488C22—44no—
489C22—44no—
490C22C2×#6693511yescharacteristic
491C22D4×Dic2212yes—
492C22D11×AGL(1,ℤ/8)11yescharacteristic
493C22#1062330612yes—
494C22#1062330811yescharacteristic
495C4—22no—
496C4—22no—
497C4—42no—
498C4—44no—
499C4—222no—
500C4—441no—
501C4—442no—
502C4—441no—
503C4—441no—
504C4—442no—
505C4—882no—
506C4C2×#6692211yescharacteristic
507C4#1062305711yescharacteristic
508C4D4×Dic2212yes—
509C2—44no—
510C2#1307226011yescharacteristic
511C2#1307786911yescharacteristic
512C2#1307789411yescharacteristic
513C2#1308151612yes—
514C2#1308156212yes—
subgroupG/H
centerC23C2×D4×D11
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC11×#1762C2
socleC23×C11C22×D4
prime radicalC22×D4×C11C23
hypercenter#1762D11
hypocenterC11#25851
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. C2×#10622315C2semidirect
4. #13073604C2semidirect
5. #13069957C2semidirect
6. #13069948C2semidirect
7. #13069573C2semidirect
8. #13069556C2semidirect
100. C11×#102Q8semidirect
198. D4×C11C4⋊Q8semidirect
356. #102Dic22semidirect
435. C11#25851semidirect
447. D4#66922semidirect

Maximal subgroups

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

subgroupindex
C2×#106230702
C2×#106223152
C2×#106226212
#130736042
#130699572
#130699482
#130695732
#130695562
#130688792
#130688652
C11×#17622
#2585111