Finite Group Databasebeta

← #14809912

Subgroups — #14809912

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×#10623110C211yesmaximal, characteristic
2C2×#10622315C211yesmaximal, characteristic, complemented
3C2×#10622455C211yesmaximal, characteristic
4#13073601C211yesmaximal, characteristic, complemented
5#13069955C212yesmaximal, complemented
6#13069950C212yesmaximal, complemented
7#13069571C211yesmaximal, characteristic, complemented
8#13068880C212yesmaximal
9#13068866C212yesmaximal
10#13068442C211yesmaximal, characteristic
11C11×#1985C211yesmaximal, characteristic, complemented
12C2×C4×Q8×C11C2211yescharacteristic
13C2×#66999—21no—
14C2×#66997—21no—
15#10623110—24no—
16#10623110—24no—
17C2×#66949—21no—
18C2×#66949—21no—
19C2×C4⋊C4×D11—21no—
20C2×C4×D44C2211yescharacteristic
21C2×C4×D44C2211yescharacteristic
22C2×C4×D44C2211yescharacteristic
23C2×#66923C2211yescharacteristic
24C2×#66923C2211yescharacteristic
25C2×#66922C2211yescharacteristic
26C2×#66922C2211yescharacteristic
27C2×C11×C4⋊C8C2211yescharacteristic
28C2×C11×#97C2211yescharacteristic
29C2×#66885C2212yes—
30C2×#66885C2212yes—
31C2×#66882C2212yes—
32C2×#66881C2212yes—
33C2×#66881C2212yes—
34#10622596—21no—
35#10622596—21no—
36#10622587—21no—
37C4×C4⋊Dic11C2211yescharacteristic
38#10622541C2212yes—
39#10622541C2212yes—
40#10622455—24no—
41#10622315—24no—
42C11×#336C2212yes—
43C11×#336C2212yes—
44#10622189C2212yes—
45#10622189C2212yes—
46#10622189C2212yes—
47C22×Q8×C11D411yescharacteristic
48C22×D44C2311yescharacteristic
49C22×C4×D11—21no—
50C22×C4×D11—21no—
51C22×C4×D11—21no—
52C22×Dic22—21no—
53C22×Dic22—21no—
54C22×C8×C11C2312yes—
55C22×C8×C11C2312yes—
56C4×Q8×C11—22no—
57C4×Q8×C11—22no—
58C2×C11×C4⋊C4D411yescharacteristic
59C2×C11×C4⋊C4D411yescharacteristic
60C2×C11×C4⋊C4D411yescharacteristic
61C2×C42×C11C2311yescharacteristic
62C2×C42×C11C2311yescharacteristic
63C2×C42×C11C2311yescharacteristic
64#66999—28no—
65#66997—44no—
66C2×(C2×D11)⋊C4—21no—
67C2×(C2×D11)⋊C4—21no—
68C2×(C2×D11)⋊C4—21no—
69C2×C4⋊Dic11—21no—
70C2×C4⋊Dic11—21no—
71C2×C4⋊Dic11C2312yes—
72C2×C4⋊Dic11C2311yescharacteristic
73C2×C4⋊Dic11C2311yescharacteristic
74C2×C4⋊Dic11C2311yescharacteristic
75C2×C4⋊Dic11C2312yes—
76C2×Dic11⋊C4—21no—
77C2×Dic11⋊C4—21no—
78C2×Dic11⋊C4—21no—
79C2×Dic11⋊C4—21no—
80C2×Dic11⋊C4—22no—
81C2×Dic11⋊C4—22no—
82C2×C4×Dic11—21no—
83#66949—28no—
84#66949—28no—
85C4⋊C4×D11—28no—
86C4×D44—22no—
87C4×D44—22no—
88C4×D44—22no—
89C4×D44—22no—
90C4×D44—22no—
91C4×D44D414yes—
92#66923—22no—
93#66923—22no—
94#66923—22no—
95#66923D414yes—
96#66922—22no—
97#66922—22no—
98#66922—22no—
99#66922—22no—
100C11×C4⋊C8—22no—
101C11×#97—22no—
102C11⋊#95—22no—
103C11⋊#95—22no—
104C11⋊#95—21no—
105#66885—24no—
106#66885Q818yescomplemented
107#66882—24no—
108#66881—24no—
109#66881—24no—
110#25851—111nomaximal, Sylow
111C23×D11—21no—
112C2×Q8×C11—22no—
113C2×Q8×C11—22no—
114C2×Q8×C11—41no—
115C22×C4×C11C2×D411yescharacteristic
116C22×C4×C11C2×D411yescharacteristic
117C22×C4×C11C2×D411yescharacteristic
118C22×C4×C11C2×D411yescharacteristic
119C22×C4×C11C2×D411yescharacteristic
120C22×C4×C11C2×D411yescharacteristic
121C22×C4×C11C2411yescharacteristic
122C22×Dic11—22no—
123C22×Dic11—21no—
124C22×Dic11—21no—
125C22×Dic11—21no—
126C22×Dic11—22no—
127C2×D44—21no—
128C2×D44—22no—
129C2×D44C2×D412yes—
130C2×D44C2×Q814yes—
131C2×C4×D11—24no—
132C2×C4×D11—24no—
133C2×C4×D11—24no—
134C2×C4×D11—42no—
135C2×C4×D11—42no—
136C2×C4×D11—82no—
137C2×C4×D11—82no—
138C2×Dic22—24no—
139C2×Dic22—42no—
140C2×Dic22—42no—
141C2×Dic22—42no—
142C2×Dic22—82no—
143C2×C8×C11—22no—
144C2×C8×C11—22no—
145C2×C8×C11—22no—
146C2×C8×C11—24no—
147C2×C8×C11C2×Q814yes—
148C11×C4⋊C4—22no—
149C11×C4⋊C4—22no—
150C11×C4⋊C4—22no—
151C42×C11—21no—
152C42×C11—21no—
153C42×C11—21no—
154C42×C11—21no—
155C42×C11C2×D412yes—
156C42×C11C2×D412yes—
157(C2×D11)⋊C4—24no—
158(C2×D11)⋊C4—24no—
159(C2×D11)⋊C4—24no—
160C4⋊Dic11—24no—
161C4⋊Dic11—21no—
162C4⋊Dic11—21no—
163C4⋊Dic11—21no—
164C4⋊Dic11—22no—
165C4⋊Dic11—22no—
166C4⋊Dic11—22no—
167C4⋊Dic11—42no—
168C4⋊Dic11C2×D412yes—
169C4⋊Dic11C2×D412yes—
170C4⋊Dic11C2×D412yes—
171C4⋊Dic11C2×Q814yes—
172Dic11⋊C4—24no—
173Dic11⋊C4—24no—
174Dic11⋊C4—24no—
175Dic11⋊C4—24no—
176Dic11⋊C4—44no—
177Dic11⋊C4—44no—
178C4×Dic11—42no—
179C2×#554—111no—
180C2×#501—111no—
181C2×#476—111no—
182#2546—111no—
183#2059—112no—
184#2058—112no—
185#1989—111no—
186#1985D1111yescharacteristic, complemented
187#1864—112no—
188#1862—112no—
189#1762—111no—
190C23×C11C22×D411yescharacteristic
191C22×D11—22no—
192C22×D11—24no—
193C22×D11—41no—
194C22×D11—41no—
195C22×D11—42no—
196Q8×C11—44no—
197C2×C4×C11—21no—
198C2×C4×C11—21no—
199C2×C4×C11—21no—
200C2×C4×C11—21no—
201C2×C4×C11—22no—
202C2×C4×C11—21no—
203C2×C4×C11—21no—
204C2×C4×C11—21no—
205C2×C4×C11—21no—
206C2×C4×C11—21no—
207C2×C4×C11—22no—
208C2×C4×C11—22no—
209C2×C4×C11#12012yes—
210C2×C4×C11#12012yes—
211C2×C4×C11AGL(1,ℤ/8)12yes—
212C2×C4×C11C22×D411yescharacteristic
213C2×C4×C11C22×D411yescharacteristic
214C2×C4×C11C22×Q812yes—
215C2×C4×C112−1+412yes—
216C2×Dic11—22no—
217C2×Dic11—22no—
218C2×Dic11—22no—
219C2×Dic11—24no—
220C2×Dic11—42no—
221C2×Dic11—41no—
222C2×Dic11—41no—
223C2×Dic11—42no—
224C2×Dic11—42no—
225C2×Dic11—42no—
226C2×Dic11—42no—
227C2×Dic11—42no—
228C2×Dic11—42no—
229D44—22no—
230D44—24no—
231D44C4⋊Q814yescomplemented
232C4×D11—48no—
233C4×D11—84no—
234C4×D11—84no—
235Dic22—84no—
236Dic22—84no—
237C8×C11—24no—
238C8×C11—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×Q8C2×D1111yescharacteristic
250C2×C4×D4—111no—
251C2×C4×D4—111no—
252C2×C4×D4—111no—
253C22×C4⋊C4—221no—
254#501—224no—
255#476—224no—
256C2×#107—112no—
257C2×#106—112no—
258C2×#106—112no—
259C2×C4⋊C8C2×D1111yescharacteristic
260C2×#102—112no—
261C2×#102—112no—
262C2×#97C2×D1111yescharacteristic
263#398—221no—
264#398—221no—
265#391—221no—
266#389—112no—
267#389—112no—
268C4×C4⋊C4—111no—
269#340—112no—
270#340—112no—
271#340—112no—
272#336C2×D1112yes—
273#336C2×D1112yes—
274C22×C11C2×#12011yescharacteristic
275C22×C11D4×Q812yes—
276C22×C11C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C11#57511yescharacteristic
278C22×C11#57712yes—
279C2×D11—24no—
280C2×D11—44no—
281C2×D11—44no—
282C2×D11—42no—
283C2×D11—42no—
284C4×C11—22no—
285C4×C11—22no—
286C4×C11—22no—
287C4×C11—41no—
288C4×C11—41no—
289C4×C11—41no—
290C4×C11C2×C4⋊Q811yescharacteristic
291C4×C11#53611yescharacteristic
292C4×C11D4×Q812yes—
293Dic11—44no—
294Dic11—42no—
295Dic11—42no—
296Dic11—42no—
297Dic11—82no—
298C22×Q8—221no—
299C22×Q8—221no—
300C22×Q8D4411yescharacteristic
301C22×D4—111no—
302C23×C4—221no—
303C23×C4—221no—
304C23×C4—221no—
305C22×C8C22×D1112yes—
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—22no—
320C4×Q8—22no—
321C4×D4—114no—
322C4×D4—222no—
323C4×D4—222no—
324C4×D4—222no—
325C4×D4—222no—
326C4×D4—222no—
327C2×C4⋊C4—111no—
328C2×C4⋊C4—111no—
329C2×C4⋊C4—111no—
330C2×C4⋊C4—112no—
331C2×C4⋊C4—112no—
332C2×C4⋊C4—228no—
333C2×C4⋊C4—221no—
334C2×C4⋊C4—221no—
335C2×C4⋊C4—221no—
336C2×C4⋊C4—221no—
337C2×C4⋊C4—221no—
338C2×C4⋊C4—221no—
339C2×C4⋊C4—222no—
340C2×C4⋊C4—222no—
341C2×C4⋊C4D4411yescharacteristic
342C2×C4⋊C4D4411yescharacteristic
343C2×C4⋊C4D4411yescharacteristic
344C2×C22⋊C4—221no—
345C2×C22⋊C4—221no—
346C2×C22⋊C4—221no—
347C2×C42—221no—
348C2×C42C22×D1111yescharacteristic
349C2×C42C22×D1111yescharacteristic
350C2×C42C22×D1111yescharacteristic
351#107—224no—
352#106—224no—
353#106—224no—
354C4⋊C8—22no—
355#102—118no—
356#102—224no—
357#97—22no—
358#95—221no—
359#95—222no—
360#95—222no—
361C2×C11#243411yescharacteristic
362C2×C11#300611yescharacteristic
363C2×C11#300911yescharacteristic
364C2×C11#338412yes—
365C2×C11#338812yes—
366D11—44no—
367C24—221no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—41no—
371C2×Q8—224no—
372C2×Q8—442no—
373C2×Q8—442no—
374C2×Q8—442no—
375C2×Q8—882no—
376C2×D4—112no—
377C2×D4—114no—
378C2×D4—221no—
379C2×D4—222no—
380C22×C4—224no—
381C22×C4—224no—
382C22×C4—222no—
383C22×C4—221no—
384C22×C4—221no—
385C22×C4—221no—
386C22×C4—224no—
387C22×C4—222no—
388C22×C4—442no—
389C22×C4—442no—
390C22×C4—882no—
391C22×C4—882no—
392C22×C4C2×D4411yescharacteristic
393C22×C4C2×D4411yescharacteristic
394C22×C4C2×D4411yescharacteristic
395C22×C4C2×D4411yescharacteristic
396C22×C4C2×D4411yescharacteristic
397C22×C4C2×D4411yescharacteristic
398C22×C4C23×D1111yescharacteristic
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—24no—
403C2×C8Q8×D1114yes—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—112no—
408C4⋊C4—112no—
409C4⋊C4—112no—
410C4⋊C4—114no—
411C4⋊C4—224no—
412C4⋊C4—224no—
413C4⋊C4—224no—
414C4⋊C4—224no—
415C4⋊C4—224no—
416C4⋊C4—221no—
417C4⋊C4—221no—
418C4⋊C4—221no—
419C4⋊C4—222no—
420C4⋊C4—222no—
421C4⋊C4—222no—
422C4⋊C4—444no—
423C4⋊C4—442no—
424C4⋊C4—444no—
425C22⋊C4—224no—
426C22⋊C4—224no—
427C22⋊C4—224no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—442no—
433C42D4×D1112yes—
434C42D4×D1112yes—
435C11#2585111yescharacteristic, complemented, Sylow
436C23—222no—
437C23—224no—
438C23—441no—
439C23—441no—
440C23—442no—
441C23C22×D4411yescharacteristic
442Q8—44no—
443Q8—884no—
444Q8—884no—
445D4—114no—
446D4—224no—
447D4—222no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—222no—
461C2×C4—222no—
462C2×C4—222no—
463C2×C4—224no—
464C2×C4—448no—
465C2×C4—442no—
466C2×C4—441no—
467C2×C4—441no—
468C2×C4—442no—
469C2×C4—442no—
470C2×C4—442no—
471C2×C4—442no—
472C2×C4—442no—
473C2×C4—442no—
474C2×C4—884no—
475C2×C4—884no—
476C2×C4#6693512yes—
477C2×C4#6693512yes—
478C2×C4#6696112yes—
479C2×C4C2×D4×D1111yescharacteristic
480C2×C4C2×D4×D1111yescharacteristic
481C2×C4C2×Q8×D1112yes—
482C2×C4#6704312yes—
483C8—24no—
484C8—42no—
485C22—224no—
486C22—442no—
487C22—442no—
488C22—444no—
489C22—444no—
490C22C2×#6693511yescharacteristic
491C22Q8×D4412yes—
492C22C2×#6696111yescharacteristic
493C22#1062327911yescharacteristic
494C22#1062328212yes—
495C4—22no—
496C4—22no—
497C4—22no—
498C4—41no—
499C4—41no—
500C4—41no—
501C4—442no—
502C4—442no—
503C4—442no—
504C4—444no—
505C4—882no—
506C4#1062316312yes—
507C4#1062320611yescharacteristic
508C4D11×C4⋊Q811yescharacteristic
509C2—444no—
510C2#1307225911yescharacteristic
511C2#1307787011yescharacteristic
512C2#1307789511yescharacteristic
513C2#1308151712yes—
514C2#1308156312yes—
subgroupG/H
centerC23C22×D44
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC11×#1985C2
socleC23×C11C22×D4
prime radicalC22×D44C23
hypercenter#1985D11
hypocenterC11#25851
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #1985

Complements

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

NH ≅ G/Ntype
2. C2×#10622315C2semidirect
4. #13073601C2semidirect
5. #13069955C2semidirect
6. #13069950C2semidirect
7. #13069571C2semidirect
11. C11×#1985C2semidirect
106. #66885Q8semidirect
186. #1985D11semidirect
231. D44C4⋊Q8semidirect
435. C11#25851semidirect

Maximal subgroups

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

subgroupindex
C2×#106231102
C2×#106223152
C2×#106224552
#130736012
#130699552
#130699502
#130695712
#130688802
#130688662
#130684422
C11×#19852
#2585111