Finite Group Databasebeta

← #14788545

Subgroups — #14788545

Order 2816.

1118 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623101C211yesmaximal, characteristic, complemented
2C2×#10622632C211yesmaximal, characteristic
3C2×C4×#4324C211yesmaximal, characteristic
4#13069862C212yesmaximal, complemented
5#13069698C212yesmaximal
6#13069553C211yesmaximal, characteristic, complemented
7#13069417C211yesmaximal, characteristic, complemented
8#13068974C212yesmaximal
9#13068883C212yesmaximal, complemented
10#13068618C211yesmaximal, characteristic, complemented
11C11×#2538C211yesmaximal, characteristic
12C2×C11×C22⋊Q8—21no—
13C2×C4×Q8×C11C2211yescharacteristic
14C2×C4×D4×C11C2211yescharacteristic
15C2×Q8×Dic11—21no—
16C22×#4324C2211yescharacteristic
17C22×#4324C2212yes—
18#10623101—24no—
19#10623101—24no—
20C2×#66943—21no—
21C2×#66943—22no—
22C2×#66922C2212yes—
23C2×#66922C2211yescharacteristic
24C2×C4×Dic22C2212yes—
25C2×C4×Dic22C2211yescharacteristic
26C11×#396—22no—
27C11×#390C2212yes—
28C11×#389C2212yes—
29C11×#384C2211yescharacteristic
30C11×#380—21no—
31C2×#66896C2211yescharacteristic
32#10622632—24no—
33C4×#4324—24no—
34C2×#66874C2212yes—
35C2×#66874C2211yescharacteristic
36C2×#66874C2212yes—
37C2×#66874C2212yes—
38C2×#66874C2212yes—
39C2×#66872C2211yescharacteristic
40C2×C4⋊(C11⋊C8)C2211yescharacteristic
41C2×C4×C11⋊C8C2211yescharacteristic
42#10622167C2212yes—
43#10622162C2212yes—
44#10622162C2212yes—
45C22×Q8×C11D411yescharacteristic
46C22×D4×C11C2311yescharacteristic
47C23×C4×C11—21no—
48C22×Dic22C2312yes—
49C22×Dic22C2311yescharacteristic
50C11×C22⋊Q8—44no—
51C4×Q8×C11—22no—
52C4×Q8×C11—22no—
53C4×D4×C11—22no—
54C4×D4×C11—22no—
55C2×C11×C4⋊C4—21no—
56C2×C11×C4⋊C4—22no—
57C2×C11×C4⋊C4—21no—
58C2×C11×C4⋊C4D411yescharacteristic
59C2×C11×C4⋊C4D412yes—
60C2×C11×C4⋊C4C2311yescharacteristic
61C2×C11×C4⋊C4C2312yes—
62C2×C11×C22⋊C4—21no—
63C2×C11×C22⋊C4—22no—
64C2×C11×C22⋊C4—22no—
65C2×C42×C11C2312yes—
66C2×C42×C11C2311yescharacteristic
67Q8×Dic11—28no—
68C2×#4324—22no—
69C2×#4324—22no—
70C2×#4324—24no—
71C2×#4324—24no—
72C2×#4324—22no—
73C2×#4324—42no—
74C2×C4⋊Dic11—22no—
75C2×C4⋊Dic11—21no—
76C2×C4⋊Dic11—22no—
77C2×C4⋊Dic11—21no—
78C2×C4⋊Dic11C2311yescharacteristic
79C2×Dic11⋊C4—22no—
80C2×Dic11⋊C4—21no—
81C2×C4×Dic11—22no—
82C2×C4×Dic11—21no—
83C22×C11⋊C8C2311yescharacteristic
84C22×C11⋊C8C2312yes—
85C22×C11⋊C8C2311yescharacteristic
86#66943—216no—
87#66943—28no—
88#66922—24no—
89#66922—24no—
90#66922—22no—
91#66922D414yes—
92C4×Dic22—24no—
93C4×Dic22—22no—
94C4×Dic22—24no—
95C4×Dic22D414yes—
96C11×#95—22no—
97#66896—22no—
98#66874—22no—
99#66874—24no—
100#66874—24no—
101#66874—42no—
102#66874—42no—
103#66872—22no—
104C4⋊(C11⋊C8)—22no—
105C4×C11⋊C8—22no—
106#24187—111nomaximal, Sylow
107C24×C11—21no—
108C2×Q8×C11—22no—
109C2×Q8×C11—22no—
110C2×Q8×C11SD1614yescomplemented
111C2×D4×C11—21no—
112C2×D4×C11—21no—
113C2×D4×C11—21no—
114C2×D4×C11—21no—
115C2×D4×C11—21no—
116C2×D4×C11#4112yes—
117C22×C4×C11—21no—
118C22×C4×C11—22no—
119C22×C4×C11—42no—
120C22×C4×C11—42no—
121C22×C4×C11—42no—
122C22×C4×C11C2×D412yes—
123C22×C4×C11C2×D411yescharacteristic
124C22×C4×C11C2×D412yes—
125C22×C4×C11C2×D411yescharacteristic
126C22×C4×C11C2411yescharacteristic
127C22×Dic11—21no—
128C22×Dic11—22no—
129C22×Dic11—21no—
130C2×Dic22—22no—
131C2×Dic22—21no—
132C2×Dic22—22no—
133C2×Dic22—21no—
134C2×Dic22—22no—
135C2×Dic22—21no—
136C2×Dic22—21no—
137C2×Dic22—22no—
138C2×Dic22—21no—
139C2×Dic22—22no—
140C2×Dic22C2×D414yes—
141C2×Dic22C2×D412yes—
142C11×C4⋊C4—24no—
143C11×C4⋊C4—22no—
144C11×C4⋊C4—21no—
145C11×C4⋊C4—21no—
146C11×C4⋊C4—22no—
147C11×C4⋊C4—22no—
148C11×C4⋊C4—42no—
149C11×C4⋊C4—44no—
150C11×C4⋊C4—81no—
151C11×C22⋊C4—42no—
152C11×C22⋊C4—44no—
153C11×C22⋊C4—82no—
154C42×C11—22no—
155C42×C11—22no—
156C42×C11C2×D412yes—
157C42×C11C2×D412yes—
158#4324—28no—
159#4324—48no—
160C4⋊Dic11—28no—
161C4⋊Dic11—28no—
162C4⋊Dic11—24no—
163C4⋊Dic11—24no—
164C4⋊Dic11—21no—
165C4⋊Dic11C2×D412yes—
166Dic11⋊C4—28no—
167Dic11⋊C4—24no—
168C4×Dic11—28no—
169C4×Dic11—24no—
170C2×C11⋊C8—21no—
171C2×C11⋊C8—21no—
172C2×C11⋊C8—22no—
173C2×C11⋊C8—21no—
174C2×C11⋊C8—22no—
175C2×C11⋊C8—21no—
176C2×C11⋊C8—22no—
177C2×C11⋊C8—21no—
178C2×C11⋊C8#4112yes—
179C2×#557—111no—
180C2×#461—111no—
181C2×C4×SD16—111no—
182#2538D1111yescharacteristic
183#2043—112no—
184#2013—112no—
185#1984—111no—
186#1956—111no—
187#1880—112no—
188#1865—112no—
189#1803—111no—
190C23×C11—22no—
191C23×C11—41no—
192C23×C11—41no—
193C23×C11—41no—
194C23×C11—41no—
195C23×C11—41no—
196C23×C11—41no—
197C23×C11C22×D411yescharacteristic
198Q8×C11—28no—
199D4×C11—22no—
200D4×C11—22no—
201D4×C11—41no—
202D4×C11—41no—
203C2×C4×C11—22no—
204C2×C4×C11—22no—
205C2×C4×C11—21no—
206C2×C4×C11—21no—
207C2×C4×C11—22no—
208C2×C4×C11—22no—
209C2×C4×C11—22no—
210C2×C4×C11—41no—
211C2×C4×C11—42no—
212C2×C4×C11—42no—
213C2×C4×C11—41no—
214C2×C4×C11—41no—
215C2×C4×C11—42no—
216C2×C4×C11—84no—
217C2×C4×C11#12012yes—
218C2×C4×C11#12012yes—
219C2×C4×C11C2×SD1614yes—
220C2×C4×C11C2×SD1612yes—
221C2×C4×C11#13712yes—
222C2×C4×C11C22×D411yescharacteristic
223C2×C4×C11C22×D411yescharacteristic
224C2×C4×C11C2×#4111yescharacteristic
225C2×C4×C11C2×#4111yescharacteristic
226C2×C4×C11H(5,𝔽2)11yescharacteristic
227C2×C4×C112−1+411yescharacteristic
228C2×Dic11—22no—
229C2×Dic11—24no—
230C2×Dic11—22no—
231C2×Dic11—41no—
232C2×Dic11—42no—
233C2×Dic11—42no—
234C2×Dic11—41no—
235C2×Dic11—41no—
236C2×Dic11—41no—
237Dic22—22no—
238Dic22—22no—
239Dic22—24no—
240Dic22—24no—
241Dic22—42no—
242Dic22—42no—
243Dic22—41no—
244Dic22—41no—
245C11⋊C8—22no—
246C11⋊C8—41no—
247C11⋊C8—42no—
248C22×SD16—111no—
249C22×SD16—112no—
250#557—224no—
251#557—224no—
252C2×C4⋊Q8—111no—
253C2×C4⋊Q8—112no—
254C2×#125—221no—
255C2×#125—222no—
256C2×C22⋊Q8—21no—
257C2×C4×Q8—111no—
258C2×C4×Q8—112no—
259C2×C4×Q8—221no—
260C2×C4×Q8C2×D1111yescharacteristic
261C2×C4×D4C2×D1111yescharacteristic
262#461—224no—
263C4×SD16—224no—
264C2×#106—111no—
265C2×C4⋊C8—111no—
266C2×#103—112no—
267C2×#103—112no—
268C2×#103—111no—
269C2×#103—112no—
270C2×#103—112no—
271C2×#102—111no—
272C2×C4×C8—111no—
273#396—22no—
274#390C2×D1112yes—
275#389C2×D1112yes—
276#384C2×D1111yescharacteristic
277#380—21no—
278#340—112no—
279#336—112no—
280#336—112no—
281C22×C11—42no—
282C22×C11—42no—
283C22×C11—42no—
284C22×C11—81no—
285C22×C11—81no—
286C22×C11—81no—
287C22×C11—81no—
288C22×C11C2×#12011yescharacteristic
289C22×C11#54811yescharacteristic
290C22×C11#55011yescharacteristic
291C22×C11C22×SD1611yescharacteristic
292C22×C11C2×#13711yescharacteristic
293C22×C11#57711yescharacteristic
294C22×C11#57811yescharacteristic
295C4×C11—24no—
296C4×C11—22no—
297C4×C11—42no—
298C4×C11—82no—
299C4×C11—81no—
300C4×C11#44814yes—
301C4×C11#52511yescharacteristic
302C4×C11#53711yescharacteristic
303C4×C11#54611yescharacteristic
304C4×C11#54711yescharacteristic
305Dic11—42no—
306Dic11—44no—
307Dic11—42no—
308C22×Q8—111no—
309C22×Q8—112no—
310C22×Q8C22⋊D1111yescharacteristic
311C22×D4C22×D1111yescharacteristic
312C23×C4—21no—
313C2×SD16—222no—
314C2×SD16—222no—
315C2×SD16—222no—
316C2×SD16—224no—
317C2×SD16—224no—
318C2×SD16—442no—
319C22×C8—111no—
320C22×C8—112no—
321C22×C8—111no—
322C4⋊Q8—114no—
323C4⋊Q8—224no—
324C4⋊Q8—224no—
325C4⋊Q8—222no—
326#125—2216no—
327#125—228no—
328C22⋊Q8—44no—
329C4×Q8—22no—
330C4×Q8—22no—
331C4×Q8—114no—
332C4×Q8—224no—
333C4×Q8—224no—
334C4×Q8—228no—
335C4×Q8—222no—
336C4×D4—22no—
337C4×D4—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—111no—
342C2×C4⋊C4—222no—
343C2×C4⋊C4—221no—
344C2×C4⋊C4—222no—
345C2×C4⋊C4—221no—
346C2×C4⋊C4—222no—
347C2×C4⋊C4—221no—
348C2×C4⋊C4C22⋊D1111yescharacteristic
349C2×C4⋊C4C22⋊D1112yes—
350C2×C4⋊C4C22×D1111yescharacteristic
351C2×C4⋊C4C22×D1112yes—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C42—222no—
356C2×C42—221no—
357C2×C42C22×D1112yes—
358C2×C42C22×D1111yescharacteristic
359#106—222no—
360C4⋊C8—222no—
361#103—224no—
362#103—222no—
363#103—224no—
364#103—442no—
365#103—442no—
366#102—222no—
367C4×C8—222no—
368#95—22no—
369C2×C11—82no—
370C2×C11#240711yescharacteristic
371C2×C11C2×#44811yescharacteristic
372C2×C11#301311yescharacteristic
373C2×C11#335911yescharacteristic
374C2×C11#337211yescharacteristic
375C2×C11#339211yescharacteristic
376C2×C11#339811yescharacteristic
377C24—21no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—112no—
381C2×Q8—114no—
382C2×Q8—222no—
383C2×Q8—221no—
384C2×Q8—221no—
385C2×Q8—222no—
386C2×Q8—221no—
387C2×Q8—222no—
388C2×Q8—222no—
389C2×Q8—221no—
390C2×Q8—221no—
391C2×Q8—222no—
392C2×Q8#432414yescomplemented
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4#434312yes—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—221no—
405C22×C4—222no—
406C22×C4—221no—
407C22×C4C2×C22⋊D1112yes—
408C22×C4C2×C22⋊D1112yes—
409C22×C4C2×C22⋊D1111yescharacteristic
410C22×C4C2×C22⋊D1111yescharacteristic
411C22×C4C23×D1111yescharacteristic
412SD16—228no—
413SD16—448no—
414C2×C8—112no—
415C2×C8—221no—
416C2×C8—221no—
417C2×C8—221no—
418C2×C8—222no—
419C2×C8—222no—
420C2×C8—221no—
421C2×C8—221no—
422C2×C8—222no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—42no—
430C4⋊C4—44no—
431C4⋊C4—81no—
432C4⋊C4—112no—
433C4⋊C4—228no—
434C4⋊C4—228no—
435C4⋊C4—228no—
436C4⋊C4—224no—
437C4⋊C4—224no—
438C4⋊C4—224no—
439C4⋊C4—221no—
440C22⋊C4—44no—
441C22⋊C4—42no—
442C22⋊C4—82no—
443C42—22no—
444C42—22no—
445C42—228no—
446C42—224no—
447C42D4×D1112yes—
448C42D4×D1112yes—
449C11#2418711yescharacteristic, complemented, Sylow
450C23—22no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23C22×C22⋊D1111yescharacteristic
458Q8—28no—
459Q8—222no—
460Q8—222no—
461Q8—224no—
462Q8—224no—
463Q8—442no—
464Q8—442no—
465Q8—441no—
466Q8—441no—
467D4—22no—
468D4—22no—
469D4—41no—
470D4—41no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—84no—
485C2×C4—222no—
486C2×C4—224no—
487C2×C4—222no—
488C2×C4—441no—
489C2×C4—442no—
490C2×C4—442no—
491C2×C4—441no—
492C2×C4—441no—
493C2×C4—441no—
494C2×C4C2×#432414yes—
495C2×C4C2×#432412yes—
496C2×C4#6699012yes—
497C2×C4#6699012yes—
498C2×C4#6700412yes—
499C2×C4C2×D4×D1111yescharacteristic
500C2×C4C2×D4×D1111yescharacteristic
501C2×C4C2×#434311yescharacteristic
502C2×C4#6704011yescharacteristic
503C2×C4#41×D1111yescharacteristic
504C2×C4#6704211yescharacteristic
505C8—222no—
506C8—441no—
507C8—442no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22C22×#432411yescharacteristic
516C22C2×#6699011yescharacteristic
517C22#1062334811yescharacteristic
518C22C2×#6700411yescharacteristic
519C22#1062336511yescharacteristic
520C22#1062336711yescharacteristic
521C22#1062336911yescharacteristic
522C4—24no—
523C4—22no—
524C4—42no—
525C4—81no—
526C4—82no—
527C4—442no—
528C4—444no—
529C4—442no—
530C4#1062265314yes—
531C4#1062315811yescharacteristic
532C4#1062316011yescharacteristic
533C4#1062316111yescharacteristic
534C4#1062316211yescharacteristic
535C2—82no—
536C2#1307196311yescharacteristic
537C2C2×#1062265311yescharacteristic
538C2#1307792811yescharacteristic
539C2#1308125811yescharacteristic
540C2#1308139711yescharacteristic
541C2#1308159711yescharacteristic
542C2#1308164711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D11
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC11×#2538C2
socleC23×C11C22×D4
prime radicalC22×D4×C11C23
hypercenter#2538D11
hypocenterC11#24187
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2538

Complements

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

NH ≅ G/Ntype
1. C2×#10623101C2semidirect
4. #13069862C2semidirect
6. #13069553C2semidirect
7. #13069417C2semidirect
9. #13068883C2semidirect
10. #13068618C2semidirect
110. C2×Q8×C11SD16semidirect
392. C2×Q8#4324semidirect
449. C11#24187semidirect

Maximal subgroups

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

subgroupindex
C2×#106231012
C2×#106226322
C2×C4×#43242
#130698622
#130696982
#130695532
#130694172
#130689742
#130688832
#130686182
C11×#25382
#2418711