Finite Group Databasebeta

← #14781596

Subgroups — #14781596

Order 2816.

1326 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#66995C211yesmaximal, characteristic
2#13077180C211yesmaximal, characteristic, complemented
3#13077133C211yesmaximal, characteristic
4C2×#10622745C211yesmaximal, characteristic
5C22×#66900C211yesmaximal, characteristic, complemented
6#13069883C214yesmaximal, complemented
7#13069876C214yesmaximal, complemented
8#13068342C211yesmaximal, characteristic, complemented
9C2×C11×#419C211yesmaximal, characteristic, complemented
10C23×Q8×C11C2211yescharacteristic
11C22×#4339C2211yescharacteristic
12C22×C11×M16C2211yescharacteristic
13C2×C11×#117C2211yescharacteristic
14C22×#4326C2212yes—
15C2×#66995—24no—
16C2×#66995—24no—
17C2×#66995—22no—
18C2×#66995—22no—
19C22×#4325C2212yes—
20C22×#4318C2211yescharacteristic
21C22×#4318C2211yescharacteristic
22C22×#4318C2211yescharacteristic
23C11×#419—22no—
24C11×#419C2214yes—
25C2×C11×#103—22no—
26C2×#66900—24no—
27C2×#66900C2214yes—
28C2×#66888C2212yes—
29#10622745—24no—
30C2×#66887C2212yes—
31#10622617C2212yes—
32#10622615—22no—
33#10622614C2212yes—
34#10622612—22no—
35#10622612C2214yes—
36#10622579—22no—
37#10622537—22no—
38#10622196C2214yes—
39C22×Q8×C11—22no—
40C22×Q8×C11—21no—
41C22×Q8×C11—21no—
42C22×Q8×C11—21no—
43C22×Q8×C11—22no—
44C22×Q8×C11—21no—
45C22×Q8×C11—21no—
46C22×Q8×C11—22no—
47C22×Q8×C11D414yes—
48C22×Q8×C11C2312yes—
49C23×C4×C11—21no—
50C23×C4×C11C2311yescharacteristic
51C22×C22⋊D11—21no—
52C2×#4339—22no—
53C2×#4339—22no—
54C2×#4339—21no—
55C2×#4339—21no—
56C2×#4339—21no—
57C2×#4339—21no—
58C2×#4339—22no—
59C2×#4339D414yes—
60C22×D44C2311yescharacteristic
61C22×C4×D11—21no—
62C22×Dic22C2311yescharacteristic
63C2×C11×M16—21no—
64C2×C11×M16—22no—
65C2×C11×M16—22no—
66C2×C11×M16C2312yes—
67C22×C8×C11—21no—
68C11×#117—21no—
69C11×#117—21no—
70C11×#117D412yes—
71C11×#117C2312yes—
72C2×C11×C4⋊C4D412yes—
73C2×C11×C22⋊C4—21no—
74C2×C42×C11—21no—
75C2×#4326—24no—
76C2×#4326—24no—
77C2×#4326—24no—
78#66995—216no—
79#66995—216no—
80C2×#4325—24no—
81C2×#4325—24no—
82C2×#4325—24no—
83C2×#4318—22no—
84C2×#4318—22no—
85C2×#4318—21no—
86C2×#4318—21no—
87C2×#4318—22no—
88C2×#4318—22no—
89C2×#4318—22no—
90C2×#4318—22no—
91C2×#4318D412yes—
92C2×#4318D412yes—
93C2×#4318C2312yes—
94C2×#4318C2312yes—
95C22×C11⋊C8—21no—
96C22×C11⋊C8—21no—
97C22×C11⋊C8C2312yes—
98C11×#103—28no—
99#66900—28no—
100#66888—24no—
101#66887—24no—
102#66874—24no—
103#66873—24no—
104C4⋊(C11⋊C8)—28no—
105#23522—111nomaximal, Sylow
106C24×C11#4111yescharacteristic
107C23×D11—21no—
108C2×Q8×C11—24no—
109C2×Q8×C11—22no—
110C2×Q8×C11—24no—
111C2×Q8×C11—24no—
112C2×Q8×C11—24no—
113C2×Q8×C11—24no—
114C2×Q8×C11—22no—
115C2×Q8×C11—22no—
116C2×Q8×C11—22no—
117C2×Q8×C11—22no—
118C2×Q8×C11—41no—
119C2×Q8×C11—41no—
120C2×Q8×C11—42no—
121C2×Q8×C11—42no—
122C2×Q8×C11—42no—
123C2×Q8×C11—42no—
124C2×Q8×C11—42no—
125C2×Q8×C11C2×D414yes—
126C22×C4×C11—24no—
127C22×C4×C11—22no—
128C22×C4×C11—22no—
129C22×C4×C11—41no—
130C22×C4×C11—41no—
131C22×C4×C11—42no—
132C22×C4×C11C2×D412yes—
133C22×C4×C11C2×D412yes—
134C22×C4×C11C2×D411yescharacteristic
135C22×C4×C11C2×D411yescharacteristic
136C22×C4×C11C2×D411yescharacteristic
137C22×C4×C11C2×D411yescharacteristic
138C22×C4×C11#4111yescharacteristic
139C22×C4×C11#4112yes—
140C22×C4×C11#4112yes—
141C22×C4×C11C2411yescharacteristic
142C2×C22⋊D11—24no—
143C2×C22⋊D11—41no—
144C2×C22⋊D11—41no—
145C2×C22⋊D11—42no—
146C22×Dic11—21no—
147#4339—24no—
148#4339—24no—
149#4339—24no—
150#4339—24no—
151#4339—42no—
152#4339—42no—
153#4339—42no—
154#4339—42no—
155C2×D44—21no—
156C2×D44—21no—
157C2×D44—21no—
158C2×D44—21no—
159C2×D44—21no—
160C2×D44C2×D412yes—
161C2×C4×D11—24no—
162C2×C4×D11—41no—
163C2×C4×D11—41no—
164C2×C4×D11—42no—
165C2×Dic22—21no—
166C2×Dic22—21no—
167C2×Dic22—21no—
168C2×Dic22—21no—
169C2×Dic22—21no—
170C2×Dic22C2×D412yes—
171C11×M16—22no—
172C11×M16—22no—
173C11×M16—42no—
174C2×C8×C11—22no—
175C2×C8×C11—42no—
176C11×C4⋊C4—22no—
177C11×C4⋊C4C2×D414yes—
178C11×C22⋊C4—22no—
179C11×C22⋊C4—22no—
180C42×C11—22no—
181C42×C11—22no—
182#4326—216no—
183#4325—216no—
184#4318—24no—
185#4318—24no—
186#4318—22no—
187#4318—22no—
188#4318—22no—
189#4318—22no—
190#4318—42no—
191#4318—42no—
192C2×C11⋊C8—22no—
193C2×C11⋊C8—22no—
194C2×C11⋊C8—22no—
195C2×C11⋊C8—22no—
196C2×C11⋊C8—42no—
197C2×C11⋊C8—42no—
198C2×C11⋊C8C2×D414yes—
199C22×#137—111no—
200#2905—111no—
201#2896—111no—
202C2×#419D1111yescharacteristic, complemented
203C2×#413—111no—
204C22×#101—111no—
205#2047—114no—
206#2046—114no—
207#1738—111no—
208C23×C11—42no—
209C23×C11#12111yescharacteristic
210C23×C11#12111yescharacteristic
211C23×C11#12312yes—
212C23×C11#12312yes—
213C23×C11C2×#4111yescharacteristic
214C22×D11—22no—
215C22×D11—41no—
216C22×D11—41no—
217C22×D11—82no—
218Q8×C11—24no—
219Q8×C11—24no—
220Q8×C11—42no—
221Q8×C11—42no—
222Q8×C11—44no—
223Q8×C11—44no—
224C2×C4×C11—24no—
225C2×C4×C11—24no—
226C2×C4×C11—42no—
227C2×C4×C11—42no—
228C2×C4×C11—44no—
229C2×C4×C11—44no—
230C2×C4×C11—82no—
231C2×C4×C11#12012yes—
232C2×C4×C11#12012yes—
233C2×C4×C11#12112yes—
234C2×C4×C11#12112yes—
235C2×C4×C11#12112yes—
236C2×C4×C11#12112yes—
237C2×C4×C11#12111yescharacteristic
238C2×C4×C11#12111yescharacteristic
239C2×C4×C11#12312yes—
240C2×C4×C11#12312yes—
241C2×C4×C11#12412yes—
242C2×C4×C11#12412yes—
243C2×C4×C11C22×D411yescharacteristic
244C2×C4×C11C22×D411yescharacteristic
245C2×C4×C11C22×D411yescharacteristic
246C2×C4×C11C22×D411yescharacteristic
247C2×C4×C11C2×#4111yescharacteristic
248C2×C4×C11C2×#4111yescharacteristic
249C22⋊D11—44no—
250C22⋊D11—44no—
251C2×Dic11—22no—
252C2×Dic11—41no—
253C2×Dic11—41no—
254D44—22no—
255D44—22no—
256D44—41no—
257D44—41no—
258C4×D11—44no—
259C4×D11—44no—
260Dic22—22no—
261Dic22—22no—
262Dic22—41no—
263Dic22—41no—
264C8×C11—42no—
265C11⋊C8—24no—
266C11⋊C8—42no—
267C11⋊C8—42no—
268C22×#41—111no—
269C23×Q8C2×D1111yescharacteristic
270C2×#137—222no—
271C2×#137—222no—
272C2×#137—224no—
273C2×#137—224no—
274C22×Q16—112no—
275C22×SD16—112no—
276C22×M16—111no—
277C22×M16—111no—
278C22×M16—111no—
279C22×M16C2×D1111yescharacteristic
280C2×#117C2×D1111yescharacteristic
281#424—114no—
282#424—222no—
283#423—222no—
284#419—22no—
285#419—112no—
286#419C2×D1114yes—
287#418—112no—
288#417—222no—
289#416—222no—
290C2×#103—22no—
291#413—224no—
292C2×#101—114no—
293C2×#101—112no—
294C2×#101—224no—
295C2×#100—112no—
296#341—114no—
297C22×C11—22no—
298C22×C11—22no—
299C22×C11—22no—
300C22×C11—42no—
301C22×C11—42no—
302C22×C11#39412yes—
303C22×C11#39412yes—
304C22×C11C2×#12111yescharacteristic
305C22×C11C2×#12311yescharacteristic
306C22×C11C2×#12311yescharacteristic
307C2×D11—42no—
308C2×D11—82no—
309C2×D11—82no—
310C2×D11—82no—
311C4×C11—44no—
312C4×C11—44no—
313C4×C11#39412yes—
314C4×C11#39412yes—
315C4×C11C2×#12011yescharacteristic
316C4×C11C2×#12111yescharacteristic
317C4×C11C2×#12111yescharacteristic
318C4×C11C2×#12411yescharacteristic
319Dic11—42no—
320C2×#41—114no—
321C2×#41—222no—
322C2×#41—222no—
323C2×#41—221no—
324C2×#41—221no—
325C2×#41—222no—
326C2×#41—221no—
327C2×#41—221no—
328C22×Q8—22no—
329C22×Q8—21no—
330C22×Q8—22no—
331C22×Q8—22no—
332C22×Q8—21no—
333C22×Q8—21no—
334C22×Q8—21no—
335C22×Q8—21no—
336C22×Q8—111no—
337C22×Q8D4414yes—
338C22×Q8C22×D1112yes—
339C22×D4—111no—
340C22×D4—221no—
341C23×C4—21no—
342C23×C4—221no—
343C23×C4C22×D1111yescharacteristic
344#137—2216no—
345#137—2216no—
346C2×Q16—224no—
347C2×Q16—224no—
348C2×Q16—224no—
349C2×SD16—224no—
350C2×SD16—224no—
351C2×SD16—224no—
352C2×M16—21no—
353C2×M16—22no—
354C2×M16—22no—
355C2×M16—112no—
356C2×M16—112no—
357C2×M16—112no—
358C2×M16—112no—
359C2×M16—222no—
360C2×M16—222no—
361C2×M16—221no—
362C2×M16—221no—
363C2×M16—222no—
364C2×M16—222no—
365C2×M16—222no—
366C2×M16—222no—
367C2×M16C22×D1112yes—
368C22×C8—21no—
369C22×C8—112no—
370C22×C8—221no—
371C22×C8—221no—
372#117—21no—
373#117—21no—
374#117C22⋊D1112yes—
375#117C22×D1112yes—
376C2×C4⋊C4C22⋊D1112yes—
377C2×C22⋊C4—21no—
378C2×C42—21no—
379C4⋊C8—228no—
380#103—28no—
381#103—224no—
382#102—224no—
383#101—224no—
384#101—228no—
385#100—224no—
386C2×C11—21no—
387C2×C11—21no—
388C2×C11—22no—
389C2×C11—22no—
390C2×C11C2×#39411yescharacteristic
391C2×C11#306912yes—
392D11—82no—
393C24—221no—
394C24#434111yescharacteristic
395#41—224no—
396#41—224no—
397#41—224no—
398#41—224no—
399#41—442no—
400#41—442no—
401#41—442no—
402#41—442no—
403C2×Q8—24no—
404C2×Q8—22no—
405C2×Q8—24no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—42no—
414C2×Q8—42no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×Q8—42no—
418C2×Q8—41no—
419C2×Q8—41no—
420C2×Q8—112no—
421C2×Q8—221no—
422C2×Q8—221no—
423C2×Q8—221no—
424C2×Q8—221no—
425C2×Q8—221no—
426C2×Q8C2×D4414yes—
427C2×D4—112no—
428C2×D4—224no—
429C2×D4—221no—
430C2×D4—221no—
431C2×D4—221no—
432C2×D4—221no—
433C2×D4—221no—
434C2×D4—442no—
435C2×D4—441no—
436C2×D4—441no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—42no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—221no—
444C22×C4—224no—
445C22×C4—442no—
446C22×C4—441no—
447C22×C4—441no—
448C22×C4C2×D4412yes—
449C22×C4#433912yes—
450C22×C4D4×D1112yes—
451C22×C4D4×D1111yescharacteristic
452C22×C4D4×D1111yescharacteristic
453C22×C4#434111yescharacteristic
454C22×C4#434112yes—
455C22×C4C2×C22⋊D1111yescharacteristic
456C22×C4C2×C22⋊D1111yescharacteristic
457C22×C4C23×D1111yescharacteristic
458Q16—2216no—
459SD16—2216no—
460M16—22no—
461M16—22no—
462M16—42no—
463M16—224no—
464M16—224no—
465M16—222no—
466M16—222no—
467M16—222no—
468M16—222no—
469M16—442no—
470M16—442no—
471C2×C8—22no—
472C2×C8—42no—
473C2×C8—114no—
474C2×C8—222no—
475C2×C8—222no—
476C2×C8—222no—
477C2×C8—222no—
478C2×C8—442no—
479C2×C8—442no—
480C4⋊C4—22no—
481C4⋊C4C2×C22⋊D1114yes—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C42—22no—
485C42—22no—
486C11#2352211yescharacteristic, complemented, Sylow
487C23—42no—
488C23—222no—
489C23—441no—
490C23—441no—
491C23—882no—
492C23#6693612yes—
493C23#6693912yes—
494C23#6699211yescharacteristic
495C23#6699211yescharacteristic
496C23C2×#434111yescharacteristic
497Q8—24no—
498Q8—24no—
499Q8—42no—
500Q8—42no—
501Q8—44no—
502Q8—44no—
503Q8—222no—
504Q8—222no—
505Q8—441no—
506Q8—441no—
507D4—222no—
508D4—222no—
509D4—444no—
510D4—444no—
511D4—441no—
512D4—441no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—82no—
520C2×C4—222no—
521C2×C4—441no—
522C2×C4—441no—
523C2×C4—444no—
524C2×C4—444no—
525C2×C4#6693512yes—
526C2×C4#6693512yes—
527C2×C4#6693612yes—
528C2×C4#6693812yes—
529C2×C4#6693812yes—
530C2×C4#6693912yes—
531C2×C4#6698312yes—
532C2×C4#6698312yes—
533C2×C4#6699112yes—
534C2×C4#6699112yes—
535C2×C4#6699211yescharacteristic
536C2×C4#6699211yescharacteristic
537C2×C4C22×D4411yescharacteristic
538C2×C4C2×#433911yescharacteristic
539C2×C4C2×D4×D1111yescharacteristic
540C2×C4C2×D4×D1111yescharacteristic
541C2×C4C2×#434111yescharacteristic
542C2×C4C22×C22⋊D1111yescharacteristic
543C8—42no—
544C8—224no—
545C8—442no—
546C8—442no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—42no—
551C22—42no—
552C22—442no—
553C22—882no—
554C22—882no—
555C22—882no—
556C22#1062257412yes—
557C22#1062257412yes—
558C22C2×#6693611yescharacteristic
559C22C2×#6693911yescharacteristic
560C22C2×#6699211yescharacteristic
561C4—44no—
562C4—44no—
563C4—442no—
564C4#1062257412yes—
565C4#1062257412yes—
566C4C2×#6693511yescharacteristic
567C4C2×#6693811yescharacteristic
568C4C2×#6698311yescharacteristic
569C4C2×#6699111yescharacteristic
570C2—21no—
571C2—21no—
572C2—22no—
573C2—22no—
574C2—882no—
575C2C2×#1062257411yescharacteristic
576C2#1307835012yes—
subgroupG/H
centerC22C2×#66992
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC2×C11×#419C2
socleC22×C11C2×#121
prime radicalC22×#4339C22
hypercenterC2×#419D11
hypocenterC11#23522
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
2. #13077180C2semidirect
5. C22×#66900C2semidirect
6. #13069883C2semidirect
7. #13069876C2semidirect
8. #13068342C2semidirect
9. C2×C11×#419C2semidirect
202. C2×#419D11semidirect
486. C11#23522semidirect

Maximal subgroups

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

subgroupindex
C22×#669952
#130771802
#130771332
C2×#106227452
C22×#669002
#130698832
#130698762
#130683422
C2×C11×#4192
#2352211