Finite Group Databasebeta

← #14757388

Subgroups — #14757388

Order 2816.

1718 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622612C211yesmaximal, characteristic, complemented
2C2×#10622847C211yesmaximal, characteristic, complemented
3C2×#10622847C211yesmaximal, characteristic, complemented
4C2×#10622820C211yesmaximal, characteristic, complemented
5C22×C22⋊(C11⋊C8)C211yesmaximal, characteristic, complemented
6C2×#10622610C211yesmaximal, characteristic, complemented
7#13069415C214yesmaximal, complemented
8#13069411C214yesmaximal, complemented
9C2×C11×#525C211yesmaximal, characteristic
10C22×C11×#41C2211yescharacteristic
11C24×C4×C11C2211yescharacteristic
12C11×#525—24no—
13C11×#525—24no—
14C2×C11×#123C412yes—
15C2×C11×C22⋊Q8C412yes—
16C2×C11×#121C412yes—
17C2×C11×#120C412yes—
18C2×C4×D4×C11C2214yes—
19C2×C11×#117C2211yescharacteristic
20C22×#4318C2211yescharacteristic
21C22×#4318C2211yescharacteristic
22C23×C11⋊C8C2211yescharacteristic
23C23×C11⋊C8C2211yescharacteristic
24#10622847—24no—
25#10622847—24no—
26#10622820—24no—
27C2×C22⋊(C11⋊C8)—24no—
28C2×C22⋊(C11⋊C8)C2212yes—
29C2×C22⋊(C11⋊C8)C2212yes—
30C2×C22⋊(C11⋊C8)C2214yes—
31C2×C22⋊(C11⋊C8)C2212yescomplemented
32C2×C22⋊(C11⋊C8)C2214yescomplemented
33C2×C22⋊(C11⋊C8)C2212yescomplemented
34#10622612—24no—
35#10622610—24no—
36C2×C4⋊(C11⋊C8)C2214yescomplemented
37C2×#66867C2212yescomplemented
38C2×C4×C11⋊C8C2212yescomplemented
39C25×C11C2×C411yescharacteristic
40C2×C11×#41—22no—
41C2×C11×#41—22no—
42C2×C11×#41—22no—
43C2×C11×#41—22no—
44C2×C11×#41—22no—
45C2×C11×#41—22no—
46C22×Q8×C11C2×C411yescharacteristic
47C22×D4×C11C2×C412yes—
48C22×D4×C11C2×C411yescharacteristic
49C22×D4×C11C2×C414yes—
50C23×C4×C11—22no—
51C23×C4×C11—22no—
52C23×C4×C11—22no—
53C23×C4×C11—22no—
54C23×C4×C11—22no—
55C23×C4×C11—22no—
56C23×C4×C11C2×C412yes—
57C23×C4×C11C2×C411yescharacteristic
58C23×C4×C11C2312yes—
59C23×C4×C11C2312yes—
60C23×C4×C11C2311yescharacteristic
61C11×#123—28no—
62C11×C22⋊Q8—28no—
63C11×#121—28no—
64C11×#120—28no—
65C4×D4×C11—28no—
66C4×D4×C11—28no—
67C11×#117D414yes—
68C11×#117D414yes—
69C2×C11×C4⋊C4C2×C414yes—
70C2×C11×C4⋊C4C2×C412yes—
71C2×C11×C22⋊C4C2×C414yes—
72C2×C11×C22⋊C4C2×C414yes—
73C2×C11×C22⋊C4C2×C412yes—
74C2×C42×C11C2312yes—
75C2×#4318—22no—
76C2×#4318—22no—
77C2×#4318—22no—
78C2×#4318—22no—
79C2×#4318—22no—
80C2×#4318D414yescomplemented
81C22×C11⋊C8—22no—
82C22×C11⋊C8—22no—
83C22×C11⋊C8—22no—
84C22×C11⋊C8—22no—
85C22×C11⋊C8—22no—
86C22×C11⋊C8D414yescomplemented
87C22×C11⋊C8C2312yes—
88C22×C11⋊C8C2312yes—
89C22×C11⋊C8C2311yescharacteristic
90C22×C11⋊C8C2312yes—
91C22×C11⋊C8C2311yescharacteristic
92C22⋊(C11⋊C8)—24no—
93C22⋊(C11⋊C8)—24no—
94C22⋊(C11⋊C8)—28no—
95C22⋊(C11⋊C8)—28no—
96C22⋊(C11⋊C8)—24no—
97C22⋊(C11⋊C8)—24no—
98C4⋊(C11⋊C8)—28no—
99#66867—24no—
100C4×C11⋊C8—24no—
101#21191—111nomaximal, Sylow
102C24×C11—22no—
103C24×C11—24no—
104C24×C11—22no—
105C24×C11—22no—
106C24×C11—42no—
107C24×C11C22×C412yes—
108C24×C11C22×C412yes—
109C24×C11C22×C411yescharacteristic
110C11×#41—44no—
111C11×#41—44no—
112C11×#41—44no—
113C11×#41—44no—
114C2×Q8×C11—22no—
115C2×Q8×C11—22no—
116C2×Q8×C11—22no—
117C2×D4×C11—24no—
118C2×D4×C11—24no—
119C2×D4×C11—24no—
120C2×D4×C11—28no—
121C2×D4×C11—28no—
122C2×D4×C11—22no—
123C2×D4×C11—22no—
124C2×D4×C11—22no—
125C2×D4×C11—44no—
126C22×C4×C11—22no—
127C22×C4×C11—22no—
128C22×C4×C11—22no—
129C22×C4×C11—24no—
130C22×C4×C11—24no—
131C22×C4×C11—24no—
132C22×C4×C11—22no—
133C22×C4×C11—22no—
134C22×C4×C11—22no—
135C22×C4×C11—22no—
136C22×C4×C11—22no—
137C22×C4×C11—22no—
138C22×C4×C11—22no—
139C22×C4×C11—22no—
140C22×C4×C11—22no—
141C22×C4×C11—22no—
142C22×C4×C11—22no—
143C22×C4×C11—22no—
144C22×C4×C11—44no—
145C22×C4×C11—44no—
146C22×C4×C11—44no—
147C22×C4×C11—44no—
148C22×C4×C11C22×C412yes—
149C22×C4×C11C22×C412yes—
150C22×C4×C11C22×C414yes—
151C22×C4×C11C22×C411yescharacteristic
152C22×C4×C11C2×D412yes—
153C22×C4×C11C2×D412yes—
154C22×C4×C11C2×D412yes—
155C22×C4×C11C2×D412yes—
156C22×C4×C11#4112yes—
157C22×C4×C11#4112yes—
158C22×C4×C11C2411yescharacteristic
159C11×C4⋊C4—28no—
160C11×C4⋊C4C22⋊C418yes—
161C11×C22⋊C4—28no—
162C11×C22⋊C4—28no—
163C11×C22⋊C4C22⋊C418yes—
164C42×C11C2×D414yes—
165C42×C11C2×D414yes—
166#4318—28no—
167#4318—44no—
168C2×C11⋊C8—28no—
169C2×C11⋊C8—21no—
170C2×C11⋊C8—22no—
171C2×C11⋊C8—21no—
172C2×C11⋊C8—22no—
173C2×C11⋊C8—22no—
174C2×C11⋊C8—22no—
175C2×C11⋊C8—22no—
176C2×C11⋊C8—22no—
177C2×C11⋊C8—22no—
178C2×C11⋊C8—21no—
179C2×C11⋊C8—21no—
180C2×C11⋊C8—22no—
181C2×C11⋊C8—44no—
182C2×C11⋊C8C2×D412yes—
183C2×C11⋊C8C2×D412yes—
184C2×C11⋊C8C2×D414yes—
185C2×#525D1111yescharacteristic
186C2×#424—111no—
187C2×#408—111no—
188C2×#408—111no—
189C2×#407—111no—
190C22×C22⋊C8—111no—
191C2×#405—111no—
192#1955—114no—
193#1954—114no—
194C23×C11—22no—
195C23×C11—22no—
196C23×C11—28no—
197C23×C11—22no—
198C23×C11—22no—
199C23×C11—22no—
200C23×C11—22no—
201C23×C11—22no—
202C23×C11—22no—
203C23×C11—22no—
204C23×C11—22no—
205C23×C11—22no—
206C23×C11—24no—
207C23×C11—24no—
208C23×C11—48no—
209C23×C11—48no—
210C23×C11—44no—
211C23×C11—44no—
212C23×C11C2×C22⋊C412yes—
213C23×C11C4×D412yes—
214C23×C11C4×D412yes—
215C23×C11#13114yes—
216C23×C11C23×C411yescharacteristic
217Q8×C11—44no—
218D4×C11—416no—
219D4×C11—48no—
220D4×C11—44no—
221C2×C4×C11—24no—
222C2×C4×C11—24no—
223C2×C4×C11—22no—
224C2×C4×C11—22no—
225C2×C4×C11—22no—
226C2×C4×C11—22no—
227C2×C4×C11—22no—
228C2×C4×C11—22no—
229C2×C4×C11—44no—
230C2×C4×C11—42no—
231C2×C4×C11—42no—
232C2×C4×C11—42no—
233C2×C4×C11—42no—
234C2×C4×C11—42no—
235C2×C4×C11—42no—
236C2×C4×C11—42no—
237C2×C4×C11—42no—
238C2×C4×C11—48no—
239C2×C4×C11C2×C22⋊C418yes—
240C2×C4×C11C2×C22⋊C412yes—
241C2×C4×C11C4×D412yes—
242C2×C4×C11C4×D412yes—
243C2×C4×C11#12114yes—
244C2×C4×C11#12114yes—
245C2×C4×C11#12414yes—
246C2×C4×C11#12714yes—
247C2×C4×C11C22×D411yescharacteristic
248C2×C4×C11C22×D411yescharacteristic
249C2×C4×C11C22×D411yescharacteristic
250C2×C4×C11C22×D411yescharacteristic
251C2×C4×C11C2×#4111yescharacteristic
252C2×C4×C11C2×#4111yescharacteristic
253C11⋊C8—22no—
254C11⋊C8—22no—
255C11⋊C8—24no—
256C11⋊C8—42no—
257C11⋊C8—42no—
258C22×#41C2×D1111yescharacteristic
259C24×C4C2×D1111yescharacteristic
260C22×M16—111no—
261C22×M16—111no—
262C23×C8—111no—
263C23×C8—111no—
264#525—24no—
265#525—24no—
266C2×#123Dic1112yes—
267C2×C22⋊Q8Dic1112yes—
268C2×#121Dic1112yes—
269C2×#120Dic1112yes—
270C2×C4×D4C2×D1114yes—
271C2×#117C2×D1111yescharacteristic
272#424—224no—
273C2×C4⋊C8—114no—
274#408—224no—
275#408—224no—
276#407—224no—
277C2×C22⋊C8—112no—
278C2×C22⋊C8—112no—
279C2×C22⋊C8—114no—
280C2×C22⋊C8—112no—
281C2×C22⋊C8—114no—
282C2×C22⋊C8—112no—
283C2×C22⋊C8—224no—
284#405—224no—
285C2×#97—112no—
286C2×C4×C8—112no—
287C22×C11—24no—
288C22×C11—22no—
289C22×C11—22no—
290C22×C11—24no—
291C22×C11—24no—
292C22×C11—24no—
293C22×C11—24no—
294C22×C11—416no—
295C22×C11—42no—
296C22×C11—42no—
297C22×C11—42no—
298C22×C11—42no—
299C22×C11—44no—
300C22×C11—42no—
301C22×C11—42no—
302C22×C11—44no—
303C22×C11—42no—
304C22×C11#39014yes—
305C22×C11C22×C22⋊C411yescharacteristic
306C22×C11C2×C4×D411yescharacteristic
307C22×C11C2×C4×D411yescharacteristic
308C22×C11C2×#13112yes—
309C22×C11#56812yes—
310C4×C11—28no—
311C4×C11—42no—
312C4×C11—42no—
313C4×C11#39014yes—
314C4×C11C2×#12111yescharacteristic
315C4×C11C2×#12111yescharacteristic
316C4×C11C2×#12411yescharacteristic
317C4×C11C2×#12711yescharacteristic
318C25C2×Dic1111yescharacteristic
319C2×#41—22no—
320C2×#41—22no—
321C2×#41—22no—
322C2×#41—22no—
323C2×#41—22no—
324C2×#41—22no—
325C22×Q8C2×Dic1111yescharacteristic
326C22×D4C2×Dic1112yes—
327C22×D4C2×Dic1111yescharacteristic
328C22×D4C2×Dic1114yes—
329C23×C4—22no—
330C23×C4—22no—
331C23×C4—22no—
332C23×C4—22no—
333C23×C4—22no—
334C23×C4—22no—
335C23×C4C2×Dic1112yes—
336C23×C4C2×Dic1111yescharacteristic
337C23×C4C22×D1112yes—
338C23×C4C22×D1112yes—
339C23×C4C22×D1111yescharacteristic
340C2×M16—114no—
341C2×M16—222no—
342C2×M16—222no—
343C2×M16—222no—
344C2×M16—222no—
345C2×M16—222no—
346C22×C8—114no—
347C22×C8—111no—
348C22×C8—111no—
349C22×C8—112no—
350C22×C8—112no—
351C22×C8—112no—
352C22×C8—222no—
353C22×C8—222no—
354C22×C8—222no—
355C22×C8—222no—
356C22×C8—222no—
357#123—28no—
358C22⋊Q8—28no—
359#121—28no—
360#120—28no—
361C4×D4—28no—
362C4×D4—28no—
363#117C22⋊D1114yes—
364#117C22⋊D1114yes—
365C2×C4⋊C4C2×Dic1114yes—
366C2×C4⋊C4C2×Dic1112yes—
367C2×C22⋊C4C2×Dic1114yes—
368C2×C22⋊C4C2×Dic1114yes—
369C2×C22⋊C4C2×Dic1112yes—
370C2×C42C22×D1112yes—
371C4⋊C8—228no—
372C22⋊C8—224no—
373C22⋊C8—224no—
374C22⋊C8—228no—
375C22⋊C8—224no—
376C22⋊C8—224no—
377C22⋊C8—228no—
378#97—224no—
379C4×C8—224no—
380C2×C11—24no—
381C2×C11—44no—
382C2×C11—82no—
383C2×C11C2×#39011yescharacteristic
384C2×C11#288012yes—
385C2×C11#292911yescharacteristic
386C2×C11#293011yescharacteristic
387C2×C11#293312yes—
388C24—22no—
389C24—24no—
390C24—22no—
391C24—22no—
392C24—42no—
393C24C22×Dic1112yes—
394C24C22×Dic1112yes—
395C24C22×Dic1111yescharacteristic
396#41—44no—
397#41—44no—
398#41—44no—
399#41—44no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×D4—24no—
404C2×D4—24no—
405C2×D4—28no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—28no—
410C2×D4—22no—
411C2×D4—44no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4D4×D1112yes—
435C22×C4D4×D1112yes—
436C22×C4#434112yes—
437C22×C4#434112yes—
438C22×C4C22×Dic1112yes—
439C22×C4C22×Dic1112yes—
440C22×C4C22×Dic1114yes—
441C22×C4C22×Dic1111yescharacteristic
442C22×C4C2×C22⋊D1112yes—
443C22×C4C2×C22⋊D1112yes—
444C22×C4C23×D1111yescharacteristic
445M16—228no—
446M16—444no—
447C2×C8—112no—
448C2×C8—112no—
449C2×C8—114no—
450C2×C8—228no—
451C2×C8—221no—
452C2×C8—221no—
453C2×C8—222no—
454C2×C8—221no—
455C2×C8—222no—
456C2×C8—221no—
457C2×C8—222no—
458C2×C8—222no—
459C2×C8—222no—
460C2×C8—222no—
461C2×C8—222no—
462C2×C8—222no—
463C2×C8—444no—
464C4⋊C4—28no—
465C4⋊C4C22⋊Dic1118yes—
466C22⋊C4—28no—
467C22⋊C4—28no—
468C22⋊C4C22⋊Dic1118yes—
469C42C2×C22⋊D1114yes—
470C42C2×C22⋊D1114yes—
471C11#2119111yescharacteristic, complemented, Sylow
472C23—22no—
473C23—22no—
474C23—28no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—24no—
484C23—24no—
485C23—22no—
486C23—48no—
487C23—48no—
488C23—44no—
489C23—44no—
490C23D4×Dic1112yes—
491C23D4×Dic1112yes—
492C23#6700114yes—
493C23C2×C22⋊Dic1112yes—
494C23C23×Dic1111yescharacteristic
495Q8—44no—
496D4—416no—
497D4—48no—
498D4—44no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—48no—
517C2×C4D4×Dic1112yes—
518C2×C4D4×Dic1112yes—
519C2×C4#6698914yes—
520C2×C4#6699114yes—
521C2×C4#6699114yes—
522C2×C4#6699314yes—
523C2×C4C2×C22⋊Dic1118yes—
524C2×C4C2×C22⋊Dic1112yes—
525C2×C4C2×D4×D1111yescharacteristic
526C2×C4C2×D4×D1111yescharacteristic
527C2×C4C2×#434111yescharacteristic
528C2×C4C2×#434111yescharacteristic
529C2×C4C22×C22⋊D1111yescharacteristic
530C2×C4C22×C22⋊D1111yescharacteristic
531C8—222no—
532C8—222no—
533C8—224no—
534C8—442no—
535C8—442no—
536C22—22no—
537C22—22no—
538C22—24no—
539C22—24no—
540C22—24no—
541C22—24no—
542C22—24no—
543C22—416no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—44no—
551C22—44no—
552C22—42no—
553C22#1062283414yes—
554C22C2×D4×Dic1111yescharacteristic
555C22C2×D4×Dic1111yescharacteristic
556C22C2×#6700112yes—
557C22#1062335012yes—
558C22C22×C22⋊Dic1111yescharacteristic
559C4—28no—
560C4—42no—
561C4—42no—
562C4#1062283414yes—
563C4C2×#6698911yescharacteristic
564C4C2×#6699111yescharacteristic
565C4C2×#6699111yescharacteristic
566C4C2×#6699311yescharacteristic
567C2—24no—
568C2—44no—
569C2—82no—
570C2C2×#1062283411yescharacteristic
571C2#1307706512yes—
572C2#1307731111yescharacteristic
573C2#1307732011yescharacteristic
574C2#1307734312yes—
subgroupG/H
center, FrattiniC22×C4C23×D11
commutator, socleC23×C11C23×C4
Fitting, prime radicalC2×C11×#525C2
cosocleC22×C4×C11C24
hypercenterC2×#525D11
hypocenterC11#21191
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C22×C4 ◁ C2×#525

Complements

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

NH ≅ G/Ntype
1. C2×#10622612C2semidirect
2. C2×#10622847C2semidirect
3. C2×#10622847C2semidirect
4. C2×#10622820C2semidirect
5. C22×C22⋊(C11⋊C8)C2semidirect
6. C2×#10622610C2semidirect
7. #13069415C2semidirect
8. #13069411C2semidirect
31. C2×C22⋊(C11⋊C8)C22semidirect
32. C2×C22⋊(C11⋊C8)C22semidirect
33. C2×C22⋊(C11⋊C8)C22semidirect
36. C2×C4⋊(C11⋊C8)C22semidirect
37. C2×#66867C22semidirect
38. C2×C4×C11⋊C8C22semidirect
80. C2×#4318D4semidirect
86. C22×C11⋊C8D4semidirect
471. C11#21191semidirect

Maximal subgroups

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

subgroupindex
C2×#106226122
C2×#106228472
C2×#106228202
C22×C22⋊(C11⋊C8)2
C2×#106226102
#130694152
#130694112
C2×C11×#5252
#2119111