Finite Group Databasebeta

← #36716052

Subgroups — #36716052

Order 22784.

2912 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C22⋊D89C211yesmaximal, characteristic
2C2×#19152074C211yesmaximal, characteristic, complemented
3#26023306C213yesmaximal, complemented
4#26023275C213yesmaximal, complemented
5#26023257C213yesmaximal, complemented
6#26023177C213yesmaximal
7#26023170C211yesmaximal, characteristic, complemented
8Q8×C22⋊Dic89C211yesmaximal, characteristic, complemented
9Q8×#13083551C211yesmaximal, characteristic, complemented
10#26022968C213yesmaximal, complemented
11#26022928C213yesmaximal, complemented
12C4×#15667991C213yesmaximal, complemented
13#26022695C213yesmaximal
14#26022617C211yesmaximal, characteristic, complemented
15C22×C4×Q8×C89C211yesmaximal, characteristic, complemented
16C22×Q8×D89C2211yescharacteristic
17Q8×C22⋊D89C4116yescomplemented
18C2×#15668010C2213yes—
19C2×#15667991C2213yes—
20C2×#15667988C2213yes—
21#19152074—28no—
22C2×Q8×Dic89C2211yescharacteristic
23C2×Q8×Dic89C2211yescharacteristic
24C2×Q8×Dic89C2211yescharacteristic
25C2×C4×Dic178C2213yes—
26C2×#15667977C2213yes—
27C2×C4×C22⋊D89C2213yes—
28C2×#15667970C2213yes—
29C2×#15667969C2213yes—
30#19151505C2213yes—
31#19151505C2213yes—
32#19151504C2213yes—
33#19151503C2213yes—
34#19151496C2213yes—
35#19151494C2213yes—
36#19151493C2216yes—
37#19151489C2213yes—
38#19151489C2213yes—
39#19151488C2216yes—
40#19151482C2216yes—
41#19151482C2213yes—
42#19151482C2213yes—
43#19151479C2216yes—
44#19151479C2216yes—
45#19151477C2213yes—
46#19151477C2211yescharacteristic
47#19151471C2213yes—
48#19151471C2213yes—
49#19151463C2213yes—
50#19151458C2213yes—
51#19151455C2213yes—
52C4⋊C4×Dic89C2213yes—
53C4⋊C4×Dic89C2213yes—
54C4×#13083553C2213yes—
55C4×#13083553C2216yes—
56C4×C4⋊Dic89C2213yes—
57C4×#13083551C2213yes—
58C4×C22⋊Dic89C2216yes—
59C4×C22⋊Dic89C2213yes—
60#19151444C2213yes—
61C23×Q8×C89C2211yescharacteristic
62C2×C4×Q8×C89—28no—
63C2×C4×Q8×C89C2211yescharacteristic
64C2×C4×Q8×C89C2213yes—
65C2×C4×Q8×C89C2213yes—
66C2×C4×Q8×C89C2211yescharacteristic
67C22×C4⋊C4×C89C2213yes—
68C22×C42×C89C2213yes—
69C22×Dic178C2313yes—
70C2×Q8×D89—24no—
71C2×Q8×D89—24no—
72C2×Q8×D89C2×C418yescomplemented
73C22×C22⋊D89C2311yescharacteristic
74C22×C4×D89C2313yes—
75#15668010C2×C4124yescomplemented
76#15667991C2×C4124yes—
77#15667988C2×C4124yescomplemented
78Q8×Dic89—24no—
79Q8×Dic89—24no—
80Q8×Dic89C2×C418yescomplemented
81C4×Dic178—212no—
82#15667977—212no—
83C4×C22⋊D89C2×C4124yes—
84#15667970—212no—
85#15667969—212no—
86C2×C4⋊Dic89C2313yes—
87C2×C4⋊Dic89C2313yes—
88C2×C4⋊Dic89C2313yes—
89C2×#13083553C2313yes—
90C2×#13083553C2313yes—
91C2×#13083553C2313yes—
92C2×#13083553C2313yes—
93C2×#13083553C2313yes—
94C2×#13083553C2316yes—
95C2×C22⋊Dic89C2313yes—
96C2×C22⋊Dic89C2311yescharacteristic
97C2×C22⋊Dic89C2313yes—
98C2×#13083551C2311yescharacteristic
99C2×#13083551C2313yes—
100C2×#13083551C2311yescharacteristic
101C2×#13083551C2313yes—
102C2×#13083551C2313yes—
103C2×C4×Dic89C2313yes—
104C2×C4×Dic89C2316yes—
105C2×C4×Dic89C2313yes—
106C2×C4×Dic89C2316yes—
107C2×C4×Dic89C2313yes—
108#15667919C2311yescharacteristic
109#15667919C2313yes—
110#15667919C2313yes—
111#15667919C2313yes—
112#15667919C2316yes—
113#15667919C2316yes—
114#15667919C2313yes—
115#15667919C2313yes—
116#15667919C2313yes—
117#15667919C2313yes—
118#15667919C2311yescharacteristic
119#15667919C2316yes—
120#15667919C2313yes—
121C22×Q8×C89—24no—
122C22×Q8×C89—24no—
123C22×Q8×C89C2×C418yes—
124C22×Q8×C89C2313yes—
125C22×Q8×C89C2311yescharacteristic
126C23×C4×C89C2311yescharacteristic
127C23×C4×C89C2313yes—
128C23×C4×C89C2313yes—
129C4×Q8×C89—24no—
130C4×Q8×C89—212no—
131C4×Q8×C89—212no—
132C4×Q8×C89D418yescomplemented
133C2×C4⋊C4×C89—212no—
134C2×C4⋊C4×C89C2313yes—
135C2×C4⋊C4×C89C2316yes—
136C2×C4⋊C4×C89C2313yes—
137C2×C42×C89—212no—
138C2×C42×C89C2313yes—
139C2×C42×C89C2313yes—
140C2×C42×C89C2313yes—
141C2×C42×C89C2313yes—
142C23×D89C2411yescharacteristic
143C2×Dic178—26no—
144C2×Dic178—26no—
145C2×Dic178C22×C4112yes—
146Q8×D89—232no—
147C2×C22⋊D89—22no—
148C2×C22⋊D89—22no—
149C2×C22⋊D89C22×C414yes—
150C22×Dic89C2411yescharacteristic
151C22×Dic89C2413yes—
152C22×Dic89C2413yes—
153C22×Dic89C2411yescharacteristic
154C22×Dic89C2413yes—
155C22×Dic89C2411yescharacteristic
156C22×Dic89C2413yes—
157C2×C4×D89—26no—
158C2×C4×D89—26no—
159C2×C4×D89C22×C4112yes—
160C4⋊Dic89—26no—
161C4⋊Dic89—26no—
162C4⋊Dic89C22×C4112yes—
163#13083553—26no—
164#13083553—26no—
165#13083553—26no—
166#13083553—43no—
167#13083553C22×C4112yes—
168#13083553C22×C4124yes—
169C22⋊Dic89—22no—
170C22⋊Dic89—26no—
171C22⋊Dic89C22×C4112yes—
172#13083551—22no—
173#13083551—26no—
174#13083551—22no—
175#13083551—43no—
176#13083551C22×C4112yes—
177C4×Dic89—26no—
178C4×Dic89—212no—
179C4×Dic89—26no—
180C4×Dic89—46no—
181C4×Dic89C22×C4112yes—
182C24×C89C2411yescharacteristic
183C2×Q8×C89—232no—
184C2×Q8×C89—26no—
185C2×Q8×C89—26no—
186C2×Q8×C89C22×C4112yes—
187C2×Q8×C89C22×C414yes—
188C2×Q8×C89C2×D414yes—
189C2×Q8×C89#4114yes—
190C22×C4×C89—22no—
191C22×C4×C89—22no—
192C22×C4×C89—26no—
193C22×C4×C89—26no—
194C22×C4×C89—22no—
195C22×C4×C89—26no—
196C22×C4×C89—26no—
197C22×C4×C89—43no—
198C22×C4×C89C22×C4112yes—
199C22×C4×C89C2413yes—
200C22×C4×C89C2413yes—
201C22×C4×C89C2411yescharacteristic
202C22×C4×C89C2413yes—
203C22×C4×C89C2411yescharacteristic
204C22×C4×C89C2413yes—
205C4⋊C4×C89—212no—
206C4⋊C4×C89—26no—
207C4⋊C4×C89C2×D4112yes—
208C42×C89—26no—
209C42×C89—26no—
210C42×C89—26no—
211C42×C89C2×D4112yes—
212C23×C89—21no—
213C23×C89—21no—
214C23×C89—21no—
215C23×C89—21no—
216C23×C89C23×C412yes—
217C23×C892−1+412yes—
218C23×C892−1+412yes—
219C23×C89C2511yescharacteristic
220C22×D89—21no—
221C22×D89—21no—
222C22×D89—21no—
223C22×D89—21no—
224C22×D89—21no—
225C22×D89—21no—
226C22×D89C23×C412yes—
227Q8×C89—224no—
228Q8×C89C4×D4116yes—
229C2×C4×C89—28no—
230C2×C4×C89—21no—
231C2×C4×C89—23no—
232C2×C4×C89—23no—
233C2×C4×C89—23no—
234C2×C4×C89—21no—
235C2×C4×C89—21no—
236C2×C4×C89—21no—
237C2×C4×C89—23no—
238C2×C4×C89—224no—
239C2×C4×C89—23no—
240C2×C4×C89—23no—
241C2×C4×C89—23no—
242C2×C4×C89—412no—
243C2×C4×C89C23×C416yes—
244C2×C4×C89C23×C416yes—
245C2×C4×C89C22×D412yes—
246C2×C4×C89C22×D416yes—
247C2×C4×C89C22×D416yes—
248C2×C4×C89C2×#4116yes—
249C2×C4×C892−1+412yes—
250C22⋊D89—28no—
251C2×Dic89—23no—
252C2×Dic89—23no—
253C2×Dic89—23no—
254C2×Dic89—21no—
255C2×Dic89—21no—
256C2×Dic89—23no—
257C2×Dic89—21no—
258C2×Dic89—23no—
259C2×Dic89—21no—
260C2×Dic89—23no—
261C2×Dic89—21no—
262C2×Dic89—21no—
263C2×Dic89—21no—
264C2×Dic89—23no—
265C2×Dic89—23no—
266C2×Dic89—23no—
267C2×Dic89—23no—
268C2×Dic89—21no—
269C2×Dic89C23×C412yes—
270C2×Dic89C23×C416yes—
271C2×Dic89C23×C416yes—
272C4×D89—224no—
273Dic178—224no—
274C22×C89—22no—
275C22×C89—22no—
276C22×C89—22no—
277C22×C89—22no—
278C22×C89—22no—
279C22×C89—22no—
280C22×C89#51914yescomplemented
281C22×C89C24×C411yescharacteristic
282C22×C89C23×D411yescharacteristic
283C22×C89C22×#4111yescharacteristic
284C22×C89C2×2−1+411yescharacteristic
285C22×C89C2×2−1+411yescharacteristic
286C22×C89C2×2−1+411yescharacteristic
287C22×C89#58511yescharacteristic
288C2×D89—22no—
289C2×D89—22no—
290C2×D89—22no—
291C2×D89—41no—
292C2×D89—41no—
293C2×D89—41no—
294C2×D89—41no—
295C4×C89—26no—
296C4×C89—22no—
297C4×C89—22no—
298C4×C89—26no—
299C4×C89—43no—
300C4×C89C2×C4×D4112yes—
301Dic89—26no—
302Dic89—22no—
303Dic89—26no—
304Dic89—43no—
305Dic89—41no—
306Dic89—43no—
307Dic89—41no—
308#34291—891nomaximal, Sylow
309C2×C89—24no—
310C2×C89C22×C4×D411yescharacteristic
311C2×C89C2×#51911yescharacteristic
312C2×C89C4×2−1+411yescharacteristic
313C2×C89#348611yescharacteristic
314C2×C89#348911yescharacteristic
315C2×C89#351611yescharacteristic
316C2×C89#352311yescharacteristic
317D89—42no—
318C2×D4×Q8—891no—
319C2×#519—891no—
320C22×C4×Q8D8911yescharacteristic, complemented
321#2365—893no—
322#2362—893no—
323#2360—893no—
324#2352—893no—
325#2351—891no—
326Q8×C22⋊C4—891no—
327Q8×C22⋊C4—891no—
328#2327—893no—
329#2323—893no—
330C4×C22⋊Q8—893no—
331#2290—893no—
332#2276—891no—
333C89#3429111yescharacteristic, complemented, Sylow
334C23×Q8—891no—
335C23×Q8C2×D8911yescharacteristic
336D4×Q8—8916no—
337C2×C4⋊Q8—893no—
338C2×C22⋊Q8—893no—
339C2×C22⋊Q8—893no—
340#519—1788no—
341C2×C4×Q8—28no—
342C2×C4×Q8—891no—
343C2×C4×Q8—891no—
344C2×C4×Q8—891no—
345C2×C4×Q8—893no—
346C2×C4×Q8—893no—
347C2×C4×Q8C2×D8913yes—
348C2×C4×Q8C2×D8911yescharacteristic
349C2×C4×Q8C2×D8913yes—
350C2×C4×Q8C2×D8911yescharacteristic
351C2×C4×D4—893no—
352C2×#117—893no—
353C2×#117—893no—
354C22×C4⋊C4C2×D8913yes—
355C22×C42C2×D8913yes—
356#391—893no—
357#391—893no—
358#391—893no—
359#391—893no—
360#389—893no—
361#389—896no—
362#389—893no—
363#388—896no—
364#388—893no—
365#388—893no—
366#387—896no—
367#387—896no—
368#387—896no—
369#387—893no—
370#387—893no—
371#386—893no—
372#386—891no—
373#385—893no—
374#385—893no—
375#382—893no—
376#380—893no—
377#380—893no—
378C4×C4⋊C4—893no—
379C4×C4⋊C4—893no—
380C4×C4⋊C4—893no—
381C4×C4⋊C4—893no—
382C4×C4⋊C4—896no—
383C4×C22⋊C4—893no—
384C4×C22⋊C4—896no—
385C4×C22⋊C4—893no—
386#376—893no—
387C22×Q8—24no—
388C22×Q8—24no—
389C22×Q8—898no—
390C22×Q8—893no—
391C22×Q8—1784no—
392C22×Q8—1784no—
393C22×Q8C4×D8918yes—
394C22×Q8C22×D8913yes—
395C22×Q8C22×D8911yescharacteristic
396C22×D4—891no—
397C23×C4—893no—
398C23×C4C22×D8911yescharacteristic
399C23×C4C22×D8913yes—
400C23×C4C22×D8913yes—
401C4⋊Q8—8924no—
402C22⋊Q8—8924no—
403C22⋊Q8—8924no—
404C4×Q8—24no—
405C4×Q8—212no—
406C4×Q8—212no—
407C4×Q8—898no—
408C4×Q8—1784no—
409C4×Q8—1784no—
410C4×Q8—17812no—
411C4×Q8—17812no—
412C4×Q8C22⋊D8918yescomplemented
413C4×D4—8924no—
414#117—17812no—
415#117—17812no—
416C2×C4⋊C4—212no—
417C2×C4⋊C4—893no—
418C2×C4⋊C4—893no—
419C2×C4⋊C4—896no—
420C2×C4⋊C4—893no—
421C2×C4⋊C4—893no—
422C2×C4⋊C4—893no—
423C2×C4⋊C4—893no—
424C2×C4⋊C4—893no—
425C2×C4⋊C4—893no—
426C2×C4⋊C4C22×D8913yes—
427C2×C4⋊C4C22×D8916yes—
428C2×C4⋊C4C22×D8913yes—
429C2×C22⋊C4—893no—
430C2×C22⋊C4—893no—
431C2×C22⋊C4—893no—
432C2×C22⋊C4—893no—
433C2×C22⋊C4—891no—
434C2×C22⋊C4—893no—
435C2×C22⋊C4—891no—
436C2×C22⋊C4—891no—
437C2×C42—212no—
438C2×C42—893no—
439C2×C42—896no—
440C2×C42—896no—
441C2×C42—893no—
442C2×C42—893no—
443C2×C42C22×D8913yes—
444C2×C42C22×D8913yes—
445C2×C42C22×D8913yes—
446C2×C42C22×D8913yes—
447#95—891no—
448#95—893no—
449#95—893no—
450#95—893no—
451#95—893no—
452#95—896no—
453#95—896no—
454#95—893no—
455#95—893no—
456#95—893no—
457#95—891no—
458#95—896no—
459#95—893no—
460C24—891no—
461C24C23×D8911yescharacteristic
462C2×Q8—232no—
463C2×Q8—26no—
464C2×Q8—26no—
465C2×Q8—8912no—
466C2×Q8—17832no—
467C2×Q8—1786no—
468C2×Q8—1786no—
469C2×Q8C2×C4×D89112yes—
470C2×Q8C2×C4×D8914yes—
471C2×Q8C2×C22⋊D8914yes—
472C2×Q8#1308357414yes—
473C2×D4—894no—
474C2×D4—1782no—
475C2×D4—1782no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—26no—
479C22×C4—26no—
480C22×C4—22no—
481C22×C4—26no—
482C22×C4—26no—
483C22×C4—43no—
484C22×C4—891no—
485C22×C4—893no—
486C22×C4—893no—
487C22×C4—8912no—
488C22×C4—891no—
489C22×C4—893no—
490C22×C4—893no—
491C22×C4—891no—
492C22×C4—1786no—
493C22×C4—1786no—
494C22×C4C2×C4×D89112yes—
495C22×C4C23×D8913yes—
496C22×C4C23×D8913yes—
497C22×C4C23×D8911yescharacteristic
498C22×C4C23×D8913yes—
499C22×C4C23×D8911yescharacteristic
500C22×C4C23×D8913yes—
501C4⋊C4—212no—
502C4⋊C4—26no—
503C4⋊C4—8912no—
504C4⋊C4—8912no—
505C4⋊C4—8924no—
506C4⋊C4—3563no—
507C4⋊C4—1786no—
508C4⋊C4—1786no—
509C4⋊C4—1786no—
510C4⋊C4—1786no—
511C4⋊C4—1786no—
512C4⋊C4C2×C22⋊D89112yes—
513C22⋊C4—8912no—
514C22⋊C4—8912no—
515C22⋊C4—3563no—
516C22⋊C4—1782no—
517C22⋊C4—1786no—
518C22⋊C4—1782no—
519C22⋊C4—1786no—
520C22⋊C4—1782no—
521C42—26no—
522C42—26no—
523C42—26no—
524C42—8912no—
525C42—3566no—
526C42—1786no—
527C42—17812no—
528C42—1786no—
529C42C2×C22⋊D89112yes—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—892no—
535C23—1781no—
536C23—1781no—
537C23—1781no—
538C23—1781no—
539C23—1781no—
540C23—1781no—
541C23C22×C4×D8912yes—
542C23#1566805912yes—
543C23#1566806012yes—
544C23C24×D8911yescharacteristic
545Q8—224no—
546Q8—17824no—
547Q8C4×C22⋊D89116yes—
548D4—1788no—
549C2×C4—28no—
550C2×C4—21no—
551C2×C4—23no—
552C2×C4—23no—
553C2×C4—23no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—23no—
558C2×C4—224no—
559C2×C4—23no—
560C2×C4—23no—
561C2×C4—23no—
562C2×C4—412no—
563C2×C4—892no—
564C2×C4—896no—
565C2×C4—896no—
566C2×C4—1781no—
567C2×C4—1783no—
568C2×C4—1781no—
569C2×C4—1783no—
570C2×C4—1781no—
571C2×C4—1781no—
572C2×C4—1783no—
573C2×C4—1781no—
574C2×C4—1783no—
575C2×C4—1783no—
576C2×C4—1783no—
577C2×C4—1781no—
578C2×C4—1783no—
579C2×C4—1783no—
580C2×C4—1781no—
581C2×C4—1783no—
582C2×C4—1781no—
583C2×C4—1783no—
584C2×C4—17824no—
585C2×C4C22×C4×D8916yes—
586C2×C4C22×C4×D8916yes—
587C2×C4C22×C22⋊D8912yes—
588C2×C4C22×C22⋊D8916yes—
589C2×C4C22×C22⋊D8916yes—
590C2×C4C2×#1308357416yes—
591C2×C4#1566806012yes—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—3561no—
599C22—3561no—
600C22—3561no—
601C22—3561no—
602C22—1782no—
603C22—1782no—
604C22—1782no—
605C22#1915207414yescomplemented
606C22C23×C4×D8911yescharacteristic
607C22C23×C22⋊D8911yescharacteristic
608C22C22×#1308357411yescharacteristic
609C22C2×#1566805911yescharacteristic
610C22C2×#1566806011yescharacteristic
611C22C2×#1566806011yescharacteristic
612C22#1915243411yescharacteristic
613C4—26no—
614C4—26no—
615C4—22no—
616C4—22no—
617C4—43no—
618C4—3561no—
619C4—3563no—
620C4—3561no—
621C4—3563no—
622C4—1786no—
623C4—1782no—
624C4—1786no—
625C4C2×C4×C22⋊D89112yes—
626C2—24no—
627C2—3562no—
628C2C22×C4×C22⋊D8911yescharacteristic
629C2C2×#1915207411yescharacteristic
630C2C4×#1566806011yescharacteristic
631C2#2603390311yescharacteristic
632C2#2603393211yescharacteristic
633C2#2603424511yescharacteristic
634C2#2603434811yescharacteristic
subgroupG/H
center, FrattiniC23C24×D89
commutatorC22×C89C24×C4
FittingC22×C4×Q8×C89C2
socle, cosocleC23×C89C25
prime radicalC22×C22⋊D89C23
hypercenterC22×C4×Q8D89
hypocenterC89#34291
LCSG ▷ C22×C89 ▷ C89
UCSC1 ◁ C23 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#19152074C2semidirect
3. #26023306C2semidirect
4. #26023275C2semidirect
5. #26023257C2semidirect
7. #26023170C2semidirect
8. Q8×C22⋊Dic89C2semidirect
9. Q8×#13083551C2semidirect
10. #26022968C2semidirect
11. #26022928C2semidirect
12. C4×#15667991C2semidirect
14. #26022617C2semidirect
15. C22×C4×Q8×C89C2semidirect
17. Q8×C22⋊D89C4semidirect
72. C2×Q8×D89C2×C4semidirect
75. #15668010C2×C4semidirect
77. #15667988C2×C4semidirect
80. Q8×Dic89C2×C4semidirect
132. C4×Q8×C89D4semidirect
280. C22×C89#519semidirect
320. C22×C4×Q8D89semidirect
333. C89#34291semidirect
412. C4×Q8C22⋊D89semidirect
605. C22#19152074semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C22⋊D892
C2×#191520742
#260233062
#260232752
#260232572
#260231772
#260231702
Q8×C22⋊Dic892
Q8×#130835512
#260229682
#260229282
C4×#156679912
#260226952
#260226172
C22×C4×Q8×C892
#3429189