Finite Group Databasebeta

← #36716082

Subgroups — #36716082

Order 22784.

2984 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19152259C211yesmaximal, characteristic
2C2×C4×#13083575C211yesmaximal, characteristic, complemented
3#26023319C213yesmaximal, complemented
4#26023290C213yesmaximal, complemented
5#26023275C213yesmaximal, complemented
6#26023202C213yesmaximal, complemented
7Q8×#13083553C211yesmaximal, characteristic, complemented
8#26023090C211yesmaximal, characteristic, complemented
9Q8×C22⋊Dic89C211yesmaximal, characteristic, complemented
10#26022959C213yesmaximal, complemented
11#26022945C213yesmaximal, complemented
12C4×#15667988C213yesmaximal, complemented
13#26022689C213yesmaximal, complemented
14#26022617C211yesmaximal, characteristic, complemented
15C2×#519×C89C211yesmaximal, characteristic, complemented
16C22×#13083575C2211yescharacteristic
17#19152259C4116yescomplemented
18C2×#15668001C2213yes—
19C2×#15667988C2213yes—
20C2×#15667986C2213yes—
21C4×#13083575—28no—
22C2×Q8×Dic89C2211yescharacteristic
23C2×Q8×Dic89C2211yescharacteristic
24C2×Q8×Dic89C2211yescharacteristic
25C2×C4×D356C2213yes—
26C2×#15667974C2213yes—
27C2×C4×C22⋊D89C2213yes—
28C2×#15667970C2213yes—
29C2×C42×D89C2213yes—
30#19151505C2213yes—
31#19151505C2213yes—
32#19151500C2213yes—
33#19151499C2213yes—
34#19151498C2216yes—
35#19151496C2213yes—
36#19151494C2213yes—
37#19151489C2216yes—
38#19151489C2216yes—
39#19151488C2216yes—
40#19151486C2211yescharacteristic
41#19151486C2213yes—
42#19151479C2213yes—
43#19151479C2213yes—
44#19151474C2213yes—
45#19151474C2213yes—
46#19151469C2213yes—
47#19151469C2213yes—
48#19151469C2216yes—
49#19151467C2213yes—
50#19151458C2213yes—
51#19151455C2213yes—
52C4⋊C4×Dic89C2213yes—
53C4⋊C4×Dic89C2213yes—
54C4×#13083553C2213yes—
55C4×#13083553C2216yes—
56C4×C4⋊Dic89C2213yes—
57C4×#13083551C2213yes—
58C4×#13083551C2216yes—
59C4×C22⋊Dic89C2213yes—
60#19151444C2213yes—
61C23×Q8×C89C2211yescharacteristic
62#519×C89—28no—
63C2×C4×Q8×C89C2213yes—
64C2×C4×Q8×C89C2213yes—
65C2×C4×Q8×C89C2211yescharacteristic
66C2×C4×Q8×C89C2211yescharacteristic
67C2×#117×C89C2213yes—
68C2×#117×C89C2213yes—
69C2×#13083575—24no—
70C2×#13083575—24no—
71C2×#13083575C2×C418yes—
72C22×D356C2313yes—
73C22×C22⋊D89C2313yes—
74C22×C4×D89C2313yes—
75C22×C4×D89C2313yes—
76C22×C4×D89C2313yes—
77#15668001C2×C4124yescomplemented
78#15667988C2×C4124yescomplemented
79#15667986C2×C4124yescomplemented
80Q8×Dic89—24no—
81Q8×Dic89—24no—
82Q8×Dic89C2×C418yescomplemented
83C4×D356—212no—
84#15667974—212no—
85C4×C22⋊D89C2×C4124yescomplemented
86#15667970—212no—
87C2×C4⋊Dic89C2313yes—
88C2×C4⋊Dic89C2313yes—
89C2×C4⋊Dic89C2313yes—
90C2×#13083553C2311yescharacteristic
91C2×#13083553C2313yes—
92C2×#13083553C2316yes—
93C2×#13083553C2313yes—
94C2×C22⋊Dic89C2311yescharacteristic
95C2×C22⋊Dic89C2313yes—
96C2×C22⋊Dic89C2311yescharacteristic
97C2×#13083551C2316yes—
98C2×#13083551C2316yes—
99C2×#13083551C2313yes—
100C2×#13083551C2313yes—
101C2×#13083551C2316yes—
102C2×#13083551C2313yes—
103C2×C4×Dic89C2313yes—
104C2×C4×Dic89C2313yes—
105C2×C4×Dic89C2313yes—
106C2×C4×Dic89C2311yescharacteristic
107C2×C4×Dic89C2316yes—
108C2×C4×Dic89C2313yes—
109C42×D89—212no—
110#15667919C2311yescharacteristic
111#15667919C2313yes—
112#15667919C2313yes—
113#15667919C2313yes—
114#15667919C2313yes—
115#15667919C2313yes—
116#15667919C2311yescharacteristic
117#15667919C2313yes—
118#15667919C2313yes—
119#15667919C2316yes—
120#15667919C2313yes—
121C22×Q8×C89—24no—
122C22×Q8×C89—24no—
123C22×Q8×C89C2×C418yescomplemented
124C22×Q8×C89C2313yes—
125C22×Q8×C89C2311yescharacteristic
126C23×C4×C89C2313yes—
127C4×Q8×C89—24no—
128C4×Q8×C89—212no—
129C4×Q8×C89—212no—
130C4×Q8×C89D418yescomplemented
131#117×C89—212no—
132#117×C89—212no—
133C2×C4⋊C4×C89C2313yes—
134C2×C4⋊C4×C89C2313yes—
135C2×C4⋊C4×C89C2313yes—
136C2×C4⋊C4×C89C2313yes—
137C2×C22⋊C4×C89C2311yescharacteristic
138C2×C22⋊C4×C89C2313yes—
139C2×C42×C89C2316yes—
140C2×C42×C89C2313yes—
141C2×C42×C89C2313yes—
142C23×D89C2413yes—
143#13083575—232no—
144C2×D356—26no—
145C2×D356—26no—
146C2×D356C22×C4112yes—
147C2×C22⋊D89—43no—
148C2×C22⋊D89—43no—
149C2×C22⋊D89C22×C4112yes—
150C22×Dic89C2413yes—
151C22×Dic89C2411yescharacteristic
152C22×Dic89C2413yes—
153C22×Dic89C2411yescharacteristic
154C22×Dic89C2413yes—
155C22×Dic89C2411yescharacteristic
156C22×Dic89C2411yescharacteristic
157C2×C4×D89—26no—
158C2×C4×D89—26no—
159C2×C4×D89—26no—
160C2×C4×D89—26no—
161C2×C4×D89—26no—
162C2×C4×D89—43no—
163C2×C4×D89—43no—
164C2×C4×D89—43no—
165C2×C4×D89C22×C4112yes—
166C4⋊Dic89—26no—
167C4⋊Dic89—26no—
168C4⋊Dic89C22×C4112yes—
169#13083553—22no—
170#13083553—43no—
171#13083553—46no—
172#13083553C22×C4112yes—
173C22⋊Dic89—22no—
174C22⋊Dic89—43no—
175C22⋊Dic89C22×C414yes—
176#13083551—212no—
177#13083551—43no—
178#13083551—43no—
179#13083551—46no—
180#13083551C22×C4112yes—
181#13083551C22×C4124yes—
182C4×Dic89—26no—
183C4×Dic89—26no—
184C4×Dic89—22no—
185C4×Dic89—26no—
186C4×Dic89—46no—
187C4×Dic89C22×C4112yes—
188C24×C89C2411yescharacteristic
189C2×Q8×C89—232no—
190C2×Q8×C89—26no—
191C2×Q8×C89—26no—
192C2×Q8×C89C22×C4112yes—
193C2×Q8×C89C22×C414yes—
194C2×Q8×C89C2×D414yes—
195C2×Q8×C89#4114yes—
196C22×C4×C89—26no—
197C22×C4×C89—26no—
198C22×C4×C89C22×C4112yes—
199C22×C4×C89C2411yescharacteristic
200C22×C4×C89C2413yes—
201C22×C4×C89C2413yes—
202C22×C4×C89C2411yescharacteristic
203C22×C4×C89C2413yes—
204C22×C4×C89C2413yes—
205C4⋊C4×C89—26no—
206C4⋊C4×C89—26no—
207C4⋊C4×C89—26no—
208C4⋊C4×C89C2×D4112yes—
209C22⋊C4×C89—26no—
210C22⋊C4×C89—22no—
211C42×C89—212no—
212C42×C89—26no—
213C42×C89C2×D4112yes—
214C23×C89—21no—
215C23×C89—21no—
216C23×C89—21no—
217C23×C89—21no—
218C23×C89—21no—
219C23×C89—21no—
220C23×C89C23×C412yes—
221C23×C89C2511yescharacteristic
222C22×D89—23no—
223C22×D89—23no—
224C22×D89—23no—
225C22×D89—23no—
226C22×D89—23no—
227C22×D89—23no—
228C22×D89C23×C416yes—
229Q8×C89—224no—
230Q8×C89C4×D4116yes—
231C2×C4×C89—23no—
232C2×C4×C89—23no—
233C2×C4×C89—23no—
234C2×C4×C89—23no—
235C2×C4×C89—21no—
236C2×C4×C89—21no—
237C2×C4×C89—224no—
238C2×C4×C89—23no—
239C2×C4×C89—21no—
240C2×C4×C89—21no—
241C2×C4×C89—21no—
242C2×C4×C89—23no—
243C2×C4×C89—23no—
244C2×C4×C89C23×C416yes—
245C2×C4×C89C23×C416yes—
246C2×C4×C89C22×D416yes—
247C2×C4×C89C22×D412yes—
248C2×C4×C89C22×D416yes—
249C2×C4×C89C2×#4116yes—
250C22⋊D89—412no—
251C2×Dic89—21no—
252C2×Dic89—21no—
253C2×Dic89—21no—
254C2×Dic89—21no—
255C2×Dic89—23no—
256C2×Dic89—21no—
257C2×Dic89—23no—
258C2×Dic89—21no—
259C2×Dic89—23no—
260C2×Dic89—21no—
261C2×Dic89—23no—
262C2×Dic89—21no—
263C2×Dic89—23no—
264C2×Dic89—21no—
265C2×Dic89—23no—
266C2×Dic89—23no—
267C2×Dic89—23no—
268C2×Dic89C23×C412yes—
269C2×Dic89C23×C416yes—
270C2×Dic89C23×C412yes—
271C2×Dic892−1+412yes—
272D356—224no—
273C4×D89—224no—
274C4×D89—412no—
275C4×D89—412no—
276C22×C89—22no—
277C22×C89—22no—
278C22×C89—22no—
279C22×C89—41no—
280C22×C89—41no—
281C22×C89—41no—
282C22×C89—41no—
283C22×C89C24×C411yescharacteristic
284C22×C89C23×D411yescharacteristic
285C22×C89C22×#4111yescharacteristic
286C22×C89C2×2−1+411yescharacteristic
287C22×C89#58511yescharacteristic
288C22×C89#58511yescharacteristic
289C22×C89#58511yescharacteristic
290C2×D89—26no—
291C2×D89—26no—
292C2×D89—26no—
293C2×D89—43no—
294C2×D89—43no—
295C2×D89—43no—
296C2×D89—43no—
297C4×C89—26no—
298C4×C89—22no—
299C4×C89—26no—
300C4×C89—43no—
301C4×C89—41no—
302C4×C89C2×C4×D4112yes—
303Dic89—22no—
304Dic89—26no—
305Dic89—22no—
306Dic89—22no—
307Dic89—43no—
308Dic89—41no—
309Dic89—43no—
310#34293—891nomaximal, Sylow
311C2×C89—42no—
312C2×C89C22×C4×D411yescharacteristic
313C2×C89#342811yescharacteristic
314C2×C89C4×2−1+411yescharacteristic
315C2×C89#348711yescharacteristic
316C2×C89#348911yescharacteristic
317C2×C89#351811yescharacteristic
318C2×C89#352411yescharacteristic
319D89—46no—
320C2×#548—891no—
321C2×#519D8911yescharacteristic, complemented
322C2×C4×#41—891no—
323#2367—893no—
324#2363—893no—
325#2362—893no—
326#2354—893no—
327Q8×C4⋊C4—891no—
328#2341—891no—
329Q8×C22⋊C4—891no—
330#2326—893no—
331#2324—893no—
332C4×C22⋊Q8—893no—
333#2290—893no—
334#2276—891no—
335C89#3429311yescharacteristic, complemented, Sylow
336C22×#41—891no—
337C23×Q8C2×D8911yescharacteristic
338#548—8916no—
339C2×#124—893no—
340C2×C22⋊Q8—893no—
341C2×#121—893no—
342#519—28no—
343C4×#41—1788no—
344C2×C4×Q8—891no—
345C2×C4×Q8—891no—
346C2×C4×Q8—891no—
347C2×C4×Q8C2×D8913yes—
348C2×C4×Q8C2×D8911yescharacteristic
349C2×C4×Q8C2×D8913yes—
350C2×C4×Q8C2×D8911yescharacteristic
351C2×C4×D4—893no—
352C2×C4×D4—893no—
353C2×C4×D4—893no—
354C2×#117—893no—
355C2×#117C2×D8913yes—
356C2×#117C2×D8913yes—
357C22×C42—893no—
358#391—893no—
359#391—893no—
360#390—893no—
361#390—896no—
362#390—893no—
363#389—893no—
364#389—893no—
365#388—896no—
366#388—893no—
367#388—891no—
368#388—896no—
369#388—896no—
370#387—893no—
371#387—893no—
372#386—893no—
373#386—893no—
374#385—896no—
375#385—893no—
376#385—893no—
377#384—893no—
378#380—893no—
379#380—893no—
380C4×C4⋊C4—893no—
381C4×C4⋊C4—893no—
382C4×C4⋊C4—893no—
383C4×C4⋊C4—893no—
384C4×C4⋊C4—896no—
385C4×C22⋊C4—893no—
386C4×C22⋊C4—893no—
387C4×C22⋊C4—896no—
388#376—893no—
389C2×#41—898no—
390C2×#41—1784no—
391C2×#41—1784no—
392C22×Q8—24no—
393C22×Q8—24no—
394C22×Q8C4×D8918yescomplemented
395C22×Q8C22×D8913yes—
396C22×Q8C22×D8911yescharacteristic
397C22×D4—893no—
398C22×D4—893no—
399C23×C4—893no—
400C23×C4—893no—
401C23×C4—893no—
402C23×C4C22×D8913yes—
403#124—8924no—
404C22⋊Q8—8924no—
405#121—8924no—
406C4×Q8—24no—
407C4×Q8—212no—
408C4×Q8—212no—
409C4×Q8—898no—
410C4×Q8—1784no—
411C4×Q8—1784no—
412C4×Q8C22⋊D8918yescomplemented
413C4×D4—8924no—
414C4×D4—17812no—
415C4×D4—17812no—
416#117—212no—
417#117—212no—
418#117—17812no—
419C2×C4⋊C4—893no—
420C2×C4⋊C4—893no—
421C2×C4⋊C4—893no—
422C2×C4⋊C4—896no—
423C2×C4⋊C4—893no—
424C2×C4⋊C4—893no—
425C2×C4⋊C4—891no—
426C2×C4⋊C4C22×D8913yes—
427C2×C4⋊C4C22×D8913yes—
428C2×C4⋊C4C22×D8913yes—
429C2×C4⋊C4C22×D8913yes—
430C2×C22⋊C4—891no—
431C2×C22⋊C4—893no—
432C2×C22⋊C4—896no—
433C2×C22⋊C4—893no—
434C2×C22⋊C4—896no—
435C2×C22⋊C4—893no—
436C2×C22⋊C4—893no—
437C2×C22⋊C4—891no—
438C2×C22⋊C4—896no—
439C2×C22⋊C4C22×D8911yescharacteristic
440C2×C22⋊C4C22×D8913yes—
441C2×C42—893no—
442C2×C42—896no—
443C2×C42—893no—
444C2×C42—893no—
445C2×C42—893no—
446C2×C42—891no—
447C2×C42—17812no—
448C2×C42C22×D8916yes—
449C2×C42C22×D8913yes—
450C2×C42C22×D8913yes—
451#95—891no—
452#95—893no—
453#95—893no—
454#95—893no—
455#95—893no—
456#95—891no—
457#95—893no—
458#95—893no—
459#95—893no—
460#95—896no—
461#95—893no—
462C24—893no—
463C24C23×D8911yescharacteristic
464#41—17832no—
465C2×Q8—232no—
466C2×Q8—26no—
467C2×Q8—26no—
468C2×Q8C2×C4×D89112yes—
469C2×Q8C2×C4×D8914yes—
470C2×Q8C2×C22⋊D8914yes—
471C2×Q8#1308357414yes—
472C2×D4—8912no—
473C2×D4—8912no—
474C2×D4—3563no—
475C2×D4—3563no—
476C2×D4—1786no—
477C2×D4—1786no—
478C22×C4—26no—
479C22×C4—26no—
480C22×C4—891no—
481C22×C4—891no—
482C22×C4—891no—
483C22×C4—893no—
484C22×C4—8912no—
485C22×C4—891no—
486C22×C4—893no—
487C22×C4—893no—
488C22×C4—3563no—
489C22×C4—3563no—
490C22×C4—3563no—
491C22×C4—1786no—
492C22×C4—1786no—
493C22×C4—1786no—
494C22×C4—1786no—
495C22×C4—1786no—
496C22×C4C2×C4×D89112yes—
497C22×C4C23×D8911yescharacteristic
498C22×C4C23×D8913yes—
499C22×C4C23×D8913yes—
500C22×C4C23×D8911yescharacteristic
501C22×C4C23×D8913yes—
502C22×C4C23×D8913yes—
503C4⋊C4—26no—
504C4⋊C4—26no—
505C4⋊C4—26no—
506C4⋊C4—8912no—
507C4⋊C4—8912no—
508C4⋊C4—3563no—
509C4⋊C4—3566no—
510C4⋊C4—1782no—
511C4⋊C4—1786no—
512C4⋊C4—1786no—
513C4⋊C4C2×C22⋊D89112yes—
514C22⋊C4—26no—
515C22⋊C4—22no—
516C22⋊C4—894no—
517C22⋊C4—8912no—
518C22⋊C4—8924no—
519C22⋊C4—3563no—
520C22⋊C4—3566no—
521C22⋊C4—3563no—
522C22⋊C4—3563no—
523C22⋊C4—1782no—
524C22⋊C4—17812no—
525C42—212no—
526C42—26no—
527C42—8912no—
528C42—3566no—
529C42—1782no—
530C42—1786no—
531C42—1786no—
532C42—1786no—
533C42C2×C22⋊D89112yes—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—896no—
541C23—1783no—
542C23—1783no—
543C23—1783no—
544C23—1783no—
545C23—1783no—
546C23—1783no—
547C23C22×C4×D8912yes—
548C23C24×D8911yescharacteristic
549Q8—224no—
550Q8C4×C22⋊D89116yes—
551D4—35612no—
552D4—17824no—
553C2×C4—23no—
554C2×C4—23no—
555C2×C4—23no—
556C2×C4—23no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—224no—
560C2×C4—21no—
561C2×C4—23no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—23no—
565C2×C4—23no—
566C2×C4—892no—
567C2×C4—896no—
568C2×C4—892no—
569C2×C4—892no—
570C2×C4—35612no—
571C2×C4—35612no—
572C2×C4—1783no—
573C2×C4—1781no—
574C2×C4—1783no—
575C2×C4—1781no—
576C2×C4—1783no—
577C2×C4—1781no—
578C2×C4—1783no—
579C2×C4—1783no—
580C2×C4—1783no—
581C2×C4—1781no—
582C2×C4—1781no—
583C2×C4—1781no—
584C2×C4—1781no—
585C2×C4—1783no—
586C2×C4—1781no—
587C2×C4—1783no—
588C2×C4—1781no—
589C2×C4—17824no—
590C2×C4C22×C4×D8916yes—
591C2×C4C22×C4×D8916yes—
592C2×C4C22×C22⋊D8916yes—
593C2×C4C22×C22⋊D8912yes—
594C2×C4C22×C22⋊D8916yes—
595C2×C4C2×#1308357416yes—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—3563no—
604C22—3563no—
605C22—3563no—
606C22—3563no—
607C22—1786no—
608C22—1786no—
609C22—1786no—
610C22C23×C4×D8911yescharacteristic
611C22C23×C22⋊D8911yescharacteristic
612C22C22×#1308357411yescharacteristic
613C222−1+4×D8911yescharacteristic
614C22#1915243511yescharacteristic
615C22#1915243511yescharacteristic
616C22#1915243511yescharacteristic
617C4—26no—
618C4—26no—
619C4—22no—
620C4—41no—
621C4—43no—
622C4—3561no—
623C4—3563no—
624C4—3563no—
625C4—1782no—
626C4—1782no—
627C4—1786no—
628C4—1782no—
629C4C2×C4×C22⋊D89112yes—
630C2—42no—
631C2—3566no—
632C2C22×C4×C22⋊D8911yescharacteristic
633C2#2603345411yescharacteristic
634C2#2603346111yescharacteristic
635C2#2603390711yescharacteristic
636C2#2603392511yescharacteristic
637C2#2603426311yescharacteristic
638C2#2603436311yescharacteristic
subgroupG/H
center, FrattiniC23C24×D89
commutatorC22×C89C24×C4
FittingC2×#519×C89C2
socle, cosocleC23×C89C25
prime radicalC2×#19152259C2
hypercenterC2×#519D89
hypocenterC89#34293
LCSG ▷ C22×C89 ▷ C89
UCSC1 ◁ C23 ◁ C2×#519

Complements

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

NH ≅ G/Ntype
2. C2×C4×#13083575C2semidirect
3. #26023319C2semidirect
4. #26023290C2semidirect
5. #26023275C2semidirect
6. #26023202C2semidirect
7. Q8×#13083553C2semidirect
8. #26023090C2semidirect
9. Q8×C22⋊Dic89C2semidirect
10. #26022959C2semidirect
11. #26022945C2semidirect
12. C4×#15667988C2semidirect
13. #26022689C2semidirect
14. #26022617C2semidirect
15. C2×#519×C89C2semidirect
17. #19152259C4semidirect
77. #15668001C2×C4semidirect
78. #15667988C2×C4semidirect
79. #15667986C2×C4semidirect
82. Q8×Dic89C2×C4semidirect
85. C4×C22⋊D89C2×C4semidirect
123. C22×Q8×C89C2×C4semidirect
130. C4×Q8×C89D4semidirect
321. C2×#519D89semidirect
335. C89#34293semidirect
394. C22×Q8C4×D89semidirect
412. C4×Q8C22⋊D89semidirect

Maximal subgroups

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

subgroupindex
C2×#191522592
C2×C4×#130835752
#260233192
#260232902
#260232752
#260232022
Q8×#130835532
#260230902
Q8×C22⋊Dic892
#260229592
#260229452
C4×#156679882
#260226892
#260226172
C2×#519×C892
#3429389