Finite Group Databasebeta

← #36715374

Subgroups — #36715374

Order 22784.

3106 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19152080C211yesmaximal, characteristic
2C2×#19152080C211yesmaximal, characteristic
3C2×C4×#13083575C211yesmaximal, characteristic
4#26023170C211yesmaximal, characteristic, complemented
5Q8×C4⋊Dic89C211yesmaximal, characteristic, complemented
6Q8×#13083553C211yesmaximal, characteristic, complemented
7#26023161C213yesmaximal, complemented
8#26023160C213yesmaximal, complemented
9C4⋊C4×D356C213yesmaximal, complemented
10#26023153C213yesmaximal, complemented
11#26022877C213yesmaximal, complemented
12#26022867C213yesmaximal, complemented
13#26022732C213yesmaximal, complemented
14#389×D89C213yesmaximal, complemented
15C89×#2351C211yesmaximal, characteristic, complemented
16C22×#13083575C2211yescharacteristic
17#19152080—28no—
18#19152080C4116yescomplemented
19C4×#13083575—28no—
20C2×Q8×Dic89C2211yescharacteristic
21C2×Q8×Dic89C2211yescharacteristic
22C2×Q8×Dic89C2211yescharacteristic
23C2×C4×D356C2213yes—
24C2×C4×D356C2213yes—
25C2×C4×D356C2213yes—
26C2×#15667974C2213yes—
27C2×#15667974C2213yes—
28C2×#15667974C2213yes—
29C2×#15667970C2213yes—
30C2×#15667969C2213yes—
31C2×#15667969C2213yes—
32C2×C4⋊C4×D89C2213yes—
33C2×C4⋊C4×D89C2216yes—
34C2×C4⋊C4×D89C2213yes—
35C2×C42×D89C2213yes—
36#19151497C2213yes—
37#19151497C2213yes—
38#19151496C2213yes—
39#19151496C2213yes—
40#19151495C2216yes—
41#19151495C2211yescharacteristic
42#19151495C2213yes—
43#19151494C2213yes—
44#19151494C2213yes—
45#19151493C2213yes—
46#19151493C2213yes—
47#19151472C2213yes—
48#19151472C2213yes—
49#19151472C2216yes—
50#19151472C2216yes—
51#19151469C2213yes—
52#19151469C2213yes—
53#19151468C2213yes—
54#19151467C2213yes—
55#19151463C2213yes—
56#19151457C2216yes—
57#19151455C2216yes—
58C4×#13083553C2213yes—
59C4×#13083553C2213yes—
60C4×C4⋊Dic89C2213yes—
61#95×D89C2216yes—
62C2×C4×Q8×C89C2211yescharacteristic
63C2×C4×Q8×C89C2211yescharacteristic
64C2×C4×Q8×C89C2211yescharacteristic
65#389×C89C2213yes—
66#389×C89C2213yes—
67#382×C89C2213yes—
68C4×C4⋊C4×C89C2213yes—
69C2×#13083575C2×C418yes—
70C2×#13083575D418yes—
71C2×#13083575Q818yes—
72C22×D356C2313yes—
73C22×C4×D89C2313yes—
74C22×C4×D89C2313yes—
75C22×C4×D89C2316yes—
76C22×C4×D89C2316yes—
77C22×C4×D89C2313yes—
78Q8×Dic89—24no—
79Q8×Dic89—24no—
80Q8×Dic89C2×C418yescomplemented
81C4×D356—212no—
82C4×D356—212no—
83C4×D356C2×C4124yescomplemented
84#15667974—212no—
85#15667974—212no—
86#15667974C2×C4124yescomplemented
87#15667970—212no—
88#15667969—212no—
89#15667969C2×C4124yescomplemented
90C2×C4⋊Dic89C2311yescharacteristic
91C2×C4⋊Dic89C2313yes—
92C2×C4⋊Dic89C2313yes—
93C2×C4⋊Dic89C2313yes—
94C2×C4⋊Dic89C2316yes—
95C2×C4⋊Dic89C2313yes—
96C2×#13083553C2311yescharacteristic
97C2×#13083553C2313yes—
98C2×#13083553C2311yescharacteristic
99C2×#13083553C2313yes—
100C2×#13083553C2313yes—
101C2×#13083553C2313yes—
102C2×#13083553C2316yes—
103C2×#13083553C2313yes—
104C2×#13083553C2311yescharacteristic
105C4⋊C4×D89—212no—
106C4⋊C4×D89—412no—
107C4⋊C4×D89C2×C4124yescomplemented
108C2×#13083551C2313yes—
109C2×#13083551C2316yes—
110C2×#13083551C2313yes—
111C2×#13083551C2313yes—
112C2×#13083551C2313yes—
113C2×C4×Dic89C2313yes—
114C2×C4×Dic89C2313yes—
115C2×C4×Dic89C2311yescharacteristic
116C2×C4×Dic89C2313yes—
117C2×C4×Dic89C2313yes—
118C42×D89—212no—
119#15667919C2313yes—
120#15667919C2313yes—
121#15667919C2316yes—
122#15667919C2313yes—
123#15667919C2311yescharacteristic
124#15667919C2313yes—
125#15667919C2313yes—
126C22×Q8×C89C2311yescharacteristic
127C4×Q8×C89—24no—
128C4×Q8×C89—24no—
129C4×Q8×C89C2×C418yescomplemented
130C2×C4⋊C4×C89C2313yes—
131C2×C4⋊C4×C89C2313yes—
132C2×C4⋊C4×C89C2313yes—
133C2×C4⋊C4×C89C2313yes—
134C2×C4⋊C4×C89C2313yes—
135C2×C42×C89C2313yes—
136C2×C42×C89C2313yes—
137C2×C42×C89C2313yes—
138#95×C89C2311yescharacteristic
139#95×C89C2313yes—
140#95×C89C2316yes—
141C23×D89C2413yes—
142#13083575C4⋊C4164yescomplemented
143C2×D356C22×C4112yes—
144C2×D356C2×D4112yes—
145C2×D356C2×Q8112yes—
146C22×Dic89C2413yes—
147C22×Dic89C2413yes—
148C22×Dic89C2411yescharacteristic
149C22×Dic89C2413yes—
150C22×Dic89C2411yescharacteristic
151C22×Dic89C2411yescharacteristic
152C22×Dic89C2411yescharacteristic
153C2×C4×D89—26no—
154C2×C4×D89—26no—
155C2×C4×D89—26no—
156C2×C4×D89—26no—
157C2×C4×D89—212no—
158C2×C4×D89—46no—
159C2×C4×D89—43no—
160C2×C4×D89—43no—
161C2×C4×D89—46no—
162C2×C4×D89—46no—
163C2×C4×D89—46no—
164C2×C4×D89C22×C4124yes—
165C2×C4×D89C22×C4112yes—
166C2×C4×D89C2×D4112yes—
167C2×C4×D89C2×Q8112yes—
168C4⋊Dic89—26no—
169C4⋊Dic89—26no—
170C4⋊Dic89—22no—
171C4⋊Dic89—46no—
172C4⋊Dic89—43no—
173C4⋊Dic89C22×C4112yes—
174#13083553—26no—
175#13083553—22no—
176#13083553—26no—
177#13083553—22no—
178#13083553—46no—
179#13083553—43no—
180#13083553—43no—
181#13083553C22×C414yes—
182#13083553C22×C4112yes—
183#13083551—26no—
184#13083551—26no—
185#13083551—212no—
186#13083551C22×C4112yes—
187#13083551C22×C4112yes—
188C4×Dic89—26no—
189C4×Dic89—26no—
190C4×Dic89—26no—
191C4×Dic89—22no—
192C4×Dic89C22×C4112yes—
193C2×Q8×C89C22×C414yes—
194C2×Q8×C89C2×D414yes—
195C2×Q8×C89C2×Q814yes—
196C22×C4×C89C2413yes—
197C22×C4×C89C2413yes—
198C22×C4×C89C2411yescharacteristic
199C22×C4×C89C2413yes—
200C22×C4×C89C2411yescharacteristic
201C22×C4×C89C2411yescharacteristic
202C22×C4×C89C2413yes—
203C4⋊C4×C89—26no—
204C4⋊C4×C89—26no—
205C4⋊C4×C89—26no—
206C4⋊C4×C89—43no—
207C4⋊C4×C89C22×C4112yes—
208C42×C89—26no—
209C42×C89—26no—
210C42×C89C22×C4112yes—
211C23×C89C2511yescharacteristic
212C22×D89—23no—
213C22×D89—23no—
214C22×D89—23no—
215C22×D89—23no—
216C22×D89C23×C416yes—
217C22×D89C22×D416yes—
218C22×D89C22×Q816yes—
219Q8×C89C2×C4⋊C4116yescomplemented
220C2×C4×C89—21no—
221C2×C4×C89—23no—
222C2×C4×C89—21no—
223C2×C4×C89—23no—
224C2×C4×C89—21no—
225C2×C4×C89—21no—
226C2×C4×C89—21no—
227C2×C4×C89—23no—
228C2×C4×C89—23no—
229C2×C4×C89—23no—
230C2×C4×C89—21no—
231C2×C4×C89—23no—
232C2×C4×C89—21no—
233C2×C4×C89—23no—
234C2×C4×C89—23no—
235C2×C4×C89C23×C412yes—
236C2×C4×C89C23×C416yes—
237C2×C4×C89C23×C416yes—
238C2×C4×C89C22×D416yes—
239C2×C4×C89C22×Q816yes—
240C2×C4×C89H(5,𝔽2)12yes—
241C2×Dic89—21no—
242C2×Dic89—23no—
243C2×Dic89—21no—
244C2×Dic89—21no—
245C2×Dic89—23no—
246C2×Dic89—21no—
247C2×Dic89—21no—
248C2×Dic89—23no—
249C2×Dic89—23no—
250C2×Dic89—23no—
251C2×Dic89—23no—
252C2×Dic89—21no—
253C2×Dic89—21no—
254C2×Dic89—23no—
255C2×Dic89—23no—
256C2×Dic89C23×C412yes—
257C2×Dic89C23×C416yes—
258C2×Dic89C23×C412yes—
259C2×Dic89C22×D412yes—
260C2×Dic89C22×Q812yes—
261C2×Dic892−1+412yes—
262D356C2×C4⋊C4148yescomplemented
263C4×D89—424no—
264C4×D89—412no—
265C4×D89—412no—
266C4×D89—812no—
267C4×D89C2×C4⋊C4148yescomplemented
268C22×C89C24×C411yescharacteristic
269C22×C89C23×D411yescharacteristic
270C22×C89C23×Q811yescharacteristic
271C22×C89C2×H(5,𝔽2)11yescharacteristic
272C22×C89C2×2−1+411yescharacteristic
273C22×C89#58511yescharacteristic
274C22×C89#58511yescharacteristic
275C2×D89—26no—
276C2×D89—26no—
277C2×D89—26no—
278C2×D89—26no—
279C2×D89—26no—
280C2×D89—26no—
281C2×D89C22×C4⋊C4112yes—
282C4×C89—26no—
283C4×C89—22no—
284C4×C89—22no—
285C4×C89—43no—
286C4×C89—43no—
287C4×C89—41no—
288C4×C89C22×C4⋊C4112yes—
289Dic89—26no—
290Dic89—22no—
291Dic89—22no—
292Dic89—43no—
293Dic89—43no—
294Dic89—41no—
295Dic89C22×C4⋊C414yes—
296#34250—891nomaximal, Sylow
297C2×C89C23×C4⋊C411yescharacteristic
298C2×C89C4×H(5,𝔽2)11yescharacteristic
299C2×C89C4×2−1+411yescharacteristic
300C2×C89#348811yescharacteristic
301C2×C89#348911yescharacteristic
302C2×C89#350311yescharacteristic
303C2×C89#350411yescharacteristic
304D89—212no—
305C2×#520—891no—
306C2×#520—891no—
307C2×C4×#41—891no—
308#2351—891no—
309#2351D8911yescharacteristic, complemented
310Q8×C4⋊C4—891no—
311Q8×C4⋊C4—891no—
312#2349—893no—
313#2349—893no—
314D4×C4⋊C4—893no—
315D4×C4⋊C4—893no—
316#2317—893no—
317#2316—893no—
318#2296—893no—
319C2×#389—893no—
320C89#3425011yescharacteristic, complemented, Sylow
321C22×#41—891no—
322#520—8916no—
323#520—1788no—
324C4×#41—1788no—
325C2×C4×Q8—891no—
326C2×C4×Q8—891no—
327C2×C4×Q8—891no—
328C2×C4×Q8C2×D8911yescharacteristic
329C2×C4×Q8C2×D8911yescharacteristic
330C2×C4×Q8C2×D8911yescharacteristic
331C2×C4×D4—893no—
332C2×C4×D4—893no—
333C2×C4×D4—893no—
334C2×C4×D4—893no—
335C2×C4×D4—893no—
336C2×C4×D4—893no—
337C2×#117—893no—
338C2×#117—893no—
339C2×#117—893no—
340C22×C4⋊C4—893no—
341C22×C4⋊C4—896no—
342C22×C4⋊C4—893no—
343C22×C42—893no—
344#389—893no—
345#389—893no—
346#389—893no—
347#389—893no—
348#389—893no—
349#389—893no—
350#389—893no—
351#389—893no—
352#389—896no—
353#389—893no—
354#389—891no—
355#389C2×D8913yes—
356#389C2×D8913yes—
357#385—893no—
358#385—893no—
359#385—896no—
360#385—896no—
361#385—893no—
362#385—893no—
363#384—893no—
364#384—893no—
365#382—893no—
366#382C2×D8913yes—
367#380—896no—
368#380—896no—
369C4×C4⋊C4—893no—
370C4×C4⋊C4—893no—
371C4×C4⋊C4—893no—
372C4×C4⋊C4C2×D8913yes—
373C2×#95—896no—
374C2×#41—898no—
375C2×#41—898no—
376C2×#41—898no—
377C22×Q8C22×D8911yescharacteristic
378C22×D4—893no—
379C23×C4—893no—
380C23×C4—896no—
381C23×C4—896no—
382C23×C4—893no—
383C23×C4—893no—
384C4×Q8—24no—
385C4×Q8—24no—
386C4×Q8—898no—
387C4×Q8—1784no—
388C4×Q8—1784no—
389C4×Q8C4×D8918yescomplemented
390C4×D4—8924no—
391C4×D4—8924no—
392C4×D4—17812no—
393C4×D4—17812no—
394C4×D4—17812no—
395C4×D4—17812no—
396#117—8924no—
397#117—17812no—
398#117—17812no—
399C2×C4⋊C4—893no—
400C2×C4⋊C4—893no—
401C2×C4⋊C4—891no—
402C2×C4⋊C4—8924no—
403C2×C4⋊C4—896no—
404C2×C4⋊C4—896no—
405C2×C4⋊C4—893no—
406C2×C4⋊C4—893no—
407C2×C4⋊C4—893no—
408C2×C4⋊C4—893no—
409C2×C4⋊C4—893no—
410C2×C4⋊C4—893no—
411C2×C4⋊C4—891no—
412C2×C4⋊C4—893no—
413C2×C4⋊C4—891no—
414C2×C4⋊C4—891no—
415C2×C4⋊C4—35612no—
416C2×C4⋊C4—17812no—
417C2×C4⋊C4C22×D8913yes—
418C2×C4⋊C4C22×D8913yes—
419C2×C4⋊C4C22×D8913yes—
420C2×C4⋊C4C22×D8913yes—
421C2×C4⋊C4C22×D8913yes—
422C2×C22⋊C4—893no—
423C2×C22⋊C4—893no—
424C2×C22⋊C4—893no—
425C2×C22⋊C4—893no—
426C2×C22⋊C4—896no—
427C2×C42—893no—
428C2×C42—893no—
429C2×C42—893no—
430C2×C42—893no—
431C2×C42—891no—
432C2×C42—17812no—
433C2×C42C22×D8913yes—
434C2×C42C22×D8913yes—
435C2×C42C22×D8913yes—
436#95—893no—
437#95—893no—
438#95—896no—
439#95—891no—
440#95—893no—
441#95—893no—
442#95—893no—
443#95C22×D8916yes—
444#95C22×D8911yescharacteristic
445#95C22×D8913yes—
446C24—893no—
447#41—8964no—
448C2×Q8C2×C4×D8914yes—
449C2×Q8D4×D8914yes—
450C2×Q8Q8×D8914yes—
451C2×D4—8912no—
452C2×D4—8912no—
453C2×D4—8912no—
454C22×C4—891no—
455C22×C4—8912no—
456C22×C4—8912no—
457C22×C4—8912no—
458C22×C4—891no—
459C22×C4—891no—
460C22×C4—893no—
461C22×C4—8924no—
462C22×C4—891no—
463C22×C4—893no—
464C22×C4—893no—
465C22×C4—3566no—
466C22×C4—3566no—
467C22×C4—3563no—
468C22×C4—3563no—
469C22×C4—3566no—
470C22×C4—3566no—
471C22×C4—1786no—
472C22×C4—1786no—
473C22×C4—1786no—
474C22×C4—17812no—
475C22×C4—1786no—
476C22×C4C23×D8913yes—
477C22×C4C23×D8911yescharacteristic
478C22×C4C23×D8913yes—
479C22×C4C23×D8913yes—
480C22×C4C23×D8911yescharacteristic
481C22×C4C23×D8911yescharacteristic
482C22×C4C23×D8913yes—
483C4⋊C4—26no—
484C4⋊C4—26no—
485C4⋊C4—26no—
486C4⋊C4—43no—
487C4⋊C4—8912no—
488C4⋊C4—8912no—
489C4⋊C4—894no—
490C4⋊C4—3566no—
491C4⋊C4—3566no—
492C4⋊C4—3563no—
493C4⋊C4—3563no—
494C4⋊C4—3563no—
495C4⋊C4—1786no—
496C4⋊C4—1782no—
497C4⋊C4—1782no—
498C4⋊C4—1782no—
499C4⋊C4—1786no—
500C4⋊C4—1786no—
501C4⋊C4—1786no—
502C4⋊C4C2×C4×D89112yes—
503C22⋊C4—8912no—
504C22⋊C4—8912no—
505C22⋊C4—1786no—
506C22⋊C4—17812no—
507C22⋊C4—1786no—
508C42—26no—
509C42—26no—
510C42—8912no—
511C42—1782no—
512C42—1786no—
513C42—1786no—
514C42—1786no—
515C42C2×C4×D89112yes—
516C23—896no—
517C23—896no—
518C23—896no—
519C23—1783no—
520C23—1783no—
521C23—1783no—
522C23—1783no—
523C23C24×D8911yescharacteristic
524Q8C4⋊C4×D89116yescomplemented
525D4—8948no—
526C2×C4—21no—
527C2×C4—23no—
528C2×C4—21no—
529C2×C4—23no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—23no—
534C2×C4—23no—
535C2×C4—23no—
536C2×C4—21no—
537C2×C4—23no—
538C2×C4—21no—
539C2×C4—23no—
540C2×C4—23no—
541C2×C4—892no—
542C2×C4—892no—
543C2×C4—892no—
544C2×C4—8948no—
545C2×C4—892no—
546C2×C4—896no—
547C2×C4—892no—
548C2×C4—35612no—
549C2×C4—35624no—
550C2×C4—35612no—
551C2×C4—1781no—
552C2×C4—1783no—
553C2×C4—1783no—
554C2×C4—1783no—
555C2×C4—1783no—
556C2×C4—1781no—
557C2×C4—1781no—
558C2×C4—1783no—
559C2×C4—1783no—
560C2×C4—1781no—
561C2×C4—1783no—
562C2×C4—1781no—
563C2×C4—1783no—
564C2×C4—1781no—
565C2×C4—1781no—
566C2×C4—71212no—
567C2×C4C22×C4×D8912yes—
568C2×C4C22×C4×D8916yes—
569C2×C4C22×C4×D8916yes—
570C2×C4C2×D4×D8916yes—
571C2×C4C2×Q8×D8916yes—
572C2×C4#1566805812yes—
573C22—8912no—
574C22—1786no—
575C22—1786no—
576C22—1786no—
577C22—1786no—
578C22—1786no—
579C22—1786no—
580C22C23×C4×D8911yescharacteristic
581C22C22×D4×D8911yescharacteristic
582C22C22×Q8×D8911yescharacteristic
583C22C2×#1566805811yescharacteristic
584C222−1+4×D8911yescharacteristic
585C22#1915243411yescharacteristic
586C22#1915243511yescharacteristic
587C4—26no—
588C4—22no—
589C4—22no—
590C4—43no—
591C4—41no—
592C4—43no—
593C4—894no—
594C4—3561no—
595C4—3563no—
596C4—3563no—
597C4—1782no—
598C4—1786no—
599C4—1782no—
600C4C2×C4⋊C4×D89112yes—
601C2—17812no—
602C2C22×C4⋊C4×D8911yescharacteristic
603C2C4×#1566805811yescharacteristic
604C2#2603346111yescharacteristic
605C2#2603391811yescharacteristic
606C2#2603393111yescharacteristic
607C2#2603406911yescharacteristic
608C2#2603407711yescharacteristic
subgroupG/H
center, FrattiniC23C24×D89
commutatorC22×C89C24×C4
FittingC89×#2351C2
socle, cosocleC23×C89C25
prime radicalC22×#13083575C22
hypercenter#2351D89
hypocenterC89#34250
LCSG ▷ C22×C89 ▷ C89
UCSC1 ◁ C23 ◁ #2351

Complements

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

NH ≅ G/Ntype
4. #26023170C2semidirect
5. Q8×C4⋊Dic89C2semidirect
6. Q8×#13083553C2semidirect
7. #26023161C2semidirect
8. #26023160C2semidirect
9. C4⋊C4×D356C2semidirect
10. #26023153C2semidirect
11. #26022877C2semidirect
12. #26022867C2semidirect
13. #26022732C2semidirect
14. #389×D89C2semidirect
15. C89×#2351C2semidirect
18. #19152080C4semidirect
80. Q8×Dic89C2×C4semidirect
83. C4×D356C2×C4semidirect
86. #15667974C2×C4semidirect
89. #15667969C2×C4semidirect
107. C4⋊C4×D89C2×C4semidirect
129. C4×Q8×C89C2×C4semidirect
142. #13083575C4⋊C4semidirect
219. Q8×C89C2×C4⋊C4semidirect
262. D356C2×C4⋊C4semidirect
267. C4×D89C2×C4⋊C4semidirect
309. #2351D89semidirect
320. C89#34250semidirect
389. C4×Q8C4×D89semidirect
524. Q8C4⋊C4×D89semidirect

Maximal subgroups

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

subgroupindex
C2×#191520802
C2×C4×#130835752
#260231702
Q8×C4⋊Dic892
Q8×#130835532
#260231612
#260231602
C4⋊C4×D3562
#260231532
#260228772
#260228672
#260227322
#389×D892
C89×#23512
#3425089