Finite Group Databasebeta

← #27400712

Subgroups — #27400712

Order 13568.

1110 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811332C211yesmaximal, characteristic
2C2×#16810752C211yesmaximal, characteristic, complemented
3#20277627C211yesmaximal, characteristic, complemented
4#20275683C211yesmaximal, characteristic, complemented
5#20275682C211yesmaximal, characteristic, complemented
6#20275674C211yesmaximal, characteristic, complemented
7#20275671C211yesmaximal, characteristic, complemented
8#20275670C211yesmaximal, characteristic, complemented
9#20275661C211yesmaximal, characteristic, complemented
10#20274713C211yesmaximal, characteristic
11#20274713C211yesmaximal, characteristic
12#20274542C211yesmaximal, characteristic
13#20274540C211yesmaximal, characteristic
14#102×Dic53C211yesmaximal, characteristic
15C53×#1802C211yesmaximal, characteristic
16#16811332—24no—
17#16811332C418yescomplemented
18C2×#13133216C2211yescharacteristic
19C2×#13133209C2211yescharacteristic
20C2×#13133208C2211yescharacteristic
21C2×#13133196—21no—
22C2×#13133194—21no—
23C2×#13133194—21no—
24C2×C4×Dic106C2211yescharacteristic
25C2×#13133182—21no—
26C2×D4×Dic53C2211yescharacteristic
27C2×D4×Dic53C2211yescharacteristic
28C22×#11719919—21no—
29#16810752—24no—
30C2×#13133146C2211yescharacteristic
31C2×#13133146C2211yescharacteristic
32C2×#13133138C2211yescharacteristic
33C2×#13133138C2211yescharacteristic
34C2×#13133128C2211yescharacteristic
35C2×#13133127C2211yescharacteristic
36C2×C8×Dic53C2211yescharacteristic
37#16810519—21no—
38#16810518—21no—
39#16810518—21no—
40#16810492C2211yescharacteristic
41#16810492C2211yescharacteristic
42C4⋊C4×Dic53C2211yescharacteristic
43C4⋊C4×Dic53C2211yescharacteristic
44C4×C4⋊Dic53C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810385C2211yescharacteristic
53#16810385C2211yescharacteristic
54#16810385C2211yescharacteristic
55C2×C4×D4×C53C2211yescharacteristic
56C2×C4⋊C8×C53C2211yescharacteristic
57C2×#102×C53C2211yescharacteristic
58C2×#102×C53C2211yescharacteristic
59C53×#384C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C53×#340C2211yescharacteristic
62C22×Dic106—21no—
63C22×Dic106D411yescharacteristic
64C23×Dic53—21no—
65C23×Dic53—21no—
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68#13133216—22no—
69#13133216C2×C414yes—
70#13133209—22no—
71#13133209C2×C414yes—
72#13133208—22no—
73#13133208C2×C414yes—
74#13133196—28no—
75#13133194—28no—
76#13133194—28no—
77C4×Dic106C2×C414yes—
78C4×Dic106D414yes—
79#13133182—44no—
80D4×Dic53—22no—
81D4×Dic53—22no—
82D4×Dic53C2×C414yes—
83D4×Dic53C2×C414yes—
84C2×C4⋊Dic53—21no—
85C2×C4⋊Dic53D411yescharacteristic
86C2×C4⋊Dic53C2311yescharacteristic
87C2×C4⋊Dic53C2311yescharacteristic
88C2×C4⋊Dic53C2311yescharacteristic
89C2×C4⋊Dic53C2311yescharacteristic
90C2×#11719919—28no—
91C2×#11719919—21no—
92C2×#11719919—21no—
93C2×#11719919—21no—
94C2×#11719919—21no—
95C2×#11719919—21no—
96C2×#11719919—21no—
97C2×#11719919—21no—
98C2×#11719919—21no—
99C2×#11719919D411yescharacteristic
100C2×#11719919D411yescharacteristic
101C2×C22⋊Dic53—21no—
102C2×C22⋊Dic53—21no—
103C2×C4×Dic53—21no—
104C2×C4×Dic53—21no—
105C2×C4×Dic53—21no—
106C2×C4×Dic53C2311yescharacteristic
107C2×C4×Dic53C2311yescharacteristic
108#13133146—22no—
109#13133146—22no—
110#13133138—22no—
111#13133138Q814yescomplemented
112#13133128—22no—
113#13133127—22no—
114C8×Dic53—22no—
115#13133124—21no—
116#13133124—21no—
117#13133124—21no—
118C22×D4×C53C2311yescharacteristic
119C23×C4×C53—21no—
120C22×C8×C53C2311yescharacteristic
121C22×C8×C53C2311yescharacteristic
122C4×D4×C53—22no—
123C4×D4×C53C2×C414yes—
124C2×C4⋊C4×C53—21no—
125C2×C4⋊C4×C53—21no—
126C2×C4⋊C4×C53C2311yescharacteristic
127C2×C4⋊C4×C53C2311yescharacteristic
128C2×C4⋊C4×C53C2311yescharacteristic
129C2×C22⋊C4×C53—21no—
130C2×C42×C53C2311yescharacteristic
131C4⋊C8×C53—22no—
132#102×C53—22no—
133#102×C53—22no—
134C24×C53—21no—
135C2×D4×C53—21no—
136C2×D4×C53—21no—
137C2×D4×C53—21no—
138C2×D4×C53C22×C412yes—
139C2×D4×C53C2×Q812yes—
140C2×D4×C53#4112yes—
141C22×C4×C53—21no—
142C22×C4×C53—24no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—81no—
146C22×C4×C53—81no—
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2411yescharacteristic
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—24no—
154C22×Dic53—24no—
155C22×Dic53—21no—
156C22×Dic53—42no—
157C22×Dic53—42no—
158C22×Dic53—81no—
159C22×Dic53—81no—
160C22×Dic53C2×D411yescharacteristic
161C22×Dic53C2×D411yescharacteristic
162C22×Dic53C2×D411yescharacteristic
163C22×Dic53C2×D411yescharacteristic
164C2×Dic106—22no—
165C2×Dic106—22no—
166C2×Dic106—24no—
167C2×Dic106—81no—
168C2×Dic106—81no—
169C2×Dic106C22⋊C414yes—
170C2×C8×C53—21no—
171C2×C8×C53—21no—
172C2×C8×C53—21no—
173C2×C8×C53—21no—
174C2×C8×C53—21no—
175C2×C8×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—42no—
182C4⋊C4×C53—81no—
183C4⋊C4×C53C22×C412yes—
184C22⋊C4×C53—24no—
185C42×C53C22×C412yes—
186C42×C53C2×D412yes—
187C22⋊Dic53—24no—
188C22⋊Dic53—24no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—24no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53C22⋊C414yes—
196C4⋊Dic53C22×C412yes—
197C4⋊Dic53C22×C412yes—
198C4⋊Dic53C2×Q812yes—
199#11719919—24no—
200#11719919—24no—
201#11719919—24no—
202#11719919—24no—
203#11719919—42no—
204#11719919—42no—
205#11719919—81no—
206#11719919—81no—
207#11719919C22⋊C414yes—
208#11719919C22⋊C414yes—
209C4×Dic53—42no—
210C4×Dic53—42no—
211C4×Dic53—42no—
212C4×Dic53C22×C412yes—
213C4×Dic53C22×C412yes—
214C4×Dic53C2×D412yes—
215C4×Dic53C2×D412yes—
216C2×#88322—21no—
217C2×#88322—21no—
218C2×#88322—21no—
219C2×#88322—21no—
220C2×#88322—21no—
221C2×#88322C2×Q812yes—
222C23×C53—22no—
223C23×C53—22no—
224C23×C53—22no—
225C23×C53—41no—
226C23×C53—41no—
227C23×C53—41no—
228C23×C53—41no—
229C23×C53C22×D411yescharacteristic
230D4×C53—22no—
231D4×C53—22no—
232D4×C53—22no—
233D4×C53C4×Q814yescomplemented
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—22no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—84no—
248C2×C4×C53C2×C22⋊C412yes—
249C2×C4×C53C2×C22⋊C412yes—
250C2×C4×C53C23×C411yescharacteristic
251C2×C4×C53C22×D411yescharacteristic
252C2×C4×C53C22×Q811yescharacteristic
253C2×C4×C53C2×#4111yescharacteristic
254C2×C4×C532−1+411yescharacteristic
255C2×C4×C532−1+411yescharacteristic
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—22no—
263C2×Dic53—22no—
264C2×Dic53—22no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—48no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—84no—
276C2×Dic53C2×C22⋊C412yes—
277C2×Dic53C2×C22⋊C412yes—
278C2×Dic53C2×C22⋊C412yes—
279C2×Dic53C2×C22⋊C412yes—
280C2×Dic53AGL(1,ℤ/8)12yes—
281C2×Dic53#13712yes—
282Dic106—28no—
283Dic106—84no—
284C8×C53—41no—
285C8×C53—41no—
286#88322—22no—
287#88322—41no—
288#22522—531nomaximal, Sylow
289C22×C53—24no—
290C22×C53—42no—
291C22×C53—42no—
292C22×C53—42no—
293C22×C53—42no—
294C22×C53—42no—
295C22×C53—42no—
296C22×C53C22×C22⋊C411yescharacteristic
297C22×C53#54811yescharacteristic
298C22×C53D4×Q811yescharacteristic
299C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C53C2×#13711yescharacteristic
301C22×C53#57611yescharacteristic
302C22×C53#57711yescharacteristic
303C4×C53—22no—
304C4×C53—22no—
305C4×C53—42no—
306C4×C53—81no—
307C4×C53—81no—
308C4×C53C2×C4×Q811yescharacteristic
309C4×C53#51911yescharacteristic
310C4×C53#54811yescharacteristic
311C4×C53D4×Q811yescharacteristic
312Dic53—22no—
313Dic53—22no—
314Dic53—22no—
315Dic53—42no—
316Dic53—42no—
317Dic53—42no—
318Dic53—81no—
319Dic53#41714yes—
320C2×#554—531no—
321C2×#446—531no—
322Q8×C4⋊C4—531no—
323#1992—531no—
324#1992—531no—
325#1990—531no—
326#1989—531no—
327#1989—531no—
328#1987—531no—
329#1802—531no—
330#1802—531no—
331#1802D5311yescharacteristic
332#1762—531no—
333#1762—531no—
334C4×#102—531no—
335C2×C53—44no—
336C2×C53Q8×C22⋊C411yescharacteristic
337C2×C53C2×#41711yescharacteristic
338C2×C53#289711yescharacteristic
339C2×C53#336611yescharacteristic
340C2×C53#337211yescharacteristic
341C2×C53#338411yescharacteristic
342C2×C53#338511yescharacteristic
343#554—538no—
344#554—1064no—
345C2×C4⋊Q8—531no—
346C2×#125—531no—
347C2×#125—531no—
348C2×C22⋊Q8—1061no—
349C2×C22⋊Q8—1061no—
350C2×C22⋊Q8—1061no—
351C2×C4×Q8—531no—
352C2×C4×Q8—1061no—
353C2×C4×D4—531no—
354C2×C4×D4—531no—
355C2×C4×D4C2×D5311yescharacteristic
356C22×C4⋊C4—1061no—
357#446—1064no—
358C2×C4⋊C8—531no—
359C2×C4⋊C8—531no—
360C2×C4⋊C8C2×D5311yescharacteristic
361C2×#102—531no—
362C2×#102—531no—
363C2×#102C2×D5311yescharacteristic
364C2×#102C2×D5311yescharacteristic
365C2×#97—531no—
366C2×#97—531no—
367C2×C4×C8—531no—
368#389—1061no—
369#389—1061no—
370#389—1061no—
371#384—531no—
372#384—531no—
373#384C2×D5311yescharacteristic
374C4×C4⋊C4—531no—
375C4×C4⋊C4—531no—
376C4×C4⋊C4—531no—
377#340—531no—
378#340—531no—
379#340—531no—
380#340—531no—
381#340—531no—
382#340—531no—
383#340—531no—
384#340—531no—
385#340—531no—
386#340—531no—
387#340C2×D5311yescharacteristic
388#340C2×D5311yescharacteristic
389C53#2252211yescharacteristic, complemented, Sylow
390C22×Q8—531no—
391C22×Q8—1061no—
392C22×D4C22×D5311yescharacteristic
393C23×C4—21no—
394C23×C4—1061no—
395C23×C4—1061no—
396C22×C8—531no—
397C22×C8—531no—
398C22×C8C22×D5311yescharacteristic
399C22×C8C22×D5311yescharacteristic
400C4⋊Q8—534no—
401C4⋊Q8—1062no—
402#125—534no—
403#125—534no—
404#125—1062no—
405#125—1062no—
406C22⋊Q8—1068no—
407C22⋊Q8—1068no—
408C22⋊Q8—1068no—
409C4×Q8—534no—
410C4×Q8—534no—
411C4×Q8—2124no—
412C4×D4—22no—
413C4×D4—534no—
414C4×D4—534no—
415C4×D4—1062no—
416C4×D4—1062no—
417C4×D4C4×D5314yes—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—531no—
421C2×C4⋊C4—531no—
422C2×C4⋊C4—531no—
423C2×C4⋊C4—531no—
424C2×C4⋊C4—531no—
425C2×C4⋊C4—531no—
426C2×C4⋊C4—531no—
427C2×C4⋊C4—1068no—
428C2×C4⋊C4—1061no—
429C2×C4⋊C4—1061no—
430C2×C4⋊C4—1061no—
431C2×C4⋊C4—1061no—
432C2×C4⋊C4—1061no—
433C2×C4⋊C4—1061no—
434C2×C4⋊C4—1061no—
435C2×C4⋊C4—1061no—
436C2×C4⋊C4—1061no—
437C2×C4⋊C4C22×D5311yescharacteristic
438C2×C4⋊C4C22×D5311yescharacteristic
439C2×C4⋊C4C22×D5311yescharacteristic
440C2×C22⋊C4—21no—
441C2×C22⋊C4—1061no—
442C2×C22⋊C4—1061no—
443C2×C42—531no—
444C2×C42—531no—
445C2×C42—1061no—
446C2×C42—1061no—
447C2×C42—1061no—
448C2×C42C22×D5311yescharacteristic
449C4⋊C8—22no—
450C4⋊C8—1062no—
451C4⋊C8—1062no—
452#102—22no—
453#102—22no—
454#102—534no—
455#102—1062no—
456#97—1062no—
457#97—1062no—
458C4×C8—1062no—
459#95—1061no—
460#95—1061no—
461#95—1061no—
462C24—21no—
463C2×Q8—534no—
464C2×Q8—1062no—
465C2×Q8—1062no—
466C2×Q8—1064no—
467C2×Q8—4241no—
468C2×Q8—4241no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4C2×C4×D5312yes—
473C2×D4#1171994712yes—
474C2×D4Q8×D5312yes—
475C22×C4—21no—
476C22×C4—24no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—81no—
480C22×C4—81no—
481C22×C4—531no—
482C22×C4—531no—
483C22×C4—531no—
484C22×C4—531no—
485C22×C4—1064no—
486C22×C4—1064no—
487C22×C4—1061no—
488C22×C4—1061no—
489C22×C4—1061no—
490C22×C4—1061no—
491C22×C4—4241no—
492C22×C4—4241no—
493C22×C4—2122no—
494C22×C4—2122no—
495C22×C4D4×D5311yescharacteristic
496C22×C4D4×D5311yescharacteristic
497C22×C4C23×D5311yescharacteristic
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—532no—
505C2×C8—1061no—
506C2×C8—1061no—
507C2×C8—1061no—
508C2×C8—1061no—
509C2×C8—1061no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—42no—
516C4⋊C4—81no—
517C4⋊C4—534no—
518C4⋊C4—534no—
519C4⋊C4—534no—
520C4⋊C4—532no—
521C4⋊C4—532no—
522C4⋊C4—532no—
523C4⋊C4—1064no—
524C4⋊C4—1064no—
525C4⋊C4—1064no—
526C4⋊C4—1064no—
527C4⋊C4—1064no—
528C4⋊C4—1061no—
529C4⋊C4—1061no—
530C4⋊C4—1061no—
531C4⋊C4—1061no—
532C4⋊C4—1061no—
533C4⋊C4—4241no—
534C4⋊C4—4241no—
535C4⋊C4—2122no—
536C4⋊C4—2122no—
537C4⋊C4C2×C4×D5312yes—
538C22⋊C4—24no—
539C22⋊C4—1064no—
540C22⋊C4—1064no—
541C42—532no—
542C42—532no—
543C42—532no—
544C42—532no—
545C42—2122no—
546C42—2122no—
547C42—2122no—
548C42C2×C4×D5312yes—
549C42D4×D5312yes—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23C2×D4×D5311yescharacteristic
558Q8—1068no—
559Q8—4244no—
560D4—22no—
561D4—22no—
562D4—22no—
563D4#1313318214yescomplemented
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—84no—
578C2×C4—532no—
579C2×C4—532no—
580C2×C4—532no—
581C2×C4—532no—
582C2×C4—532no—
583C2×C4—532no—
584C2×C4—1061no—
585C2×C4—1061no—
586C2×C4—1061no—
587C2×C4—1061no—
588C2×C4—1061no—
589C2×C4—1061no—
590C2×C4—1062no—
591C2×C4—1062no—
592C2×C4—1062no—
593C2×C4—4244no—
594C2×C4—2128no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4—2121no—
603C2×C4—2121no—
604C2×C4C22⋊C4×D5312yes—
605C2×C4C22⋊C4×D5312yes—
606C2×C4C22×C4×D5311yescharacteristic
607C2×C4C2×D4×D5311yescharacteristic
608C2×C4C2×Q8×D5311yescharacteristic
609C2×C4C2×#1171994711yescharacteristic
610C2×C4#1313326411yescharacteristic
611C2×C4#1313326411yescharacteristic
612C8—41no—
613C8—41no—
614C8—1062no—
615C8—2121no—
616C22—24no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22C2×C22⋊C4×D5311yescharacteristic
624C22#1681127911yescharacteristic
625C22#1681128611yescharacteristic
626C22AGL(1,ℤ/8)×D5311yescharacteristic
627C22#137×D5311yescharacteristic
628C22#1681142411yescharacteristic
629C22#1681142911yescharacteristic
630C4—22no—
631C4—22no—
632C4—42no—
633C4—81no—
634C4—81no—
635C4—534no—
636C4—1062no—
637C4—1062no—
638C4—1062no—
639C4—4241no—
640C4—2122no—
641C4—2122no—
642C4—2122no—
643C4C2×#1313318211yescharacteristic
644C4#1681110011yescharacteristic
645C4#1681127911yescharacteristic
646C4#1681128611yescharacteristic
647C2—44no—
648C2#2027754711yescharacteristic
649C2#417×D5311yescharacteristic
650C2#2028323611yescharacteristic
651C2#2028741711yescharacteristic
652C2#2028749111yescharacteristic
653C2#2028760911yescharacteristic
654C2#2028762011yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1802C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
cosocleC22×C4×C53C24
hypercenter#1802D53
hypocenterC53#22522
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. C2×#16810752C2semidirect
3. #20277627C2semidirect
4. #20275683C2semidirect
5. #20275682C2semidirect
6. #20275674C2semidirect
7. #20275671C2semidirect
8. #20275670C2semidirect
9. #20275661C2semidirect
17. #16811332C4semidirect
111. #13133138Q8semidirect
233. D4×C53C4×Q8semidirect
389. C53#22522semidirect
563. D4#13133182semidirect

Maximal subgroups

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

subgroupindex
C2×#168113322
C2×#168107522
#202776272
#202756832
#202756822
#202756742
#202756712
#202756702
#202756612
#202747132
#202745422
#202745402
#102×Dic532
C53×#18022
#2252253