Finite Group Databasebeta

← #27320673

Subgroups — #27320673

Order 13568.

1822 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133216C211yesmaximal, characteristic
2C2×#16810544C212yesmaximal
3C2×#16810520C212yesmaximal
4C2×C4⋊C4×Dic53C211yesmaximal, characteristic
5#20272985C212yesmaximal
6#20272980C211yesmaximal, characteristic
7#20272978C211yesmaximal, characteristic
8#20272968C211yesmaximal, characteristic
9#20272943C211yesmaximal, characteristic
10#20272943C212yesmaximal
11C53×#1441C211yesmaximal, characteristic
12C23×Dic106C2212yes—
13C2×#13133216—28no—
14C2×#13133216—24no—
15C22×C4⋊Dic53C2211yescharacteristic
16C22×C4⋊Dic53C2212yes—
17C22×C4⋊Dic53C2211yescharacteristic
18C22×#11719919C2212yes—
19C22×#11719919C2212yes—
20C22×#11719919C2212yes—
21C22×C4×Dic53C2211yescharacteristic
22C22×C4×Dic53C2211yescharacteristic
23C22×C4×Dic53C2211yescharacteristic
24#16810544—28no—
25#16810520—28no—
26C4⋊C4×Dic53—24no—
27C2×#13133124C2211yescharacteristic
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2211yescharacteristic
31C2×#13133124C2211yescharacteristic
32C2×#13133124C2212yes—
33C2×#13133124C2212yes—
34C2×#13133124C2212yes—
35C22×C4⋊C4×C53C2211yescharacteristic
36C22×C4⋊C4×C53C2211yescharacteristic
37C2×#95×C53C2211yescharacteristic
38C2×#95×C53C2212yes—
39C2×#95×C53C2212yes—
40C22×Dic106—24no—
41C22×Dic106—24no—
42C22×Dic106—24no—
43C22×Dic106—24no—
44C22×Dic106—24no—
45C22×Dic106—42no—
46C22×Dic106—42no—
47C23×Dic53C2311yescharacteristic
48C23×Dic53C2312yes—
49C23×Dic53C2311yescharacteristic
50C23×Dic53C2312yes—
51C23×Dic53C2311yescharacteristic
52C23×Dic53C2311yescharacteristic
53#13133216—232no—
54C2×C4⋊Dic53—22no—
55C2×C4⋊Dic53—24no—
56C2×C4⋊Dic53—22no—
57C2×C4⋊Dic53—24no—
58C2×C4⋊Dic53—24no—
59C2×C4⋊Dic53—22no—
60C2×C4⋊Dic53—42no—
61C2×C4⋊Dic53D414yes—
62C2×#11719919—24no—
63C2×#11719919—24no—
64C2×#11719919—24no—
65C2×#11719919—24no—
66C2×#11719919—24no—
67C2×#11719919—24no—
68C2×#11719919—42no—
69C2×#11719919D418yes—
70C2×#11719919Q818yescomplemented
71C2×C4×Dic53—24no—
72C2×C4×Dic53—22no—
73C2×C4×Dic53—24no—
74C2×C4×Dic53—22no—
75C2×C4×Dic53—22no—
76C2×C4×Dic53D414yes—
77C2×C4×Dic53Q814yescomplemented
78#13133124—24no—
79#13133124—22no—
80#13133124—22no—
81#13133124—24no—
82#13133124—24no—
83#13133124—22no—
84#13133124—42no—
85#13133124—42no—
86C23×C4×C53C2312yes—
87C23×C4×C53C2311yescharacteristic
88C23×C4×C53C2312yes—
89C23×C4×C53C2311yescharacteristic
90C23×C4×C53C2311yescharacteristic
91C2×C4⋊C4×C53—24no—
92C2×C4⋊C4×C53—22no—
93C2×C4⋊C4×C53—22no—
94C2×C4⋊C4×C53—22no—
95C2×C4⋊C4×C53Q814yescomplemented
96#95×C53—24no—
97#95×C53—24no—
98#95×C53—22no—
99C24×C53C2411yescharacteristic
100C22×C4×C53—22no—
101C22×C4×C53—21no—
102C22×C4×C53—21no—
103C22×C4×C53—21no—
104C22×C4×C53—21no—
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—21no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—21no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53C2×D414yes—
123C22×C4×C53C2×D412yes—
124C22×C4×C53C2×D412yes—
125C22×C4×C53C2×Q812yes—
126C22×C4×C53C2×Q814yes—
127C22×C4×C53C2×Q812yes—
128C22×C4×C53C2×Q812yes—
129C22×C4×C53C2×Q812yes—
130C22×C4×C53#4112yes—
131C22×Dic53—22no—
132C22×Dic53—22no—
133C22×Dic53—21no—
134C22×Dic53—22no—
135C22×Dic53—21no—
136C22×Dic53—22no—
137C22×Dic53—22no—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—21no—
143C22×Dic53—22no—
144C22×Dic53—21no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53C2×D414yes—
154C22×Dic53C2×D412yes—
155C22×Dic53C2×D412yes—
156C22×Dic53C2×D412yes—
157C22×Dic53C2×D412yes—
158C22×Dic53C2×D414yes—
159C22×Dic53C2×Q812yes—
160C22×Dic53C2×Q814yes—
161C22×Dic53C2×Q812yes—
162C22×Dic53C2×Q814yes—
163C22×Dic53#4112yes—
164C22×Dic53#4112yes—
165C2×Dic106—216no—
166C2×Dic106—48no—
167C2×Dic106—48no—
168C2×Dic106—48no—
169C2×Dic106—48no—
170C2×Dic106—48no—
171C2×Dic106—48no—
172C4⋊C4×C53—28no—
173C4⋊C4×C53—28no—
174C4⋊Dic53—28no—
175C4⋊Dic53—28no—
176C4⋊Dic53—48no—
177#11719919—216no—
178#11719919—216no—
179#11719919—48no—
180C4×Dic53—28no—
181C4×Dic53—28no—
182C4×Dic53—44no—
183C23×C53C22×D411yescharacteristic
184C23×C53C22×D412yes—
185C23×C53C22×D411yescharacteristic
186C23×C53C22×Q811yescharacteristic
187C23×C53C22×Q812yes—
188C23×C53C22×Q811yescharacteristic
189C23×C53C2×#4111yescharacteristic
190C23×C53C2×#4111yescharacteristic
191C23×C53C2×#4111yescharacteristic
192C23×C53H(5,𝔽2)12yes—
193C23×C532−1+412yes—
194C2×C4×C53—22no—
195C2×C4×C53—22no—
196C2×C4×C53—22no—
197C2×C4×C53—22no—
198C2×C4×C53—22no—
199C2×C4×C53—22no—
200C2×C4×C53—22no—
201C2×C4×C53—22no—
202C2×C4×C53—24no—
203C2×C4×C53—24no—
204C2×C4×C53—24no—
205C2×C4×C53—24no—
206C2×C4×C53—22no—
207C2×C4×C53—24no—
208C2×C4×C53—24no—
209C2×C4×C53—24no—
210C2×C4×C53—24no—
211C2×C4×C53—24no—
212C2×C4×C53—41no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53—42no—
216C2×C4×C53—41no—
217C2×C4×C53—42no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
220C2×C4×C53—42no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×C4×C53C22⋊Q814yes—
224C2×C4×C53C4⋊Q814yescomplemented
225C2×Dic53—22no—
226C2×Dic53—22no—
227C2×Dic53—24no—
228C2×Dic53—24no—
229C2×Dic53—24no—
230C2×Dic53—22no—
231C2×Dic53—22no—
232C2×Dic53—24no—
233C2×Dic53—24no—
234C2×Dic53—24no—
235C2×Dic53—24no—
236C2×Dic53—24no—
237C2×Dic53—24no—
238C2×Dic53—24no—
239C2×Dic53—41no—
240C2×Dic53—42no—
241C2×Dic53—42no—
242C2×Dic53—42no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—42no—
247C2×Dic53—42no—
248C2×Dic53—42no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251C2×Dic53—42no—
252C2×Dic53—42no—
253C2×Dic53#12014yes—
254C2×Dic53#12114yes—
255C2×Dic53C22⋊Q814yes—
256C2×Dic53C22⋊Q818yes—
257C2×Dic53C4⋊Q814yes—
258C2×Dic53C4⋊Q818yes—
259Dic106—432no—
260#15128—531nomaximal, Sylow
261C22×C53C2×#12011yescharacteristic
262C22×C53C2×#12111yescharacteristic
263C22×C53C2×C22⋊Q811yescharacteristic
264C22×C53C2×C22⋊Q812yes—
265C22×C53C2×C22⋊Q811yescharacteristic
266C22×C53C2×C4⋊Q811yescharacteristic
267C22×C53C2×C4⋊Q812yes—
268C22×C53C2×C4⋊Q811yescharacteristic
269C22×C53#53211yescharacteristic
270C22×C53#53311yescharacteristic
271C22×C53#53311yescharacteristic
272C22×C53#53511yescharacteristic
273C22×C53#53611yescharacteristic
274C22×C53#54212yes—
275C22×C53#54412yes—
276C22×C53D4212yes—
277C22×C53#54612yes—
278C22×C53#54812yes—
279C22×C53D4×Q812yes—
280C22×C53D4×Q812yes—
281C22×C53#55412yes—
282C22×C53#55712yes—
283C22×C53Q8212yes—
284C4×C53—24no—
285C4×C53—24no—
286C4×C53—44no—
287C4×C53—42no—
288C4×C53—44no—
289Dic53—24no—
290Dic53—42no—
291Dic53—44no—
292Dic53—42no—
293Dic53—44no—
294Dic53#39518yescomplemented
295C22×C4⋊Q8—531no—
296C2×#395—532no—
297C2×#389—532no—
298C2×C4×C4⋊C4—531no—
299#1446—532no—
300#1444—531no—
301#1444—531no—
302#1441—531no—
303#1441D5311yescharacteristic
304C4⋊#95—532no—
305C4⋊#95—531no—
306C2×C53C2×#39511yescharacteristic
307C2×C53#240811yescharacteristic
308C2×C53#240911yescharacteristic
309C2×C53#245112yes—
310C2×C53#245212yes—
311C2×C53#254611yescharacteristic
312C2×C53#255012yes—
313C2×C53#255112yes—
314C2×C53#256711yescharacteristic
315C2×C53#262811yescharacteristic
316C2×C53#285711yescharacteristic
317C23×Q8—532no—
318C2×C4⋊Q8—1064no—
319C2×C4⋊Q8—1068no—
320C22×C4⋊C4—532no—
321C22×C4⋊C4—532no—
322C22×C4⋊C4—531no—
323C22×C4⋊C4—531no—
324C22×C4⋊C4—532no—
325C22×C4⋊C4—532no—
326C22×C4⋊C4C2×D5311yescharacteristic
327C22×C4⋊C4C2×D5311yescharacteristic
328C22×C42—531no—
329C22×C42—531no—
330C22×C42—531no—
331#395—1068no—
332#389—1068no—
333C4×C4⋊C4—1064no—
334C2×#95—532no—
335C2×#95—532no—
336C2×#95—531no—
337C2×#95—532no—
338C2×#95—531no—
339C2×#95—531no—
340C2×#95—532no—
341C2×#95—532no—
342C2×#95C2×D5312yes—
343C2×#95C2×D5311yescharacteristic
344C2×#95C2×D5312yes—
345C53#1512811yescharacteristic, complemented, Sylow
346C22×Q8—1064no—
347C22×Q8—1064no—
348C22×Q8—1064no—
349C22×Q8—1064no—
350C22×Q8—1064no—
351C22×Q8—2122no—
352C22×Q8—2122no—
353C23×C4—531no—
354C23×C4—531no—
355C23×C4—531no—
356C23×C4—532no—
357C23×C4—532no—
358C23×C4—531no—
359C23×C4C22×D5311yescharacteristic
360C23×C4C22×D5312yes—
361C23×C4C22×D5312yes—
362C23×C4C22×D5311yescharacteristic
363C23×C4C22×D5311yescharacteristic
364C4⋊Q8—10632no—
365C2×C4⋊C4—24no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—538no—
370C2×C4⋊C4—534no—
371C2×C4⋊C4—538no—
372C2×C4⋊C4—1064no—
373C2×C4⋊C4—1064no—
374C2×C4⋊C4—1064no—
375C2×C4⋊C4—1064no—
376C2×C4⋊C4—1064no—
377C2×C4⋊C4—1062no—
378C2×C4⋊C4—1062no—
379C2×C4⋊C4—1062no—
380C2×C4⋊C4—1064no—
381C2×C4⋊C4—1064no—
382C2×C4⋊C4—1064no—
383C2×C4⋊C4—1064no—
384C2×C4⋊C4—2122no—
385C2×C4⋊C4—2122no—
386C2×C4⋊C4Dic10614yescomplemented
387C2×C42—534no—
388C2×C42—534no—
389C2×C42—1064no—
390C2×C42—1062no—
391C2×C42—1062no—
392C2×C42—1062no—
393C2×C42—1064no—
394#95—24no—
395#95—24no—
396#95—22no—
397#95—1062no—
398#95—1064no—
399#95—1064no—
400#95—1062no—
401#95—1064no—
402#95—1062no—
403#95—2122no—
404#95—2122no—
405C24C23×D5311yescharacteristic
406C2×Q8—10616no—
407C2×Q8—2128no—
408C2×Q8—2128no—
409C2×Q8—2128no—
410C2×Q8—2128no—
411C2×Q8—2128no—
412C2×Q8—2128no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—532no—
436C22×C4—534no—
437C22×C4—532no—
438C22×C4—534no—
439C22×C4—532no—
440C22×C4—532no—
441C22×C4—532no—
442C22×C4—532no—
443C22×C4—534no—
444C22×C4—532no—
445C22×C4—532no—
446C22×C4—534no—
447C22×C4—1062no—
448C22×C4—1062no—
449C22×C4—1061no—
450C22×C4—1062no—
451C22×C4—1061no—
452C22×C4—1062no—
453C22×C4—1062no—
454C22×C4—1062no—
455C22×C4—1062no—
456C22×C4—1062no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—1062no—
460C22×C4—1062no—
461C22×C4—1062no—
462C22×C4—1062no—
463C22×C4—1062no—
464C22×C4—1062no—
465C22×C4—1062no—
466C22×C4—1062no—
467C22×C4—1062no—
468C22×C4—1062no—
469C22×C4C2×Dic10612yes—
470C22×C4C2×Dic10612yes—
471C22×C4C2×Dic10612yes—
472C22×C4D4×D5314yes—
473C22×C4D4×D5312yes—
474C22×C4D4×D5312yes—
475C22×C4#1171994712yes—
476C22×C4Q8×D5314yes—
477C22×C4Q8×D5312yes—
478C4⋊C4—28no—
479C4⋊C4—28no—
480C4⋊C4—10616no—
481C4⋊C4—10616no—
482C4⋊C4—1068no—
483C4⋊C4—1068no—
484C4⋊C4—2128no—
485C4⋊C4—2128no—
486C42—1068no—
487C42—1068no—
488C42—2124no—
489C23C2×D4×D5312yes—
490C23C2×D4×D5311yescharacteristic
491C23C2×D4×D5311yescharacteristic
492C23C2×Q8×D5312yes—
493C23C2×Q8×D5311yescharacteristic
494C23C22×Dic10611yescharacteristic
495C23#41×D5311yescharacteristic
496C23#41×D5311yescharacteristic
497C23C2×#1171994711yescharacteristic
498C23#1313326312yes—
499C23#1313326412yes—
500Q8—21232no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—534no—
531C2×C4—534no—
532C2×C4—538no—
533C2×C4—534no—
534C2×C4—538no—
535C2×C4—534no—
536C2×C4—1064no—
537C2×C4—1062no—
538C2×C4—1062no—
539C2×C4—1064no—
540C2×C4—1064no—
541C2×C4—1064no—
542C2×C4—1064no—
543C2×C4—1064no—
544C2×C4—1062no—
545C2×C4—1062no—
546C2×C4—1064no—
547C2×C4—1064no—
548C2×C4—1064no—
549C2×C4—1064no—
550C2×C4—2122no—
551C2×C4—2122no—
552C2×C4—2122no—
553C2×C4—2121no—
554C2×C4—2122no—
555C2×C4—2122no—
556C2×C4—2121no—
557C2×C4—2121no—
558C2×C4—2122no—
559C2×C4—2121no—
560C2×C4—2122no—
561C2×C4—2122no—
562C2×C4—2122no—
563C2×C4—2122no—
564C2×C4#1313319414yes—
565C2×C4#1313321614yescomplemented
566C22#120×D5311yescharacteristic
567C22#121×D5311yescharacteristic
568C22C22⋊Q8×D5312yes—
569C22C22⋊Q8×D5311yescharacteristic
570C22C2×#1313319411yescharacteristic
571C22C4⋊Q8×D5311yescharacteristic
572C22C4⋊Q8×D5312yes—
573C22C2×#1313321611yescharacteristic
574C22#1681116811yescharacteristic
575C22#1681117511yescharacteristic
576C22#1681117611yescharacteristic
577C22#1681119111yescharacteristic
578C22#1681119511yescharacteristic
579C22#1681123912yes—
580C22#1681125312yes—
581C22#1681125712yes—
582C22#1681126512yes—
583C22#1681127912yes—
584C22#1681128612yes—
585C22D4×Dic10612yes—
586C22#1681133112yes—
587C22#1681135012yes—
588C22Q8×Dic10612yes—
589C4—24no—
590C4—24no—
591C4—44no—
592C4—44no—
593C4—42no—
594C4—538no—
595C4—1064no—
596C4—2124no—
597C4—2122no—
598C4—2124no—
599C4—2122no—
600C2#395×D5311yescharacteristic
601C2#2027806611yescharacteristic
602C2#2027807711yescharacteristic
603C2#2027855312yes—
604C2#2027855812yes—
605C2#2027969911yescharacteristic
606C2#2027973112yes—
607C2#2027974312yes—
608C2#2027992511yescharacteristic
609C2#2028054311yescharacteristic
610C2#2028306511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutator, socle, prime radical, cosocleC24×C53C24
FittingC53×#1441C2
hypercenter#1441D53
hypocenterC53#15128
LCSG ▷ C24×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
70. C2×#11719919Q8semidirect
77. C2×C4×Dic53Q8semidirect
95. C2×C4⋊C4×C53Q8semidirect
224. C2×C4×C53C4⋊Q8semidirect
294. Dic53#395semidirect
345. C53#15128semidirect
386. C2×C4⋊C4Dic106semidirect
565. C2×C4#13133216semidirect

Maximal subgroups

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

subgroupindex
C22×#131332162
C2×#168105442
C2×#168105202
C2×C4⋊C4×Dic532
#202729852
#202729802
#202729782
#202729682
#202729432
C53×#14412
#1512853