Finite Group Databasebeta

← #27279464

Subgroups — #27279464

Order 13568.

1136 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×C22⋊#88322C211yesmaximal, characteristic, complemented
2#20274625C211yesmaximal, characteristic, complemented
3C2×#16810347C212yesmaximal, complemented
4#20272558C212yesmaximal, complemented
5C2×C53×#386C211yesmaximal, characteristic
6C23×#88322—21no—
7C2×C22⋊#88322—24no—
8C2×C22⋊#88322—21no—
9C2×C22⋊#88322—21no—
10C2×C22⋊#88322—21no—
11C2×C22⋊#88322C2212yes—
12C2×C22⋊#88322C2212yes—
13#16810347—24no—
14C23×D4×C53C411yescharacteristic
15C24×C4×C53C2211yescharacteristic
16C22×C22⋊C4×C53C411yescharacteristic
17C22×C22⋊C4×C53C411yescharacteristic
18C22×C22⋊C4×C53C2211yescharacteristic
19C53×#386—21no—
20C53×#386—21no—
21C53×#386—21no—
22C53×#386—21no—
23C2×#95×C53C411yescharacteristic
24C2×#95×C53C2211yescharacteristic
25C22×#88322—24no—
26C22×#88322—22no—
27C22×#88322—22no—
28C22×#88322—42no—
29C22×#88322—42no—
30C22⋊#88322—24no—
31C22⋊#88322—22no—
32C22⋊#88322—22no—
33C22⋊#88322—22no—
34C22⋊#88322—42no—
35C22⋊#88322—42no—
36C22⋊#88322D414yescomplemented
37C25×C53C2×C411yescharacteristic
38C25×C53C2×C411yescharacteristic
39C22×D4×C53—21no—
40C22×D4×C53—21no—
41C22×D4×C53—22no—
42C22×D4×C53—42no—
43C22×D4×C53—42no—
44C22×D4×C53C814yescomplemented
45C23×C4×C53—22no—
46C23×C4×C53—22no—
47C23×C4×C53—42no—
48C23×C4×C53—42no—
49C23×C4×C53C2×C411yescharacteristic
50C23×C4×C53C2×C411yescharacteristic
51C23×C4×C53C2×C411yescharacteristic
52C23×C4×C53C2×C411yescharacteristic
53C23×C4×C53D411yescharacteristic
54C23×C4×C53D411yescharacteristic
55C23×C4×C53D411yescharacteristic
56C23×C4×C53D411yescharacteristic
57C23×C4×C53C2311yescharacteristic
58C2×C22⋊C4×C53—21no—
59C2×C22⋊C4×C53—21no—
60C2×C22⋊C4×C53—21no—
61C2×C22⋊C4×C53—21no—
62C2×C22⋊C4×C53—22no—
63C2×C22⋊C4×C53—21no—
64C2×C22⋊C4×C53—21no—
65C2×C22⋊C4×C53—22no—
66C2×C22⋊C4×C53—42no—
67C2×C22⋊C4×C53—42no—
68#95×C53—21no—
69#95×C53—21no—
70#95×C53—21no—
71#95×C53—21no—
72C24×C53—22no—
73C24×C53—22no—
74C24×C53—22no—
75C24×C53—22no—
76C24×C53—21no—
77C24×C53—21no—
78C24×C53—22no—
79C24×C53—22no—
80C24×C53—41no—
81C24×C53—41no—
82C24×C53—41no—
83C24×C53—41no—
84C24×C53—41no—
85C24×C53—41no—
86C24×C53C22⋊C411yescharacteristic
87C24×C53C22⋊C411yescharacteristic
88C24×C53C2×C812yes—
89C24×C53C2×C812yescomplemented
90C24×C53M1612yes—
91C24×C53C22×C411yescharacteristic
92C2×D4×C53—24no—
93C2×D4×C53—24no—
94C2×D4×C53—48no—
95C2×D4×C53—42no—
96C2×D4×C53—42no—
97C2×D4×C53—42no—
98C2×D4×C53—42no—
99C2×D4×C53—48no—
100C22×C4×C53—22no—
101C22×C4×C53—22no—
102C22×C4×C53—21no—
103C22×C4×C53—21no—
104C22×C4×C53—22no—
105C22×C4×C53—21no—
106C22×C4×C53—22no—
107C22×C4×C53—21no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—21no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—21no—
114C22×C4×C53—21no—
115C22×C4×C53—21no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—41no—
119C22×C4×C53—41no—
120C22×C4×C53—41no—
121C22×C4×C53—41no—
122C22×C4×C53—41no—
123C22×C4×C53—41no—
124C22×C4×C53—41no—
125C22×C4×C53—48no—
126C22×C4×C53—48no—
127C22×C4×C53—41no—
128C22×C4×C53—41no—
129C22×C4×C53—41no—
130C22×C4×C53—41no—
131C22×C4×C53—41no—
132C22×C4×C53—41no—
133C22×C4×C53C22⋊C411yescharacteristic
134C22×C4×C53C22⋊C411yescharacteristic
135C22×C4×C53C2×C812yescomplemented
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D412yes—
139C22×C4×C53#4112yes—
140C22⋊C4×C53—42no—
141C22⋊C4×C53—42no—
142C22⋊C4×C53—42no—
143C22⋊C4×C53—42no—
144C22⋊C4×C53—41no—
145C22⋊C4×C53—42no—
146C22⋊C4×C53—41no—
147C2×#88322—24no—
148C2×#88322—48no—
149C2×#88322—42no—
150C2×#88322—42no—
151C2×#88322—42no—
152C2×#88322—42no—
153C2×#88322—42no—
154C23×C53—21no—
155C23×C53—21no—
156C23×C53—21no—
157C23×C53—21no—
158C23×C53—21no—
159C23×C53—21no—
160C23×C53—21no—
161C23×C53—21no—
162C23×C53—24no—
163C23×C53—24no—
164C23×C53—21no—
165C23×C53—21no—
166C23×C53—21no—
167C23×C53—21no—
168C23×C53—21no—
169C23×C53—21no—
170C23×C53—24no—
171C23×C53—24no—
172C23×C53—22no—
173C23×C53—24no—
174C23×C53—42no—
175C23×C53—42no—
176C23×C53—44no—
177C23×C53—44no—
178C23×C53—44no—
179C23×C53—44no—
180C23×C53—42no—
181C23×C53—42no—
182C23×C53—42no—
183C23×C53—42no—
184C23×C53—41no—
185C23×C53—41no—
186C23×C53—41no—
187C23×C53—41no—
188C23×C53—41no—
189C23×C53—41no—
190C23×C53—42no—
191C23×C53—82no—
192C23×C53—82no—
193C23×C53—82no—
194C23×C53—82no—
195C23×C53C22⋊C812yes—
196C23×C53C22⋊C812yes—
197C23×C53#9912yes—
198C23×C53#10012yes—
199C23×C53C2×C22⋊C411yescharacteristic
200C23×C53C4×D412yes—
201C23×C53C22×C811yescharacteristic
202C23×C53C2×M1611yescharacteristic
203C23×C53#13112yes—
204D4×C53—48no—
205D4×C53—48no—
206C2×C4×C53—24no—
207C2×C4×C53—24no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—22no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
220C2×C4×C53—44no—
221C2×C4×C53—44no—
222C2×C4×C53—44no—
223C2×C4×C53—42no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—42no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—82no—
236C2×C4×C53—82no—
237C2×C4×C53#9912yes—
238C2×C4×C53#12011yescharacteristic
239C2×C4×C53#12112yes—
240C2×C4×C53#12311yescharacteristic
241#88322—44no—
242#88322—82no—
243#9079—531nomaximal, Sylow
244C22×C53—22no—
245C22×C53—22no—
246C22×C53—22no—
247C22×C53—22no—
248C22×C53—22no—
249C22×C53—21no—
250C22×C53—21no—
251C22×C53—21no—
252C22×C53—21no—
253C22×C53—22no—
254C22×C53—22no—
255C22×C53—22no—
256C22×C53—22no—
257C22×C53—22no—
258C22×C53—22no—
259C22×C53—22no—
260C22×C53—22no—
261C22×C53—48no—
262C22×C53—48no—
263C22×C53—41no—
264C22×C53—41no—
265C22×C53—41no—
266C22×C53—41no—
267C22×C53—42no—
268C22×C53—42no—
269C22×C53—42no—
270C22×C53—84no—
271C22×C53—84no—
272C22×C53—81no—
273C22×C53—81no—
274C22×C53—81no—
275C22×C53—81no—
276C22×C53#32314yescomplemented
277C22×C53#38611yescharacteristic
278C22×C53C2×C22⋊C811yescharacteristic
279C22×C53#40811yescharacteristic
280C22×C53C2×#9911yescharacteristic
281C22×C53C2×#9911yescharacteristic
282C22×C53C2×#10011yescharacteristic
283C22×C53#41311yescharacteristic
284C22×C53C8×D412yes—
285C22×C53#43512yes—
286C4×C53—21no—
287C4×C53—22no—
288C4×C53—22no—
289C4×C53—21no—
290C4×C53—41no—
291C4×C53—42no—
292C4×C53—81no—
293C4×C53—81no—
294C22×C22⋊C8—531no—
295C2×#386D5311yescharacteristic
296#1779—531no—
297C2×#323—532no—
298#1280—532no—
299C2×C53—22no—
300C2×C53—22no—
301C2×C53—24no—
302C2×C53—42no—
303C2×C53—82no—
304C2×C53C2×#32311yescharacteristic
305C2×C53#145911yescharacteristic
306C2×C53#146811yescharacteristic
307C2×C53#147111yescharacteristic
308C2×C53#186911yescharacteristic
309C2×C53#189811yescharacteristic
310C2×C53#190011yescharacteristic
311C23×D4Dic5311yescharacteristic
312C24×C4C2×D5311yescharacteristic
313C23×C8—1061no—
314C22×C22⋊C4Dic5311yescharacteristic
315C22×C22⋊C4Dic5311yescharacteristic
316C22×C22⋊C4C2×D5311yescharacteristic
317C2×C22⋊C8—532no—
318C2×C22⋊C8—532no—
319C2×C22⋊C8—1064no—
320C2×C22⋊C8—1061no—
321C2×C22⋊C8—1061no—
322C2×C22⋊C8—1061no—
323#386—21no—
324#386—21no—
325#386—21no—
326#386—21no—
327C2×#95Dic5311yescharacteristic
328C2×#95C2×D5311yescharacteristic
329#323—1064no—
330C53#907911yescharacteristic, complemented, Sylow
331C25C2×Dic5311yescharacteristic
332C25C2×Dic5311yescharacteristic
333C22×D4—22no—
334C22×D4—21no—
335C22×D4—21no—
336C22×D4—42no—
337C22×D4—42no—
338C22×D4#8832214yescomplemented
339C23×C4—22no—
340C23×C4—22no—
341C23×C4—42no—
342C23×C4—42no—
343C23×C4C2×Dic5311yescharacteristic
344C23×C4C2×Dic5311yescharacteristic
345C23×C4C2×Dic5311yescharacteristic
346C23×C4C2×Dic5311yescharacteristic
347C23×C4C22⋊D5311yescharacteristic
348C23×C4C22⋊D5311yescharacteristic
349C23×C4C22⋊D5311yescharacteristic
350C23×C4C22⋊D5311yescharacteristic
351C23×C4C22×D5311yescharacteristic
352C22×C8—1064no—
353C22×C8—1062no—
354C22×C8—1062no—
355C22×C8—2122no—
356C22×C8—2122no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—42no—
366C2×C22⋊C4—42no—
367C22⋊C8—534no—
368C22⋊C8—1064no—
369C22⋊C8—1062no—
370C22⋊C8—1062no—
371C22⋊C8—1062no—
372C22⋊C8—2122no—
373C22⋊C8—2122no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—21no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—21no—
383C24—21no—
384C24—22no—
385C24—22no—
386C24—41no—
387C24—41no—
388C24—41no—
389C24—41no—
390C24—41no—
391C24—41no—
392C24C2×#8832212yes—
393C24C2×#8832212yescomplemented
394C24#1171991712yes—
395C24C22⋊Dic5311yescharacteristic
396C24C22⋊Dic5311yescharacteristic
397C24C22×Dic5311yescharacteristic
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—48no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—48no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4C2×#8832212yescomplemented
440C22×C4C22⋊Dic5311yescharacteristic
441C22×C4C22⋊Dic5311yescharacteristic
442C22×C4D4×D5312yes—
443C22×C4#1171994712yes—
444C22×C4C2×C22⋊D5311yescharacteristic
445C22×C4C2×C22⋊D5311yescharacteristic
446C2×C8—1064no—
447C2×C8—2122no—
448C2×C8—2122no—
449C2×C8—2122no—
450C2×C8—2128no—
451C2×C8—2122no—
452C2×C8—2122no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—41no—
458C22⋊C4—42no—
459C22⋊C4—41no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—24no—
474C23—24no—
475C23—21no—
476C23—24no—
477C23—24no—
478C23—22no—
479C23—24no—
480C23—42no—
481C23—42no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—82no—
498C23—82no—
499C23—82no—
500C23—82no—
501C23C22⋊#8832212yes—
502C23C22⋊#8832212yes—
503C23#1313313212yes—
504C23#1313313412yes—
505C23C2×C22⋊Dic5311yescharacteristic
506C23D4×Dic5312yes—
507C23C22×#8832211yescharacteristic
508C23C2×#1171991711yescharacteristic
509C23#1313322412yes—
510D4—48no—
511D4—48no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4#1313313212yes—
544C2×C4#1313318511yescharacteristic
545C2×C4#1313318912yes—
546C2×C4#1313319811yescharacteristic
547C8—4242no—
548C8—2124no—
549C22—22no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—48no—
567C22—48no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—84no—
576C22—84no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22#1681034714yescomplemented
582C22#1681050311yescharacteristic
583C22C2×C22⋊#8832211yescharacteristic
584C22#1681059011yescharacteristic
585C22C2×#1313313211yescharacteristic
586C22C2×#1313313211yescharacteristic
587C22C2×#1313313411yescharacteristic
588C22#1681060711yescharacteristic
589C22D4×#8832212yes—
590C22#1681069512yes—
591C4—22no—
592C4—21no—
593C4—22no—
594C4—21no—
595C4—42no—
596C4—41no—
597C4—81no—
598C4—81no—
599C2—22no—
600C2—22no—
601C2—24no—
602C2—42no—
603C2—82no—
604C2C2×#1681034711yescharacteristic
605C2#2027303311yescharacteristic
606C2#2027306311yescharacteristic
607C2#2027307511yescharacteristic
608C2#2027500811yescharacteristic
609C2#2027516911yescharacteristic
610C2#2027517911yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic53
commutatorC23×C53C22×C8
FrattiniC23×C4C22×D53
FittingC2×C53×#386C2
socleC23×C53C2×C22⋊C4
prime radicalC23×D4×C53C4
cosocleC23×C4×C53C23
hypercenterC2×#386D53
hypocenterC53#9079
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#386

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊#88322C2semidirect
2. #20274625C2semidirect
3. C2×#16810347C2semidirect
4. #20272558C2semidirect
36. C22⋊#88322D4semidirect
44. C22×D4×C53C8semidirect
89. C24×C53C2×C8semidirect
135. C22×C4×C53C2×C8semidirect
276. C22×C53#323semidirect
330. C53#9079semidirect
338. C22×D4#88322semidirect
393. C24C2×#88322semidirect
439. C22×C4C2×#88322semidirect
581. C22#16810347semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊#883222
#202746252
C2×#168103472
#202725582
C2×C53×#3862
#907953