Finite Group Databasebeta

← #29709560

Subgroups — #29709560

Order 15616.

876 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11758200C211yesmaximal, characteristic, complemented
2C2×#16839824C211yesmaximal, characteristic, complemented
3#21448506C211yesmaximal, characteristic
4#21448502C211yesmaximal, characteristic, complemented
5#21447323C211yesmaximal, characteristic, complemented
6#21447258C211yesmaximal, characteristic, complemented
7#21447201C211yesmaximal, characteristic, complemented
8#21447189C211yesmaximal, characteristic, complemented
9#21447135C211yesmaximal, characteristic, complemented
10#21447018C211yesmaximal, characteristic, complemented
11#21447006C211yesmaximal, characteristic
12#21446823C211yesmaximal, characteristic
13C4×#14519708C211yesmaximal, characteristic, complemented
14#21445650C211yesmaximal, characteristic, complemented
15C61×#1762C211yesmaximal, characteristic, complemented
16C22×#11758200C2211yescharacteristic
17C2×#14519757C2211yescharacteristic
18C2×#14519748C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20C2×#14519734C2211yescharacteristic
21C4×#11758200—24no—
22#16839824—24no—
23C2×#14519708C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×#14519706C2211yescharacteristic
27C2×#14519706C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×#14519689C2211yescharacteristic
33C2×#14519689C2211yescharacteristic
34C2×C4×#92418C2211yescharacteristic
35#16839667C2211yescharacteristic
36#16839665C2211yescharacteristic
37#16839662—21no—
38#16839661C2211yescharacteristic
39#16839660—21no—
40#16839658C2211yescharacteristic
41#16839656—21no—
42#16839654—21no—
43#16839653—21no—
44#16839647—21no—
45#16839641—21no—
46#16839635—21no—
47#16839619C2211yescharacteristic
48#16839619C2211yescharacteristic
49C4×C4⋊Dic61C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54C2×C4×D4×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×#97×C61C2211yescharacteristic
58C4×C4⋊C4×C61C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C22×C22⋊D61—21no—
63C22×C4×D61C2311yescharacteristic
64C2×#11758200—22no—
65C2×#11758200—22no—
66C2×#11758200—22no—
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69#14519757—22no—
70#14519757—22no—
71#14519748—22no—
72#14519748—22no—
73C4×Dic122—22no—
74C4×Dic122—22no—
75#14519734—22no—
76#14519734—22no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196D411yescharacteristic
85C2×#11758196D411yescharacteristic
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198D411yescharacteristic
93C2×#11758198D411yescharacteristic
94C2×C4×Dic61—21no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61C2311yescharacteristic
97#14519708—22no—
98#14519708—22no—
99#14519708—22no—
100#14519706—22no—
101#14519706—22no—
102#14519702—22no—
103#14519702—22no—
104#14519700—22no—
105#14519698—22no—
106#14519689—22no—
107#14519689—22no—
108C4×#92418D414yescomplemented
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116C22×D4×C61C2311yescharacteristic
117C23×C4×C61—21no—
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C4×D4×C61—22no—
121C4×D4×C61—22no—
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C22⋊C4×C61—21no—
126C2×C42×C61—21no—
127C2×C42×C61C2311yescharacteristic
128#102×C61—22no—
129#102×C61—22no—
130#97×C61—22no—
131#95×C61—21no—
132C24×C61—21no—
133C23×D61C2×D411yescharacteristic
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61#4112yes—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—42no—
144C22×C4×C61—81no—
145C22×C4×C61—81no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2411yescharacteristic
149C2×C22⋊D61—24no—
150C2×C22⋊D61—81no—
151C2×C22⋊D61—81no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×C4×D61—21no—
162C2×C4×D61—21no—
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122#4112yes—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61#4112yes—
183C22⋊C4×C61—42no—
184C42×C61—81no—
185C42×C61C2×D412yes—
186C42×C61#4112yes—
187C22⋊Dic61—24no—
188C22⋊Dic61—81no—
189C22⋊Dic61—81no—
190#11758200—28no—
191#11758198—24no—
192#11758198—22no—
193#11758198—22no—
194#11758198—81no—
195#11758198—81no—
196C4⋊Dic61—24no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—81no—
203C4⋊Dic61#4112yes—
204#11758196—22no—
205#11758196—22no—
206#11758196—42no—
207#11758196—81no—
208C4×Dic61—21no—
209C4×Dic61—21no—
210C4×Dic61—42no—
211C4×Dic61—81no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418C2×D412yes—
217C2×#92418C2×D412yes—
218C23×C61—22no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61C22×D411yescharacteristic
226C22×D61—21no—
227C22×D61—21no—
228C22×D61—21no—
229C22×D61—21no—
230C22×D61—21no—
231C22×D61—21no—
232C22×D61—21no—
233D4×C61—22no—
234D4×C61—22no—
235D4×C61—41no—
236D4×C61—41no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—22no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—84no—
251C2×C4×C61#12112yes—
252C2×C4×C61#12112yes—
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C22⋊D61—84no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—22no—
265C2×Dic61—22no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C4×D61—41no—
282C4×D61—41no—
283C4×D61—41no—
284C4×D61—41no—
285Dic122—22no—
286Dic122—22no—
287Dic122—41no—
288Dic122—41no—
289C8×C61—41no—
290C8×C61—41no—
291#92418—22no—
292#92418—22no—
293#23707—611nomaximal, Sylow
294C22×C61—42no—
295C22×C61—42no—
296C22×C61—42no—
297C22×C61—81no—
298C22×C61—81no—
299C22×C61—81no—
300C22×C61—81no—
301C22×C61C2×#12111yescharacteristic
302C22×C61#52511yescharacteristic
303C22×C61#52511yescharacteristic
304C22×C61#57711yescharacteristic
305C22×C61#57711yescharacteristic
306C22×C61#57711yescharacteristic
307C22×C61#57711yescharacteristic
308C2×D61—41no—
309C2×D61—41no—
310C2×D61—41no—
311C2×D61—41no—
312C2×D61—41no—
313C2×D61—41no—
314C2×D61—41no—
315C4×C61—22no—
316C4×C61—22no—
317C4×C61—42no—
318C4×C61—81no—
319C4×C61—81no—
320C4×C61#52511yescharacteristic
321C4×C61#52911yescharacteristic
322C4×C61#52911yescharacteristic
323C4×C61#53211yescharacteristic
324Dic61—42no—
325Dic61—42no—
326Dic61—41no—
327Dic61—41no—
328Dic61—41no—
329Dic61—81no—
330Dic61—81no—
331C2×C4×SD16—611no—
332C2×#433—611no—
333#2324—611no—
334#2323—611no—
335#2088—611no—
336#2077—611no—
337#2066—611no—
338#2065—611no—
339#2055—611no—
340#2035—611no—
341#2034—611no—
342#2000—611no—
343C4×#106—611no—
344#1763—611no—
345#1762D6111yescharacteristic, complemented
346C2×C61—82no—
347C2×C61#239211yescharacteristic
348C2×C61#305711yescharacteristic
349C2×C61#306011yescharacteristic
350C2×C61#312311yescharacteristic
351C2×C61#312511yescharacteristic
352C2×C61#312511yescharacteristic
353C2×C61#312611yescharacteristic
354D61—81no—
355C22×SD16—611no—
356C2×C22⋊Q8—611no—
357C2×#121—611no—
358C2×C4×Q8—611no—
359C2×C4×D4C2×D6111yescharacteristic
360C2×#117—611no—
361C4×SD16—1224no—
362#433—1224no—
363C2×#106—611no—
364C2×#106—611no—
365C2×#106—611no—
366C2×C4⋊C8—611no—
367C2×C4⋊C8—611no—
368C2×#103—611no—
369C2×#103—611no—
370C2×#103—611no—
371C2×#102—611no—
372C2×#102C2×D6111yescharacteristic
373C2×#102C2×D6111yescharacteristic
374C2×C22⋊C8—611no—
375C2×C22⋊C8—611no—
376C2×#97C2×D6111yescharacteristic
377C2×C4×C8—611no—
378#391—611no—
379#390—611no—
380#390—1221no—
381#389—611no—
382#389—611no—
383#389—1221no—
384#388—1221no—
385#388—1221no—
386#388—1221no—
387#387—1221no—
388#386—1221no—
389#385—1221no—
390#380—611no—
391#380—611no—
392C4×C4⋊C4—611no—
393C4×C4⋊C4C2×D6111yescharacteristic
394#340—611no—
395#340—611no—
396#340—611no—
397#340—611no—
398#340C2×D6111yescharacteristic
399#340C2×D6111yescharacteristic
400C61#2370711yescharacteristic, complemented, Sylow
401C22×Q8—611no—
402C22×D4—1221no—
403C22×D4C22×D6111yescharacteristic
404C23×C4—21no—
405C23×C4—611no—
406C2×SD16—1222no—
407C2×SD16—1222no—
408C2×SD16—1222no—
409C22×C8—611no—
410C22×C8—611no—
411C22×C8C22×D6111yescharacteristic
412C22×C8C22×D6111yescharacteristic
413C22⋊Q8—1222no—
414C22⋊Q8—1222no—
415#121—1222no—
416#121—1222no—
417C4×Q8—1222no—
418C4×Q8—1222no—
419C4×D4—22no—
420C4×D4—22no—
421#117—1222no—
422#117—1222no—
423C2×C4⋊C4—611no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—611no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4—1221no—
431C2×C4⋊C4—1221no—
432C2×C4⋊C4C22×D6111yescharacteristic
433C2×C4⋊C4C22×D6111yescharacteristic
434C2×C4⋊C4C22×D6111yescharacteristic
435C2×C22⋊C4—21no—
436C2×C22⋊C4—611no—
437C2×C22⋊C4—611no—
438C2×C22⋊C4—1221no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1221no—
443C2×C22⋊C4—1221no—
444C2×C42—21no—
445C2×C42—611no—
446C2×C42—1221no—
447C2×C42—1221no—
448C2×C42C22×D6111yescharacteristic
449#106—1222no—
450#106—1222no—
451#106—1222no—
452C4⋊C8—1222no—
453C4⋊C8—1222no—
454#103—1222no—
455#103—1222no—
456#103—1222no—
457#102—22no—
458#102—22no—
459#102—1222no—
460C22⋊C8—1222no—
461C22⋊C8—1222no—
462#97—22no—
463C4×C8—614no—
464#95—21no—
465#95—1221no—
466#95—1221no—
467#95—1221no—
468#95—1221no—
469#95—1221no—
470#95—1221no—
471#95—1221no—
472C24—21no—
473C24—611no—
474C2×Q8—612no—
475C2×Q8—1221no—
476C2×Q8—1221no—
477C2×Q8—1221no—
478C2×Q8—1221no—
479C2×Q8—1221no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—1224no—
486C2×D4—4881no—
487C2×D4—4881no—
488C2×D4#1175822212yes—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—42no—
493C22×C4—81no—
494C22×C4—81no—
495C22×C4—611no—
496C22×C4—611no—
497C22×C4—611no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4—1221no—
505C22×C4—1221no—
506C22×C4—1221no—
507C22×C4—1221no—
508C22×C4D4×D6111yescharacteristic
509C22×C4D4×D6111yescharacteristic
510C22×C4C23×D6111yescharacteristic
511SD16—1228no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—612no—
519C2×C8—612no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—1221no—
523C2×C8—1221no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—612no—
530C4⋊C4—1224no—
531C4⋊C4—1221no—
532C4⋊C4—1222no—
533C4⋊C4—1222no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—4881no—
539C4⋊C4—4881no—
540C4⋊C4—2442no—
541C4⋊C4#1175822212yes—
542C22⋊C4—42no—
543C22⋊C4—1224no—
544C22⋊C4—1224no—
545C22⋊C4—1222no—
546C22⋊C4—1222no—
547C22⋊C4—4881no—
548C22⋊C4—4881no—
549C22⋊C4—4881no—
550C22⋊C4—4881no—
551C42—81no—
552C42—1221no—
553C42—1221no—
554C42—4881no—
555C42—2442no—
556C42#1175822212yes—
557C42D4×D6112yes—
558C23—22no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—1221no—
566C23—1221no—
567C23—1221no—
568C23—1221no—
569C23—1221no—
570C23—1221no—
571C23—1221no—
572C23C2×D4×D6111yescharacteristic
573Q8—1222no—
574Q8—1222no—
575Q8—2441no—
576Q8—2441no—
577D4—22no—
578D4—22no—
579D4—41no—
580D4—41no—
581D4—4884no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—84no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1222no—
601C2×C4—1222no—
602C2×C4—1221no—
603C2×C4—1221no—
604C2×C4—1221no—
605C2×C4—1221no—
606C2×C4—1221no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4#1451975012yes—
622C2×C4#1451975012yes—
623C2×C4C2×D4×D6111yescharacteristic
624C2×C4#41×D6111yescharacteristic
625C2×C4#41×D6111yescharacteristic
626C2×C4C2×#1175822211yescharacteristic
627C2×C4C2×#1175822211yescharacteristic
628C2×C4C2×#1175822211yescharacteristic
629C8—41no—
630C8—41no—
631C8—1222no—
632C8—1222no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—2441no—
641C22—2441no—
642C22—2441no—
643C22—2441no—
644C22—2441no—
645C22—2441no—
646C22—2441no—
647C22C2×#1451975011yescharacteristic
648C22#1684027111yescharacteristic
649C22#1684027111yescharacteristic
650C22#1684057111yescharacteristic
651C22#1684057211yescharacteristic
652C22#1684057211yescharacteristic
653C22#1684057211yescharacteristic
654C4—22no—
655C4—22no—
656C4—42no—
657C4—81no—
658C4—81no—
659C4—4881no—
660C4—4881no—
661C4—2441no—
662C4—2441no—
663C4—2442no—
664C4—2441no—
665C4—2442no—
666C4#1684027111yescharacteristic
667C4#1684029011yescharacteristic
668C4#1684029611yescharacteristic
669C4#1684030911yescharacteristic
670C2—82no—
671C2—4881no—
672C2#2144906311yescharacteristic
673C2#2145545711yescharacteristic
674C2#2145548011yescharacteristic
675C2#2145607011yescharacteristic
676C2#2145610011yescharacteristic
677C2#2145610011yescharacteristic
678C2#2145610911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1762C2
socleC23×C61C22×D4
prime radicalC2×#14519748C22
hypercenter#1762D61
hypocenterC61#23707
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
1. C2×C4×#11758200C2semidirect
2. C2×#16839824C2semidirect
4. #21448502C2semidirect
5. #21447323C2semidirect
6. #21447258C2semidirect
7. #21447201C2semidirect
8. #21447189C2semidirect
9. #21447135C2semidirect
10. #21447018C2semidirect
13. C4×#14519708C2semidirect
14. #21445650C2semidirect
15. C61×#1762C2semidirect
108. C4×#92418D4semidirect
345. #1762D61semidirect
400. C61#23707semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117582002
C2×#168398242
#214485062
#214485022
#214473232
#214472582
#214472012
#214471892
#214471352
#214470182
#214470062
#214468232
C4×#145197082
#214456502
C61×#17622
#2370761