Finite Group Databasebeta

← #29742896

Subgroups — #29742896

Order 15616.

860 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840061C211yesmaximal, characteristic, complemented
2C2×#16839803C211yesmaximal, characteristic, complemented
3#21449546C211yesmaximal, characteristic, complemented
4#21448859C211yesmaximal, characteristic
5#21447343C211yesmaximal, characteristic, complemented
6#21447323C211yesmaximal, characteristic, complemented
7#21447192C211yesmaximal, characteristic, complemented
8#21447156C211yesmaximal, characteristic, complemented
9#21446390C211yesmaximal, characteristic, complemented
10#21446256C211yesmaximal, characteristic
11#21446135C211yesmaximal, characteristic
12#21446063C211yesmaximal, characteristic, complemented
13#21446017C211yesmaximal, characteristic, complemented
14#21446000C211yesmaximal, characteristic, complemented
15C61×#1949C211yesmaximal, characteristic, complemented
16C22×#11758189C2211yescharacteristic
17C2×#14519768C2211yescharacteristic
18C2×#14519752C2211yescharacteristic
19C2×#14519739C2211yescharacteristic
20C2×#14519735C2211yescharacteristic
21C2×C4⋊C4×D61C2211yescharacteristic
22#16840061—24no—
23#16839803—24no—
24C2×#14519710C2211yescharacteristic
25C2×#14519710C2211yescharacteristic
26C2×#14519709C2211yescharacteristic
27C2×#14519708C2211yescharacteristic
28C2×#14519706C2211yescharacteristic
29C2×#14519699C2211yescharacteristic
30C2×#14519699C2211yescharacteristic
31C2×#14519697C2211yescharacteristic
32C2×#14519697C2211yescharacteristic
33C2×#14519689C2211yescharacteristic
34#16839703—21no—
35#16839696—21no—
36#16839664—21no—
37#16839664C2211yescharacteristic
38#16839661C2211yescharacteristic
39#16839660C2211yescharacteristic
40#16839658C2211yescharacteristic
41#16839654—21no—
42#16839640—21no—
43#16839636—21no—
44#16839621—21no—
45#16839619C2211yescharacteristic
46C4×#11758198—21no—
47C4×#11758198C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839517C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×#107×C61C2211yescharacteristic
55C2×#106×C61C2211yescharacteristic
56C2×C4⋊C8×C61C2211yescharacteristic
57C61×#382C2211yescharacteristic
58C4×C4⋊C4×C61C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×D244—21no—
62C22×D244C2311yescharacteristic
63C22×C4×D61—21no—
64C22×C4×D61—21no—
65C22×C4×D61C2311yescharacteristic
66C2×#11758189—22no—
67C2×#11758189—22no—
68C2×#11758189—22no—
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71#14519768—22no—
72#14519768—41no—
73#14519752—22no—
74#14519752Q814yes—
75#14519739—22no—
76#14519739—41no—
77#14519735—22no—
78#14519735—41no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196—21no—
88C4⋊C4×D61—22no—
89C4⋊C4×D61—41no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198—21no—
94C2×#11758198—21no—
95C2×#11758198—21no—
96C2×#11758198D411yescharacteristic
97C2×#11758198D411yescharacteristic
98C2×C4×Dic61—21no—
99C2×C4×Dic61C2311yescharacteristic
100#14519710—41no—
101#14519710D414yescomplemented
102#14519709—22no—
103#14519708—22no—
104#14519706—22no—
105#14519699—22no—
106#14519699—22no—
107#14519697—22no—
108#14519697—41no—
109#14519689—22no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684D411yescharacteristic
115#14519684D411yescharacteristic
116C22×C8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C42×C61—21no—
123C2×C42×C61C2311yescharacteristic
124#107×C61—22no—
125#106×C61—22no—
126C4⋊C8×C61—22no—
127#95×C61—21no—
128#95×C61—21no—
129#95×C61—21no—
130C23×D61—21no—
131C23×D61C2×D411yescharacteristic
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2411yescharacteristic
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61C2×D411yescharacteristic
143C22×Dic61C2×D411yescharacteristic
144C22×Dic61C2×D411yescharacteristic
145C2×D244—21no—
146C2×D244—21no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—24no—
151C2×D244—81no—
152C2×D244—81no—
153C2×D244C2×Q812yes—
154C2×C4×D61—21no—
155C2×C4×D61—21no—
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—42no—
159C2×C4×D61—42no—
160C2×C4×D61—42no—
161C2×C4×D61—42no—
162C2×C4×D61—81no—
163C2×C4×D61—81no—
164C2×C4×D61C2×Q812yes—
165C2×C4×D61#4112yes—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61#4112yes—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61C2×D412yes—
179C4⋊C4×C61C2×Q812yes—
180C42×C61—21no—
181C42×C61—21no—
182C42×C61—81no—
183#11758198—24no—
184#11758198—41no—
185#11758198—41no—
186#11758198—42no—
187#11758198—42no—
188#11758198—42no—
189#11758198—42no—
190#11758198—81no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—42no—
197C4⋊Dic61—81no—
198C4⋊Dic61C2×D412yes—
199#11758196—42no—
200#11758196—42no—
201#11758196—81no—
202#11758196—81no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61—42no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418C2×D412yes—
212#11758189—28no—
213C23×C61C22×D411yescharacteristic
214C22×D61—21no—
215C22×D61—21no—
216C22×D61—21no—
217C22×D61—21no—
218C22×D61—21no—
219C22×D61—21no—
220C22×D61—22no—
221C22×D61—41no—
222C22×D61—41no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61C22⋊Q812yes—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C22×Q811yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61H(5,𝔽2)11yescharacteristic
251C2×C4×C612−1+411yescharacteristic
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—22no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61C22⋊Q812yes—
270D244—22no—
271D244—22no—
272D244—41no—
273D244—41no—
274D244—84no—
275C4×D61—22no—
276C4×D61—22no—
277C4×D61—22no—
278C4×D61—22no—
279C4×D61—84no—
280C4×D61—84no—
281C8×C61—22no—
282C8×C61—41no—
283#92418—22no—
284#92418—41no—
285#26363—611nomaximal, Sylow
286C22×C61C2×C22⋊Q811yescharacteristic
287C22×C61D4211yescharacteristic
288C22×C61#54711yescharacteristic
289C22×C61#57511yescharacteristic
290C22×C61#57511yescharacteristic
291C22×C61#57611yescharacteristic
292C22×C61#57711yescharacteristic
293C2×D61—22no—
294C2×D61—22no—
295C2×D61—22no—
296C2×D61—41no—
297C2×D61—41no—
298C2×D61—41no—
299C2×D61—41no—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61—42no—
303C2×D61—81no—
304C2×D61—81no—
305C2×D61—81no—
306C2×D61—81no—
307C4×C61—42no—
308C4×C61—41no—
309C4×C61—41no—
310C4×C61—42no—
311C4×C61—81no—
312C4×C61—81no—
313C4×C61#52511yescharacteristic
314C4×C61#53911yescharacteristic
315C4×C61D4×Q811yescharacteristic
316C4×C61#55411yescharacteristic
317Dic61—22no—
318Dic61—41no—
319Dic61—41no—
320Dic61—42no—
321Dic61—81no—
322Dic61—81no—
323C2×#478—611no—
324C2×#428—611no—
325#2442—611no—
326#2363—611no—
327#2092—611no—
328#2088—611no—
329#2065—611no—
330#2059—611no—
331#1949D6111yescharacteristic, complemented
332#1918—611no—
333#1895—611no—
334#1874—611no—
335#1862—611no—
336#1854—611no—
337#1851—611no—
338C2×C61#244911yescharacteristic
339C2×C61#308011yescharacteristic
340C2×C61#308111yescharacteristic
341C2×C61#327111yescharacteristic
342C2×C61#327311yescharacteristic
343C2×C61#334811yescharacteristic
344C2×C61#335511yescharacteristic
345D61—42no—
346D61—82no—
347C22×M16—611no—
348C2×#125—611no—
349C2×#121—611no—
350C2×C4×D4—611no—
351C2×#117—611no—
352C22×C4⋊C4—611no—
353#478—1224no—
354#428—1224no—
355C2×#107—611no—
356C2×#107—611no—
357C2×#107C2×D6111yescharacteristic
358C2×#106—611no—
359C2×#106—611no—
360C2×#106C2×D6111yescharacteristic
361C2×C4⋊C8—611no—
362C2×C4⋊C8C2×D6111yescharacteristic
363C2×#102—611no—
364C2×#102—611no—
365C2×#102—611no—
366C2×#102—611no—
367C2×C22⋊C8—611no—
368#400—1221no—
369#398—1221no—
370#390—611no—
371#390—1221no—
372#389—611no—
373#389—611no—
374#389—611no—
375#388—1221no—
376#386—1221no—
377#385—1221no—
378#382C2×D6111yescharacteristic
379#380—611no—
380#380—1221no—
381C4×C4⋊C4C2×D6111yescharacteristic
382C4×C22⋊C4—611no—
383C4×C22⋊C4—1221no—
384#340—611no—
385#340—611no—
386#340—611no—
387#340—611no—
388#340C2×D6111yescharacteristic
389#340C2×D6111yescharacteristic
390#336—611no—
391#336—611no—
392C61#2636311yescharacteristic, complemented, Sylow
393C22×D4—611no—
394C22×D4—1221no—
395C23×C4—611no—
396C23×C4—1221no—
397C23×C4—1221no—
398C2×M16—1222no—
399C2×M16—1222no—
400C2×M16—1222no—
401C22×C8—611no—
402C22×C8—611no—
403C22×C8C22×D6111yescharacteristic
404C22×C8C22×D6111yescharacteristic
405#125—1222no—
406#125—2441no—
407#121—614no—
408#121—1222no—
409C4×D4—1222no—
410C4×D4—2441no—
411#117—1222no—
412#117—2441no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—1222no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—2441no—
424C2×C4⋊C4C22×D6111yescharacteristic
425C2×C4⋊C4C22×D6111yescharacteristic
426C2×C4⋊C4C22×D6111yescharacteristic
427C2×C4⋊C4C22×D6111yescharacteristic
428C2×C22⋊C4—611no—
429C2×C22⋊C4—611no—
430C2×C22⋊C4—1221no—
431C2×C22⋊C4—1221no—
432C2×C22⋊C4—1221no—
433C2×C22⋊C4—1221no—
434C2×C22⋊C4—1221no—
435C2×C22⋊C4—1221no—
436C2×C42—21no—
437C2×C42—611no—
438C2×C42—1221no—
439C2×C42C22×D6111yescharacteristic
440#107—22no—
441#107—614no—
442#107—2441no—
443#106—22no—
444#106—1222no—
445#106—1222no—
446C4⋊C8—22no—
447C4⋊C8—1222no—
448#102—1222no—
449#102—1222no—
450#102—1222no—
451#102—2441no—
452C22⋊C8—1222no—
453#95—21no—
454#95—21no—
455#95—21no—
456#95—611no—
457#95—611no—
458#95—1221no—
459#95—1221no—
460#95—1221no—
461#95—1221no—
462C24—611no—
463C24—1221no—
464C2×D4—612no—
465C2×D4—1224no—
466C2×D4—1221no—
467C2×D4—1221no—
468C2×D4—1221no—
469C2×D4—1221no—
470C2×D4—1221no—
471C2×D4—4881no—
472C2×D4—4881no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—611no—
478C22×C4—612no—
479C22×C4—612no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—4881no—
490C22×C4—4881no—
491C22×C4—2442no—
492C22×C4—2442no—
493C22×C4—2442no—
494C22×C4—2442no—
495C22×C4D4×D6111yescharacteristic
496C22×C4D4×D6111yescharacteristic
497C22×C4C23×D6111yescharacteristic
498M16—1228no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—612no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8#1175822212yes—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—612no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—4881no—
524C4⋊C4—4881no—
525C4⋊C4—4881no—
526C4⋊C4—2442no—
527C4⋊C4—2442no—
528C4⋊C4—2442no—
529C4⋊C4D4×D6112yes—
530C4⋊C4Q8×D6112yes—
531C22⋊C4—1224no—
532C22⋊C4—4881no—
533C22⋊C4—2441no—
534C22⋊C4—2441no—
535C22⋊C4—2442no—
536C22⋊C4—2442no—
537C22⋊C4—2442no—
538C22⋊C4—2442no—
539C42—21no—
540C42—21no—
541C42—81no—
542C42—1221no—
543C42—1221no—
544C42—2442no—
545C23—612no—
546C23—1221no—
547C23—1221no—
548C23—1221no—
549C23—1221no—
550C23—1221no—
551C23—1221no—
552C23—1222no—
553C23—2441no—
554C23—2441no—
555C23—2441no—
556C23—2441no—
557C23—2441no—
558C23—2441no—
559C23C2×D4×D6111yescharacteristic
560D4—1222no—
561D4—1222no—
562D4—4884no—
563D4—2441no—
564D4—2441no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—612no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—1222no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1222no—
597C2×C4—4884no—
598C2×C4—4884no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4C2×D4×D6111yescharacteristic
608C2×C4C2×Q8×D6111yescharacteristic
609C2×C4#41×D6111yescharacteristic
610C2×C4C2×#1175822211yescharacteristic
611C2×C4#1451982311yescharacteristic
612C2×C4#1451982511yescharacteristic
613C8—22no—
614C8—41no—
615C8—1222no—
616C8—2441no—
617C22—1222no—
618C22—1222no—
619C22—1222no—
620C22—4881no—
621C22—4881no—
622C22—4881no—
623C22—4881no—
624C22—2442no—
625C22—2442no—
626C22—2442no—
627C22—2441no—
628C22—2441no—
629C22—2441no—
630C22—2441no—
631C22C22⋊Q8×D6111yescharacteristic
632C22#1684039611yescharacteristic
633C22#1684041311yescharacteristic
634C22#1684055911yescharacteristic
635C22#1684056111yescharacteristic
636C22#1684056611yescharacteristic
637C22#1684056911yescharacteristic
638C4—42no—
639C4—42no—
640C4—41no—
641C4—41no—
642C4—81no—
643C4—81no—
644C4—1222no—
645C4—4881no—
646C4—4881no—
647C4—2441no—
648C4—2441no—
649C4—2442no—
650C4#1684027111yescharacteristic
651C4#1684036011yescharacteristic
652C4#1684042711yescharacteristic
653C4#1684047211yescharacteristic
654C2—4882no—
655C2—2442no—
656C2#2144963511yescharacteristic
657C2#2145563811yescharacteristic
658C2#2145564911yescharacteristic
659C2#2145741011yescharacteristic
660C2#2145742511yescharacteristic
661C2#2145833011yescharacteristic
662C2#2145842111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1949C2
socleC23×C61C22×D4
prime radicalC2×#14519752C22
hypercenter#1949D61
hypocenterC61#26363
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
1. C2×#16840061C2semidirect
2. C2×#16839803C2semidirect
3. #21449546C2semidirect
5. #21447343C2semidirect
6. #21447323C2semidirect
7. #21447192C2semidirect
8. #21447156C2semidirect
9. #21446390C2semidirect
12. #21446063C2semidirect
13. #21446017C2semidirect
14. #21446000C2semidirect
15. C61×#1949C2semidirect
101. #14519710D4semidirect
331. #1949D61semidirect
392. C61#26363semidirect

Maximal subgroups

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

subgroupindex
C2×#168400612
C2×#168398032
#214495462
#214488592
#214473432
#214473232
#214471922
#214471562
#214463902
#214462562
#214461352
#214460632
#214460172
#214460002
C61×#19492
#2636361