Finite Group Databasebeta

← #29738885

Subgroups — #29738885

Order 15616.

782 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840171C211yesmaximal, characteristic, complemented
2C2×#16839804C211yesmaximal, characteristic
3#21449543C211yesmaximal, characteristic
4#21449510C211yesmaximal, characteristic
5#21447302C211yesmaximal, characteristic, complemented
6#21447302C211yesmaximal, characteristic, complemented
7#21447158C211yesmaximal, characteristic, complemented
8#21447148C211yesmaximal, characteristic, complemented
9#21446557C211yesmaximal, characteristic, complemented
10#21446552C211yesmaximal, characteristic, complemented
11#21446020C211yesmaximal, characteristic
12#21446009C211yesmaximal, characteristic
13#21445983C211yesmaximal, characteristic, complemented
14#21445852C211yesmaximal, characteristic
15C61×#1753C211yesmaximal, characteristic
16C2×#14519777C2211yescharacteristic
17C2×#14519769C2211yescharacteristic
18C2×#14519736C2211yescharacteristic
19C22×C4⋊Dic61C2211yescharacteristic
20#16840171—24no—
21#16839804—24no—
22C2×#14519711C2211yescharacteristic
23C2×#14519711C2211yescharacteristic
24C2×#14519711C2211yescharacteristic
25C2×#14519711C2211yescharacteristic
26C2×#14519711C2211yescharacteristic
27C2×#14519711C2211yescharacteristic
28C2×#14519709C2211yescharacteristic
29C2×#14519709C2211yescharacteristic
30C2×#14519709C2211yescharacteristic
31C2×#14519709C2211yescharacteristic
32#16839704—21no—
33#16839696—21no—
34#16839685—21no—
35#16839674—21no—
36#16839666—21no—
37#16839660C2211yescharacteristic
38#16839660C2211yescharacteristic
39#16839660C2211yescharacteristic
40#16839660C2211yescharacteristic
41#16839649—21no—
42#16839634—21no—
43#16839628C2211yescharacteristic
44#16839628C2211yescharacteristic
45#16839622C2211yescharacteristic
46#16839622C2211yescharacteristic
47#16839620—21no—
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53#16839525C2211yescharacteristic
54C22×M16×C61C2211yescharacteristic
55C2×C22⋊C8×C61C2211yescharacteristic
56C2×#97×C61C2211yescharacteristic
57C2×C4×C8×C61C2211yescharacteristic
58C4×C22⋊C4×C61C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×Dic122—21no—
62C23×Dic61—21no—
63#14519777—22no—
64#14519777—41no—
65#14519769—22no—
66#14519769—41no—
67#14519736—22no—
68#14519736—41no—
69C2×C4⋊Dic61—22no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—41no—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61—21no—
90C2×C4×Dic61—21no—
91#14519711—22no—
92#14519711—22no—
93#14519711—22no—
94#14519711—41no—
95#14519711—41no—
96#14519711D414yes—
97#14519709—22no—
98#14519709—22no—
99#14519709—22no—
100#14519709—22no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111C23×C4×C61C2311yescharacteristic
112C2×M16×C61—22no—
113C2×M16×C61—22no—
114C2×M16×C61Q814yescomplemented
115C22×C8×C61C2311yescharacteristic
116C22×C8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118C22×C8×C61C2311yescharacteristic
119C2×C22⋊C4×C61D411yescharacteristic
120C2×C22⋊C4×C61D411yescharacteristic
121C2×C42×C61C2311yescharacteristic
122C2×C42×C61C2311yescharacteristic
123C22⋊C8×C61Q814yescomplemented
124#97×C61—22no—
125C4×C8×C61—22no—
126#95×C61D411yescharacteristic
127#95×C61D411yescharacteristic
128C24×C61C2×D411yescharacteristic
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×Q812yes—
139C22×C4×C61C2×Q812yes—
140C22×C4×C61C2411yescharacteristic
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—42no—
150C22×Dic61—42no—
151C22×Dic61—81no—
152C2×Dic122—42no—
153C2×Dic122—42no—
154C2×Dic122—81no—
155M16×C61—28no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61C2×D412yes—
163C2×C8×C61C2×Q812yes—
164C2×C8×C61C2×Q812yes—
165C2×C8×C61C2×Q812yes—
166C2×C8×C61C2×Q812yes—
167C2×C8×C61#4112yes—
168C22⋊C4×C61—41no—
169C22⋊C4×C61—41no—
170C42×C61—21no—
171C42×C61—21no—
172C42×C61—21no—
173C42×C61—21no—
174C22⋊Dic61—42no—
175C22⋊Dic61—42no—
176C22⋊Dic61—81no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—42no—
192C4⋊Dic61—42no—
193C4⋊Dic61C2×D412yes—
194C4⋊Dic61C2×D412yes—
195#11758196—42no—
196#11758196—42no—
197#11758196—42no—
198#11758196—81no—
199#11758196—81no—
200#11758196—81no—
201C4×Dic61—42no—
202C23×C61—21no—
203C23×C61—21no—
204C23×C61—21no—
205C23×C61—21no—
206C23×C61—21no—
207C23×C61—21no—
208C23×C61C4⋊Q812yes—
209C23×C61C22×D411yescharacteristic
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61C4⋊Q812yes—
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C22×Q811yescharacteristic
231C2×C4×C61C22×Q811yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61H(5,𝔽2)11yescharacteristic
234C2×C4×C612−1+411yescharacteristic
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—84no—
260Dic122—84no—
261C8×C61—22no—
262C8×C61—22no—
263C8×C61—22no—
264C8×C61C22⋊Q814yes—
265#26079—611nomaximal, Sylow
266C22×C61—22no—
267C22×C61—22no—
268C22×C61—22no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61C2×C4⋊Q811yescharacteristic
274C22×C61D4211yescharacteristic
275C22×C61#54711yescharacteristic
276C22×C61#57511yescharacteristic
277C22×C61#57511yescharacteristic
278C22×C61#57611yescharacteristic
279C22×C61#57811yescharacteristic
280C4×C61—22no—
281C4×C61—41no—
282C4×C61—41no—
283C4×C61—41no—
284C4×C61—41no—
285C4×C61C2×C22⋊Q811yescharacteristic
286C4×C61#54411yescharacteristic
287C4×C61D4×Q811yescharacteristic
288C4×C61#55411yescharacteristic
289Dic61—42no—
290Dic61—42no—
291Dic61—81no—
292Dic61—81no—
293Dic61—81no—
294Dic61—81no—
295Dic61—81no—
296Dic61—81no—
297C2×#501—611no—
298C2×#428—611no—
299#2442—611no—
300#2439—611no—
301#2085—611no—
302#2085—611no—
303#2059—611no—
304#2057—611no—
305#1943—611no—
306#1942—611no—
307#1854—611no—
308#1852—611no—
309#1846—611no—
310#1809—611no—
311#1753D6111yescharacteristic
312C2×C61—42no—
313C2×C61#244011yescharacteristic
314C2×C61#316411yescharacteristic
315C2×C61#316611yescharacteristic
316C2×C61#327311yescharacteristic
317C2×C61#327711yescharacteristic
318C2×C61#334711yescharacteristic
319C2×C61#334911yescharacteristic
320C22×M16C2×D6111yescharacteristic
321C2×C4⋊Q8—611no—
322C2×#125—611no—
323C2×#117—611no—
324C22×C4⋊C4—611no—
325#501—1224no—
326#428—1224no—
327C2×#107—611no—
328C2×#107—611no—
329C2×#107—611no—
330C2×#107—611no—
331C2×#107—611no—
332C2×#107—611no—
333C2×#106—611no—
334C2×#106—611no—
335C2×#106—611no—
336C2×#106—611no—
337C2×C22⋊C8C2×D6111yescharacteristic
338C2×#97C2×D6111yescharacteristic
339C2×C4×C8C2×D6111yescharacteristic
340#400—1221no—
341#398—1221no—
342#396—1221no—
343#393—1221no—
344#391—1221no—
345#389—611no—
346#389—611no—
347#389—611no—
348#389—611no—
349#387—1221no—
350#385—1221no—
351#382—611no—
352#382—611no—
353#380—611no—
354#380—611no—
355#380—1221no—
356C4×C22⋊C4C2×D6111yescharacteristic
357#340—611no—
358#340—611no—
359#340—611no—
360#340—611no—
361#340—611no—
362#340—611no—
363#336C2×D6111yescharacteristic
364#336C2×D6111yescharacteristic
365C61#2607911yescharacteristic, complemented, Sylow
366C22×Q8—1221no—
367C23×C4—1221no—
368C23×C4C22×D6111yescharacteristic
369C2×M16—22no—
370C2×M16—22no—
371C2×M16Dic12214yescomplemented
372C22×C8C22×D6111yescharacteristic
373C22×C8C22×D6111yescharacteristic
374C22×C8C22×D6111yescharacteristic
375C22×C8C22×D6111yescharacteristic
376C4⋊Q8—1222no—
377C4⋊Q8—2441no—
378#125—1222no—
379#125—2441no—
380#117—1222no—
381#117—2441no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—611no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—611no—
390C2×C4⋊C4—1222no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4—1221no—
395C2×C4⋊C4—1221no—
396C2×C4⋊C4—1221no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—2441no—
400C2×C22⋊C4—1221no—
401C2×C22⋊C4—1221no—
402C2×C22⋊C4—1221no—
403C2×C22⋊C4D24411yescharacteristic
404C2×C22⋊C4D24411yescharacteristic
405C2×C42—1221no—
406C2×C42C22×D6111yescharacteristic
407C2×C42C22×D6111yescharacteristic
408#107—614no—
409#107—1222no—
410#107—1222no—
411#107—1222no—
412#107—2441no—
413#107—2441no—
414#106—1222no—
415#106—1222no—
416#106—1222no—
417#106—1222no—
418C22⋊C8Dic12214yescomplemented
419#97—22no—
420C4×C8—22no—
421#95—1221no—
422#95—1221no—
423#95—1221no—
424#95—1221no—
425#95—1221no—
426#95—1221no—
427#95—1221no—
428#95—1221no—
429#95—1221no—
430#95—1221no—
431#95D24411yescharacteristic
432#95D24411yescharacteristic
433C24C2×D24411yescharacteristic
434C2×Q8—4881no—
435C2×Q8—2442no—
436C2×Q8—2442no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—1221no—
442C22×C4—1221no—
443C22×C4—1221no—
444C22×C4—1221no—
445C22×C4—1221no—
446C22×C4—1221no—
447C22×C4—1221no—
448C22×C4—1221no—
449C22×C4—4881no—
450C22×C4—2442no—
451C22×C4—2442no—
452C22×C4C2×Dic12212yes—
453C22×C4C2×Dic12212yes—
454C22×C4C2×D24411yescharacteristic
455C22×C4C2×D24411yescharacteristic
456C22×C4C2×D24411yescharacteristic
457C22×C4C2×D24411yescharacteristic
458C22×C4C2×D24411yescharacteristic
459C22×C4C23×D6111yescharacteristic
460M16—28no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8C2×Dic12212yes—
468C2×C8C2×Dic12212yes—
469C2×C8C2×Dic12212yes—
470C2×C8C2×Dic12212yes—
471C2×C8D4×D6112yes—
472C2×C8#1175822412yes—
473C4⋊C4—612no—
474C4⋊C4—612no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—4881no—
490C4⋊C4—4881no—
491C4⋊C4—4881no—
492C4⋊C4—2442no—
493C4⋊C4—2442no—
494C4⋊C4—2442no—
495C4⋊C4—2442no—
496C4⋊C4—2442no—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—4881no—
500C22⋊C4—2442no—
501C22⋊C4—2442no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—2442no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23#1451977712yes—
514C23C22×D24411yescharacteristic
515Q8—4884no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—1222no—
535C2×C4—1222no—
536C2×C4—4884no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4#1451977712yes—
560C2×C4C2×D4×D6111yescharacteristic
561C2×C4C22×Dic12211yescharacteristic
562C2×C4C22×Dic12211yescharacteristic
563C2×C4C2×#1175822411yescharacteristic
564C2×C4#1451982311yescharacteristic
565C2×C4#1451982411yescharacteristic
566C8—22no—
567C8—22no—
568C8—22no—
569C8#1451975414yes—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22C2×#1451977711yescharacteristic
578C22D4×D24411yescharacteristic
579C22#1684041611yescharacteristic
580C22#1684056211yescharacteristic
581C22#1684056211yescharacteristic
582C22#1684056711yescharacteristic
583C22#1684057911yescharacteristic
584C4—22no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—4881no—
590C4—4881no—
591C4—4881no—
592C4—4881no—
593C4—4881no—
594C4—4881no—
595C4—2442no—
596C4—2442no—
597C4C2×#1451975411yescharacteristic
598C4#1684039011yescharacteristic
599C4D4×Dic12211yescharacteristic
600C4#1684046911yescharacteristic
601C2—42no—
602C2#2144952111yescharacteristic
603C2#2145635811yescharacteristic
604C2#2145637111yescharacteristic
605C2#2145743111yescharacteristic
606C2#2145747711yescharacteristic
607C2#2145832111yescharacteristic
608C2#2145835111yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1753C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#1753D61
hypocenterC61#26079
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
1. C2×#16840171C2semidirect
5. #21447302C2semidirect
6. #21447302C2semidirect
7. #21447158C2semidirect
8. #21447148C2semidirect
9. #21446557C2semidirect
10. #21446552C2semidirect
13. #21445983C2semidirect
114. C2×M16×C61Q8semidirect
123. C22⋊C8×C61Q8semidirect
365. C61#26079semidirect
371. C2×M16Dic122semidirect
418. C22⋊C8Dic122semidirect

Maximal subgroups

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

subgroupindex
C2×#168401712
C2×#168398042
#214495432
#214495102
#214473022
#214471582
#214471482
#214465572
#214465522
#214460202
#214460092
#214459832
#214458522
C61×#17532
#2607961