Finite Group Databasebeta

← #29730092

Subgroups — #29730092

Order 15616.

840 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840122C211yesmaximal, characteristic
2C2×#16839985C211yesmaximal, characteristic
3#21451027C211yesmaximal, characteristic, complemented
4#21448888C211yesmaximal, characteristic
5#21447229C211yesmaximal, characteristic, complemented
6#21447224C211yesmaximal, characteristic
7#21447142C211yesmaximal, characteristic, complemented
8#21447137C211yesmaximal, characteristic
9#21446770C211yesmaximal, characteristic, complemented
10#21446159C211yesmaximal, characteristic, complemented
11#21446152C211yesmaximal, characteristic
12#21445825C211yesmaximal, characteristic, complemented
13#21445820C211yesmaximal, characteristic
14#21445650C211yesmaximal, characteristic, complemented
15C61×#1940C211yesmaximal, characteristic, complemented
16C22×Dic244C2211yescharacteristic
17C22×#11758190C2211yescharacteristic
18C2×#14519777C2211yescharacteristic
19C2×#14519765C2211yescharacteristic
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×Dic122C2211yescharacteristic
22C2×C4×D244C2211yescharacteristic
23#16840122—24no—
24#16839985—24no—
25C2×#14519702C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519699C2211yescharacteristic
31C2×#14519699C2211yescharacteristic
32C2×#14519699C2211yescharacteristic
33#16839683—21no—
34#16839683—21no—
35#16839668C2211yescharacteristic
36#16839668C2211yescharacteristic
37#16839668C2211yescharacteristic
38#16839664C2211yescharacteristic
39#16839661—21no—
40#16839661—21no—
41#16839652—21no—
42#16839651—21no—
43#16839633—21no—
44#16839632C2211yescharacteristic
45#16839632C2211yescharacteristic
46C4×C4⋊Dic61C2211yescharacteristic
47#16839609—21no—
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53#16839525C2211yescharacteristic
54C2×C4⋊C8×C61C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×C4⋊C8×C61C2211yescharacteristic
57C2×#97×C61C2211yescharacteristic
58C4×C4⋊C4×C61C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×Dic122—21no—
62C22×Dic122C2311yescharacteristic
63C22×Dic122C2311yescharacteristic
64C22×D244C2311yescharacteristic
65C22×C4×D61—21no—
66C2×Dic244—22no—
67C2×Dic244—22no—
68C2×Dic244—41no—
69C2×#11758190—22no—
70C2×#11758190—22no—
71C2×#11758190—41no—
72#14519777—22no—
73#14519777—22no—
74#14519765—22no—
75#14519765D414yes—
76C4×Dic122—22no—
77C4×Dic122—22no—
78C4×Dic122—22no—
79C4×Dic122—41no—
80C4×D244—22no—
81C4×D244—41no—
82C2×C4⋊Dic61—21no—
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61—21no—
85C2×C4⋊Dic61—21no—
86C2×C4⋊Dic61—21no—
87C2×C4⋊Dic61C2311yescharacteristic
88C2×C4⋊Dic61C2311yescharacteristic
89C2×C4⋊Dic61C2311yescharacteristic
90C2×C4⋊Dic61C2311yescharacteristic
91C2×C4⋊Dic61C2311yescharacteristic
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758198—21no—
95C2×#11758198—21no—
96C2×#11758198—21no—
97C2×#11758198—21no—
98C2×#11758198—21no—
99C2×C4×Dic61—21no—
100C2×C4×Dic61—21no—
101C2×C4×Dic61—21no—
102#14519702—22no—
103#14519702—22no—
104#14519702—22no—
105#14519702—22no—
106#14519702—22no—
107#14519699—22no—
108#14519699—22no—
109#14519699—22no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117C22×C8×C61C2311yescharacteristic
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C2×C4⋊C4×C61D411yescharacteristic
122C2×C4⋊C4×C61D411yescharacteristic
123C2×C42×C61C2311yescharacteristic
124C2×C42×C61C2311yescharacteristic
125C2×C42×C61C2311yescharacteristic
126C4⋊C8×C61—22no—
127C4⋊C8×C61—41no—
128C4⋊C8×C61—41no—
129#97×C61—22no—
130#95×C61D411yescharacteristic
131#95×C61D411yescharacteristic
132C23×D61—21no—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2411yescharacteristic
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244C2×D412yes—
153C2×C4×D61—42no—
154C2×C4×D61—42no—
155C2×C4×D61—81no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—42no—
167C2×Dic122—81no—
168C2×Dic122—81no—
169C2×Dic122C2×D412yes—
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—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—22no—
185C42×C61—21no—
186C42×C61—21no—
187C42×C61—21no—
188C42×C61—21no—
189C42×C61C2×D412yes—
190C42×C61#4112yes—
191#11758198—24no—
192#11758198—24no—
193#11758198—42no—
194#11758198—42no—
195#11758198—81no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—42no—
206C4⋊Dic61—81no—
207C4⋊Dic61—81no—
208C4⋊Dic61—81no—
209C4⋊Dic61—81no—
210C4⋊Dic61#4112yes—
211#11758196—42no—
212#11758196—42no—
213C4×Dic61—42no—
214C4×Dic61—42no—
215C4×Dic61—81no—
216Dic244—44no—
217#11758190—44no—
218C23×C61C22×D411yescharacteristic
219C22×D61—22no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61—41no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—41no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
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—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61#12112yes—
241C2×C4×C61#12112yes—
242C2×C4×C61#13712yes—
243C2×C4×C61#13712yes—
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C2×C4×C61C2×#4111yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C2×C4×C61H(5,𝔽2)11yescharacteristic
249C2×C4×C612−1+411yescharacteristic
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—22no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
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×Dic61—41no—
270C2×Dic61—41no—
271D244—22no—
272D244—22no—
273D244—41no—
274D244—41no—
275C4×D61—84no—
276Dic122—22no—
277Dic122—22no—
278Dic122—22no—
279Dic122—22no—
280Dic122—41no—
281Dic122—41no—
282Dic122—41no—
283Dic122—41no—
284Dic122—84no—
285C8×C61—41no—
286C8×C61—41no—
287C8×C61—41no—
288C8×C61—41no—
289#25336—611nomaximal, Sylow
290C22×C61C2×#12111yescharacteristic
291C22×C61#54611yescharacteristic
292C22×C61#54811yescharacteristic
293C22×C61C2×#13711yescharacteristic
294C22×C61C2×#13711yescharacteristic
295C22×C61#57611yescharacteristic
296C22×C61#57711yescharacteristic
297C2×D61—42no—
298C2×D61—42no—
299C2×D61—42no—
300C2×D61—81no—
301C2×D61—81no—
302C2×D61—81no—
303C2×D61—81no—
304C4×C61—22no—
305C4×C61—22no—
306C4×C61—22no—
307C4×C61—22no—
308C4×C61—41no—
309C4×C61—41no—
310C4×C61#52911yescharacteristic
311C4×C61#54211yescharacteristic
312C4×C61#54611yescharacteristic
313C4×C61#54711yescharacteristic
314Dic61—42no—
315Dic61—42no—
316Dic61—42no—
317Dic61—81no—
318Dic61—81no—
319Dic61—81no—
320Dic61—81no—
321C2×#489—611no—
322C2×#464—611no—
323#2567—611no—
324#2367—611no—
325#2072—611no—
326#2071—611no—
327#2056—611no—
328#2055—611no—
329#1988—611no—
330#1940D6111yescharacteristic, complemented
331#1878—611no—
332#1877—611no—
333#1803—611no—
334#1802—611no—
335#1763—611no—
336C2×C61#242711yescharacteristic
337C2×C61#303911yescharacteristic
338C2×C61#304511yescharacteristic
339C2×C61#331411yescharacteristic
340C2×C61#331611yescharacteristic
341C2×C61#336911yescharacteristic
342C2×C61#337511yescharacteristic
343D61—82no—
344C22×Q16—611no—
345C22×SD16—611no—
346C2×C4⋊Q8—611no—
347C2×#124—611no—
348C2×C4×Q8—611no—
349C2×C4×Q8—611no—
350C2×C4×D4—611no—
351#489—1224no—
352#464—1224no—
353C2×C4⋊C8C2×D6111yescharacteristic
354C2×C4⋊C8C2×D6111yescharacteristic
355C2×C4⋊C8C2×D6111yescharacteristic
356C2×#103—611no—
357C2×#103—611no—
358C2×#103—611no—
359C2×#103—611no—
360C2×#103—611no—
361C2×#102—611no—
362C2×#102—611no—
363C2×#102—611no—
364C2×#97C2×D6111yescharacteristic
365#395—1221no—
366#395—1221no—
367#391—611no—
368#391—611no—
369#391—611no—
370#390—611no—
371#389—1221no—
372#389—1221no—
373#388—1221no—
374#388—1221no—
375#385—1221no—
376#384—611no—
377#384—611no—
378C4×C4⋊C4—611no—
379C4×C4⋊C4C2×D6111yescharacteristic
380C4×C22⋊C4—1221no—
381#340—611no—
382#340—611no—
383#340—611no—
384#340—611no—
385#340—611no—
386#340—611no—
387#336C2×D6111yescharacteristic
388#336C2×D6111yescharacteristic
389C61#2533611yescharacteristic, complemented, Sylow
390C22×Q8—611no—
391C22×Q8—611no—
392C22×Q8—1221no—
393C22×D4—611no—
394C23×C4—1221no—
395C2×Q16—1222no—
396C2×Q16—1222no—
397C2×Q16—2441no—
398C2×SD16—1222no—
399C2×SD16—1222no—
400C2×SD16—2441no—
401C22×C8C22×D6111yescharacteristic
402C22×C8C22×D6111yescharacteristic
403C22×C8C22×D6111yescharacteristic
404C22×C8C22×D6111yescharacteristic
405C4⋊Q8—1222no—
406C4⋊Q8—1222no—
407#124—614no—
408#124—1222no—
409C4×Q8—1222no—
410C4×Q8—1222no—
411C4×Q8—1222no—
412C4×Q8—2441no—
413C4×D4—1222no—
414C4×D4—2441no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4D24411yescharacteristic
428C2×C4⋊C4D24411yescharacteristic
429C2×C22⋊C4—1221no—
430C2×C22⋊C4—1221no—
431C2×C22⋊C4—1221no—
432C2×C22⋊C4—1221no—
433C2×C22⋊C4—1221no—
434C2×C42—1221no—
435C2×C42—1221no—
436C2×C42—1221no—
437C2×C42C22×D6111yescharacteristic
438C2×C42C22×D6111yescharacteristic
439C2×C42C22×D6111yescharacteristic
440C4⋊C8—22no—
441C4⋊C8—41no—
442C4⋊C8—41no—
443#103—1222no—
444#103—1222no—
445#103—1222no—
446#103—1222no—
447#103—1222no—
448#102—1222no—
449#102—1222no—
450#102—1222no—
451#97—22no—
452#95—1221no—
453#95—1221no—
454#95—1221no—
455#95—1221no—
456#95—1221no—
457#95—1221no—
458#95—1221no—
459#95D24411yescharacteristic
460#95D24411yescharacteristic
461C24—1221no—
462C2×Q8—612no—
463C2×Q8—612no—
464C2×Q8—1221no—
465C2×Q8—1221no—
466C2×Q8—1221no—
467C2×Q8—1221no—
468C2×Q8—1221no—
469C2×Q8—1221no—
470C2×Q8—1221no—
471C2×Q8—1221no—
472C2×Q8—1221no—
473C2×Q8—1221no—
474C2×Q8—4881no—
475C2×Q8—4881no—
476C2×Q8—2442no—
477C2×D4—612no—
478C2×D4—1221no—
479C2×D4—1221no—
480C2×D4—1221no—
481C2×D4—1221no—
482C2×D4—1221no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—4881no—
491C22×C4—2442no—
492C22×C4—2442no—
493C22×C4C2×D24411yescharacteristic
494C22×C4C2×D24411yescharacteristic
495C22×C4C2×D24411yescharacteristic
496C22×C4C2×D24411yescharacteristic
497C22×C4C2×D24411yescharacteristic
498C22×C4C2×D24411yescharacteristic
499C22×C4C23×D6111yescharacteristic
500Q16—2444no—
501SD16—2444no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—612no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4—4881no—
527C4⋊C4—4881no—
528C4⋊C4—4881no—
529C4⋊C4—4881no—
530C4⋊C4—2442no—
531C4⋊C4—2442no—
532C4⋊C4—2442no—
533C22⋊C4—1224no—
534C22⋊C4—1224no—
535C22⋊C4—4881no—
536C22⋊C4—2442no—
537C22⋊C4—2442no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—4881no—
543C42—2442no—
544C42—2442no—
545C42D4×D6112yes—
546C42#1175822612yes—
547C23—1222no—
548C23—2441no—
549C23—2441no—
550C23—2441no—
551C23—2441no—
552C23—2441no—
553C23—2441no—
554C23C22×D24411yescharacteristic
555Q8—1222no—
556Q8—1222no—
557Q8—1222no—
558Q8—1222no—
559Q8—4884no—
560Q8—2441no—
561Q8—2441no—
562Q8—2441no—
563Q8—2441no—
564D4—1222no—
565D4—1222no—
566D4—2441no—
567D4—2441no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—1222no—
583C2×C4—1222no—
584C2×C4—1222no—
585C2×C4—4884no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4#1451975212yes—
605C2×C4#1451975212yes—
606C2×C4#1451981012yes—
607C2×C4#1451981012yes—
608C2×C4C2×D4×D6111yescharacteristic
609C2×C4#41×D6111yescharacteristic
610C2×C4#41×D6111yescharacteristic
611C2×C4C2×#1175822611yescharacteristic
612C2×C4#1451982211yescharacteristic
613C2×C4#1451982511yescharacteristic
614C8—41no—
615C8—41no—
616C8—41no—
617C8—41no—
618C22—4881no—
619C22—4881no—
620C22—4881no—
621C22—4881no—
622C22—2442no—
623C22—2442no—
624C22—2442no—
625C22C2×#1451975211yescharacteristic
626C22#1684040711yescharacteristic
627C22#1684042411yescharacteristic
628C22C2×#1451981011yescharacteristic
629C22C2×#1451981011yescharacteristic
630C22#1684056711yescharacteristic
631C22#1684057411yescharacteristic
632C4—22no—
633C4—22no—
634C4—22no—
635C4—22no—
636C4—41no—
637C4—41no—
638C4—4881no—
639C4—4881no—
640C4—4881no—
641C4—4881no—
642C4—2442no—
643C4—2442no—
644C4—2442no—
645C4#1684029711yescharacteristic
646C4#1684038611yescharacteristic
647C4#1684040911yescharacteristic
648C4#1684041411yescharacteristic
649C2—4882no—
650C2#2144939411yescharacteristic
651C2#2145530711yescharacteristic
652C2#2145536011yescharacteristic
653C2#2145790211yescharacteristic
654C2#2145793211yescharacteristic
655C2#2145856111yescharacteristic
656C2#2145863511yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1940C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1940D61
hypocenterC61#25336
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1940

Complements

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

NH ≅ G/Ntype
3. #21451027C2semidirect
5. #21447229C2semidirect
7. #21447142C2semidirect
9. #21446770C2semidirect
10. #21446159C2semidirect
12. #21445825C2semidirect
14. #21445650C2semidirect
15. C61×#1940C2semidirect
330. #1940D61semidirect
389. C61#25336semidirect

Maximal subgroups

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

subgroupindex
C2×#168401222
C2×#168399852
#214510272
#214488882
#214472292
#214472242
#214471422
#214471372
#214467702
#214461592
#214461522
#214458252
#214458202
#214456502
C61×#19402
#2533661