Finite Group Databasebeta

← #29647217

Subgroups — #29647217

Order 15616.

1102 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459870C211yesmaximal, characteristic
2#21454853C211yesmaximal, characteristic, complemented
3#21448181C211yesmaximal, characteristic, complemented
4#21444344C211yesmaximal, characteristic, complemented
5#21444337C212yesmaximal, complemented
6#21444330C211yesmaximal, characteristic, complemented
7#21444320C212yesmaximal, complemented
8#21444319C211yesmaximal, characteristic, complemented
9#21444307C212yesmaximal, complemented
10#328×D61C212yesmaximal, complemented
11C61×#1500C211yesmaximal, characteristic, complemented
12#16840485C411yescharacteristic, complemented
13#16840469—21no—
14#16840450C412yescomplemented
15#16840435—22no—
16Q8×D244C411yescharacteristic, complemented
17#16840420C412yescomplemented
18#16840417C411yescharacteristic, complemented
19#16840416C411yescharacteristic, complemented
20#16840408—21no—
21#16840383—21no—
22#16840353—22no—
23#16840319C2211yescharacteristic
24#16840272C2212yes—
25C22⋊Q8×D61C2212yes—
26#16840242—21no—
27C2×C4×D244—21no—
28C2×C4×D244C2211yescharacteristic
29#16839841—21no—
30#16839832—22no—
31C8×D244—21no—
32#16839819C2211yescharacteristic
33#16839724C2212yes—
34C22⋊C8×D61C2212yes—
35C4×C4⋊Dic61—21no—
36C4×C4⋊Dic61C2211yescharacteristic
37#16839609—22no—
38#16839609C2212yes—
39#95×D61C2212yes—
40#16839500C2212yes—
41#16839500C2212yes—
42#16839499C2212yes—
43#16839498C2212yes—
44#16839498C2212yes—
45#16839496C2212yes—
46C61×#529C2211yescharacteristic
47C61×#431C2211yescharacteristic
48C61×#376C2211yescharacteristic
49C61×#328C2211yescharacteristic
50C61×#328C2211yescharacteristic
51C61×#327C2211yescharacteristic
52C61×#327C2211yescharacteristic
53C2×#11758222—21no—
54C2×#11758224C2×C412yescomplemented
55C2×Q8×D61C2×C412yescomplemented
56C22×D244D411yescharacteristic
57C22×C4×D61—22no—
58C22×C4×D61C2312yes—
59C2×#11758189—22no—
60C2×C8×D61—22no—
61#14519777—21no—
62#14519777C2×C411yescharacteristic, complemented
63#14519777C2×C411yescharacteristic, complemented
64#14519772—22no—
65#14519772—22no—
66#14519769—21no—
67#14519768—22no—
68#14519765—21no—
69#14519763—22no—
70#14519762—22no—
71#14519761—22no—
72#14519759—22no—
73#14519759—22no—
74#14519757—22no—
75#14519757C2×C412yescomplemented
76#14519757C2×C412yescomplemented
77#14519756—21no—
78#14519756—21no—
79#14519756C2312yes—
80#14519755—22no—
81#14519754—22no—
82#14519754—22no—
83#14519753C2×C412yescomplemented
84#14519751—21no—
85#14519750—22no—
86#14519748C2×C412yescomplemented
87#14519745—22no—
88C4×Dic122—21no—
89C4×Dic122—21no—
90C4×Dic122C2×C412yescomplemented
91C4×D244—28no—
92C4×D244—21no—
93C4×D244C2×C412yes—
94C4×D244C2×C411yescharacteristic
95C4×D244C2×C411yescharacteristic
96C4×D244D411yescharacteristic
97C4×D244D412yes—
98C4×D244D411yescharacteristic
99D4×Dic61—22no—
100#14519737—22no—
101#14519737—22no—
102#14519736—21no—
103#14519735—22no—
104#14519735C2×C412yescomplemented
105C2×C4⋊Dic61—21no—
106C2×C4⋊Dic61—21no—
107C2×C4⋊Dic61D411yescharacteristic
108C4⋊C4×D61—22no—
109C4⋊C4×D61C2×C412yescomplemented
110C2×#11758198—22no—
111C2×#11758198D412yes—
112C22⋊C4×D61—22no—
113C2×C4×Dic61—21no—
114C2×C4×Dic61—22no—
115C4⋊#92418—21no—
116C22⋊#92418C2312yes—
117#14519689—22no—
118#14519689—22no—
119#14519688—21no—
120#14519684—22no—
121#14519684C2312yes—
122#126×C61—21no—
123#125×C61C2×C411yescharacteristic, complemented
124#124×C61C2×C411yescharacteristic, complemented
125#123×C61—21no—
126C22⋊Q8×C61C2311yescharacteristic
127#121×C61C2311yescharacteristic
128C4×Q8×C61C2×C411yescharacteristic, complemented
129C4×D4×C61—21no—
130C4×D4×C61C2×C411yescharacteristic, complemented
131#117×C61—21no—
132C2×C42×C61—21no—
133C2×C42×C61C2311yescharacteristic
134C4⋊C8×C61—21no—
135C22⋊C8×C61C2311yescharacteristic
136C22⋊C8×C61C2311yescharacteristic
137C4×C8×C61—21no—
138#95×C61—21no—
139#95×C61C2311yescharacteristic
140#95×C61C2311yescharacteristic
141C23×D61C2×D412yes—
142C2×Q8×C61C22×C411yescharacteristic
143C2×D4×C61—21no—
144C2×D4×C61C22×C411yescharacteristic
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C2×C22⋊D61—22no—
152C2×C22⋊D61—22no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155C22×Dic61C2×D412yes—
156Q8×D61—82no—
157#11758224—82no—
158#11758222—44no—
159C2×D244—24no—
160C2×D244C22⋊C412yes—
161C2×D244C22⋊C411yescharacteristic
162C2×D244C22⋊C411yescharacteristic
163C2×C4×D61—22no—
164C2×C4×D61—28no—
165C2×C4×D61—22no—
166C2×C4×D61—82no—
167C2×C4×D61—82no—
168C2×C4×D61—82no—
169C2×C4×D61C22×C412yes—
170C2×C4×D61C22×C412yes—
171C2×C4×D61C2×D412yes—
172C2×C4×D61C2×D412yes—
173C2×Dic122—21no—
174C2×Dic122—22no—
175C2×Dic122C22×C412yes—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61C22×C411yescharacteristic
183C4⋊C4×C61C22×C411yescharacteristic
184C22⋊C4×C61—21no—
185C22⋊C4×C61—21no—
186C22⋊C4×C61—21no—
187C22⋊C4×C61—21no—
188C22⋊C4×C61—21no—
189C42×C61—24no—
190C42×C61—21no—
191C42×C61C22×C411yescharacteristic
192C42×C61C22×C411yescharacteristic
193C42×C61C2×D411yescharacteristic
194C42×C61C2×D411yescharacteristic
195C22⋊Dic61—22no—
196C22⋊Dic61—22no—
197#11758198—28no—
198#11758198—22no—
199#11758198—22no—
200#11758198—22no—
201#11758198—22no—
202#11758198C22⋊C412yes—
203#11758198C22⋊C412yes—
204#11758198C22⋊C412yes—
205#11758198C22⋊C412yes—
206C4⋊Dic61—24no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—22no—
211C4⋊Dic61—22no—
212C4⋊Dic61C22⋊C411yescharacteristic
213C4⋊Dic61C22⋊C412yes—
214C4⋊Dic61C22⋊C411yescharacteristic
215C4⋊Dic61C22×C412yes—
216#11758196—22no—
217#11758196—22no—
218#11758196—22no—
219#11758196—22no—
220C4×Dic61—22no—
221C4×Dic61—22no—
222C4×Dic61—81no—
223C4×Dic61—82no—
224C2×#92418—22no—
225#11758189—82no—
226C8×D61—82no—
227C23×C61—21no—
228C23×C61C22×D411yescharacteristic
229C22×D61—22no—
230C22×D61—22no—
231C22×D61—24no—
232C22×D61—42no—
233C22×D61C2×C22⋊C412yes—
234C22×D61C2×C22⋊C412yes—
235Q8×C61—41no—
236D4×C61—41no—
237D4×C61—41no—
238D4×C61—41no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—42no—
253C2×C4×C61C2×C22⋊C411yescharacteristic
254C2×C4×C61C2×C22⋊C411yescharacteristic
255C2×C4×C61C2×C22⋊C411yescharacteristic
256C2×C4×C61C2×C22⋊C411yescharacteristic
257C2×C4×C61C23×C411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C22⋊D61—44no—
260C22⋊D61—82no—
261C2×Dic61—24no—
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—42no—
265C2×Dic61—42no—
266C2×Dic61—42no—
267C2×Dic61—82no—
268C2×Dic61C2×C22⋊C412yes—
269C2×Dic61C2×C22⋊C412yes—
270D244—24no—
271D244—22no—
272D244#10414yes—
273C4×D61—42no—
274C4×D61—42no—
275C4×D61—42no—
276C4×D61—42no—
277C4×D61—44no—
278C4×D61—82no—
279C4×D61—88no—
280Dic122—42no—
281Dic122—42no—
282Dic122—82no—
283C8×C61—41no—
284C8×C61—41no—
285#92418—42no—
286#17548—611nomaximal, Sylow
287C22×C61—21no—
288C22×C61—21no—
289C22×C61—41no—
290C22×C61—41no—
291C22×C61—41no—
292C22×C61C22×C22⋊C411yescharacteristic
293C22×C61#57711yescharacteristic
294C22×C61#57811yescharacteristic
295C2×D61—22no—
296C2×D61—22no—
297C2×D61—22no—
298C2×D61—22no—
299C2×D61—22no—
300C2×D61—44no—
301C2×D61—44no—
302C2×D61C2×#10414yes—
303C4×C61—21no—
304C4×C61—21no—
305C4×C61—21no—
306C4×C61—21no—
307C4×C61—21no—
308C4×C61—42no—
309C4×C61—42no—
310C4×C61—41no—
311C4×C61—41no—
312C4×C61—41no—
313C4×C61C2×#10412yes—
314Dic61—22no—
315Dic61—22no—
316Dic61—44no—
317Dic61—42no—
318Dic61—42no—
319#3520—611no—
320#2974—611no—
321#2275—611no—
322#1509—611no—
323#1508—612no—
324#1506—611no—
325#1504—612no—
326#1503—611no—
327#1501—612no—
328#1500D6111yescharacteristic, complemented
329C2×#328—612no—
330C2×C61—21no—
331C2×C61—21no—
332C2×C61—81no—
333C2×C61#288311yescharacteristic
334C2×C61#289811yescharacteristic
335C2×C61C22×#10411yescharacteristic
336D61—24no—
337D61—42no—
338#556—611no—
339#554—1221no—
340#552—612no—
341#551—1222no—
342D4×Q8—611no—
343#548—612no—
344#547—611no—
345#547—611no—
346#546—1221no—
347#542—1221no—
348#539—1222no—
349#533—611no—
350#529C2×D6111yescharacteristic
351#525—612no—
352C2×C22⋊Q8—612no—
353#520—1221no—
354C2×C4×D4—611no—
355C2×C4×D4—1221no—
356#436—1221no—
357#435—1222no—
358C8×D4—1221no—
359#432—611no—
360#431C2×D6111yescharacteristic
361#407—612no—
362C2×C22⋊C8—612no—
363C4×C4⋊C4—611no—
364C4×C4⋊C4—1221no—
365C4×C22⋊C4—612no—
366C4×C22⋊C4—1222no—
367#376C2×D6111yescharacteristic
368C2×#95—612no—
369#328—612no—
370#328—612no—
371#328—612no—
372#328—612no—
373#328—612no—
374#328C2×D6111yescharacteristic
375#328C2×D6111yescharacteristic
376#327—612no—
377#327C2×D6111yescharacteristic
378#327C2×D6111yescharacteristic
379C61#1754811yescharacteristic, complemented, Sylow
380C2×#41—612no—
381C2×#41—1221no—
382C22×Q8—612no—
383C22×D4—611no—
384C23×C4—612no—
385C23×C4—1222no—
386C2×M16—1222no—
387C22×C8—1222no—
388C4⋊Q8—611no—
389C4⋊Q8—611no—
390C4⋊Q8—1221no—
391#126—21no—
392#126—1222no—
393#126—1222no—
394#125—1222no—
395#125—1221no—
396#125C4×D6111yescharacteristic, complemented
397#124—1221no—
398#124—1222no—
399#124C4×D6111yescharacteristic, complemented
400#123—21no—
401#123—1222no—
402#123—1222no—
403#123—1222no—
404#123—1222no—
405C22⋊Q8—612no—
406C22⋊Q8—612no—
407C22⋊Q8—612no—
408C22⋊Q8—612no—
409C22⋊Q8—1222no—
410C22⋊Q8—1222no—
411C22⋊Q8—1222no—
412C22⋊Q8—1222no—
413C22⋊Q8—1221no—
414C22⋊Q8—1221no—
415C22⋊Q8C22×D6111yescharacteristic
416#121—612no—
417#121—1222no—
418#121—1221no—
419#121C22×D6111yescharacteristic
420#120—1222no—
421C4×Q8—612no—
422C4×Q8—1221no—
423C4×Q8—1221no—
424C4×Q8C4×D6111yescharacteristic, complemented
425C4×D4—21no—
426C4×D4—611no—
427C4×D4—611no—
428C4×D4—612no—
429C4×D4—611no—
430C4×D4—612no—
431C4×D4—611no—
432C4×D4—1221no—
433C4×D4—1228no—
434C4×D4—1222no—
435C4×D4—1222no—
436C4×D4—1222no—
437C4×D4C4×D6111yescharacteristic, complemented
438#117—21no—
439#117—612no—
440#117—1222no—
441#117—1221no—
442C2×C4⋊C4—611no—
443C2×C4⋊C4—612no—
444C2×C4⋊C4—1222no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4—1221no—
447C2×C22⋊C4—612no—
448C2×C22⋊C4—1222no—
449C2×C22⋊C4—1222no—
450C2×C42—21no—
451C2×C42—1222no—
452C2×C42—1221no—
453C2×C42C22×D6111yescharacteristic
454C4⋊C8—21no—
455C4⋊C8—1221no—
456C22⋊C8—612no—
457C22⋊C8—1222no—
458C22⋊C8—1222no—
459C22⋊C8C22×D6111yescharacteristic
460C22⋊C8C22×D6111yescharacteristic
461#97—1221no—
462C4×C8—21no—
463#95—21no—
464#95—612no—
465#95—1222no—
466#95C22×D6111yescharacteristic
467#95C22×D6111yescharacteristic
468C24—612no—
469#41—4882no—
470#41—2444no—
471C2×Q8—612no—
472C2×Q8—1222no—
473C2×Q8—1221no—
474C2×Q8—4882no—
475C2×Q8C2×C4×D6111yescharacteristic
476C2×D4—21no—
477C2×D4—611no—
478C2×D4—612no—
479C2×D4—611no—
480C2×D4—1224no—
481C2×D4—1222no—
482C2×D4—1222no—
483C2×D4C2×C4×D6111yescharacteristic
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—612no—
488C22×C4—612no—
489C22×C4—612no—
490C22×C4—612no—
491C22×C4—612no—
492C22×C4—1222no—
493C22×C4—1228no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—1222no—
497C22×C4—4882no—
498C22×C4—4882no—
499C22×C4—4882no—
500C22×C4D4×D6111yescharacteristic
501C22×C4D4×D6111yescharacteristic
502C22×C4C23×D6111yescharacteristic
503M16—4882no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—1222no—
507C2×C8—4882no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—611no—
513C4⋊C4—612no—
514C4⋊C4—611no—
515C4⋊C4—612no—
516C4⋊C4—1222no—
517C4⋊C4—1222no—
518C4⋊C4—1222no—
519C4⋊C4—1222no—
520C4⋊C4—1224no—
521C4⋊C4—1222no—
522C4⋊C4—1221no—
523C4⋊C4—1222no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4C2×C4×D6111yescharacteristic
527C4⋊C4C2×C4×D6111yescharacteristic
528C22⋊C4—21no—
529C22⋊C4—21no—
530C22⋊C4—21no—
531C22⋊C4—21no—
532C22⋊C4—21no—
533C22⋊C4—612no—
534C22⋊C4—612no—
535C22⋊C4—612no—
536C22⋊C4—612no—
537C22⋊C4—1222no—
538C22⋊C4—1222no—
539C22⋊C4—1222no—
540C22⋊C4—1222no—
541C22⋊C4—1228no—
542C22⋊C4—1222no—
543C22⋊C4—1222no—
544C42—24no—
545C42—21no—
546C42—1222no—
547C42—1222no—
548C42—4881no—
549C42—4882no—
550C42C2×C4×D6111yescharacteristic
551C42C2×C4×D6111yescharacteristic
552C42D4×D6111yescharacteristic
553C42D4×D6111yescharacteristic
554C23—21no—
555C23—612no—
556C23—612no—
557C23—1224no—
558C23—1222no—
559C23—1222no—
560C23—2442no—
561C23C2×D4×D6111yescharacteristic
562Q8—41no—
563Q8—4882no—
564Q8—2442no—
565Q8—2442no—
566D4—41no—
567D4—41no—
568D4—41no—
569D4—614no—
570D4—1224no—
571D4—1222no—
572D4—4882no—
573D4—2444no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—612no—
589C2×C4—612no—
590C2×C4—1224no—
591C2×C4—1222no—
592C2×C4—1222no—
593C2×C4—4882no—
594C2×C4—4888no—
595C2×C4—4882no—
596C2×C4—2442no—
597C2×C4—2442no—
598C2×C4—2442no—
599C2×C4—2442no—
600C2×C4—2444no—
601C2×C4—2442no—
602C2×C4—2442no—
603C2×C4—2442no—
604C2×C4C22⋊C4×D6111yescharacteristic
605C2×C4C22⋊C4×D6111yescharacteristic
606C2×C4C22⋊C4×D6111yescharacteristic
607C2×C4C22⋊C4×D6111yescharacteristic
608C2×C4C22×C4×D6111yescharacteristic
609C2×C4C2×D4×D6111yescharacteristic
610C8—41no—
611C8—41no—
612C8—2442no—
613C22—21no—
614C22—21no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—614no—
619C22—1222no—
620C22—1222no—
621C22—1222no—
622C22—1222no—
623C22—1222no—
624C22—2444no—
625C22—2444no—
626C22C2×C22⋊C4×D6111yescharacteristic
627C22#1684056911yescharacteristic
628C22#1684057611yescharacteristic
629C4—21no—
630C4—21no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—42no—
635C4—42no—
636C4—41no—
637C4—41no—
638C4—41no—
639C4—1222no—
640C4—1222no—
641C4—2444no—
642C4—2442no—
643C4—2442no—
644C4#104×D6112yes—
645C2—21no—
646C2—21no—
647C2—81no—
648C2—1224no—
649C2—2442no—
650C2#2145427811yescharacteristic
651C2#2145434811yescharacteristic
652C2C2×#104×D6111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1500C2
socleC22×C61C22×C22⋊C4
prime radical#21459870C2
cosocleC22×C4×C61C24
hypercenter#1500D61
hypocenterC61#17548
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1500

Complements

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

NH ≅ G/Ntype
2. #21454853C2semidirect
3. #21448181C2semidirect
4. #21444344C2semidirect
5. #21444337C2semidirect
6. #21444330C2semidirect
7. #21444320C2semidirect
8. #21444319C2semidirect
9. #21444307C2semidirect
10. #328×D61C2semidirect
11. C61×#1500C2semidirect
12. #16840485C4semidirect
14. #16840450C4semidirect
16. Q8×D244C4semidirect
17. #16840420C4semidirect
18. #16840417C4semidirect
19. #16840416C4semidirect
54. C2×#11758224C2×C4semidirect
55. C2×Q8×D61C2×C4semidirect
62. #14519777C2×C4semidirect
63. #14519777C2×C4semidirect
75. #14519757C2×C4semidirect
76. #14519757C2×C4semidirect
83. #14519753C2×C4semidirect
86. #14519748C2×C4semidirect
90. C4×Dic122C2×C4semidirect
104. #14519735C2×C4semidirect
109. C4⋊C4×D61C2×C4semidirect
123. #125×C61C2×C4semidirect
124. #124×C61C2×C4semidirect
128. C4×Q8×C61C2×C4semidirect
130. C4×D4×C61C2×C4semidirect
328. #1500D61semidirect
379. C61#17548semidirect
396. #125C4×D61semidirect
399. #124C4×D61semidirect
424. C4×Q8C4×D61semidirect
437. C4×D4C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214598702
#214548532
#214481812
#214443442
#214443372
#214443302
#214443202
#214443192
#214443072
#328×D612
C61×#15002
#1754861