Finite Group Databasebeta

← #30615232

Subgroups — #30615232

Order 15616.

1336 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462496C211yesmaximal, characteristic, complemented
2#21462266C211yesmaximal, characteristic, complemented
3#21461725C212yesmaximal, complemented
4#21461692C214yesmaximal, complemented
5#21461662C212yesmaximal
6#21461583C214yesmaximal, complemented
7C2×C4×#11758226C211yesmaximal, characteristic
8#16840784—21no—
9#16840783—21no—
10#16840781—21no—
11#16840778—21no—
12#16840766C2212yescomplemented
13C2×#14519842C2218yes—
14#16840733C2212yes—
15C4×#11758213C2212yescomplemented
16#16840707—21no—
17#16840706—21no—
18#16840705—21no—
19#16840704—21no—
20C4×C4⋊#6923C2212yescomplemented
21C4×C22⋊#6923C2212yes—
22#16840698C2212yescomplemented
23#16840632C2218yescomplemented
24C22×#11758226C2211yescharacteristic
25C4×#11758226—21no—
26C4×#11758226—21no—
27C4×#11758226—21no—
28C4×#11758226—21no—
29C4×#11758226C412yescomplemented
30C4×#11758226C412yescomplemented
31C4×#11758226C2212yes—
32C4×#11758226C2212yes—
33C2×Q8×Dic61C411yescharacteristic, complemented
34C2×C4×D244—21no—
35C2×C4×D244C411yescharacteristic, complemented
36C2×#14519739—21no—
37C2×#14519739C411yescharacteristic, complemented
38C2×#14519735—21no—
39C2×#14519735C2211yescharacteristic
40C2×C42×D61—21no—
41C2×C42×D61C2211yescharacteristic
42C2×C4×Q8×C61C411yescharacteristic, complemented
43C2×#11758213C2314yes—
44C2×C4⋊#6923—22no—
45C2×C22⋊#6923—22no—
46C2×C4×#6923C2314yes—
47C4⋊#92420—22no—
48C4⋊#92420—22no—
49#14519842—216no—
50#14519834—22no—
51C22⋊#92420—22no—
52#14519832—22no—
53C4×#92420—22no—
54(C2×#6923)⋊C4—22no—
55(C2×#6923)⋊C4—22no—
56(C2×#6923)⋊C4—22no—
57(C2×#6923)⋊C4—22no—
58C2×#11758226—28no—
59C2×#11758226—21no—
60C2×#11758226—21no—
61C2×#11758226C2×C412yes—
62C2×#11758226C2312yes—
63C22×D244—21no—
64C22×D244C2×C411yescharacteristic
65C22×C4×D61—21no—
66C22×C4×D61—21no—
67C22×C4×D61—21no—
68C22×C4×D61D411yescharacteristic
69C22×C4×D61D411yescharacteristic
70C22×C4×D61C2311yescharacteristic
71Q8×Dic61—21no—
72Q8×Dic61—21no—
73Q8×Dic61C2×C412yes—
74Q8×Dic61C2×C412yes—
75C4×D244—22no—
76C4×D244—22no—
77C4×D244—22no—
78C4×D244—22no—
79C4×D244—21no—
80C4×D244—21no—
81C4×D244C2×C412yes—
82C4×D244C2×C412yes—
83#14519739—22no—
84#14519739—22no—
85#14519739—22no—
86#14519739—22no—
87#14519739—21no—
88#14519739—21no—
89#14519739C2×C412yes—
90#14519739C2×C412yes—
91#14519735—21no—
92#14519735—21no—
93#14519735—22no—
94#14519735—22no—
95#14519735—22no—
96#14519735—22no—
97#14519735C2×C412yes—
98#14519735C2312yes—
99C2×C4⋊Dic61—21no—
100C2×C4⋊Dic61C2×C411yescharacteristic
101C2×#11758198—21no—
102C2×#11758198—21no—
103C2×#11758198—21no—
104C2×C4×Dic61—21no—
105C2×C4×Dic61—21no—
106C2×C4×Dic61C2×C411yescharacteristic
107C2×C4×Dic61D411yescharacteristic
108C2×C4×Dic61D411yescharacteristic
109C42×D61—22no—
110C42×D61—22no—
111C42×D61—22no—
112C42×D61—22no—
113C42×D61C2×C411yescharacteristic
114C42×D61C2×C411yescharacteristic
115C42×D61D411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61D411yescharacteristic
118C42×D61D411yescharacteristic
119C42×D61C2311yescharacteristic
120C42×D61C2311yescharacteristic
121C22×Q8×C61C2×C411yescharacteristic
122C4×Q8×C61—21no—
123C4×Q8×C61—21no—
124C4×Q8×C61C2×C412yes—
125C4×Q8×C61C2×C412yes—
126C2×C4⋊C4×C61—21no—
127C2×C4⋊C4×C61C2×C411yescharacteristic
128C2×C42×C61—21no—
129C2×C42×C61C2×C411yescharacteristic
130C23×D61—21no—
131C23×D61C2×D411yescharacteristic
132C22×#6923—24no—
133C2×Q8×C61—24no—
134C2×Q8×C61—21no—
135C2×Q8×C61C22×C412yes—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61C22⋊C411yescharacteristic
139C22×C4×C61C22⋊C411yescharacteristic
140C22×C4×C61C22×C411yescharacteristic
141C22×Dic61—21no—
142C22×Dic61C22⋊C411yescharacteristic
143C22×Dic61C22⋊C411yescharacteristic
144C22×Dic61C2×D411yescharacteristic
145#11758226—28no—
146#11758226—28no—
147#11758226—24no—
148#11758226—24no—
149#11758226—24no—
150#11758226#4118yes—
151C2×D244—28no—
152C2×D244—22no—
153C2×D244—22no—
154C2×D244—21no—
155C2×D244—24no—
156C2×D244C22×C412yes—
157C2×C4×D61—28no—
158C2×C4×D61—22no—
159C2×C4×D61—22no—
160C2×C4×D61—24no—
161C2×C4×D61—22no—
162C2×C4×D61—22no—
163C2×C4×D61—22no—
164C2×C4×D61—22no—
165C2×C4×D61—42no—
166C2×C4×D61—42no—
167C2×C4×D61—44no—
168C2×C4×D61—44no—
169C2×C4×D61C22⋊C411yescharacteristic
170C2×C4×D61C22⋊C411yescharacteristic
171C2×C4×D61C22⋊C411yescharacteristic
172C2×C4×D61C22⋊C411yescharacteristic
173C2×C4×D61C22×C411yescharacteristic
174C2×C4×D61C2×D411yescharacteristic
175C2×C4×D61C2×D411yescharacteristic
176C2×C4×D61C2×D411yescharacteristic
177C2×C4×D61C2×D411yescharacteristic
178C2×C4×D61C2×D411yescharacteristic
179C2×C4×D61C2×D411yescharacteristic
180C2×C4×D61C2411yescharacteristic
181C22⋊#6923—44no—
182C2×#92420—24no—
183C4⋊#6923—44no—
184C4×#6923—28no—
185#11758213—28no—
186C4⋊C4×C61—22no—
187C4⋊C4×C61—22no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61C22×C412yes—
190C42×C61—22no—
191C42×C61—22no—
192C42×C61C22⋊C411yescharacteristic
193C42×C61C22⋊C411yescharacteristic
194C42×C61C22×C411yescharacteristic
195C42×C61C22×C411yescharacteristic
196#11758198—22no—
197#11758198—22no—
198#11758198—22no—
199#11758198—22no—
200#11758198—22no—
201#11758198—22no—
202C4⋊Dic61—21no—
203C4⋊Dic61—22no—
204C4⋊Dic61—22no—
205C4⋊Dic61C22×C412yes—
206C4×Dic61—22no—
207C4×Dic61—22no—
208C4×Dic61—22no—
209C4×Dic61—22no—
210C4×Dic61C22⋊C411yescharacteristic
211C4×Dic61C22⋊C411yescharacteristic
212C4×Dic61C22⋊C411yescharacteristic
213C4×Dic61C22⋊C411yescharacteristic
214C4×Dic61C22⋊C411yescharacteristic
215C4×Dic61C22⋊C411yescharacteristic
216C4×Dic61C22×C411yescharacteristic
217C4×Dic61C22×C411yescharacteristic
218C4×Dic61C2×D411yescharacteristic
219C4×Dic61C2×D411yescharacteristic
220C4×Dic61C2×D411yescharacteristic
221C4×Dic61C2×D411yescharacteristic
222C23×C61C2×C22⋊C411yescharacteristic
223C22×D61—24no—
224C22×D61—22no—
225C22×D61—22no—
226C22×D61—22no—
227C22×D61—41no—
228C22×D61—42no—
229C22×D61—42no—
230C22×D61C2×C22⋊C411yescharacteristic
231C22×D61C22×D411yescharacteristic
232C2×#6923—44no—
233C2×#6923—44no—
234C2×#6923—44no—
235Q8×C61—24no—
236Q8×C61—22no—
237Q8×C61#11714yes—
238C2×C4×C61—24no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—42no—
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61C2×C22⋊C411yescharacteristic
247C2×C4×C61C2×C22⋊C411yescharacteristic
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61C23×C411yescharacteristic
251C2×Dic61—22no—
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—41no—
255C2×Dic61—42no—
256C2×Dic61C2×C22⋊C411yescharacteristic
257C2×Dic61C2×C22⋊C411yescharacteristic
258C2×Dic61C2×C22⋊C411yescharacteristic
259C2×Dic61C2×C22⋊C411yescharacteristic
260C2×Dic61C2×C22⋊C411yescharacteristic
261C2×Dic61C22×D411yescharacteristic
262D244—28no—
263D244—24no—
264D244—28no—
265D244—22no—
266D244#11714yes—
267C4×D61—28no—
268C4×D61—22no—
269C4×D61—22no—
270C4×D61—22no—
271C4×D61—22no—
272C4×D61—28no—
273C4×D61—22no—
274C4×D61—41no—
275C4×D61—41no—
276C4×D61—41no—
277C4×D61—41no—
278C4×D61—41no—
279C4×D61—41no—
280C4×D61—44no—
281C4×D61—44no—
282C4×D61—44no—
283C4×D61—44no—
284C4×D61C2×#4114yes—
285#92420—44no—
286#20748—611nomaximal, Sylow
287C22×C61—21no—
288C22×C61—21no—
289C22×C61#42112yes—
290C22×C61C22×C22⋊C411yescharacteristic
291C2×D61—21no—
292C2×D61—21no—
293C2×D61—21no—
294C2×D61—21no—
295C2×D61—21no—
296C2×D61—24no—
297C2×D61—44no—
298C2×D61—48no—
299C2×D61—42no—
300C2×D61—42no—
301C2×D61#52512yes—
302#6923—84no—
303C4×C61—21no—
304C4×C61—21no—
305C4×C61—21no—
306C4×C61—24no—
307C4×C61—21no—
308C4×C61—42no—
309C4×C61#42112yes—
310C4×C61C2×#11712yes—
311Dic61—21no—
312Dic61—21no—
313Dic61—21no—
314Dic61—21no—
315Dic61—21no—
316Dic61—42no—
317Dic61#52512yes—
318C2×C4×#41—611no—
319#2920—611no—
320#2283—611no—
321#1826—612no—
322#1806—614no—
323#1789—612no—
324#1759—614no—
325C2×C61—22no—
326C2×C61#228011yescharacteristic
327C2×C61C2×#42111yescharacteristic
328C2×C61C2×#42111yescharacteristic
329D61—22no—
330D61—41no—
331D61—44no—
332C22×#41—611no—
333C4×#41—612no—
334C4×#41—612no—
335C4×#41—612no—
336C4×#41—612no—
337C4×#41—1221no—
338C4×#41—1221no—
339C4×#41—1221no—
340C4×#41—1221no—
341C2×C4×Q8—611no—
342C2×C4×Q8#692311yescharacteristic, complemented
343C2×C4×D4—611no—
344C2×C4×D4—611no—
345C2×C4×D4—1221no—
346C2×C4×D4—1221no—
347C2×#117—611no—
348C2×#117—1221no—
349C22×C42—611no—
350C22×C42—1221no—
351#432—1221no—
352#432—1221no—
353#431—1221no—
354#431—1221no—
355#423—612no—
356C2×#104—618no—
357#407—612no—
358C4×M16—612no—
359#383—1221no—
360#382—1221no—
361#381—1221no—
362#380—1221no—
363C4×C4⋊C4—612no—
364C4×C22⋊C4—612no—
365#376—612no—
366#339—618no—
367C61#2074811yescharacteristic, complemented, Sylow
368C2×#41—612no—
369C2×#41—612no—
370C2×#41—1221no—
371C2×#41—1221no—
372C2×#41—1228no—
373C22×Q8C2×#692311yescharacteristic
374C22×D4—611no—
375C22×D4—1221no—
376C23×C4—611no—
377C23×C4—611no—
378C23×C4—611no—
379C23×C4—1221no—
380C23×C4—1221no—
381C23×C4—1221no—
382C2×M16—614no—
383C4×Q8—21no—
384C4×Q8—21no—
385C4×Q8—612no—
386C4×Q8—612no—
387C4×Q8—1221no—
388C4×Q8—1221no—
389C4×Q8C2×#692312yes—
390C4×Q8C2×#692312yes—
391C4×D4—612no—
392C4×D4—612no—
393C4×D4—612no—
394C4×D4—612no—
395C4×D4—1222no—
396C4×D4—1222no—
397C4×D4—1222no—
398C4×D4—1222no—
399C4×D4—1221no—
400C4×D4—1221no—
401C4×D4—1222no—
402C4×D4—1222no—
403C4×D4—1222no—
404C4×D4—1222no—
405C4×D4—1221no—
406C4×D4—1221no—
407#117—612no—
408#117—612no—
409#117—1222no—
410#117—1222no—
411#117—1222no—
412#117—1222no—
413#117—1221no—
414#117—1221no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1222no—
419C2×C4⋊C4C2×#692311yescharacteristic
420C2×C22⋊C4—1221no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1222no—
424C2×C42—21no—
425C2×C42—611no—
426C2×C42—611no—
427C2×C42—611no—
428C2×C42—611no—
429C2×C42—611no—
430C2×C42—611no—
431C2×C42—611no—
432C2×C42—611no—
433C2×C42—611no—
434C2×C42—611no—
435C2×C42—611no—
436C2×C42—614no—
437C2×C42—1222no—
438C2×C42—1222no—
439C2×C42—1222no—
440C2×C42—1222no—
441C2×C42—1221no—
442C2×C42—1221no—
443C2×C42C2×#692311yescharacteristic
444C4⋊C8—1222no—
445C4⋊C8—1222no—
446#104—12216no—
447C22⋊C8—1222no—
448C22⋊C8—1222no—
449#97—1222no—
450C4×C8—1222no—
451#95—1222no—
452#95—1222no—
453#95—1222no—
454#95—1222no—
455C24—611no—
456C24—1221no—
457#41—618no—
458#41—1224no—
459#41—1224no—
460#41—1228no—
461#41—1228no—
462#41—1224no—
463C2×Q8—24no—
464C2×Q8—21no—
465C2×Q8C22×#692312yes—
466C2×D4—612no—
467C2×D4—1224no—
468C2×D4—1222no—
469C2×D4—1222no—
470C2×D4—1228no—
471C2×D4—1221no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—611no—
475C22×C4—611no—
476C22×C4—611no—
477C22×C4—611no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—611no—
486C22×C4—611no—
487C22×C4—611no—
488C22×C4—611no—
489C22×C4—1222no—
490C22×C4—1222no—
491C22×C4—1224no—
492C22×C4—1222no—
493C22×C4—1222no—
494C22×C4—1228no—
495C22×C4—1221no—
496C22×C4—1222no—
497C22×C4—1222no—
498C22×C4—1224no—
499C22×C4—2442no—
500C22×C4—2442no—
501C22×C4—2444no—
502C22×C4—2444no—
503C22×C4C22⋊#692311yescharacteristic
504C22×C4C22⋊#692311yescharacteristic
505C22×C4C22×#692311yescharacteristic
506M16—1228no—
507C2×C8—1224no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—21no—
511C4⋊C4—612no—
512C4⋊C4—1222no—
513C4⋊C4—1222no—
514C4⋊C4—1221no—
515C4⋊C4—2444no—
516C4⋊C4C22×#692312yes—
517C22⋊C4—1222no—
518C22⋊C4—1222no—
519C22⋊C4—1222no—
520C22⋊C4—1222no—
521C22⋊C4—1222no—
522C22⋊C4—1222no—
523C22⋊C4—2444no—
524C42—22no—
525C42—22no—
526C42—611no—
527C42—611no—
528C42—611no—
529C42—611no—
530C42—611no—
531C42—611no—
532C42—611no—
533C42—611no—
534C42—611no—
535C42—611no—
536C42—611no—
537C42—611no—
538C42—1222no—
539C42—1222no—
540C42—1222no—
541C42—1222no—
542C42—1228no—
543C42C22⋊#692311yescharacteristic
544C42C22⋊#692311yescharacteristic
545C42C22×#692311yescharacteristic
546C42C22×#692311yescharacteristic
547C23—611no—
548C23—611no—
549C23—1222no—
550C23—1224no—
551C23—1222no—
552C23—1222no—
553C23—2441no—
554C23—2442no—
555C23—2442no—
556C23C2×C22⋊#692311yescharacteristic
557Q8—24no—
558Q8—22no—
559Q8#1451985614yes—
560D4—614no—
561D4—1228no—
562D4—1228no—
563D4—1224no—
564D4—1222no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—42no—
572C2×C4—611no—
573C2×C4—614no—
574C2×C4—611no—
575C2×C4—611no—
576C2×C4—611no—
577C2×C4—611no—
578C2×C4—611no—
579C2×C4—1222no—
580C2×C4—1228no—
581C2×C4—1222no—
582C2×C4—1228no—
583C2×C4—1222no—
584C2×C4—1222no—
585C2×C4—1222no—
586C2×C4—1222no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2444no—
596C2×C4—2444no—
597C2×C4—2444no—
598C2×C4—2441no—
599C2×C4—2444no—
600C2×C4—2444no—
601C2×C4—2444no—
602C2×C4—2442no—
603C2×C4—2444no—
604C2×C4C2×C22⋊#692311yescharacteristic
605C2×C4C2×C22⋊#692311yescharacteristic
606C2×C4C2×C22⋊#692311yescharacteristic
607C2×C4C2×C22⋊#692311yescharacteristic
608C2×C4C2×C22⋊#692311yescharacteristic
609C2×C4C23×#692311yescharacteristic
610C8—2444no—
611C22—21no—
612C22—21no—
613C22—612no—
614C22—1221no—
615C22—1221no—
616C22—1221no—
617C22—1221no—
618C22—1224no—
619C22—1221no—
620C22—2444no—
621C22—2448no—
622C22—2442no—
623C22—2442no—
624C22#1684075912yes—
625C22C22×C22⋊#692311yescharacteristic
626C4—21no—
627C4—21no—
628C4—21no—
629C4—24no—
630C4—21no—
631C4—42no—
632C4—612no—
633C4—1221no—
634C4—1221no—
635C4—1221no—
636C4—1221no—
637C4—1221no—
638C4—4884no—
639C4—2442no—
640C4#1684076112yes—
641C4C2×#1451985612yes—
642C2—22no—
643C2—1222no—
644C2—2441no—
645C2—2444no—
646C2#2146226311yescharacteristic
647C2C2×#1684075911yescharacteristic
648C2C2×#1684076111yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC2×C4×Q8×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC22×#11758226C22
cosocleC2×C4×D61C24
hypercenterC2×C4×Q8#6923
hypocenterC61#20748
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #21462496C2semidirect
2. #21462266C2semidirect
3. #21461725C2semidirect
4. #21461692C2semidirect
6. #21461583C2semidirect
12. #16840766C22semidirect
15. C4×#11758213C22semidirect
20. C4×C4⋊#6923C22semidirect
22. #16840698C22semidirect
23. #16840632C22semidirect
29. C4×#11758226C4semidirect
30. C4×#11758226C4semidirect
33. C2×Q8×Dic61C4semidirect
35. C2×C4×D244C4semidirect
37. C2×#14519739C4semidirect
42. C2×C4×Q8×C61C4semidirect
342. C2×C4×Q8#6923semidirect
367. C61#20748semidirect

Maximal subgroups

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

subgroupindex
#214624962
#214622662
#214617252
#214616922
#214616622
#214615832
C2×C4×#117582262
#2074861