Finite Group Databasebeta

← #30617843

Subgroups — #30617843

Order 15616.

1352 conjugacy classes of subgroups.

The subgroup lattice has 690 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#21461984C214yesmaximal, complemented
4#21461956C214yesmaximal, complemented
5#21461662C212yesmaximal, complemented
6#21461660C212yesmaximal, complemented
7C2×#16840311C211yesmaximal, characteristic
8#16840784—21no—
9#16840783—21no—
10#16840781—21no—
11#16840778—21no—
12#16840766C2212yescomplemented
13C2×#14519842C2218yescomplemented
14C2×#14519841C2218yescomplemented
15#16840733C2212yes—
16C4×#11758213C2212yescomplemented
17#16840707—21no—
18#16840706—21no—
19#16840705—21no—
20#16840704—21no—
21C4×C4⋊#6923C2212yescomplemented
22C4×C22⋊#6923C2212yes—
23#16840698C2212yescomplemented
24C22×#11758226C2211yescharacteristic
25C22×#11758224C2211yescharacteristic
26#16840311—21no—
27#16840311—21no—
28#16840311—21no—
29#16840311—21no—
30#16840311C412yescomplemented
31#16840311C412yescomplemented
32#16840311C2212yes—
33#16840311C2212yes—
34C2×#14519775C411yescharacteristic, complemented
35C2×#14519773C411yescharacteristic, complemented
36C2×#14519765C411yescharacteristic, complemented
37C2×#14519750—21no—
38C2×#14519750—21no—
39C2×#14519737—21no—
40C2×#14519737—21no—
41C2×C42×D61C2211yescharacteristic
42C2×#124×C61C411yescharacteristic, complemented
43C2×#11758213C2314yes—
44C2×C4⋊#6923—22no—
45C2×C22⋊#6923—22no—
46C2×C4×#6923C2314yes—
47C4⋊#92420—22no—
48C4⋊#92420—22no—
49#14519842—48no—
50#14519841—48no—
51#14519834—22no—
52C22⋊#92420—22no—
53#14519832—22no—
54C4×#92420—22no—
55(C2×#6923)⋊C4—22no—
56(C2×#6923)⋊C4—22no—
57(C2×#6923)⋊C4—22no—
58(C2×#6923)⋊C4—22no—
59C2×#11758226—21no—
60C2×#11758226—21no—
61C2×#11758226—44no—
62C2×#11758226C2×C412yes—
63C2×#11758226C2312yes—
64C2×#11758224—21no—
65C2×#11758224—21no—
66C2×#11758224—44no—
67C2×#11758224C2×C412yes—
68C2×#11758224C2312yes—
69C22×Dic122C2×C411yescharacteristic
70C22×D244—21no—
71C22×D244C2×C411yescharacteristic
72C22×C22⋊D61—21no—
73C22×C22⋊D61—21no—
74C22×C22⋊D61—21no—
75C23×Dic61—21no—
76C22×C4×D61—21no—
77C22×C4×D61D411yescharacteristic
78C22×C4×D61D411yescharacteristic
79C22×C4×D61C2311yescharacteristic
80#14519775—21no—
81#14519775—21no—
82#14519775C2×C412yes—
83#14519775C2×C412yes—
84#14519773—21no—
85#14519773—21no—
86#14519773C2×C412yes—
87#14519773C2×C412yes—
88#14519765—21no—
89#14519765—21no—
90#14519765C2×C412yes—
91#14519765C2×C412yes—
92#14519750—22no—
93#14519750—22no—
94#14519750—22no—
95#14519750—22no—
96#14519750—22no—
97#14519750—22no—
98#14519750—22no—
99#14519750—22no—
100#14519737—22no—
101#14519737—22no—
102#14519737—22no—
103#14519737—22no—
104#14519737—22no—
105#14519737—22no—
106#14519737—22no—
107#14519737—22no—
108C2×#11758196—21no—
109C2×#11758196—21no—
110C2×#11758198—21no—
111C2×#11758198—21no—
112C2×C4×Dic61C2×C411yescharacteristic
113C2×C4×Dic61D411yescharacteristic
114C2×C4×Dic61D411yescharacteristic
115C42×D61C2×C411yescharacteristic
116C42×D61C2×C411yescharacteristic
117C42×D61D411yescharacteristic
118C42×D61D411yescharacteristic
119C42×D61D411yescharacteristic
120C42×D61D411yescharacteristic
121C42×D61C2311yescharacteristic
122C42×D61C2311yescharacteristic
123C22×Q8×C61C2×C411yescharacteristic
124C22×D4×C61C2×C411yescharacteristic
125#124×C61—21no—
126#124×C61—21no—
127#124×C61C2×C412yes—
128#124×C61C2×C412yes—
129C2×C22⋊C4×C61—21no—
130C2×C22⋊C4×C61—21no—
131C2×C42×C61C2×C411yescharacteristic
132C24×C61—21no—
133C23×D61—21no—
134C23×D61C2×D411yescharacteristic
135C22×#6923—24no—
136C2×Q8×C61—21no—
137C2×Q8×C61—42no—
138C2×Q8×C61C22×C412yes—
139C2×D4×C61—21no—
140C2×D4×C61—42no—
141C2×D4×C61C22×C412yes—
142C22×C4×C61—21no—
143C22×C4×C61C22⋊C411yescharacteristic
144C22×C4×C61C22⋊C411yescharacteristic
145C22×C4×C61C22×C411yescharacteristic
146C2×C22⋊D61—22no—
147C2×C22⋊D61—22no—
148C2×C22⋊D61—22no—
149C2×C22⋊D61—22no—
150C2×C22⋊D61—22no—
151C2×C22⋊D61—22no—
152C2×C22⋊D61—44no—
153C2×C22⋊D61—44no—
154C2×C22⋊D61—44no—
155C22×Dic61—21no—
156C22×Dic61—22no—
157C22×Dic61—22no—
158C22×Dic61—44no—
159C22×Dic61C22⋊C411yescharacteristic
160C22×Dic61C22⋊C411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162#11758226—28no—
163#11758226—44no—
164#11758226—44no—
165#11758226—44no—
166#11758224—28no—
167#11758224—44no—
168#11758224—44no—
169#11758224—44no—
170C2×D244—22no—
171C2×D244—22no—
172C2×D244—21no—
173C2×D244—42no—
174C2×D244—44no—
175C2×D244C22×C412yes—
176C2×C4×D61—24no—
177C2×C4×D61—22no—
178C2×C4×D61—22no—
179C2×C4×D61—42no—
180C2×C4×D61—42no—
181C2×C4×D61—44no—
182C2×C4×D61C22⋊C411yescharacteristic
183C2×C4×D61C22⋊C411yescharacteristic
184C2×C4×D61C22⋊C411yescharacteristic
185C2×C4×D61C22⋊C411yescharacteristic
186C2×C4×D61C22×C411yescharacteristic
187C2×C4×D61C2×D411yescharacteristic
188C2×C4×D61C2×D411yescharacteristic
189C2×C4×D61C2×D411yescharacteristic
190C2×C4×D61C2×D411yescharacteristic
191C2×C4×D61C2×D411yescharacteristic
192C2×C4×D61C2×D411yescharacteristic
193C2×C4×D61C2411yescharacteristic
194C2×Dic122—21no—
195C2×Dic122—42no—
196C2×Dic122C22×C412yes—
197C22⋊#6923—44no—
198C2×#92420—24no—
199C4⋊#6923—44no—
200C4×#6923—28no—
201#11758213—28no—
202C22⋊C4×C61—22no—
203C22⋊C4×C61—22no—
204C22⋊C4×C61—22no—
205C22⋊C4×C61—22no—
206C42×C61C22⋊C411yescharacteristic
207C42×C61C22⋊C411yescharacteristic
208C42×C61C22×C411yescharacteristic
209C42×C61C22×C411yescharacteristic
210#11758198—22no—
211#11758198—22no—
212#11758198—22no—
213#11758198—22no—
214#11758196—22no—
215#11758196—22no—
216#11758196—22no—
217#11758196—22no—
218C4×Dic61C22⋊C411yescharacteristic
219C4×Dic61C22⋊C411yescharacteristic
220C4×Dic61C22⋊C411yescharacteristic
221C4×Dic61C22⋊C411yescharacteristic
222C4×Dic61C22⋊C411yescharacteristic
223C4×Dic61C22⋊C411yescharacteristic
224C4×Dic61C22×C411yescharacteristic
225C4×Dic61C22×C411yescharacteristic
226C4×Dic61C2×D411yescharacteristic
227C4×Dic61C2×D411yescharacteristic
228C4×Dic61C2×D411yescharacteristic
229C4×Dic61C2×D411yescharacteristic
230C23×C61—22no—
231C23×C61—42no—
232C23×C61—42no—
233C23×C61—42no—
234C23×C61C2×C22⋊C411yescharacteristic
235C22×D61—22no—
236C22×D61—22no—
237C22×D61—22no—
238C22×D61—41no—
239C22×D61—42no—
240C22×D61—42no—
241C22×D61—42no—
242C22×D61C2×C22⋊C411yescharacteristic
243C22×D61C22×D411yescharacteristic
244C2×#6923—44no—
245C2×#6923—44no—
246C2×#6923—44no—
247Q8×C61—24no—
248Q8×C61—42no—
249D4×C61—24no—
250D4×C61—42no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
253C2×C4×C61—22no—
254C2×C4×C61—41no—
255C2×C4×C61—42no—
256C2×C4×C61C2×C22⋊C411yescharacteristic
257C2×C4×C61C2×C22⋊C411yescharacteristic
258C2×C4×C61C2×C22⋊C411yescharacteristic
259C2×C4×C61C2×C22⋊C411yescharacteristic
260C2×C4×C61C2×C22⋊C411yescharacteristic
261C2×C4×C61C23×C411yescharacteristic
262C22⋊D61—44no—
263C22⋊D61—44no—
264C22⋊D61—44no—
265C22⋊D61—44no—
266C22⋊D61—44no—
267C22⋊D61—44no—
268C2×Dic61—22no—
269C2×Dic61—22no—
270C2×Dic61—22no—
271C2×Dic61—22no—
272C2×Dic61—44no—
273C2×Dic61—42no—
274C2×Dic61—44no—
275C2×Dic61C2×C22⋊C411yescharacteristic
276C2×Dic61C2×C22⋊C411yescharacteristic
277C2×Dic61C2×C22⋊C411yescharacteristic
278C2×Dic61C2×C22⋊C411yescharacteristic
279C2×Dic61C2×C22⋊C411yescharacteristic
280C2×Dic61C22×D411yescharacteristic
281D244—24no—
282D244—42no—
283D244—44no—
284D244—44no—
285C4×D61—22no—
286C4×D61—22no—
287C4×D61—22no—
288C4×D61—22no—
289C4×D61—22no—
290C4×D61—41no—
291C4×D61—41no—
292C4×D61—41no—
293C4×D61—41no—
294C4×D61—41no—
295C4×D61—41no—
296C4×D61—44no—
297C4×D61—44no—
298C4×D61H(5,𝔽2)14yes—
299Dic122—24no—
300Dic122—42no—
301#92420—44no—
302#21816—611nomaximal, Sylow
303C22×C61—21no—
304C22×C61—21no—
305C22×C61—42no—
306C22×C61—42no—
307C22×C61—42no—
308C22×C61—84no—
309C22×C61#42112yes—
310C22×C61C22×C22⋊C411yescharacteristic
311C2×D61—21no—
312C2×D61—21no—
313C2×D61—21no—
314C2×D61—21no—
315C2×D61—21no—
316C2×D61—44no—
317C2×D61—42no—
318C2×D61—42no—
319C2×D61—42no—
320C2×D61—84no—
321C2×D61#53512yes—
322#6923—84no—
323C4×C61—21no—
324C4×C61—21no—
325C4×C61—21no—
326C4×C61—21no—
327C4×C61—21no—
328C4×C61—42no—
329C4×C61#51812yes—
330Dic61—21no—
331Dic61—21no—
332Dic61—21no—
333Dic61—21no—
334Dic61—42no—
335Dic61#42112yes—
336Dic61#53512yes—
337C2×#532—611no—
338#2920—611no—
339#2283—611no—
340#1980—614no—
341#1963—614no—
342#1789—612no—
343#1789—612no—
344C2×C61—22no—
345C2×C61—82no—
346C2×C61#230911yescharacteristic
347C2×C61C2×#42111yescharacteristic
348C2×C61C2×#42111yescharacteristic
349D61—22no—
350D61—41no—
351D61—82no—
352C22×#41—611no—
353C22×#41—611no—
354#532—612no—
355#532—612no—
356#532—612no—
357#532—612no—
358#532—1221no—
359#532—1221no—
360#532—1221no—
361#532—1221no—
362C2×C4⋊Q8—611no—
363C2×#127—611no—
364C2×#124—611no—
365C2×#124#692311yescharacteristic, complemented
366C2×#121—1221no—
367C2×#121—1221no—
368C2×C4×D4—1221no—
369C2×C4×D4—1221no—
370C22×C42—611no—
371#432—1221no—
372#432—1221no—
373#431—1221no—
374#431—1221no—
375#423—612no—
376C2×#104—618no—
377C2×#104—618no—
378#407—612no—
379C4×M16—612no—
380#383—1221no—
381#382—1221no—
382#381—1221no—
383#380—1221no—
384C4×C4⋊C4—612no—
385C4×C22⋊C4—612no—
386#376—612no—
387C61#2181611yescharacteristic, complemented, Sylow
388C2×#41—612no—
389C2×#41—612no—
390C2×#41—612no—
391C2×#41—612no—
392C2×#41—1221no—
393C2×#41—1221no—
394C2×#41—1221no—
395C2×#41—1221no—
396C2×#41—2444no—
397C2×#41—2444no—
398C22×Q8—611no—
399C22×Q8C2×#692311yescharacteristic
400C22×D4—611no—
401C22×D4—1221no—
402C22×D4—1221no—
403C22×D4—1221no—
404C22×D4—1221no—
405C22×D4C2×#692311yescharacteristic
406C23×C4—611no—
407C23×C4—611no—
408C23×C4—611no—
409C23×C4—1221no—
410C23×C4—1221no—
411C2×M16—614no—
412C4⋊Q8—612no—
413C4⋊Q8—612no—
414C4⋊Q8—1221no—
415C4⋊Q8—1221no—
416#127—612no—
417#127—612no—
418#127—1221no—
419#127—1221no—
420#124—21no—
421#124—21no—
422#124—612no—
423#124—612no—
424#124—1221no—
425#124—1221no—
426#124C2×#692312yes—
427#124C2×#692312yes—
428#121—1222no—
429#121—1222no—
430#121—1222no—
431#121—1222no—
432#121—1222no—
433#121—1222no—
434#121—1222no—
435#121—1222no—
436C4×D4—1222no—
437C4×D4—1222no—
438C4×D4—1222no—
439C4×D4—1222no—
440C4×D4—1222no—
441C4×D4—1222no—
442C4×D4—1222no—
443C4×D4—1222no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4—1222no—
447C2×C22⋊C4—21no—
448C2×C22⋊C4—21no—
449C2×C22⋊C4—1221no—
450C2×C22⋊C4—1221no—
451C2×C22⋊C4—1222no—
452C2×C42—611no—
453C2×C42—611no—
454C2×C42—611no—
455C2×C42—611no—
456C2×C42—611no—
457C2×C42—611no—
458C2×C42—611no—
459C2×C42—611no—
460C2×C42—611no—
461C2×C42—611no—
462C2×C42—611no—
463C2×C42—614no—
464C2×C42C2×#692311yescharacteristic
465C4⋊C8—1222no—
466C4⋊C8—1222no—
467#104—2448no—
468#104—2448no—
469C22⋊C8—1222no—
470C22⋊C8—1222no—
471#97—1222no—
472C4×C8—1222no—
473#95—1222no—
474#95—1222no—
475#95—1222no—
476#95—1222no—
477C24—21no—
478C24—611no—
479C24—1221no—
480#41—1228no—
481#41—1228no—
482#41—2444no—
483#41—2444no—
484#41—2444no—
485#41—2444no—
486#41—2444no—
487#41—2444no—
488C2×Q8—21no—
489C2×Q8—42no—
490C2×Q8—612no—
491C2×Q8—1221no—
492C2×Q8—2442no—
493C2×Q8C22×#692312yes—
494C2×D4—21no—
495C2×D4—42no—
496C2×D4—612no—
497C2×D4—1222no—
498C2×D4—1222no—
499C2×D4—1222no—
500C2×D4—1222no—
501C2×D4—1222no—
502C2×D4—1222no—
503C2×D4—1222no—
504C2×D4—1222no—
505C2×D4—1221no—
506C2×D4—2442no—
507C2×D4—2444no—
508C2×D4—2444no—
509C2×D4—2444no—
510C2×D4—2444no—
511C2×D4C22×#692312yes—
512C22×C4—21no—
513C22×C4—611no—
514C22×C4—611no—
515C22×C4—611no—
516C22×C4—611no—
517C22×C4—611no—
518C22×C4—611no—
519C22×C4—611no—
520C22×C4—611no—
521C22×C4—611no—
522C22×C4—611no—
523C22×C4—611no—
524C22×C4—611no—
525C22×C4—611no—
526C22×C4—611no—
527C22×C4—611no—
528C22×C4—1224no—
529C22×C4—1222no—
530C22×C4—1222no—
531C22×C4—1222no—
532C22×C4—1222no—
533C22×C4—1221no—
534C22×C4—1224no—
535C22×C4—2442no—
536C22×C4—2442no—
537C22×C4—2444no—
538C22×C4—2444no—
539C22×C4C22⋊#692311yescharacteristic
540C22×C4C22⋊#692311yescharacteristic
541C22×C4C22×#692311yescharacteristic
542M16—1228no—
543C2×C8—1224no—
544C4⋊C4—1222no—
545C4⋊C4—1222no—
546C4⋊C4—1222no—
547C4⋊C4—1222no—
548C4⋊C4—2444no—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C22⋊C4—22no—
552C22⋊C4—22no—
553C22⋊C4—1222no—
554C22⋊C4—1222no—
555C22⋊C4—1222no—
556C22⋊C4—1222no—
557C22⋊C4—2444no—
558C42—611no—
559C42—611no—
560C42—611no—
561C42—611no—
562C42—611no—
563C42—611no—
564C42—611no—
565C42—611no—
566C42—611no—
567C42—611no—
568C42—611no—
569C42—611no—
570C42—1228no—
571C42C22⋊#692311yescharacteristic
572C42C22⋊#692311yescharacteristic
573C42C22×#692311yescharacteristic
574C42C22×#692311yescharacteristic
575C23—22no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—611no—
580C23—611no—
581C23—1222no—
582C23—1222no—
583C23—1222no—
584C23—2441no—
585C23—2442no—
586C23—2442no—
587C23—2442no—
588C23C2×C22⋊#692311yescharacteristic
589Q8—24no—
590Q8—42no—
591Q8—1224no—
592Q8—2442no—
593D4—24no—
594D4—42no—
595D4—1224no—
596D4—2442no—
597D4—2444no—
598D4—2444no—
599D4—2444no—
600D4—2444no—
601D4—2444no—
602D4—2444no—
603D4—2444no—
604D4—2444no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—611no—
611C2×C4—611no—
612C2×C4—614no—
613C2×C4—611no—
614C2×C4—611no—
615C2×C4—611no—
616C2×C4—611no—
617C2×C4—1222no—
618C2×C4—1222no—
619C2×C4—1222no—
620C2×C4—1222no—
621C2×C4—1222no—
622C2×C4—1222no—
623C2×C4—1222no—
624C2×C4—1222no—
625C2×C4—1222no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2444no—
633C2×C4—2444no—
634C2×C4—2444no—
635C2×C4—2444no—
636C2×C4—2444no—
637C2×C4—2444no—
638C2×C4—2444no—
639C2×C4—2442no—
640C2×C4C2×C22⋊#692311yescharacteristic
641C2×C4C2×C22⋊#692311yescharacteristic
642C2×C4C2×C22⋊#692311yescharacteristic
643C2×C4C2×C22⋊#692311yescharacteristic
644C2×C4C2×C22⋊#692311yescharacteristic
645C2×C4C23×#692311yescharacteristic
646C8—2444no—
647C22—21no—
648C22—21no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—84no—
653C22—612no—
654C22—1221no—
655C22—1221no—
656C22—1221no—
657C22—1221no—
658C22—1221no—
659C22—4884no—
660C22—2444no—
661C22—2442no—
662C22—2442no—
663C22—2442no—
664C22#1684075912yes—
665C22C22×C22⋊#692311yescharacteristic
666C4—21no—
667C4—21no—
668C4—21no—
669C4—21no—
670C4—21no—
671C4—42no—
672C4—612no—
673C4—612no—
674C4—1221no—
675C4—1221no—
676C4—1221no—
677C4—1221no—
678C4—4884no—
679C4—2442no—
680C4#1684082912yes—
681C2—22no—
682C2—82no—
683C2—1222no—
684C2—4882no—
685C2—2441no—
686C2#2146229211yescharacteristic
687C2D61⋊#42111yescharacteristic
688C2C2×#1684075911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC2×#124×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840311C2
cosocleC2×C4×D61C24
hypercenterC2×#124#6923
hypocenterC61#21816
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #21462496C2semidirect
2. #21462266C2semidirect
3. #21461984C2semidirect
4. #21461956C2semidirect
5. #21461662C2semidirect
6. #21461660C2semidirect
12. #16840766C22semidirect
13. C2×#14519842C22semidirect
14. C2×#14519841C22semidirect
16. C4×#11758213C22semidirect
21. C4×C4⋊#6923C22semidirect
23. #16840698C22semidirect
30. #16840311C4semidirect
31. #16840311C4semidirect
34. C2×#14519775C4semidirect
35. C2×#14519773C4semidirect
36. C2×#14519765C4semidirect
42. C2×#124×C61C4semidirect
365. C2×#124#6923semidirect
387. C61#21816semidirect

Maximal subgroups

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

subgroupindex
#214624962
#214622662
#214619842
#214619562
#214616622
#214616602
C2×#168403112
#2181661