Finite Group Databasebeta

← #29696169

Subgroups — #29696169

Order 15616.

1322 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840413C211yesmaximal, characteristic
2M16×C22⋊D61C212yesmaximal, complemented
3#21446806C212yesmaximal, complemented
4#21446282C212yesmaximal, complemented
5#21446281C212yesmaximal, complemented
6#21445899C211yesmaximal, characteristic, complemented
7#21445898C211yesmaximal, characteristic, complemented
8#21445810C211yesmaximal, characteristic, complemented
9#21445610C211yesmaximal, characteristic, complemented
10#343×D61C211yesmaximal, characteristic, complemented
11C2×C61×#343C211yesmaximal, characteristic, complemented
12C22×#11758222C412yes—
13C2×M16×D61C2212yes—
14#16840413—22no—
15#16840413—22no—
16#16840413—22no—
17#16840413—22no—
18C2×#14519776C2211yescharacteristic
19C2×#14519762C412yes—
20C2×#14519760C412yes—
21C2×#14519755C412yes—
22C2×#14519752C2211yescharacteristic
23C2×#14519751C2211yescharacteristic
24C2×#14519739C2211yescharacteristic
25C2×C4×C22⋊D61C2211yescharacteristic
26C2×C4⋊C4×D61C2211yescharacteristic
27#16839838—22no—
28#16839833—22no—
29#16839831—22no—
30C8×C22⋊D61—22no—
31#16839785C2212yes—
32C2×#14519693C2212yes—
33C2×#14519693C2212yes—
34#16839726C2212yes—
35#16839725C2212yes—
36M16×Dic61C2212yes—
37#16839531C2211yescharacteristic
38#16839531C2212yes—
39#16839531C2211yescharacteristic
40#16839530C2212yes—
41#16839530C2212yes—
42C22×M16×C61C2212yes—
43C22×C4⋊C4×C61C2211yescharacteristic
44C61×#343C2211yescharacteristic
45C61×#343C2212yes—
46C61×#343C2211yescharacteristic
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222—22no—
50C2×#11758222—22no—
51C2×#11758222—48no—
52C22×Dic122C2×C412yes—
53C22×D244C2×C412yes—
54C22×D244C2311yescharacteristic
55C22×C22⋊D61C2×C411yescharacteristic
56C22×C22⋊D61C2×C412yes—
57C22×C4×D61C2×C412yes—
58C22×C4×D61C2×C412yes—
59C22×C4×D61C2311yescharacteristic
60C2×#11758194C2312yes—
61C2×#11758194C2312yes—
62C2×#11758189—22no—
63M16×D61—24no—
64M16×D61—42no—
65C2×C8×D61—22no—
66#14519776—22no—
67#14519776—22no—
68#14519762—24no—
69#14519762—24no—
70#14519760—24no—
71#14519760—24no—
72#14519755—24no—
73#14519755—24no—
74#14519752—22no—
75#14519752—22no—
76#14519751—22no—
77#14519751—22no—
78#14519739—22no—
79#14519739—22no—
80C4×C22⋊D61D412yes—
81C4×C22⋊D61D412yes—
82C4×C22⋊D61D412yes—
83C4×C22⋊D61D412yes—
84C2×C4⋊Dic61C2311yescharacteristic
85C2×#11758196C2×C412yes—
86C2×#11758196C2×C412yes—
87C2×#11758196C2×C411yescharacteristic
88C4⋊C4×D61—22no—
89C4⋊C4×D61—22no—
90C2×C22⋊Dic61C2×C411yescharacteristic
91C2×#11758198C2×C412yes—
92C2×#11758198C2×C412yes—
93C2×#11758198C2×C412yes—
94C2×#11758198C2×C411yescharacteristic
95C2×C4×Dic61C2311yescharacteristic
96#14519706—22no—
97#14519706—22no—
98#14519693—24no—
99#14519693—42no—
100C22⋊#92418—22no—
101C22⋊#92418—22no—
102#14519689—22no—
103#14519689—22no—
104#14519687—22no—
105C8×Dic61—22no—
106C23×C4×C61C2×C412yes—
107C23×C4×C61C2311yescharacteristic
108C2×M16×C61—22no—
109C2×M16×C61—24no—
110C2×M16×C61—42no—
111C2×M16×C61C2312yes—
112C2×M16×C61C2312yes—
113C22×C8×C61—22no—
114C2×C4⋊C4×C61—22no—
115C2×C4⋊C4×C61—22no—
116C2×C4⋊C4×C61C2×C412yes—
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C24×C61C22×C411yescharacteristic
120C23×D61C22×C411yescharacteristic
121C23×D61C22×C412yes—
122C22×C4×C61—22no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—44no—
128C22×C4×C61C22×C411yescharacteristic
129C22×C4×C61C22×C412yes—
130C22×C4×C61C22×C412yes—
131C22×C4×C61C2×D412yes—
132C22×C4×C61C2×D412yes—
133C22×C4×C61C2411yescharacteristic
134C2×C22⋊D61—24no—
135C2×C22⋊D61—22no—
136C2×C22⋊D61—22no—
137C2×C22⋊D61—44no—
138C2×C22⋊D61C22⋊C411yescharacteristic
139C2×C22⋊D61C22⋊C411yescharacteristic
140C2×C22⋊D61C22⋊C411yescharacteristic
141C2×C22⋊D61C22⋊C411yescharacteristic
142C22×Dic61C22×C411yescharacteristic
143C22×Dic61C22×C411yescharacteristic
144C22×Dic61C22×C412yes—
145C22×Dic61C22×C411yescharacteristic
146#11758222—48no—
147#11758222—48no—
148#11758222—48no—
149#11758222—48no—
150C2×D244—22no—
151C2×D244—22no—
152C2×D244—22no—
153C2×D244—22no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—44no—
157C2×C4×D61—22no—
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—22no—
161C2×C4×D61—22no—
162C2×C4×D61—22no—
163C2×C4×D61—22no—
164C2×C4×D61—44no—
165C2×C4×D61—44no—
166C2×C4×D61C2×D412yes—
167C2×C4×D61C2×D412yes—
168C2×Dic122—22no—
169C2×Dic122—22no—
170C2×Dic122—44no—
171M16×C61—22no—
172M16×C61—22no—
173M16×C61—22no—
174M16×C61—44no—
175M16×C61#4114yescomplemented
176C2×C8×C61—22no—
177C2×C8×C61—22no—
178C2×C8×C61—44no—
179C4⋊C4×C61—24no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—22no—
182C22⋊Dic61C22⋊C412yes—
183C22⋊Dic61C22⋊C412yes—
184#11758198—24no—
185#11758198—24no—
186#11758198—24no—
187#11758198C22⋊C412yes—
188#11758198C22⋊C412yes—
189C4⋊Dic61—22no—
190#11758196—24no—
191#11758196—24no—
192#11758196C22⋊C412yes—
193#11758196C22⋊C412yes—
194C4×Dic61C2×D412yes—
195C4×Dic61C2×D412yes—
196#11758194—22no—
197#11758194—22no—
198#11758194—22no—
199#11758194—22no—
200C2×#92418—22no—
201C2×#92418—22no—
202#11758189—44no—
203C8×D61—44no—
204C23×C61—21no—
205C23×C61—21no—
206C23×C61—22no—
207C23×C61—41no—
208C23×C61C2×C22⋊C411yescharacteristic
209C23×C61C2×C22⋊C411yescharacteristic
210C23×C61C23×C411yescharacteristic
211C22×D61—21no—
212C22×D61—21no—
213C22×D61—22no—
214C22×D61—22no—
215C22×D61—22no—
216C22×D61—22no—
217C22×D61—41no—
218C22×D61—42no—
219C22×D61—42no—
220C22×D61C2×C22⋊C411yescharacteristic
221C22×D61C2×C22⋊C411yescharacteristic
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—44no—
231C2×C4×C61—44no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61C2×C22⋊C412yes—
235C2×C4×C61C22×D412yes—
236C2×C4×C61C2×#4112yes—
237C2×C4×C61H(5,𝔽2)11yescharacteristic
238C2×C4×C612−1+411yescharacteristic
239C22⋊D61—22no—
240C22⋊D61—22no—
241C22⋊D61—22no—
242C22⋊D61—44no—
243C22⋊D61—44no—
244C22⋊D61#10014yescomplemented
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—22no—
248C2×Dic61—22no—
249C2×Dic61—41no—
250C2×Dic61—42no—
251C2×Dic61C2×C22⋊C412yes—
252C2×Dic61C2×C22⋊C411yescharacteristic
253C2×Dic61C2×C22⋊C412yes—
254C2×Dic61C2×C22⋊C411yescharacteristic
255D244—44no—
256D244—44no—
257D244—42no—
258D244—42no—
259C4×D61—22no—
260C4×D61—22no—
261C4×D61—22no—
262C4×D61—22no—
263C4×D61—44no—
264C4×D61—44no—
265C4×D61—44no—
266C4×D61—44no—
267Dic122—44no—
268Dic122—44no—
269C8×C61—24no—
270C8×C61—42no—
271#92418—42no—
272#92418—42no—
273#22479—611nomaximal, Sylow
274C22×C61—21no—
275C22×C61—21no—
276C22×C61—21no—
277C22×C61—21no—
278C22×C61—21no—
279C22×C61—21no—
280C22×C61—21no—
281C22×C61—42no—
282C22×C61—42no—
283C22×C61C2×#10012yes—
284C22×C61C22×C22⋊C411yescharacteristic
285C22×C61C4×#4111yescharacteristic
286C22×C61#52011yescharacteristic
287C2×D61—21no—
288C2×D61—21no—
289C2×D61—21no—
290C2×D61—21no—
291C2×D61—21no—
292C2×D61—42no—
293C2×D61—42no—
294C2×D61—42no—
295C2×D61—42no—
296C2×D61—42no—
297C2×D61—82no—
298C2×D61—82no—
299C2×D61C2×#10012yes—
300C4×C61—22no—
301C4×C61—42no—
302C4×C61—42no—
303C4×C61#54612yes—
304C4×C61#54712yes—
305Dic61—22no—
306Dic61—21no—
307Dic61—22no—
308Dic61—42no—
309Dic61C2×#10012yes—
310C2×#547—611no—
311D4×M16—612no—
312#1997—612no—
313#1900—612no—
314#1900—612no—
315#1821—611no—
316#1821—611no—
317#1800—611no—
318C4×#100—611no—
319C2×#343—611no—
320C2×#343D6111yescharacteristic, complemented
321C2×C61—21no—
322C2×C61—21no—
323C2×C61—22no—
324C2×C61—41no—
325C2×C61#233911yescharacteristic
326C2×C61C22×#10011yescharacteristic
327C2×C61#288911yescharacteristic
328D61—22no—
329D61—41no—
330D61—82no—
331C22×#41—612no—
332C22×M16—612no—
333C22×M16C2×D6112yes—
334#547—1222no—
335#547—1222no—
336#547—1222no—
337#547—1222no—
338C2×C4⋊Q8—611no—
339C2×#123—612no—
340C2×#123—612no—
341C2×C22⋊Q8—612no—
342C2×#121—611no—
343C2×#121—611no—
344C2×C4×D4—611no—
345C2×C4×D4—611no—
346C22×C4⋊C4—611no—
347C22×C4⋊C4C2×D6111yescharacteristic
348#436—1222no—
349#435—1222no—
350#435—1222no—
351C8×D4—1222no—
352#423—612no—
353C2×#100—612no—
354C2×#100—612no—
355#407—612no—
356#407—612no—
357C4×M16—612no—
358#343—612no—
359#343—612no—
360#343—611no—
361#343—611no—
362#343—612no—
363#343C2×D6111yescharacteristic
364#343C2×D6112yes—
365#343C2×D6111yescharacteristic
366C61#2247911yescharacteristic, complemented, Sylow
367C2×#41—1222no—
368C2×#41—1222no—
369C2×#41—1222no—
370C2×#41—1222no—
371C2×#41—2448no—
372C22×Q8—612no—
373C22×D4—611no—
374C22×D4—612no—
375C22×D4—612no—
376C22×D4—611no—
377C23×C4—611no—
378C23×C4—612no—
379C23×C4—612no—
380C23×C4C4×D6112yes—
381C23×C4C22×D6111yescharacteristic
382C2×M16—22no—
383C2×M16—24no—
384C2×M16—42no—
385C2×M16—612no—
386C2×M16—612no—
387C2×M16—1222no—
388C2×M16—1224no—
389C2×M16—2442no—
390C2×M16C22×D6112yes—
391C2×M16C22×D6112yes—
392C22×C8—22no—
393C22×C8—1222no—
394C4⋊Q8—1222no—
395C4⋊Q8—1222no—
396#123—1224no—
397#123—1224no—
398#123—1224no—
399#123—1224no—
400C22⋊Q8—1224no—
401C22⋊Q8—1224no—
402#121—1222no—
403#121—1222no—
404#121—1222no—
405#121—1222no—
406C4×D4—612no—
407C4×D4—612no—
408C4×D4—612no—
409C4×D4—612no—
410C4×D4—1222no—
411C4×D4—1222no—
412C2×C4⋊C4—22no—
413C2×C4⋊C4—22no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—612no—
416C2×C4⋊C4—612no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—1222no—
419C2×C4⋊C4—1222no—
420C2×C4⋊C4C4×D6112yes—
421C2×C4⋊C4C22×D6111yescharacteristic
422C2×C4⋊C4C22×D6111yescharacteristic
423C2×C22⋊C4—611no—
424C2×C22⋊C4—611no—
425C2×C22⋊C4—612no—
426C2×C22⋊C4—612no—
427C2×C22⋊C4—612no—
428C2×C42—611no—
429C4⋊C8—1222no—
430C4⋊C8—1222no—
431#100—1224no—
432#100—2442no—
433C22⋊C8—1222no—
434C22⋊C8—1222no—
435C22⋊C8—1222no—
436C22⋊C8—1222no—
437#97—1222no—
438C4×C8—1222no—
439C24—611no—
440C24—612no—
441C24C2×C4×D6111yescharacteristic
442#41—2448no—
443#41—2448no—
444#41—2448no—
445#41—2448no—
446C2×Q8—1222no—
447C2×Q8—1222no—
448C2×Q8—2444no—
449C2×D4—611no—
450C2×D4—611no—
451C2×D4—611no—
452C2×D4—611no—
453C2×D4—1222no—
454C2×D4—1222no—
455C2×D4—1224no—
456C2×D4—1222no—
457C2×D4—1222no—
458C2×D4—1222no—
459C2×D4—1222no—
460C2×D4—1221no—
461C2×D4—1221no—
462C2×D4—2444no—
463C2×D4—2444no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—44no—
470C22×C4—611no—
471C22×C4—611no—
472C22×C4—611no—
473C22×C4—612no—
474C22×C4—612no—
475C22×C4—612no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—2444no—
484C22×C4—2444no—
485C22×C4C2×C4×D6111yescharacteristic
486C22×C4C2×C4×D6112yes—
487C22×C4C2×C4×D6112yes—
488C22×C4D4×D6112yes—
489C22×C4D4×D6112yes—
490C22×C4C23×D6111yescharacteristic
491M16—22no—
492M16—22no—
493M16—22no—
494M16—44no—
495M16—1222no—
496M16—1222no—
497M16—1222no—
498M16—1222no—
499M16—2444no—
500M16#1175822214yescomplemented
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—44no—
504C2×C8—1222no—
505C2×C8—1222no—
506C2×C8—2444no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—24no—
510C4⋊C4—612no—
511C4⋊C4—612no—
512C4⋊C4—1224no—
513C4⋊C4—1224no—
514C4⋊C4—1222no—
515C22⋊C4—612no—
516C22⋊C4—612no—
517C22⋊C4—612no—
518C22⋊C4—612no—
519C22⋊C4—1224no—
520C22⋊C4—1224no—
521C22⋊C4—1224no—
522C42—612no—
523C42—612no—
524C23—21no—
525C23—21no—
526C23—22no—
527C23—41no—
528C23—611no—
529C23—611no—
530C23—1222no—
531C23—1221no—
532C23—1221no—
533C23—1222no—
534C23—1222no—
535C23—1222no—
536C23—2441no—
537C23—2442no—
538C23—2442no—
539C23C22⋊C4×D6111yescharacteristic
540C23C22⋊C4×D6111yescharacteristic
541C23C22×C4×D6111yescharacteristic
542Q8—2444no—
543Q8—2444no—
544D4—614no—
545D4—1222no—
546D4—1222no—
547D4—1222no—
548D4—2444no—
549D4—2444no—
550D4—2444no—
551D4—2444no—
552D4—2442no—
553D4—2442no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—611no—
567C2×C4—612no—
568C2×C4—611no—
569C2×C4—612no—
570C2×C4—1222no—
571C2×C4—1221no—
572C2×C4—1221no—
573C2×C4—1222no—
574C2×C4—1222no—
575C2×C4—1222no—
576C2×C4—1222no—
577C2×C4—1222no—
578C2×C4—2444no—
579C2×C4—2444no—
580C2×C4—2444no—
581C2×C4—2444no—
582C2×C4—2441no—
583C2×C4—2442no—
584C2×C4C22⋊C4×D6112yes—
585C2×C4C2×D4×D6112yes—
586C2×C4C2×#1175822212yes—
587C2×C4#1451982211yescharacteristic
588C2×C4#1451982511yescharacteristic
589C8—24no—
590C8—42no—
591C8—2442no—
592C8—2442no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—42no—
601C22—42no—
602C22—612no—
603C22—1221no—
604C22—1221no—
605C22—1221no—
606C22—1221no—
607C22—1221no—
608C22—4882no—
609C22—4882no—
610C22—2442no—
611C22—2442no—
612C22—2442no—
613C22—2442no—
614C22—2442no—
615C22#100×D6112yes—
616C22C2×C22⋊C4×D6111yescharacteristic
617C22C4×#1175822211yescharacteristic
618C22#1684024311yescharacteristic
619C4—22no—
620C4—42no—
621C4—42no—
622C4—612no—
623C4—1222no—
624C4—1221no—
625C4—1222no—
626C4—2442no—
627C4#1684040212yes—
628C4#1684041312yes—
629C2—21no—
630C2—21no—
631C2—22no—
632C2—41no—
633C2—1222no—
634C2—4882no—
635C2—2441no—
636C2#2144864611yescharacteristic
637C2C2×#100×D6111yescharacteristic
638C2#2145429911yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#343C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840413C2
cosocleC22×C4×C61C24
hypercenterC2×#343D61
hypocenterC61#22479
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
2. M16×C22⋊D61C2semidirect
3. #21446806C2semidirect
4. #21446282C2semidirect
5. #21446281C2semidirect
6. #21445899C2semidirect
7. #21445898C2semidirect
8. #21445810C2semidirect
9. #21445610C2semidirect
10. #343×D61C2semidirect
11. C2×C61×#343C2semidirect
175. M16×C61#41semidirect
244. C22⋊D61#100semidirect
320. C2×#343D61semidirect
366. C61#22479semidirect
500. M16#11758222semidirect

Maximal subgroups

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

subgroupindex
C2×#168404132
M16×C22⋊D612
#214468062
#214462822
#214462812
#214458992
#214458982
#214458102
#214456102
#343×D612
C2×C61×#3432
#2247961