Finite Group Databasebeta

← #29696171

Subgroups — #29696171

Order 15616.

1322 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840415C211yesmaximal, characteristic
2M16×D244C211yesmaximal, characteristic, complemented
3#21454554C211yesmaximal, characteristic, complemented
4#21446809C211yesmaximal, characteristic, complemented
5#21446281C212yesmaximal, complemented
6#21446281C212yesmaximal, complemented
7#21445898C212yesmaximal, complemented
8#21445808C211yesmaximal, characteristic, complemented
9C4×#14519693C211yesmaximal, characteristic, complemented
10#343×D61C212yesmaximal, complemented
11C61×#1997C211yesmaximal, characteristic, complemented
12C22×#11758226C412yes—
13C2×M16×D61C2212yes—
14C2×M16×D61C2212yes—
15#16840415—21no—
16#16840415—21no—
17#16840415—22no—
18#16840415—22no—
19#16840415—21no—
20#16840415—21no—
21C2×#14519762C414yes—
22C2×#14519753C412yes—
23C2×#14519752C2212yes—
24C2×C4×D244C2211yescharacteristic
25C2×C4×D244C2211yescharacteristic
26C2×C4⋊C4×D61C2212yes—
27#16839841—21no—
28#16839839—21no—
29#16839835—22no—
30#16839832—22no—
31C8×D244—21no—
32#16839827—21no—
33#16839786C2211yescharacteristic
34C2×#14519693C2211yescharacteristic
35C2×#14519693C2211yescharacteristic
36C2×#14519693C2212yes—
37#16839725C2212yes—
38#16839725C2212yes—
39C4×#11758194C2211yescharacteristic
40#16839531C2212yes—
41#16839531C2212yes—
42#16839530C2211yescharacteristic
43#16839530C2212yes—
44#16839530C2211yescharacteristic
45C2×C4⋊Q8×C61C2211yescharacteristic
46C61×#423C2211yescharacteristic
47C4×M16×C61C2211yescharacteristic
48C61×#343C2212yes—
49C61×#343C2212yes—
50C2×#11758226—22no—
51C2×#11758226—24no—
52C2×#11758226—22no—
53C2×#11758226—48no—
54C22×D244C2×C411yescharacteristic
55C22×D244C2×C414yes—
56C22×D244C2311yescharacteristic
57C22×C4×D61C2×C414yes—
58C22×C4×D61C2×C412yes—
59C22×C4×D61C2312yes—
60C2×#11758194C2311yescharacteristic
61C2×#11758194C2312yes—
62C2×#11758194C2311yescharacteristic
63C2×#11758189—22no—
64C2×#11758189—22no—
65M16×D61—24no—
66M16×D61—24no—
67M16×D61—42no—
68M16×D61—42no—
69C2×C8×D61—22no—
70C2×C8×D61—22no—
71#14519762—24no—
72#14519762—24no—
73#14519762—24no—
74#14519762—24no—
75#14519753—24no—
76#14519753—22no—
77#14519753—22no—
78#14519752—22no—
79#14519752—22no—
80#14519752—22no—
81#14519752—22no—
82C4×D244—21no—
83C4×D244—21no—
84C4×D244—21no—
85C4×D244—21no—
86C4×D244D411yescharacteristic
87C4×D244D411yescharacteristic
88C4×D244D412yes—
89C4×D244D412yes—
90C4×D244D411yescharacteristic
91C4×D244D411yescharacteristic
92C2×C4⋊Dic61C2×C411yescharacteristic
93C2×C4⋊Dic61C2311yescharacteristic
94C2×#11758196C2×C414yes—
95C4⋊C4×D61—22no—
96C4⋊C4×D61—22no—
97C4⋊C4×D61—22no—
98C4⋊C4×D61—22no—
99C2×#11758198C2×C412yes—
100C2×#11758198C2×C414yes—
101C2×#11758198C2×C412yes—
102C4⋊#92418—21no—
103C4⋊#92418—21no—
104#14519693—22no—
105#14519693—22no—
106#14519693—42no—
107#14519689—22no—
108#14519689—22no—
109#14519689—22no—
110#14519689—22no—
111#14519688—21no—
112C4×#92418—21no—
113C22×Q8×C61C2×C412yes—
114C2×M16×C61C2311yescharacteristic
115C2×M16×C61C2312yes—
116C2×M16×C61C2311yescharacteristic
117C4⋊Q8×C61—21no—
118C4⋊Q8×C61—21no—
119C4⋊Q8×C61—21no—
120C4⋊Q8×C61—21no—
121C2×C4⋊C4×C61C2×C412yes—
122C2×C4⋊C4×C61C2312yes—
123C2×C42×C61C2311yescharacteristic
124C4⋊C8×C61—21no—
125C4⋊C8×C61—21no—
126#97×C61—21no—
127C4×C8×C61—21no—
128C23×D61C22×C412yes—
129C23×D61C22×C412yes—
130C2×Q8×C61—22no—
131C2×Q8×C61—22no—
132C2×Q8×C61—44no—
133C22×C4×C61C22×C411yescharacteristic
134C22×C4×C61C22×C414yes—
135C22×C4×C61C22×C411yescharacteristic
136C22×C4×C61C2411yescharacteristic
137C22×Dic61C22×C412yes—
138C22×Dic61C22×C412yes—
139#11758226—416no—
140#11758226—48no—
141#11758226—48no—
142C2×D244—24no—
143C2×D244—24no—
144C2×D244—24no—
145C2×D244—21no—
146C2×D244—21no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244—48no—
152C2×D244C22⋊C412yes—
153C2×D244C22⋊C411yescharacteristic
154C2×D244C22⋊C411yescharacteristic
155C2×C4×D61—22no—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61—24no—
160C2×C4×D61—24no—
161C2×C4×D61—22no—
162C2×C4×D61—22no—
163C2×C4×D61—44no—
164C2×C4×D61—48no—
165C2×C4×D61C2×D412yes—
166C2×C4×D61C2×D412yes—
167C2×C4×D61C2×D412yes—
168C2×C4×D61C2×D412yes—
169M16×C61—21no—
170M16×C61—21no—
171M16×C61—22no—
172M16×C61—22no—
173M16×C61—21no—
174M16×C61#4112yescomplemented
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—22no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—22no—
182C42×C61C2×D411yescharacteristic
183C42×C61C2×D411yescharacteristic
184C42×C61C2×D411yescharacteristic
185C42×C61C2×D411yescharacteristic
186#11758198—22no—
187#11758198—22no—
188#11758198—24no—
189#11758198—24no—
190#11758198C22⋊C412yes—
191#11758198C22⋊C412yes—
192#11758198C22⋊C412yes—
193#11758198C22⋊C412yes—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61C22⋊C412yes—
197C4⋊Dic61C22⋊C411yescharacteristic
198C4⋊Dic61C22⋊C411yescharacteristic
199#11758196—24no—
200#11758196—24no—
201#11758194—21no—
202#11758194—21no—
203#11758194—22no—
204#11758194—22no—
205#11758194—21no—
206#11758194#4112yescomplemented
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—22no—
210#11758189—44no—
211#11758189—44no—
212C8×D61—44no—
213C8×D61—44no—
214C23×C61C23×C411yescharacteristic
215C22×D61—22no—
216C22×D61—22no—
217C22×D61—24no—
218C22×D61—22no—
219C22×D61—22no—
220C22×D61—22no—
221C22×D61—42no—
222C22×D61—42no—
223C22×D61—42no—
224C22×D61C2×C22⋊C412yes—
225C22×D61C2×C22⋊C412yes—
226Q8×C61—44no—
227Q8×C61—44no—
228C2×C4×C61—22no—
229C2×C4×C61—24no—
230C2×C4×C61—41no—
231C2×C4×C61—44no—
232C2×C4×C61C2×C22⋊C411yescharacteristic
233C2×C4×C61C2×C22⋊C411yescharacteristic
234C2×C4×C61C2×C22⋊C411yescharacteristic
235C2×C4×C61C2×C22⋊C411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×C4×C61H(5,𝔽2)11yescharacteristic
241C2×C4×C612−1+411yescharacteristic
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—42no—
246C2×Dic61C2×C22⋊C412yes—
247C2×Dic61C2×C22⋊C412yes—
248D244—24no—
249D244—22no—
250D244—48no—
251D244—48no—
252D244—41no—
253D244—41no—
254D244—41no—
255D244—41no—
256D244#10014yescomplemented
257C4×D61—22no—
258C4×D61—22no—
259C4×D61—22no—
260C4×D61—22no—
261C4×D61—22no—
262C4×D61—22no—
263C4×D61—22no—
264C4×D61—22no—
265C4×D61—44no—
266C4×D61—48no—
267C4×D61—48no—
268C4×D61—44no—
269C8×C61—22no—
270C8×C61—42no—
271C8×C61—41no—
272#92418—22no—
273#92418—42no—
274#92418—41no—
275#22479—611nomaximal, Sylow
276C22×C61—21no—
277C22×C61—21no—
278C22×C61C22×C22⋊C411yescharacteristic
279C22×C61C4×#4111yescharacteristic
280C22×C61#52011yescharacteristic
281C2×D61—22no—
282C2×D61—22no—
283C2×D61—22no—
284C2×D61—22no—
285C2×D61—22no—
286C2×D61—44no—
287C2×D61—44no—
288C2×D61—42no—
289C2×D61—42no—
290C2×D61—42no—
291C2×D61—82no—
292C2×D61—82no—
293C2×D61C2×#10014yes—
294C4×C61—21no—
295C4×C61—21no—
296C4×C61—21no—
297C4×C61—44no—
298C4×C61C2×#10012yes—
299C4×C61#54611yescharacteristic
300C4×C61#54611yescharacteristic
301C4×C61#54711yescharacteristic
302C4×C61#54711yescharacteristic
303Dic61—22no—
304Dic61—22no—
305Dic61—42no—
306C2×#547—611no—
307D4×M16—611no—
308D4×M16—611no—
309#1997—611no—
310#1997D6111yescharacteristic, complemented
311#1900—612no—
312#1900—612no—
313#1821—612no—
314#1800—611no—
315C4×#100—611no—
316C2×#343—612no—
317C2×C61—21no—
318C2×C61—21no—
319C2×C61#233911yescharacteristic
320C2×C61C22×#10011yescharacteristic
321C2×C61#288911yescharacteristic
322D61—24no—
323D61—42no—
324D61—82no—
325C22×#41—612no—
326C22×M16—612no—
327C22×M16—612no—
328#547—1221no—
329#547—1221no—
330#547—1222no—
331#547—1222no—
332#547—1221no—
333#547—1221no—
334C2×C4⋊Q8C2×D6111yescharacteristic
335C2×#123—614no—
336C2×C22⋊Q8—612no—
337C2×#121—612no—
338C2×C4×D4—611no—
339C2×C4×D4—611no—
340C22×C4⋊C4—612no—
341#436—1221no—
342#436—1221no—
343#435—1222no—
344#435—1222no—
345C8×D4—1221no—
346C8×D4—1221no—
347#423—611no—
348#423C2×D6111yescharacteristic
349C2×#100—611no—
350C2×#100—611no—
351C2×#100—612no—
352#407—612no—
353#407—612no—
354C4×M16—611no—
355C4×M16C2×D6111yescharacteristic
356#343—612no—
357#343—611no—
358#343—611no—
359#343—612no—
360#343—612no—
361#343C2×D6112yes—
362#343C2×D6112yes—
363C61#2247911yescharacteristic, complemented, Sylow
364C2×#41—1222no—
365C2×#41—1224no—
366C2×#41—1222no—
367C2×#41—2448no—
368C22×Q8C4×D6112yes—
369C22×D4—611no—
370C22×D4—614no—
371C22×D4—611no—
372C23×C4—612no—
373C23×C4—614no—
374C23×C4—612no—
375C2×M16—611no—
376C2×M16—612no—
377C2×M16—611no—
378C2×M16—1222no—
379C2×M16—1222no—
380C2×M16—1224no—
381C2×M16—1224no—
382C2×M16—2442no—
383C2×M16—2442no—
384C2×M16C22×D6112yes—
385C2×M16C22×D6111yescharacteristic
386C2×M16C22×D6111yescharacteristic
387C22×C8—1222no—
388C22×C8—1222no—
389C4⋊Q8—21no—
390C4⋊Q8—21no—
391C4⋊Q8—21no—
392C4⋊Q8—21no—
393#123—1224no—
394#123—1224no—
395#123—1224no—
396#123—1224no—
397C22⋊Q8—1224no—
398C22⋊Q8—1222no—
399C22⋊Q8—1222no—
400#121—1222no—
401#121—1222no—
402#121—1222no—
403#121—1222no—
404C4×D4—611no—
405C4×D4—611no—
406C4×D4—612no—
407C4×D4—612no—
408C4×D4—611no—
409C4×D4—611no—
410C4×D4—1221no—
411C4×D4—1221no—
412C4×D4—1221no—
413C4×D4—1221no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—614no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—1222no—
418C2×C4⋊C4—1222no—
419C2×C4⋊C4—1222no—
420C2×C4⋊C4—1222no—
421C2×C4⋊C4C4×D6112yes—
422C2×C4⋊C4C22×D6112yes—
423C2×C22⋊C4—612no—
424C2×C22⋊C4—612no—
425C2×C22⋊C4—614no—
426C2×C42C22×D6111yescharacteristic
427C4⋊C8—21no—
428C4⋊C8—21no—
429C4⋊C8—1221no—
430C4⋊C8—1221no—
431#100—1222no—
432#100—1222no—
433#100—2442no—
434C22⋊C8—1222no—
435C22⋊C8—1222no—
436C22⋊C8—1222no—
437C22⋊C8—1222no—
438#97—21no—
439#97—1221no—
440C4×C8—21no—
441C4×C8—1221no—
442C24—612no—
443C24—612no—
444#41—2448no—
445#41—24416no—
446#41—2448no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—44no—
450C2×D4—611no—
451C2×D4—612no—
452C2×D4—611no—
453C2×D4—1224no—
454C2×D4—1224no—
455C2×D4—1224no—
456C2×D4—1221no—
457C2×D4—1221no—
458C2×D4—1221no—
459C2×D4—1221no—
460C2×D4—1221no—
461C2×D4—1221no—
462C2×D4—2448no—
463C22×C4—612no—
464C22×C4—612no—
465C22×C4—612no—
466C22×C4—612no—
467C22×C4—612no—
468C22×C4—612no—
469C22×C4—1224no—
470C22×C4—1224no—
471C22×C4—1222no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—2444no—
478C22×C4—2448no—
479C22×C4C2×C4×D6111yescharacteristic
480C22×C4C2×C4×D6114yes—
481C22×C4C2×C4×D6111yescharacteristic
482C22×C4C23×D6111yescharacteristic
483M16—21no—
484M16—21no—
485M16—22no—
486M16—22no—
487M16—21no—
488M16—612no—
489M16—1221no—
490M16—1221no—
491M16—1222no—
492M16—1222no—
493M16—1221no—
494M16—2444no—
495M16—2444no—
496M16#1175822612yescomplemented
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—22no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1222no—
503C2×C8—2444no—
504C2×C8—2444no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—612no—
510C4⋊C4—611no—
511C4⋊C4—611no—
512C4⋊C4—1224no—
513C4⋊C4—1224no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C22⋊C4—612no—
517C22⋊C4—612no—
518C22⋊C4—612no—
519C22⋊C4—612no—
520C22⋊C4—1224no—
521C22⋊C4—1224no—
522C22⋊C4—1222no—
523C22⋊C4—1222no—
524C42D4×D6111yescharacteristic
525C42D4×D6111yescharacteristic
526C42D4×D6111yescharacteristic
527C42D4×D6111yescharacteristic
528C23—612no—
529C23—612no—
530C23—1224no—
531C23—1222no—
532C23—1222no—
533C23—1222no—
534C23—1222no—
535C23—1222no—
536C23—2442no—
537C23—2442no—
538C23—2442no—
539C23C22×C4×D6111yescharacteristic
540Q8—44no—
541Q8—44no—
542D4—614no—
543D4—1224no—
544D4—1222no—
545D4—2448no—
546D4—2448no—
547D4—2441no—
548D4—2441no—
549D4—2441no—
550D4—2441no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—41no—
554C2×C4—44no—
555C2×C4—612no—
556C2×C4—612no—
557C2×C4—1222no—
558C2×C4—1222no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—1222no—
565C2×C4—1222no—
566C2×C4—1222no—
567C2×C4—1222no—
568C2×C4—2444no—
569C2×C4—2448no—
570C2×C4—2448no—
571C2×C4—2444no—
572C2×C4—2442no—
573C2×C4C22⋊C4×D6111yescharacteristic
574C2×C4C22⋊C4×D6111yescharacteristic
575C2×C4C22⋊C4×D6111yescharacteristic
576C2×C4C22⋊C4×D6111yescharacteristic
577C2×C4C2×D4×D6111yescharacteristic
578C2×C4C2×D4×D6111yescharacteristic
579C2×C4#41×D6111yescharacteristic
580C2×C4C2×#1175822611yescharacteristic
581C2×C4#1451982311yescharacteristic
582C2×C4#1451982511yescharacteristic
583C8—22no—
584C8—42no—
585C8—41no—
586C8—1222no—
587C8—2442no—
588C8—2441no—
589C22—21no—
590C22—21no—
591C22—614no—
592C22—1222no—
593C22—1222no—
594C22—1222no—
595C22—1222no—
596C22—1222no—
597C22—4882no—
598C22—4882no—
599C22—2444no—
600C22—2444no—
601C22—2442no—
602C22—2442no—
603C22—2442no—
604C22C2×C22⋊C4×D6111yescharacteristic
605C22C4×#1175822611yescharacteristic
606C22#1684024411yescharacteristic
607C4—21no—
608C4—21no—
609C4—21no—
610C4—44no—
611C4—1222no—
612C4—1222no—
613C4—2442no—
614C4#100×D6112yes—
615C4#1684040311yescharacteristic
616C4#1684040511yescharacteristic
617C4#1684041411yescharacteristic
618C4#1684041511yescharacteristic
619C2—21no—
620C2—21no—
621C2—1224no—
622C2—4882no—
623C2—2442no—
624C2#2144864811yescharacteristic
625C2C2×#100×D6111yescharacteristic
626C2#2145429911yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1997C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840415C2
cosocleC22×C4×C61C24
hypercenter#1997D61
hypocenterC61#22479
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1997

Complements

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

NH ≅ G/Ntype
2. M16×D244C2semidirect
3. #21454554C2semidirect
4. #21446809C2semidirect
5. #21446281C2semidirect
6. #21446281C2semidirect
7. #21445898C2semidirect
8. #21445808C2semidirect
9. C4×#14519693C2semidirect
10. #343×D61C2semidirect
11. C61×#1997C2semidirect
174. M16×C61#41semidirect
206. #11758194#41semidirect
256. D244#100semidirect
310. #1997D61semidirect
363. C61#22479semidirect
496. M16#11758226semidirect

Maximal subgroups

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

subgroupindex
C2×#168404152
M16×D2442
#214545542
#214468092
#214462812
#214458982
#214458082
C4×#145196932
#343×D612
C61×#19972
#2247961