Finite Group Databasebeta

← #29711660

Subgroups — #29711660

Order 15616.

1814 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459463C211yesmaximal, characteristic, complemented
2C2×#16839939C211yesmaximal, characteristic, complemented
3C2×#16839929C211yesmaximal, characteristic, complemented
4#21455061C212yesmaximal, complemented
5#21446855C212yesmaximal, complemented
6#21446841C212yesmaximal, complemented
7#21446262C212yesmaximal, complemented
8#21446218C211yesmaximal, characteristic, complemented
9#21446208C211yesmaximal, characteristic, complemented
10#21445747C211yesmaximal, characteristic, complemented
11C2×C61×#532C211yesmaximal, characteristic, complemented
12C2×#14519822C2211yescharacteristic
13C2×#14519808C2212yes—
14C2×#14519803C2212yes—
15#16840411—24no—
16#16840400—24no—
17#16840336C2212yes—
18#16840321—22no—
19#16840273C2212yes—
20C2×#14519760—22no—
21#16840237C2211yescharacteristic
22#16839939—24no—
23#16839929—24no—
24#16839924—24no—
25#16839907—24no—
26C2×#14519705C2212yes—
27C2×#14519705C2212yes—
28#16839771C2212yescomplemented
29#16839767C2212yescomplemented
30C2×#14519696C2212yescomplemented
31C2×#14519694C2211yescharacteristic
32C2×#14519694C2211yescharacteristic
33#16839726C2212yescomplemented
34#16839522C2212yescomplemented
35#16839522C2212yescomplemented
36C22×#41×C61C2211yescharacteristic
37C22×#41×C61C2211yescharacteristic, complemented
38C61×#532—24no—
39C61×#532—24no—
40C2×C4⋊Q8×C61C2211yescharacteristic, complemented
41C2×#127×C61C2211yescharacteristic
42C2×#124×C61C2211yescharacteristic
43C2×#124×C61C2211yescharacteristic, complemented
44C2×#121×C61—22no—
45C2×C4×D4×C61—22no—
46C22×C42×C61C2211yescharacteristic, complemented
47#14519822—22no—
48#14519822—22no—
49#14519822—42no—
50C2×#11758222C2312yes—
51C2×#11758224—22no—
52C22×C22⋊D61—22no—
53C2×D4×D61—22no—
54#14519808—24no—
55#14519808—42no—
56#14519803—24no—
57#14519803—42no—
58C2×#11758202—22no—
59C2×#11758200—22no—
60C2×#11758201—22no—
61C2×#11758199—22no—
62C2×#11758194C2312yes—
63C2×#11758194C2312yes—
64#14519775—22no—
65#14519766—22no—
66#14519760—28no—
67#14519760—22no—
68#14519760—24no—
69#14519758—24no—
70#14519756D414yescomplemented
71#14519753—24no—
72#14519751D414yescomplemented
73#14519749—28no—
74#14519746—22no—
75#14519745—24no—
76D4×Dic61—24no—
77C4×C22⋊D61—24no—
78#14519736C2312yes—
79C2×#11758196—24no—
80C2×C22⋊Dic61—22no—
81C2×#11758198—24no—
82#14519705—24no—
83#14519705—24no—
84#14519701—24no—
85#14519698—24no—
86#14519696—42no—
87#14519694—22no—
88#14519694—22no—
89C22⋊#92418—24no—
90C2×#41×C61—22no—
91C2×#41×C61—22no—
92C2×#41×C61—22no—
93C2×#41×C61—22no—
94C2×#41×C61—24no—
95C2×#41×C61—24no—
96C2×#41×C61—42no—
97C2×#41×C61—42no—
98C22×Q8×C61C2311yescharacteristic
99C22×Q8×C61C2311yescharacteristic
100C22×D4×C61—22no—
101C22×D4×C61—21no—
102C22×D4×C61D412yes—
103C22×D4×C61C2311yescharacteristic
104C22×D4×C61C2311yescharacteristic
105C23×C4×C61—21no—
106C23×C4×C61—21no—
107C23×C4×C61D412yes—
108C23×C4×C61C2311yescharacteristic
109C4⋊Q8×C61—22no—
110C4⋊Q8×C61—22no—
111#127×C61—22no—
112#127×C61D414yescomplemented
113#124×C61—22no—
114#124×C61—22no—
115#124×C61—22no—
116#124×C61D414yescomplemented
117#121×C61—216no—
118C4×D4×C61—216no—
119C2×C4⋊C4×C61—22no—
120C2×C22⋊C4×C61—22no—
121C2×C22⋊C4×C61—22no—
122C2×C42×C61—21no—
123C2×C42×C61—22no—
124C2×C42×C61—22no—
125C2×C42×C61C2311yescharacteristic
126C2×C42×C61C2311yescharacteristic
127C24×C61—21no—
128C24×C61C2×D412yes—
129C24×C61C2×D411yescharacteristic
130C23×D61—22no—
131#41×C61—48no—
132#41×C61—48no—
133#41×C61—48no—
134#41×C61—48no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—22no—
139C2×Q8×C61—22no—
140C2×Q8×C61—22no—
141C2×Q8×C61—22no—
142C2×Q8×C61C2×D412yes—
143C2×D4×C61—24no—
144C2×D4×C61—28no—
145C2×D4×C61—24no—
146C2×D4×C61—24no—
147C2×D4×C61—22no—
148C2×D4×C61—22no—
149C2×D4×C61—21no—
150C2×D4×C61—22no—
151C2×D4×C61—22no—
152C2×D4×C61—42no—
153C2×D4×C61—48no—
154C2×D4×C61—42no—
155C2×D4×C61—42no—
156C2×D4×C61C2×D412yes—
157C2×D4×C61C2×D412yes—
158C2×D4×C61C2×D412yes—
159C22×C4×C61—21no—
160C22×C4×C61—21no—
161C22×C4×C61—21no—
162C22×C4×C61—22no—
163C22×C4×C61—24no—
164C22×C4×C61—24no—
165C22×C4×C61—24no—
166C22×C4×C61—21no—
167C22×C4×C61—21no—
168C22×C4×C61—24no—
169C22×C4×C61—42no—
170C22×C4×C61—44no—
171C22×C4×C61—42no—
172C22×C4×C61—42no—
173C22×C4×C61C2×D412yes—
174C22×C4×C61C2×D411yescharacteristic
175C22×C4×C61C2×D414yes—
176C22×C4×C61C2411yescharacteristic
177C2×C22⋊D61—24no—
178C2×C22⋊D61—22no—
179C2×C22⋊D61—24no—
180C2×C22⋊D61—44no—
181C2×C22⋊D61—44no—
182C22×Dic61—22no—
183C22×Dic61—24no—
184#11758224—44no—
185#11758224—44no—
186D4×D61—44no—
187D4×D61—44no—
188#11758222—22no—
189#11758222—22no—
190#11758222—22no—
191#11758222—22no—
192C2×D244C2×D412yes—
193C2×C4×D61—22no—
194C2×Dic122C2×D412yes—
195C4⋊C4×C61—28no—
196C22⋊C4×C61—28no—
197C22⋊C4×C61—28no—
198C42×C61—24no—
199C42×C61—21no—
200C42×C61—21no—
201C42×C61C2×D412yes—
202C42×C61C2×D412yes—
203C22⋊Dic61—24no—
204C22⋊Dic61—24no—
205C22⋊Dic61—22no—
206#11758202—44no—
207#11758201—44no—
208#11758200—44no—
209#11758199—44no—
210#11758198—28no—
211#11758198—24no—
212C4⋊Dic61C2×D414yes—
213#11758196—28no—
214#11758196—24no—
215C4×Dic61—22no—
216#11758194—22no—
217#11758194—22no—
218#11758194—22no—
219#11758194—22no—
220C2×#92418—22no—
221C2×#92418—22no—
222C23×C61—22no—
223C23×C61—21no—
224C23×C61—21no—
225C23×C61—22no—
226C23×C61—22no—
227C23×C61—42no—
228C23×C61—44no—
229C23×C61—44no—
230C23×C61—41no—
231C23×C61—41no—
232C23×C61#12012yes—
233C23×C61#12014yes—
234C23×C61C22×D411yescharacteristic
235C22×D61—22no—
236C22×D61—44no—
237C22×D61—84no—
238Q8×C61—24no—
239Q8×C61—42no—
240Q8×C61—42no—
241Q8×C61—42no—
242D4×C61—24no—
243D4×C61—24no—
244D4×C61—24no—
245D4×C61—416no—
246D4×C61—48no—
247D4×C61—48no—
248D4×C61—42no—
249C2×C4×C61—24no—
250C2×C4×C61—21no—
251C2×C4×C61—22no—
252C2×C4×C61—21no—
253C2×C4×C61—24no—
254C2×C4×C61—24no—
255C2×C4×C61—22no—
256C2×C4×C61—22no—
257C2×C4×C61—22no—
258C2×C4×C61—22no—
259C2×C4×C61—22no—
260C2×C4×C61—48no—
261C2×C4×C61—48no—
262C2×C4×C61—48no—
263C2×C4×C61—42no—
264C2×C4×C61—41no—
265C2×C4×C61—41no—
266C2×C4×C61#12012yes—
267C2×C4×C61C22×D412yes—
268C2×C4×C61C22×D411yescharacteristic
269C2×C4×C61C22×D411yescharacteristic
270C2×C4×C61H(5,𝔽2)12yes—
271C22⋊D61—48no—
272C22⋊D61—42no—
273C22⋊D61—42no—
274C22⋊D61—84no—
275C2×Dic61—24no—
276C2×Dic61—22no—
277C2×Dic61—44no—
278C2×Dic61—84no—
279D244—22no—
280D244—22no—
281C4×D61—42no—
282C4×D61—42no—
283Dic122—22no—
284Dic122—22no—
285#92418—42no—
286#92418—42no—
287#23908—611nomaximal, Sylow
288C22×C61—22no—
289C22×C61—24no—
290C22×C61—22no—
291C22×C61—24no—
292C22×C61—44no—
293C22×C61—42no—
294C22×C61—42no—
295C22×C61—44no—
296C22×C61—44no—
297C22×C61—42no—
298C22×C61—82no—
299C22×C61—82no—
300C22×C61—82no—
301C22×C61C2×#12011yescharacteristic
302C22×C61C2×#12011yescharacteristic
303C22×C61#53411yescharacteristic
304C2×D61—42no—
305C2×D61—44no—
306C2×D61—164no—
307C4×C61—24no—
308C4×C61—22no—
309C4×C61—42no—
310C4×C61—42no—
311C4×C61—42no—
312C4×C61D4212yes—
313C4×C61D4212yes—
314Dic61—42no—
315Dic61—44no—
316#3485—611no—
317C2×#532D6111yescharacteristic, complemented
318C2×#455—611no—
319C2×#453—611no—
320#3005—612no—
321#2005—612no—
322#2003—612no—
323#1896—612no—
324#1887—611no—
325#1885—611no—
326#1785—611no—
327C2×C61—21no—
328C2×C61—21no—
329C2×C61—44no—
330C2×C61—42no—
331C2×C61—82no—
332C2×C61#240311yescharacteristic
333C2×C61#301812yes—
334D61—82no—
335C2×H(5,𝔽2)—611no—
336C22×#41C2×D6111yescharacteristic
337C22×#41C2×D6111yescharacteristic, complemented
338C2×#137—612no—
339C2×AGL(1,ℤ/8)—612no—
340#547—1224no—
341#546—1224no—
342#536—612no—
343#534—1222no—
344#532—24no—
345#532—24no—
346C2×C4⋊Q8C2×D6111yescharacteristic, complemented
347C2×#127C2×D6111yescharacteristic
348C2×#124C2×D6111yescharacteristic
349C2×#124C2×D6111yescharacteristic, complemented
350#525—612no—
351C2×#123—1222no—
352C2×#121—22no—
353#518—611no—
354C2×C4×D4—22no—
355C22×C42C2×D6111yescharacteristic, complemented
356#455—1224no—
357#453—1224no—
358#452—1224no—
359#449—1224no—
360C2×#104—612no—
361C2×#104—612no—
362#419—612no—
363#418—612no—
364C2×#101—612no—
365C2×#100—611no—
366C2×#100—611no—
367#407—612no—
368#339—612no—
369#339—612no—
370C61#2390811yescharacteristic, complemented, Sylow
371H(5,𝔽2)—1222no—
372H(5,𝔽2)—1222no—
373H(5,𝔽2)—2442no—
374C2×#41—22no—
375C2×#41—22no—
376C2×#41—22no—
377C2×#41—22no—
378C2×#41—24no—
379C2×#41—24no—
380C2×#41—42no—
381C2×#41—42no—
382C2×#41—612no—
383C2×#41—1222no—
384C22×Q8C22×D6111yescharacteristic
385C22×Q8C22×D6111yescharacteristic
386C22×D4—22no—
387C22×D4—21no—
388C22×D4—1222no—
389C22×D4—1222no—
390C22×D4C22⋊D6112yes—
391C22×D4C22×D6111yescharacteristic
392C22×D4C22×D6111yescharacteristic
393C23×C4—21no—
394C23×C4—21no—
395C23×C4C22⋊D6112yes—
396C23×C4C22×D6111yescharacteristic
397#137—1224no—
398#137—2442no—
399AGL(1,ℤ/8)—1224no—
400AGL(1,ℤ/8)—2442no—
401C2×Q16—1222no—
402C2×SD16—1222no—
403C2×SD16—1222no—
404C2×D8—1222no—
405C2×M16—612no—
406C2×M16—612no—
407C4⋊Q8—22no—
408C4⋊Q8—22no—
409C4⋊Q8—1222no—
410#127—22no—
411#127C22⋊D6114yescomplemented
412#124—22no—
413#124—22no—
414#124—22no—
415#124—1222no—
416#124C22⋊D6114yescomplemented
417#123—1228no—
418#123—1224no—
419#123—1222no—
420#123—1224no—
421C22⋊Q8—614no—
422C22⋊Q8—1224no—
423#121—216no—
424#121—614no—
425#121—1228no—
426#120—1224no—
427#120—1222no—
428C4×D4—216no—
429C4×D4—1224no—
430C4×D4—1224no—
431#117—612no—
432C2×C4⋊C4—22no—
433C2×C4⋊C4—1224no—
434C2×C22⋊C4—22no—
435C2×C22⋊C4—22no—
436C2×C22⋊C4—1224no—
437C2×C22⋊C4—1222no—
438C2×C42—21no—
439C2×C42—22no—
440C2×C42—22no—
441C2×C42C22×D6111yescharacteristic
442C2×C42C22×D6111yescharacteristic
443#104—1224no—
444#104—1224no—
445#103—1224no—
446#102—1224no—
447#101—2442no—
448#100—1222no—
449#100—1222no—
450C22⋊C8—1224no—
451C24—21no—
452C24—1222no—
453C24C2×C22⋊D6111yescharacteristic
454C24C2×C22⋊D6112yes—
455#41—48no—
456#41—48no—
457#41—48no—
458#41—48no—
459#41—1222no—
460#41—1222no—
461#41—1222no—
462#41—1222no—
463#41—2444no—
464#41—2444no—
465C2×Q8—22no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—22no—
470C2×Q8—22no—
471C2×Q8—22no—
472C2×Q8—612no—
473C2×Q8C2×C22⋊D6112yes—
474C2×D4—24no—
475C2×D4—24no—
476C2×D4—28no—
477C2×D4—24no—
478C2×D4—22no—
479C2×D4—22no—
480C2×D4—21no—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—42no—
484C2×D4—48no—
485C2×D4—42no—
486C2×D4—42no—
487C2×D4—612no—
488C2×D4—1224no—
489C2×D4—1224no—
490C2×D4—1222no—
491C2×D4—2444no—
492C2×D4—2444no—
493C2×D4—2444no—
494C2×D4—2444no—
495C2×D4C2×C22⋊D6112yes—
496C2×D4C2×C22⋊D6112yes—
497C2×D4C2×C22⋊D6112yes—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—22no—
502C22×C4—24no—
503C22×C4—24no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—24no—
507C22×C4—24no—
508C22×C4—42no—
509C22×C4—42no—
510C22×C4—44no—
511C22×C4—42no—
512C22×C4—1224no—
513C22×C4—1222no—
514C22×C4—1222no—
515C22×C4D4×D6114yes—
516C22×C4C2×C22⋊D6112yes—
517C22×C4C2×C22⋊D6111yescharacteristic
518C22×C4C23×D6111yescharacteristic
519Q16—2444no—
520SD16—2444no—
521SD16—2444no—
522D8—2444no—
523M16—1222no—
524M16—1222no—
525M16—1222no—
526M16—1222no—
527C2×C8—1222no—
528C2×C8—1222no—
529C4⋊C4—28no—
530C4⋊C4—614no—
531C4⋊C4—1228no—
532C4⋊C4—1224no—
533C22⋊C4—28no—
534C22⋊C4—28no—
535C22⋊C4—1228no—
536C22⋊C4—1224no—
537C22⋊C4—1222no—
538C22⋊C4—1224no—
539C22⋊C4—1224no—
540C42—24no—
541C42—21no—
542C42—21no—
543C42—1222no—
544C42C2×C22⋊D6112yes—
545C42C2×C22⋊D6112yes—
546C23—22no—
547C23—21no—
548C23—21no—
549C23—22no—
550C23—22no—
551C23—42no—
552C23—44no—
553C23—44no—
554C23—41no—
555C23—41no—
556C23—1222no—
557C23—4884no—
558C23—2444no—
559C23#1451974512yes—
560C23#1451974614yes—
561C23C22×C22⋊D6111yescharacteristic
562Q8—24no—
563Q8—42no—
564Q8—42no—
565Q8—42no—
566Q8—1222no—
567Q8—1222no—
568D4—24no—
569D4—24no—
570D4—24no—
571D4—416no—
572D4—48no—
573D4—48no—
574D4—42no—
575D4—1222no—
576D4—1222no—
577D4—4884no—
578D4—2448no—
579D4—2442no—
580D4—2442no—
581C2×C4—24no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—48no—
593C2×C4—48no—
594C2×C4—48no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—1224no—
599C2×C4—1222no—
600C2×C4—4884no—
601C2×C4—2444no—
602C2×C4—2442no—
603C2×C4—2442no—
604C2×C4#1451974512yes—
605C2×C4C2×D4×D6112yes—
606C2×C4C22×C22⋊D6111yescharacteristic
607C2×C4C22×C22⋊D6111yescharacteristic
608C2×C4#1451982212yes—
609C8—2442no—
610C8—2442no—
611C22—22no—
612C22—24no—
613C22—22no—
614C22—24no—
615C22—44no—
616C22—42no—
617C22—42no—
618C22—44no—
619C22—44no—
620C22—42no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—9764no—
625C22—2444no—
626C22—2442no—
627C22C2×#1451974511yescharacteristic
628C22C2×#1451974611yescharacteristic
629C22#1684032111yescharacteristic
630C4—24no—
631C4—22no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—2444no—
636C4—2442no—
637C4D4×C22⋊D6112yes—
638C4D4×C22⋊D6112yes—
639C2—21no—
640C2—21no—
641C2—42no—
642C2—44no—
643C2—82no—
644C2—4882no—
645C2#2144912911yescharacteristic
646C2#2145516712yes—
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#532C2
socleC22×C61C2×#120
hypercenterC2×#532D61
hypocenterC61#23908
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
1. #21459463C2semidirect
2. C2×#16839939C2semidirect
3. C2×#16839929C2semidirect
4. #21455061C2semidirect
5. #21446855C2semidirect
6. #21446841C2semidirect
7. #21446262C2semidirect
8. #21446218C2semidirect
9. #21446208C2semidirect
10. #21445747C2semidirect
11. C2×C61×#532C2semidirect
28. #16839771C22semidirect
29. #16839767C22semidirect
30. C2×#14519696C22semidirect
33. #16839726C22semidirect
34. #16839522C22semidirect
35. #16839522C22semidirect
37. C22×#41×C61C22semidirect
40. C2×C4⋊Q8×C61C22semidirect
43. C2×#124×C61C22semidirect
46. C22×C42×C61C22semidirect
70. #14519756D4semidirect
72. #14519751D4semidirect
112. #127×C61D4semidirect
116. #124×C61D4semidirect
317. C2×#532D61semidirect
337. C22×#41C2×D61semidirect
346. C2×C4⋊Q8C2×D61semidirect
349. C2×#124C2×D61semidirect
355. C22×C42C2×D61semidirect
370. C61#23908semidirect
411. #127C22⋊D61semidirect
416. #124C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214594632
C2×#168399392
C2×#168399292
#214550612
#214468552
#214468412
#214462622
#214462182
#214462082
#214457472
C2×C61×#5322
#2390861