Finite Group Databasebeta

← #29707615

Subgroups — #29707615

Order 15616.

1450 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459148C211yesmaximal, characteristic
2#21454408C211yesmaximal, characteristic, complemented
3C22×#14519705C211yesmaximal, characteristic, complemented
4#21454336C211yesmaximal, characteristic, complemented
5C2×#16839746C211yesmaximal, characteristic, complemented
6#21447036C214yesmaximal, complemented
7#21447029C214yesmaximal, complemented
8C2×#16839522C211yesmaximal, characteristic, complemented
9C61×#2297C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12C22×#11758194C2211yescharacteristic
13#16840340—22no—
14#16840336C2212yes—
15#16840333—21no—
16#16840329C2212yes—
17#16840326—21no—
18#16840321—22no—
19#16840308—22no—
20#16840273—22no—
21#16840269—22no—
22C2×#14519736C2211yescharacteristic
23#16839805—22no—
24#16839802C2214yescomplemented
25#16839798—22no—
26C2×#14519705—24no—
27C2×#14519705C2214yes—
28#16839770—21no—
29#16839766—21no—
30#16839761C2214yes—
31#16839756—22no—
32#16839746—24no—
33C2×#14519696C2212yescomplemented
34C2×#14519694C2212yescomplemented
35#16839522C2214yescomplemented
36C22×#41×C61C2211yescharacteristic
37C2×#117×C61C2211yescharacteristic
38C22×C42×C61C2211yescharacteristic
39C61×#391C2212yescomplemented
40C61×#390C2212yescomplemented
41C61×#386—22no—
42C4×C22⋊C4×C61—22no—
43C2×#11758222—21no—
44C2×#11758222—21no—
45C2×#11758222—22no—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222D412yes—
50C2×#11758222C2312yes—
51C22×Dic122D411yescharacteristic
52C22×D244D411yescharacteristic
53C22×C22⋊D61—21no—
54C22×C4×D61—21no—
55C2×#11758194—21no—
56C2×#11758194—22no—
57C2×#11758194—21no—
58C2×#11758194—22no—
59C2×#11758194—22no—
60C2×#11758194D414yescomplemented
61C2×#11758194C2312yes—
62C2×#11758194C2312yes—
63C22×#92418—21no—
64C22×#92418—21no—
65#14519775—22no—
66#14519773—22no—
67#14519766—22no—
68#14519764—22no—
69#14519760—24no—
70#14519758—24no—
71#14519756—24no—
72#14519753—24no—
73#14519751—24no—
74#14519749—28no—
75#14519748—24no—
76#14519746—24no—
77#14519745—24no—
78C4×C22⋊D61—28no—
79#14519736—21no—
80#14519736—21no—
81#14519736D412yes—
82#14519736C2312yes—
83C2×C4⋊Dic61D412yes—
84C2×C22⋊Dic61—21no—
85C2×C4×Dic61—21no—
86#14519710—24no—
87#14519708—24no—
88#14519705—28no—
89#14519700—24no—
90#14519697—24no—
91#14519696—24no—
92#14519694—24no—
93C2×#41×C61—22no—
94C2×#41×C61—21no—
95C2×#41×C61—21no—
96C2×#41×C61—21no—
97C2×#41×C61—21no—
98C2×#41×C61—22no—
99C2×#41×C61—22no—
100C2×#41×C61—22no—
101C22×Q8×C61C2311yescharacteristic
102C22×D4×C61D412yes—
103C22×D4×C61C2311yescharacteristic
104C23×C4×C61—21no—
105C23×C4×C61D412yes—
106C23×C4×C61C2311yescharacteristic
107#117×C61—21no—
108#117×C61—21no—
109#117×C61—21no—
110#117×C61—21no—
111C2×C4⋊C4×C61C2312yes—
112C2×C22⋊C4×C61—24no—
113C2×C22⋊C4×C61—21no—
114C2×C42×C61—22no—
115C2×C42×C61—22no—
116C2×C42×C61—21no—
117C2×C42×C61—21no—
118C2×C42×C61C2312yes—
119#95×C61—24no—
120C24×C61C2×D412yes—
121C24×C61C2×D411yescharacteristic
122C23×D61—21no—
123#41×C61—42no—
124#41×C61—42no—
125#41×C61—42no—
126#41×C61—42no—
127#41×C61—42no—
128#41×C61—42no—
129#41×C61—42no—
130#41×C61—42no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×D4×C61—22no—
138C2×D4×C61—22no—
139C2×D4×C61—24no—
140C2×D4×C61—24no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C22×C4×C61—22no—
148C22×C4×C61—22no—
149C22×C4×C61—22no—
150C22×C4×C61—24no—
151C22×C4×C61—24no—
152C22×C4×C61—22no—
153C22×C4×C61—44no—
154C22×C4×C61—42no—
155C22×C4×C61C2×D412yes—
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61C2×D412yes—
162C22×C4×C61#4112yes—
163C22×C4×C61#4112yes—
164C22×C4×C61#4112yes—
165C22×C4×C61C2411yescharacteristic
166C2×C22⋊D61—22no—
167C2×C22⋊D61—22no—
168C2×C22⋊D61—28no—
169C2×C22⋊D61—42no—
170C2×C22⋊D61—42no—
171C22×Dic61—21no—
172C22×Dic61—22no—
173#11758222—24no—
174#11758222—24no—
175#11758222—22no—
176#11758222—22no—
177#11758222—22no—
178#11758222—22no—
179#11758222—42no—
180#11758222—42no—
181#11758222—42no—
182#11758222—42no—
183C2×D244—22no—
184C2×D244—22no—
185C2×D244—21no—
186C2×D244C2×D412yes—
187C2×C4×D61—22no—
188C2×C4×D61—22no—
189C2×C4×D61—42no—
190C2×C4×D61—42no—
191C2×Dic122—22no—
192C2×Dic122—22no—
193C2×Dic122—21no—
194C2×Dic122C2×D412yes—
195C4⋊C4×C61—22no—
196C4⋊C4×C61—22no—
197C22⋊C4×C61—48no—
198C22⋊C4×C61—41no—
199C22⋊C4×C61—41no—
200C42×C61—22no—
201C42×C61—22no—
202C42×C61—41no—
203C42×C61—41no—
204C42×C61—42no—
205C22⋊Dic61—28no—
206C22⋊Dic61—22no—
207C22⋊Dic61—22no—
208#11758198—28no—
209C4⋊Dic61—22no—
210C4⋊Dic61C2×D414yes—
211#11758196—28no—
212C4×Dic61—22no—
213C4×Dic61—22no—
214#11758194—24no—
215#11758194—22no—
216#11758194—22no—
217#11758194—22no—
218#11758194—42no—
219#11758194C2×D414yes—
220C2×#92418—24no—
221C2×#92418—22no—
222C2×#92418—22no—
223C2×#92418—42no—
224C23×C61—22no—
225C23×C61—22no—
226C23×C61—22no—
227C23×C61—42no—
228C23×C61—44no—
229C23×C61#12011yescharacteristic
230C23×C61#12011yescharacteristic
231C23×C61#12112yes—
232C23×C61#12312yes—
233C23×C61C22×D411yescharacteristic
234C22×D61—22no—
235C22×D61—42no—
236C22×D61—82no—
237Q8×C61—41no—
238Q8×C61—41no—
239Q8×C61—41no—
240Q8×C61—41no—
241D4×C61—44no—
242D4×C61—44no—
243D4×C61—41no—
244D4×C61—41no—
245D4×C61—41no—
246D4×C61—41no—
247C2×C4×C61—24no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—22no—
251C2×C4×C61—42no—
252C2×C4×C61—48no—
253C2×C4×C61—42no—
254C2×C4×C61—44no—
255C2×C4×C61—44no—
256C2×C4×C61—44no—
257C2×C4×C61—42no—
258C2×C4×C61#12011yescharacteristic
259C2×C4×C61#12011yescharacteristic
260C2×C4×C61#12112yes—
261C2×C4×C61#12112yes—
262C2×C4×C61#12112yes—
263C2×C4×C61#12112yes—
264C2×C4×C61#12112yes—
265C2×C4×C61#12312yes—
266C2×C4×C61#12312yes—
267C2×C4×C61#12312yes—
268C2×C4×C61#12412yes—
269C2×C4×C61#12412yes—
270C2×C4×C61C22×D411yescharacteristic
271C2×C4×C61C22×D411yescharacteristic
272C2×C4×C61C22×D411yescharacteristic
273C2×C4×C61C2×#4111yescharacteristic
274C2×C4×C61C2×#4111yescharacteristic
275C2×C4×C61C2×#4111yescharacteristic
276C22⋊D61—42no—
277C22⋊D61—42no—
278C22⋊D61—44no—
279C22⋊D61—88no—
280C2×Dic61—24no—
281C2×Dic61—22no—
282C2×Dic61—42no—
283C2×Dic61—82no—
284D244—22no—
285D244—22no—
286D244—42no—
287C4×D61—42no—
288C4×D61—42no—
289C4×D61—44no—
290Dic122—22no—
291Dic122—22no—
292Dic122—42no—
293#92418—24no—
294#92418—42no—
295#23492—611nomaximal, Sylow
296C22×C61—22no—
297C22×C61—22no—
298C22×C61—22no—
299C22×C61—42no—
300C22×C61—42no—
301C22×C61—44no—
302C22×C61—42no—
303C22×C61—44no—
304C22×C61—82no—
305C22×C61#39412yes—
306C22×C61#39412yes—
307C22×C61C2×#12011yescharacteristic
308C22×C61C2×#12111yescharacteristic
309C22×C61C2×#12311yescharacteristic
310C2×D61—42no—
311C2×D61—44no—
312C2×D61—84no—
313C4×C61—44no—
314C4×C61—42no—
315C4×C61—82no—
316C4×C61#39412yes—
317C4×C61#39412yes—
318C4×C61C2×#12111yescharacteristic
319C4×C61C2×#12111yescharacteristic
320C4×C61C2×#12311yescharacteristic
321C4×C61C2×#12411yescharacteristic
322Dic61—42no—
323Dic61—44no—
324#3449—611no—
325#2911—611no—
326C22×#104—611no—
327#2896—611no—
328C2×#413—611no—
329#2297D6111yescharacteristic, complemented
330#2038—614no—
331#2037—614no—
332C2×#339—611no—
333C2×C61—21no—
334C2×C61—22no—
335C2×C61—21no—
336C2×C61—22no—
337C2×C61—82no—
338C2×C61C2×#39411yescharacteristic
339C2×C61#301812yes—
340D61—82no—
341C22×#41—611no—
342C22×#41C2×D6111yescharacteristic
343C22×M16—611no—
344C22×M16—611no—
345#537—1222no—
346#536—612no—
347#536—1221no—
348#535—612no—
349#535—1221no—
350#534—1222no—
351#532—1222no—
352#525—1222no—
353#525—1222no—
354C2×#117—611no—
355C2×#117C2×D6111yescharacteristic
356C22×C42C2×D6111yescharacteristic
357#428—614no—
358#428—1222no—
359#427—1222no—
360C2×#104—614no—
361C2×#104—1224no—
362#419—1221no—
363#418—1221no—
364#417—614no—
365#416—1222no—
366#413—1224no—
367C2×#101—612no—
368C2×#100—612no—
369#391C2×D6112yescomplemented
370#390C2×D6112yescomplemented
371#386—22no—
372C4×C22⋊C4—22no—
373#339—614no—
374C61#2349211yescharacteristic, complemented, Sylow
375C2×#41—22no—
376C2×#41—22no—
377C2×#41—21no—
378C2×#41—21no—
379C2×#41—21no—
380C2×#41—21no—
381C2×#41—22no—
382C2×#41—22no—
383C2×#41—612no—
384C2×#41—612no—
385C2×#41—1221no—
386C2×#41—1221no—
387C2×#41—1222no—
388C2×#41—1222no—
389C2×#41—1222no—
390C2×#41—1222no—
391C22×Q8—611no—
392C22×Q8C22×D6111yescharacteristic
393C22×D4—611no—
394C22×D4—1221no—
395C22×D4C22⋊D6112yes—
396C22×D4C22×D6111yescharacteristic
397C23×C4—21no—
398C23×C4—1221no—
399C23×C4C22⋊D6112yes—
400C23×C4C22×D6111yescharacteristic
401C2×M16—614no—
402C2×M16—612no—
403C2×M16—612no—
404C2×M16—1222no—
405C2×M16—1221no—
406C2×M16—1221no—
407C2×M16—1222no—
408C2×M16—1222no—
409C22×C8—1221no—
410C22×C8—1221no—
411C4⋊Q8—1222no—
412#127—1222no—
413#124—1222no—
414#124—1222no—
415#123—1224no—
416#123—1224no—
417C22⋊Q8—1224no—
418C22⋊Q8—1224no—
419#121—1224no—
420#121—1224no—
421#121—1228no—
422#120—1224no—
423#120—1224no—
424C4×D4—1228no—
425#117—21no—
426#117—21no—
427#117—21no—
428#117—21no—
429#117—612no—
430#117—612no—
431#117—1221no—
432#117—1221no—
433C2×C4⋊C4—612no—
434C2×C4⋊C4C22×D6112yes—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—24no—
437C2×C22⋊C4—1221no—
438C2×C42—22no—
439C2×C42—22no—
440C2×C42—21no—
441C2×C42—21no—
442C2×C42—1221no—
443C2×C42C22×D6112yes—
444#107—1224no—
445#106—1224no—
446#104—1228no—
447#103—1224no—
448#102—1224no—
449#101—1224no—
450#100—1224no—
451#95—24no—
452C24—1221no—
453C24C2×C22⋊D6112yes—
454C24C2×C22⋊D6111yescharacteristic
455#41—42no—
456#41—42no—
457#41—42no—
458#41—42no—
459#41—42no—
460#41—42no—
461#41—42no—
462#41—42no—
463#41—1222no—
464#41—1222no—
465#41—1222no—
466#41—1222no—
467#41—1224no—
468#41—1224no—
469#41—2442no—
470#41—2442no—
471#41—2442no—
472#41—2442no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—612no—
480C2×Q8—1221no—
481C2×Q8—1222no—
482C2×Q8—1222no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—24no—
486C2×D4—24no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—612no—
494C2×D4—1228no—
495C2×D4—1222no—
496C2×D4—1222no—
497C2×D4—1221no—
498C2×D4—1222no—
499C2×D4—1222no—
500C2×D4—2442no—
501C2×D4—2442no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—24no—
506C22×C4—24no—
507C22×C4—22no—
508C22×C4—44no—
509C22×C4—42no—
510C22×C4—1222no—
511C22×C4—1221no—
512C22×C4—1222no—
513C22×C4—1222no—
514C22×C4—2442no—
515C22×C4—2442no—
516C22×C4#1175822212yes—
517C22×C4#1175822212yes—
518C22×C4D4×D6111yescharacteristic
519C22×C4D4×D6111yescharacteristic
520C22×C4D4×D6111yescharacteristic
521C22×C4D4×D6111yescharacteristic
522C22×C4D4×D6112yes—
523C22×C4#1175822412yes—
524C22×C4C2×C22⋊D6112yes—
525C22×C4C2×C22⋊D6111yescharacteristic
526C22×C4C23×D6111yescharacteristic
527M16—614no—
528M16—1222no—
529M16—1224no—
530M16—1222no—
531M16—1222no—
532M16—2442no—
533C2×C8—1222no—
534C2×C8—1224no—
535C2×C8—1222no—
536C2×C8—2442no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—614no—
540C4⋊C4—1228no—
541C4⋊C4—1222no—
542C22⋊C4—48no—
543C22⋊C4—41no—
544C22⋊C4—41no—
545C22⋊C4—1228no—
546C22⋊C4—1228no—
547C22⋊C4—1222no—
548C22⋊C4—1222no—
549C42—22no—
550C42—22no—
551C42—41no—
552C42—41no—
553C42—42no—
554C42—1222no—
555C42—1222no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—42no—
560C23—44no—
561C23—1222no—
562C23—4882no—
563C23—2442no—
564C23#1451974511yescharacteristic
565C23#1451974511yescharacteristic
566C23#1451974912yes—
567C23#1451976012yes—
568C23C22×C22⋊D6111yescharacteristic
569Q8—41no—
570Q8—41no—
571Q8—41no—
572Q8—41no—
573Q8—1222no—
574Q8—1222no—
575Q8—2442no—
576D4—44no—
577D4—44no—
578D4—41no—
579D4—41no—
580D4—41no—
581D4—41no—
582D4—1222no—
583D4—1222no—
584D4—4888no—
585D4—2442no—
586D4—2442no—
587D4—2444no—
588D4—2442no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—42no—
594C2×C4—48no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—1224no—
601C2×C4—1222no—
602C2×C4—4882no—
603C2×C4—2442no—
604C2×C4—2442no—
605C2×C4—2442no—
606C2×C4—2444no—
607C2×C4#1451974511yescharacteristic
608C2×C4#1451974511yescharacteristic
609C2×C4#1451974912yes—
610C2×C4#1451975012yes—
611C2×C4#1451975012yes—
612C2×C4#1451975012yes—
613C2×C4#1451975012yes—
614C2×C4#1451975912yes—
615C2×C4#1451975912yes—
616C2×C4#1451976012yes—
617C2×C4#1451976312yes—
618C2×C4#1451976312yes—
619C2×C4C2×D4×D6111yescharacteristic
620C2×C4C2×D4×D6111yescharacteristic
621C2×C4C2×D4×D6111yescharacteristic
622C2×C4C2×#1175822411yescharacteristic
623C2×C4C2×#1175822211yescharacteristic
624C2×C4C2×#1175822211yescharacteristic
625C8—1224no—
626C8—2442no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—42no—
631C22—42no—
632C22—44no—
633C22—42no—
634C22—44no—
635C22—82no—
636C22—4884no—
637C22—2442no—
638C22—2444no—
639C22#1683967712yes—
640C22#1683967712yes—
641C22C2×#1451974511yescharacteristic
642C22C2×#1451974911yescharacteristic
643C22C2×#1451976011yescharacteristic
644C4—44no—
645C4—42no—
646C4—82no—
647C4—2442no—
648C4—2444no—
649C4#1683967712yes—
650C4#1683967712yes—
651C4C2×#1451975011yescharacteristic
652C4C2×#1451975011yescharacteristic
653C4C2×#1451975911yescharacteristic
654C4C2×#1451976311yescharacteristic
655C2—21no—
656C2—21no—
657C2—22no—
658C2—22no—
659C2—82no—
660C2—4882no—
661C2C2×#1683967711yescharacteristic
662C2#2145516712yes—
subgroupG/H
centerC2×C4#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2297C2
socleC22×C61C2×#120
prime radical#21459148C2
hypercenter#2297D61
hypocenterC61#23492
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2297

Complements

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

NH ≅ G/Ntype
2. #21454408C2semidirect
3. C22×#14519705C2semidirect
4. #21454336C2semidirect
5. C2×#16839746C2semidirect
6. #21447036C2semidirect
7. #21447029C2semidirect
8. C2×#16839522C2semidirect
9. C61×#2297C2semidirect
24. #16839802C22semidirect
33. C2×#14519696C22semidirect
34. C2×#14519694C22semidirect
35. #16839522C22semidirect
39. C61×#391C22semidirect
40. C61×#390C22semidirect
60. C2×#11758194D4semidirect
329. #2297D61semidirect
369. #391C2×D61semidirect
370. #390C2×D61semidirect
374. C61#23492semidirect

Maximal subgroups

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

subgroupindex
#214591482
#214544082
C22×#145197052
#214543362
C2×#168397462
#214470362
#214470292
C2×#168395222
C61×#22972
#2349261