Finite Group Databasebeta

← #29708700

Subgroups — #29708700

Order 15616.

1178 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839785C211yesmaximal, characteristic, complemented
2C2×#16839764C211yesmaximal, characteristic
3C2×#16839761C211yesmaximal, characteristic
4#21448307C211yesmaximal, characteristic
5C2×#16839661C211yesmaximal, characteristic, complemented
6#21447209C212yesmaximal, complemented
7#21447202C212yesmaximal, complemented
8#21447195C212yesmaximal, complemented
9#21447188C212yesmaximal, complemented
10#21445540C211yesmaximal, characteristic, complemented
11C2×C61×#428C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C22×#11758194C2211yescharacteristic
14C22×C4⋊Dic61—21no—
15C22×C4⋊Dic61C2211yescharacteristic
16C22×C4×Dic61C2211yescharacteristic
17#16839785—24no—
18C2×#14519706C2212yes—
19C2×#14519706C2212yes—
20#16839764—24no—
21#16839761—24no—
22C2×#14519702C2212yes—
23C2×#14519700C2212yes—
24C2×#14519699C2212yes—
25C2×#14519697C2212yes—
26#16839667C2211yescharacteristic
27#16839667C2211yescharacteristic
28#16839663C2211yescharacteristic
29#16839663C2211yescharacteristic
30#16839661—21no—
31#16839661—21no—
32#16839661C2212yes—
33#16839661C2212yes—
34#16839639—21no—
35#16839639—21no—
36C4×#11758198—21no—
37C4×#11758198—21no—
38C2×#14519684—21no—
39#16839524C2212yes—
40#16839524C2212yes—
41C22×M16×C61C2211yescharacteristic
42C2×#117×C61C2211yescharacteristic
43C22×C4⋊C4×C61C2211yescharacteristic
44C61×#428—24no—
45C2×#107×C61C2212yes—
46C2×#106×C61C2212yes—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222—22no—
50C2×#11758222—22no—
51C2×#11758222—22no—
52C2×#11758222Q814yes—
53C22×Dic122C2311yescharacteristic
54C22×D244C2311yescharacteristic
55C22×C22⋊D61—21no—
56C23×Dic61—21no—
57C23×Dic61D411yescharacteristic
58C23×Dic61D411yescharacteristic
59C22×C4×D61—21no—
60C2×#11758194—22no—
61C2×#11758194—22no—
62C2×#11758194—22no—
63C22×#92418C2312yes—
64C2×C4⋊Dic61—22no—
65C2×C4⋊Dic61—22no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—22no—
68C2×C4⋊Dic61—22no—
69C2×C4⋊Dic61—44no—
70C2×C4⋊Dic61C2312yes—
71C2×#11758198—21no—
72C2×#11758198—21no—
73C2×#11758198—21no—
74C2×#11758198—21no—
75C2×C4×Dic61—22no—
76C2×C4×Dic61—22no—
77C2×C4×Dic61D411yescharacteristic
78C2×C4×Dic61D411yescharacteristic
79C2×C4×Dic61C2311yescharacteristic
80C2×C4×Dic61C2311yescharacteristic
81#14519706—24no—
82#14519706—24no—
83#14519702—24no—
84#14519700—24no—
85#14519699—24no—
86#14519697—24no—
87#14519684—22no—
88#14519684—22no—
89#14519684—21no—
90#14519684—21no—
91#14519684—21no—
92#14519684—21no—
93C23×C4×C61—21no—
94C23×C4×C61C2311yescharacteristic
95C2×M16×C61—22no—
96C2×M16×C61—22no—
97C2×M16×C61D414yes—
98C22×C8×C61C2312yes—
99#117×C61—22no—
100#117×C61—22no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—22no—
103C2×C4⋊C4×C61D414yes—
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C4⋊C4×C61C2312yes—
107C2×C22⋊C4×C61—21no—
108C2×C42×C61—21no—
109#107×C61—24no—
110#106×C61—24no—
111C24×C61C2×D411yescharacteristic
112C23×D61—21no—
113C22×C4×C61—24no—
114C22×C4×C61—22no—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—42no—
118C22×C4×C61—42no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D412yes—
121C22×C4×C61C2×D412yes—
122C22×C4×C61C2×Q812yes—
123C22×C4×C61#4112yes—
124C22×C4×C61#4112yes—
125C22×C4×C61#4112yes—
126C22×C4×C61C2411yescharacteristic
127C2×C22⋊D61—24no—
128C2×C22⋊D61—42no—
129C2×C22⋊D61—42no—
130C22×Dic61—22no—
131C22×Dic61—22no—
132C22×Dic61—21no—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—41no—
137C22×Dic61—41no—
138C22×Dic61—42no—
139C22×Dic61—42no—
140C22×Dic61—42no—
141C22×Dic61C2×D411yescharacteristic
142C22×Dic61C2×D411yescharacteristic
143C22×Dic61C2×D411yescharacteristic
144C22×Dic61C2×D411yescharacteristic
145#11758222—28no—
146#11758222—28no—
147#11758222—44no—
148#11758222—44no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244C2×Q812yes—
155C2×C4×D61—24no—
156C2×C4×D61—42no—
157C2×C4×D61—42no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122C2×Q812yes—
164M16×C61—28no—
165C2×C8×C61—22no—
166C2×C8×C61—22no—
167C2×C8×C61C2×D414yes—
168C4⋊C4×C61—24no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61C2×D414yes—
175C22⋊C4×C61—42no—
176C42×C61—42no—
177#11758198—42no—
178#11758198—42no—
179#11758198—42no—
180#11758198—42no—
181C4⋊Dic61—22no—
182C4⋊Dic61—22no—
183C4⋊Dic61—44no—
184C4⋊Dic61—42no—
185C4⋊Dic61—44no—
186C4×Dic61—22no—
187C4×Dic61—22no—
188C4×Dic61—21no—
189C4×Dic61—21no—
190C4×Dic61—21no—
191C4×Dic61—21no—
192#11758194—44no—
193C2×#92418—22no—
194C2×#92418—22no—
195C2×#92418—22no—
196C23×C61—21no—
197C23×C61—21no—
198C23×C61—21no—
199C23×C61—21no—
200C23×C61#12112yes—
201C23×C61#12112yes—
202C23×C61C22⋊Q812yes—
203C23×C61C22×D411yescharacteristic
204C22×D61—22no—
205C22×D61—41no—
206C22×D61—41no—
207C22×D61—41no—
208C22×D61—41no—
209C22×D61—41no—
210C22×D61—41no—
211C2×C4×C61—24no—
212C2×C4×C61—48no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61#12114yes—
222C2×C4×C61#12112yes—
223C2×C4×C61#12112yes—
224C2×C4×C61C22⋊Q814yes—
225C2×C4×C61C22⋊Q814yes—
226C2×C4×C61C22⋊Q812yes—
227C2×C4×C61#12614yes—
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C22×Q811yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C22⋊D61—48no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—22no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—44no—
247C2×Dic61—44no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—42no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—84no—
254C2×Dic61AGL(1,ℤ/8)12yes—
255C2×Dic61#13712yes—
256D244—22no—
257D244—22no—
258D244—41no—
259D244—41no—
260C4×D61—48no—
261Dic122—22no—
262Dic122—22no—
263Dic122—41no—
264Dic122—41no—
265C8×C61—24no—
266#92418—42no—
267#23617—611nomaximal, Sylow
268C22×C61—22no—
269C22×C61—22no—
270C22×C61—22no—
271C22×C61—22no—
272C22×C61—22no—
273C22×C61—22no—
274C22×C61#39614yes—
275C22×C61C2×#12111yescharacteristic
276C22×C61C2×#12111yescharacteristic
277C22×C61C2×C22⋊Q811yescharacteristic
278C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C61C2×#13711yescharacteristic
280C22×C61#57511yescharacteristic
281C22×C61#57511yescharacteristic
282C2×D61—42no—
283C2×D61—42no—
284C2×D61—42no—
285C2×D61—81no—
286C2×D61—81no—
287C2×D61—81no—
288C2×D61—81no—
289C4×C61—44no—
290C4×C61—81no—
291C4×C61—81no—
292C4×C61#39614yes—
293C4×C61C2×#12111yescharacteristic
294C4×C61C2×C22⋊Q811yescharacteristic
295C4×C61C2×C22⋊Q811yescharacteristic
296C4×C61C2×#12611yescharacteristic
297Dic61—22no—
298Dic61—22no—
299Dic61—42no—
300Dic61—41no—
301Dic61—41no—
302Dic61—82no—
303C2×#428D6111yescharacteristic, complemented
304C2×#423—611no—
305C2×#417—611no—
306C2×#417—611no—
307#2297—611no—
308C2×#389—611no—
309#2068—612no—
310#2067—612no—
311#2066—612no—
312#2065—612no—
313#1736—611no—
314C2×C61—24no—
315C2×C61C2×#39611yescharacteristic
316C2×C61#303611yescharacteristic
317C2×C61#303711yescharacteristic
318C2×C61#305511yescharacteristic
319C2×C61#305611yescharacteristic
320C2×C61#307911yescharacteristic
321C2×C61#308011yescharacteristic
322D61—82no—
323C22×#41—611no—
324C22×M16—611no—
325C22×M16C2×D6111yescharacteristic
326C2×#117C2×D6111yescharacteristic
327C22×C4⋊C4—611no—
328C22×C4⋊C4—1221no—
329C22×C4⋊C4C2×D6111yescharacteristic
330C22×C42—611no—
331#428—24no—
332C2×#107C2×D6112yes—
333C2×#106C2×D6112yes—
334#423—1224no—
335C2×C4⋊C8—612no—
336C2×C4⋊C8—612no—
337#417—1224no—
338#417—1224no—
339C2×#103—612no—
340C2×#103—612no—
341C2×#102—612no—
342C2×#102—612no—
343#391—611no—
344#391—611no—
345#390—611no—
346#390—611no—
347#389—612no—
348#389—612no—
349#389—1221no—
350#389—1221no—
351#386—1221no—
352#386—1221no—
353C4×C22⋊C4—1221no—
354C4×C22⋊C4—1221no—
355C2×#95—1221no—
356#340—612no—
357#340—612no—
358C61#2361711yescharacteristic, complemented, Sylow
359C2×#41—614no—
360C2×#41—1222no—
361C2×#41—1222no—
362C2×#41—1222no—
363C2×#41—1222no—
364C2×#41—1222no—
365C22×Q8—611no—
366C22×D4—611no—
367C22×D4—1221no—
368C23×C4—21no—
369C23×C4—611no—
370C23×C4—611no—
371C23×C4—1221no—
372C23×C4—1221no—
373C23×C4C22×D6111yescharacteristic
374C2×M16—22no—
375C2×M16—22no—
376C2×M16—1222no—
377C2×M16—1222no—
378C2×M16—1222no—
379C2×M16D24414yes—
380C22×C8—612no—
381C22×C8C22×D6112yes—
382#117—22no—
383#117—22no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—612no—
387C2×C4⋊C4—1222no—
388C2×C4⋊C4—1222no—
389C2×C4⋊C4—1222no—
390C2×C4⋊C4—1222no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—2444no—
393C2×C4⋊C4C22⋊D6114yes—
394C2×C4⋊C4C22×D6111yescharacteristic
395C2×C4⋊C4C22×D6111yescharacteristic
396C2×C4⋊C4C22×D6112yes—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1221no—
401C2×C22⋊C4—1221no—
402C2×C42—21no—
403C2×C42—611no—
404C2×C42—611no—
405C2×C42—611no—
406C2×C42—611no—
407C2×C42—1222no—
408C2×C42—1222no—
409#107—24no—
410#106—24no—
411C4⋊C8—1224no—
412C4⋊C8—1224no—
413#103—1224no—
414#103—1224no—
415#102—1224no—
416#102—1224no—
417#95—1222no—
418#95—1222no—
419#95—1221no—
420#95—1221no—
421#95—1221no—
422#95—1221no—
423C24—1221no—
424C24D4×D6111yescharacteristic
425#41—1228no—
426#41—1228no—
427#41—2444no—
428#41—2444no—
429C2×Q8—612no—
430C2×Q8—1221no—
431C2×Q8—1221no—
432C2×Q8—1221no—
433C2×Q8—1221no—
434C2×Q8—1221no—
435C2×D4—612no—
436C2×D4—1224no—
437C2×D4—1221no—
438C2×D4—1221no—
439C2×D4—1221no—
440C2×D4—1221no—
441C2×D4—1221no—
442C2×D4—2442no—
443C2×D4—2442no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—1221no—
457C22×C4—1222no—
458C22×C4—1222no—
459C22×C4—1224no—
460C22×C4—1222no—
461C22×C4—2441no—
462C22×C4—2441no—
463C22×C4—2442no—
464C22×C4—2442no—
465C22×C4—2442no—
466C22×C4—2442no—
467C22×C4—2442no—
468C22×C4C2×D24412yes—
469C22×C4#1175822212yes—
470C22×C4D4×D6111yescharacteristic
471C22×C4#1175822412yes—
472C22×C4Q8×D6112yes—
473C22×C4#1175822612yes—
474C22×C4C2×C22⋊D6112yes—
475C22×C4C23×D6111yescharacteristic
476M16—28no—
477M16—2444no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—1222no—
481C2×C8—1222no—
482C2×C8—1222no—
483C2×C8C2×D24414yes—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—1222no—
491C4⋊C4—1222no—
492C4⋊C4—2444no—
493C4⋊C4—2444no—
494C4⋊C4—2442no—
495C4⋊C4C2×C22⋊D6114yes—
496C22⋊C4—42no—
497C22⋊C4—2442no—
498C22⋊C4—2442no—
499C22⋊C4—2442no—
500C22⋊C4—2442no—
501C42—42no—
502C42—1221no—
503C42—1221no—
504C42—1221no—
505C42—1221no—
506C42—1222no—
507C42—1222no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—1222no—
513C23—2441no—
514C23—2441no—
515C23—2441no—
516C23—2441no—
517C23—2441no—
518C23—2441no—
519C23#1451974912yes—
520C23#1451975212yes—
521C23#1451975512yes—
522C23C2×D4×D6111yescharacteristic
523Q8—1222no—
524Q8—1222no—
525Q8—2441no—
526Q8—2441no—
527D4—1222no—
528D4—1222no—
529D4—2448no—
530D4—2441no—
531D4—2441no—
532C2×C4—24no—
533C2×C4—48no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—612no—
543C2×C4—612no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1222no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—4884no—
556C2×C4—2444no—
557C2×C4—2444no—
558C2×C4—2442no—
559C2×C4—2442no—
560C2×C4—2442no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2448no—
564C2×C4#1451974912yes—
565C2×C4#1451975114yes—
566C2×C4#1451975212yes—
567C2×C4#1451975314yes—
568C2×C4#1451975512yes—
569C2×C4#1451975714yes—
570C2×C4#1451977214yes—
571C2×C4C22×C22⋊D6111yescharacteristic
572C2×C4C22×D24411yescharacteristic
573C2×C4C2×Q8×D6111yescharacteristic
574C2×C4C2×#1175822411yescharacteristic
575C2×C4C2×#1175822211yescharacteristic
576C2×C4C2×#1175822611yescharacteristic
577C8—24no—
578C8—2442no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—4881no—
586C22—4881no—
587C22—4881no—
588C22—4881no—
589C22—2442no—
590C22—2442no—
591C22—2442no—
592C22#1683968414yes—
593C22C2×#1451974911yescharacteristic
594C22C2×#1451975211yescharacteristic
595C22C2×#1451975511yescharacteristic
596C22AGL(1,ℤ/8)×D6111yescharacteristic
597C22#137×D6111yescharacteristic
598C22#1684055811yescharacteristic
599C22#1684056111yescharacteristic
600C4—44no—
601C4—81no—
602C4—81no—
603C4—1222no—
604C4—1222no—
605C4—4882no—
606C4—2442no—
607C4—2441no—
608C4—2441no—
609C4#1683968414yes—
610C4C2×#1451975111yescharacteristic
611C4C2×#1451975311yescharacteristic
612C4C2×#1451975711yescharacteristic
613C4C2×#1451977211yescharacteristic
614C2—24no—
615C2—4882no—
616C2C2×#1683968411yescharacteristic
617C2#2145528011yescharacteristic
618C2#2145528811yescharacteristic
619C2#2145543911yescharacteristic
620C2#2145545011yescharacteristic
621C2#2145563211yescharacteristic
622C2#2145564011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#428C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
hypercenterC2×#428D61
hypocenterC61#23617
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
1. C2×#16839785C2semidirect
5. C2×#16839661C2semidirect
6. #21447209C2semidirect
7. #21447202C2semidirect
8. #21447195C2semidirect
9. #21447188C2semidirect
10. #21445540C2semidirect
11. C2×C61×#428C2semidirect
303. C2×#428D61semidirect
358. C61#23617semidirect

Maximal subgroups

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

subgroupindex
C2×#168397852
C2×#168397642
C2×#168397612
#214483072
C2×#168396612
#214472092
#214472022
#214471952
#214471882
#214455402
C2×C61×#4282
#2361761