Finite Group Databasebeta

← #29709827

Subgroups — #29709827

Order 15616.

1098 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839803C211yesmaximal, characteristic
2C2×#16839801C211yesmaximal, characteristic
3C2×#16839769C211yesmaximal, characteristic
4C2×#16839769C211yesmaximal, characteristic
5#391×D61C211yesmaximal, characteristic
6#21448225C211yesmaximal, characteristic
7#21447370C211yesmaximal, characteristic, complemented
8#21447367C211yesmaximal, characteristic, complemented
9#21447366C211yesmaximal, characteristic, complemented
10#21447360C211yesmaximal, characteristic, complemented
11#21447359C211yesmaximal, characteristic, complemented
12#21447355C211yesmaximal, characteristic, complemented
13#21447354C211yesmaximal, characteristic, complemented
14#21445511C211yesmaximal, characteristic
15C61×#2097C211yesmaximal, characteristic, complemented
16C22×Q8×D61C2211yescharacteristic
17C22×#11758189C2211yescharacteristic
18C22×#11758189C2211yescharacteristic
19C2×#14519735C2211yescharacteristic
20C2×#14519735C2211yescharacteristic
21C2×C4⋊C4×D61C2211yescharacteristic
22C2×C42×D61C2211yescharacteristic
23#16839803—24no—
24#16839801—24no—
25C2×#14519711C2211yescharacteristic
26C2×#14519710C2211yescharacteristic
27C2×#14519710C2211yescharacteristic
28C2×#14519709C2211yescharacteristic
29C2×#14519708C2211yescharacteristic
30C2×#14519708C2211yescharacteristic
31#16839769—24no—
32#16839769—24no—
33C2×#14519702C2211yescharacteristic
34C2×#14519702C2211yescharacteristic
35C2×#14519701C2211yescharacteristic
36C2×#14519701C2211yescharacteristic
37C2×#14519700C2211yescharacteristic
38C2×#14519700C2211yescharacteristic
39#16839669C2211yescharacteristic
40#16839668C2211yescharacteristic
41#16839667C2211yescharacteristic
42#16839666—22no—
43#16839631C2211yescharacteristic
44#16839631C2211yescharacteristic
45#16839628C2211yescharacteristic
46#16839621—22no—
47#16839613—22no—
48#95×D61—22no—
49#16839517C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×#107×C61C2211yescharacteristic
53C2×#106×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C61×#391C2211yescharacteristic
57C61×#382C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×Dic122—21no—
60C22×Dic122C2311yescharacteristic
61C2×Q8×D61—22no—
62C2×Q8×D61—22no—
63C2×Q8×D61—22no—
64C2×Q8×D61—22no—
65C2×Q8×D61—22no—
66C2×Q8×D61Q814yes—
67C22×C4×D61—21no—
68C22×C4×D61—21no—
69C22×C4×D61D412yes—
70C22×C4×D61C2311yescharacteristic
71C2×#11758189—22no—
72C2×#11758189—22no—
73C2×#11758189—22no—
74C2×#11758189—22no—
75C2×#11758189D414yes—
76C2×#11758189D414yes—
77C22×#92418C2311yescharacteristic
78C22×#92418C2311yescharacteristic
79#14519735—22no—
80#14519735—22no—
81#14519735—22no—
82#14519735—22no—
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×#11758196—21no—
87C2×#11758196—22no—
88C2×#11758196—22no—
89C4⋊C4×D61—22no—
90C4⋊C4×D61—22no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61D412yes—
96C2×C4×Dic61C2311yescharacteristic
97C42×D61—22no—
98C42×D61—22no—
99#14519711—22no—
100#14519710—22no—
101#14519710—22no—
102#14519709—22no—
103#14519708—22no—
104#14519708—22no—
105#14519702—22no—
106#14519702—22no—
107#14519701—22no—
108#14519701—22no—
109#14519700—22no—
110#14519700—22no—
111#14519684—22no—
112#14519684—22no—
113#14519684—22no—
114#14519684—22no—
115C22×Q8×C61C2311yescharacteristic
116C22×C8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C42×C61C2311yescharacteristic
122#107×C61—22no—
123#106×C61—22no—
124#103×C61—22no—
125#103×C61—22no—
126#95×C61—22no—
127#95×C61—22no—
128C23×D61C2×D411yescharacteristic
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61C2×Q812yes—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61C2×D412yes—
140C22×C4×C61C2411yescharacteristic
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61C2×D412yes—
146C22×Dic61C2×D411yescharacteristic
147Q8×D61—28no—
148Q8×D61—28no—
149Q8×D61—44no—
150Q8×D61—44no—
151C2×C4×D61—24no—
152C2×C4×D61—24no—
153C2×C4×D61—24no—
154C2×C4×D61—42no—
155C2×C4×D61—42no—
156C2×C4×D61—42no—
157C2×C4×D61—44no—
158C2×C4×D61—42no—
159C2×C4×D61C2×D412yes—
160C2×C4×D61C2×D412yes—
161C2×C4×D61C2×Q812yes—
162C2×C4×D61#4112yes—
163C2×C4×D61#4112yes—
164C2×C4×D61#4112yes—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—24no—
171C2×Dic122—42no—
172C2×Dic122—42no—
173C2×Dic122C2×Q812yes—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61C2×D412yes—
179C2×C8×C61C2×D412yes—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C42×C61—21no—
187C42×C61—21no—
188#11758198—42no—
189#11758198—42no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196#11758196—44no—
197#11758196—42no—
198#11758196—44no—
199C4×Dic61—21no—
200C4×Dic61—21no—
201C4×Dic61—24no—
202C4×Dic61—42no—
203C4×Dic61—42no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418C2×D412yes—
209C2×#92418C2×D412yes—
210#11758189—28no—
211#11758189—28no—
212C23×C61C22×D411yescharacteristic
213C22×D61—22no—
214C22×D61—22no—
215C22×D61C22⋊Q812yes—
216C22×D61#12312yes—
217C22×D61#12712yes—
218Q8×C61—22no—
219Q8×C61—22no—
220Q8×C61—41no—
221Q8×C61—41no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×Q811yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—42no—
249C2×Dic61—42no—
250C2×Dic61—42no—
251C2×Dic61C22⋊Q812yes—
252C2×Dic61#12312yes—
253C2×Dic61#12712yes—
254C2×Dic61#13714yes—
255C4×D61—48no—
256C4×D61—48no—
257C4×D61—84no—
258C4×D61C22⋊Q814yes—
259C4×D61C22⋊Q814yes—
260C4×D61#12314yes—
261C4×D61#12314yes—
262Dic122—22no—
263Dic122—22no—
264Dic122—41no—
265Dic122—41no—
266Dic122—48no—
267C8×C61—22no—
268C8×C61—22no—
269#92418—22no—
270#92418—22no—
271#23732—611nomaximal, Sylow
272C22×C61C2×C22⋊Q811yescharacteristic
273C22×C61C2×#12311yescharacteristic
274C22×C61C2×#12711yescharacteristic
275C22×C61C2×#13712yes—
276C22×C61#57512yes—
277C2×D61—24no—
278C2×D61—24no—
279C2×D61—22no—
280C2×D61—22no—
281C2×D61#39914yes—
282C4×C61—42no—
283C4×C61—42no—
284C4×C61—81no—
285C4×C61—81no—
286C4×C61—81no—
287C4×C61C2×C22⋊Q811yescharacteristic
288C4×C61C2×C22⋊Q811yescharacteristic
289C4×C61C2×#12311yescharacteristic
290C4×C61C2×#12311yescharacteristic
291Dic61—24no—
292Dic61—42no—
293Dic61—81no—
294Dic61—81no—
295Dic61—81no—
296Dic61#39914yes—
297C2×#428—611no—
298C2×#428—611no—
299C2×#419—611no—
300C2×#419—611no—
301C2×#391—611no—
302#2285—611no—
303#2097—611no—
304#2097—611no—
305#2097—611no—
306#2097D6111yescharacteristic, complemented
307#2096—611no—
308#2096—611no—
309#2095—611no—
310#2095—611no—
311#1728—611no—
312C2×C61C2×#39911yescharacteristic
313C2×C61#307812yes—
314C2×C61#309512yes—
315C2×C61#315211yescharacteristic
316C2×C61#315311yescharacteristic
317D61—24no—
318C23×Q8—611no—
319C22×M16—611no—
320C22×M16—611no—
321C2×#117—611no—
322C2×#117—611no—
323C22×C4⋊C4—611no—
324C22×C42—611no—
325#428—1224no—
326#428—1224no—
327C2×#107—611no—
328C2×#107—611no—
329C2×#107—611no—
330C2×#107C2×D6111yescharacteristic
331C2×#106—611no—
332C2×#106—611no—
333C2×#106—611no—
334C2×#106C2×D6111yescharacteristic
335#419—1224no—
336#419—1224no—
337C2×#103—611no—
338C2×#103—611no—
339C2×#103—611no—
340C2×#103—611no—
341C2×#103—611no—
342C2×#103—611no—
343C2×#103C2×D6111yescharacteristic
344C2×#103C2×D6111yescharacteristic
345#391—611no—
346#391—611no—
347#391—611no—
348#391—1222no—
349#391C2×D6111yescharacteristic
350#384—611no—
351#384—611no—
352#382—611no—
353#382C2×D6111yescharacteristic
354#380—1222no—
355C4×C4⋊C4—1222no—
356C2×#95—1222no—
357#336—611no—
358#336—611no—
359#336—611no—
360#336C2×D6111yescharacteristic
361C61#2373211yescharacteristic, complemented, Sylow
362C22×Q8—614no—
363C22×Q8—611no—
364C22×Q8—1221no—
365C22×Q8—1222no—
366C22×Q8—1222no—
367C22×Q8—1222no—
368C22×Q8—1222no—
369C22×Q8—1222no—
370C22×Q8C22×D6111yescharacteristic
371C23×C4—611no—
372C23×C4—612no—
373C23×C4—1221no—
374C23×C4—1221no—
375C2×M16—614no—
376C2×M16—614no—
377C2×M16—1222no—
378C2×M16—1222no—
379C2×M16—1222no—
380C2×M16—1222no—
381C22×C8—611no—
382C22×C8—611no—
383C22×C8C22×D6111yescharacteristic
384C22×C8C22×D6111yescharacteristic
385#117—1222no—
386#117—1222no—
387#117—1222no—
388#117—1222no—
389C2×C4⋊C4—611no—
390C2×C4⋊C4—611no—
391C2×C4⋊C4—611no—
392C2×C4⋊C4—1222no—
393C2×C4⋊C4—1222no—
394C2×C4⋊C4—1221no—
395C2×C4⋊C4—1222no—
396C2×C4⋊C4—1222no—
397C2×C4⋊C4C22×D6111yescharacteristic
398C2×C4⋊C4C22×D6111yescharacteristic
399C2×C4⋊C4C22×D6111yescharacteristic
400C2×C22⋊C4—1221no—
401C2×C22⋊C4—1221no—
402C2×C42—612no—
403C2×C42—611no—
404C2×C42—1222no—
405C2×C42—1222no—
406C2×C42—1221no—
407C2×C42—1221no—
408C2×C42C22×D6111yescharacteristic
409#107—22no—
410#107—1222no—
411#107—1222no—
412#107—1222no—
413#106—22no—
414#106—1222no—
415#106—1222no—
416#106—1222no—
417#103—22no—
418#103—22no—
419#103—1222no—
420#103—1222no—
421#103—1222no—
422#103—1222no—
423#103—1222no—
424#103—1222no—
425#95—22no—
426#95—22no—
427#95—1222no—
428#95—1222no—
429#95—1222no—
430#95—1222no—
431C24—611no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—612no—
438C2×Q8—1224no—
439C2×Q8—1228no—
440C2×Q8—1228no—
441C2×Q8—1221no—
442C2×Q8—1221no—
443C2×Q8—1221no—
444C2×Q8—1221no—
445C2×Q8—1221no—
446C2×Q8—2442no—
447C2×Q8—2442no—
448C2×Q8—2444no—
449C2×Q8—2444no—
450C2×Q8Q8×D6112yes—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—611no—
456C22×C4—612no—
457C22×C4—612no—
458C22×C4—612no—
459C22×C4—612no—
460C22×C4—612no—
461C22×C4—612no—
462C22×C4—612no—
463C22×C4—1224no—
464C22×C4—1224no—
465C22×C4—1224no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—2442no—
471C22×C4—2442no—
472C22×C4—2444no—
473C22×C4—2442no—
474C22×C4—2442no—
475C22×C4D4×D6112yes—
476C22×C4C23×D6111yescharacteristic
477M16—1228no—
478M16—1228no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—612no—
484C2×C8—612no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8D4×D6112yes—
490C2×C8D4×D6112yes—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—2442no—
504C4⋊C4—2444no—
505C4⋊C4—2444no—
506C22⋊C4—2442no—
507C22⋊C4—2442no—
508C42—21no—
509C42—21no—
510C42—1224no—
511C42—1221no—
512C42—1221no—
513C42—2442no—
514C42—2442no—
515C23—612no—
516C23—612no—
517C23—612no—
518C23—1222no—
519C23—1222no—
520C23C2×D4×D6111yescharacteristic
521Q8—22no—
522Q8—22no—
523Q8—41no—
524Q8—41no—
525Q8—1222no—
526Q8—1222no—
527Q8—2448no—
528Q8—2441no—
529Q8—2441no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—612no—
543C2×C4—612no—
544C2×C4—612no—
545C2×C4—614no—
546C2×C4—614no—
547C2×C4—614no—
548C2×C4—614no—
549C2×C4—614no—
550C2×C4—1222no—
551C2×C4—1222no—
552C2×C4—1222no—
553C2×C4—4884no—
554C2×C4—2448no—
555C2×C4—2448no—
556C2×C4—2442no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4—2442no—
561C2×C4—2442no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4C2×D4×D6111yescharacteristic
565C2×C4C2×D4×D6111yescharacteristic
566C2×C4C2×Q8×D6111yescharacteristic
567C2×C4#41×D6111yescharacteristic
568C2×C4#41×D6111yescharacteristic
569C2×C4#41×D6111yescharacteristic
570C8—22no—
571C8—22no—
572C8—1222no—
573C8—1222no—
574C22—614no—
575C22—1224no—
576C22—1222no—
577C22—1224no—
578C22—1222no—
579C22C22⋊Q8×D6111yescharacteristic
580C22#123×D6111yescharacteristic
581C22#127×D6111yescharacteristic
582C22#137×D6112yes—
583C22#1684055912yes—
584C4—42no—
585C4—42no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4—614no—
590C4—1224no—
591C4—4881no—
592C4—4881no—
593C4—4881no—
594C4—2442no—
595C4C22⋊Q8×D6111yescharacteristic
596C4C22⋊Q8×D6111yescharacteristic
597C4#123×D6111yescharacteristic
598C4#123×D6111yescharacteristic
599C2—1224no—
600C2#399×D6111yescharacteristic
601C2#2145562212yes—
602C2#2145576612yes—
603C2#2145627511yescharacteristic
604C2#2145628011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2097C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#2097D61
hypocenterC61#23732
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2097

Complements

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

NH ≅ G/Ntype
7. #21447370C2semidirect
8. #21447367C2semidirect
9. #21447366C2semidirect
10. #21447360C2semidirect
11. #21447359C2semidirect
12. #21447355C2semidirect
13. #21447354C2semidirect
15. C61×#2097C2semidirect
306. #2097D61semidirect
361. C61#23732semidirect

Maximal subgroups

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

subgroupindex
C2×#168398032
C2×#168398012
C2×#168397692
#391×D612
#214482252
#214473702
#214473672
#214473662
#214473602
#214473592
#214473552
#214473542
#214455112
C61×#20972
#2373261