Finite Group Databasebeta

← #29646893

Subgroups — #29646893

Order 15616.

994 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459397C211yesmaximal, characteristic
2C8×#11758222C211yesmaximal, characteristic
3#21454388C211yesmaximal, characteristic
4#21444272C212yesmaximal, complemented
5#21444247C211yesmaximal, characteristic, complemented
6#21444246C212yesmaximal, complemented
7#21444222C211yesmaximal, characteristic, complemented
8#21444220C212yesmaximal, complemented
9C2×#16839494C211yesmaximal, characteristic, complemented
10Q8⋊C8×D61C212yesmaximal, complemented
11C61×#1493C211yesmaximal, characteristic, complemented
12C2×#14519785—21no—
13#16840492C411yescharacteristic
14#16840489—21no—
15#16840487C411yescharacteristic
16#16840473C411yescharacteristic
17#16840470C412yes—
18#16840468—21no—
19#16840466—22no—
20#16840427C411yescharacteristic
21#16840425C412yes—
22#16840316—22no—
23#16840316C2212yes—
24#16840315—21no—
25C4×#11758222C2211yescharacteristic
26#16840230—21no—
27C2×C4×Dic122C2211yescharacteristic
28C4×Q8×D61C2212yes—
29C8×Dic122—21no—
30C8×D244—21no—
31C8×C22⋊D61—22no—
32#16839815C2211yescharacteristic
33#16839813C2212yes—
34#16839790—21no—
35#16839787—22no—
36#16839784C2212yes—
37C2×C4⋊#92418C2211yescharacteristic
38C4⋊C8×D61C2212yes—
39C4×C8×D61C2212yes—
40#16839494C2211yescharacteristic
41#16839494C2211yescharacteristic
42#16839494C2211yescharacteristic
43#16839494C2211yescharacteristic
44Q8⋊#92418C2212yes—
45Q8⋊#92418C2212yes—
46#16839492C2212yes—
47#16839492C2212yes—
48C61×#533C2211yescharacteristic
49C61×#423C2211yescharacteristic
50C2×C4×C8×C61C2211yescharacteristic
51C61×Q8⋊C8C2212yes—
52C61×Q8⋊C8C2212yes—
53C2×#11758222D411yescharacteristic
54C22×Dic122C2×C411yescharacteristic
55C2×Q8×D61C2×C412yes—
56#14519785—44no—
57C2×#11758189—22no—
58C22×#92418—21no—
59C2×C8×D61—22no—
60C2×C8×D61—22no—
61#14519776—21no—
62#14519776—21no—
63#14519776—22no—
64#14519776C2×C412yes—
65#14519775—22no—
66#14519775C2×C412yes—
67#14519768—21no—
68#14519768—21no—
69#14519767—22no—
70#14519767—22no—
71#14519767—22no—
72#14519757—22no—
73#14519757C2×C412yes—
74#14519755—22no—
75#14519755—22no—
76#14519755—22no—
77#14519754—22no—
78#14519754—22no—
79#14519753—22no—
80#14519753C2×C412yes—
81Q8×Dic61C2×C412yes—
82C4×Dic122—21no—
83C4×Dic122—21no—
84C4×Dic122—22no—
85C4×Dic122—22no—
86C4×Dic122C2×C411yescharacteristic
87C4×Dic122C2×C411yescharacteristic
88C4×Dic122C2×C411yescharacteristic
89C4×Dic122D411yescharacteristic
90C4×Dic122C2311yescharacteristic
91C4×Dic122C2311yescharacteristic
92#14519742—21no—
93#14519742—21no—
94#14519742—22no—
95#14519742C2×C412yes—
96C4×D244C2×C411yescharacteristic
97C4×D244D411yescharacteristic
98#14519739—21no—
99#14519739—21no—
100C4×C22⋊D61C2×C412yes—
101C4×C22⋊D61D412yes—
102#14519737—22no—
103#14519737—22no—
104#14519736D411yescharacteristic
105#14519734—22no—
106#14519734D412yes—
107C2×C4⋊Dic61C2×C411yescharacteristic
108C2×#11758196—21no—
109C4⋊C4×D61—22no—
110C4⋊C4×D61C2×C412yes—
111C2×C4×Dic61—21no—
112C42×D61—22no—
113C42×D61C2312yes—
114C4⋊#92418—21no—
115C4⋊#92418—21no—
116C4⋊#92418C2311yescharacteristic
117C4⋊#92418C2311yescharacteristic
118#14519706—22no—
119#14519706—22no—
120#14519706—22no—
121C22⋊#92418—21no—
122#14519689—22no—
123C4×#92418C2312yes—
124C8×Dic61—22no—
125C2×M16×C61—21no—
126C22×C8×C61—21no—
127C4⋊Q8×C61C2×C411yescharacteristic
128C4⋊Q8×C61C2×C411yescharacteristic
129#125×C61—21no—
130C22⋊Q8×C61—21no—
131C22⋊Q8×C61C2×C412yes—
132C4×Q8×C61—21no—
133C4×Q8×C61C2312yes—
134#117×C61—21no—
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61—21no—
137C4⋊C8×C61C2312yes—
138C4×C8×C61—21no—
139C4×C8×C61C2311yescharacteristic
140C4×C8×C61C2311yescharacteristic
141C2×Q8×C61—21no—
142C2×Q8×C61C22×C412yes—
143C22×C4×C61C22×C411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C2×C22⋊D61C22⋊C412yes—
147C22×Dic61—21no—
148C22×Dic61—21no—
149Q8×D61—24no—
150Q8×D61—42no—
151#11758222SD1614yes—
152#11758222Q1614yes—
153C2×D244C22⋊C411yescharacteristic
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61C22×C412yes—
157C2×C4×D61C2×D412yes—
158C2×C4×D61C2×D412yes—
159C2×Dic122—22no—
160C2×Dic122—21no—
161C2×Dic122—22no—
162C2×Dic122—22no—
163C2×Dic122—41no—
164C2×Dic122C22⋊C411yescharacteristic
165C2×Dic122C22×C411yescharacteristic
166C2×Dic122C22×C411yescharacteristic
167M16×C61—42no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—22no—
171C2×C8×C61—42no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61C22×C412yes—
178C22⋊C4×C61—22no—
179C42×C61—22no—
180C42×C61C22×C411yescharacteristic
181C42×C61C2×D411yescharacteristic
182C42×C61C2×D411yescharacteristic
183C42×C61C2411yescharacteristic
184C22⋊Dic61C22⋊C412yes—
185#11758198—22no—
186#11758198—22no—
187#11758198C22⋊C412yes—
188#11758198C22⋊C412yes—
189C4⋊Dic61—21no—
190C4⋊Dic61—22no—
191C4⋊Dic61—22no—
192C4⋊Dic61C22⋊C411yescharacteristic
193C4⋊Dic61C22⋊C411yescharacteristic
194C4⋊Dic61C22×C411yescharacteristic
195C4⋊Dic61C22×C411yescharacteristic
196#11758196—21no—
197#11758196—21no—
198#11758196—21no—
199#11758196—21no—
200#11758196—22no—
201#11758196—22no—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206#11758196C22⋊C412yes—
207#11758196C22⋊C412yes—
208C4×Dic61—21no—
209C4×Dic61—21no—
210C4×Dic61—21no—
211C4×Dic61—21no—
212C4×Dic61—22no—
213C4×Dic61—22no—
214C4×Dic61C22×C412yes—
215C4×Dic61C2×D412yes—
216C4×Dic61C2×D412yes—
217C2×#92418—22no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—42no—
221#11758189—44no—
222C8×D61—44no—
223C8×D61—44no—
224C23×C61C2×C22⋊C411yescharacteristic
225C22×D61C2×C22⋊C412yes—
226Q8×C61—22no—
227Q8×C61—22no—
228Q8×C61—81no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—41no—
234C2×C4×C61C2×C22⋊C411yescharacteristic
235C2×C4×C61C2×C22⋊C411yescharacteristic
236C2×C4×C61C2×C22⋊C411yescharacteristic
237C2×C4×C61C2×SD1612yes—
238C2×C4×C61C2×Q1612yes—
239C2×C4×C61C23×C411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C22⋊D61#10318yes—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—42no—
248C2×Dic61—81no—
249C2×Dic61C2×C22⋊C412yes—
250C2×Dic61C2×C22⋊C412yes—
251C2×Dic61C2×C22⋊C412yes—
252D244#10314yes—
253C4×D61—22no—
254C4×D61—22no—
255C4×D61—42no—
256C4×D61—44no—
257C4×D61—82no—
258C4×D61C2×SD1614yes—
259C4×D61C2×Q1614yes—
260Dic122—21no—
261Dic122—21no—
262Dic122—21no—
263Dic122—44no—
264Dic122—42no—
265Dic122—82no—
266Dic122#10314yes—
267Dic122#13112yescomplemented
268C8×C61—22no—
269C8×C61—41no—
270C8×C61—42no—
271#92418—42no—
272#92418—41no—
273#92418—41no—
274#17509—611nomaximal, Sylow
275C22×C61—21no—
276C22×C61—21no—
277C22×C61C2×#10312yes—
278C22×C61C22×C22⋊C411yescharacteristic
279C2×D61—22no—
280C2×D61—22no—
281C2×D61C2×#10314yes—
282C4×C61—21no—
283C4×C61—21no—
284C4×C61—21no—
285C4×C61—42no—
286C4×C61—42no—
287C4×C61C2×#10312yes—
288C4×C61C2×#13111yescharacteristic
289C4×C61#56811yescharacteristic
290C4×C61C22×SD1611yescharacteristic
291C4×C61C22×Q1611yescharacteristic
292Dic61—22no—
293Dic61—22no—
294Dic61—22no—
295Dic61—41no—
296Dic61—41no—
297Dic61—41no—
298Dic61C2×#10314yes—
299Q8×#41—611no—
300C8×#41—611no—
301#2906—611no—
302#1493—612no—
303#1493D6111yescharacteristic, complemented
304#1487—611no—
305#1487—612no—
306#1481—612no—
307#1481—611no—
308C2×Q8⋊C8—611no—
309C2×Q8⋊C8—612no—
310C2×C61—22no—
311C2×C61#288011yescharacteristic
312C2×C61C22×#10311yescharacteristic
313C2×C61#290111yescharacteristic
314D61—24no—
315C2×#131—1221no—
316Q82—611no—
317#557—611no—
318#557—1221no—
319#554—611no—
320#554—612no—
321#554—1222no—
322#554—1221no—
323D4×Q8—611no—
324D4×Q8—612no—
325#533—612no—
326#533—1221no—
327#533—1222no—
328#533C2×D6111yescharacteristic
329C4×#41—611no—
330C4×#41—1221no—
331C2×C4×Q8—612no—
332C2×C4×Q8—611no—
333C8×Q8—1221no—
334C8×D4—1221no—
335C8×D4—1222no—
336#431—611no—
337#431—612no—
338#424—1222no—
339#424—1221no—
340#423—612no—
341#423C2×D6111yescharacteristic
342C2×C4⋊C8—611no—
343C2×C4⋊C8—612no—
344C2×C4×C8—612no—
345C2×C4×C8C2×D6111yescharacteristic
346Q8⋊C8—612no—
347Q8⋊C8—612no—
348Q8⋊C8—612no—
349Q8⋊C8—612no—
350Q8⋊C8—611no—
351Q8⋊C8—611no—
352Q8⋊C8—611no—
353Q8⋊C8—611no—
354Q8⋊C8C2×D6112yes—
355Q8⋊C8C2×D6112yes—
356C61#1750911yescharacteristic, complemented, Sylow
357C2×#41—611no—
358C22×Q8—612no—
359C22×Q8—611no—
360#131—2444no—
361C2×M16—21no—
362C2×M16—1222no—
363C22×C8—21no—
364C22×C8—1222no—
365C22×C8—1222no—
366C22×C8—1221no—
367C4⋊Q8—612no—
368C4⋊Q8—612no—
369C4⋊Q8—1222no—
370C4⋊Q8—1221no—
371C4⋊Q8—1221no—
372C4⋊Q8—1222no—
373C4⋊Q8C4×D6111yescharacteristic
374C4⋊Q8C4×D6111yescharacteristic
375#125—21no—
376#125—1222no—
377#125—1222no—
378#125—1221no—
379#125—1222no—
380#125—1221no—
381C22⋊Q8—21no—
382C22⋊Q8—612no—
383C22⋊Q8—612no—
384C22⋊Q8—1222no—
385C22⋊Q8—1222no—
386C22⋊Q8—1222no—
387C22⋊Q8—1222no—
388C22⋊Q8—1222no—
389C22⋊Q8—1222no—
390C22⋊Q8—1222no—
391C22⋊Q8C4×D6112yes—
392C4×Q8—21no—
393C4×Q8—611no—
394C4×Q8—611no—
395C4×Q8—612no—
396C4×Q8—612no—
397C4×Q8—611no—
398C4×Q8—611no—
399C4×Q8—611no—
400C4×Q8—611no—
401C4×Q8—1222no—
402C4×Q8—1222no—
403C4×Q8—1222no—
404C4×Q8—1221no—
405C4×Q8—1221no—
406C4×Q8—1221no—
407C4×Q8—1221no—
408C4×Q8C22×D6112yes—
409C4×D4—612no—
410C4×D4—612no—
411C4×D4—611no—
412C4×D4—611no—
413C4×D4—1222no—
414C4×D4—1222no—
415C4×D4—1221no—
416C4×D4—1221no—
417#117—21no—
418#117—611no—
419#117—612no—
420#117—1222no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—612no—
423C2×C4⋊C4—1222no—
424C2×C4⋊C4—1221no—
425C2×C42—612no—
426C2×C42—1222no—
427C2×C42—1221no—
428C2×C42C22×D6111yescharacteristic
429C4⋊C8—21no—
430C4⋊C8—611no—
431C4⋊C8—611no—
432C4⋊C8—1222no—
433C4⋊C8—1222no—
434C4⋊C8—1222no—
435C4⋊C8—1221no—
436C4⋊C8—1221no—
437C4⋊C8C22×D6112yes—
438C22⋊C8—1222no—
439C22⋊C8—1221no—
440C4×C8—21no—
441C4×C8—612no—
442C4×C8—1222no—
443C4×C8C22×D6111yescharacteristic
444C4×C8C22×D6111yescharacteristic
445#41—614no—
446#41—614no—
447C2×Q8—21no—
448C2×Q8—611no—
449C2×Q8—611no—
450C2×Q8—611no—
451C2×Q8—1222no—
452C2×Q8—1222no—
453C2×Q8—1222no—
454C2×Q8—1221no—
455C2×Q8—1224no—
456C2×Q8—2441no—
457C2×Q8—2442no—
458C2×Q8C2×C4×D6112yes—
459C2×D4—612no—
460C2×D4—611no—
461C22×C4—612no—
462C22×C4—612no—
463C22×C4—612no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4C2×C4×D6111yescharacteristic
469C22×C4D4×D6111yescharacteristic
470C22×C4D4×D6111yescharacteristic
471M16—42no—
472M16—2444no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—22no—
476C2×C8—42no—
477C2×C8—1222no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—2444no—
481C2×C8—2444no—
482C2×C8—2442no—
483C4⋊C4—22no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—612no—
489C4⋊C4—612no—
490C4⋊C4—611no—
491C4⋊C4—611no—
492C4⋊C4—611no—
493C4⋊C4—611no—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C4⋊C4—1222no—
497C4⋊C4—1222no—
498C4⋊C4—1222no—
499C4⋊C4—1222no—
500C4⋊C4—1222no—
501C4⋊C4—1222no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4C2×C4×D6112yes—
508C22⋊C4—22no—
509C22⋊C4—612no—
510C22⋊C4—612no—
511C22⋊C4—612no—
512C22⋊C4—1222no—
513C22⋊C4—1222no—
514C42—22no—
515C42—612no—
516C42—612no—
517C42—612no—
518C42—1222no—
519C42—1222no—
520C42—1221no—
521C42—1221no—
522C42—1221no—
523C42—1221no—
524C42C2×C4×D6111yescharacteristic
525C42D4×D6111yescharacteristic
526C42D4×D6111yescharacteristic
527C42C23×D6111yescharacteristic
528C23—612no—
529C23C22⋊C4×D6111yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—81no—
533Q8—614no—
534Q8—612no—
535Q8—1221no—
536Q8—1221no—
537Q8—1221no—
538Q8—4882no—
539Q8—2442no—
540Q8—2444no—
541D4—618no—
542D4—614no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—41no—
548C2×C4—612no—
549C2×C4—614no—
550C2×C4—614no—
551C2×C4—612no—
552C2×C4—612no—
553C2×C4—1221no—
554C2×C4—1222no—
555C2×C4—1222no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—4882no—
560C2×C4—4881no—
561C2×C4—2442no—
562C2×C4—2444no—
563C2×C4—2442no—
564C2×C4C22⋊C4×D6111yescharacteristic
565C2×C4C22⋊C4×D6111yescharacteristic
566C2×C4C22⋊C4×D6111yescharacteristic
567C2×C4SD16×D6112yes—
568C2×C4Q16×D6112yes—
569C2×C4C22×C4×D6111yescharacteristic
570C2×C4C2×D4×D6111yescharacteristic
571C2×C4C2×D4×D6111yescharacteristic
572C8—22no—
573C8—41no—
574C8—42no—
575C8—2442no—
576C8—2441no—
577C8—2441no—
578C22—21no—
579C22—21no—
580C22—614no—
581C22—1222no—
582C22—1222no—
583C22#103×D6112yes—
584C22C2×C22⋊C4×D6111yescharacteristic
585C4—21no—
586C4—21no—
587C4—21no—
588C4—42no—
589C4—42no—
590C4—614no—
591C4—1222no—
592C4—1222no—
593C4—1222no—
594C4—2441no—
595C4—2441no—
596C4—2441no—
597C4#103×D6112yes—
598C4#131×D6111yescharacteristic
599C4#1684052711yescharacteristic
600C4C2×SD16×D6111yescharacteristic
601C4C2×Q16×D6111yescharacteristic
602C2—22no—
603C2—1224no—
604C2#2145426711yescharacteristic
605C2C2×#103×D6111yescharacteristic
606C2#2145436311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1493C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#11758222D4
cosocleC42×C61C24
hypercenter#1493D61
hypocenterC61#17509
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1493

Complements

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

NH ≅ G/Ntype
4. #21444272C2semidirect
5. #21444247C2semidirect
6. #21444246C2semidirect
7. #21444222C2semidirect
8. #21444220C2semidirect
9. C2×#16839494C2semidirect
10. Q8⋊C8×D61C2semidirect
11. C61×#1493C2semidirect
267. Dic122#131semidirect
303. #1493D61semidirect
356. C61#17509semidirect

Maximal subgroups

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

subgroupindex
#214593972
C8×#117582222
#214543882
#214442722
#214442472
#214442462
#214442222
#214442202
C2×#168394942
Q8⋊C8×D612
C61×#14932
#1750961