Finite Group Databasebeta

← #29646952

Subgroups — #29646952

Order 15616.

926 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459397C211yesmaximal, characteristic
2#21454785C211yesmaximal, characteristic
3#21454487C211yesmaximal, characteristic
4#21444290C212yesmaximal, complemented
5#21444264C212yesmaximal, complemented
6#21444247C211yesmaximal, characteristic, complemented
7#21444238C212yesmaximal, complemented
8#21444222C211yesmaximal, characteristic, complemented
9#21444212C212yesmaximal, complemented
10C2×#16839494C211yesmaximal, characteristic, complemented
11C61×#1493C211yesmaximal, characteristic, complemented
12#16840492C411yescharacteristic
13#16840489—21no—
14#16840487C411yescharacteristic
15#16840473C411yescharacteristic
16#16840470C412yes—
17#16840468—21no—
18#16840466—22no—
19#16840427C411yescharacteristic
20#16840425C412yes—
21#16840316—22no—
22#16840316C2212yes—
23#16840315—21no—
24C4×#11758222C2211yescharacteristic
25#16840230—21no—
26C2×C4×Dic122C2211yescharacteristic
27C4×Q8×D61C2212yes—
28#16839892—21no—
29#16839841—21no—
30#16839833—22no—
31#16839822—22no—
32#16839818C2212yes—
33#16839815C2211yescharacteristic
34#16839814C2212yes—
35#16839813C2212yes—
36C2×C4⋊#92418C2211yescharacteristic
37#16839730—21no—
38#16839720—21no—
39C4×#11758189C2212yes—
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—
56C2×#11758189—22no—
57C22×#92418—21no—
58#14519776—21no—
59#14519776—21no—
60#14519776—22no—
61#14519776C2×C412yes—
62#14519775—22no—
63#14519775C2×C412yes—
64#14519768—21no—
65#14519768—21no—
66#14519767—22no—
67#14519767—22no—
68#14519767—22no—
69#14519757—22no—
70#14519757C2×C412yes—
71#14519755—22no—
72#14519755—22no—
73#14519755—22no—
74#14519754—22no—
75#14519754—22no—
76#14519753—22no—
77#14519753C2×C412yes—
78Q8×Dic61C2×C412yes—
79C4×Dic122—21no—
80C4×Dic122—21no—
81C4×Dic122—22no—
82C4×Dic122—22no—
83C4×Dic122C2×C411yescharacteristic
84C4×Dic122C2×C411yescharacteristic
85C4×Dic122C2×C411yescharacteristic
86C4×Dic122D411yescharacteristic
87C4×Dic122C2311yescharacteristic
88C4×Dic122C2311yescharacteristic
89#14519742—21no—
90#14519742—21no—
91#14519742—22no—
92#14519742C2×C412yes—
93C4×D244C2×C411yescharacteristic
94C4×D244D411yescharacteristic
95#14519739—21no—
96#14519739—21no—
97C4×C22⋊D61C2×C412yes—
98C4×C22⋊D61D412yes—
99#14519737—22no—
100#14519737—22no—
101#14519736D411yescharacteristic
102#14519734—22no—
103#14519734D412yes—
104C2×C4⋊Dic61C2×C411yescharacteristic
105C2×#11758196—21no—
106C4⋊C4×D61—22no—
107C4⋊C4×D61C2×C412yes—
108C2×C4×Dic61—21no—
109C42×D61—22no—
110C42×D61C2312yes—
111C4⋊#92418—21no—
112C4⋊#92418—21no—
113C4⋊#92418C2311yescharacteristic
114C4⋊#92418C2311yescharacteristic
115#14519706—22no—
116C22⋊#92418—21no—
117#14519689—22no—
118#14519689—22no—
119#14519689—22no—
120#14519687—22no—
121#14519687—22no—
122C4×#92418C2312yes—
123C8×Dic61—22no—
124C2×M16×C61—21no—
125C22×C8×C61—21no—
126C4⋊Q8×C61C2×C411yescharacteristic
127C4⋊Q8×C61C2×C411yescharacteristic
128#125×C61—21no—
129C22⋊Q8×C61—21no—
130C22⋊Q8×C61C2×C412yes—
131C4×Q8×C61—21no—
132C4×Q8×C61C2312yes—
133#117×C61—21no—
134C2×C42×C61C2311yescharacteristic
135C4⋊C8×C61—21no—
136C4⋊C8×C61C2312yes—
137C4×C8×C61—21no—
138C4×C8×C61C2311yescharacteristic
139C4×C8×C61C2311yescharacteristic
140C2×Q8×C61—21no—
141C2×Q8×C61C22×C412yes—
142C22×C4×C61C22×C411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C2×C22⋊D61C22⋊C412yes—
146C22×Dic61—21no—
147C22×Dic61—21no—
148Q8×D61—24no—
149Q8×D61—42no—
150#11758222—22no—
151#11758222—22no—
152C2×D244C22⋊C411yescharacteristic
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61C22×C412yes—
156C2×C4×D61C2×D412yes—
157C2×C4×D61C2×D412yes—
158C2×Dic122—22no—
159C2×Dic122—21no—
160C2×Dic122—22no—
161C2×Dic122—22no—
162C2×Dic122—41no—
163C2×Dic122C22⋊C411yescharacteristic
164C2×Dic122C22×C411yescharacteristic
165C2×Dic122C22×C411yescharacteristic
166M16×C61—42no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—22no—
170C2×C8×C61—42no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61C22×C412yes—
177C22⋊C4×C61—22no—
178C42×C61—22no—
179C42×C61C22×C411yescharacteristic
180C42×C61C2×D411yescharacteristic
181C42×C61C2×D411yescharacteristic
182C42×C61C2411yescharacteristic
183C22⋊Dic61C22⋊C412yes—
184#11758198—22no—
185#11758198—22no—
186#11758198C22⋊C412yes—
187#11758198C22⋊C412yes—
188C4⋊Dic61—21no—
189C4⋊Dic61—22no—
190C4⋊Dic61—22no—
191C4⋊Dic61C22⋊C411yescharacteristic
192C4⋊Dic61C22⋊C411yescharacteristic
193C4⋊Dic61C22×C411yescharacteristic
194C4⋊Dic61C22×C411yescharacteristic
195#11758196—21no—
196#11758196—21no—
197#11758196—21no—
198#11758196—21no—
199#11758196—22no—
200#11758196—22no—
201#11758196—22no—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196C22⋊C412yes—
206#11758196C22⋊C412yes—
207C4×Dic61—21no—
208C4×Dic61—21no—
209C4×Dic61—21no—
210C4×Dic61—21no—
211C4×Dic61—22no—
212C4×Dic61—22no—
213C4×Dic61C22×C412yes—
214C4×Dic61C2×D412yes—
215C4×Dic61C2×D412yes—
216C2×#92418—21no—
217C2×#92418—22no—
218C2×#92418—21no—
219C2×#92418—42no—
220#11758189—44no—
221C23×C61C2×C22⋊C411yescharacteristic
222C22×D61C2×C22⋊C412yes—
223Q8×C61—22no—
224Q8×C61—22no—
225Q8×C61—81no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—41no—
231C2×C4×C61C2×C22⋊C411yescharacteristic
232C2×C4×C61C2×C22⋊C411yescharacteristic
233C2×C4×C61C2×C22⋊C411yescharacteristic
234C2×C4×C61#13712yes—
235C2×C4×C61#13712yes—
236C2×C4×C61C23×C411yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C22⋊D61—24no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—42no—
245C2×Dic61—81no—
246C2×Dic61C2×C22⋊C412yes—
247C2×Dic61C2×C22⋊C412yes—
248C2×Dic61C2×C22⋊C412yes—
249D244—22no—
250C4×D61—22no—
251C4×D61—22no—
252C4×D61—22no—
253C4×D61—22no—
254C4×D61—42no—
255C4×D61—44no—
256C4×D61—82no—
257Dic122—21no—
258Dic122—22no—
259Dic122—21no—
260Dic122—21no—
261Dic122—44no—
262Dic122—42no—
263Dic122—82no—
264Dic122#13112yes—
265C8×C61—22no—
266C8×C61—41no—
267C8×C61—42no—
268#92418—42no—
269#92418—41no—
270#92418—41no—
271#17514—611nomaximal, Sylow
272C22×C61—21no—
273C22×C61—21no—
274C22×C61#41912yes—
275C22×C61C22×C22⋊C411yescharacteristic
276C2×D61—22no—
277C2×D61—22no—
278C2×D61—22no—
279C4×C61—21no—
280C4×C61—21no—
281C4×C61—21no—
282C4×C61—42no—
283C4×C61—42no—
284C4×C61#41912yes—
285C4×C61C2×#13111yescharacteristic
286C4×C61#56811yescharacteristic
287C4×C61C2×#13711yescharacteristic
288C4×C61C2×#13711yescharacteristic
289Dic61—22no—
290Dic61—22no—
291Dic61—22no—
292Dic61—22no—
293Dic61—41no—
294Dic61—41no—
295Dic61—41no—
296Q8×#41—611no—
297#2965—611no—
298#2925—611no—
299#1497—612no—
300#1493D6111yescharacteristic, complemented
301#1491—612no—
302#1487—611no—
303#1485—612no—
304#1481—611no—
305#1479—612no—
306C2×Q8⋊C8—611no—
307C2×C61—22no—
308C2×C61#288011yescharacteristic
309C2×C61C2×#41911yescharacteristic
310C2×C61#290111yescharacteristic
311D61—42no—
312Q82—611no—
313#557—611no—
314#557—1221no—
315#554—611no—
316#554—612no—
317#554—1222no—
318#554—1221no—
319D4×Q8—611no—
320D4×Q8—612no—
321#533—612no—
322#533—1221no—
323#533—1222no—
324#533C2×D6111yescharacteristic
325C4×#41—611no—
326C4×#41—1221no—
327C2×C4×Q8—612no—
328C2×C4×Q8—611no—
329#446—1221no—
330#436—1221no—
331#435—1222no—
332#433—1222no—
333#432—612no—
334#431—611no—
335#431—612no—
336#431—612no—
337#423C2×D6111yescharacteristic
338C2×C4⋊C8—611no—
339#408—1221no—
340#405—1221no—
341C4×M16—612no—
342C2×C4×C8C2×D6111yescharacteristic
343Q8⋊C8—612no—
344Q8⋊C8—612no—
345Q8⋊C8—612no—
346Q8⋊C8—612no—
347Q8⋊C8—611no—
348Q8⋊C8—611no—
349Q8⋊C8—611no—
350Q8⋊C8—611no—
351Q8⋊C8C2×D6112yes—
352Q8⋊C8C2×D6112yes—
353C61#1751411yescharacteristic, complemented, Sylow
354C2×#41—611no—
355C22×Q8—612no—
356C22×Q8—611no—
357C2×M16—21no—
358C2×M16—1222no—
359C22×C8—21no—
360C22×C8—1221no—
361C4⋊Q8—612no—
362C4⋊Q8—612no—
363C4⋊Q8—1222no—
364C4⋊Q8—1221no—
365C4⋊Q8—1221no—
366C4⋊Q8—1222no—
367C4⋊Q8C4×D6111yescharacteristic
368C4⋊Q8C4×D6111yescharacteristic
369#125—21no—
370#125—1222no—
371#125—1222no—
372#125—1221no—
373#125—1222no—
374#125—1221no—
375C22⋊Q8—21no—
376C22⋊Q8—612no—
377C22⋊Q8—612no—
378C22⋊Q8—1222no—
379C22⋊Q8—1222no—
380C22⋊Q8—1222no—
381C22⋊Q8—1222no—
382C22⋊Q8—1222no—
383C22⋊Q8—1222no—
384C22⋊Q8—1222no—
385C22⋊Q8C4×D6112yes—
386C4×Q8—21no—
387C4×Q8—611no—
388C4×Q8—611no—
389C4×Q8—612no—
390C4×Q8—612no—
391C4×Q8—611no—
392C4×Q8—611no—
393C4×Q8—611no—
394C4×Q8—611no—
395C4×Q8—1222no—
396C4×Q8—1222no—
397C4×Q8—1222no—
398C4×Q8—1221no—
399C4×Q8—1221no—
400C4×Q8—1221no—
401C4×Q8—1221no—
402C4×Q8C22×D6112yes—
403C4×D4—612no—
404C4×D4—612no—
405C4×D4—611no—
406C4×D4—611no—
407C4×D4—1222no—
408C4×D4—1222no—
409C4×D4—1221no—
410C4×D4—1221no—
411#117—21no—
412#117—611no—
413#117—612no—
414#117—1222no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—612no—
417C2×C4⋊C4—1222no—
418C2×C4⋊C4—1221no—
419C2×C42—612no—
420C2×C42—1222no—
421C2×C42—1221no—
422C2×C42C22×D6111yescharacteristic
423C4⋊C8—21no—
424C4⋊C8—611no—
425C4⋊C8—611no—
426C4⋊C8—1222no—
427C4⋊C8—1221no—
428C4⋊C8—1221no—
429C4⋊C8C22×D6112yes—
430C22⋊C8—1222no—
431C22⋊C8—1222no—
432C22⋊C8—1222no—
433C22⋊C8—1221no—
434#97—1222no—
435#97—1222no—
436C4×C8—21no—
437C4×C8—612no—
438C4×C8—1222no—
439C4×C8C22×D6111yescharacteristic
440C4×C8C22×D6111yescharacteristic
441#41—1222no—
442#41—1222no—
443C2×Q8—21no—
444C2×Q8—611no—
445C2×Q8—611no—
446C2×Q8—611no—
447C2×Q8—1222no—
448C2×Q8—1222no—
449C2×Q8—1222no—
450C2×Q8—1221no—
451C2×Q8—1224no—
452C2×Q8—2441no—
453C2×Q8—2442no—
454C2×Q8C2×C4×D6112yes—
455C2×D4—612no—
456C2×D4—611no—
457C22×C4—612no—
458C22×C4—612no—
459C22×C4—612no—
460C22×C4—1222no—
461C22×C4—1222no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4C2×C4×D6111yescharacteristic
465C22×C4D4×D6111yescharacteristic
466C22×C4D4×D6111yescharacteristic
467M16—42no—
468M16—2444no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—22no—
472C2×C8—42no—
473C2×C8—1222no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8—2442no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—612no—
483C4⋊C4—612no—
484C4⋊C4—611no—
485C4⋊C4—611no—
486C4⋊C4—611no—
487C4⋊C4—611no—
488C4⋊C4—1222no—
489C4⋊C4—1222no—
490C4⋊C4—1222no—
491C4⋊C4—1222no—
492C4⋊C4—1222no—
493C4⋊C4—1222no—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4C2×C4×D6112yes—
502C22⋊C4—22no—
503C22⋊C4—612no—
504C22⋊C4—612no—
505C22⋊C4—612no—
506C22⋊C4—1222no—
507C22⋊C4—1222no—
508C42—22no—
509C42—612no—
510C42—612no—
511C42—612no—
512C42—1222no—
513C42—1222no—
514C42—1221no—
515C42—1221no—
516C42—1221no—
517C42—1221no—
518C42C2×C4×D6111yescharacteristic
519C42D4×D6111yescharacteristic
520C42D4×D6111yescharacteristic
521C42C23×D6111yescharacteristic
522C23—612no—
523C23C22⋊C4×D6111yescharacteristic
524Q8—22no—
525Q8—22no—
526Q8—81no—
527Q8—612no—
528Q8—1221no—
529Q8—1222no—
530Q8—1221no—
531Q8—1221no—
532Q8—4882no—
533Q8—2442no—
534Q8—2444no—
535D4—1224no—
536D4—1222no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—41no—
542C2×C4—612no—
543C2×C4—612no—
544C2×C4—612no—
545C2×C4—1221no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1222no—
549C2×C4—1222no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—4882no—
554C2×C4—4881no—
555C2×C4—2442no—
556C2×C4—2444no—
557C2×C4—2442no—
558C2×C4C22⋊C4×D6111yescharacteristic
559C2×C4C22⋊C4×D6111yescharacteristic
560C2×C4C22⋊C4×D6111yescharacteristic
561C2×C4#1451980612yes—
562C2×C4#1451980912yes—
563C2×C4C22×C4×D6111yescharacteristic
564C2×C4C2×D4×D6111yescharacteristic
565C2×C4C2×D4×D6111yescharacteristic
566C8—22no—
567C8—41no—
568C8—42no—
569C8—2442no—
570C8—2441no—
571C8—2441no—
572C22—21no—
573C22—21no—
574C22—1222no—
575C22—1222no—
576C22—1222no—
577C22#1683976912yes—
578C22C2×C22⋊C4×D6111yescharacteristic
579C4—21no—
580C4—21no—
581C4—21no—
582C4—42no—
583C4—42no—
584C4—1222no—
585C4—1222no—
586C4—1222no—
587C4—1222no—
588C4—2441no—
589C4—2441no—
590C4—2441no—
591C4#1683976912yes—
592C4#131×D6111yescharacteristic
593C4#1684052711yescharacteristic
594C4C2×#1451980611yescharacteristic
595C4C2×#1451980911yescharacteristic
596C2—22no—
597C2—2442no—
598C2#2145426711yescharacteristic
599C2C2×#1683976911yescharacteristic
600C2#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#17514
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. #21444290C2semidirect
5. #21444264C2semidirect
6. #21444247C2semidirect
7. #21444238C2semidirect
8. #21444222C2semidirect
9. #21444212C2semidirect
10. C2×#16839494C2semidirect
11. C61×#1493C2semidirect
300. #1493D61semidirect
353. C61#17514semidirect

Maximal subgroups

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

subgroupindex
#214593972
#214547852
#214544872
#214442902
#214442642
#214442472
#214442382
#214442222
#214442122
C2×#168394942
C61×#14932
#1751461