Finite Group Databasebeta

← #29671312

Subgroups — #29671312

Order 15616.

738 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457634C211yesmaximal, characteristic, complemented
2#21457465C211yesmaximal, characteristic, complemented
3#21457439C211yesmaximal, characteristic, complemented
4#21445423C211yesmaximal, characteristic, complemented
5#21445306C211yesmaximal, characteristic, complemented
6#21445296C211yesmaximal, characteristic, complemented
7#21445263C211yesmaximal, characteristic, complemented
8#21445253C211yesmaximal, characteristic, complemented
9#21445012C211yesmaximal, characteristic, complemented
10#21444997C211yesmaximal, characteristic, complemented
11#21444922C211yesmaximal, characteristic, complemented
12#21444907C211yesmaximal, characteristic, complemented
13#21444704C211yesmaximal, characteristic, complemented
14C8×#11758201C211yesmaximal, characteristic, complemented
15C61×#3358C211yesmaximal, characteristic, complemented
16C2×#14519808—21no—
17C2×#14519802—21no—
18#16840417C2211yescharacteristic
19D4×D244C2211yescharacteristic
20#16840141C2211yescharacteristic, complemented
21#16840138C2211yescharacteristic
22#16840110C2211yescharacteristic
23#16840109C2211yescharacteristic
24#16840108C2211yescharacteristic, complemented
25#16840077—21no—
26#16840042C2211yescharacteristic
27#16840018—21no—
28#16840013—21no—
29#16839997—21no—
30#16839972C2211yescharacteristic, complemented
31#16839964C2211yescharacteristic
32#16839962C2211yescharacteristic
33#16839957C2211yescharacteristic
34#16839954C2211yescharacteristic
35#16839922—21no—
36#16839918—21no—
37#16839907—21no—
38#16839895—21no—
39#16839894—21no—
40C4×#11758201C2211yescharacteristic
41#16839840C2211yescharacteristic
42#16839840C2211yescharacteristic, complemented
43C8×D244C2211yescharacteristic
44#102×D61—21no—
45#16839515C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839508C2211yescharacteristic, complemented
48#16839507C2211yescharacteristic, complemented
49#16839506C2211yescharacteristic, complemented
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic, complemented
54#16839489C2211yescharacteristic
55C8×#92418C2211yescharacteristic, complemented
56C61×#550C2211yescharacteristic
57D4×Q8×C61C2211yescharacteristic, complemented
58C61×#486—21no—
59C61×#486C2211yescharacteristic, complemented
60C61×#474—21no—
61C61×#474C2211yescharacteristic
62C61×#459—21no—
63C61×#459C2211yescharacteristic, complemented
64C8×Q8×C61C2211yescharacteristic, complemented
65C4×D8×C61—21no—
66C4×D8×C61C2211yescharacteristic
67C2×#11758222—21no—
68C22×D244—21no—
69C2×D4×D61—21no—
70#14519808—44no—
71#14519802—44no—
72C2×#11758202—21no—
73C2×#11758202—21no—
74C2×#11758202—21no—
75C2×#11758201—21no—
76C2×#11758201—21no—
77C2×#11758201D411yescharacteristic
78C2×#11758199—21no—
79C2×#11758199—21no—
80C2×#11758199—21no—
81C2×#11758199—21no—
82C2×#11758194—21no—
83C2×#11758194—21no—
84C2×C8×D61—21no—
85#14519777C2311yescharacteristic
86#14519774C2311yescharacteristic
87#14519759—41no—
88#14519756—21no—
89#14519752—21no—
90#14519751—21no—
91#14519748—21no—
92#14519747—41no—
93C4×D244C2311yescharacteristic
94C4⋊C4×D61—21no—
95#14519710—21no—
96#14519708D411yescharacteristic
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100#14519701—21no—
101#14519701D411yescharacteristic
102#14519700—21no—
103#14519699—21no—
104#14519699—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698—41no—
108#14519697—21no—
109#14519697—21no—
110#14519697—41no—
111#14519697D411yescharacteristic
112C22⋊#92418—21no—
113C22⋊#92418—21no—
114#14519689—21no—
115C4×#92418C2311yescharacteristic
116C4×#92418C2311yescharacteristic
117C2×#41×C61—21no—
118C22×Q8×C61—21no—
119C2×D8×C61—21no—
120C2×D8×C61D411yescharacteristic
121C4⋊Q8×C61—21no—
122C4⋊Q8×C61C2311yescharacteristic
123#127×C61—21no—
124#127×C61C2311yescharacteristic
125C22⋊Q8×C61—21no—
126C22⋊Q8×C61—41no—
127#121×C61—21no—
128#121×C61—41no—
129C4×Q8×C61C2311yescharacteristic
130C4×D4×C61—21no—
131C4×D4×C61—21no—
132C4×D4×C61C2311yescharacteristic
133C4×D4×C61C2311yescharacteristic
134#107×C61—21no—
135#107×C61D411yescharacteristic
136C4⋊C8×C61—21no—
137C4⋊C8×C61C2311yescharacteristic
138#102×C61—21no—
139#102×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142#102×C61D411yescharacteristic
143#102×C61D411yescharacteristic
144C4×C8×C61—21no—
145C4×C8×C61C2311yescharacteristic
146C23×D61—41no—
147#41×C61—42no—
148#41×C61—81no—
149C2×Q8×C61—21no—
150C2×Q8×C61—41no—
151C2×Q8×C61—42no—
152C2×Q8×C61—81no—
153C2×Q8×C61C2×D411yescharacteristic
154C2×D4×C61—21no—
155C2×D4×C61—21no—
156C2×D4×C61—21no—
157C2×D4×C61—41no—
158C2×D4×C61—41no—
159C2×D4×C61C2×D411yescharacteristic
160C2×D4×C61C2×D411yescharacteristic
161C22×C4×C61—21no—
162C22×C4×C61—21no—
163C22×C4×C61—41no—
164C22×C4×C61—41no—
165C2×C22⋊D61—41no—
166C2×C22⋊D61—41no—
167D4×D61—42no—
168D4×D61—81no—
169#11758222—42no—
170#11758222—81no—
171C2×D244—21no—
172C2×D244—21no—
173C2×D244—21no—
174C2×D244—42no—
175C2×D244—41no—
176C2×D244—81no—
177C2×D244C2×D411yescharacteristic
178C2×C4×D61—21no—
179C2×C4×D61—41no—
180C2×Dic122—21no—
181D8×C61—41no—
182D8×C61—42no—
183C2×C8×C61—21no—
184C2×C8×C61C2×D411yescharacteristic
185C2×C8×C61C2×D411yescharacteristic
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—41no—
190C4⋊C4×C61C2×D411yescharacteristic
191C4⋊C4×C61C2×D411yescharacteristic
192C4⋊C4×C61C2×D411yescharacteristic
193C22⋊C4×C61—21no—
194C22⋊C4×C61—21no—
195C22⋊C4×C61—41no—
196C22⋊C4×C61—41no—
197C42×C61—21no—
198C42×C61C2411yescharacteristic
199C22⋊Dic61—41no—
200#11758202—42no—
201#11758202—42no—
202#11758202—81no—
203#11758201—42no—
204#11758201—42no—
205#11758201D814yescomplemented
206#11758199—42no—
207#11758199—42no—
208#11758199—81no—
209#11758199—81no—
210#11758198—21no—
211#11758198—41no—
212C4⋊Dic61—21no—
213C4⋊Dic61—21no—
214C4⋊Dic61C2×D411yescharacteristic
215#11758196—41no—
216#11758194—42no—
217#11758194—42no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418C2×D411yescharacteristic
223C2×#92418C2×D411yescharacteristic
224C8×D61—81no—
225C23×C61—21no—
226C23×C61—21no—
227C23×C61—41no—
228C22×D61—21no—
229C22×D61—41no—
230C22×D61—82no—
231C22×D61—82no—
232C22×D61—161no—
233Q8×C61—21no—
234Q8×C61—41no—
235Q8×C61—41no—
236Q8×C61—82no—
237Q8×C61C2×D812yes—
238D4×C61—21no—
239D4×C61—21no—
240D4×C61—21no—
241D4×C61—21no—
242D4×C61—41no—
243D4×C61—41no—
244D4×C61—41no—
245D4×C61—41no—
246D4×C61—82no—
247D4×C61—81no—
248D4×C61—161no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—82no—
259C2×C4×C61—82no—
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C22×D411yescharacteristic
262C2×C4×C61H(5,𝔽2)11yescharacteristic
263C22⋊D61—82no—
264C22⋊D61—82no—
265C2×Dic61—21no—
266C2×Dic61—41no—
267D244—21no—
268D244—41no—
269D244—41no—
270D244—41no—
271D244—41no—
272D244—81no—
273D244—82no—
274D244C2×D812yes—
275C4×D61—41no—
276C4×D61—41no—
277C4×D61—82no—
278Dic122—41no—
279Dic122—41no—
280C8×C61—21no—
281C8×C61—21no—
282C8×C61—41no—
283#92418—21no—
284#92418—41no—
285#92418—41no—
286#92418—41no—
287#92418—41no—
288#92418C2×D812yes—
289#19886—611nomaximal, Sylow
290C22×C61—41no—
291C22×C61—41no—
292C22×C61—41no—
293C22×C61—41no—
294C22×C61—41no—
295C22×C61—41no—
296C22×C61—81no—
297C22×C61—81no—
298C22×C61—81no—
299C22×C61D4211yescharacteristic
300C2×D61—22no—
301C2×D61—41no—
302C2×D61—41no—
303C2×D61—81no—
304C2×D61—81no—
305C2×D61—81no—
306C2×D61—164no—
307C4×C61—22no—
308C4×C61—21no—
309C4×C61—41no—
310C4×C61—41no—
311C4×C61—41no—
312C4×C61—81no—
313C4×C61C22×D811yescharacteristic
314C4×C61#57511yescharacteristic
315C4×C61#57611yescharacteristic
316C4×C61#57711yescharacteristic
317Dic61—41no—
318Dic61—81no—
319#3358D6111yescharacteristic, complemented
320#3294—611no—
321#3276—611no—
322#3274—611no—
323#1709—611no—
324#1689—611no—
325#1687—611no—
326#1682—611no—
327#1681—611no—
328#1641—611no—
329#1639—611no—
330#1627—611no—
331#1625—611no—
332#1589—611no—
333C8×SD16—611no—
334C2×C61—81no—
335C2×C61—81no—
336C2×C61—161no—
337C2×C61#327511yescharacteristic
338C2×C61D4×D811yescharacteristic
339C2×C61#329011yescharacteristic
340D61—42no—
341D61—161no—
342C2×#137—1221no—
343C2×AGL(1,ℤ/8)—1221no—
344#550C2×D6111yescharacteristic
345D4×Q8C2×D6111yescharacteristic, complemented
346#547—611no—
347D42—611no—
348#494—611no—
349#493—611no—
350#486—21no—
351#486—611no—
352#486—611no—
353#486—611no—
354#486C2×D6111yescharacteristic, complemented
355#480—1221no—
356#474—21no—
357#474—611no—
358#474C2×D6111yescharacteristic
359#470—1221no—
360#469—1221no—
361#466—1221no—
362#462—611no—
363#461—611no—
364#460—611no—
365#459—21no—
366#459—611no—
367#459—611no—
368#459C2×D6111yescharacteristic, complemented
369#452—1221no—
370#451—1221no—
371#449—1221no—
372#447—1221no—
373#447—1221no—
374C8×Q8C2×D6111yescharacteristic, complemented
375C4×SD16—611no—
376C4×D8—21no—
377C4×D8C2×D6111yescharacteristic
378#436—611no—
379#436—611no—
380C8×D4—611no—
381C2×#102—1221no—
382#335—611no—
383#334—611no—
384#332—611no—
385#332—611no—
386#331—611no—
387#331—611no—
388Q8⋊C8—611no—
389D4⋊C8—611no—
390D4⋊C8—611no—
391D4⋊C8—611no—
392C82—611no—
393C61#1988611yescharacteristic, complemented, Sylow
394C2×#41—21no—
395C2×#41—1221no—
396C22×Q8—21no—
397C22×D4—1221no—
398C22×D4—1221no—
399#137—2444no—
400AGL(1,ℤ/8)—2444no—
401C2×Q16—1221no—
402C2×Q16—1221no—
403C2×Q16—1221no—
404C2×SD16—611no—
405C2×SD16—1221no—
406C2×SD16—1221no—
407C2×D8—21no—
408C2×D8—1221no—
409C2×D8—1221no—
410C2×D8—1221no—
411C2×D8—1221no—
412C2×D8C22⋊D6111yescharacteristic
413C2×M16—1221no—
414C2×M16—1221no—
415C22×C8—1221no—
416C4⋊Q8—21no—
417C4⋊Q8—611no—
418C4⋊Q8C22×D6111yescharacteristic
419#127—21no—
420#127—611no—
421#127C22×D6111yescharacteristic
422#123—2441no—
423C22⋊Q8—21no—
424C22⋊Q8—41no—
425C22⋊Q8—1221no—
426#121—21no—
427#121—41no—
428#121—1221no—
429#121—1221no—
430#121—1221no—
431#120—2441no—
432C4×Q8C22×D6111yescharacteristic
433C4×D4—21no—
434C4×D4—21no—
435C4×D4—611no—
436C4×D4C22×D6111yescharacteristic
437C4×D4C22×D6111yescharacteristic
438C2×C4⋊C4—1221no—
439#107—21no—
440#107—1221no—
441#107C22⋊D6111yescharacteristic
442#106—611no—
443C4⋊C8—21no—
444C4⋊C8—611no—
445C4⋊C8—611no—
446C4⋊C8—611no—
447C4⋊C8C22×D6111yescharacteristic
448#103—611no—
449#103—1221no—
450#103—1221no—
451#102—21no—
452#102—21no—
453#102—21no—
454#102—21no—
455#102—611no—
456#102—1221no—
457#102—1221no—
458#102—1221no—
459#102—1221no—
460#102—1221no—
461#102—1221no—
462#102—2441no—
463#102—2441no—
464#102C22⋊D6111yescharacteristic
465#102C22⋊D6111yescharacteristic
466C22⋊C8—1221no—
467C22⋊C8—1221no—
468C22⋊C8—1221no—
469C4×C8—21no—
470C4×C8—611no—
471C4×C8—611no—
472C4×C8C22×D6111yescharacteristic
473C24—2441no—
474#41—42no—
475#41—81no—
476#41—4881no—
477#41—2442no—
478C2×Q8—21no—
479C2×Q8—41no—
480C2×Q8—42no—
481C2×Q8—81no—
482C2×Q8—1221no—
483C2×Q8D4×D6111yescharacteristic
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—41no—
488C2×D4—41no—
489C2×D4—611no—
490C2×D4—1221no—
491C2×D4—1221no—
492C2×D4—1221no—
493C2×D4—4881no—
494C2×D4—4881no—
495C2×D4—2441no—
496C2×D4—2442no—
497C2×D4—2441no—
498C2×D4—2442no—
499C2×D4—2441no—
500C2×D4C2×C22⋊D6111yescharacteristic
501C2×D4C2×C22⋊D6111yescharacteristic
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—41no—
505C22×C4—41no—
506C22×C4—1221no—
507C22×C4—2441no—
508Q16—4881no—
509Q16—2442no—
510Q16—2442no—
511SD16—614no—
512SD16—2442no—
513SD16—2442no—
514D8—41no—
515D8—42no—
516D8—4881no—
517D8—4881no—
518D8—2442no—
519D8—2442no—
520M16—2442no—
521M16—2442no—
522C2×C8—21no—
523C2×C8—611no—
524C2×C8—611no—
525C2×C8—1221no—
526C2×C8—1221no—
527C2×C8—1221no—
528C2×C8—1221no—
529C2×C8—4881no—
530C2×C8C2×C22⋊D6111yescharacteristic
531C2×C8C2×C22⋊D6111yescharacteristic
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—41no—
536C4⋊C4—611no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4—2441no—
540C4⋊C4D4×D6111yescharacteristic
541C4⋊C4C2×C22⋊D6111yescharacteristic
542C4⋊C4C2×C22⋊D6111yescharacteristic
543C22⋊C4—21no—
544C22⋊C4—21no—
545C22⋊C4—41no—
546C22⋊C4—41no—
547C22⋊C4—1221no—
548C22⋊C4—2441no—
549C22⋊C4—2441no—
550C42—21no—
551C42C23×D6111yescharacteristic
552C23—21no—
553C23—21no—
554C23—41no—
555C23—1221no—
556C23—9761no—
557C23—4882no—
558C23—4882no—
559C23—2441no—
560Q8—21no—
561Q8—41no—
562Q8—41no—
563Q8—82no—
564Q8—2441no—
565Q8—2441no—
566Q8D8×D6112yes—
567D4—21no—
568D4—21no—
569D4—21no—
570D4—21no—
571D4—41no—
572D4—41no—
573D4—41no—
574D4—41no—
575D4—82no—
576D4—81no—
577D4—161no—
578D4—612no—
579D4—1221no—
580D4—4881no—
581D4—4882no—
582D4—4882no—
583D4—4882no—
584D4—2441no—
585D4—2441no—
586D4—2441no—
587D4—2441no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—1221no—
600C2×C4—4882no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4C2×D4×D6111yescharacteristic
605C2×C4C22×C22⋊D6111yescharacteristic
606C2×C4#1451982211yescharacteristic
607C8—21no—
608C8—21no—
609C8—41no—
610C8—612no—
611C8—1221no—
612C8—2441no—
613C8—2441no—
614C8—2441no—
615C8—2441no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—1222no—
626C22—9764no—
627C22—4881no—
628C22—4881no—
629C22—4881no—
630C22—2441no—
631C22—2441no—
632C22D4×C22⋊D6111yescharacteristic
633C4—22no—
634C4—21no—
635C4—41no—
636C4—41no—
637C4—41no—
638C4—81no—
639C4—4881no—
640C4—2441no—
641C4C2×D8×D6111yescharacteristic
642C4#1684056011yescharacteristic
643C4#1684056311yescharacteristic
644C4#1684057011yescharacteristic
645C2—81no—
646C2—81no—
647C2—161no—
648C2—9761no—
649C2—2442no—
650C2#2145745111yescharacteristic
651C2D8×C22⋊D6111yescharacteristic
652C2#2145759911yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3358C2
socleC22×C61D42
hypercenter#3358D61
hypocenterC61#19886
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3358

Complements

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

NH ≅ G/Ntype
1. #21457634C2semidirect
2. #21457465C2semidirect
3. #21457439C2semidirect
4. #21445423C2semidirect
5. #21445306C2semidirect
6. #21445296C2semidirect
7. #21445263C2semidirect
8. #21445253C2semidirect
9. #21445012C2semidirect
10. #21444997C2semidirect
11. #21444922C2semidirect
12. #21444907C2semidirect
13. #21444704C2semidirect
14. C8×#11758201C2semidirect
15. C61×#3358C2semidirect
20. #16840141C22semidirect
24. #16840108C22semidirect
30. #16839972C22semidirect
42. #16839840C22semidirect
45. #16839515C22semidirect
46. #16839512C22semidirect
47. #16839508C22semidirect
48. #16839507C22semidirect
49. #16839506C22semidirect
51. Q8⋊#92418C22semidirect
53. D4⋊#92418C22semidirect
55. C8×#92418C22semidirect
57. D4×Q8×C61C22semidirect
59. C61×#486C22semidirect
63. C61×#459C22semidirect
64. C8×Q8×C61C22semidirect
205. #11758201D8semidirect
319. #3358D61semidirect
345. D4×Q8C2×D61semidirect
354. #486C2×D61semidirect
368. #459C2×D61semidirect
374. C8×Q8C2×D61semidirect
393. C61#19886semidirect

Maximal subgroups

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

subgroupindex
#214576342
#214574652
#214574392
#214454232
#214453062
#214452962
#214452632
#214452532
#214450122
#214449972
#214449222
#214449072
#214447042
C8×#117582012
C61×#33582
#1988661