Finite Group Databasebeta

← #30617098

Subgroups — #30617098

Order 15616.

938 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
1AGL(1,ℤ/8)×#6923C212yesmaximal, complemented
2C2×#16840803C211yesmaximal, characteristic, complemented
3#21462389C211yesmaximal, characteristic, complemented
4#21461980C211yesmaximal, characteristic, complemented
5#21461936C211yesmaximal, characteristic, complemented
6#21461905C212yesmaximal, complemented
7#21461858C212yesmaximal, complemented
8#21461707C211yesmaximal, characteristic, complemented
9#21461640C211yesmaximal, characteristic, complemented
10#100×#6923C212yesmaximal, complemented
11#472×D61C211yesmaximal, characteristic
12D61⋊#131C2211yescharacteristic
13#41×#6923C2211yescharacteristic
14C2×D4×#6923C2212yes—
15#16840803—22no—
16#16840803—21no—
17#16840803—21no—
18#16840803—21no—
19#16840803—21no—
20#16840801—22no—
21SD16×#6923—22no—
22D8×#6923—22no—
23D61⋊#108C2211yescharacteristic
24#16840773C2212yes—
25C2×#14519845C2211yescharacteristic
26#16840754C2212yes—
27#16840753C2212yes—
28C2×#14519839C2212yes—
29D61⋊#100C2212yes—
30#16840728—21no—
31#16840728C2211yescharacteristic
32#16840725—21no—
33#16840725C2211yescharacteristic
34M16×#6923—21no—
35M16×#6923C2212yes—
36M16×#6923C2211yescharacteristic
37C2×#14519831—21no—
38C2×#14519831C2211yescharacteristic
39#16840641C2212yes—
40#16840641C2212yes—
41#16840637C2212yes—
42AGL(1,ℤ/8)×D61C2212yes—
43C2×D8×D61C2211yescharacteristic
44#131×D61C2211yescharacteristic
45#16840033C411yescharacteristic, complemented
46#16840032C411yescharacteristic, complemented
47#16840031C412yescomplemented
48#16840030C412yescomplemented
49#16840029C411yescharacteristic, complemented
50#108×D61C2211yescharacteristic
51#100×D61C2212yes—
52C61×#472C411yescharacteristic, complemented
53C23×#6923—22no—
54#14519865—21no—
55#14519864—21no—
56#14519864—22no—
57D61⋊M16—21no—
58D61⋊M16C2312yes—
59D61⋊M16C2311yescharacteristic
60C2×D61⋊C8—21no—
61C2×D61⋊C8C2311yescharacteristic
62Q8×#6923D412yes—
63D4×#6923—22no—
64D4×#6923—22no—
65D4×#6923—22no—
66D4×#6923—22no—
67D4×#6923—22no—
68D4×#6923D412yes—
69#14519856—21no—
70#14519856C2311yescharacteristic
71C2×C4⋊#6923C2312yes—
72C2×C22⋊#6923—22no—
73C2×C4×#6923—21no—
74C2×C4×#6923C2311yescharacteristic
75#14519848—21no—
76#14519847—21no—
77#14519846—22no—
78#14519845—22no—
79#14519845—21no—
80#14519845—21no—
81#14519840—22no—
82#14519839—22no—
83#14519839—22no—
84#14519839—22no—
85#14519839—22no—
86#14519839—22no—
87#14519837—24no—
88#14519831—21no—
89#14519831—22no—
90#14519831—21no—
91#14519831—21no—
92#14519831D411yescharacteristic
93#14519831D411yescharacteristic
94#14519831D411yescharacteristic
95#14519831D411yescharacteristic
96C8×#6923—21no—
97C8×#6923—22no—
98C8×#6923—21no—
99C8×#6923—21no—
100#41×D61C2311yescharacteristic
101C2×D4×D61C2312yes—
102#14519805C2×C412yescomplemented
103#14519804—22no—
104#14519803C2×C412yescomplemented
105#14519802C2×C412yescomplemented
106#14519801—22no—
107SD16×D61—22no—
108C2×D488C2×C411yescharacteristic, complemented
109C2×#11758199C2×C412yescomplemented
110D8×D61—22no—
111D8×D61—21no—
112D8×D61—21no—
113D8×D61—21no—
114D8×D61—21no—
115#14519785—21no—
116#14519785C2×C411yescharacteristic, complemented
117#14519784C2×C411yescharacteristic, complemented
118#14519783—21no—
119#14519783C2×C411yescharacteristic, complemented
120M16×D61—21no—
121M16×D61C2311yescharacteristic
122M16×D61C2312yes—
123C2×C8×D61—21no—
124C2×C8×D61C2311yescharacteristic
125#14519713C2×C411yescharacteristic, complemented
126#14519712C2×C412yescomplemented
127#14519694C2×C412yescomplemented
128#14519693C2×C412yescomplemented
129#14519693C2×C412yescomplemented
130AGL(1,ℤ/8)×C61C2×C412yescomplemented
131C2×D8×C61C2×C411yescharacteristic, complemented
132#131×C61C2×C411yescharacteristic, complemented
133#108×C61C2×C411yescharacteristic, complemented
134#100×C61C2×C412yescomplemented
135C23×D61—22no—
136C22×#6923—22no—
137C22×#6923—22no—
138C22×#6923—42no—
139C22×#6923—42no—
140C22×#6923—84no—
141C22×#6923#4112yes—
142#41×C61C22×C411yescharacteristic
143C2×D4×C61C22×C412yes—
144C2×C22⋊D61—22no—
145#11758226#4111yescharacteristic
146Q8×D61C2×D411yescharacteristic
147#11758224—21no—
148#11758224#4111yescharacteristic
149D4×D61—21no—
150D4×D61—22no—
151D4×D61—22no—
152D4×D61—22no—
153D4×D61—22no—
154D4×D61C2×D411yescharacteristic
155#11758222—21no—
156#11758222C22×C411yescharacteristic
157C2×D244C22×C412yes—
158C2×C4×D61—21no—
159C2×C4×D61C2411yescharacteristic
160C22⋊#6923—21no—
161C22⋊#6923—22no—
162C22⋊#6923—42no—
163C22⋊#6923—42no—
164#11758217—22no—
165#11758217—22no—
166#11758217#4112yes—
167C2×#92420—22no—
168C2×#92420—21no—
169C4⋊#6923—22no—
170C4⋊#6923—22no—
171C4⋊#6923—22no—
172C4⋊#6923C2×D412yes—
173C4×#6923—22no—
174C4×#6923—21no—
175C4×#6923C2×D412yes—
176C4×#6923C2×D411yescharacteristic
177C4×#6923C2×D411yescharacteristic
178#11758213—22no—
179#11758213—21no—
180#11758213—22no—
181D61⋊C8—22no—
182D61⋊C8—21no—
183D61⋊C8—21no—
184D61⋊C8—22no—
185D61⋊C8C2×D411yescharacteristic
186D61⋊C8C2×D411yescharacteristic
187SD16×C61—22no—
188D8×C61—22no—
189D8×C61—21no—
190D8×C61—21no—
191M16×C61—21no—
192M16×C61C22×C411yescharacteristic
193M16×C61C22×C412yes—
194C2×C8×C61—21no—
195C2×C8×C61C22×C411yescharacteristic
196#11758201—22no—
197#11758200—22no—
198#11758199—22no—
199#11758199—22no—
200#11758199—22no—
201#11758199—22no—
202#11758194—21no—
203#11758194C22×C411yescharacteristic
204#11758194C22×C412yes—
205C2×#92418—21no—
206C2×#92418C22×C411yescharacteristic
207D488—22no—
208D488—21no—
209D488—21no—
210#11758190—22no—
211#11758189—21no—
212#11758189—22no—
213#11758189—21no—
214#11758189—21no—
215C8×D61—21no—
216C8×D61—22no—
217C8×D61—21no—
218C8×D61—21no—
219C8×D61C2×D411yescharacteristic
220C8×D61C2×D411yescharacteristic
221C8×D61#4111yescharacteristic
222C8×D61#4111yescharacteristic
223C23×C61—22no—
224C22×D61—21no—
225C22×D61—22no—
226C22×D61—42no—
227C22×D61—42no—
228C22×D61—82no—
229C22×D61—82no—
230C22×D61C2×#4111yescharacteristic
231C2×#6923—22no—
232C2×#6923—42no—
233C2×#6923—42no—
234C2×#6923—82no—
235C2×#6923—84no—
236C2×#6923—84no—
237C2×#6923—82no—
238C2×#6923#12112yes—
239C2×#6923#12112yes—
240Q8×C61C4×D411yescharacteristic
241D4×C61—21no—
242D4×C61—22no—
243D4×C61—22no—
244D4×C61C4×D411yescharacteristic
245C2×C4×C61—21no—
246C2×C4×C61C23×C411yescharacteristic
247C22⋊D61—21no—
248C22⋊D61—21no—
249C22⋊D61—21no—
250C22⋊D61—42no—
251C22⋊D61—42no—
252C2×Dic61—21no—
253C2×Dic61C2×#4111yescharacteristic
254D244—21no—
255D244—22no—
256D244—22no—
257D244C4×D411yescharacteristic
258C4×D61—21no—
259C4×D61—21no—
260C4×D61—21no—
261C4×D61C22×D411yescharacteristic
262C4×D61C22×D411yescharacteristic
263C4×D61C2×#4111yescharacteristic
264C4×D61C2×#4111yescharacteristic
265Dic122—21no—
266Dic122C4×D411yescharacteristic
267#92420—21no—
268#92420—22no—
269#92420—21no—
270#92420—42no—
271C8×C61—21no—
272C8×C61—22no—
273C8×C61C4×D411yescharacteristic
274C8×C61C4×D411yescharacteristic
275#92418—21no—
276#92418—22no—
277#92418C4×D411yescharacteristic
278#92418C4×D411yescharacteristic
279#21555—611nomaximal, Sylow
280C22×C61—21no—
281C22×C61—42no—
282C22×C61—82no—
283C22×C61C4×#4111yescharacteristic
284C2×D61—22no—
285C2×D61—21no—
286C2×D61—42no—
287C2×D61—42no—
288C2×D61—84no—
289C2×D61—82no—
290C2×D61—82no—
291C2×D61—82no—
292C2×D61C2×#12111yescharacteristic
293C2×D61#52911yescharacteristic
294#6923—22no—
295#6923—22no—
296#6923—42no—
297#6923—82no—
298C4×C61—21no—
299C4×C61C2×C4×D411yescharacteristic
300C4×C61C2×C4×D411yescharacteristic
301Dic61—21no—
302Dic61#52511yescharacteristic
303Dic61#53211yescharacteristic
304C2×#472—611no—
305C4×AGL(1,ℤ/8)—612no—
306C2×#442—611no—
307#2875—611no—
308#1978—611no—
309#1951—611no—
310#1931—612no—
311#1907—612no—
312#1815—611no—
313#1778—611no—
314C4×#100—612no—
315C2×C61—21no—
316C2×C61—41no—
317C2×C61—82no—
318C2×C61C4×#12111yescharacteristic
319D61—21no—
320D61—41no—
321D61—82no—
322D61#306812yes—
323C2×AGL(1,ℤ/8)—612no—
324C22×D8—611no—
325C2×#131—611no—
326C2×#131—611no—
327C4×#41—611no—
328C2×C4×D4—612no—
329#472—612no—
330#472—611no—
331#472—611no—
332#472—611no—
333#472—612no—
334#472#692311yescharacteristic, complemented
335#442—1221no—
336#442—1221no—
337#442—1221no—
338#442—1221no—
339#442—1222no—
340#440—1222no—
341C4×SD16—1222no—
342C4×D8—1222no—
343C2×#108—611no—
344C2×#108—611no—
345#428—612no—
346C2×#106—611no—
347#418—612no—
348#417—612no—
349C2×#102—612no—
350C2×#100—612no—
351C2×#100—612no—
352#405—611no—
353#405—611no—
354#405—1221no—
355#405—1221no—
356C4×M16—611no—
357C4×M16—612no—
358C4×M16—1221no—
359C2×#97—611no—
360C2×#97—1221no—
361#343—612no—
362#343—612no—
363#341—612no—
364C61#2155511yescharacteristic, complemented, Sylow
365C2×#41—611no—
366C22×D4—612no—
367C23×C4—1222no—
368AGL(1,ℤ/8)—612no—
369AGL(1,ℤ/8)—612no—
370AGL(1,ℤ/8)—612no—
371AGL(1,ℤ/8)—1222no—
372AGL(1,ℤ/8)—1222no—
373AGL(1,ℤ/8)C2×#692312yescomplemented
374C2×SD16—1222no—
375C2×D8—612no—
376C2×D8—611no—
377C2×D8—1221no—
378C2×D8—1221no—
379C2×D8—1221no—
380C2×D8—1221no—
381C2×D8—1222no—
382C2×D8C2×#692311yescharacteristic, complemented
383#131—611no—
384#131—611no—
385#131—611no—
386#131—1221no—
387#131—1221no—
388#131—1222no—
389#131—1221no—
390#131—1221no—
391#131C2×#692311yescharacteristic, complemented
392C2×M16—611no—
393C2×M16—611no—
394C2×M16—612no—
395C2×M16—612no—
396C2×M16—1221no—
397C2×M16—1221no—
398C22×C8—611no—
399C22×C8—611no—
400C22×C8—1221no—
401C22×C8—1221no—
402C4×Q8—612no—
403C4×D4—612no—
404C4×D4—1222no—
405C4×D4—1222no—
406C4×D4—1222no—
407C4×D4—1222no—
408C4×D4—1222no—
409#117—611no—
410#117—1221no—
411C2×C4⋊C4—612no—
412C2×C22⋊C4—1222no—
413C2×C42—611no—
414C2×C42—1221no—
415#108—612no—
416#108—611no—
417#108—1221no—
418#108—1221no—
419#108C2×#692311yescharacteristic, complemented
420#107—1222no—
421#106—1222no—
422#106—1221no—
423#106—1221no—
424#103—1222no—
425#102—1222no—
426#102—1222no—
427#102—1222no—
428#102—1222no—
429#102—1222no—
430#100—612no—
431#100—612no—
432#100—612no—
433#100—1224no—
434#100C2×#692312yescomplemented
435#97—611no—
436#97—611no—
437#97—611no—
438#97—611no—
439#97—1221no—
440#97—1222no—
441#97—1221no—
442#97—1221no—
443C4×C8—1221no—
444C4×C8—1222no—
445C4×C8—1221no—
446C4×C8—1221no—
447C24—1222no—
448#41—611no—
449#41—611no—
450#41—611no—
451#41—1221no—
452#41—1221no—
453#41C22×#692311yescharacteristic
454C2×Q8—611no—
455C2×D4—611no—
456C2×D4—612no—
457C2×D4—1222no—
458C2×D4—1222no—
459C2×D4—1222no—
460C2×D4—1222no—
461C2×D4—1222no—
462C2×D4—1221no—
463C2×D4C22×#692312yes—
464C22×C4—611no—
465C22×C4—612no—
466C22×C4—1221no—
467C22×C4—1222no—
468C22×C4—1222no—
469C22×C4—4884no—
470C22×C4—2442no—
471C22×C4—2442no—
472SD16—22no—
473SD16—1222no—
474SD16—1222no—
475SD16—1222no—
476D8—22no—
477D8—21no—
478D8—21no—
479D8—1222no—
480D8—1222no—
481D8—1222no—
482D8—1222no—
483D8—1222no—
484D8—1221no—
485D8—1221no—
486M16—21no—
487M16—612no—
488M16—611no—
489M16—612no—
490M16—1221no—
491M16—1221no—
492M16—1222no—
493M16—1221no—
494M16—1221no—
495M16—1222no—
496M16—1222no—
497M16—1221no—
498M16—1222no—
499M16—1222no—
500M16C22×#692311yescharacteristic
501M16C22×#692312yes—
502C2×C8—21no—
503C2×C8—611no—
504C2×C8—611no—
505C2×C8—611no—
506C2×C8—611no—
507C2×C8—611no—
508C2×C8—611no—
509C2×C8—611no—
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8—1222no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1222no—
518C2×C8—1222no—
519C2×C8—1221no—
520C2×C8—1222no—
521C2×C8C22×#692311yescharacteristic
522C4⋊C4—612no—
523C4⋊C4—1222no—
524C4⋊C4—1222no—
525C4⋊C4—1222no—
526C22⋊C4—1222no—
527C22⋊C4—1221no—
528C22⋊C4—2442no—
529C22⋊C4—2442no—
530C42—611no—
531C42—612no—
532C42—611no—
533C42—1221no—
534C42—1222no—
535C23—22no—
536C23—611no—
537C23—1221no—
538C23—1222no—
539C23—4882no—
540C23—4882no—
541C23—2442no—
542C23—2442no—
543Q8—611no—
544Q8—1221no—
545Q8D4×#692311yescharacteristic
546D4—21no—
547D4—22no—
548D4—22no—
549D4—611no—
550D4—1221no—
551D4—1221no—
552D4—1221no—
553D4—1221no—
554D4—1222no—
555D4—1222no—
556D4—2442no—
557D4—2442no—
558D4D4×#692311yescharacteristic
559C2×C4—21no—
560C2×C4—611no—
561C2×C4—611no—
562C2×C4—611no—
563C2×C4—611no—
564C2×C4—611no—
565C2×C4—612no—
566C2×C4—612no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—1222no—
572C2×C4—4884no—
573C2×C4—4884no—
574C2×C4—4882no—
575C2×C4—4882no—
576C2×C4—2442no—
577C2×C4—2442no—
578C2×C4C23×#692311yescharacteristic
579C8—21no—
580C8—22no—
581C8—611no—
582C8—611no—
583C8—1221no—
584C8—1221no—
585C8—1222no—
586C8—1221no—
587C8—1222no—
588C8—2442no—
589C8D4×#692311yescharacteristic
590C8D4×#692311yescharacteristic
591C22—21no—
592C22—42no—
593C22—82no—
594C22—611no—
595C22—611no—
596C22—1222no—
597C22—1221no—
598C22—4882no—
599C22—4882no—
600C22—4882no—
601C22—4884no—
602C22—2442no—
603C22—2442no—
604C22#41×#692311yescharacteristic
605C4—21no—
606C4—611no—
607C4—611no—
608C4—1222no—
609C4—1222no—
610C4—1221no—
611C4—4882no—
612C4—2442no—
613C4C2×D4×#692311yescharacteristic
614C4C2×D4×#692311yescharacteristic
615C2—21no—
616C2—41no—
617C2—82no—
618C2—612no—
619C2—1221no—
620C2—4882no—
621C2—2441no—
622C2#121×#692311yescharacteristic
subgroupG/H
centerC2#121×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#472C4
socleC2×C61C4×#121
prime radical#472×D61C2
cosocleC2×C4×D61C24
hypercenter#472#6923
hypocenterC61#21555
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. AGL(1,ℤ/8)×#6923C2semidirect
2. C2×#16840803C2semidirect
3. #21462389C2semidirect
4. #21461980C2semidirect
5. #21461936C2semidirect
6. #21461905C2semidirect
7. #21461858C2semidirect
8. #21461707C2semidirect
9. #21461640C2semidirect
10. #100×#6923C2semidirect
45. #16840033C4semidirect
46. #16840032C4semidirect
47. #16840031C4semidirect
48. #16840030C4semidirect
49. #16840029C4semidirect
52. C61×#472C4semidirect
102. #14519805C2×C4semidirect
104. #14519803C2×C4semidirect
105. #14519802C2×C4semidirect
108. C2×D488C2×C4semidirect
109. C2×#11758199C2×C4semidirect
116. #14519785C2×C4semidirect
117. #14519784C2×C4semidirect
119. #14519783C2×C4semidirect
125. #14519713C2×C4semidirect
126. #14519712C2×C4semidirect
127. #14519694C2×C4semidirect
128. #14519693C2×C4semidirect
129. #14519693C2×C4semidirect
130. AGL(1,ℤ/8)×C61C2×C4semidirect
131. C2×D8×C61C2×C4semidirect
132. #131×C61C2×C4semidirect
133. #108×C61C2×C4semidirect
134. #100×C61C2×C4semidirect
334. #472#6923semidirect
364. C61#21555semidirect
373. AGL(1,ℤ/8)C2×#6923semidirect
382. C2×D8C2×#6923semidirect
391. #131C2×#6923semidirect
419. #108C2×#6923semidirect
434. #100C2×#6923semidirect

Maximal subgroups

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

subgroupindex
AGL(1,ℤ/8)×#69232
C2×#168408032
#214623892
#214619802
#214619362
#214619052
#214618582
#214617072
#214616402
#100×#69232
#472×D612
#2155561