Finite Group Databasebeta

← #29693918

Subgroups — #29693918

Order 15616.

796 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456295C211yesmaximal, characteristic
2#21454737C211yesmaximal, characteristic, complemented
3#21454680C211yesmaximal, characteristic, complemented
4#21446722C211yesmaximal, characteristic, complemented
5#21446717C211yesmaximal, characteristic, complemented
6#21446355C211yesmaximal, characteristic, complemented
7#21446350C211yesmaximal, characteristic, complemented
8#21446345C211yesmaximal, characteristic, complemented
9#21446340C211yesmaximal, characteristic, complemented
10#21446314C211yesmaximal, characteristic, complemented
11#21446307C211yesmaximal, characteristic, complemented
12#21446300C211yesmaximal, characteristic, complemented
13#21446293C211yesmaximal, characteristic, complemented
14#21445943C211yesmaximal, characteristic, complemented
15C61×#1747C211yesmaximal, characteristic, complemented
16C2×#14519810C2211yescharacteristic
17C2×#14519805C2211yescharacteristic
18C2×#14519799C2211yescharacteristic
19C22×#11758190C2211yescharacteristic
20#16840528C2211yescharacteristic
21#16840528C2211yescharacteristic
22#16840338C2211yescharacteristic
23#16840331C2211yescharacteristic
24#16840239C2211yescharacteristic
25#16840236C2211yescharacteristic
26#16840156C412yescomplemented
27#16840151C412yescomplemented
28#16840146C412yescomplemented
29#16840135C412yescomplemented
30#16839885—22no—
31#16839880—22no—
32#16839864—22no—
33C4×#11758190—22no—
34C2×#14519713C2211yescharacteristic
35#16839800C2211yescharacteristic
36#16839780C2211yescharacteristic
37#16839780C2211yescharacteristic
38#16839778C2211yescharacteristic
39#16839778C2211yescharacteristic
40#16839772C2211yescharacteristic
41#16839770C2211yescharacteristic
42#16839768C2211yescharacteristic
43#16839766C2211yescharacteristic
44C2×#14519695C2211yescharacteristic
45C2×#14519693C2211yescharacteristic
46#16839735C2211yescharacteristic
47#16839735C2211yescharacteristic
48#16839533C2211yescharacteristic
49#16839533C2211yescharacteristic
50#16839518C2211yescharacteristic
51#16839518C2211yescharacteristic
52C61×#405C2211yescharacteristic
53C61×#405C2211yescharacteristic
54C61×#405C2211yescharacteristic
55C61×#338C2211yescharacteristic
56C61×H(3,ℤ/4)C2211yescharacteristic
57C61×H(3,ℤ/4)C2211yescharacteristic
58C61×H(3,ℤ/4)C2211yescharacteristic
59C2×#11758222C2311yescharacteristic
60C2×#11758222C2311yescharacteristic
61C22×Dic122C2311yescharacteristic
62C22×D244C2311yescharacteristic
63#14519810—41no—
64#14519810—41no—
65#14519805—41no—
66#14519805—41no—
67#14519799—21no—
68#14519799—21no—
69#14519799—21no—
70#14519799—21no—
71C2×Dic244C2×C411yescharacteristic, complemented
72C2×Dic244C2×C412yescomplemented
73C2×#11758190—21no—
74C2×#11758190—21no—
75C2×#11758190—21no—
76C2×#11758190—21no—
77C2×#11758190C2×C412yes—
78C2×#11758190C2×C412yes—
79C2×#11758190C2×C412yes—
80C2×#11758190C2×C411yescharacteristic, complemented
81C2×#11758190C2×C411yescharacteristic, complemented
82C2×#11758190C2×C412yescomplemented
83C2×D488C2×C412yes—
84C2×D488C2×C411yescharacteristic, complemented
85#14519785—41no—
86#14519785—41no—
87#14519783—22no—
88#14519783—22no—
89C2×#11758194C2311yescharacteristic
90C2×#11758194C2311yescharacteristic
91M16×D61—41no—
92M16×D61—41no—
93#14519777C2×C412yescomplemented
94#14519774C2×C412yes—
95#14519765C2×C412yes—
96#14519765C2×C412yescomplemented
97#14519761—41no—
98#14519757—41no—
99#14519752—41no—
100#14519747—41no—
101C4×Dic122—22no—
102#14519742—41no—
103C4×D244—22no—
104#14519739—41no—
105#14519736C2311yescharacteristic
106#14519736C2311yescharacteristic
107#14519733—41no—
108C22⋊C4×D61—41no—
109#14519713—21no—
110#14519713—21no—
111#14519711—21no—
112#14519709—21no—
113#14519705—22no—
114#14519705—22no—
115#14519703—41no—
116#14519703—41no—
117#14519702—22no—
118#14519700—41no—
119#14519699—22no—
120#14519697—41no—
121#14519695—41no—
122#14519693—41no—
123#14519691—41no—
124#14519691—41no—
125C2×M16×C61C2311yescharacteristic
126C2×M16×C61C2311yescharacteristic
127C2×M16×C61C2311yescharacteristic
128C22×C8×C61C2311yescharacteristic
129#117×C61C2311yescharacteristic
130#117×C61C2311yescharacteristic
131#117×C61C2311yescharacteristic
132#97×C61—21no—
133#97×C61C2×C412yescomplemented
134#97×C61D412yescomplemented
135C4×C8×C61—21no—
136C4×C8×C61C2×C412yescomplemented
137C4×C8×C61D412yescomplemented
138C23×D61—21no—
139C22×C4×C61C2411yescharacteristic
140C2×C22⋊D61—21no—
141C2×C22⋊D61—21no—
142C22×Dic61—21no—
143#11758222—21no—
144#11758222—21no—
145#11758222—21no—
146#11758222—21no—
147#11758222—21no—
148#11758222—21no—
149#11758222—21no—
150#11758222—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—42no—
157C2×D244C22×C412yes—
158C2×D244C22×C411yescharacteristic
159C2×D244C22×C411yescharacteristic
160C2×C4×D61—21no—
161C2×C4×D61—21no—
162C2×C4×D61—42no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122C22×C412yes—
169C2×Dic122C22×C411yescharacteristic
170C2×Dic122C22×C411yescharacteristic
171M16×C61—21no—
172M16×C61—21no—
173M16×C61—21no—
174M16×C61—21no—
175M16×C61—21no—
176M16×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61C22×C412yes—
179C2×C8×C61C22×C411yescharacteristic
180C2×C8×C61C22×C411yescharacteristic
181C2×C8×C61C2×D412yes—
182C2×C8×C61C2×D411yescharacteristic
183C2×C8×C61C2×D411yescharacteristic
184C2×C8×C61#4111yescharacteristic
185C2×C8×C61#4111yescharacteristic
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C22⋊C4×C61—21no—
190C22⋊C4×C61—21no—
191C22⋊C4×C61—21no—
192C42×C61—21no—
193C42×C61C22×C412yes—
194C42×C61C2×D412yes—
195C22⋊Dic61—21no—
196C22⋊Dic61—21no—
197#11758198—42no—
198#11758198—42no—
199#11758198—42no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—42no—
203#11758196—42no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C4×Dic61—42no—
207#11758194—21no—
208#11758194—21no—
209#11758194—21no—
210#11758194—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213Dic244—21no—
214Dic244—21no—
215Dic244—42no—
216D488—21no—
217D488—21no—
218D488—42no—
219#11758190—22no—
220#11758190—22no—
221#11758190—22no—
222#11758190—22no—
223#11758190—21no—
224#11758190—21no—
225#11758190—21no—
226#11758190—21no—
227#11758190—42no—
228#11758190—42no—
229#11758189—42no—
230#11758189—42no—
231C8×D61—42no—
232C8×D61—42no—
233C23×C61H(5,𝔽2)11yescharacteristic
234C22×D61—21no—
235C22×D61—22no—
236C22×D61—21no—
237C22×D61—41no—
238C22×D61—41no—
239C22×D61—161no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—41no—
243C2×C4×C61C23×C411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C2×C4×C61H(5,𝔽2)11yescharacteristic
247C2×C4×C61H(5,𝔽2)11yescharacteristic
248C2×C4×C61H(5,𝔽2)11yescharacteristic
249C22⋊D61—41no—
250C22⋊D61—41no—
251C22⋊D61—41no—
252C22⋊D61—41no—
253C2×Dic61—21no—
254C2×Dic61—22no—
255C2×Dic61—21no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260D244—22no—
261D244—21no—
262D244—21no—
263D244—21no—
264D244—22no—
265D244—21no—
266D244—41no—
267D244—41no—
268D244—162no—
269C4×D61—41no—
270C4×D61—41no—
271C4×D61—41no—
272C4×D61—41no—
273C4×D61—162no—
274Dic122—22no—
275Dic122—21no—
276Dic122—21no—
277Dic122—21no—
278Dic122—22no—
279Dic122—21no—
280Dic122—41no—
281Dic122—41no—
282C8×C61—22no—
283C8×C61—22no—
284C8×C61—41no—
285C8×C61C4×D411yescharacteristic
286C8×C61C4×D411yescharacteristic
287C8×C61C4×D411yescharacteristic
288C8×C61C4×D411yescharacteristic
289#92418—41no—
290#92418—41no—
291#22284—611nomaximal, Sylow
292C22×C61—41no—
293C22×C61#51811yescharacteristic
294C22×C61#53511yescharacteristic
295C22×C61#54011yescharacteristic
296C2×D61—41no—
297C2×D61—41no—
298C2×D61—42no—
299C2×D61—82no—
300C2×D61—81no—
301C2×D61—82no—
302C2×D61—81no—
303C2×D61—161no—
304C4×C61—42no—
305C4×C61—42no—
306C4×C61—81no—
307C4×C61C2×C4×D411yescharacteristic
308C4×C61#51811yescharacteristic
309C4×C61#53511yescharacteristic
310C4×C61#54011yescharacteristic
311Dic61—42no—
312Dic61—41no—
313Dic61—41no—
314Dic61—81no—
315Dic61—81no—
316#3154—611no—
317#2957—611no—
318#2949—611no—
319#1977—611no—
320#1976—611no—
321#1912—611no—
322#1911—611no—
323#1910—611no—
324#1909—611no—
325#1905—611no—
326#1904—611no—
327#1903—611no—
328#1902—611no—
329#1833—611no—
330#1747D6111yescharacteristic, complemented
331C2×C61—21no—
332C2×C61—21no—
333C2×C61—21no—
334C2×C61#232811yescharacteristic
335D61—81no—
336D61—81no—
337D61—82no—
338C2×#137—611no—
339C2×AGL(1,ℤ/8)—611no—
340C2×#135—611no—
341C22×SD16—611no—
342#568—611no—
343#568—611no—
344#536—611no—
345#535—611no—
346#519—611no—
347#518—611no—
348#497—612no—
349#496—612no—
350#495—612no—
351#492—612no—
352#444—1222no—
353#443—1222no—
354#440—1222no—
355C4×SD16—1222no—
356C2×#108—611no—
357#427—611no—
358#421—611no—
359#421—611no—
360#421—611no—
361#421—611no—
362#419—611no—
363#419—611no—
364#418—611no—
365#418—611no—
366C2×#101—611no—
367C2×#100—611no—
368#410—611no—
369#410—611no—
370#405C2×D6111yescharacteristic
371#405C2×D6111yescharacteristic
372#405C2×D6111yescharacteristic
373#344—611no—
374#344—611no—
375#338C2×D6111yescharacteristic
376H(3,ℤ/4)—611no—
377H(3,ℤ/4)—611no—
378H(3,ℤ/4)C2×D6111yescharacteristic
379H(3,ℤ/4)C2×D6111yescharacteristic
380H(3,ℤ/4)C2×D6111yescharacteristic
381C61#2228411yescharacteristic, complemented, Sylow
382C2×#41—611no—
383C2×#41—611no—
384C22×Q8—611no—
385C22×D4—611no—
386#137—2441no—
387#137—2441no—
388AGL(1,ℤ/8)—2441no—
389AGL(1,ℤ/8)—2441no—
390#135—1221no—
391#135—1221no—
392#135—1221no—
393#135—1221no—
394C2×Q16—611no—
395C2×Q16—612no—
396C2×SD16—612no—
397C2×SD16—612no—
398C2×SD16—611no—
399C2×SD16—611no—
400C2×SD16—612no—
401C2×SD16—612no—
402C2×SD16—1221no—
403C2×SD16—1221no—
404C2×SD16—1221no—
405C2×SD16—1221no—
406C2×D8—611no—
407C2×D8—612no—
408#131—1222no—
409#131—1222no—
410#131—2441no—
411#131—2441no—
412C2×M16—611no—
413C2×M16—611no—
414C2×M16—2441no—
415C2×M16—2441no—
416C2×M16C22×D6111yescharacteristic
417C2×M16C22×D6111yescharacteristic
418C2×M16C22×D6111yescharacteristic
419C22×C8C22×D6111yescharacteristic
420C4⋊Q8—612no—
421#127—612no—
422#124—612no—
423#124—612no—
424#123—2441no—
425C22⋊Q8—2441no—
426#121—2441no—
427#120—2441no—
428C4×Q8—1222no—
429C4×Q8—2441no—
430C4×D4—1222no—
431C4×D4—2441no—
432#117—611no—
433#117—611no—
434#117—2441no—
435#117C22×D6111yescharacteristic
436#117C22×D6111yescharacteristic
437#117C22×D6111yescharacteristic
438C2×C22⋊C4—2441no—
439#108—1221no—
440#108—1221no—
441#107—1221no—
442#106—1221no—
443#104—1222no—
444#104—1222no—
445#104—2441no—
446#104—2441no—
447#103—1222no—
448#103—2441no—
449#102—1222no—
450#102—2441no—
451#101—2441no—
452#100—2441no—
453#99—2441no—
454#99—2441no—
455#97—21no—
456#97C4×D6112yescomplemented
457#97D24412yescomplemented
458C4×C8—21no—
459C4×C8C4×D6112yescomplemented
460C4×C8D24412yescomplemented
461C24—1221no—
462#41—1221no—
463#41—1221no—
464#41—1221no—
465#41—1221no—
466#41—1221no—
467#41—1221no—
468#41—1221no—
469#41—1221no—
470C2×Q8—611no—
471C2×Q8—612no—
472C2×Q8—611no—
473C2×Q8—1221no—
474C2×Q8—1221no—
475C2×Q8—1221no—
476C2×Q8—1221no—
477C2×Q8—1221no—
478C2×D4—611no—
479C2×D4—612no—
480C2×D4—611no—
481C2×D4—1221no—
482C2×D4—1221no—
483C2×D4—1221no—
484C2×D4—1221no—
485C2×D4—1221no—
486C2×D4—1221no—
487C2×D4—1221no—
488C2×D4—2442no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—2442no—
493C22×C4C23×D6111yescharacteristic
494Q16—1221no—
495Q16—1221no—
496Q16—2442no—
497SD16—1222no—
498SD16—1222no—
499SD16—1222no—
500SD16—1222no—
501SD16—1221no—
502SD16—1221no—
503SD16—1221no—
504SD16—1221no—
505SD16—2442no—
506SD16—2442no—
507D8—1221no—
508D8—1221no—
509D8—2442no—
510M16—21no—
511M16—21no—
512M16—21no—
513M16—21no—
514M16—21no—
515M16—21no—
516M16—1221no—
517M16—1221no—
518M16—1221no—
519M16—1221no—
520M16—2442no—
521M16—2442no—
522C2×C8—21no—
523C2×C8—1221no—
524C2×C8—1221no—
525C2×C8—2442no—
526C2×C8—2442no—
527C2×C8C2×C4×D6112yes—
528C2×C8C2×C4×D6111yescharacteristic
529C2×C8C2×C4×D6111yescharacteristic
530C2×C8C2×D24412yes—
531C2×C8C2×D24411yescharacteristic
532C2×C8C2×D24411yescharacteristic
533C2×C8#1175822211yescharacteristic
534C2×C8#1175822211yescharacteristic
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4—2442no—
541C4⋊C4—2442no—
542C22⋊C4—21no—
543C22⋊C4—21no—
544C22⋊C4—21no—
545C22⋊C4—1221no—
546C22⋊C4—1221no—
547C22⋊C4—2442no—
548C22⋊C4—2442no—
549C22⋊C4—2442no—
550C42—21no—
551C42—1221no—
552C42—1221no—
553C42—2442no—
554C42C2×C4×D6112yes—
555C42C2×D24412yes—
556C23—1221no—
557C23—1222no—
558C23—1221no—
559C23—9761no—
560C23—2441no—
561C23—2441no—
562C23#1451982311yescharacteristic
563Q8—1222no—
564Q8—1221no—
565Q8—1221no—
566Q8—1221no—
567Q8—1222no—
568Q8—1221no—
569Q8—2441no—
570Q8—2441no—
571D4—1221no—
572D4—1221no—
573D4—1222no—
574D4—1221no—
575D4—1222no—
576D4—1221no—
577D4—9762no—
578D4—2441no—
579D4—2441no—
580D4—2441no—
581D4—2441no—
582D4—2441no—
583D4—2441no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—41no—
587C2×C4—1221no—
588C2×C4—1222no—
589C2×C4—1221no—
590C2×C4—9762no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4C22×C4×D6111yescharacteristic
600C2×C4C22×D24411yescharacteristic
601C2×C4C2×#1175822211yescharacteristic
602C2×C4#1451982311yescharacteristic
603C2×C4#1451982311yescharacteristic
604C2×C4#1451982311yescharacteristic
605C8—22no—
606C8—22no—
607C8—41no—
608C8—2441no—
609C8—2441no—
610C8C4×D24411yescharacteristic
611C8C4×D24411yescharacteristic
612C8C4×D24411yescharacteristic
613C8C4×D24411yescharacteristic
614C22—41no—
615C22—9761no—
616C22—4881no—
617C22—4882no—
618C22—4881no—
619C22—4882no—
620C22—2441no—
621C22—2441no—
622C22—2442no—
623C22#1684023611yescharacteristic
624C22#1684033111yescharacteristic
625C22#1684036711yescharacteristic
626C4—42no—
627C4—42no—
628C4—81no—
629C4—4881no—
630C4—4881no—
631C4—2441no—
632C4—2441no—
633C4—2442no—
634C4C2×C4×D24411yescharacteristic
635C4#1684023611yescharacteristic
636C4#1684033111yescharacteristic
637C4#1684036711yescharacteristic
638C2—21no—
639C2—21no—
640C2—21no—
641C2—4882no—
642C2—4881no—
643C2—4881no—
644C2#2144854411yescharacteristic
subgroupG/H
centerC2#21448544
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1747C2
socleC2×C61#2328
prime radical#21456295C2
cosocleC22×C4×C61C24
hypercenter#1747D61
hypocenterC61#22284
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1747

Complements

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

NH ≅ G/Ntype
2. #21454737C2semidirect
3. #21454680C2semidirect
4. #21446722C2semidirect
5. #21446717C2semidirect
6. #21446355C2semidirect
7. #21446350C2semidirect
8. #21446345C2semidirect
9. #21446340C2semidirect
10. #21446314C2semidirect
11. #21446307C2semidirect
12. #21446300C2semidirect
13. #21446293C2semidirect
14. #21445943C2semidirect
15. C61×#1747C2semidirect
26. #16840156C4semidirect
27. #16840151C4semidirect
28. #16840146C4semidirect
29. #16840135C4semidirect
71. C2×Dic244C2×C4semidirect
72. C2×Dic244C2×C4semidirect
80. C2×#11758190C2×C4semidirect
81. C2×#11758190C2×C4semidirect
82. C2×#11758190C2×C4semidirect
84. C2×D488C2×C4semidirect
93. #14519777C2×C4semidirect
96. #14519765C2×C4semidirect
133. #97×C61C2×C4semidirect
134. #97×C61D4semidirect
136. C4×C8×C61C2×C4semidirect
137. C4×C8×C61D4semidirect
330. #1747D61semidirect
381. C61#22284semidirect
456. #97C4×D61semidirect
457. #97D244semidirect
459. C4×C8C4×D61semidirect
460. C4×C8D244semidirect

Maximal subgroups

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

subgroupindex
#214562952
#214547372
#214546802
#214467222
#214467172
#214463552
#214463502
#214463452
#214463402
#214463142
#214463072
#214463002
#214462932
#214459432
C61×#17472
#2228461