Finite Group Databasebeta

← #29663103

Subgroups — #29663103

Order 15616.

746 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457126C211yesmaximal, characteristic, complemented
2#21456245C211yesmaximal, characteristic, complemented
3#21454531C211yesmaximal, characteristic, complemented
4#21445286C212yesmaximal, complemented
5#21445282C212yesmaximal, complemented
6#21445182C212yesmaximal, complemented
7#21445172C212yesmaximal, complemented
8C2×#16839514C211yesmaximal, characteristic, complemented
9#21444496C211yesmaximal, characteristic, complemented
10#21444494C211yesmaximal, characteristic, complemented
11C61×#3245C211yesmaximal, characteristic, complemented
12C2×#14519797—21no—
13C2×#14519797—21no—
14C2×#14519784—21no—
15#16840312C2211yescharacteristic
16#16840144—21no—
17#16840144—21no—
18#16840141C2211yescharacteristic
19#16840138C2211yescharacteristic
20#16840136C2211yescharacteristic
21#16840134C2211yescharacteristic
22#16840042C2212yes—
23#16840039—21no—
24#16839996—22no—
25#16839957C2212yes—
26#16839954—21no—
27#16839840—21no—
28#16839840C2212yes—
29D4×#92418—21no—
30D4×#92418C2212yes—
31#16839786C2211yescharacteristic
32C2×C4⋊#92418C2211yescharacteristic
33#16839729—21no—
34C2×C4×#92418C2211yescharacteristic
35#16839514C2212yes—
36#16839514C2212yes—
37#16839508C2212yes—
38#16839507C2212yes—
39#16839506C2212yes—
40#16839505C2212yes—
41C61×#532C2211yescharacteristic
42C61×#532C2211yescharacteristic
43C61×#474—21no—
44C61×#474C2212yes—
45C61×#466—22no—
46C61×#459—21no—
47C61×#459C2212yes—
48C2×C4⋊C8×C61C2211yescharacteristic
49C2×#11758222—21no—
50#14519797—44no—
51#14519797—44no—
52C2×#11758202—21no—
53C2×#11758202—21no—
54C2×#11758200—21no—
55C2×#11758200—21no—
56C2×#11758201—21no—
57C2×#11758201—21no—
58C2×#11758199—22no—
59C2×#11758199—21no—
60C2×#11758199—21no—
61#14519784—44no—
62C2×#11758194—21no—
63C2×#11758194—22no—
64C22×#92418—21no—
65C22×#92418—22no—
66C22×#92418D411yescharacteristic
67C22×#92418D411yescharacteristic
68#14519777C2311yescharacteristic
69#14519774C2311yescharacteristic
70#14519765—21no—
71#14519751—21no—
72#14519751—21no—
73C4×D244—21no—
74C4×D244—21no—
75#14519710—22no—
76C4⋊#92418—21no—
77C4⋊#92418—21no—
78C4⋊#92418C2312yes—
79C4⋊#92418C2312yes—
80#14519698—22no—
81#14519697—22no—
82C22⋊#92418—22no—
83C22⋊#92418—22no—
84C4×#92418D411yescharacteristic
85C4×#92418D411yescharacteristic
86C4×#92418C2311yescharacteristic
87C4×#92418C2311yescharacteristic
88C2×#41×C61—21no—
89C2×#41×C61—21no—
90C2×D8×C61—22no—
91C22×C8×C61—21no—
92C4⋊Q8×C61D411yescharacteristic
93C4⋊Q8×C61C2311yescharacteristic
94#127×C61D411yescharacteristic
95#127×C61C2311yescharacteristic
96#124×C61—21no—
97#124×C61—21no—
98#121×C61—21no—
99#121×C61—21no—
100#121×C61—21no—
101#121×C61D412yes—
102C4×D4×C61—21no—
103C4×D4×C61—21no—
104C4×D4×C61—21no—
105C4×D4×C61C2312yes—
106C2×C42×C61C2311yescharacteristic
107#107×C61—22no—
108C4⋊C8×C61—21no—
109C4⋊C8×C61C2312yes—
110#102×C61—22no—
111#102×C61—22no—
112#41×C61—42no—
113#41×C61—42no—
114#41×C61—42no—
115#41×C61—42no—
116C2×Q8×C61—21no—
117C2×Q8×C61—21no—
118C2×D4×C61—22no—
119C2×D4×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61—21no—
122C2×D4×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61—41no—
125C2×D4×C61C2×D412yes—
126C22×C4×C61—22no—
127C22×C4×C61—41no—
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61#4111yescharacteristic
131C2×C22⋊D61—41no—
132#11758222—42no—
133#11758222—42no—
134C2×D244—21no—
135C2×D244—21no—
136C2×D244—21no—
137C2×C4×D61—41no—
138C2×Dic122—21no—
139D8×C61—82no—
140C2×C8×C61—22no—
141C2×C8×C61—81no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61C2×D412yes—
146C22⋊C4×C61—22no—
147C22⋊C4×C61—22no—
148C22⋊C4×C61—41no—
149C22⋊C4×C61—41no—
150C42×C61C2×D411yescharacteristic
151C42×C61C2×D411yescharacteristic
152C42×C61#4111yescharacteristic
153C42×C61C2411yescharacteristic
154#11758202—42no—
155#11758202—42no—
156#11758201—42no—
157#11758201—42no—
158#11758200—42no—
159#11758200—42no—
160#11758199—42no—
161#11758199—42no—
162#11758199—82no—
163#11758198—41no—
164#11758198—41no—
165C4⋊Dic61—21no—
166C4⋊Dic61—21no—
167#11758194—42no—
168#11758194—44no—
169C2×#92418—22no—
170C2×#92418—22no—
171C2×#92418—22no—
172C2×#92418—44no—
173C2×#92418—81no—
174C2×#92418D814yescomplemented
175C2×#92418C2×D411yescharacteristic
176C2×#92418C2×D411yescharacteristic
177C2×#92418C2×D411yescharacteristic
178C2×#92418C2×D411yescharacteristic
179C23×C61—22no—
180C23×C61—41no—
181C23×C61#12111yescharacteristic
182C22×D61—41no—
183Q8×C61—42no—
184Q8×C61—41no—
185Q8×C61—41no—
186D4×C61—22no—
187D4×C61—22no—
188D4×C61—42no—
189D4×C61—44no—
190D4×C61—41no—
191D4×C61—41no—
192D4×C61—41no—
193D4×C61—41no—
194D4×C61—42no—
195D4×C61—81no—
196D4×C61—82no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—22no—
200C2×C4×C61—41no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61—41no—
204C2×C4×C61—82no—
205C2×C4×C61#12111yescharacteristic
206C2×C4×C61#12111yescharacteristic
207C2×C4×C61#12111yescharacteristic
208C2×C4×C61C2×D812yes—
209C2×C4×C61AGL(1,ℤ/8)12yes—
210C2×C4×C61C22×D411yescharacteristic
211C2×C4×C61C22×D411yescharacteristic
212C2×C4×C61C2×#4111yescharacteristic
213C22⋊D61—82no—
214C2×Dic61—41no—
215D244—41no—
216D244—41no—
217D244—41no—
218D244—41no—
219D244—81no—
220C4×D61—82no—
221Dic122—41no—
222Dic122—41no—
223C8×C61—42no—
224#92418—21no—
225#92418—21no—
226#92418—42no—
227#92418—42no—
228#92418C2×D812yes—
229#92418C2×D812yes—
230#19190—611nomaximal, Sylow
231C22×C61—21no—
232C22×C61—21no—
233C22×C61—42no—
234C22×C61—42no—
235C22×C61—42no—
236C22×C61—81no—
237C22×C61—81no—
238C22×C61—81no—
239C22×C61#45912yes—
240C22×C61C2×#12111yescharacteristic
241C2×D61—81no—
242C2×D61—81no—
243C2×D61—81no—
244C4×C61—21no—
245C4×C61—21no—
246C4×C61—21no—
247C4×C61—42no—
248C4×C61—81no—
249C4×C61#45912yes—
250C4×C61C22×D811yescharacteristic
251C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
252C4×C61#57611yescharacteristic
253C4×C61#57811yescharacteristic
254Dic61—81no—
255#3245—611no—
256#3245D6111yescharacteristic, complemented
257#3147—611no—
258#2931—611no—
259#1685—612no—
260#1685—612no—
261#1669—612no—
262#1667—612no—
263C2×#335—611no—
264#1547—611no—
265#1547—611no—
266C2×C61—22no—
267C2×C61—82no—
268C2×C61—161no—
269C2×C61C2×#45911yescharacteristic
270C2×C61#306111yescharacteristic
271C2×C61#306711yescharacteristic
272D61—161no—
273C2×#135—1221no—
274C2×#135—1221no—
275C2×#131—1221no—
276#532—611no—
277#532C2×D6111yescharacteristic
278#532C2×D6111yescharacteristic
279#495—1221no—
280#495—1221no—
281#494—611no—
282#493—611no—
283#492—611no—
284#492—611no—
285#474—21no—
286#474—612no—
287#474—1221no—
288#474C2×D6112yes—
289#466—22no—
290#466—1222no—
291#459—21no—
292#459—612no—
293#459—1221no—
294#459C2×D6112yes—
295#436—612no—
296#436—1221no—
297C8×D4—612no—
298C8×D4—1221no—
299#423—611no—
300C2×C4⋊C8—611no—
301C2×C4⋊C8C2×D6111yescharacteristic
302#408—1221no—
303C2×C4×C8—611no—
304#335—612no—
305#335—612no—
306#332—612no—
307#332—612no—
308#331—612no—
309#331—612no—
310C61#1919011yescharacteristic, complemented, Sylow
311C2×#41—21no—
312C2×#41—21no—
313C2×#41—1221no—
314#135—2444no—
315#135—2444no—
316C2×Q16—1221no—
317C2×Q16—1221no—
318C2×SD16—1221no—
319C2×SD16—1221no—
320C2×SD16—1221no—
321C2×SD16—1221no—
322C2×D8—22no—
323C2×D8—1221no—
324C2×D8—1221no—
325C2×D8—1222no—
326#131—2444no—
327C2×M16—1222no—
328C2×M16—1221no—
329C22×C8—21no—
330C22×C8—611no—
331C22×C8—611no—
332C22×C8—1222no—
333C22×C8—1221no—
334C4⋊Q8—611no—
335C4⋊Q8C22⋊D6111yescharacteristic
336C4⋊Q8C22×D6111yescharacteristic
337#127—611no—
338#127C22⋊D6111yescharacteristic
339#127C22×D6111yescharacteristic
340#124—21no—
341#124—21no—
342#124—1221no—
343#121—21no—
344#121—21no—
345#121—21no—
346#121—1221no—
347#121—1221no—
348#121C22⋊D6112yes—
349C4×D4—21no—
350C4×D4—21no—
351C4×D4—21no—
352C4×D4—1221no—
353C4×D4—1221no—
354C4×D4C22×D6112yes—
355C2×C42C22×D6111yescharacteristic
356#107—22no—
357#107—1222no—
358C4⋊C8—21no—
359C4⋊C8—612no—
360C4⋊C8—612no—
361C4⋊C8—1221no—
362C4⋊C8—1221no—
363C4⋊C8C22×D6112yes—
364#102—22no—
365#102—22no—
366#102—1222no—
367#102—1222no—
368C22⋊C8—1222no—
369C22⋊C8—1222no—
370C4×C8—611no—
371C4×C8—611no—
372C4×C8—611no—
373C4×C8—611no—
374#41—42no—
375#41—42no—
376#41—42no—
377#41—42no—
378#41—2442no—
379#41—2442no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—1221no—
383C2×D4—22no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—41no—
390C2×D4—1221no—
391C2×D4—1221no—
392C2×D4—1221no—
393C2×D4—2441no—
394C2×D4C2×C22⋊D6112yes—
395C22×C4—22no—
396C22×C4—41no—
397C22×C4—2441no—
398C22×C4D4×D6111yescharacteristic
399C22×C4#1175822411yescharacteristic
400C22×C4C2×C22⋊D6111yescharacteristic
401Q16—2442no—
402Q16—2442no—
403SD16—2442no—
404SD16—2442no—
405SD16—2442no—
406SD16—2442no—
407D8—82no—
408D8—4882no—
409D8—2442no—
410D8—2442no—
411M16—2444no—
412M16—2442no—
413C2×C8—22no—
414C2×C8—81no—
415C2×C8—611no—
416C2×C8—611no—
417C2×C8—614no—
418C2×C8—611no—
419C2×C8—611no—
420C2×C8—1222no—
421C2×C8—1222no—
422C2×C8—1222no—
423C2×C8—4881no—
424C2×C8—2444no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—1221no—
429C4⋊C4—1221no—
430C4⋊C4C2×C22⋊D6112yes—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—2441no—
436C22⋊C4—2441no—
437C42D4×D6111yescharacteristic
438C42#1175822411yescharacteristic
439C42C2×C22⋊D6111yescharacteristic
440C42C23×D6111yescharacteristic
441C23—22no—
442C23—41no—
443C23—2441no—
444C23#1451974911yescharacteristic
445Q8—42no—
446Q8—41no—
447Q8—41no—
448Q8—2441no—
449Q8—2441no—
450D4—22no—
451D4—22no—
452D4—42no—
453D4—44no—
454D4—42no—
455D4—41no—
456D4—41no—
457D4—41no—
458D4—41no—
459D4—81no—
460D4—82no—
461D4—4881no—
462D4—4882no—
463D4—2441no—
464D4—2441no—
465D4—2441no—
466D4—2441no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—82no—
475C2×C4—4882no—
476C2×C4—2441no—
477C2×C4#1451974911yescharacteristic
478C2×C4#1451974911yescharacteristic
479C2×C4#1451974911yescharacteristic
480C2×C4D8×D6112yes—
481C2×C4#1451980112yes—
482C2×C4C2×D4×D6111yescharacteristic
483C2×C4C22×C22⋊D6111yescharacteristic
484C2×C4C2×#1175822411yescharacteristic
485C8—42no—
486C8—612no—
487C8—612no—
488C8—1221no—
489C8—1221no—
490C8—2442no—
491C8—2442no—
492C22—21no—
493C22—21no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—81no—
498C22—81no—
499C22—81no—
500C22—4881no—
501C22—4881no—
502C22—4881no—
503C22#1683995512yes—
504C22C2×#1451974911yescharacteristic
505C4—21no—
506C4—21no—
507C4—21no—
508C4—42no—
509C4—81no—
510C4—4881no—
511C4#1683995512yes—
512C4C2×D8×D6111yescharacteristic
513C4C2×#1451980111yescharacteristic
514C4#1684056511yescharacteristic
515C4#1684057711yescharacteristic
516C2—22no—
517C2—82no—
518C2—161no—
519C2—9761no—
520C2C2×#1683995511yescharacteristic
521C2#2145548611yescharacteristic
522C2#2145553611yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3245C2
socleC22×C61C2×#121
hypercenter#3245D61
hypocenterC61#19190
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3245

Complements

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

NH ≅ G/Ntype
1. #21457126C2semidirect
2. #21456245C2semidirect
3. #21454531C2semidirect
4. #21445286C2semidirect
5. #21445282C2semidirect
6. #21445182C2semidirect
7. #21445172C2semidirect
8. C2×#16839514C2semidirect
9. #21444496C2semidirect
10. #21444494C2semidirect
11. C61×#3245C2semidirect
174. C2×#92418D8semidirect
256. #3245D61semidirect
310. C61#19190semidirect

Maximal subgroups

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

subgroupindex
#214571262
#214562452
#214545312
#214452862
#214452822
#214451822
#214451722
C2×#168395142
#214444962
#214444942
C61×#32452
#1919061