Finite Group Databasebeta

← #29837008

Subgroups — #29837008

Order 15616.

580 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457562C211yesmaximal, characteristic, complemented
2#21448058C211yesmaximal, characteristic, complemented
3#21448021C211yesmaximal, characteristic, complemented
4#21447999C211yesmaximal, characteristic, complemented
5#21447968C211yesmaximal, characteristic, complemented
6#21447930C211yesmaximal, characteristic, complemented
7#21447903C211yesmaximal, characteristic, complemented
8#21447874C211yesmaximal, characteristic, complemented
9#21447740C211yesmaximal, characteristic, complemented
10#21447735C211yesmaximal, characteristic, complemented
11#21447683C211yesmaximal, characteristic, complemented
12#21447645C211yesmaximal, characteristic, complemented
13#21447529C211yesmaximal, characteristic, complemented
14C2×#16839565C211yesmaximal, characteristic, complemented
15C61×#3334C211yesmaximal, characteristic, complemented
16C2×#14519799C2211yescharacteristic
17C22×Dic244C2211yescharacteristic
18#16840421—21no—
19C2×#14519721C2211yescharacteristic
20C2×#14519721C2211yescharacteristic
21C2×#14519718C2211yescharacteristic
22C2×#14519717C2211yescharacteristic
23C22×#11758185C2211yescharacteristic
24#16840146C2211yescharacteristic
25#16840003C2211yescharacteristic
26#16839998C2211yescharacteristic
27#16839985—21no—
28#16839926—21no—
29#16839903—21no—
30C4×Dic244C2211yescharacteristic
31#16839587C2211yescharacteristic
32#16839587C2211yescharacteristic
33#16839587C2211yescharacteristic
34C4⋊#11758185C2211yescharacteristic
35#16839565C2211yescharacteristic
36#16839565C2211yescharacteristic
37#16839565C2211yescharacteristic
38#16839565C2211yescharacteristic
39#16839565C2211yescharacteristic
40#16839565C2211yescharacteristic
41#16839563C2211yescharacteristic
42#16839562C2211yescharacteristic
43#16839562C2211yescharacteristic
44#16839562C2211yescharacteristic
45C22⋊#11758185C2211yescharacteristic
46C22⋊#11758185C2211yescharacteristic
47#16839538C2211yescharacteristic
48C61×#547—21no—
49C61×#500C2211yescharacteristic
50C61×#480—21no—
51C61×#480—21no—
52C61×#474—21no—
53C61×#466C2211yescharacteristic
54C61×#466C2211yescharacteristic
55C4×D8×C61C2211yescharacteristic
56C8×D4×C61C2211yescharacteristic
57C2×#107×C61C2211yescharacteristic
58C2×#107×C61C2211yescharacteristic
59C2×#11758222—21no—
60C22×Dic122—21no—
61#14519799—22no—
62#14519799—41no—
63C2×Dic244—22no—
64C2×Dic244—21no—
65C2×Dic244—41no—
66C2×Dic244C2311yescharacteristic
67C2×Dic244C2311yescharacteristic
68C2×#11758190—21no—
69C2×#11758190—21no—
70C2×D488C2311yescharacteristic
71#14519765—21no—
72#14519763—41no—
73#14519756—21no—
74#14519755—41no—
75#14519751—21no—
76#14519749—41no—
77C4×Dic122—21no—
78#14519737—41no—
79#14519721—22no—
80#14519721—41no—
81#14519718—41no—
82#14519717—41no—
83C2×#11758185—41no—
84C2×#11758185C2311yescharacteristic
85C2×#11758185C2311yescharacteristic
86C2×#11758185C2311yescharacteristic
87C2×#11758185C2311yescharacteristic
88#14519711C2311yescharacteristic
89#14519702—21no—
90#14519702—21no—
91#14519699—21no—
92C2×#41×C61—41no—
93C2×D8×C61C2311yescharacteristic
94C22×C8×C61C2311yescharacteristic
95C22×C8×C61C2311yescharacteristic
96C4⋊Q8×C61—21no—
97#123×C61—41no—
98#123×C61—41no—
99C22⋊Q8×C61—41no—
100#121×C61—21no—
101#121×C61—21no—
102C4×D4×C61—21no—
103C4×D4×C61D411yescharacteristic
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61—21no—
106#107×C61—21no—
107#107×C61—21no—
108#107×C61—21no—
109#107×C61C2311yescharacteristic
110#107×C61C2311yescharacteristic
111#107×C61C2311yescharacteristic
112C4⋊C8×C61D411yescharacteristic
113#102×C61—21no—
114#102×C61—21no—
115#102×C61—21no—
116C22⋊C8×C61D411yescharacteristic
117C22⋊C8×C61D411yescharacteristic
118C4×C8×C61C2311yescharacteristic
119#41×C61—84no—
120C2×Q8×C61—41no—
121C2×D4×C61—21no—
122C2×D4×C61—41no—
123C2×D4×C61—41no—
124C2×D4×C61D811yescharacteristic
125C22×C4×C61—41no—
126C22×C4×C61—41no—
127C22×C4×C61—41no—
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C2×C22⋊D61—41no—
131C2×C22⋊D61—41no—
132C22×Dic61—41no—
133#11758222—42no—
134#11758222—81no—
135C2×D244—21no—
136C2×C4×D61—41no—
137C2×Dic122—21no—
138C2×Dic122—21no—
139C2×Dic122—42no—
140C2×Dic122—41no—
141C2×Dic122—81no—
142D8×C61—21no—
143D8×C61—21no—
144C2×C8×C61—21no—
145C2×C8×C61—21no—
146C2×C8×C61—21no—
147C2×C8×C61—21no—
148C2×C8×C61C2×D411yescharacteristic
149C2×C8×C61C2×D411yescharacteristic
150C2×C8×C61C2×D411yescharacteristic
151C2×C8×C61C2411yescharacteristic
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—41no—
156C4⋊C4×C61—41no—
157C4⋊C4×C61—41no—
158C4⋊C4×C61D811yescharacteristic
159C22⋊C4×C61—41no—
160C22⋊C4×C61—41no—
161C22⋊C4×C61—41no—
162C22⋊C4×C61D811yescharacteristic
163C22⋊C4×C61D811yescharacteristic
164C42×C61C2×D411yescharacteristic
165C22⋊Dic61—41no—
166#11758198—41no—
167#11758198—41no—
168#11758198—41no—
169C4⋊Dic61—21no—
170#11758196—41no—
171#11758196—41no—
172C4×Dic61—41no—
173Dic244—21no—
174Dic244—21no—
175Dic244—21no—
176Dic244—42no—
177Dic244#4112yescomplemented
178D488—21no—
179D488—21no—
180#11758190—42no—
181#11758190—81no—
182#11758185—21no—
183#11758185—21no—
184#11758185—21no—
185#11758185—21no—
186C23×C61—41no—
187C23×C61C2×D811yescharacteristic
188C23×C61C2×D811yescharacteristic
189C22×D61—41no—
190Q8×C61—82no—
191D4×C61—22no—
192D4×C61—41no—
193D4×C61—41no—
194D4×C61—82no—
195D4×C61—82no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—82no—
204C2×C4×C61—82no—
205C2×C4×C61—82no—
206C2×C4×C61C2×D811yescharacteristic
207C2×C4×C61C2×D811yescharacteristic
208C2×C4×C61C2×D811yescharacteristic
209C2×C4×C61C2×D811yescharacteristic
210C2×C4×C61C22×D411yescharacteristic
211C22⋊D61—82no—
212C22⋊D61—82no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215C2×Dic61—41no—
216C2×Dic61—82no—
217D244—41no—
218D244—41no—
219C4×D61—82no—
220Dic122—22no—
221Dic122—41no—
222Dic122—41no—
223Dic122—41no—
224Dic122—82no—
225C8×C61—21no—
226C8×C61—21no—
227C8×C61—21no—
228C8×C61C2×#4111yescharacteristic
229C8×C61H(5,𝔽2)11yescharacteristic
230#33875—611nomaximal, Sylow
231C22×C61—21no—
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—21no—
235C22×C61—21no—
236C22×C61—81no—
237C22×C61—81no—
238C22×C61—81no—
239C22×C61#50912yes—
240C22×C61C22×D811yescharacteristic
241C2×D61—81no—
242C2×D61—81no—
243C2×D61—81no—
244C4×C61—21no—
245C4×C61—21no—
246C4×C61—21no—
247C4×C61—21no—
248C4×C61—81no—
249C4×C61—81no—
250C4×C61—81no—
251C4×C61#54611yescharacteristic
252C4×C61#57711yescharacteristic
253Dic61—42no—
254Dic61—81no—
255Dic61—81no—
256#3334D6111yescharacteristic, complemented
257#3285—611no—
258#2243—611no—
259#2236—611no—
260#2232—611no—
261#2227—611no—
262#2220—611no—
263#2215—611no—
264#2210—611no—
265#2186—611no—
266#2185—611no—
267#2175—611no—
268#2168—611no—
269#2139—611no—
270C2×#358—611no—
271C2×C61—22no—
272C2×C61—41no—
273C2×C61—161no—
274C2×C61#329411yescharacteristic
275C2×C61C2×#50911yescharacteristic
276C2×C61#341711yescharacteristic
277D61—161no—
278C2×#135—611no—
279C22×Q16—611no—
280#548—1221no—
281#547—21no—
282C2×SD32—611no—
283C2×SD32—611no—
284C2×D16—611no—
285C2×M32—611no—
286C22×C16—611no—
287#500C2×D6111yescharacteristic
288#495—611no—
289#480—21no—
290#480—21no—
291#474—21no—
292#467—611no—
293#466—611no—
294#466C2×D6111yescharacteristic
295#466C2×D6111yescharacteristic
296#464—1221no—
297#452—1221no—
298#448—1221no—
299C4×Q16—611no—
300C4×D8C2×D6111yescharacteristic
301C8×D4C2×D6111yescharacteristic
302C2×#107C2×D6111yescharacteristic
303C2×#107C2×D6111yescharacteristic
304#366—611no—
305#366—611no—
306#366—611no—
307C4⋊C16—611no—
308#358—611no—
309#358—611no—
310#358—611no—
311#358—611no—
312#358—611no—
313#358—611no—
314#357—611no—
315#357—611no—
316#357—611no—
317#357—611no—
318C22⋊C16—611no—
319C22⋊C16—611no—
320#346—611no—
321C61#3387511yescharacteristic, complemented, Sylow
322C2×#41—41no—
323C2×#41—1221no—
324C22×Q8—1221no—
325#135—1222no—
326#135—2441no—
327C2×Q16—611no—
328C2×Q16—611no—
329C2×Q16—1222no—
330C2×Q16—1221no—
331C2×Q16—2441no—
332C2×SD16—1221no—
333C2×SD16—1221no—
334C2×D8—611no—
335C2×D8C22×D6111yescharacteristic
336C22×C8C22×D6111yescharacteristic
337C22×C8C22×D6111yescharacteristic
338C4⋊Q8—21no—
339#124—1221no—
340#124—2441no—
341#123—41no—
342#123—41no—
343C22⋊Q8—41no—
344C22⋊Q8—1221no—
345C22⋊Q8—2441no—
346#121—21no—
347#121—21no—
348#121—1221no—
349#121—2441no—
350C4×Q8—1221no—
351C4×D4—21no—
352C4×D4—2441no—
353C4×D4C22⋊D6111yescharacteristic
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356SD32—1222no—
357SD32—2441no—
358D16—2441no—
359M32—2441no—
360C2×C16—611no—
361C2×C16—611no—
362C2×C16—611no—
363C2×C16—611no—
364C2×C16—2441no—
365#107—21no—
366#107—21no—
367#107—21no—
368#107—611no—
369#107C22×D6111yescharacteristic
370#107C22×D6111yescharacteristic
371#107C22×D6111yescharacteristic
372C4⋊C8C22⋊D6111yescharacteristic
373#103—1221no—
374#103—1221no—
375#102—21no—
376#102—21no—
377#102—21no—
378#102—1221no—
379C22⋊C8C22⋊D6111yescharacteristic
380C22⋊C8C22⋊D6111yescharacteristic
381C4×C8C22×D6111yescharacteristic
382#41—84no—
383#41—4881no—
384#41—2442no—
385C2×Q8—41no—
386C2×Q8—1221no—
387C2×Q8—1221no—
388C2×Q8—4881no—
389C2×Q8—2442no—
390C2×Q8—2441no—
391C2×D4—21no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—1221no—
395C2×D4—2441no—
396C2×D4—2441no—
397C2×D4#1175819911yescharacteristic
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—2441no—
402C22×C4—2441no—
403C22×C4C2×C22⋊D6111yescharacteristic
404C22×C4C2×C22⋊D6111yescharacteristic
405Q16—612no—
406Q16—1221no—
407Q16—1221no—
408Q16—1221no—
409Q16—2442no—
410SD16—4881no—
411SD16—2442no—
412D8—21no—
413D8—21no—
414D8—1221no—
415D8—1221no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8C2×C22⋊D6111yescharacteristic
421C2×C8C2×C22⋊D6111yescharacteristic
422C2×C8C2×C22⋊D6111yescharacteristic
423C2×C8C23×D6111yescharacteristic
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—1221no—
431C4⋊C4—2441no—
432C4⋊C4—2441no—
433C4⋊C4#1175819911yescharacteristic
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—2441no—
438C22⋊C4—2441no—
439C22⋊C4—2441no—
440C22⋊C4—2441no—
441C22⋊C4#1175819911yescharacteristic
442C22⋊C4#1175819911yescharacteristic
443C42—2441no—
444C42C2×C22⋊D6111yescharacteristic
445C16—1221no—
446C16—1221no—
447C16—1221no—
448C16—1221no—
449C23—41no—
450C23—2441no—
451C23C2×#1175819911yescharacteristic
452C23C2×#1175819911yescharacteristic
453Q8—82no—
454Q8—1222no—
455Q8—4882no—
456Q8—2441no—
457Q8—2441no—
458Q8—2441no—
459D4—22no—
460D4—41no—
461D4—41no—
462D4—82no—
463D4—82no—
464D4—4882no—
465D4—4882no—
466D4—2441no—
467D4—2441no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—4882no—
479C2×C4—4882no—
480C2×C4—2441no—
481C2×C4—2441no—
482C2×C4—2441no—
483C2×C4C2×#1175819911yescharacteristic
484C2×C4C2×#1175819911yescharacteristic
485C2×C4C2×#1175819911yescharacteristic
486C2×C4C2×#1175819911yescharacteristic
487C2×C4C22×C22⋊D6111yescharacteristic
488C8—21no—
489C8—21no—
490C8—21no—
491C8#41×D6111yescharacteristic
492C8#1451982311yescharacteristic
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—4881no—
502C22—4881no—
503C22—4881no—
504C22#1684019812yes—
505C22C22×#1175819911yescharacteristic
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—81no—
511C4—81no—
512C4—81no—
513C4—4881no—
514C4—4881no—
515C4—2442no—
516C4#1684040011yescharacteristic
517C4#1684057011yescharacteristic
518C2—22no—
519C2—41no—
520C2—161no—
521C2—9761no—
522C2#2145762611yescharacteristic
523C2C2×#1684019811yescharacteristic
524C2#2145897011yescharacteristic
subgroupG/H
centerC22C22×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3334C2
socleC22×C61C22×D8
hypercenter#3334D61
hypocenterC61#33875
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3334

Complements

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

NH ≅ G/Ntype
1. #21457562C2semidirect
2. #21448058C2semidirect
3. #21448021C2semidirect
4. #21447999C2semidirect
5. #21447968C2semidirect
6. #21447930C2semidirect
7. #21447903C2semidirect
8. #21447874C2semidirect
9. #21447740C2semidirect
10. #21447735C2semidirect
11. #21447683C2semidirect
12. #21447645C2semidirect
13. #21447529C2semidirect
14. C2×#16839565C2semidirect
15. C61×#3334C2semidirect
177. Dic244#41semidirect
256. #3334D61semidirect
321. C61#33875semidirect

Maximal subgroups

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

subgroupindex
#214575622
#214480582
#214480212
#214479992
#214479682
#214479302
#214479032
#214478742
#214477402
#214477352
#214476832
#214476452
#214475292
C2×#168395652
C61×#33342
#3387561