Finite Group Databasebeta

← #29836927

Subgroups — #29836927

Order 15616.

558 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458222C211yesmaximal, characteristic, complemented
2#21457562C211yesmaximal, characteristic, complemented
3#21448051C211yesmaximal, characteristic, complemented
4#21448028C211yesmaximal, characteristic, complemented
5#21448005C211yesmaximal, characteristic, complemented
6#21447984C211yesmaximal, characteristic, complemented
7#21447915C211yesmaximal, characteristic, complemented
8#21447905C211yesmaximal, characteristic, complemented
9#21447875C211yesmaximal, characteristic, complemented
10#21447744C211yesmaximal, characteristic, complemented
11#21447738C211yesmaximal, characteristic, complemented
12C4×#14519722C211yesmaximal, characteristic, complemented
13#21447527C211yesmaximal, characteristic, complemented
14C2×#16839567C211yesmaximal, characteristic, complemented
15D4×C16×C61C211yesmaximal, characteristic, complemented
16C2×#14519799C2211yescharacteristic
17C22×Dic244C2211yescharacteristic
18#16840458—21no—
19#16840421—21no—
20C2×#14519722C2211yescharacteristic
21C2×#14519722C2211yescharacteristic
22C2×D976C2211yescharacteristic
23#16840164C2211yescharacteristic
24#16840146C2211yescharacteristic
25#16840087—21no—
26#16840081—21no—
27#16840063—21no—
28#16840003C2211yescharacteristic
29#16839998C2211yescharacteristic
30#16839998C2211yescharacteristic
31#16839998C2211yescharacteristic
32#16839985—21no—
33#16839926—21no—
34#16839903—21no—
35C4×Dic244C2211yescharacteristic
36C4×D488C2211yescharacteristic
37#16839800C2211yescharacteristic
38C2×#14519711C2211yescharacteristic
39#16839590C2211yescharacteristic
40#16839590C2211yescharacteristic
41#16839588C2211yescharacteristic
42#16839567C2211yescharacteristic
43#16839567C2211yescharacteristic
44#16839567C2211yescharacteristic
45#16839567C2211yescharacteristic
46#16839567C2211yescharacteristic
47#16839567C2211yescharacteristic
48#16839564C2211yescharacteristic
49#16839564C2211yescharacteristic
50#16839564C2211yescharacteristic
51#16839564C2211yescharacteristic
52C22×C16×C61C2211yescharacteristic
53C22×C16×C61C2211yescharacteristic
54C8×D4×C61C2211yescharacteristic
55C61×C4⋊C16C2211yescharacteristic
56C61×C22⋊C16C2211yescharacteristic
57C61×C22⋊C16C2211yescharacteristic
58C4×C16×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
71C2×D488C2311yescharacteristic
72#14519772—41no—
73#14519769—21no—
74#14519765—21no—
75#14519763—41no—
76#14519759—41no—
77#14519756—21no—
78#14519755—41no—
79#14519751—21no—
80#14519751—21no—
81#14519751—21no—
82#14519749—41no—
83#14519749—41no—
84C4×Dic122—21no—
85C4×D244—21no—
86#14519737—41no—
87#14519737—41no—
88#14519736—21no—
89C2×C4⋊Dic61—21no—
90#14519722—22no—
91#14519722—41no—
92D976—41no—
93#14519711—21no—
94#14519711C2311yescharacteristic
95#14519711C2311yescharacteristic
96#14519711C2311yescharacteristic
97#14519711C2311yescharacteristic
98#14519709—21no—
99#14519709—21no—
100#14519702—21no—
101#14519702—21no—
102#14519699—21no—
103#14519699—21no—
104#14519699—21no—
105#14519699—21no—
106C22×C8×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108C4×D4×C61D411yescharacteristic
109C2×C16×C61—41no—
110C2×C16×C61—41no—
111C2×C16×C61C2311yescharacteristic
112C2×C16×C61C2311yescharacteristic
113C2×C16×C61C2311yescharacteristic
114C2×C16×C61C2311yescharacteristic
115C4⋊C8×C61D411yescharacteristic
116C22⋊C8×C61D411yescharacteristic
117C22⋊C8×C61D411yescharacteristic
118C4×C8×C61C2311yescharacteristic
119C2×D4×C61D811yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C2×C22⋊D61—41no—
123C2×C22⋊D61—41no—
124C2×C22⋊D61—41no—
125C2×C22⋊D61—41no—
126C22×Dic61—41no—
127C22×Dic61—41no—
128#11758222—42no—
129#11758222—81no—
130C2×D244—21no—
131C2×D244—21no—
132C2×C4×D61—41no—
133C2×C4×D61—41no—
134C2×Dic122—21no—
135C2×Dic122—21no—
136C2×Dic122—42no—
137C2×Dic122—41no—
138C2×Dic122—81no—
139C2×C8×C61—21no—
140C2×C8×C61—21no—
141C2×C8×C61—21no—
142C2×C8×C61—21no—
143C2×C8×C61C2×D411yescharacteristic
144C2×C8×C61C2×D411yescharacteristic
145C2×C8×C61C2×D411yescharacteristic
146C2×C8×C61C2411yescharacteristic
147C4⋊C4×C61D811yescharacteristic
148C22⋊C4×C61D811yescharacteristic
149C22⋊C4×C61D811yescharacteristic
150C42×C61C2×D411yescharacteristic
151C22⋊Dic61—41no—
152C22⋊Dic61—41no—
153#11758198—41no—
154#11758198—41no—
155#11758198—41no—
156#11758198—41no—
157#11758198—41no—
158#11758198—41no—
159C4⋊Dic61—21no—
160C4⋊Dic61—21no—
161C4⋊Dic61—21no—
162C4⋊Dic61—21no—
163C4⋊Dic61—41no—
164#11758196—41no—
165#11758196—41no—
166#11758196—41no—
167#11758196—41no—
168C4×Dic61—41no—
169C4×Dic61—41no—
170Dic244—21no—
171Dic244—21no—
172Dic244—21no—
173Dic244—42no—
174Dic244#4112yes—
175D488—21no—
176D488—21no—
177D488—21no—
178D488—21no—
179#11758190—42no—
180#11758190—81no—
181C16×C61—21no—
182C16×C61—21no—
183C16×C61—21no—
184C16×C61—21no—
185C23×C61C2×D811yescharacteristic
186C23×C61C2×D811yescharacteristic
187C22×D61—41no—
188C22×D61—41no—
189D4×C61—22no—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61C2×D811yescharacteristic
195C2×C4×C61C2×D811yescharacteristic
196C2×C4×C61C2×D811yescharacteristic
197C2×C4×C61C2×D811yescharacteristic
198C2×C4×C61C22×D411yescharacteristic
199C22⋊D61—82no—
200C22⋊D61—82no—
201C22⋊D61—82no—
202C22⋊D61—161no—
203C2×Dic61—41no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—82no—
210C2×Dic61—161no—
211D244—41no—
212D244—41no—
213D244—41no—
214D244—41no—
215C4×D61—82no—
216C4×D61—161no—
217Dic122—22no—
218Dic122—41no—
219Dic122—41no—
220Dic122—41no—
221Dic122—82no—
222C8×C61—21no—
223C8×C61—21no—
224C8×C61—21no—
225C8×C61C2×#4111yescharacteristic
226C8×C61H(5,𝔽2)11yescharacteristic
227#33868—611nomaximal, Sylow
228C22×C61—21no—
229C22×C61—21no—
230C22×C61—21no—
231C22×C61—21no—
232C22×C61—21no—
233C22×C61#50812yes—
234C22×C61C22×D811yescharacteristic
235C2×D61—81no—
236C2×D61—81no—
237C2×D61—81no—
238C2×D61—81no—
239C2×D61—81no—
240C2×D61—81no—
241C4×C61—21no—
242C4×C61—21no—
243C4×C61—21no—
244C4×C61—21no—
245C4×C61#54611yescharacteristic
246C4×C61#57711yescharacteristic
247Dic61—42no—
248Dic61—81no—
249Dic61—81no—
250Dic61—81no—
251Dic61—81no—
252Dic61—81no—
253#3339—611no—
254#3285—611no—
255#2242—611no—
256#2237—611no—
257#2233—611no—
258#2229—611no—
259#2217—611no—
260#2215—611no—
261#2210—611no—
262#2186—611no—
263#2185—611no—
264C4×SD32—611no—
265D4×C16D6111yescharacteristic, complemented
266#2138—611no—
267C2×#358—611no—
268C2×C61—22no—
269C2×C61—41no—
270C2×C61#329411yescharacteristic
271C2×C61C2×#50811yescharacteristic
272C2×C61#341611yescharacteristic
273D61—161no—
274D61—161no—
275C2×#135—611no—
276C22×Q16—611no—
277#553—1221no—
278#548—1221no—
279C2×SD32—611no—
280C2×SD32—611no—
281C2×D16—611no—
282C22×C16C2×D6111yescharacteristic
283C22×C16C2×D6111yescharacteristic
284#499—611no—
285#495—611no—
286#482—1221no—
287#481—1221no—
288#478—1221no—
289#467—611no—
290#466—611no—
291#466—611no—
292#466—611no—
293#464—1221no—
294#452—1221no—
295#448—1221no—
296C4×Q16—611no—
297C4×D8—611no—
298C8×D4C2×D6111yescharacteristic
299#427—611no—
300C2×#107—611no—
301#367—611no—
302#367—611no—
303#366—611no—
304C4⋊C16C2×D6111yescharacteristic
305#358—611no—
306#358—611no—
307#358—611no—
308#358—611no—
309#358—611no—
310#358—611no—
311#357—611no—
312#357—611no—
313#357—611no—
314#357—611no—
315C22⋊C16C2×D6111yescharacteristic
316C22⋊C16C2×D6111yescharacteristic
317C4×C16C2×D6111yescharacteristic
318C61#3386811yescharacteristic, complemented, Sylow
319C2×#41—1221no—
320C22×Q8—1221no—
321#135—1222no—
322#135—2441no—
323C2×Q16—611no—
324C2×Q16—611no—
325C2×Q16—1222no—
326C2×Q16—1221no—
327C2×Q16—2441no—
328C2×SD16—1221no—
329C2×SD16—1221no—
330C2×D8—611no—
331C2×D8—611no—
332C22×C8C22×D6111yescharacteristic
333C22×C8C22×D6111yescharacteristic
334#126—2441no—
335#125—1221no—
336#124—1221no—
337#124—2441no—
338#123—2441no—
339C22⋊Q8—1221no—
340C22⋊Q8—2441no—
341#121—1221no—
342#121—1221no—
343#121—1221no—
344#121—2441no—
345#121—2441no—
346C4×Q8—1221no—
347C4×D4—1221no—
348C4×D4—2441no—
349C4×D4—2441no—
350C4×D4D24411yescharacteristic
351#117—1221no—
352C2×C4⋊C4—1221no—
353SD32—1222no—
354SD32—2441no—
355D16—2441no—
356C2×C16—41no—
357C2×C16—41no—
358C2×C16C22×D6111yescharacteristic
359C2×C16C22×D6111yescharacteristic
360C2×C16C22×D6111yescharacteristic
361C2×C16C22×D6111yescharacteristic
362#107—611no—
363#107—611no—
364#107—611no—
365#107—611no—
366#107—1221no—
367#106—1221no—
368#106—1221no—
369C4⋊C8D24411yescharacteristic
370#103—1221no—
371#103—1221no—
372#102—1221no—
373#102—1221no—
374#102—1221no—
375#102—1221no—
376C22⋊C8D24411yescharacteristic
377C22⋊C8D24411yescharacteristic
378C4×C8C22×D6111yescharacteristic
379#41—4881no—
380#41—2442no—
381C2×Q8—1221no—
382C2×Q8—1221no—
383C2×Q8—4881no—
384C2×Q8—2442no—
385C2×Q8—2441no—
386C2×D4—1221no—
387C2×D4—1221no—
388C2×D4—2441no—
389C2×D4—2441no—
390C2×D4—2441no—
391C2×D4—2441no—
392C2×D4D48811yescharacteristic
393C22×C4—2441no—
394C22×C4—2441no—
395C22×C4—2441no—
396C22×C4—2441no—
397C22×C4C2×D24411yescharacteristic
398C22×C4C2×D24411yescharacteristic
399Q16—612no—
400Q16—1221no—
401Q16—1221no—
402Q16—1221no—
403Q16—2442no—
404SD16—4881no—
405SD16—2442no—
406D8—1221no—
407D8—1221no—
408D8—1221no—
409D8—1221no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8C2×D24411yescharacteristic
415C2×C8C2×D24411yescharacteristic
416C2×C8C2×D24411yescharacteristic
417C2×C8C23×D6111yescharacteristic
418C4⋊C4—1221no—
419C4⋊C4—1221no—
420C4⋊C4—1221no—
421C4⋊C4—1221no—
422C4⋊C4—2441no—
423C4⋊C4—2441no—
424C4⋊C4—2441no—
425C4⋊C4—2441no—
426C4⋊C4—2441no—
427C4⋊C4D48811yescharacteristic
428C22⋊C4—2441no—
429C22⋊C4—2441no—
430C22⋊C4—2441no—
431C22⋊C4—2441no—
432C22⋊C4—2441no—
433C22⋊C4—2441no—
434C22⋊C4—2441no—
435C22⋊C4—2441no—
436C22⋊C4D48811yescharacteristic
437C22⋊C4D48811yescharacteristic
438C42—2441no—
439C42—2441no—
440C42C2×D24411yescharacteristic
441C16—21no—
442C16—21no—
443C16—21no—
444C16—21no—
445C23—2441no—
446C23—2441no—
447C23C2×D48811yescharacteristic
448C23C2×D48811yescharacteristic
449Q8—1222no—
450Q8—4882no—
451Q8—2441no—
452Q8—2441no—
453Q8—2441no—
454D4—22no—
455D4—9761no—
456D4—4882no—
457D4—4882no—
458D4—4882no—
459D4—2441no—
460D4—2441no—
461D4—2441no—
462D4—2441no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—9761no—
468C2×C4—9761no—
469C2×C4—4882no—
470C2×C4—4882no—
471C2×C4—2441no—
472C2×C4—2441no—
473C2×C4—2441no—
474C2×C4—2441no—
475C2×C4—2441no—
476C2×C4—2441no—
477C2×C4C2×D48811yescharacteristic
478C2×C4C2×D48811yescharacteristic
479C2×C4C2×D48811yescharacteristic
480C2×C4C2×D48811yescharacteristic
481C2×C4C22×D24411yescharacteristic
482C8—21no—
483C8—21no—
484C8—21no—
485C8#41×D6111yescharacteristic
486C8#1451982211yescharacteristic
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—4881no—
493C22—4881no—
494C22—4881no—
495C22—4881no—
496C22—4881no—
497C22—4881no—
498C22#1684019412yes—
499C22C22×D48811yescharacteristic
500C4—21no—
501C4—21no—
502C4—21no—
503C4—21no—
504C4—4881no—
505C4—4881no—
506C4—4881no—
507C4—4881no—
508C4—4881no—
509C4—2442no—
510C4#1684040711yescharacteristic
511C4#1684057411yescharacteristic
512C2—22no—
513C2—41no—
514C2—9761no—
515C2—9761no—
516C2#2145763311yescharacteristic
517C2C2×#1684019411yescharacteristic
518C2#2145896711yescharacteristic
subgroupG/H
centerC22C22×D488
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingD4×C16×C61C2
socleC22×C61C22×D8
hypercenterD4×C16D61
hypocenterC61#33868
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×C16

Complements

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

NH ≅ G/Ntype
1. #21458222C2semidirect
2. #21457562C2semidirect
3. #21448051C2semidirect
4. #21448028C2semidirect
5. #21448005C2semidirect
6. #21447984C2semidirect
7. #21447915C2semidirect
8. #21447905C2semidirect
9. #21447875C2semidirect
10. #21447744C2semidirect
11. #21447738C2semidirect
12. C4×#14519722C2semidirect
13. #21447527C2semidirect
14. C2×#16839567C2semidirect
15. D4×C16×C61C2semidirect
265. D4×C16D61semidirect
318. C61#33868semidirect

Maximal subgroups

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

subgroupindex
#214582222
#214575622
#214480512
#214480282
#214480052
#214479842
#214479152
#214479052
#214478752
#214477442
#214477382
C4×#145197222
#214475272
C2×#168395672
D4×C16×C612
#3386861