Finite Group Databasebeta

← #29837029

Subgroups — #29837029

Order 15616.

564 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458222C211yesmaximal, characteristic, complemented
2#21448060C211yesmaximal, characteristic, complemented
3#21448007C211yesmaximal, characteristic, complemented
4#21447997C211yesmaximal, characteristic, complemented
5#21447983C211yesmaximal, characteristic, complemented
6#21447920C211yesmaximal, characteristic, complemented
7#21447903C211yesmaximal, characteristic, complemented
8#21447851C211yesmaximal, characteristic, complemented
9#21447742C211yesmaximal, characteristic, complemented
10#21447742C211yesmaximal, characteristic, complemented
11#21447681C211yesmaximal, characteristic, complemented
12#21447645C211yesmaximal, characteristic, complemented
13#21447531C211yesmaximal, characteristic, complemented
14#21447526C211yesmaximal, characteristic, complemented
15C61×#3287C211yesmaximal, characteristic, complemented
16#16840458—21no—
17C2×#14519720C2211yescharacteristic
18C2×#14519718C2211yescharacteristic
19C2×#14519718C2211yescharacteristic
20C2×#14519717C2211yescharacteristic
21C22×#11758185C2211yescharacteristic
22#16840164C2211yescharacteristic
23#16840087—21no—
24#16840081—21no—
25#16840063—21no—
26#16839998C2211yescharacteristic
27#16839998C2211yescharacteristic
28C4×D488C2211yescharacteristic
29#16839800C2211yescharacteristic
30C2×#14519711C2211yescharacteristic
31#16839589C2211yescharacteristic
32#16839587C2211yescharacteristic
33#16839587C2211yescharacteristic
34C4⋊#11758185C2211yescharacteristic
35#16839566C2211yescharacteristic
36#16839566C2211yescharacteristic
37#16839566C2211yescharacteristic
38#16839563C2211yescharacteristic
39#16839563C2211yescharacteristic
40#16839563C2211yescharacteristic
41#16839563C2211yescharacteristic
42#16839563C2211yescharacteristic
43#16839563C2211yescharacteristic
44#16839562C2211yescharacteristic
45C22⋊#11758185C2211yescharacteristic
46C22⋊#11758185C2211yescharacteristic
47#16839538C2211yescharacteristic
48C2×#135×C61C2211yescharacteristic
49C2×#135×C61C2211yescharacteristic
50C61×#550—21no—
51C61×#493C2211yescharacteristic
52C61×#466C2211yescharacteristic
53C61×#466C2211yescharacteristic
54C61×#460—21no—
55C61×#449—21no—
56C61×#449—21no—
57C4×Q16×C61C2211yescharacteristic
58C8×D4×C61C2211yescharacteristic
59C2×D488C2311yescharacteristic
60#14519772—41no—
61#14519769—21no—
62#14519759—41no—
63#14519751—21no—
64#14519751—21no—
65#14519749—41no—
66C4×D244—21no—
67#14519737—41no—
68#14519736—21no—
69C2×C4⋊Dic61—21no—
70#14519720—22no—
71#14519718—41no—
72#14519718—41no—
73#14519717—41no—
74C2×#11758185—41no—
75C2×#11758185C2311yescharacteristic
76C2×#11758185C2311yescharacteristic
77C2×#11758185C2311yescharacteristic
78C2×#11758185C2311yescharacteristic
79#14519711—21no—
80#14519711C2311yescharacteristic
81#14519711C2311yescharacteristic
82#14519711C2311yescharacteristic
83#14519709—21no—
84#14519709—21no—
85#14519699—21no—
86#14519699—21no—
87#14519699—21no—
88C2×#41×C61—21no—
89C2×#41×C61—21no—
90#135×C61—22no—
91#135×C61—22no—
92#135×C61—41no—
93#135×C61—41no—
94C2×Q16×C61C2311yescharacteristic
95C2×SD16×C61—21no—
96C2×SD16×C61—21no—
97C2×D8×C61—21no—
98C2×D8×C61C2311yescharacteristic
99C2×D8×C61C2311yescharacteristic
100C22×C8×C61C2311yescharacteristic
101C22×C8×C61C2311yescharacteristic
102#127×C61—21no—
103#127×C61—41no—
104#121×C61—21no—
105#121×C61—21no—
106#121×C61—41no—
107#121×C61—41no—
108C4×Q8×C61—21no—
109C4×D4×C61—41no—
110C4×D4×C61D411yescharacteristic
111#107×C61C2311yescharacteristic
112C4⋊C8×C61D411yescharacteristic
113#103×C61—21no—
114#102×C61—21no—
115#102×C61—21no—
116C22⋊C8×C61D411yescharacteristic
117C22⋊C8×C61D411yescharacteristic
118C4×C8×C61C2311yescharacteristic
119#41×C61—42no—
120#41×C61—42no—
121#41×C61—81no—
122#41×C61—81no—
123C2×Q8×C61—21no—
124C2×D4×C61—21no—
125C2×D4×C61—21no—
126C2×D4×C61—41no—
127C2×D4×C61—41no—
128C2×D4×C61—41no—
129C2×D4×C61—41no—
130C2×D4×C61—41no—
131C2×D4×C61—41no—
132C2×D4×C61D811yescharacteristic
133C22×C4×C61—41no—
134C22×C4×C61—41no—
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C2×C22⋊D61—41no—
138C2×C22⋊D61—41no—
139C22×Dic61—41no—
140C2×D244—21no—
141C2×C4×D61—41no—
142Q16×C61—21no—
143Q16×C61#4112yes—
144SD16×C61—42no—
145SD16×C61—42no—
146D8×C61—21no—
147D8×C61—21no—
148D8×C61—21no—
149D8×C61—21no—
150D8×C61—81no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61C2×D411yescharacteristic
156C2×C8×C61C2×D411yescharacteristic
157C2×C8×C61C2×D411yescharacteristic
158C2×C8×C61C2411yescharacteristic
159C4⋊C4×C61—21no—
160C4⋊C4×C61—41no—
161C4⋊C4×C61D811yescharacteristic
162C22⋊C4×C61—41no—
163C22⋊C4×C61—41no—
164C22⋊C4×C61D811yescharacteristic
165C22⋊C4×C61D811yescharacteristic
166C42×C61—41no—
167C42×C61C2×D411yescharacteristic
168C22⋊Dic61—41no—
169#11758198—41no—
170#11758198—41no—
171#11758198—41no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—41no—
176#11758196—41no—
177#11758196—41no—
178C4×Dic61—41no—
179D488—21no—
180D488—21no—
181#11758185—21no—
182#11758185—21no—
183#11758185—21no—
184#11758185—21no—
185C23×C61—41no—
186C23×C61—41no—
187C23×C61C2×D811yescharacteristic
188C23×C61C2×D811yescharacteristic
189C22×D61—41no—
190Q8×C61—22no—
191Q8×C61—41no—
192D4×C61—41no—
193D4×C61—41no—
194D4×C61—41no—
195D4×C61—41no—
196D4×C61—41no—
197D4×C61—82no—
198D4×C61—82no—
199D4×C61—82no—
200D4×C61—161no—
201D4×C61—161no—
202D4×C61—161no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—82no—
210C2×C4×C61—82no—
211C2×C4×C61C2×D811yescharacteristic
212C2×C4×C61C2×D811yescharacteristic
213C2×C4×C61C2×D811yescharacteristic
214C2×C4×C61C2×D811yescharacteristic
215C2×C4×C61C22×D411yescharacteristic
216C22⋊D61—82no—
217C22⋊D61—161no—
218C2×Dic61—41no—
219C2×Dic61—41no—
220C2×Dic61—41no—
221C2×Dic61—161no—
222D244—41no—
223D244—41no—
224C4×D61—161no—
225C8×C61—21no—
226C8×C61—21no—
227C8×C61—21no—
228C8×C61C2×#4111yescharacteristic
229C8×C61H(5,𝔽2)11yescharacteristic
230#33876—611nomaximal, Sylow
231C22×C61—21no—
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—21no—
235C22×C61—21no—
236C22×C61—21no—
237C22×C61—81no—
238C22×C61—81no—
239C22×C61—81no—
240C22×C61—81no—
241C22×C61—81no—
242C22×C61—81no—
243C22×C61C22×D811yescharacteristic
244C2×D61—81no—
245C2×D61—81no—
246C2×D61—81no—
247C4×C61—21no—
248C4×C61—21no—
249C4×C61—21no—
250C4×C61—21no—
251C4×C61—42no—
252C4×C61—81no—
253C4×C61#54611yescharacteristic
254C4×C61#57711yescharacteristic
255Dic61—81no—
256Dic61—81no—
257Dic61—81no—
258#3339—611no—
259#3287D6111yescharacteristic, complemented
260#2244—611no—
261#2234—611no—
262#2232—611no—
263#2229—611no—
264#2218—611no—
265#2215—611no—
266#2206—611no—
267#2186—611no—
268#2186—611no—
269#2175—611no—
270#2168—611no—
271#2139—611no—
272#2138—611no—
273C2×C61—41no—
274C2×C61—41no—
275C2×C61—161no—
276C2×C61—161no—
277C2×C61#329411yescharacteristic
278C2×C61#341411yescharacteristic
279C2×C61#341511yescharacteristic
280D61—161no—
281C2×#135C2×D6111yescharacteristic
282C2×#135C2×D6111yescharacteristic
283#553—1221no—
284#550—21no—
285C2×SD32—611no—
286C2×D16—611no—
287C2×D16—611no—
288C2×M32—611no—
289C22×C16—611no—
290#499—611no—
291#493C2×D6111yescharacteristic
292#482—1221no—
293#481—1221no—
294#478—1221no—
295#466—611no—
296#466—611no—
297#466C2×D6111yescharacteristic
298#466C2×D6111yescharacteristic
299#460—21no—
300#449—21no—
301#449—21no—
302C4×Q16C2×D6111yescharacteristic
303C4×D8—611no—
304C8×D4C2×D6111yescharacteristic
305#427—611no—
306C2×#107—611no—
307#367—611no—
308#366—611no—
309#366—611no—
310C4⋊C16—611no—
311#358—611no—
312#358—611no—
313#358—611no—
314#357—611no—
315#357—611no—
316#357—611no—
317#357—611no—
318#357—611no—
319#357—611no—
320#357—611no—
321C22⋊C16—611no—
322C22⋊C16—611no—
323#346—611no—
324C61#3387611yescharacteristic, complemented, Sylow
325C2×#41—21no—
326C2×#41—21no—
327#135—22no—
328#135—22no—
329#135—41no—
330#135—41no—
331C2×Q16C22×D6111yescharacteristic
332C2×SD16—21no—
333C2×SD16—21no—
334C2×D8—21no—
335C2×D8—611no—
336C2×D8C22×D6111yescharacteristic
337C2×D8C22×D6111yescharacteristic
338C22×C8C22×D6111yescharacteristic
339C22×C8C22×D6111yescharacteristic
340#127—21no—
341#127—41no—
342#126—2441no—
343#125—1221no—
344#123—2441no—
345#121—21no—
346#121—21no—
347#121—41no—
348#121—41no—
349#121—1221no—
350#121—1221no—
351#121—2441no—
352C4×Q8—21no—
353C4×D4—41no—
354C4×D4—1221no—
355C4×D4—2441no—
356C4×D4C22⋊D6111yescharacteristic
357#117—1221no—
358C2×C4⋊C4—1221no—
359SD32—1222no—
360D16—2441no—
361D16—2441no—
362M32—2441no—
363C2×C16—611no—
364C2×C16—611no—
365C2×C16—611no—
366C2×C16—611no—
367C2×C16—2441no—
368#107—611no—
369#107—611no—
370#107—611no—
371#107—1221no—
372#107C22×D6111yescharacteristic
373#106—1221no—
374#106—1221no—
375C4⋊C8C22⋊D6111yescharacteristic
376#103—21no—
377#102—21no—
378#102—21no—
379#102—1221no—
380#102—1221no—
381#102—1221no—
382C22⋊C8C22⋊D6111yescharacteristic
383C22⋊C8C22⋊D6111yescharacteristic
384C4×C8C22×D6111yescharacteristic
385#41—42no—
386#41—42no—
387#41—81no—
388#41—81no—
389C2×Q8—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4—41no—
398C2×D4—1221no—
399C2×D4—2441no—
400C2×D4—2441no—
401C2×D4#1175819911yescharacteristic
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—2441no—
405C22×C4—2441no—
406C22×C4C2×C22⋊D6111yescharacteristic
407C22×C4C2×C22⋊D6111yescharacteristic
408Q16—21no—
409Q16#1175822412yes—
410SD16—42no—
411SD16—42no—
412D8—21no—
413D8—21no—
414D8—21no—
415D8—21no—
416D8—81no—
417D8—1221no—
418D8—1221no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8C2×C22⋊D6111yescharacteristic
424C2×C8C2×C22⋊D6111yescharacteristic
425C2×C8C2×C22⋊D6111yescharacteristic
426C2×C8C23×D6111yescharacteristic
427C4⋊C4—21no—
428C4⋊C4—41no—
429C4⋊C4—1221no—
430C4⋊C4—1221no—
431C4⋊C4—1221no—
432C4⋊C4—2441no—
433C4⋊C4—2441no—
434C4⋊C4—2441no—
435C4⋊C4#1175819911yescharacteristic
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—2441no—
439C22⋊C4—2441no—
440C22⋊C4—2441no—
441C22⋊C4—2441no—
442C22⋊C4#1175819911yescharacteristic
443C22⋊C4#1175819911yescharacteristic
444C42—41no—
445C42—2441no—
446C42C2×C22⋊D6111yescharacteristic
447C16—1221no—
448C16—1221no—
449C16—1221no—
450C16—1221no—
451C23—41no—
452C23—41no—
453C23—2441no—
454C23C2×#1175819911yescharacteristic
455C23C2×#1175819911yescharacteristic
456Q8—22no—
457Q8—41no—
458D4—41no—
459D4—41no—
460D4—41no—
461D4—41no—
462D4—41no—
463D4—82no—
464D4—82no—
465D4—82no—
466D4—161no—
467D4—161no—
468D4—161no—
469D4—9761no—
470D4—4882no—
471D4—2441no—
472D4—2441no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—9761no—
482C2×C4—9761no—
483C2×C4—2441no—
484C2×C4—2441no—
485C2×C4—2441no—
486C2×C4C2×#1175819911yescharacteristic
487C2×C4C2×#1175819911yescharacteristic
488C2×C4C2×#1175819911yescharacteristic
489C2×C4C2×#1175819911yescharacteristic
490C2×C4C22×C22⋊D6111yescharacteristic
491C8—21no—
492C8—21no—
493C8—21no—
494C8C2×#1175822411yescharacteristic
495C8#1451982211yescharacteristic
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—4881no—
509C22—4881no—
510C22—4881no—
511C22C22×#1175819911yescharacteristic
512C4—21no—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—42no—
517C4—81no—
518C4—4881no—
519C4—4881no—
520C4—4881no—
521C4#1684040611yescharacteristic
522C4#1684057311yescharacteristic
523C2—41no—
524C2—41no—
525C2—161no—
526C2—161no—
527C2—9761no—
528C2#2145763211yescharacteristic
529C2#2145895311yescharacteristic
530C2#2145895811yescharacteristic
subgroupG/H
centerC22C22×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3287C2
socleC22×C61C22×D8
hypercenter#3287D61
hypocenterC61#33876
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3287

Complements

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

NH ≅ G/Ntype
1. #21458222C2semidirect
2. #21448060C2semidirect
3. #21448007C2semidirect
4. #21447997C2semidirect
5. #21447983C2semidirect
6. #21447920C2semidirect
7. #21447903C2semidirect
8. #21447851C2semidirect
9. #21447742C2semidirect
10. #21447742C2semidirect
11. #21447681C2semidirect
12. #21447645C2semidirect
13. #21447531C2semidirect
14. #21447526C2semidirect
15. C61×#3287C2semidirect
259. #3287D61semidirect
324. C61#33876semidirect

Maximal subgroups

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

subgroupindex
#214582222
#214480602
#214480072
#214479972
#214479832
#214479202
#214479032
#214478512
#214477422
#214476812
#214476452
#214475312
#214475262
C61×#32872
#3387661