Finite Group Databasebeta

← #29837014

Subgroups — #29837014

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#21458167C211yesmaximal, characteristic, complemented
2#21448055C211yesmaximal, characteristic, complemented
3#21448019C211yesmaximal, characteristic, complemented
4#21447997C211yesmaximal, characteristic, complemented
5#21447971C211yesmaximal, characteristic, complemented
6#21447934C211yesmaximal, characteristic, complemented
7#21447903C211yesmaximal, characteristic, complemented
8#21447871C211yesmaximal, characteristic, complemented
9#21447742C211yesmaximal, characteristic, complemented
10#21447736C211yesmaximal, characteristic, complemented
11#21447681C211yesmaximal, characteristic, complemented
12#21447645C211yesmaximal, characteristic, complemented
13#21447531C211yesmaximal, characteristic, complemented
14C2×#16839566C211yesmaximal, characteristic, complemented
15C61×#3285C211yesmaximal, characteristic, complemented
16#16840416—21no—
17C2×#14519720C2211yescharacteristic
18C2×#14519720C2211yescharacteristic
19C2×#14519718C2211yescharacteristic
20C2×#14519717C2211yescharacteristic
21C22×#11758185C2211yescharacteristic
22#16840167C2211yescharacteristic
23#16840076—21no—
24#16840076—21no—
25#16840043—21no—
26#16839998C2211yescharacteristic
27#16839998C2211yescharacteristic
28C4×D488C2211yescharacteristic
29C2×#14519711C2211yescharacteristic
30C2×#14519711C2211yescharacteristic
31#16839587C2211yescharacteristic
32#16839587C2211yescharacteristic
33#16839587C2211yescharacteristic
34C4⋊#11758185C2211yescharacteristic
35#16839566C2211yescharacteristic
36#16839566C2211yescharacteristic
37#16839566C2211yescharacteristic
38#16839566C2211yescharacteristic
39#16839566C2211yescharacteristic
40#16839566C2211yescharacteristic
41#16839563C2211yescharacteristic
42#16839563C2211yescharacteristic
43#16839563C2211yescharacteristic
44#16839562C2211yescharacteristic
45C22⋊#11758185C2211yescharacteristic
46C22⋊#11758185C2211yescharacteristic
47#16839538C2211yescharacteristic
48C2×#135×C61C2211yescharacteristic
49C22×Q16×C61C2211yescharacteristic
50C61×#548—21no—
51C61×#495C2211yescharacteristic
52C61×#467C2211yescharacteristic
53C61×#466C2211yescharacteristic
54C61×#464—21no—
55C61×#452—21no—
56C61×#448—21no—
57C4×Q16×C61C2211yescharacteristic
58C8×D4×C61C2211yescharacteristic
59C2×#11758224—41no—
60C2×D488C2311yescharacteristic
61#14519777—21no—
62#14519761—41no—
63#14519761—41no—
64#14519757—41no—
65#14519751—21no—
66#14519751—21no—
67C4×D244—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70#14519720—22no—
71#14519720—41no—
72#14519718—41no—
73#14519717—41no—
74C2×#11758185—41no—
75C2×#11758185C2311yescharacteristic
76C2×#11758185C2311yescharacteristic
77C2×#11758185C2311yescharacteristic
78C2×#11758185C2311yescharacteristic
79#14519711—21no—
80#14519711—21no—
81#14519711—21no—
82#14519711C2311yescharacteristic
83#14519711C2311yescharacteristic
84#14519711C2311yescharacteristic
85#14519699—21no—
86#14519699—21no—
87#14519699—21no—
88C2×#41×C61—21no—
89C22×Q8×C61—21no—
90#135×C61—22no—
91#135×C61—41no—
92C2×Q16×C61—22no—
93C2×Q16×C61—21no—
94C2×Q16×C61—41no—
95C2×Q16×C61C2311yescharacteristic
96C2×Q16×C61C2311yescharacteristic
97C2×SD16×C61—21no—
98C2×SD16×C61—21no—
99C2×D8×C61C2311yescharacteristic
100C22×C8×C61C2311yescharacteristic
101C22×C8×C61C2311yescharacteristic
102#124×C61—21no—
103#124×C61—41no—
104C22⋊Q8×C61—21no—
105C22⋊Q8×C61—41no—
106#121×C61—21no—
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#103×C61—21no—
115#102×C61—21no—
116C22⋊C8×C61D411yescharacteristic
117C22⋊C8×C61D411yescharacteristic
118C4×C8×C61C2311yescharacteristic
119#41×C61—42no—
120#41×C61—81no—
121C2×Q8×C61—21no—
122C2×Q8×C61—21no—
123C2×Q8×C61—42no—
124C2×Q8×C61—41no—
125C2×Q8×C61—81no—
126C2×D4×C61—21no—
127C2×D4×C61—41no—
128C2×D4×C61—41no—
129C2×D4×C61D811yescharacteristic
130C22×C4×C61—41no—
131C22×C4×C61—41no—
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C2×C22⋊D61—41no—
135C2×C22⋊D61—41no—
136C22×Dic61—41no—
137C22×Dic61—41no—
138#11758224—84no—
139C2×D244—21no—
140C2×C4×D61—41no—
141C2×Dic122—41no—
142Q16×C61—21no—
143Q16×C61—21no—
144Q16×C61—21no—
145Q16×C61—42no—
146Q16×C61#4112yescomplemented
147SD16×C61—42no—
148SD16×C61—81no—
149D8×C61—21no—
150D8×C61—21no—
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×C61—41no—
162C4⋊C4×C61D811yescharacteristic
163C22⋊C4×C61—41no—
164C22⋊C4×C61—41no—
165C22⋊C4×C61—41no—
166C22⋊C4×C61—41no—
167C22⋊C4×C61D811yescharacteristic
168C22⋊C4×C61D811yescharacteristic
169C42×C61—41no—
170C42×C61C2×D411yescharacteristic
171#11758198—41no—
172#11758198—41no—
173#11758198—41no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—41no—
178C4⋊Dic61—41no—
179C4⋊Dic61—41no—
180D488—21no—
181D488—21no—
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—22no—
191Q8×C61—41no—
192Q8×C61—41no—
193Q8×C61—41no—
194Q8×C61—82no—
195D4×C61—22no—
196D4×C61—41no—
197D4×C61—41no—
198D4×C61—82no—
199D4×C61—82no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—82no—
208C2×C4×C61—82no—
209C2×C4×C61C2×D811yescharacteristic
210C2×C4×C61C2×D811yescharacteristic
211C2×C4×C61C2×D811yescharacteristic
212C2×C4×C61C2×D811yescharacteristic
213C2×C4×C61C22×D411yescharacteristic
214C22⋊D61—82no—
215C22⋊D61—82no—
216C2×Dic61—41no—
217C2×Dic61—41no—
218C2×Dic61—41no—
219C2×Dic61—82no—
220C2×Dic61—82no—
221D244—41no—
222D244—41no—
223C4×D61—82no—
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—42no—
249C4×C61—81no—
250C4×C61—81no—
251C4×C61#54611yescharacteristic
252C4×C61#57711yescharacteristic
253Dic61—81no—
254Dic61—81no—
255Dic61—81no—
256#3334—611no—
257#3285D6111yescharacteristic, complemented
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×#135C2×D6111yescharacteristic
279C22×Q16C2×D6111yescharacteristic
280#548—21no—
281#547—1221no—
282C2×SD32—611no—
283C2×SD32—611no—
284C2×D16—611no—
285C2×M32—611no—
286C22×C16—611no—
287#500—611no—
288#495C2×D6111yescharacteristic
289#480—1221no—
290#480—1221no—
291#474—1221no—
292#467C2×D6111yescharacteristic
293#466—611no—
294#466—611no—
295#466C2×D6111yescharacteristic
296#464—21no—
297#452—21no—
298#448—21no—
299C4×Q16C2×D6111yescharacteristic
300C4×D8—611no—
301C8×D4C2×D6111yescharacteristic
302C2×#107—611no—
303C2×#107—611no—
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—21no—
323C2×#41—2441no—
324C22×Q8—21no—
325#135—22no—
326#135—41no—
327C2×Q16—22no—
328C2×Q16—21no—
329C2×Q16—41no—
330C2×Q16C22×D6111yescharacteristic
331C2×Q16C22×D6111yescharacteristic
332C2×SD16—21no—
333C2×SD16—21no—
334C2×D8—611no—
335C2×D8C22×D6111yescharacteristic
336C22×C8C22×D6111yescharacteristic
337C22×C8C22×D6111yescharacteristic
338C4⋊Q8—1221no—
339#124—21no—
340#124—41no—
341#123—2441no—
342#123—2441no—
343C22⋊Q8—21no—
344C22⋊Q8—41no—
345C22⋊Q8—2441no—
346#121—21no—
347#121—41no—
348#121—1221no—
349#121—1221no—
350C4×Q8—21no—
351C4×D4—41no—
352C4×D4—1221no—
353C4×D4C22⋊D6111yescharacteristic
354C2×C4⋊C4—1221no—
355C2×C4⋊C4—1221no—
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—611no—
366#107—611no—
367#107—611no—
368#107—1221no—
369#107—1221no—
370#107—1221no—
371#107C22×D6111yescharacteristic
372C4⋊C8C22⋊D6111yescharacteristic
373#103—21no—
374#103—21no—
375#102—21no—
376#102—1221no—
377#102—1221no—
378#102—1221no—
379C22⋊C8C22⋊D6111yescharacteristic
380C22⋊C8C22⋊D6111yescharacteristic
381C4×C8C22×D6111yescharacteristic
382#41—42no—
383#41—81no—
384#41—4884no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—42no—
388C2×Q8—41no—
389C2×Q8—81no—
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—2441no—
401C22×C4—2441no—
402C22×C4—2441no—
403C22×C4C2×C22⋊D6111yescharacteristic
404C22×C4C2×C22⋊D6111yescharacteristic
405Q16—21no—
406Q16—21no—
407Q16—21no—
408Q16—42no—
409Q16#1175822412yescomplemented
410SD16—42no—
411SD16—81no—
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—41no—
426C4⋊C4—41no—
427C4⋊C4—1221no—
428C4⋊C4—1221no—
429C4⋊C4—1221no—
430C4⋊C4—2441no—
431C4⋊C4—2441no—
432C4⋊C4—2441no—
433C4⋊C4#1175819911yescharacteristic
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—2441no—
439C22⋊C4—2441no—
440C22⋊C4—2441no—
441C22⋊C4#1175819911yescharacteristic
442C22⋊C4#1175819911yescharacteristic
443C42—41no—
444C42C2×C22⋊D6111yescharacteristic
445C16—1221no—
446C16—1221no—
447C16—1221no—
448C16—1221no—
449C23—41no—
450C23—2441no—
451C23C2×#1175819911yescharacteristic
452C23C2×#1175819911yescharacteristic
453Q8—22no—
454Q8—41no—
455Q8—41no—
456Q8—41no—
457Q8—82no—
458Q8—4882no—
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—4882no—
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—
491C8C2×#1175822411yescharacteristic
492C8#1451982211yescharacteristic
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—4881no—
502C22—4881no—
503C22—4881no—
504C22#1684019712yes—
505C22C22×#1175819911yescharacteristic
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—42no—
511C4—81no—
512C4—81no—
513C4—4881no—
514C4—4881no—
515C4—4881no—
516C4#1684040611yescharacteristic
517C4#1684057311yescharacteristic
518C2—22no—
519C2—41no—
520C2—161no—
521C2—9761no—
522C2#2145763211yescharacteristic
523C2C2×#1684019711yescharacteristic
524C2#2145897011yescharacteristic
subgroupG/H
centerC22C22×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3285C2
socleC22×C61C22×D8
hypercenter#3285D61
hypocenterC61#33875
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3285

Complements

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

NH ≅ G/Ntype
1. #21458167C2semidirect
2. #21448055C2semidirect
3. #21448019C2semidirect
4. #21447997C2semidirect
5. #21447971C2semidirect
6. #21447934C2semidirect
7. #21447903C2semidirect
8. #21447871C2semidirect
9. #21447742C2semidirect
10. #21447736C2semidirect
11. #21447681C2semidirect
12. #21447645C2semidirect
13. #21447531C2semidirect
14. C2×#16839566C2semidirect
15. C61×#3285C2semidirect
146. Q16×C61#41semidirect
257. #3285D61semidirect
321. C61#33875semidirect
409. Q16#11758224semidirect

Maximal subgroups

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

subgroupindex
#214581672
#214480552
#214480192
#214479972
#214479712
#214479342
#214479032
#214478712
#214477422
#214477362
#214476812
#214476452
#214475312
C2×#168395662
C61×#32852
#3387561