Finite Group Databasebeta

← #29836921

Subgroups — #29836921

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#21458217C211yesmaximal, characteristic, complemented
2#21457560C211yesmaximal, characteristic, complemented
3#21448052C211yesmaximal, characteristic, complemented
4#21448026C211yesmaximal, characteristic, complemented
5#21448004C211yesmaximal, characteristic, complemented
6#21447985C211yesmaximal, characteristic, complemented
7#21447918C211yesmaximal, characteristic, complemented
8#21447905C211yesmaximal, characteristic, complemented
9#21447877C211yesmaximal, characteristic, complemented
10#21447745C211yesmaximal, characteristic, complemented
11#21447736C211yesmaximal, characteristic, complemented
12#21447668C211yesmaximal, characteristic, complemented
13C16×C22⋊D61C211yesmaximal, characteristic, complemented
14#21447525C211yesmaximal, characteristic, complemented
15C2×C61×#358C211yesmaximal, characteristic, complemented
16C2×#14519800C2211yescharacteristic
17#16840460—21no—
18#16840423—21no—
19C2×#14519720C2211yescharacteristic
20C2×#14519720C2211yescharacteristic
21C2×D976C2211yescharacteristic
22C2×C16×D61C2211yescharacteristic
23#16840163C2211yescharacteristic
24#16840147C2211yescharacteristic
25#16840089—21no—
26#16840080—21no—
27#16840065—21no—
28#16840000C2211yescharacteristic
29#16839999C2211yescharacteristic
30#16839998C2211yescharacteristic
31#16839998C2211yescharacteristic
32#16839986—21no—
33#16839923—21no—
34#16839901—21no—
35Q16×Dic61C2211yescharacteristic
36#16839844C2211yescharacteristic
37C8×C22⋊D61C2211yescharacteristic
38#16839799C2211yescharacteristic
39#16839589C2211yescharacteristic
40#16839589C2211yescharacteristic
41#16839588C2211yescharacteristic
42#16839580C2211yescharacteristic
43#16839566C2211yescharacteristic
44#16839566C2211yescharacteristic
45#16839564C2211yescharacteristic
46#16839564C2211yescharacteristic
47#16839562C2211yescharacteristic
48#16839562C2211yescharacteristic
49C22⋊#11758185C2211yescharacteristic
50#16839541C2211yescharacteristic
51C16×Dic61C2211yescharacteristic
52C22×Q16×C61C2211yescharacteristic
53C22×C16×C61C2211yescharacteristic
54C2×#107×C61C2211yescharacteristic
55C61×#358C2211yescharacteristic
56C61×#358C2211yescharacteristic
57C61×#358C2211yescharacteristic
58C61×#358C2211yescharacteristic
59C2×#11758226—21no—
60#14519800—22no—
61#14519800—41no—
62C2×#11758201—21no—
63C2×#11758201—21no—
64C2×D488C2311yescharacteristic
65C2×D488C2311yescharacteristic
66C2×C8×D61C2311yescharacteristic
67#14519772—41no—
68#14519768—21no—
69#14519766—21no—
70#14519766—41no—
71#14519762—41no—
72#14519753—21no—
73#14519753—41no—
74#14519752—21no—
75#14519751—21no—
76#14519751—21no—
77#14519751—41no—
78#14519751—41no—
79Q8×Dic61—21no—
80#14519739—21no—
81C4×C22⋊D61—41no—
82C4×C22⋊D61—41no—
83C4×C22⋊D61D411yescharacteristic
84#14519735—21no—
85#14519720—22no—
86#14519720—41no—
87D976—41no—
88C2×#11758185C2311yescharacteristic
89C2×#11758185C2311yescharacteristic
90C16×D61—41no—
91#14519711C2311yescharacteristic
92#14519711C2311yescharacteristic
93#14519708—21no—
94#14519708—21no—
95#14519706D411yescharacteristic
96#14519701—21no—
97#14519701—21no—
98#14519699—21no—
99#14519699—21no—
100#14519697—21no—
101#14519697—21no—
102C22⋊#92418D411yescharacteristic
103#14519689D411yescharacteristic
104C8×Dic61C2311yescharacteristic
105C22×Q8×C61—21no—
106C2×Q16×C61—22no—
107C2×Q16×C61—21no—
108C2×Q16×C61—41no—
109C2×Q16×C61C2311yescharacteristic
110C2×Q16×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112C2×C4⋊C4×C61—21no—
113C2×C16×C61—41no—
114C2×C16×C61C2311yescharacteristic
115C2×C16×C61C2311yescharacteristic
116#107×C61—21no—
117#107×C61C2311yescharacteristic
118#107×C61C2311yescharacteristic
119C2×Q8×C61—21no—
120C2×Q8×C61—21no—
121C2×Q8×C61—42no—
122C2×Q8×C61—41no—
123C2×Q8×C61—81no—
124C22×C4×C61—41no—
125C22×C4×C61—41no—
126C22×C4×C61C2×D411yescharacteristic
127C2×C22⋊D61—41no—
128C2×C22⋊D61—41no—
129C2×C22⋊D61D811yescharacteristic
130#11758226—42no—
131#11758226—81no—
132C2×D244—21no—
133C2×D244—21no—
134C2×D244—41no—
135C2×D244—41no—
136C2×C4×D61—41no—
137C2×C4×D61—41no—
138C2×C4×D61C2×D411yescharacteristic
139Q16×C61—21no—
140Q16×C61—21no—
141Q16×C61—21no—
142Q16×C61—42no—
143Q16×C61#4112yes—
144C2×C8×C61—21no—
145C2×C8×C61—21no—
146C2×C8×C61C2×D411yescharacteristic
147C2×C8×C61C2411yescharacteristic
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—41no—
151C22⋊Dic61D811yescharacteristic
152#11758201—42no—
153#11758201—81no—
154#11758198—41no—
155#11758198—41no—
156#11758198—41no—
157#11758198—41no—
158#11758198—41no—
159#11758198—41no—
160#11758198—41no—
161#11758198—41no—
162#11758198D811yescharacteristic
163C4⋊Dic61—21no—
164C4⋊Dic61—21no—
165C4⋊Dic61—41no—
166C4⋊Dic61—41no—
167#11758196—41no—
168#11758196—41no—
169#11758196D811yescharacteristic
170C4×Dic61—41no—
171C4×Dic61—41no—
172C4×Dic61C2×D411yescharacteristic
173C2×#92418C2×D411yescharacteristic
174C2×#92418C2×D411yescharacteristic
175D488—21no—
176D488—21no—
177D488—21no—
178D488—21no—
179C8×D61—21no—
180C8×D61—21no—
181C16×C61—21no—
182C16×C61—21no—
183#11758185—21no—
184#11758185—21no—
185C23×C61C2×D811yescharacteristic
186C22×D61—41no—
187C22×D61—41no—
188C22×D61C2×D811yescharacteristic
189Q8×C61—22no—
190Q8×C61—41no—
191Q8×C61—41no—
192Q8×C61—41no—
193Q8×C61—82no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—82no—
201C2×C4×C61—161no—
202C2×C4×C61C2×D811yescharacteristic
203C2×C4×C61C22×D411yescharacteristic
204C22⋊D61—22no—
205C22⋊D61—82no—
206C22⋊D61—82no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61C2×D811yescharacteristic
210C2×Dic61C2×D811yescharacteristic
211C2×Dic61C2×D811yescharacteristic
212D244—41no—
213D244—41no—
214D244—41no—
215D244—41no—
216D244—82no—
217D244—161no—
218C4×D61—21no—
219C4×D61—21no—
220C4×D61—82no—
221C4×D61—161no—
222C8×C61—21no—
223C8×C61C2×#4111yescharacteristic
224C8×C61H(5,𝔽2)11yescharacteristic
225#92418—21no—
226#92418—21no—
227#33868—611nomaximal, Sylow
228C22×C61—21no—
229C22×C61—21no—
230C22×C61#50812yes—
231C22×C61C22×D811yescharacteristic
232C2×D61—21no—
233C2×D61—21no—
234C2×D61—21no—
235C2×D61—81no—
236C2×D61—81no—
237C2×D61—81no—
238C2×D61—81no—
239C2×D61—81no—
240C2×D61—81no—
241C4×C61—21no—
242C4×C61—42no—
243C4×C61—81no—
244C4×C61—81no—
245C4×C61—81no—
246C4×C61#54611yescharacteristic
247C4×C61#57711yescharacteristic
248Dic61—21no—
249Dic61—21no—
250Dic61—21no—
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×C16—611no—
266#2138—611no—
267C2×#358D6111yescharacteristic, complemented
268C2×C61—22no—
269C2×C61#329411yescharacteristic
270C2×C61C2×#50811yescharacteristic
271C2×C61#341611yescharacteristic
272D61—41no—
273D61—161no—
274D61—161no—
275C2×#135—611no—
276C22×Q16C2×D6111yescharacteristic
277#553—1221no—
278#548—1221no—
279C2×SD32—611no—
280C2×SD32—611no—
281C2×D16—611no—
282C22×C16—611no—
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×D4—611no—
299#427—611no—
300C2×#107C2×D6111yescharacteristic
301#367—611no—
302#367—611no—
303#366—611no—
304C4⋊C16—611no—
305#358—611no—
306#358—611no—
307#358C2×D6111yescharacteristic
308#358C2×D6111yescharacteristic
309#358C2×D6111yescharacteristic
310#358C2×D6111yescharacteristic
311#357—611no—
312#357—611no—
313#357—611no—
314#357—611no—
315C22⋊C16—611no—
316C22⋊C16—611no—
317C4×C16—611no—
318C61#3386811yescharacteristic, complemented, Sylow
319C2×#41—1221no—
320C22×Q8—21no—
321#135—1222no—
322#135—2441no—
323C2×Q16—22no—
324C2×Q16—21no—
325C2×Q16—41no—
326C2×Q16C22×D6111yescharacteristic
327C2×Q16C22×D6111yescharacteristic
328C2×SD16—1221no—
329C2×SD16—1221no—
330C2×D8—611no—
331C2×D8—611no—
332C22×C8—611no—
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—611no—
348C4×D4—1221no—
349C4×D4—2441no—
350C4×D4—2441no—
351#117—1221no—
352C2×C4⋊C4—21no—
353SD32—1222no—
354SD32—2441no—
355D16—2441no—
356C2×C16—41no—
357C2×C16—611no—
358C2×C16—611no—
359C2×C16—2441no—
360C2×C16C22×D6111yescharacteristic
361C2×C16C22×D6111yescharacteristic
362#107—21no—
363#107—611no—
364#107—611no—
365#107C22×D6111yescharacteristic
366#107C22×D6111yescharacteristic
367#106—1221no—
368#106—1221no—
369C4⋊C8—611no—
370#103—1221no—
371#103—1221no—
372#102—1221no—
373#102—1221no—
374#102—1221no—
375#102—1221no—
376C22⋊C8—611no—
377C22⋊C8—611no—
378C4×C8—611no—
379#41—4881no—
380#41—2442no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—42no—
384C2×Q8—41no—
385C2×Q8—81no—
386C2×D4—611no—
387C2×D4—1221no—
388C2×D4—1221no—
389C2×D4—2441no—
390C2×D4—2441no—
391C2×D4—2441no—
392C2×D4—2441no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—611no—
396C22×C4—2441no—
397C22×C4—2441no—
398C22×C4D4×D6111yescharacteristic
399Q16—21no—
400Q16—21no—
401Q16—21no—
402Q16—42no—
403Q16#1175822212yes—
404SD16—4881no—
405SD16—2442no—
406D8—1221no—
407D8—1221no—
408D8—1221no—
409D8—1221no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—611no—
413C2×C8—611no—
414C2×C8—1221no—
415C2×C8—1221no—
416C2×C8D4×D6111yescharacteristic
417C2×C8C23×D6111yescharacteristic
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—41no—
421C4⋊C4—611no—
422C4⋊C4—1221no—
423C4⋊C4—1221no—
424C4⋊C4—2441no—
425C4⋊C4—2441no—
426C4⋊C4—2441no—
427C4⋊C4—2441no—
428C22⋊C4—611no—
429C22⋊C4—611no—
430C22⋊C4—2441no—
431C22⋊C4—2441no—
432C22⋊C4—2441no—
433C22⋊C4—2441no—
434C22⋊C4—2441no—
435C22⋊C4—2441no—
436C22⋊C4—2441no—
437C22⋊C4—2441no—
438C42—611no—
439C42—2441no—
440C42—2441no—
441C16—21no—
442C16—21no—
443C16—1221no—
444C16—1221no—
445C23—611no—
446C23—2441no—
447C23—2441no—
448C23D8×D6111yescharacteristic
449Q8—22no—
450Q8—41no—
451Q8—41no—
452Q8—41no—
453Q8—82no—
454D4—1222no—
455D4—9761no—
456D4—4882no—
457D4—4882no—
458D4—4882no—
459D4—2441no—
460D4—2441no—
461D4—2441no—
462D4—2441no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—82no—
470C2×C4—161no—
471C2×C4—611no—
472C2×C4—611no—
473C2×C4—611no—
474C2×C4—1221no—
475C2×C4—1221no—
476C2×C4—9761no—
477C2×C4—4882no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4D8×D6111yescharacteristic
481C2×C4C2×D4×D6111yescharacteristic
482C8—21no—
483C8—1221no—
484C8—1221no—
485C8C2×#1175822211yescharacteristic
486C8#1451982211yescharacteristic
487C22—21no—
488C22—21no—
489C22—1221no—
490C22—1221no—
491C22—1221no—
492C22—4881no—
493C22—4881no—
494C22—4881no—
495C22—4881no—
496C22—4881no—
497C22—4881no—
498C22#1684019512yes—
499C22C2×D8×D6111yescharacteristic
500C4—21no—
501C4—42no—
502C4—81no—
503C4—81no—
504C4—81no—
505C4—1221no—
506C4—1221no—
507C4—1221no—
508C4—4881no—
509C4—4881no—
510C4#1684040211yescharacteristic
511C4#1684056811yescharacteristic
512C2—22no—
513C2—9761no—
514C2—9761no—
515C2—2441no—
516C2#2145762811yescharacteristic
517C2C2×#1684019511yescharacteristic
518C2#2145896111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#358C2
socleC22×C61C22×D8
hypercenterC2×#358D61
hypocenterC61#33868
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#358

Complements

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

NH ≅ G/Ntype
1. #21458217C2semidirect
2. #21457560C2semidirect
3. #21448052C2semidirect
4. #21448026C2semidirect
5. #21448004C2semidirect
6. #21447985C2semidirect
7. #21447918C2semidirect
8. #21447905C2semidirect
9. #21447877C2semidirect
10. #21447745C2semidirect
11. #21447736C2semidirect
12. #21447668C2semidirect
13. C16×C22⋊D61C2semidirect
14. #21447525C2semidirect
15. C2×C61×#358C2semidirect
267. C2×#358D61semidirect
318. C61#33868semidirect

Maximal subgroups

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

subgroupindex
#214582172
#214575602
#214480522
#214480262
#214480042
#214479852
#214479182
#214479052
#214478772
#214477452
#214477362
#214476682
C16×C22⋊D612
#214475252
C2×C61×#3582
#3386861