Finite Group Databasebeta

← #29836920

Subgroups — #29836920

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#21458216C211yesmaximal, characteristic, complemented
2#21457563C211yesmaximal, characteristic, complemented
3#21448052C211yesmaximal, characteristic, complemented
4#21448024C211yesmaximal, characteristic, complemented
5#21448006C211yesmaximal, characteristic, complemented
6#21447986C211yesmaximal, characteristic, complemented
7#21447904C211yesmaximal, characteristic, complemented
8#21447876C211yesmaximal, characteristic, complemented
9#21447740C211yesmaximal, characteristic, complemented
10#21447734C211yesmaximal, characteristic, complemented
11#21447674C211yesmaximal, characteristic, complemented
12C16×C22⋊D61C211yesmaximal, characteristic, complemented
13#21447527C211yesmaximal, characteristic, complemented
14#358×D61C211yesmaximal, characteristic, complemented
15C61×#2217C211yesmaximal, characteristic, complemented
16C2×#14519799C2211yescharacteristic
17C2×Q16×D61C2211yescharacteristic
18#16840459—21no—
19#16840422—21no—
20C2×#14519721C2211yescharacteristic
21C2×#14519718C2211yescharacteristic
22C2×C16×D61C2211yescharacteristic
23#16840163C2211yescharacteristic
24#16840147C2211yescharacteristic
25#16840083—21no—
26#16840079—21no—
27#16840064—21no—
28#16839999C2211yescharacteristic
29#16839995C2211yescharacteristic
30#16839987—21no—
31#16839922—21no—
32#16839902—21no—
33#16839856C2211yescharacteristic
34D8×Dic61C2211yescharacteristic
35C8×C22⋊D61C2211yescharacteristic
36#16839800C2211yescharacteristic
37#107×D61C2211yescharacteristic
38#16839589C2211yescharacteristic
39#16839587C2211yescharacteristic
40#16839580C2211yescharacteristic
41#16839567C2211yescharacteristic
42#16839567C2211yescharacteristic
43#16839566C2211yescharacteristic
44#16839566C2211yescharacteristic
45#16839565C2211yescharacteristic
46#16839564C2211yescharacteristic
47#16839564C2211yescharacteristic
48#16839562C2211yescharacteristic
49C22⋊#11758185C2211yescharacteristic
50#16839541C2211yescharacteristic
51C16×Dic61C2211yescharacteristic
52C2×SD32×C61C2211yescharacteristic
53C22×C16×C61C2211yescharacteristic
54C61×#467C2211yescharacteristic
55C61×#466C2211yescharacteristic
56C61×#367C2211yescharacteristic
57C61×#358C2211yescharacteristic
58C61×#357C2211yescharacteristic
59C2×#11758222—21no—
60C2×Q8×D61—21no—
61#14519799—22no—
62#14519799—41no—
63C2×Dic244C2311yescharacteristic
64C2×#11758202—21no—
65Q16×D61—22no—
66Q16×D61—41no—
67C2×#11758190—21no—
68C2×#11758201—21no—
69C2×D488C2311yescharacteristic
70C2×C8×D61C2311yescharacteristic
71#14519770—41no—
72#14519768—21no—
73#14519766—21no—
74#14519763—41no—
75#14519758—41no—
76#14519755—41no—
77#14519752—21no—
78#14519750—41no—
79#14519748—21no—
80#14519748—41no—
81#14519742—21no—
82D4×Dic61—21no—
83#14519739—41no—
84C4×C22⋊D61D411yescharacteristic
85#14519737—41no—
86#14519736—21no—
87C4⋊C4×D61—21no—
88#14519721—22no—
89#14519718—41no—
90C2×#11758185C2311yescharacteristic
91C2×#11758185C2311yescharacteristic
92C16×D61—41no—
93#14519711C2311yescharacteristic
94#14519711C2311yescharacteristic
95#14519710—21no—
96#14519709—21no—
97#14519708—21no—
98#14519706D411yescharacteristic
99#14519700—21no—
100#14519698—21no—
101#14519698—21no—
102#14519697—21no—
103C22⋊#92418D411yescharacteristic
104#14519689D411yescharacteristic
105C8×Dic61C2311yescharacteristic
106C2×Q16×C61C2311yescharacteristic
107C2×D8×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109C22⋊Q8×C61—21no—
110#121×C61—21no—
111SD32×C61—41no—
112C2×C16×C61—41no—
113C2×C16×C61C2311yescharacteristic
114C2×C16×C61C2311yescharacteristic
115#107×C61C2311yescharacteristic
116#107×C61C2311yescharacteristic
117#103×C61—21no—
118#102×C61—21no—
119C2×Q8×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61—41no—
122C22×C4×C61C2×D411yescharacteristic
123C2×C22⋊D61—41no—
124C2×C22⋊D61—41no—
125C2×C22⋊D61D811yescharacteristic
126C22×Dic61—41no—
127Q8×D61—42no—
128Q8×D61—81no—
129#11758222—42no—
130#11758222—81no—
131C2×D244—21no—
132C2×D244—41no—
133C2×C4×D61—41no—
134C2×C4×D61—41no—
135C2×C4×D61—41no—
136C2×C4×D61C2×D411yescharacteristic
137C2×Dic122—21no—
138C2×Dic122—41no—
139Q16×C61—21no—
140Q16×C61—21no—
141D8×C61—21no—
142D8×C61—21no—
143C2×C8×C61—21no—
144C2×C8×C61—21no—
145C2×C8×C61C2×D411yescharacteristic
146C2×C8×C61C2411yescharacteristic
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—41no—
150C22⋊C4×C61—41no—
151C22⋊C4×C61—41no—
152C22⋊Dic61—41no—
153C22⋊Dic61—41no—
154C22⋊Dic61—41no—
155C22⋊Dic61D811yescharacteristic
156#11758202—42no—
157#11758201—81no—
158#11758198—41no—
159#11758198—41no—
160#11758198—41no—
161#11758198D811yescharacteristic
162C4⋊Dic61—21no—
163C4⋊Dic61—21no—
164C4⋊Dic61—41no—
165#11758196—41no—
166#11758196—41no—
167#11758196—41no—
168#11758196D811yescharacteristic
169C4×Dic61—41no—
170C4×Dic61—41no—
171C4×Dic61C2×D411yescharacteristic
172C2×#92418C2×D411yescharacteristic
173C2×#92418C2×D411yescharacteristic
174Dic244—21no—
175Dic244#4112yes—
176D488—21no—
177D488—21no—
178#11758190—42no—
179C8×D61—21no—
180C8×D61—21no—
181C16×C61—21no—
182C16×C61—21no—
183#11758185—21no—
184#11758185—21no—
185C23×C61—41no—
186C23×C61C2×D811yescharacteristic
187C22×D61—41no—
188C22×D61C2×D811yescharacteristic
189Q8×C61—41no—
190Q8×C61—41no—
191D4×C61—41no—
192D4×C61—41no—
193D4×C61—161no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61C2×D811yescharacteristic
200C2×C4×C61C22×D411yescharacteristic
201C22⋊D61—22no—
202C22⋊D61—82no—
203C22⋊D61—82no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61—161no—
208C2×Dic61C2×D811yescharacteristic
209C2×Dic61C2×D811yescharacteristic
210C2×Dic61C2×D811yescharacteristic
211D244—41no—
212D244—41no—
213D244—82no—
214C4×D61—21no—
215C4×D61—21no—
216C4×D61—82no—
217C4×D61—82no—
218C4×D61—161no—
219Dic122—22no—
220Dic122—41no—
221Dic122—82no—
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—21no—
231C22×C61—81no—
232C22×C61—81no—
233C22×C61—81no—
234C22×C61C22×D811yescharacteristic
235C2×D61—21no—
236C2×D61—21no—
237C2×D61—81no—
238C2×D61—81no—
239C2×D61—81no—
240C2×D61#50812yes—
241C4×C61—21no—
242C4×C61—81no—
243C4×C61—81no—
244C4×C61—81no—
245C4×C61#54611yescharacteristic
246C4×C61#57711yescharacteristic
247Dic61—21no—
248Dic61—21no—
249Dic61—21no—
250Dic61—42no—
251Dic61—81no—
252Dic61—81no—
253#3339—611no—
254#3285—611no—
255#2242—611no—
256#2237—611no—
257#2233—611no—
258#2229—611no—
259#2217D6111yescharacteristic, complemented
260#2215—611no—
261#2210—611no—
262#2186—611no—
263#2185—611no—
264C4×SD32—611no—
265D4×C16—611no—
266#2138—611no—
267C2×#358—611no—
268C2×C61—41no—
269C2×C61—161no—
270C2×C61#329411yescharacteristic
271C2×C61C2×#50811yescharacteristic
272C2×C61#341611yescharacteristic
273D61—22no—
274D61—161no—
275C2×#135—611no—
276C22×Q16—611no—
277#553—1221no—
278#548—1221no—
279C2×SD32—611no—
280C2×SD32C2×D6111yescharacteristic
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#467C2×D6111yescharacteristic
290#466—611no—
291#466—611no—
292#466C2×D6111yescharacteristic
293#464—1221no—
294#452—1221no—
295#448—1221no—
296C4×Q16—611no—
297C4×D8—611no—
298C8×D4—611no—
299#427—611no—
300C2×#107—611no—
301#367—611no—
302#367C2×D6111yescharacteristic
303#366—611no—
304C4⋊C16—611no—
305#358—611no—
306#358—611no—
307#358—611no—
308#358—611no—
309#358—611no—
310#358C2×D6111yescharacteristic
311#357—611no—
312#357—611no—
313#357—611no—
314#357C2×D6111yescharacteristic
315C22⋊C16—611no—
316C22⋊C16—611no—
317C4×C16—611no—
318C61#3386811yescharacteristic, complemented, Sylow
319C2×#41—1221no—
320C22×Q8—1221no—
321#135—1222no—
322#135—2441no—
323C2×Q16—611no—
324C2×Q16—1221no—
325C2×Q16—1222no—
326C2×Q16—2441no—
327C2×Q16C22×D6111yescharacteristic
328C2×SD16—1221no—
329C2×SD16—1221no—
330C2×D8—611no—
331C2×D8C22×D6111yescharacteristic
332C22×C8—611no—
333C22×C8C22×D6111yescharacteristic
334#126—2441no—
335#125—1221no—
336#124—1221no—
337#124—2441no—
338#123—2441no—
339C22⋊Q8—21no—
340C22⋊Q8—2441no—
341#121—21no—
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—1221no—
353SD32—41no—
354SD32—1222no—
355D16—2441no—
356C2×C16—41no—
357C2×C16—611no—
358C2×C16—611no—
359C2×C16—2441no—
360C2×C16C22×D6111yescharacteristic
361C2×C16C22×D6111yescharacteristic
362#107—611no—
363#107—611no—
364#107—1221no—
365#107C22×D6111yescharacteristic
366#107C22×D6111yescharacteristic
367#106—1221no—
368#106—1221no—
369C4⋊C8—611no—
370#103—21no—
371#103—1221no—
372#102—21no—
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—1221no—
383C2×Q8—4881no—
384C2×Q8—2441no—
385C2×Q8—2442no—
386C2×D4—21no—
387C2×D4—41no—
388C2×D4—611no—
389C2×D4—1221no—
390C2×D4—2441no—
391C2×D4—2441no—
392C2×D4—2441no—
393C22×C4—611no—
394C22×C4—2441no—
395C22×C4—2441no—
396C22×C4—2441no—
397C22×C4—2441no—
398C22×C4D4×D6111yescharacteristic
399Q16—21no—
400Q16—21no—
401Q16—612no—
402Q16—1221no—
403Q16—2442no—
404SD16—4881no—
405SD16—2442no—
406D8—21no—
407D8—21no—
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—41no—
429C22⋊C4—41no—
430C22⋊C4—611no—
431C22⋊C4—611no—
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—41no—
446C23—611no—
447C23—2441no—
448C23D8×D6111yescharacteristic
449Q8—41no—
450Q8—41no—
451Q8—1222no—
452Q8—4882no—
453Q8—2441no—
454D4—41no—
455D4—41no—
456D4—161no—
457D4—1222no—
458D4—4882no—
459D4—4882no—
460D4—4882no—
461D4—2441no—
462D4—2441no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—611no—
469C2×C4—611no—
470C2×C4—611no—
471C2×C4—1221no—
472C2×C4—1221no—
473C2×C4—9761no—
474C2×C4—9761no—
475C2×C4—4882no—
476C2×C4—4882no—
477C2×C4—2441no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4D8×D6111yescharacteristic
481C2×C4C2×D4×D6111yescharacteristic
482C8—21no—
483C8—1221no—
484C8—1221no—
485C8#41×D6111yescharacteristic
486C8#1451982211yescharacteristic
487C22—21no—
488C22—21no—
489C22—21no—
490C22—81no—
491C22—81no—
492C22—81no—
493C22—612no—
494C22—1221no—
495C22—1221no—
496C22—4881no—
497C22—4881no—
498C22—4881no—
499C22C2×D8×D6111yescharacteristic
500C4—21no—
501C4—81no—
502C4—81no—
503C4—81no—
504C4—1221no—
505C4—1221no—
506C4—1221no—
507C4—4881no—
508C4—4881no—
509C4—2442no—
510C4#1684040111yescharacteristic
511C4#1684056911yescharacteristic
512C2—41no—
513C2—161no—
514C2—1222no—
515C2—9761no—
516C2#2145762711yescharacteristic
517C2#508×D6111yescharacteristic
518C2#2145896111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2217C2
socleC22×C61C22×D8
hypercenter#2217D61
hypocenterC61#33868
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2217

Complements

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

NH ≅ G/Ntype
1. #21458216C2semidirect
2. #21457563C2semidirect
3. #21448052C2semidirect
4. #21448024C2semidirect
5. #21448006C2semidirect
6. #21447986C2semidirect
7. #21447904C2semidirect
8. #21447876C2semidirect
9. #21447740C2semidirect
10. #21447734C2semidirect
11. #21447674C2semidirect
12. C16×C22⋊D61C2semidirect
13. #21447527C2semidirect
14. #358×D61C2semidirect
15. C61×#2217C2semidirect
259. #2217D61semidirect
318. C61#33868semidirect

Maximal subgroups

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

subgroupindex
#214582162
#214575632
#214480522
#214480242
#214480062
#214479862
#214479042
#214478762
#214477402
#214477342
#214476742
C16×C22⋊D612
#214475272
#358×D612
C61×#22172
#3386861