Finite Group Databasebeta

← #29836960

Subgroups — #29836960

Order 15616.

556 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#21458247C211yesmaximal, characteristic, complemented
2#21457576C211yesmaximal, characteristic
3#21448041C211yesmaximal, characteristic, complemented
4#21448013C211yesmaximal, characteristic, complemented
5#21447960C211yesmaximal, characteristic, complemented
6#21447916C211yesmaximal, characteristic, complemented
7#21447913C211yesmaximal, characteristic, complemented
8#21447858C211yesmaximal, characteristic, complemented
9#21447761C211yesmaximal, characteristic, complemented
10#21447760C211yesmaximal, characteristic, complemented
11#21447677C211yesmaximal, characteristic, complemented
12#21447636C211yesmaximal, characteristic, complemented
13#21447524C211yesmaximal, characteristic, complemented
14#21447523C211yesmaximal, characteristic, complemented
15C61×#2234C211yesmaximal, characteristic, complemented
16C2×#14519799C2211yescharacteristic
17C2×#14519796C2211yescharacteristic
18#16840480—21no—
19#16840430—21no—
20C2×Dic488C2211yescharacteristic
21C2×#14519722C2211yescharacteristic
22C2×#14519721C2211yescharacteristic
23C22×#11758185C2211yescharacteristic
24C2×C16×D61C2211yescharacteristic
25#16840166C2211yescharacteristic
26#16840137C2211yescharacteristic
27#16840106—21no—
28#16840103—21no—
29#16840046—21no—
30#16840004C2211yescharacteristic
31#16840003C2211yescharacteristic
32#16839999C2211yescharacteristic
33#16839961—21no—
34#16839908—21no—
35#16839905—21no—
36#16839856C2211yescharacteristic
37#16839856C2211yescharacteristic
38C8×C22⋊D61C2211yescharacteristic
39#16839799C2211yescharacteristic
40#16839589C2211yescharacteristic
41#16839580C2211yescharacteristic
42#16839567C2211yescharacteristic
43#16839565C2211yescharacteristic
44#16839565C2211yescharacteristic
45#16839565C2211yescharacteristic
46#16839564C2211yescharacteristic
47#16839563C2211yescharacteristic
48#16839562C2211yescharacteristic
49#16839562C2211yescharacteristic
50#16839541C2211yescharacteristic
51C16×Dic61C2211yescharacteristic
52C61×#466C2211yescharacteristic
53C61×#427C2211yescharacteristic
54C61×#367C2211yescharacteristic
55C61×#366C2211yescharacteristic
56C61×#357C2211yescharacteristic
57C61×#357C2211yescharacteristic
58C61×C22⋊C16C2211yescharacteristic
59C2×#11758222—21no—
60C2×#11758224—21no—
61#14519799—22no—
62#14519799—41no—
63#14519796—22no—
64#14519796—41no—
65C2×Dic244C2311yescharacteristic
66C2×Dic244C2311yescharacteristic
67C2×#11758190—21no—
68C2×#11758200—21no—
69C2×D488C2311yescharacteristic
70C2×#11758199—21no—
71C2×C8×D61C2311yescharacteristic
72#14519776—21no—
73#14519773—21no—
74#14519773—41no—
75#14519771—41no—
76#14519770—41no—
77#14519763—41no—
78#14519757—21no—
79#14519756—21no—
80#14519752—21no—
81#14519750—41no—
82#14519749—41no—
83C4×Dic122—41no—
84#14519742—21no—
85#14519742—21no—
86C4×C22⋊D61D411yescharacteristic
87#14519737—41no—
88#14519735—21no—
89Dic488—22no—
90#14519722—41no—
91#14519721—41no—
92C2×#11758185—41no—
93C2×#11758185C2311yescharacteristic
94C2×#11758185C2311yescharacteristic
95C16×D61—41no—
96#14519711C2311yescharacteristic
97#14519710—21no—
98#14519708—21no—
99#14519706D411yescharacteristic
100#14519702—21no—
101#14519700—21no—
102#14519700—21no—
103#14519700—21no—
104#14519697—21no—
105C22⋊#92418D411yescharacteristic
106#14519689D411yescharacteristic
107C8×Dic61C2311yescharacteristic
108C2×D8×C61C2311yescharacteristic
109C22×C8×C61C2311yescharacteristic
110#121×C61—21no—
111#117×C61—21no—
112C2×C16×C61C2311yescharacteristic
113C2×C16×C61C2311yescharacteristic
114#107×C61C2311yescharacteristic
115#107×C61C2311yescharacteristic
116#107×C61C2311yescharacteristic
117#106×C61—21no—
118#102×C61—21no—
119C2×D4×C61—21no—
120C2×D4×C61—41no—
121C22×C4×C61C2×D411yescharacteristic
122C2×C22⋊D61—41no—
123C2×C22⋊D61—41no—
124C2×C22⋊D61—41no—
125C2×C22⋊D61—41no—
126C2×C22⋊D61D811yescharacteristic
127C22×Dic61—41no—
128#11758224—42no—
129#11758224—81no—
130#11758222—42no—
131#11758222—81no—
132C2×D244—21no—
133C2×D244—41no—
134C2×C4×D61—41no—
135C2×C4×D61C2×D411yescharacteristic
136C2×Dic122—21no—
137C2×Dic122—21no—
138C2×Dic122—41no—
139D8×C61—21no—
140D8×C61—21no—
141C2×C8×C61—21no—
142C2×C8×C61—21no—
143C2×C8×C61C2×D411yescharacteristic
144C2×C8×C61C2411yescharacteristic
145C4⋊C4×C61—21no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C22⋊C4×C61—41no—
149C22⋊C4×C61—41no—
150C42×C61—41no—
151C22⋊Dic61—41no—
152C22⋊Dic61D811yescharacteristic
153#11758200—42no—
154#11758199—81no—
155#11758198—41no—
156#11758198—41no—
157#11758198—41no—
158#11758198D811yescharacteristic
159C4⋊Dic61—21no—
160C4⋊Dic61—41no—
161#11758196—41no—
162#11758196—41no—
163#11758196—41no—
164#11758196—41no—
165#11758196D811yescharacteristic
166C4×Dic61—41no—
167C4×Dic61—41no—
168C4×Dic61—41no—
169C4×Dic61C2×D411yescharacteristic
170C2×#92418C2×D411yescharacteristic
171C2×#92418C2×D411yescharacteristic
172Dic244—21no—
173Dic244—21no—
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
189D4×C61—41no—
190D4×C61—41no—
191D4×C61—161no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61C2×D811yescharacteristic
198C2×C4×C61C22×D411yescharacteristic
199C22⋊D61—22no—
200C22⋊D61—82no—
201C22⋊D61—82no—
202C22⋊D61—82no—
203C22⋊D61—161no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61—82no—
208C2×Dic61C2×D811yescharacteristic
209C2×Dic61C2×D811yescharacteristic
210C2×Dic61C2×D811yescharacteristic
211D244—41no—
212D244—41no—
213D244—161no—
214C4×D61—21no—
215C4×D61—21no—
216C4×D61—82no—
217Dic122—22no—
218Dic122—41no—
219Dic122—41no—
220Dic122—41no—
221Dic122—82no—
222C8×C61—21no—
223C8×C61C2×#4111yescharacteristic
224C8×C61H(5,𝔽2)11yescharacteristic
225#92418—21no—
226#92418—21no—
227#33871—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—21no—
238C2×D61—81no—
239C2×D61—81no—
240C2×D61—81no—
241C4×C61—21no—
242C4×C61—81no—
243C4×C61—81no—
244C4×C61—81no—
245C4×C61#54611yescharacteristic
246C4×C61#57711yescharacteristic
247Dic61—21no—
248Dic61—21no—
249Dic61—42no—
250Dic61—81no—
251Dic61—81no—
252Dic61#50812yes—
253#3342—611no—
254#3287—611no—
255#2240—611no—
256#2234—611no—
257#2234D6111yescharacteristic, complemented
258#2225—611no—
259#2217—611no—
260#2217—611no—
261#2207—611no—
262#2189—611no—
263#2189—611no—
264C4×Q32—611no—
265D4×C16—611no—
266#2138—611no—
267#2138—611no—
268C2×C61—41no—
269C2×C61—161no—
270C2×C61#329411yescharacteristic
271C2×C61C2×#50811yescharacteristic
272C2×C61#341611yescharacteristic
273D61—41no—
274D61—161no—
275C2×#135—611no—
276C2×#135—611no—
277#556—1221no—
278#550—1221no—
279C2×Q32—611no—
280C2×SD32—611no—
281C2×SD32—611no—
282C22×C16—611no—
283C22×C16—611no—
284#500—611no—
285#493—611no—
286#485—1221no—
287#485—1221no—
288#475—1221no—
289#467—611no—
290#467—611no—
291#466—611no—
292#466C2×D6111yescharacteristic
293#460—1221no—
294#449—1221no—
295#449—1221no—
296C4×Q16—611no—
297C4×Q16—611no—
298C8×D4—611no—
299#427—611no—
300#427C2×D6111yescharacteristic
301#367—611no—
302#367C2×D6111yescharacteristic
303#366C2×D6111yescharacteristic
304C4⋊C16—611no—
305#358—611no—
306#358—611no—
307#358—611no—
308#358—611no—
309#357—611no—
310#357—611no—
311#357—611no—
312#357—611no—
313#357C2×D6111yescharacteristic
314#357C2×D6111yescharacteristic
315C22⋊C16—611no—
316C22⋊C16C2×D6111yescharacteristic
317C4×C16—611no—
318C61#3387111yescharacteristic, complemented, Sylow
319C2×#41—1221no—
320C2×#41—1221no—
321#135—1222no—
322#135—1222no—
323#135—2441no—
324#135—2441no—
325C2×Q16—611no—
326C2×Q16—611no—
327C2×SD16—1221no—
328C2×SD16—1221no—
329C2×D8—611no—
330C2×D8—1221no—
331C2×D8C22×D6111yescharacteristic
332C22×C8—611no—
333C22×C8C22×D6111yescharacteristic
334C4⋊Q8—1221no—
335#127—1221no—
336#127—2441no—
337#126—2441no—
338#126—2441no—
339#124—2441no—
340C22⋊Q8—1221no—
341C22⋊Q8—1221no—
342#121—21no—
343#121—1221no—
344#121—2441no—
345#121—2441no—
346C4×Q8—1221no—
347C4×Q8—1221no—
348C4×Q8—2441no—
349C4×D4—611no—
350C4×D4—2441no—
351#117—21no—
352#117—1221no—
353Q32—1222no—
354SD32—2441no—
355SD32—2441no—
356C2×C16—611no—
357C2×C16—611no—
358C2×C16—2441no—
359C2×C16—2441no—
360C2×C16C22×D6111yescharacteristic
361C2×C16C22×D6111yescharacteristic
362#107—611no—
363#107—1221no—
364#107C22×D6111yescharacteristic
365#107C22×D6111yescharacteristic
366#107C22×D6111yescharacteristic
367#106—21no—
368#106—1221no—
369C4⋊C8—611no—
370#103—1221no—
371#103—1221no—
372#103—1221no—
373#103—1221no—
374#102—21no—
375#102—1221no—
376C22⋊C8—611no—
377C22⋊C8—611no—
378C4×C8—611no—
379#41—4881no—
380#41—4881no—
381#41—2442no—
382#41—2442no—
383C2×Q8—1221no—
384C2×Q8—1221no—
385C2×Q8—2441no—
386C2×D4—21no—
387C2×D4—41no—
388C2×D4—611no—
389C2×D4—1221no—
390C2×D4—2441no—
391C2×D4—2441no—
392C2×D4—2441no—
393C2×D4—2441no—
394C2×D4—2441no—
395C22×C4—611no—
396C22×C4—2441no—
397C22×C4—2441no—
398C22×C4D4×D6111yescharacteristic
399Q16—612no—
400Q16—1221no—
401Q16—1221no—
402Q16—1221no—
403SD16—2442no—
404SD16—2442no—
405D8—21no—
406D8—21no—
407D8—1221no—
408D8—1221no—
409D8—4881no—
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—21no—
421C4⋊C4—611no—
422C4⋊C4—1221no—
423C4⋊C4—2441no—
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—
436C42—41no—
437C42—611no—
438C42—2441no—
439C42—2441no—
440C42—2441no—
441C16—21no—
442C16—21no—
443C16—1221no—
444C16—1221no—
445C23—41no—
446C23—611no—
447C23—2441no—
448C23D8×D6111yescharacteristic
449Q8—1222no—
450Q8—4882no—
451Q8—2441no—
452Q8—2441no—
453Q8—2441no—
454D4—41no—
455D4—41no—
456D4—161no—
457D4—1222no—
458D4—9761no—
459D4—9761no—
460D4—4882no—
461D4—4882no—
462D4—4882no—
463D4—2441no—
464D4—2441no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—611no—
471C2×C4—611no—
472C2×C4—611no—
473C2×C4—1221no—
474C2×C4—1221no—
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#1451982311yescharacteristic
487C22—21no—
488C22—21no—
489C22—21no—
490C22—81no—
491C22—81no—
492C22—81no—
493C22—1221no—
494C22—1221no—
495C22—1221no—
496C22—4881no—
497C22—4881no—
498C22—4881no—
499C22C2×D8×D6111yescharacteristic
500C4—21no—
501C4—81no—
502C4—81no—
503C4—81no—
504C4—612no—
505C4—1221no—
506C4—1221no—
507C4—4881no—
508C4—4881no—
509C4—2442no—
510C4#1684040411yescharacteristic
511C4#1684057111yescharacteristic
512C2—41no—
513C2—161no—
514C2—9761no—
515C2—2441no—
516C2#2145763011yescharacteristic
517C2#508×D6111yescharacteristic
518C2#2145896511yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2234C2
socleC22×C61C22×D8
prime radical#21457576C2
hypercenter#2234D61
hypocenterC61#33871
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2234

Complements

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

NH ≅ G/Ntype
1. #21458247C2semidirect
3. #21448041C2semidirect
4. #21448013C2semidirect
5. #21447960C2semidirect
6. #21447916C2semidirect
7. #21447913C2semidirect
8. #21447858C2semidirect
9. #21447761C2semidirect
10. #21447760C2semidirect
11. #21447677C2semidirect
12. #21447636C2semidirect
13. #21447524C2semidirect
14. #21447523C2semidirect
15. C61×#2234C2semidirect
257. #2234D61semidirect
318. C61#33871semidirect

Maximal subgroups

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

subgroupindex
#214582472
#214575762
#214480412
#214480132
#214479602
#214479162
#214479132
#214478582
#214477612
#214477602
#214476772
#214476362
#214475242
#214475232
C61×#22342
#3387161