Finite Group Databasebeta

← #29834411

Subgroups — #29834411

Order 15616.

706 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458953C211yesmaximal, characteristic, complemented
2#21456300C211yesmaximal, characteristic, complemented
3#21454736C211yesmaximal, characteristic, complemented
4#21447893C214yesmaximal, complemented
5#21447886C214yesmaximal, complemented
6#21447584C211yesmaximal, characteristic, complemented
7#21447569C211yesmaximal, characteristic, complemented
8C2×#16839575C211yesmaximal, characteristic, complemented
9C2×C61×#509C211yesmaximal, characteristic, complemented
10C2×#14519803—21no—
11C2×#14519799C2211yescharacteristic
12#16840529—21no—
13#16840329—21no—
14#16840203—21no—
15#16840202C2212yescomplemented
16#16840198—21no—
17#16840197C2212yescomplemented
18#16840192—22no—
19C2×#14519717C2211yescharacteristic
20#16840150—22no—
21#16840145C2212yescomplemented
22#16840137C2212yescomplemented
23#16839884—22no—
24#16839879C2214yescomplemented
25C2×#14519713C2211yescharacteristic
26#16839779—21no—
27#16839717C2211yescharacteristic
28#16839582C2214yescomplemented
29#16839579C2212yescomplemented
30#16839576C2212yescomplemented
31#16839575C2214yescomplemented
32C2×#135×C61C2211yescharacteristic
33C22×D8×C61C2211yescharacteristic
34C61×#509—22no—
35C61×#509C2214yes—
36C2×SD32×C61—21no—
37C2×D16×C61—21no—
38C2×M32×C61C2211yescharacteristic
39C2×#11758222—21no—
40#14519803—42no—
41#14519803—42no—
42#14519799—21no—
43#14519799—21no—
44#14519799—21no—
45#14519799—21no—
46C2×Dic244C2311yescharacteristic
47C2×#11758190—21no—
48C2×#11758200—22no—
49C2×D488C2311yescharacteristic
50C2×#11758199—22no—
51#14519784—24no—
52#14519784—42no—
53C2×#11758194—21no—
54C2×#11758194—42no—
55C2×#11758194D411yescharacteristic
56#14519773—22no—
57#14519764—22no—
58#14519748—42no—
59#14519746—42no—
60#14519736D411yescharacteristic
61#14519723—22no—
62#14519721—22no—
63#14519720—22no—
64#14519718—22no—
65#14519717—21no—
66#14519717C2312yes—
67C2×#11758185—21no—
68#14519713—21no—
69#14519713C2312yes—
70#14519704—24no—
71#14519687D412yes—
72C8×Dic61C2312yes—
73C2×#41×C61—21no—
74C22×D4×C61—21no—
75#135×C61—21no—
76#135×C61—21no—
77#135×C61D412yescomplemented
78#135×C61C2312yes—
79C2×Q16×C61D411yescharacteristic, complemented
80C2×SD16×C61—21no—
81C2×D8×C61—21no—
82C2×D8×C61—21no—
83C2×D8×C61—21no—
84C2×D8×C61—21no—
85C2×D8×C61—21no—
86C2×D8×C61D411yescharacteristic, complemented
87C2×D8×C61C2312yes—
88C22×C8×C61C2311yescharacteristic
89SD32×C61—24no—
90D16×C61—24no—
91M32×C61—21no—
92M32×C61C2312yes—
93C2×C16×C61—21no—
94C24×C61—41no—
95#41×C61—22no—
96#41×C61—22no—
97#41×C61—41no—
98#41×C61—41no—
99C2×Q8×C61—21no—
100C2×D4×C61—21no—
101C2×D4×C61—22no—
102C2×D4×C61—41no—
103C2×D4×C61—41no—
104C2×D4×C61—41no—
105C2×D4×C61—41no—
106C2×D4×C61—41no—
107C2×D4×C61—41no—
108C22×C4×C61—41no—
109C22×C4×C61C2×D411yescharacteristic
110C2×C22⋊D61—44no—
111C2×C22⋊D61—41no—
112#11758222—41no—
113#11758222—41no—
114#11758222—41no—
115#11758222—41no—
116C2×D244—21no—
117C2×C4×D61—41no—
118C2×Dic122—21no—
119Q16×C61—21no—
120Q16×C61C2×D412yes—
121SD16×C61—22no—
122SD16×C61—22no—
123D8×C61—21no—
124D8×C61—22no—
125D8×C61—22no—
126D8×C61—41no—
127D8×C61—41no—
128D8×C61C2×D412yes—
129C2×C8×C61C2×D411yescharacteristic
130C2×C8×C61C2×D411yescharacteristic
131C2×C8×C61C2×D411yescharacteristic
132C2×C8×C61#4111yescharacteristic
133C2×C8×C61#4111yescharacteristic
134C2×C8×C61C2411yescharacteristic
135C22⋊Dic61—21no—
136C22⋊Dic61—44no—
137#11758200—44no—
138#11758199—44no—
139C4⋊Dic61—21no—
140C4×Dic61C2×D412yes—
141#11758194—22no—
142#11758194—41no—
143#11758194—44no—
144#11758194D812yes—
145#11758194D812yes—
146C2×#92418—41no—
147C2×#92418—44no—
148C2×#92418C2×D412yes—
149Dic244—21no—
150Dic244—21no—
151D488—21no—
152D488—21no—
153#11758190—41no—
154#11758190—41no—
155C16×C61—22no—
156#11758185—22no—
157C23×C61—41no—
158C23×C61—42no—
159C23×C61—81no—
160C23×C61—81no—
161C23×C61—162no—
162C23×C61AGL(1,ℤ/8)11yescharacteristic
163C22×D61—41no—
164Q8×C61—22no—
165Q8×C61—41no—
166D4×C61—22no—
167D4×C61—42no—
168D4×C61—42no—
169D4×C61—42no—
170D4×C61—42no—
171D4×C61—41no—
172D4×C61—81no—
173D4×C61—81no—
174C2×C4×C61—41no—
175C2×C4×C61—42no—
176C2×C4×C61—42no—
177C2×C4×C61#12111yescharacteristic
178C2×C4×C61#12111yescharacteristic
179C2×C4×C61C2×D811yescharacteristic
180C2×C4×C61C2×D811yescharacteristic
181C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
182C2×C4×C61C22×D411yescharacteristic
183C22⋊D61—81no—
184C22⋊D61—81no—
185C22⋊D61—164no—
186C2×Dic61—22no—
187C2×Dic61—41no—
188D244—41no—
189D244—41no—
190C4×D61—81no—
191C4×D61—81no—
192Dic122—41no—
193Dic122—41no—
194C8×C61#12111yescharacteristic
195C8×C61#12111yescharacteristic
196C8×C61C22×D411yescharacteristic
197C8×C61C2×#4111yescharacteristic
198#92418—42no—
199#92418C2×D814yes—
200#33595—611nomaximal, Sylow
201C22×C61—41no—
202C22×C61—42no—
203C22×C61—84no—
204C22×C61—82no—
205C22×C61—161no—
206C22×C61—162no—
207C22×C61—162no—
208C22×C61#45911yescharacteristic
209C22×C61#45911yescharacteristic
210C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
211C2×D61—82no—
212C2×D61—81no—
213C4×C61—42no—
214C4×C61#45911yescharacteristic
215C4×C61#45911yescharacteristic
216C4×C61C2×#12111yescharacteristic
217C4×C61C22×D811yescharacteristic
218Dic61—42no—
219Dic61—81no—
220#3414—611no—
221C2×#509D6111yescharacteristic, complemented
222#3155—611no—
223#2957—611no—
224#2213—614no—
225#2212—614no—
226#2153—611no—
227#2148—611no—
228C2×#361—611no—
229C2×C61—21no—
230C2×C61—21no—
231C2×C61—21no—
232C2×C61—82no—
233C2×C61—162no—
234C2×C61C2×#45911yescharacteristic
235D61—161no—
236C2×AGL(1,ℤ/8)—1221no—
237C2×#135—611no—
238C2×#135C2×D6111yescharacteristic
239C22×D8C2×D6111yescharacteristic
240#568—1221no—
241#535—1221no—
242#510—612no—
243#510—1221no—
244#509—22no—
245#509—612no—
246#509—1221no—
247#509C2×D6114yes—
248#508—1222no—
249C2×SD32—21no—
250C2×D16—21no—
251C2×M32—611no—
252C2×M32C2×D6111yescharacteristic
253#496—1222no—
254#495—612no—
255#493—612no—
256#444—1222no—
257#443—614no—
258C2×#108—611no—
259#421—1221no—
260#405—611no—
261#364—614no—
262#362—612no—
263#361—614no—
264#361—612no—
265C61#3359511yescharacteristic, complemented, Sylow
266C2×#41—21no—
267C2×#41—1221no—
268C22×D4—21no—
269AGL(1,ℤ/8)—2442no—
270AGL(1,ℤ/8)—2442no—
271#135—21no—
272#135—21no—
273#135—1221no—
274#135—1221no—
275#135—1221no—
276#135—1221no—
277#135C22⋊D6112yescomplemented
278#135C22×D6112yes—
279C2×Q16—611no—
280C2×Q16C22⋊D6111yescharacteristic, complemented
281C2×SD16—21no—
282C2×SD16—1222no—
283C2×SD16—1221no—
284C2×D8—21no—
285C2×D8—21no—
286C2×D8—21no—
287C2×D8—21no—
288C2×D8—21no—
289C2×D8—611no—
290C2×D8—1222no—
291C2×D8C22⋊D6111yescharacteristic, complemented
292C2×D8C22×D6112yes—
293#131—1224no—
294#131—2442no—
295C2×M16—611no—
296C2×M16—1221no—
297C2×M16—2442no—
298C22×C8C22×D6111yescharacteristic
299#127—1222no—
300#124—1222no—
301#121—2442no—
302#120—2442no—
303#117—611no—
304Q32—1222no—
305SD32—24no—
306SD32—1222no—
307SD32—1222no—
308D16—24no—
309D16—1222no—
310M32—21no—
311M32—612no—
312M32—1221no—
313M32C22×D6112yes—
314C2×C16—21no—
315C2×C16—1221no—
316#108—612no—
317#108—1221no—
318#104—1224no—
319#97—612no—
320C4×C8—612no—
321C24—41no—
322#41—22no—
323#41—22no—
324#41—41no—
325#41—41no—
326#41—2441no—
327#41—2441no—
328#41—2441no—
329#41—2441no—
330C2×Q8—21no—
331C2×Q8—1221no—
332C2×D4—22no—
333C2×D4—21no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—1221no—
341C2×D4—2444no—
342C2×D4—2441no—
343C22×C4—41no—
344C22×C4—2441no—
345C22×C4D4×D6111yescharacteristic
346Q16—21no—
347Q16—1221no—
348Q16—1221no—
349Q16C2×C22⋊D6112yes—
350SD16—22no—
351SD16—22no—
352SD16—2444no—
353SD16—2441no—
354SD16—2441no—
355D8—22no—
356D8—22no—
357D8—21no—
358D8—41no—
359D8—41no—
360D8—1221no—
361D8—1221no—
362D8—2444no—
363D8C2×C22⋊D6112yes—
364M16—612no—
365M16—612no—
366M16—1222no—
367M16—2444no—
368M16—2441no—
369C2×C8—612no—
370C2×C8—2444no—
371C2×C8—2441no—
372C2×C8D4×D6111yescharacteristic
373C2×C8#1175822411yescharacteristic
374C2×C8#1175822411yescharacteristic
375C2×C8C2×C22⋊D6111yescharacteristic
376C2×C8C2×C22⋊D6111yescharacteristic
377C2×C8C23×D6111yescharacteristic
378C4⋊C4—1221no—
379C22⋊C4—1221no—
380C22⋊C4—2444no—
381C42—612no—
382C16—22no—
383C16—1222no—
384C23—41no—
385C23—42no—
386C23—81no—
387C23—81no—
388C23—162no—
389C23—2441no—
390C23#1451980111yescharacteristic
391Q8—22no—
392Q8—41no—
393Q8—2441no—
394Q8—2441no—
395D4—22no—
396D4—42no—
397D4—42no—
398D4—41no—
399D4—42no—
400D4—42no—
401D4—81no—
402D4—81no—
403D4—9764no—
404D4—4881no—
405D4—4881no—
406D4—2441no—
407D4—2441no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—1222no—
412C2×C4—4881no—
413C2×C4—4881no—
414C2×C4—2441no—
415C2×C4#1451974911yescharacteristic
416C2×C4#1451974911yescharacteristic
417C2×C4D8×D6111yescharacteristic
418C2×C4D8×D6111yescharacteristic
419C2×C4#1451980111yescharacteristic
420C2×C4C2×D4×D6111yescharacteristic
421C8—614no—
422C8—2442no—
423C8#1451974911yescharacteristic
424C8#1451974911yescharacteristic
425C8C22×C22⋊D6111yescharacteristic
426C8C2×#1175822411yescharacteristic
427C22—41no—
428C22—42no—
429C22—84no—
430C22—82no—
431C22—161no—
432C22—162no—
433C22—162no—
434C22—4882no—
435C22—4881no—
436C22#1683995511yescharacteristic
437C22#1683995511yescharacteristic
438C22C2×#1451980111yescharacteristic
439C4—42no—
440C4—4881no—
441C4—2442no—
442C4#1683995511yescharacteristic
443C4#1683995511yescharacteristic
444C4C2×#1451974911yescharacteristic
445C4C2×D8×D6111yescharacteristic
446C2—21no—
447C2—21no—
448C2—21no—
449C2—82no—
450C2—162no—
451C2—9761no—
452C2C2×#1683995511yescharacteristic
subgroupG/H
centerC2C2×#16839955
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#509C2
socleC2×C61C2×#459
hypercenterC2×#509D61
hypocenterC61#33595
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×C8 ◁ C2×#509

Complements

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

NH ≅ G/Ntype
1. #21458953C2semidirect
2. #21456300C2semidirect
3. #21454736C2semidirect
4. #21447893C2semidirect
5. #21447886C2semidirect
6. #21447584C2semidirect
7. #21447569C2semidirect
8. C2×#16839575C2semidirect
9. C2×C61×#509C2semidirect
15. #16840202C22semidirect
17. #16840197C22semidirect
21. #16840145C22semidirect
22. #16840137C22semidirect
24. #16839879C22semidirect
28. #16839582C22semidirect
29. #16839579C22semidirect
30. #16839576C22semidirect
31. #16839575C22semidirect
77. #135×C61D4semidirect
79. C2×Q16×C61D4semidirect
86. C2×D8×C61D4semidirect
221. C2×#509D61semidirect
265. C61#33595semidirect
277. #135C22⋊D61semidirect
280. C2×Q16C22⋊D61semidirect
291. C2×D8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214589532
#214563002
#214547362
#214478932
#214478862
#214475842
#214475692
C2×#168395752
C2×C61×#5092
#3359561