Finite Group Databasebeta

← #29834414

Subgroups — #29834414

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
2C2×#16840197C211yesmaximal, characteristic, complemented
3#21456305C211yesmaximal, characteristic, complemented
4#21447895C214yesmaximal, complemented
5#21447888C214yesmaximal, complemented
6#21447585C211yesmaximal, characteristic, complemented
7#21447566C211yesmaximal, characteristic, complemented
8C2×#16839574C211yesmaximal, characteristic, complemented
9C61×#2957C211yesmaximal, characteristic, complemented
10C2×#14519805—21no—
11C2×#14519799C2211yescharacteristic
12C22×D488C2211yescharacteristic
13#16840331—21no—
14#16840203—21no—
15#16840202C2212yescomplemented
16#16840198—21no—
17#16840197—22no—
18#16840197C2214yes—
19#16840197C2212yescomplemented
20#16840192—22no—
21C2×#14519720—21no—
22C2×#14519718—21no—
23C2×#14519717C2211yescharacteristic
24C2×#14519717C2211yescharacteristic
25#16840151—22no—
26#16840146C2212yescomplemented
27#16840139C2212yescomplemented
28#16839583C2214yescomplemented
29#16839577C2212yescomplemented
30#16839574C2214yescomplemented
31#16839574C2212yescomplemented
32C2×#135×C61C2211yescharacteristic
33C61×#568—21no—
34C61×#444—22no—
35C61×#443C2214yescomplemented
36C2×#108×C61C2211yescharacteristic
37C61×#421—21no—
38C61×#405C2211yescharacteristic
39C2×#11758222—21no—
40C22×D244—21no—
41#14519805—42no—
42#14519805—42no—
43#14519799—21no—
44#14519799—21no—
45#14519799—21no—
46#14519799—21no—
47C2×Dic244C2311yescharacteristic
48C2×#11758190—21no—
49C2×#11758190—22no—
50C2×D488—21no—
51C2×D488—21no—
52C2×D488—21no—
53C2×D488—21no—
54C2×D488—21no—
55C2×D488—22no—
56C2×D488C2312yes—
57C2×D488C2311yescharacteristic
58#14519774—22no—
59#14519765—22no—
60#14519752—42no—
61#14519747—42no—
62#14519723—22no—
63#14519721—22no—
64#14519720—22no—
65#14519720—24no—
66#14519718—22no—
67#14519718—24no—
68#14519717—21no—
69#14519717—21no—
70#14519717C2312yes—
71#14519717C2312yes—
72C2×#11758185—21no—
73C2×#11758185—21no—
74C2×#41×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×C61D411yescharacteristic, complemented
82#131×C61—24no—
83#131×C61—42no—
84C2×M16×C61—21no—
85C2×M16×C61—42no—
86C2×M16×C61D411yescharacteristic
87C22×C8×C61C2311yescharacteristic
88#117×C61D411yescharacteristic
89#108×C61—21no—
90#108×C61C2312yes—
91#104×C61—24no—
92#97×C61D412yes—
93C4×C8×C61C2312yes—
94C23×D61—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—41no—
102C22×C4×C61—41no—
103C22×C4×C61C2×D411yescharacteristic
104C2×C22⋊D61—41no—
105#11758222—41no—
106#11758222—41no—
107#11758222—41no—
108#11758222—41no—
109C2×D244—22no—
110C2×D244—21no—
111C2×D244—41no—
112C2×D244—41no—
113C2×D244—41no—
114C2×D244—41no—
115C2×D244—41no—
116C2×D244—44no—
117C2×C4×D61—41no—
118C2×Dic122—21no—
119Q16×C61—21no—
120Q16×C61C2×D412yes—
121SD16×C61—22no—
122SD16×C61—22no—
123D8×C61—21no—
124D8×C61C2×D412yes—
125M16×C61—22no—
126M16×C61—41no—
127M16×C61—44no—
128M16×C61D812yes—
129M16×C61D812yes—
130C2×C8×C61—41no—
131C2×C8×C61—44no—
132C2×C8×C61C2×D411yescharacteristic
133C2×C8×C61C2×D411yescharacteristic
134C2×C8×C61C2×D411yescharacteristic
135C2×C8×C61C2×D412yes—
136C2×C8×C61#4111yescharacteristic
137C2×C8×C61#4111yescharacteristic
138C2×C8×C61C2411yescharacteristic
139C4⋊C4×C61—21no—
140C22⋊C4×C61—21no—
141C42×C61C2×D412yes—
142#11758198—44no—
143Dic244—21no—
144Dic244—21no—
145D488—22no—
146D488—21no—
147D488—22no—
148D488—21no—
149D488—41no—
150D488—41no—
151D488—44no—
152#11758190—41no—
153#11758190—41no—
154#11758190—44no—
155#11758185—22no—
156#11758185—22no—
157C23×C61—41no—
158C23×C61AGL(1,ℤ/8)11yescharacteristic
159C22×D61—42no—
160C22×D61—41no—
161C22×D61—81no—
162C22×D61—81no—
163C22×D61—162no—
164Q8×C61—22no—
165Q8×C61—41no—
166D4×C61—22no—
167D4×C61—42no—
168D4×C61—42no—
169D4×C61—41no—
170C2×C4×C61—22no—
171C2×C4×C61—41no—
172C2×C4×C61—42no—
173C2×C4×C61—42no—
174C2×C4×C61#12111yescharacteristic
175C2×C4×C61#12111yescharacteristic
176C2×C4×C61C2×D811yescharacteristic
177C2×C4×C61C2×D811yescharacteristic
178C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
179C2×C4×C61C22×D411yescharacteristic
180C22⋊D61—81no—
181C22⋊D61—81no—
182C2×Dic61—41no—
183D244—41no—
184D244—42no—
185D244—42no—
186D244—41no—
187D244—81no—
188D244—81no—
189D244—164no—
190C4×D61—81no—
191C4×D61—81no—
192Dic122—41no—
193Dic122—41no—
194C8×C61—42no—
195C8×C61#12111yescharacteristic
196C8×C61#12111yescharacteristic
197C8×C61C2×D814yes—
198C8×C61C22×D411yescharacteristic
199C8×C61C2×#4111yescharacteristic
200#33595—611nomaximal, Sylow
201C22×C61—41no—
202C22×C61—42no—
203C22×C61—161no—
204C22×C61#45911yescharacteristic
205C22×C61#45911yescharacteristic
206C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
207C2×D61—82no—
208C2×D61—84no—
209C2×D61—81no—
210C2×D61—82no—
211C2×D61—162no—
212C2×D61—162no—
213C4×C61—42no—
214C4×C61—42no—
215C4×C61#45911yescharacteristic
216C4×C61#45911yescharacteristic
217C4×C61C2×#12111yescharacteristic
218C4×C61C22×D811yescharacteristic
219Dic61—81no—
220#3414—611no—
221C2×#509—611no—
222#3155—611no—
223#2957D6111yescharacteristic, complemented
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×C61C2×#45911yescharacteristic
234D61—161no—
235D61—162no—
236C2×AGL(1,ℤ/8)—1221no—
237C2×#135—611no—
238C2×#135C2×D6111yescharacteristic
239C22×D8—611no—
240#568—21no—
241#535—1221no—
242#510—612no—
243#510—1221no—
244#509—614no—
245#509—612no—
246#509—1222no—
247#509—1221no—
248#508—1222no—
249C2×SD32—1221no—
250C2×D16—1221no—
251C2×M32—611no—
252C2×M32—611no—
253#496—1222no—
254#495—612no—
255#493—612no—
256#444—22no—
257#443C2×D6114yescomplemented
258C2×#108C2×D6111yescharacteristic
259#421—21no—
260#405C2×D6111yescharacteristic
261#364—614no—
262#362—612no—
263#361—614no—
264#361—612no—
265C61#3359511yescharacteristic, complemented, Sylow
266C2×#41—21no—
267C2×#41—1221no—
268C22×D4—1221no—
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#135D24412yescomplemented
278#135C22×D6112yes—
279C2×Q16—611no—
280C2×Q16D24411yescharacteristic, complemented
281C2×SD16—21no—
282C2×SD16—1221no—
283C2×SD16—1222no—
284C2×D8—612no—
285C2×D8—611no—
286C2×D8—1221no—
287C2×D8—1221no—
288C2×D8—1221no—
289C2×D8—1221no—
290C2×D8—1221no—
291C2×D8—1222no—
292C2×D8D24411yescharacteristic, complemented
293#131—24no—
294#131—42no—
295C2×M16—21no—
296C2×M16—42no—
297C2×M16C22⋊D6111yescharacteristic
298C22×C8C22×D6111yescharacteristic
299#127—1222no—
300#124—1222no—
301#121—2442no—
302#120—2442no—
303#117C22⋊D6111yescharacteristic
304Q32—1222no—
305SD32—1224no—
306SD32—1222no—
307SD32—1222no—
308D16—1224no—
309D16—1222no—
310M32—612no—
311M32—612no—
312M32—1221no—
313M32—1221no—
314C2×C16—1221no—
315C2×C16—1221no—
316#108—21no—
317#108C22×D6112yes—
318#104—24no—
319#97C22⋊D6112yes—
320C4×C8C22×D6112yes—
321C24—2441no—
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—21no—
333C2×D4—41no—
334C2×D4—1222no—
335C2×D4—1221no—
336C2×D4—2444no—
337C2×D4—2441no—
338C2×D4—2441no—
339C2×D4—2441no—
340C2×D4—2441no—
341C2×D4—2441no—
342C2×D4—2441no—
343C22×C4—41no—
344C22×C4—2441no—
345C22×C4C2×C22⋊D6111yescharacteristic
346Q16—21no—
347Q16—1221no—
348Q16—1221no—
349Q16C2×D24412yes—
350SD16—22no—
351SD16—22no—
352SD16—2441no—
353SD16—2441no—
354SD16—2444no—
355D8—21no—
356D8—1222no—
357D8—1221no—
358D8—1222no—
359D8—1221no—
360D8—2441no—
361D8—2441no—
362D8—2444no—
363D8C2×D24412yes—
364M16—22no—
365M16—41no—
366M16—44no—
367M16#1175819912yes—
368M16#1175819912yes—
369C2×C8—41no—
370C2×C8—44no—
371C2×C8C2×D24411yescharacteristic
372C2×C8C2×D24411yescharacteristic
373C2×C8#1175822211yescharacteristic
374C2×C8#1175822211yescharacteristic
375C2×C8C2×C22⋊D6111yescharacteristic
376C2×C8C2×C22⋊D6112yes—
377C2×C8C23×D6111yescharacteristic
378C4⋊C4—21no—
379C22⋊C4—21no—
380C22⋊C4—2444no—
381C42C2×C22⋊D6112yes—
382C16—1222no—
383C16—1222no—
384C23—41no—
385C23—9762no—
386C23—4881no—
387C23—4881no—
388C23—2442no—
389C23—2441no—
390C23#1451980211yescharacteristic
391Q8—22no—
392Q8—41no—
393Q8—2441no—
394Q8—2441no—
395D4—22no—
396D4—42no—
397D4—42no—
398D4—41no—
399D4—9764no—
400D4—4881no—
401D4—4881no—
402D4—4881no—
403D4—4881no—
404D4—2442no—
405D4—2441no—
406D4—2441no—
407D4—2442no—
408C2×C4—22no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—4881no—
413C2×C4—4881no—
414C2×C4—2441no—
415C2×C4#1451975111yescharacteristic
416C2×C4#1451975111yescharacteristic
417C2×C4C2×#1175819911yescharacteristic
418C2×C4C2×#1175819911yescharacteristic
419C2×C4#1451980211yescharacteristic
420C2×C4C22×C22⋊D6111yescharacteristic
421C8—42no—
422C8#1451975111yescharacteristic
423C8#1451975111yescharacteristic
424C8C2×#1175819914yes—
425C8C22×D24411yescharacteristic
426C8C2×#1175822211yescharacteristic
427C22—41no—
428C22—42no—
429C22—161no—
430C22—9762no—
431C22—9762no—
432C22—4882no—
433C22—4884no—
434C22—4881no—
435C22—4882no—
436C22#1683995711yescharacteristic
437C22#1683995711yescharacteristic
438C22C2×#1451980211yescharacteristic
439C4—42no—
440C4—42no—
441C4—4881no—
442C4#1683995711yescharacteristic
443C4#1683995711yescharacteristic
444C4C2×#1451975111yescharacteristic
445C4C22×#1175819911yescharacteristic
446C2—21no—
447C2—21no—
448C2—21no—
449C2—82no—
450C2—9762no—
451C2—9761no—
452C2C2×#1683995711yescharacteristic
subgroupG/H
centerC2C2×#16839957
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2957C2
socleC2×C61C2×#459
hypercenter#2957D61
hypocenterC61#33595
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2957

Complements

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

NH ≅ G/Ntype
1. #21458953C2semidirect
2. C2×#16840197C2semidirect
3. #21456305C2semidirect
4. #21447895C2semidirect
5. #21447888C2semidirect
6. #21447585C2semidirect
7. #21447566C2semidirect
8. C2×#16839574C2semidirect
9. C61×#2957C2semidirect
15. #16840202C22semidirect
19. #16840197C22semidirect
26. #16840146C22semidirect
27. #16840139C22semidirect
28. #16839583C22semidirect
29. #16839577C22semidirect
30. #16839574C22semidirect
31. #16839574C22semidirect
35. C61×#443C22semidirect
77. #135×C61D4semidirect
79. C2×Q16×C61D4semidirect
81. C2×D8×C61D4semidirect
223. #2957D61semidirect
257. #443C2×D61semidirect
265. C61#33595semidirect
277. #135D244semidirect
280. C2×Q16D244semidirect
292. C2×D8D244semidirect

Maximal subgroups

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

subgroupindex
#214589532
C2×#168401972
#214563052
#214478952
#214478882
#214475852
#214475662
C2×#168395742
C61×#29572
#3359561