Finite Group Databasebeta

← #29834426

Subgroups — #29834426

Order 15616.

674 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458953C211yesmaximal, characteristic
2C2×#16840205C211yesmaximal, characteristic, complemented
3#21456314C211yesmaximal, characteristic, complemented
4#21454735C211yesmaximal, characteristic, complemented
5#21447899C214yesmaximal, complemented
6#21447887C214yesmaximal, complemented
7#21447584C211yesmaximal, characteristic, complemented
8C2×#16839577C211yesmaximal, characteristic, complemented
9C61×#2148C211yesmaximal, characteristic, complemented
10C2×#14519807—21no—
11C2×#14519799C2211yescharacteristic
12C22×Dic244C2211yescharacteristic
13#16840528—21no—
14#16840333—21no—
15#16840205—22no—
16#16840205C2214yes—
17#16840203C2212yes—
18#16840202—21no—
19#16840198C2212yes—
20#16840197—21no—
21#16840192—22no—
22C2×Dic488—21no—
23C2×#14519722—21no—
24C2×#14519717C2211yescharacteristic
25#16840153—22no—
26#16840145C2212yescomplemented
27#16840140C2212yescomplemented
28#16839883—22no—
29#16839878C2214yes—
30#16839778—21no—
31#16839717C2211yescharacteristic
32#16839582C2214yescomplemented
33#16839577C2214yescomplemented
34C2×#135×C61C2211yescharacteristic
35C2×M32×C61C2211yescharacteristic
36C2×#108×C61C2211yescharacteristic
37C61×#362C2212yescomplemented
38C61×#361C2212yescomplemented
39C2×#11758222—21no—
40C22×Dic122—21no—
41#14519807—42no—
42#14519807—42no—
43#14519799—21no—
44#14519799—21no—
45#14519799D412yes—
46#14519799C2312yes—
47C2×Dic244—21no—
48C2×Dic244—21no—
49C2×Dic244—21no—
50C2×Dic244—21no—
51C2×Dic244—21no—
52C2×Dic244D411yescharacteristic
53C2×Dic244C2312yes—
54C2×#11758202—22no—
55C2×#11758190—21no—
56C2×#11758200—22no—
57C2×D488D411yescharacteristic
58#14519785—24no—
59#14519783—42no—
60C2×#11758194D411yescharacteristic
61M16×D61—42no—
62#14519775—22no—
63#14519764—22no—
64#14519758—42no—
65#14519756—42no—
66#14519736D411yescharacteristic
67Dic488—24no—
68#14519723—22no—
69#14519722—24no—
70#14519721—22no—
71#14519720—22no—
72#14519718—22no—
73#14519717—21no—
74#14519717C2312yes—
75C2×#11758185—21no—
76#14519703—24no—
77#14519687D412yes—
78C8×Dic61C2312yes—
79C2×#41×C61—21no—
80#135×C61—21no—
81#135×C61—21no—
82#135×C61—21no—
83#135×C61—21no—
84C2×Q16×C61C2311yescharacteristic
85C2×SD16×C61—21no—
86C2×D8×C61C2311yescharacteristic
87C2×M16×C61—21no—
88C22×C8×C61C2311yescharacteristic
89M32×C61—21no—
90M32×C61C2312yes—
91C2×C16×C61—21no—
92#108×C61—21no—
93#108×C61C2312yes—
94#41×C61—41no—
95#41×C61—41no—
96#41×C61—41no—
97#41×C61—41no—
98C2×Q8×C61—21no—
99C2×D4×C61—21no—
100C2×D4×C61—41no—
101C22×C4×C61—41no—
102C22×C4×C61C2×D411yescharacteristic
103C2×C22⋊D61—41no—
104C22×Dic61—41no—
105#11758222—22no—
106#11758222—22no—
107#11758222—41no—
108#11758222—41no—
109C2×D244—21no—
110C2×C4×D61—41no—
111C2×Dic122—21no—
112C2×Dic122—22no—
113C2×Dic122—41no—
114C2×Dic122—41no—
115C2×Dic122—41no—
116C2×Dic122—41no—
117C2×Dic122—41no—
118Q16×C61—21no—
119Q16×C61—21no—
120SD16×C61—41no—
121SD16×C61—41no—
122D8×C61—21no—
123D8×C61—21no—
124M16×C61—22no—
125M16×C61—41no—
126C2×C8×C61—41no—
127C2×C8×C61C2×D411yescharacteristic
128C2×C8×C61C2×D411yescharacteristic
129C2×C8×C61C2×D411yescharacteristic
130C2×C8×C61#4111yescharacteristic
131C2×C8×C61#4111yescharacteristic
132C2×C8×C61C2411yescharacteristic
133C22⋊Dic61—21no—
134C22⋊Dic61—44no—
135#11758202—44no—
136#11758200—44no—
137C4⋊Dic61—21no—
138#11758196—44no—
139C4×Dic61C2×D412yes—
140#11758194D812yes—
141#11758194D812yes—
142C2×#92418C2×D412yes—
143Dic244—21no—
144Dic244—22no—
145Dic244—22no—
146Dic244—41no—
147Dic244—41no—
148Dic244C2×D412yes—
149D488—21no—
150D488C2×D412yes—
151#11758190—22no—
152#11758190—22no—
153#11758189—44no—
154C8×D61—44no—
155C16×C61—22no—
156#11758185—22no—
157C23×C61—41no—
158C23×C61AGL(1,ℤ/8)11yescharacteristic
159C22×D61—41no—
160Q8×C61—41no—
161Q8×C61—41no—
162D4×C61—41no—
163D4×C61—41no—
164D4×C61—81no—
165D4×C61—81no—
166C2×C4×C61—41no—
167C2×C4×C61—81no—
168C2×C4×C61—81no—
169C2×C4×C61#12111yescharacteristic
170C2×C4×C61#12111yescharacteristic
171C2×C4×C61C2×D811yescharacteristic
172C2×C4×C61C2×D811yescharacteristic
173C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
174C2×C4×C61C22×D411yescharacteristic
175C22⋊D61—42no—
176C22⋊D61—42no—
177C2×Dic61—22no—
178C2×Dic61—41no—
179C2×Dic61—42no—
180C2×Dic61—81no—
181C2×Dic61—81no—
182D244—22no—
183D244—41no—
184C4×D61—42no—
185C4×D61—42no—
186Dic122—22no—
187Dic122—41no—
188Dic122—42no—
189Dic122—42no—
190Dic122—81no—
191Dic122—81no—
192C8×C61—42no—
193C8×C61#12111yescharacteristic
194C8×C61#12111yescharacteristic
195C8×C61C22×D411yescharacteristic
196C8×C61C2×#4111yescharacteristic
197#92418C2×D814yes—
198#33596—611nomaximal, Sylow
199C22×C61—41no—
200C22×C61—82no—
201C22×C61—81no—
202C22×C61#45911yescharacteristic
203C22×C61#45911yescharacteristic
204C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
205C2×D61—42no—
206C2×D61—161no—
207C4×C61—81no—
208C4×C61#45911yescharacteristic
209C4×C61#45911yescharacteristic
210C4×C61C2×#12111yescharacteristic
211C4×C61C22×D811yescharacteristic
212Dic61—42no—
213Dic61—42no—
214Dic61—82no—
215#3414—611no—
216C2×#510—611no—
217#3156—611no—
218#2957—611no—
219#2214—614no—
220#2212—614no—
221#2153—611no—
222#2148D6111yescharacteristic, complemented
223C2×#362—611no—
224C2×C61—21no—
225C2×C61—21no—
226C2×C61—21no—
227C2×C61—161no—
228C2×C61C2×#45911yescharacteristic
229D61—82no—
230C2×#137—1221no—
231C2×#135—611no—
232C2×#135C2×D6111yescharacteristic
233C22×Q16—611no—
234#568—1221no—
235#536—1221no—
236#510—612no—
237#510—614no—
238#510—1222no—
239#510—1221no—
240#509—612no—
241#509—1221no—
242#508—1222no—
243C2×Q32—1221no—
244C2×SD32—1221no—
245C2×M32—611no—
246C2×M32C2×D6111yescharacteristic
247#497—1222no—
248#495—612no—
249#494—612no—
250#444—1222no—
251#443—614no—
252C2×#108C2×D6111yescharacteristic
253#421—1221no—
254#405—611no—
255#364—614no—
256#362—614no—
257#362C2×D6112yescomplemented
258#361C2×D6112yescomplemented
259C61#3359611yescharacteristic, complemented, Sylow
260C2×#41—21no—
261C2×#41—1221no—
262C22×Q8—1221no—
263#137—2442no—
264#137—2442no—
265#135—21no—
266#135—21no—
267#135—21no—
268#135—21no—
269#135—612no—
270#135—612no—
271#135—1221no—
272#135—1221no—
273C2×Q16—611no—
274C2×Q16—612no—
275C2×Q16—1222no—
276C2×Q16—1221no—
277C2×Q16—1221no—
278C2×Q16—1221no—
279C2×Q16—1221no—
280C2×Q16—1221no—
281C2×Q16C22×D6111yescharacteristic
282C2×SD16—21no—
283C2×SD16—1222no—
284C2×SD16—1221no—
285C2×D8—611no—
286C2×D8C22×D6111yescharacteristic
287#131—1224no—
288#131—2442no—
289C2×M16—21no—
290C2×M16—611no—
291C2×M16—2442no—
292C22×C8C22×D6111yescharacteristic
293C4⋊Q8—1222no—
294#124—1222no—
295#123—2442no—
296C22⋊Q8—2442no—
297#117—611no—
298Q32—1224no—
299Q32—1222no—
300SD32—1224no—
301SD32—1222no—
302SD32—1222no—
303D16—1222no—
304M32—21no—
305M32—612no—
306M32—1221no—
307M32C22×D6112yes—
308C2×C16—21no—
309C2×C16—1221no—
310#108—21no—
311#108C22×D6112yes—
312#104—1224no—
313#97—612no—
314C4×C8—612no—
315#41—41no—
316#41—41no—
317#41—41no—
318#41—41no—
319#41—1222no—
320#41—1222no—
321#41—2441no—
322#41—2441no—
323C2×Q8—21no—
324C2×Q8—1222no—
325C2×Q8—1221no—
326C2×Q8—2441no—
327C2×Q8—2441no—
328C2×Q8—2441no—
329C2×Q8—2441no—
330C2×Q8—2441no—
331C2×D4—21no—
332C2×D4—41no—
333C2×D4—1221no—
334C2×D4—2441no—
335C22×C4—41no—
336C22×C4—2441no—
337C22×C4—2441no—
338C22×C4D4×D6111yescharacteristic
339Q16—21no—
340Q16—21no—
341Q16—612no—
342Q16—1222no—
343Q16—1222no—
344Q16—1221no—
345Q16—2444no—
346Q16—2441no—
347Q16—2441no—
348SD16—41no—
349SD16—41no—
350SD16—1222no—
351SD16—1222no—
352SD16—2444no—
353D8—21no—
354D8—21no—
355D8—612no—
356D8—1221no—
357M16—22no—
358M16—41no—
359M16—612no—
360M16—612no—
361M16—2444no—
362C2×C8—41no—
363C2×C8—612no—
364C2×C8—2444no—
365C2×C8#1175822211yescharacteristic
366C2×C8#1175822211yescharacteristic
367C2×C8D4×D6111yescharacteristic
368C2×C8D4×D6111yescharacteristic
369C2×C8D4×D6111yescharacteristic
370C2×C8C23×D6111yescharacteristic
371C4⋊C4—1221no—
372C4⋊C4—2444no—
373C22⋊C4—1221no—
374C22⋊C4—2444no—
375C42—612no—
376C16—22no—
377C16—1222no—
378C23—41no—
379C23—2441no—
380C23#1451980411yescharacteristic
381Q8—41no—
382Q8—41no—
383Q8—1222no—
384Q8—4881no—
385Q8—4881no—
386Q8—2442no—
387Q8—2442no—
388Q8—2441no—
389D4—41no—
390D4—41no—
391D4—81no—
392D4—81no—
393D4—1222no—
394D4—2442no—
395D4—2442no—
396D4—2441no—
397C2×C4—41no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—1222no—
401C2×C4—4881no—
402C2×C4—4881no—
403C2×C4—2442no—
404C2×C4—2441no—
405C2×C4—2442no—
406C2×C4—2442no—
407C2×C4#1451975011yescharacteristic
408C2×C4#1451975011yescharacteristic
409C2×C4D8×D6111yescharacteristic
410C2×C4D8×D6111yescharacteristic
411C2×C4#1451980411yescharacteristic
412C2×C4C2×D4×D6111yescharacteristic
413C8—42no—
414C8—614no—
415C8#1451975011yescharacteristic
416C8#1451975011yescharacteristic
417C8C2×D4×D6111yescharacteristic
418C8C2×#1175822211yescharacteristic
419C22—41no—
420C22—82no—
421C22—81no—
422C22—9761no—
423C22—2442no—
424C22#1683995611yescharacteristic
425C22#1683995611yescharacteristic
426C22C2×#1451980411yescharacteristic
427C4—81no—
428C4—4882no—
429C4—2442no—
430C4—2442no—
431C4#1683995611yescharacteristic
432C4#1683995611yescharacteristic
433C4C2×#1451975011yescharacteristic
434C4C2×D8×D6111yescharacteristic
435C2—21no—
436C2—21no—
437C2—21no—
438C2—161no—
439C2—4882no—
440C2C2×#1683995611yescharacteristic
subgroupG/H
centerC2C2×#16839956
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2148C2
socleC2×C61C2×#459
prime radical#21458953C2
hypercenter#2148D61
hypocenterC61#33596
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×C8 ◁ #2148

Complements

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

NH ≅ G/Ntype
2. C2×#16840205C2semidirect
3. #21456314C2semidirect
4. #21454735C2semidirect
5. #21447899C2semidirect
6. #21447887C2semidirect
7. #21447584C2semidirect
8. C2×#16839577C2semidirect
9. C61×#2148C2semidirect
26. #16840145C22semidirect
27. #16840140C22semidirect
32. #16839582C22semidirect
33. #16839577C22semidirect
37. C61×#362C22semidirect
38. C61×#361C22semidirect
222. #2148D61semidirect
257. #362C2×D61semidirect
258. #361C2×D61semidirect
259. C61#33596semidirect

Maximal subgroups

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

subgroupindex
#214589532
C2×#168402052
#214563142
#214547352
#214478992
#214478872
#214475842
C2×#168395772
C61×#21482
#3359661