Finite Group Databasebeta

← #29834420

Subgroups — #29834420

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
2#21456308C211yesmaximal, characteristic, complemented
3#21454736C211yesmaximal, characteristic, complemented
4#21447898C214yesmaximal, complemented
5#21447890C214yesmaximal, complemented
6#21447584C211yesmaximal, characteristic, complemented
7#21447569C211yesmaximal, characteristic, complemented
8C2×#16839578C211yesmaximal, characteristic, complemented
9C2×C61×#510C211yesmaximal, characteristic, complemented
10C2×#14519808—21no—
11C2×#14519799C2211yescharacteristic
12#16840529—21no—
13#16840336—21no—
14#16840203—21no—
15#16840202C2212yes—
16#16840198—21no—
17#16840197C2212yes—
18#16840192—22no—
19C2×#14519717C2211yescharacteristic
20#16840155—22no—
21#16840147C2212yescomplemented
22#16840140C2212yescomplemented
23#16839884—22no—
24#16839879C2214yes—
25C2×#14519713C2211yescharacteristic
26#16839779—21no—
27#16839717C2211yescharacteristic
28#16839582C2214yescomplemented
29#16839579C2212yescomplemented
30#16839578C2214yescomplemented
31#16839576C2212yescomplemented
32C2×#135×C61C2211yescharacteristic
33C22×Q16×C61C2211yescharacteristic
34C61×#510—22no—
35C61×#510C2214yes—
36C2×Q32×C61—21no—
37C2×SD32×C61—21no—
38C2×M32×C61C2211yescharacteristic
39C2×#11758222—21no—
40#14519808—42no—
41#14519808—42no—
42#14519799—21no—
43#14519799—21no—
44#14519799—21no—
45#14519799—21no—
46C2×Dic244C2311yescharacteristic
47C2×#11758202—22no—
48C2×#11758190—21no—
49C2×#11758201—22no—
50C2×D488C2311yescharacteristic
51#14519784—24no—
52#14519784—42no—
53C2×#11758194—21no—
54C2×#11758194—42no—
55C2×#11758194D411yescharacteristic
56#14519775—22no—
57#14519766—22no—
58#14519760—42no—
59#14519753—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×Q8×C61—21no—
75#135×C61—21no—
76#135×C61—21no—
77#135×C61D412yes—
78#135×C61C2312yes—
79C2×Q16×C61—21no—
80C2×Q16×C61—21no—
81C2×Q16×C61—21no—
82C2×Q16×C61—21no—
83C2×Q16×C61—21no—
84C2×Q16×C61D411yescharacteristic
85C2×Q16×C61C2312yes—
86C2×SD16×C61—21no—
87C2×D8×C61D411yescharacteristic
88C22×C8×C61C2311yescharacteristic
89Q32×C61—24no—
90SD32×C61—24no—
91M32×C61—21no—
92M32×C61C2312yes—
93C2×C16×C61—21no—
94#41×C61—22no—
95#41×C61—22no—
96#41×C61—41no—
97#41×C61—41no—
98C2×Q8×C61—21no—
99C2×Q8×C61—22no—
100C2×Q8×C61—41no—
101C2×Q8×C61—41no—
102C2×Q8×C61—41no—
103C2×Q8×C61—41no—
104C2×Q8×C61—41no—
105C2×D4×C61—21no—
106C2×D4×C61—41no—
107C22×C4×C61—41no—
108C22×C4×C61—41no—
109C22×C4×C61C2×D411yescharacteristic
110C2×C22⋊D61—41no—
111#11758222—41no—
112#11758222—41no—
113#11758222—41no—
114#11758222—41no—
115C2×D244—21no—
116C2×C4×D61—41no—
117C2×Dic122—21no—
118Q16×C61—21no—
119Q16×C61—22no—
120Q16×C61—22no—
121Q16×C61—41no—
122Q16×C61—41no—
123Q16×C61C2×D412yes—
124SD16×C61—22no—
125SD16×C61—22no—
126D8×C61—21no—
127D8×C61C2×D412yes—
128C2×C8×C61C2×D411yescharacteristic
129C2×C8×C61C2×D411yescharacteristic
130C2×C8×C61C2×D411yescharacteristic
131C2×C8×C61#4111yescharacteristic
132C2×C8×C61#4111yescharacteristic
133C2×C8×C61C2411yescharacteristic
134C22⋊Dic61—21no—
135#11758202—44no—
136#11758201—44no—
137#11758198—44no—
138C4⋊Dic61—21no—
139#11758196—44no—
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×C61AGL(1,ℤ/8)11yescharacteristic
159C22×D61—41no—
160Q8×C61—22no—
161Q8×C61—41no—
162Q8×C61—42no—
163Q8×C61—42no—
164Q8×C61—81no—
165Q8×C61—81no—
166D4×C61—22no—
167D4×C61—42no—
168D4×C61—42no—
169D4×C61—41no—
170C2×C4×C61—41no—
171C2×C4×C61—42no—
172C2×C4×C61—42no—
173C2×C4×C61—42no—
174C2×C4×C61—81no—
175C2×C4×C61—81no—
176C2×C4×C61#12111yescharacteristic
177C2×C4×C61#12111yescharacteristic
178C2×C4×C61C2×D811yescharacteristic
179C2×C4×C61C2×D811yescharacteristic
180C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
181C2×C4×C61C22×D411yescharacteristic
182C22⋊D61—81no—
183C22⋊D61—81no—
184C2×Dic61—22no—
185C2×Dic61—41no—
186D244—41no—
187D244—41no—
188C4×D61—81no—
189C4×D61—81no—
190Dic122—41no—
191Dic122—41no—
192C8×C61#12111yescharacteristic
193C8×C61#12111yescharacteristic
194C8×C61C22×D411yescharacteristic
195C8×C61C2×#4111yescharacteristic
196#92418—42no—
197#92418C2×D814yes—
198#33596—611nomaximal, Sylow
199C22×C61—41no—
200C22×C61—42no—
201C22×C61—161no—
202C22×C61#45911yescharacteristic
203C22×C61#45911yescharacteristic
204C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
205C2×D61—82no—
206C2×D61—81no—
207C4×C61—42no—
208C4×C61—82no—
209C4×C61#45911yescharacteristic
210C4×C61#45911yescharacteristic
211C4×C61C2×#12111yescharacteristic
212C4×C61C22×D811yescharacteristic
213Dic61—42no—
214Dic61—81no—
215#3414—611no—
216C2×#510D6111yescharacteristic, complemented
217#3156—611no—
218#2957—611no—
219#2214—614no—
220#2212—614no—
221#2153—611no—
222#2148—611no—
223C2×#362—611no—
224C2×C61—21no—
225C2×C61—21no—
226C2×C61—21no—
227C2×C61—82no—
228C2×C61C2×#45911yescharacteristic
229D61—161no—
230C2×#137—1221no—
231C2×#135—611no—
232C2×#135C2×D6111yescharacteristic
233C22×Q16C2×D6111yescharacteristic
234#568—1221no—
235#536—1221no—
236#510—22no—
237#510—612no—
238#510—1221no—
239#510C2×D6114yes—
240#509—612no—
241#509—1221no—
242#508—1222no—
243C2×Q32—21no—
244C2×SD32—21no—
245C2×M32—611no—
246C2×M32C2×D6111yescharacteristic
247#497—1222no—
248#495—612no—
249#494—612no—
250#444—1222no—
251#443—614no—
252C2×#108—611no—
253#421—1221no—
254#405—611no—
255#364—614no—
256#362—614no—
257#362—612no—
258#361—612no—
259C61#3359611yescharacteristic, complemented, Sylow
260C2×#41—21no—
261C2×#41—1221no—
262C22×Q8—21no—
263#137—2442no—
264#137—2442no—
265#135—21no—
266#135—21no—
267#135—1221no—
268#135—1221no—
269#135—1221no—
270#135—1221no—
271#135C22⋊D6112yes—
272#135C22×D6112yes—
273C2×Q16—21no—
274C2×Q16—21no—
275C2×Q16—21no—
276C2×Q16—21no—
277C2×Q16—21no—
278C2×Q16—611no—
279C2×Q16—1222no—
280C2×Q16C22⋊D6111yescharacteristic
281C2×Q16C22×D6112yes—
282C2×SD16—21no—
283C2×SD16—1222no—
284C2×SD16—1221no—
285C2×D8—611no—
286C2×D8C22⋊D6111yescharacteristic
287#131—1224no—
288#131—2442no—
289C2×M16—611no—
290C2×M16—1221no—
291C2×M16—2442no—
292C22×C8C22×D6111yescharacteristic
293C4⋊Q8—1222no—
294#124—1222no—
295#123—2442no—
296C22⋊Q8—2442no—
297#117—611no—
298Q32—24no—
299Q32—1222no—
300SD32—24no—
301SD32—1222no—
302SD32—1222no—
303D16—1222no—
304M32—21no—
305M32—612no—
306M32—1221no—
307M32C22×D6112yes—
308C2×C16—21no—
309C2×C16—1221no—
310#108—612no—
311#108—1221no—
312#104—1224no—
313#97—612no—
314C4×C8—612no—
315#41—22no—
316#41—22no—
317#41—41no—
318#41—41no—
319#41—2441no—
320#41—2441no—
321#41—2441no—
322#41—2441no—
323C2×Q8—22no—
324C2×Q8—21no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—1221no—
331C2×D4—21no—
332C2×D4—41no—
333C2×D4—1221no—
334C2×D4—2441no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—2441no—
338C22×C4D4×D6111yescharacteristic
339Q16—22no—
340Q16—22no—
341Q16—21no—
342Q16—41no—
343Q16—41no—
344Q16—1221no—
345Q16—1221no—
346Q16—2444no—
347Q16C2×C22⋊D6112yes—
348SD16—22no—
349SD16—22no—
350SD16—2444no—
351SD16—2441no—
352SD16—2441no—
353D8—21no—
354D8—1221no—
355D8—1221no—
356D8C2×C22⋊D6112yes—
357M16—612no—
358M16—612no—
359M16—1222no—
360M16—2444no—
361M16—2441no—
362C2×C8—612no—
363C2×C8—2444no—
364C2×C8—2441no—
365C2×C8D4×D6111yescharacteristic
366C2×C8#1175822411yescharacteristic
367C2×C8#1175822411yescharacteristic
368C2×C8C2×C22⋊D6111yescharacteristic
369C2×C8C2×C22⋊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#1451980111yescharacteristic
381Q8—22no—
382Q8—41no—
383Q8—42no—
384Q8—42no—
385Q8—81no—
386Q8—81no—
387Q8—2441no—
388Q8—2441no—
389D4—22no—
390D4—42no—
391D4—42no—
392D4—41no—
393D4—4881no—
394D4—4881no—
395D4—2441no—
396D4—2441no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—1222no—
404C2×C4—4881no—
405C2×C4—4881no—
406C2×C4—2441no—
407C2×C4#1451974911yescharacteristic
408C2×C4#1451974911yescharacteristic
409C2×C4D8×D6111yescharacteristic
410C2×C4D8×D6111yescharacteristic
411C2×C4#1451980111yescharacteristic
412C2×C4C2×D4×D6111yescharacteristic
413C8—614no—
414C8—2442no—
415C8#1451974911yescharacteristic
416C8#1451974911yescharacteristic
417C8C22×C22⋊D6111yescharacteristic
418C8C2×#1175822411yescharacteristic
419C22—41no—
420C22—42no—
421C22—161no—
422C22—4882no—
423C22—4881no—
424C22#1683995511yescharacteristic
425C22#1683995511yescharacteristic
426C22C2×#1451980111yescharacteristic
427C4—42no—
428C4—82no—
429C4—4881no—
430C4—2442no—
431C4#1683995511yescharacteristic
432C4#1683995511yescharacteristic
433C4C2×#1451974911yescharacteristic
434C4C2×D8×D6111yescharacteristic
435C2—21no—
436C2—21no—
437C2—21no—
438C2—82no—
439C2—9761no—
440C2C2×#1683995511yescharacteristic
subgroupG/H
centerC2C2×#16839955
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#510C2
socleC2×C61C2×#459
prime radical#21458953C2
hypercenterC2×#510D61
hypocenterC61#33596
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×C8 ◁ C2×#510

Complements

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

NH ≅ G/Ntype
2. #21456308C2semidirect
3. #21454736C2semidirect
4. #21447898C2semidirect
5. #21447890C2semidirect
6. #21447584C2semidirect
7. #21447569C2semidirect
8. C2×#16839578C2semidirect
9. C2×C61×#510C2semidirect
21. #16840147C22semidirect
22. #16840140C22semidirect
28. #16839582C22semidirect
29. #16839579C22semidirect
30. #16839578C22semidirect
31. #16839576C22semidirect
216. C2×#510D61semidirect
259. C61#33596semidirect

Maximal subgroups

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

subgroupindex
#214589532
#214563082
#214547362
#214478982
#214478902
#214475842
#214475692
C2×#168395782
C2×C61×#5102
#3359661