Finite Group Databasebeta

← #29663788

Subgroups — #29663788

Order 15616.

744 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456374C211yesmaximal, characteristic, complemented
2#21455616C211yesmaximal, characteristic, complemented
3C2×D4×#92418C211yesmaximal, characteristic, complemented
4#21445244C212yesmaximal, complemented
5#21445226C212yesmaximal, complemented
6#21445192C212yesmaximal, complemented
7#21445158C212yesmaximal, complemented
8#21444574C211yesmaximal, characteristic, complemented
9#21444510C211yesmaximal, characteristic, complemented
10#21444496C211yesmaximal, characteristic, complemented
11C61×#3036C211yesmaximal, characteristic
12C23×#92418—21no—
13#16840319C2211yescharacteristic
14#16840158C2211yescharacteristic, complemented
15#16840134C2211yescharacteristic, complemented
16#16840114C2211yescharacteristic, complemented
17#16840108C2211yescharacteristic, complemented
18#16840062—22no—
19#16840052C2212yes—
20#16840042C2212yes—
21#16839989—21no—
22#16839989—21no—
23C4×#11758200—21no—
24C4×#11758200—21no—
25D4×#92418—21no—
26D4×#92418—21no—
27D4×#92418C2212yes—
28D4×#92418C2212yes—
29#16839802—21no—
30#16839786C2211yescharacteristic
31C2×C4⋊#92418C2211yescharacteristic
32#16839767—21no—
33C2×C22⋊#92418—21no—
34C2×C4×#92418C2211yescharacteristic
35#16839512C2212yescomplemented
36#16839512C2212yescomplemented
37#16839510C2212yescomplemented
38#16839508C2212yescomplemented
39#16839507C2212yescomplemented
40#16839506C2212yescomplemented
41C2×AGL(1,ℤ/8)×C61—21no—
42C61×#532C2211yescharacteristic
43C2×C4×D4×C61C2211yescharacteristic
44C61×#463—22no—
45C61×#461C2212yes—
46C61×#459C2212yes—
47C61×#423C2211yescharacteristic
48C61×#417—21no—
49C2×#11758200—21no—
50C2×#11758200—21no—
51C2×#11758194—21no—
52C22×#92418—21no—
53C22×#92418—22no—
54C22×#92418—22no—
55C22×#92418—44no—
56C22×#92418D411yescharacteristic
57C22×#92418D411yescharacteristic
58#14519777C2311yescharacteristic
59#14519777C2311yescharacteristic
60#14519769—21no—
61#14519756—21no—
62#14519756—21no—
63C4×Dic122—21no—
64C4×Dic122—21no—
65#14519736—21no—
66#14519710—22no—
67#14519708—22no—
68#14519708—21no—
69#14519708—21no—
70C4⋊#92418—21no—
71C4⋊#92418—21no—
72C4⋊#92418C2312yes—
73C4⋊#92418C2312yes—
74#14519700—21no—
75#14519700—21no—
76#14519698—22no—
77#14519698—22no—
78#14519698—21no—
79#14519698—21no—
80C22⋊#92418—22no—
81C22⋊#92418—22no—
82C4×#92418D411yescharacteristic
83C4×#92418D411yescharacteristic
84C4×#92418C2311yescharacteristic
85C4×#92418C2311yescharacteristic
86C2×#41×C61—21no—
87C22×D4×C61D411yescharacteristic
88C23×C4×C61—21no—
89AGL(1,ℤ/8)×C61—44no—
90C2×SD16×C61—22no—
91C2×D8×C61—22no—
92C2×M16×C61—21no—
93C4⋊Q8×C61C2311yescharacteristic
94#127×C61C2311yescharacteristic
95#124×C61—21no—
96#121×C61—21no—
97#121×C61—21no—
98C4×D4×C61—21no—
99C4×D4×C61—21no—
100C4×D4×C61—21no—
101C4×D4×C61—21no—
102C4×D4×C61D412yes—
103C4×D4×C61C2312yes—
104C2×C4⋊C4×C61D411yescharacteristic
105C2×C22⋊C4×C61—21no—
106C2×C42×C61C2311yescharacteristic
107C4⋊C8×C61—21no—
108C4⋊C8×C61C2312yes—
109#103×C61—22no—
110#102×C61—22no—
111C24×C61—21no—
112#41×C61—42no—
113#41×C61—42no—
114C2×Q8×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—22no—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61—21no—
120C2×D4×C61—41no—
121C2×D4×C61SD1614yescomplemented
122C2×D4×C61C2×D412yes—
123C22×C4×C61—21no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—42no—
127C22×C4×C61—42no—
128C22×C4×C61—41no—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61#4111yescharacteristic
132C2×Dic122—21no—
133C2×Dic122—21no—
134SD16×C61—44no—
135D8×C61—44no—
136M16×C61—42no—
137C2×C8×C61—22no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61C2×D412yes—
142C22⋊C4×C61—22no—
143C22⋊C4×C61—22no—
144C22⋊C4×C61—41no—
145C22⋊C4×C61—41no—
146C42×C61C2×D411yescharacteristic
147C42×C61C2×D411yescharacteristic
148C42×C61#4111yescharacteristic
149C42×C61C2411yescharacteristic
150C22⋊Dic61—41no—
151#11758200—42no—
152#11758200—42no—
153C4⋊Dic61—21no—
154C4⋊Dic61—21no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157#11758196—41no—
158#11758196—41no—
159C4×Dic61—41no—
160#11758194—42no—
161C2×#92418—22no—
162C2×#92418—22no—
163C2×#92418—22no—
164C2×#92418—41no—
165C2×#92418—44no—
166C2×#92418—42no—
167C2×#92418—44no—
168C2×#92418C2×D411yescharacteristic
169C2×#92418C2×D411yescharacteristic
170C2×#92418C2×D411yescharacteristic
171C2×#92418C2×D411yescharacteristic
172C23×C61—22no—
173C23×C61—24no—
174C23×C61—42no—
175C23×C61—42no—
176C23×C61—41no—
177C23×C61#12111yescharacteristic
178Q8×C61—41no—
179Q8×C61—41no—
180D4×C61—22no—
181D4×C61—24no—
182D4×C61—41no—
183D4×C61—41no—
184D4×C61—41no—
185D4×C61—41no—
186D4×C61—81no—
187D4×C61—82no—
188D4×C61C2×SD1614yes—
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—22no—
192C2×C4×C61—41no—
193C2×C4×C61—44no—
194C2×C4×C61—42no—
195C2×C4×C61—42no—
196C2×C4×C61—42no—
197C2×C4×C61—41no—
198C2×C4×C61—82no—
199C2×C4×C61#12111yescharacteristic
200C2×C4×C61#12111yescharacteristic
201C2×C4×C61#12111yescharacteristic
202C2×C4×C61C2×SD1612yes—
203C2×C4×C61#13512yes—
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C2×#4111yescharacteristic
207C2×Dic61—41no—
208C2×Dic61—41no—
209Dic122—41no—
210Dic122—41no—
211Dic122—81no—
212C8×C61—42no—
213#92418—21no—
214#92418—21no—
215#92418—21no—
216#92418—42no—
217#92418—42no—
218#92418AGL(1,ℤ/8)12yescomplemented
219#19254—611nomaximal, Sylow
220C22×C61—21no—
221C22×C61—21no—
222C22×C61—24no—
223C22×C61—48no—
224C22×C61—42no—
225C22×C61—42no—
226C22×C61—81no—
227C22×C61—81no—
228C22×C61—81no—
229C22×C61#46512yes—
230C22×C61C2×#12111yescharacteristic
231C4×C61—21no—
232C4×C61—21no—
233C4×C61—21no—
234C4×C61—42no—
235C4×C61—81no—
236C4×C61#46512yes—
237C4×C61C22×SD1611yescharacteristic
238C4×C61C2×#13511yescharacteristic
239C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
240C4×C61#57511yescharacteristic
241Dic61—81no—
242Dic61—81no—
243#3167—611no—
244#3077—611no—
245#3036D6111yescharacteristic
246C2×C8×D4—611no—
247#1679—612no—
248#1677—612no—
249#1671—612no—
250#1665—612no—
251#1566—611no—
252#1550—611no—
253#1547—611no—
254C2×C61—22no—
255C2×C61—44no—
256C2×C61—161no—
257C2×C61#303611yescharacteristic
258C2×C61C2×#46511yescharacteristic
259C2×C61#306711yescharacteristic
260C2×AGL(1,ℤ/8)—21no—
261C23×C8—1221no—
262#533—611no—
263#532C2×D6111yescharacteristic
264C2×C4×D4C2×D6111yescharacteristic
265#498—611no—
266#492—611no—
267#487—611no—
268#486—611no—
269#478—1222no—
270#476—612no—
271#474—612no—
272#465—1221no—
273#465—1221no—
274#463—22no—
275#461C2×D6112yes—
276#459C2×D6112yes—
277C4×SD16—1221no—
278C4×SD16—1221no—
279C8×D4—612no—
280C8×D4—612no—
281C8×D4—1221no—
282C8×D4—1221no—
283#428—1221no—
284#423—611no—
285#423C2×D6111yescharacteristic
286C2×C4⋊C8—611no—
287#418—1221no—
288#417—21no—
289C2×C22⋊C8—1221no—
290C2×C4×C8—611no—
291#334—612no—
292#334—612no—
293#333—612no—
294#332—612no—
295#332—612no—
296#331—612no—
297C61#1925411yescharacteristic, complemented, Sylow
298C2×#41—21no—
299C22×D4C22⋊D6111yescharacteristic
300C23×C4—21no—
301AGL(1,ℤ/8)—44no—
302C2×SD16—22no—
303C2×SD16—1221no—
304C2×SD16—1221no—
305C2×D8—22no—
306C2×M16—21no—
307C2×M16—1221no—
308C22×C8—611no—
309C22×C8—611no—
310C22×C8—1222no—
311C22×C8—1222no—
312C22×C8—1221no—
313C22×C8—2444no—
314C4⋊Q8—611no—
315C4⋊Q8—611no—
316C4⋊Q8C22×D6111yescharacteristic
317#127C22×D6111yescharacteristic
318#125—1221no—
319#124—21no—
320C22⋊Q8—1221no—
321C22⋊Q8—1221no—
322#121—21no—
323#121—21no—
324C4×Q8—1221no—
325C4×Q8—1221no—
326C4×D4—21no—
327C4×D4—21no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4C22⋊D6112yes—
331C4×D4C22×D6112yes—
332#117—1221no—
333C2×C4⋊C4C22⋊D6111yescharacteristic
334C2×C22⋊C4—21no—
335C2×C42C22×D6111yescharacteristic
336#107—1222no—
337#106—1221no—
338#106—1221no—
339#106—1222no—
340C4⋊C8—21no—
341C4⋊C8—612no—
342C4⋊C8—612no—
343C4⋊C8—1221no—
344C4⋊C8—1221no—
345C4⋊C8C22×D6112yes—
346#103—22no—
347#103—1221no—
348#103—1221no—
349#102—22no—
350#102—1221no—
351#102—1221no—
352#102—1222no—
353#102—1222no—
354C22⋊C8—1222no—
355C22⋊C8—1222no—
356C4×C8—611no—
357C4×C8—611no—
358C4×C8—611no—
359C4×C8—611no—
360C24—21no—
361#41—42no—
362#41—42no—
363C2×Q8—21no—
364C2×Q8—1221no—
365C2×Q8—1221no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—22no—
371C2×D4—41no—
372C2×D4#1175820014yescomplemented
373C2×D4C2×C22⋊D6112yes—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4D4×D6111yescharacteristic
381C22×C4#1175822411yescharacteristic
382C22×C4C2×C22⋊D6111yescharacteristic
383SD16—44no—
384SD16—2442no—
385SD16—2442no—
386D8—44no—
387M16—42no—
388M16—2442no—
389C2×C8—22no—
390C2×C8—611no—
391C2×C8—611no—
392C2×C8—611no—
393C2×C8—611no—
394C2×C8—1222no—
395C2×C8—1222no—
396C2×C8—1222no—
397C2×C8—2444no—
398C2×C8—2441no—
399C2×C8—2444no—
400C2×C8—2442no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—1221no—
405C4⋊C4—1221no—
406C4⋊C4—1221no—
407C4⋊C4—1221no—
408C4⋊C4—2441no—
409C4⋊C4—2441no—
410C4⋊C4C2×C22⋊D6112yes—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—2441no—
416C42—2441no—
417C42D4×D6111yescharacteristic
418C42#1175822411yescharacteristic
419C42C2×C22⋊D6111yescharacteristic
420C42C23×D6111yescharacteristic
421C23—24no—
422C23—22no—
423C23—42no—
424C23—42no—
425C23—41no—
426C23#1451974911yescharacteristic
427Q8—41no—
428Q8—41no—
429Q8—4881no—
430Q8—2441no—
431Q8—2441no—
432D4—24no—
433D4—22no—
434D4—41no—
435D4—41no—
436D4—41no—
437D4—41no—
438D4—81no—
439D4—82no—
440D4C2×#1175820014yes—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—41no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—82no—
451C2×C4—2441no—
452C2×C4—2441no—
453C2×C4#1451974911yescharacteristic
454C2×C4#1451974911yescharacteristic
455C2×C4#1451974911yescharacteristic
456C2×C4C2×#1175820012yes—
457C2×C4#1451979712yes—
458C2×C4C2×D4×D6111yescharacteristic
459C2×C4C22×C22⋊D6111yescharacteristic
460C2×C4C2×#1175822411yescharacteristic
461C8—42no—
462C8—612no—
463C8—1221no—
464C8—1221no—
465C8—1221no—
466C8—2442no—
467C8—2442no—
468C22—21no—
469C22—21no—
470C22—24no—
471C22—48no—
472C22—42no—
473C22—42no—
474C22—81no—
475C22—81no—
476C22—81no—
477C22#1683998912yes—
478C22C2×#1451974911yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—42no—
483C4—81no—
484C4—4881no—
485C4—4881no—
486C4#1683998912yes—
487C4C22×#1175820011yescharacteristic
488C4C2×#1451979711yescharacteristic
489C4AGL(1,ℤ/8)×D6111yescharacteristic
490C4#1684055811yescharacteristic
491C2—22no—
492C2—44no—
493C2—161no—
494C2#2145528011yescharacteristic
495C2C2×#1683998911yescharacteristic
496C2#2145553611yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3036C2
socleC22×C61C2×#121
hypercenter#3036D61
hypocenterC61#19254
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3036

Complements

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

NH ≅ G/Ntype
1. #21456374C2semidirect
2. #21455616C2semidirect
3. C2×D4×#92418C2semidirect
4. #21445244C2semidirect
5. #21445226C2semidirect
6. #21445192C2semidirect
7. #21445158C2semidirect
8. #21444574C2semidirect
9. #21444510C2semidirect
10. #21444496C2semidirect
14. #16840158C22semidirect
15. #16840134C22semidirect
16. #16840114C22semidirect
17. #16840108C22semidirect
35. #16839512C22semidirect
36. #16839512C22semidirect
37. #16839510C22semidirect
38. #16839508C22semidirect
39. #16839507C22semidirect
40. #16839506C22semidirect
121. C2×D4×C61SD16semidirect
218. #92418AGL(1,ℤ/8)semidirect
297. C61#19254semidirect
372. C2×D4#11758200semidirect

Maximal subgroups

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

subgroupindex
#214563742
#214556162
C2×D4×#924182
#214452442
#214452262
#214451922
#214451582
#214445742
#214445102
#214444962
C61×#30362
#1925461