Finite Group Databasebeta

← #29656087

Subgroups — #29656087

Order 15616.

1048 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456276C211yesmaximal, characteristic, complemented
2#21455941C211yesmaximal, characteristic, complemented
3#21455910C211yesmaximal, characteristic
4#21444953C214yesmaximal, complemented
5#21444891C214yesmaximal, complemented
6#21444545C211yesmaximal, characteristic, complemented
7#21444266C211yesmaximal, characteristic, complemented
8#21444262C211yesmaximal, characteristic, complemented
9C2×C61×#554C211yesmaximal, characteristic, complemented
10C2×#14519808—22no—
11#16840312C2211yescharacteristic
12#16840154—22no—
13#16840141C2212yescomplemented
14#16840136C2212yescomplemented
15#16840080—21no—
16#16840075—21no—
17#16840057C2212yescomplemented
18#16840052C2212yes—
19#16839962C2212yescomplemented
20#16839957C2212yes—
21#16839901—21no—
22#16839896—21no—
23#16839872—22no—
24#16839860—22no—
25#16839819C2211yescharacteristic
26#16839819C2211yescharacteristic
27C4×#11758194C2211yescharacteristic
28#16839511C2214yescomplemented
29Q8⋊#92418C2214yescomplemented
30D4⋊#92418C2214yes—
31C61×#554—22no—
32C61×#554—22no—
33C61×#554—22no—
34C61×#554C2214yes—
35C2×C4⋊Q8×C61C2211yescharacteristic
36C2×#125×C61—21no—
37C2×C22⋊Q8×C61—21no—
38C2×C22⋊Q8×C61—22no—
39C2×C4×Q8×C61C2211yescharacteristic
40C2×C4×D4×C61—21no—
41C2×C4×D4×C61C2211yescharacteristic
42C22×C4⋊C4×C61—21no—
43C2×#11758222—21no—
44#14519808—48no—
45C2×#11758202—24no—
46C2×#11758201—22no—
47C2×#11758201—24no—
48C2×#11758199—22no—
49C2×#11758194—21no—
50#14519777C2311yescharacteristic
51#14519774C2311yescharacteristic
52#14519765—21no—
53#14519751—21no—
54#14519751—21no—
55C4×D244—21no—
56C4×D244—21no—
57#14519710—22no—
58#14519708—22no—
59C4⋊#92418C2312yes—
60C4⋊#92418C2312yes—
61#14519701—22no—
62#14519698—22no—
63#14519697—22no—
64#14519697—22no—
65C22⋊#92418—21no—
66C22⋊#92418—21no—
67#14519688—21no—
68#14519688—21no—
69#14519688—21no—
70C4×#92418C2312yes—
71C22×Q8×C61D412yes—
72C22×Q8×C61D411yescharacteristic
73C22×D4×C61D411yescharacteristic
74C23×C4×C61—22no—
75C23×C4×C61—21no—
76C4⋊Q8×C61—21no—
77C4⋊Q8×C61—21no—
78C4⋊Q8×C61D412yes—
79C4⋊Q8×C61C2312yes—
80#125×C61—24no—
81#125×C61—24no—
82C22⋊Q8×C61—24no—
83C22⋊Q8×C61—28no—
84C22⋊Q8×C61—24no—
85C22⋊Q8×C61—28no—
86C4×Q8×C61—21no—
87C4×Q8×C61—21no—
88C4×Q8×C61D412yes—
89C4×Q8×C61C2312yes—
90C4×D4×C61—24no—
91C4×D4×C61—24no—
92C4×D4×C61—21no—
93C4×D4×C61—21no—
94C4×D4×C61D412yes—
95C4×D4×C61C2312yes—
96C2×C4⋊C4×C61—24no—
97C2×C4⋊C4×C61—22no—
98C2×C4⋊C4×C61—22no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—24no—
103C2×C4⋊C4×C61D411yescharacteristic
104C2×C4⋊C4×C61D411yescharacteristic
105C2×C22⋊C4×C61—22no—
106C2×C22⋊C4×C61—21no—
107C2×C42×C61—21no—
108C2×C42×C61C2311yescharacteristic
109C24×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—22no—
112C2×Q8×C61—42no—
113C2×Q8×C61—44no—
114C2×Q8×C61C2×D412yes—
115C2×Q8×C61C2×D414yes—
116C2×D4×C61—24no—
117C2×D4×C61—21no—
118C2×D4×C61C2×D412yes—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—22no—
123C22×C4×C61—28no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—44no—
127C22×C4×C61—84no—
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C2×C22⋊D61—41no—
132#11758222—44no—
133C2×D244—21no—
134C2×D244—21no—
135C2×D244—21no—
136C2×C4×D61—41no—
137C2×Dic122—21no—
138C4⋊C4×C61—28no—
139C4⋊C4×C61—28no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61C2×D412yes—
149C4⋊C4×C61C2×D412yes—
150C22⋊C4×C61—28no—
151C22⋊C4×C61—22no—
152C22⋊C4×C61—22no—
153C42×C61—22no—
154C42×C61—22no—
155C42×C61C2×D411yescharacteristic
156C42×C61C2×D411yescharacteristic
157C42×C61C2×D411yescharacteristic
158C42×C61C2411yescharacteristic
159#11758202—48no—
160#11758201—48no—
161#11758201—44no—
162#11758199—44no—
163#11758198—41no—
164#11758198—41no—
165C4⋊Dic61—21no—
166C4⋊Dic61—21no—
167#11758194—24no—
168C2×#92418—22no—
169C2×#92418—22no—
170C2×#92418—22no—
171C23×C61—24no—
172C23×C61—22no—
173C23×C61—42no—
174C23×C61—42no—
175C23×C61#12011yescharacteristic
176C22×D61—41no—
177Q8×C61—22no—
178Q8×C61—22no—
179Q8×C61—42no—
180Q8×C61—44no—
181Q8×C61—82no—
182D4×C61—24no—
183D4×C61—22no—
184D4×C61#13714yescomplemented
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—24no—
192C2×C4×C61—44no—
193C2×C4×C61—42no—
194C2×C4×C61—42no—
195C2×C4×C61—42no—
196C2×C4×C61—44no—
197C2×C4×C61—44no—
198C2×C4×C61—88no—
199C2×C4×C61#12011yescharacteristic
200C2×C4×C61#12011yescharacteristic
201C2×C4×C61#12011yescharacteristic
202C2×C4×C61C22×D411yescharacteristic
203C2×C4×C61C22×D411yescharacteristic
204C2×C4×C61C22×D411yescharacteristic
205C22⋊D61—82no—
206C2×Dic61—41no—
207D244—42no—
208D244—42no—
209D244—42no—
210C4×D61—82no—
211Dic122—42no—
212#92418—24no—
213#92418—42no—
214#92418—42no—
215#18525—611nomaximal, Sylow
216C22×C61—21no—
217C22×C61—21no—
218C22×C61—24no—
219C22×C61—48no—
220C22×C61—42no—
221C22×C61—42no—
222C22×C61#45312yes—
223C22×C61C2×#12011yescharacteristic
224C2×D61—81no—
225C2×D61—81no—
226C2×D61—81no—
227C4×C61—21no—
228C4×C61—21no—
229C4×C61—21no—
230C4×C61—21no—
231C4×C61—42no—
232C4×C61—42no—
233C4×C61—42no—
234C4×C61—44no—
235C4×C61C2×#13712yes—
236C4×C61#57512yes—
237Dic61—81no—
238C2×#554D6111yescharacteristic, complemented
239#3152—611no—
240#3112—611no—
241#3109—611no—
242#1632—614no—
243#1622—614no—
244#1558—611no—
245#1491—611no—
246#1490—611no—
247C2×C61—22no—
248C2×C61—44no—
249C2×C61#300411yescharacteristic
250C2×C61#300511yescharacteristic
251C2×C61C2×#45311yescharacteristic
252D61—161no—
253C2×#137—1222no—
254#554—22no—
255#554—22no—
256#554—22no—
257#554C2×D6114yes—
258#532—611no—
259C2×C4⋊Q8C2×D6111yescharacteristic
260C2×#125—21no—
261C2×C22⋊Q8—21no—
262C2×C22⋊Q8—22no—
263C2×C4×Q8C2×D6111yescharacteristic
264C2×C4×D4—21no—
265C2×C4×D4C2×D6111yescharacteristic
266C22×C4⋊C4—21no—
267#497—1222no—
268#494—612no—
269#492—612no—
270#481—1221no—
271#480—1221no—
272#477—612no—
273#476—612no—
274#460—612no—
275#459—612no—
276#448—1221no—
277#447—1221no—
278#442—1222no—
279#440—1222no—
280#432—611no—
281#432—611no—
282C4×M16—611no—
283#333—614no—
284Q8⋊C8—614no—
285D4⋊C8—614no—
286C61#1852511yescharacteristic, complemented, Sylow
287C2×#41—1221no—
288C22×Q8C22⋊D6111yescharacteristic
289C22×Q8C22⋊D6112yes—
290C22×D4C22⋊D6111yescharacteristic
291C23×C4—22no—
292C23×C4—21no—
293#137—2448no—
294C2×Q16—1224no—
295C2×SD16—1224no—
296C2×SD16—1222no—
297C2×D8—1222no—
298C2×M16—1221no—
299C4⋊Q8—21no—
300C4⋊Q8—21no—
301C4⋊Q8—611no—
302C4⋊Q8C22⋊D6112yes—
303C4⋊Q8C22×D6112yes—
304#127—611no—
305#125—24no—
306#125—24no—
307#124—1221no—
308C22⋊Q8—24no—
309C22⋊Q8—28no—
310C22⋊Q8—24no—
311C22⋊Q8—28no—
312#121—1221no—
313#121—1221no—
314C4×Q8—21no—
315C4×Q8—21no—
316C4×Q8C22⋊D6112yes—
317C4×Q8C22×D6112yes—
318C4×D4—24no—
319C4×D4—24no—
320C4×D4—21no—
321C4×D4—21no—
322C4×D4—1221no—
323C4×D4—1221no—
324C4×D4C22⋊D6112yes—
325C4×D4C22×D6112yes—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4C22⋊D6111yescharacteristic
334C2×C4⋊C4C22⋊D6111yescharacteristic
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C42—21no—
338C2×C42C22×D6111yescharacteristic
339#107—1222no—
340#106—1222no—
341C4⋊C8—612no—
342C4⋊C8—612no—
343#103—1222no—
344#102—1222no—
345#102—1222no—
346#102—1222no—
347C22⋊C8—1221no—
348C22⋊C8—1221no—
349#97—1221no—
350#97—1221no—
351#97—1221no—
352C4×C8—612no—
353C24—21no—
354#41—2444no—
355C2×Q8—21no—
356C2×Q8—22no—
357C2×Q8—42no—
358C2×Q8—44no—
359C2×Q8—1221no—
360C2×Q8C2×C22⋊D6112yes—
361C2×Q8C2×C22⋊D6114yes—
362C2×D4—24no—
363C2×D4—21no—
364C2×D4—1221no—
365C2×D4—1221no—
366C2×D4—1221no—
367C2×D4—2441no—
368C2×D4C2×C22⋊D6112yes—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—28no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—44no—
377C22×C4—84no—
378C22×C4—2441no—
379C22×C4C2×C22⋊D6111yescharacteristic
380C22×C4C2×C22⋊D6111yescharacteristic
381C22×C4C2×C22⋊D6111yescharacteristic
382Q16—2448no—
383SD16—2448no—
384SD16—2444no—
385D8—2444no—
386M16—1224no—
387C2×C8—1222no—
388C2×C8—1222no—
389C2×C8—1222no—
390C4⋊C4—28no—
391C4⋊C4—28no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—21no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—21no—
400C4⋊C4—1221no—
401C4⋊C4—1221no—
402C4⋊C4C2×C22⋊D6112yes—
403C4⋊C4C2×C22⋊D6112yes—
404C22⋊C4—28no—
405C22⋊C4—22no—
406C22⋊C4—22no—
407C22⋊C4—2441no—
408C22⋊C4—2441no—
409C42—22no—
410C42—22no—
411C42C2×C22⋊D6111yescharacteristic
412C42C2×C22⋊D6111yescharacteristic
413C42C2×C22⋊D6111yescharacteristic
414C42C23×D6111yescharacteristic
415C23—24no—
416C23—22no—
417C23—42no—
418C23—42no—
419C23—2441no—
420C23#1451974611yescharacteristic
421Q8—22no—
422Q8—22no—
423Q8—44no—
424Q8—42no—
425Q8—82no—
426Q8—2442no—
427D4—24no—
428D4—22no—
429D4—4882no—
430D4—2442no—
431D4—2442no—
432D4—2442no—
433D4#1451980814yescomplemented
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—88no—
448C2×C4—4882no—
449C2×C4—2441no—
450C2×C4#1451974611yescharacteristic
451C2×C4#1451974611yescharacteristic
452C2×C4#1451974611yescharacteristic
453C2×C4C22×C22⋊D6111yescharacteristic
454C2×C4C22×C22⋊D6111yescharacteristic
455C2×C4C22×C22⋊D6111yescharacteristic
456C8—1224no—
457C8—2442no—
458C8—2442no—
459C22—21no—
460C22—21no—
461C22—24no—
462C22—48no—
463C22—42no—
464C22—42no—
465C22—4881no—
466C22—4881no—
467C22—4881no—
468C22#1683993012yes—
469C22C2×#1451974611yescharacteristic
470C4—21no—
471C4—21no—
472C4—21no—
473C4—21no—
474C4—42no—
475C4—44no—
476C4—42no—
477C4—42no—
478C4—4881no—
479C4C2×#1451980812yes—
480C4#1684056012yes—
481C2—22no—
482C2—44no—
483C2—9761no—
484C2#2145505311yescharacteristic
485C2#2145506111yescharacteristic
486C2C2×#1683993011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#554C2
socleC22×C61C2×#120
prime radical#21455910C2
hypercenterC2×#554D61
hypocenterC61#18525
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. #21456276C2semidirect
2. #21455941C2semidirect
4. #21444953C2semidirect
5. #21444891C2semidirect
6. #21444545C2semidirect
7. #21444266C2semidirect
8. #21444262C2semidirect
9. C2×C61×#554C2semidirect
13. #16840141C22semidirect
14. #16840136C22semidirect
17. #16840057C22semidirect
19. #16839962C22semidirect
28. #16839511C22semidirect
29. Q8⋊#92418C22semidirect
184. D4×C61#137semidirect
238. C2×#554D61semidirect
286. C61#18525semidirect
433. D4#14519808semidirect

Maximal subgroups

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

subgroupindex
#214562762
#214559412
#214559102
#214449532
#214448912
#214445452
#214442662
#214442622
C2×C61×#5542
#1852561