Finite Group Databasebeta

← #29658536

Subgroups — #29658536

Order 15616.

1020 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456744C211yesmaximal, characteristic, complemented
2#21455819C211yesmaximal, characteristic
3#21455291C211yesmaximal, characteristic, complemented
4#21445033C214yesmaximal, complemented
5#21444891C214yesmaximal, complemented
6#21444500C211yesmaximal, characteristic, complemented
7#21444244C211yesmaximal, characteristic, complemented
8#21444214C211yesmaximal, characteristic, complemented
9C2×C61×#554C211yesmaximal, characteristic, complemented
10C2×#14519808—21no—
11#16840312C2211yescharacteristic
12#16840136C2212yescomplemented
13#16840112C2212yescomplemented
14#16840075—21no—
15#16840042C2212yes—
16#16839984—22no—
17#16839972C2212yescomplemented
18#16839964—22no—
19#16839962C2212yescomplemented
20#16839957C2212yes—
21#16839901—22no—
22#16839896—21no—
23C4×#11758199—22no—
24#16839815C2211yescharacteristic
25#16839815C2211yescharacteristic
26#16839786C2211yescharacteristic
27#16839761—21no—
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—44no—
45C2×#11758202—22no—
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—
58C4⋊#92418—21no—
59C4⋊#92418—21no—
60C4⋊#92418C2312yes—
61C4⋊#92418C2312yes—
62#14519701—24no—
63#14519700—22no—
64#14519698—22no—
65#14519697—22no—
66#14519697—22no—
67C22⋊#92418—21no—
68C22⋊#92418—21no—
69C4×#92418—21no—
70C4×#92418C2312yes—
71C22×Q8×C61D411yescharacteristic
72C22×Q8×C61D412yes—
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—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61—82no—
115C2×Q8×C61C2×D412yes—
116C2×Q8×C61C2×D414yes—
117C2×D4×C61—24no—
118C2×D4×C61—21no—
119C2×D4×C61C2×D412yes—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—22no—
124C22×C4×C61—28no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—42no—
128C22×C4×C61—42no—
129C22×C4×C61—84no—
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C2×C22⋊D61—41no—
134#11758222—42no—
135#11758222—42no—
136C2×D244—21no—
137C2×D244—21no—
138C2×D244—21no—
139C2×C4×D61—41no—
140C2×Dic122—21no—
141C4⋊C4×C61—28no—
142C4⋊C4×C61—28no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61C2×D412yes—
152C4⋊C4×C61C2×D412yes—
153C22⋊C4×C61—28no—
154C22⋊C4×C61—22no—
155C22⋊C4×C61—22no—
156C42×C61—22no—
157C42×C61—22no—
158C42×C61C2×D411yescharacteristic
159C42×C61C2×D411yescharacteristic
160C42×C61C2×D411yescharacteristic
161C42×C61C2411yescharacteristic
162#11758202—44no—
163#11758201—44no—
164#11758201—84no—
165#11758199—44no—
166#11758198—41no—
167#11758198—41no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170#11758194—42no—
171C2×#92418—22no—
172C2×#92418—22no—
173C2×#92418—22no—
174C23×C61—24no—
175C23×C61—22no—
176C23×C61—42no—
177C23×C61—42no—
178C23×C61#12011yescharacteristic
179C22×D61—41no—
180Q8×C61—22no—
181Q8×C61—22no—
182Q8×C61—44no—
183Q8×C61—42no—
184Q8×C61—82no—
185D4×C61—24no—
186D4×C61—22no—
187D4×C61#13714yescomplemented
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—24no—
193C2×C4×C61—41no—
194C2×C4×C61—42no—
195C2×C4×C61—44no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61—42no—
200C2×C4×C61—44no—
201C2×C4×C61—88no—
202C2×C4×C61#12011yescharacteristic
203C2×C4×C61#12011yescharacteristic
204C2×C4×C61#12011yescharacteristic
205C2×C4×C61#13512yes—
206C2×C4×C61AGL(1,ℤ/8)12yes—
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C22×D411yescharacteristic
209C2×C4×C61C22×D411yescharacteristic
210C22⋊D61—82no—
211C2×Dic61—41no—
212D244—42no—
213D244—41no—
214D244—41no—
215D244—42no—
216C4×D61—82no—
217Dic122—41no—
218Dic122—41no—
219#92418—42no—
220#92418—42no—
221#92418—42no—
222#18752—611nomaximal, Sylow
223C22×C61—21no—
224C22×C61—21no—
225C22×C61—24no—
226C22×C61—48no—
227C22×C61—42no—
228C22×C61—42no—
229C22×C61#44912yes—
230C22×C61C2×#12011yescharacteristic
231C2×D61—81no—
232C2×D61—81no—
233C2×D61—81no—
234C4×C61—21no—
235C4×C61—21no—
236C4×C61—21no—
237C4×C61—42no—
238C4×C61—42no—
239C4×C61—42no—
240C4×C61—44no—
241C4×C61#44912yes—
242C4×C61C2×#13511yescharacteristic
243C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
244C4×C61C2×#13712yes—
245Dic61—81no—
246C2×#554D6111yescharacteristic, complemented
247#3206—611no—
248#3100—611no—
249#3037—611no—
250#1644—614no—
251#1622—614no—
252#1548—611no—
253#1486—611no—
254#1479—611no—
255C2×C61—22no—
256C2×C61—44no—
257C2×C61C2×#44911yescharacteristic
258C2×C61#300411yescharacteristic
259C2×C61#301911yescharacteristic
260D61—161no—
261C2×#137—1221no—
262#554—22no—
263#554—22no—
264#554—22no—
265#554C2×D6114yes—
266#532—611no—
267C2×C4⋊Q8C2×D6111yescharacteristic
268C2×#125—21no—
269C2×C22⋊Q8—21no—
270C2×C22⋊Q8—22no—
271C2×C4×Q8C2×D6111yescharacteristic
272C2×C4×D4—21no—
273C2×C4×D4C2×D6111yescharacteristic
274C22×C4⋊C4—21no—
275#492—612no—
276#487—612no—
277#480—1221no—
278#474—612no—
279#464—1222no—
280#462—612no—
281#461—1222no—
282#460—612no—
283#459—612no—
284#448—1222no—
285#447—1221no—
286C4×D8—1222no—
287#431—611no—
288#431—611no—
289#423—611no—
290#417—1221no—
291#333—614no—
292Q8⋊C8—614no—
293D4⋊C8—614no—
294C61#1875211yescharacteristic, complemented, Sylow
295C2×#41—1221no—
296C22×Q8C22⋊D6111yescharacteristic
297C22×Q8C22⋊D6112yes—
298C22×D4C22⋊D6111yescharacteristic
299C23×C4—22no—
300C23×C4—21no—
301#137—2444no—
302C2×Q16—1222no—
303C2×SD16—1224no—
304C2×SD16—1222no—
305C2×D8—1222no—
306C2×M16—1221no—
307C4⋊Q8—21no—
308C4⋊Q8—21no—
309C4⋊Q8—611no—
310C4⋊Q8C22⋊D6112yes—
311C4⋊Q8C22×D6112yes—
312#127—611no—
313#125—24no—
314#125—24no—
315#124—1221no—
316C22⋊Q8—24no—
317C22⋊Q8—28no—
318C22⋊Q8—24no—
319C22⋊Q8—28no—
320#121—1221no—
321#121—1221no—
322C4×Q8—21no—
323C4×Q8—21no—
324C4×Q8C22⋊D6112yes—
325C4×Q8C22×D6112yes—
326C4×D4—24no—
327C4×D4—24no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—1221no—
331C4×D4—1221no—
332C4×D4C22⋊D6112yes—
333C4×D4C22×D6112yes—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4C22⋊D6111yescharacteristic
342C2×C4⋊C4C22⋊D6111yescharacteristic
343C2×C22⋊C4—22no—
344C2×C22⋊C4—21no—
345C2×C42—21no—
346C2×C42C22×D6111yescharacteristic
347#107—1222no—
348C4⋊C8—612no—
349C4⋊C8—612no—
350C4⋊C8—1221no—
351C4⋊C8—1221no—
352#103—1224no—
353#103—1222no—
354#102—1222no—
355#102—1222no—
356#102—1222no—
357C22⋊C8—1221no—
358C22⋊C8—1221no—
359C4×C8—612no—
360C4×C8—1221no—
361C24—21no—
362#41—2442no—
363#41—2442no—
364C2×Q8—21no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—82no—
369C2×Q8—1221no—
370C2×Q8C2×C22⋊D6112yes—
371C2×Q8C2×C22⋊D6114yes—
372C2×D4—24no—
373C2×D4—21no—
374C2×D4—1221no—
375C2×D4—1221no—
376C2×D4—1221no—
377C2×D4—2441no—
378C2×D4C2×C22⋊D6112yes—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—28no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—84no—
389C22×C4—2441no—
390C22×C4C2×C22⋊D6111yescharacteristic
391C22×C4C2×C22⋊D6111yescharacteristic
392C22×C4C2×C22⋊D6111yescharacteristic
393Q16—2444no—
394SD16—4884no—
395SD16—2444no—
396D8—2444no—
397M16—2442no—
398C2×C8—1222no—
399C2×C8—1222no—
400C2×C8—1222no—
401C4⋊C4—28no—
402C4⋊C4—28no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—21no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—21no—
411C4⋊C4—1221no—
412C4⋊C4—1221no—
413C4⋊C4C2×C22⋊D6112yes—
414C4⋊C4C2×C22⋊D6112yes—
415C22⋊C4—28no—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C22⋊C4—2441no—
419C22⋊C4—2441no—
420C42—22no—
421C42—22no—
422C42C2×C22⋊D6111yescharacteristic
423C42C2×C22⋊D6111yescharacteristic
424C42C2×C22⋊D6111yescharacteristic
425C42C23×D6111yescharacteristic
426C23—24no—
427C23—22no—
428C23—42no—
429C23—42no—
430C23—2441no—
431C23#1451974611yescharacteristic
432Q8—22no—
433Q8—22no—
434Q8—44no—
435Q8—42no—
436Q8—82no—
437Q8—2441no—
438Q8—2441no—
439D4—24no—
440D4—22no—
441D4—4882no—
442D4—2442no—
443D4—2441no—
444D4—2441no—
445D4—2442no—
446D4#1451980814yescomplemented
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—88no—
461C2×C4—4882no—
462C2×C4—2441no—
463C2×C4#1451974611yescharacteristic
464C2×C4#1451974611yescharacteristic
465C2×C4#1451974611yescharacteristic
466C2×C4#1451979712yes—
467C2×C4#1451980212yes—
468C2×C4C22×C22⋊D6111yescharacteristic
469C2×C4C22×C22⋊D6111yescharacteristic
470C2×C4C22×C22⋊D6111yescharacteristic
471C8—2442no—
472C8—2442no—
473C8—2442no—
474C22—21no—
475C22—21no—
476C22—24no—
477C22—48no—
478C22—42no—
479C22—42no—
480C22—4881no—
481C22—4881no—
482C22—4881no—
483C22#1683990712yes—
484C22C2×#1451974611yescharacteristic
485C4—21no—
486C4—21no—
487C4—21no—
488C4—42no—
489C4—44no—
490C4—42no—
491C4—42no—
492C4—4881no—
493C4#1683990712yes—
494C4C2×#1451979711yescharacteristic
495C4C2×#1451980211yescharacteristic
496C4C2×#1451980812yes—
497C2—22no—
498C2—44no—
499C2—9761no—
500C2C2×#1683990711yescharacteristic
501C2#2145505311yescharacteristic
502C2#2145517611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#554C2
socleC22×C61C2×#120
prime radical#21455819C2
hypercenterC2×#554D61
hypocenterC61#18752
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. #21456744C2semidirect
3. #21455291C2semidirect
4. #21445033C2semidirect
5. #21444891C2semidirect
6. #21444500C2semidirect
7. #21444244C2semidirect
8. #21444214C2semidirect
9. C2×C61×#554C2semidirect
12. #16840136C22semidirect
13. #16840112C22semidirect
17. #16839972C22semidirect
19. #16839962C22semidirect
28. #16839511C22semidirect
29. Q8⋊#92418C22semidirect
187. D4×C61#137semidirect
246. C2×#554D61semidirect
294. C61#18752semidirect
446. D4#14519808semidirect

Maximal subgroups

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

subgroupindex
#214567442
#214558192
#214552912
#214450332
#214448912
#214445002
#214442442
#214442142
C2×C61×#5542
#1875261