Finite Group Databasebeta

← #29652299

Subgroups — #29652299

Order 15616.

1012 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459224C211yesmaximal, characteristic, complemented
2#21455827C212yesmaximal, complemented
3#21444826C212yesmaximal, complemented
4#21444809C212yesmaximal, complemented
5#21444758C212yesmaximal, complemented
6#21444741C212yesmaximal
7#21444317C212yesmaximal, complemented
8#21444133C211yesmaximal, characteristic, complemented
9C61×#2392C211yesmaximal, characteristic
10#16840393—21no—
11#16840393C2212yes—
12#16840390—22no—
13#16840390—22no—
14#16840388—21no—
15#16840388C2212yes—
16#16840319—21no—
17#16840319C2212yes—
18#16840315—24no—
19C2×#14519736—21no—
20#16840092C2214yes—
21#16840052—22no—
22#16839967—22no—
23#16839913C2214yes—
24C4×#11758200—24no—
25#16839815C2212yes—
26#16839498C2214yes—
27#16839498C2214yes—
28#16839488C2212yes—
29#16839486C2212yes—
30C2×C4×D4×C61—21no—
31C2×C4×D4×C61C2211yescharacteristic
32C2×#117×C61C2211yescharacteristic
33C61×#398C2211yescharacteristic
34C61×#397C2211yescharacteristic
35C61×#396—21no—
36C61×#394—21no—
37C61×#394C2211yescharacteristic
38C61×#392C2211yescharacteristic
39C4×C4⋊C4×C61C2211yescharacteristic
40C4×C22⋊C4×C61—21no—
41C2×#11758200—24no—
42#14519777—21no—
43#14519777—21no—
44#14519777D412yescomplemented
45#14519777D412yescomplemented
46#14519776—24no—
47#14519776—24no—
48#14519769—22no—
49#14519769—22no—
50#14519768—24no—
51#14519768—24no—
52#14519756—22no—
53#14519756—22no—
54#14519756—22no—
55#14519756C2314yes—
56#14519754—24no—
57#14519754—24no—
58#14519754—24no—
59#14519754—24no—
60C4×Dic122—22no—
61C4×Dic122—22no—
62C4×Dic122—22no—
63C4×Dic122D414yescomplemented
64#14519736—22no—
65#14519736—22no—
66#14519736—22no—
67#14519736—22no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C22⋊Dic61—22no—
71C2×C4×Dic61—22no—
72#14519708—24no—
73C4⋊#92418—22no—
74#14519700—24no—
75#14519698—24no—
76C22⋊#92418C2314yes—
77C4×#92418—22no—
78C22×D4×C61—21no—
79C22×D4×C61C2311yescharacteristic
80C23×C4×C61—21no—
81C23×C4×C61D411yescharacteristic
82C4×D4×C61—22no—
83C4×D4×C61—22no—
84C4×D4×C61—44no—
85#117×C61—22no—
86#117×C61—22no—
87C2×C4⋊C4×C61—21no—
88C2×C4⋊C4×C61—21no—
89C2×C4⋊C4×C61D411yescharacteristic
90C2×C4⋊C4×C61C2311yescharacteristic
91C2×C4⋊C4×C61C2311yescharacteristic
92C2×C22⋊C4×C61—21no—
93C2×C22⋊C4×C61—21no—
94C2×C22⋊C4×C61—21no—
95C2×C22⋊C4×C61—21no—
96C2×C22⋊C4×C61—21no—
97C2×C22⋊C4×C61D411yescharacteristic
98C2×C22⋊C4×C61C2311yescharacteristic
99C2×C42×C61—21no—
100C2×C42×C61D411yescharacteristic
101C2×C42×C61C2311yescharacteristic
102#95×C61—21no—
103#95×C61—21no—
104#95×C61C2311yescharacteristic
105#95×C61C2311yescharacteristic
106C24×C61—21no—
107C24×C61C2×D411yescharacteristic
108C2×D4×C61—22no—
109C2×D4×C61—42no—
110C2×D4×C61—42no—
111C2×D4×C61—42no—
112C2×D4×C61—42no—
113C22×C4×C61—21no—
114C22×C4×C61—21no—
115C22×C4×C61—22no—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—41no—
120C22×C4×C61—41no—
121C22×C4×C61—42no—
122C22×C4×C61—42no—
123C22×C4×C61SD1614yescomplemented
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2411yescharacteristic
130C22×Dic61—22no—
131C22×Dic61—22no—
132C2×Dic122—22no—
133C2×Dic122—22no—
134C2×Dic122—22no—
135C2×Dic122C2×D414yes—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—42no—
140C4⋊C4×C61—42no—
141C22⋊C4×C61—22no—
142C22⋊C4×C61—22no—
143C22⋊C4×C61—42no—
144C22⋊C4×C61—42no—
145C22⋊C4×C61—42no—
146C22⋊C4×C61—42no—
147C22⋊C4×C61—42no—
148C42×C61—21no—
149C42×C61—21no—
150C42×C61—42no—
151C42×C61C2×D412yes—
152C42×C61C2×D412yes—
153C22⋊Dic61—24no—
154C22⋊Dic61—24no—
155#11758200—48no—
156C4⋊Dic61—22no—
157C4⋊Dic61—22no—
158C4⋊Dic61—22no—
159C4⋊Dic61—22no—
160C4⋊Dic61—22no—
161C4⋊Dic61—22no—
162C4⋊Dic61—22no—
163C4⋊Dic61C2×D414yes—
164#11758196—24no—
165#11758196—24no—
166#11758196—24no—
167#11758196—24no—
168C4×Dic61—24no—
169C4×Dic61—24no—
170C2×#92418—24no—
171C23×C61—21no—
172C23×C61—21no—
173C23×C61—22no—
174C23×C61—41no—
175C23×C61—41no—
176C23×C61—41no—
177C23×C61—41no—
178C23×C61—41no—
179C23×C61—41no—
180C23×C61—41no—
181C23×C61—41no—
182C23×C61C2×SD1612yes—
183C23×C61C22×D411yescharacteristic
184D4×C61—24no—
185D4×C61—42no—
186D4×C61—48no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—24no—
190C2×C4×C61—24no—
191C2×C4×C61—22no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—82no—
207C2×C4×C61—82no—
208C2×C4×C61#12012yes—
209C2×C4×C61#12011yescharacteristic
210C2×C4×C61#12011yescharacteristic
211C2×C4×C61C2×SD1612yes—
212C2×C4×C61C2×SD1612yes—
213C2×C4×C61#13712yes—
214C2×C4×C61C22×D412yes—
215C2×Dic61—24no—
216C2×Dic61—24no—
217C2×Dic61—82no—
218C2×Dic61—82no—
219Dic122—44no—
220Dic122—82no—
221Dic122—82no—
222Dic122—82no—
223#92418—44no—
224#18124—611nomaximal, Sylow
225C22×C61—21no—
226C22×C61—21no—
227C22×C61—22no—
228C22×C61—42no—
229C22×C61—42no—
230C22×C61—42no—
231C22×C61—41no—
232C22×C61—41no—
233C22×C61—42no—
234C22×C61—81no—
235C22×C61—81no—
236C22×C61—81no—
237C22×C61—81no—
238C22×C61—81no—
239C22×C61C2×#12011yescharacteristic
240C22×C61C22×SD1611yescharacteristic
241C22×C61C2×#13711yescharacteristic
242C4×C61—22no—
243C4×C61—22no—
244C4×C61—22no—
245C4×C61—42no—
246C4×C61—41no—
247C4×C61—41no—
248C4×C61—81no—
249C4×C61—81no—
250C4×C61#44812yes—
251C4×C61#44812yes—
252Dic61—44no—
253Dic61—44no—
254#3458—611no—
255#3101—612no—
256#2392D6111yescharacteristic
257#1611—612no—
258#1608—612no—
259#1599—612no—
260#1596—612no—
261#1503—612no—
262#1458—611no—
263C2×C61—21no—
264C2×C61—21no—
265C2×C61—42no—
266C2×C61—82no—
267C2×C61C2×#44811yescharacteristic
268C2×C61#301811yescharacteristic
269C2×C61#302511yescharacteristic
270#544—612no—
271#544—1222no—
272#544—1222no—
273#544—1221no—
274#543—612no—
275#543—1221no—
276#533—612no—
277#533—1224no—
278#533—1221no—
279C2×C4×D4—21no—
280C2×C4×D4C2×D6111yescharacteristic
281C2×#117—1221no—
282C2×#117C2×D6111yescharacteristic
283#483—614no—
284#476—1222no—
285#461—1222no—
286#450—614no—
287C4×SD16—1224no—
288#431—612no—
289#398C2×D6111yescharacteristic
290#397C2×D6111yescharacteristic
291#396—21no—
292#394—21no—
293#394C2×D6111yescharacteristic
294#392C2×D6111yescharacteristic
295C4×C4⋊C4C2×D6111yescharacteristic
296C4×C22⋊C4—21no—
297#328—614no—
298#328—614no—
299#324—612no—
300#323—612no—
301C61#1812411yescharacteristic, complemented, Sylow
302C22×D4—21no—
303C22×D4C22×D6111yescharacteristic
304C23×C4—21no—
305C23×C4C22⋊D6111yescharacteristic
306C2×SD16—1224no—
307C4⋊Q8—612no—
308C4⋊Q8—612no—
309C4⋊Q8—1221no—
310C4⋊Q8—1221no—
311C4⋊Q8—1224no—
312C4⋊Q8—1224no—
313#125—1222no—
314#125—1224no—
315#125—1224no—
316#125—1222no—
317C22⋊Q8—614no—
318C22⋊Q8—1224no—
319C22⋊Q8—1224no—
320C22⋊Q8—1224no—
321C22⋊Q8—1224no—
322C22⋊Q8—1222no—
323C22⋊Q8—1222no—
324C22⋊Q8—1222no—
325C4×Q8—614no—
326C4×Q8—1222no—
327C4×Q8—1222no—
328C4×Q8—1222no—
329C4×D4—22no—
330C4×D4—22no—
331C4×D4—44no—
332#117—22no—
333#117—22no—
334#117—1222no—
335#117—1222no—
336#117—1222no—
337#117—1222no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—1221no—
341C2×C4⋊C4—1221no—
342C2×C4⋊C4C22⋊D6111yescharacteristic
343C2×C4⋊C4C22×D6111yescharacteristic
344C2×C4⋊C4C22×D6111yescharacteristic
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—1222no—
351C2×C22⋊C4C22⋊D6111yescharacteristic
352C2×C22⋊C4C22×D6111yescharacteristic
353C2×C42—21no—
354C2×C42—1222no—
355C2×C42C22⋊D6111yescharacteristic
356C2×C42C22×D6111yescharacteristic
357#106—1224no—
358C4⋊C8—1222no—
359#103—1224no—
360#102—1224no—
361C22⋊C8—614no—
362C4×C8—1222no—
363#95—21no—
364#95—21no—
365#95C22×D6111yescharacteristic
366#95C22×D6111yescharacteristic
367C24—21no—
368C24C2×C22⋊D6111yescharacteristic
369C2×Q8—614no—
370C2×Q8—1222no—
371C2×Q8—1222no—
372C2×Q8—1222no—
373C2×D4—22no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—42no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—1222no—
389C22×C4—1222no—
390C22×C4#1175820014yescomplemented
391C22×C4C2×C22⋊D6111yescharacteristic
392C22×C4C2×C22⋊D6111yescharacteristic
393C22×C4C2×C22⋊D6111yescharacteristic
394C22×C4C2×C22⋊D6111yescharacteristic
395C22×C4C2×C22⋊D6111yescharacteristic
396C22×C4C23×D6111yescharacteristic
397SD16—2448no—
398C2×C8—1224no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—614no—
405C4⋊C4—1222no—
406C4⋊C4—1222no—
407C4⋊C4—1224no—
408C4⋊C4—1224no—
409C4⋊C4—1222no—
410C4⋊C4—1222no—
411C4⋊C4—1224no—
412C4⋊C4—1222no—
413C4⋊C4—1222no—
414C4⋊C4—1224no—
415C4⋊C4—1222no—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C22⋊C4—42no—
419C22⋊C4—42no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—42no—
423C22⋊C4—1224no—
424C22⋊C4—1224no—
425C42—21no—
426C42—21no—
427C42—42no—
428C42—1224no—
429C42—1224no—
430C42D4×D6112yes—
431C42D4×D6112yes—
432C23—21no—
433C23—21no—
434C23—22no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23C2×#1175820012yes—
444C23C22×C22⋊D6111yescharacteristic
445Q8—4882no—
446Q8—4882no—
447Q8—4882no—
448Q8—2444no—
449D4—24no—
450D4—42no—
451D4—48no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—1224no—
474C2×C4—1224no—
475C2×C4—4882no—
476C2×C4—4882no—
477C2×C4#1451974512yes—
478C2×C4#1451974511yescharacteristic
479C2×C4#1451974511yescharacteristic
480C2×C4C2×#1175820012yes—
481C2×C4C2×#1175820012yes—
482C2×C4#1451980712yes—
483C2×C4C2×D4×D6112yes—
484C8—2444no—
485C22—21no—
486C22—21no—
487C22—22no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—41no—
492C22—41no—
493C22—42no—
494C22—81no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22—81no—
499C22C2×#1451974511yescharacteristic
500C22C22×#1175820011yescharacteristic
501C22C2×#1451980711yescharacteristic
502C4—22no—
503C4—22no—
504C4—22no—
505C4—42no—
506C4—41no—
507C4—41no—
508C4—81no—
509C4—81no—
510C4—2444no—
511C4—2444no—
512C4#1683990012yes—
513C4#1683990012yes—
514C2—21no—
515C2—21no—
516C2—42no—
517C2—82no—
518C2C2×#1683990011yescharacteristic
519C2#2145516711yescharacteristic
520C2#2145521011yescharacteristic
subgroupG/H
centerC2×C4#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2392C2
socleC22×C61C2×#120
prime radicalC61×#392C22
hypercenter#2392D61
hypocenterC61#18124
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2392

Complements

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

NH ≅ G/Ntype
1. #21459224C2semidirect
2. #21455827C2semidirect
3. #21444826C2semidirect
4. #21444809C2semidirect
5. #21444758C2semidirect
7. #21444317C2semidirect
8. #21444133C2semidirect
44. #14519777D4semidirect
45. #14519777D4semidirect
63. C4×Dic122D4semidirect
123. C22×C4×C61SD16semidirect
301. C61#18124semidirect
390. C22×C4#11758200semidirect

Maximal subgroups

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

subgroupindex
#214592242
#214558272
#214448262
#214448092
#214447582
#214447412
#214443172
#214441332
C61×#23922
#1812461