Finite Group Databasebeta

← #29658282

Subgroups — #29658282

Order 15616.

940 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456757C211yesmaximal, characteristic, complemented
2#21456734C211yesmaximal, characteristic
3#21456195C211yesmaximal, characteristic, complemented
4#21444929C214yesmaximal, complemented
5#21444922C214yesmaximal, complemented
6#21444484C211yesmaximal, characteristic, complemented
7#21444240C211yesmaximal, characteristic, complemented
8#21444236C211yesmaximal, characteristic, complemented
9C61×#3550C211yesmaximal, characteristic, complemented
10C2×#14519774C2211yescharacteristic
11#16840138—21no—
12#16840136—21no—
13#16840128—22no—
14#16840110—21no—
15#16840110—21no—
16#16840110C2212yes—
17#16840110C2212yes—
18#16839962C2214yes—
19#16839957C2214yes—
20#16839912—22no—
21#16839895—22no—
22#16839815C2211yescharacteristic
23#16839815C2211yescharacteristic
24#16839766—22no—
25C4×#11758194C2211yescharacteristic
26#16839508C2212yescomplemented
27#16839508C2212yescomplemented
28Q8⋊#92418C2214yescomplemented
29D4⋊#92418C2214yescomplemented
30C61×#563—22no—
31C61×#559—21no—
32Q82×C61—21no—
33C61×#556—21no—
34C61×#556C2212yescomplemented
35D4×Q8×C61—21no—
36D4×Q8×C61C2212yescomplemented
37C61×#542—22no—
38C61×#542—22no—
39C61×#536—22no—
40C61×#533C2211yescharacteristic
41C61×#533C2211yescharacteristic
42C61×#532C2211yescharacteristic
43C22×D244—21no—
44C22×D244—21no—
45C22×D244—21no—
46C2×#11758201—24no—
47C2×#11758199—24no—
48C2×#11758194—21no—
49#14519774—21no—
50#14519774—21no—
51#14519774—21no—
52#14519774C2312yes—
53C4⋊#92418C2312yes—
54C4⋊#92418C2312yes—
55#14519697—24no—
56#14519697—24no—
57#14519697—24no—
58#14519697—24no—
59C22⋊#92418—21no—
60C22⋊#92418—21no—
61#14519688—21no—
62C4×#92418—21no—
63C4×#92418—21no—
64C4×#92418C2312yes—
65C2×#41×C61—21no—
66C22×Q8×C61—22no—
67C4⋊Q8×C61—24no—
68C4⋊Q8×C61—22no—
69C4⋊Q8×C61—22no—
70C4⋊Q8×C61D411yescharacteristic
71C4⋊Q8×C61D411yescharacteristic
72C4⋊Q8×C61D411yescharacteristic
73C4⋊Q8×C61C2311yescharacteristic
74C4⋊Q8×C61C2311yescharacteristic
75#127×C61D411yescharacteristic
76#126×C61—24no—
77#126×C61—24no—
78#125×C61—21no—
79#125×C61D412yes—
80#124×C61—24no—
81#124×C61—22no—
82#124×C61—22no—
83#124×C61—21no—
84#123×C61—24no—
85#123×C61—24no—
86C22⋊Q8×C61—24no—
87C22⋊Q8×C61—22no—
88C22⋊Q8×C61—21no—
89C22⋊Q8×C61—21no—
90C22⋊Q8×C61—21no—
91C22⋊Q8×C61—22no—
92C22⋊Q8×C61—24no—
93C22⋊Q8×C61D412yes—
94#121×C61—21no—
95#121×C61D412yes—
96C4×Q8×C61—22no—
97C4×Q8×C61—22no—
98C4×Q8×C61—21no—
99C4×Q8×C61—21no—
100C4×Q8×C61—21no—
101C4×Q8×C61C2312yes—
102C4×D4×C61—22no—
103C4×D4×C61—22no—
104C4×D4×C61—21no—
105C4×D4×C61C2312yes—
106#117×C61—21no—
107#117×C61—22no—
108#117×C61D412yes—
109C2×C42×C61C2311yescharacteristic
110C23×D61—41no—
111#41×C61—82no—
112C2×Q8×C61—24no—
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×Q8×C61—21no—
116C2×Q8×C61—21no—
117C2×Q8×C61—84no—
118C2×Q8×C61C2×D412yes—
119C2×D4×C61—22no—
120C2×D4×C61—21no—
121C2×D4×C61—21no—
122C2×D4×C61C2×D412yes—
123C22×C4×C61—24no—
124C22×C4×C61—22no—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C2×D244—22no—
129C2×D244—22no—
130C2×D244—22no—
131C2×D244—44no—
132C2×D244—41no—
133C2×D244—41no—
134C2×D244—41no—
135C2×D244—82no—
136C2×D244—82no—
137C4⋊C4×C61—24no—
138C4⋊C4×C61—24no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61C2×D412yes—
150C4⋊C4×C61C2×D412yes—
151C4⋊C4×C61C2×D412yes—
152C4⋊C4×C61C2×D412yes—
153C22⋊C4×C61—24no—
154C22⋊C4×C61—24no—
155C22⋊C4×C61—24no—
156C22⋊C4×C61—22no—
157C22⋊C4×C61—22no—
158C22⋊C4×C61—22no—
159C22⋊C4×C61—22no—
160C42×C61—24no—
161C42×C61—22no—
162C42×C61—22no—
163C42×C61C2×D411yescharacteristic
164C42×C61C2×D411yescharacteristic
165C42×C61C2×D411yescharacteristic
166C42×C61C2411yescharacteristic
167#11758201—84no—
168#11758199—84no—
169#11758194—42no—
170C2×#92418—22no—
171C2×#92418—22no—
172C2×#92418—22no—
173C23×C61—22no—
174C23×C61#12011yescharacteristic
175C22×D61—42no—
176C22×D61—82no—
177C22×D61—82no—
178C22×D61—82no—
179Q8×C61—24no—
180Q8×C61—42no—
181Q8×C61—42no—
182Q8×C61—42no—
183Q8×C61—81no—
184Q8×C61—84no—
185D4×C61—24no—
186D4×C61—42no—
187D4×C61—81no—
188D4×C61—82no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—44no—
198C2×C4×C61—84no—
199C2×C4×C61#12011yescharacteristic
200C2×C4×C61#12011yescharacteristic
201C2×C4×C61#12011yescharacteristic
202C2×C4×C61C22×D411yescharacteristic
203C2×C4×C61C22×D411yescharacteristic
204C2×C4×C61C22×D411yescharacteristic
205D244—44no—
206D244—44no—
207D244—44no—
208D244—82no—
209D244—82no—
210D244—82no—
211#92418—42no—
212#92418—42no—
213#92418—42no—
214#18729—611nomaximal, Sylow
215C22×C61—21no—
216C22×C61—21no—
217C22×C61—21no—
218C22×C61—42no—
219C22×C61—42no—
220C22×C61—42no—
221C22×C61C2×#12011yescharacteristic
222C2×D61—82no—
223C2×D61—82no—
224C2×D61—82no—
225C2×D61—164no—
226C4×C61—21no—
227C4×C61—21no—
228C4×C61—21no—
229C4×C61—42no—
230C4×C61—42no—
231C4×C61—42no—
232C4×C61—42no—
233C4×C61—42no—
234C4×C61#45412yes—
235C4×C61#57612yes—
236C4×C61#57712yes—
237#3550D6111yescharacteristic, complemented
238#3208—611no—
239#3205—611no—
240#3138—611no—
241#1628—614no—
242#1627—614no—
243#1544—611no—
244#1485—611no—
245#1484—611no—
246C2×C61—41no—
247C2×C61—82no—
248C2×C61#300811yescharacteristic
249C2×C61#301011yescharacteristic
250C2×C61C2×#45411yescharacteristic
251D61—162no—
252#563—22no—
253#559—21no—
254Q82—21no—
255#556—21no—
256#556C2×D6112yescomplemented
257D4×Q8—21no—
258D4×Q8C2×D6112yescomplemented
259#542—22no—
260#542—22no—
261#536—22no—
262#533C2×D6111yescharacteristic
263#533C2×D6111yescharacteristic
264#532C2×D6111yescharacteristic
265C2×#127—611no—
266#493—1221no—
267#492—1221no—
268#490—1222no—
269#486—612no—
270#486—612no—
271#486—1221no—
272#486—1221no—
273#460—614no—
274#459—614no—
275#450—1222no—
276#447—1222no—
277#431—611no—
278#431—611no—
279#418—1222no—
280C4×M16—611no—
281#332—612no—
282#332—612no—
283Q8⋊C8—614no—
284D4⋊C8—614no—
285C61#1872911yescharacteristic, complemented, Sylow
286C2×#41—21no—
287C22×Q8—22no—
288C22×D4—1221no—
289C22×D4—1221no—
290C22×D4—1221no—
291C2×SD16—1224no—
292C2×D8—1224no—
293C2×M16—1221no—
294C4⋊Q8—24no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8C22⋊D6111yescharacteristic
298C4⋊Q8C22⋊D6111yescharacteristic
299C4⋊Q8C22⋊D6111yescharacteristic
300C4⋊Q8C22×D6111yescharacteristic
301C4⋊Q8C22×D6111yescharacteristic
302#127—612no—
303#127—1221no—
304#127—1221no—
305#127—1221no—
306#127C22⋊D6111yescharacteristic
307#126—24no—
308#126—24no—
309#125—21no—
310#125C22⋊D6112yes—
311#124—24no—
312#124—22no—
313#124—22no—
314#124—21no—
315#123—24no—
316#123—24no—
317C22⋊Q8—24no—
318C22⋊Q8—22no—
319C22⋊Q8—21no—
320C22⋊Q8—21no—
321C22⋊Q8—21no—
322C22⋊Q8—22no—
323C22⋊Q8—24no—
324C22⋊Q8C22⋊D6112yes—
325#121—21no—
326#121C22⋊D6112yes—
327C4×Q8—22no—
328C4×Q8—22no—
329C4×Q8—21no—
330C4×Q8—21no—
331C4×Q8—21no—
332C4×Q8C22×D6112yes—
333C4×D4—22no—
334C4×D4—22no—
335C4×D4—21no—
336C4×D4C22×D6112yes—
337#117—21no—
338#117—22no—
339#117C22⋊D6112yes—
340C2×C42C22×D6111yescharacteristic
341C4⋊C8—612no—
342C4⋊C8—612no—
343#102—1224no—
344#102—1224no—
345#102—1224no—
346#102—1224no—
347C22⋊C8—1221no—
348C22⋊C8—1221no—
349#97—1221no—
350C4×C8—612no—
351C4×C8—1221no—
352C4×C8—1221no—
353C24—2441no—
354#41—82no—
355C2×Q8—24no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8—84no—
361C2×Q8C2×C22⋊D6112yes—
362C2×D4—22no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—1222no—
366C2×D4—1222no—
367C2×D4—1222no—
368C2×D4—4882no—
369C2×D4—4882no—
370C2×D4—2444no—
371C2×D4—2441no—
372C2×D4—2441no—
373C2×D4—2441no—
374C2×D4C2×C22⋊D6112yes—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4C2×C22⋊D6111yescharacteristic
378C22×C4C2×C22⋊D6111yescharacteristic
379C22×C4C2×C22⋊D6111yescharacteristic
380SD16—4884no—
381D8—4884no—
382M16—2442no—
383C2×C8—1222no—
384C2×C8—1222no—
385C2×C8—1222no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—22no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4C2×C22⋊D6112yes—
399C4⋊C4C2×C22⋊D6112yes—
400C4⋊C4C2×C22⋊D6112yes—
401C4⋊C4C2×C22⋊D6112yes—
402C22⋊C4—24no—
403C22⋊C4—24no—
404C22⋊C4—24no—
405C22⋊C4—22no—
406C22⋊C4—22no—
407C22⋊C4—22no—
408C22⋊C4—22no—
409C42—24no—
410C42—22no—
411C42—22no—
412C42C2×C22⋊D6111yescharacteristic
413C42C2×C22⋊D6111yescharacteristic
414C42C2×C22⋊D6111yescharacteristic
415C42C23×D6111yescharacteristic
416C23—22no—
417C23—4882no—
418C23—4882no—
419C23—4882no—
420C23—2442no—
421C23#1451974611yescharacteristic
422Q8—24no—
423Q8—42no—
424Q8—42no—
425Q8—42no—
426Q8—81no—
427Q8—84no—
428D4—24no—
429D4—42no—
430D4—81no—
431D4—82no—
432D4—4882no—
433D4—4882no—
434D4—4882no—
435D4—2444no—
436D4—2444no—
437D4—2444no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—44no—
447C2×C4—84no—
448C2×C4#1451974611yescharacteristic
449C2×C4#1451974611yescharacteristic
450C2×C4#1451974611yescharacteristic
451C2×C4C22×C22⋊D6111yescharacteristic
452C2×C4C22×C22⋊D6111yescharacteristic
453C2×C4C22×C22⋊D6111yescharacteristic
454C8—2442no—
455C8—2442no—
456C8—2442no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—9764no—
464C22—4882no—
465C22—4882no—
466C22—4882no—
467C22C2×#1451974611yescharacteristic
468C4—21no—
469C4—21no—
470C4—21no—
471C4—42no—
472C4—42no—
473C4—42no—
474C4—42no—
475C4—42no—
476C4#1683993512yes—
477C4#1684056512yes—
478C4#1684057012yes—
479C2—41no—
480C2—82no—
481C2—9762no—
482C2#2145508711yescharacteristic
483C2#2145510511yescharacteristic
484C2C2×#1683993511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3550C2
socleC22×C61C2×#120
prime radical#21456734C2
hypercenter#3550D61
hypocenterC61#18729
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3550

Complements

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

NH ≅ G/Ntype
1. #21456757C2semidirect
3. #21456195C2semidirect
4. #21444929C2semidirect
5. #21444922C2semidirect
6. #21444484C2semidirect
7. #21444240C2semidirect
8. #21444236C2semidirect
9. C61×#3550C2semidirect
26. #16839508C22semidirect
27. #16839508C22semidirect
28. Q8⋊#92418C22semidirect
29. D4⋊#92418C22semidirect
34. C61×#556C22semidirect
36. D4×Q8×C61C22semidirect
237. #3550D61semidirect
256. #556C2×D61semidirect
258. D4×Q8C2×D61semidirect
285. C61#18729semidirect

Maximal subgroups

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

subgroupindex
#214567572
#214567342
#214561952
#214449292
#214449222
#214444842
#214442402
#214442362
C61×#35502
#1872961