Finite Group Databasebeta

← #29665506

Subgroups — #29665506

Order 15616.

694 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456940C211yesmaximal, characteristic, complemented
2#21455302C211yesmaximal, characteristic, complemented
3#21455276C211yesmaximal, characteristic
4#21454928C211yesmaximal, characteristic, complemented
5#21445266C212yesmaximal, complemented
6#21445259C212yesmaximal, complemented
7#21445212C212yesmaximal, complemented
8#21445207C212yesmaximal, complemented
9#21444593C211yesmaximal, characteristic, complemented
10#21444494C211yesmaximal, characteristic, complemented
11C2×C61×#332C211yesmaximal, characteristic, complemented
12C2×#14519808—21no—
13C2×#14519799—21no—
14#16840312C2211yescharacteristic
15C4×#11758222C2211yescharacteristic
16#16840165C2212yes—
17#16840110C2212yes—
18#16840075—22no—
19#16840039—21no—
20#16840039—21no—
21#16839985—21no—
22#16839978—21no—
23#16839973C2211yescharacteristic
24#16839969—21no—
25#16839969C2212yes—
26#16839968C2211yescharacteristic
27#16839964—21no—
28#16839964C2212yes—
29#16839963C2211yescharacteristic
30#16839958C2211yescharacteristic
31#16839886C2212yes—
32#16839827C2212yes—
33C8×C22⋊D61—22no—
34#16839815C2211yescharacteristic
35#16839814—21no—
36#16839786C2211yescharacteristic
37C4⋊C8×D61—21no—
38#16839771—21no—
39#16839759—21no—
40#16839514C2212yes—
41#16839514C2212yes—
42#16839507C2212yes—
43#16839505C2212yes—
44C2×C4⋊Q8×C61C2211yescharacteristic
45C2×C4⋊C8×C61C2211yescharacteristic
46C2×C4⋊C8×C61C2211yescharacteristic
47C61×#332C2212yes—
48C61×#332C2212yes—
49C2×#11758222—21no—
50C2×#11758222D411yescharacteristic
51#14519808—44no—
52#14519799—44no—
53C2×Dic244—21no—
54C2×#11758202—22no—
55C2×#11758190—21no—
56C2×#11758190—21no—
57C2×#11758201—22no—
58C2×D488—21no—
59C2×#11758194—21no—
60C2×C8×D61—22no—
61#14519777C2311yescharacteristic
62#14519774C2311yescharacteristic
63#14519765—21no—
64#14519751—21no—
65#14519751—21no—
66C4×Dic122D411yescharacteristic
67C4×Dic122C2311yescharacteristic
68C4×D244—21no—
69C4×D244—21no—
70C4×D244D411yescharacteristic
71C4×D244C2311yescharacteristic
72C4×C22⋊D61—21no—
73C4×C22⋊D61—21no—
74#14519736D411yescharacteristic
75#14519734—21no—
76C42×D61—21no—
77#14519710—22no—
78#14519710—22no—
79C4⋊#92418—21no—
80C4⋊#92418D411yescharacteristic
81C4⋊#92418D411yescharacteristic
82C4⋊#92418C2312yes—
83#14519706—22no—
84#14519702—21no—
85#14519702—21no—
86#14519701—22no—
87#14519701—22no—
88#14519699—21no—
89#14519699—21no—
90#14519697—22no—
91#14519697—22no—
92C22⋊#92418D412yes—
93#14519689—22no—
94C4×#92418C2312yes—
95C8×Dic61—22no—
96C22×Q8×C61—21no—
97C22×C8×C61—21no—
98C22×C8×C61—21no—
99C4⋊Q8×C61—21no—
100C4⋊Q8×C61—21no—
101C4⋊Q8×C61—21no—
102C4⋊Q8×C61C2312yes—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61—21no—
105C2×C42×C61C2311yescharacteristic
106C4⋊C8×C61—21no—
107C4⋊C8×C61—21no—
108C4⋊C8×C61C2311yescharacteristic
109C4⋊C8×C61C2311yescharacteristic
110C4⋊C8×C61C2312yes—
111C2×Q8×C61—22no—
112C2×Q8×C61—41no—
113C2×Q8×C61—42no—
114C2×Q8×C61—42no—
115C22×C4×C61—41no—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61#4111yescharacteristic
119C2×C22⋊D61—21no—
120C2×C22⋊D61—41no—
121#11758222—22no—
122#11758222—42no—
123#11758222—42no—
124#11758222D814yes—
125C2×D244—21no—
126C2×D244—21no—
127C2×D244—21no—
128C2×D244C2×D411yescharacteristic
129C2×C4×D61—21no—
130C2×C4×D61—21no—
131C2×C4×D61—21no—
132C2×C4×D61—41no—
133C2×Dic122—21no—
134C2×Dic122C2×D411yescharacteristic
135C2×C8×C61—22no—
136C2×C8×C61—21no—
137C2×C8×C61—21no—
138C2×C8×C61—42no—
139C2×C8×C61—81no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—41no—
143C4⋊C4×C61—41no—
144C42×C61C2×D411yescharacteristic
145C42×C61C2×D411yescharacteristic
146C42×C61#4111yescharacteristic
147C42×C61C2411yescharacteristic
148C22⋊Dic61—21no—
149#11758202—44no—
150#11758201—44no—
151#11758198—21no—
152#11758198—21no—
153#11758198—41no—
154#11758198—41no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61C2×D411yescharacteristic
158C4⋊Dic61C2×D411yescharacteristic
159#11758196—21no—
160#11758196—21no—
161C4×Dic61—21no—
162C4×Dic61—21no—
163C4×Dic61—21no—
164#11758194—42no—
165C2×#92418—22no—
166C2×#92418C2×D412yes—
167C2×#92418C2×D412yes—
168Dic244—42no—
169D488—42no—
170#11758190—42no—
171#11758190—42no—
172C8×D61—82no—
173C23×C61#12111yescharacteristic
174C22×D61—21no—
175C22×D61—41no—
176Q8×C61—42no—
177Q8×C61—42no—
178Q8×C61—82no—
179C2×C4×C61—21no—
180C2×C4×C61—21no—
181C2×C4×C61—41no—
182C2×C4×C61—42no—
183C2×C4×C61—82no—
184C2×C4×C61#12111yescharacteristic
185C2×C4×C61#12111yescharacteristic
186C2×C4×C61#12111yescharacteristic
187C2×C4×C61C2×D812yes—
188C2×C4×C61#13512yes—
189C2×C4×C61C22×D411yescharacteristic
190C2×C4×C61C22×D411yescharacteristic
191C2×C4×C61C2×#4111yescharacteristic
192C22⋊D61—24no—
193C22⋊D61—82no—
194C2×Dic61—21no—
195C2×Dic61—21no—
196C2×Dic61—21no—
197C2×Dic61—41no—
198D244—21no—
199D244—41no—
200D244—41no—
201D244—41no—
202D244—41no—
203D244—81no—
204D244C2×D812yes—
205C4×D61—24no—
206C4×D61—41no—
207C4×D61—41no—
208C4×D61—81no—
209C4×D61—82no—
210Dic122—21no—
211Dic122—41no—
212Dic122—41no—
213Dic122C2×D812yes—
214C8×C61—41no—
215C8×C61—41no—
216C8×C61—42no—
217#92418—22no—
218#92418—22no—
219#92418—42no—
220#19416—611nomaximal, Sylow
221C22×C61—21no—
222C22×C61—21no—
223C22×C61#46612yes—
224C22×C61C2×#12111yescharacteristic
225C2×D61—22no—
226C2×D61—41no—
227C2×D61—41no—
228C2×D61—81no—
229C2×D61—81no—
230C2×D61—81no—
231C4×C61—21no—
232C4×C61—21no—
233C4×C61—21no—
234C4×C61—82no—
235C4×C61#46612yes—
236C4×C61C22×D811yescharacteristic
237C4×C61C2×#13511yescharacteristic
238C4×C61#57711yescharacteristic
239C4×C61#57811yescharacteristic
240Dic61—22no—
241Dic61—41no—
242Dic61—41no—
243Dic61—81no—
244#3226—611no—
245#3039—611no—
246#3035—611no—
247#2988—611no—
248#1682—612no—
249#1681—612no—
250#1674—612no—
251#1673—612no—
252#1571—611no—
253#1547—611no—
254C2×#332D6111yescharacteristic, complemented
255C2×C61—22no—
256C2×C61#304611yescharacteristic
257C2×C61C2×#46611yescharacteristic
258C2×C61#306911yescharacteristic
259D61—42no—
260D61—161no—
261C2×#137—1221no—
262C2×#135—1221no—
263#532—611no—
264C2×C4⋊Q8C2×D6111yescharacteristic
265C4×#41—611no—
266#500—612no—
267#486—612no—
268#480—1222no—
269#474—1221no—
270#474—1221no—
271#464—1221no—
272#463—1221no—
273#462—612no—
274#462—611no—
275#462—1221no—
276#461—612no—
277#461—611no—
278#461—1221no—
279#460—611no—
280#459—611no—
281C8×Q8—612no—
282C8×D4—612no—
283C8×D4—1222no—
284#431—611no—
285#431—1221no—
286#423—611no—
287C2×C4⋊C8—1221no—
288C2×C4⋊C8C2×D6111yescharacteristic
289C2×C4⋊C8C2×D6111yescharacteristic
290#419—1221no—
291#416—1221no—
292#335—612no—
293#335—612no—
294#332—612no—
295#332C2×D6112yes—
296#332C2×D6112yes—
297#331—612no—
298C61#1941611yescharacteristic, complemented, Sylow
299C2×#41—611no—
300C2×#41—1221no—
301C22×Q8—21no—
302#137—2444no—
303#135—2444no—
304C2×Q16—1221no—
305C2×Q16—1222no—
306C2×SD16—1221no—
307C2×SD16—1221no—
308C2×SD16—1222no—
309C2×D8—1221no—
310C2×M16—1221no—
311C22×C8—21no—
312C22×C8—21no—
313C22×C8—1222no—
314C4⋊Q8—21no—
315C4⋊Q8—21no—
316C4⋊Q8—21no—
317C4⋊Q8—611no—
318C4⋊Q8C22×D6112yes—
319#127—611no—
320#124—1221no—
321#121—1221no—
322#121—1221no—
323C4×Q8—611no—
324C4×Q8—611no—
325C4×D4—611no—
326C4×D4—611no—
327C4×D4—1221no—
328C4×D4—1221no—
329C4×D4—1221no—
330C4×D4—1221no—
331#117—611no—
332#117—1221no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C42—1221no—
336C2×C42C22×D6111yescharacteristic
337#107—1222no—
338#107—1222no—
339C4⋊C8—21no—
340C4⋊C8—21no—
341C4⋊C8—611no—
342C4⋊C8—611no—
343C4⋊C8—612no—
344C4⋊C8—1222no—
345C4⋊C8—1221no—
346C4⋊C8C22×D6112yes—
347C4⋊C8C22×D6111yescharacteristic
348C4⋊C8C22×D6111yescharacteristic
349#103—1221no—
350#103—1221no—
351#103—1222no—
352#103—1222no—
353#102—1222no—
354#102—1222no—
355#102—1221no—
356#102—1221no—
357C22⋊C8—612no—
358C22⋊C8—1222no—
359C4×C8—612no—
360C4×C8—1222no—
361#41—614no—
362#41—1222no—
363#41—2442no—
364#41—2442no—
365C2×Q8—22no—
366C2×Q8—41no—
367C2×Q8—42no—
368C2×Q8—42no—
369C2×Q8—611no—
370C2×Q8—1221no—
371C2×D4—611no—
372C2×D4—1221no—
373C2×D4—1221no—
374C2×D4—1221no—
375C2×D4—1221no—
376C2×D4—2441no—
377C22×C4—41no—
378C22×C4—1221no—
379C22×C4—1221no—
380C22×C4—1221no—
381C22×C4—2441no—
382C22×C4#1175822211yescharacteristic
383C22×C4D4×D6111yescharacteristic
384C22×C4D4×D6111yescharacteristic
385Q16—2442no—
386Q16—2444no—
387SD16—2442no—
388SD16—2442no—
389SD16—2444no—
390D8—2442no—
391M16—2442no—
392C2×C8—22no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—42no—
396C2×C8—81no—
397C2×C8—612no—
398C2×C8—612no—
399C2×C8—1222no—
400C2×C8—4882no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—611no—
406C4⋊C4—611no—
407C4⋊C4—1221no—
408C4⋊C4—1221no—
409C4⋊C4—1221no—
410C4⋊C4—1221no—
411C22⋊C4—1221no—
412C22⋊C4—1221no—
413C22⋊C4—1221no—
414C22⋊C4—2441no—
415C22⋊C4—2441no—
416C42—1221no—
417C42—1221no—
418C42—1221no—
419C42#1175822211yescharacteristic
420C42D4×D6111yescharacteristic
421C42D4×D6111yescharacteristic
422C42C23×D6111yescharacteristic
423C23—1221no—
424C23—2441no—
425C23#1451975011yescharacteristic
426Q8—42no—
427Q8—42no—
428Q8—82no—
429Q8—612no—
430Q8—1221no—
431Q8—2441no—
432Q8—2441no—
433D4—612no—
434D4—1224no—
435D4—1221no—
436D4—4881no—
437D4—4882no—
438D4—2441no—
439D4—2441no—
440D4—2441no—
441D4—2441no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—82no—
447C2×C4—1221no—
448C2×C4—1224no—
449C2×C4—1221no—
450C2×C4—1221no—
451C2×C4—4881no—
452C2×C4—4882no—
453C2×C4—2441no—
454C2×C4—2441no—
455C2×C4—2441no—
456C2×C4#1451975011yescharacteristic
457C2×C4#1451975011yescharacteristic
458C2×C4#1451975011yescharacteristic
459C2×C4D8×D6112yes—
460C2×C4#1451979812yes—
461C2×C4C2×D4×D6111yescharacteristic
462C2×C4C2×D4×D6111yescharacteristic
463C2×C4C2×#1175822211yescharacteristic
464C8—41no—
465C8—41no—
466C8—42no—
467C8—1222no—
468C8—1222no—
469C8—2442no—
470C22—21no—
471C22—21no—
472C22—1222no—
473C22—4881no—
474C22—4881no—
475C22—4881no—
476C22—2441no—
477C22—2441no—
478C22#1683999712yes—
479C22C2×#1451975011yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—82no—
484C4—1222no—
485C4—4881no—
486C4—2441no—
487C4—2441no—
488C4#1683999712yes—
489C4C2×D8×D6111yescharacteristic
490C4C2×#1451979811yescharacteristic
491C4#1684056811yescharacteristic
492C4#1684057511yescharacteristic
493C2—22no—
494C2—9761no—
495C2—2442no—
496C2#2145537411yescharacteristic
497C2C2×#1683999711yescharacteristic
498C2#2145555511yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#332C2
socleC22×C61C2×#121
prime radical#21455276C2
hypercenterC2×#332D61
hypocenterC61#19416
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
1. #21456940C2semidirect
2. #21455302C2semidirect
4. #21454928C2semidirect
5. #21445266C2semidirect
6. #21445259C2semidirect
7. #21445212C2semidirect
8. #21445207C2semidirect
9. #21444593C2semidirect
10. #21444494C2semidirect
11. C2×C61×#332C2semidirect
254. C2×#332D61semidirect
298. C61#19416semidirect

Maximal subgroups

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

subgroupindex
#214569402
#214553022
#214552762
#214549282
#214452662
#214452592
#214452122
#214452072
#214445932
#214444942
C2×C61×#3322
#1941661