Finite Group Databasebeta

← #29665501

Subgroups — #29665501

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#21456942C211yesmaximal, characteristic, complemented
2#21455306C211yesmaximal, characteristic, complemented
3#21455274C211yesmaximal, characteristic
4#21445263C212yesmaximal, complemented
5#21445256C212yesmaximal, complemented
6#21445213C212yesmaximal, complemented
7#21445208C212yesmaximal, complemented
8#21444594C211yesmaximal, characteristic, complemented
9#21444494C211yesmaximal, characteristic, complemented
10C2×#16839507C211yesmaximal, characteristic, complemented
11C61×#2988C211yesmaximal, characteristic, complemented
12C2×#14519807—21no—
13C2×#14519797—21no—
14#16840312C2211yescharacteristic
15C2×#14519777C2211yescharacteristic
16#16840167C2212yes—
17#16840109C2212yes—
18#16840076—22no—
19#16840043—21no—
20#16840043—21no—
21#16839984—21no—
22#16839977—21no—
23#16839972—21no—
24#16839972C2211yescharacteristic
25#16839972C2212yes—
26#16839967—21no—
27#16839967C2211yescharacteristic
28#16839967C2212yes—
29#16839962C2211yescharacteristic
30#16839957C2211yescharacteristic
31#16839786C2211yescharacteristic
32C2×C4⋊#92418C2211yescharacteristic
33#16839770—21no—
34#16839756—21no—
35#16839515C2212yes—
36#16839515C2212yes—
37#16839507C2212yes—
38#16839507C2212yes—
39#16839507C2212yes—
40#16839505C2212yes—
41C4×#41×C61C2211yescharacteristic
42C8×Q8×C61C2212yes—
43C8×D4×C61—22no—
44C8×D4×C61C2212yes—
45C61×#431—21no—
46C61×#431C2211yescharacteristic
47C2×C4⋊C8×C61—21no—
48C2×C4⋊C8×C61C2211yescharacteristic
49C2×#11758222—21no—
50C22×Dic122—21no—
51#14519807—44no—
52#14519797—44no—
53C2×#11758202—22no—
54C2×#11758202—21no—
55C2×#11758200—22no—
56C2×#11758200—21no—
57C2×#11758201—21no—
58C2×#11758199—21no—
59C2×#11758194—21no—
60C22×#92418—21no—
61#14519777—21no—
62#14519777—21no—
63#14519777—21no—
64#14519777C2312yes—
65#14519777C2311yescharacteristic
66#14519774C2311yescharacteristic
67#14519765—21no—
68#14519751—21no—
69#14519751—21no—
70C4×D244—21no—
71C4×D244—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74#14519711—22no—
75#14519711—22no—
76C4⋊#92418—21no—
77C4⋊#92418—21no—
78C4⋊#92418C2311yescharacteristic
79C4⋊#92418C2311yescharacteristic
80C4⋊#92418C2312yes—
81#14519700—22no—
82#14519700—22no—
83#14519700—21no—
84#14519700—21no—
85#14519699—22no—
86#14519699—22no—
87#14519697—21no—
88#14519697—21no—
89C2×#41×C61D411yescharacteristic
90C22×C8×C61—21no—
91C22×C8×C61—22no—
92C4×Q8×C61D411yescharacteristic
93C4×Q8×C61C2311yescharacteristic
94C4×D4×C61—21no—
95C4×D4×C61—21no—
96C4×D4×C61D411yescharacteristic
97C4×D4×C61C2311yescharacteristic
98#117×C61—21no—
99#117×C61D411yescharacteristic
100C2×C42×C61—21no—
101C2×C42×C61C2311yescharacteristic
102C4⋊C8×C61—22no—
103C4⋊C8×C61—21no—
104C4⋊C8×C61D411yescharacteristic
105C4⋊C8×C61D411yescharacteristic
106C4⋊C8×C61C2312yes—
107C22⋊C8×C61—22no—
108C22⋊C8×C61D412yes—
109C4×C8×C61—22no—
110C4×C8×C61C2312yes—
111#41×C61—22no—
112#41×C61D814yes—
113C2×Q8×C61C2×D411yescharacteristic
114C2×D4×C61—21no—
115C2×D4×C61C2×D411yescharacteristic
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61#4111yescharacteristic
122C2×C22⋊D61—41no—
123C22×Dic61—41no—
124#11758222—42no—
125#11758222—42no—
126C2×D244—21no—
127C2×D244—21no—
128C2×D244—21no—
129C2×C4×D61—41no—
130C2×Dic122—22no—
131C2×Dic122—21no—
132C2×Dic122—42no—
133C2×Dic122—42no—
134C2×Dic122—41no—
135C2×C8×C61—22no—
136C2×C8×C61—82no—
137C2×C8×C61—81no—
138C2×C8×C61C2×D412yes—
139C2×C8×C61C2×D412yes—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61C2×D411yescharacteristic
143C4⋊C4×C61C2×D411yescharacteristic
144C22⋊C4×C61—21no—
145C22⋊C4×C61—21no—
146C22⋊C4×C61—21no—
147C42×C61—21no—
148C42×C61—21no—
149C42×C61—21no—
150C42×C61C2×D411yescharacteristic
151C42×C61C2×D411yescharacteristic
152C42×C61#4111yescharacteristic
153C42×C61C2411yescharacteristic
154#11758202—42no—
155#11758202—44no—
156#11758201—42no—
157#11758200—42no—
158#11758200—44no—
159#11758199—42no—
160#11758198—41no—
161#11758198—41no—
162C4⋊Dic61—22no—
163C4⋊Dic61—21no—
164C4⋊Dic61—22no—
165C4⋊Dic61—21no—
166C4⋊Dic61—41no—
167C4⋊Dic61—41no—
168#11758194—42no—
169C2×#92418—21no—
170C2×#92418—21no—
171C2×#92418—22no—
172C2×#92418—42no—
173C23×C61—21no—
174C23×C61#12111yescharacteristic
175C22×D61—41no—
176Q8×C61—21no—
177Q8×C61C2×D812yes—
178D4×C61—24no—
179D4×C61—21no—
180D4×C61C2×D812yes—
181C2×C4×C61—21no—
182C2×C4×C61—21no—
183C2×C4×C61—21no—
184C2×C4×C61—24no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—81no—
191C2×C4×C61#12111yescharacteristic
192C2×C4×C61#12111yescharacteristic
193C2×C4×C61#12111yescharacteristic
194C2×C4×C61C2×D812yes—
195C2×C4×C61#13512yes—
196C2×C4×C61C22×D411yescharacteristic
197C2×C4×C61C22×D411yescharacteristic
198C2×C4×C61C2×#4111yescharacteristic
199C22⋊D61—82no—
200C2×Dic61—42no—
201C2×Dic61—41no—
202C2×Dic61—82no—
203D244—41no—
204D244—41no—
205D244—41no—
206D244—41no—
207D244—81no—
208C4×D61—82no—
209Dic122—42no—
210Dic122—41no—
211Dic122—42no—
212Dic122—41no—
213Dic122—82no—
214C8×C61—22no—
215C8×C61—22no—
216C8×C61—42no—
217#92418—41no—
218#92418—41no—
219#92418—42no—
220#19416—611nomaximal, Sylow
221C22×C61—21no—
222C22×C61—21no—
223C22×C61—22no—
224C22×C61—41no—
225C22×C61—41no—
226C22×C61#46612yes—
227C22×C61C2×#12111yescharacteristic
228C2×D61—81no—
229C2×D61—81no—
230C2×D61—81no—
231C4×C61—21no—
232C4×C61—22no—
233C4×C61—21no—
234C4×C61—21no—
235C4×C61—41no—
236C4×C61—41no—
237C4×C61#46612yes—
238C4×C61C22×D811yescharacteristic
239C4×C61C2×#13511yescharacteristic
240C4×C61#57711yescharacteristic
241C4×C61#57811yescharacteristic
242Dic61—82no—
243Dic61—81no—
244#3226—611no—
245#3039—611no—
246#3035—611no—
247#2988D6111yescharacteristic, complemented
248#1682—612no—
249#1681—612no—
250#1674—612no—
251#1673—612no—
252#1571—611no—
253#1547—611no—
254C2×#332—611no—
255C2×C61—22no—
256C2×C61—42no—
257C2×C61#304611yescharacteristic
258C2×C61C2×#46611yescharacteristic
259C2×C61#306911yescharacteristic
260D61—161no—
261C2×#137—1221no—
262C2×#135—1221no—
263#532—611no—
264C2×C4⋊Q8—611no—
265C4×#41C2×D6111yescharacteristic
266#500—612no—
267#486—612no—
268#480—1222no—
269#474—1221no—
270#474—1221no—
271#464—1221no—
272#463—1221no—
273#462—611no—
274#462—612no—
275#462—1221no—
276#461—611no—
277#461—612no—
278#461—1221no—
279#460—611no—
280#459—611no—
281C8×Q8C2×D6112yes—
282C8×D4—22no—
283C8×D4C2×D6112yes—
284#431—21no—
285#431C2×D6111yescharacteristic
286#423—611no—
287C2×C4⋊C8—21no—
288C2×C4⋊C8—611no—
289C2×C4⋊C8C2×D6111yescharacteristic
290#419—1221no—
291#416—1221no—
292#335—612no—
293#335—612no—
294#332—612no—
295#332—612no—
296#332—612no—
297#331—612no—
298C61#1941611yescharacteristic, complemented, Sylow
299C2×#41—1221no—
300C2×#41D24411yescharacteristic
301C22×Q8—1221no—
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—22no—
313C22×C8—1221no—
314C4⋊Q8—612no—
315C4⋊Q8—611no—
316C4⋊Q8—1221no—
317C4⋊Q8—1221no—
318C4⋊Q8—1221no—
319#127—611no—
320#124—1221no—
321#121—1221no—
322#121—1221no—
323C4×Q8D24411yescharacteristic
324C4×Q8C22×D6111yescharacteristic
325C4×D4—21no—
326C4×D4—21no—
327C4×D4—1221no—
328C4×D4—1221no—
329C4×D4D24411yescharacteristic
330C4×D4C22×D6111yescharacteristic
331#117—21no—
332#117D24411yescharacteristic
333C2×C4⋊C4—1221no—
334C2×C4⋊C4—1221no—
335C2×C42—21no—
336C2×C42C22×D6111yescharacteristic
337#107—1222no—
338#107—1222no—
339C4⋊C8—22no—
340C4⋊C8—21no—
341C4⋊C8—612no—
342C4⋊C8—611no—
343C4⋊C8—611no—
344C4⋊C8—1221no—
345C4⋊C8—1221no—
346C4⋊C8C22⋊D6111yescharacteristic
347C4⋊C8C22⋊D6111yescharacteristic
348C4⋊C8C22×D6112yes—
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—22no—
358C22⋊C8C22⋊D6112yes—
359C4×C8—22no—
360C4×C8C22×D6112yes—
361#41—22no—
362#41—2442no—
363#41—2442no—
364#41D48814yes—
365C2×Q8—1221no—
366C2×Q8—1222no—
367C2×Q8—2442no—
368C2×Q8—2442no—
369C2×Q8—2441no—
370C2×Q8C2×D24411yescharacteristic
371C2×D4—21no—
372C2×D4—1221no—
373C2×D4—1221no—
374C2×D4—1221no—
375C2×D4—2441no—
376C2×D4C2×D24411yescharacteristic
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—2441no—
381C22×C4—2441no—
382C22×C4C2×D24411yescharacteristic
383C22×C4#1175822211yescharacteristic
384C22×C4C2×C22⋊D6111yescharacteristic
385Q16—2442no—
386Q16—2444no—
387SD16—2442no—
388SD16—2442no—
389SD16—2444no—
390D8—2442no—
391M16—2442no—
392C2×C8—22no—
393C2×C8—82no—
394C2×C8—81no—
395C2×C8—1221no—
396C2×C8—1221no—
397C2×C8—1222no—
398C2×C8—2442no—
399C2×C8C2×C22⋊D6112yes—
400C2×C8C2×C22⋊D6112yes—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—1222no—
404C4⋊C4—1221no—
405C4⋊C4—1222no—
406C4⋊C4—1221no—
407C4⋊C4—2441no—
408C4⋊C4—2441no—
409C4⋊C4C2×D24411yescharacteristic
410C4⋊C4C2×D24411yescharacteristic
411C22⋊C4—21no—
412C22⋊C4—21no—
413C22⋊C4—21no—
414C22⋊C4—2441no—
415C22⋊C4—2441no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42C2×D24411yescharacteristic
420C42#1175822211yescharacteristic
421C42C2×C22⋊D6111yescharacteristic
422C42C23×D6111yescharacteristic
423C23—21no—
424C23—2441no—
425C23#1451975111yescharacteristic
426Q8—21no—
427Q8—4882no—
428Q8—2442no—
429Q8—2441no—
430Q8—2442no—
431Q8—2441no—
432Q8C2×D48812yes—
433D4—24no—
434D4—21no—
435D4—4881no—
436D4—4882no—
437D4—2441no—
438D4—2441no—
439D4—2441no—
440D4—2441no—
441D4C2×D48812yes—
442C2×C4—21no—
443C2×C4—24no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—81no—
452C2×C4—4882no—
453C2×C4—4882no—
454C2×C4—2441no—
455C2×C4—2442no—
456C2×C4#1451975111yescharacteristic
457C2×C4#1451975111yescharacteristic
458C2×C4#1451975111yescharacteristic
459C2×C4C2×D48812yes—
460C2×C4#1451979912yes—
461C2×C4C22×C22⋊D6111yescharacteristic
462C2×C4C22×D24411yescharacteristic
463C2×C4C2×#1175822211yescharacteristic
464C8—22no—
465C8—22no—
466C8—42no—
467C8—2441no—
468C8—2441no—
469C8—2442no—
470C22—21no—
471C22—21no—
472C22—22no—
473C22—41no—
474C22—41no—
475C22—4881no—
476C22—4881no—
477C22—4881no—
478C22#1683999812yes—
479C22C2×#1451975111yescharacteristic
480C4—21no—
481C4—22no—
482C4—21no—
483C4—21no—
484C4—41no—
485C4—41no—
486C4—4881no—
487C4—4882no—
488C4#1683999812yes—
489C4C22×D48811yescharacteristic
490C4C2×#1451979911yescharacteristic
491C4#1684057311yescharacteristic
492C4#1684057711yescharacteristic
493C2—22no—
494C2—42no—
495C2—9761no—
496C2#2145536911yescharacteristic
497C2C2×#1683999811yescharacteristic
498C2#2145555611yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#2988C2
socleC22×C61C2×#121
prime radical#21455274C2
hypercenter#2988D61
hypocenterC61#19416
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2988

Complements

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

NH ≅ G/Ntype
1. #21456942C2semidirect
2. #21455306C2semidirect
4. #21445263C2semidirect
5. #21445256C2semidirect
6. #21445213C2semidirect
7. #21445208C2semidirect
8. #21444594C2semidirect
9. #21444494C2semidirect
10. C2×#16839507C2semidirect
11. C61×#2988C2semidirect
247. #2988D61semidirect
298. C61#19416semidirect

Maximal subgroups

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

subgroupindex
#214569422
#214553062
#214552742
#214452632
#214452562
#214452132
#214452082
#214445942
#214444942
C2×#168395072
C61×#29882
#1941661