Finite Group Databasebeta

← #29665497

Subgroups — #29665497

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#21455302C211yesmaximal, characteristic, complemented
2#21455268C211yesmaximal, characteristic
3#21454927C211yesmaximal, characteristic, complemented
4#21445260C212yesmaximal, complemented
5#21445253C212yesmaximal, complemented
6#21445210C212yesmaximal, complemented
7#21445205C212yesmaximal, complemented
8#21444594C211yesmaximal, characteristic, complemented
9#21444496C211yesmaximal, characteristic, complemented
10C2×#16839508C211yesmaximal, characteristic, complemented
11C61×#3226C211yesmaximal, characteristic, complemented
12C2×#14519808—21no—
13C2×#14519797—21no—
14C4×#11758222C2211yescharacteristic
15#16839981—21no—
16#16839974—21no—
17#16839969—21no—
18#16839969C2211yescharacteristic
19#16839969C2212yes—
20#16839964—21no—
21#16839964C2211yescharacteristic
22#16839964C2212yes—
23#16839959C2211yescharacteristic
24#16839954C2211yescharacteristic
25C8×Dic122C2212yes—
26C8×D244C2212yes—
27#16839825—22no—
28#16839814—21no—
29#16839786C2211yescharacteristic
30C2×C4⋊#92418C2211yescharacteristic
31C2×C4⋊#92418C2211yescharacteristic
32C4⋊C8×D61—21no—
33#16839771—21no—
34#16839758—21no—
35#16839515C2212yes—
36#16839515C2212yes—
37#16839508C2212yes—
38#16839508C2212yes—
39#16839508C2212yes—
40#16839506C2212yes—
41C61×#532C2211yescharacteristic
42C2×C4⋊Q8×C61C2211yescharacteristic
43C61×#500C2212yes—
44C61×#486C2212yes—
45C61×#480—22no—
46C61×#474—21no—
47C61×#474—21no—
48C61×#431C2211yescharacteristic
49C2×#11758222D411yescharacteristic
50#14519808—44no—
51#14519797—44no—
52C2×#11758202—22no—
53C2×#11758202—21no—
54C2×#11758200—21no—
55C2×#11758201—22no—
56C2×#11758201—21no—
57C2×#11758199—21no—
58C2×#11758194—21no—
59C22×#92418—21no—
60C22×#92418—21no—
61C2×C8×D61—22no—
62C4×Dic122D411yescharacteristic
63C4×Dic122C2311yescharacteristic
64C4×D244D411yescharacteristic
65C4×D244C2311yescharacteristic
66C4×C22⋊D61—21no—
67C4×C22⋊D61—21no—
68#14519736D411yescharacteristic
69#14519734—21no—
70C42×D61—21no—
71C4⋊#92418—21no—
72C4⋊#92418—21no—
73C4⋊#92418—21no—
74C4⋊#92418C2311yescharacteristic
75C4⋊#92418C2311yescharacteristic
76C4⋊#92418C2312yes—
77C4⋊#92418C2312yes—
78#14519706—22no—
79#14519701—22no—
80#14519701—22no—
81#14519701—21no—
82#14519701—21no—
83#14519698—21no—
84#14519698—21no—
85#14519689—22no—
86C8×Dic61—22no—
87C2×#41×C61—21no—
88C22×Q8×C61—21no—
89C4⋊Q8×C61—21no—
90C4⋊Q8×C61—21no—
91C4⋊Q8×C61—21no—
92C4⋊Q8×C61C2312yes—
93C4⋊Q8×C61C2311yescharacteristic
94#127×C61C2311yescharacteristic
95#124×C61—21no—
96#121×C61—21no—
97#121×C61—21no—
98C4×D4×C61—21no—
99C4×D4×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C42×C61C2311yescharacteristic
103#107×C61—22no—
104#107×C61—22no—
105C4⋊C8×C61D411yescharacteristic
106C4⋊C8×C61D411yescharacteristic
107#102×C61—22no—
108#102×C61—22no—
109C22⋊C8×C61D412yes—
110C4×C8×C61C2312yes—
111#41×C61—42no—
112#41×C61—42no—
113C2×Q8×C61—21no—
114C2×Q8×C61—22no—
115C2×Q8×C61—41no—
116C2×Q8×C61—42no—
117C2×Q8×C61—42no—
118C2×D4×C61—21no—
119C2×D4×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61—41no—
122C22×C4×C61—41no—
123C22×C4×C61—41no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61#4111yescharacteristic
127C2×C22⋊D61—21no—
128#11758222—22no—
129#11758222D814yes—
130C2×D244C2×D411yescharacteristic
131C2×C4×D61—21no—
132C2×C4×D61—21no—
133C2×C4×D61—21no—
134C2×Dic122C2×D411yescharacteristic
135C2×C8×C61C2×D412yes—
136C2×C8×C61C2×D412yes—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—41no—
142C4⋊C4×C61—41no—
143C22⋊C4×C61—41no—
144C22⋊C4×C61—41no—
145C42×C61C2×D411yescharacteristic
146C42×C61C2×D411yescharacteristic
147C42×C61#4111yescharacteristic
148C42×C61C2411yescharacteristic
149C22⋊Dic61—21no—
150#11758202—44no—
151#11758202—42no—
152#11758201—44no—
153#11758201—42no—
154#11758200—42no—
155#11758199—42no—
156#11758198—21no—
157#11758198—21no—
158C4⋊Dic61C2×D411yescharacteristic
159C4⋊Dic61C2×D411yescharacteristic
160#11758196—21no—
161#11758196—21no—
162C4×Dic61—21no—
163C4×Dic61—21no—
164C4×Dic61—21no—
165#11758194—42no—
166C2×#92418—22no—
167C2×#92418—21no—
168C2×#92418—21no—
169C2×#92418—22no—
170C2×#92418—42no—
171C2×#92418—81no—
172C8×D61—82no—
173C23×C61—41no—
174C23×C61#12111yescharacteristic
175C22×D61—21no—
176Q8×C61—42no—
177Q8×C61—41no—
178Q8×C61—42no—
179Q8×C61—41no—
180Q8×C61—82no—
181D4×C61—41no—
182D4×C61—41no—
183D4×C61—41no—
184D4×C61—41no—
185D4×C61—81no—
186D4×C61—82no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61—42no—
192C2×C4×C61—82no—
193C2×C4×C61—82no—
194C2×C4×C61#12111yescharacteristic
195C2×C4×C61#12111yescharacteristic
196C2×C4×C61#12111yescharacteristic
197C2×C4×C61C2×D812yes—
198C2×C4×C61#13512yes—
199C2×C4×C61C22×D411yescharacteristic
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C2×#4111yescharacteristic
202C22⋊D61—24no—
203C2×Dic61—21no—
204C2×Dic61—21no—
205C2×Dic61—21no—
206D244—21no—
207D244C2×D812yes—
208C4×D61—24no—
209C4×D61—41no—
210C4×D61—41no—
211C4×D61—81no—
212Dic122—21no—
213Dic122C2×D812yes—
214C8×C61—22no—
215C8×C61—22no—
216#92418—41no—
217#92418—41no—
218#92418—42no—
219#92418—42no—
220#19416—611nomaximal, Sylow
221C22×C61—21no—
222C22×C61—21no—
223C22×C61—81no—
224C22×C61—81no—
225C22×C61—81no—
226C22×C61#46612yes—
227C22×C61C2×#12111yescharacteristic
228C2×D61—22no—
229C2×D61—41no—
230C2×D61—41no—
231C4×C61—21no—
232C4×C61—21no—
233C4×C61—21no—
234C4×C61—81no—
235C4×C61—82no—
236C4×C61#46612yes—
237C4×C61C22×D811yescharacteristic
238C4×C61C2×#13511yescharacteristic
239C4×C61#57711yescharacteristic
240C4×C61#57811yescharacteristic
241Dic61—22no—
242Dic61—41no—
243Dic61—41no—
244#3226D6111yescharacteristic, complemented
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×#332—611no—
255C2×C61—22no—
256C2×C61—161no—
257C2×C61#304611yescharacteristic
258C2×C61C2×#46611yescharacteristic
259C2×C61#306911yescharacteristic
260D61—42no—
261C2×#137—1221no—
262C2×#135—1221no—
263#532C2×D6111yescharacteristic
264C2×C4⋊Q8C2×D6111yescharacteristic
265C4×#41—611no—
266#500C2×D6112yes—
267#486C2×D6112yes—
268#480—22no—
269#474—21no—
270#474—21no—
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×Q8—612no—
282C8×D4—612no—
283C8×D4—1222no—
284#431—1221no—
285#431C2×D6111yescharacteristic
286#423—611no—
287C2×C4⋊C8—611no—
288C2×C4⋊C8—611no—
289C2×C4⋊C8—1221no—
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—21no—
300C2×#41—611no—
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—1221no—
312C22×C8—1221no—
313C22×C8—1222no—
314C4⋊Q8—21no—
315C4⋊Q8—21no—
316C4⋊Q8—21no—
317C4⋊Q8C22×D6112yes—
318C4⋊Q8C22×D6111yescharacteristic
319#127C22×D6111yescharacteristic
320#124—21no—
321#121—21no—
322#121—21no—
323C4×Q8—611no—
324C4×Q8—611no—
325C4×D4—21no—
326C4×D4—21no—
327C4×D4—611no—
328C4×D4—611no—
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—22no—
338#107—22no—
339C4⋊C8—612no—
340C4⋊C8—612no—
341C4⋊C8—611no—
342C4⋊C8—611no—
343C4⋊C8—1222no—
344C4⋊C8—1221no—
345C4⋊C8—1221no—
346C4⋊C8—1221no—
347C4⋊C8C22⋊D6111yescharacteristic
348C4⋊C8C22⋊D6111yescharacteristic
349#103—1221no—
350#103—1221no—
351#103—1222no—
352#103—1222no—
353#102—22no—
354#102—22no—
355#102—1221no—
356#102—1221no—
357C22⋊C8—1222no—
358C22⋊C8C22⋊D6112yes—
359C4×C8—1222no—
360C4×C8C22×D6112yes—
361#41—42no—
362#41—42no—
363#41—614no—
364#41—1222no—
365C2×Q8—21no—
366C2×Q8—22no—
367C2×Q8—41no—
368C2×Q8—42no—
369C2×Q8—42no—
370C2×Q8—611no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—41no—
375C2×D4—611no—
376C2×D4—1221no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—1221no—
380C22×C4—1221no—
381C22×C4—1221no—
382C22×C4D4×D6111yescharacteristic
383C22×C4#1175822411yescharacteristic
384C22×C4C2×C22⋊D6111yescharacteristic
385Q16—2442no—
386Q16—2444no—
387SD16—2442no—
388SD16—2442no—
389SD16—2444no—
390D8—2442no—
391M16—2442no—
392C2×C8—1221no—
393C2×C8—1221no—
394C2×C8—1222no—
395C2×C8—1222no—
396C2×C8—4882no—
397C2×C8—4881no—
398C2×C8—2442no—
399C2×C8C2×C22⋊D6112yes—
400C2×C8C2×C22⋊D6112yes—
401C4⋊C4—22no—
402C4⋊C4—21no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—611no—
408C4⋊C4—611no—
409C4⋊C4—1221no—
410C4⋊C4—1221no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—1221no—
414C22⋊C4—1221no—
415C22⋊C4—1221no—
416C42—1221no—
417C42—1221no—
418C42—1221no—
419C42D4×D6111yescharacteristic
420C42#1175822411yescharacteristic
421C42C2×C22⋊D6111yescharacteristic
422C42C23×D6111yescharacteristic
423C23—41no—
424C23—1221no—
425C23#1451974811yescharacteristic
426Q8—42no—
427Q8—41no—
428Q8—42no—
429Q8—41no—
430Q8—82no—
431Q8—612no—
432Q8—1221no—
433D4—41no—
434D4—41no—
435D4—41no—
436D4—41no—
437D4—81no—
438D4—82no—
439D4—612no—
440D4—1224no—
441D4—1221no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—1221no—
450C2×C4—1224no—
451C2×C4—1221no—
452C2×C4—1221no—
453C2×C4—4881no—
454C2×C4—2441no—
455C2×C4—2441no—
456C2×C4#1451974811yescharacteristic
457C2×C4#1451974811yescharacteristic
458C2×C4#1451974811yescharacteristic
459C2×C4D8×D6112yes—
460C2×C4#1451979612yes—
461C2×C4C2×D4×D6111yescharacteristic
462C2×C4C22×C22⋊D6111yescharacteristic
463C2×C4C2×#1175822411yescharacteristic
464C8—22no—
465C8—22no—
466C8—2441no—
467C8—2441no—
468C8—2442no—
469C8—2442no—
470C22—21no—
471C22—21no—
472C22—81no—
473C22—81no—
474C22—81no—
475C22—1222no—
476C22—2441no—
477C22—2441no—
478C22#1683999512yes—
479C22C2×#1451974811yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—81no—
484C4—82no—
485C4—1222no—
486C4—2441no—
487C4—2441no—
488C4#1683999512yes—
489C4C2×D8×D6111yescharacteristic
490C4C2×#1451979611yescharacteristic
491C4#1684057011yescharacteristic
492C4#1684057711yescharacteristic
493C2—22no—
494C2—161no—
495C2—2442no—
496C2#2145536511yescharacteristic
497C2C2×#1683999511yescharacteristic
498C2#2145555211yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3226C2
socleC22×C61C2×#121
prime radical#21455268C2
hypercenter#3226D61
hypocenterC61#19416
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3226

Complements

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

NH ≅ G/Ntype
1. #21455302C2semidirect
3. #21454927C2semidirect
4. #21445260C2semidirect
5. #21445253C2semidirect
6. #21445210C2semidirect
7. #21445205C2semidirect
8. #21444594C2semidirect
9. #21444496C2semidirect
10. C2×#16839508C2semidirect
11. C61×#3226C2semidirect
244. #3226D61semidirect
298. C61#19416semidirect

Maximal subgroups

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

subgroupindex
#214553022
#214552682
#214549272
#214452602
#214452532
#214452102
#214452052
#214445942
#214444962
C2×#168395082
C61×#32262
#1941661