Finite Group Databasebeta

← #29837071

Subgroups — #29837071

Order 15616.

536 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#21458186C211yesmaximal, characteristic
2Q16×D244C211yesmaximal, characteristic
3#21448066C211yesmaximal, characteristic, complemented
4#21448029C211yesmaximal, characteristic, complemented
5#21448018C211yesmaximal, characteristic, complemented
6#21447928C211yesmaximal, characteristic, complemented
7#21447911C211yesmaximal, characteristic, complemented
8#21447870C211yesmaximal, characteristic, complemented
9#21447754C211yesmaximal, characteristic, complemented
10#21447754C211yesmaximal, characteristic, complemented
11#21447688C211yesmaximal, characteristic, complemented
12#21447643C211yesmaximal, characteristic
13#21447537C211yesmaximal, characteristic, complemented
14#358×D61C211yesmaximal, characteristic, complemented
15C61×#2229C211yesmaximal, characteristic, complemented
16C2×Q16×D61C2211yescharacteristic
17C2×Q16×D61C2211yescharacteristic
18#16840450—21no—
19Q8×D244—21no—
20C2×Dic488C2211yescharacteristic
21C2×Dic488C2211yescharacteristic
22C2×#14519723C2211yescharacteristic
23C2×#14519716C2211yescharacteristic
24C2×C16×D61C2211yescharacteristic
25#16840142C2211yescharacteristic
26#16840106—21no—
27#16840094—21no—
28#16840066—21no—
29#16840004C2211yescharacteristic
30#16840004C2211yescharacteristic
31#16840004C2211yescharacteristic
32#16840004C2211yescharacteristic
33#16839972—21no—
34#16839919—21no—
35#16839919—21no—
36C4×Dic244C2211yescharacteristic
37C8×D244C2211yescharacteristic
38#16839799C2211yescharacteristic
39#107×D61C2211yescharacteristic
40#16839589C2211yescharacteristic
41#16839567C2211yescharacteristic
42#16839567C2211yescharacteristic
43#16839566C2211yescharacteristic
44#16839565C2211yescharacteristic
45#16839565C2211yescharacteristic
46#16839565C2211yescharacteristic
47#16839565C2211yescharacteristic
48#16839565C2211yescharacteristic
49#16839541C2211yescharacteristic
50#16839541C2211yescharacteristic
51#16839538C2211yescharacteristic
52C61×#499C2211yescharacteristic
53C4×Q16×C61C2211yescharacteristic
54C61×#367C2211yescharacteristic
55C61×#366C2211yescharacteristic
56C61×C4⋊C16C2211yescharacteristic
57C61×#358C2211yescharacteristic
58C61×#358C2211yescharacteristic
59C2×Q8×D61—21no—
60C2×Q8×D61—21no—
61C2×Dic244—21no—
62C2×Dic244C2311yescharacteristic
63C2×Dic244C2311yescharacteristic
64C2×Dic244C2311yescharacteristic
65C2×#11758202—21no—
66C2×#11758202—21no—
67Q16×D61—22no—
68Q16×D61—22no—
69Q16×D61—41no—
70Q16×D61—41no—
71C2×C8×D61C2311yescharacteristic
72C2×C8×D61C2311yescharacteristic
73#14519777—21no—
74#14519777—41no—
75#14519772—41no—
76#14519768—41no—
77#14519762—41no—
78#14519757—21no—
79#14519757—21no—
80#14519757—21no—
81#14519757—21no—
82#14519757—41no—
83#14519757—41no—
84C4×Dic122—21no—
85C4×D244—41no—
86C4×D244D411yescharacteristic
87#14519735—21no—
88#14519735—41no—
89C4⋊C4×D61—21no—
90Dic488—41no—
91Dic488—41no—
92#14519723—22no—
93#14519716—41no—
94C2×#11758185C2311yescharacteristic
95C2×#11758185C2311yescharacteristic
96C16×D61—41no—
97#14519711C2311yescharacteristic
98#14519710—21no—
99#14519708—21no—
100C4⋊#92418D411yescharacteristic
101#14519702—21no—
102#14519700—21no—
103#14519700—21no—
104#14519700—21no—
105#14519700—21no—
106#14519689D411yescharacteristic
107#14519689D411yescharacteristic
108C2×Q16×C61C2311yescharacteristic
109#125×C61—21no—
110C4×Q8×C61—21no—
111C2×C16×C61C2311yescharacteristic
112C2×C16×C61C2311yescharacteristic
113#107×C61C2311yescharacteristic
114#107×C61C2311yescharacteristic
115#107×C61C2311yescharacteristic
116#106×C61—21no—
117#103×C61—21no—
118C4×C8×C61C2311yescharacteristic
119C2×Q8×C61—21no—
120Q8×D61—42no—
121Q8×D61—42no—
122Q8×D61—81no—
123Q8×D61—81no—
124C2×D244D811yescharacteristic
125C2×C4×D61—41no—
126C2×C4×D61—41no—
127C2×C4×D61—41no—
128C2×C4×D61C2×D411yescharacteristic
129C2×C4×D61C2×D411yescharacteristic
130C2×Dic122—21no—
131C2×Dic122—21no—
132C2×Dic122—21no—
133C2×Dic122—41no—
134C2×Dic122—41no—
135Q16×C61—21no—
136Q16×C61#4112yes—
137C2×C8×C61C2×D411yescharacteristic
138C2×C8×C61C2411yescharacteristic
139C4⋊C4×C61—21no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—41no—
143C4⋊C4×C61—41no—
144C4⋊C4×C61—41no—
145C42×C61—41no—
146C42×C61C2×D411yescharacteristic
147#11758202—42no—
148#11758202—42no—
149#11758198—41no—
150#11758198—41no—
151#11758198—41no—
152#11758198—41no—
153#11758198—41no—
154#11758198D811yescharacteristic
155#11758198D811yescharacteristic
156C4⋊Dic61—21no—
157C4⋊Dic61—41no—
158C4⋊Dic61—41no—
159C4⋊Dic61—41no—
160C4⋊Dic61—41no—
161C4⋊Dic61—41no—
162C4⋊Dic61—41no—
163C4⋊Dic61D811yescharacteristic
164#11758196—41no—
165#11758196—41no—
166C4×Dic61—41no—
167C4×Dic61—41no—
168C2×#92418C2×D411yescharacteristic
169C2×#92418C2×D411yescharacteristic
170Dic244—21no—
171Dic244—21no—
172Dic244—21no—
173Dic244—21no—
174Dic244—21no—
175Dic244—21no—
176Dic244—81no—
177C8×D61—21no—
178C8×D61—21no—
179C8×D61—21no—
180C8×D61—21no—
181C16×C61—21no—
182C16×C61—21no—
183#11758185—21no—
184#11758185—21no—
185C22×D61C2×D811yescharacteristic
186C22×D61C2×D811yescharacteristic
187Q8×C61—22no—
188Q8×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61C2×D811yescharacteristic
194C2×C4×C61C2×D811yescharacteristic
195C2×C4×C61C22×D411yescharacteristic
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61C2×D811yescharacteristic
201C2×Dic61C2×D811yescharacteristic
202D244—22no—
203C4×D61—21no—
204C4×D61—21no—
205C4×D61—21no—
206C4×D61—21no—
207C4×D61—82no—
208C4×D61—82no—
209C4×D61—82no—
210Dic122—41no—
211Dic122—41no—
212Dic122—41no—
213Dic122—41no—
214Dic122—41no—
215Dic122—41no—
216Dic122—82no—
217Dic122—82no—
218C8×C61—21no—
219C8×C61C2×#4111yescharacteristic
220C8×C61H(5,𝔽2)11yescharacteristic
221#92418—21no—
222#92418—21no—
223#33879—611nomaximal, Sylow
224C22×C61C22×D811yescharacteristic
225C2×D61—21no—
226C2×D61—21no—
227C2×D61—21no—
228C2×D61—21no—
229C2×D61—21no—
230C2×D61#51012yes—
231C4×C61—21no—
232C4×C61—21no—
233C4×C61—42no—
234C4×C61—81no—
235C4×C61—81no—
236C4×C61—81no—
237C4×C61#54611yescharacteristic
238C4×C61#57711yescharacteristic
239Dic61—21no—
240Dic61—21no—
241Dic61—81no—
242Dic61—81no—
243Dic61—81no—
244Dic61—81no—
245#3336—611no—
246D4×Q16—611no—
247#2245—611no—
248#2237—611no—
249#2235—611no—
250#2229D6111yescharacteristic, complemented
251#2219—611no—
252#2216—611no—
253#2209—611no—
254#2188—611no—
255#2188—611no—
256#2176—611no—
257#2168—611no—
258#2141—611no—
259C2×#358—611no—
260C2×C61#329411yescharacteristic
261C2×C61C2×#51011yescharacteristic
262C2×C61#341711yescharacteristic
263D61—22no—
264D61—41no—
265C22×Q16—611no—
266C22×Q16—611no—
267#552—1221no—
268D4×Q8—1221no—
269C2×Q32—611no—
270C2×Q32—611no—
271C2×Q32—611no—
272C2×M32—611no—
273C22×C16—611no—
274#499C2×D6111yescharacteristic
275#494—611no—
276#485—1221no—
277#483—1221no—
278#479—1221no—
279#467—611no—
280#467—611no—
281#467—611no—
282#467—611no—
283#462—1221no—
284#451—1221no—
285#451—1221no—
286C4×Q16—611no—
287C4×Q16C2×D6111yescharacteristic
288C8×D4—611no—
289#427—611no—
290C2×#107—611no—
291#367—611no—
292#367C2×D6111yescharacteristic
293#366C2×D6111yescharacteristic
294C4⋊C16C2×D6111yescharacteristic
295#358—611no—
296#358—611no—
297#358—611no—
298#358—611no—
299#358—611no—
300#358—611no—
301#358—611no—
302#358—611no—
303#358C2×D6111yescharacteristic
304#358C2×D6111yescharacteristic
305C22⋊C16—611no—
306C22⋊C16—611no—
307#346—611no—
308C61#3387911yescharacteristic, complemented, Sylow
309C22×Q8—1221no—
310C22×Q8—1221no—
311C2×Q16—611no—
312C2×Q16—611no—
313C2×Q16—611no—
314C2×Q16—1221no—
315C2×Q16—1221no—
316C2×Q16—1222no—
317C2×Q16—1222no—
318C2×Q16—1221no—
319C2×Q16—2441no—
320C2×Q16—2441no—
321C2×Q16C22×D6111yescharacteristic
322C22×C8—611no—
323C22×C8—611no—
324C4⋊Q8—1221no—
325C4⋊Q8—2441no—
326#126—2441no—
327#125—21no—
328#125—2441no—
329#123—2441no—
330C22⋊Q8—1221no—
331C22⋊Q8—1221no—
332C22⋊Q8—1221no—
333C22⋊Q8—1221no—
334C22⋊Q8—2441no—
335C22⋊Q8—2441no—
336C4×Q8—21no—
337C4×Q8—1221no—
338C4×D4—611no—
339C4×D4—2441no—
340#117—1221no—
341#117—2441no—
342C2×C4⋊C4—1221no—
343Q32—1222no—
344Q32—2441no—
345Q32—2441no—
346M32—2441no—
347C2×C16—611no—
348C2×C16—611no—
349C2×C16—2441no—
350C2×C16C22×D6111yescharacteristic
351C2×C16C22×D6111yescharacteristic
352#107—611no—
353#107—1221no—
354#107C22×D6111yescharacteristic
355#107C22×D6111yescharacteristic
356#107C22×D6111yescharacteristic
357#106—21no—
358#106—1221no—
359C4⋊C8—611no—
360#103—21no—
361#103—1221no—
362#103—1221no—
363#103—1221no—
364#103—1221no—
365#103—1221no—
366C22⋊C8—611no—
367C22⋊C8—611no—
368C4×C8C22×D6111yescharacteristic
369C2×Q8—21no—
370C2×Q8—1221no—
371C2×Q8—1221no—
372C2×Q8—1221no—
373C2×Q8—4881no—
374C2×Q8—4881no—
375C2×Q8—2441no—
376C2×Q8—2441no—
377C2×Q8—2442no—
378C2×Q8—2442no—
379C2×D4—611no—
380C22×C4—611no—
381C22×C4—611no—
382C22×C4—2441no—
383C22×C4—2441no—
384C22×C4—2441no—
385Q16—21no—
386Q16—1221no—
387Q16—1221no—
388Q16—1221no—
389Q16—1221no—
390Q16—1221no—
391Q16—1221no—
392Q16—4881no—
393Q16—2442no—
394Q16—2442no—
395Q16#1175822612yes—
396C2×C8—611no—
397C2×C8—611no—
398C2×C8—1221no—
399C2×C8—1221no—
400C2×C8—1221no—
401C2×C8—1221no—
402C2×C8D4×D6111yescharacteristic
403C2×C8C23×D6111yescharacteristic
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—611no—
411C4⋊C4—1221no—
412C4⋊C4—2441no—
413C4⋊C4—2441no—
414C4⋊C4—2441no—
415C4⋊C4—2441no—
416C4⋊C4—2441no—
417C4⋊C4—2441no—
418C4⋊C4—2441no—
419C4⋊C4—2441no—
420C22⋊C4—611no—
421C22⋊C4—611no—
422C22⋊C4—2441no—
423C22⋊C4—2441no—
424C22⋊C4—2441no—
425C22⋊C4—2441no—
426C22⋊C4—2441no—
427C42—41no—
428C42—2441no—
429C42—2441no—
430C42D4×D6111yescharacteristic
431C16—21no—
432C16—21no—
433C16—1221no—
434C16—1221no—
435C23—611no—
436C23—611no—
437Q8—22no—
438Q8—41no—
439Q8—4882no—
440Q8—4882no—
441Q8—2441no—
442Q8—2441no—
443Q8—2441no—
444Q8—2441no—
445Q8—2441no—
446Q8—2441no—
447D4—1222no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—611no—
453C2×C4—611no—
454C2×C4—1221no—
455C2×C4—1221no—
456C2×C4—1221no—
457C2×C4—1221no—
458C2×C4—4882no—
459C2×C4—4882no—
460C2×C4—4882no—
461C2×C4—2441no—
462C2×C4—2441no—
463C2×C4—2441no—
464C2×C4—2441no—
465C2×C4D8×D6111yescharacteristic
466C2×C4D8×D6111yescharacteristic
467C2×C4C2×D4×D6111yescharacteristic
468C8—21no—
469C8—1221no—
470C8—1221no—
471C8C2×#1175822611yescharacteristic
472C8#1451982311yescharacteristic
473C22—612no—
474C22—1221no—
475C22—1221no—
476C22—1221no—
477C22—1221no—
478C22—1221no—
479C22C2×D8×D6111yescharacteristic
480C4—21no—
481C4—21no—
482C4—42no—
483C4—81no—
484C4—81no—
485C4—81no—
486C4—1221no—
487C4—1221no—
488C4—4881no—
489C4—4881no—
490C4—4881no—
491C4—4881no—
492C4#1684040311yescharacteristic
493C4#1684057111yescharacteristic
494C2—1222no—
495C2—2441no—
496C2#2145762911yescharacteristic
497C2#510×D6111yescharacteristic
498C2#2145896911yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2229C2
socleC22×C61C22×D8
prime radicalC2×D244D8
hypercenter#2229D61
hypocenterC61#33879
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2229

Complements

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

NH ≅ G/Ntype
3. #21448066C2semidirect
4. #21448029C2semidirect
5. #21448018C2semidirect
6. #21447928C2semidirect
7. #21447911C2semidirect
8. #21447870C2semidirect
9. #21447754C2semidirect
10. #21447754C2semidirect
11. #21447688C2semidirect
13. #21447537C2semidirect
14. #358×D61C2semidirect
15. C61×#2229C2semidirect
250. #2229D61semidirect
308. C61#33879semidirect

Maximal subgroups

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

subgroupindex
#214581862
Q16×D2442
#214480662
#214480292
#214480182
#214479282
#214479112
#214478702
#214477542
#214476882
#214476432
#214475372
#358×D612
C61×#22292
#3387961