Finite Group Databasebeta

← #29836907

Subgroups — #29836907

Order 15616.

546 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458186C211yesmaximal, characteristic, complemented
2#21457561C211yesmaximal, characteristic
3#21448053C211yesmaximal, characteristic, complemented
4#21448030C211yesmaximal, characteristic, complemented
5#21448001C211yesmaximal, characteristic, complemented
6#21447916C211yesmaximal, characteristic, complemented
7#21447911C211yesmaximal, characteristic, complemented
8#21447882C211yesmaximal, characteristic, complemented
9#21447762C211yesmaximal, characteristic, complemented
10#21447754C211yesmaximal, characteristic, complemented
11C4×#14519723C211yesmaximal, characteristic, complemented
12#21447638C211yesmaximal, characteristic, complemented
13#21447525C211yesmaximal, characteristic, complemented
14#358×D61C211yesmaximal, characteristic, complemented
15C61×#2229C211yesmaximal, characteristic, complemented
16C2×#14519800C2211yescharacteristic
17C2×Q16×D61C2211yescharacteristic
18#16840450—21no—
19#16840424—21no—
20C2×Dic488C2211yescharacteristic
21C2×#14519723C2211yescharacteristic
22C2×#14519722C2211yescharacteristic
23C2×C16×D61C2211yescharacteristic
24C2×C16×D61C2211yescharacteristic
25#16840146C2211yescharacteristic
26#16840106—21no—
27#16840094—21no—
28#16840066—21no—
29#16840004C2211yescharacteristic
30#16840004C2211yescharacteristic
31#16840004C2211yescharacteristic
32#16839999C2211yescharacteristic
33#16839984—21no—
34#16839927—21no—
35#16839902—21no—
36C4×Dic244C2211yescharacteristic
37C8×D244C2211yescharacteristic
38#16839799C2211yescharacteristic
39#107×D61C2211yescharacteristic
40#16839587C2211yescharacteristic
41#16839567C2211yescharacteristic
42#16839566C2211yescharacteristic
43#16839565C2211yescharacteristic
44#16839565C2211yescharacteristic
45#16839565C2211yescharacteristic
46#16839564C2211yescharacteristic
47#16839562C2211yescharacteristic
48#16839562C2211yescharacteristic
49#16839541C2211yescharacteristic
50#16839541C2211yescharacteristic
51C4×#11758185C2211yescharacteristic
52C61×#499C2211yescharacteristic
53C4×Q16×C61C2211yescharacteristic
54C61×#367C2211yescharacteristic
55C61×#366C2211yescharacteristic
56C61×C4⋊C16C2211yescharacteristic
57C61×#358C2211yescharacteristic
58C61×#358C2211yescharacteristic
59C2×#11758226—21no—
60C2×Q8×D61—21no—
61#14519800—22no—
62#14519800—41no—
63C2×Dic244C2311yescharacteristic
64C2×Dic244C2311yescharacteristic
65C2×#11758202—21no—
66Q16×D61—22no—
67Q16×D61—41no—
68C2×#11758190—21no—
69C2×#11758201—21no—
70C2×D488C2311yescharacteristic
71C2×C8×D61C2311yescharacteristic
72C2×C8×D61C2311yescharacteristic
73#14519772—41no—
74#14519768—41no—
75#14519765—21no—
76#14519765—41no—
77#14519762—41no—
78#14519757—21no—
79#14519757—21no—
80#14519757—21no—
81#14519757—41no—
82#14519752—21no—
83#14519752—41no—
84C4×Dic122—21no—
85C4×D244—41no—
86C4×D244D411yescharacteristic
87#14519735—21no—
88#14519735—41no—
89C4⋊C4×D61—21no—
90Dic488—41no—
91#14519723—22no—
92#14519722—41no—
93C2×#11758185C2311yescharacteristic
94C2×#11758185C2311yescharacteristic
95C16×D61—41no—
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#14519697—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—
120#11758226—42no—
121#11758226—81no—
122Q8×D61—42no—
123Q8×D61—81no—
124C2×D244—21no—
125C2×D244—41no—
126C2×D244—41no—
127C2×D244D811yescharacteristic
128C2×C4×D61—41no—
129C2×C4×D61—41no—
130C2×C4×D61—41no—
131C2×C4×D61C2×D411yescharacteristic
132C2×C4×D61C2×D411yescharacteristic
133C2×Dic122—21no—
134C2×Dic122—21no—
135C2×Dic122—41no—
136Q16×C61—21no—
137Q16×C61#4112yes—
138C2×C8×C61C2×D411yescharacteristic
139C2×C8×C61C2411yescharacteristic
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—41no—
144C4⋊C4×C61—41no—
145C4⋊C4×C61—41no—
146C42×C61—41no—
147C42×C61C2×D411yescharacteristic
148#11758202—42no—
149#11758201—42no—
150#11758198—41no—
151#11758198—41no—
152#11758198—41no—
153#11758198—41no—
154#11758198—41no—
155#11758198—41no—
156#11758198—41no—
157#11758198D811yescharacteristic
158#11758198D811yescharacteristic
159C4⋊Dic61—21no—
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—
174D488—21no—
175D488—21no—
176#11758190—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×D61—41no—
186C22×D61C2×D811yescharacteristic
187C22×D61C2×D811yescharacteristic
188Q8×C61—22no—
189Q8×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61C2×D811yescharacteristic
195C2×C4×C61C2×D811yescharacteristic
196C2×C4×C61C22×D411yescharacteristic
197C2×Dic61—41no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61C2×D811yescharacteristic
201C2×Dic61C2×D811yescharacteristic
202D244—22no—
203D244—41no—
204D244—41no—
205D244—82no—
206D244—82no—
207C4×D61—21no—
208C4×D61—21no—
209C4×D61—21no—
210C4×D61—21no—
211C4×D61—82no—
212C4×D61—82no—
213C4×D61—82no—
214Dic122—41no—
215Dic122—41no—
216Dic122—41no—
217Dic122—41no—
218Dic122—82no—
219C8×C61—21no—
220C8×C61C2×#4111yescharacteristic
221C8×C61H(5,𝔽2)11yescharacteristic
222#92418—21no—
223#92418—21no—
224#33867—611nomaximal, Sylow
225C22×C61C22×D811yescharacteristic
226C2×D61—21no—
227C2×D61—21no—
228C2×D61—21no—
229C2×D61—21no—
230C2×D61—21no—
231C2×D61—81no—
232C2×D61—81no—
233C2×D61—81no—
234C2×D61#50812yes—
235C4×C61—21no—
236C4×C61—21no—
237C4×C61—42no—
238C4×C61—81no—
239C4×C61—81no—
240C4×C61—81no—
241C4×C61#54611yescharacteristic
242C4×C61#57711yescharacteristic
243Dic61—21no—
244Dic61—21no—
245Dic61—81no—
246Dic61—81no—
247Dic61—81no—
248#3336—611no—
249#3285—611no—
250#2242—611no—
251#2237—611no—
252#2232—611no—
253#2229D6111yescharacteristic, complemented
254#2217—611no—
255#2216—611no—
256#2211—611no—
257#2189—611no—
258#2188—611no—
259C4×Q32—611no—
260D4×C16—611no—
261#2138—611no—
262C2×#358—611no—
263C2×C61#329411yescharacteristic
264C2×C61C2×#50811yescharacteristic
265C2×C61#341711yescharacteristic
266D61—22no—
267D61—41no—
268D61—161no—
269C2×#135—611no—
270C22×Q16—611no—
271#552—1221no—
272#548—1221no—
273C2×Q32—611no—
274C2×Q32—611no—
275C2×SD32—611no—
276C22×C16—611no—
277C22×C16—611no—
278#499C2×D6111yescharacteristic
279#495—611no—
280#485—1221no—
281#483—1221no—
282#479—1221no—
283#467—611no—
284#467—611no—
285#467—611no—
286#466—611no—
287#464—1221no—
288#452—1221no—
289#448—1221no—
290C4×Q16—611no—
291C4×Q16C2×D6111yescharacteristic
292C8×D4—611no—
293#427—611no—
294C2×#107—611no—
295#367C2×D6111yescharacteristic
296#366—611no—
297#366C2×D6111yescharacteristic
298C4⋊C16C2×D6111yescharacteristic
299#358—611no—
300#358—611no—
301#358—611no—
302#358—611no—
303#358—611no—
304#358C2×D6111yescharacteristic
305#358C2×D6111yescharacteristic
306#357—611no—
307#357—611no—
308#357—611no—
309C22⋊C16—611no—
310C22⋊C16—611no—
311C4×C16—611no—
312C61#3386711yescharacteristic, complemented, Sylow
313C2×#41—1221no—
314C22×Q8—1221no—
315#135—1222no—
316#135—2441no—
317C2×Q16—611no—
318C2×Q16—611no—
319C2×Q16—1221no—
320C2×Q16—1222no—
321C2×Q16—2441no—
322C2×Q16C22×D6111yescharacteristic
323C2×SD16—1221no—
324C2×SD16—1221no—
325C2×D8—611no—
326C22×C8—611no—
327C22×C8—611no—
328#126—2441no—
329#125—21no—
330#125—2441no—
331#124—1221no—
332#124—2441no—
333#123—2441no—
334C22⋊Q8—1221no—
335C22⋊Q8—1221no—
336C22⋊Q8—1221no—
337C22⋊Q8—2441no—
338#121—1221no—
339#121—2441no—
340C4×Q8—21no—
341C4×Q8—1221no—
342C4×D4—611no—
343C4×D4—2441no—
344#117—1221no—
345#117—2441no—
346C2×C4⋊C4—1221no—
347Q32—1222no—
348Q32—2441no—
349SD32—2441no—
350C2×C16—611no—
351C2×C16—611no—
352C2×C16—2441no—
353C2×C16—2441no—
354C2×C16C22×D6111yescharacteristic
355C2×C16C22×D6111yescharacteristic
356#107—611no—
357#107—1221no—
358#107C22×D6111yescharacteristic
359#107C22×D6111yescharacteristic
360#107C22×D6111yescharacteristic
361#106—21no—
362#106—1221no—
363C4⋊C8—611no—
364#103—21no—
365#103—1221no—
366#103—1221no—
367#103—1221no—
368#103—1221no—
369#102—1221no—
370C22⋊C8—611no—
371C22⋊C8—611no—
372C4×C8C22×D6111yescharacteristic
373#41—4881no—
374#41—2442no—
375C2×Q8—21no—
376C2×Q8—1221no—
377C2×Q8—1221no—
378C2×Q8—4881no—
379C2×Q8—2441no—
380C2×Q8—2442no—
381C2×D4—611no—
382C2×D4—1221no—
383C2×D4—2441no—
384C2×D4—2441no—
385C22×C4—611no—
386C22×C4—611no—
387C22×C4—2441no—
388C22×C4—2441no—
389C22×C4—2441no—
390Q16—21no—
391Q16—1221no—
392Q16—1221no—
393Q16—1221no—
394Q16—1221no—
395Q16—2442no—
396Q16#1175822612yes—
397SD16—4881no—
398SD16—2442no—
399D8—1221no—
400D8—1221no—
401C2×C8—611no—
402C2×C8—611no—
403C2×C8—1221no—
404C2×C8—1221no—
405C2×C8—1221no—
406C2×C8—1221no—
407C2×C8D4×D6111yescharacteristic
408C2×C8C23×D6111yescharacteristic
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—611no—
416C4⋊C4—1221no—
417C4⋊C4—2441no—
418C4⋊C4—2441no—
419C4⋊C4—2441no—
420C4⋊C4—2441no—
421C4⋊C4—2441no—
422C22⋊C4—611no—
423C22⋊C4—611no—
424C22⋊C4—2441no—
425C22⋊C4—2441no—
426C22⋊C4—2441no—
427C22⋊C4—2441no—
428C22⋊C4—2441no—
429C22⋊C4—2441no—
430C22⋊C4—2441no—
431C42—41no—
432C42—2441no—
433C42—2441no—
434C42D4×D6111yescharacteristic
435C16—21no—
436C16—21no—
437C16—1221no—
438C16—1221no—
439C23—611no—
440C23—611no—
441C23—2441no—
442Q8—22no—
443Q8—41no—
444Q8—4882no—
445Q8—2441no—
446Q8—2441no—
447Q8—2441no—
448Q8—2441no—
449D4—1222no—
450D4—4882no—
451D4—4882no—
452D4—2441no—
453D4—2441no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—611no—
459C2×C4—611no—
460C2×C4—1221no—
461C2×C4—1221no—
462C2×C4—1221no—
463C2×C4—1221no—
464C2×C4—4882no—
465C2×C4—4882no—
466C2×C4—4882no—
467C2×C4—2441no—
468C2×C4—2441no—
469C2×C4—2441no—
470C2×C4D8×D6111yescharacteristic
471C2×C4D8×D6111yescharacteristic
472C2×C4C2×D4×D6111yescharacteristic
473C8—21no—
474C8—1221no—
475C8—1221no—
476C8C2×#1175822611yescharacteristic
477C8#1451982311yescharacteristic
478C22—612no—
479C22—1221no—
480C22—1221no—
481C22—1221no—
482C22—1221no—
483C22—1221no—
484C22—4881no—
485C22—4881no—
486C22—4881no—
487C22C2×D8×D6111yescharacteristic
488C4—21no—
489C4—21no—
490C4—42no—
491C4—81no—
492C4—81no—
493C4—81no—
494C4—1221no—
495C4—1221no—
496C4—4881no—
497C4—4881no—
498C4—4881no—
499C4#1684040311yescharacteristic
500C4#1684057111yescharacteristic
501C2—1222no—
502C2—9761no—
503C2—2441no—
504C2#2145762911yescharacteristic
505C2#508×D6111yescharacteristic
506C2#2145897111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2229C2
socleC22×C61C22×D8
prime radical#21457561C2
hypercenter#2229D61
hypocenterC61#33867
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
1. #21458186C2semidirect
3. #21448053C2semidirect
4. #21448030C2semidirect
5. #21448001C2semidirect
6. #21447916C2semidirect
7. #21447911C2semidirect
8. #21447882C2semidirect
9. #21447762C2semidirect
10. #21447754C2semidirect
11. C4×#14519723C2semidirect
12. #21447638C2semidirect
13. #21447525C2semidirect
14. #358×D61C2semidirect
15. C61×#2229C2semidirect
253. #2229D61semidirect
312. C61#33867semidirect

Maximal subgroups

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

subgroupindex
#214581862
#214575612
#214480532
#214480302
#214480012
#214479162
#214479112
#214478822
#214477622
#214477542
C4×#145197232
#214476382
#214475252
#358×D612
C61×#22292
#3386761