Finite Group Databasebeta

← #29837747

Subgroups — #29837747

Order 15616.

518 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458792C211yesmaximal, characteristic
2#21458710C211yesmaximal, characteristic, complemented
3#21448060C211yesmaximal, characteristic, complemented
4#21448059C211yesmaximal, characteristic, complemented
5#21448041C211yesmaximal, characteristic, complemented
6#21447978C211yesmaximal, characteristic, complemented
7#21447977C211yesmaximal, characteristic, complemented
8#21447954C211yesmaximal, characteristic, complemented
9#21447857C211yesmaximal, characteristic, complemented
10#21447857C211yesmaximal, characteristic, complemented
11#21447726C211yesmaximal, characteristic, complemented
12#21447684C211yesmaximal, characteristic, complemented
13#21447681C211yesmaximal, characteristic, complemented
14SD32×Dic61C211yesmaximal, characteristic, complemented
15C61×#2207C211yesmaximal, characteristic, complemented
16#16840488—21no—
17#16840432—21no—
18C2×#14519720C2211yescharacteristic
19#16840166C2211yescharacteristic
20#16840164C2211yescharacteristic
21#16840163C2211yescharacteristic
22#16840137C2211yescharacteristic
23#16840137C2211yescharacteristic
24#16840069—21no—
25#16840067—21no—
26#16840045—21no—
27#16839955—21no—
28#16839955—21no—
29#16839954—21no—
30C8×Dic122C2211yescharacteristic
31Q16×Dic61C2211yescharacteristic
32Q16×Dic61C2211yescharacteristic
33C4×D488C2211yescharacteristic
34D8×Dic61C2211yescharacteristic
35D8×Dic61C2211yescharacteristic
36#16839590C2211yescharacteristic
37#16839588C2211yescharacteristic
38#16839587C2211yescharacteristic
39#16839580C2211yescharacteristic
40#16839580C2211yescharacteristic
41#16839566C2211yescharacteristic
42#16839564C2211yescharacteristic
43#16839564C2211yescharacteristic
44#16839563C2211yescharacteristic
45#16839563C2211yescharacteristic
46#16839563C2211yescharacteristic
47#16839563C2211yescharacteristic
48#16839562C2211yescharacteristic
49#16839538C2211yescharacteristic
50#16839537C2211yescharacteristic
51C16×Dic61C2211yescharacteristic
52C2×SD32×C61C2211yescharacteristic
53C2×SD32×C61C2211yescharacteristic
54C61×#493C2211yescharacteristic
55C4×Q16×C61C2211yescharacteristic
56C61×C4⋊C16C2211yescharacteristic
57C61×#357C2211yescharacteristic
58C61×#357C2211yescharacteristic
59C2×#11758224—41no—
60C2×D488C2311yescharacteristic
61C2×#11758199—21no—
62C2×#11758199—21no—
63#14519776—21no—
64#14519776—41no—
65#14519773—21no—
66#14519773—21no—
67#14519769—21no—
68#14519769—41no—
69#14519768—21no—
70#14519768—41no—
71#14519749—41no—
72#14519749—41no—
73#14519748—41no—
74Q8×Dic61—21no—
75Q8×Dic61—21no—
76C4×Dic122—41no—
77C4×Dic122D411yescharacteristic
78C4×D244—21no—
79D4×Dic61—21no—
80D4×Dic61—21no—
81#14519720—22no—
82C2×#11758185C2311yescharacteristic
83C2×#11758185C2311yescharacteristic
84#14519711C2311yescharacteristic
85#14519711C2311yescharacteristic
86#14519711C2311yescharacteristic
87#14519710—21no—
88#14519709—21no—
89#14519708—21no—
90C4⋊#92418D411yescharacteristic
91#14519706D411yescharacteristic
92#14519706D411yescharacteristic
93#14519701—21no—
94#14519701—21no—
95#14519699—21no—
96#14519698—21no—
97#14519698—21no—
98C8×Dic61C2311yescharacteristic
99C8×Dic61C2311yescharacteristic
100C2×Q16×C61C2311yescharacteristic
101C2×D8×C61—21no—
102C2×D8×C61C2311yescharacteristic
103C2×D8×C61C2311yescharacteristic
104#127×C61—21no—
105C4×Q8×C61—21no—
106SD32×C61—22no—
107SD32×C61—22no—
108C2×C16×C61C2311yescharacteristic
109C2×C16×C61C2311yescharacteristic
110#107×C61C2311yescharacteristic
111#103×C61—21no—
112C4×C8×C61C2311yescharacteristic
113C2×Q8×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—41no—
117C2×D4×C61—41no—
118C2×C22⋊D61—41no—
119C2×C22⋊D61—41no—
120C2×C22⋊D61—41no—
121C2×C22⋊D61—41no—
122C22×Dic61—41no—
123C22×Dic61—41no—
124#11758224—84no—
125C2×D244—21no—
126C2×C4×D61—41no—
127C2×Dic122—41no—
128C2×Dic122D811yescharacteristic
129Q16×C61—21no—
130Q16×C61#4112yescomplemented
131D8×C61—21no—
132D8×C61—21no—
133D8×C61—21no—
134D8×C61—21no—
135D8×C61—81no—
136C2×C8×C61C2×D411yescharacteristic
137C2×C8×C61C2411yescharacteristic
138C4⋊C4×C61—21no—
139C4⋊C4×C61—41no—
140C42×C61—41no—
141C42×C61C2×D411yescharacteristic
142C22⋊Dic61—41no—
143C22⋊Dic61—41no—
144#11758199—81no—
145#11758199—81no—
146#11758198—41no—
147C4⋊Dic61—21no—
148C4⋊Dic61—21no—
149C4⋊Dic61—21no—
150C4⋊Dic61—41no—
151C4⋊Dic61—41no—
152C4⋊Dic61—41no—
153C4⋊Dic61D811yescharacteristic
154#11758196—41no—
155#11758196—41no—
156#11758196—41no—
157#11758196—41no—
158#11758196D811yescharacteristic
159#11758196D811yescharacteristic
160C4×Dic61—41no—
161C4×Dic61—41no—
162C4×Dic61C2×D411yescharacteristic
163C4×Dic61C2×D411yescharacteristic
164C2×#92418C2×D411yescharacteristic
165C2×#92418C2×D411yescharacteristic
166D488—21no—
167D488—21no—
168C16×C61—21no—
169C16×C61—21no—
170#11758185—21no—
171#11758185—21no—
172C23×C61—41no—
173C23×C61—41no—
174C22×D61—41no—
175Q8×C61—22no—
176Q8×C61—41no—
177D4×C61—41no—
178D4×C61—41no—
179D4×C61—41no—
180D4×C61—41no—
181D4×C61—82no—
182D4×C61—161no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61C2×D811yescharacteristic
186C2×C4×C61C2×D811yescharacteristic
187C2×C4×C61C22×D411yescharacteristic
188C22⋊D61—82no—
189C22⋊D61—82no—
190C22⋊D61—161no—
191C22⋊D61—161no—
192C2×Dic61—41no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195C2×Dic61—82no—
196C2×Dic61—82no—
197C2×Dic61C2×D811yescharacteristic
198C2×Dic61C2×D811yescharacteristic
199C2×Dic61C2×D811yescharacteristic
200C2×Dic61C2×D811yescharacteristic
201D244—41no—
202D244—41no—
203C4×D61—82no—
204Dic122—22no—
205Dic122—82no—
206C8×C61—21no—
207C8×C61C2×#4111yescharacteristic
208C8×C612−1+411yescharacteristic
209#92418—21no—
210#92418—21no—
211#33935—611nomaximal, Sylow
212C22×C61—81no—
213C22×C61—81no—
214C22×C61—81no—
215C22×C61—81no—
216C22×C61—81no—
217C22×C61—81no—
218C22×C61C22×D811yescharacteristic
219C2×D61—81no—
220C2×D61—81no—
221C2×D61—81no—
222C4×C61—21no—
223C4×C61—21no—
224C4×C61—42no—
225C4×C61—81no—
226C4×C61#54811yescharacteristic
227C4×C61#57711yescharacteristic
228Dic61—21no—
229Dic61—21no—
230Dic61—21no—
231Dic61—81no—
232Dic61—81no—
233Dic61—81no—
234Dic61#50912yes—
235#3391—611no—
236#3383—611no—
237#2244—611no—
238#2244—611no—
239#2240—611no—
240#2228—611no—
241#2228—611no—
242#2224—611no—
243#2207—611no—
244#2207—611no—
245#2207D6111yescharacteristic, complemented
246#2183—611no—
247#2175—611no—
248#2175—611no—
249C4×SD32—611no—
250C2×C61—161no—
251C2×C61—161no—
252C2×C61#335811yescharacteristic
253C2×C61C2×#50911yescharacteristic
254C2×C61#341611yescharacteristic
255D61—161no—
256#557—1221no—
257#550—1221no—
258C2×SD32—611no—
259C2×SD32C2×D6111yescharacteristic
260C2×SD32C2×D6111yescharacteristic
261#500—611no—
262#499—611no—
263#499—611no—
264#493—611no—
265#493—611no—
266#493C2×D6111yescharacteristic
267#479—1221no—
268#479—1221no—
269#475—1221no—
270#459—1221no—
271#459—1221no—
272#459—1221no—
273C8×Q8—611no—
274C4×Q16—611no—
275C4×Q16—611no—
276C4×Q16C2×D6111yescharacteristic
277C4×D8—611no—
278C4×D8—611no—
279C4×D8—611no—
280#367—611no—
281#366—611no—
282#366—611no—
283C4⋊C16—611no—
284C4⋊C16—611no—
285C4⋊C16C2×D6111yescharacteristic
286#358—611no—
287#357—611no—
288#357—611no—
289#357—611no—
290#357—611no—
291#357—611no—
292#357—611no—
293#357—611no—
294#357C2×D6111yescharacteristic
295#357C2×D6111yescharacteristic
296#346—611no—
297#346—611no—
298C4×C16—611no—
299C61#3393511yescharacteristic, complemented, Sylow
300C2×#41—2441no—
301C2×Q16C22×D6111yescharacteristic
302C2×D8—21no—
303C2×D8—611no—
304C2×D8—1221no—
305C2×D8—1221no—
306C2×D8C22×D6111yescharacteristic
307C2×D8C22×D6111yescharacteristic
308C4⋊Q8—1221no—
309C4⋊Q8—2441no—
310#127—21no—
311#127—1221no—
312#127—1221no—
313#125—1221no—
314#125—1221no—
315#125—2441no—
316#125—2441no—
317#121—2441no—
318#121—2441no—
319#121—2441no—
320C4×Q8—21no—
321C4×Q8—611no—
322C4×Q8—1221no—
323C4×Q8—1221no—
324C4×Q8—2441no—
325C4×D4—1221no—
326C4×D4—1221no—
327C4×D4—1221no—
328SD32—22no—
329SD32—22no—
330SD32—1222no—
331C2×C16—611no—
332C2×C16—611no—
333C2×C16C22×D6111yescharacteristic
334C2×C16C22×D6111yescharacteristic
335#107—611no—
336#107—611no—
337#107—611no—
338#107—1221no—
339#107C22×D6111yescharacteristic
340#106—1221no—
341#106—1221no—
342C4⋊C8—611no—
343C4⋊C8—611no—
344C4⋊C8—611no—
345#103—21no—
346#103—1221no—
347#103—1221no—
348#102—1221no—
349#102—1221no—
350#102—1221no—
351C4×C8—611no—
352C4×C8—611no—
353C4×C8C22×D6111yescharacteristic
354#41—4884no—
355C2×Q8—21no—
356C2×Q8—611no—
357C2×Q8—2441no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—1221no—
363C2×D4—2441no—
364C2×D4—2441no—
365C2×D4—2441no—
366C2×D4—2441no—
367C22×C4—2441no—
368C22×C4—2441no—
369C22×C4—2441no—
370Q16—21no—
371Q16#1175822412yescomplemented
372D8—21no—
373D8—21no—
374D8—21no—
375D8—21no—
376D8—81no—
377D8—1221no—
378D8—1221no—
379D8—4881no—
380D8—4881no—
381C2×C8—611no—
382C2×C8—611no—
383C2×C8D4×D6111yescharacteristic
384C2×C8C23×D6111yescharacteristic
385C4⋊C4—21no—
386C4⋊C4—41no—
387C4⋊C4—611no—
388C4⋊C4—611no—
389C4⋊C4—611no—
390C4⋊C4—1221no—
391C4⋊C4—1221no—
392C4⋊C4—1221no—
393C4⋊C4—2441no—
394C4⋊C4—2441no—
395C4⋊C4—2441no—
396C4⋊C4—2441no—
397C4⋊C4—2441no—
398C4⋊C4—2441no—
399C4⋊C4—2441no—
400C22⋊C4—2441no—
401C22⋊C4—2441no—
402C22⋊C4—2441no—
403C42—41no—
404C42—611no—
405C42—611no—
406C42—2441no—
407C42—2441no—
408C42D4×D6111yescharacteristic
409C16—21no—
410C16—21no—
411C16—1221no—
412C16—1221no—
413C23—41no—
414C23—41no—
415C23—2441no—
416Q8—22no—
417Q8—41no—
418Q8—1222no—
419Q8—4882no—
420D4—41no—
421D4—41no—
422D4—41no—
423D4—41no—
424D4—82no—
425D4—161no—
426D4—9761no—
427D4—9761no—
428D4—4882no—
429D4—4882no—
430D4—2441no—
431D4—2441no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—611no—
435C2×C4—611no—
436C2×C4—611no—
437C2×C4—611no—
438C2×C4—4882no—
439C2×C4—4882no—
440C2×C4—4882no—
441C2×C4—2441no—
442C2×C4—2441no—
443C2×C4—2441no—
444C2×C4D8×D6111yescharacteristic
445C2×C4D8×D6111yescharacteristic
446C2×C4C2×D4×D6111yescharacteristic
447C8—21no—
448C8—1221no—
449C8—1221no—
450C8C2×#1175822411yescharacteristic
451C8#1451982411yescharacteristic
452C22—81no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—81no—
457C22—81no—
458C22—4881no—
459C22—4881no—
460C22—4881no—
461C22C2×D8×D6111yescharacteristic
462C4—21no—
463C4—21no—
464C4—42no—
465C4—81no—
466C4—612no—
467C4—1221no—
468C4—1221no—
469C4—1221no—
470C4—4881no—
471C4—4881no—
472C4—4881no—
473C4#1684041811yescharacteristic
474C4#1684057211yescharacteristic
475C2—161no—
476C2—161no—
477C2—9761no—
478C2#2145844911yescharacteristic
479C2#509×D6111yescharacteristic
480C2#2145896111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2207C2
socleC22×C61C22×D8
prime radical#21458792C2
hypercenter#2207D61
hypocenterC61#33935
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2207

Complements

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

NH ≅ G/Ntype
2. #21458710C2semidirect
3. #21448060C2semidirect
4. #21448059C2semidirect
5. #21448041C2semidirect
6. #21447978C2semidirect
7. #21447977C2semidirect
8. #21447954C2semidirect
9. #21447857C2semidirect
10. #21447857C2semidirect
11. #21447726C2semidirect
12. #21447684C2semidirect
13. #21447681C2semidirect
14. SD32×Dic61C2semidirect
15. C61×#2207C2semidirect
130. Q16×C61#41semidirect
245. #2207D61semidirect
299. C61#33935semidirect
371. Q16#11758224semidirect

Maximal subgroups

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

subgroupindex
#214587922
#214587102
#214480602
#214480592
#214480412
#214479782
#214479772
#214479542
#214478572
#214477262
#214476842
#214476812
SD32×Dic612
C61×#22072
#3393561