Finite Group Databasebeta

← #29837736

Subgroups — #29837736

Order 15616.

508 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458804C211yesmaximal, characteristic, complemented
2D8×Dic122C211yesmaximal, characteristic, complemented
3#21448055C211yesmaximal, characteristic, complemented
4#21448054C211yesmaximal, characteristic, complemented
5#21448045C211yesmaximal, characteristic, complemented
6#21447972C211yesmaximal, characteristic, complemented
7#21447969C211yesmaximal, characteristic, complemented
8#21447954C211yesmaximal, characteristic, complemented
9#21447872C211yesmaximal, characteristic
10#21447862C211yesmaximal, characteristic, complemented
11#21447726C211yesmaximal, characteristic, complemented
12#21447684C211yesmaximal, characteristic, complemented
13#21447683C211yesmaximal, characteristic, complemented
14D16×Dic61C211yesmaximal, characteristic
15C61×#2210C211yesmaximal, characteristic
16#16840481—21no—
17D4×Dic122—21no—
18C2×#14519721C2211yescharacteristic
19#16840167C2211yescharacteristic
20#16840166C2211yescharacteristic
21#16840166C2211yescharacteristic
22#16840145C2211yescharacteristic
23#16840140C2211yescharacteristic
24#16840042—21no—
25#16840040—21no—
26#16840040—21no—
27#16839982—21no—
28#16839981—21no—
29#16839965—21no—
30C8×Dic122C2211yescharacteristic
31C4×Dic244C2211yescharacteristic
32Q16×Dic61C2211yescharacteristic
33D8×Dic61C2211yescharacteristic
34D8×Dic61C2211yescharacteristic
35D8×Dic61C2211yescharacteristic
36#16839588C2211yescharacteristic
37#16839588C2211yescharacteristic
38#16839587C2211yescharacteristic
39#16839580C2211yescharacteristic
40#16839580C2211yescharacteristic
41#16839567C2211yescharacteristic
42#16839567C2211yescharacteristic
43#16839566C2211yescharacteristic
44#16839566C2211yescharacteristic
45#16839565C2211yescharacteristic
46#16839563C2211yescharacteristic
47#16839563C2211yescharacteristic
48#16839563C2211yescharacteristic
49#16839538C2211yescharacteristic
50#16839537C2211yescharacteristic
51C16×Dic61C2211yescharacteristic
52C2×SD32×C61C2211yescharacteristic
53C2×D16×C61C2211yescharacteristic
54C61×#495C2211yescharacteristic
55C4×D8×C61C2211yescharacteristic
56C61×C4⋊C16C2211yescharacteristic
57C61×#358C2211yescharacteristic
58C61×#357C2211yescharacteristic
59C22×Dic122—41no—
60C2×Dic244C2311yescharacteristic
61C2×#11758202—21no—
62C2×#11758200—21no—
63#14519777—21no—
64#14519776—21no—
65#14519776—21no—
66#14519775—21no—
67#14519770—41no—
68#14519770—41no—
69#14519764—21no—
70#14519756—41no—
71#14519754—41no—
72#14519754—41no—
73#14519754—41no—
74Q8×Dic61—21no—
75C4×Dic122—21no—
76C4×Dic122D411yescharacteristic
77D4×Dic61—21no—
78D4×Dic61—21no—
79D4×Dic61—21no—
80#14519733—41no—
81#14519721—22no—
82C2×#11758185C2311yescharacteristic
83C2×#11758185C2311yescharacteristic
84#14519711—21no—
85#14519711C2311yescharacteristic
86#14519711C2311yescharacteristic
87#14519711C2311yescharacteristic
88#14519710—21no—
89#14519710—21no—
90C4⋊#92418D411yescharacteristic
91#14519706D411yescharacteristic
92#14519706D411yescharacteristic
93#14519702—21no—
94#14519701—21no—
95#14519698—21no—
96#14519698—21no—
97#14519698—21no—
98C8×Dic61C2311yescharacteristic
99C8×Dic61C2311yescharacteristic
100C2×Q16×C61C2311yescharacteristic
101C2×SD16×C61—21no—
102C2×D8×C61C2311yescharacteristic
103C2×D8×C61C2311yescharacteristic
104#124×C61—21no—
105C4×D4×C61—21no—
106SD32×C61—22no—
107D16×C61—22no—
108C2×C16×C61C2311yescharacteristic
109C2×C16×C61C2311yescharacteristic
110#107×C61C2311yescharacteristic
111#102×C61—21no—
112C4×C8×C61C2311yescharacteristic
113C2×Q8×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C22×C4×C61—41no—
117C22×Dic61—41no—
118C22×Dic61—41no—
119C22×Dic61—41no—
120C2×Dic122—21no—
121C2×Dic122—41no—
122C2×Dic122—41no—
123C2×Dic122—41no—
124C2×Dic122—84no—
125C2×Dic122D811yescharacteristic
126Q16×C61—21no—
127Q16×C61—21no—
128SD16×C61—81no—
129D8×C61—21no—
130D8×C61—21no—
131D8×C61—21no—
132D8×C61#4112yes—
133C2×C8×C61C2×D411yescharacteristic
134C2×C8×C61C2411yescharacteristic
135C4⋊C4×C61—21no—
136C22⋊C4×C61—41no—
137C22⋊C4×C61—41no—
138C22⋊C4×C61—41no—
139C42×C61C2×D411yescharacteristic
140C22⋊Dic61—41no—
141C22⋊Dic61—41no—
142C22⋊Dic61—41no—
143C22⋊Dic61—41no—
144C22⋊Dic61—41no—
145#11758202—81no—
146#11758200—81no—
147C4⋊Dic61—21no—
148C4⋊Dic61—21no—
149C4⋊Dic61—21no—
150C4⋊Dic61—41no—
151C4⋊Dic61—41no—
152C4⋊Dic61D811yescharacteristic
153#11758196—41no—
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
166Dic244—21no—
167Dic244—21no—
168C16×C61—21no—
169C16×C61—21no—
170#11758185—21no—
171#11758185—21no—
172C23×C61—41no—
173C23×C61—41no—
174Q8×C61—41no—
175Q8×C61—41no—
176D4×C61—22no—
177D4×C61—41no—
178D4×C61—41no—
179D4×C61—41no—
180C2×C4×C61—41no—
181C2×C4×C61—41no—
182C2×C4×C61—82no—
183C2×C4×C61C2×D811yescharacteristic
184C2×C4×C61C2×D811yescharacteristic
185C2×C4×C61C22×D411yescharacteristic
186C2×Dic61—41no—
187C2×Dic61—41no—
188C2×Dic61—41no—
189C2×Dic61—41no—
190C2×Dic61—82no—
191C2×Dic61—82no—
192C2×Dic61—82no—
193C2×Dic61C2×D811yescharacteristic
194C2×Dic61C2×D811yescharacteristic
195C2×Dic61C2×D811yescharacteristic
196C2×Dic61C2×D811yescharacteristic
197Dic122—22no—
198Dic122—41no—
199Dic122—41no—
200Dic122—82no—
201Dic122—82no—
202Dic122—82no—
203C8×C61—21no—
204C8×C61C2×#4111yescharacteristic
205C8×C612−1+411yescharacteristic
206#92418—21no—
207#92418—21no—
208#33934—611nomaximal, Sylow
209C22×C61—42no—
210C22×C61—81no—
211C22×C61—81no—
212C22×C61—81no—
213C22×C61—81no—
214C22×C61—81no—
215C22×C61C22×D811yescharacteristic
216C4×C61—21no—
217C4×C61—21no—
218C4×C61—81no—
219C4×C61—81no—
220C4×C61#54811yescharacteristic
221C4×C61#57711yescharacteristic
222Dic61—21no—
223Dic61—21no—
224Dic61—21no—
225Dic61—81no—
226Dic61—81no—
227Dic61—81no—
228Dic61—81no—
229Dic61#50912yes—
230#3393—611no—
231Q8×D8—611no—
232#2243—611no—
233#2243—611no—
234#2241—611no—
235#2227—611no—
236#2227—611no—
237#2224—611no—
238#2210—611no—
239#2210D6111yescharacteristic
240#2208—611no—
241#2183—611no—
242#2175—611no—
243#2175—611no—
244C4×D16—611no—
245C2×C61—82no—
246C2×C61—161no—
247C2×C61#335811yescharacteristic
248C2×C61C2×#50911yescharacteristic
249C2×C61#341711yescharacteristic
250#556—1221no—
251D4×Q8—1221no—
252C2×SD32—611no—
253C2×SD32C2×D6111yescharacteristic
254C2×D16C2×D6111yescharacteristic
255#500—611no—
256#500—611no—
257#500—611no—
258#495—611no—
259#495C2×D6111yescharacteristic
260#494—611no—
261#474—1221no—
262#474—1221no—
263#474—1221no—
264#464—1221no—
265#464—1221no—
266#461—1221no—
267C8×Q8—611no—
268C4×Q16—611no—
269C4×Q16—611no—
270C4×D8—611no—
271C4×D8—611no—
272C4×D8—611no—
273C4×D8C2×D6111yescharacteristic
274#366—611no—
275#366—611no—
276#366—611no—
277C4⋊C16—611no—
278C4⋊C16—611no—
279C4⋊C16C2×D6111yescharacteristic
280#358—611no—
281#358—611no—
282#358—611no—
283#358—611no—
284#358—611no—
285#358C2×D6111yescharacteristic
286#357—611no—
287#357—611no—
288#357—611no—
289#357C2×D6111yescharacteristic
290#346—611no—
291#346—611no—
292C4×C16—611no—
293C61#3393411yescharacteristic, complemented, Sylow
294C22×Q8—2441no—
295C2×Q16—611no—
296C2×Q16—1221no—
297C2×Q16C22×D6111yescharacteristic
298C2×SD16—21no—
299C2×SD16—1221no—
300C2×D8C22×D6111yescharacteristic
301C2×D8C22×D6111yescharacteristic
302C4⋊Q8—1221no—
303C4⋊Q8—1221no—
304C4⋊Q8—1221no—
305C4⋊Q8—1221no—
306#126—2441no—
307#126—2441no—
308#124—21no—
309#124—1221no—
310C22⋊Q8—2441no—
311C22⋊Q8—2441no—
312C22⋊Q8—2441no—
313C22⋊Q8—2441no—
314C4×Q8—611no—
315C4×Q8—1221no—
316C4×Q8—1221no—
317C4×D4—21no—
318C4×D4—1221no—
319C4×D4—1221no—
320C4×D4—1221no—
321#117—2441no—
322SD32—22no—
323SD32—1222no—
324D16—22no—
325C2×C16—611no—
326C2×C16—611no—
327C2×C16C22×D6111yescharacteristic
328C2×C16C22×D6111yescharacteristic
329#107—611no—
330#107—611no—
331#107—611no—
332#107—1221no—
333#107—1221no—
334#107—1221no—
335#107C22×D6111yescharacteristic
336C4⋊C8—611no—
337C4⋊C8—611no—
338C4⋊C8—611no—
339#103—1221no—
340#103—1221no—
341#102—21no—
342#102—1221no—
343#102—1221no—
344#102—1221no—
345C4×C8—611no—
346C4×C8—611no—
347C4×C8C22×D6111yescharacteristic
348C2×Q8—21no—
349C2×Q8—611no—
350C2×Q8—1221no—
351C2×Q8—4884no—
352C2×Q8—2441no—
353C2×Q8—2441no—
354C2×Q8—2441no—
355C2×D4—21no—
356C2×D4—21no—
357C22×C4—41no—
358C22×C4—2441no—
359C22×C4—2441no—
360C22×C4—2441no—
361Q16—21no—
362Q16—21no—
363Q16—1221no—
364Q16—1221no—
365Q16—4881no—
366SD16—81no—
367SD16—4881no—
368D8—21no—
369D8—21no—
370D8—21no—
371D8#1175822412yes—
372C2×C8—611no—
373C2×C8—611no—
374C2×C8D4×D6111yescharacteristic
375C2×C8C23×D6111yescharacteristic
376C4⋊C4—21no—
377C4⋊C4—611no—
378C4⋊C4—611no—
379C4⋊C4—611no—
380C4⋊C4—1221no—
381C4⋊C4—1221no—
382C4⋊C4—1221no—
383C4⋊C4—2441no—
384C4⋊C4—2441no—
385C4⋊C4—2441no—
386C4⋊C4—2441no—
387C4⋊C4—2441no—
388C4⋊C4—2441no—
389C4⋊C4—2441no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—2441no—
394C22⋊C4—2441no—
395C22⋊C4—2441no—
396C22⋊C4—2441no—
397C22⋊C4—2441no—
398C42—611no—
399C42—611no—
400C42—2441no—
401C42—2441no—
402C42D4×D6111yescharacteristic
403C16—21no—
404C16—21no—
405C16—1221no—
406C16—1221no—
407C23—41no—
408C23—41no—
409Q8—41no—
410Q8—41no—
411Q8—1222no—
412Q8—4882no—
413Q8—4882no—
414Q8—4882no—
415Q8—2441no—
416Q8—2441no—
417D4—22no—
418D4—41no—
419D4—41no—
420D4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—82no—
424C2×C4—611no—
425C2×C4—611no—
426C2×C4—611no—
427C2×C4—611no—
428C2×C4—4882no—
429C2×C4—4882no—
430C2×C4—4882no—
431C2×C4—2441no—
432C2×C4—2441no—
433C2×C4—2441no—
434C2×C4—2441no—
435C2×C4D8×D6111yescharacteristic
436C2×C4D8×D6111yescharacteristic
437C2×C4C2×D4×D6111yescharacteristic
438C8—21no—
439C8—1221no—
440C8—1221no—
441C8C2×#1175822411yescharacteristic
442C8#1451982411yescharacteristic
443C22—42no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22C2×D8×D6111yescharacteristic
450C4—21no—
451C4—21no—
452C4—81no—
453C4—81no—
454C4—612no—
455C4—1221no—
456C4—1221no—
457C4—1221no—
458C4—4881no—
459C4—4881no—
460C4—4881no—
461C4—4881no—
462C4#1684041811yescharacteristic
463C4#1684057211yescharacteristic
464C2—82no—
465C2—161no—
466C2#2145844911yescharacteristic
467C2#509×D6111yescharacteristic
468C2#2145896911yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2210C2
socleC22×C61C22×D8
prime radicalC2×D16×C61C22
hypercenter#2210D61
hypocenterC61#33934
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2210

Complements

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

NH ≅ G/Ntype
1. #21458804C2semidirect
2. D8×Dic122C2semidirect
3. #21448055C2semidirect
4. #21448054C2semidirect
5. #21448045C2semidirect
6. #21447972C2semidirect
7. #21447969C2semidirect
8. #21447954C2semidirect
10. #21447862C2semidirect
11. #21447726C2semidirect
12. #21447684C2semidirect
13. #21447683C2semidirect
293. C61#33934semidirect

Maximal subgroups

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

subgroupindex
#214588042
D8×Dic1222
#214480552
#214480542
#214480452
#214479722
#214479692
#214479542
#214478722
#214478622
#214477262
#214476842
#214476832
D16×Dic612
C61×#22102
#3393461